Understanding MacroeconomicTheory
At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activitycan be summarized by aggregate variables such as an economyrsquos total productionof various goods and services the aggregate level of unemployment the generallevel of interest rates and the overall level of prices Macroeconomics is the studyof movements in such economy-wide variables as output employment and prices
The focus of this book is on developing simple theoretical models that provideinsight into the reasons for fluctuations in such aggregate variables These modelsexplore how shocks or ldquoimpulsesrdquo to the economy impact individualsrsquo behaviorin specific markets and the resulting implications in terms of changes in aggregatevariables
Understanding Macroeconomic Theory will provide the reader with anin-depth understanding of standard theoretical models Walrasian Keynesian andneoclassical It is written in a concise accessible style and will be an indispensabletool for all students who wish to gain firm grounding in the complexities ofmacroeconomic theories
John M Barron is the Loeb Professor of Economics in the Krannert School ofManagement at Purdue University
Bradley T Ewing is the Jerry S Rawls Endowed Professor in OperationsManagement in the Rawls College of Business at Texas Tech University
Gerald J Lynch is Professor of Economics in the Krannert School of Managementat Purdue University
Routledge Advanced Texts in Economics and Finance
Financial EconometricsPeijie Wang
Macroeconomics for Developing Countries 2nd editionRaghbendra Jha
Advanced Mathematical EconomicsRakesh V Vohra
Advanced Econometric TheoryJohn S Chipman
Understanding Macroeconomic TheoryJohn M Barron Bradley T Ewing and Gerald J Lynch
Understanding MacroeconomicTheory
John M Barron Bradley T Ewingand Gerald J Lynch
First published 2006by Routledge270 Madison Ave New York NY 10016
Simultaneously published in the UKby Routledge2 Park Square Milton Park Abingdon Oxon OX14 4RN
Routledge is an imprint of the Taylor amp Francis Group an informa business
copy 2006 John M Barron Bradley T Ewing and Gerald J Lynch
All rights reserved No part of this book may be reprinted orreproduced or utilized in any form or by any electronicmechanical or other means now known or hereafterinvented including photocopying and recording or in anyinformation storage or retrieval system without permission inwriting from the publishers
Library of Congress Cataloging in Publication DataBarron John M
Understanding macroeconomic theory John M BarronBradley T Ewing and Gerald J Lynch
p cmIncludes bibliographical references and index1 Macroeconomics I Ewing Bradley T II Lynch Gerald J III Title
HB1725B3753 2006339ndashdc22 2005026372
British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library
ISBN10 0ndash415ndash70195ndash3 ISBN13 978ndash0ndash415ndash70195ndash2 (hbk)ISBN10 0ndash415ndash70196ndash1 ISBN13 978ndash0ndash415ndash70196ndash9 (pbk)ISBN10 0ndash203ndash08822ndash0 ISBN13 978ndash0ndash203ndash08822ndash7 (ebk)
This edition published in the Taylor amp Francis e-Library 2006
ldquoTo purchase your own copy of this or any of Taylor amp Francis or Routledgersquoscollection of thousands of eBooks please go to wwweBookstoretandfcoukrdquo
Contents
1 Introduction 1
2 Walrasian economy 7
3 Firms as market participants 18
4 Households as market participants 39
5 Summarizing the behavior and constraints of firms andhouseholds 64
6 The simple neoclassical macroeconomic model (withoutgovernment or depository institutions) 78
7 Empirical macroeconomics traditional approaches and timeseries models 96
8 The neoclassical model 116
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model 130
10 The Lucas model 146
11 Policy 167
12 Open economy 186
Notes 202References 223Index 228
1 Introduction
The topics of macroeconomics
At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activity canbe summarized by aggregate variables such as an economyrsquos total production ofvarious goods and services the aggregate level of employment and unemploymentthe general level of interest rates and the overall level of prices1 Macroeconomicsis the study of movements in such economy-wide variables as output employmentand prices
The focus of this book will be on developing simple theoretical models thatprovide insight into the reasons for fluctuations in such aggregate variables Thesemodels explore how shocks or ldquoimpulsesrdquo to the economy (eg changes to tech-nology the money supply or government policy) impact individualsrsquo behavior inspecific markets and the resulting implications in terms of changes in aggregatevariables
An overview of some facets of theoreticalmacroeconomic analysis
Given the breadth of economic activity in an economy the study of macroeco-nomics must involve an examination of a variety of different markets For instanceit is common for macroeconomic analysis to consider exchanges of labor servicesin the labor markets of consumption and capital goods in the output marketsand of financial assets in the financial markets The fact that macroeconomicssimultaneously analyses exchanges of different goods in different markets meansthat macroeconomic theory is a general equilibrium theory That is macro-economic theory must by necessity incorporate the links across markets that arefundamental to general equilibrium analysis As we will see throughout this booka key reflection of the links across markets is Walrasrsquo law named in honor ofthe nineteenth-century French economist Leon Walras2 Simply put Walrasrsquo lawnotes that the budget constraints faced by individual agents in the economy sug-gest that if n minus 1 of the n markets in the economy are in equilibrium then the nthmarket must be in equilibrium We will repeatedly rely on Walrasrsquo law or variantsof it to simplify macroeconomic analysis
2 Introduction
While macroeconomic theories have in common (a) an attempt to explainfluctuations in aggregate variables and (b) a general equilibrium character thereremain wide differences among macroeconomic models Below we break downthese differences across macroeconomic models in several ways in order to makesome sense of what passes for simple theoretical macroeconomic analysis
Static dynamic and stationary analysis
One way of breaking down macroeconomic analyses is into static models dynamicmodels and stationary analysis of dynamic models Static macroeconomic modelsanalyze the economy at a point in time They consider the determination of pro-duction exchange and prices of various goods only for the markets that currentlyexist John Hicks (1939) sketched out an analysis of ldquospotrdquo or ldquotemporaryrdquo equi-librium The advantage to such an approach is that it provides for rather simpleldquocomparative staticrdquo analysis of the effects of changes in a variety of exogenousvariables on the endogenous variables3 Such static analysis is useful in providinginsight into a variety of questions of interest
Static macroeconomic analysis can be viewed as a modification of a Walrasiangeneral equilibrium analysis or what is commonly referred to as ldquoArrowndashDebreutheoryrdquo (Arrow and Hahn 1971 Debreu 1959) In ArrowndashDebreu theory eachcommodity is described by its physical characteristics its location and its dateof availability It is assumed there are a complete set of spot and forwardmarkets Prices adjust to clear all markets However if one restricts attentionto just spot markets then one moves from traditional Walrasian generalequilibrium to an analysis of ldquotemporary equilibriumrdquo a phrase coined by Hicks(1939) This restriction to spot markets is one element of static macroeconomicanalysis4
A second element of static models is that if there is a future then staticmacroeconomic analysis simply assumes given expectations of future prices andenvironment How expectations of future events are formed is left unspecified sothat expectations of future prices become simply an element in the set of exogenousvariables5
While static analysis provides insights there are several disadvantages of staticanalysis severe enough that it alone does not provide an adequate grounding inmacroeconomic analysis The key disadvantage of static analysis is that it breaksties between current events and future events To show the limitations of staticanalysis let us suppose that underlying a simple static macroeconomic analy-sis of current markets is a microeconomic analysis of individualsrsquo decisions thatidentifies the anticipated future level of prices as one of the exogenous variablesaffecting current behavior As we have seen static analysis takes expectations ofsuch variables as future prices as exogenous variables Doing so however resultsin (a) an incomplete enumeration of exogenous variables that can impact currenteconomic activity and (b) a potentially incomplete accounting of the effects of theimpact on current economic activity of a change in those exogenous variables thatare identified by the analysis
Introduction 3
To illustrate the first point of an incomplete listing of exogenous variables letus suppose that the static model identifies changes in the current money supplyas one factor that influences current prices This suggests that if we replicate thestatic analysis in future periods changes in the money supply in the future would beshown to affect prices at that time It seems natural to then presume that individualsrsquoanticipation of future prices would incorporate this link between changes in thefuture money supply and future prices in forming their expectations of futureprice levels so that the anticipated future money supply becomes a determinantof current activity6 Yet static analysis since it does not analyze markets beyondthe current period will not identify the potential impact of future changes in themoney supply on current activity7
To illustrate the second point of an incomplete accounting of the effects of achange in an exogenous variable let us suppose that underlying the static macro-economic analysis of current markets is a microeconomic analysis of firmsrsquo currentinvestment behavior that identifies the anticipated future tax levels as well as futureprices as two exogenous variables affecting investment decisions Thus staticanalysis would suggest that a change in future tax levels will impact current activitythrough the direct effect on current investment It is not hard to see however that(a) the current change in investment means a different future capital stock and(b) the change in future tax levels could affect future as well as current investmentEither or both of these changes would likely impact future prices and if such animpact were anticipated be a second way that future tax changes impact currentactivity8
An obvious way to avoid the above problems is to introduce forward marketsso that the macroeconomic analysis determines the prices of goods to be tradedin the future along with the prices of goods traded at the current time9 In doingso we have moved from static to dynamic analysis That is the macroeconomicmodels now determine the paths of variables (such as prices) over time rather thanprices (and other variables) at only one point in time
In a deterministic setting this expansion of dynamic analysis incorporates thenotion of ldquoperfect foresightrdquo in which individuals correctly anticipate all futureprices If there were uncertainty the analysis indexes goods by both the date oftrade and the ldquostate of naturerdquo with trades contingent on the realized state ofnature10 The result is that at each date there is a distribution of potential prices atwhich trade for a good could occur and given common knowledge of likelihood ofthe states of nature expectations of future prices would be defined by the analysis(ldquorational expectationsrdquo)
Once dynamic analysis is introduced we can consider a special limiting form ofdynamic analysis termed stationary analysis The aim of stationary analysis is toidentify in the context of a dynamic model the limiting tendencies of endogenousvariables such as the capital stock or the rate of growth in prices given that theexogenous variables remain constant or stationary over time11
While stationary analysis is distinct from static analysis in some cases one canthink of static analysis as a form of stationary analysis That is static analysis insome cases can be viewed as the outcome that would emerge each period given
4 Introduction
that the exogenous variables remain constant (or in some cases grow at a steadyrate over time) and given that one picks the correct fixed level of certain keyexogenous variables (eg the capital stock and the rate of change in the moneysupply) Note however that this implies that for static analysis to perfectly mimicstationary analysis one must to all intents and purposes have first executed theunderlying dynamic analysis
Period (discrete) versus continuous-time analysis
Macroeconomic analysis can be broken down into period or discrete-time macro-economic models and continuous-time macroeconomic models Substantivedifferences in terms of theoretical predictions do not exist between these two typesof analyses if one is careful to assure identical underlying assumptions Yet the twoanalyses do differ in the analytical techniques used For instance while discretemacroeconomics relies on the techniques of dynamic programming and differenceequations to characterize elements of the model in similar circumstances con-tinuous macroeconomic analysis turns to the techniques of optimal control anddifferential equations
Although substantive issues are not raised by the discrete- versus continuous-time dichotomy it is sometimes argued that one is preferred to the other Forinstance an attractive feature of the continuous-time analysis is that it highlightsquite clearly the distinctions between stocks and flows something that is not soclearly discernable in discrete analysis On the other hand an attractive feature ofdiscrete analysis is that it makes more transparent the link between the theoreticalanalysis and empirical testing since such analysis coincides with the obvious factthat empirical data on macroeconomic variables is discrete
New classical economics versus non-market-clearing
Classical analysis refers to the widely adopted view of how the macroeconomyshould be modeled that existed prior to the experience of the Great Depressionand John Maynard Keynesrsquo General Theory of Employment Interest and Money(1936) In classical theory the real side of the economy was separate from themoney side Classical analysis of the ldquorealrdquo side of the economy is aimed atdetermining such variables as total production relative prices the real rate ofinterest and the distribution of output Classical analysis of the money side ofthe economy meant analysis is aimed at determining money prices and nominalinterest rates
The separation of the monetary side from the real side in classical or neoclassicalanalysis led to the prediction that monetary changes do not have any effect onreal variables such as total output12 A similar prediction is often obtained bymore recent macroeconomic analysis and this is one reason why this more recentanalysis is referred to as the new classical economics13 Alternative labels ofthese new classical models include rational expectations models with market
Introduction 5
clearing neoclassical models of business fluctuations and equilibrium businesscycle models
A common feature of the analyses of new classical economics besides thefact that it suggests a divorcement of monetary changes from the real side ofthe economy is that prices are determined in the analysis so as to clear marketsThis view that prices serve to equate demands and supplies a view common tomicroeconomics is taken as an important strength of the analysis for it meansthat the models have consistent ldquomicroeconomic foundationsrdquo One implicationof the market-clearing assumption is the same as in microeconomics ndash the analysissuggests that all gains to exchange have been extracted
Contrasting the new classical economics with what preceded it helps one putthis rebirth of classical analysis into perspective Following the Great Depressionmacroeconomic analysis took as its main premise the idea that markets did notclear ndash in particular that prices did not adjust In this context the business cyclewas defined by ldquomarket failurerdquo and the role of government to stabilize the eco-nomy was clear There are a number of different types of non-market-clearing orKeynesian models One version of such Keynesian models that popularized byPatinkin (1965 chapters 13ndash14) Clower (1965) and Barro and Grossman (1971)takes as given output prices such that the output market fails to clear A secondmodel popularized by Fischer (1977) Phelps and Taylor (1977) and Sargent(1987a) as an alternative formalization of the Keynesian model takes as given theprice of labor such that the labor market fails to clear
The common theme of these non-market-clearing analyses is that for variousreasons prices do not clear markets and concepts such as excess demand and supplyplay a role in the analysis Yet no concise reason is given as to why there is marketfailure other than suggesting such items as ldquocoordination problemsrdquo and ldquotrans-action costsrdquo14 The result is that such analysis is challenged by the new classicaleconomics as lacking the microeconomic foundations for price determination AsHowitt (1986 108) suggests such a view ldquoforces the proponent of active stabi-lization policy to explain the precise nature of the impediments of transacting andcommunicating that prevent private arrangements from exhausting all gains fromtraderdquo15 This is not an easy task according to Howitt since ldquoimpediments to com-munication in a model simple enough for an economist to understand will typicallyalso be simple enough that the economist can think of institutional changes thatwould overcome themrdquo (1989 108)
Microeconomic foundations and aggregation issues
An important feature of macroeconomic analysis is that it reflects the aggre-gation of individual decisions A common approach to such aggregation is toassume ldquorepresentativerdquo agents characterize their optimal behavior then use suchbehavioral specifications in building the macroeconomic model Thus much ofmacroeconomic analysis entails looking at individualsrsquo decisions such as house-holdsrsquo decisions to work consume and save or firmsrsquo decisions to produceborrow and invest in capital
6 Introduction
These characterizations of optimizing individual behavior make up part of thebuilding blocks or ldquomicroeconomic foundationsrdquo of macroeconomic analysisYet microeconomic foundations of macroeconomics are not restricted to suchanalysis For instance such foundations also include a characterization of howprices in individual markets are determined as we saw in our discussion of newclassical economics
In developing the microeconomic foundations of macroeconomic models wewill often be struck by the extent to which the analysis restricts any role forheterogeneity or diversity among the individual agents in the economy Yet suchdiversity can in certain instances be critical to the analysis One attempt to introducediverse or heterogeneous agents into macroeconomic analysis is represented by theoverlapping generations models These models also have the advantage of beinggenuinely dynamic in nature and as such represent one area of macroeconomicsthat has recently received significant attention
Deterministic versus stochastic
In recent years an important element to macroeconomic models has been to intro-duce stochastic elements The rationale is clear the presence of uncertainty as tofuture events is real As noted by Lucas (1981 286)
the idea that speculative elements play a key role in business cycles that theseevents seem to involve agents reacting to imperfect signals in a way whichafter the fact appears inappropriate has been commonplace in the verbaltradition of business cycle theory at least since Mitchell It is now entirelypractical to view price and quantity paths that follow complicated stochasticprocesses as equilibrium ldquopointsrdquo in an appropriately specified space
As the quote suggests in dynamic models especially for new classical economicswhere market clearing is presumed stochastic elements are incorporated into theanalysis so that the role played by shocks to an economy in a dynamic setting canbe well defined
2 Walrasian economy
Introduction
This chapter develops a competitive model of the economy The key assumptionsneeded for this model are spelled out in detail One important aspect of this modelis the ldquonumerairerdquo or the commodity price that is used as a reference in the modelA distinction is made between accounting prices and relative prices and it isseen that traditional general equilibrium analysis does not determine the levelof accounting prices but rather simply relative prices A number of modificationsto the model are mentioned and add a sense of ldquorealismrdquo to the framework Thesemodifications include the introduction of futures markets quantity constraintsand the costs associated with carrying out a transaction
The chapter continues by considering individual decision-making and the theoryof the consumer and how this relates to the determination of market demand Thegeneral equilibrium conditions are stated in terms of relative prices and allocationsA theme carried throughout this book is emphasized and that is the use of theaggregate budget constraint Finally it is shown that explicitly excluding moneyas a ldquomarketrdquo allows one to understand how Sayrsquos conclusion that ldquosupply createsits own demandrdquo is arrived at in an economy composed of a single aggregatecommodity
A simple Walrasian model
As discussed previously the idea that prices adjust to clear markets is commonto much of new classical macroeconomics Thus we begin our examination ofmacroeconomic analysis by considering an economy consisting of perfectly com-petitive markets This means that individuals take prices at which exchanges canbe made as parametric and prices adjust to eliminate excess demands so that indi-vidualsrsquo plans at given prices are feasible As the title to this section suggests sucha characterization is sometimes referred to as being indicative of a ldquoWalrasianrdquoeconomy Walras described the process by which prices adjust to excess demand orsupply as a groping or tatonnement process (see Walras 1954)1 A fictitious auc-tioneer calls out different prices for the various markets and no exchange occursuntil equilibrium prices are reached2
8 Walrasian economy
Our analysis also begins at a very simple level The term ldquosimplerdquo reflects atleast the following three characteristics of the economy that we consider
1 It is a barter economy That is any commodity can be freely traded for anyother commodity There is no role for a medium of exchange (money) inreducing the costs of arranging exchanges
2 It is an exchange economy That is there is no production Instead individualshave initial fixed endowments of various commodities
3 It is a timeless economy Goods are indexed by physical characteristics andlocation but not dated according to availability This rules out futures mar-kets or the formation of expectations of future events and planning Such aneconomy was suggested by Hicks (1939) Patinkin (1965 Chapter 1) andHansen (1970 Chapter 4) provide a more detailed view of such an economyAn actual example of a pure barter exchange economy is offered by Radford(1945) The simple model of the economy developed below is useful in high-lighting such concepts as relative prices the numeraire individual versusmarket experiments aggregation issues conditions for general equilibriumand Walrasrsquo and Sayrsquos laws
The first model developed below also takes an approach to modeling the econ-omy that is in vogue in current theoretical macroeconomics As Sargent (1987b)states the ldquoattraction of (such) general equilibrium models is their internal con-sistency one is assured the agentsrsquo choices are derived from a common set ofassumptionsrdquo
Yet this advantage of general equilibrium analysis is not fully exploited until theelements of time money and production are introduced and so we will expand thediscussion in subsequent sections by introducing such features Below we intro-duce in more detail some of the key assumptions underlying the simple Walrasianmodel we start with
Key assumptions underlying a simple Walrasian model
As noted by Debreu (1959 74) ldquoan economy is defined by m consumers (charac-terized by their consumption sets and their preferences) n producers (characterizedby their production sets) and the total resources (the available quantities of thevarious commodities which are a priori given)rdquo As discussed above we considera special case of Debreursquos ldquoconcept of an economyrdquo one in which productionis absent As the following set of assumptions makes clear we also restrict ouranalysis to private ownership economies with a price system In particular assume(partial listing)
Assumption 21 There are m individuals (agents) in the economy indexed bya = 1 m There are T commodities indexed by i = 1 T 3 Agent arsquos initial
Walrasian economy 9
endowment of commodity i is denoted by cai cai ge 0 i = 1 T Naturally
msuma=1
cai = ci gt 0
Note that this assumption reflects an exchange economy in which private propertyrights exist ldquoPrivate property rightsrdquo means that for each unit of each good theexclusive right to determine use has been assigned to a particular individual
Assumption 22 All exchanges occur at a single point in time
Assumption 23 Each individual confronts the same known set of prices at whichexchange can occur4 A relative (purchase) price of commodity i indicates the unitsof commodity j required to purchase one unit of commodity i A relative (sale)price of commodity i indicates units of commodity j received when one unit ofcommodity i is sold
Assumption 24 Purchase and sale prices are identical for each commodity Thismeans that there are no ldquoprice spreadsrdquo which would suggest either a gain to anindividual buying and selling the same commodity or the presence of costs tomaking an exchange5 Thus for the T commodities there are T 2 exchange ratesor relative prices taking two commodities at a time
The numeraire
While there are T 2 exchange rates the complete set of exchange rates can bededuced directly or indirectly by the set of T minus 1 relative prices
(π1j πjminus1 j πj+1 j πTj)
where the πij denotes the price of commodity i in terms of commodity j6 In thelisting of relative prices (π1j πjminus1 j πj+1 j πTj) commodity j is referredto as the ldquonumerairerdquo
To see how the set of relative prices reduces to T minus 1 we rely on the factthat πhh = πhjπhj for all h j and k Let us see what this means for a simpleexample of three commodities h j and k There are then T 2 or nine differentrelative prices which are πhh πjj πkk πhj πjh πhk πkh πjk and πkj But we canuse the relationship πhh = πhjπhj to reduce this to T minus 1 = 2 relative prices withinformational content In particular we know that
1 πhh πhkπhj and similarly for πjj and πkk when h = j = k In other wordsthe exchange rate of a commodity with itself is unity This reduces from nineto six the number of relative prices for which information is required
2 πjh = 1πhj when k = j (such that πkj = πkk = 1) Similarly πhk = 1πkhand πjk = 1πkj For example if the jth commodity is pears and the hth
10 Walrasian economy
commodity is oranges then if πjh = 3 (3 oranges = 1 pear) πhj = 13(13 pear = 1 orange) This reduces from six to three the number of relativeprices for which information is required to reconstruct the complete set ofrelative prices
3 Finally πkh = πkjπhj (for k = j = h) For example if the jth commodity ispears the kth commodity is apples and the hth commodity is oranges thenif πkj = 3 (3 pears = 1 apple) and πhj = 13 (13 pears = 1 orange) thenπkh = 9 (9 oranges = 1 apple) That is with 9 oranges you can get 3 pearswhich in turn will purchase 1 apple This drops us from three to two relativeprices required to reconstruct the complete set of relative prices Since thenumber of commodities T = 3 we have shown how the T 2 relative pricescan be constructed from T minus 1 relative prices
In subsequent discussions we will arbitrarily let commodity T be the numerairesuch that the set of relative prices can be summarized by
(π1T πTminus1T )
For simplicity let us change notation such that πiT = πi i = 1 T Thus theset of relative prices can be rewritten as
(π1 πTminus1)
Note that πT = 1 since we are assuming the T th good is the numeraireIn traditional general equilibrium theory there is a concept of ldquoaccounting pricesrdquo
as well as the concept of relative prices Accounting prices can be represented bya set of real numbers (say pi i = 1 T ) attached to the T commodities7 Therelationship between these accounting prices and the set of relative prices that doimpinge on behavior is that πi = pipT i = 1 T (for Pi = 0) As we will seetraditional general equilibrium analysis does not determine the level of accountingprices but rather simply relative prices8
Anticipating future modifications
Before continuing it might be useful to anticipate some of the subsequent changeswe will make in the characterization of the economy Besides the introduction ofproduction we will
bull introduce time implying either forward (futures) markets or an important rolefor expectations of future spot prices
bull introduce quantity constraints that can arise if prices are fixed at non-market-clearing levels (ie depart from a Walrasian framework) and
bull introduce the cost of carrying out an exchange
Walrasian economy 11
With respect to point (c) such costs have been characterized by Coase (1960) asldquotransaction costsrdquo arising from the costs ldquonecessary to discover who it is that onewishes to deal with and to inform people that one wishes to dealrdquo the costs ofldquoconduct[ing] negotiations leading up to a bargain and to draw up a contractrdquoand the costs ldquoto undertake the inspection needed to make sure that the terms ofthe contract are being observedrdquo9 Such costs will alter the nature of contracts(exchange agreements) formed and may provide the reason for ldquoprice rigiditiesrdquoSuch costs also suggest a role for money
Transaction costs are assumed to be zero in the simple Walrasian sys-tem outlined above Sometimes this is referred to as a situation wherethere is ldquoperfect informationrdquo or where markets are complete and ldquoperfectlycompetitiverdquo
Individual experiments
General equilibrium analysis can be divided into what Patinkin (1965) refers to asldquoindividual experimentsrdquo and ldquomarket experimentsrdquo In the context of the simpleWalrasian barter exchange economy individual experiments consider the behaviorof individual agents given an initial endowment and preferences when confrontedwith a set of prices Market experiments consider the resulting determination ofprices
In the simple Walrasian model under consideration individual experimentsreplicate standard microeconomic analysis of consumer behavior In particularassume
Assumption 25 Individual arsquos preferences are described by his utility functionua(ca1 caT ) where ca1 caT denote agent arsquos consumption bundle cai ge 0i = 1 T ua maps the set of all T -tuples of non-negative numbers into the setof all real numbers (ua RT+ rarr R) We make the appropriate assumptions withrespect to individualsrsquo preferences such that a utility function exists and is wellbehaved10
Assumption 26 Individual a will choose the most preferred consumption bundlefrom the set of feasible alternatives (rationality) Given the possibility of cost-less exchange at the set of relative prices represented by (π1 πTminus1) feasibleconsumption bundles or sets (ca1 caT ) are defined by
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0
wheresumT
i=1 πi cai denotes the initial endowment of individual a in terms ofcommodity T The above expression defines the budget set
12 Walrasian economy
The consumer problem
From Assumptions 25 and 26 individual arsquos optimum consumption bundle is thesolution to the problem
maxCa1CaT
ua(ca1 caT )
subject to
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0 cai ge 0 i = 1 T
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λ
L(ca1 caT λ) = ua(ca1 caT ) + λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
with first-order (necessary) conditions being11
partL
partcaile 0 i = 1 T
partL
partcaicai = 0 i = 1 T
cai ge 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
λ ge 0
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λmicro1microT
L(ca1 caT λ micro1 microT ) = ua(ca1 caT )
+ λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
Walrasian economy 13
with the necessary conditions being
partL
partcai= 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
partL
partmicroige 0 i = 1 T
microipartL
partmicroi= 0 i = 1 T
λ ge 0
microi ge 0 i = 1 T
Individual demands and excess demands
The optimal consumption bundle for agent a is defined by the above first-orderconditions and will be denoted by the (demand) set
(ca1 caT ) cai ge 0 i = 1 T
Individual arsquos demand functions will be of the form
cdai
(π1 πTminus1
Tsumi=1
πi cai
) i = 1 T
That is individual arsquos demand (consumption) of commodity i depends on the T minus1relative prices of commodities and the initial endowment Note that the form ofthe utility function implies the utility-maximizing consumption bundle meets thebudget constraint with equality Thus at the optimal bundle we have
partL
partλ=
Tsumi=1
πi cai minusTsum
i=1
πicdai = 0
An important point to note about demand functions is that they are homogeneousof degree zero in what might be called accounting prices12 Accounting prices aredefined such that pi = πi middot pT i = 1 T so it is clear that if all prices increaseby the multiple θ relative prices and the initial endowment are unchanged
Individual arsquos excess demand function for commodity i is defined byzai = cd
ai minus cai If zai is positive agent a is a net buyer of commodity i whileif zai is negative the agent is a net seller of commodity i The market value
14 Walrasian economy
(in terms of the numeraire) of the quantity of the ith commodity that individuala seeks to exchange (buy or sell) is then given by πizai From the budget con-straint we know that
sumTi=1 πi(cd
ai minus cai) = 0 orsumT
i=1 πizai = 0 In other words foreach individual the market value (in terms of commodity T ) of individual excessdemands must sum to zero This rather obvious finding generates what is referredto as Walrasrsquo law as we will see
Market experiments
In the previous section we reviewed the nature of individual demand functions andindividual excess demand functions Now consider the collection of m individualsAggregating or summing individual demand functions we obtain an aggregate orldquomarketrdquo demand function for commodity i of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci1 Tsum
i=1
πi
)equiv
msuma=1
cdai(middot)
Similarly summing agentsrsquo excess demand functions for commodity i gives usthe aggregate or ldquomarketrdquo excess demand function for commodity i of the form
zi equivmsum
a=1
(zai) equivmsum
a=1
(cdai minus cai)
Note that a zero aggregate excess demand for commodity i does not imply thatno exchange of commodity i occurs among the m agents However as we haveseen a zero individual excess demand for commodity i does imply no exchangeof commodity i by that particular individual
Aggregation issues
So far our aggregations have remained true to the underlying microeconomicanalysis Yet this is rarely the case in macroeconomic analysis which typicallyabstracts from what might be termed ldquodistributionalrdquo effects An example of thisin the above context as we will see later is to ignore the effects of the distributionof initial endowments across individuals on market demands such that the marketdemand function for commodity i is assumed to be of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci
)
where
Tsumi=1
πicai equivmsum
a=1
(Tsum
i=1
πi cai
)
Walrasian economy 15
As you can see with heterogeneity (either in initial endowments or preferences)such a posited aggregate market demand function is unlikely to follow exactlyfrom the underlying microeconomic analysis One should keep in mind suchapproximations when interpreting macroeconomic analysis
Equilibrium an isolated market
With respect to a single market equilibrium is characterized by an accounting pricepi and implied relative price πi = pipT such that cd
i = ci (demand equals fixedendowment) or equivalently zi = 0 (excess demand equals zero) The tatonnementprocess is the description of how prices change to clear the market In the Walrasianmodel movement toward equilibrium the tatonnement process involves twofacets
1 The Walrasian excess demand hypothesis which indicates that the accountingprice of commodity i rises if there is excess demand and falls if there is excesssupply That is
dpi = fi(zi) i = 1 T fi(0) = 0dfidzi
gt 0
In terms of relative prices
dπi = dpi
pT= f (zi)
pT i = 1 T minus 1
Note that the change in price is not across time since each market is assumedto clear instantaneously at the same point in time
2 The recontracting assumption which states that offers to buy or sell at var-ious relative prices are not binding unless market(s) clear Only when theequilibrium price (or price vector) is obtained are contracts then made final
General equilibrium (conditions)
A general equilibrium will be characterized by a set of T minus 1 relative prices(πlowast
1 πlowasttminus1) and allocations (clowast
a1 clowastaT ) for individual a a = 1 m such
that
clowastai = cd
ai
(πlowast
1 πlowasttminus1
Tsumi=1
πi cai
) i = 1 T a = 1 m (21)
rArr clowasti equiv
msuma=1
clowastai =
msuma=1
cdai equiv cd
i
clowasti = ci i = 1 T (22)
16 Walrasian economy
Equation (21) indicates that the equilibrium allocation must be optimal in thatit must satisfy all demands for commodities at the specified set of pricesEquation (22) indicates that such an allocation must be feasible that is sumto total resource endowment Together these two conditions imply a set of relativeprices such that excess demands are zero or
cdi = ci i = 1 T
For questions concerning the existence uniqueness and stability of generalequilibrium consult Varian (1992) Debreu (1959) and Arrow and Hahn (1971)
Walrasrsquo law and Sayrsquos law
Note that the above statement of general equilibrium involves setting T excessdemand equations equal to zero but there are only T minus 1 unknowns (relativeprices) Walras solved this problem by showing that one of the equations arbitrarilychosen can be deduced from the other T minus 1 equations In other words there areonly T minus 1 independent equations To show the dependency remember that thebudget constraint for each individual is given by
Tsumi=1
πicdai minus
Tsumi=1
πi cai = 0
Summing across all individuals it must then be the case that
msuma=1
(Tsum
i=1
πi cai minusTsum
i=1
πicdai
)= 0
Rearranging and substituting in cdi for
summaminus1 cd
ai and ci forsumm
aminus1 cai we obtain
Tsumi=1
πi(cdai minus cai) = 0 or
Tsumi=1
πizi = 0
The above is an explicit statement of Walrasrsquo law Walrasrsquo law states that the sumof the excess demands across all markets must be zero Note that in summing theexcess demand of each commodity is weighted by its relative price so that we aresumming common units (ie all excess demands are in units of the numeraire)The above aggregate budget constraint is sometimes referred to as Sayrsquos law orSayrsquos identity If there is a distinction between the two it is that Sayrsquos law explicitlyexcluded money as a ldquomarketrdquo In this setting one can understand how Sayrsquosconclusion that ldquosupply creates its own demandrdquo is arrived at in an economycomposed of a single aggregate commodity
Walrasian economy 17
Conclusion
This chapter has developed a general equilibrium framework that sets the stagefor a thorough understanding of how the macroeconomy works Particular atten-tion has been paid to the development of relative prices and the development ofaggregate demand through a process of many individual consumers operating in anenvironment in which they set out to maximize their own utility This frameworkand the tool of constrained optimization is used throughout this book
3 Firms as market participants
Introduction
In this chapter the simple Walrasian model is discussed in the context of moneyfinancial assets and production The chapter clearly illustrates the firmrsquos objectivethat is to maximize profits However the firm is constrained in that it must financepurchases of capital and equipment as well as pay its workers Moreover attentionis paid to all the costs faced by the firm not just the obvious ones The investmentand financing decisions of firms are discussed and issues related to Tobinrsquos Q anddebt-to-equity are explored This chapter provides a detailed examination of therole that firms play in the macroeconomy
A simple Walrasian model with money financialassets and production
In the exchange economy we just considered endowments of the commodity goodwere magically bestowed on individuals each period We now introduce productionas the source of commodities At the start of each period there now exists a ldquolaborrdquomarket in which labor services are exchanged New agents denoted ldquofirmsrdquo hirethe labor services provided by households During the period firms combine thelabor services with an existing capital stock to produce output (commodities)which is sold in the output market net of output retained to replace capital usedup during production Revenues from the sale of output are distributed to ldquohouse-holdsrdquo in the form of wages during the period At the end of the period interestpayments and dividends are made to households out of revenues Each period firmsalso enter the output market to augment their capital stock with such purchasesfinanced by the issue of bonds and a new financial asset denoted ldquoequity sharesrdquoIn particular we assume
Assumption 31 There are new agents in the economy denoted as ldquofirmsrdquo Theseagents are initially endowed at time t with a capital stock K and a technologyfor transforming capital services from a capital stock and labor services into a
Firms as market participants 19
single ldquocompositerdquo commodity The technology is summarized by the productionfunction
yt = f (Nt K)
where K denotes the capital stock the firm inherits at time t Nt denotes the employ-ment of labor services arranged at time t for period t (from time t to time t + 1)and yt denotes the constant rate of production of the commodity for period t (fromtime t to time t + 1)1 Similarly for period i (i = t + 1 t + 2 ) which runsfrom time i to time i + 1 output produced is given by2
yi = f (Ni Ki)
Assumption 32 During each period firms sell the output produced in the outputmarket For output produced during period t (from time t to time t+1) let pt denoteits price when it is sold during the period up to and including time t + 1 Let pt+1denote the price of output produced during period t + 1 that is sold beyond timet +1 up to and including time t +2 and so on At time t the price level associatedwith the prior period is denoted by p
Assumption 33 At the start of each period households rent their labor servicesto firms for the period At the start of period t agreements to exchange the Ntlabor services during period t (from time t to t + 1) are entered into at the moneywage rate denoted by wt Similarly at the start of period t + 1 Nt+1 labor servicesare exchanged at the money wage rate wt+1 and so on3
Assumption 34 At the end of each period two types of financial assets areexchanged bonds (in the form of perpetuities) and equity shares Bonds promiseto pay a fixed money (coupon) payment z each future period in perpetuity Let pbpbt and pbt+1 denote the money price of such bonds in markets at the end of periodstminus1 t and t+1 respectively the gross (nominal) interest rates over period t (fromtime t to time t + 1) and over period t + 1 (from time t + 1 to time t + 2) are thengiven by4
1 + r = (z + pbt)pb
1 + rt = (z + pbt+1)pbt
Note that if ri = rt i = t + 1 then successive substitution for the priceof bonds in future periods will result in the following expression for the price ofbonds at time t
pb = 1
1 + r(z + pbt) = 1
1 + r
[z +
infinsumi=1
z
(1 + rt)i
]
= 1
1 + r
(z + z
rt
)
20 Firms as market participants
where pbt = zrt The number of previously issued bonds outstanding at time t isdenoted by B5
Assumption 35 Equity shares are the second type of financial asset exchangedat the end of each period Equity shares (ldquostocksrdquo) are contracts that obligate theissuer (firms) to pay to bearers at the end of each future period the income from thesale of output produced during the period net of other contractual obligations ofthe firms (eg wage payments to suppliers of labor services and interest paymentsto holders of bonds issued by firms) Let S denote the number of previously issuedequity shares outstanding at time t Holders of these equity shares are the ldquoownersof the firmsrdquo Let pe pet and pet+1 denote the money price of equity sharesexchanged in markets at the end of periods t minus 1 t and t + 1 respectively Thegross (nominal) rate of return on equity shares over period t (from time t to timet + 1) and period t + 1 (from time t + 1 to time t + 2) is then given by6
1 + re = [( ptdtS) + pet)]pe
1 + ret = [( pt+1dt+1St) + pet+1)]pet
where ptdt and pt+1dt+1 denote total nominal dividend payments made at the endof periods t and t + 1 (at time t + 1 and time t + 2) respectively St denotesthe anticipated number of equity shares outstanding after the equity market at theend of period t7 Note that the price of an equity share indicates the fact that thepurchase of an equity share entitles the holder to a portion of future (not current)dividends
Assumption 36 Households view bonds and equity shares as perfect substitutesldquoPerfect substitutesrdquo means that if equality in yields did not hold households wouldrefuse to purchase the asset with the lower yield forcing an adjustment in its pricethat would result in equivalent yields8 With bonds and equity shares as perfectsubstitutes we can speak of a single ldquofinancial asset marketrdquo that incorporatesboth bonds and equity shares and determines a single ldquointerest raterdquo
Assumption 37 There are incomplete markets Let pei i = t + 1 t + 2
denote the expectation formed in period t concerning the price of the consumptiongood over period i9 Similarly in period t we have pe
ei i = t + 1 and wei
i = t + 1 Such expectations are assumed to be held with subjective certaintyallowing us to abstract from risk considerations for the moment
Assumption 38 There are positive transaction costs to arranging exchanges ofthe consumption commodity during each period t Money holdings serve to reducethe transaction costs of arranging exchanges during a period10
Assumption 39 It is prohibitively costly for individuals to directly store theldquocompositerdquo commodity for consumption in future periods However output notconsumed during the period can be transformed (by ldquofirmsrdquo) into output in thesubsequent periods through the augmentation of the capital stock which permitshigher rates of production of output in future periods
Firms as market participants 21
Individual experiments firms
As always we start our analysis at the individual level The behavior of two typesof agents must now be considered ndash firms and households We start with firmsIn doing so we consider a ldquorepresentativerdquo agent a unit whose behavior exceptfor scale is identical to the behavior of the aggregate of such units Thus the samenotation will be used to represent both the individual unit and the aggregate of allunits In addition we consider an infinite time horizon
You should now recognize that an analysis of a representative unit neglectscertain potentially important ldquodistributionalrdquo aspects of the problem For instancefor households the ldquoreal indebtedness effectsrdquo of a price change on demand maynot be offsetting in the aggregate but that potential impact is ignored11 For firmsthe distribution of the initial capital stock can given adjustment costs affect totalemployment and output but that too is ignored12
To consider the behavior of a firm (or more specifically the manager whodirects production for the representative firm) we assume
Assumption 310 Technology is represented by the concave production function
yt = f (Nt K)
where yt denotes the firmrsquos planned (at time t) constant rate of output for the timeperiod from time t to t +1 to be sold during the period and at time t +1 Nt denotesthe firmrsquos planned (at time t) rate of employment of labor during period (t t + 1)with labor services purchased in the labor market at time t and K denotes thefirmrsquos planned capital stock for period t Recall that to simplify matters we takethe capital stock for the current period K as fixed at the individual firm levelThis would be the case given appropriate capital adjustment costs13
Assumption 311 The representative firm will choose the most preferred inputcombination and implied output given technology and prices (both current andanticipated future prices) At time t the objective of the firm is to maximize theexpected real market value of the S equity shares
Vt = peS
p
where pe is the price of equity shares at the end of period t minus 1 (at time t)14
A restatement of the firmrsquos objective
We have indicated that the objective of the firm at time t is to maximize the realmarket value of the S equity shares as given by
Vt = peS
p
22 Firms as market participants
To understand what underlies this market value of the firm we have to definethe elements underlying the price of equity shares and dividends We start byexamining what lies behind the price of equity shares
The assumption that equity shares and bonds are perfect substitutes means thatthe price of an equity share can be expressed as
pe = [ptdtS + pet](1 + r)
where dt denotes real dividends at the end of period t so that ptdt denotes nominaldividends S denotes the number of equity shares outstanding at time t pet is theprice of equity shares at the end of period t and r denotes the interest rate overperiod t (ie from time t to t + 1)
By successively substituting in a similar expression for the price of equity sharesin the next period we obtain for an infinite horizon that15
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
That is the price of an equity share at the end of period t minus1 (at time t) pe reflectsthe anticipated discounted future stream of nominal dividends per share16
Since the real value of the firm is given by Vt = peSp we can now expressthe value of the firm as
Vt = [Sp(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
which means that the objective of the firm can be stated in terms of maximizingthe discounted stream of current and future dividends Before examining whatdetermines dividends each period let us simplify the above expression for Vtby putting it in terms of real dividends each period To do so note that bydefinition
pt equiv p(1 + π)
pt+k equiv p(1 + π)
⎛⎝ kprod
j=1
(1 + rt+jminus1)
⎞⎠ k = 1 2 3
where πt+j denotes the rate of change in the price level between period t + j andt + j + 1 Thus we have
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
Firms as market participants 23
where R = (1 + r)(1 + π) and Ri = (1 + ri)(1 + πi) denotes the real gross rateof interest for period i (from time i + 1 to time i + 2 i = t t + 1 ) Our nextstep in outlining the firmrsquos problem is to obtain an expression for real dividendsin each period
Dividends and the firm distribution constraint
In general we may denote real dividends at the end of any period as the differencebetween a firmrsquos total real revenues during the period and the total costs incurredduring the period Total revenues derive from the sale of output produced duringthe period In addition one could add revenues from a change in the numberof equity shares outstanding or a change in the number of bonds outstanding atthe end of the period17 Total costs to the firm in any period include the agreedupon wage payments to the labor hired during the period payments to replacedepreciated capital plus coupon payments at the end of the period to holders ofpreviously issued bonds Payments at the end of the period for the purchase ofcapital and associated adjustment costs could be counted as well18 That is firmsare constrained to have
dividends = revenue from sale of output
minus wages
minus interest payments
+ funds from change in outstanding bonds and stocks
minus costs to replace depreciated capital add new capital
and capital adjustment costs
The above constraint is typically divided into two separate constraints Oneconstraint earmarks funds raised from the change in outstanding bonds andequity shares at the end of the period to pay for or ldquofinancerdquo the installationof new capital stock during the period plus any capital adjustment costs Thisis the ldquofirm financing constraintrdquo The remaining revenues minus expendituresthen determine the level of dividends This part is called the ldquofirm distributionconstraintrdquo
The firm distribution constraint simply states that the revenues from the saleof output that exceed expenditures to meet wage payments purchases of capitalto replace that used up in the production process and interest payments to bondholders at the end of the period are distributed at the end of the period to householdsas dividends Thus real dividends at the end of period t are given by
dt = yt minus (wtpt)Nt minus zBpt minus δK
According to this expression real dividends at the end of period t equal realrevenues derived from the sale of output yt produced during period t minus costs
24 Firms as market participants
to the firm during period t that reflect the real wage wtpt times the quantity oflabor hired Nt less real coupon payments at the end of the period on previouslyissued bonds zBpt plus purchases of capital during the prior period to replacethat used up in the production process δK 19
In similar manner real dividends for periods t + 1 and t + 2 (paid at the end ofeach period) are given by
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
We can rewrite the above definition of dividends to see more clearly why itis also termed the ldquofirm distribution constraintrdquo This constraint simply says thatrevenue from the sale of output net of that retained to replace capital used up inthe production process (ie ldquonet productrdquo) is distributed to households either aswage payments interest payments or dividends That is
dt + (wtpt)Nt + zBpt = yt minus δK
dt+1 + (wt+1pt+1)Nt+1 + zBtpt+1 = yt+1 minus δKt+1
The firm financing constraint
The second part of the general constraint that firmsrsquo total expenditures equalrevenues is that changes in the firmrsquos holdings of capital as well as any asso-ciated adjustment costs are financed by a change in outstanding equity sharesandor bonds This linking of funding for capital purchases to the issuing of equityshares and bonds is denoted the ldquofirm financing constraintrdquo For instance at theend of periods t + 1 t + 2 we have the following firm financing constraints
Kt+1 minus K + ψ(Int) = [ pet middot (St minus S) + pbt middot (Bt minus B)]pt
Kt+2 minus Kt+1 + ψ(Int+1) = [ pet+1 middot (St+1 minus St) + pbt+1 middot (Bt+1 minus Bt)]pt+1
where ψ(Ii) denotes the costs of installing new capital at rate Ii during period i(between time i and time i + 1) a cost that depends directly on the rate of netinvestment (Ini = Ki+1 minusKi) planned at time t to occur between time i and i+120
Recall that we assume that firmsrsquo plans with respect to the number of stocks andbonds that will be outstanding following the financial market at the end of period i(time i + 1) mirror householdsrsquo expectations concerning the number of bonds andstocks that will be outstanding so we do not distinguish between firmsrsquo plans andhouseholdsrsquo expectations with respect to these variables
Firms as market participants 25
Since bonds and equity shares are perfect substitutes we can rewrite the abovefirm financing constraints in the simpler form
Int + ψ(Int) = At minus At = net At
Int+1 + ψ(Int+1) = At+1 minus At+1 = net At+1
where Ini denotes net real investment (ie Ki+1 minusKi) Ai denotes the real planned(at time t) value of total equity shares and bonds to be outstanding after the financialmarket at the end of period i and Ai denotes the initial real value of equity sharesand bonds for period i reflecting financing and capital decisions in prior periodsbut period i prices For instance
At = [ petSt + pbtBt]pt and At = [ petS + pbtB]pt
At+1 = [ pet+1St+1 + pbt+1Bt+1]pt+1 and
At+1 = [ pet+1St + pbt+1Bt]pt+1
There are several aspects of interest with respect to the firm financingconstraints First note that net capital purchases planned for period i to be installedbetween time i and time i + 1 are paid for at the end of period i when completelyinstalled from the sale of financial assets at that time
Second note that firms purchase the output of the composite good to augmentthe capital stock That is we have a ldquoone-sectorrdquo model in which the same goodserves both households (for consumption) and firms (for investment) There isonly a single commodity price A typical extension is a two-sector model in whichtwo goods are produced a consumption good and a capital good In such casesa new variable the relative price of the capital good in terms of the consumptiongood is introduced
Third note that the firm financing constraint holds whether the firm financescapital with new bonds new equity shares or ldquoretained earningsrdquo Suppose forinstance that a firm plans to add 100 units to its capital stock by buying a newpiece of machinery If the firm issues a bond with real value of 100 to pay for themachinery then there is a direct 100 unit increase (the new bond) in the value of thefinancial assets issued by the firm Note that the value of the current shareholdersrsquostock is unchanged in this case of bond financed investment While it is true thatthe tangible assets of the firm have increased by the 100 addition to capital thisbenefit to shareholders is exactly offset by the fact that the firmrsquos debt has alsoincreased by 10021
If the firm finances the 100 net investment by issuing new shares of stock equalto 100 again there is a 100 unit increase (the new equity shares) in the real valueof the financial assets issued by the firm As with bond financing however thereal value of the initial shareholdersrsquo stock is unchanged when the firm finances itscapital purchases by issuing new equity shares The new shares do not dilute thevalue of the shares of the initial shareholders since the capital purchase increases
26 Firms as market participants
the firmrsquos tangible assets by 100 which is exactly the real value of the new equityshares issued
Finally if the firm finances the 100 net investment through retained earningsthere is in essence a 100 unit increase in the value of the financial assets issued bythe firm for the following reason When a firm retains earnings in order to financea capital purchase the current stockholders own the right to the income generatedfrom the additional capital As a consequence the value of their equity shares risesto reflect the value of the new capital owned by the firm We could equivalentlyview this as the firm paying out 100 units in dividends to its initial shareholderswho then use the dividends to buy additional ldquoconstant valuerdquo equity shares equalto the value of the capital purchased by the firm In other words when the firmuses retained earnings to finance its investment spending it is implicitly issuingnew financial assets ndash equity shares
The nature of capital adjustment costs
An important aspect of the above financing constraint is that it incorporatespotential adjustment costs to purchases of capital as captured by the terms ψ(middot)which depend on Int Int+1 where Ini denotes the planned (at time t) net rateof investment for period i22 The total cost of capital purchases is thus the sum of(a) the real payments (or receipts if negative) involved in the purchase (or sale ifnegative) of capital in the output market and (b) potential real payments denotedldquoinstallationrdquo or adjustment costs associated with new capital acquisitions Forperiod t adjustment costs are given by ψ(Int) where Int denotes net investmentbetween time t and t + 1 Gross investment for period t is given by Int + δK Notethat we can thus decompose gross investment over the period into the change inthe capital stock Kt+1 minus K which is termed ldquoplanned net investmentrdquo and thereplacement of capital used up in the production process δK which is termedldquodepreciationrdquo23
To understand the conversion of the above analysis to continuous time we notethat in general adjustment costs over period t of length h are given by hψ(Inth)where the limit of the term Inth defines the rate of net investment That is incontinuous time the planned rate of investment would be defined by the rate ofgross investment
it = limhrarr0
It
h= lim
hrarr0
Kt+h minus K + hδK
h= Kt + δKt
and the adjustment cost function in continuous time would be ψ(int)For the aggregate rate of gross investment (a flow) to be defined by the above
expression K must equal K To achieve this one of two approaches is typicallytaken One approach assumes zero adjustment costs in that ψ equiv 0 This situationsometimes referred to as the case of ldquoperfect malleabilityrdquo means that the rateof investment may not be defined at the level of the individual firm That is ifthe existing capital stock were higher or lower than the planned level investment
Firms as market participants 27
would be infinitely positive or negative However it can be shown that withzero adjustment costs the output market in a continuous-time model at a pointin time is simply a ldquocapital marketrdquo and the expression Kt = K emerges as anequilibrium condition with respect to the capital market at time t24 Thus wecan apply LrsquoHospitalrsquos rule to define the aggregate rate of (net) investment in thecontinuous-time model with zero adjustment costs as25
it = limhrarr0
Kt+h minus K
h= limhrarr0 d(Kt+h minus K)dh
limhrarr0 dhdh= K
where it is rate of investmentIn contrast to the case of zero adjustment cost one can assume adjustment costs
that take the following form
ψ(0) = 0
ψ(β) gt 0 if β gt 0 ψ primeprime gt 0 and limβrarrinfin ψ(β) = infin
ψ(β) lt 0 if β lt 0
The above set of assumptions reflects the presumption that adjustment costsincrease at an increasing rate with the rate of change in capital and that it isinfinitely costly to change the capital stock arbitrarily fast The result of suchadjustment costs in both discrete-time analysis and continuous-time analysis isthat at time t the firm chooses Kt = K That is at time t the firm views the inher-ited capital stock as optimal since it is prohibitively costly to change the capitalstock at a point in time given such adjustment costs26
As we will see with ldquocosts of installing a unit of new capitalrdquo there will be adifference between the market value of capital goods in place and their replacementcost In particular the ratio of these two values known as ldquoTobinrsquos Qrdquo will exceedone In addition adjustment costs will mean that the firmrsquos decision with respectto investment will not be myopic (ie plans will not be based on forecasts thatextend only one period into the future) Rather the firm will consider all futureperiods in making current investment decisions
The firm problem a general statement
One way to state the optimization problem faced by the firm is to say that attime t the firm makes plans with respect to current and future employment oflabor (Nt Nt+1 ) the future employment of capital (Kt+1 Kt+2 ) the stockof outstanding bonds (Bt Bt+1 ) and the stock of outstanding equity shares(St St+1 ) in order to maximize the real value of the previously issued equityshares outstanding at time t with that real value given in general form by
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
28 Firms as market participants
Such plans are subject to the combined distribution and financing constraints listedabove as well as to the production function That is in general the firmrsquos problemcan be stated as27
max(SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
subject to the financing constraints
minus [Kt+1 minus K + ψ(Kt+1 minus K)] + (petpt) middot [Bt minus B]+ (pbtpt) middot (St minus S)] = 0
minus [Kt+2 minus Kt+1 + ψ(Kt+2 minus Kt+1)] + (pet+1pt+1) middot [Bt+1 minus Bt]+ (pbt+1pt+1) middot (St+1 minus St)] = 0
and the distribution constraints
dt = yt minus (wtpt)Nt minus zBpt minus δK
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
and given the production functions
yt = f (Nt K)
yt+1 = f (Nt+1 Kt+1)
for i = t t + 1 The above problem has a recursive nature to it At the start of any given period
the firm inherits a stock of capital an outstanding stock of bonds and an outstand-ing stock of equity shares28 These variables are state variables Each period thefirm chooses a set of the ldquocontrolrdquo variables ndash employment investment Thesechoices in conjunction with the production function result in outcomes in termsof (a) a one-period return (dividends) at the end of the period and (b) a new setof ldquostaterdquo variables ndash capital stock and stock of financial assets (equity shares andbonds) ndash inherited in the subsequent period Further note that the objective of thefirm is additive in these one-period returns (dividends) Thus the problem is oneto which we can apply Bellmanrsquos dynamic programming technique
To reformulate the problem facing the firm as a dynamic programming prob-lem we use the conventional notation of dynamic programming problems29
Firms as market participants 29
Specifically the above problem can be viewed as involving
(a) A set of ldquocontrolrdquo variables each period
zt = Nt Int zt+1 = Nt+1 Int+1 etc
(b) A set of ldquostaterdquo variables each period
xt = K B S xt+1 = Kt+1 Bt St etc
(c) ldquoTransition functionsrdquo that link the choices specifically the choice of netinvestment during the period to the capital stock available at the start ofthe next period as well as the stock of financial assets outstanding in thesubsequent period For instance the choice of the net investment rate Int during period t dictates Kt+1 given K since
Kt+1 = Int + K
From the firm financing constraint we know that the choice of the investmentrate also determines the real stock of financial assets
( petpt)[Bt minus B] + ( pbtpt)[St minus S] = Int + ψ(Int)
(d) A set of one-period return functions (evaluated at the end of period t)30
rt(K B S) = dt rt+1(Kt+1 Bt St) = Sdt+1StRt etc
Since bonds and equity shares are perfect substitutes the above problem cannotbe solved for a unique optimal number of bonds or equity shares to have outstandingeach period Thus without any loss of generality we may restrict our focus toeither bond or equity share financing That is we can hold constant either bonds(ie Bi = B i = t t + 1 ) or equity shares (ie Si = S i = t t + 1 )Alternatively we can combine the distribution and financing constraints into asingle expression for dividends and hold constant both equity shares and bondsIn this case we have ldquoretained earningsrdquo financing of changes in the capital stockIt is this case that we consider below31
Simplifying the firm problem ldquoretained earnings financingrdquo
If we assume that capital expenditures are financed from ldquoretained earningsrdquo thereis a single state variable the stock of capital The problem facing the firm thencan be simply stated as follows The Bellman equation for period t given inheritedcapital stock K is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
30 Firms as market participants
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
Substituting the above definitions for real dividends and output into the Bellmanequation and substituting in the transition function the first-order conditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
Equation (31) is the standard condition that labor is employed up to the pointwhere the real marginal gain for an additional unit of labor in terms of the increasein output attained in the current period (ie the marginal product of labor partftpartNt)equals the real marginal cost as reflected by the real wage wtpt Equation (32)indicating the optimal choice of investment is discussed in the next section
Optimal investment (and the future capital stock) zeroadjustment costs
To express the optimal condition for investment and thus the future capital stockin a more transparent form we need to expand upon the effect of an increase inthe capital stock on the value function for period t + 1 In other words we need toclarify the nature of the term dW (Kt+1)dKt+1 in Equation (32) To do so let usconsider the Bellman equation for period t + 1 To simplify matters we initiallyfocus on the case of zero adjustment costs (ie that ψ equiv 0 implying that ψ prime
i = 0i = t t +1 ) Given the inherited capital stock Kt+1 for period t +1 and a fixedstock of equity shares the Bellman equation for period t + 1 is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1(Rt)minus1 + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and (assuming retained earnings financing of capital changes and zero adjustmentcosts) the following definitions for real dividends and output for period t + 1
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBtpt+1 minus Int+1
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 31
Thus we have32
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (33)
We can use first-order conditions for the Bellman equation for period t + 1 toclarify the nature of dW (Kt+2)dKt+2 in Equation (33) In particular we havethat the optimal choice of investment in period t + 1 satisfies
minus 1
Rt+ dW (Kt+2)
dKt+2= 0 (34)
Substituting (34) into (33) we obtain the following expression for the effect of achange in the inherited capital stock on the value function for period t + 1
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt (35)
Substituting (35) into equation (32) and recalling our assumption that ψ prime = 0we thus have that the optimal choice of investment in period t satisfies
minus1 + 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt= 0 (36)
The above expression can be rearranged to obtain
partft+1
partKt+1= mt minus δ (37)
where mt Fisherrsquos expected real rate of interest equals Rt minus1 or (rt minusπt)(1+πt)The interpretation of (37) is fairly straightforward Each period the firm chooseslabor and capital such that the marginal gain in the subsequent period in terms ofincreased output equals the real marginal cost For capital the marginal cost is therate of depreciation plus the expected real rate of interest An explanation of thisreal ldquouserrdquo or ldquorentalrdquo cost of capital follows
Over period t the firm pays for one unit of capital at price pt Since the firmcould have instead used these funds to reduce the outstanding stock of bonds by pt the cost of this capital (reduced dividends) in nominal terms is pt(1 + rt) In realterms the cost one period later is anticipated to be pt(1 + rt)pt+1 After oneperiod 1 minus δ of the capital remains so that the sale of the remaining capital afterone period of use (or the reduced purchases of new capital) reaps a nominal returnof (1 minus δ)pt+1 and real return (1 minus δ) The real rental cost of the unit of capital isthus33
pt(1 + rt)pt+1 minus (1 minus δ) = (1 + rt)(1 + πt+1) minus 1 + δ
= (rt minus πt)(1 + πt) + δ
= mt + δ
32 Firms as market participants
Summarizing our discussion in the case of zero adjustment costs the optimalbehavior of firms in periods t and t +1 is given by the following demand functionsfor period t and t + 1
N dt = N d(wtpt K)
I dnt = I d
n (mt + δ wt+1pt+1 K)
where I dnt = Kd
t+1minusK and Kdt+1 = Kd
t (mt+δ wt+1pt+1) Note that the anticipatedreal wage next period affects Id
nt since changes in the real wage affect the employ-ment of labor and thus assuming part2f partNpartK does not equal zero the marginalproduct of capital A similar statement explains why K enters as an argument inthe labor demand function
An important feature of the above is that planned investment demand when thereare zero adjustment costs simply depends on adjacent expected real user costs ofcapital and real wages This reflects the fact that with zero adjustment cost capitaldemand is a function of the expected real user cost of capital and the real wageover the next period alone
Financing choices and different debt-to-equityratios a digression
We have characterized the above choice of the capital stock under the presumptionthat the firm finances capital purchases through retained earnings Yet we can showthat the planned (at time t) choice of the optimal capital stock at time t+1 t+2 is independent of the method of financing given that (a) bonds and equity sharesare assumed to be perfect substitutes and (b) there is no cost to arranging theexchange of financial assets (otherwise the retained earnings financing method ispreferred) This result is sometimes referred to as the ldquoModiglianindashMiller theoremrdquowhich states that the total value of the firm is independent of its financial structureThat is the present value of the stream of dividends to the initial owners is inde-pendent of how liabilities are divided between bonds and equity shares The resultis that the capital structure is indeterminant
The view that the method of financing capital purchases is largely irrelevant isa very useful simplification for macroeconomic analysis However you should beaware of some complicating factors that we are ignoring factors that can causefirms to care about the method by which they finance their capital purchases
When a firm issues bonds the value of its outstanding debt rises When it issuesstocks the value of its outstanding equity shares increases Thus the method offinancing capital purchases affects what is known as the firmrsquos debt-to-equityratio34 Financing capital purchases with bonds will increase the firmrsquos debt-to-equity ratio while financing capital purchases with equity shares (eitherexplicitly or implicitly by using retained earnings) will reduce the firmrsquos debt-to-equity ratio Two factors that can influence a firmrsquos desired ldquocapital structurerdquoare tax considerations and bankruptcy costs35
Firms as market participants 33
The corporate taxes that firms pay are calculated as a percentage of earningsFor tax purposes corporate earnings are equal to total revenue net of costs wherecosts are calculated as including not only wages and payments for raw materi-als and intermediate goods but also interest payments to bondholders If a firmfinances its purchases of capital using bonds the interest it pays in the futurewill reduce its taxable earnings and thus the taxes that it has to pay This meansthat a firm can lower its future tax liability by raising its debt-to-equity ratio ndashthat is by financing new capital purchases with new bonds rather than equityshares
Raising the debt-to-equity ratio however is generally not without costs whichare typically referred to as ldquobankruptcy costsrdquo Unlike equity shares which promiseshareholders dividend payments if profits are sufficiently high bonds promise fixedpayments to their holders Greater debt thus increases the fixed obligations thatfirms must meet in the future This means that a fall in future revenues is morelikely to force the firm into bankruptcy
Bankruptcy occurs when a firmrsquos revenues do not cover its costs and it isforced to default on its obligations to bondholders Associated with bankruptcyare bankruptcy costs the most obvious being the hefty legal costs associated witheither reorganizing or undergoing a court-supervised liquidation The existence ofbankruptcy costs serves to limit the amount of borrowing a firm will undertake Itwill hesitate to increase its debt-to-equity ratio beyond some level since the gainin tax savings will be offset by the costs associated with an increased likelihoodof incurring bankruptcy costs
To summarize a firm can be viewed as having an optimal debt-to-equity ratiothat reflects a tradeoff of tax and bankruptcy cost considerations Table 31 liststhe general level of debt-to-equity for a sample of industries in US manufacturingNote that the debt-to-equity ratios vary widely among the industries in the sampleranging from significantly over one to significantly less than one The ratio ishighest in the steel industry where the value of debt is close to 17 times the
Table 31 Debt-to-equity ratios across select industries
Book value Market valueof equity of equity
Steel 1973 1665Petroleum refining 1548 1117Textiles 1405 1296Motor vehicles 0922 0594Plastics 0843 0792Machine tools 0472 0425Pharmaceuticals 0194 0079
Source Kester (1986) The book value of equity is computed fromaccounting sources while the market value of equity is obtained bymultiplying the number of outstanding shares by the current marketprice of the outstanding shares
34 Firms as market participants
market value of outstanding market shares In contrast for pharmaceuticals thedebt-to-equity ratio is only 0079 indicating that the industry uses bond financingvery little instead financing its investment activities almost exclusively throughthe issuance of equity
Adjustment costs for capital and Tobinrsquos Q
Let us now consider the choice of capital when there exist adjustment costs Tokeep the maximization problem simple we shall continue to assume ldquoretainedearningsrdquo financing of changes in the capital stock we would however obtainidentical results with bond or equity share financing As we have seen in the caseof retained earnings financing the problem facing the firm is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t36
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
As before substituting the above definitions for real dividends and output intothe Bellman equation and substituting in the transition function the first-orderconditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
The problem facing the firm in period t+1 assuming retained earnings financingof capital changes is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1Rt + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and given the following definitions for real dividends and output for period t37
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBpt+1 minus Int+1 minus ψ(Int+1)
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 35
Again we can substitute the above definitions for real dividends and output intothe Bellman equation for period t+1 and substituting in the transition function (inparticular note the fact that dKt+2dInt+1 = 1) obtain the following first-orderconditions
partft+1
partNt+1minus wt+1
pt+1= 0 (38)
minus (1 + ψ primet+1) + dW (Kt+2)
dKt+2= 0 where ψ prime
t+1 = dψ(Int+1)
dInt+1 (39)
Finally from our expression for the value function at time t + 1 W (Kt+1) wehave that
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (310)
Substituting (39) into (310) and then substituting the resulting expression fordW (Kt+1)dKt+1 into the first-order condition for Int (equation (32)) we obtain
minus(1 + ψ primet ) + 1
Rt
(partft+1
partKt+1 minus δd
)+ 1 + ψ prime
t+1
Rt= 0 (311)
Rearranging and simplifying we have that
partft+1
partKt+1= mt + δ + (mt + 1)ψ prime
t minus ψ primet+1
where ψ primet = ψ prime(I d
nt) and ψ primet+1 = ψ prime(I d
nt+1) An important feature of adjustmentcosts that is highlighted by the above equation is that the choice of investment inperiod t is now linked to the optimal choice of investment next period Since thisholds for each period in the future the choice of investment today is linked toinvestment decisions over all subsequent periods
We can simplify and rearrange the above first-order condition for investment inperiod t to obtain what is known as ldquoTobinrsquos Qrdquo To do so let us first assume anidentical real rate of return over time in particular we then have Ri = Rt = 1+mi = t + 1 t + 2 where (1 + m) equiv (1 + r)(1 + π) Let us also assume thefirm has attained its optimal capital stock so that Kd
t+1 = K and I dnt = 0 If the
production function is separable into capital and labor then the assumption of aninvariant real interest rate implies that I d
ni i = t +1 t +2 equal zero as well38
In this case ψ primet and ψ prime
t+1 can be replaced by a common ψ prime(0) gt 0 Then we maywrite the first-order condition as
partft+1partKt+1 minus m minus δ = ψ primem
Dividing by m and adding one to both sides we have
Tobinrsquos ldquomarginalrdquo Q equiv 1 + [partft+1partKt+1 minus m minus δ]m = 1 + ψ prime gt 1(312)
36 Firms as market participants
The above provides the definition for Tobinrsquos Q39 More precisely we have Tobinrsquosldquomarginalrdquo Q for it represents the ratio of the market value of an additional unitof capital to its replacement cost
Given adjustment costs the market value of an additional unit of capital exceedsits replacement cost so Tobinrsquos marginal Q is greater than one Since investmentdemand planned over the coming period determines marginal adjustment costsψ prime(I d
nt) we see that investment can be written in terms of Tobinrsquos Q40 The optimalrate of investment is that rate for which Q minus 1 is equal to the marginal cost ofinstallation Thus net investment demand is sometimes expressed as
Idnt = I d
nt(Q minus 1)dId
nt
d(Q minus 1)gt 0
The Q theory of investment is not operational as long as Q is not observableWhile marginal Q is not typically apparent with some additional assumptionswe can show that the expression known as Tobinrsquos ldquoaveragerdquo Q is identical toldquomarginalrdquo Q Tobinrsquos ldquoaveragerdquo Q is defined as the ratio of the total value of thefirmrsquos existing capital (the market value of its equity shares) to its total replacementcost and these variables are more easily measured41 In particular for a one-sectormodel in which the cost of capital and output are identical42
Tobinrsquos ldquoaveragerdquo Q equiv V
K
Hayashi (1982) has shown that if the firm is a price-taker if the production func-tion is linear homogeneous in K and N and if expectations of future real interestrates and real wages are static then the marginal and average Q are identical43 Tosee why this is the case note that if the real interest rate and real wage are invariantand the firm is at the optimal level of capital so that the capital stock is invariantover time then omitting time identifiers we have
V =infinsum
k=1
dRk
where (since ψ(Idnt) = ψ(0) = 0)
d = y minus (wp)N minus δK
Thus
V = [y minus (wp)N minus δK]m
where m equiv R minus 1 By Eulerrsquos theorem for a linear homogeneous production func-tion (ie K(partf partK) = y minus N (partf partN )) and the marginal productivity condition
Firms as market participants 37
for labor (ie wp = partf partN ) which reflects the price-taker assumption the firsttwo terms in the expression become K(partf partK)44 Dividing by K we thus obtain
Tobinrsquos ldquoaveragerdquo Q equiv V K = [partf partK minus m minus δ]m + 1
which as we can see is the same expression as that for Tobinrsquos marginal QAlternatively we can write the above as
V = [partf partK minus m minus δ]m + 1 = (Q minus 1) ([partf partK minus m minus δ]m)
Adjustment costs for labor labor as a ldquoquasi-fixed factorrdquo
Our prior characterization of the optimal choice of labor reflects an underlyingproduction process that incorporates a very simple view of labor For instancelabor markets are restricted to be only spot markets That is we rule out multiperiodlabor contracts Yet there is an extensive body of literature that investigates variousrationales for and the implications of such multiperiod (implicit) labor contractsOne reason why long-term contracts might emerge is an attempt by firms who areless risk-averse than workers to smooth out income over time45
A second reason why multiperiod labor contracts might emerge is if there areldquoadjustment costsrdquo to changes in the size of the labor force Adjustment costs couldreflect the fact that in order to hire new workers firms must incur hiring and trainingcosts Adjustment costs mean that a firm would view potential new hires andpreviously employed workers as imperfect substitutes and this would provide animpetus for multiperiod labor contracts The absence of adjustment costs simplifiesthe analysis by eliminating a rationale for multiperiod labor contracts and a choiceof labor given adjustment costs It also simplifies the analysis in two other ways
First the absence of adjustment costs suggests that we can measure labor ser-vices as the product of the fraction of the period each labor supplier works andthe number of individuals hired That is given no adjustment costs differences inldquohours workedrdquo and ldquonumber employedrdquo that leave total work hours unchanged areviewed by the firm as equivalent in terms of production In contrast with positiveadjustment costs firms would have a preference for meeting temporary changesin output by changing hours rather than by changing the number employed
A second implication of the absence of adjustment costs is that the employerviews as equivalent two workers working at ldquohalf-speedrdquo (and receiving ldquohalf-wagesrdquo) and one worker working at ldquofull-speedrdquo for ldquofull-wagesrdquo In contrastgiven adjustment costs firms would have a preference for meeting temporaryoutput changes by altering not only hours per worker but also the intensity that eachemployee was asked to work In fact output per work hour or ldquolabor productivityrdquodoes typically increase more rapidly during a recovery suggesting a more intensiveuse of labor
In contrast labor productivity growth is typically less rapid when the growth intotal output slackens This phenomenon is due in part to employersrsquo hoarding laborin slack times so as not to lose trained employees whom they will want when there
38 Firms as market participants
is an upturn in demand (That is to reduce subsequent adjustment costs given afuture upturn in production) Hoarding labor means that employers keep on moreworkers than necessary to produce the current output so that each worker has lesswork to do than normal The labor hoarding phenomenon also referred to as thelabor reserve hypothesis is the formal term for changes in the ldquointensityrdquo at whichlabor is used and explains lower output per work hour during periods of slackdemand46
Conclusion
The nature of the firm was discussed with the emphasis on profit maximizationDecisions of firm owners facing a variety of constraints and costs were analyzedwith particular attention paid to how financing constraints and adjustment costsaffected firm profits and the ability to adjust production levels A link was madebetween households and firms that will lead us into the next chapter Firms hireworkers who of course constitute households in the economy Workers are paidwages as costs to the firm but are an integral part of the production processMoreover firms may face costs of adjustment and other costly phenomena asso-ciated with decisions to alter the use of workers in the production process Takentogether then we see a link between firms and households as firm decisions havethe propensity to affect consumer income
4 Households as marketparticipants
Introduction
This chapter brings the household into our model of the macroeconomySpecifically the householdrsquos ability to obtain utility through consumption andthe labor supply decision is modeled within a choice framework The solutionis then used to formulate predictions about labor supply The concept of time iscritical to a thorough understanding of household behavior in the marketplace anda good deal of this chapter is spent analyzing intertemporal choices
The life-cycle and permanent income hypotheses are introduced and a theory ofportfolio choice is developed Finally the chapter ties many of these issues togetherand addresses the macroeconomic questions of absence of money illusion the realbalance effect and the real indebtedness effect
Individual experiments households
Two agents inhabit our expanded macroeconomic model with production firmsand households Having just discussed the nature of decisions confronting firmswe turn now to those confronting households These decisions can be broken downinto three types consumptionsaving portfolio and labor supply Consider nowthe first two decisions which should be familiar
The ldquoconsumptionsavingrdquo decision The representative household must deter-mine at time t the consumption purchases over each period at the implied ratect ct+1 We term this problem the ldquoFisherianrdquo problem
The ldquoportfoliordquo decision The individual must determine at time t the collectionof assets to hold at the end of each period In our expanded economy there areostensibly three types of assets
1 nominal money balances planned at time t to be held at the end of periodi Mi i = t t + 1
2 the nominal value of bonds planned at time t to be held at the end ofperiod i pbiBi i = t t + 1 where Bi denotes the planned number ofbonds held and pbi denotes the money price of bonds at the end of periodi and
40 Households as market participants
3 the nominal value of equity shares planned at time t to be held at the endof period i peiSi i = t t + 1 where pei denotes the money price ofan equity share at the end of period i and Si denotes the number of equityshares planned at time t to be held at the end of period i
Since bonds and equity shares are perfect substitutes we can consider them asa single entity with respect to householdsrsquo portfolio decisions We will let Aidenote the real holdings of financial assets planned at time t to be held at theend of period i i = t t + 1 That is for period t
At = [petSt + pbtBt]pt
and so on Excluding current dividend and interest payments the real value ofinherited financial assets at the end of period i reflecting portfolio decisions inthe prior period will be denoted by Ai i = t t + 1 For instance
At = [petS + pbtB]pt
At+1 = [pet+1St + pbt+1Bt]pt+1
The household problem
We start our analysis of the representative household as usual by discussing thehouseholdrsquos preferences constraints and objectives In particular we make thefollowing assumptions
Assumption 41 The representative householdrsquos preferences are described bythe utility function
u(ct ct+1 Mtpt Mt+1pt+1 1minusNt 1minusNt+1 )
where ci denotes the householdrsquos planned (at time t) rate of consumption dur-ing period i (from time i to time i + 1) Mi denotes the representative householdrsquosplanned (at time t) nominal holdings of money at the end of period i and Ni denotesthe householdrsquos planned (at time t) rate of supply of labor services during period i(from time i to time i +1) such that 1minusNi denotes the planned rate of leisure dur-ing period i It is assumed that partupartci gt 0 partupart(Mipi) gt 0 and partupart(1 minus Ni)
gt 0 i = t t + 1 Macroeconomics often assumes a time-separable utilityfunction a form of the utility function that ensures ldquotime consistencyrdquo1 In par-ticular following the tradition of macroeconomics we will assume that the totalutility for the representative household at time t with an infinite planning horizonis given by
infinsumi=t
β iminustu(ci Mipi 1 minus Ni)
Households as market participants 41
where β denotes the fixed personal or ldquoutilityrdquo discount factor with 0 lt β lt 12
In the above note that the one-period utility function for period t (time t to t + 1)is u(ct Mtpt 1minusNt) for period t +1 (time t +1 to t +2) it is u(ct+1 Mt+1pt+11 minus Nt+1) and so on Further note the infinite time horizon
Assumption 42 Individuals will choose the most preferred sequence of con-sumption money holdings and labor supply from the set of feasible alternatives(rationality) The feasible set of consumption money holdings and leisure
(ct ct+1 Mtpt Mt+1pt+1 1 minus Nt 1 minus Nt+1 )
is defined by the set of equalities
(wtpt)Nt + zBpt + At + Mpt minus [ct + Mtpt + At] = 0
(wt+1pt+1)Nt+1 + zBtpt+1 + At+1 + Mtpt+1
minus [ct+1 + Mt+1pt+1 + At+1] = 0
Note that we assume the budget constraints are met with equality Further note thatthe sum of dividends wage payments and interest payments equals total outputminus depreciation For instance for period t
(wtpt)Nt + dt + z middot Bpt = yt minus δK
and so on This is simply the firm distribution constraint for period t
Several aspects of the above problem deserve further elaboration First a wordon notation for future variables It is common in macroeconomics to derive themicroeconomic theoretical restrictions for the aggregate model under the conditionof certainty even though the analysis is then applied to situations that involvepotential stochastic elements One obvious way to eliminate considerations ofuncertainty from the analysis is to assume perfect foresight A second way is toassume that individual expectations of future events are point estimates held withsubjective certainty Note that either approach simplifies the analysis and in manycases this simplification gives us results that are not overturned if risk were to besystematically incorporated into the analysis
In the analysis of individual behavior below we will often for notationalsimplicity not distinguish between future prices and the expectations of futureprices Assuming expectations are held with subjective certainty this lack of dis-tinction will not be serious in discussing the result of the optimization problemsThat is the findings for perfect foresight can be made identical to those withoutthe assumption of perfect foresight by switching actual future prices for expectedprices Sometimes for clarity we will explicitly denote expected future prices(point estimates held with subjective certainty) by the superscript ldquoerdquo
42 Households as market participants
A second aspect of the above analysis that may initially appear odd concerningthe above one-period utility function for period i (from time i to time i+1) is that itseems that we are mixing money balances that occur at one time with consumptionand leisure that occur at an earlier time The reason for this is that money balancesare a stock variable and we are recording their value at the end of each period ithat is at time i + 13 The following scenario for the discrete-time analysis mayhelp clarify what is going on
At time t the labor market takes place and agreements are made to exchangelabor services at rate Nt over the period (t t + 1) for the money wage wt Duringthe period an output market operates in which firms sell output produced at rate yt During the period households receive money wages wt At the end of the period(time t + 1) households anticipate real interest payments zBpt from their priorpurchases of bonds (B) and real dividends dt from their prior purchase of equityshares stock (S)4 Given the above income sources as well as inherited nominalholdings of money (M ) and the anticipated value of inherited financial asset At atthe end of the period households plan an average rate of consumption ct duringthe period
At the end of period t after all income is received and final planned pur-chases of consumption goods are made the remainder reflects householdsrsquo planned(at time t) end-of-the-period change in real money balances ((Mt minus M )pt) andplanned changes in real financial assets holdings (At minus At) For financial assetsthe real price for bonds at the end of period t is pbtpt and the real price for equityshares is petpt
According to the above scenario the sale of labor services at rate Nt and rateof consumption ct over the period from time t to t + 1 tend to coincide whilereal money balances Mtpt and real financial asset holdings At can be viewed asthe planned (at time t) real stocks of such assets to be held at the end of period tIn continuous-time analysis as the length of the period h goes to zero the rate atwhich leisure is lost from supplying labor services during the period (Nt) the rateof consumption (ct) and the stocks of real money and real financial asset holdingswould coincide
A third aspect of the above analysis is that we have interpreted 1 minus Ni as theportion of the period of length 1 that the individual spends at leisure given thesupply of labor at rate Ni This is a simplification however for at the same timewe have suggested that the ldquoutility yieldrdquo of money is derived from its ability toreduce the transaction costs in arranging exchanges with such transaction costsreflecting at least in part a loss of leisure To explicitly incorporate such a viewof money leisure during a period i of length h given the sale of labor services Niand the real money balances Mipi held at the end of the period would be givenby h(1 minus Ni minus (Mipi)) where the function (Mipi) reflects transactions costsin terms of the loss of leisure The fact that prime lt 0 indicates that increased moneyholdings raise utility by reducing leisure lost in arranging transactions In this casethe one-period utility function would formally be given by u(ci 1minusNiminus(Mipi))with partupartci gt 0 and partupart(1 minus Ni minus (Mipi)) gt 0
Households as market participants 43
The general solution to the household problem
We can express the household problem in terms of a set of Bellman equationsAssuming perfect foresight (or equivalently interpreting future prices dividendetc as expectations of such variables held with subjective certainty) we thus havefor period t (time t to t + 1)
W (xt) = maxct Nt
Mtpt xt+1
[u(ct Mtpt 1 minus Nt) + W (xt+1)]
subject to the transition function
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt] Mtpt
and given xt Rmt is the real gross rate of interest on money that is the gross realrate of return on money and equals one divided by one plus the rate of inflation(1(1+πt)) The term xt is the total value in period t derived from the ldquoinheritedrdquoholdings of money bonds and stocks This total value is the sum of currentdividends and interest (received at the end of period t) on stock and bond holdingsacquired previously the real value of these financial assets at the end of period texclusive of these current interest and dividend payments and the real value ofpreviously acquired money holdings
xt equiv dt + zBpt + At + Mpt
where
At equiv [petS + pbtB]pt
The difference between the total real value derived in period t from inheritedbonds stocks and money balances plus real wage income xt + (wtpt)Nt andconsumption in period t ct reflects the acquisition of bonds equity shares andmoney holdings at the end of period t by the representative household Letting Atdenote the planned holdings of financial assets at the end of period t we thus havefrom the household budget constraint that
At + Mtpt = xt + (wtpt)Nt minus ct
where
At equiv [petSt + pbtBt]pt
Recall that Rt the gross real rate of return on financial assets equals one plus thenominal interest rate divided by one plus the rate of inflation ((1 + rt)(1 + πt))Thus in period t + 1 and given our definition of Rmt the inherited real value of
44 Households as market participants
bonds stocks and money balances including dividends and interest payments isgiven by
xt+1 = RtAt + RmtMtpt
Substituting in the expression for At derived from the household budget constraint(ie At = xt + (wtpt)Nt minus ct minus Mtpt) and rearranging we obtain the transitionfunction
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt]Mtpt
Substituting the transition function into Bellmanrsquos equation for period t wehave the following first-order conditions for ct Nt and Mtpt assuming interiorsolutions (ie ct gt 0 1 gt Nt gt 0 and Mtpt gt 0)
partutpartct minus (partW (xt+1)partxt+1)Rt = 0 (41)
partutpart(1 minus Nt) + (partW (xt+1)partxt+1)Rt(wtpt) = 0 (42)
partutpart(Mtpt) minus (partW (xt+1)partxt+1)(Rt minus Rmt) = 0 (43)
The above conditions indicate that for period t (time t to time t + 1) we havefrom equations (41) and (42) that
partutpart(1 minus Nt)
partutpartct= wt
pt (44)
In words the optimal choice of leisure is such that the marginal value of leisurein terms of consumption that is the marginal rate of substitution between leisureand consumption as given by
partutpart(1 minus Nt)
partutpartct
equals the marginal cost of leisure in terms of consumption forgone in the currentperiod as given by the real wage wtpt
From equations (41) and (43) we have for period t that
partutpart(Mtpt)
partutpartct= (Rt)
minus1(Rt minus Rmt)
In words the optimal choice of real money balances is such that the marginal rateof substitution between real money balances and consumption as given by
partutpart(Mtpt)
partutpartct
Households as market participants 45
equals the marginal cost in terms of the present value of the loss in interest incomein the subsequent period due to the holding of money balances instead of financialassets as given by the expression (Rt)
minus1(Rt minus Rmt) Recall that Rt minus Rmt equals(1 + rt)(1 + πt) minus (1(1 + πt)) which is simply rt(1 + πt) or essentially theanticipated nominal rate of interest Given that Rt = (1 + rt)(1 + πt) we thushave that (Rt)
minus1(Rt minus Rmt) = rt(1 + rt)We can expand upon the above discussion of the first-order conditions for
period t by first noting that W (xt+1) is defined by
W (xt+1) = maxct+1Nt+1
Mt+1pt+1xt+2
[βu(ct+1 Mt+1pt+1 1 minus Nt+1) + W (xt+2)]
where
xt+2 = Rt+1[xt+1 + (wt+1pt+1)Nt+1 minus ct+1] minus [Rt+1 minus Rmt+1]Mt+1pt+1
given
xt+1 equiv dt+1 + zBtpt+1 + At+1 + Mtpt+1
At+1 equiv [pet+1St + pbt+1Bt]pt+1
Again substituting the transition function into the Bellman equation for periodt + 1 we have the following first-order conditions for period t + 1
βpartut+1partct+1 minus (partW (xt+2)partxt+2)Rt+1 = 0 (41prime)minus βpartut+1part(1 minus Nt+1) + (partW (xt+2)partxt+2)Rt+1wt+1pt+1 = 0 (42prime)βpartut+1part(Mt+1pt+1) minus (partW (xt+2)partxt+2)(Rt+1 minus Rmt+1) = 0 (43prime)
Now consider the impact of the change in xt+1 on the value function W (xt+1)The above first-order conditions imply that the indirect effects of the change inxt+1 on W (xt+1) through the effect of such a change on the choice of the optimalvalues of consumption labor supply and real money balances for period t + 1 arezero5 This is simply an application of the envelope theorem which states thatthe change in the objective function adjusting the choice variables optimally isequal to the change in the objective function when one does not adjust the choicevariables This fact along with the transition function for xt+2 gives us
partW (xt+1)partxt+1 = (partW (ct+2)partxt+2)Rt+1 (45)
Substituting equation (41prime) into (45) we obtain
partW (xt+1)partxt+1 = βpartut+1partct+1 (46)
46 Households as market participants
Alternatively by substituting in (42prime) we can obtain6
partW (xt+1)partxt+1 = β[partut+1part(1 minus Nt+1)]pt+1wt+1 (47)
Combining equations (41) and (46) gives us the standard Fisherian solutionfor the optimal allocation of consumption between period t (time t to t + 1) andperiod t + 1 (time t + 1 to t + 2)
partutpartct
βpartut+1partct+1= Rt
Combining equations (43) and (46) we obtain the standard expression for theoptimal portfolio choice of money
partutpart(Mtpt)
βpartut+1partct+1= Rt minus Rmt
Combining equations (42) and (47) gives us an expression for the optimalallocation of labor supply over time
partutpart(1 minus Nt)
βpartut+1part(1 minus Nt+1)= pt+1
wt+1Rt
wt
pt (48)
Having discussed the ldquoFisherian problemrdquo and the ldquoportfolio problemrdquo con-fronting the household we turn our attention in the next section to the ldquolaborsupply problemrdquo as captured by equations (44) and (48) Before doing so how-ever a general comment should be made with respect to the discussions to followas well as the preceding discussions of the Fisherian problem and the portfoliodecision
In focusing on first-order conditions with respect to the particular variablesat issue (ie first-order conditions for consumption now and next period forthe Fisherian problem first-order conditions for real money holdings and futureconsumption for the portfolio problem and first-order conditions for labor sup-ply now and next period for the labor supply decision) one has a tendency toforget that the optimizing problem involves the simultaneous choice of consump-tion portfolio and leisure In general this means that the analysis is often notas straightforward as it may first appear For instance a change in the currentreal wage or an expected real interest rate can affect the first-order conditionconcerning the choice of labor supply through its impact on the choice of consump-tion if part2upartcpartN = 0 for then the change in consumption alters the ldquomarginalutilityrdquo of leisure One simple way to abstract from these ldquoindirectrdquo effects is toassume that the utility function is separable not only across time but also withrespect to consumption leisure and real money balances each period such thatpart2upartcpartN = part2upartcpart(Mp) = part2upartNpart(Mp) = 0
Households as market participants 47
The choice of hours within a period
Equation (44) indicates that at the optimal labor supply the household cannot bemade better off by trading consumption for leisure within periods at the expectedreal wage for the period Equation (48) indicates that at the optimum the householdcannot be made better off by trading leisure across periods given the relevant realwages and the real interest rate7 Figure 41 captures the first situation that is theoptimal choice of consumption and leisure within a period
For the moment let us hold anticipated real interest rates constant Further letus assume unit elastic expectations with respect to wages as well as prices so thata change in the current wage or price level that alters the current real wage changesfuture expected real wages as well so that there is no change in the current realwage relative to future expected real wages In addition let us hold constant for themoment the effect on current real money balances of a change in the current realwage These assumptions help us to mimic the traditional ldquostaticrdquo or single-periodanalysis (eg Patinkin) of the effect of a change in the current periodrsquos anticipatedreal wage on individualsrsquo labor supply during period t Under such circumstancesan increase in the real wage for period t can have ambiguous effects As Patinkin(1965) states ldquofor simplicity it is assumed that [labor] supply is an increasingfunction of the real wage though there are well known reservations on this scorerdquo
To understand what Patinkin is referring to consider an increase in the real wagedue to a rise in the money wage wt Consider one possible result on the householdrsquoslabor supply decision The increase in the net real wage means a steeper budgetline as the householdrsquos optimal leisurendashconsumption combination changes from1 minus N s
t and cdt (call this choice A) to (1 minus N s
t )prime and (cdt )prime (choice C)
An increase in the real wage has two conceptually distinct effects on the house-holdrsquos labor supply decision an income effect and a substitution effect The incomeeffect refers to the fact that an increase in the real wage makes the household betteroff because it leads to an increase in the householdrsquos feasible consumption set in
$
Leisure
Figure 41 Consumption and leisure
48 Households as market participants
the current period The higher real wage means that the household if it so desirescan increase both its leisure and consumption Thus the household is able to reacha higher indifference curve which has associated with it a higher utility levelThe substitution effect refers to the fact that the increase in the real wage makesan hour of leisure relatively more expensive in terms of consumption that must beforgone
To isolate the substitution and income effects of the change in the net real wagesuppose that after the real wage increases we temporarily take away just enoughof the householdrsquos nonlabor income so that it is just able to attain its originalindifference curve The householdrsquos choice of leisure and income would then be(1 minus N s
t )primeprime and (cdt )primeprime (call this choice B) Thus if we hold the householdrsquos utility
level constant the increase in the real wage leads unambiguously to a lower levelof leisure since an hour of leisure is relatively more expensive the householdsubstitutes away from leisure choosing to work more hours The movement fromchoice A to choice B constitutes the pure ldquosubstitution effectrdquo of the higher netreal wage8
Now suppose that we give the household back the nonlabor income that wetemporarily took away This causes an outward shift in the budget line and thehouseholdrsquos new choice of leisure and consumption would be (1 minus N s
t )prime and (cdt )prime
(choice C) The movement from B to C constitutes the ldquoincome effectrdquo of thechange in the net real wage In the present case the income effect on the choiceof leisure is positive reflecting the assumption that leisure is a normal good
Note that the substitution and income effects on leisure work in opposite direc-tions The substitution effect of the higher real wage causes leisure to fall andhours worked to rise while the income effect causes leisure to rise and hoursworked to fall The net effect on leisure and hours worked depends on whicheffect dominates9 If the substitution effect dominates then a higher real wageresults in a decrease in desired leisure and an increase in desired working hoursIf the income effect dominates the opposite is true and the individualrsquos laborsupply curve is ldquobackward bendingrdquo when plotted against the real wage
The available evidence suggests that for many workers the income effect tendsto dominate slightly Estimates are that for men an increase of 10 percent in the realwage results in approximately a 15 percent reduction in the hours worked Thisreduction in hours worked reflects an income effect of approximately minus25 percentand a substitution effect of about 1 percent Other evidence suggests a similarpattern for working women10
The choice of participation within a period
Thus far we have considered a household representative of those who are in thelabor force working a positive number of hours Yet this masks the unambiguouseffect of a higher real wage on the labor supply of those households not in the laborforce To show this we consider a corner solution with respect to labor supply inparticular a household denoted a that has chosen not to participate in the labormarket For such a household let us return to the Bellman equation for period t and
Households as market participants 49
introduce explicitly the nonnegativity constraint for labor supply (ie Nt ge 0)Letting micron denote the multiplier associated with this constraint we have as first-order conditions for household arsquos consumption and labor supply11
partuatpartcat minus (partW (xat+1)partxat+1)Rt = 0
minus partuatpart(1 minus Nat) + micron + (partW (xat+1)partxat+1)Rtwtpt = 0
Nat ge 0 micronNat = 0
Substituting the first equation into the second and rearranging gives
minuspartuatpart(1 minus Nat)
partuatpartcat= wt
pt+ micron
partuatpartcat
For individuals not participating in the labor force micron ge 0 The ldquocorner solutionrdquo isa case in which the marginal rate of substitution of leisure in terms of consumptionis greater than the real wage In other words the absolute value of the indifferencecurve is equal to or greater than the absolute value of the budget line at the pointwhere N s
at = 0Note that at the optimal choice the corresponding indifference curve is more
steeply sloped than the budget line Thus even when all hours are devoted toleisure and none to work the individualrsquos marginal rate of substitution of leisurein terms of consumption still exceeds the real wage rate In other words theindividualrsquos valuation of leisure exceeds the marketrsquos valuation of leisure As aresult individual a does not find it worthwhile to participate in the labor market
The greater the real wage the greater is the probability that a given individualwill choose to participate in the labor market A higher real wage rotates thebudget line outward Since the individual is not working an increase in the netreal wage does not make him better off and thus has no income effect There isonly a substitution effect Thus if the real wage rises sufficiently the individualcan be induced to enter the labor market
According to the above analysis the economy-wide labor supply response toan increase in the current real wage is a combination of an ambiguous effecton the labor supply of those currently working but an unambiguous increase inthe labor supply among those not working12 It is the net of these two effects theldquohoursrdquo decision and the ldquoparticipationrdquo decision that is captured by the aggregatelabor supply function it is commonly assumed that this net effect is such that theaggregate quantity of labor supplied is an increasing function of the current realwage
The labor supply intertemporal substitution hypothesis
Our discussion has yet to consider ldquothe labor market intertemporal substitutionhypothesis (ISH) which states that labor supply responds positively to transitoryincreases in real wages and increases in the real interest rate a central hypoth-esis of modern competitive models of the business cyclerdquo (Alogoskoufis 1987)
50 Households as market participants
To do so we simply expand our focus to the inherent intertemporal decisionconfronting the household In particular recall our expression (48) for the optimalallocation of labor supply over time The view of labor supply embedded in (48)has life-cycle as well as business-cycle implications With respect to life-cycleimplications the theory predicts that workers will concentrate their labor supplyin years of peak earnings consuming leisure in larger than average amounts duringchildhood and old age
With respect to the business cycle the above helps explain an apparent contra-diction in the static theory of labor supply ndash the observed wage inelasticity of laborsupply in the long run with short-run fluctuations in employment which requirean elastic labor supply if one takes a ldquomarket-clearingrdquo approach with respect tothe labor market13 It does so by introducing a distinction between a permanentchange in the real wage and a temporary or transitory change in the real wage
To show the intertemporal substitution effect with respect to labor supply spec-ify wlowastplowast as the permanent or ldquonormalrdquo real wage with the anticipated real wagenext period equal to this value such that
wipi = wlowastplowast i = t + 1 t + 2
Further we assume that
Ri = Rlowast i = t t + 1 t + 2
In this case equation (48) becomes
partutpart(1 minus Nt)
βpartut+1part(1 minus N lowast)= Rlowast wtpt
wlowastplowast (48prime)
where N lowast denotes the ldquolong-runrdquo supply of labor at ldquonormalrdquo wages Further letus assume that for the representative household βRlowast = 1 Then equation (48prime)becomes
partutpart(1 minus Nt)
partut+1part(1 minus N lowast)= wtpt
wlowastplowast (48primeprime)
Equation (48primeprime) indicates that if the current real wage is higher than the normal realwage then ldquomore labor is supplied than would be implied by the long-run laborsupply functionrdquo That is this theory views suppliers of labor as reacting primarilyto three variables an anticipated ldquonormalrdquo or ldquopermanentrdquo real wage rate whichcorresponds to the wage rate in the usual one-period analysis of the laborndashleisurechoice and has a negligible effect on labor supply the deviation of the current realwage from this normal rate which has a strong positive effect on labor supplyand the expected real rate of interest (Lucas and Rapping 1970 284ndash285)14
The above theory provides the underlying microtheoretical basis for the fol-lowing statement ldquomeasured unemployment (more exactly its nonfrictionalcomponent) is then viewed as consisting of persons who regard the wage ratesat which they can currently be employed as temporarily low and who therefore
Households as market participants 51
choose to wait or search for improved conditions rather than invest in moving oroccupational changerdquo (Lucas and Rapping 1970 285)
Empirical tests seem to provide some support for this intertemporal substitutionhypothesis15 Alogoskoufis (1987 950) finds that for measures of the total numberof employees the real wage and interest rate elasticities are high and relativelywell determined ldquoThe elasticity of labor supply to transitory changes in real wagesis around unity and is statistically significant at conventional significance levelswith one exception The real interest rate always has a significant independentinfluencerdquo Note that as suggested by equation (48) Alogoskoufis finds that arise in the real interest rate increases current labor supply
Note that equation (48) does not explicitly identify the initial asset holdings asa variable that affects the relative choice of labor supply across periods Similarlythe condition for the optimal choice of consumption across periods did not have theinitial value of assets affecting the relative consumption purchases across periodsThis is a property of time-separable preferences
Special topics in intertemporal choices
As we have seen the intertemporal problem confronting the individual involvessimultaneous decisions with respect to consumption versus saving and with respectto the composition of the asset portfolio To make some sense of what is involvedwe start by considering what is known as the ldquoFisherianrdquo problem which focuseson the individualrsquos choice of consumption across time
Fisherian analysis
The Fisherian problem typically deals with the allocation of consumption acrosstime when there is a single means by which income can be allocated across time16
To restrict our analysis to such a case we can simply omit real money holdingsfrom the utility function Further we consider only interior solutions with respectto consumption (ie cai gt 0 i = t t + T )17
Thus the maximization problem becomes18
maxcat cat+T
xat+1xat+T+1
t+Tsumi=t
β iminustua(cai)
subject to
minus xat+1 + Rt[xat + cat minus cat] = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1] = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2] = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ] = 0
xat+T+1 ge 0
52 Households as market participants
Recall that Ri = (1 + ri)(1 + πi) i = t t + T denotes the ldquogross real rateof returnrdquo on agent arsquos portfolio between the end of period i and the end of periodi + 1 and asset holdings are solely in the form of bonds such that
xat equiv (1 + r)pbBapt
xat+1 equiv (1 + rt)pbtBatpt+1 = [(1 + rt)(1 + πt)] pbtBatpt
xat+i equiv (1 + rt+iminus1)pbt+iminus1Bat+iminus1pt+i
= [(1 + rt+iminus1)(1 + πt+iminus1)] pbt+iminus1Bat+iminus1pt+iminus1
i = 2 T + 1
Let λi i = t t +T denote the Lagrange multipliers for the constraints linkingthe total value of real asset holdings at the end of period i with the total value ofreal asset holdings at the end of period i + 1 Let microT denote the multiplier for thenonnegativity condition that xat+T+1 ge 0 Then the first-order conditions for theimplied Lagrangian L include
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai) = 0 i = t t + T 19
partLpartmicroT = xat+T+1 ge 0
microT ge 0
where uai = ua(cai) i = t t + T 20 Note that the above set of first-order
conditions consist of 3(T + 1) + 1 equations to determine 3(T + 1) + 1 variablesThe variables to be determined are cai i = t t + T xai i = t t + T + 1and microT Assuming continuity and strict concavity in ua(middot) and given a convex setof constraints
xai+1 xai cai| minus xai+1 + Ri[xai + cai minus cai] ge 0there is a unique solution to the problem
There are several implications of the above first-order conditions First theconditions imply that the desired total real value of assets (bonds) inherited at timet + T + 1 xd
at+T+1 will equal zero if duat+T dcat+T gt 0 In particular from the
condition
partLpartcat+T = βT (duat+T dcat+T ) minus λt+T Rt+T = 0
we see that if duat+T dcat+T gt 0 then λt+T gt 0 From the condition
partLpartxat+T+1 = minusλt+T + microT = 0
Households as market participants 53
we thus have that microT gt 0 This in turn implies from the condition
microT partLpartmicroT = microT xat+T+1 = 0
that xat+T+1 = 0 This finding should not be surprising With a time horizonof t + T agent a perceives no gain (utility) from acquiring assets at time t + Tto finance consumption in period t + T + 1 and a clear loss from doing so attime t + T in terms of consumption forgone given the assumption of nonsatiation(dua
t+T dcat+T gt 0) Second from the first set of equations we know that betweenany two periods i and i + 1
β iminustduai dcai
β iminust+1duai+1dcai+1
= λiRi
λi+1Ri+1 i = t t + T minus 1
This expression can be simplified to obtain
duai dcai
βduai+1dcai+1
= Ri i = t t + T minus 1
where Ri the real gross return between the end of periods i and i + 1 is given by(1+ri)(1+πi) Ri has been called Fisherrsquos ldquo(gross) real interest raterdquo since he wasone of the first to provide a lucid account of its role in determining consumptionacross time
Fisherrsquos ldquo(net) real interest raterdquo denoted by mi is then defined by
1 + mi equiv Ri = (1 + ri)(1 + πi)
Subtracting one from both sides and rearranging we have
mi = (ri minus πi)(1 + πi)
Thus for small expected rates of inflation we have the approximation21
mi = ri minus πi
In words the real interest rate is approximately equal to the nominal interest rateminus the rate of inflation The expected real interest rate is then the nominalinterest rate minus the expected rate of inflation
There are several features of the above that should be noted First if an individ-ualrsquos discount factor (β lt 1) equals the reciprocal of the real gross rate of interest([Ri]minus1 = (1 + πi)(1 + ri)) between periods i and i + 1 so that
βRi = 1
then the above expression of the first-order conditions for cai and cai+1 indicatesthat dua
i dcai = duai+1dcai+1 Given the concavity of the single-period utility
function (ua(middot)) and our assumption of a time-invariant one-period utility function
54 Households as market participants
it then follows that the individual will choose the same rate of consumption acrossperiods i and i + 1 in such a case22 This constant path of consumption betweenthe two periods can be said to emerge if an individualrsquos ldquorate of time preferencerdquoequals the (real) interest rate
If the expected gross return between two periods were higher (or for an individualwith a higher discount factor) the fact that β gt (Ri)
minus1 or equivalently βRi gt 1means from the first-order conditions that dua
i dcai gt duai+1dcai+1 Given the
assumed concavity of the one-period utility function the implication is that agentarsquos consumption during period i would be less than during period i + 1 That isβRi gt 1 rArr cd
ai lt cdai+1 Conversely if the expected real gross return were to be
lower (or for an individual with a lower discount factor) the fact that β lt (Ri)minus1
or equivalently βRi lt 1 means that the individualrsquos consumption during period iwould be greater than during period i + 1 That is βRi lt 1 rArr cd
ai gt cdai+1
For the two-period case (say periods t and t + 1) the optimal consumption ineach of the two periods can be shown graphically by the point of tangency betweenan indifference curve with slope minus(dua
i dcai)β(duai+1dcai+1) and a budget line
with slope minusRt 23 If one were to place cat+1 on the vertical axis and cat on thehorizontal axis then it would be possible to determine whether an individual is aborrower or a lender in any period For example if disposable income were lessthan consumption in the first period t then the individual is a lender at time tNote that points on the same indifference curve are such that
d[ua(cat) + βua(cat+1)
] = (duat dcat+1)dcat + β(dua
t+1dcat+1)dcat+1
= 0
Rearranging we have the slope of an indifference curve given by
dcat+1
dcat= minus dua
t dcat
βduat+1dcat+1
Note that points on the budget line satisfy the present value constraint
cat + (Rt)minus1cat+1 = cat + xat + (Rt)
minus1cat+1
Rearranging we have
cat+1 = Rt[cat + xat minus cat] + cat+1
such that the slope of the budget line is given by
dcat+1dcat = minusRt
Thus at the point of tangency between the budget line and an indifference curve
duat dcat
βduat+1dcat+1
= Rt
Households as market participants 55
which is the expression we obtained previously concerning the optimal choice ofconsumption between periods i and i + 1
Note that the above analysis is for an individual consumer Thus aggregate con-sumption need not behave as that predicted above for the individual For instancean aging population could lead to variations in aggregate consumption that reflectthe aggregation at different times across agents with differing characteristics
Life-cycle and permanent income hypotheses
We have seen how a householdrsquos consumption in any period is not constrained bythe income it receives during that period but rather that the discounted value oflifetime consumption is constrained by the discounted stream of income accruingto the household over its lifetime plus initial asset holdings While income tendsto rise and fall during the lifetime of an individual through appropriate saving andborrowing the individual can maintain a smooth or constant rate of consumptionover his lifetime This smoothing of consumption across time plays a critical rolein Franco Modiglianirsquos ldquolife-cycle hypothesisrdquo of consumption24
A stylized pattern of income and consumption expenditures over an individ-ualrsquos lifetime is the following Prior to retirement income exceeds consumptionand saving is positive During this period saving increases household wealth Onretirement consumption is financed by dissaving During the retirement periodhousehold wealth falls as people draw on their accumulated savings to financeconsumption Implied in this discussion is an inverted U-shape wealthndashage pro-file (save during pre-retirement years and dissave in years following retirementrunning down the stock of accumulated wealth) A number of studies of aggregatehousehold consumption and saving behavior support this wealthndashage pattern25
To make clear the implications of consumption smoothing for the demand forthe consumption good at time t let us make the simplifying assumptions that
(a) for any period i = t t + T Ri = R and(b) the individualrsquos personal discount rate β equals the constant real ldquomarketrdquo
discount rate Rminus126
From the first-order conditions we thus have the result that agent a willcompletely smooth out consumption spending across time so that consumptioncd
ai = cda i = t t + T In this case we can use the prior combined budget
constraint to obtain
cdat =
xat +
t+Tsumi=t
(caiRiminust)
where which equals 1sumt+T
i=t (1Riminust) is what Modigliani has calledthe ldquoproportionality factorrdquo and indicates the proportion of householdsrsquo totalresources ndash consisting of initial assets current income and anticipated futureincome ndash devoted to consumption each year
56 Households as market participants
An important implication of Modiglianirsquos life-cycle hypothesis is that thefraction of an increase in current income (cat) that goes toward increased cur-rent consumption (the ldquomarginal propensity to consumerdquo) will vary depending onwhether the increase in current disposable income is accompanied by an equivalentincrease in anticipated future income (cat i = t + 1 t + T )27 If a change incurrent income is viewed as ldquotransitoryrdquo most of the increase in income will goto saving in order to finance increased consumption during future years
This idea that the effect of a change in current disposable income on consump-tion demand depends on the degree to which the change in income is viewed astemporary or permanent lies at the heart of Milton Friedmanrsquos permanent incomehypothesis The permanent income hypothesis is like the life-cycle hypothesis inthat it emphasizes the fact that consumption demand in period t depends not onlyon current income but also on anticipated income in the future periods Permanentincome is that income which if received each year over a householdrsquos time hori-zon would yield an income stream with present value exactly equal to the presentvalue of the householdrsquos anticipated income stream That is permanent income cpis defined by the following equation
t+Tsumi=t
(cpRiminust) =t+Tsumi=t
(caiRiminust) + xat (49)
Factoring out cp on the left-hand side of (49) and rearranging we have
cp =
xat +
t+Tsumi=t
(caiRiminust)
(410)
Equation (410) indicates that permanent income is simply a weighted average ofcurrent and future incomes but in this case income in the more distant future isweighted less heavily since it is discounted more highly
Comparing permanent income to agent arsquos consumption demand in period tcd
at if there is complete smoothing of consumption spending across time then weobtain
cdat = cp
The implication of this equation is that the marginal propensity to consume out ofa change in current income that is perceived as permanent is equal to one whilethe marginal propensity to consume out of a change in current disposable incomethat is perceived as entirely transitory (having little impact on permanent income)is small Changes in transitory components of income are almost entirely saved ifpositive or borrowed if negative28
Our discussion so far of the impact of changes in income on current consump-tion demand has been restricted to what might be referred to as the effects ofldquotransitoryrdquo versus ldquopermanentrdquo income changes In doing so we have assumed adeterministic world in which individuals have perfect foresight concerning future
Households as market participants 57
income streams But what happens if individuals do not have perfect foresight Inparticular what if we introduce stochastic elements so that realized future incomeis a random variable Then the above theories suggest a difference in the responseof consumption demand to income changes that are anticipated or expected versusunanticipated changes In particular the life-cyclepermanent income hypothesiswould predict that previously anticipated (or expected) changes in income wouldhave no effect on consumption demand since consumption plans have alreadyincorporated this income29
Portfolio choice
Now let us consider the more general case in which agent a chooses not onlyconsumption across time but the portfolio of assets (money and bonds) That isconsider the following problem
maxcat cat+T
xat+1 xat+T+1Matpt Mat+T pt+T
t+Tsumi=t
β iminustua(cai Maipi)
subject to
minus xat+1 + Rt[xat + cat minus cat] minus [Rt minus Rmt]Matpt = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1]minus [Rt+1 minus Rmt+1]Mat+1pt+1 = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2]minus [Rt+2 minus Rmt+2]Mat+2pt+2 = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ]minus [Rt+T minus Rmt+T ]Mat+T pt+T = 0
xat+T+1 ge 0
As before to simplify the problem we assume interior solutions in this case notonly with respect to the consumption good but also with respect to money holdingsAgain let λi i = t t + T denote the Lagrange multipliers for the constraintslinking the real asset holdings at the end of period i with their real value at the endof period i + 1 and let microT denote the multiplier for the nonnegativity conditionthat xat+T+1 ge 0 The first-order conditions are
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpart(Maipi) = β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0 i = t t + T
58 Households as market participants
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai)
minus [Ri minus Rmi]Maipi = 0 i = t t + T 30
partLpartmicroT = xat+T+1 ge 0
microT partLpartmicroT = microT xat+T+1 = 0
λi ge 0 i = t t + T
microT ge 0
where uai = ua(cai Maipi) i = t t + T To isolate the portfolio choice
consider the portfolio choice of money and bond holdings for a given level ofcurrent consumption That is let us look at the optimal choice of Maipi given thatcai is held constant From the second set of conditions
β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0
Substituting the condition for the optimal choice of the total value of assets in thatperiod
λi = λi+1Ri+1
we obtain
β iminustpartuai part(Maipi) minus λi+1Ri+1[Ri minus Rmi] = 0
Now substituting the condition for the optimal choice of consumption next period(i + 1) as given by
β iminust+1duai+1dcai+1 minus λi+1Ri+1 = 0
we obtain
β iminustpartuai part(Maipi) minus β iminust+1(dua
i+1dcai+1)[Ri minus Rmi] = 0
Rearranging gives
duai d(Maipi)
βduai+1dcai+1
= Ri minus Rmi
Recalling that the expected gross real return on bonds in period i Ri equals(1 + ri)(1 + πi) and that the expected gross real return on money Rmi equals1(1 + πi) the above expression can be written as
duai d(Maipi)
βduai+1dcai+1
= ri
1 minus πi
Households as market participants 59
Note that in the limit (as the length of the period goes to zero) the expected rate ofinflation term would vanish The implication is that the optimal division of assetsbetween money and bonds depends primarily on the money interest rate aloneWhat is essentially being shown is that an increase in money holdings with nochange in current consumption means a reduction in bond holdings and thus theloss of nominal interest income ri or real interest income ri(1 + πi)
Absence of money illusion real balance effects and realindebtedness effects
There are two aspects of agent arsquos demand function that should be noted Firstagent arsquos demand functions at time t can be shown to be homogeneous of degree 0 inthe current price level pt initial money balances M a and initial bond holdings BaIn this economy this is said to reflect the ldquoabsence of money illusionrdquo One criticalreason for this is the assumption of unit elastic expectations with respect to futureprices so that changes in the current price level leave unchanged the expectedrates of change in the price level in subsequent periods Also note that the currentand expected future money payments attached to bonds are being held constantso that given the fixed money payment x on maturity interest rates are unchangedAlternatively one could have money prices and the fixed future money paymentattached to one-period bonds rise by the same proportion
The above implies that individual arsquos demand for the consumption good andreal money balances at time t can be represented by
cdat = cd
at(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
M datpt = M d
atpt(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
where xat is the individualrsquos real wealth at the end of period t as given by
Wat = xat + cat +t+Tminus1sum
j=t
⎡⎣ jprod
i=t
(Ri)
⎤⎦
minus1
(caj+1)
From the budget constraint for period t we know that the above two demandconditions imply a real demand for bonds of a similar form since
pbtBdatpt = cat + xat minus
[cd
at + M datpt
]
Note that the above demand functions do not depend solely on wealth and thepattern of expected real (gross) rates of interest since given the portfolio choice agiven pattern of expected real (gross) interest rates could alter demand dependingon the underlying values of the money interest rate As before individual arsquos excessdemand function for the consumption good and money in period t are defined byzat = cd
at minus cat zam = M datpt minus M apt and zab = pbtBd
atpt 31
60 Households as market participants
The ldquoreal balance effectrdquo indicates the effect of a change in real balances (M apt)
on individual demand for goods other than money As before there is a real balanceeffect that reflects a wealth effect That is a decrease in initial real balances leadsto a reduction in real money demand M d
atpt and a reduction in the real demandfor bonds pbtBd
atpt There is also what might be referred to as a ldquoreal indebtednesseffectrdquo in that a change in prices alters not only real money balances but also realinitial debt If Ba gt 0 an increase in pt reduces wealth while if Ba lt 0 anincrease in pt increases wealth This is why the characterization of the absence ofmoney illusion has been expanded to include changes in Ba
It is typical in macroeconomics to adopt the convention of the ldquorepresentativeagentrdquo to reduce notational clutter Recall that the ldquorepresentative agentrdquo is essen-tially the average agent For instance if we let cd
at denote the demand for theconsumption good in period t by representative agent a and cd
t market demandat the time then cd
at = cdt Thus depending on the context we can interpret cd
tas demand by the representative agent or market demand Recall that in doingso we essentially ignore distribution effects such as effects on market demandof changes in the distribution of initial endowments of commodities or moneybalances or of changes in the distribution of future endowments In the context ofthe real indebtedness effect since in the aggregate B = 0 this effect is removedfrom our analysis
Intertemporal substitution the evidence
We have focused above on the behavior of an individual with respect to the plannedpath of consumption across time As Robert Hall (1988 340) indicates
The essential idea is that consumers plan to change their consumptionfrom one year to the next by an amount that depends on their expectationsof real interest rates Actual movements of consumption differ from plannedmovements by a completely unpredictable random variable that indexes allinformation available next year that was not incorporated in the planningprocess the year before If expectations of real interest rates shift then thereshould be a corresponding shift in the rate of change of consumption Themagnitude of the response of consumption to a change in real interest rateexpectations measures the intertemporal elasticity of substitution32
Hall (1988 340ndash341) goes on to state that
the basic model of the joint distribution of consumption and the return earnedby one asset that has emerged is the following The joint distribution ofthe log of consumption in period t log ct and the (real) return earned by theassets from period t minus 1 to period t mtminus1 is normal with a covariance matrixthat is unchanging over time The means obey the linear relation
E(log ct) = k + ctminus1 + σE(mtminus1) (411)
Households as market participants 61
That is the expected change in the log of consumption is a parameter σ times the expected real return plus a constant If the expected real interestrate E(mtminus1) is observed directly then the key parameter σ can be estimatedsimply by regressing the change in the log of consumption on the expectedreal rate That regression also has the property that no other variable knownin period t minus 1 belongs in the regression
Hall proceeds to estimate the parameter using aggregate data on consumptionand finds that there is ldquolittle basis for a conclusion that the behavior of aggregateconsumption in the United States in the twentieth century reveals an importantpositive value of the intertemporal elasticity of substitutionrdquo (1988 356)
Hallrsquos empirical finding is of importance to macroeconomic analysis Thework by Hall and others is also of interest as an example of how theoreticalmacroeconomic analysis specifically Fisherian analysis can be tested To seethe link between the theory developed in the prior section and the proposed test(equation (411)) assume the following
Assumption 43 The path of aggregate consumption reflects agent arsquos decisionsconcerning the optimal allocation of consumption across time That is we treatagent a as the ldquorepresentative consumerrdquo Thus cai i = t t+T which denotesconsumption in period i of the representative agent a differs only in scale from cii = t t + T which denotes the aggregate level of consumption This assump-tion that an aggregate variable can be viewed as reflecting decisions of a representa-tive agent is not innocuous For instance the actual path of aggregate consumptioncould well differ from that predicted by an analysis of individualsrsquo optimaldecisions due to changes across time in the composition of individuals in theeconomy33
Assumption 44 Individualsrsquo expectations in period t minus 1 of future real interestrates incorporate all information available as of period t minus 1 New informationoccurring in period t that alters consumption from what was planned results inthe distribution of consumption being ldquolog normal conditional on informationavailable last period that is log ct is normal with mean E(ct)rdquo (Hall 1988 342)
Assumption 45 In period t minus 1 the representative agentrsquos utility function takesthe following form
t+Tsumi=tminus1
expminusδi + ((δ minus 1)δ) log(ci)
where c gt 0 σ gt 0 and δ gt 0 This exponential utility function has the followingdesired properties
1 It is time-separable2 If consumption were equal across any two periods the individual would place
greater value on an increase in consumption in the earlier period ndash that is if
62 Households as market participants
ctminus1 = ct then
expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)gt expminusδt + ((δ minus 1)δ) log(ct)
3 Given 1 gt σ ge 0 utility increases in any period with increased consumptionbut at a decreasing rate ndash that is
du(ctminus1) = dctminus1
= [(1 minus δ)(δctminus1)] middot expminusδt(t minus 1)
+ ((δ minus 1)δ) log(ctminus1)gt 0 d2u(ctminus1)dc2
tminus1 lt 0
Note that the ldquointertemporal elasticity of substitutionrdquo will be given by σ As σ approaches 0 substitution of consumption across time in response tochanges in the real interest rate will approach zero as well
Assumption 46 Individualsrsquo forecasts of future variables are held with subjec-tive certainty This last assumption is a departure from Hallrsquos analysis that allowsus for the moment to maintain the ldquodeterministicrdquo aspect of the prior optimizationproblem That is we continue to assume that individualrsquos expectations of futurevariables such as expected rates of inflation and future interest rates are held withsubjective certainty
Given the above assumptions we know from our prior discussion that thechoice of consumption for periods t minus1 and t must satisfy the following first-ordercondition
dudctminus1
dudct= Rtminus1
Substituting in the appropriate expressions for the marginal utility of consumptionin periods t minus 1 and t we obtain
[(1 minus δ)δctminus1] middot expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)[(1 minus δ)δct] middot expδ + ((δ minus 1)δ) log(ct)
or
(ctctminus1) expδ + ((δ minus 1)δ) log(ctminus1ct) = Rtminus1
Taking the logarithm of both sides of the above expression and rearranging theabove first-order condition becomes
log(ctctminus1) + δ minus ((δ minus 1)δ) log(ctminus1ct) = log(Rtminus1)
Households as market participants 63
which simplifies to
δ + (1δ) log(ctminus1ct) = log(Rtminus1)
or
log ct = minusδσ + log ctminus1 + σ log(Rtminus1) (412)
which is similar in form to (411) Note that the ldquointertemporal elasticity ofsubstitutionrdquo is given by
σ = (dctdRtminus1)ctRtminus1
The form of equation (42) can be made closer to that of equation (411) if wenote that we can define the ldquoinstantaneous real rate of interestrdquo associated withcontinuous compounding mtminus1 by the expression
exp(mtminus1) = Rtminus1
Then (412) becomes
log ct = minusδσ + log ctminus1 + σmtminus1
Conclusion
This chapter has developed an in-depth understanding of household behaviorA good deal of the discussion has dealt with intertemporal choices and the tradeoffsinherent in consuming today versus consuming in the future Many policy-relatedissues in macroeconomics are related to decisions made today that are not indepen-dent of future states or activities This issue will arise again and again throughoutthis book and it is imperative that one comprehend the nature of decision-makingand time
5 Summarizing the behavior andconstraints of firms andhouseholds
Introduction
In this chapter we summarize our discussion of the behavior of firms andhouseholds in the simple Walrasian model with money and production In doingso we consider first the nature of constraints faced by the participants in the econ-omy with respect to decisions during period t and then their behavior in termsof demand andor supply Along the way we will try to simplify the notationand introduce various expectations and assumptions of different macroeconomicmodels We start our discussion with firms
Summarizing firmsrsquo constraints
We have seen how we can divide the general constraint facing firms that totalrevenues from all sources just exhausts expenditures each period into two sepa-rate constraints One is the ldquofirm financing constraintrdquo which states that desiredchanges in the capital stock as well as any capital adjustment costs are financedby issuing new bonds or equity shares That is for period t
I dnt + ψ(I d
nt) minus net Ast = 0 (51)
where
net Ast equiv As
t minus At
Ast equiv [
pbtBst + petS
st
]pt
At equiv [pbtB + petS
]pt
Idnt equiv Kd
t+1 minus K
Note that we implicitly assume that firmsrsquo plans for purchasing capital during theperiod correctly anticipate the price of output (capital) during the period and theprices of bonds and equity shares to be issued at the end of the period to financesuch purchases
Behavior and constraints 65
The second constraint the ldquofirm distribution constraintrdquo is that all revenuesfrom the sale of output net of that required to replace capital used up in the produc-tion process during the period be distributed to households either as wages interestpayments or dividends At time t firmsrsquo anticipated distribution constraint isgiven by
dt + (wtpt)Ndt + zBpt minus (ys
t minus δK) = 0 (52)
where z is the coupon payment and planned output supply is related to labordemand by the production function
yst = f (N d
t K) (53)
Equation (52) implicitly assumes that firms at time t have perfect foresight withrespect to the price of output during period t Alternatively the firm financingconstraint would take the above form if we presumed there were futures marketsat time t for the exchange of output during period t and financial assets at the endof period t
The labor market at time t determines employment for the period at a level N lowastt
and an associated rate of output denoted by ylowastt At the realized price of output the
firm distribution constraint for period t will turn out to be
dt + (wtpt)Nlowastt + zBpt minus ( ylowast
t minus δK) = 0 (54)
As (54) indicates actual real output during the period net of that used to replacedepreciated capital will be distributed to households1
Summarizing householdsrsquo constraints
With respect to households there is a single budget constraint for period t Like thatof the firm distribution constraint its form changes depending on what is assumedconcerning the correctness of expectations or the timing of markets As we haveseen at time t households make plans with respect to labor supply consumptiondemand and desired additions to their real holdings of financial assets and moneybalances based on a perceived constraint of the form
cdt + (M d
t minus M )pet + net Ad
t minus (wtpt)eN s
t minus (dt + zBpt) = 0 (55)
where
net Adt equiv Ad
t minus At
Adt equiv
[pbtB
dt + petS
dt
]pt
At equiv [pbtB + petS
]pt
66 Behavior and constraints
We presume that households have perfect foresight at time t with respect to thereal value of financial assets and the real value of dividends plus interest paymentsreceived from firms at the end of period t We leave open the possibility of errorsin expectations (held with subjective certainty) concerning the price level as itaffects real money balances and the real wage The term pt would replace pe
t if wepresumed perfect foresight on the part of households concerning the price level orequivalently presumed that there were futures markets at time t for the exchangeof output during period t
The labor market at time t determines employment and output for the periodAs before we let N lowast
t and ylowastt denote the actual rate of employment and production
of output during period t In such a case the actual firm distribution constraint(54) can be substituted into the household budget constraint Since prices will beknown by households at this point we may also replace the expected prices ofoutput bonds and equity shares by their actual prices Thus during period t thehousehold budget constraint for period t can then be expressed as
cdt + (M d
t minus M )pt + net Adt minus ( ylowast
t minus δK) = 0 (56)
As discussed below householdsrsquo behavior in the output and financial markets willbe based on realized income and prices and will differ from their plans made attime t based on expected prices and a labor supply decision unless they possessperfect foresight at time t concerning the prices that will prevail over period t andthe labor market clears with actual employment equal to labor supply2
Walrasrsquo law labor market and other markets at time t
Recall that Walrasrsquo law reflects the summing up of the constraints faced by individ-ual agents in the economy Since our preceding analysis concerned the constraintsof the ldquorepresentativerdquo firm and household we need only sum the constraints ofsuch representative agents to obtain Walrasrsquo law There still remains a potentialproblem however as to which of the different versions of the constraints enumer-ated above for firms and households to use The choice as one would suspectdepends on whether the market for labor effectively occurs at the same time as themarkets for output and financial assets or at different times
One version of Walrasrsquo law essentially combines the market for labor at time twith a futures market at time t for the exchange of output during period t andfinancial assets at the end of period t Equivalently this version of Walrasrsquo lawassumes limited perfect foresight at time t by both firms and households withrespect to prices for the period3 In such a case we would sum constraints (51)(52) and (55) (with pt replacing pe
t in (55)) to obtain
[cd
t + I dnt + δK + ψ(I d
nt) minus yst
]+[net Ad
t minus net Ast
]+ (wtpt)
[N d
t minus N st
]+ [M d
t minus M ]pt = 0 (57)
Behavior and constraints 67
Thus we have that the sum of excess demand across four markets ndash the laboroutput financial and money markets ndash must equal zero4 Note that one of thesefour markets the money ldquomarketrdquo reflects the equality between the demand forand supply of money The money ldquomarketrdquo is not of course like other marketsof an economy ndash which is why the word ldquomarketrdquo is in quotes That is unlike theother markets the money ldquomarketrdquo is not a place where the exchange of goods(eg labor financial assets or output) takes place5
Walrasrsquo law sequential markets and potential lackof perfect foresight
There is a modification to make with respect to the above that is required if marketsoccur sequentially and if there is not perfect foresight on the part of all agents attime t concerning prices for period t The sequential nature of markets is clearin that we have the labor market occurring at time t while the markets for outputand financial assets occur during the period In addition households in particularmay not correctly foresee at time t the price of output for period t Under suchcircumstances the prior version of Walrasrsquo law no longer holds for it would thensum constraints that are only anticipated not realized Instead given the sequentialnature of the markets we must break the analysis down into an analysis of thelabor market and an analysis of the other three markets
At time t the labor market occurs Assuming a competitive equilibrium for thelabor market we have a money wage determined at time t such that
N st = N d
t
Underlying the supply of labor at time t are householdsrsquo plans with respect toconsumption demand and saving (either in the form of financial assets or money)during the period These plans are influenced at time t by the anticipated pricelevel for commodities pe
t among other variables and as such these plans may notbe feasible given realized prices during period t
Once the labor market ends employment and output are determined for theperiod at levels N lowast
t and ylowastt respectively At that point households make plans with
respect to consumption and saving in light of the realized prices and the resultingeffective household budget constraint That realized household budget constraintis simply equation (56) which incorporates the actual firm distribution constraintAdding the firm financing constraint (51) we obtain a modified Walrasrsquo law forthe markets during period t of the form
[cd
t + I dnt + δK + ψ(I d
nt) minus ylowastt
]+[net Ad
t minus net Ast
]+ [M d
t minus M ]pt = 0
In the absence of perfect foresight the demands for consumption money balancesand financial assets during the period expressed in the above equation can differfrom the plans made at time t
68 Behavior and constraints
Summarizing firm behavior with limited perfect foresight
Consider firmsrsquo optimal plans at time t Note that at time t firms have an expectationof the price of output for the period We shall continue to assume that firms haveperfect foresight at time t with respect to the price of output over the periodWith respect to labor demand a diminishing marginal product of labor implies aninverse relationship between the real wage and labor demand
N st = N d
t (wtpt K)
It is typical to assume that the labor market is such that firms achieve employ-ment equal to that demanded In this case given the production function and thenature of the labor demand function we have an output supply function for periodt of the form6
yst = yd
t (wtpt K)
An increase in the real wage reduces labor demand and thus output supply so thatwe have
partN dt part(wtpt) lt 0 and partys
t part(wtpt) lt 0
Now consider firmsrsquo behavior with respect to investment and consequent finan-cial asset supply Given a diminishing marginal product of capital capital demandis inversely related to the expected real user cost of capital7
Assuming labor and capital are complements in the production process(part2f partKpartN gt 0) capital demand will be inversely related to the expected realwage in the subsequent period as well8 In particular in the absence of adjustmentcosts (for both capital and labor) we have the following capital demand function
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (58)
with
partKdt+1part(me
t + δ) lt 0 and partKdt+1part(we
t+1pet+1) lt 09
The above demand for capital stock at the end of period t (in place at time t +1)implies a net investment demand function for period t of the form
I dnt = I d
nt(met + δ we
t+1pet+1 K)
where net investment demand is inversely related to the expected real user cost ofcapital me
t + δ the anticipated real wage in the next period wet+1pe
t+1 and theexisting capital stock at time t K
Recall that the firm financing constraint in the absence of capital adjustmentcosts equates firmsrsquo net real financial asset supply to net investment demandThus given the nature of the net investment demand function the net real financial
Behavior and constraints 69
asset supply function for firms at the end of period t (at time t + 1) is identical tonet investment demand or
net Ast = net As
t (met + δ we
t+1pet+1 K)
where net real financial asset supply for period t like net investment demandduring period t is inversely related to the expected real user cost of capital theanticipated real wage in the next period and the existing capital stock at time t
With convex adjustment costs the optimal capital stock (as well as investmentdemand) depends on the entire future path of the expected real user cost of capitaland real wages That is with adjustment costs
Kdt+1 = Kd
t+1(met + δ me
t+1 + δ wet+1pe
t+1 wet+2pe
t+2 ) (59)
We can rewrite the above demand function for capital given adjustment costs soas to collapse future periods into essentially a single subsequent period10 To doso recall that the expected real rate of interest for period i me
i equals (ri minus πei )
(1+πei ) Let us assume static expectations concerning future interest rates so that
ri = rt i = t + 1 t + 2 Further let us assume static expectations concerningfuture expected inflation so that πe
i = πet i = t + 1 t + 2 The result is that
the expected constant real rate of interest between periods t and t + 1 is expectedto prevail in the future so that me
i = met i = t + 1 t + 2 We can then rewrite
the demand function for capital accumulated at the end of period t in the simplerform
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1 we
t+2pet+2 )
Now note that we can decompose the anticipated wage for period i and theexpected price level for period i into two components the wage or price levelin period i minus 1 and the expected rate of change in wages or prices respectivelyIn particular the anticipated money wage and price level for period t + 2 can beexpressed by
wet+2 equiv we
t+1(1 + πewt+1) and pe
t+2 equiv pet+1(1 + πe
t+1)
Let us now assume static expectations with respect to the rate of change in wagesbeyond the next period so that πe
wi = πewt+1 i = t + 2 t + 3 Recall that
we have already assumed a constant rate of price inflation in subsequent periodsIf we then add the assumption that beyond the next period the (constant) rate ofinflation in wages (πe
wt+1) equals the expected rate of inflation in prices (πet+1)
we have that wei pe
i = wet+1pe
t+1 i = t + 2 t + 3 In words the real wagein the subsequent period we
t+1pet+1 i = t + 2 t + 3 is anticipated to persist
indefinitely11 We can now rewrite the capital demand function given adjustmentcost (equation (59) as
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (510)
70 Behavior and constraints
Note that given our expectation assumptions the capital demand function withadjustment costs (equation (54)) has the same form as the capital demand functionin the absence of capital adjustment costs However the actual response to changesin the real user cost of capital or in the anticipated real wage in the subsequentperiod would typically be less with adjustment costs as such costs would leadfirms to only gradually move toward a new optimal capital stock
Equation (510) captures the idea that the demand for capital to be in placeat the end of period t (at time t + 1) depends inversely on the expected realuser cost of capital and that assuming capital and labor are complements capitaldemand depends inversely on the anticipated future real wage Similarly sincenet investment demand during period t simply reflects the difference betweencapital demand at the end of the period and the initial capital stock we haveinvestment demand being inversely related to the expected real user (or rental) costof capital and to the anticipated real wage for the subsequent period Finally fromthe firm financing constraint that relates firmsrsquo net real financial asset supply tonet investment demand we have firmsrsquo net financial asset supply being inverselyrelated to the expected real user cost of capital and the expected future real wageThus in summary we have
partKdt+1part(me
t + δ) lt 0 partI dntpart(me
t + δ) lt 0
partKdt+1part(we
t+1pet+1) lt 0 partI d
ntpart(wet+1pe
t+1) lt 0
partnet Ast part(me
t + δ) gt 0 partnet Ast part(we
t+1pet+1) lt 0
where mei equiv (rt minus πe
t )(1 + πet ) Note that gross investment demand is given by
Idt equiv Kd
t+1 minus K + δK = I dnt + δK
Summarizing household behavior with limitedperfect foresight
Householdsrsquo plans at time t concerning the labor supply choice the consump-tionsaving choice and the portfolio choice for period t are constrained by theanticipated household budget constraint as given by equation (55) From ourintertemporal analysis of these optimal choices we know that in general at time twe thus have labor supply at time t
N st = N s
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
consumption demand planned at time t
cdt = cd
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
Behavior and constraints 71
and money demand planned at time t
Ldt = Ld
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
where for notational ease we have defined planned real money demand by the termLd
t From the anticipated budget constraint as given by
cdt + (M d
t minus M )pet + net Ad
t minus (wtpet )N
st minus (dt + zBpt) = 0
we can infer net real financial asset demand planned at time t net Adt from
householdsrsquo plans concerning money demand consumption demand and laborsupply
With perfect foresight at time t concerning the prices for period t house-holdsrsquo plans made at time t concerning consumption money demand andnet real financial asset demand during period t will be fulfilled Thus lettingxt = dt + zBpt + At + M tpt we can express the above behavior conditions asfollows labor supply at time t is
N st = N s
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
consumption demand for period t is
cdt = cd
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
and money demand for period t is
Ldt = Ld
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
where Ldt equiv M d
t pt As we did with respect to firmsrsquo capital demand function given adjustment
costs we now introduce expectation assumptions that essentially collapse theentire sequence of future periods into a single future period First we assume staticexpectations concerning future interest rates so that ri = rt i = t + 1 t + 2 Next we assume static expectations with respect to future rates of inflation suchthat πe
i = πet i = t + 1 t + 2 Finally we assume that the expected rate of
wage inflation beyond the next period is constant and equal to the expected rate ofinflation (ie πe
wi = πewt+1 i = t + 2 t + 3 and πe
wt+1 = πet+1) so that the
real wage anticipated for the next period is expected to prevail indefinitely Giventhe above restrictive expectation assumptions we can rewrite the householdsrsquobehavioral functions in the following form the labor supply at time t becomes
N st = N s
t (wtpt wet+1pe
t+1 rt πet xt)
consumption demand for period t becomes
cdt = cd
t (wtpt wet+1pe
t+1 rt πet xt)
72 Behavior and constraints
and money demand for period t becomes
Ldt = Ld
t (wtpt wet+1pe
t+1 rt πet xt)
where real money demand at the end of period t is given by Ldt equiv M d
t pt Our discussion of the labor supply decision suggests that labor supply is directly
related to the real wage and inversely related to the real wage next period (whichreflects the real wage in all subsequent periods given our expectation assumptions)This response to changes in the real wages incorporates the intertemporal substitu-tion hypothesis The intertemporal substitution hypothesis also suggests that laborsupply is directly related to the interest rate and inversely related to the expectedrate of inflation in that either change implies a higher expected real rate of inter-est and thus a substitution from leisure to increased labor supply today Finallyhigher real initial holdings of financial assets real money balances or income inthe current period from bond and equity shares holdings will reduce labor supplyif leisure is a normal good In summary we thus have
partN st part(wtpt) gt 0 partN s
t part(wet+1pe
t+1) lt 0 partN st partrt gt 0
partN st partπe
t lt 0 partN st part xt lt 0
Our discussion of the consumptionsaving decision suggests that consumptiondemand is directly related to both the current and anticipated future real wagesince an increase in either implies an increase in the discounted stream of incomeFocusing on the Fisherian analysis of the allocation of consumption across timeconsumption demand would be inversely related to the money interest rate anddirectly related to the expected rate of inflation since either change implies ahigher expected real rate of interest Finally an increase in xt (reflecting sayhigher real initial holdings of financial assets increased real money balancesor higher income in the current period from bond and equity share holdings) willraise consumption demand in the current period In summary we thus have
partcdt part(wtpt) gt 0 partcd
t part(wet+1pe
t+1) gt 0 partcdt partrt lt 0
partcdt partπe
t gt 0 partcdt part xt gt 0
Our discussion of the portfolio decision suggests that real money demand isdirectly related to both the current and anticipated future real wage since an increasein either implies an increase in the discounted stream of income Focusing on theportfolio analysis money demand would be inversely related to the money interestrate as households shift from money to financial asset holdings Any effect of achange in the expected rate of inflation is indirect and will be ignored Finallyan increase in xt (reflecting say higher real initial holdings of financial assetsincreased real money balances or higher income in the current period from bondand equity share holdings) will likely raise money demand in the current period
Behavior and constraints 73
In summary we thus have
partLdt part(wtpt) gt 0 partLd
t part(wet+1pe
t+1) gt 0 partLdt partrt lt 0
partLdt part xt gt 0
So far our discussion has not included householdsrsquo real net financial assetdemand To see what determines this we can simply use the budget constraintand money labor and commodity demand functions Specifically rewriting thebudget constraint for period t
net Adt = (wtpt)N
st + dt + zBpt minus cd
t minus (M dt minus M )pt
An increase in the current real wage initial real money balances or dividendand interest payments for the current period is presumed to increase not onlycurrent consumption and money demand but also future consumption and moneydemand and thus increase net financial asset demand by households From theintertemporal substitution hypothesis an increase in the expected future real wagewill reduce current labor supply as well as increase current consumption demandboth changes imply a fall in net real financial asset demand for households
From the Fisherian analysis a higher expected rate of inflation will reducethe expected real rate of interest the above constraint indicates that the resultingincrease in current consumption demand will reduce householdsrsquo acquisition offinancial assets Similarly an increase in real initial financial asset holdings byraising both current consumption and money demand will lead to a fall in real netfinancial asset demand On the other hand a higher money interest rate due toboth the Fisherian effect on current consumption demand and the portfolio effecton money demand implies a higher net financial asset demand In summary wethus have
partnet Adt part(wtpt) lt (or gt or =) 0 partnet Ad
t part(wet+1pe
t+1) lt 0
partnet Adt partrt gt 0
partnet Adt partπe
t lt 0 partnet Adt partAt lt 0 partnet Ad
t partMpt) gt 0
partnet Adt part(dt + zBpt) lt (or gt or =) 0
where there are ambiguous effects on net financial asset demand of a change in thereal wage and of a change in anticipated dividend and interest payments becausean increase in either raises both consumption demand and money demand Notethat in the limit as the length of the period goes to zero the household budgetconstraint at time t becomes
net Adt + (M d
t minus M )pt = 0
74 Behavior and constraints
as all the flow terms go to zero at a point in time In this case assuming real moneydemand is directly related to income we obtain the unambiguous effect of
partnet Adt part(dt + zBpt) lt 0 partnet Ad
t part(wtpt) lt 0
However in the period analysis net Adt reflects net real financial asset demand at the
end of the period Thus assuming any increase in income is not fully reflected inan increased rate of consumption we have an offsetting effect and thus ambiguity
Summarizing household behavior without perfectforesight
If we presume that households learn of prices that will exist for period t aftertime t and they differ from what was expected then the actual demands for outputmoney and financial assets can differ from those reflecting plans made at time tThe key reason for this is that the actual constraint faced by households will differfrom that anticipated In particular using the actual firm distribution constraintwe replace anticipated real income from wages dividends and interest paymentsfrom firms with the actual real income net of depreciation ( ylowast
t minusδK) The resultingrealized household budget constraint after time t is then
cdt + (M d
t minus M )pet + net Ad
t minus ( ylowastt minus δK) = 0
If anticipations by households were incorrect at time t concerning prices or div-idends during the period then revisions in plans for consumption and saving willbe made in light of the actual budget constraint faced In this case the actual house-hold demand functions for output and money are written as follows consumptiondemand during period t is
cdt = cd
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
money demand during period t is
Ldt = Ld
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
and we replace partcdt part(wtpt) gt 0 and partcd
t part(dt + zBpt) gt 0 with
partcdt partylowast
t gt 0
Note that the term partcdt partylowast
t is referred to as the ldquomarginal propensity toconsumerdquo12 Similarly we replace partLd
t part(wtpt) gt 0 and partLdt part(dt + zBpt) gt 0
with
partLdt partylowast
t gt 0
Behavior and constraints 75
Money illusion and the real balance effect
Let us consider (sufficient) assumptions under which demands and supplies arehomogeneous of degree 0 in current wages prices and the nominal stock ofmoney ndash that is there is the absence of money illusion Note that in consideringwhether or not there exists money illusion we must now look at the behavior notonly of households but also of firms Consider firms first
Assuming perfect foresight on the part of firms it is clear that current labordemand is homogeneous of degree 0 in current prices (the wage rate wt and the pricelevel pt) Thus so also current output supply Assuming unit elastic expectationswith respect to wages and prices in all future periods and expectations of futureinterest rates that are invariant to changes in current prices and wages we havethat capital demand and thus also investment demand and firmsrsquo net real financialasset supply are homogeneous of degree 0 in current prices
We already assumed static expectations concerning future interest rates and unitelastic expectations with respect to prices beyond period t + 1 in terms of nextperiodrsquos price to obtain a simple form for the demand functions for investmentThus we need only add (a) the assumption of unit elastic expectations concerningthe price of output between period t and t+1 (implying an expected rate of inflationπe
t and thus an expected real rate of interest that is independent of a change in theprice level pt)
13 and (b) unit elastic expectations concerning next periodrsquos wageand price level (implying an anticipated real wage next period independent of anequiproportionate change in wages and prices in the current period) to obtain theabsence of money illusion with respect to firmsrsquo investment demand14
To obtain the absence of money illusion with respect to households we mustshow that each of the arguments in their demand and supply functions is invariantto an equiproportional change in current prices wages and the nominal stockof money Given perfect foresight the current real wage and initial real moneybalances meet this condition But what about the expected real wage next periodthe expected rate of inflation current dividends and interest payments and the realvalue of initial financial assets holdings As it turns out the assumption of unitelastic expectations with respect to all future prices and wages is again critical inshowing these to be invariant as it was in deriving a capital demand homogeneousof degree 0 in current wages and prices
First it is clear that the assumption of unit elastic expectations with respectto future wages and prices makes future real wages invariant to equiproportionalchanges in the current wages and prices But note that in so doing we have elimi-nated the ldquointertemporal substitution hypothesisrdquo effect on labor supply of a changein the current real wage Similarly the assumption of unit elastic expectations withrespect to the future price level eliminates any effect of a change in the price levelon the expected rate of inflation Finally assuming that expectations of nominalfuture interest rates are invariant to an equiproportionate change in current priceswages and the money supply the expected real rate of interest will not be affectedby equiproportionate changes in wages prices and the money supply But notethat the ldquointertemporal substitution hypothesisrdquo impact on labor supply of a change
76 Behavior and constraints
in the expected real rate of interest initiated by a change in the current price levelis now absent as well15
What is left in order to obtain the absence of money illusion for households isto show that changes in current prices leave current dividend payments and thereal value of initial bond and stock holdings unchanged Once again as we seebelow the assumption of unit elastic expectations concerning next periodrsquos wagesand prices will be invoked to achieve this What we are looking for are sufficientassumptions that will result in the terms dt + zBpt and At being homogeneous ofdegree 0 in wt and pt
From the firm distribution constraint (52) and assuming firmsrsquo labor demandis satisfied (ie N lowast
t = N dt and thus ylowast
t = yst ) we know that
dt + zBpt = yst minus δK minus (wtpt)N
dt
Since N dt and ys
t are homogeneous of degree 0 in prices it is clear that the sumof current dividends plus interest payments is not affected by equiproportionatechanges in both the current wage and price level A higher price level does alterthe composition of payments however as real dividends rise and real interestpayments fall
Now consider At To show that this can be homogeneous of degree 0 in thecurrent wage and price level note from the firm distribution constraint and fromthe assumption that firmsrsquo demand for labor is satisfied in subsequent periods that
dt + zBpet+1 = f (N d
t+1 Kdt+1) minus (we
t+1pet+1)N
dt+1 minus δKd
t+1
A similar equation holds for future periods as well At simply reflects the presentvalue of such future real payments using the appropriate expected real interest ratesfor discounting The assumption of unit elastic expectations concerning prices inall future periods coupled with expectations of future nominal interest rates thatare unaffected by an equiproportionate change in money prices and the moneysupply thus means that A is homogeneous of degree 0 with regard to a change inprices (price level and wages) and the money supply in period t16
A special case of the above is if we assume static expectations concerning futureinterest rates (ie ri = rt i = t + 1 t + 2 ) and zero adjustment costs In thiscase Kd
i i = t + 1 t + 2 would be the same in each future period Therewould be a constant labor demand (N d
i = N dt+1 i = t + 2 t + 3 ) as well given
the invariant real wage in conjunction with no change in their capital stock Nowrecall that At is defined by
At equiv [pbtB + petS]pt
where pbtBt is the present value of future interest payments and petS is the presentvalue of future dividends Since At is simply the present value of the now constantfuture stream of dividends and interest payments discounted using an invariant
Behavior and constraints 77
expected real rate of interest we thus have
At = [dt+1 + zBtpet+1]me
t
Since dt+1 +zBtpet+1 = f (N d
t+1 Kdt+1)minus (we
t+1pet+1)N
dt+1 minusδKd
t+1 from the firmdistribution constraint we can rewrite the initial real holdings of financial assetsas
At = [ f (N dt+1 Kd
t+1) minus (wet+1pe
t+1)Ndt+1 minus δKd
t+1]met
which is homogeneous of degree 0 in current wages and prices if we assume that theanticipated real wage next period is independent of an equiproportionate changein the current wages and prices In other words to obtain the absence of moneyillusion for households we assume as we did with firms that there are unit elasticexpectations concerning wages and prices in the next period
Note that an increase in the money interest rate and the expected rate of inflationthat leaves the expected real rate of interest unchanged will alter the compositionof financial asset holdings although not the total To see this recall that with staticexpectations concerning the interest rate the price of bonds at the end of period tis given by
pbt =infinsum
i=1
z(1 + rt)i = zrt
An increase in the money interest rate and expected rate of inflation such that thereal rate of interest is unchanged means a fall in the price of bonds but an offsettingrise in the price of stock as over time real dividend payments will be rising morerapidly while real interest payments will be falling more rapidly Similarly anincrease in the current level of prices although leaving At unaffected wouldreduce the real value of bond holdings and lead to an exactly offsetting increasein the real value of equity share holdings
The real balance effect is apparent from the nature of the demand and supplyfunctions In particular an increase in nominal money balances or fall in prices withno change in nominal money balances will increase initial real money holdingsand in general lead to an increase in consumption demand real money demandand real net financial asset demand In general labor supply would fall
6 The simple neoclassicalmacroeconomic model (withoutgovernment or depositoryinstitutions)
Introduction
We have now covered a substantial part of the underlying structure for a simpleaggregate model of an economy with production The specific elements of theldquomicroeconomic foundationsrdquo of this aggregate model developed so far have dealtwith the optimizing behavior of individuals (ldquorepresentativerdquo firms and house-holds) in a setting in which individuals take prices as given Implicit in thesediscussions is another part of the microeconomic foundations the way in whichindividual markets operate We have been assuming that prices adjust so that thepresumption that buyers and sellers are price-takers is justified In other wordsequilibrium within individual markets entails price adjustment to equate suppliesand demands In addition we will assume that all individuals in the economy cor-rectly foresee period trsquos output prices when input supply and production decisionsare made at the start of the period As discussed below however there are otheroptions
Static macroeconomic models the options
Grandmont (1977 542) notes that
one way to look at the evolution of an economic system is to view it as asuccession of temporary or short-run competitive equilibrium That is onepostulates that at each date prices move fast enough to match supply anddemand Although one assumes equilibrium in each period the economicsystem displays a disequilibrium feature along a sequence of temporary com-petitive equilibrium at each date the plans of the agents for the future arenot coordinated and thus will be in general incompatible this is to becontrasted with the perfect foresight approach where by definition such adisequilibrium phenomenon cannot occur
This temporary equilibrium view of the economy is characteristic of the simplestatic neoclassical model a model in which all prices adjust to maintainequilibrium1 The second key assumption of the neoclassical model is that agents
Simple neoclassical macroeconomic model 79
are informed about prices within the period In particular when making their laborsupply and demand decisions at the start of the period households and firms areassumed to correctly anticipate the prices they will have to pay to purchase outputduring the period As it turns out this element of ldquoperfect foresightrdquo with respectto markets through time t +1 is a critical feature of the analysis Before examiningthe implications of these assumptions however some history on the origin of theneoclassical model might be helpful
The phrase ldquoneoclassical macroeconomic modelrdquo is a descendant of ldquoclassicalrdquoeconomic theory as reflected in the work of Sir William Petty during the 1600sIn Das Kapital Karl Marx (1976 85) stated that ldquoby classical Political EconomyI understand that economy which since the time of W Petty has investigated thereal relations of production in bourgeois societyrdquo As Marx suggested early classi-cal economists focused on the determinants of the economyrsquos productive capacityThe neoclassical macroeconomic model shares this focus on the productive capac-ity of the economy as the determinant of total output It also turns out to be thestatic precursor to much of the current analysis in the macroeconomic literaturethat falls under the heading of ldquoreal business cycle theoryrdquo
While the simple static neoclassical model along with its dynamic and stochas-tic counterparts is one popular approach to macroeconomic analysis there areother approaches In fact even though static analysis is restricted to markets in thecurrent period there remains enough flexibility to introduce at least four ways ofcharacterizing macroeconomic analysis
1 ldquoNeoclassical modelrdquo competitive equilibria are assumed to exist in cur-rent and future markets and limited perfect foresight is assumed for allparticipants
2 ldquoIllusion modelrdquo competitive equilibria are assumed to exist in current andfuture markets but imperfect foresight is assumed on the part of some agentsThe result is like the ldquoLucas supply functionrdquo popularized by Lucas (1973)in which output can respond directly to increases in the actual output prices
3 ldquoKeynesian modelrdquo a competitive equilibrium is assumed not to exist inthe labor market as the money wage is fixed and employment is demand-determined However other prices in particular the prices of output arepresumed to reflect competitive equilibria A rational expectation version ofthis model is developed by Fisher
4 ldquoNon-market-clearing modelrdquo a competitive equilibrium is assumed not toexist in the output market as the price of output is fixed above the competitiveequilibrium level This model forms the basis of much of what appears inundergraduate macroeconomic analysis including the IS-LM model
The neoclassical model with its assumptions of flexible prices and informedagents provides a benchmark against which we can compare the predictions ofother (static) macroeconomic models2 It also provides insight into the nature of thestationary states for dynamic macroeconomic models that presume market-clearingprices and accurate forecasts of prices
80 Simple neoclassical macroeconomic model
Hicksian temporary equilibrium and Walrasrsquo law
For the market for any good ldquocompetitiverdquo equilibrium is defined by equalitybetween market demand and supply3 A temporary competitive equilibrium for theeconomy during period t will be characterized by a money wage for labor (wlowast
t )a money price for the consumption commodity ( plowast
t ) a money price for bonds( plowast
bt) a money price for equity shares ( plowastet) allocations to households in terms
of employment consumption bond holdings equity share holdings and moneybalances (N lowast
t clowastt Blowast
t Slowastt and M lowast
t ) and allocations to firms in terms of employmentoutput investment bond issues and equity share issues (N lowast
t ylowastt Ilowast
t Blowastt Slowast
t ) suchthat
bull these allocations are in the demand (supply) set of each agentbull these allocations are feasible
Together these two conditions imply prices determined in the labor output bondand equity shares markets for period t that result in zero excess aggregate demandfor labor output bonds equity shares and money Thus we may rewrite theconditions for a general equilibrium as a money wage a price of output a price ofbonds and a price of equity shares such that
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Bst = Bd
t
Sst = Sd
t
Mpt = M dt pt
where
I dnt = Kd
t+1 minus K
yst = f (N d
t K)
Given our assumption that bonds and equity shares are perfect substitutes ingeneral there will not be a unique equilibrium in terms of the number of bondsand equity shares supplied or demanded although the total value of financialassets supplied or demanded will be determinant Thus we replace the bond andequity share markets with a single market the financial market The equilibriumconditions then become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Simple neoclassical macroeconomic model 81
Ast = Ad
t
Mpt = M dt pt
where
Ast equiv [ pbtB
st + petS
st ]pt
Adt equiv [ pbtB
dt + petS
dt ]pt
We can view the financial market as simultaneously determining the price ofbonds pbt the price of equity shares pet and the interest rate rt That is once oneof these is known the other two are implied For instance from our definition ofthe interest rate
1 + rt equiv [z + pbt+1]pt
we see that given the coupon rate and expected price of bonds in the subsequentperiod the interest rate rt implies a price of bonds pbt As perfect substitutesbonds and equity shares must offer the identical expected gross return Thus wehave that
1 + rt = 1 + ret equiv [dt+1 + pet+1]pet
As you can see given expectations of future dividends and the future price ofequity shares an interest rate rt also implies a price of equity shares pet We oftentalk of the financial market in terms of an equilibrium interest rate The aboveshould make it clear that associated with such an equilibrium interest rate areprices of bonds and equity shares And a rise (fall) in the interest rate means a fall(rise) in the prices of bonds and equity shares
We will make one additional change in the characterization of the financialmarket to put it in terms of additional demands and supplies that is put it in netrather than gross terms The reason for this is that net financial asset demands andsupplies are what correspond to household saving in the form of financial assetsand firm investment Thus the equilibrium conditions become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
net Ast = net Ad
t
Mpt = M dt pt
where
net Ast equiv As
t minus At
net Adt equiv Ad
t minus At
At equiv [pbtB + petS]pt
82 Simple neoclassical macroeconomic model
According to the above we have four equilibrium conditions but only threeprices ndash the money wage rate the level of output prices and the interest rate ndashto be determined As usual Walrasrsquo law is invoked to show that only n minus 1 ofthe n equilibrium conditions are independent However the nature of Walrasrsquo lawdepends on whether we assume limited perfect foresight or not
Walrasrsquo law for limited perfect foresight sums up the constraints faced by theindividual agents in the economy at time t to obtain
[cdt + I d
nt + δK + ψ(I dnt) minus ys
t ] + [net Adt minus net As
t ]+ (wtpt)[N d
t minus N st ] + M d
t pt minus Mpt = 0
Thus the sum of excess demands for output financial assets labor and moneymust equal zero
When there is not perfect foresight at time t concerning prices for period t inthe output and financial markets we have the equilibrium condition for the labormarket
N st minus N d
t = 0
and the modified Walrasrsquo law based on the resulting employment and output ofthe form
[cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t ] + [net Adt minus net As
t ] + M dt pt minus Mpt = 0
In this case the money wage employment and thus output are determined inthe labor market and the modified Walrasrsquo law indicates that the price level andinterest rate are determined by any two of the remaining three markets
As it turns out most macroeconomic analysis takes this second approach tosolving for equilibrium That is the analysis focuses on the labor (and other input)markets and determines the effect of changes in output price (and potentially othervariables such as the interest rate) on equilibrium employment and thus outputThis generates an ldquoaggregate supply equationrdquo which is then combined with twoof the remaining three equilibrium equations ndash typically the commodity and moneymarket equilibrium conditions ndash to determine the equilibrium price level interestrate and output The modified Walrasrsquo law is invoked to ensure equilibrium inthe financial market at this point Working backward one can infer from theaggregate supply equation the equilibrium money wage and employment impliedby the analysis
The advantage of the above approach is that it can be used whether or not thereexists limited perfect foresight at time t with respect to the price level and whetheror not prices adjust to clear markets A disadvantage of the analysis is that inthe case of the neoclassical model with limited perfect foresight and competitiveequilibrium it arbitrarily breaks up the analysis of markets In doing so it requiresthat demand functions for such goods as money and commodities be specifiedwith income as an argument This form of the demand functions obscures the fact
Simple neoclassical macroeconomic model 83
that as we have seen in standard general equilibrium analysis demand functionsdepend on prices (including the real wage) and income is a variable determinedby the choice of labor supply
With these qualifications in mind we take the standard approach of macroeco-nomics and separate out the analysis of the labor (and other input) markets (theldquoaggregate supplyrdquo part) from the other markets (the ldquoaggregate demandrdquo part)The next section considers the labor market at time t
Labor market equilibrium
At the start of period t the labor market takes place and a rate of production of outputis determined From our analysis of firm behavior at time t we know that behindlabor demand is an expected price of output over the period and associated expectedreal wage an existing capital stock and existing technology as incorporated in theproduction function We shall assume that firms correctly anticipate at time t theprice of output for the period so that firms confront the actual real wage wtpt
From our analysis of household decision-making at time t we know that behindlabor supply at time t is not only the expected price of output and implied expectedreal wage but also such factors as the relationship of the anticipated current realwage to anticipated future real wages the expected real rate of interest anticipatedwealth in the form of financial asset and real money holdings and anticipated cur-rent nonlabor income Like firms we will assume households have limited perfectforesight in that at time t they correctly foresee the price level for period tAssuming a Walrasian or ldquocompetitive equilibriumrdquo view of the labor marketsthe money wage wt adjusts to achieve equilibrium in the labor market under thesecircumstances
We have already seen how static expectations concerning future rates of wageand price inflation along with the assumption of unit elastic expectations concern-ing wages and prices next period simplify the labor supply function by removingexpected future real wages as explicit arguments Patinkin (1965) among oth-ers goes several steps beyond these simplifying assumptions and assumes that allother variables excepting the real wage do not have a significant impact on laborsupply4 Thus equilibrium in the labor market is given by a level of employmentNt and money wage wt such that
Nt = N dt (wtpt K) and Nt = N s
t (wtpt)
As Patinkin (1965 264) notes
it will immediately be recognized that we have greatly oversimplified theanalysis of this market Both the demand and supply functions for labor shouldactually be presented as dependent on the real value of bond and moneyholdings as well as on the real wage rate5 Further if we were to permit thefirm to vary its input of capital its demand for labor would depend also onthe rate of interest Finally a full utility analysis of individual behavior wouldshow the supply of labor also to depend on this rate6
84 Simple neoclassical macroeconomic model
Besides a competitive labor market and the above simplified labor supplyfunction the neoclassical model assumes that suppliers correctly anticipate theaggregate price level As we will see the assumption of the neoclassical modelthat suppliers like firms have perfect foresight at time t with respect to the priceof output for the period means that changes in the price of output have no effect onoutput supply This characteristic of the neoclassical model implies an underlyingldquoblock recursiverdquo nature to the analysis as described below
At the start of the period the labor market occurs with employment and thusoutput and the money wage being determined for the period Employment andoutput are determined based on individualsrsquo expectations of subsequent variablessuch as the price of output And the level of employment and output influencethe remaining variables to be determined But in the neoclassical model the levelof employment and output are not influenced by the remaining variables to bedetermined We can thus solve the equilibrium sequentially looking first at thelabor market and the determination of employment and output then looking atthe output financial and money markets and the determination of the price leveland interest rate
Financial market equilibrium
With regard to the financial market Patinkin (1965 215) states
a decrease in the price of bonds (and equity shares) decreases the amountdemanded of consumption commodities it will also be assumed that itdecreases the amount demanded of money balances hence by the house-holdrsquos budget constraint their total expenditures on [net] bond holdings mustincrease
Thus Patinkin invokes a ldquoFisherian effectrdquo and a ldquoportfolio effectrdquo of a change inthe interest rate to obtain
part(net Adt )partpbt lt 0
From the firm financing constraint and the fact that a firmrsquos net investment demandis inversely related to the interest rate we have
part(net Ast )partpbt gt 0
The above analysis differs slightly from that found in Patinkin (1965 214) Forinstance Patinkin uses the assumption of static expectations concerning interestrates and a coupon rate z equal to 1 to express the price of bonds as the reciprocalof the interest rate that is
pbt = 1rt
Simple neoclassical macroeconomic model 85
Further Patinkin assumes no equity shares so his financial market consists only ofbonds Finally Patinkin does not graph net real bond demand and supply Ratherhe graphs the total number of bonds demanded and supplied Thus Patinkin hasdemands and supplies of bonds of the form
Bs = rtptAst equiv Bs
t
Bd = rtptAdt equiv Bd
t
As might be expected Patinkinrsquos bond demand and supply curves differ in naturefrom net real financial asset demand and supply curves For instance it is clearfrom the analysis of investment demand that a rise in the interest rate (a fall inthe price of bonds) will lead to reduced investment demand and thus reduced netreal financial asset supply This relationship between investment and firmsrsquo net realfinancial asset supply follows directly from the firm financing constraint whichgiven pbt = 1rt is of the form
net At equiv 1ptrt[Bst minus B] = I d
nt + ψ(I dbt)
Naturally if net investment were initially zero and there were zero adjustment costs(so that Bs
t = B) the fall in investment that results from a rise in the interest ratewould imply a similar decrease in the number of bonds supplied (Bs
t ) Howeverif initially Bs
t gt B then a higher interest rate even though it decreases invest-ment could at the same time increase the number of bonds supplied As Patinkin(1965 217) observes
consider the effect of an increase in the rate of interest (fall in 1rt) The inter-nal consistency of our model requires that this decrease the amount of realbonds supplied
However this need not reduce the number of bonds supplied As Patinkin(1965 217) continues
a rise in the interest rate ( pbt falls) has lowered the price received for bondsand so may increase the number of bonds necessary to finance the firmrsquosexpenditures on investment commodities (Bs
t gt B initially) even thoughthese expenditures have decreased
A similar analysis would apply to a comparison of householdsrsquo demand for bondsin terms of numbers with household net real financial asset demand
Money ldquomarketrdquo equilibrium
As we know from Walrasrsquo law having depicted equilibrium in the labor andfinancial markets we need only look at one of the other two remaining markets
86 Simple neoclassical macroeconomic model
the output market or the money ldquomarketrdquo Consider a money market in whichnominal money supply M and nominal money demand M d
t equiv ptLdt are plotted
against the reciprocal of the price level which indicates the ldquorelativerdquo price of oneunit of money in terms of output If there is no real balance effect with respect toreal money holdings then the money demand curve is invariant to a change in theprice level (elasticity of demand equals 1) If there is a real balance effect withrespect to money demand then a fall in the price level (a rise in the relative price ofmoney (1pt)) would lead to a less than proportionate decrease in nominal moneydemand (elasticity of demand less than 1)
Aggregate supply and demand an introduction
Temporary equilibrium for the economy can be characterized in several ways Aswe alluded to above one way common in journal articles and textbooks is to dividethe analysis under the headings of ldquoaggregate supplyrdquo and ldquoaggregate demandrdquo
Under the heading of ldquoaggregate supplyrdquo is an analysis of the input markets inparticular the labor market One aim is to determine input prices (in particular themoney wage) the employment of inputs and the implied production of output thatoccur at different levels of output prices (and potentially different levels of interestrates) The term ldquoaggregate supplyrdquo is applied to this analysis for the simple reasonthat it determines the ldquosupplyrdquo of total output at different prices
Under the heading of ldquoaggregate demandrdquo is an analysis of the other marketsin the economy during period t in particular the output financial and moneymarkets The aim is to determine the level of output prices and the interest ratethat occur at different levels of output The term ldquoaggregate demandrdquo attached tothis analysis reflects the fact that the analysis determines how the price level andinterest rate adjust to equate the ldquodemandrdquo for total output to different levels ofproduction
Combining the aggregate supply and demand analysis we can determine theoutput price level and interest rate associated with temporary equilibrium as wellas the underlying equilibrium money wage real wage employment consumptioninvestment and real money balances To understand more clearly what is involvedin aggregate supply and demand analysis and how they can be combined we con-sider below the specific case of the neoclassical model starting with the aggregatesupply
Equilibrium and aggregate supply
As we have said behind aggregate supply is an analysis of various input marketsto determine the response of total output to changes in such variables as theprice level7 In the neoclassical model changes in prices lead to equiproportionatechanges in money wages with no change in the equilibrium level of employmentand thus no change in aggregate supply To formally show this let us start withthe following statement of equilibrium in the labor market in terms of a money
Simple neoclassical macroeconomic model 87
wage wt and level of employment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpt) minus Nt = 0
A critical aspect of the above is the fact that suppliers in particular suppliers oflabor correctly anticipate the price level that will exist with respect to output sothat wtpt replaces wtpe
t in the labor supply functionTotally differentiating the above two equations with respect to wt Nt and pt
one obtains[(partN d
t part(wtpt))(1pt)
(partN st part(wtpt))(1pt)
minus1minus1
] [dwtdNt
]=[(partN d
t part(wtpt))(wtdpt( pt)2)
(partN st part(wtpt))(wtdpt( pt)
2)
]
Applying Cramerrsquos rule one obtains
dwtdpt = wtpt and dNtdpt = 0
Thus in the neoclassical model the real wage and employment level determinedin the labor market are independent of changes in the price level8
The ldquoaggregate supply equationrdquo combines the analysis of the labor market (andother input markets) and resulting determination of employment of various inputswith the production function to determine the resulting output supplied For theneoclassical model the aggregate supply equation is of the form
ylowastt = ylowast
t (K ) (61)
What is important about this equation is that the price level and interest rateare not arguments in the supply equation Of course changes in the capital stockchanges in technology or changes in the supply of other inputs (eg changes inthe oil supply) can affect output Similarly a change in the composition of thelabor force or government policies that affect labor supply can affect equilibriumemployment and thus output9
We may summarize the above findings graphically Consider an increase in pt Given perfect foresight both firms and households at time t would anticipate thishigher price level In the labor market the result would be an increase in theequilibrium money wage in the same proportion as the increase in the expectedprice level so that the anticipated real wage would remain the same as wouldemployment and output This outcome for the labor market is shown in Figure 61Note that the result of no change in employment or output in light of a higher outputprice simply requires that both labor demand and supply curves shift vertically bythe same amount Such equal shifts reflect the fact that the same increase in themoney wage leaves both firms and households anticipating the same real wage asbefore
In undergraduate textbooks the fact that changes in the price of output leaveemployment and real output unaffected in the neoclassical model is often shown
88 Simple neoclassical macroeconomic model
Labor
N demand
N supply
Figure 61 Labor market equilibrium
in ( pt yt) space by a vertical ldquoaggregate supply curverdquo Such a curve summarizesthe underlying behavior for the economy-wide labor market Later we will see thatunder other assumptions such as embedded in Lucasrsquos model (ldquomoney illusionrdquo)and the Keynesian model (fixed nominal wage) the aggregate supply curve willbe upward sloping
It is important to realize that the aggregate supply shown in Figure 62 is not thetypical market supply curve of microeconomics In microeconomics a higher priceof good x and consequent increase in the demand for inputs by firms producing thatgood draws inputs away from the production of other goods so that the higherprice of good x induces increased output of that good and associated increasedemployment of inputs (such as labor) in the production of the good Implicit inthis analysis is that there is reduced production of other goods in the economyHowever as the above analysis makes clear if the focus is on the aggregation ofall commodity markets a higher price level no longer induces increased aggregateoutput unless the quantity of total inputs supplies rises which will not be the caseunder neoclassical assumptions10
The natural rate of unemployment
The key feature of the above analysis of aggregate supply is the assumption thatprices adjust to continuously maintain equilibrium in the various markets and thatindividuals are perfectly informed concerning prices The result is a level of realoutput sometimes called the ldquofull employment levelrdquo So far missing from theanalysis however is any mention of unemployment If one expands the model tointroduce unemployment the rate of unemployment is called the ldquonaturalrdquo rate11
By ldquonaturalrdquo is meant that it is the rate of unemployment that the economy will
Simple neoclassical macroeconomic model 89
Output
Price
Aggregate supply
Figure 62 Aggregate supply
gravitate to as prices adjust to clear markets and individuals become fully informedconcerning prices (ie under the assumptions of the neoclassical model)
To introduce unemployment into the analysis when the labor market is in equilib-rium at full employment requires recognition of two facts First the labor marketis in a constant state of flux Not only do individuals enter and leave the laborforce continually but labor demand varies continuously across firms as they expe-rience variations in the relative demand for their output This instability of jobsthemselves has been estimated to account for roughly one-quarter of the averageunemployment rate as in an average year one in every nine jobs disappears and onein every eight is newly created (Leonard 1988) Second information is costly Ittakes time for new workers entering the labor force and for workers who have beenlaid off or have quit previous jobs to discover which employers have vacanciesand how wages vary across employers
When we take into account the continuous flows to unemployment together withworkersrsquo imperfect information about job vacancies we see that unemploymentis no longer inconsistent with the neoclassical model At any moment there existnew entrants into the labor market who are spending time searching for acceptablejobs There also exist laid-off workers who are either searching for alternativejobs or awaiting recall And there are workers who have quit their jobs and aresearching for other jobs This kind of unemployment is generally referred to asfrictional unemployment
Sometimes part of frictional unemployment is called structural unemploymentwith structural unemployment occurring because of a change in the compositionor ldquostructurerdquo of aggregate output across firms For example the replacement ofsteel with plastic in automobiles led to a shift in employment from steel factoriesto firms making plastic During this transition some steel workers experiencedstructural unemployment
90 Simple neoclassical macroeconomic model
To summarize when the labor market is in equilibrium there exists a positiveunemployment rate because workers continuously move into and out of the laborforce and between jobs12 To signify that this unemployment rate is ldquonaturalrdquo orconsistent with equilibrium in the labor market it is generally called the naturalrate of unemployment13 The corresponding level of employment is then oftenreferred to as the full employment level
Like equilibrium output and employment in the neoclassical model ldquosupplyfactorsrdquo determine the natural rate of unemployment as well14 Among these isthe demographic composition of the labor force but also unemployment insuran-ce and minimum wage legislation In recent years a large body of literature hasanalyzed labor markets and the sources of unemployment with the focus on searchand labor contracts Note that an understanding of the natural rate of unemploy-ment is important in determining the effects of government policies aimed at theunemployed
Equilibrium and aggregate demand
The aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) The ldquoISrdquo equation since it is simply the expression forequilibrium during period t with respect to the output market is given by15
cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t = 0 (62)
Note that the equation is termed the ldquoISrdquo equation because we can rewrite it toobtain
Idnt + δK + ψ(I d
nt) = ylowastt minus cd
t
indicating that equilibrium in the output market is equivalent to the equal-ity between Investment expenditures (the left-hand side of the equation) andhousehold Saving (the right-hand side of the equation)16
The assumption of unit elastic expectations concerning wages and prices givesthe following simple form for householdsrsquo consumption demand and firmsrsquoinvestment demand functions during period t17
cdt = cd
t (rt πet At Mpt ylowast
t )
and
Idnt = I d
nt(met + δ K)
The ldquoLMrdquo equation since it is simply the expression for equilibrium in period twith respect to the ldquomoneyrdquo market is given by
Ldt = Mpt (63)
Simple neoclassical macroeconomic model 91
This equation is termed the ldquoLMrdquo equation for it reflects the equality betweenLiquidity preference or demand and the Money supply We have seen that ourassumption of unit elastic expectations concerning wages and prices gives us thefollowing simple form for the real money demand function
Ldt = Ld
t (rt πet At Mpt ylowast
t )
From the modified Walrasrsquo law it follows that a price level and interest rate thatsatisfy the ldquoISrdquo equation and ldquoLMrdquo equation for a given level of output ylowast
t willalso result in equilibrium in the financial market
We thus have three equations (the aggregate supply equation (61) the ldquoISrdquoequation (62) and the ldquoLMrdquo equation (63)) that can be solved for the equilib-rium levels of output price and interest rate Looking at what underlies thesethree equations we can then infer the changes in money wages and employment(the labor market) as well as changes in the components of output demand (invest-ment and consumption demand) Equilibrium can be depicted in terms of a pricelevel and interest rate (Patinkinrsquos CCndashLLndashBB curves) or in terms of a price leveland output (ie aggregate demand and supply curves) We start with Patinkinrsquosdepiction of equilibrium
Depiction of equilibrium the Patinkin analysis
The neoclassical model aggregate supply is independent of changes in the pricelevel and interest rate18 This means that for any given level of output we canfocus on equations (62) and (63) to determine the equilibrium interest rate andprice level This is a reflection of the ldquoblock recursiverdquo nature of the solution tothe neoclassical model mentioned earlier
Patinkin (1965) suggests a graphical way of showing such an equilibrium com-bination of price level and interest rate using any two of three curves denoted theCC LL and BB curves The CC curve depicts combinations of the price level andinterest rate that satisfy the equilibrium condition for output (62) the LL curvedepicts combinations that satisfy the equilibrium condition for money (63) Wherethese two lines so constructed intersect it must then be the case that this price com-bination satisfies two of the three equilibrium conditions simultaneously It followsfrom the modified Walrasrsquo law that the curve indicating various combinations ofthe price level and interest rate that satisfy the equilibrium condition with respectto the financial market (the BB curve) goes through this point as well19
For the market for output equilibrium in period t is characterized by (62) Recallthat real output ylowast
t denotes that output reflecting the capital stock K technologyand the employment of labor determined in the labor market at time t From theneoclassical assumptions with respect to labor demand and supply equilibriumemployment and thus output are unchanged for any change in the price level orinterest rate
Let us presume that the pair ( plowastt rlowast
t ) is associated with equilibrium in the outputmarket In (price interest rate) space this combination is identified by a unique
92 Simple neoclassical macroeconomic model
point on the CC curve that identifies combinations of pt and rt associated withequilibrium in the commodity market (Figure 63)
To understand what lies behind the shape of the CC curve depicted above totallydifferentiate the market-clearing condition for the commodity market with respectto pt and rt Doing so we obtain20
[partcdt part(Mpt)] minus [Mp2
t ]dpt + [partcdt partrt + (1 + ψ prime)partI d
ntpartrt]drt = 0
Rearranging we have that the slope of the CC curve is given by
drtdpt | ydt minus ylowast
t = 0= [partcd
t part(Mpt)][Mp2t ][partcd
t partrt + (1 + ψ prime)partI dntpartrt] lt 0
The negative slope of the CC curve can be explained in the following way Thepositive numerator of the expression for the slope reflects the real balance effectwith respect to commodities this term indicates the fall in consumption demandthat would accompany a rise in the price level for such a rise reduces agentsrsquowealth in the form of initial real money balances21 The negative denominatorindicates the effect of a change in the interest rate on consumption and investmentdemand
Now let us consider the money market As before there is a unique point thatindicates an interest rate and a price of output at which there is equilibrium inthe economy and this point on the LL curve identifies combinations of pt andrt associated with equilibrium in the money market (Figure 64) Recall that theLL curve identifies combinations of pt and rt associated with equilibrium in themoney ldquomarketrdquo as given by (63)
The slope of the LL curve is given by totally differentiating the zero excessdemand condition with respect to money Doing so and rearranging one obtains
drtdpt |Ldt minus Mpt = 0 = [1 minus partLd
t part(Mpt)][Mp2t ][minuspartLd
t partrt] gt 0
r
CC
p
Figure 63 Commodity market
Simple neoclassical macroeconomic model 93
rLL
p
Figure 64 Money market
Note that the numerator of this term is positive The change in real money balanceswill be greater than the consequent change in real money demand given real balanceeffects with respect to financial assets andor commodities The denominator ispositive as well reflecting the fact that an increase in the interest causes householdsto shift their portfolio out of money holdings into bonds
Putting together the CC and LL curves we have equilibrium in the economyat the point where these two curves intersect From the modified Walrasrsquo law weknow that at this point demand for financial assets equals supply as well In factthere is a corresponding BB curve that goes through this same intersection
As with the CC curve to understand what lies behind the shape of the BB curvewe can totally differentiate the market-clearing condition for the financial marketwith respect to pt and rt That market-clearing condition is
net Adt = net As
t = 0
where in simplest form
net Ast = net As
t (met + δ K)
net Adt = net Ad
t (rt πet At Mpt ylowast
t )
Differentiating we obtain
[partnet Adt part(Mpt)] minus [Mp2
t ]dpt + [partnet Adt partrt minus partnet As
t partrt]drt = 0
Rearranging we have that the slope of the BB curve is given by
drtdpt |net Adt minus net As
t = 0= [partnet Ad
t part(Mpt)][Mp2t ][partnet Ad
t rt minus partnet Ast partrt] gt 0
94 Simple neoclassical macroeconomic model
The positive slope of the BB curve can be explained in the following way Thepositive numerator of the slope expression reflects the real balance effect withrespect to financial assets The term indicates the fall in net real financial assetdemand (and planned decrease in future consumption) that would accompany arise in the price level for such a rise reduces agentsrsquo wealth in the form of initialreal money balances The positive denominator indicates the effect of a change inthe interest rate on household lending and firm borrowing A rise in the interestrate would tend to decrease current consumption demand (ldquoFisherianrdquo effect) andmoney demand (ldquoportfoliordquo effect) and thus increase household net financial assetdemand On the other hand a rise in the interest rate would tend to reduce firminvestment demand by raising the expected real user cost of capital and thusreduce firm net real financial asset supply
While both the slope of the BB curve and the LL curve are positive it is thecase that the LL curve is steeper
A depiction of equilibrium aggregate demand andsupply curves
Using Patinkinrsquos analysis a higher output would require a lower price level tomaintain equilibrium in the output financial and money markets Alternativelyone can show the effect of a change in ylowast
t on pt and rt by totally differentiatingequations (62) and (63) with respect to pt rt and ylowast
t The aggregate demand curve summarizes this inverse relationship between the
price level and output arising from an analysis of the output financial and moneymarkets Specifically such a curve depicts combinations of output and price levelassociated with equilibrium in the output financial and money markets It isdownward sloping indicating that an increase in output requires a lower pricelevel to clear the output financial and money markets Behind a movement downthe aggregate demand curve are larger real balances that stimulate output demandeither directly (through a real balance effect on consumption demand) or indirectly(the resulting increase in real money balances leads to an increase in net realfinancial asset demand by households and thus a lower interest rate)22
It is important to realize that the phrase ldquoequilibrium in the output mar-ketrdquo in this context abstracts from supply-side considerations At each pricelevel the aggregate demand curve indicates the output that if produced wouldequal output demand (along with satisfying the equilibrium conditions withrespect to the financial and money markets) Production of output equal tothat demanded would occur if firms sought simply to produce to meet mar-ket demand and if workers were readily available for employment so thatfirms could hire to achieve production equal to what was demanded The termldquoequilibrium in the output marketrdquo does not imply equality between outputdemand and the output that our analysis of the labor market suggests would besupplied
Simple neoclassical macroeconomic model 95
Output
Price
Aggregate supply
Aggregate demand
Figure 65 Macroeconomic equilibrium
Figure 65 combines the aggregate demand and aggregate supply curves Wherethey intersect we know by construction that the resulting price level ( pt)0 andoutput ( yt)0 are such that
bull the labor market is in equilibrium (the economy is at a point on the aggregatesupply curve) and
bull the output financial and money markets are in equilibrium (the economy isat a point on the aggregate demand curve)
Conclusion
Using a very simple framework this chapter has developed a powerful macroeco-nomic model of the economy This model is grounded in the general equilibriumtheory set forth in earlier chapters Moreover this model highlights that degree ofconnectedness that exists among the output labor financial and money marketsThe microfoundations of macroeconomics have been emphasized and it has beenshown how market forces work in such a way as to lead to aggregate supply anddemand two key elements in any macroeconomic model
7 Empirical macroeconomicsTraditional approaches and timeseries models
Introduction
Quantitative approaches to analyzing economic data provide meaningful anduseful insight for understanding how variables interact and how they might beexpected to behave in a variety of circumstances including the future This chapteroutlines the traditional econometric-based method and the relatively simple butoften more elegant time series method for analyzing economic data in the timedomain The stochastic nature of economic data is discussed and the now com-mon ARIMA model found in much of the empirical macroeconomic literature isdeveloped in parts The chapter provides a solid background for understandingldquomacroeconometricsrdquo and time series analysis
Traditional approaches
Empirical macroeconomics can be roughly divided into two approaches ndash a tradi-tional approach that draws heavily on macroeconomic theory and a more recentapproach advocated for forecasting that does not rely to any great extent on the-ory To consider the former we start by presenting a simple theoretical model ofthe macroeconomy The model is used to illustrate traditional empirical analysesreflecting (a) tests of behavioral hypotheses (b) tests of reduced-form expressionsfor various economic aggregates and (c) the construction of econometric modelsfor policy simulations and forecasting The more recent empirical macroeconomicsapproach of using time series models for forecasting aggregate variables whichplaces less reliance on macroeconomic theory is then briefly considered
A simple theoretical macroeconomic model
Consider the following linear approximation of the simple static classical model(closed economy) such as that popularized by Sargent (1987a 20)1 The economyis divided into four markets a labor market (where wages w and total employ-ment n are determined) an output market (where total output y and the pricelevel p are determined) a financial market (where the interest rate r is deter-mined) and a money ldquomarketrdquo Our goal is to construct a model that will determine
Empirical macroeconomics 97
such endogenous variables as the level of employment wages output prices andthe interest rate
From Walrasrsquo law (as we will see more clearly later on) we need only explicitlyconsider three of the above four markets in the analysis For the labor market letus assume the following linear approximations for the key behavioral relations2
nd = f minus g(wp) (71)
ns = h + j(wp) (72)
These two linear equations indicate that firmsrsquo labor demand nd is inversely relatedto the real wage (wp) and householdsrsquo labor supply ns is directly related to thereal wage3
For the output market the key underlying behavioral and technological relationsare in linear form
cd = a + b( y minus T ) minus cr (73)
id = d minus er (74)
ys = f (n K) (75)
Equation (73) indicates that householdsrsquo consumption demand cd is directlyrelated to real disposable income (income y minus lump-sum taxes T ) andinversely related to the interest rate r4 Equation (74) indicates that firmsrsquo invest-ment demand id is inversely related to the interest rate r Equation (75) is theaggregate production function relating employment n and capital stock k tototal output supplied ys
For the money ldquomarketrdquo the key underlying behavioral relation is
Ld = l middot y minus m middot r (76)
Equation (76) indicates that householdsrsquo real money demand Ld is directly relatedto real income and inversely related to the interest rate Nominal money supplyM s is exogenous
General equilibrium in this economy means an equilibrium level of employmentnlowast wage wlowast price level plowast output ylowast and interest rate rlowast such that the labor marketis in equilibrium
ns minus n = 0 (77)
nd minus n = 0 (78)
the output market is in equilibrium
ys minus y = 0 (79)
cd + id + gd minus y = 0 (710)
98 Empirical macroeconomics
and the money ldquomarketrdquo is in equilibrium
Ld minus M sp = 0 (711)
Note that the component of output demand gd in (710) reflects exogenous realgovernment demand for output
The above macroeconomic model consists of 11 equations Macroeconomicmodels often can be represented by a system of equations This system of equationsis sometimes said to be a ldquostructural modelrdquo because the form is given from theunderlying theory As we will see below we can solve this system of equationsfor each of the ldquoendogenousrdquo variables as a function solely of the predeter-mined or ldquoexogenousrdquo variables5 Such solutions can be called the ldquoreduced-formrdquosolutions
By substituting the behavioral equations (71)ndash(76) into the equilibrium condi-tions we obtain the following set of five equilibrium conditions that can be solvedfor the five variables nlowast wlowast plowast ylowast and rlowast6
f minus g(wp) minus n = 0
h + j(wp) minus n = 0
f (n K) minus y = 0
a + by minus bT minus cr + d minus er + gd minus y = 0
ly minus mr minus M sp = 0
(712)
Note that the first three equations in the system (712) can be solved to obtainwlowast nlowast and ylowastas a function of the price level plowast In particular we obtain7
wlowast = [(f minus h)(g + j)]plowast (713)
nlowast = [1( j + g)][ jf + gh] (714)
ylowast = f (nlowast K) = f ([1( j + g)][ jf + gh] K) (715)
Equation (715) is an example of a reduced-form expression for output for itexpresses output solely in terms of the exogenous variables It is a special caseof what is called the ldquoaggregate supply equationrdquo in which the price level doesnot affect the level of output produced8 Note that equation (713) indicates thatchanges in the price level lead to equiproportionate changes in the equilibriummoney wage so that changes in the price level do not lead to changes in theequilibrium real wage
To obtain a reduced-form expression for the price level we can use the reduced-form expression for output in conjunction with the last two equations in the system(712) These last two equations are sometimes termed the ldquoIS equationrdquo andthe ldquoLM equationrdquo respectively9 Solving the LM equation (the last equation ofthe system (712)) for r and substituting both this expression for r and the priorexpression for equilibrium output ylowast (715) into the IS equation (the penultimate
Empirical macroeconomics 99
equation of the system (712)) we obtain the reduced-form expression for theequilibrium price level
plowast = M scprimem
[1 minus b + cprimem] f ([1( j + g)][ jf + gh] K) minus aprime (716)
where aprime = a + d + gd minus b middot T and cprime = c + eChanges in the variable aprime indicate changes in the autonomous component of
consumption (the term a in (73)) autonomous investment demand (d in (74))and government spending or taxing (gd or T respectively) By ldquoautonomousrdquo wemean that part of householdsrsquo and firmsrsquo output demand that is independent of thevariables to be determined by the analysis particularly income and the interestrate The variable cprime indicates the combined response of consumption demand (cin (73)) and investment demand (d in (74)) to a unit change in the interest rate
Note that the procedure to obtain equation (716) could be alternatively describedas follows First combine the last two equations in the system (712) to eliminatethe interest rate r The result is termed the ldquoaggregate demand equationrdquo Thisrelates the price level to the level of output at which the money and output markets(and thus by a modified version of Walrasrsquo law the financial market) are inequilibrium In this context equilibrium in the output market is defined as thelevel of production that if produced would equal output demand This aggregatedemand equation would then be combined with the aggregate supply equation(715) to determine the equilibrium price level or output10
Tests of behavioral hypotheses
Theoretical macroeconomic models embody predictions concerning factors thatinfluence the behavior of various groups in the economy For instance considerthe behavioral equations (71)ndash(76) in the prior simple macroeconomic modelWe could test the prediction that investment is inversely related to the interest rate(see (74)) or that money demand depends inversely on the interest rate (see (76))Or we could expand our theory of consumption behavior (equation (73)) to testa particular behavioral relationship between aggregate household consumptionand permanent income11 Or we could expand our view of labor supply behavior(equation (72)) as Stuart (1981) did in an examination of Swedish data to test theprediction that sufficiently high marginal tax rates will reduce the economy-widelabor supply
Tests of reduced-form hypotheses
Theoretical macroeconomic models also generate predictions concerning thereasons for fluctuations in such aggregate variables as real output unemploy-ment and the level of prices These predictions typically reflect the reduced-formsolutions of macroeconomic models Examples of reduced forms in our simplemacroeconomic model are equations (715) for output and (716) for price
100 Empirical macroeconomics
One example of an empirical test of macroeconomic theory that focuses ontesting the reduced-form predictions is Taylor (1979) Specifically Taylor teststhe reduced-form relationships between the money supply and the logarithmof income from trend and between the money supply and the rate of infla-tion using quarterly US data from 1953 to 197512 Similarly Christopher Sims(1972) uses quarterly US data on nominal output and the money supply to testwhether changes in the money supply lead to changes in the current dollar valueof national income (see Ewing 2001)
Large-scale econometric models and forecasting
Besides testing macroeconomic theories empirical macroeconomic analysis oftenseeks to forecast the future paths of economic aggregates Interest in forecast-ing stems not only from an obvious curiosity about the future path of aggregatevariables such as real output and prices but also from the fact that many ifnot all macroeconomic theories suggest that expectations of future events influ-ence current activity In this context forecasts can be used to proxy individualsrsquoexpectations of future events in tests of various aspects of macroeconomic models
One approach to making forecasts of future aggregate variables is to rely ontheoretical macroeconomics as a guide in the construction of large-scale econo-metric models Behavioral equations that are more detailed disaggregated versionsof equations (71)ndash(76) are estimated and coefficients are checked to make surethey agree with theory These models reflect attempts to produce large systems thatfaithfully represent the interrelationships in a complex national economy Givenpostulated paths of the exogenous variables the actual estimated equations arethen used to generate forecasts of the various aggregate variables such as outputand its components (eg consumption and investment) prices and interest rates
While large-scale econometric models as forecasting devices have their advo-cates (and many individuals reveal they have a positive value by willinglypaying for their forecasts) Granger and Newbold (1986 292ndash293) among oth-ers question the value of such a forecasting approach They note that ldquoteamsof macroeconomists have constructed forecasting models involving hundreds ofsimultaneous equations fitted to data that time series analysts would view as neitherplentiful nor of especially high qualityrdquo
An alternative to the large-scale macroeconomic models that is suggested areldquotime series modelsrdquo As summarized by Granger and Newbold (1986) ldquoin theiranxiety econometricians have failed to touch some very important basesrdquo thatinclude
bull the fact that there are many areas in which ldquoeconomic theory is not terriblywell developedrdquo
bull the fact that even where the theory is satisfactory it is ldquoalmost invariablyinsufficiently precise about dynamic specifications in the sense that it is clearthat one structure must be appropriaterdquo
Empirical macroeconomics 101
bull the fact that even after appeal to economic theory there will be error termssince ldquono theory provides a completely accurate description of the behaviorof economic agents so that any postulated equation necessarily includes astochastic error termrdquo
Given these problems many macroeconomists suggest that the appropriate start-ing point to forecasting future macroeconomic variables is the use of time seriesmodels One result is that such time series modelling terms as AR ARMA andARIMA now abound in the macroeconomic literature It is thus useful to brieflyreview the nature of time series models However at the outset it should be madeclear that the discussion below is not complete but rather is provided simply asan introduction to some terms and concepts Proofs of various propositions andrigorous definitions of various properties of time series models can be found intime series texts such as Enders (2004) and Mills (1999)
Time series models
Let us take as our premise the idea that the actual observed time series of somevariable yt t = 0 1 T (eg the logarithm of economy-wide output for thelast 30 years) is the realization of some theoretical process which can be called aldquostochastic processrdquo As phrased by Harvey (1993) ldquoeach observation in a stochas-tic process is a random variable and the observations evolve in time according tocertain probabilistic laws Thus a stochastic process may be defined as a collectionof random variables which are ordered in timerdquo To forecast future values of yt one needs a model that defines the mechanisms by which the observations aregenerated
A distinguishing feature of a pure univariate time series model is that move-ments in yt are ldquoexplainedrdquo solely in terms of its own past or by its position inrelation to time13 That is time series models look for patterns in the past move-ments of a particular variable and use that information to predict future movementsof the variable In general a time series modelrsquos forecast of y based on knownvalues yT+1 is given by
yT+1 = E( yT+1|y0 yT )
As Pindyck and Rubinfeld (1991) suggest ldquoin a sense a time-series model is just asophisticated method of extrapolation yet it may often provide a very effec-tive tool for forecastingrdquo In this sense time series models are more along the linesof empirical analyses that ldquolet the data speak for themselvesrdquo rather than empiricalanalyses that (strictly speaking) ldquotest economic theoriesrdquo As Harvey (1993) statesldquoan essential feature of time series models is that they do not involve behaviouralrelationshipsrdquo Time series models reflect a ldquostatisticalrdquo approach to forecastingHowever the patterns in the data discovered by time series models do influencetheoretical discussions of the macroeconomy and tests of macroeconomic theo-ries concerning behavior have incorporated time series models14 An example of
102 Empirical macroeconomics
how time series analysis has influenced theoretical macroeconomic discussions isgiven at the end of this section
Some properties of stochastic processes
There are several key properties of stochastic processes for time series that wecan introduce One is ldquostationarityrdquo For a stochastic process to be stationary thefollowing conditions must be satisfied for all t
E( yt) = micro
E[( yt minus micro)2] = σ 2y
E[( yt minus micro)( ytminusk minus micro)] = γk
Note that γ0 = σ 2y In words if a series is stationary the mean of the series is
invariant to time the variance of the series is invariant to time and the covarianceof the series is invariant to time As Pindyck and Rubinfeld (1991) note ldquoif astochastic process is stationary the probability distribution p( yt) is the same forall time t and its shape (or at least some of its properties) can be inferred by lookingat a histogram of the observations y1 yT that make up the observed seriesrdquo15
Also an estimate of the mean micro of the process can be obtained from the samplemean of the series
y = 1
T
Tsumj=0
yt
and an estimate of the variance σ 2y can be obtained from the sample variance16
σ 2y = 1
T
Tsumj=0
( yt minus y)2
As Granger and Newbold (1986 4) phrase it ldquoa stationarity assumption is equiv-alent to saying that the generating mechanism of the process is time-invariantso that neither the form nor the parameter values of the generation procedurechange through timerdquo The simplest example of a stationary stochastic process isa sequence of uncorrelated random variables with constant mean and variance
A second property of a stochastic process is the ldquoautocorrelation functionrdquo Theautocorrelation function provides us with a measure of how much correlation thereis (and by implication how much interdependency there is) between neighboringdata points in the series yt For stationary processes the autocorrelation with lagk is given by17
ρk = γkγ0
= E[( yt minus micro)( ytminusk minus micro)]σ 2y
Empirical macroeconomics 103
For any stochastic process ρ0 = 1 If the stochastic process is simple ldquowhitenoiserdquo (ie yt = εt where εt is an independently distributed random variablewith zero mean and finite variance) then the autocorrelation function for thisprocess is given by
ρ0 = 1 ρk = 0 for k gt 018
A simple example of a stochastic process is the ldquorandom walkrdquo process inwhich each successive change in yt is drawn independently from a probabilitydistribution with zero mean19 Thus yt is determined by
yt = ytminus1 + εt (717)
where εt is a sequence of uncorrelated random variables (E(εtεs) = 0 for t = s)with mean zero (E(εt) = 0) and constant variance (E(ε2
t ) = σ 2e for all t) Recall
that a sequence εt of this kind is typically called ldquowhite noiserdquo If the stochasticprocess is a random walk the one-period-ahead forecast of yt+1 is simply yt
A simple extension of the random walk process is to incorporate a trend in theseries yt We then obtain the following stochastic process known as a random walkwith drift20
yt = ytminus1 + d + εt (718)
The one-period-ahead forecast of yt+1 is now yt + d By repeatedly substitutingfor past values of yt into (718) we obtain
yt =tminus1sumj=0
εtminusj + td + y0 (719)
For the random walk process without drift (d = 0) we see from (719) that thefirst requirement for stationarity namely that the mean be constant over time issatisfied if y0 is fixed21 That is E( yt) = E( y0) Nevertheless the process is notstationary since Var( yt) = tσ 2
e The random walk process tends to meander awayfrom its starting value but exhibits no particular trend in doing so22
Autoregressive processes
A process similar to the random walk that is stationary is called the ldquofirst-orderautoregressive processrdquo or AR(1)23
The AR(1) process is given by
yt = φytminus1 + (1 minus φ)micro + εt (720)
A necessary condition for stationarity is that |φ| lt 1 in which case E( yt) equalsthe constant term micro in (720)24
104 Empirical macroeconomics
One possible example of an autoregressive process is the total numberunemployed each month (Granger and Newbold 1986) Let the total number unem-ployed in one month be yt This number might be thought to consist of a fixedproportion φ of those unemployed in the previous month (the others having foundemployment) plus a new group of workers seeking jobs If the new additionsare considered to form a white noise series with positive mean micro(1 minus φ) thenthe unemployment series is a first-order autoregressive process expressed byequation (720)
For convenience only it is often assumed that the process has a zero mean iemicro = 0 Note that we can always define a new variable yprime
t equiv yt minus micro such that yprimet
has a zero mean and is given by the AR(1) process
yprimet = φyprime
tminus1 + εt (721)
with |φ| lt 1 Setting micro = 0 thus simply means that the variable yt (egemployment real output etc) is measured in terms of deviations from its mean
By successive substitution for yt in (720) we obtain (assuming micro = 0)
yt =tminus1sumj=0
φjεtminusj + φty0 (722)
If the process is regarded as having started at some point in the remote past and|φ| lt 1 then we can write
yt =infinsum
j=0
φjεtminusj (723)
Given that εt is white noise we thus have that E( yt) = micro = 0 Assumingstationarity (|φ| lt 1) we know that the variance and covariances are constantRecall that εt is white noise such that E(εtεs) = 0 for s = t We thus have25
γ0 = σ 2y = E[( yt minus micro)2] = E
⎡⎢⎣⎡⎣ infinsum
j=0
φj(εtminusj)
⎤⎦
2⎤⎥⎦ = σ 2
e (1 minus φ2)
γ1 = E[( yt minus micro)( yt+1 minus micro)] = φσ 2e (1 minus φ2)
γ2 = E[( yt minus micro)( yt+2 minus micro)] = φ2σ 2e (1 minus φ2) etc
The autocorrelation function for AR(1) is thus particularly simple ndash it begins atρ0 = 1 and then declines geometrically ρk = φk Note that this process hasan infinite memory The current value of the process depends on all past valuesalthough the magnitude of this dependence declines with time
In general an autoregressive process of order p is generated by a weighted aver-age of past observations going back p periods together with a random disturbance
Empirical macroeconomics 105
in the current period The AR( p) process is thus given by26
yt = φ1ytminus1 + φ2ytminus2 + φ3ytminus3 + middot middot middot + φpytminusp
+ (1 minus φ1 minus φ2 minus middot middot middot minus φp)micro + εt
= micro + εt +psum
j=1
φj( ytminusj minus micro)
(724)
with a necessary condition for stationarity being that φ1 + φ2 + middot middot middot + φp lt 127
The next section discusses necessary and sufficient conditions for stationarity
A brief digression on necessary and sufficient conditionsfor stationarity
To get some idea of what are necessary and sufficient conditions for stationaritylet us consider two specific cases AR(1) and AR(2) In the AR(1) case
yprimet minus φyprime
tminus1 = εt
where yprimet equiv yt minus micro Note that εt can be termed the ldquostochasticrdquo component of the
expression and the remainder the ldquodeterministicrdquo component28
Focusing on the deterministic component and assuming for convenience thatmicro = 0 we have a homogeneous first-order difference equation of the form29
yt minus φytminus1 = 0 (725)
To solve this equation let us try the general solution
yt = Abt (726)
which naturally implies ytminus1 = Abtminus130 The problem then is to find the valuesof A and b Substituting the trial solution into the above difference equation weobtain
Abt minus φAbtminus1 = 0
which by multiplying through by b1minustA can be rewritten as
b minus φ = 0 (727)
Equation (727) is called the ldquoauxiliaryrdquo or characteristic equation of (725) Thisequation provides us with the solution for b which is simply that b = φ Thissolution value is sometimes referred to as the ldquorootrdquo of the characteristic equation
Using (727) to substitute for b in (726) we thus have as the solution to thefirst-order difference equation (725) the expression
yt = Aφt (728)
106 Empirical macroeconomics
The ldquoequilibriumrdquo solution for yt given a homogeneous difference equation suchas (725) is yt = 0 That is if yt equiv 0 then yt will not change over time Thisequilibrium solution is ldquostablerdquo if as t rarr infin yt rarr 0 That is deviations fromthe equilibrium yt will return toward that equilibrium value Obviously given oursolution (728) a necessary and sufficient condition for stability is that |φ| lt 1
In the context of an AR(1) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the first-order homogeneous difference equation (725)that is the deterministic component of yt for an AR(1) process The condition ofstationarity is thus that |φ| lt 1 The effect of a given shock will mitigate overtime if this stationarity condition is met A special nonstationary case is the ldquounitrootrdquo case of the random walk process in which φ = 1 In that case the effect ofa given shock will not dampen with the passage of time
Now let us consider the AR(2) case where
yprimet minus φ1yprime
tminus1 minus φ2yprimetminus2 = εt
in which yprimet equiv yt minus micro As before εt can be termed the ldquostochasticrdquo component of
the expression Assuming again for convenience that micro = 0 we can express thedeterministic component as a homogeneous second-order difference equation ofthe form
yt minus φ1ytminus1 minus φ2ytminus2 = 0 (729)
To solve this equation let us try yt = Abt which naturally implies ytminus1 = Abtminus1
and ytminus2 = Abtminus2 The problem then is to find the values of A and b Substitutingthe trial solution into the above difference equation we obtain
Abt minus φ1Abtminus1 minus φ2Abtminus2 = 0
which by multiplying through by b2minustA can be rewritten as
b2 minus φ1b minus φ2 = 0 (730)
The quadratic equation (730) is called the ldquoauxiliaryrdquo or ldquocharacteristic equationrdquoof the second-order difference equation (729) The roots of this equation will bethe solutions of (730)31 The two ldquocharacteristicrdquo roots m1 and m2 may be foundin the usual way from the formula32
m1 m2 = [φ1 plusmn (φ21 + 4φ2)
12]2 (731)
each of which is acceptable in the solution Abt Note that for the quadraticequation (730) the roots m1 and m2 satisfy
(b minus m1)(b minus m2) = 0
where φ1 = m1 + m2 and φ2 = minusm1m2
Empirical macroeconomics 107
Expression (731) provides us with two values for b and thus two potentialsolutions for yt
yt = A1mt1 and yt = A2mt
2
The general solution to a second-order difference equation combines these twoby taking the sum Thus the general solution of the second-order homogeneousdifference equation (729) is33
yt = A1mt1 + A2mt
2 (732)
Inspection of (732) suggests that stability for the linear second-order homo-geneous difference equation (729) requires that both roots of the characteristicequation have an absolute value less than one This reflects the fact that the gen-eral solution of the second-order homogeneous difference equation includes bothroots m1 and m2 The root with the higher absolute value is sometimes termed theldquodominant rootrdquo If both m1 and m2 are less than unity in absolute value yt willbe close to zero if t is large Equivalently if the dominant root is less than one inabsolute value convergence will occur
In the context of an AR(2) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the second-order homogeneous difference equation (729)that is the deterministic component of the process The condition of stationarityis thus that |m1| lt 1 and |m2 lt 1| That is for the AR(2) process stationarityrequires that the roots of the characteristic equation (730) are less than one inabsolute value ndash that is that they all lie inside the unit circle34
In terms of φ1 and φ2 the conditions for stationarity may thus be defined asfollows35
φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 φ2 gt minus1
As you can see the prior necessary condition that φ1 + φ2 lt 1 is augmented withtwo other conditions
We have seen that stability for the second-order homogeneous differenceequation requires that the maximum of (|m1| |m2|) ndash that is the dominant root ndash beless than 1 To see how one obtains the three necessary conditions listed above forstabilitywe assume this is the case and determine the resulting restrictions placedon the coefficients φ1 and φ236 Recall that
m1 m2 = [φ1 plusmn (φ21 plusmn 4φ2)
12]2
Suppose φ21 + 4φ2 gt 0 so that the roots are real A maximum of (|m1| |m2|) less
than one means that
2 minus φ1 gt (φ21 + 4φ2)
12 gt minus2 minus φ1
108 Empirical macroeconomics
and
2 minus φ1 gt minus(φ21 + 4φ2)
12 gt minus2 minus φ1
The sum of the roots is φ1 and since we are assuming each root is between minus1and 1 it must be the case that 2 gt φ1 gt minus2 or 0 gt minus2minusφ1 and 2minusφ1 gt 0 Thusthe second and third inequalities are always true and hence place no restrictionson φ1 and φ2
The first and fourth inequalities squared read
(2 minus φ1)2 gt φ2
1 + 4φ2 and φ21 + 4φ2 lt (minus2 minus φ1)
2
respectively These two expressions can be rewritten to obtain
φ1 + φ2 lt 1 and minus φ1 + φ2 lt 1
which are the first two conditions for stability The third condition for stability(φ2 gt minus1) is obtained from the case of complex conjugate roots37 The threenecessary conditions for the dominant root being less than one in absolute value arealso sufficient conditions This can be verified by showing using these inequalitiesthat it is possible to reverse the steps in the above calculations and arrive at themaximum of (|m1| |m2|) being less than one
In general a necessary and sufficient condition for stationarity of an AR( p)process is if the roots of the characteristic equation
b p minus φ1b pminus1 minus middot middot middot minus φp = 0 (733)
are less than one in absolute valueNote that for the AR(2) case there are three possible situations If φ2
1 +4φ2 gt 0the square root in (731) is a real number and m1 and m2 will be real and distinctIn this case the solution to (729) is given by
yt = A1mt1 + A2mt
2
where A1 and A2 are constants which depend on the starting values y0 and yminus1The second situation is that of repeated roots where φ2
1 + 4φ2 = 0 such thatm1 = m2 = m In this case the solution to (729) takes the form
yt = A3mt + A4tmt
If |m| lt 1 the damping force of mt will dominate both termsThe third situation for the AR(2) process is when φ2
1 + 4φ2 lt 0 in which casethe roots are a pair of complex conjugates (ie m1 m2 = h plusmn vi where h = φ12v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12) Thus the solutionto (729) is given by
yt = A5(h + vi)t + A6(h minus vi)t
Empirical macroeconomics 109
Appealing to De Moivrersquos theorem this expression can be transformed intotrigonometric terms The general form of the solution is
yt = pt(A7 cos λt + A8 sin λt)
where p is the modulus of the roots = (minusφ2)12) and λ satisfies the two conditions
cos λ = hp and sin λ = vp As in the other two cases if the absolute valueof the conjugate complex roots |h plusmn vi| lt 1 is less than one the process isstable The time path followed by yt in response to a shock is cyclical but theperiodic fluctuation will mitigate as time passes (ldquosinusoidal decayrdquo) if the processis stationary
Moving average processes
In a moving average process the process yt is described completely by a weightedsum of current and lagged random variables The simplest moving average processthe moving average process of order 1 or MA(1) takes the form
yt = micro + εt + θεtminus1 (734)
The term ldquomoving averagerdquo reflects the fact that a series so characterized will besmoother than the original white noise series εt In general such a moving averageprocess of order q MA(q) is written as38
yt = micro + εt + θ1εtminus1 + θ2εtminus2 + middot middot middot + θqεtminusq
= micro + εt +qsum
j=1
θjεtminusj (735)
As before if yt has nonzero mean we can focus with no loss of generality on thetransformed series yprime
t = yt minus micro which has zero meanOne possible example of a moving average process is economy-wide output
(Granger and Newbold 1986) Output yt could be in equilibrium (at mean micro)but is potentially moved from its equilibrium position each period by a seriesof unpredictable events such as periods of exceptional weather or strikes If thesystem is such that the effects of such events are not immediately assimilated butexert an influence on output for q periods then a moving average model can arise
Note that by repeatedly substituting for lagged valued of εt into an MA(1)process (equation (734)) one obtains39
yt =infinsum
j=1
(minusθ)jytminusj + εt (736)
If yt is not to depend on a shock to the system arising at some point in the remotepast θ must be less than one in absolute value Comparing (736) to (724) we
110 Empirical macroeconomics
see that assuming what was previously denoted the stationarity condition but whatin this context is called the ldquoinvertibility conditionrdquo namely that |θ | lt 1 then anMA(1) process can be represented by an AR(infin) process40 The weights on pastvalues of yt for this AR(infin) process decline exponentially
In general if similar invertibility conditions are met then any finite-ordermoving average process has an equivalent autoregressive process of infinite orderLikewise we have already shown (see (723)) that if the AR(1) process is station-ary it is equivalent to a moving average process of infinite order MA(infin) In factfor any stationary autoregressive process of any order there exists an equivalentmoving average process of infinite order so that the autoregressive process isldquoinvertiblerdquo into a moving average process
Mixed autoregressive moving average processes
An obvious generalization of the above discussion is to combine an AR( p) process(ie (724)) and an MA(q) process (ie (735)) An autoregressive moving averageprocess of order ( p q) or ARMA( p q) can be written as41
yt = micro + εt +psum
j=1
φj( ytminusj minus micro) +qsum
j=1
θjεtminusj (737)
where as before εt is a zero-mean white noise For simplicity consider the casewhere micro = 0
Whether or not a mixed process is stationary depends solely on its autoregres-sive part If the ARMA process is stationary then there is an equivalent MA(qprime)process Similarly provided invertibility conditions hold there is an AR( pprime) pro-cess equivalent to the above ARMA( p q) process It thus follows that a stationaryARMA process can always be well approximated by a high-order MA process andthat if the process obeys the invertibility condition it can also be well approxi-mated by a high-order AR process However an ARMA process has the advantageof ldquoparsimonyrdquo in that the mixed model ARMA( p q) often can achieve as gooda fit as say an AR( pprime) but uses fewer parameters (ie p + q lt pprime)
As an example of an ARMA( p q) process we need only consider a case wherethe variable yt is the sum of a ldquotrue seriesrdquo that is AR( p) plus a white noiseobservation error Thus an ARMA( p q) series results42
Integrated processes
In series arising in economics the assumption of stationarity is often veryrestrictive That is often the characteristics of the underlying stochastic processgenerating a time series appear to change over time With economic data some-times a transformation (such as taking the logarithm of the variable) can result in astationary series In other cases we can obtain a stationary series by differencingone or more times We say that yt is a ldquohomogeneous nonstationary process of
Empirical macroeconomics 111
order drdquo if
wt = dyt
is a stationary series Here denotes differencing that is
yt = yt minus ytminus1 2yt = yt minus ytminus1 = yt minus 2ytminus1 + ytminus2
The process yt is an ldquointegratedrdquo process if after a series yt has been differencedto produce a stationary series wt the series wt can be modelled as an ARMAprocess If wt = dyt and wt is an ARMA( p q) process then we say that yt is anldquointegrated autoregressive moving average process of order ( p d q)rdquo or simplyARIMA ( p d q)
An application of time series models
To gain a better understanding of how time series models are used in macroeco-nomic empirical work let us consider the time series for the logarithm of the levelof real output to be denoted by yt As we stated above theoretical macroeconomicmodels typically postulate shocks or ldquoimpulsesrdquo to the economy (eg shocks totechnology money supply and government policy) that in conjunction with aprediction of how various markets in the economy adjust to these shocks offersan explanation of the actual fluctuations in aggregate variables One importantquestion that can arise is whether such shocks are ldquotransitoryrdquo in that their effectsdo not persist or ldquopermanentrdquo where the economy moves to a new level of realoutput
The ldquotraditionalrdquo answer to the above question of transitory versus permanenteffects of shocks has been according to Campbell and Mankiw (1987a 111) thatfluctuations in real output ldquoprimarily reflect temporary deviations of productionfrom trendrdquo That is the traditional view has been that quarterly real output couldbe represented by
yt = Tt + St + Xt (738)
where Tt is a deterministic component representing trend St is a determinis-tic seasonal component and Xt is a stationary autoregressive process with nodeterministic component43
Trend and seasonal components were typically estimated in various fashionsand then eliminated The focus of empirical work had then been on examiningvariations in the detrended seasonally adjusted real output series For exampleBlanchard (1981) estimated the following second-order autoregressive process fordeviations of the log of seasonally adjusted real output from its estimated trendylowast
t and obtained the equation
ylowastt = 134ylowast
tminus1 minus 042ylowasttminus2 + εt (739)
112 Empirical macroeconomics
Note that the necessary conditions for stationarity with respect to the series ylowastt
(namely φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 and φ2 gt minus1) are metThe traditional view is embodied in (739) A shock to output increases for
a few quarters but the effect ultimately dies out In fact only 8 percent of ashock remains after 20 quarters Assuming an ARMA(22) process Blanchardand Fischer (1989 9) obtain a similar result
ylowastt = 131ylowast
tminus1 minus 042ylowasttminus2 + εt minus 006εtminus1 + 025εtminus2 (740)
where by construction εt is that part of the deviation of current output from trendthat cannot be predicted from past output As Blanchard and Fischer note
A shock has an effect on GNP that increases initially and then decreases overtime After 10 quarters the effect is still 40 of the initial impact after 20quarters all but 3 of the effect has disappeared The view that reversiblecyclical fluctuations account for most of the short-term movements of realGNP and unemployment has been dominant for most of the last century
However as Granger and Newbold (1986 37) note ldquothe more modern view is thatas far as possible the trend seasonal and lsquoirregularrsquo components should be han-dled simultaneously in a single model aimed at depicting as faithfully as possiblethe behavior of a given time seriesrdquo Seasonality can be treated through a gener-alization of ARMA models44 More importantly for the question at hand trend isgenerally treated by differencing leading to the consideration of the ldquointegratedprocessesrdquo suggested above In fact Campbell and Mankiw (1987b) indicate thatthe answer to the question of the importance of temporary versus permanent shocksto output is biased if detrended data are used45 Campbell and Mankiw go on topoint out that examining the differences in the logarithm of real output does notprejudge the issue of whether shocks to the economy are transitory or permanentFor instance suppose that yt follows an IMA(11) process so that
yt minus ytminus1 = d + εt minus θεtminus1
Then a unit impulse in yt changes the forecast of yt+n by 1 minus θ regardless of nldquoHence depending on the value of θ news about current GNP could have a largeor small effect on onersquos forecast of GNP in ten yearsrdquo (Campbell and Mankiw1987b 860)46
Campbell and Mankiw consider ARMA( p q) processes for the difference in thelog of real output for p = 0 1 2 3 and for q = 0 1 2 347 Interestingly they finda high level of persistence in shocks One intriguing suggestion of Campbell andMankiw (1987b 868) is that ldquowhen we examine postwar annual data we cannotreject the hypothesis that the log of real GNP is a random walk with drift In thiscase the impulse response is unity at all horizonsrdquo Recall that the random walkprocess with drift (718) is an example of a nonstationary process that is first-order
Empirical macroeconomics 113
homogeneous nonstationary To see this simply consider the series wt that resultsfrom differencing the random walk ndash that is the series
wt = yt minus ytminus1 = d + εt (741)
Since εt are assumed independent over time wt is clearly a stationary process Theterm d + εt is a white noise process In this example yt is an ARIMA(010) andyt is an ARMA(00) The impulse response to a shock is unity in such a case
As we will see later one way to interpret the above finding of persistence isto place weight on ldquoaggregate supplyrdquo shocks such as technological disturbancesrather than on ldquoaggregate demandrdquo shocks such as changes in the money supplyin explaining fluctuations in real output That is the finding gives credence tothe view of Nelson and Plosser (1982) among others that real (ie permanent)shocks dominate as a source of output fluctuations48
However as pointed out by Campbell and Mankiw (1987b 877) the Nelsonand Plosser conclusion is an extreme
one can attribute a major role to supply shocks without completely abandoninga role for demand shocks For example suppose that output Y (= log y) isthe sum of two components a supply-driven ldquotrendrdquo Y T and demand-driveldquocyclerdquo Y c that are uncorrelated at all leads and lags Suppose further thatY T is a first-order autoregressive process with parameter φ and that Y c issome stationary process If trend output is approximately a random walk sothat φ is small then the finding of great persistence implies that fluctuationsin the cycle are small relative to fluctuations in the trend If the change inthe trend is highly serially correlated (φ is large) however the finding ofpersistence is consistent with a substantial cyclical component
Campbell and Mankiw (1987b 877) go on to suggest that
a second way to interpret the finding of persistence is to abandon the
natural rate hypothesis Models of multiple equilibria might explain a long-lasting effect of aggregate demand shocks if shocks to aggregate demand canmove the economy between equilibria Shocks to aggregate demand couldhave permanent effects if technological innovation is affected by the businesscycle
Note that so far we have considered only a single or univariate time series Theconcepts involved however can be extended to multivariate series For instancea simple vector autoregression model (VAR) could take the form
yt = ytminus1 + εt
where now yt is considered an N times 1 vector reflecting N variables the randomdisturbance term εt is also an N times 1 vector and is an N times N matrix of parame-ters The disturbances are uncorrelated over time but may be contemporaneously
114 Empirical macroeconomics
correlated As in the univariate case such a model is usually only fitted to vari-ables which are stationary (possibly obtained by logarithmic transformations orby taking first or second differences) As in the univariate case the objectivewould be to find a model that transforms a vector of time series into a whitenoise vector As Harvey (1993) notes ldquoalthough a model of (this) form is oftenused for forecasting in econometrics economic theory will typically place a priorirestrictions on elements of rdquo There are also questions concerning the correlationbetween innovations across different series of data for example the link betweeninnovations in money supply and output But given that our aim is to review basicmacroeconomic theory we will not pursue such topics
Conclusion
This chapter has emphasized the empirical nature of macroeconomics Inparticular time series analysis and econometric methods have been shown tobe useful tools for analyzing macroeconomic data Many of the policy conclu-sions that will be developed in the remainder of the book can be tested using thesemethods The results obtained from a thorough understanding of the time seriesproperties of important macroeconomic variables such as interest rates employ-ment numbers real output measures and the like together with the underlyingtheory may be used to provide policy recommendations Moreover more andmore businesses are relying on empirical macroeconomics when making operatingdecisions
Appendix translation of higher-order difference equationsinto lower order
Any higher-order difference equation can be interpreted in terms of an equivalentsystem of first-order difference equations To see this consider equation (729)
yt minus φ1ytminus1 minus φ2ytminus2 = 0
To facilitate matters we start by shifting the origin of this second-order homo-geneous difference equation from t = minus2 to t = minus1 such that we may rewrite(729) as
yt+1 minus φ1yt minus φ2ytminus1 = 0 (7A1)
Now let us introduce the new variable xt defined as
xt = ytminus1
which means that xt+1 = yt We may then express the second-order differenceequation (7A1) by means of two first-order simultaneous equations
yt+1 minus φ1yt minus φ2xt = 0
xt+1 minus yt = 0(7A2)
Empirical macroeconomics 115
In matrix notation (7A2) becomes[yt+1xt+1
]= A
[ytxt
]
where A is the square nonsingular matrix defined as
A =[φ1 φ21 0
]
The ldquocharacteristic equation of a square matrixrdquo is defined by the determinant
|A minus λI | = 0
where I is the unit matrix[
1 00 1
]and λ is a scalar variable
Letting b = λ the ldquocharacteristic equation of matrix Ardquo
|A minus λI | =∣∣∣∣φ1 minus b φ2
1 minusb
∣∣∣∣ = b2 minus φ1b minus φ2 = 0
is identical to equation (730) which was termed the ldquocharacteristic equation of thesecond-order homogeneous difference equationrdquo The values for the λs (or bs) arethe roots of the characteristic equation One reason for restating the higher-orderdifference equation in matrix form is that necessary and sufficient conditions forstability can be expressed in terms of conditions on the matrix A
8 The neoclassical model
Introduction
This chapter turns our attention to one of the most popular and sometimescontroversial models used by macroeconomists the neoclassical model Themodel in its purest form is often used as a benchmark or starting point for addingldquorealismrdquo to the structure of the economy However one views the neoclassicalmodel there is no denying that it is a powerful tool for generating predictionsabout movements in economic aggregates As such this chapter works throughthe comparative static exercise of a change in the money supply This exerciseprovides a great deal of insight into how the monetary authority might influencethe economy or whether it can influence the economy at all A number of importantissues are raised for example money neutrality and money illusion Additionallythe chapter formally derives the aggregate supply curve in the neoclassical contextThe model has serious implications for the existence of a natural rate and also forthe formation of expectations
Comparative statics for the neoclassical model
We can use our previous method of analysis known as ldquocomparative staticrdquoanalysis to examine the effects of various shocks on the equilibrium level ofprices As the name suggests comparative static analysis is concerned with thecomparison of different equilibrium states associated with different sets of valuesof parameters and exogenous variables In the current context of the neoclassicalmodel the labor market can be isolated from the other markets (ie there is aldquoblock recursiverdquo character to the equilibrium solution) In such a context we candistinguish two types of exogenous variables
One type ldquosupply-siderdquo variables alter the level of output at any given pricelevel Such variables could include changes in technology and the existing capitalstock in the current version of the model or we could expand the analysis toinclude changes in the supply of other inputs (such as oil) changes in governmentpolicies that affect incentives to supply labor and changes in government policiesthat affect the incentives to invest and thus affect the productive capacity of theeconomy over time
Neoclassical model 117
The second type of exogenous variables ldquodemand-siderdquo variables do not alterthe current supply of output at prevailing prices but instead impact equilibriumprices and interest rates as determined in the output financial and money marketsBelow we consider the impact of demand-side ldquoshocksrdquo such as changes in initialmoney balances and the expected inflation rate
The macroeconomic approach to general equilibriumhow it can obscure
The neoclassical model can be viewed as a special case of a Walrasian generalequilibrium system distinguished by the existence of money1 As Clower (1965)notes
income magnitudes do not appear as independent variables in demand andsupply functions of the (Walrasian) general equilibrium model for incomesare defined in terms of quantities as well as prices and quantity variablesnever appear explicitly in the market excess demand functions of traditionaltheory To be sure income variables could be introduced by taking factorsupplies as given parameters but this would preclude the formulation of ageneral equilibrium model containing supply functions of all marketable factorservices
Referring to Chapter 9 of Patinkin (1965) Clower goes on to note that this point
was apparently overlooked by Patinkin when he formulated his ldquogeneral the-oryrdquo of macroeconomics It is instructive to notice that this chapter is notsupplemented by a mathematical appendix I do not mean to suggest thatauthors may not put such variables as they please into their models My pointis that such variables that can be shown to be fundamentally dependent onothers should not then be manipulated independently
What Clower is referring to is the fact that the neoclassical model with limitedperfect foresight effectively determines at time t the money wage for the labormarket and the (futures) price of output and interest rate Given perfect foresightindividual optimizing behavior will generate planned consumption and moneydemand functions at time t for period t that include the real wage rather thanincome That is since labor supply is a choice variable at time t labor income (theproduct of the real wage times labor supply) should not appear as an argument inhouseholdsrsquo demand and supply functions (Note that labor income along withdividend and interest payments equals output minus depreciation)
Formally we can discover how a ldquosmallrdquo change in one (or more) of the exoge-nous variables affects equilibrium values by totally differentiating the equilibriumconditions with respect to the prices to be determined and the exogenous variableand then solving for the implied change in equilibrium prices required to maintainequilibrium2 The resulting changes in equilibrium prices then imply changes in
118 Neoclassical model
the equilibrium levels of employment output consumption investment and realmoney holdings
However we choose instead the standard macroeconomic approach of arbitrar-ily separating the analysis into an analysis of the labor market and the resultingemployment and output (the ldquoaggregate supply equationrdquo) and then an analysisof the other markets and the detemination of the price level and the interest rate(the ldquoISrdquo and ldquoLMrdquo equations) While Clower is right in that this obscures thetraditional ldquogeneral equilibriumrdquo nature of the neoclassical model the approach isuseful in that it allows us to more easily extend the analysis to situations in whichprices are not market-clearing or to situations in which there is not limited perfectforesight
Given the assumption of perfect foresight on the part of both firms and workersconcerning the price level pt we have seen that the aggregate supply equation isindependent of the price level or the interest rate We know from the modifiedWalrasrsquo law that any two of the three excess demand conditions with respect tomoney financial assets and output can be used in the comparative static analysisUsing the equilibrium conditions with respect to output and money (the ldquoISrdquo andldquoLMrdquo equations) we thus have the three equations
yt = yt(K ) (81)
cdt (rt πe
t At Mpt yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (82)
Ldt (rt πe
t At Mpt yt) minus Mpt = 0 (83)
to determine equilibrium output yt price level pt and interest rate rt Recallthat from the modified Walrasrsquo law we know that there is a fourth equation theequilibrium condition for the financial market that is implied by (82) and (83)This fourth equilibrium condition is
net Adt (rt πe
t At Mpt yt) minus net Ast (m
et + δ K) = 0 (84)
A change in the money supply the comparative statics
It is clear from the above that the neoclassical aggregate supply equation (81)determines output while the IS and LM equations (82) and (83) determine theprice level and interest rate given the equilibrium level of output Focusing on thelatter two equations and the determination of the price level and interest rate fora given level of output total differentiation with respect to the equilibrium prices( pt and rt) and the money supply change gives the following system of linearequations in matrix form3
[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpartrpartLdpartr
] [dpdr
]=[ minus(partcdpart(Mp))dMp(1 minus partLdpart(Mp))dMp
]
Neoclassical model 119
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm4 Solving the above linear-equationsystem for dp and dr using Cramerrsquos rule we obtain
dp
dM= minus(partcdpart(Mp))(1p)partLdpartr minus (1 minus partLdpart(Mp))(1p)partydpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpartr
= p
M
dr
dM=
minus[(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)minus(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)](1p)
minus(partcdpart(Mp))(Mp2)partLdpartrminus(1minuspartLdpart(Mp))(Mp2)partydpartr
= 0
As you can see dpp = dMM and drdM = 0 That is a change in the moneysupply leads to the same proportional change in the price of the consumption goodand no change in the interest rate With regard to the latter result the interest ratedoes not change as the only thing that changes the interest rate is the rate of changein prices Since the price level adjusts there is no change in the interest rate and theLM curve does not change Note that from the labor market equilibrium conditionthat underlies the aggregate supply equation we know that the change in the pricelevel results in an equiproportionate change in the money wage The result is thatwe have ldquoneutrality of moneyrdquo In other words if individuals correctly anticipatethe effect of a change in the money supply on the price level as is the case inthe deterministic model under the assumption of limited perfect foresight thenmonetary changes have no ldquorealrdquo effects Real output the real wage the expectedreal rate of interest real consumption real investment and the real money supplyare all unaffected by the monetary change
The basic reason why changes in the money supply are ldquoneutralrdquo is the absenceof money illusion on the part of both firms and households In particular house-hold demand and supply functions indicate that if a change in money balancesis accompanied by an equiproportionate change in the price of output then thereis no change in any demands That is household demand and supply functionsare homogeneous of degree zero in money balances and prices This absence ofldquomoney illusionrdquo occurs assuming
bull perfect foresight at time t for price during period t so that a change in priceresults in no change in the real wage or employment (when coupled with thesimilar assumption of limited perfect foresight on the part of firms)
bull ldquoneutral distribution effectsrdquo such that the shift in wealth from prior creditorsto debtors that would accompany a rise in the price level leaves aggregatedemands unchanged
bull unit elastic expectations so that changes in the current price level can beviewed as leaving unaffected the expected rates of change in the pricelevel
120 Neoclassical model
As we will see later this money neutrality result of the neoclassical model formsthe basis of what has become known as the ldquopolicy ineffectiveness propositionrdquo(see McCallum 1979)
The ldquodynamicsrdquo of the system and the neutrality of money a review
As we discussed earlier the ldquostoryrdquo often told with respect to the dynamics of theabove situation is a ldquoloanable funds theoryrdquo of interest rate determination in whichthe interest rate moves to clear the financial market5 In this case the ldquotatonnementprocessrdquo or movement toward equilibrium involves
dpt = f ( ydt minus yt) f (0) = 0 df d( yd
t minus yt) gt 0
dpbt = fb(net Adt minus As
t ) fb(0) = 0 dfbd(net Adt minus net As
t ) gt 0
Note that we put ldquostoryrdquo in quotes since the analysis itself simply identifies variousequilibrium points with adjustments in prices to reach a different equilibrium pointgiven a shock essentially occurring without any passage of time Neverthelessconsider the following story with respect to a rise in the money supply
In the Patinkin analysis described earlier if the money supply were to doublefrom M to 2M the LL BB and CC curves would shift to the right so that the newequilibrium would occur at the original interest rate but at a price level double theoriginal one Such a once-and-for-all change in the initial money balances leavesthe money interest rate and real demands unchanged (the neutrality of money)
Superneutrality an informal review
Superneutrality of money occurs when changes in the rate of growth of the moneysupply leave the paths of capital and real output unaffected Although the aboveanalysis is static in nature it does provide some insight into the issue of thesuperneutrality of money To see how suppose each period the economy can bereplicated in every way That is the money supply equilibrium price level andinterest rate are identical each period If expectations of price changes are correctexpected inflation would equal zero For the economy to replicate itself it mustalso be the case that the initial capital stock is optimal such that the capital stockand thus output does not change over time With zero adjustment costs such asituation would imply a marginal product of capital equal to the expected real usercost of capital (me
t + δ where met equiv (rt minus πe
t )(1 + πet )) net investment demand
equal to zero each period and gross investment demand equal to δK Now let us compare this situation to an alternative sequence of temporary equi-
librium in which the economy is identical in every way except one the moneysupply is increased each period by a constant percentage from its level in the priorperiod Other things being equal the above analysis would suggest that one dif-ference across periods would be a rise in prices due to the positive growth in themoney supply Let us further assume that expectations of inflation adjust to reflect
Neoclassical model 121
what Sargent and Wallace (1975) would characterize as a new ldquosystematic moneysupply rulerdquo
Without fully developing the appropriate dynamic analysis it is easy to see thatone obvious adjustment of the static analysis given a growing money supply (asopposed to one that does not change across periods) is thus a higher expectedinflation rate (positive as opposed to zero) In fact the analysis of the static modelcan mimic to some extent the effects of a higher rate of growth in the money supplyby considering the impact of an increase in exogenous inflationary expectations
Following Sargent and Wallace (1975) among others assume that consump-tion demand depends on the expected real rate of interest r minus π not separatelyon its components (the money interest rate and the expected rate of inflation)6
Further assume that changes in expected inflation do not affect real moneydemand7 The comparative static results are then[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpart(r minus π)
partLdpartr
] [dpdr
]=[minus(partydpart(r minus π))dπ
0dπ
]
where partydpart(r minusπ) = partcdpart(r minusπ)+ (1 +ψ prime)partI dpart(r minusπ) Solving the abovelinear equation system for dp and dr using Cramerrsquos rule we obtain
dp
dπ=
minus(partydpart(r minus π))partLdpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
dr
dπ=
minus(partLdpart(Mp))(Mp2)partydpart(r minus π)
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
As we have discussed before if there is no real balance effect with respect to con-sumption demand (partcdpart(Mp) = 0) or if real money demand does not respondto changes in the interest rate (partLdpartr = 0) then we can see from the above that
dr = minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)dπ
minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)= dπ
so that the change in the expected rate of inflation results in no change in theexpected real rate of interest (r minus π ) In this case money is superneutral in thata change in the growth of the money supply (although it alters inflation and thusgiven perfect foresight expected inflation) leaves the expected real rate of interestunchanged and thus does not affect investment and the future size of the capitalstock which depend inversely on the expected real interest rate
122 Neoclassical model
Recall that Sargent and Wallace (1975) have argued for superneutrality ofmoney In contrast Begg (1980) has noted that steady-state analyses of growthmodels with money like the analysis above do find that different rates of growthin the money supply have real effects (through changes in the expected real grossinterest rate)8 As the above analysis makes clear there are two conditions eitherof which is sufficient that will result in money being superneutral Begg (1980293) describes them as follows ldquoThe first condition is that the level of real moneybalances is not an argument in the consumption function and it is this conditionwhich distinguishes the rational expectations model of Sargent and Wallace fromthe analysis of growth models with money The second condition is that the demandfor money is independent of the nominal interest raterdquo (as well as π ) Otherwisea higher expected inflation will result in a fall in the expected real rate of interestand a higher price level
In the Patinkin framework with consumption demand and investment demanddepending on the expected real interest rate at a given price level and higherexpected inflation the CC curve must shift up vertically by the amount of theincrease in expected inflation so as to maintain the same expected real rate ofinterest But the LL curve does not shift with a change in expected inflationThus given a downward-sloping CC curve and an upward-sloping LL curve thenew equilibrium money interest rate does not rise by the extent of the increase inexpected inflation Superneutrality of money seems not to hold
In such a case the static analysis thus predicts a lower real stock of money eachperiod higher investment and a greater capital stock next period This suggests anew steady state in which the capital stock is greater In fact a complete dynamicanalysis confirms these predictions a higher steady-state capital stock is associatedwith an increase in the rate of growth of the money supply and consequent increasedexpected inflation
The role of the key assumptions of the neoclassical model
At this point it might be useful to review the role played by the two key assumptionsof the neoclassical model namely price flexibility and complete information onprices In most cases and this is no exception one can gain an understanding ofthe role of a particular assumption by exploring how the analysis would proceedif the assumption were not made Consider first the implications of dropping theneoclassical modelrsquos assumption that prices are perfectly flexible
Suppose that the demand for output decreases and firms cannot sell all theydesire at prevailing prices With flexible prices output prices will fall and thefalling price level restores output demand to its previous level However if outputprices are inflexible and do not adjust downward in response to a reduced demandfor output firms will respond by reducing production Reduced production willlead to a lower level of labor demand and thus a fall in employment Further labordemand will no longer depend upon the real wage but will be determined by whatcan be sold in the output market In short if output prices are inflexible then theoverall level of demand for goods and services is paramount in determining thelevel of employment
Neoclassical model 123
A second modification of the neoclassical model is to assume that output pricesare flexible but money wages are not If wages are ldquostickyrdquo relative to outputprices then changes in the price level will alter the real wage For instance inthe late 1970s high inflation rates led workers and employers in certain industriesto bargain for long-term contracts with a high rate of growth of the money wageThe high expected inflation did not materialize in the early 1980s The lower rateof inflation with no change in the rate of increase in wages meant a rise in thereal wage The resulting fall in the demand for labor led to lower employmentand output Given that wages are not perfectly flexible (eg ldquomultiperiod laborcontracts without complete indexationrdquo) output prices below that anticipated whenlabor contracts are signed lead to a fall in output and employment With inflexiblemoney wages the aggregate supply equation includes the price level of output asa determinant of output supply
Let us consider one more modification of the neoclassical model Suppose thatworkers have incomplete information on output prices and the real wage A firmdetermines its relevant real wage by dividing the money wage it pays its workersby the price it anticipates for the particular product its workers produce On theother hand workers must anticipate prices for a variety of different goods to bepurchased in order to determine their relevant real wage Thus firms may moreaccurately anticipate changes in prices and thus real wages than workers
Now suppose that there is an increase in output prices Given our currentassumption of incomplete information this will not only lead to an increase infirmsrsquo demand for labor and to higher wages as firms anticipate the fall in realwages but may also lead to an increase in the quantity of labor supplied for thefollowing reason Workers who have not anticipated the rise in output priceswill perceive the higher money wages as implying a rise in the real wage andwill increase their supply of labor accordingly As a result equilibrium employ-ment will rise with an increase in output prices Once again the aggregate supplyequation will incorporate the current price level as a potential determinant
The illusion model one modification of the neoclassical model
Our first departure from the neoclassical model is to introduce the potential forldquoimperfectrdquo foresight on the part of suppliers at time t concerning the price levelfor period t As a consequence the ldquonotionalrdquo or planned demands made at time tbased on anticipated prices for the period can differ from ldquoeffectiverdquo or realizeddemands andor supplies at the actual prevailing prices Further realized or effec-tive demands now depend on the quantity constraints experienced in other marketsas well as prices That is realized output now becomes a determinant of actualconsumption and money demand on the part of households
Real wage illusion the labor market and aggregate supply
As we have seen equilibrium employment is determined at the start of each periodin the labor market To formally show this let us start with the following statement
124 Neoclassical model
of equilibrium in the labor market in terms of a money wage wt and level ofemployment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpe
t ) minus Nt = 0
A critical aspect of the above is the fact that suppliers ndash in particular suppliers oflabor ndash may not correctly anticipate the price level that will exist with respect tooutput In particular we let pe
t denote suppliersrsquo expectation formed at the start ofperiod t of the price level for period t We assume this expectation is held withsubjective certainty so that we may express the real wage expected by supplierssimply by wtpe
t It is this anticipated real wage not the realized real wage wtpt that appears in the labor supply function
Firms on the other hand are presumed to correctly forecast the actual realwage9 Note that we retain the presumption that the money wage adjusts to clearthe labor market Similarly we implicitly have assumed that the price level adjuststo clear the output market such that firms are price-takers in the output marketand thus labor demand depends on the real wage rather than on a sales constraint
Initially let us assume that householdsrsquo anticipated level of prices is correctThat is we start with a money wage and level of employment consistent with theneoclassical model But we now assume that any change in the price level willnot be fully anticipated In particular we assume that
pet = g( pt) 1 gt gprime ge 0
The fact that gprime lt 1 implies imperfect foresight at time t on the part of householdsconcerning the price of output for period t If gprime gt 0 it indicates that householdsto some extent but not completely (gprime lt 1) anticipate changes in the equilibriumlevel of prices that will prevail for period t
Totally differentiating the above two equations representing equilibrium in thelabor market with respect to wt Nt pt and noting our prior assumption that pe
t = ptinitially one obtains[minus(partN d
t part(wtpt))pt(partN s
t part(wtpet ))pt
minus1minus1
] [dwtdNt
]=[
(partN dt part(wtpt))wtdpt( pt)
2
(partN st part(wtpe
t ))wtgprimedpt( pt)2
]
Applying Cramerrsquos rule gives
dNt
dpt= (partN d
t part(wtpt))(partN st part(wtpe
t ))(wt( pt)2)(gprime minus 1)
minus(partN dt part(wtpt))pt + (partN s
t part(wtpet ))pe
t
gt 0
dwt
dpt= minuspartN d
t part(wtpt) + (partN st part(wtpe
t ))gprime
minuspartN dt part(wtpt) + partN s
t part(wtpet )
wt
ptgt 0
Note that if gprime = 1 then we have the standard neoclassical result that dNtdpt = 0and dwtwt = dptpt so that a change in the price level results in no change in
Neoclassical model 125
either employment or the real wage However with gprime lt 1 we have that anincrease in the price level leads to a rise in employment and an increase in themoney wage less than proportional to the increase in the price level such thatthe real wage falls (ie dptpt gt dwtwt gt 0) Given the aggregate produc-tion function yt = f (Nt K) we thus have an ldquoaggregate supply functionrdquo ofthe form
yt = yt( pt minus pet K ) (81prime)
so that aggregate supply depends directly on the difference between the price levelfor period t and the price level anticipated by suppliers at time t
Consider the above findings with respect to the labor market and suppose thatthere is an increase in pt Given perfect foresight on the part of firms at time t thelabor demand curve shifts up vertically so that at the higher money wage associatedwith the same real wage demand would be the same However given 1 gt gprime ge 0the vertical shift upward in the supply curve is less than this with the result thatequilibrium employment rises as the money wage rises by proportionately lessthan the rise in prices
In undergraduate textbooks the fact that changes in the price of output can nowaffect real output is shown in ( pt yt) by an upward-sloping ldquoaggregate supplycurverdquo as in Figure 81 Recall that in the neoclassical model the aggregate supplycurve is vertical In either model such a curve summarizes the underlying eventsin the labor market
The above character of the money illusion model is sometimes said to reflect theldquonatural rate hypothesisrdquo The natural rate hypothesis posits that fully anticipatedincreases in prices have no effect on the rate of real economic activity ndash specificallyreal output employment and thus unemployment Thus we will refer to the abovemodel as a static version of a ldquonatural rate modelrdquo
p
y
Figure 81 Upward-sloping aggregate supply
126 Neoclassical model
Equilibrium aggregate supply and demand
An important feature of macroeconomic theories is that to a large extent they aredistinguished by their different treatment of labor markets What this means isthat the aggregate demand side is typical of macroeconomic models Recall thatthe aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) Thus the equilibrium output price level and interest rate aregiven by equations (81prime) (82) and (83) Equation (81prime) is the aggregate supplyequation of a natural rate model (82) is the ldquoISrdquo equation depicting equilibriumbetween output demand and production and (83) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market
At this point we will simplify equations (82) and (83) by removing the realbalance effect representing them as follows
cdt (rt πe
t At yt) + I dnt(m
et + δ K) + δK + ψ(I d
t ) minus yt = 0 (82prime)
Ldt (rt πe
t At yt) minus Mpt = 0 (83prime)
In our model this implies that
partnet Adt part(Mpt) = 1
As we will see one justification for this form is if real money balances are notpart of household wealth which can be the case when we introduce depositoryinstitutions into the analysis In the meantime the above assumption makes theanalysis not only simpler but also more in line with traditional macroeconomicanalysis
Graphically the equilibrium price level and output can be shown using theaggregate demand and supply curves In Figure 82 the equilibrium output andprice level are thus given by plowast
t and ylowastt Looking at what underlies these curves
we can then infer the changes in money wages and employment (specifically froman analysis of the labor market that underlies the aggregate supply curve) as wellas changes in the interest rate and the components of investment and consumptioncomponents of output demand (specifically from an analysis of the output andmoney markets that underlie the aggregate demand curve)
Money supply change comparative statics for a naturalrate model
Collecting the aggregate supply IS and LM equations for the natural rate modelunder consideration we have (81prime) (82prime) and (83prime)
These three equations determine the equilibrium output the price level andthe interest rate Substituting (81prime) into (82prime) and (83prime) in order to focus on the
Neoclassical model 127
p
y
p
y
Aggregate supply
Aggregate demand
Figure 82 Macroeconomic equilibrium with upward-sloping supply
determination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form10
[ minus(partcdparty minus 1)partyp(partLdparty)partypartp + Mp2
partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + c1 + ψ prime)partI dpartm11 The term partypartp gt 0 reflectsthe direct effect of the price level on output as implied by the aggregate supplyequation (81prime)12 It is important to note that the equilibrium condition with respectto the labor market is incorporated into the above analysis in the form of thisaggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp)minus 1)(1p)
(partcdparty minus 1)partypartp(partLdpartr)minus((partLdparty)(partypartp)+ Mp2)(partydpartr)
lt 0
As you can see we no longer have drdM = 0 Further letting x denote thedenominator for the expression for dp we have
dp
p= dM
M
1
x
128 Neoclassical model
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater13
Let us now consider the effect of the money supply shock on other variablesFrom our analysis of the labor market we know that
dp
pgt
dw
wgt 0
so that while the money wage rises the real wage falls We also know from thelabor market that the rise in the price level leads to higher employment and thusan increase in output From the demand functions we can derive the effects of thechange in the money supply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Graphically one can show the effect of the money supply shock in terms ofthe aggregate demand and aggregate supply curves Note that this exercise simplyinvolves a shift out of the aggregate demand curve
The natural rate hypothesis and expectation formation a preview
An important feature of the above analysis one already noted is that real activity ndashin particular employment output and unemployment ndash changes only to the extentthat price changes are not fully anticipated As we have just seen this ldquonatural ratehypothesisrdquo introduces the logical foundations for a monetary change to have realeffects
However note that monetary changes have real effects only to the extent that theresulting changes in the price level are not fully anticipated To understand whenthis might occur we first have to indicate why individuals may err in their formationof expectations concerning the price level The Lucas model suggests one way ofexplaining errors in forecasts such that suppliers only partially anticipate a changein the price level (ie 1 gt gprime ge 0) This model introduces the assumption ofrational expectations
Combining rational expectations with the natural rate hypothesis results in a verypowerful statement concerning monetary policy which has so far not been madeclear If the actions of monetary authorities are predictable under the presumptionof rational expectations individuals will correctly predict the consequences onprices The result in the context of a natural rate model is that such predictable
Neoclassical model 129
monetary changes will have no real effects In the deterministic world we are backto the neoclassical model The analysis above then refers only to an ldquounexpectedrdquoincrease in the money supply or to monetary ldquosurprisesrdquo Only such random shocksto the money supply will have real effects
Conclusion
A formal neoclassical model of the macroeconomy has been introduced and fullydeveloped The issues of money neutrality and money illusion have been discussedand it has been seen that money supply changes have no effect on real economicactivity when the assumptions of the neoclassical model hold However a numberof issues have been raised namely the existence of a natural rate and the potentialeffect of unanticipated money supply changes on economic activity
9 The ldquoKeynesian modelrdquo withfixed money wageModifying the neoclassical model
Introduction
The first modification of the neoclassical model is presented in this chapterTo begin we introduce the very realistic assumption that nominal wages are fixedat least for a period of time The ramifications of this change in the model aredeveloped in the context of the aggregate supply and demand model As with theneoclassical model we perform a comparative statics exercise in which the mone-tary authority changes the money supply and we trace out the effects of this actionon the economic aggregates in the model The model is then made slightly morecomplete and issues associated with sticky wages and the natural rate hypothesisare discussed We introduce the concept of rational expectations and the first ldquoover-lappingrdquo model and show that the Keynesian model has important implicationsfor the conduct of monetary policy
The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
In the standard neoclassical model it is assumed that prices adjust in all marketsto equate demand and supply With respect to the labor market this implies a spotmarket at the start of each period in which one-period labor contracts are enteredinto and an associated one-period wage set Yet employment contracts are likelyto be multiperiod in the presence of hiring and training costs That is to minimizehiring and training costs firms seek long-term relationships with their employees
Firms promote long-term relationships with their employees by offering higherwages to their experienced workers As a consequence long-time employeesbecome attached or ldquoloyalrdquo to their employers since the wages they receive aregreater than those that other firms would offer them In essence employers aresharing the returns to their hiring and training investment with their workers inorder to reduce the number who quit A long-term attachment of workers to par-ticular firms could also stem from the high cost to workers of finding alternativeemployment as obtaining such employment means that workers must generallyinterview various employers visit employment agencies and spend valuable timesimply waiting for decisions on job applications to be made
Keynesian model 131
Given long-term employment contracts between firms and their workers wagesare typically specified for extended periods of time These long-term wage agree-ments are sometimes explicit as with many labor union contracts1 In other casesonly an implicit understanding exists on the wages that a firm will pay its employeesover some extended period of time If these contracts or understandings specifywages in money terms and if modifying these agreements is costly then thereexists an inherent inflexibility in money wages ndash that is there are ldquostickyrdquo wages2
This assumption of ldquostickyrdquo nominal wages is often viewed as the critical aspectof what has been termed the ldquoKeynesianrdquo macroeconomic model
If money wages are ldquostickyrdquo relative to prices then changes in the price levelwill alter the real wage For instance in the late 1970s high inflation rates ledworkers and employers in certain industries to bargain for long-term contractswith a high rate of growth of the money wage The high expected inflation did notmaterialize in the early 1980s The lower rate of inflation with no change in therate of increase in wages meant a rise in the real wage The resulting fall in thedemand for labor led to lower employment and output In this section we formallydevelop these results in the context of a static neoclassical macroeconomic modelwith the additional assumption of a fixed money wage The subsequent section thendevelops a linear rational expectations version of this model in which overlappingmultiperiod employment contracts introduce an element of nominal wage rigidity
Fixed money wage the labor market and aggregate supply
As we have seen in the competitive (spot) labor market of the neoclassical model(or in the Lucas-type macroeconomic model) the money wage and employmentare determined at the start of each period in the labor market If we accept theneoclassical modelrsquos assumption of limited perfect foresight on the part of bothlabor suppliers and firms we have equilibrium in the labor market in terms of amoney wage wt and level of employment Nt determined such that
N dt (wtpt K) minus Nt = 0 (91)
N st (wtpt) minus Nt = 0 (92)
In this case a change in the price level pt leads to an equiproportionate change inthe money wage wt and no change in employment Nt
We now seek to modify this analysis by assuming a fixed money wage wt = wfor period t As Sargent (1987a 21) states
the essential difference between the classical model and the Keynesian modelis the absence from the latter of the classical labor supply curve combined withthe labor market equilibrium condition Since there is one fewer equation inthe Keynesian model it can determine only six endogenous variables insteadof the seven determined in the classical model3 To close the Keynesianmodel the money wage is regarded as an exogenous variable one that atany point in time can be regarded as being given from outside the model
132 Keynesian model
perhaps from the past behavior of itself and other endogenous or exogenousvariables It bears emphasizing that the equation that we have deletedin moving from the classical to the Keynesian model [equation (92)] is acombination of a supply schedule (and) an equilibrium condition Notethat we continue to require that employment satisfy the labor demand schedule[equation (91)]
Sargent goes on to say that
we shall think of the labor supply schedule as being satisfied and helping todetermine the unemployment rate Usually the model is assumed to reachequilibrium in a position satisfying Nt lt N s
t so that there is an excess supplyof labor
Totally differentiating the labor demand condition (91) that determines the levelof employment with respect to the price level and employment we have
minus(partN dt (partwpt))(wp2
t )dpt minus dNt = 0 (93)
which can be rearranged to give
dNtdpt = minus(partN dt part(wpt))(wp2
t ) gt 0 (94)
where the sign reflects the presumption that partN dt part(wtpt) lt 0
In the simple case of no labor adjustment costs labor demand is defined bythe equality between the marginal product of labor and the real wage that ispartf (Nt K)partNt = wpt 4 Differentiating this implies that
[part2ftpartN 2t ]dNt = minus(wp2
t )dpt
or rearranging
dNtdpt = minus(wp2t )[part2ftpartN 2
t ] gt 0 (95)
given diminishing returns to the labor input (ie part2ftpartN 2t lt 0)
Combining the above analysis with the aggregate production function yt =f (Nt K) we thus have the ldquoaggregate supply equationrdquo
yt = yt( ptw K ) with partytpart( ptw) gt 0 (96)
Thus (as in a Lucas-type model) we have an aggregate supply that can dependdirectly on the price level for period t
The above findings can be understood with respect to the labor market Considera decrease in pt Given limited perfect foresight on the part of both firms andhouseholds at time t there is a downward (vertical) shift in labor demand so thatat the lower money wage wlowast
t associated with the same real wage demand would
Keynesian model 133
be the same Similarly the labor supply curve shifts down vertically so that atthis lower money wage wlowast
t labor supply would be the same as well Howevermultiperiod labor contracts fix the money wage at w so that the lower price level(and implied higher real wage) results in a fall in employment (which is nowdemand-determined) and an excess supply of labor
In the opposite case of a rise in the price level that can lead to an excessdemand in the labor market at the fixed money wage the presumption remainsthat employment is demand-determined This presumption reflects the view thatat least temporarily firms can direct workers with whom they have long-termemployment contracts to work overtime or extra shifts which the workers wouldotherwise not volunteer for
The above story provides a rationale for an upward-sloping ldquoaggregate sup-ply curverdquo Both contrast with the neoclassical model in which the aggregatesupply curve is vertical as a fall in the price level results in an equiproportionatefall in the money wage so that the real wage and employment remain unchangedIn the fixed wage model the underlying events in the labor market summarized bythe aggregate supply curve are the change in the real wage and thus labor demandand employment that accompany a price change when the money wage is fixedSuch an aggregate supply curve is upward-sloping
Equilibrium aggregate supply and demand
As we have noted before an important feature of macroeconomic theories is that toa large extent they are distinguished by their different treatment of labor marketsWhat this means is that the aggregate demand side is similar across macroeconomicmodels Recall that the aggregate demand side of macroeconomic models typicallyconsiders the equilibrium conditions of two of the remaining three markets inparticular the output market (reflected by an ldquoISrdquo equation) and the money market(reflected by an ldquoLMrdquo or ldquoportfoliordquo equation) Thus for the Keynesian modelwith fixed money wage the equilibrium output price level and interest rate aregiven by the following three equations
yt = yt( ptw K ) (96)
cdt (rt πe
t+1 At yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (97)
Ldt (rt πe
t+1 At yt) minus Mpt = 0 (98)
Equation (96) is the aggregate supply equation of a Keynesian model with fixedmoney wage (97) is the ldquoISrdquo equation depicting equilibrium between outputdemand and production and equation (98) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market Note that we havesimplified the IS and LM equations by removing the real balance effect for con-sumption demand and money demand5 Equations (97) and (98) can be combinedto eliminate the interest rate The resulting equation is referred to as the ldquoaggregatedemand equationrdquo
134 Keynesian model
The equilibrium price level and output ( plowastt and ylowast
t ) can be shown graphicallyusing aggregate demand and supply curves Looking at what underlies thesecurves we can then infer the change in employment (specifically from an analysisof the labor market that underlies the aggregate supply curve) as well as changes inthe interest rate and the investment and consumption components of output demand(specifically from an analysis of the output and money markets that underlie theaggregate demand curve)
A change in the money supply the comparative statics for theKeynesian model
The aggregate supply IS and LM equations (96)ndash(98) for the static Keynesianmodel under consideration determine the equilibrium output the price level andthe interest rate Substituting (96) into (97) and (98) in order to focus on thedetermination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form6[ minus(partcdparty minus 1)partypartp
(partLdparty)(partypartp) + Mp2partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm7 The term partypartp gt 0 reflects thedirect effect on the price level as implied by the aggregate supply equation (96)8
Note that the equilibrium condition with respect to the labor market is incorporatedinto the analysis in the form of this aggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp) minus 1)p
(partcdparty minus 1)partypartp(partLdpartr) minus (partLdparty)(partypartp) + (Mp2)(partydpartr)
lt 0
In contrast to the neoclassical model we no longer have drdM = 0 Furtherletting x denote the denominator for the expression for dp we have
dp
p= dM
M
1
x
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Keynesian model 135
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater
Consider now the effect of the money supply shock on other variables Fromour analysis of the labor market we know that w is fixed so that the increase inthe price level means a fall in the real wage and thus increased labor demandemployment and thus an increase in output From the demand functions for con-sumption and investment we can derive the effects of the change in the moneysupply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
and
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Note that the effect of the money supply shock is to increase the aggregate demandwhile not affecting aggregate supply
Sticky wages and the natural rate hypothesis
Expectations play an important role in the two modifications of the neoclassicalmodel In the modification with fixed wages the level at which negotiators fixfuture money wages depends on the expectation formed when wages were set con-cerning future prices The higher the expectation of future prices the higher thelevel of wages set in the labor agreements between workers and firms The pre-sumption is that workers and firms attempt to set future wages at their anticipatedmarket-clearing levels Associated with these anticipated market-clearing wagesis a particular real wage a natural rate of unemployment and a full employmentor natural rate of output
If price expectations turn out to be incorrect then output will vary from itsnatural rate For instance a shock that causes actual output prices to fall belowthose expected means that the money wage is fixed at a level that is too highfor full employment Consequently employment and output fall below the fullemployment level
In the typical Lucas-type model firms and workers set wages for the currentperiod based on incomplete information as well As we saw if suppliersrsquo expec-tations are incorrect then output will deviate from the full employment level Forexample a shock that causes actual output prices to fall below those expectedmeans lower employment and output as workers mistake lower money wages forlower real wages In fact higher real wages accompany the lower price level andthis is the source of the reduced demand for labor and employment
The two modifications of the neoclassical model have a second common ele-ment Both predict that a macroeconomic demand shock ultimately affects onlythe level of prices Even though money wages in the Keynesian model are fixed
136 Keynesian model
for the current period we know that money wages are not fixed forever Over timelabor agreements are renegotiated and money wages change to once again equatethe ldquoexpectedrdquo future demand for and supply of labor Over time in the absenceof further shocks the economy would thus tend to behave as neoclassical analysispredicts money wages and output prices would adjust to restore equilibrium tothe various markets in the economy
While the Keynesian model with fixed money wage admits the tendency foroutput to approach its natural level over time it does introduce a potential rolefor monetary policy to play in dampening fluctuations in output In so doing itchallenges the policy ineffectiveness view of Sargent and Wallace The best-knownexamples employing the Keynesian model to demonstrate the potential stabilizingpowers of monetary policy under rational expectations are the dual papers byFischer (1977) and Phelps and Taylor (1977)
A linear rational expectations version of the Keynesian model
Counting the above discussion we have so far considered three different modelsthat can be used to assess monetary policy One model is along the lines of theneoclassical model with limited perfect foresight Since this view of the economypredicts the neutrality of money a role for monetary policy either as an instigatorof output fluctuations or as an instrument to dampen output fluctuations is missingAs Mankiw (1987) suggests people who adopt this model ldquoview economic fluctu-ations through the lens of real business cycle theoryrdquo in which output fluctuationsare traced to ldquosupply-siderdquo disturbances
As Mankiw goes on to note however ldquothere are surely readers who believe thatmonetary policy has real short-run effects because of temporary misperceptions ornominal rigiditiesrdquo Mankiw is referring to individuals who adopt either the Lucas-type model or the ldquoKeynesianrdquo fixed money wage model9 Either one as we haveseen introduces a role for monetary policy as an instigator of output fluctuationsHowever these two models do differ as to whether monetary policy can be aninstrument to dampen output fluctuations in the context of rational expectations
A Lucas-type model built on ldquotemporary misperceptionsrdquo when coupled withrational expectations leaves little if any room for countercyclical monetary policyIn fact following the analysis of Sargent and Wallace it can be shown that whilerandom monetary shocks can impact output deterministic monetary policy basedon a set of policy rules is ineffective in counteracting fluctuations in output givenrational expectation10 Further attempts at discretionary monetary policy in thiscontext only result in a suboptimal (ldquotoo highrdquo) rate of inflation (Barro and Gordon1983) Thus in this model there remains a ldquostochasticrdquo neutrality of money
As the analysis in the previous section suggests however a ldquoKeynesian-typerdquomodel built on ldquonominal rigiditiesrdquo might introduce a role for monetary policyin stabilizing output even in the context of rational expectations The reasoningfor this is that wages (or as we will see later prices) can be set prior to thereceipt of information by the monetary authority that enters into the money supplyrule In this context as Phelps and Taylor (1977) state ldquoeven systematic and
Keynesian model 137
correctly anticipated policy can make a difference for the stability of output in arational expectations model with sticky prices and wagesrdquo11 Below we considerone example of such a model that counters the Sargent and Wallace ineffectivenessproposition a model proposed by Fischer that assumes ldquostickyrdquo wages
The supply equation with overlapping two-period labor contracts
The Lucas aggregate supply equation with adjustment costs can be expressed as
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (99)
where Yt = ln yt minus ln yn denotes difference between the logarithm of output forperiod t and the logarithm of the natural rate of output (which we have normalizedto equal zero) Pt = ln Pt is the logarithm of the price level for period t Etminus1Pt isthe expectation of the logarithm of the price level for period t using all informationavailable up to the end of period t minus 1 (at time t) and γ θ is a positive constant12
The supply of output as expressed by equation (99) satisfies the conditionthat employment equals labor demand (92) The fact that a higher price level Ptinduces firms to increase employment and thus output reflects the underlying lowerequilibrium real wage that accompanies the higher price level when suppliers donot anticipate the higher price level Thus we could express (99) in the form13
Yt = (Pt minus Wt + φ) + λYtminus1 (99prime)
where Wt is the logarithm of the equilibrium nominal wage for period t Theterm φ in (99prime) is defined such that if Etminus1Pt = Pt then the resulting log ofthe equilibrium real wage (ie ln(wtpt)) equals φ Ignoring the lagged outputterm in equation (99prime) this equilibrium real wage is the one associated with thenatural level of output and employment14 We will follow others and assume forconvenience that φ = 0 implying an equilibrium real wage with no surprisesequal to one
According to (99prime) if an increase in the price level is accompanied by a lessthan proportionate increase in the equilibrium money wage Pt minusWt rises (the realwage falls) and employment and output increase In the Lucas-type model such anevent occurs if suppliers do not forecast the price increase However Sargent andWallacersquos ineffectiveness proposition eliminates deterministic monetary policyrules as a source of such a price rise not matched by a similar rise in wages if(a) wages are set each period to equate labor demand and supply and (b) rationalexpectations are assumed In this case individuals and the monetary authoritiesare assumed to have a common set of information based on events up to the endof period t minus 1 (at time t) Individuals are also privy to the monetary policy ruleand they know the structure of the economy Thus they have knowledge of thedeterministic monetary policy to be followed during period t and its effect on theprice level for period t as predicted by the model Assuming flexible wages thispredicted effect of monetary policy on prices will be factored into the setting of themoney wage for period t As a consequence such deterministic monetary policy
138 Keynesian model
cannot change the real wage and thus leaves employment and output unaffectedas well
Obviously this chain of reasoning breaks down if money wages for period twere set prior to time t For example this policy ineffectiveness doctrine disap-pears if some wages are set at the end of period t minus 2 for period t In this casenew information that arrives during period t minus 1 can be incorporated by the mon-etary authorities into their money supply rule Even with rational expectationsthe implications of this cannot be used by individuals to adjust the money wagefor period t since by assumption the money wage is fixed Thus deterministicmonetary policy based on information revealed during period t minus 1 can alter thereal wage employment and output
To formally develop this potential stabilizing role of monetary policy in a moreldquoelegantrdquo fashion let us consider Fischerrsquos model The model disaggregates theeconomy into two sectors and assumes that the sectors alternate in setting multi-period employment contracts that fix nominal wages In particular ldquosuppose thatall labor contracts run for two periods and that the contract drawn up at the endof period t minus 2 specifies nominal wages for periods t minus 1 and t [Assume] thatcontracts are drawn up to maintain constancy of the real wagerdquo (Fischer 1977198) In other words
tminusiWt = EtminusiPt i = 1 2 (910)
where tminusiWt is the logarithm of the wage set at the end of period tminus i for period t15
The idea embodied in (910) that wages are set for more than one period iscritical to Fischerrsquos finding It essentially means that in any period half of thelabor contracts have fixed money wages16 Given that the wage is predeterminedfor each firm the aggregate supply equation is given by
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911)
where ut is a stochastic ldquorealrdquo disturbance or ldquosupply shockrdquo that impinges onproduction in each period17 Substituting (910) into (911) we can rewrite theaggregate supply equation as
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911prime)
A complete model except for specifying the source of expectations
Equation (911prime) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To close the model we require LM and ISequations The explicit derivation of these is left to the next chapter but let usassume for the time being that the LM equation is given by
mt minus Pt = α1Yt minus α2 middot rt minus εt
Keynesian model 139
and the IS equation by
Yt = Xt minus β1(rt minus πet+1) + ut
Here mt = ln Mt mt = mt + εt such that the deterministic component mt is setby government authorities (ie the monetary authority) according to a monetaryrule Further rt minus πe
t+1 represents the expected real rate of interest Xt denotesa vector of exogenous variables that affect output demand εt and ut are randomterms associated with output demand and money supply respectively and assumedindependent (ie E(εtut) = 0) and well behaved
Combining the LM and IS equations to eliminate the interest rate rt we obtain
Yt = Xt + ut minus (β1α2)(minusmt minus εt + Pt minus α1Yt) + β1πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = [α2(α2 + α1β1)][Xt + ut + (β1α2)(mt + εt minus Pt) + β1πet+1] (912)
Note that if α2 = 0 (changes in the interest rate do not affect money demand) andα1 = 1 (the income elasticity of real money demand is one) then this simplifies towhat Fischer refers to as a ldquovelocity equationrdquo
Yt = mt minus Pt + vt (913)
where vt = εt is now to be interpreted as a money demand disturbance termaffecting the ldquovelocityrdquo of money18
To see why (913) is called a ldquovelocity equationrdquo note that the assumption ofmoney demand being independent of the interest rate allows us to capture therelationship between income and the price level summarized by the aggregatedemand equation by looking solely at the LM equation (ie neglecting the ISequation) In particular if we assume that real money demand can be expressedby the equation
Ldt = yt(exp(minusvt))
then equating real money demand to real money supply (Mtpt) gives us
yt(exp(minusvt)) = Mtpt (914)
Taking the logarithm of the equilibrium condition with respect to the money market(914) we obtain
ln yt minus vt = ln Mt minus ln pt
Given Pt = ln pt Yt = ln yt and mt = ln(Mt) this is simply equation (913)
140 Keynesian model
Note that equilibrium velocity is defined as the ratio of nominal output to themoney supply Thus rearranging (914) we have that
Equilibrium velocity equiv ptyt
Mt= exp(vt)
which explains the interpretation of vt as a ldquovelocityrdquo disturbance Fischer assumesthat vt has a zero mean so that expected velocity is one A vt above zero meansa decrease in real money demand relative to real output and thus an increase inequilibrium velocity As (914) makes clear for a given Mt a higher vt implies ahigher pt andor a higher yt to maintain equilibrium with respect to the demandfor and supply of money
As Fischer states (913) is ldquothe simplest way of taking demand consid-erations into accountrdquo In sum then the macroeconomics model considered byFischer is given by the aggregate supply equation (911prime) and the aggregate demandequation (913)
Combining (911prime) and (913) to eliminate Yt we have
12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut = mt minus Pt + vt
This can be solved to give the reduced-form equation for the price level Pt
Pt = 12
[(12 Etminus1Pt + 1
2 Etminus2Pt
)minus ut + mt + vt
] (915)
Combining (911prime) and (913) to eliminate Pt we have
Yt = 12 (mt minus Yt + vt minus Etminus1Pt) + 1
2 (mt minus Yt + vt minus Etminus2Pt) + ut
This can be solved for the reduced-form equation for output Yt
Yt = 12
[mt + vt + ut minus 1
2 Etminus1Pt minus 12 Etminus2Pt
] (916)
According to equations (915) and (916) an increase in the ldquorealrdquo disturbanceterm ut leads to higher equilibrium output and a reduced price level Intuitivelythis corresponds to a shift to the right in the ldquoaggregate supply curverdquo On the otherhand an increase in the ldquovelocityrdquo disturbance term vt corresponds to a shift to theright in the ldquoaggregate demand curverdquo and thus leads to a higher output and pricelevel given an upward-sloping aggregate supply curve Note that an increase in vtmeans a lower real money demand at each level of income The shift in the aggre-gate demand curve reflects the fact that a higher price level andor higher output isrequired to restore equilibrium in the money commodity and financial markets
If expectations can be taken as exogenous with respect to money supply changesthen (916) indicates that money supply changes can affect output But this wasalso the case for a Lucas-type model The next step is thus to see what happens
Keynesian model 141
when we assume rational expectations As will become clear even with rationalexpectations expectations formed at the end of period t minus 2 can be viewed asexogenous with respect to monetary changes planned for period t based on infor-mation obtained during period t minus 1 Thus monetary policy has the potential tooffset the persistent effect of disturbances that originate during period t minus 1
Introducing rational expectations
Let us now assume that individuals form their expectations Etminus1Pt and Etminus2Ptldquorationallyrdquo in that
Etminus2Pt = E(Pt |tminus2) (917)
Etminus1Pt = E(Pt |tminus1) (918)
indicating that EtminusiPt is the mathematical expectation of Pt conditional on theinformation set tminusi which is all information available at the end of period t minus ii = 1 2 Taking the expectation of (915) at the end of period t minus 2 we thus have
Etminus2Pt = Etminus2
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (919)
Note that Etminus2Etminus1Pt = Etminus2Pt Thus (919) becomes
Etminus2Pt = Etminus2minusut + vt + mt (920)
Taking the expectation of (915) at the end of period t minus1 (at time t) we then have
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (921)
Substituting (920) into (921) gives
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2minusut + vt + mt minus ut + vt + mt
]
(922)
Rearranging
34 Etminus1Pt = 1
4 Etminus2minusut + vt + mt + 12 Etminus1minusut + vt + mt
or
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + mt (923)
which is Fischerrsquos (1977) equation (16)19
142 Keynesian model
Let the money supply be determined by the simple linear rule
mt = a1utminus1 + b1vtminus1
Since mt is a function only of information available up to the end of period t minus 1(at time t) Etminus1mt = mt Accordingly (923) can be written as
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + 23 mt (924)
Substituting (920) and (924) into the reduced-form equation for output (916)we obtain
Yt = 12 [ut + vt + mt] minus 1
4
[13 Etminus2minusut + vt + mt
+ 23 Etminus1minusut + vt + 2
3 mt
]minus 1
4 [Etminus2minusut + vt + mt](925)
Equation (925) simplifies to
Yt = 13 (mt minus Etminus2mt) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt
(926)
which is Fischerrsquos (1977) equation (18)As Fischer (1977 196) notes
disturbances aside this very simple macro model would be assumed in equi-librium to have the real wage set at its full employment level would imply theneutrality of money and would obviously have no role for monetary policyin affecting the level of output A potential role for monetary policy is createdby the presence of the disturbances ut and vt that are assumed to affect thelevel of output each period Each of the disturbances is assumed to follow afirst-order autoregressive scheme
ut = ρ1 middot utminus1 + εt where |ρ1| lt 1 (927)
vt = ρ2 middot vtminus1 + ηt where |ρ2| lt 1 (928)
where εt and ηt are mutually and serially uncorrelated stochastic terms withexpectation zero and finite variances σ 2
e and σ 2n respectively
Given equations (927) and (928) and the money supply rule
mt = a1utminus1 + b1vtminus1 (929)
Etminus2mt = a1ρ1utminus2 + b1ρ2vtminus2 (930)
Keynesian model 143
so that
mt minus Etminus2mt = a1utminus1 + b1vtminus1 minus [a1ρ1utminus2 + b1ρ2vtminus2]= a1εtminus1 + b1ηtminus1
(931)
According to (931)
the difference between the actual money stock in period t and that stockas predicted two periods earlier arises from the reactions of the monetaryauthority to the disturbances εtminus1 and ηtminus1 occurring in the interim It isprecisely these disturbances that cannot influence the nominal wage for thesecond period of wage contracts entered into at time t minus 2
(Fischer 1977 199)
Substituting (931) into (926)
Yt = 13 (a1εtminus1 + b1ηtminus1) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt (932)
From (927) and (928) we know that
ut + vt = (ρ1utminus1 + εt) + (ρ2vtminus1 + ηt)
= (ρ1utminus1 + εt) + (ρ22vtminus2 + ρ2ηtminus1 + ηt)
since by substitution vt = ρ22vtminus2 + ρ2ηtminus1 + ηt we also have that
Etminus1ut minus vt = ρ1utminus1 minus ρ2vtminus1 = ρ1utminus1 minus ρ22vtminus2 minus ρ2ηtminus1
and
Etminus2ut minus vt = ρ21utminus2 minus ρ2
2vtminus2
Thus we can rewrite (932) as
Yt = 12 (εtminus1 + ηtminus1) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
(933)
which is Fischerrsquos (1977) equation (21)20
Fischer (1977 199) notes that
before we examine the variance of output as a function of the parameters a1and b1 it is worth explaining why the values of those parameters affect thebehavior of output even when the parameters are fully known The essential
144 Keynesian model
reason is that between the time the two-year contract is drawn up and the lastyear of operation of that contract there is time for the monetary authorityto react to new information about recent economic disturbances Given thenegotiated second-period nominal wage the way the monetary authority reactsto disturbances will affect the real wage for the second period of the contractand thus output
Optimal monetary policy rules the effectiveness of policy
As in our discussion of Sargent and Wallace let us presume that the goal of themonetary authority focuses solely on output In particular suppose that the mone-tary authority desires to set the money supply in order to minimize the fluctuationin the log of output around some desired level Then the objective can be expressedas to
min Etminus1(Yt minus Y lowast)2
Let us assume that Y lowast = Yn = 0 so that the objective becomes to
min Etminus1(Yt)2 (934)
From (933) we have that
(Yt)2 =
[12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21 utminus2
]times[
12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
]
(935)
Note that E(εi) = E(ηi) = 0 E(ε2i ) = σ 2
e E(η2i ) = σ 2
n and that our independenceassumptions imply that E(εiηi) = 0 and for i = s E(εiεs) = 0 and E(εiηs) = 0Thus substituting (935) into (934) we have the following explicit form for theobjective
min Etminus1(Yt)2 = σ 2
e
[14 + 1
9 (a1 + 2ρ1)2]
+ (ρ21utminus2)
2
+ σ 2n
[14 + 1
9 (b1 + ρ2)2]
(936)
Given (936) the optimal monetary rule is to choose values for a1 and b1 suchthat
a1 = minus2ρ1 b1 = minusρ2 (937)
The above findings correspond to Fischerrsquos (1977) equation (23)21
Keynesian model 145
As Fischer (1977 200) states
to interpret the monetary rule examine [equation (933)] It can be seen therethat the level of output is affected by current disturbances (εt + ηt) that can-not be offset by monetary policy by disturbances (εtminus1 and ηtminus1) that haveoccurred since the signing of the older of the existing labor contracts andby a lagged real disturbance utminus2 The disturbances εtminus1 and ηtminus1 can bewholly offset by monetary policy and that is precisely what equation [(937)]indicates The utminus2 disturbance on the other hand was known when the olderlabor contract was drawn up and cannot be offset by monetary policy becauseit is taken into account in wage setting Note however that the stabilizationis achieved by affecting the real wage of those in the second year of laborcontracts and thus should not be expected to be available to attain arbitrarylevels of output ndash the use of too active a policy would lead to a change in thestructure of contracts
[A] more general interpretation of the monetary rule is to accommodatereal disturbances that tend to increase the price level and to counteract nominaldisturbances that tend to increase the price level
Fischer concludes by noting that
given a structure of contracts there is some room for maneuver by the mon-etary authorities ndash which is to say that their policies can though will notnecessarily be stabilizing
Conclusion
This chapter presented the sticky money wage or Keynesian model of the macro-economy We find that in contrast to the neoclassical model changes in the pricelevel affect real variables and the amount of labor employed in the economy Thusmoney has real effects The development of an upward-sloping aggregate supplycurve has dramatic implications for the conduct of monetary policy However it isshown that the expectations of agents in the economy also play an important rolein whether or not monetary policy is effective
10 The Lucas model
Introduction
As we have seen anticipated changes in prices have no impact on real vari-ables in the neoclassical model A key element of this model is the ldquoessentialpresumption that nominal output is determined on the aggregate demand sideof the economy with the division into real output and the price level largely depen-dent on the behavior of suppliers of labor and goodsrdquo (Lucas 1973) As such thismodel implies no link between price changes and real output
We have also seen how the natural rate model allows one to introduce a linkbetween unanticipated price changes and real output The seminal paper by Lucas(1973) formally develops a more complete model of the potential for ldquoshort-runsupply behavior (resulting) from suppliersrsquo lack of information on some of theprices relevant to their decisionsrdquo Lucasrsquos explanation of a tradeoff between unem-ployment and inflation ldquois that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relative pricechangesrdquo
As with the ldquoillusion modelrdquo Lucas postulates ldquorational agents whose deci-sions depend on relative prices only placed in an economic setting where theycannot distinguish relative from general price movementsrdquo That is we retain thehypothesis that prices adjust to clear markets
Lucas adds to the simple (static) natural rate model so far discussed by explicitlymodeling the source of forecast errors In doing so he assumes that ldquoinferenceson these relevant unobserved prices are made optimally (or lsquorationallyrsquo) in lightof the stochastic nature of the economyrdquo Below we outline Lucasrsquos model1
The ldquoislandrdquo paradigm
Lucasrsquos model begins by disaggregating the economy into a number of what havebeen called ldquosectorsrdquo ldquomarketsrdquo or ldquoislandsrdquo As Lucas says ldquowe imagine sup-pliers as located in a large number of scattered competitive markets Demand forgoods in each period is distributed unevenly over markets leading to relative aswell as general price movementsrdquo In terms of our previous analysis one couldthink of each of the n sectors in the economy as inhabited by firms producing the
Lucas model 147
ith commodity (i = 1 n) Associated with each sector or ldquoislandrdquo is a set ofworkers and thus a labor market
For firms producing commodity i in period t the key relative price is the priceof their output relative to the wage paid in sector i or pitwit where pit is themoney price of commodity i (produced in sector i) and wit is the money wage forlabor in sector i It is assumed that individuals (firms and workers) in sector irsquoslabor market know the money wage and price of commodity i For labor suppliersin sector i however the key relative price is witpt where pt is the economy-wideprice level reflecting the fact that suppliers plan to use money wages to purchasea bundle of goods consisting of all n commodities2
It is this setup of ldquodispersed marketsrdquo and ldquoinformational discrepanciesrdquo thatLucas uses to generate a correlation between price changes and output ndash the famousLucas supply equation
The supply function for a particular sector
The Lucas model assumes a competitive labor market for sector or ldquoislandrdquo i suchthat the equilibrium level of employment and money wage equate market demandand supply With respect to the labor demand function let us start by assumingthe simple CobbndashDouglas production function such that the marginal product oflabor is given by
a(Nit)minusα where a gt 03
The profit-maximizing condition for the representative firm producingcommodity i is to equate the money wage to the marginal product of labor multi-plied by the money price of output The resulting optimal labor demand can thusbe defined by the equation4
wit = pita(N dit )
minusα
Taking the natural log of this equation and rearranging we have
ln N dit = 1
α[ln pit minus ln wit + ln a] (101)
With respect to labor supply let us assume for the moment that the real wage isknown Further let us assume that the labor supply function takes the followinglogarithmic form
ln N sit = 1
βln
wit
pit
where β is a positive constantEmployment contracts entered into at time t in sector i specify the money
wage wit so that element of the real wage is known However if the price level is
148 Lucas model
unknown then the expected real wage based on information available at time t isequal to witEt(1pt) The associated expected logarithm of the real wage is then5
Et ln(witpt) = ln wit minus Et(ln pt)
Given the assumed log-linear labor supply function and ignoring the implicationsof uncertainty for labor supply we thus have
ln N sit =
(1
β
)(ln wit minus Et(ln pt)) (102)
Equilibrium in the labor market for sector i entails a level of employment Nitand money wage wit such that the demand for labor equals the supply In loga-rithmic form and using the specific labor demand and supply functions given byequations (101) and (102) equilibrium requires that the log of the money wageln wit and the log of employment ln Nit be such that
ln Nit = 1
α(ln pit minus ln wit + ln a)
and
ln Nit = 1
β(ln wit minus Et(ln pt))
Substituting the first expression into the second to eliminate the logarithm of themoney wage we have
ln Nit = 1
α(ln pit + ln a) minus 1
α(β ln Nit + Et(ln pt))
which upon rearranging becomes
ln Nit = ln Nni + [1 + (α + β)][ln pit minus Et(ln pt)] (103)
where ln Nni = (ln a)(α + β)Equation (103) indicates that the logarithm of equilibrium employment in the ith
sector and thus the production of commodity i depends directly on the expectationof logarithm of the ratio of the price of commodity i(pit) to the general level ofprices pt The term Nni can be viewed as the ldquonormalrdquo level of employment Notethat we have abstracted from population growth and other factors that would resultin this ldquonormalrdquo level of employment varying across time
Given the assumption of a simple CobbndashDouglas production function of theform yit = (Nit)
1minusα(Ki)α we thus have
ln yit = ln yni + γ [ln pit minus Et(ln pt)] (104)
where γ = (1 minus α)(α + β) and ln yni = (1 minus α) ln Nni + α ln Ki The term yni isdenoted by Lucas as the ldquonormalrdquo level of output in the particular sector or market i
Lucas model 149
under consideration As Lucas (1973 327) states the ldquoquantity supplied in eachmarket will be viewed as the product of a normal (or secular) component commonto all markets and a cyclical component which varies from market to market
The cyclical component varies with perceived relative prices and with its ownlagged valuerdquo Note that for the moment we do not include the lagged value ofoutput in the above supply function One can justify the inclusion of such byassuming adjustment costs
The source of forecasting errors
According to (104) output of commodity i depends critically on suppliersrsquo fore-cast of the log of the general level of prices Et(ln pt) Now consider how sucha forecast may be obtained in a stochastic environment First it is assumed thatagents in sector i ndash in particular suppliers of labor involved in the production ofcommodity i ndash know commodity irsquos price pit However the exact extent to whichany change in the money price of commodity i reflects a change in the overalllevel of money prices as opposed to a change in commodity irsquos price relative toother prices is unknown It is this uncertainty that leads suppliers in sector i tomisinterpret a change in the general price level in terms of a change in a relativeprice
To be concrete suppose Etminus1(ln pt) incorporates all information available atthe end of period t minus 1 The logarithm of the actual price level will vary from thelogarithm of this expected price level to the extent that there are ldquosurprisesrdquo withrespect to the aggregate price level Letting ξt denote this ldquosurpriserdquo for period twe have
ln pt = Etminus1(ln pt) + ξt (105)
We assume that ξt which is that part of the price level that cannot be predictedfrom past data is a normally distributed random variable with zero expectationand variance σ 26
At the start of period t suppliers in market i receive one additional piece ofinformation the logarithm of the price of commodity i ln pit This signal isassumed to contain some information about the logarithm of the overall pricelevel in that
ln pit = ln pt + zit (106)
where zit is a normally distributed random variable with zero mean and varianceσ 2
z Thus using equation (105) to substitute for ln pt
ln pit = Etminus1(ln pt) + ξt + zit (107)
In words the logarithm of the nominal price for the sector ln pit is assumed toinform the supplier of the sum of the current ldquowhite noiserdquo innovations to the
150 Lucas model
relative price process in that sector (zit) and the innovations to aggregate demandand thus the economy-wide level of prices (ξt)7
We can then express the expectation of the logarithm of the price level at time tfor suppliers in sector i given the observed logarithm of the price of commodity iln pit by
Et(ln pt) equiv Eln pt |Etminus1(ln pt) ln pit (108)
Formally we have the joint distribution of two random variables f (ln pit ln pt)where one of them ln pit is known to take a particular value The problem is thebasic one of ldquobivariate regressionrdquo in that we have to determine the conditionalexpectation Eln pt | ln pit namely the ldquoaveragerdquo value of ln pt for the givenvalue of ln pit 8 As we shall see in the next section the resulting expression for theexpected general price level (in logs) given ln pit is observed can be expressedin linear form as
Et(ln pt) = Etminus1(ln pt) + (1 minus θ)(ln pit minus Etminus1(ln pt)) where 0 le θ le 1(109)
A digression on linear regression analysis
Let us assume a linear regression equation that links the observed logarithm of theprice of commodity i to the logarithm of the general level of prices of the form9
Etln pt | ln pit = a0 + a1(ln pit) (1010)
We can express the regression coefficients a0 and a1 in terms of some of the lowermoments of the joint distribution of ln pt and ln pit namely in terms of10
Eln pit = Etminus1(ln pt) (from (107))
Eln pt = Etminus1(ln pt) (from (105))11
Varln pit = σ 2 + σ 2z (from (107))
Cov(ln pit ln pt) = E(minusξt minus zit)(minusξt) = σ 2 + Ezit middot ξtIn general Ezit middot ξt = Cov(zit ξt) + EzitEξt However given that Ezit =Eξt = 0 and given the assumption that zit and ut are independent variables sothat Cov(zit ξt) = 0 we have12
Cov(ln pit ln pt) = σ 2
From (1010) we have that13
E(ln pt | ln pit) equivint
(ln pt)φ(ln pt | ln pit)d ln pt = a0 + a1(ln pit) (1011)
Lucas model 151
where φ(middot) is the conditional density function of ln pt given ln pit If we thenmultiply the expression on both sides of (1011) by the marginal density functionof ln pit denoted by g(ln pit) and integrate on ln pit we obtainintint
(ln pt)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0g(ln pit)d ln pit +int
a1(ln pit)g(ln pit)d ln pit
or
Eln pt = a0 + a1Eln pit (1012)
since φ(ln pt | ln pit)g ln(pit) = f (ln pit ln pt) Had we multiplied the expressionon both sides of (1011) also by ln pit before integrating on ln pit we would haveobtainedintint
(ln pt)(ln pit)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0(ln pit)g(ln pit)d ln pit +int
a1(ln pit)2g(ln pit)d ln pit
or
E(ln pit)(ln pt) = a0Eln pit + a1E(ln pit)2 (1013)
Solving (1012) and (1013) for a0 and a1 and making use of the fact that
E(ln pit)(ln pt) = Cov(ln pit ln pt) + E(ln pit)Eln ptand
E(ln pit)2 = Var(ln pit) + [Eln pit]2
we find that
a0 = Eln pt minus [Cov(ln pit ln pt)Eln pit](Var(ln pit))
a1 = (Cov(ln pit ln pt))(Var(ln pit))
Hence we can write equation (1010) as
Et(ln pt) equiv E(ln pt | ln pit)
= E(ln pt) + [Cov(ln pit ln pt)Var(ln pit)](ln pt minus Eln pit)Substituting in the above expressions for means variance and covariance we havethus derived (109) with
1 minus θ = σ 2(σ 2 + σ 2z )
152 Lucas model
Equation (109) indicates that agentsrsquo rational expectation of the current pricelevel is a ldquolinear least-squares projectionrdquo That is one could rewrite (109) as
Et(ln pt) = θEtminus1(ln pt) + (1 minus θ) ln pit (1014)
where θ = σ 2z (σ 2 + σ 2
z ) To see why this is called ldquolinear least-squaresrdquo notethat we could start with (1014) (the ldquolinearrdquo part of the projection) and thenpick θ to minimize the variance in this forecast or projection of ln pt (the ldquoleast-squaresrdquo part of the projection) In particular substituting in (105) for Etminus1(ln pt)
(ie Etminus1(ln pt) = ln pt minus ξt) and (106) for ln pit (ie ln pit = ln pt + zit)(1014) becomes
Et(ln pt) = ln pt minus θξt + (1 minus θ)zit (1014prime)
The problem of picking θ to minimize the variance of this projection can then beexpressed as14
minθ
Eln pt minus θξt + (1 minus θ)zit minus ln pt2 = θσ 2 + (1 minus θ)σ 2z
Taking the derivative of the above expression with respect to θ setting it equal tozero (ldquoleast squaresrdquo) and solving for θ we verify that θ = σ 2
z (σ 2 + σ 2z )
As Sargent (1987a 442) points out
the parameter θ is the fraction of the conditional variance in ln pit due torelative price variation The larger is this fraction the smaller is the weightplaced on ln pit in revising Etminus1(ln pt) to form Et(ln pt) This makes sensesince the larger is θ the more likely it is that a change in ln pit reflects arelative rather than a general price change15
Equation (1014) can be substituted into the supply function for commodity i(104) to obtain
ln yit = ln yni + γ θ(ln pit minus Etminus1(ln pt)) (1015)
where as before ln yni = (1 minus α)(ln Nni) + α(ln Ki) As noted above if weassumed adjustment costs then a lagged output term could be added to (1015)In this case we would have16
ln yni = ln yni + γ θ [ln pit minus Etminus1(ln pt)] + λ(ln yitminus1 minus ln yni) (1015prime)
If suppliers were able to observe the actual value of the price level so thatEt(ln pt) = ln pt then going back to (104) one could express the resulting ldquofullinformationrdquo output produced in sector i by
ln ylowastit = ln yni + γ [ln pit minus ln pt]
Lucas model 153
which given ln pit = ln pt + zit simply becomes
ln ylowastit = ln yni + γ zit
As you can see since the expectation of the random shock to relative prices zit iszero ln yni has the natural interpretation as the expected output of sector i givenfull information
The Lucas aggregate supply function
Equation (1015) is close to what is known as the ldquoLucas aggregate supplyfunctionrdquo Without adjustment costs the Lucas supply function takes the form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) (1016)
With adjustment costs the Lucas supply function takes the general form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) + λ(ln ytminus1 minus ln yn) (1016prime)
where yn denotes the natural rate of output17 For simplicity we have assumed thatthe natural rate of total output is constant across periods
The term ln yn can be interpreted either as the logarithm of output for theldquorepresentativerdquo sector or as the logarithm of total output across the n sectorsLet us assume the former interpretation The average level of real output can bedefined by
yt equiv 1
pt
⎡⎣ nprod
i=1
pityit
⎤⎦
1n
where [prodni=1 pityit]1n is the geometric mean of nominal output across the n markets
or sectors Taking logs we have the following definition for the logarithm ofaverage output
ln yt equiv minus ln pt + 1
n
nsumi=1
(ln pit + ln yit) (1017)
We will assume that the overall price level is constructed as a geometric mean ofindividual prices such that
pt equiv⎡⎣ nprod
i=1
pit
⎤⎦
1n
Taking logs
ln pt equiv 1
n
nsumi=1
ln pit
154 Lucas model
Substituting the above into (1017) we have the following definition for thelogarithm of average output
ln yt equiv 1
n
nsumi=1
ln yit (1018)
Substituting into (1018) the supply functions for the individual sectors as givenby (1015) we thus have18
ln yt equiv 1
n
nsumi=1
[ln yni + γ θ(ln pit minus Etminus1(ln pt))] (1019)
Recall that ln pit = ln pt + zit where zit is a normal random variable indepen-dently distributed across markets with a mean of zero and variance σ 2
z Substitutingthis into (1015) and rearranging we have
ln yt = 1
n
nsumi=1
[ln yni] + γ θ(ln pt minus Etminus1(ln pt)) + 1
n
nsumi=1
γ θzit (1020)
As the number of markets n approaches infinity from the law of large numberswe know that the sum of the zit divided by n approaches zero19 Thus for a largenumber of markets we may approximate (1020) by
ln yt = 1
n
⎡⎣ nsum
i=1
ln yni + γ θ(ln pt minus Etminus1(ln pt))
⎤⎦ (1021)
By definition the logarithm of the geometric average of ldquonormalrdquo output acrossmarkets ln yn is given by nminus1 sumn
i=1 ln yni Thus we can rewrite (1021) as (1016)in which ln yn is the logarithm of ldquonormalrdquo output that would occur if there were nosurprises with respect to the aggregate price level that is when ln pt = Etminus1(ln pt)Note that for simplicity we assume the natural rate is constant over time
Equation (1016) is the Lucas aggregate supply equation with the last termmissing As noted above if we include adjustment costs then we obtain (1016prime)indicating that the deviation of real output from its ldquonaturalrdquo level or trend isassociated with a deviation in the price level from that expected and past deviationsof output from the natural rate This last term makes output serially correlatedover time
The Lucas supply function and the Phillips curve
The Lucas supply function predicts a direct correlation between unanticipatedprice changes and output and thus a potential tradeoff between price changesand unemployment if one assumes that unemployment and output are inverselyrelated This potential inverse relationship between unemployment and inflation
Lucas model 155
is sometimes referred to as the ldquoPhillips curverdquo after AW Phillips who noted theempirical relationship between wage inflation and unemployment for the Britisheconomy for the 100 years up to 1957 (see Phillips 1958) Later depictions of thePhillips curve replaced the rate of change in wages with the inflation rate
To see this Phillips relationship more clearly rearrange the aggregate Lucassupply function without adjustment costs (1016) to obtain
ln pt = (ln yt minus ln yn)γ θ + Etminus1(ln pt)
Subtracting ln ptminus1 from both sides of this aggregate supply equation we have
ln(ptptminus1) = (ln yt minus ln yn)γ θ + Etminus1(ln(ptptminus1)) (1022)
Let the term πt denote the rate of inflation between periods t minus 1 and t20
πt equiv (pt minus ptminus1)ptminus1 = (ptptminus1) minus 1
It is common in macroeconomics to approximate the above rate of change inprices by the log of the ratio of the two prices If the ratio equals one then thelog equals zero which is the rate of inflation If the ratio is 1 + x and x is a smallproportion then the log of this ratio approximately equals the actual inflation rateFor instance if ptptminus1 = 105 so that inflation is 005 or 5 percent then the logof 105 is 00488 which approximates this 005 rate of inflation Thus we have
πt asymp ln(ptptminus1) = ln pt minus ln ptminus1
Using the above approximation for the rate of inflation we can rewrite(1022) as
πt = (ln yt minus ln yn)γ θ + Etminus1πt (1023)
which as Sargent (1987a 443) states
is in the form of a standard natural rate Phillips curve relating inflation (πt)
directly to output (ln yt) and to expected inflation (Etminus1πt) According to[(1023)] the Phillips curve shifts up in the (πt yt) plane by the exact amountof any increase in expected inflation This characteristic of equation [(1023)]is often taken as the hallmark of the natural unemployment rate hypothesisIt seems to offer an explanation for why the Phillips curve tradeoff worsenedas average inflation rates increased over the 1970s in many western countries
If we assumed that due to adjustment cost the lagged deviation in output fromthe natural level affects the current deviation as the Lucas aggregate supplyequation (1016prime) suggests then in terms of rates of change in prices we wouldhave
πt = (ln yt minus ln yn)γ θ + Etminus1πt minus (λγ θ)(ln ytminus1 minus ln yn) (1023prime)
156 Lucas model
We could instead express (1023) in terms of unemployment by assuming thereis a linear inverse relationship between deviations in output from the natural rateand deviations in the actual level of unemployment Ut from its natural rate Unsuch that
ln ytminus1 minus ln yn = minus(Ut minus Un)
where is a positive constant Substituting the above into (1023) we thus have
πt = minus(γ θ)(Ut minus Un) + Etminus1(πt) (1023primeprime)
indicating the inverse relationship between unanticipated price changes and theactual level of unemployment Rearranging (1023primeprime) we have that
Ut = Un minus (γ θ)[πt minus Etminus1(πt)] (1023primeprimeprime)
where γ θ gt 0 Equation (1023primeprimeprime) is the typical expression of the Phillips curvefound in the literature It indicates that deviations in the unemployment rate belowits natural level must be accompanied by deviations in the actual rate of inflationabove that expected It reflects the ldquonatural rate of unemployment hypothesisrdquo asoriginally coined by Friedman (1968 11)
There is always a temporary trade-off between inflation and unemploymentthere is no permanent trade-off The temporary trade-off comes not frominflation per se but from unanticipated inflation which generally means froma rising rate of inflation
Recall that as Barro and Gordon (1983 592) observed the term Etminus1(πt) in(1023primeprimeprime) is the
prior expectation of inflation for period t [which is] distinguished from theexpectation that is conditional on partial information about current pricesThis distinction arises in models (eg Lucas 1972 1973 Barro 1976) inwhich people operate in localized markets with incomplete information aboutcontemporaneous nominal aggregates In this setting the Phillips curve slopecoefficient (γ θ) turns out to depend on the relative variances for generaland market-specific shocks
Variability in prices and the tradeoff
As Lucas (1973 333) states
demand policies [can] tend to move inflation rates and output (relative totrend) in the same direction or alternatively unemployment and inflation inopposite directions The conventional Phillips curve account of this observedco-movement says that the terms of the tradeoff arise from the relatively stable
Lucas model 157
structural features of the economy and are thus independent of the nature ofthe aggregate demand policy pursued The alternative explanation of the sameobserved tradeoff is that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relativeprice changes
Taking Lucasrsquos alternative viewpoint two aspects concerning the tradeoff aresuggested First as Lucas states ldquochanges in average inflation rates will notincrease average outputrdquo As we have seen if we compare the expected pricelevel for period t with the price level for the prior period the difference wouldincorporate individualsrsquo expectation of this average rate of inflation (along with anumber of other potentially relevant variables) Second ldquothe higher the variancein average prices the less lsquofavorablersquo will be the observed tradeoffrdquo We considerthis second point below by referring back to the simple Lucas supply functionwithout lagged output (1016)
Recall that the term ξt given by (105) denotes that part of the price level thatcannot be predicted from past data We have assumed that this ldquosurpriserdquo term ξtis a normally distributed random variable with zero expectation and variance σ 2Substituting this into (1016) we have that
ln yt minus ln yn = γ θξt (1024)
where θ = σ 2z (σ 2 + σ 2
z ) and γ = (1 minus α)(α + β) Recall that θ is the weightattached to the expected price level prior to observing pit
Equation (1024) indicates that deviations in output from the natural level dependsolely on surprises In his statement concerning the variance of prices Lucas ispointing out that the impact of ldquosurprisesrdquo on output relative to its natural leveldepends on the ldquosloperdquo term λθ which is given by
λθ = 1 minus α
α + β
σ 2z
σ 2 + σ 2z
As Sargent (1987a 444) notes
a ldquofavorablerdquo tradeoff between output and unexpected inflation (that is a largevalue of γ θ ) will exist only when σ 2 is small relative to σ 2
z An attempt byauthorities to exploit the tradeoff between output and unexpected inflationmore fully by changing aggregate demand regimes might increase the vari-ance σ 2 relative to σ 2
z and thus change the slope γ θ This is yet anotherexample of how agentsrsquo optimal decision rules change in response to changesin the random processes governing the exogenous variables they base theirdecisions on
Sargentrsquos last point is another example of the ldquoLucas critiquerdquo in this contextwith respect to the validity of using past econometric estimates of a tradeoff inpredicting future tradeoffs
158 Lucas model
Note that although the tradeoff worsens with higher variability in prices theeffect of higher variability in prices on the variance of output about the natural rateis unclear In particular from (1016) we know that the variance in the differencebetween output and the natural rate is simply (γ θ)2σ 2 Given our definition ofγ θ the variance of the logarithm of output becomes
Var(ln yt) = γ 2(
σ 2z
σ 2 + σ 2z
)2
σ 2
Differentiating with respect to σ 2 we have
partVar(ln yt)
partσ 2 = γ 2(
σ 2z
σ 2 + σ 2z
)2
minus 2γ 2σ 2 σ 2z
(σ 2 + σ 2z )
σ 2z
(σ 2 + σ 2z )2
=(
γ 2σ 2z
σ 2 + σ 2z
)2 (1 minus 2σ 2
σ 2 + σ 2z
)
As the above expression indicates by itself an increase in the variation in theaverage price level (σ 2) will increase the variation in the logarithm of output for agiven ldquosloperdquo (γ θ) On the other hand as Lucas pointed out such an increase inthe variation in price level will result in a reduction in the effect of any given pricechange on output which by itself would decrease the variation in the logarithm ofoutput
Substituting (105) into the expanded Lucas supply function with lagged outputwe have the Lucas supply function of the form
ln yt minus ln yn = γ θξt + λ(ln ytminus1 minus ln yn)
where as before θ = σ 2z (σ 2 + σ 2
z ) Substituting for prior differences in outputfrom its natural level we thus have
ln yt minus ln yn = γ θ
infinsumi=0
λiξtminusi (1025)
which shows that the deviation of output from its natural rate depends on thecurrent and all previous values of the ldquoaggregate demand shockrdquo that affects theequilibrium price level
A complete model except for specifying thesource of expectations
Equation (1016) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To simplify the analysis we will normalize outputso that the natural level of real output is equal to 1 We have already assumed that
Lucas model 159
the natural level of real output is constant over time These two assumptions allowus to write (1016) in the more compact form
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (1026)
where Yt denotes log of real output supply for period t (or equivalently the deviationin output from its natural level for period t) Pt denotes the log of the price leveland Etminus1Pt denotes the expectation of log of the price level What we now requireis a characterization of the aggregate demand side of the economy as typicallysummarized by the LM and IS equations
To obtain an explicit form for the portfolio or LM equation we start by assuminga real money demand function for the end of period t of the form
Ldt = yα1
t exp[minus(α2rt)] (1027)
where yt is real output α1 gt 0 and α2 gt 0 indicating that real money demand isdirectly related to real output but inversely related to the nominal interest rate21
Let Mt denote the nominal money supply at the end of period t (previously thishas been denoted by M ) and let mt denote the logarithm of this money supplyfor period t such that mt = ln Mt Further let us assume a logarithmic supply ofmoney function of the form
mt = mt + εt (1028)
Equation (1028) separates the logarithm of the money supply into two compo-nents a deterministic component mt set by government authorities according toa rule tying money supply changes to past variables and a random component εt which is assumed to be normally distributed with zero mean This random term isalso assumed to be serially independent (ie E(εtεs) = 0 for s = t)
The LM equation is simply the money market equilibrium condition equatingthe real supply of money to real money demand and thus is given by
Mtpt = Ldt (1029)
Taking logs of the equilibrium condition (1029) and substituting the logarithm ofthe money demand function (1027) and the money supply function (1028) wehave
mt minus Pt = α1Yt minus α2rt minus εt (1030)
where Pt = ln pt Equation (1030) is the standard log-linear form of the portfolioor LM equation22
To obtain an explicit form for the IS equation which is the equilibrium conditionin terms of equating output production to the demand for output we must postulatea specific form for output demand One common assumption is to include the
160 Lucas model
expected real rate of interest rt minus πet+1 as a determinant of output demand To do
so let
πet+1 equiv (pe
t+1 minus pt)pt = (pet+1pt) minus 1
As before we can approximate the expected rate of change in prices by the expecta-tion of the log of the ratio of the future to current price level (ie πe
t+1 asymp Pet+1minusPt
where Pet+1 is the expected log of the price level for period t+1) Thus the expected
real rate of interest becomes rt minus πet+1
Letting the term Xt denote a vector of exogenous variables that also affectsoutput demand we have in log-linear form the following equilibrium conditionfor the output market
Yt = Xt minus β1(rt minus πet+1) + ut (1031)
where ut is a serially independent stationary random process with mean zeroand finite variance equal to σ 2
u The random terms for output demand and moneysupply εt and ut respectively are assumed independent (ie E(εtut) = 0)
Summarizing we have a model consisting of the aggregate supply equation(1026) the LM equation (1030) and the IS equation (1031) which can be solvedfor the equilibrium output price and interest rate
In particular combining the LM and IS equations to eliminate the interest ratert we obtain
Yt = Xt + ut minus (β1α2) middot (minusmt minus εt + Pt minus α1Yt) + β1 middot πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
] (1032)
Or in terms of the price level we have an ldquoaggregate demand equationrdquo of theform
Pt = mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1) (1033)
Equations (1032) and (1033) indicate the inverse relationship between the pricelevel and output that is shown graphically by a downward-sloping aggregatedemand curve
With respect to (1033) note that the expected rate of inflation for the nextperiod πe
t+1 is viewed as a distinct entity Our prior assumption of unit elasticexpectations concerning the expected log of the future price level Pe
t+1 wouldimply that this term is in fact independent of changes in the current price levelNote also that if output were unchanged then an x percent change in the moneysupply would result in an x percent change in the price level This is the standardresult of the ldquoneoclassicalrdquo model23
Lucas model 161
Combining the above aggregate demand equation (1032) with the aggregatesupply equation to eliminate the output term Yt we have24
α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
]
= γ θ(Pt minus Etminus1Pt) + λYtminus1
Solving the above for the equilibrium price level one obtains
Pt = 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1)
+ J0
(Etminus1Pt minus λYtminus1
γ θ
)] (1034)
where J0 = γ θ(α2 + β1α1) Expression (1034) is sometimes called the reduced-form equation for the price level
Rearranging (1034) we have
Pt minus J0
J0 + β1Etminus1Pt
= 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1) minus J0
λYtminus1
γ θ
] (1034prime)
Let us assume perfect foresight meaning that Etminus1Pt = Pt Noting that1 minus J0(J0 + β1) = β1(J0 + β1) we can solve (1034prime) for the equilibriumprice level under this hypothesis of perfect foresight obtaining
Pt = mt + εtα2
β1(Xt + ut + β1π
et+1) minus J0
λ
β1γ θYtminus1 (1034primeprime)
As equation (1034primeprime) makes clear under the presumption of limited perfectforesight the predictions are those of the neoclassical model
(a) a change in the money supply (in log form given by mt + εt) results in anequiproportionate change in the price level (in log form given by Pt)
(b) an increase in expected inflation πet+1 raises the price level
(c) a higher level of lagged output (Ytminus1) lowers the price level(d) an increase in output demand (Xt + ut) raises prices
Following a procedure similar to that used to derive (1034) if we combine theaggregate demand equation (1033) and aggregate supply equation to eliminatethe price level we have
Yt = λYtminus1 minus γ θEtminus1Pt
+ γ θ
mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1)
162 Lucas model
Solving for the equilibrium real output we have
Yt = β1
J0 + β1
[λYtminus1 + γ θ
[mt + εt minus Etminus1Pt + α2
β1(Xt + ut + β1π
et+1)
]]
(1035)
The above expression is sometimes call the reduced-form equation for real outputAccording to (1035) changes in the money supply (in logs given by mt = mt +εt)will affect real output to the extent that the impact of such changes on prices is notfully anticipated
Note that with perfect foresight we have Etminus1Pt = Pt In this case substituting(1034primeprime) into (1035) for Etminus1Pt we obtain
Yt = β1
J0 + β1
(λYtminus1 + J0
λ
β1Ytminus1
)= λYtminus1 (1035primeprime)
Thus we have the standard neoclassical result that output in the current period isindependent of demand-side changes such as changes in expected inflation or themoney supply
One source of expectations autoregressive expectations
As Shiller (1978) has noted
one of the most difficult problems which confronts builders of macroeco-nomic models is the need to model the mechanism by which the public formsits expectations of future economic variables Many of the most importanttheoretical macroeconomic behavioral relations (eg the supply equationinvestment saving) depend critically on public expectations of future eco-nomic variables yet we often do not even have any data on what theseexpectations are
This and the next section suggest two approaches that have been taken to modelexpectations in particular the expected price level that enters into the aggre-gate supply equations These two approaches to expectation formation are thedistributed lag (or adaptive) scheme and the rational expectations scheme
To understand the ideas behind distributed lag schemes as the source of expec-tations we start by noting that the price level pt can be broken down into acombination of the price level for the previous period ptminus1 multiplied by theratio of the price level this period to last period
pt equiv ptminus1(ptptminus1)
Taking logs and recalling that ln(ptptminus1) is approximately equal to the rate ofinflation πt we thus have
Pt equiv ln pt = ln ptminus1 + πt
Lucas model 163
Taking expectations at time t assuming that at a minimum the price level for theprior period is known we have
Etminus1Pt equiv ln ptminus1 + Etminus1πt (1036)
Until the 1970s the approach to modeling the source of the expected rate of infla-tion embedded in (1036) was to assume individuals forecast the rate of inflationby looking at past inflation rates A common quantitative representation of thishypothesis originated by Fisher (1930) was to have individualsrsquo expectation ofthe inflation rate behave like a weighted average or ldquodistributed lagrdquo of recent pastinflation rates That is
Etminus1πt =qsum
i=1
ηiπtminusi (1037)
where the ηi are fixed numbers A typical idea behind this distributed lag approachto anticipated inflation was that individuals have ldquoadaptive expectationsrdquo whichmeant that individuals adjusted or ldquoadaptedrdquo their expectations of the rate of infla-tion in light of the actual forecast error made concerning the prior periodrsquos inflationrate Specifically adaptive expectations can be expressed as
Etminus1πt = Etminus2πtminus1 + δ(πtminus1 minus Etminus2πtminus1)
= δπtminus1 + (1 minus δ)Etminus1πtminus1(1038)
where 1 gt δ gt 0 Successive substitution allows us to rewrite (1038) as
Etminus1πt = δπtminus1 + (1 minus δ)δπtminus2 + (1 minus δ)2δπtminus3 + middot middot middotor
Etminus1πt =infinsum
i=1
δ(1 minus δ)iminus1πtminusi (1039)
As you can see (1039) is simply a specific form of equation (1037) in whichthe ηi place declining weight on past inflation rates the more distant they are andq = infin Given declining weights we can obtain a reasonable approximation of(1039) even if we truncate the distributed lag on past inflation after q periods aslong as q is reasonably large andor δ is reasonably large
Now let us place the above discussion not in terms of past rates of inflation butinstead in terms of past price levels Recalling the approximation
πtminusi asymp ln ptminusi minus ln ptminusiminus1
we can rewrite (1037) in the form
Etminus1πt =qsum
i=1
ηi(ln ptminusi minus ln ptminusiminus1) (1040)
164 Lucas model
Writing this out we have
Etminus1πt = η1(ln ptminus1 minus ln ptminus2) + η2(ln ptminus2 minus ln ptminus3)
+ η3(ln ptminus3 minus ln ptminus4) + middot middot middot + ηq(ln ptminusq minus ln ptminusqminus1)
Thus we may rewrite (1040) as
Etminus1πt = η1 ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
(1041)
Combining (1041) with equation (1036) we obtain the following expression forthe expectation formed at time t concerning the log of the price level
Etminus1Pt = (1 + η1) ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
or
Etminus1Pt =q+1sumi=1
vi ln ptminusi (1042)
Equation (1042) is what Sargent and Wallace (1975) refer to as ldquoautoregressiveexpectationsrdquo
A second source of expectations rational expectations
The papers by Lucas (1972 1973) and Sargent and Wallace (1975) suggestedthat in macroeconomic model building a different approach to specifying thesource of expectation is preferred As Cukierman (1986) summarizes this ldquorationalexpectationsrdquo approach to the modeling of inflationary expectations is
based on the maintained hypothesis that individuals know the structure of theeconomy and of governmentrsquos decision rule and that they use this structurein conjunction with the available information in order to form an optimalpredictor of future inflation [this approach] requires a precise specificationof the model of the economy as well as of the information sets of individualsEmpirical tests of this hypothesis are therefore joint tests of the validity of theexpectational hypothesis as well as of the postulated structure of the economyand of the particular assumptions made about the information possessed byindividuals
A ldquostructure of the economyrdquo was derived based on the Lucas aggregate sup-ply equation that included the assumption of market-clearing wages and pricesSuppose that individuals know this model and accept it as reflecting the structureof the economy As we saw above this model was solved to obtain (1034) the
Lucas model 165
reduced form for the equilibrium price level in period t We assume that at time tindividuals form their expectations Etminus1Pt ldquorationallyrdquo in that
Etminus1Pt = E(Pt |tminus1) (1043)
indicating that Etminus1Pt is the mathematical expectation of Pt conditional on theinformation set tminus1 which is all information available at period t minus1 As Sargentnotes (1987a 440)
Lucas assumed that tminus1 included information on all lagged values of ln pitand lagged values of real output in all markets One could equally well con-ceive of less comprehensive definitions of tminus1 For now along with Lucaswe suppose that tminus1 includes a comprehensive list of variables includinglagged outputs and prices in all markets
At the end of period t minus 1 individuals of course know Etminus1Pt as well as Ytminus1and πe
t+1 Thus taking the expectation of (1034) and subtracting it from Pt wehave
Pt minus Etminus1Pt = β1
J0 + β1(mt + εt minus Etminus1mt) + α2
J0 + β1(Xt minus Etminus1Xt + ut)
(1044)
In Sargent and Wallace (1975 244) the deterministic part of the money supplymt is assumed to reflect a ldquolinear feedback rulerdquo of the form
mt = Gθlowastt (1045)
where ldquoθlowastt represents the set of current and past values of all of the endogenous and
exogenous variables in the system as of the end of period t minus 1 and G is a vectorof parameters conformable to θlowast
t rdquo A simple example of a monetary feedback rulewould be
mt = a0 + a1Ytminus1 (1046)
where a0 and a1 are positive constantsLet us assume that individualsrsquo information set tminus1 includes not only the
structure of the economy (as summarized by the above linear macroeconomicmodel) but also the money supply rule In the particular example of (1046) theyknow a0 a1 and Ytminus1 and thus mt Then the assumption of rational expectationsimplies that Etminus1mt = mt Furthermore since Xt represents the deterministicpart of the vector of exogenous variables affecting output demand we have thatEtminus1Xt = Xt Thus (1044) becomes
Pt minus Etminus1Pt = 1
J0 + β1(β1εt + α2ut) (1047)
166 Lucas model
Substituting (1047) into the aggregate supply equation one obtains
Yt = γ θ1
J0 + β1(β1εt + α2ut) + λYtminus1 (1048)
An important feature of (1048) is that a deviation in output from its natural levelwhich is represented by the term Yt different from zero given our assumption thatthe natural output level is normalized to equal one is determined only by pastdeviations and ldquosurprisesrdquo with respect to the money supply (the term εt) andoutput demand (the term ut) The ldquodeterministicrdquo or predictable component of anymoney supply change has no real effects Equation (1048) should look familiarGiven λ lt 1 it suggests that the times series for deviations of the logarithmof output from its natural rate is a stationary autoregressive process of order 1or AR(1)
The above analysis is an example of a ldquolinear rational expectations modelrdquoThe result that ldquopredictablerdquo monetary policy has no real effects reflects the twinassumptions of the natural rate hypothesis and rational expectations It should notbe surprising that deterministic monetary policy has no effect for the model ishomogeneous of degree 0 in mt Pt and Etminus1Pt This is a critical characteristic ofa ldquonatural rate modelrdquo
Conclusion
The main focus of this chapter has been on the development of the Lucas supplyfunction The model is often discussed in the context of the ldquoislandrdquo paradigmin which we specify the supply function for a particular sector in the economyThe role of forecasting errors is introduced and from that the Lucas aggregatesupply function is constructed and the relationship of this function to the Phillipscurve is discussed A number of other issues were discussed involving variabilityin prices and the corresponding economic tradeoff and then a complete modelwas introduced except for specifying the source of expectations Two sources ofexpectations were then described autoregressive expectations and rational expec-tations The implications of expectations were then discussed in their historicalcontext
11 Policy
Introduction
This chapter extends the earlier discussions about the actions of the monetaryauthority and how these actions affect the macroeconomy Perhaps the most inter-esting issue of monetary economics is addressed here that is the optimal roleof monetary policy The chapter highlights the differences in model results thatdepend on what type of expectations are assumed Particular attention is givento the Sargent and Wallace ldquoineffectiveness propositionsrdquo and the Phillips curveOther issues are then introduced including the ldquorule versus discretionrdquo debatetime inconsistency and the role of credibility and enforcement
Optimal monetary policy
In the 1970s the articles by Sargent and Wallace (1975 1976) and Lucas (19721973) altered the view of how one should assess the impact of monetary policy onthe economy and by implication what is optimal monetary policy The discussionstarts with the premise that monetary policy should be conducted according to arule or set of rules As Sargent and Wallace (1976 169) state
It is widely agreed that monetary policy should obey a rule that is a scheduleexpressing the setting of the monetary authorityrsquos instrument (eg the moneysupply) as a function of all the information it has received up through thecurrent moment Such a rule has the happy characteristic that in any givenset of circumstances the optimal setting for policy is unique If by remotechance the same circumstances should prevail at two different dates theappropriate settings for monetary policy would be identical1
The SargentndashWallace premise that monetary rules are preferred leads them toexplore the form of the optimal rule But as we will discover in going over thepaper by Barro and Gordon (1983) there is a question of whether monetary rulescan be enforced over time If not then what is typically left in these models isa ldquosecond bestrdquo solution involving the determination of optimal ldquodiscretionaryrdquo
168 Policy
policy Note that we use the term ldquosecond bestrdquo because enforceable rules tend todominate discretion in these models2
Accepting the premise that monetary policy can adopt enforceable rules stillleaves open the specification of the optimal set of rules The simplest rule sug-gested by Friedman (1959) would increase the money supply at a constant rateeach year perhaps 3 percent3 More complex rules known as ldquoreactive rulesrdquowould specify in advance how the growth of the money supply will change basedon new information on the state of the economy One such rule suggested is thatthe growth in the monetary base and thus the money supply automatically adjustwhenever the growth of nominal GNP deviates from its trend (McCallum 1985)Another reactive rule suggests that the government commit itself to holding theCPI to a preannounced target and adjust the monetary base and thus the moneysupply accordingly (Hall 1982)4
Below we begin our discussion of optimal monetary policy by reviewing theanalysis of optimal enforceable rules as suggested by the Sargent and Wallace(1975 1976) papers and reviewed in Sargent (1987a Chapter 17) This discussionis in the context of a natural rate model without rational expectations and then inthe context of a model which assumes rational expectations
Optimal monetary policy exogenous expectations
As we saw previously the reduced form for the log of the output in period t canbe expressed as
Yt = H0
[λYtminus1 + γ θ
[minusEtminus1Pt + mt + εt + α2
β1(Xt + ut + β1π
et+1)
]]
(111)
where H0 = β1(J0 +β1) and J0 = γ θ(α2 +β1α1) Recall that Yt is the differencein period t between the logarithm of output and the logarithm of the natural levelof output Normalizing so that the natural level of output equals one we canequivalently interpret Yt in equation (111) as total output As Sargent and Wallace(1976) state ldquoYt can be thought of as the unemployment rate or the deviation ofreal GNP from lsquopotentialrsquo GNP This equation should be thought of as the reducedform of a simple econometric modelrdquo
Recall that the log of the money supply in period t mt is the sum of adeterministic component mt and the random component εt with variance σ 2
e 5
To understand the impact of monetary changes on real GNP we must firstconsider how the expected log of the price level Etminus1Pt varies with changes inmonetary policy One approach in the spirit of ldquoautoregressive expectationsrdquo isto assume that the expected price level is independent of the current monetarypolicy This essentially means viewing the expectation of the log of the price level(Etminus1Pt) as exogenous6 Given this assumption we may rewrite the reduced-form
Policy 169
equation for output (111) as
Yt = a0 + a1Ytminus1 + a2mt + vt (112)
where vt = H0γ θ(α2β1)ut is a serially independent normally distributed randomvariable with variance σ 2
v and mean zero mt is the log of the money supply forperiod t where mt = mt + εt Ytminus1 is lagged output and a0 a1 and a2 areparameters
Suppose that the monetary authority desires to set the money supply in orderto minimize the fluctuation in the log of output around some desired level Let usassume that the log of this desired level denoted Y lowast is above the log of the naturallevel of output Yn7 Then the objective can be expressed as
min Etminus1(Yt minus Y lowast)2
We can break this expression into two terms in that the objective can beequivalently expressed as
min Etminus1(Yt minus Etminus1Yt)2 + (Etminus1Yt minus Y lowast)2 (113)
The second way of expressing the objective allows us to see the objective asminimizing the sum of two terms the variance of Yt conditional on informationup to the end of period t minus 1 and the ldquobias squaredrdquo around Y lowast The secondterm the bias squared around Y lowast is the reason for an ldquoactivistrdquo monetary policyEquation (113) indicates that the optimal monetary policy entails
1 minimizing the variance in the random component of the money supply Thisfollows since the first term in equation (113) the variance of Yt is given bya2
2σ2e +σ 2
v 8 If feasible complete elimination of the random component to themoney supply (ie a purely ldquodeterministicrdquo money supply) is optimal suchthat εt = 0 for all t and thus σ 2
e = 02 setting Etminus1Yt = Y lowast so as to make the second term in equation (113) equal
to zero From equation (112) this means a monetary policy such that
Etminus1(a0 + a1 middot Ytminus1 + a2 middot mt + vt) = Y lowast
Noting that Etminus1Yt = 0 and that Etminus1mt = mt we see that this deterministic partof the optimal monetary policy is defined by the equation
a0 + a1Ytminus1 + a2mt = Y lowast
which can be solved for the optimal deterministic monetary policy rule
mt = g0 minus g1Ytminus1 (114)
where g0 = (Y lowast minus a0)a2 and g1 = a1a2
170 Policy
An equivalent expression for the optimal monetary rule (114) is derived inSargent (1987a Chapter 17) under the presumption of exogenous expectations Inparticular Sargent uses equation (112) to substitute out for Ytminus1 in (114) so thatthe optimal (deterministic) monetary rule (114) becomes
mt = g0 minus g1[a0 + a1Ytminus2 + a2mtminus1 + vtminus1] (115)
Now substituting into (115) the expression for Ytminus2 suggested by equation (112)we have
mt = g0 minus g1[a0 + a2mtminus1 + vtminus1] minus g1a1[a0 + a1Ytminus3 + a2mtminus2 + vtminus2](116)
Continuing to successively substitute for Ytminusi i = 3 4 we have Sargentrsquosequivalent expression for the optimal (deterministic) policy rule as given by9
mt = g0 minus g1a0 minus⎛⎝g1a2
infinsumiminus1
aiminus11 mtminusi
⎞⎠ minus
(g1
infinsumi=1
aiminus11 + vtminusi
) (117)
Following the above optimal monetary policy (ie reducing any random compo-nent to the money supply to its minimum level and establishing the rule for thedeterministic component of the money supply as specified by (114) or (117)) wehave by construction that
Etminus1Yt = Y lowast and Yt = Y lowast + a2εt + vt
where vt = H0γ θ(α2β1)ut and the variance of the random component of themoney supply εt is set at its lowest feasible level Thus optimal monetary policyin essence sets output each period equal to Y lowast plus irreducible noise As Sargentand Wallace (1976 171) note
the application of the rule eliminates all serial correlation in output since thisis the way to minimize the variance in output The basic idea is that wherethe effects of shocks to a goal variable (like GNP) display a stable pattern ofpersistence (serial correlation) and hence are predictable the authority canimprove the behavior of the goal variable by inducing offsetting movementsin its instruments
Note that without the lag term for output g1 in (114) equals zero
Adaptive expectations and the accelerationist result
The well-known ldquoaccelerationist outcomerdquo concerning the path of inflation isimplied by the above analysis if expectations are adaptive and if the aim of mone-tary policy is to keep output above its natural level To see this let us go back to the
Policy 171
aggregate supply equation that underlies the reduced form for output To simplifylet us abstract from the stochastic elements in demand εt and ut (as well as anysupply-side disturbances) and also from adjustment costs (ie omit the laggedoutput term) Then the aggregate supply equation
Yt = γ θ [πt minus Etminus1πt] (118)
reflects the actual path that output will take10
To derive the accelerationist result assume that Y lowast gt Yn By normalizationYn = 0 so that to have Etminus1Yt = Y lowast gt Yn we must have Etminus1Yt gt 0Equation (118) suggests that to achieve an expected level of output greater thanits natural level the government must pursue a monetary policy that results inthe actual rate of inflation typically being above that expected In particular thedesire to keep output above its natural level means that monetary policy in period tresults in the actual inflation rate πt such that
πt minus Etminus1πt = Y lowastγ θ gt 0
If expectations are adaptive then we have that
Etminus1πt minus Etminus2πtminus1 = δ(πtminus1 minus Etminus2πtminus1)
The assumption of adaptive expectations coupled with Y lowast gt Yn = 0 thus impliesthat
Etπt+1 = Etminus1πt + δ(πt minus Etminus1πt) = Etminus1πt + δY lowastγ θ
In words the fact that individuals underestimate inflation this period (by theamount Y lowastγ θ ) leads them to adjust (ldquoadaptrdquo) their expectations of inflationupward (by the amount δY lowastγ θ ) The result is that to keep expected output atthe level Y lowast gt Yn next period means an increase in the inflation rate by theamount δY lowastγ θ each period As Blanchard and Fischer (1989 572) note
this is the famous accelerationist result derived by Friedman (1968) andPhelps (1968) using their Phillips curve together with the adaptive expec-tations assumption The explanation is simple if the government is trying tokeep output above the natural rate it has to produce inflation at a higher ratethan expected each period Since the expected inflation is a weighted averageof past inflation rates the actual rate must be increasing
Now let us assume instead that Y lowast = 0 To provide a role for an activist monetarypolicy let us reintroduce the lagged output term λYtminus1 into the right-hand sideof (118) Thus monetary policy can eliminate the effect of past deviations inoutput from the natural rate on current output In other words a policy aimed
172 Policy
at setting Etminus1Yt = Y lowast would imply altering inflation relative to expected onlywhen lagged output deviated from the natural level Rather than the prior result ofan ever-increasing inflation with Y lowast gt Yn with Y lowast = Yn we have that inflationsimply wanders11 For instance if logged lagged output Ytminus1 fell below the log ofthe natural level of output Yn = 0 then the difference Ytminus1 would be negativeOther things being equal this would imply a lower output in the current periodTo counteract this the government would pursue a monetary policy that results inthe actual rate of inflation being above that expected
The above discussion helps us understand the comment of Hall (1976) that
the benefits of inflation derive from the use of expansionary power to trickeconomic agents into behaving in socially preferable ways even thoughtheir behavior is not in their own interests The gap between actual andexpected inflation measures the extent of the trickery the optimal pol-icy is not nearly as expansionary when expectations adjust rapidly andmost of the effect of an inflationary policy is dissipated in costly anticipatedinflation
The above extract raises the following question Can the monetary authoritiessystematically trick the public in order to exploit the link between inflation andoutput For Sargent and Wallace and others the answer is no due to the existenceof rational expectations
Rational expectations and the SargentndashWallace ineffectivenessproposition
As Sargent (1987a) notes a critical aspect of the simple example of an optimalmonetary rule as given by (114) (or equivalently (117)) ldquois the implicit assumptionthat agentrsquos decision rules remain unchanged in the face of alternative stochas-tic processes for the control variable that different feedback rules implyrdquo WhatSargent means in this context is that the optimal monetary rule has been derivedunder the presumption that private agents do not take this rule into account informing their expectation of the price level Under this assumption one couldestimate the parameters of the reduced-form equation output (a0 a1 and a2 in(112)) independently of the feedback rule (114)
However Sargent and Wallace (1976) criticize this view In particular theyargue that ldquoin the reduced forms are embedded the responses of expectations tothe way policy is formed Changes in the way policy is made then ought not toleave the parameters of estimated reduced forms unchangedrdquo12 In other wordsrational individuals would clearly seek out and use information on how monetaryauthorities act as well as on the structure of the economy in forming expectationsof prices
Let us now consider the following version of the reduced-form equation foroutput (112) that explicitly includes the potential role of expected monetary policy
Policy 173
when individuals form expectations on prices13
Yt = a0 + a1Ytminus1 + a2(mt minus Etminus1) + vt (119)
For a given anticipated log of the money supply Etminus1mt we have as before theoptimal (deterministic) monetary rule of the form
mt = g0 minus g1Ytminus1 (1110)
so that
mt = g0 minus g1Ytminus1 + εt (1111)
where εt is the irreducible random element in the money supply determinationprocess Now assume that the public knows the monetary authoritiesrsquo feedbackrule Then our assumption of rational expectation (ie individuals use all availableinformation in forming expectations) implies that
Etminus1(mt) = g0 minus g1Ytminus1 (1112)
Combining (119) (1111) and (1112) the reduced form for output is now given by
Yt = a0 + a1Ytminus1 + a2εt + vt (1113)
so that the biased squared term in the objective of the monetary authorities(Etminus1Yt minus Y lowast)2 equals (a0 + a1Ytminus1 minus Y lowast)2
As is clear from (1113) there is no role for systematic monetary policy to affectreal output As Sargent (1987a 459) notes ldquothe bias squared is independent ofthe parameters of the money supply rulerdquo The optimal policy is then to makemonetary policy deterministic if feasible for then the variance of output (given bya2
2σ2e + σ 2
v ) is minimized by setting σ 2e = 0 Until we add an inflation objective
any deterministic rule will be equally as good for none will have any impact onoutput This is once again an example of the neutrality of money
As Sargent (1987a 458) notes ldquopolicy rules should be deterministic and involveno surprisesrdquo He goes on to argue that we
have therefore established the following stochastic neutrality theorem thatcharacterizes our model one deterministic feedback rule on the basis of theinformation set tminus1 which is common to the public and to the authority is asgood as any other deterministic feedback rule Via deterministic feedbackrules the monetary authority is powerless to combat the business cycle (theserial correlation in Yt)
Naturally if one abandons rational expectations or the natural rate hypothesis thenthis ldquostochastic neutralityrdquo result need not hold
174 Policy
The SargentndashWallace ineffectiveness proposition in thecontext of the Phillips curve
The above finding of the ldquoneutrality of moneyrdquo in the context of a stochasticlinear natural rate model with rational expectations is viewed by Sargent (1987a459) as
the antithesis of our earlier result rationalizing the activist Keynesian policyrules The reader is invited to verify that the truth of the neutrality theoremis not dependent on the particular information set assumed It will continueto hold for any specification of tminus1 so long as the public and the authorityshare the same information set
He concludes
The preceding results provide a [weak] defense for following rules with-out feedback Simple x-percent growth rules do as well as any deterministicfeedback rule and dominate rules with a stochastic component
Below we recast the SargentndashWallace ineffectiveness proposition in terms ofthe expectational Phillips curve This makes the discussion more in line with thenext sectionrsquos review of some implications of non-enforceable monetary rules Inaddition we add to the government goalrsquos an inflation objective In particular wemodify our analysis in the following four ways
1 we alter the objective to be in terms of unemployment rather than output2 we expand our objective function to include inflation3 we link inflation to unemployment via a modified Lucas supply equation4 we incorporate rational expectations
Our first task is to convert the objective of the government into unemploymentterms Before we assumed that the government simply sought to minimize thefluctuations in output about a particular level In particular if we let Zt denote thecost incurred in period t we assumed the objective was to
min Etminus1Zt where Zt equiv (Yt minus Y lowast)2 (1114)
We have previously assumed that the deviation of unemployment from its naturalrate is linearly related to the deviation of the log of output from the log of its naturallevel In particular we assumed
minus(Ut minus Un) = Yt
Policy 175
given the normalization of the natural level of output such that Yn equiv ln yn = 0Substituting the above into (1114) the problem facing the government policy-maker becomes
min Etminus1Zt where Zt = a(Ut minus kUn)2 a = 2 gt 0
k = 1 minus Y lowastUn (1115)
Note in (1115) that we assume k lt 1 which is equivalent to assuming thatthe log of optimal level of output Y lowast is greater than the log of the natural levelof output which by normalization has been set equal to zero As Blanchard andFischer (1989 596ndash597) suggest
The most plausible justification [for k lt 1] is the presence of distortions orimperfections that causes the natural rate of unemployment to be too high Thisjustification allows the loss function to be consistent with the single-periodutility function of private agents Another is that the governmentrsquos objectivefunction as shaped by the electoral process leads the government to seek toraise output above the natural level
Having converted the objective function into unemployment terms the next stepis to expand the objective function to include an inflation goal In particular let usassume that the cost in period t includes a term reflecting differences between theactual inflation rate πt and an optimal rate of inflation πlowast Assuming a simplequadratic form we have14
Zt = a(Ut minus kUn)2 + b(πt minus πlowast)2 (1116)
The problem of the government policy-maker is then
min Etminus1Zt where Zt equiv a(Ut minus kUn)2 + b(πt minus πlowast)2
As before we can decompose this objective to obtain the following equivalentexpression for the object of the government policy-maker
min a[Etminus1(Ut minus kUn)2 + (Etminus1Ut minus kUn)
2]+ b[Etminus1(πt minus πlowast)2 + (Etπt minus πlowast)2] (1117)
Our third task is to link inflation to unemployment Recall that if we ignore thelagged term with respect to output in the Lucas supply equation assume unem-ployment is linearly related to real output and approximate inflation by the log ofthe ratio of the price level this period to the price level last period then we canmanipulate the Lucas supply equation to obtain the expression
Ut = Un minus α(πt minus Etminus1πt) (1118)
176 Policy
where α = (γ θ) gt 015 Substituting (1118) into (1116) the governmentrsquosobjective becomes
min Etminus1a(1 minus k)Un minus α(πt minus Etminus1πt)2 + b middot (πt minus πlowast)2
Our fourth and final task is to introduce rational expectations As we saw earlierwe obtain the reduced form for the deviation of the price level from that expectedin period t
Pt minus Etminus1Pt = 1
J0 + β0(β1εt + α2ut) (1119)
if (a) the government follows a particular rule in determining monetary policy(b) that rule is known to the public and (c) there exist rational expectations
An algebraic manipulation ndash simultaneously subtracting and adding the log ofthe price level for period t minus 1 to the left-hand side of equation (1119) ndash bringsus closer to having an expression that may be interpreted in terms of inflation
ln( ptptminus1) minus Etminus1(ln( ptptminus1) = 1
J0 + β1(β1εt + α2ut) (1120)
Using the log of the price ratio as an approximation for inflation we thus canapproximate (1120) as
πt minus Etminus1πt = 1
J0 + β1(β1εt + α2ut) (1121)
Substituting the above which reflects the rational expectations approach tomodeling expectations into the new government objective we have
min Etminus1
a
[(1 minus k)Un minus α
J0 + β1(β1εt + α2ut)
]2
+ b(πt minus πlowast)2
or
min a[(1 minus k)Un]2 +(
α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1 b(πt minus πlowast)2 (1122)
since Etminus1εt = Etminus1ut = Etminus1εtut = 0It is clear from (1122) that with rational expectations monetary policy can play
no role in helping the government meet its objective concerning unemploymentIn other words in the context of the Lucas model the assumption of rationalexpectations means that the expected loss from deviations in unemployment fromits desired level and thus production from its desired level is independent ofthe deterministic monetary rule As a consequence the minimization problem asgiven by the first half of equation (1117) becomes simply one of specifying any
Policy 177
deterministic monetary policy rule and eliminating any random changes in themoney supply (ie setting σ 2
e = 0 if feasible)But the objective of the government now also includes an objective concerning
the rate of inflation so we need an expression for the equilibrium rate of changein prices πt that incorporates rational expectations To do so we start with thereduced form for the log of the price level obtained previously which is given by
Pt = 1
J0 + β1
[β1mt + εt + α2(Xt + ut + β1π
et+1) + J0
(Etminus1Pt minus λYtminus1
γ θ
)]
(1123)
Taking the difference between the reduced form for the log of the price level forperiod t and for period tminus1 we can obtain an expression for πt the rate of inflationbetween period t and t minus 1 of the form
πt = 1
J0 + β1[β1mt + α2(Xt minus Xtminus1 + ut minus utminus1) + α2β1(π
et+1 minus πw
t )
+ J0(Etminus1Pt minus Etminus2Ptminus1) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1124)
where πmt approximates the rate of change in the money supply (ie πmt =mt minus mtminus1 = ln(MtMtminus1)) Assuming rational expectations we have fromequation (1119) that
Etminus1Pt = Pt minus 1
J0 + β1(β1εt + α2ut)
and similarly
Etminus2Ptminus1 = Ptminus1 minus 1
J0 + β1(β1εtminus1 + α2utminus1)
so that
Etminus1Pt minus Etminus2Ptminus1 = πt + 1
J0 + β1[β1(εtminus1 minus εt) + α2(utminus1 minus ut)]
(1125)
Substituting this expression into equation (1124) one obtains
πtβ1
J0 + β1= 1
J0 + β1[β1πmt + α2(Xt minus Xtminus1 + ut minus utminus1)
+ α2β1 middot (πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1126)
178 Policy
where we use the fact that 1 minus J0(J0 + β1) = β1(J0 + β1) Solving for πt we have
πt = πmt + α2
β1(Xt minus Xtminus1 + ut minus utminus1)
+ α2(πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)
(1127)
which given πmt = mt minus mtminus1 = mt + εt minus (mtminus1 + εtminus1) can be rearranged toobtain
πt = πmt + εt minus εtminus1 + α2
β1(Xt minus Xtminus1 + ut minus utminus1) + α2(π
et+1 minus πe
t )
minus J0λ
β1γ θ(Ytminus1 minus Ytminus2) (1128)
where πmt = mt minus mtminus1If we assume no change in exogenous (nonrandom) demand factors this period
compared to the last period (ie Xt = Xtminus1) no change in the expected future infla-tion between last period and this period (ie πe
t+1 = πet ) and ignore adjustment
costs (ie λ = 0) we can simplify equation (1128) to obtain
πt = πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) (1129)
The three assumptions we made to derive equation (1129) from equation (1128)largely limit any differences between period t and t minus 1 to differences in the sizeof the money supply In fact the two periods differ only by the deterministiccomponent of the money supply and by random factors where these randomfactors ndash the term ut minus utminus1 with respect to output demand and the term εt minus εtminus1with respect to the irreducible random component of the money supply ndash havemean zero In other words all potential changes in aggregate demand or productionexcept money supply changes that would lead to different expected price levels inthe two periods have been removed
Substituting equation (1129) into (1122) we thus obtain the following completegovernment objective function under rational expectations
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1b[πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) minus πlowast]2
]
from which we see that constant monetary growth will not achieve a constant rateof inflation unless we neglect the lagged disturbance terms16 To obtain the resultof an optimal monetary rule in the form of a constant rate of growth in the moneysupply we must further simplify and neglect the lagged disturbance terms If we
Policy 179
for the moment ignore the lagged stochastic terms εtminus1 and utminus1 we have theobjective
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ b(πmt minus πlowast)2 + σ 2e +
(α2
β1
)2
σ 2u
]
As before to minimize the loss requires that one reduces the random variation inthe money supply to zero (if feasible) so that σ 2
e = 0 That is the optimal monetaryrule is ldquodeterministicrdquo Further given an inflation objective and no reason otherthan monetary policy for prices to change the obvious optimal rule is simply toset the determinant rate of change in the money supply equal to the desired rate ofinflation That is the optimal policy is
πmt = πlowast
Note that this rule assumes no shocks to the economy If there were a steady rateof increase in output (and thus the real demand for output) then that would raisethe optimal constant rate of change in the money supply to achieve a given rate ofchange in prices Thus Friedmanrsquos ldquo3 percentrdquo rule for monetary growth presumeda 3 percent growth in real output so as to be consistent with a zero rate of inflation
Rules versus discretion monetary policy and timeinconsistency
The SargentndashWallace ineffectiveness proposition has been used by many to supportarguments for the government not adopting activist policy rules to offset fluctua-tions ndash particularly downturns ndash in an economyrsquos real output reflecting demand-sidedisturbances The reason as we have seen is that such policies have no real effectsonce one assumes rational expectations although the attempt can lead to higherinflation
Yet as Barro and Gordon (1983) point out
empirical studies indicate the presence of countercyclical monetary policyat least for the post-World War II United States ndash rises in the unemploymentrate appear to generate subsequent expansions in monetary growth Within thenatural rate framework it is difficult to reconcile this countercyclical behaviorwith rationality of the policymaker
As Barro and Gordon go on to say ldquoa principal object of our analysis is to achievethis reconciliationrdquo That is rather than saying what policy rules governmentshould follow Barro and Gordon want to explain why government acts the way itdoes ndash thus the term ldquopositive theoryrdquo in the title of their paper
180 Policy
The BarrondashGordon approach combines a number of topics that we haveconsidered before First they utilize a natural rate model like the Lucas modelSecond they assume rational expectations And third and most interestingly theyprovide us with a nice example of the phenomenon of ldquotime inconsistencyrdquo in thecontext of optimal monetary policy when there exists the potential Phillips curvetradeoff
Contrasting the BarrondashGordon and SargentndashWallace policyenvironments
In the SargentndashWallace view of the optimal monetary policy choice it is assumedthat the policy-maker makes a once-and-for-all decision with respect to the partic-ular monetary policy rule (reactive or not) Under certain assumptions as we haveseen this leads to the natural conclusion that optimal monetary policy entails thesimple rule of a constant rate of growth in the money supply so that the resultingaverage rate of inflation equals the desired level If the desired level were zerothen inflation would be set equal to zero through appropriate monetary policy
Barro and Gordon (1983 598) have questioned this result on the basis that ldquotheremay be no mechanism in place to constrain the policymaker to stick to the rule astime evolvesrdquo The result is that the policy-maker decides each period the optimalmonetary policy to follow In other words ldquothough the objective function anddecision rules of private agents are identicalrdquo Barro and Gordon obtain differentresults from Sargent and Wallace because ldquothe problems differ in the opportunitysets of the policymakerrdquo Below we illustrate the exact nature of this differenceby first showing how Barro and Gordonrsquos setup provides an example of the ldquotimeinconsistency problemrdquo not present in the Sargent and Wallace problem and thenderive the equilibrium for an economy characterized by the policy-makers whoare allowed each period to pick a potentially new optimal monetary policy
Time inconsistency an example in the context of optimalmonetary policy
In an important paper on the ldquotime inconsistencyrdquo problem of optimal policyKydland and Prescott (1977) point out situations in which the optimal policiesdecided at time t would be changed at time t + 117 In the context of monetarypolicy as Kydland and Prescott note
the reason that such policies are suboptimal is not due to myopia The effect of(monetary policy) upon the entire future is taken into consideration Rather thesuboptimality arises because there is no mechanism to induce future policy-makers to take into consideration the effect of their policy via the expectationsmechanism upon current decisions of agents
Let us see what this means by way of a concrete example
Policy 181
A simple example of ldquotime inconsistencyrdquo following Kydland and Prescott isconstructed below To simplify the discussion somewhat for the moment we ignoreuncertainty and restrict our attention to two periods In this setting the govern-ment through monetary policy can determine each period the actual inflation raterather than the expected rate of inflation For periods 0 and 1 the choice variablesfacing the monetary policy-maker are thus π0 the actual inflation in period 0 andπ1 the actual inflation in period 1 The inflation choice impacts the unemploymentrate in that U0 and U1 the unemployment rates in periods 0 and 1 are given bythe Phillips curve formulation (1118)
U0 = Un minus α(π0 minus Eminus1π0)
U1 = Un minus α(π1 minus E0π1)
where the state variables Eminus1π0 and E0π1 denote the expected rate of inflation inperiods 0 and 1 respectively In periods 0 and 1 the objective for periods 0 and 1is to minimize the simple quadratic forms
Z0 = a(U0 minus kUn)2 + b(π0 minus πlowast)2
Z1 = a(U1 minus kUn)2 + b(π1 minus πlowast)2
where the constants a and b are positive It is assumed that k lt 1 to capture theidea that distortions exist in the economy that an activist policy can address18
Substituting in the ldquoPhillips curverdquo relations in period 0 the present value of theobjective function is
Z = Z0 + β middot Z1 = a[(1 minus k)Un minus α(π0 minus Eminus1π0)]2 + b[π0 minus πlowast]2
+ βa[(1 minus k)Un minus α(π1 minus E0π1)]2 + βb[π1 minus πlowast]2
where β is the constant discount factorLet us presume that the expectations of inflation the ldquostaterdquo variables are
exogenous and equal to the desired level each period (ie Eminus1π0 = E0π1 = πlowast)In period 0 the optimal inflation rates for periods 0 and 1 (as determined bymonetary policy) are then
partZpartπ0 = minus2aαq0 + 2b(π0 minus πlowast) = 0
partZpartπ1 = β[minus2aαq1 + 2b(π1 minus πlowast)] = 0
where qi = (1 minus k)Un minus α(πi minus πlowast) i = 0 1 In period 1 the objective is
Z1 = a[(1 minus k)Un minus α(π1 minus E0π1)]2 + b[π0 minus πlowast]2
and the optimal solution given π0 and E0π1 = πlowast is thus
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
182 Policy
Comparing this first-order condition to partZ0partπ1 it is clear that in the case ofexogenous expectations the optimal solution is time-consistent Further it impliesa rate of inflation greater than the expected (optimal) rate of πlowast each period toachieve an unemployment rate below the natural This follows given that one ofthe objectives is to approach a level of unemployment equal to kUn with k lt 1
What happens however if individualsrsquo expectations of inflation for period 1 isa forecast that correctly anticipates the optimal future policy decision That is wehave ldquorational expectationsrdquo such that in a deterministic world
E0π1 = h(0) = π1
where 0 is the information set at the end of period zeroWhat the optimal inflation policy is now depends on whether policy-makers
can ldquocommitrdquo to future policy actions Let us start by assuming they can placeconstraints on future policy We know that in period 0 the optimal solution for π1is then given by
βpartZ1
partπ1+ partZ
partE0π1
dh(middot)dπ1
= β[2b(π1 minus πlowast)] = 0
Since dh(middot)partπ1 = 1 the optimal planned (at time 0) rate of inflation for period 1is equal to the desired level πlowast This is the SargentndashWallace ineffectivenessproposition
However once period 1 occurs E0π1 is set (let us assume it equals πlowast) If thepolicy-maker was not then constrained by the prior specification of a monetarypolicy to achieve a rate of inflation equal to πlowast they would choose π1 such that
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
which implies π1 gt πlowast The ldquotime inconsistencyrdquo arises as you can see becauseparth(middot)partπ1 = 0 As Barro and Gordon state ldquothe term lsquotime inconsistencyrsquo refersto the policymakerrsquos incentives to deviate from the rule when private agents expectit to be followedrdquo
Equilibrium when monetary rules are not enforceable
As Barro and Gordon note in the time inconsistency problem ldquoconstraints onfuture policy actions are infeasible by assumptionrdquo In contrast in the SargentndashWallace view
rules are enforceable so that the policymaker can commit the course of futurepolicy (and thus of expectations) In the former case the time-inconsistentsolution is not an equilibrium given the problem facing the policymaker Inthe latter case the incentives to deviate from the rule are irrelevant sincecommitments are assumed to be binding
(Barro and Gordon 1983 599)
Policy 183
As the above extract suggests in the ldquotime inconsistency caserdquo we have not yetcharacterized a rational expectationsrsquo equilibrium since in our example individu-alsrsquo expectations of inflation for period 1 were incorrect According to Barro andGordon there are three features of an equilibrium The first is ldquoa decision rulefor private agents which determines their actions as a function of their currentinformationrdquo These actions of private agents based on current information aresummarized by the Phillips curve
Ut = Un + ζt minus α middot (πt minus Etminus1πt) (1130)
where ζt a random variable with zero mean and variance σ 2z has been added
to denote a real shock that affects the natural unemployment rate for the currentperiod only19
The second feature of an equilibrium is ldquoa policy rule which specifies the behav-ior of policy instruments as a function of the policymakerrsquos current informationsetrdquo This policy rule is given by the choice of inflation rates πt+i i = 0 1 2 by the monetary authorities with the following objective
min Et
⎡⎣ infinsum
i=0
β iZt+i|It
⎤⎦
where Zt+i equiv a(Ut+i minus kUn)2 + b(πt+i minus πlowast)2 a gt 0 b gt 0 and 0 le k le 1
The term It denotes the initial state of information and 1 gt β gt 0 is the constantdiscount factor (β = 1(1+rreal) where rreal is the exogenous real rate of interest)For simplicity we will assume that the optimal level of inflation is zero so thatπlowast = 0
The third feature is an expectations function which determines the expectationsof private agents as a function of their current information Assuming that ldquothepublic understands the nature of the policymakerrsquos optimization problem in eachperiodrdquo then Etminus1πt = πt where πt is the optimal choice of inflation by thepolicy-maker for period t
Combining the information contained in our discussion of the first and secondfeatures of equilibrium the policy-makerrsquos optimal choice of πt minimizes
EtZt = Et[a(Ut minus kUn)2 + b(πt)
2]= Et[a((1 minus k)Un minus α(πt minus Etminus1πt))
2 + b(πt)2]
Given that Eξt = 0 the expression to be minimized by the policy-maker can bewritten as
EtZt = [a((1 minus k)Un minus α(πt minus Etminus1πt))2 + b(πt)
2]A critical point to note is that each period the policy-maker inherits Etminus1πt andtakes that expected rate of inflation as given in the above optimal choice of πt
184 Policy
The first-order condition for period t is thus
partEtZtpartπt = 2a((1 minus k)Un minus α(πt minus Etminus1πt))(minusα) + 2b(πt) = 0
which can be simplified and rearranged to obtain the following expression for theoptimal inflation rate
πt = aα
b[(1 minus k)Un minus α(πt minus Etminus1πt)] (1131)
From the third feature of equilibrium we know that the public realizes the problemfaced by the policy-maker in terms of choosing the optimal rate of inflation forperiod t as defined by (1131) and thus Etminus1πt = πt so that the second term dropsout and we have
πt = aα
b(1 minus k)Un = Etminus1πt gt πlowast = 0 (1132)
as long as k lt 1 We thus have that expectations are rational and individualsoptimize subject to these expectations Since Etminus1πt = πt we have that EtUt = UnThus ldquothe equilibrium solution delivers the same unemployment rate and a higherrate of inflation at each daterdquo than is the case in the SargentndashWallace problem wherethere is a ldquorules-type equilibriumrdquo Given the optimal rate of inflation πlowast = 0a rules-type equilibrium with rational expectations would have the actual rate ofinflation equal to zero
As Barro and Gordon (1983 608) conclude
under a discretionary regime the policymaker performs optimally subject toan assumed inability to commit future actions The framework assumes ratio-nality within the given institutional mode Excessive inflation apparentlyunrewarding countercyclical policy responses and reactions of monetarygrowth and inflation to other exogenous influences can be viewed as prod-ucts of rational calculation under a regime where long-term commitments areprecluded
The model stresses the importance of monetary institutions which deter-mine the underlying rules of the game A purely discretionary environmentcontrasts with regimes such as a gold standard or paper-money constitutionin which monetary growth and inflation are determined via choices amongalternative rules (the SargentWallace approach) The rule of law or equivalentcommitments about future governmental behavior are important for inflationjust as they are for other areas that are influenced by possibly shifting publicpolicies
An alternative to the ldquorule of lawrdquo is reputation As Blanchard and Fischer(1989 599) state
reputation is the most interesting and persuasive explanation of how gov-ernments avoid dynamic inconsistency Governments know that they can do
Policy 185
better than the shortsighted solution over the long run They hope by actingconsistently over long periods to build a reputation that will cause the privatesector to believe their announcements The key to the answer [to the ques-tion of whether reputation can sustain the optimal policy] is the specificationof private sector expectations of how the public reacts to broken promises
Conclusion
This chapter has presented an overview of many of the major themes and issuesfaced in macroeconomics in terms of the conduct of monetary policy The roles ofexpectations the ldquoineffectiveness propositionrdquo the modified Phillips curve rulesversus discretion time inconsistency credibility and enforcement are all dealt within this chapter Perhaps the major contribution of this chapter is that it highlightsmany of the issues and problems that real-world monetary authorities face whendeciding what course of action to take
12 Open economy
Introduction
The field of international economics can be roughly categorized as concernedwith either the real side or the finance side of international issues The ldquorealrdquoside focuses on such basic questions as why trade occurs between countries whatdetermines the terms of trade and how government policies such as tariffs do orquotas affect trade The ldquofinancerdquo side makes explicit the fact that countries differin currencies in order to focus on such questions as what determines exchangerates and how macroeconomic shocks in one country (eg a change in the supplyof money) affect its economy and the economies of the countries with which ittrades
In the discussion below we examine questions more like those considered by theldquofinancerdquo side Namely we extend simple macroeconomic analysis to an ldquoopenrdquoeconomy that is an economy that incorporates a foreign sector This analysisdiffers from traditional macroeconomic analysis of a ldquoclosedrdquo economy in thefollowing respects
1 There is trade of composite commodities and financial assets between twocountries For the moment we assume not only that the two countries producedifferentiated output but also that the financial assets issued by the firms andgovernment of one country are not perfect substitutes for the financial assetsissued by firms and government of the second country
2 The two economies are isolated in that individuals in each country can onlypurchase or sell labor services in their own labor markets
3 The two economies are differentiated in that each has its own media ofexchange This means that an individual in the domestic economy whoseeks to purchase foreign goods must exchange domestic money for foreignmoney1 Similarly an individual in the foreign country must exchange hisforeign money for the domestic money to purchase the domestic goods Suchexchanges take place in the foreign exchange market at the prevailing ldquoforeignexchange raterdquo or the price of one currency in terms of the second currencyWe will let et denote the exchange rate for domestic money For instanceif the domestic country is the USA and the foreign country is Japan then
Open economy 187
et on December 1 1989 was 1434 yen in that 1434 yen = 1 dollar Thisimplies that 1et the price of dollars in terms of yen was 0006973 dollarson December 1 1989
4 There is distinct government policy (fiscal and monetary) in each country inthat each countryrsquos government determines spending taxation and monetarypolicy for its economy
To understand how macroeconomic analysis is altered in the above ldquoopen econ-omyrdquo setting it is instructive to first consider the nature of the constraints faced bythe various participants in an open economy We then examine how the behaviorof these participants is affected by changes in such variables as exchange ratesWith that background we are ready to consider some simple examples of openmacroeconomic analysis such as the effect of a change in the money supply in thecontext of an open economy neoclassical model
Open economy participants constraints and Walrasrsquo law
To begin our task of modeling an open economy recall that it is typical ofmacroeconomic models to simplify the economy by grouping markets into broadcategories In a closed economy there were three important markets the outputfinancial and labor markets One could add to this the money ldquomarketrdquo sinceequilibrium required that the demand for money equaled supply In an open econ-omy we add another market the foreign exchange market where the currencyof one country is traded for that of the other country In a closed economy theparticipants in the various markets in the economy could be placed in one of fivecategories households firms government (fiscal side) the central bank and pri-vate depository institutions In an open economy we add one more participantforeigners
As we have seen a common theme of macroeconomic models is their emphasison the interdependencies among markets Macroeconomics recognizes that eventsin one market imply changes in other markets as well This ldquogeneral equilibriumrdquoapproach contrasts with the ldquopartial equilibriumrdquo approach of microeconomicswhich is less concerned with how changes in one market affect all other marketsTo fully understand the links across markets it is useful to specify the ldquofinancingconstraintsrdquo faced by the participants in the various markets Our discussion ofopen economy macroeconomics thus begins by introducing these financing con-straints for firms households government (fiscal side) the central bank privatedepository institutions and foreigners We then sum these constraints to obtain amodified Walrasrsquo law
The financing constraints in an open economy
We start our discussion of the constraints faced by the participants in an openeconomy by considering the financing constraint faced by the new participant for-eigners Foreigners purchase domestic output and financial assets Let X d
t denote
188 Open economy
foreignersrsquo demand for domestic output (exports) and let net Adft denote foreignersrsquo
desired real change in their holdings of domestic financial assetsTo purchase domestic output and financial assets foreigners must first acquire
domestic money That is foreigners must finance these purchases either fromincome generated from their previously acquired holdings of domestic financialassets or by supplying foreign currency in exchange for the domestic currencyin the foreign exchange markets In particular we have the following foreignerfinancing constraint
X dt + net Ad
ft minus αf (zBf pt + dt) minus FCst etpt = 0 (121)
Note that for simplicity we limit foreignersrsquo holdings of domestic financial assetsto private financial assets (ie we do not have them holding government bonds)Further we assume foreignersrsquo portfolio of holdings of domestic financial assetsis identical to2 that of domestic households and that they own αf 1 gt αf ge 0 ofthe total value of financial assets issued by domestic firms Thus αf (zBf pt + dt)
is the real income foreigners gain from their holdings of domestic financial assets3
In equation (121) the term FCst etpt denotes foreignersrsquo real supply of foreign
currency in the foreign exchange market FCst is foreignersrsquo supply in units of the
foreign currency in period t Multiplying by 1et the price of the foreign currencyin terms of the domestic currency puts this in terms of the domestic currencyDividing by the price level pt then puts it in real terms (ie in terms of the domesticcomposite commodity) Embedded in the desired change in foreignersrsquo holdingsof domestic financial assets are changes in the desired holdings by foreign centralbanks (ie changes in foreign central banks ldquointernational reservesrdquo)
In addition to the above new constraint that accompanies the introduction ofa new participant to the economy foreigners we have constraints faced by thedomestic households the central bank private depository institutions government(fiscal side) and firms In the case of households and the central bank we modifythe constraints introduced in previous chapters to incorporate the exchanges ofcommodities and financial assets with foreigners In particular for householdsthe budget constraint in an open economy can be expressed by
bdt + cd
t + zdt + (M d
t minus M )pt + net Adht + net AFd
ht minus [yt minus αf (zBf pt + dt)
+ α(zBff pft + pftdft)etpt minus δK minus Tnt] = 0 (122)
where the new term zdt denotes real imports net Ad
ht denotes householdsrsquo desiredchange in their real holdings of foreign assets and α(zBff pft + pftdft) denotes theincome (in terms of the foreign currency) gained from holding the proportion aof the financial assets issued by foreign firms4 This income (in foreign currency)times the price of foreign currency in terms of domestic currency (1et) gives thedomestic currency value of income from foreign asset holdings Dividing by theprice level pt puts the income in terms of the composite commodity (ie in realterms)
Open economy 189
Total consumption of commodities in period t is now the sum of cdt purchase
of output and zdt imports of commodities produced abroad5 Similarly the total
desired change in financial assets is the sum of net Adht the desired change in hold-
ings of domestic financial assets and net AFdht the desired change in holdings of
foreign financial assets Note that in incorporating the firm distribution constraintinto the household budget constraint we have taken into account the fact that notall domestic output yt is income to domestic households since foreigners own theshare αf of domestic firms On the other hand households have an additionalsource of income from the ownership of foreign financial assets
For the central bank the (stock) financing constraint equates the sum of thechange in real (domestic) financial asset holdings and international assets to thereal change in the monetary base or
net Adct + net AFd
ct minus (MBst minus MB)pt = 0 (123)
where net Adct = pbt(Bd
gct minus Bgc)pt MB = R + C and MBst = Rs
t + Cst 6 We
have added the term net AFdct to denote the real demand for additional international
(foreign currency denominated) assets by the central bank In particular
net AFdct = pfbt(B
dfct minus Bfc)etpt
The quantity pfbt(Bdfct minusBfc) is the change in the amount of foreign assets demanded
by the central bank in terms of the foreign currency pfbt denotes the price of foreignbonds in terms of foreign currency and the numbers of such bonds demanded andinitially held by the domestic central bank are denoted respectively by Bd
fct and
Bfc Multiplying this quantity by the price of foreign currency in domestic currencyterms (1et) gives the domestic currency value of foreign assets demanded by thecentral bank Then dividing by the price level pt puts the net demand for interna-tional assets by the central bank net AFd
ct in terms of the composite commodity(ie in real terms)
For private depository institutions the (stock) financing constraint indicates thatprivate depository institutions can be viewed as financing additions to reserves andto financial assets holdings by creating deposits or
net Atpb + (Rd
t minus R)pt minus (Dst minus D)pt = 0 (124)
where net reserves demanded or initially held are denoted by Rdt and R respec-
tively and checkable deposits supplied or initially outstanding are denoted byDs
t and D respectively For simplicity we assume that all private demands forforeign financial assets as well as for foreign commodities are captured in thehousehold budget constraint To the extent that private banks purchase foreignfinancial assets they can be viewed as acting as financial intermediaries forhouseholds
190 Open economy
For the domestic government (fiscal side) the financing constraint is
gdct minus T lowast
nt minus net Asgt = 0 (125)
where T lowastnt = Tt minustrt minusz(Bgh+Bgp)pt tr denotes transfer payments and net As
gt =Pbt(Bs
gt minusBg)pt Equation (125) incorporates the ldquoflowrdquo financing constraint forthe central bank In doing so it is assumed that the central bank claims on realresources just exhaust its interest payments on government debt holdings plus anyincome associated with central bank holdings of foreign assets Finally for firmsthe financing constraint on capital purchases is given by
Idnt + ψ(I d
nt) minus net Asft = 0 (126)
Note that we assume for simplicity that neither firms nor the government purchaseforeign commodities
Walrasrsquo law and the balance of payments
We now sum the constraints faced by the six participants in an open economy(foreigners households the central bank private banks the government andfirms) as given by equations (121)ndash(126) In doing so assume equilibrium withrespect to the demand and supply of bank reserves (ie the Rs
t part of MBst in the
central bank constraint equals Rdt in the private banks constraint) and note that
the initial monetary base MB equals R + C that the money supply is defined byM s
t = Dst + Cs
t and that the initial money supply is given by M = D + C Weobtain
[Bdt + Cd
t + X dt + gd
ct + δK + I dnt + ψ(I d
nt) minus yt] + [M dt pt minus Mpt]
+ [net Adht + net Ad
ft + net Adct + net Ad
pt minus net Asgt minus net As
ft]+ [zdlowast
t + net AFdht + net AFd
ct minus FCst etpt] = 0 (127)
where zdlowastt + net AFd
ht + net AFdct minus FCs
t etpt denotes householdsrsquo imports notfinanced out of income generated from foreign asset holdings Equation (127) isan example of the modified Walrasrsquo law for an open economy7 As we discussbelow (127) can be viewed as stating that the sum of excess demands in fourmarkets must equal zero
The first term in (127) reflects excess demand in the output market where thedemand now includes foreignersrsquo demand for domestic output (exports) Note thatby adding and subtracting import demand we could express the excess demandfor output in the form
bdt + cdlowast
t + (xdt minus zd
t ) + gdct + δK + I d
nt + ψ(I dnt) minus yt (128)
where the term cdlowastt denotes householdsrsquo total consumption in terms of both domes-
tic and foreign output (ie cdlowastt = cd
t + zdt ) Excess demand in the output market
Open economy 191
is often expressed this way with the consumption term denoting total householdconsumption such that the ldquonetrdquo export demand term (xd
t minus zdt ) appears
Setting the excess demand for output term in (127) to zero gives us the ISequation in an open economy The second term in (127) reflects the excess demandfor money Note that we assume that only domestic households desire to hold thedomestic money (foreigners seek the domestic money in the foreign exchangemarket not to hold but as a means to purchase the domestic output or financialassets) Setting this second term in (127) to zero gives us the standard LM equation
The third term in (127) reflects excess demand in the financial market In goingto an open economy we add to the demand side of the financial market a demandfor domestic financial assets by foreigners (which could include private foreignagents as well as foreign central banks) In addition note that householdsrsquo demandfor domestic financial assets (net Ad
ht) no longer reflects householdsrsquo total demandfor financial assets since they now have the option of purchasing foreign financialassets (net AFd
t )The fourth and final term in (127) reflects excess real demand for foreign
currency in the foreign exchange market8 As before we can view output ( yt)or the price of output ( pt) as determined in the output market and the domesticinterest rate (rt) as determined in the financial market Now we can view the foreignexchange rate (et) as determined in the foreign exchange market The demand forforeign currency reflects zdlowast
t the demand associated with householdsrsquo purchases offoreign commodities that could not be financed from the foreign currency earningsof their holdings of foreign financial assets net AFd
ht the demand for foreigncurrency associated with householdsrsquo purchases of additional foreign financialassets and net AFd
ct the demand associated with the central bankrsquos desired changein international reserves Note that the real demand for the foreign currency couldinstead be stated as the real supply of the domestic currency in the foreign exchangemarket
FCst etpt denotes the real supply of foreign currency or real demand for the
domestic currency in the foreign exchange market which from (121) reflectsforeignersrsquo purchases of financial assets and exports net of those financed throughdomestic currency earnings on financial assets held by foreigners Foreignersrsquopurchases of financial assets include both foreign private (ie foreign householdsrsquo)and foreign public (ie foreign central bank) purchases
The balance of payments accounts
Let us assume for now that the domestic economy discussed above is the USA TheUS Department of Commerce actually measures the various sources of the demandfor and supply of dollars in the foreign exchange markets cited above These data ofinternational transactions are presented as the US balance of payments accountsTable 121 summarizes its major components Transactions are categorized aseither sources of the real demand for or the real supply of dollars in the foreignexchange market for the US dollar9 Equivalently they reflect the real supply ofor real demand for foreign currency in the foreign exchange market
192 Open economy
Table 121 The US balance of payments accounts (in ldquorealrdquo terms)
Real demand for dollars Real supply of dollars
1 US exports of goods and services xt US imports of goods and services zt2 Transfers (interest and dividends to US
holders of foreign financial assetsgovernment grants and gifts)
Transfers (interest and dividends toforeign holders of US financial assetsgovernment grants and gifts)
α(zBff pft + pftdft)etpt αf (zBf pt + dt)
3 Foreign purchases of US financialassets (capital inflow) net Ad
ft
US purchases of foreign financial assets(capital outflow) net AFd
h + net AFdct
The net demand for dollars associated with the first component of the balanceof payments accounts (exports minus imports) is called the balance of trade Ifthe balance of trade is negative as it has been recently for the USA then the USAis said to experience a balance of trade deficit If it is positive as was the case for106 consecutive years from the end of the Civil War to 1971 as well as throughmost of the 1970s then a balance of trade surplus is said to exist
The third component of the demand for and supply of dollars in the balance ofpayments accounts is the capital account This capital account can be divided intoa private part and a public part On the demand side the private part measures thereal dollars demanded by foreigners other than foreign central banks to financepurchases of US financial assets On the supply side the private part measuresthe real dollars supplied by US households to buy foreign financial assets Thesecurrency exchanges are called private international capital flows Private inter-national capital flows associated with the demand for dollars are referred to asUS private international capital inflows since they reflect the inflow of foreigncurrency due to private foreignersrsquo purchases of US financial assets Private inter-national capital flows associated with the supply of dollars are referred to as USprivate international capital outflows since they reflect the outflow of dollars dueto US householdsrsquo purchases of foreign financial assets
Summing the net demand (demand minus supply) for dollars associated withthe first two components plus the private international capital flows and adjustingfor measurement errors (the discrepancy term) we obtain what is called the USbalance of payments10 In years in which the balance of payments is negative it isreferred to as a balance of payments deficit On the other hand a positive balanceof payments is called a balance of payments surplus
When there is a surplus or deficit in the balance of payments accounts thenequality between the real demand for and real supply of dollars is brought about byan offsetting deficit or surplus on what is known as ldquothe official reserve transactionbalancerdquo The official reserve transaction balance reflects the intervention into theforeign exchange market by the US central bank (the Fed) andor by foreign centralbanks This is the ldquopublicrdquo part of international capital flows Whenever there existsa balance of payments deficit in the USA then (on net) central banks demand USdollars in the foreign exchange markets If this were solely the US central bank
Open economy 193
intervening this means that net AFdct wasis negative (the Fed wasis a net supplier
of dollars in the foreign exchange market) and the US central bank would havelostlose international reserves
Behavior in an open economy
With an understanding of the constraints faced by the various participants in anopen economy we now consider the behavior of these participants in particularthe determinants of imports (zd
t ) exports (xdt ) private international capital inflows
(a part of net Adft) and private international capital outflows (net AFd
ht) In the firstpart of this section we examine how a change in the foreign exchange rate for thedomestic currency (to be concrete the US dollar) can alter the relative price offoreign goods and thus lead to a change in the division of household consumptionbetween purchases of domestic goods and purchases of foreign goods11 We alsoexamine other factors that influence US imports of goods and services and thusthe real demand for foreign currency (real supply of US dollars) in the foreignexchange markets
In the second part of this section we consider the factors that influence howhouseholds divide their real accumulation of financial assets between US stocksand bonds and the financial assets of foreign countries As we saw above house-hold purchases of foreign financial assets constitute capital outflows since inorder to make these purchases households must demand foreign currency (supplydollars) in the foreign exchange market We will see how differences in foreignand domestic interest rates and the expected appreciation or depreciation of theUS dollar determine the relative returns on foreign and domestic financial assetsThese relative returns in turn affect householdsrsquo portfolio choices between foreignand domestic financial assets and the real demand for foreign currency (real supplyof dollars) in the foreign exchange market
In the third and fourth parts of this section we turn to the other side of theforeign exchange market to examine determinants of foreignersrsquo purchases of USgoods and services and of US financial assets In particular we consider how suchfactors as exchange rates affect foreignersrsquo demand for US goods (US exports)and how such factors as relative interest rates and the expected rate of changein the exchange rate affect foreignersrsquo demand for US financial assets We finishthis section by illustrating graphically the behavior discussed in terms of the realdemand for and supply of the domestic currency (the US dollar)
Householdsrsquo demand for imports
When deciding whether to purchase foreign or domestic goods households look attheir relative prices To be concrete consider two countries The domestic countryis the USA and the foreign country is Japan The relative price of Japanese goods isthen the real quantity of US goods that must be sacrificed to purchase the foreigngood For example if the price of a Japanese car is $6000 and the price of aUS computer is $1500 then the relative price of a Japanese car in terms of US
194 Open economy
computers is 4 computers If the relative price of Japanese cars rises the USAwill import fewer Japanese cars The relative price of foreign goods sometimesreferred to as the terms of trade is an important determinant of the quantity ofimports The relative prices of foreign goods depend on the dollar prices of USgoods the prices of foreign goods in their own currency and foreign exchangerates (the price of one currency in terms of a second currency) The simple examplethat follows illustrates this point Suppose that the price of a US computer is $1500and that in Japan the price of a Japanese car is 600000 yen The third ldquopricerdquo thatwe need to know in order to compute the relative price of a Japanese car in terms ofUS computers is the foreign exchange rate in particular the price of a yen in termsof dollars Suppose that it takes et yen to buy one dollar in the foreign exchangemarkets Then it takes 1et dollars to buy one yen Returning to our example ofJapanese cars and US computers if et = 100 yen per dollar and the Japanese carhad a yen price of 600000 then its dollar price would be 6000
In general the calculation of the relative price of Japanese goods is thus
Relative price of Japanese goods (in terms of US goods)
= Yen price of Japanese goods ( pft)
times Price of yen in terms of dollars (1et)Dollar price of US goods ( pt)
= pft
etpt
According to this expression a rise in the yen price of Japanese goods ( pft)
raises their relative price Similarly a fall in the dollar price of US commodities( pt) raises the relative price of Japanese goods Finally a rise in the price of yen interms of dollars (1et) also increases the relative price of Japanese goods12 Sinceall three changes mean a higher relative price of Japanese goods and thus anincrease in the cost to US buyers of Japanese goods in terms of US goods forgoneall three changes reduce US imports of Japanese goods The above relative priceexpression for a basket of foreign goods is sometimes termed the real exchangerate for foreign goods ndash that is
Real exchange rates (foreign goods)
= Dollar price of foreign goods
Dollar price of US goods= pft(1et)
pt= pft
ptet
While a rise in the relative price of foreign goods will lead to a reduction in thequantity of foreign goods bought we have to be careful not to infer from this thatUS import demand will necessarily be inversely related to the relative or real priceof imports The reason for this is that our measure of US real import demand is interms of the US good and services not in terms of the foreign good An examplewill highlight this distinction
Open economy 195
Suppose that the dollar depreciates (et falls) With the implied appreciation offoreign currency (1et rises) the price of the foreign good in terms of US goodsbecomes greater as our expression for the real exchange rate for foreign goods( pftpt et) indicates With a higher price fewer foreign goods will be purchased ndashthis is clear This by itself would suggest a fall in the value of imports into theUSA But the higher price also means that each foreign good purchased will costmore in terms of US goods that must be sacrificed This by itself would suggest arise in the value of US imports (measured in terms of US goods that must be paidto obtain the imports) The net impact of a change in the relative price of foreigngoods on US imports measured in terms of US goods depends on which of thesetwo effects is stronger
It is typically assumed that over time the effect of a change in the relative pricesof imports on the quantity of imports purchased dominates so that the value ofimports in terms of US goods will fall with a rise in the relative price of importsThis is what we will assume13 Formally this condition requires that the priceelasticity of demand for foreign goods be greater than one That is a 1 percentincrease in the price of foreign goods must cause a greater than 1 percent reductionin the amount of foreign goods that US households demand14
Besides the relative prices of imports real disposable income affects US importdemand An increase in disposable income can lead to a rise in household con-sumption demand In an open economy with foreign trade a rise in consumptiondemand means an increase in purchases not only of domestically produced goodsbut also of foreign goods Thus householdsrsquo import demand is directly related totheir disposable income To summarize household real import demand zd
t dependsinversely on the relative price of foreign goods pftetpt and directly on disposableincome yt minus δK minus Tnt
zdt = zd
t ( pftetpt yt minus δK minus Tnt ) (129)
Capital outflows householdsrsquo demand for foreign financial assets
When deciding whether to purchase US or foreign financial assets householdscompare domestic and foreign rates of return The nominal rate of return on USfinancial assets is simply the money interest rate rt The comparable nominal rateof return on foreign financial assets is not so simple to identify To explain howto compute this return which we will denote rlowast
t let us suppose that a householdlends one dollar in the foreign financial market
If the price of a dollar is et units of the foreign currency say yen then in termsof the foreign currency the household lends et yen If foreign financial assets offerthe interest rate rft then one period from now the household will have et(1 + rft)
yen At that time the household can convert these yen holdings back to dollars atthe exchange rate then existing At the time the money is lent this future exchangerate may be uncertain15 Let householdsrsquo expectation of this future exchange ratebe denoted by ee
t+116 Then the household expects to convert its et(1 + rft) yennext year into et(1 + rft)ee
t+1 dollars Subtracting the one dollar with which the
196 Open economy
household started the rate of return to lending in foreign financial markets rlowast isgiven by
rlowastt = (et(1 + rft)ee
t+1) minus 1 = [et(1 + rft) minus eet+1]ee
t+1
We can simplify the above equation by noting that the expected future dollarexchange rate ee
t+1 yen per dollar equals the current exchange rate of et yen times(1 + θe
t+1) where θet+1 is the expected rate of change in the price of a dollar in
terms of yen Substituting the expression et(1+θet+1) for the expected future dollar
exchange rate eet+1 into the above expression for rlowast
t we have
rlowastt = [et(1 + rft) minus et(1 + θe
t+1)]et(1 + θet+1) = (rft minus θe
t+1)(1 + θet+1)
Since the expected rate of appreciation of a dollar is typically small we canapproximate the above expression by
rlowastt = rft minus θe
t+1 (1210)
Equation (1210) has a straightforward interpretation The return to lending in theforeign financial market equals the difference between the foreign interest rateand the expected rate of change in the price of the dollar The return to lendingin foreign financial markets increases with a higher foreign interest rate rft anddecreases with a higher expected rate of increase in the price of the dollar (θe
t+1)The higher the expected rate of increase in the price of the dollar the lower theexpected return to lending in foreign financial markets since for a given numberof dollars sold for foreign currency at the start of the period fewer dollars can bebought back at the end of the period
When households choose between purchasing domestic and foreign financialassets they compare the domestic interest rate rt with the rate of return to lendingin the foreign financial markets rlowast
t We can thus express household real demandfor additional foreign financial assets as
net AFdht = net AFd
ht(rt rft minus θet+1 ) (1211)
In (1211) an increase in the US interest rate or a fall in the rate of return onforeign financial assets implies a reduction in householdsrsquo real demand for foreignfinancial assets The three dots in (1211) reflect other factors that have been leftunspecified For instance changes in the political stability of foreign governmentsare one unspecified factor that would likely impact on US householdsrsquo demandfor foreign financial assets Equation (1211) suggests that we should add anotherfacet to our previous discussion on householdsrsquo demand for US financial assetsnet Ad
ht In addition to such factors as real income taxes the US money interestrate and the expected rate of inflation household demand for US financial assetsdepends on the expected return to lending abroad rft minus θe
t+1
Open economy 197
Foreignersrsquo demand for exports
Just as US demand for foreign goods depends on relative prices so too isforeignersrsquo demand for US goods based on the relative prices of those goodsThe relative prices of US goods to foreigners measure what foreigners have togive up of their own goods in order to purchase US goods As we have seenthese relative prices depend on the money prices of US goods the money pricesof foreign goods and the exchange rate Considering ldquocompositerdquo goods for eachcountry the relative price of US goods to foreigners or the real exchange rate forforeign goods is given by
Real exchange rates (US goods)
= Dollar price of US goods
Dollar price of foreign goods= pt
pft(1et)= ptet
pft
Note that the real exchange rate for US goods is simply the inverse of the previouslyobtained real exchange rate for foreign goods The price of a dollar in terms ofan index of foreign currencies rose by 70 percent in 1984 and 1985 The resultingrise in the relative prices of US goods contributed significantly to a reduction inforeign demand for US goods and the large US trade deficit of the mid-1980sSimilarly the dramatic fall in the price of a dollar in the subsequent period fromlate 1985 to 1988 led to an increase in US exports Thus we have
xdt = xd
t ( ptetpft ) (1212)
Equation (1212) indicates that export demand falls with a rise in the relative priceof US goods to foreigners
We know from our previous discussion that an increase in US disposable incomeleads to a rise in household purchases of both domestically produced output andforeign goods and services By the same token an increase in foreignersrsquo dispos-able income leads to a rise in their purchases of US goods Thus among the itemsmissing in (1212) that determine foreignersrsquo real export demand xd
t is foreigndisposable income
Capital inflows foreignersrsquo demand for financial assets
In 1960 purchases of US financial assets by foreigners were approximately one-half the amount of purchases of foreign financial assets by US citizens In the USfinancial markets foreign purchases of new US financial assets were less than5 percent of household and depository institution purchases Twenty-five yearslater foreigners were purchasing four times as many US financial assets than theUSA was purchasing abroad In the US financial market close to 30 percent ofnew US financial assets were being purchased by foreigners This dramatic changein capital inflows to the USA is one indication of the growing importance to theUS economy of international trade not only in goods but also in financial assets
198 Open economy
As with US households we will assume that foreigners decide to purchase eitherUS assets or financial assets of their own country by comparing the rates of returnon the two types of financial assets For foreigners the nominal rate of return ondomestic assets is the money interest rate in their own country (rft) The expectedrate of return to foreigners on US financial assets equals the US interest rate plusthe expected change in the price of the dollar in the foreign exchange market (iert + θe
t+1)Not surprisingly the expected return to foreigners lending in US financial mar-
kets increases when the US interest rate rt increases Not as obvious is that thereturn also increases with an increase in the expected rate of change in the price ofthe dollar θe
t+1 This is because foreigners lending in US financial markets converttheir currency to dollars to make the loans When the loans are repaid they thenconvert dollars back to their own currency If the dollar is anticipated to appreciateduring the course of the year then part of their expected return to lending in theUSA is the increase in the value of the dollars (in terms of their own currency)
Summarizing we can express the foreign demand for US financial assetsnet Ad
ft as
net Adft = net Ad
ft(rft rt θet+1 ) (1213)
Equation (1213) indicates that foreignersrsquo demand for US financial assetsincreases if the US interest rate (rt) rises or the expected rate of change in theprice of the dollar (θe
t+1) increases or the foreign interest rate (rft) falls
The foreign exchange market the real demand and supply of dollars
The change in the price of a dollar (in terms of a second currency) affects the realquantity of dollars supplied and demanded in the foreign exchange market Letus start with the price of a dollar set at the equilibrium level of (et)0 let us say100 yen If the price of a dollar now falls to (et)1 say 50 yen then this depreciationof the dollar (appreciation of the yen) leads to a reduction in the real quantity ofdollars supplied from Q0 to Q1
A fall in the price of a dollar from 100 to 50 yen means a rise in the dollar priceof a yen from 001 to 002 dollars or from 1 cent to 2 cents Even though there isno increase in the yen price of Japanese goods the dollar price of Japanese goodsrises For example a 600000 yen Japanese car that formerly cost 6000 dollars(600000 times 001) now costs 12000 dollars (600000 times 002) If the dollar prices ofUS goods have not changed then the relative or real prices of Japanese cars haverisen In our example this means that US households must give up an increasedamount of US goods to obtain one more Japanese car
The depreciation of the dollar and resulting higher relative price for Japanesegoods leads US households to reduce the quantity of Japanese goods demandedHowever as we discussed above the fact that the quantity of Japanese goodspurchased falls does not necessarily mean that the quantity of dollars suppliedin the foreign exchange market also falls There are two countervailing forces
Open economy 199
at work here While the purchase of fewer Japanese goods would reduce thequantity of dollars supplied the fact that each Japanese good has a higher pricewould increase the quantity of dollars supplied By assuming that the first effectoutweighs the second effect we conclude that a depreciation of the dollar causesthe real quantity of dollars supplied in the foreign exchange market to fall Thusthere is an upward-sloping supply of dollars curve17
Naturally behind the supply of dollars in the foreign exchange market is notonly US real import demand but also the real demand by households and the UScentral bank for additional foreign financial assets (net AFd
ht + net AFdct) As we
saw above this sum of the US import demand and demand for foreign financialassets can be interpreted as representing not only a supply of dollars but also ademand for foreign currency18
The above discussion also highlights the effect of a change in the price ofa dollar (in terms of a second currency) on the quantity of dollars demandedin the foreign exchange market The depreciation of the dollar (or appreciationof the yen) from (et)0 to (et)1 leads to an increase in the quantity of dol-lars demanded US in real terms The fall in the price increases the quantity of dollarsdemanded because it lowers the relative price of US goods to foreigners A fall inthe price of a dollar from 100 yen to 50 yen causes a rise in the dollar price of theyen from 1 cent to 2 cents Even though there has been no increase in the dollarprice of US goods the yen price of US goods falls and the Japanese increase theirdemand for US goods As a consequence the real quantity of dollars demandedin the foreign exchange market increases
Naturally behind the demand for dollars in the foreign exchange market isnot only foreignersrsquo export demand but also their demand for US financial assets(net AFd
ht) As we have seen this sum of the export demand and foreignersrsquo demandfor US financial assets can be interpreted as representing not only a demand fordollars but also a supply of foreign currency
Simple examples of open economy (static) macroeconomicanalysis
The statement of Walrasrsquo law for an open economy indicates that for static macro-economic analysis of an open economy we need look at only three of the fourexcess demand conditions reflecting the output financial foreign exchange andmoney markets Standard practice is to look at the output money and foreignexchange markets In the neoclassical model these three equations would besolved to obtain the equilibrium price level interest rate and exchange rate ( pt rt and et respectively) In the BarrondashGrossman disequilibrium analysis thesethree equilibrium conditions could be solved to obtain the equilibrium outputinterest rate and exchange rate ( yt rt and et respectively) In the Lucas modelor the Keynesian fixed money wage model these three equilibrium conditionsalong with the appropriate aggregate supply equation could be solved for theequilibrium price level output interest rate and exchange rate ( pt yt rt and et respectively)
200 Open economy
With flexible exchange rates (ie exchange rates determined without theintervention of central banks) the basic change in the macroeconomic analysisis to recognize that the net export component of output demand (seeequation (128)) is sensitive to interest rate changes which implies the IS curve(in (interest rate output) space) is flatter The reason why a higher US interest ratereduces net export demand is that a higher interest rate increases capital inflows(net Ad
ft) and reduces capital outflows (net AFdht) The first change increases the
demand for dollars in the foreign exchange market while the second reduces thesupply of dollars in the foreign exchange market The result is an appreciationof the dollar that reduces exports and increases imports Note that with flexibleexchange rates a change in either the price level or income does not affect netexport demand
Money supply changes in the neoclassical model purchasingpower parity
In general when considering two countries the country with the lower inflationrate will tend to have an exchange rate that is appreciating at a rate approximatelyequal to the difference in inflation rates between the two countries This patternof changes in foreign exchange rates is sometimes said to reflect the purchasingpower parity condition The condition of purchasing power parity means that thepurchasing power of each countryrsquos currency is the same whether the currency isused to purchase domestic goods or foreign goods Purchasing power parity existsfor a monetary shock in the neoclassical model since a money supply change willlead to changes not only in domestic prices but also in foreign exchange rates suchthat relative prices remain constant
Interest-rate parity
There are of course a number of variations to the above analysis For instancewe could assume that domestic and foreign financial assets are perfect substitutesand that the domestic country is sufficiently small in its interactions with foreignfinancial markets that it takes the foreign interest rate rft as a given In this caseof ldquointerest rate parityrdquo since the return to lending at home rt must equal that oflending abroad rlowast
t = rft minus θet+1 we have that
rft = rt + θet+1 (a constant) (1214)
One use of (1214) is in the well-known Dornbusch model (see Dornbusch 1976)Assume that output and the price of output are fixed initially Assume θe
t+1 = 0initially such that rt = rft Now consider an increase in the money supply Tomaintain equilibrium with respect to the demand and supply of money the interestrate rt must decrease According to (1214) the resulting increase in internationalcapital outflows and reduction in international capital inflows must reduce theexchange rate such that it is expected to appreciate at a rate equal to the difference
Open economy 201
between xdt and the new lower rt This is Dornbuschrsquos famous ldquoexchange rate
overshootingrdquo That is the analysis implies a fall in the exchange rate below whatit will ultimately be after income or prices adjust
A second use of (1214) is to combine it with the assumption that exchangerates are fixed (through appropriate central bank intervention in the foreignexchange markets) An example taking this approach is the MundellndashFlemingmodel (Fleming 1962 Mundell 1968)
Conclusion
This chapter has brought together many of the issues associated with analyzinga macroeconomy that participates in the open or global economy In keepingwith the basic model framework of this book the household is now viewed ashaving a demand for imported products as well as domestically produced goodsand services Additionally firms are allowed to sell to agents in foreign coun-tries This flow of goods and services across borders logically leads to a flow offunds Moreover this interconnectedness among trading partners implies that theactions of one country in particular those of the monetary authority may influenceconditions in the other country
Notes
1 Introduction
1 Empirically an economyrsquos total output is measured by real gross domestic product (orindustrial production) employment is estimated from company records or householdsurveys estimates of unemployment are compiled from household surveys or statisticson recipients of unemployment benefits and price indexes are computed to measurechanges in the overall level of prices
2 Leon Walras first derived the result in his book Elements drsquoEconomie Politique Purefirst published in 1874ndash77 (see Walras 1954)
3 An endogenous variable is one that is determined by the model An exogenous variableis one that is taken as given by the model
4 An element that distinguishes both static and dynamic macroeconomic analysis fromArrowndashDebreu analysis is the incorporation of money The exceptions to this in macro-economic analysis are real business cycle theories which for the most part are purelyldquorealrdquo in the sense that money is not an intrinsic part of the analysis
5 Note that an alternative to exogenous expectations is to specify an ldquoexpectation func-tionrdquo If this function relates expectations in certain specific ways to all currentinformation such that the expectation function reflects ldquorational expectationsrdquo thenstatic analysis is converted to dynamic analysis
6 In dynamic models this would be termed ldquoperfect foresightrdquo in a deterministic settingand ldquorational expectationsrdquo in a stochastic setting
7 One could say that the effect of such a variable is implicit in the exogenous level ofexpected future prices of the static analysis
8 A similar example of an incomplete listing of effects is a change in current governmentpolicies A change in current government actions may require changes in governmentactions in subsequent periods that will impact future markets Yet the construction ofstatic analysis does not require such effects to be spelled out
9 Forward markets are markets in which agreements are made that specify the prices atwhich goods will be exchanged in the future Goods are thus in essence indexed by thedate of trade
10 This is the ArrowndashDebreu contingent-claim interpretation of a competition equilibriummodel (see Arrow 1964 Debreu 1959)
11 In some cases one can attain the stationary state under the less restrictive requirementthat certain exogenous variables simply grow at a steady rate over time
12 While classical economists did not fully articulate their model Patinkin (1965) is widelycited as providing a comprehensive review and formalization of many of the key ideasunderlying the classical economistsrsquo views The result may be denoted the prototypeof the neoclassical (static) model Dynamic neoclassical macroeconomic models arebroadly based on neoclassical growth models with the addition of shocks of variouskinds Among the classic works developing neoclassical growth models are Solow(1956) Cass (1965) Koopmans (1965) and Sidrauski (1967a 1967b)
Notes 203
13 Two of the most widely known proponents of new classical economics are Robert Lucasand Thomas Sargent It has been suggested by some however that this line of analysisis less the extension of classical analysis than it is the antithesis of Keynesian analysisas discussed below See Niehans (1987) for this interpretation
14 This is changing as indicated by Howitt (1985) Shapiro and Stiglitz (1984) Weitzman(1985) and by the papers cited in the May 1988 American Economic Review
15 Howitt goes on to say that there is the ldquoreciprocal Keynesian question of how exactlythe economic system manages to overcome all the obvious coordination problems thatstand in the way of attaining the state of equilibrium common to new classical economicsmodelsrdquo
2 Walrasian economy
1 A brief description of the theory is given in Patinkin (1965 note B)2 In reality of course there is no auctioneer In fact some think that Walras would have
been the first to question such a description of the economy as realistically capturingthe true dynamic process by which the economy reaches the equilibrium described bysupply and demand curves See Walker (1987) who points out that Walras promoted aldquodisequilibrium-production model of tatonnementrdquo as more representative of Walrasrsquothought than the tatonnement model with an ldquoauctioneerrdquo
3 We use the letter T to denote the number of commodities because later we distinguishthe T commodities according to time of availability (ie from period 1 to period T )
4 One could replace the assumption of known prices by the assumption that individualsform expectations at time t about prices at time t and that such expectations are correctThis has been referred to as a situation in which expectations satisfy the assumption ofldquoweak consistencyrdquo
5 With zero transactions costs equality of purchase and sale prices could be viewed asforced by arbitrage conditions
6 Walras was one of the first to note this point Note that πjj = 17 See Debreu (1959 Chapter 2) for a discussion of such accounting prices8 A distinctive feature of macroeconomics is to alter traditional ArrowndashDebreu general
equilibrium analysis in such a way that we can reinterpret accounting prices as moneyprices and determine the level of money prices
9 Others characterize transactions costs in similar fashion For instance Alchian and Dem-setz (1972) cite the costs of ldquoformingrdquo ldquonegotiatingrdquo and ldquoenforcing contractsrdquo whileDahlman (1979) writes of ldquosearch and information costsrdquo ldquobargaining and decisioncostsrdquo and ldquopolicing and enforcingrdquo costs
10 See Varian (1992) for a discussion on these points By ldquowell behavedrdquo we mean thatpartuapartcai gt 0 and ua is strictly quasi-concave
11 In this case the Lagrangian is written ignoring the non-negativity constraints onconsumption of commodity i i = 1 T Below we provide an equivalent character-ization of the constrained maximization problem that incorporates these constraints inthe Lagrangian
12 A function y = f (x1 xn xn+1 xm) is said to be homogeneous of degree k inthe arguments x1 through xn if
f (λx1 λxn xn+1 xm) = λk f (x1 xn xn+1 xm)
3 Firms as market participants
1 With zero ldquoadjustment costsrdquo there could exist a market for capital at time t such thatthe capital employed during the initial period Kt is conceptually distinct from capitalinherited K In this case however equilibrium in the capital market at time t wouldthen imply that Kt = K
204 Notes
2 Note that output is a ldquoflow variablerdquo That is for a period i of length h the outputproduced is given by hyi The term yi is thus the ldquorate of outputrdquo over period i Sinceoutput is a flow variable at a point in time the rate of production is not defined (Asyou can see for any rate of output the limit of production as the length of the periodgoes to zero is zero)
3 In general for a period i of length h labor services are denoted by hNi such that totalwage payments at the end of the period are wihNi Like the labor input one shouldthink of the capital input in flow terms with the stock of capital determining the rateor flow of capital services
4 For simplicity of notation these expressions presume perfect foresight5 In general for a period i of length h the nominal interest rate from the end of that period
to the end of the next period is given by hri = hzpbi + (pbi+h minus pbi)pbi 6 For simplicity of notation these expressions assume perfect foresight with respect to
future prices of equity shares In addition the expressions presume future dividends areknown
7 To minimize notational clutter we have chosen to not denote such anticipations withthe expectation operator We do presume that agents (firms and households) have com-mon expectations (assumed to be held with subjective certainty) concerning plans withrespect to the issuing of equity shares
8 For instance if ri lt rei for period i there would be zero demand for bonds Theresulting excess supply of bonds would lead to a fall in the price of bonds and thus arise in the return on bonds until equality across rates of return held
9 We assume for the moment perfect foresight at time t with respect to prices at the endof the period (at time t + 1)
10 As in the prior models including money balances in the utility function reflectshow money can save (leisure) time required to make exchanges within a period Forsimplicity we limit money holdings to agents labelled ldquohouseholdsrdquo
11 For the representative household holdings of bonds issued by other households mustbe zero so that there is no real indebtedness effect with respect to the bonds exchangedamong representative households
12 In general if we assume there are ni firms producing commodity i and that there are mdifferent commodities then
yt =msum
i=1
⎡⎣ nisum
j=1
( pip)fij(Nijt Kijt)
⎤⎦
The above is a stylized view of how the empirical counterpart to total output real grossdomestic product is actually computed by the Commerce Department
13 Note that it is assumed that the capital stock K generates a fixed rate of capital servicesThat is we do not consider variation in the ldquoutilization of capitalrdquo If we did so thenvariation in the services flowing from the capital stock could be considered an additionalchoice variable with capital utilization presumably affecting the extent of depreciationin the capital stock over time
14 Fama and Miller (1972) cite conditions under which the ldquoowners of the firmrdquo iethe holders of the S equity shares will direct the managers of the firm at time t tomaximize Vt
15 In continuous time
petminus1 =int infin
t[psdsSs]eminusr(sminust)ds
where the interest rate r in the above expression is formally rs
Notes 205
16 Note that Stminus1 equiv S Assuming ri = rt i = t + 1 we have
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[pt+k dt+kSt+kminus1](1 + rt)k
⎤⎦
17 If the change is negative net revenues available for distribution as dividends would bereduced
18 We will explain more fully below the nature of these ldquoadjustment costsrdquo19 As suggested earlier if the utilization of capital were viewed as a choice variable then
it would be natural to have δ directly related to utilization of capital during the period20 Later we will say more about investment and the nature of adjustment costs21 Note that this discussion assumes for simplicity zero adjustment costs22 We follow Sargent and implicitly assume adjustment costs are related to net not gross
investment Net investment measures the change in the capital stock As we will seelater the result is a slightly different expression for Tobinrsquos Q than found elsewherewhen adjustment costs depend on gross investment We could also expand the natureof ψ so that adjustment costs depend on the size of the capital stock as is done in suchpapers as Lucas (1967) Uzawa (1969) and Gould (1968)
23 In the National Income and Product Accounts of the USA that report various measuresof the activity in the economy depreciation is measured by what is called the ldquocapitalconsumption allowancerdquo
24 The presence of such markets reflects the existence of ldquoperfect capital marketsrdquo in themacroeconomics literature Remember that with respect to the choice of investment forthe individual firm zero adjustment costs imply a potential discrete jump in the capitalstock at a point in time so that investment for the individual firm may not be defined
25 LrsquoHospitalrsquos rule states that if a is a number if f (x) and g(x) are differentiable and g(x)does not equal zero for all x on some interval 0 lt |x minusa| lt ε if the limit of f (x) equalszero as x approaches a and if the limit of g(x) is zero as x approaches zero then whenthe limit of the ratio f (x)gprime(x) as x approaches a exists or is infinite it equals the limitof f (x)g(x) as x approaches a
26 In continuous-time or discrete-time models such adjustment costs mean that the ldquocapitalmarketrdquo at time t is eliminated
27 Note that since bonds and equity shares are perfect substitutes the optimization problemwill not provide a breakdown into the optimal number of bonds versus equity shares
28 Recall that we are holding the stock of equity shares outstanding constant29 Note that we ignore the potential choice of capital at time t Kt and associated choice
of bonds setting Kt = K This in fact would be the case with capital adjustment costs30 If we evaluated returns from the start of a period each of the return functions would be
multiplied by 1R31 The equivalence of bond equity share and retained earnings financing of changes in
the capital stock can be shown32 This expression reflects the envelope theorem In particular we have that
dW (Kt+1)
dInt+1
dInt+1
dKt+1= dW (Kt+1)
dNt+1
dNt+1
dKt+1= 0
33 If the price of capital differed from the price of output but both prices were expected tochange at the same rate so that the relative price of capital was assumed to be constantover time then the expected real user or rental cost of capital would be (pkp)(mt + δ)where pkp denotes the relative price of capital
206 Notes
34 Formally at time t the inherited debt-to-equity ratio is given by pbBpeS35 Naturally there are other factors not discussed36 Note that during period t when the length of the period between planned purchases of
capital (at time t) and final installation (at time t + 1) is 1 then net investment Int isgiven by Int = Kt+1 minus K and adjustment costs are given by ψ(Int)
37 During period t+1 net investment is defined by Int+1 = Kt+2 minusKt+1 and adjustmentcosts are given by ψ(Int+1)
38 We can see from the general nature of the investment demand functions that if theproduction function were not separable then the assumption of a constant real wageover time as well as a constant expected real rate of return over time would obtain thisresult
39 See Sargent (1987a 11) for a similar expression in continuous time Note that thetwo expressions would be exact if we take the limit as the length of the period goesto zero The equality between ψ prime
t+h and ψ primet+2h that typically would be an approx-
imation in discrete time holds exactly in the limit In addition the definition ofthe real interest rate for a period of length h 1 + hm = (1 + hr)(1 + hπ) orm = (r minus π)(1 + hπ) indicates that in the limit (as h goes to zero) m = r minus π If we had assumed that adjustment costs were based on gross not net investmentthen the fraction on the left-hand side of (312) would include the term minusδψ in thedenominator
40 The Q theory of investment demand was suggested by Tobin (1969) Sargent is oneauthor who expresses investment demand in this way
41 In fact empirical measures of Tobinrsquos average Q have been constructed although suchmeasures are more complex than those discussed here since they must incorporateinfluences of the tax system (such as investment tax credits accelerated depreciationallowances and the like) on the optimal choice of the capital stock
42 In a two-sector model Tobinrsquos average Q is given by pV p1K where pV is the nominalvalue of the firm and p1 denotes the price of capital which differs from p the price ofoutput
43 In our case there is a single argument of the adjustment function ndash net investment Ifone follows Hayashi (1982) and assumes as he does that adjustment costs depend ongross investment then one must add the assumption that there is a linear homogeneousadjustment function such that ψ(δk) = ψ primeδk over the appropriate range Hayashirsquosadjustment cost function (like others) includes the stock of capital as well
44 Eulerrsquos theorem (or law) is that if the function f (middot) is a differentiable functionhomogeneous of degree 1 (linear homogeneous) with f Rn rarr R then
f (x) equivnsum
i=1
[partf (x)partxi]xi
Replacing the real wage with the marginal product of labor reflects the assumption thatthe firm is a price-taker in both the output and labor markets
45 See for example Azariadis (1976) A second reason is that new workers and previouslyemployed workers may not be considered perfect substitutes (eg see Oi 1962)
46 Taylor (1972) and Hall (1980) are among those who have examined thisphenomenon
4 Households as market participants
1 Time consistency means in this context that households will follow through on priorplans as the starting date advances Strotz (1955ndash56) discussed this point in terms of autility maximizing problem
Notes 207
2 An example of nonseparable preferences is a ldquohabit persistence modelrdquo in which utilityis given by
infinsumi=t
βiminustu(ci ciminus1 )
so that consumption at time t depends on prior consumption Alternatively one couldhave utility given by
infinsumi=t
βiminustu(ci 1 minus Ni 1 minus Niminus1)
such that past work becomes a pertinent state variable for the current period Kydlandand Prescott (1982) is one important exception to the macroeconomic literature inassuming nonseparable preferences
3 This is sometimes referred to as an ldquoend-of-periodrdquo equilibrium specification in themarket for assets Alternative asset specifications for discrete-time analysis have beendiscussed by among others Foley (1975) As Edi Karni (1978) pointed out Patinkinrsquosmodel is an end-of-period model
4 For simplicity we ignore the financial asset markets at time t If we assumed portfolioadjustment costs then it would be the case that at time t desired bond and moneyholdings would be B and S respectively
5 That is (partW (xt+1)partcdt+1)(partcd
t+1partxt+1) = 0 since partW (xt+1)partcdt+1 = 0 similarly
the indirect effects of a change in xt+1 on partW (xt+1) through its impact on optimallabor supply and real money balance holdings are zero
6 For completeness note that by substituting equation (43prime) into (45) we could have theequivalent expression
partW (xt+1)partxt+1 = β[partut+1part(Mt+1pt+1)]Rt+1[Rt+1 minus Rmt+1]7 These conditions appear in various forms throughout the macroeconomics literature
see for instance Mankiw et al (1985) or Barro and King (1984)8 The resulting demand for leisure function is termed a ldquoHicksian or compensated demand
functionrdquo as it is constructed by varying the price of leisure (the real wage) and incomeso as to keep the individual at a fixed level of utility
9 The resulting demand function for leisure that incorporates both the income and sub-stitution effects of changes in the real wage is an example of a ldquoMarshallianrdquo demandfunction
10 Borjas and Heckman (1978) See also Pencavel (1985) for a survey of estimates Forevidence on the effect of wage increases on the labor supply of working women seeNakamura and Nakamura (1981) and Robinson and Tomes (1985)
11 Note that we continue to assume nonsatiation such that partutpartct gt 012 The second statement presumes sufficient dispersion in preferences so that at each real
wage there are some individuals who are just indifferent between a zero and positivelabor supply Thus any rise in the real wage will increase labor force participation
13 This point was first formally developed in the classic paper by Lucas and Rapping(1970) The role of ldquomarket clearingrdquo in macroeconomic analysis will be clearer laterwhen we consider alternative characterizations of the labor market
14 To be exact this result assumes that utility is separable and concave in leisure That isit is assumed that part2upartcpartN = part2upartNpart(Mp) = 0 and part2upartN 2 lt 0
15 Alogoskoufis (1987) provides a good review of the empirical analysis in this area
208 Notes
16 Fisherian ndash or in Patinkinrsquos (1965) terminology ldquoFisherinerdquo ndash analysis takes itsname from the classic ldquotime-preferencerdquo analysis of Fisher see in particular Fisher(1930)
17 To assure this one could let duadcai gt 0 with limcairarr0(duadcai) rarr infin andlimcairarrinfin(duadcai) rarr 0
18 Equivalently one could assume partuapart(Maip) equiv 0 for all i Given the nonnegativityconstraint on Maipi i = t t + T and the presumption of a positive nominalinterest rate ri i = t t +T minus1 the optimal solution would be M d
aipi = 0 for all iThat is bonds will dominate money as an asset and only bonds will be held if assetholdings are positive The reason is simple ndash the unique attribute of money holdingsas a way to reduce ldquotransaction costsrdquo has not been introduced by providing a ldquoutilityyieldrdquo to holding money
19 Note that the equality sign here rather than the ge sign reflects the fact that the first-order conditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
20 Note that we assume a time-invariant one-period utility function The notation uai
reflects the fact that utility of agent a in period i depends on consumption inperiod i cai
21 In general for a period of length h we have that 1 + hm = (1 + hri)(1 + hπi)Solving for hmi and then dividing through by h we have that mi = (ri minusπi)(1+hπi)Thus in the limit as the length of each period goes to zero mi = ri minus πi Thus incontinuous-time analysis the real rate of interest is exactly defined by ri minus πi
22 In the discussion to follow we maintain the assumption of a time-invariant utilityfunction such that ua
i (cai) = ua(cai) for i = t t + T 23 At the point where cat = cat+1 the slope of the indifference curve is minus1β Assuming
cat = cat+1 and no initial assets or debt (ie zBat = 0) if 1β = Rt then the resultof the same consumption in each period would imply the individual would be neither alender nor a borrower
24 As Modigliani (1996) has stated ldquothe consumption and saving decisions of householdsat each point of time reflect a more or less conscious attempt at achieving the preferreddistribution of consumption over the life cycle subject to the constraint imposed bythe resources accruing to the household over the lifetimerdquo Modigliani summarizes hiscontribution to the analysis of consumption behavior in his Nobel lecture of December1985 (see Modigliani 1986)
25 Examples of such discussions are Diamond and Hausman (1984) and Hurd (1987)However what happens in the aggregate does mask different behavior among subgroupsof the populations For instance Burbidge and Robb (1985) find for Canadian data thatwhile an inverted U-shaped profile exists for the ldquoaveragerdquo Canadian household ldquowhitecollarrdquo households do appear to continue to accumulate wealth years after both husbandand wife have left the labor force
26 If individual a were the representative agent then consumption smoothing would implythat the aggregate endowment of the consumption good is identical across periods ieci = ci+1 i = t t + T minus 1
27 Note that in the case of multiperiod bonds agent arsquos future income could includepayments derived from the initial holdings of assets Since the present value of suchfuture ldquoincomerdquo is incorporated in the current value of the assets operationally futureincome cat i = t+1 t+T is defined as income other than derived from initial assetholdings In a production context the source of such income would be compensationfor labor services sold
28 In an economy with production this transitory component of current income can reflectsuch events as a temporary layoff a short-run opportunity to work overtime or atemporary tax rebate
29 This discussion ignores the effects of uncertainty on optimal consumption plans
Notes 209
30 Note that the equality sign here rather than the ge sign reflects the fact that the first-orderconditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
31 Note that with one-period bonds individual a has a zero initial endowment of bondsthat continue into the future at time t
32 Other papers on this topic include Hansen and Singleton (1983) and Mankiw et al(1985)
33 An example of a compositional change that could affect aggregate consumption butwould not be accounted for in the analysis to follow is a change in the proportionof retired individuals in the economy Thus on this ground at least Hallrsquos empiricalfindings cannot be taken as the last word on consumption behavior at the level of theindividual
5 Summarizing the behavior and constraints of firms and households
1 Caballero and Engel (1999) provide an empirical study of investment dynamics in thecontext of manufacturing firms
2 Weber (1998) provides empirical evidence on the link between the financial marketsand consumption spending
3 We refer to this perfect foresight as ldquolimited perfect foresightrdquo since it concerns only thecurrent period This focus on current markets alone typical of static analysis impliesldquoexpectations functions of the agentsrdquo with respect to prices in subsequent periodsthat do not reflect the underlying analysis of markets beyond the current period Thusbeyond the current period expectations do not have the property of perfect foresight (orrational expectations in a nondeterministic setting)
4 In an open economy ndash that is one which admits a foreign sector ndash we would add afourth market the market for foreign exchange
5 Note that in a fully monetized economy money enters on one side of every exchange ndashpurchase or sale ndash in these three markets
6 Note that money holdings and money demand arise only for ldquohouseholdsrdquo To the extentfirms do hold money and make choices with respect to the size of such holdings wepresume their behavior would be similar to that of households and so lump firms withhouseholds with respect to such activity Thus the behavior of the firm is restricted tolabor demand output supply investment demand and financial asset supply
7 Recall that the expected real user cost of capital is mt + δ where δ is the rate ofdepreciation of capital and mt is the expected real rate of interest (ie mt equiv (rt minusπe
t )(1+πet ) where rt is the money interest rate and πe
t is the expected rate of inflationbetween periods t and t + 1)
8 As previously for simplicity we continue to assume that expectations of future pricesare held with subjective certainty
9 If part2f partKpartN = 0 then the real wage would not enter as an argument in the capitaldemand function nor would the existing capital stock affect labor demand
10 That is we can express a demand function in a form similar to one that would beobtained if the analysis were to consider only two periods
11 Note that we do allow the expected wage inflation between period t and t + 1 πewt
to differ from subsequent wage inflation so that the expected real wage next periodwe
t+1pet+1 equiv wt(1 + πe
wt)pt(1 + πet ) can differ from the current real wage This
introduces the possibility of intertemporal substitution of labor supply in response to achange in the current real wage
12 Assuming less than unit elastic expectations with respect to future income streams wouldassure from the standard Fisherian problem that the marginal propensity to consumewould be less than one
13 Recall that πet equiv ( pe
t+1 minus pt)pt
210 Notes
14 Recall that firms are presumed not to hold money balances so there are no real balanceeffects to concern us with respect to firmsrsquo demands or supplies
15 As Lucas and Rapping (1970) point out introducing other expectation assumptionscan retain the ldquointertemporal substitution hypothesisrdquo in the context of changes in thecurrent price level but in doing so money illusion is introduced In their own wordsthe
labor-supply equation is not homogeneous in current wages and current prices(such that) there is ldquomoney illusionrdquo in the supply of labor ldquomoney illusionrdquoresults not from a myopic concentration on money values but from our assumptionthat the suppliers of labor are adaptive on the level of prices expecting a returnto normal price levels regardless of current prices and from the empirical factthat the nominal interest rate does not change in proportion to the actual rate ofinflation With these expectations it is to a supplierrsquos advantage to increase hiscurrent supply of labor and his current money savings when prices rise
(Lucas and Rapping 1970 268ndash269)
Lucas and Rapping are particularly looking at the effect of a change in the current pricelevel on the expected real rate of interest Their assumption of ldquoadaptiverdquo expectationsimplies that an increase in pt results in a fall in πe
t equiv (pet+1 minus pt)pt a rise in the
expected real rate of interest and thus a rise in labor supply16 That is future technology capital demand labor demand the real wage the rate of
depreciation and future real interest rates are all unchanged by such a change in pricesand the money supply
6 The simple neoclassical macroeconomic model (without governmentor depository institutions)
1 In particular it is assumed that the money wage rate adjusts to continuously maintainequality between the demand for and supply of labor the price of output adjusts tomaintain equality between the demand for and supply of output and the price of financialassets (and thus the interest rate) adjusts to maintain equality between the demand for andsupply of financial assets Patinkin (1965) provides one of the first complete accountingsof this model
2 That is we would expect prices in the various markets to eventually adjust to eliminateany possible excess demands or supplies in the economy We would also expect agentsultimately to correctly anticipate the price level The neoclassical model can be modifiedto explain the workings of the economy in the face of incomplete information and priceinflexibility
3 As before the ldquolaws of motionrdquo dictating how prices change to reach equilibrium aregiven by Walrasrsquo excess demand hypothesis and we maintain the assumption that noexchange occurs until an equilibrium is reached (the recontracting assumption)
4 Alternatively one could assume that expectations at time t concerning these futurevariables are constant
5 Note that Patinkin has firms as well as households managing a portfolio of financialassets and money balances which is why he includes the demand function for labor inthe above statement In our analysis this statement applies to the labor supply functionalone
6 This last sentence anticipates the intertemporal substitution hypothesis7 We ignore the potential effect of changes in the interest rate on labor supply and thus
employment8 This reflects the assumption that households and firms share common expectations
concerning the price level (in fact for both pet = pt)
Notes 211
9 Reasons such as these for changes in output form the basis of much of the currentanalysis in the literature with respect to ldquoreal business cyclesrdquo
10 This is perhaps too extreme a statement To the extent that a higher price level isanticipated the resulting lower real money balances could lead to an increase in laborsupply at any given real wage and consequently increased employment and output Alsothere is a potential effect of changes in the price level on aggregate labor supply throughthe impact of such changes on the expected real rate of interest if unit elastic expecta-tions concerning future prices are not assumed Recall that we follow macroeconomictradition and abstract from these possibilities
11 In a recent study Ewing et al (2002) develop a model of the equilibrium unemploymentrate and examine how it responds to unanticipated changes in real output
12 Fairlie and Kletzer (1998) discuss the issues revolving around job displacement13 Unemployment may also result if prices in the economy do not adjust quickly enough to
ensure that all markets (particularly the labor market) are continuously in equilibriumUnemployment associated with labor market disequilibrium is sometimes referred toas ldquoinvoluntaryrdquo unemployment We analyze such unemployment later
14 See for instance the paper by Evans (1989) that examines the relationship betweenoutput and unemployment in the United States
15 The ldquoISrdquo equation is sometimes referred to as the aggregate demand equation indicatingthat it reflects equality between total or aggregate demand for output and productionNote that equilibrium in the output market is being described in terms of demand equalto what is produced ylowast
t The aggregate supply equation indicates what will be produced16 This is the case only for this simple aggregate model without government17 Such an assumption removes the anticipated real wage next period as an argument
in these demand functions Recall that earlier ldquostaticrdquo assumptions concerning futureinterest rates and rates of inflation have already simplified the form of these functions
18 However ldquorealrdquo or ldquosupplyrdquo shocks such as the above-mentioned changes in technologycapital stock supply of other inputs such as oil or in labor supply at prevailing realwages can affect real output
19 This analysis should be familiar since we performed a similar analysis for the economywithout production
20 Note that unit elastic expectations imply that partπet partpt = 0
21 Note that without a real balance effect the CC curve would be horizontal22 The lower interest rate abstracts from a real balance effect in the output market so that
the CC curve is horizontal
7 Empirical macroeconomics traditional approaches and time series models
1 To reduce notational clutter we suppress time subscripts All variables are period tvariables
2 Note that in Sargent (1987a 20) equation (71) is replaced by the ldquorepresentativerdquofirmrsquos first-order condition for the optimal use of the labor input given a competitivelabor market that is the condition that the real wage equals the marginal product oflabor wp = Fn(n K) Note that if the production function F(n k) is separable in thelabor and capital inputs such that f (n K) = v(n)+u(K) and v(n) = (1g)(fnminusn22)then equation (71) is identical to Sargentrsquos equation since fn(n K) = (1g)(f minus n)
3 Unless otherwise noted all parameters in this model such as f g h and j in equations(71) and (72) are assumed to be positive
4 In this context ldquolump-sumrdquo taxes are taxes independent of income Equation (73) isan example of the ldquoconsumption functionrdquo
5 Endogenous variables are variables whose values are determined by the analysis6 Sargent adds equations (73) and (74) to the system (712) in order to determine
consumption and investment demand as well
212 Notes
7 To derive wlowast start with the equilibrium condition nd = ns Substituting the first twoequations of 712 the equilibrium money wage satisfies
f minus g middot (wlowastp) = h + j middot (wlowastp)
Then one can simply solve this equation for the equilibrium money wage wlowast We canthen obtain the equilibrium employment level nlowast by substituting the expression for wlowastinto either of the first two equations of 712
8 This is a property of ldquoclassicalrdquo macroeconomic models in that monetary changesthat alter the price level do not affect real variables such as the real wage output andemployment
9 The IS equation indicates combinations of the interest rate r and output y at which ifthe output were produced and that interest rate prevailed output demand would equaloutput produced The LM equation indicates combinations of r and y that will equatethe demand for and supply of money
10 For the model under consideration the ldquoaggregate demand curverdquo (a plot in ( p y) spaceof the aggregate demand equation) slopes downward and the ldquoaggregate supply curverdquo(a plot in ( p y) space of the aggregate supply curve) is vertical The intersection of theaggregate demand and supply curves graphically determines the equilibrium output andprice level
11 See Altonji and Siow (1987) Ewing and Payne (1998) examined the relation-ship between the personal savings rate and consumer sentiment in the context of aconsumption model
12 Note that Taylor assumes that certain demands for instance consumption demand maydepend on past as well as current values of output and the money supply so that thereduced-form expression estimated includes lagged values of income and the real moneysupply
13 Time series analysis can be viewed as primarily the art of specifying the most likelystochastic process that could have generated an observed time series
14 That is forecasts generated by time series models have been used to proxy individualsrsquoexpectations of future events in tests of various theoretical macroeconomic models
15 A histogram is a plot of the frequency distribution of a set of observations16 If the process is also ldquoergodicrdquo these statistics give consistent estimates of the mean and
variance Ergodicity basically requires that observations sufficiently far apart should bealmost uncorrelated Then by averaging a series through time one is continually addingnew and useful information to the average For a rigorous explanation of this conceptsee Hannan (1970 201)
17 The variable γk k = 0 1 2 is termed the autocovariance function18 Or more generally k = 0 Note that ρk = ρminusk 19 Some have suggested that stock market prices follow a random walk See Campbell
et al (1997)20 Note that if yt is taken to be the logarithm of real output then the trend reflects a constant
rate of growth of output equal to d in the absence of shocks21 This assumes y0 is the initial value of the function yt 22 A random walk is an example of a class of nonstationary processes known as ldquointe-
gratedrdquo processes that can be made stationary by the application of a time-invariantldquofilterrdquo As defined by Granger and Newbold (1986) ldquoif a series wt is formed by a linearcombination of terms of a series yt so that wt = summ
j=minuss cjytminusj then wt is called a
lsquofilteredrsquo version of yt If only past terms of yt are involved so that wt = summj=0 cjytminusj
then wt might be called a one-sided or backward-looking filterrdquo23 This is a special case of a class of stochastic processes known as Markov processes24 Recall that for the random walk process φ = 1 in which case the process was not
stationary as the variance of the process becomes larger and larger with time
Notes 213
25 To obtain this result note that
E
⎡⎣ infinsum
j=0
(φ2)jε2tminusj
⎤⎦ =
infinsumj=0
(φ2)jσ 2e = σ 2
e (1 minus φ2)
26 Often the ldquolag operatorrdquo L or equivalently the ldquobackward shift operatorrdquo B is usedto express this equation Lτ yt (or Bτ yt) = ytminusτ τ = 1 2 3 There are associatedpolynomials in the lag operator such that
d(L) = d0 + d1L + d2L2 + d3L3 + middot middot middot + dpLp
Letting
φ(L) = 1 minus φ1L minus φ2L2 minus φ3L3 minus middot middot middot minus φpLp
we can thus express an AR( p) process for yt as φ(L)yt = εt 27 If yt is an AR( p) process it may be described in the following way ldquoan appropriate
finite backward-looking filter applied to yt will produce a white noise seriesrdquo (Grangerand Newbold 1986 32)
28 If there were a single shock to yt at time 0 (ie εt = 0 for all t gt 0) then the deterministiccomponent would signify the deviation of the time path from its equilibrium level Aswe will see in this case stationarity would imply ldquostability of equilibriumrdquo in that ytwould converge to its equilibrium value over time
29 A linear difference equation of order s is of the form
yt =ssum
j=1
ajytminusj + c
30 Successive substitution (the ldquoiterative method of solutionrdquo) reveals this essential natureof the solution In particular substituting for past values of yt in (725) results inyt = φty0
31 The appendix to this chapter shows how one can reinterpret higher-order differenceequations as a system of first-order difference equations
32 Note that in general a quadratic equation of the form
ax2 + bx + c = 0
can be solved using the quadratic formula
x1 x2 = [minusb plusmn (b2 minus 4ac)12]2a
In our case a = 1 b = minus(m1 + m2) = minusφ1 and c = m1m2 = minusφ233 This assumes m1 = m2The solution for repeated roots is discussed briefly below34 A variable x is said to be inside the unit circle if x lt |1| An alternative way of
expressing this condition is in terms of the associated polynomial 1 minusφ1L minusφ2L2 = 0This polynomial equation is similar to (730) except that b is replaced by 1L and thewhole equation is multiplied through by L2 The stationarity condition is that the rootsof this polynomial equation should lie outside the unit circle
35 Recall that φ1 = m1 + m2 and φ2 = minusm1m236 The following discussion follows Goldberg (1958 171ndash172)
214 Notes
37 As discussed below in this case the two roots are m1 = h + vi and m2 = h minus vi whereh = φ12 v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12 The product
of these two roots is (φ21 minus 4φ2 minus φ2
1)4 = minusφ2 which is the square of the modulus
of the roots We are assuming (minusφ2)12 lt 1 so we have the condition that φ2 lt 1 orφ2 gt minus1
38 Note that if yt is an MA process then it is a ldquobackward looking filterrdquo applied to a whitenoise process As before we can use a lag operator L (or backward shift operator B) toexpress an MA(q) process for yt as yt = micro + θ(L)εt where the polynomial in the lagoperator θ(L) is given by
θ(L) = 1 + θ1L + θ2L2 + middot middot middot + θqLq
39 Recall that as discussed above we assume for simplicity zero mean for yt that ismicro = 0
40 For a more detailed description of invertibility see Granger and Newbold (1986) orBox and Jenkins (1970)
41 Note that ARMA( p 0) equiv AR( p) and ARMA(0 q) equiv MA(q) Using the lag operatorthe ARMA model (in the case of a zero mean) can be simply expressed as φ(L)Yt =θ(L)εt
42 In general if yt is ARMA( p1 q1) and xt is ARMA( p2 q2) the sum zt = yt + xt isARMA( p3 q3) where p3 le p1 + p2 and q3 le max(p1 + q2 p2 + q1) A proof of thisis found in Granger and Morris (1976)
43 A time sequence T (t) is called ldquodeterministicrdquo if there exists a function of past andpresent values gt = g(T (t minus j)) j = 0 1 such that E[(Tt+1 minus gt)
2] = 0 If thefunction gt is a linear function of Ttminusj j ge 0 then Tt is called ldquolinear deterministicrdquo
44 For instance for a stationary series of quarterly data one could postulate a simplefourth-order seasonal AR process of the form
yt = φ4ytminus4 + εt
This is a special case of AR(4) with φ1 = φ2 = φ3 = 0 The model could be extendedto include both AR and MA terms at other seasonal lags for instance the followingARMA(21)
yt = φ4ytminus4 + φ8ytminus8 + θ4εtminus4 + εt
To add other than seasonal components one could simply fill in the gaps (eg addterms such as φ1ytminus1 θ1εtminus1 and the like to the above) Other options are discussedby Harvey (1993) and others
45 Campbell and Mankiw argue that even if the log of output followed a random walk withdrift indicating that the effect of any shock persists indefinitely into the future estimatesusing the detrended series would be biased and erroneously conclude otherwise
46 Note that if the log of real output is an ARMA( p q) process then the differencedprocess will be an ARMA( p q + 1) process This means that to allow for stationaritywith respect to the level of real output requires at least one moving average process forthe differenced series
47 Equivalently for the logarithm of real output they are considering ARIMA(p 1 q)processes for p = 0 1 2 3 and for q = 0 1 2 3
48 Such a finding is often termed as supportive of real business cycle theories
Notes 215
8 The neoclassical model
1 The presence of money reflects the introduction of ldquoimperfectrdquo or costly informationA medium of exchange can arise to minimize costs incurred by participants in theeconomy when there exists imperfect information on potential exchange partners
2 In so doing we assume that the new equilibrium like the initial one exists Further wedisregard the process of adjustment of the variables to the new equilibrium Alterna-tively we could introduce ldquolaws of motionrdquo for the equilibrium values (eg the excessdemand hypothesis for price changes) and examine whether the equilibriums are stable
3 For notational simplicity we let Ld = Ldt cd = cd
t Id = Idnt p = pt r = rt
yd = cdt + Id
nt + δK + ψ(Idt ) and M = M
4 For simplicity we let the expected real rate of interest component of the expected realuser cost of capital ((rt minus πe
t )(1 + πet )) be approximated by rt minus πe
t as representedby r minus π
5 One alternative is for the interest rate to adjust to equate the demand for and supply ofmoney
6 That is we assume that consumption demand is a function of the expected gross realrate of interest Rt not its components (rt and πe
t ) This would be the case if one couldview the consumption decision from the point of view of the pure ldquoFisherian problemrdquoin essence separating the allocation of consumption across time decisions from theldquoportfolio problemrdquo Note that for notational ease we not only let rt denote the moneyinterest rate rt but also let π denote the expected rate of inflation πe
t 7 This would be the case if our focus was solely on the portfolio choice problem8 Examples of growth models with money include Tobin (1965) and Sidrauski (1967b) As
Begg (1980 293) notes ldquoin a steady state any expectations generating mechanism willyield correct predictions thus the steady state analysis of growth models with moneymay be viewed as a special case of the rational expectations model with systematicmonetary policyrdquo
9 This asymmetry in information can reflect an aggregation across labor markets in whicheach firm determines labor demand based on its correct anticipation of the price of theparticular commodity it produces while suppliers of labor determine labor supply basedon their potentially incorrect anticipation of the overall level of prices reflecting theidea that suppliers are concerned with the purchasing power of wages in terms ofcommodities not restricted to the particular commodity that they produce
10 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt r = rt y = yt and M = M
11 Recall that for simplicity we let the expected real rate of interest component of theexpected real user cost of capital (rt minus πe
t )(1 + πet ) be approximated by rt minus πe
t asrepresented by r minus π
12 Note that partypartp = 0 for the neoclassical model given limited perfect foresight13 This is obvious from the graph of aggregate demand and supply curves if one compares
the vertical aggregate supply curve of the neoclassical model dpp = dMM with theupward-sloping aggregate supply curve of the model with real wage illusion Note thatthe shift in the aggregate demand curve for a given change in the money supply isidentical in either case
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
1 Typically a union contract runs for three years Often however there are provisions thatpermit parts of the agreement to be renegotiated at specific times during the three-yearcontract period
2 As a general rule labor agreements are specified in money terms An exception to thisis the cost of living agreements (COLAs) as part of union wage contracts in the United
216 Notes
States which became popular during the 1960s and 1970s COLAs adjust money wagesautomatically to changes in prices typically using changes in the Consumer Price Indexto measure price changes However the percentage of all workers who have contractswith COLAs is fairly small as less than 20 percent of the total labor force is covered bycollective bargaining agreements Further not all COLA clauses offer full protectionagainst general price increases as wages may rise by only some fraction of the increaseof the CPI Considering these qualifications for the time being we simplify by assumingthat all labor agreements are specified in money terms
3 These seven variables for the classical model are made up of three ldquopricesrdquo along withfour variables implied from the behavioral equations The three ldquopricesrdquo are moneywage price level and interest rate determined by the equilibrium conditions for the labormarket and two of the three other markets (output financial andor money markets)The variable implied from the behavioral equations are employment (labor demandfunction) output (production function) consumption (consumption demand function)and investment (investment demand function)
4 Note that this analysis is inconsistent with the idea that long-term employment contractsthat fix the money wage for several periods arise due to adjustment costs with respectto the labor input
5 As discussed before one justification for this form is if real money balances are not partof household wealth which can be the case when we introduce depository institutionsinto the analysis
6 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt Idnt = Id r = rt
y = yt and M = M 7 Recall that for simplicity we let the expected real rate of interest component of the
expected real user cost of capital (rt minus πet )(1 + πe
t ) be approximated by rt minus πet as
represented by r minus π 8 Note that partypartp = 0 for the neoclassical model given limited perfect foresight9 The Lucas model was introduced in Chapter 8 and is covered in more depth in
Chapter 1010 An exception to this statement occurs if monetary authorities can react to period t
disturbances that is if monetary authoritiesrsquo information set includes the values of therandom shocks in period t which are not known to private agents
11 Phelps and Taylor (1977) go on to state that they ldquodo not pretend to have a rigorousunderstanding of [why prices andor wages are set in advance] In the ancient andhonorable tradition of Keynesians past we take it for granted that there are disadvantagesfrom too-frequent or too-precipitate revisions of price lists and wage schedulesrdquo
12 Note that θ is affected by the variability of relative price shocks in relation to generalprice shocks
13 We assume for simplicity that the coefficient on Pt minus Wt + φ is one14 Note that if Pt minus Wt + φ = 0 then equation (99prime) is a first-order linear difference
equation of the form YtminusλYtminus1 = 0 with solution Yt = c0λt In the limit as t approachesinfinity Yt equals zero Recall that the logarithm of the natural level of output is zero
15 Recall our assumption that the scale factor in the determination of the real wage is equalto zero for convenience
16 As Fischer shows if wages were set only for the current period t that is tminusiWt = φ +EtminusiPt then his results would be like those obtained by the Sargent and Wallace modelwith rational expectations for similar reasons This can be seen clearly on substituting(910) into (99prime) in which case one obtains the standard Lucas-like aggregate supplyequation
17 Later it will be assumed that ut and utminus1 are correlated so that information obtainedduring period t minus 1 will help predict variations in output for period t Like the laggedoutput term in the Lucas equation this introduction of serial correlation in output isnecessary (but not sufficient) if monetary policy rules that dictate the money supply
Notes 217
for period t based on information obtained up to the end of period t minus 1 are to serve astabilizing role
18 Note that in Fischerrsquos paper minusvt rather than vt is used in (913) Our vt is more in linewith other models
19 With the one exception noted before that Fischer has minusvt replacing vt 20 Again note that Fischer has minusηt and minusρ2 in the above equation since our vt is his minusvt 21 Fischer has b1 = minusρ2 since our vt is his minusvt
10 The Lucas model
1 The discussion follows that found in Lucas (1973) and Sargent (1987a 438ndash446)2 Or as Sargent (1987a 483) states ldquoan employee cares about his prospective wage
measured not in terms of own-market goods but in terms of an economy-wide averagebundle of goods the assumption is that the labor supplier works in one market butshops in many other marketsrdquo
3 The CobbndashDouglas production function would be given by yit = (Nit)1minusα(Ki)
α whereKi denotes the inherited capital stock for sector or ldquoislandrdquo i Nit is the employment oflabor for sector i and yit is the production of commodity i by sector i during period t Inthis case the marginal product of labor is given by partyitpartNit = (1 minus α)(Nit)
minusα(Ki)α
which implies that a = (1 minus α)(Ki)α
4 Solving this expression for the demand for labor and differentiating with respect to thereal wage relevant for a firm producing commodity i we have
partN dit part(witpit) = (1 minus α)minus1α(minus1α)(witpit)
(minus1minusα)α lt 0
Note that our particular form for the labor demand function implies a constant elasticityIn particular the elasticity of demand for labor is given by
partN dit
part(witpit)
witpit
N dit
= minus 1
α
5 Up to this point we have assumed that expectations are held with ldquosubjective certaintyrdquoso that Et(1pt) = 1Etpt and the expected real wage can be represented as witEtpt However we now assume that individuals view the price level as a random variableIn this setting the expression for the expected real wage is Et(witpt) or witEt(1pt)which is not the same as witEtpt In fact the relationship between witEt(1pt) andwitEtpt is shown by Jensenrsquos inequality witEt(1pt) ge wit(1Etpt) This inequalityreflects the fact that the function f ( pt) = 1pt is convex rather than linear in pt Forinstance if pt = p0 with probability t and p1 with probability 1 minus t then we haveEt(1pt) = tf ( p0) + (1 minus t)f ( p1) gt f (tp0 + (1 minus t)p1) = 1Etpt where 1 gt t gt 0On the other hand ln(1pt) is linear in ln pt so the log of the real wage is linear in thelog of the price level Thus we express the labor supply function in logarithmic formand consider the expectation of the log of the price level
6 The term ξt is assumed to be serially independent which means that E(ξtξs) = 0 fort = s
7 We assume that zit and ξt are statistically independent8 In problems involving more than two random variables ndash that is a ldquomultivariate regres-
sionrdquo ndash we are correspondingly concerned with the term E(z|x y) the expected valueof z for given values of x and y and so on
9 The discussion that follows is standard statistical theory Note that if the joint distributionof the two variables is a bivariate normal density function then the regression of ln pton ln pit is linear
218 Notes
10 In general if x1 x2 xn are random variables a1 a2 an are constants andq = a1x1 + a2x2 + middot middot middot + anxn then
E(q) =nsum
i=1
aiE(xi)
Var(q) =nsum
i=1
a2i Var(xi) + 2
sumiltj
aiajCov(xixj)
wheresum
iltj means that the summation extends over all values of i and j from 1 to nfor which i lt j This expression is derived from the definition of the variance
Var(q) equiv E([q minus E(q)]2) = E
⎛⎜⎝⎡⎣ nsum
i=1
ai(xi minus microi)
⎤⎦
2⎞⎟⎠
which can be expanded by means of the multinomial theorem according to which forexample (a+b+ c +d)2 = a2 +b2 + c2 +d2 +2ab+2ac +2ad +2bc +2bd +2cdNote that
Cov(xixj) = E[(xi minus microi)(xj minus microj)]
If the xi i = 1 n are independent then
Var(q) =nsum
i=1
a2i middot Var(xi)
11 Note that these first two expectations are taken without information on ln pit 12 While it is true that if two random variables are independent they are also uncorre-
lated the converse does not necessarily hold That is two random variables that areuncorrelated are not necessarily independent However two random variables havingthe bivariate normal distribution are independent if and only if they are uncorrelated
13 In general for any two random variables x1 and x2 with joint density function f (x1 x2)the marginal density of x2 g(x2) is obtained by integrating out from minusinfin to +infinthe other variable Thus g(x2) = int
f (x1 x2)dx1 Similarly we can obtain the marginaldensity of x1 by integrating out x2 The conditional probability density function of therandom variable x1 given that the random variable x2 takes on the value x2 is definedby φ(x1|x2) = f (x1 x2)g(x2) assuming that g(x2) does not equal zero
14 To obtain the following expression we use the fact that the random variables ξt and zitare independent with zero means and variances σ 2 and σ 2
z respectively15 Note that this ldquosignal extractionrdquo problem appears in a number of different contexts
such as in statistical theories of discrimination in labor economics (Aigner and Cain1977 Lundberg and Startz 1983) and in the industrial organization literature
16 If there were a trend in the natural level of output then ln ynit would replace the firstln yni and in the lagged term ln ynitminus1 would replace the second ln yni
17 The last term in (1016prime) indicates the deviation of output in the prior period from itsldquonormalrdquo level It is presumed that λ lt 1
18 Recall that this supply function assumes no adjustment costs and thus does not have alagged output term in it
Notes 219
19 In general if z1 z2 zn are independent random variables having the same distribu-tion with mean micro and variance σ 2
z and if z = (z1 + z2 + + zn)n then E(z) = micro
and Var(z) = σ 2z n Note that as n goes to infinity the variance of z goes to zero
20 Recall that we have let π(πtminus1) denote ( pt minus ptminus1)ptminus1 This was done so thatthe terms making up the expected real interest rate rt minus πe
t would have the sametime subscript πe
t = ( pet+1 minus pt)pt In the discussions to follow however we shift
time subscripts so that the previously denoted π = π and the previously denoted πet
becomes πet+1
21 Note that the natural log of real output is ln( yt) = Yt 22 Sometimes the money demand function is simplified by assuming α2 = 0 so that there
are no effects of interest rate changes on real money demand23 This follows given mt = mt +εt so that dPtdmt = 1 Recall that Pt and mt are logs of
the price level and the money supply respectively so that dPt = d(ln pt) = (1pt)dptand dmt = d(ln Mt) = (1Mt)dMt
24 Note that we are somewhat imprecise in our statement that we are eliminating the termfor output from the equation Recall that the exact interpretation of Yt would be as thedifference between the logarithm of output and the logarithm of the natural level ofoutput Equivalently we may call Yt the log of the ratio of output to its natural level
11 Policy
1 The alternative to a rule is called ldquopurely discretionary monetary policyrdquo in whichmoney supply changes are made purely at the discretion of the government dependingon its current reading of the economy and current set of objectives
2 This view is not universally accepted For instance according to the Nobel prize winnerJames Tobin (1985) the government should be free to do as it sees fit and he sees manyreasons ldquofor the Fedrsquos reluctance to tie its own hands as much as lsquorulesrsquo advocates wishrdquo Sargent and Wallace (1976) identify as ldquoKeynesiansrdquo individuals who believegovernment monetary policy should attempt to ldquolean against the windrdquo in an effort toattenuate the business cycle In the view of Sargent and Wallace and others the monetaryauthority has no scope to conduct countercyclical policy Not surprisingly those whodisagree with this view suggest alternative models of the economy such as those withldquostickyrdquo prices more favorable to their alternative views
3 Those who argue for a rule such as this are sometimes called ldquomonetaristsrdquo Someadvocates of a constant growth rate in the money supply would restrict the length oftime during which a particular rate of growth was fixed For instance William Pooleformer member of the Presidentrsquos Council of Economic Advisers (1982ndash1985) andnow with the Fed suggests that monetary rules should be adopted but that the ruleshould be ldquosubject to change at any time upon presentation of a convincing case withsupporting evidencerdquo (Poole 1985)
4 Hallrsquos suggestion echoes Simonsrsquo (1936) proposal for ldquoa monetary rule of maintainingthe constancy of some price indexrdquo Wayne Angell a 1986 appointee to the Board ofGovernors of the Federal Reserve (the monetary authorities for the USA) has arguedfor a monetary policy that will stabilize a price index constructed from a basket of basiccommodities (perhaps including gold)
5 Recall also that we assume the random variables εt and ut (the random variable asso-ciated with output demand) are independent with zero mean and respective variancesσ 2
e and σ 2u
6 An example of such ldquoexogenousrdquo price expectations would be the autoregressiveexpectations
7 Recall that the natural level of output was normalized to equal one so that Yn equiv ln yn = 0Thus if the optimal level of output is the natural level then the objective would be tominimize Etminus1(Yt)
2
220 Notes
8 Recall that we are assuming that εt and vt are independent random variables From(112)
Etminus1Yt = a0 + a1Ytminus1 + a2mt
so that Yt minus Etminus1Yt = a2εt + vt Squaring this expression and noting that E(εt) = 0E(vt) = 0 and given independence E(εt middot vt) = 0 we obtain
Etminus1(Yt minus Etminus1Yt)2 = a2
2σ 2e + σ 2
v
9 See Sargentrsquos (1987a 453) equation (13) for the corresponding expression Recall thatg0 = (Y lowast minus a0)a2
10 Note that ln yn = 0 due to normalization11 To tie down the inflation rate we would have to add an inflation objective to the goals
of the government12 Note that this is another example of the Lucas critique in which econometric estimates
of a specific set of parameters based on past policies cannot be used to project the impactof new different monetary rules to be followed in the future for with these new policiesthe parameters will change
13 This equation is derived from combining the price error expression with the aggregatesupply equation given earlier
14 The assumptions leading up to (1116) have been chosen so that (1116) matches thecriterion found in Barro and Gordon (1983) We will contrast the results of that paperwith our findings here shortly
15 The form of this equation mirrors that in Barro and Gordon16 Consider what would happen if we did not neglect the lagged disturbance terms For
instance let us say one of the disturbance terms (εtminus1 or utminus1) is positive rather thanzero while the other is held equal to zero According to the reduced-form equation forthe price level for period t minus 1 the result would be a higher price level in period t minus 1with the positive lagged disturbance term If the rate of growth in the money supplybetween period t minus 1 and t was the same in both cases then the inflation rate wouldbe higher in the case when both lagged disturbance terms are zero Thus to maintain aconstant inflation rate a lower money supply growth is implied if lagged disturbanceterms are zero instead of positive
17 Kydland and Prescott were awarded the 2004 Nobel prize in economics for their work18 This form of the objective function introduced above is discussed in Barro and Gordon
(1983) Without the assumption of k lt 1 a zero average rate of inflation would beoptimal regardless of the nature of expectation formation
19 For simplicity we assume there is no persistence in real shocks Thus we assume λ = 0in the original Barro and Gordon model
12 Open economy
1 For discussion purposes the terms ldquodomesticrdquo and ldquoforeignrdquo are used with respect tothe domestic perspective
2 For simplicity we assume foreigners do not desire to hold domestic money3 We assume for simplicity that foreigners are not taxed by the domestic government4 That is the domestic households own the remaining share of bonds issued by foreign
firms Bff and α of the equity share issues by foreign firms The total dividends (interms of the foreign currency) paid by foreign firms in period t are denoted by pftdftwhere pft is the price level of the foreign country and dft are real dividends of foreignfirms
Notes 221
5 One might also include bdt as purchase of output by private depository institutions is
included in the household budget constraint This reflects the replacement of dividendspaid by private depository institutions to their shareholders (households) by the differ-ence between banksrsquo interest income and their purchases of the composite commodityThis definition of dividends for private banks has been termed the ldquoflowrdquo constraint forprivate depository institutions
6 Recall that MB denotes the monetary base R denotes reserves and C denotes currencyin the hands of the nonbank public
7 Recall that we use the term ldquomodifiedrdquo as we have substituted the firm distribution con-straint into the household budget constraint for labor income We thus have effectivelysuppressed the labor market from the markets under consideration This modified lawis useful in understanding how the aggregate demand equation is derived
8 By ldquoreal demandrdquo we mean in units of the domestic commodity9 The term ldquorealrdquo means in terms of the domestic composite commodity
10 In measuring the exports imports and international capital flows a number of itemsare often missed For instance the clandestine transfer of funds from the Philippines toUS bank accounts would generate a demand for dollars On the other hand secretiveimports of heroin from Turkey result in a supply of dollars in international marketsThe net of such unmeasured transactions are lumped under the heading of ldquostatisticaldiscrepancyrdquo in the balance of payments accounts We have omitted this item fromTable 121
11 For simplicity we assume that the exchange rate affects only the division of totalconsumption between imports and purchases of the domestic output total consumptionis assumed to be unaffected by such exchange rate changes
12 Note that an increase in the price of the yen or an ldquoappreciationrdquo of the yen means afall in the price of a dollar in terms of yen or a ldquodepreciationrdquo of the dollar
13 If the price elasticity with respect to imports was less than one in short run a rise in therelative price of imports due to a depreciation of the dollar could in fact lead to a risein the value of imports as well This short-run phenomenon when applied to the path ofnet exports over time is referred to as the ldquoJ-curve effectrdquo
14 Actually for much of the analysis to follow we need only assume the weaker ldquoMarshallndashLernerrdquo condition that the sum of the price elasticity of demand for imports and theprice elasticity of demand for exports exceeds one This assures that a price of the dollarbelow its equilibrium level will be associated with an excess demand for dollars in theforeign exchange markets while a price of the dollar above its equilibrium level willbe associated with an excess supply of dollars
15 Given future markets for foreign currency this is not always the case16 We assume that such expectation is held with subjective certainty17 There are several reasons why in the short run the supply curve may not be upward-
sloping First it often takes time for US purchasers to adjust purchases in light of achange in relative prices Second the prices of domestic goods that are close substitutesto the imported goods can rise significantly in the short run as domestic producers hitshort-run production constraints The third complication that has the effect of makingthe supply of dollars curve less likely to be upward-sloping is that foreign producersat least in the short run often adjust the foreign currency prices of goods they exportto partially offset the impact of exchange rate changes on the prices of their goodsin foreign markets For instance Knetter (1987) found that with a depreciation of thedollar (appreciation of the West German Mark) West German exporters often reducedthe Mark price of their exports so as to minimize the rise in the dollar price of Germangoods that would result from the appreciation of the Mark For the time being weabstract from such short-run considerations although this is not to lessen the importanceof this phenomenon as the experience during the 1985ndash1987 period indicates With adepreciation of the dollar the dollar value of imports grew as the USA had to supply
222 Notes
more dollars for each unit of imported goods and there was initially little reduction inthe quantity of goods imported
18 It is important to remember that while we talk of households as being the only privatedemanders of foreign goods and financial assets this is purely a simplifying deviceFirms also demand foreign goods and private depository institutions demand foreignfinancial assets The analysis would be more complex if we explicitly recognized thesedemands but our conclusions would be unchanged since we can subsume in householdactions the actions of firms and depository institutions in foreign markets
References
Aigner DJ and Cain GC (1977) Statistical theories of discrimination in labor marketsIndustrial and Labor Relations Review 30(2) 175ndash187
Alchian A and Demsetz H (1972) Production information costs and economicorganization American Economic Review 62 777ndash795
Alogoskoufis GS (1987) On intertemporal substitution and aggregate labor supplyJournal of Political Economy 95 938ndash960
Altonji J and Siow A (1987) Testing the response of consumption to income changeswith (noisy) panel data Quarterly Journal of Economics 102 293ndash328
Arrow K (1964) The role of securities in the optimal allocation of risk-bearing Review ofEconomic Studies 31 (April) 91ndash96
Arrow K and Hahn FH (1971) General Competitive Analysis San Francisco CAHolden-Day
Azariadis C (1976) On the incidence of unemployment Review of Economic Studies 43115ndash125
Barro RJ (1976) Rational expectations and the role of monetary policy Journal ofMonetary Economics 2 1ndash32
Barro RJ and Gordon D (1983) A positive theory of monetary policy in a natural ratemodel Journal of Political Economy 91 589ndash610
Barro RJ and Grossman HI (1971) A general disequilibrium model of income andemployment American Economic Review 61 82ndash93
Barro RJ and King RG (1984) Time-separable preferences and intertemporal substitu-tions models of business cycles Quarterly Journal of Economics 99 817ndash839
Begg DKH (1980) Rational expectations and the non-neutrality of systematic monetarypolicy Review of Economic Studies 47 293ndash303
Blanchard OJ (1981) What is left of the multiplier accelerator American EconomicReview 71 150ndash154
Blanchard OJ and Fischer S (1989) Lectures on Macroeconomics Cambridge MA MITPress
Borjas GJ and Heckman JJ (1978) Labor supply estimates for public policy evaluationNBER Working Paper No W0299 November
Box GEP and Jenkins GM (1970) Time Series Analysis San Francisco CAHolden-Day
Burbidge JB and Robb AL (1985) Evidence on wealthndashage profiles in Canadian cross-section data Canadian Journal of Economics 18 854ndash875
Caballero R and Engel E (1999) Explaining investment dynamics in US manufacturingA generalized (S s) approach Econometrica 67 783ndash826
224 References
Campbell JY and Mankiw NG (1987a) Permanent and transitory components inmacroeconomic fluctuations American Economic Review 77 111ndash117
Campbell JY and Mankiw NG (1987b) Are output fluctuations transitory QuarterlyJournal of Economics 102 857ndash880
Campbell JY Lo AW and MacKinlay AC (1997) The Econometrics of FinancialMarkets Princeton NJ Princeton University Press
Cass D (1965) Optimal growth in an aggregate model of capital accumulation Review ofEconomic Studies 32 233ndash240
Clower RW (1965) The Keynesian Counter-revolution A theoretical appraisal InFH Hahn and FPR Brechling (eds) The Theory of Interest Rates London Macmillan
Coase RH (1960) The problem of social cost Journal of Law and Economics 31ndash44
Cukierman A (1986) Measuring inflationary expectations Journal of Monetary Eco-nomics 17 315ndash324
Dahlman C (1979) The problem of externality Journal of Law and Economics 22141ndash162
Debreu G (1959) The Theory of Value New York WileyDiamond PA and Hausman JA (1984) Individual retirement and savings behaviour
Journal of Public Economics 23 81ndash114Dornbusch R (1976) Expectations and exchange rate dynamics Journal of Political
Economy 84 1161ndash1176Enders W (2004) Applied Econometric Time Series 2nd edn Hoboken NJ WileyEvans G (1989) Output and unemployment dynamics in the United States 1950ndash1985
Journal of Applied Econometrics 4 213ndash237Ewing BT (2001) Cross-effects of fundamental state variables Journal of
Macroeconomics 23 633ndash645Ewing BT and Payne JE (1998) The long-run relation between the personal savings rate
and consumer sentiment Financial Counseling and Planning 9(1) 89ndash96Ewing BT Levernier W and Malik F (2002) Differential effects of output shocks on
unemployment rates by race and gender Southern Economic Journal 68 584ndash599Fama EF and Miller MH (1972) The Theory of Finance New York Holt Rinehart and
WinstonFischer S (1977) Long-term contracts rational expectations and the optimal money supply
rule Journal of Political Economy 85 191ndash205Fisher I (1930) The Theory of Interest New York MacmillanFleming MJ (1962) Domestic financial policies under fixed and under floating exchange
rates IMF Staff Papers 9 369ndash379Foley KD (1975) On two specifications of asset equilibrium in macroeconomic models
Journal of Political Economy 83 303ndash324Friedman M (1959) A Program for Monetary Stability New York Fordham University
PressFriedman M (1968) The role of monetary policy American Economic Review 58 1ndash17Goldberg S (1958) Difference Equations New York WileyGould JP (1968) Adjustment cost in the theory of investment of the firm Review of
Economic Studies 35 47ndash56Grandmont JM (1977) Temporary general equilibrium theory Econometrica 45
535ndash572Granger CWJ and Morris M (1976) Time series modelling and interpretation Journal
of the Royal Statistical Society Series A 139 246ndash257
References 225
Granger CWJ and Newbold P (1986) Forecasting Economic Time Series 2nd ednOrlando FL Academic Press
Hall R (1976) The Phillips curve and macroeconomic policy In K Brunner andAH Meltzer (eds) The Phillips Curve and Labor Markets Carnegie-RochesterConference Series on Public Policy Amsterdam North-Holland
Hall R (1980) Employment fluctuation and wage rigidity In GL Perry (ed) BrookingsPapers on Economic Activity pp 91ndash123 Washington DC Brookings Institution
Hall RE (1982) Explorations in the Gold Standard and related policies for stabilizingthe dollar In RE Hall (ed) Inflation Causes and Effects Chicago IL University ofChicago Press
Hall RE (1988) Intertemporal substitution in consumption Journal of Political Economy96 339ndash357
Hannan EJ (1970) Multiple Time Series New York WileyHansen B (1970) A Survey of General Equilibrium Systems New York McGraw-HillHansen LP and Singleton KJ (1983) Stochastic consumption risk aversion and the
temporal behavior of asset returns Journal of Political Economy 91 249ndash265Harvey AC (1993) Time Series Models Cambridge MA MIT PressHayashi F (1982) Tobinrsquos marginal q and average q A neoclassical interpretation
Econometrica 50(1) 213ndash224Hicks J (1939) Value and Capital Oxford Clarendon PressHowitt P (1985) Transaction costs in the theory of unemployment American Economic
Review 75 88ndash100Howitt P (1986) Conversations with economists A review essay Journal of Monetary
Economics 18 103ndash118Hurd M (1987) Savings of the elderly and desired bequests American Economic Review
77 298ndash312Karni E (1978) Period analysis and continuous analysis in Patinkinrsquos macroeconomic
model Journal of Economic Theory 17 134ndash140Kester WC (1986) Capital ownership structure A comparison of the United States and
Japanese manufacturing corporations Financial Management 15(1) 5ndash16Keynes JM (1936) General Theory of Employment Interest and Money London
MacmillanKletzer LG and Fairlie R (1998) Jobs lost jobs regained An analysis of blackwhite
differences in job displacement in the 1980s Industrial Relations 37 460ndash477Knetter M (1987) Export prices and exchange rates Theory and evidence Working paper
Stanford University NovemberKoopmans T (1965) On the concept of optimal economic growth In Proceedings ndash
Study Week on the Econometric Approach to Development Planning Chicago ILRand-McNally
Kydland FE and Prescott EC (1977) Rules rather than discretion The inconsistency ofoptimal plans Journal of Political Economy 85 473ndash491
Kydland FE and Prescott EC (1982) Time to build and aggregate fluctuationsEconometrica 50 1345ndash1370
Leonard JS (1988) In the wrong place at the wrong time The extent of frictional andstructural unemployment NBER Working Paper No 1979
Lucas RE (1967) Adjustment costs and the theory of supply Journal of Political Economy75 321ndash334
Lucas RE (1972) Expectations and the neutrality of money Journal of Economic Theory4 103ndash124
226 References
Lucas RE (1973) Some international evidence on outputndashinflation tradeoffs AmericanEconomic Review 63 326ndash334
Lucas R Jr (1981) Methods and problems in business cycle theory In Studies in Business-Cycle Theory Cambridge MA MIT Press First published in Journal of Money Creditand Banking 12 November 1980
Lucas RE and Rapping LA (1970) Real wages employment and inflation InES Phelps (ed) Microeconomic Foundations of Employment and Inflation TheoryNew York WW Norton
Lundberg S and Startz R (1983) Private discrimination and social intervention incompetitive labor markets American Economic Review 73(3) 340ndash347
McCallum B (1979) The current state of the policy-ineffectiveness debate AmericanEconomic Review 69 240ndash245
McCallum B (1985) On consequences and criticisms of monetary targeting Journal ofMoney Credit and Banking 17 570ndash597
Mankiw NG (1987) The optimal collection of seigniorage Theory and evidence Journalof Monetary Economics 20 327ndash341
Mankiw NG Rotemberg JJ and Summers LH (1985) Intertemporal substitution inmacroeconomics Quarterly Journal of Economics 100 225ndash251
Marx K (1976) Capital Vol 1 A Critique of Political Economy HarmondsworthPenguin
Mills TC (1999) The Econometric Modelling of Financial Time Series 2nd ednCambridge Cambridge University Press
Modigliani F (1966) Life cycle hypothesis of saving the demand for wealth and the supplyof capital Social Research 33 160ndash217
Modigliani F (1986) Life cycle individual thrift and the wealth of nations AmericanEconomic Review 76 297ndash313
Mundell RA (1968) International Economics New York MacmillanNakamura A and Nakamura M (1981) A comparison of the labor force behavior of
married women in the US and Canada with special attention to the impact of incometaxes Econometrica 49 451ndash489
Nelson CR and Plosser CI (1982) Trends and random walks in macroeconomictime series Some evidence and implications Journal of Monetary Economics 10139ndash162
Niehans J (1987) Classical monetary theory new and old Journal of Money Credit andBanking 19 409ndash424
Oi WY (1962) Labor as a quasi-fixed factor Journal of Political Economy 70 538ndash555Patinkin D (1965) Money Interest and Prices New York Harper amp RowPencavel J (1985) Labor supply of men A survey In Orley Ashenfelter (ed) Handbook
of Labor Economics Amsterdam North-HollandPhelps ES (1968) Money-wage dynamics and labor market equilibrium Journal of
Political Economy 76 678ndash711Phelps ES and Taylor JB (1977) Stabilizing powers of monetary policy under rational
expectations Journal of Political Economy 85 163ndash190Phillips AW (1958) The relationship between unemployment and the rate of change in
money wage rates in the United Kingdom 1861ndash1957 Economica 25 283ndash299Pindyck RS and Rubinfeld DL (1991) Econometric Models and Economic Forecasts
3rd edn New York McGraw-HillPoole W (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo
Journal of Money Credit and Banking 17 602ndash605
References 227
Radford RA (1945) The economic organization of a prisoner of war camp Economica12 189ndash201
Robinson C and Tomes N (1985) More on the labour supply of Canadian womenCanadian Journal of Economics 18 156ndash163
Sargent T (1987a) Macroeconomic Theory 2nd edn Boston MA Academic PressSargent T (1987b) Dynamic Macroeconomic Analysis Cambridge MA Harvard
University PressSargent TJ and Wallace N (1975) ldquoRationalrdquo expectations the optimal monetary
instrument and the optimal money supply rule Journal of Political Economy 83241ndash254
Sargent TJ and Wallace N (1976) Rational expectations and the theory of economicpolicy Journal of Monetary Economics 2 169ndash183
Shapiro C and Stiglitz JE (1984) Equilibrium unemployment as a worker disciplinedevice American Economic Review 74 433ndash444
Shiller RJ (1978) Rational expectations and the dynamic structure of macroeconomicmodels Journal of Monetary Economics 4 1ndash44
Sidrauski M (1967a) Rational choice and patterns of growth in a monetary economyAmerican Economic Review 57 534ndash544
Sidrauski M (1967b) Inflation and economic growth Journal of Political Economy 75797ndash810
Simons HC (1936) Rules versus authorities in monetary policy Journal of PoliticalEconomy 44 1ndash30
Sims C (1972) Money income and causality American Economic Review 62 540ndash552Solow R (1956) A contribution to the theory of economic growth Quarterly Journal of
Economics 70 65ndash94Strotz RH (1955ndash1956) Myopia and inconsistency in dynamic utility maximization
Review of Economic Studies 23 165ndash180Stuart CE (1981) Swedish tax rates labor supply and tax revenues Journal of Political
Economy 89 1020ndash1038Taylor J (1972) The behaviour of unemployment and unfilled vacancies Great Britain
1958ndash71 An alternative view Economic Journal 82 1352ndash1365Taylor JB (1979) Estimation and control of a macroeconomic model with rational
expectations Econometrica 47 1267ndash1286Tobin J (1965) Money and economic growth Econometrica 33 671ndash684Tobin J (1969) A general equilibrium approach to monetary theory Journal of Money
Credit and Banking 1(1) 15ndash29Tobin J (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo or
Monetary targeting Dead at last Journal of Money Credit and Banking 17 605ndash610Uzawa H (1969) Time preference and the Penrose effect in a two-class model of economic
growth Journal of Political Economy 77 628ndash652Varian H (1992) Microeconomic Analysis 3rd edn New York WW NortonWalker DA (1987) Walrasrsquo theories of tatonnement Journal of Political Economy 95
758ndash774Walras L (1954) Elements of Pure Economics trans W Jaffeacute London George Allen amp
UnwinWeber CE (1998) Consumption spending and the paper-bill spread Theory and evidence
Economic Inquiry 36 575ndash589Weitzman M (1985) The simple macroeconomics of profit sharing American Economic
Review 75 937ndash953
Index
accelerationist outcome 170ndash2aggregate demand 7 86 90ndash1 94ndash5 99
Keynesian model 133ndash4 135 139 140Lucas model 160ndash1 neoclassical model126 127 128 146 shocks 158 seealso demand
aggregate supply 82 86ndash90 91 94ndash5 98113 Keynesian model 132 133ndash4 135138 140 145 Lucas model 137153ndash4 155 158 160ndash1 164 166monetary policy 170ndash1 neoclassicalmodel 118 123 125ndash6 127 128 133see also supply
aggregation issues 5ndash6 8 14ndash15Alchian A 203n9Alogoskoufis GS 51Angell Wayne 219n4Arrow-Debreu theory 2 202n4 n10
203n8assets 51 52 53 portfolio decision
39ndash40 57ndash9 tangible 25ndash6 see alsofinancial assets
autocorrelation function 102ndash3autoregressive expectations 162ndash4 168autoregressive processes 103ndash5 110 111
113 166
balance of payments 190ndash3bankruptcy 32 33Barro RJ 156 167 179ndash80 182ndash4Begg DKH 122 215n8behavioral hypotheses 96 98 99 100Bellman equation firms 29 30 31 34
35 households 43 44 45 48ndash9Blanchard OJ 111 112 171 175 184ndash5bonds 19ndash20 23 27ndash9 32ndash3 financial
market equilibrium 84ndash5 firmfinancing constraint 24ndash6 64 Fisherian
problem 52 foreign 189 households39 40 43ndash4 72 204n11 money illusion59 portfolio choice 57 58 59 93 realvalue 77 temporary equilibrium 80 81
budget constraint 7 16 household 43 4455 65ndash6 67 70ndash1 73ndash4 open economy188 189
Burbidge JB 208n25business cycle 5 6 49 50 173 real
business cycle theory 79 136 202n4technological innovation 113
Caballero R 209n1Campbell JY 111 112 113 214n45capital 23 24 31 68 69ndash70 cost of
31ndash2 36 69 70 94 120 209n7 firmfinancing constraint 24 25 26 64international flows 192 193 195ndash6197ndash8 200 Tobinrsquos Q 36 see alsocapital stock
capital account 192capital adjustment costs 21 26ndash7 34ndash7
69ndash70capital markets 27 203n1 205n24capital stock 3 18ndash19 20ndash1 23 25 27ndash9
optimal investment 30 31 69 70retained earnings financing 29ndash30 32superneutrality of money 120 121 122Tobinrsquos Q 35 36
central banks 187 188 189 190191 192ndash3
classical economics 4 5 79 212n8Clower RW 117 118Coase RH 11CobbndashDouglas production function 147
148 217n3commodities 8 9ndash10 14 16 18ndash19 20
individual experiments 11 13ndash14
Index 229
Lucas model 146ndash7 market equilibrium92 open economy 188ndash9
comparative static analysis 116ndash23consumer behavior 11 12ndash13consumption 20 39 40ndash6 47ndash8 67
70ndash4 aggregate 55 61 demand 97 99195 215n6 financial asset demand 94Fisherian problem 51ndash5 individualexperiments 11 12ndash13 intertemporalsubstitution hypothesis 51 60ndash3Keynesian model 134 135 laborparticipation 49 life-cycle hypothesis50 55ndash6 57 money supply shocks 128neoclassical model 117ndash18 123 126neutrality of money 119 open economy190ndash1 193 permanent incomehypothesis 56ndash7 portfolio choice 5758 59 real balance effect 77 92superneutrality of money 121 122
continuous-time analysis 4cost of capital 31ndash2 36 69 70 94 120
209n7cost of living agreements (COLAs) 215n2costs bankruptcy 33 capital adjustment
21 26ndash7 34ndash7 69ndash70 dividends 23ndash4firm financing constraint 24
Cramerrsquos rule 87 119 121 127 134Cukierman A 164currency depreciation 193 195 198ndash9
221n17
Dahlman C 203n9De Moivrersquos theorem 109Debreu G 8debt 32 60debt-to-equity ratio 32ndash4decision-making behavior 5ndash6 38 51 see
also portfolio decisiondemand excess 13ndash14 15ndash16 67 118
133 190ndash1 199 individual demandfunctions 13 14 labor marketequilibrium 83 Lucas model 146156ndash7 market demand functions 1415 money 86 neoclassical model82ndash3 119 122 123 new classicaleconomics 5 price elasticity of 195real balance effect 60 77 Sayrsquos law 716 temporary equilibrium 80Walrasian model 7 15 see alsoaggregate demand
Demsetz H 203n9depository institutions 126 187 188 189
221n5 222n18
depreciation 26 31 74 205n23deterministic feedback rules 172 173 174deterministic models 6disequilibrium 78 199distributed lag schemes 162ndash4dividends depository institutions 221n5
firms 20 22 23ndash4 29 30 32 34ndash5future 76ndash7 households 43 44 66
Dornbusch model 200ndash1dynamic analysis 2 3ndash4 6 79 122dynamic programming 28ndash9
econometric models 96 100ndash1employment 1 19 128 148 full
employment model 88 142 Keynesianmodel 130ndash1 132ndash3 134 135 137neoclassical model 82 83 84 91 122123ndash5 126 see also labor labor marketlabor supply
Engel E 209n1equilibrium 1 5 7 15 199 aggregate
demand 90ndash1 94ndash5 99 aggregatesupply 86ndash8 94ndash5 employment123ndash5 financial markets 80ndash1 84ndash6illusion model 79 Keynesian model79 133ndash4 137 139ndash40 labor market83ndash4 88 90 119 123ndash5 127 131 148loanable funds theory 120 Lucas model159 160 162 monetary policy 182ndash4money market 85ndash6 92ndash3 126 159multiple equilibria models 113 naturalrate of unemployment 89 90neoclassical model 78 79 117ndash18 126non- market-clearing model 79 partial187 Patinkin analysis 91ndash4 stationarity106 stochastic models 6 temporary 278 80ndash3 86 120 velocity 139ndash40 seealso general equilibrium
equity shares 18 19ndash20 21ndash2 23 27ndash932ndash3 firm financing constraint 24ndash664 households 40 43 72 real value77 temporary equilibrium 80 81
ergodicity 212n16Eulerrsquos theorem 206n44Ewing BT 211n11 212n11exchange rate 186ndash7 191 193ndash6 197
199 200ndash1expectations 41 59 84 119 202n5
adaptive 163 170ndash2 210n15 aggregatedemand 90ndash1 autoregressive 162ndash4168 inflation 62 69 71 77 120ndash2163 164 interest rate 60 61 62 6971 75ndash7 Keynesian model 135 136ndash8
230 Index
expectations (Continued)140 141ndash4 145 labor marketequilibrium 83 Lucas model 157 160optimal monetary policy 168ndash70 180182 183ndash4 output 68 rational 3128ndash9 136ndash8 141ndash4 164ndash6 172ndash4176ndash8 182ndash4 suppliers 124 135 weakconsistency 203n4 see also forecasting
exports 190ndash1 193 197 200
financial assets 77 85 93ndash4 118 firms19ndash20 25 26 28 29 68ndash9 70households 40 42 43ndash4 66 71 72ndash4labor market equilibrium 83 openeconomy 186 187ndash8 189 191 192193 195ndash6 197ndash9 temporaryequilibrium 81ndash2 see also bondsequity shares
financial market 1 36 67 96ndash7equilibrium 80ndash1 84ndash6 93 94 95 118open economy 191 195ndash6 197 199
firm distribution constraint 23ndash4 28 4165 67 74 76ndash7 189
firm financing constraint 23 24ndash6 28ndash964 67 68ndash70 84ndash5
firms 18ndash38 64ndash5 68ndash70 130 135aggregate supply 87 capital adjustmentcosts 26ndash7 34ndash7 dividends 23ndash4investment demand 97 laboradjustment costs 37ndash8 money illusion75 119 neoclassical model 79objectives 21ndash3 open economy 187188 189 190 222n18 optimalinvestment 30ndash2 real wage illusion124 125 retained earnings financing29ndash30 34
Fischer S 112 136 137 138ndash45 171175 184ndash5
Fisher I 79 163 208n16Fisherian problem 39 46 51ndash5 209n12
215n6forecasting 96 100ndash1 101ndash14 128
149ndash50 see also expectationsforeign exchange market 186ndash7 188 191
193ndash5 198ndash9 200foreigners 187ndash90 191 197ndash8Friedman Milton 56 156 168 171 179futures market 3 7 8 10 65 66
general equilibrium 1 2 11 95 97 187neoclassical model 117 118 prices 710 15ndash16 Walrasian model 8 see alsoequilibrium
GNP see gross national productGordon D 156 167 179ndash80 182ndash4government 187 190Grandmont JM 78Granger CWJ 100 102 112 212n22gross national product (GNP) 112 168
habit persistence model 207n2Hall Robert 60ndash1 62 172Hansen B 8Harvey AC 101 114Hayashi F 36 206n43Hicks John 2 8households 18 19 39ndash63 65ndash6 70ndash4
aggregate supply 87 bonds and equityshares 20 24 consumption demand97 dividends 23 24 Fisherian problem39 46 51ndash5 imperfect foresight 124intertemporal substitution hypothesis49ndash51 60ndash3 labor supply problem46ndash51 life-cycle hypothesis 55ndash6money illusion 75ndash6 77 119neoclassical model 79 open economy187 188 189 191 193ndash6 222n18permanent income hypothesis 56ndash7portfolio decision 39ndash40 46 57ndash9price changes 21
Howitt P 5 203n15
imperfect foresight 123 124imports 193ndash5 199 200income 59 74 97 208n27 foreign goods
195 197 life-cycle hypothesis 55ndash657 neoclassical model 117 permanentincome hypothesis 56ndash7 see also wages
income effects 47ndash8 49individual experiments 11ndash14 21 39inflation 53 59 72 196 200
expectations 62 69 71 77 120ndash2 163164 Lucas model 146 154ndash6 157160 161 monetary policy 136 170ndash2174ndash9 180 181ndash2 183ndash4superneutrality of money 121 122wages 123 131 see also prices
integrated processes 110ndash11 112interest rate 1 4 20 31 36 59
consumption demand 97 equilibrium91ndash4 equity shares 22 23 expectations60 61 62 69 71 75ndash7 financialmarket equilibrium 84 85 94Fisherian problem 53 54 foreignfinancial assets 195 196 198household problem 43 45 46 47
Index 231
intertemporal substitution hypothesis49 50 51 63 72 Keynesian model133 134 labor demand 83 loanablefunds theory 120 Lucas model159ndash60 neoclassical model 118 119126ndash7 neutrality of money 119 parity200ndash1 superneutrality of money 121ndash2temporary equilibrium 81 82
intertemporal substitution hypothesis (ISH)49ndash51 60ndash3 72 73 75ndash6
invertibility condition 110investment 3 29 84 85 128 capital
adjustment costs 26ndash7 35 36 demand32 68ndash70 75 90 94 97 99 120206n40 Keynesian model 134 135neoclassical model 118 126 neutralityof money 119 optimal 30ndash2superneutrality of money 121 122supply-side variables 116
IS equation 79 90ndash1 98ndash9 118 126 191export demand 200 Keynesian model133 134 138ndash9 Lucas model 159 160
ISH see intertemporal substitutionhypothesis
lsquoislandrsquo paradigm 146ndash9
J-curve effect 221n13
Karni Edi 207n3Keynes John Maynard 4Keynesian model 5 79 88 130ndash45 199Knetter M 221n17Kydland FE 180ndash1 207n2 220n17
labor 19 21 27 42 66 adjustment costs37ndash8 demand 68 76 97 132ndash3 147215n9 217n4 Keynesian models 5marginal product of 30 optimalinvestment 31 participation 48ndash9temporary equilibrium 80 see alsoemployment labor supply wages
labor market 1 18 37 65 96ndash7 186aggregate supply 86ndash8 94 95 119intertemporal substitution hypothesis49ndash51 Keynesian model 131ndash3 134Lucas model 147 148 natural rate ofunemployment 89 90 neoclassicalmodel 83ndash4 118 124ndash5 127 Walrasrsquolaw 66ndash7 82 see also employment
labor reserve hypothesis 37ndash8labor supply 39 40 41 45 46ndash9 215n9
intertemporal substitution hypothesis49ndash51 72 73 75ndash6 210n15 Keynesian
model 132 133 Lucas model 147ndash8neoclassical model 79 83ndash4 117 123124 perfect foresight 66 70 71 realbalance effect 77 supply-side variables116 tax rates 99 see also employmentlabor
leisure 40ndash1 42 44 46ndash8 49 50 72life-cycle hypothesis 50 55ndash6 57linear regression analysis 150ndash3LM equation 79 90ndash1 98ndash9 118 126
191 Keynesian model 133 134 138ndash9Lucas model 159 160
loanable funds theory 120Lucas RE 6 50ndash1 146 149 156ndash7
164 167 203n13 210n15Lucas model 88 128 135ndash6 140 146ndash66
176 199Lucas supply function 79 137 153ndash6
158 174 175
Mankiw NG 111 112 113 136 214n45market clearing 4ndash5 6 7 15 50 79
commodity market 92 financial market93 Lucas model 146 164 wages 135
market experiments 11 14ndash16markets 1 2 20 78 187 aggregate
demand 86 90ndash1 capital 27 203n1205n24 foreign exchange 186ndash7 188191 193ndash5 198ndash9 200 futures 3 7 810 65 66 Lucas model 146 149 154sequential 67 see also financial marketlabor market money market
Marx Karl 79microeconomics 5 6 78 88 187mixed autoregressive moving average
processes 110Modigliani Franco 55ndash6 208n24ModiglianindashMiller theorem 32monetary policy 166 167ndash85 187
activist 169 171 174 discretionary136 167ndash8 184 219n1 Keynesianmodel 136ndash8 141 142 144ndash5 policyineffectiveness proposition 120 136137ndash8 172ndash9 182
money classical analysis 4 comparativestatic analysis 118 demand 71ndash4 7797 121 122 139ndash40 159 financialmarket equilibrium 84 householdproblem 41 42 43ndash4 45 46 66neoclassical model 117ndash18 neutralityof 119ndash20 136 142 173 174portfolio decision 39 57 58 59 72superneutrality of 120ndash2
232 Index
money (Continued)temporary equilibrium 80 82transaction costs 11 utility yield of 42see also money market money supply
money illusion 59 75ndash6 77 88 119 125210n15
money market 7 16 67 96ndash9 199aggregate demand 90 94 95equilibrium 85ndash6 92ndash3 126 159Keynesian model 133 134
money supply 3 76 100 119 168 190Keynesian model 134ndash5 142 Lucasmodel 159ndash60 161 162 monetarypolicy 144 169 170 173 178ndash9 180money market equilibrium 86 91neoclassical model 116 118ndash20 126ndash9purchasing power parity 200superneutrality of money 120ndash2 seealso money
moving average processes 109ndash10multiple equilibria models 113MundellndashFleming model 201
natural rate hypothesis 125 128ndash9 135166
Nelson CR 113neoclassical model 4ndash5 116ndash29 146 199
202n12 Keynesian model comparison133 136 Lucas model comparison 160162 modification of 130ndash1 purchasingpower parity 200 simple 78ndash95
new classical economics 4ndash5 6 7Newbold P 100 102 112 212n22numeraire 7 8 9ndash10
official reserve transaction balance 192open economy 186ndash201output 1 18 19 20 66ndash7 202n1
aggregate demand 86 90 94 95 139aggregate supply 86 87ndash8 classicalanalysis 4 comparative static analysis118 Dornbusch model 200 equilibrium80 82 91ndash2 94 126 expectations 6875 firm distribution constraint 65Keynesian model 5 133ndash4 135 137ndash8143ndash4 145 labor market equilibrium83 84 Lucas model 79 146 147148ndash9 153ndash5 157ndash62 165ndash6 monetarypolicy 144 168 170ndash2 173 174ndash5179 money supply shocks 128 movingaverage processes 109 neoclassicalmodel 117ndash18 122 123 125 126neutrality of money 119 open economy
187ndash8 189 190ndash1 199 simpletheoretical model 96ndash9 supply-sidevariables 116 136 time series model111ndash14
overlapping generations models 6
Patinkin D 8 11 120 202n12 critiqueof 117 equilibrium 83 84ndash5 91ndash4firms 210n5 labor supply 47
Payne JE 212n11perfect competition 7 11perfect foresight 3 87 firms 65 68ndash70
75 households 41 43 56ndash7 66 70ndash475 labor market equilibrium 83 125Lucas model 161 162 neoclassicalmodel 78 79 84 117 118 119 131Walrasrsquo law 66 67 82
perfect substitution 20 22 29 32period (discrete) analysis 4permanent income hypothesis 56ndash7Petty William 79Phelps ES 136ndash7 171 216n11Phillips AW 155Phillips curve 154ndash5 156ndash7 171 174ndash9
180 181 183Pindyck RS 101 102Plosser CI 113policy see monetary policypolicy ineffectiveness proposition 120
136 137ndash8 172ndash9 182Poole William 219n3portfolio decision 39ndash40 46 51 57ndash9 72
93 193preferences 11 40Prescott EC 180ndash1 207n2 220n17price elasticity of demand 195prices 1 2 3 19 68 accounting 7 10
13 15 aggregate demand 90 91 9495 aggregate supply 86 98autoregressive expectations 162ndash4classical analysis 4 comparative staticanalysis 116 demand-side variables117 equity shares 20 22 forecastingerrors 149ndash50 foreign goods 193ndash5197 future 41 59 75 76 householdbudget constraint 66 74 individualexperiments 11 Keynesian model 133134 135 137 145 labor marketequilibrium 83 84 Lucas model 146147 149 153 154 156ndash8 159ndash64market experiments 11 microeconomicfoundations 6 monetary policy 172ndash3money market equilibrium 86 natural
Index 233
rate hypothesis 125 natural rate ofunemployment 88ndash9 neoclassicalmodel 78ndash9 117ndash20 122ndash3 124ndash5126ndash8 130 136 146 new classicaleconomics 5 7 perfect foresight 71purchasing power parity 200 realindebtedness effect 21 relative 7 89ndash10 11 13 15ndash16 sequential markets67 superneutrality of money 120tatonnement process 15 temporaryequilibrium 80 82 Walrasian model 79ndash10 15 see also inflation
private international capital flows 192193 195ndash6 197ndash8 200
production 18ndash19productivity of labor 37purchasing power parity 200
Radford RA 8lsquorandom walkrsquo process 103 106 112ndash13Rapping LA 50ndash1 210n15rational expectations 3 128ndash9 136ndash8
141ndash4 164ndash6 172ndash4 176ndash8 182ndash4 seealso expectations
real balance effect 60 77 86 92ndash3 94121
real indebtedness effect 60real user cost of capital 31ndash2 36 69 70
94 120 209n7recontracting assumption 15reduced-form expressions 96 98 99ndash100
Lucas model 161 162 monetary policy168ndash9 172ndash3 177
reputation 184ndash5retained earnings financing 29ndash30 32 34risk 41Robb AL 208n25Rubinfeld DL 101 102
Sargent T 8 96 131ndash2 211n2 n6autoregressive expectations 164employees 217n2 investment demand206n40 Keynesians 219n2 linearregression analysis 152 monetarypolicy 136ndash7 167 168 170 172ndash4180 182 money supply 165 newclassical economics 203n13outputinflation tradeoff 157 Phillipscurve 155 superneutrality of money121 122
Say Jean-Baptiste 7Sayrsquos law 8 16seasonality 111 112
shareholders 25ndash6shares see equity sharesShiller RJ 162shocks 1 6 153 158 186 demand-side
117 monetary policy 170 moneysupply 128 129 135 Phillips curve156 time-series models 111ndash13 wages135ndash6
Simons HC 219n4Sims Christopher 100spot markets 2 37 130 131spot prices 10static analysis 2ndash4 47 50 79 116ndash23stationarity 102 103 104 105ndash9 110ndash11
113 114stationary analysis 2 3ndash4stochastic processes 6 41 101 102ndash3
110 172 173Stuart CE 99substitution effects 47ndash8 49 see also
intertemporal substitution hypothesissupply labor market equilibrium 83
Lucas supply function 79 137 153ndash6158 174 175 neoclassical model 119new classical economics 5 real balanceeffect 77 Sayrsquos law 7 16 temporaryequilibrium 80 86 Walrasian model 715 see also aggregate supply laborsupply money supply
tatonnement process 7 15 120tax issues 3 32 33 99Taylor JB 100 136ndash7 212n12 216n11technology 18ndash19 21terms of trade 194lsquotime inconsistency problemrsquo 180ndash2time series models 96 100 101ndash14Tobin James 206n40 219n2Tobinrsquos Q 27 35ndash7transaction costs 5 7 10ndash11 20
42 208n18
uncertainty 6 41unemployment 1 50ndash1 104 125
frictional 89 involuntary 211n13Lucas model 146 154ndash6 monetarypolicy 174ndash5 176 179 181ndash2 183184 natural rate of 88ndash90 135 156174ndash5
utility 13 17 40ndash1 42 46 48 61ndash2
velocity equations 139ndash40
234 Index
wages 18 19 23 24 32 36 aggregatedemand 90 91 aggregate supply 86ndash7anticipated 69 70 71 75ndash6 77 124household problem 42 43 44 47ndash866 intertemporal substitution hypothesis50 51 72 73 Keynesian model 130ndash3135ndash6 137ndash8 142ndash4 145 labordemand 68 97 labor marketequilibrium 83 84 labor participation48ndash9 Lucas model 147ndash8 moneysupply shocks 128 neoclassical model117 122ndash3 126 neutrality ofmoney 119 real wage illusion 123ndash5sticky 123 131 135ndash6 137temporary equilibrium 80 82 see alsoincome
Wallace N autoregressive expectations164 Keynesians 219n2 monetarypolicy 136ndash7 167 168 170 172 180182 money supply 165 superneutralityof money 121 122
Walrasrsquo law 1 8 14 16 85 91 balance ofpayments 190ndash1 excess demand 118financial assets 93 labor market 66ndash7market equilibrium 97 99 openeconomy 187 199 perfect foresight82 sequential markets 67
Walras Leon 1 16 202n2 203n2Walrasian models 7ndash11 15 18ndash20wealth 55 59 60 83Weber CE 209n2working hours 47ndash8
Understanding MacroeconomicTheory
At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activitycan be summarized by aggregate variables such as an economyrsquos total productionof various goods and services the aggregate level of unemployment the generallevel of interest rates and the overall level of prices Macroeconomics is the studyof movements in such economy-wide variables as output employment and prices
The focus of this book is on developing simple theoretical models that provideinsight into the reasons for fluctuations in such aggregate variables These modelsexplore how shocks or ldquoimpulsesrdquo to the economy impact individualsrsquo behaviorin specific markets and the resulting implications in terms of changes in aggregatevariables
Understanding Macroeconomic Theory will provide the reader with anin-depth understanding of standard theoretical models Walrasian Keynesian andneoclassical It is written in a concise accessible style and will be an indispensabletool for all students who wish to gain firm grounding in the complexities ofmacroeconomic theories
John M Barron is the Loeb Professor of Economics in the Krannert School ofManagement at Purdue University
Bradley T Ewing is the Jerry S Rawls Endowed Professor in OperationsManagement in the Rawls College of Business at Texas Tech University
Gerald J Lynch is Professor of Economics in the Krannert School of Managementat Purdue University
Routledge Advanced Texts in Economics and Finance
Financial EconometricsPeijie Wang
Macroeconomics for Developing Countries 2nd editionRaghbendra Jha
Advanced Mathematical EconomicsRakesh V Vohra
Advanced Econometric TheoryJohn S Chipman
Understanding Macroeconomic TheoryJohn M Barron Bradley T Ewing and Gerald J Lynch
Understanding MacroeconomicTheory
John M Barron Bradley T Ewingand Gerald J Lynch
First published 2006by Routledge270 Madison Ave New York NY 10016
Simultaneously published in the UKby Routledge2 Park Square Milton Park Abingdon Oxon OX14 4RN
Routledge is an imprint of the Taylor amp Francis Group an informa business
copy 2006 John M Barron Bradley T Ewing and Gerald J Lynch
All rights reserved No part of this book may be reprinted orreproduced or utilized in any form or by any electronicmechanical or other means now known or hereafterinvented including photocopying and recording or in anyinformation storage or retrieval system without permission inwriting from the publishers
Library of Congress Cataloging in Publication DataBarron John M
Understanding macroeconomic theory John M BarronBradley T Ewing and Gerald J Lynch
p cmIncludes bibliographical references and index1 Macroeconomics I Ewing Bradley T II Lynch Gerald J III Title
HB1725B3753 2006339ndashdc22 2005026372
British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library
ISBN10 0ndash415ndash70195ndash3 ISBN13 978ndash0ndash415ndash70195ndash2 (hbk)ISBN10 0ndash415ndash70196ndash1 ISBN13 978ndash0ndash415ndash70196ndash9 (pbk)ISBN10 0ndash203ndash08822ndash0 ISBN13 978ndash0ndash203ndash08822ndash7 (ebk)
This edition published in the Taylor amp Francis e-Library 2006
ldquoTo purchase your own copy of this or any of Taylor amp Francis or Routledgersquoscollection of thousands of eBooks please go to wwweBookstoretandfcoukrdquo
Contents
1 Introduction 1
2 Walrasian economy 7
3 Firms as market participants 18
4 Households as market participants 39
5 Summarizing the behavior and constraints of firms andhouseholds 64
6 The simple neoclassical macroeconomic model (withoutgovernment or depository institutions) 78
7 Empirical macroeconomics traditional approaches and timeseries models 96
8 The neoclassical model 116
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model 130
10 The Lucas model 146
11 Policy 167
12 Open economy 186
Notes 202References 223Index 228
1 Introduction
The topics of macroeconomics
At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activity canbe summarized by aggregate variables such as an economyrsquos total production ofvarious goods and services the aggregate level of employment and unemploymentthe general level of interest rates and the overall level of prices1 Macroeconomicsis the study of movements in such economy-wide variables as output employmentand prices
The focus of this book will be on developing simple theoretical models thatprovide insight into the reasons for fluctuations in such aggregate variables Thesemodels explore how shocks or ldquoimpulsesrdquo to the economy (eg changes to tech-nology the money supply or government policy) impact individualsrsquo behavior inspecific markets and the resulting implications in terms of changes in aggregatevariables
An overview of some facets of theoreticalmacroeconomic analysis
Given the breadth of economic activity in an economy the study of macroeco-nomics must involve an examination of a variety of different markets For instanceit is common for macroeconomic analysis to consider exchanges of labor servicesin the labor markets of consumption and capital goods in the output marketsand of financial assets in the financial markets The fact that macroeconomicssimultaneously analyses exchanges of different goods in different markets meansthat macroeconomic theory is a general equilibrium theory That is macro-economic theory must by necessity incorporate the links across markets that arefundamental to general equilibrium analysis As we will see throughout this booka key reflection of the links across markets is Walrasrsquo law named in honor ofthe nineteenth-century French economist Leon Walras2 Simply put Walrasrsquo lawnotes that the budget constraints faced by individual agents in the economy sug-gest that if n minus 1 of the n markets in the economy are in equilibrium then the nthmarket must be in equilibrium We will repeatedly rely on Walrasrsquo law or variantsof it to simplify macroeconomic analysis
2 Introduction
While macroeconomic theories have in common (a) an attempt to explainfluctuations in aggregate variables and (b) a general equilibrium character thereremain wide differences among macroeconomic models Below we break downthese differences across macroeconomic models in several ways in order to makesome sense of what passes for simple theoretical macroeconomic analysis
Static dynamic and stationary analysis
One way of breaking down macroeconomic analyses is into static models dynamicmodels and stationary analysis of dynamic models Static macroeconomic modelsanalyze the economy at a point in time They consider the determination of pro-duction exchange and prices of various goods only for the markets that currentlyexist John Hicks (1939) sketched out an analysis of ldquospotrdquo or ldquotemporaryrdquo equi-librium The advantage to such an approach is that it provides for rather simpleldquocomparative staticrdquo analysis of the effects of changes in a variety of exogenousvariables on the endogenous variables3 Such static analysis is useful in providinginsight into a variety of questions of interest
Static macroeconomic analysis can be viewed as a modification of a Walrasiangeneral equilibrium analysis or what is commonly referred to as ldquoArrowndashDebreutheoryrdquo (Arrow and Hahn 1971 Debreu 1959) In ArrowndashDebreu theory eachcommodity is described by its physical characteristics its location and its dateof availability It is assumed there are a complete set of spot and forwardmarkets Prices adjust to clear all markets However if one restricts attentionto just spot markets then one moves from traditional Walrasian generalequilibrium to an analysis of ldquotemporary equilibriumrdquo a phrase coined by Hicks(1939) This restriction to spot markets is one element of static macroeconomicanalysis4
A second element of static models is that if there is a future then staticmacroeconomic analysis simply assumes given expectations of future prices andenvironment How expectations of future events are formed is left unspecified sothat expectations of future prices become simply an element in the set of exogenousvariables5
While static analysis provides insights there are several disadvantages of staticanalysis severe enough that it alone does not provide an adequate grounding inmacroeconomic analysis The key disadvantage of static analysis is that it breaksties between current events and future events To show the limitations of staticanalysis let us suppose that underlying a simple static macroeconomic analy-sis of current markets is a microeconomic analysis of individualsrsquo decisions thatidentifies the anticipated future level of prices as one of the exogenous variablesaffecting current behavior As we have seen static analysis takes expectations ofsuch variables as future prices as exogenous variables Doing so however resultsin (a) an incomplete enumeration of exogenous variables that can impact currenteconomic activity and (b) a potentially incomplete accounting of the effects of theimpact on current economic activity of a change in those exogenous variables thatare identified by the analysis
Introduction 3
To illustrate the first point of an incomplete listing of exogenous variables letus suppose that the static model identifies changes in the current money supplyas one factor that influences current prices This suggests that if we replicate thestatic analysis in future periods changes in the money supply in the future would beshown to affect prices at that time It seems natural to then presume that individualsrsquoanticipation of future prices would incorporate this link between changes in thefuture money supply and future prices in forming their expectations of futureprice levels so that the anticipated future money supply becomes a determinantof current activity6 Yet static analysis since it does not analyze markets beyondthe current period will not identify the potential impact of future changes in themoney supply on current activity7
To illustrate the second point of an incomplete accounting of the effects of achange in an exogenous variable let us suppose that underlying the static macro-economic analysis of current markets is a microeconomic analysis of firmsrsquo currentinvestment behavior that identifies the anticipated future tax levels as well as futureprices as two exogenous variables affecting investment decisions Thus staticanalysis would suggest that a change in future tax levels will impact current activitythrough the direct effect on current investment It is not hard to see however that(a) the current change in investment means a different future capital stock and(b) the change in future tax levels could affect future as well as current investmentEither or both of these changes would likely impact future prices and if such animpact were anticipated be a second way that future tax changes impact currentactivity8
An obvious way to avoid the above problems is to introduce forward marketsso that the macroeconomic analysis determines the prices of goods to be tradedin the future along with the prices of goods traded at the current time9 In doingso we have moved from static to dynamic analysis That is the macroeconomicmodels now determine the paths of variables (such as prices) over time rather thanprices (and other variables) at only one point in time
In a deterministic setting this expansion of dynamic analysis incorporates thenotion of ldquoperfect foresightrdquo in which individuals correctly anticipate all futureprices If there were uncertainty the analysis indexes goods by both the date oftrade and the ldquostate of naturerdquo with trades contingent on the realized state ofnature10 The result is that at each date there is a distribution of potential prices atwhich trade for a good could occur and given common knowledge of likelihood ofthe states of nature expectations of future prices would be defined by the analysis(ldquorational expectationsrdquo)
Once dynamic analysis is introduced we can consider a special limiting form ofdynamic analysis termed stationary analysis The aim of stationary analysis is toidentify in the context of a dynamic model the limiting tendencies of endogenousvariables such as the capital stock or the rate of growth in prices given that theexogenous variables remain constant or stationary over time11
While stationary analysis is distinct from static analysis in some cases one canthink of static analysis as a form of stationary analysis That is static analysis insome cases can be viewed as the outcome that would emerge each period given
4 Introduction
that the exogenous variables remain constant (or in some cases grow at a steadyrate over time) and given that one picks the correct fixed level of certain keyexogenous variables (eg the capital stock and the rate of change in the moneysupply) Note however that this implies that for static analysis to perfectly mimicstationary analysis one must to all intents and purposes have first executed theunderlying dynamic analysis
Period (discrete) versus continuous-time analysis
Macroeconomic analysis can be broken down into period or discrete-time macro-economic models and continuous-time macroeconomic models Substantivedifferences in terms of theoretical predictions do not exist between these two typesof analyses if one is careful to assure identical underlying assumptions Yet the twoanalyses do differ in the analytical techniques used For instance while discretemacroeconomics relies on the techniques of dynamic programming and differenceequations to characterize elements of the model in similar circumstances con-tinuous macroeconomic analysis turns to the techniques of optimal control anddifferential equations
Although substantive issues are not raised by the discrete- versus continuous-time dichotomy it is sometimes argued that one is preferred to the other Forinstance an attractive feature of the continuous-time analysis is that it highlightsquite clearly the distinctions between stocks and flows something that is not soclearly discernable in discrete analysis On the other hand an attractive feature ofdiscrete analysis is that it makes more transparent the link between the theoreticalanalysis and empirical testing since such analysis coincides with the obvious factthat empirical data on macroeconomic variables is discrete
New classical economics versus non-market-clearing
Classical analysis refers to the widely adopted view of how the macroeconomyshould be modeled that existed prior to the experience of the Great Depressionand John Maynard Keynesrsquo General Theory of Employment Interest and Money(1936) In classical theory the real side of the economy was separate from themoney side Classical analysis of the ldquorealrdquo side of the economy is aimed atdetermining such variables as total production relative prices the real rate ofinterest and the distribution of output Classical analysis of the money side ofthe economy meant analysis is aimed at determining money prices and nominalinterest rates
The separation of the monetary side from the real side in classical or neoclassicalanalysis led to the prediction that monetary changes do not have any effect onreal variables such as total output12 A similar prediction is often obtained bymore recent macroeconomic analysis and this is one reason why this more recentanalysis is referred to as the new classical economics13 Alternative labels ofthese new classical models include rational expectations models with market
Introduction 5
clearing neoclassical models of business fluctuations and equilibrium businesscycle models
A common feature of the analyses of new classical economics besides thefact that it suggests a divorcement of monetary changes from the real side ofthe economy is that prices are determined in the analysis so as to clear marketsThis view that prices serve to equate demands and supplies a view common tomicroeconomics is taken as an important strength of the analysis for it meansthat the models have consistent ldquomicroeconomic foundationsrdquo One implicationof the market-clearing assumption is the same as in microeconomics ndash the analysissuggests that all gains to exchange have been extracted
Contrasting the new classical economics with what preceded it helps one putthis rebirth of classical analysis into perspective Following the Great Depressionmacroeconomic analysis took as its main premise the idea that markets did notclear ndash in particular that prices did not adjust In this context the business cyclewas defined by ldquomarket failurerdquo and the role of government to stabilize the eco-nomy was clear There are a number of different types of non-market-clearing orKeynesian models One version of such Keynesian models that popularized byPatinkin (1965 chapters 13ndash14) Clower (1965) and Barro and Grossman (1971)takes as given output prices such that the output market fails to clear A secondmodel popularized by Fischer (1977) Phelps and Taylor (1977) and Sargent(1987a) as an alternative formalization of the Keynesian model takes as given theprice of labor such that the labor market fails to clear
The common theme of these non-market-clearing analyses is that for variousreasons prices do not clear markets and concepts such as excess demand and supplyplay a role in the analysis Yet no concise reason is given as to why there is marketfailure other than suggesting such items as ldquocoordination problemsrdquo and ldquotrans-action costsrdquo14 The result is that such analysis is challenged by the new classicaleconomics as lacking the microeconomic foundations for price determination AsHowitt (1986 108) suggests such a view ldquoforces the proponent of active stabi-lization policy to explain the precise nature of the impediments of transacting andcommunicating that prevent private arrangements from exhausting all gains fromtraderdquo15 This is not an easy task according to Howitt since ldquoimpediments to com-munication in a model simple enough for an economist to understand will typicallyalso be simple enough that the economist can think of institutional changes thatwould overcome themrdquo (1989 108)
Microeconomic foundations and aggregation issues
An important feature of macroeconomic analysis is that it reflects the aggre-gation of individual decisions A common approach to such aggregation is toassume ldquorepresentativerdquo agents characterize their optimal behavior then use suchbehavioral specifications in building the macroeconomic model Thus much ofmacroeconomic analysis entails looking at individualsrsquo decisions such as house-holdsrsquo decisions to work consume and save or firmsrsquo decisions to produceborrow and invest in capital
6 Introduction
These characterizations of optimizing individual behavior make up part of thebuilding blocks or ldquomicroeconomic foundationsrdquo of macroeconomic analysisYet microeconomic foundations of macroeconomics are not restricted to suchanalysis For instance such foundations also include a characterization of howprices in individual markets are determined as we saw in our discussion of newclassical economics
In developing the microeconomic foundations of macroeconomic models wewill often be struck by the extent to which the analysis restricts any role forheterogeneity or diversity among the individual agents in the economy Yet suchdiversity can in certain instances be critical to the analysis One attempt to introducediverse or heterogeneous agents into macroeconomic analysis is represented by theoverlapping generations models These models also have the advantage of beinggenuinely dynamic in nature and as such represent one area of macroeconomicsthat has recently received significant attention
Deterministic versus stochastic
In recent years an important element to macroeconomic models has been to intro-duce stochastic elements The rationale is clear the presence of uncertainty as tofuture events is real As noted by Lucas (1981 286)
the idea that speculative elements play a key role in business cycles that theseevents seem to involve agents reacting to imperfect signals in a way whichafter the fact appears inappropriate has been commonplace in the verbaltradition of business cycle theory at least since Mitchell It is now entirelypractical to view price and quantity paths that follow complicated stochasticprocesses as equilibrium ldquopointsrdquo in an appropriately specified space
As the quote suggests in dynamic models especially for new classical economicswhere market clearing is presumed stochastic elements are incorporated into theanalysis so that the role played by shocks to an economy in a dynamic setting canbe well defined
2 Walrasian economy
Introduction
This chapter develops a competitive model of the economy The key assumptionsneeded for this model are spelled out in detail One important aspect of this modelis the ldquonumerairerdquo or the commodity price that is used as a reference in the modelA distinction is made between accounting prices and relative prices and it isseen that traditional general equilibrium analysis does not determine the levelof accounting prices but rather simply relative prices A number of modificationsto the model are mentioned and add a sense of ldquorealismrdquo to the framework Thesemodifications include the introduction of futures markets quantity constraintsand the costs associated with carrying out a transaction
The chapter continues by considering individual decision-making and the theoryof the consumer and how this relates to the determination of market demand Thegeneral equilibrium conditions are stated in terms of relative prices and allocationsA theme carried throughout this book is emphasized and that is the use of theaggregate budget constraint Finally it is shown that explicitly excluding moneyas a ldquomarketrdquo allows one to understand how Sayrsquos conclusion that ldquosupply createsits own demandrdquo is arrived at in an economy composed of a single aggregatecommodity
A simple Walrasian model
As discussed previously the idea that prices adjust to clear markets is commonto much of new classical macroeconomics Thus we begin our examination ofmacroeconomic analysis by considering an economy consisting of perfectly com-petitive markets This means that individuals take prices at which exchanges canbe made as parametric and prices adjust to eliminate excess demands so that indi-vidualsrsquo plans at given prices are feasible As the title to this section suggests sucha characterization is sometimes referred to as being indicative of a ldquoWalrasianrdquoeconomy Walras described the process by which prices adjust to excess demand orsupply as a groping or tatonnement process (see Walras 1954)1 A fictitious auc-tioneer calls out different prices for the various markets and no exchange occursuntil equilibrium prices are reached2
8 Walrasian economy
Our analysis also begins at a very simple level The term ldquosimplerdquo reflects atleast the following three characteristics of the economy that we consider
1 It is a barter economy That is any commodity can be freely traded for anyother commodity There is no role for a medium of exchange (money) inreducing the costs of arranging exchanges
2 It is an exchange economy That is there is no production Instead individualshave initial fixed endowments of various commodities
3 It is a timeless economy Goods are indexed by physical characteristics andlocation but not dated according to availability This rules out futures mar-kets or the formation of expectations of future events and planning Such aneconomy was suggested by Hicks (1939) Patinkin (1965 Chapter 1) andHansen (1970 Chapter 4) provide a more detailed view of such an economyAn actual example of a pure barter exchange economy is offered by Radford(1945) The simple model of the economy developed below is useful in high-lighting such concepts as relative prices the numeraire individual versusmarket experiments aggregation issues conditions for general equilibriumand Walrasrsquo and Sayrsquos laws
The first model developed below also takes an approach to modeling the econ-omy that is in vogue in current theoretical macroeconomics As Sargent (1987b)states the ldquoattraction of (such) general equilibrium models is their internal con-sistency one is assured the agentsrsquo choices are derived from a common set ofassumptionsrdquo
Yet this advantage of general equilibrium analysis is not fully exploited until theelements of time money and production are introduced and so we will expand thediscussion in subsequent sections by introducing such features Below we intro-duce in more detail some of the key assumptions underlying the simple Walrasianmodel we start with
Key assumptions underlying a simple Walrasian model
As noted by Debreu (1959 74) ldquoan economy is defined by m consumers (charac-terized by their consumption sets and their preferences) n producers (characterizedby their production sets) and the total resources (the available quantities of thevarious commodities which are a priori given)rdquo As discussed above we considera special case of Debreursquos ldquoconcept of an economyrdquo one in which productionis absent As the following set of assumptions makes clear we also restrict ouranalysis to private ownership economies with a price system In particular assume(partial listing)
Assumption 21 There are m individuals (agents) in the economy indexed bya = 1 m There are T commodities indexed by i = 1 T 3 Agent arsquos initial
Walrasian economy 9
endowment of commodity i is denoted by cai cai ge 0 i = 1 T Naturally
msuma=1
cai = ci gt 0
Note that this assumption reflects an exchange economy in which private propertyrights exist ldquoPrivate property rightsrdquo means that for each unit of each good theexclusive right to determine use has been assigned to a particular individual
Assumption 22 All exchanges occur at a single point in time
Assumption 23 Each individual confronts the same known set of prices at whichexchange can occur4 A relative (purchase) price of commodity i indicates the unitsof commodity j required to purchase one unit of commodity i A relative (sale)price of commodity i indicates units of commodity j received when one unit ofcommodity i is sold
Assumption 24 Purchase and sale prices are identical for each commodity Thismeans that there are no ldquoprice spreadsrdquo which would suggest either a gain to anindividual buying and selling the same commodity or the presence of costs tomaking an exchange5 Thus for the T commodities there are T 2 exchange ratesor relative prices taking two commodities at a time
The numeraire
While there are T 2 exchange rates the complete set of exchange rates can bededuced directly or indirectly by the set of T minus 1 relative prices
(π1j πjminus1 j πj+1 j πTj)
where the πij denotes the price of commodity i in terms of commodity j6 In thelisting of relative prices (π1j πjminus1 j πj+1 j πTj) commodity j is referredto as the ldquonumerairerdquo
To see how the set of relative prices reduces to T minus 1 we rely on the factthat πhh = πhjπhj for all h j and k Let us see what this means for a simpleexample of three commodities h j and k There are then T 2 or nine differentrelative prices which are πhh πjj πkk πhj πjh πhk πkh πjk and πkj But we canuse the relationship πhh = πhjπhj to reduce this to T minus 1 = 2 relative prices withinformational content In particular we know that
1 πhh πhkπhj and similarly for πjj and πkk when h = j = k In other wordsthe exchange rate of a commodity with itself is unity This reduces from nineto six the number of relative prices for which information is required
2 πjh = 1πhj when k = j (such that πkj = πkk = 1) Similarly πhk = 1πkhand πjk = 1πkj For example if the jth commodity is pears and the hth
10 Walrasian economy
commodity is oranges then if πjh = 3 (3 oranges = 1 pear) πhj = 13(13 pear = 1 orange) This reduces from six to three the number of relativeprices for which information is required to reconstruct the complete set ofrelative prices
3 Finally πkh = πkjπhj (for k = j = h) For example if the jth commodity ispears the kth commodity is apples and the hth commodity is oranges thenif πkj = 3 (3 pears = 1 apple) and πhj = 13 (13 pears = 1 orange) thenπkh = 9 (9 oranges = 1 apple) That is with 9 oranges you can get 3 pearswhich in turn will purchase 1 apple This drops us from three to two relativeprices required to reconstruct the complete set of relative prices Since thenumber of commodities T = 3 we have shown how the T 2 relative pricescan be constructed from T minus 1 relative prices
In subsequent discussions we will arbitrarily let commodity T be the numerairesuch that the set of relative prices can be summarized by
(π1T πTminus1T )
For simplicity let us change notation such that πiT = πi i = 1 T Thus theset of relative prices can be rewritten as
(π1 πTminus1)
Note that πT = 1 since we are assuming the T th good is the numeraireIn traditional general equilibrium theory there is a concept of ldquoaccounting pricesrdquo
as well as the concept of relative prices Accounting prices can be represented bya set of real numbers (say pi i = 1 T ) attached to the T commodities7 Therelationship between these accounting prices and the set of relative prices that doimpinge on behavior is that πi = pipT i = 1 T (for Pi = 0) As we will seetraditional general equilibrium analysis does not determine the level of accountingprices but rather simply relative prices8
Anticipating future modifications
Before continuing it might be useful to anticipate some of the subsequent changeswe will make in the characterization of the economy Besides the introduction ofproduction we will
bull introduce time implying either forward (futures) markets or an important rolefor expectations of future spot prices
bull introduce quantity constraints that can arise if prices are fixed at non-market-clearing levels (ie depart from a Walrasian framework) and
bull introduce the cost of carrying out an exchange
Walrasian economy 11
With respect to point (c) such costs have been characterized by Coase (1960) asldquotransaction costsrdquo arising from the costs ldquonecessary to discover who it is that onewishes to deal with and to inform people that one wishes to dealrdquo the costs ofldquoconduct[ing] negotiations leading up to a bargain and to draw up a contractrdquoand the costs ldquoto undertake the inspection needed to make sure that the terms ofthe contract are being observedrdquo9 Such costs will alter the nature of contracts(exchange agreements) formed and may provide the reason for ldquoprice rigiditiesrdquoSuch costs also suggest a role for money
Transaction costs are assumed to be zero in the simple Walrasian sys-tem outlined above Sometimes this is referred to as a situation wherethere is ldquoperfect informationrdquo or where markets are complete and ldquoperfectlycompetitiverdquo
Individual experiments
General equilibrium analysis can be divided into what Patinkin (1965) refers to asldquoindividual experimentsrdquo and ldquomarket experimentsrdquo In the context of the simpleWalrasian barter exchange economy individual experiments consider the behaviorof individual agents given an initial endowment and preferences when confrontedwith a set of prices Market experiments consider the resulting determination ofprices
In the simple Walrasian model under consideration individual experimentsreplicate standard microeconomic analysis of consumer behavior In particularassume
Assumption 25 Individual arsquos preferences are described by his utility functionua(ca1 caT ) where ca1 caT denote agent arsquos consumption bundle cai ge 0i = 1 T ua maps the set of all T -tuples of non-negative numbers into the setof all real numbers (ua RT+ rarr R) We make the appropriate assumptions withrespect to individualsrsquo preferences such that a utility function exists and is wellbehaved10
Assumption 26 Individual a will choose the most preferred consumption bundlefrom the set of feasible alternatives (rationality) Given the possibility of cost-less exchange at the set of relative prices represented by (π1 πTminus1) feasibleconsumption bundles or sets (ca1 caT ) are defined by
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0
wheresumT
i=1 πi cai denotes the initial endowment of individual a in terms ofcommodity T The above expression defines the budget set
12 Walrasian economy
The consumer problem
From Assumptions 25 and 26 individual arsquos optimum consumption bundle is thesolution to the problem
maxCa1CaT
ua(ca1 caT )
subject to
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0 cai ge 0 i = 1 T
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λ
L(ca1 caT λ) = ua(ca1 caT ) + λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
with first-order (necessary) conditions being11
partL
partcaile 0 i = 1 T
partL
partcaicai = 0 i = 1 T
cai ge 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
λ ge 0
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λmicro1microT
L(ca1 caT λ micro1 microT ) = ua(ca1 caT )
+ λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
Walrasian economy 13
with the necessary conditions being
partL
partcai= 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
partL
partmicroige 0 i = 1 T
microipartL
partmicroi= 0 i = 1 T
λ ge 0
microi ge 0 i = 1 T
Individual demands and excess demands
The optimal consumption bundle for agent a is defined by the above first-orderconditions and will be denoted by the (demand) set
(ca1 caT ) cai ge 0 i = 1 T
Individual arsquos demand functions will be of the form
cdai
(π1 πTminus1
Tsumi=1
πi cai
) i = 1 T
That is individual arsquos demand (consumption) of commodity i depends on the T minus1relative prices of commodities and the initial endowment Note that the form ofthe utility function implies the utility-maximizing consumption bundle meets thebudget constraint with equality Thus at the optimal bundle we have
partL
partλ=
Tsumi=1
πi cai minusTsum
i=1
πicdai = 0
An important point to note about demand functions is that they are homogeneousof degree zero in what might be called accounting prices12 Accounting prices aredefined such that pi = πi middot pT i = 1 T so it is clear that if all prices increaseby the multiple θ relative prices and the initial endowment are unchanged
Individual arsquos excess demand function for commodity i is defined byzai = cd
ai minus cai If zai is positive agent a is a net buyer of commodity i whileif zai is negative the agent is a net seller of commodity i The market value
14 Walrasian economy
(in terms of the numeraire) of the quantity of the ith commodity that individuala seeks to exchange (buy or sell) is then given by πizai From the budget con-straint we know that
sumTi=1 πi(cd
ai minus cai) = 0 orsumT
i=1 πizai = 0 In other words foreach individual the market value (in terms of commodity T ) of individual excessdemands must sum to zero This rather obvious finding generates what is referredto as Walrasrsquo law as we will see
Market experiments
In the previous section we reviewed the nature of individual demand functions andindividual excess demand functions Now consider the collection of m individualsAggregating or summing individual demand functions we obtain an aggregate orldquomarketrdquo demand function for commodity i of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci1 Tsum
i=1
πi
)equiv
msuma=1
cdai(middot)
Similarly summing agentsrsquo excess demand functions for commodity i gives usthe aggregate or ldquomarketrdquo excess demand function for commodity i of the form
zi equivmsum
a=1
(zai) equivmsum
a=1
(cdai minus cai)
Note that a zero aggregate excess demand for commodity i does not imply thatno exchange of commodity i occurs among the m agents However as we haveseen a zero individual excess demand for commodity i does imply no exchangeof commodity i by that particular individual
Aggregation issues
So far our aggregations have remained true to the underlying microeconomicanalysis Yet this is rarely the case in macroeconomic analysis which typicallyabstracts from what might be termed ldquodistributionalrdquo effects An example of thisin the above context as we will see later is to ignore the effects of the distributionof initial endowments across individuals on market demands such that the marketdemand function for commodity i is assumed to be of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci
)
where
Tsumi=1
πicai equivmsum
a=1
(Tsum
i=1
πi cai
)
Walrasian economy 15
As you can see with heterogeneity (either in initial endowments or preferences)such a posited aggregate market demand function is unlikely to follow exactlyfrom the underlying microeconomic analysis One should keep in mind suchapproximations when interpreting macroeconomic analysis
Equilibrium an isolated market
With respect to a single market equilibrium is characterized by an accounting pricepi and implied relative price πi = pipT such that cd
i = ci (demand equals fixedendowment) or equivalently zi = 0 (excess demand equals zero) The tatonnementprocess is the description of how prices change to clear the market In the Walrasianmodel movement toward equilibrium the tatonnement process involves twofacets
1 The Walrasian excess demand hypothesis which indicates that the accountingprice of commodity i rises if there is excess demand and falls if there is excesssupply That is
dpi = fi(zi) i = 1 T fi(0) = 0dfidzi
gt 0
In terms of relative prices
dπi = dpi
pT= f (zi)
pT i = 1 T minus 1
Note that the change in price is not across time since each market is assumedto clear instantaneously at the same point in time
2 The recontracting assumption which states that offers to buy or sell at var-ious relative prices are not binding unless market(s) clear Only when theequilibrium price (or price vector) is obtained are contracts then made final
General equilibrium (conditions)
A general equilibrium will be characterized by a set of T minus 1 relative prices(πlowast
1 πlowasttminus1) and allocations (clowast
a1 clowastaT ) for individual a a = 1 m such
that
clowastai = cd
ai
(πlowast
1 πlowasttminus1
Tsumi=1
πi cai
) i = 1 T a = 1 m (21)
rArr clowasti equiv
msuma=1
clowastai =
msuma=1
cdai equiv cd
i
clowasti = ci i = 1 T (22)
16 Walrasian economy
Equation (21) indicates that the equilibrium allocation must be optimal in thatit must satisfy all demands for commodities at the specified set of pricesEquation (22) indicates that such an allocation must be feasible that is sumto total resource endowment Together these two conditions imply a set of relativeprices such that excess demands are zero or
cdi = ci i = 1 T
For questions concerning the existence uniqueness and stability of generalequilibrium consult Varian (1992) Debreu (1959) and Arrow and Hahn (1971)
Walrasrsquo law and Sayrsquos law
Note that the above statement of general equilibrium involves setting T excessdemand equations equal to zero but there are only T minus 1 unknowns (relativeprices) Walras solved this problem by showing that one of the equations arbitrarilychosen can be deduced from the other T minus 1 equations In other words there areonly T minus 1 independent equations To show the dependency remember that thebudget constraint for each individual is given by
Tsumi=1
πicdai minus
Tsumi=1
πi cai = 0
Summing across all individuals it must then be the case that
msuma=1
(Tsum
i=1
πi cai minusTsum
i=1
πicdai
)= 0
Rearranging and substituting in cdi for
summaminus1 cd
ai and ci forsumm
aminus1 cai we obtain
Tsumi=1
πi(cdai minus cai) = 0 or
Tsumi=1
πizi = 0
The above is an explicit statement of Walrasrsquo law Walrasrsquo law states that the sumof the excess demands across all markets must be zero Note that in summing theexcess demand of each commodity is weighted by its relative price so that we aresumming common units (ie all excess demands are in units of the numeraire)The above aggregate budget constraint is sometimes referred to as Sayrsquos law orSayrsquos identity If there is a distinction between the two it is that Sayrsquos law explicitlyexcluded money as a ldquomarketrdquo In this setting one can understand how Sayrsquosconclusion that ldquosupply creates its own demandrdquo is arrived at in an economycomposed of a single aggregate commodity
Walrasian economy 17
Conclusion
This chapter has developed a general equilibrium framework that sets the stagefor a thorough understanding of how the macroeconomy works Particular atten-tion has been paid to the development of relative prices and the development ofaggregate demand through a process of many individual consumers operating in anenvironment in which they set out to maximize their own utility This frameworkand the tool of constrained optimization is used throughout this book
3 Firms as market participants
Introduction
In this chapter the simple Walrasian model is discussed in the context of moneyfinancial assets and production The chapter clearly illustrates the firmrsquos objectivethat is to maximize profits However the firm is constrained in that it must financepurchases of capital and equipment as well as pay its workers Moreover attentionis paid to all the costs faced by the firm not just the obvious ones The investmentand financing decisions of firms are discussed and issues related to Tobinrsquos Q anddebt-to-equity are explored This chapter provides a detailed examination of therole that firms play in the macroeconomy
A simple Walrasian model with money financialassets and production
In the exchange economy we just considered endowments of the commodity goodwere magically bestowed on individuals each period We now introduce productionas the source of commodities At the start of each period there now exists a ldquolaborrdquomarket in which labor services are exchanged New agents denoted ldquofirmsrdquo hirethe labor services provided by households During the period firms combine thelabor services with an existing capital stock to produce output (commodities)which is sold in the output market net of output retained to replace capital usedup during production Revenues from the sale of output are distributed to ldquohouse-holdsrdquo in the form of wages during the period At the end of the period interestpayments and dividends are made to households out of revenues Each period firmsalso enter the output market to augment their capital stock with such purchasesfinanced by the issue of bonds and a new financial asset denoted ldquoequity sharesrdquoIn particular we assume
Assumption 31 There are new agents in the economy denoted as ldquofirmsrdquo Theseagents are initially endowed at time t with a capital stock K and a technologyfor transforming capital services from a capital stock and labor services into a
Firms as market participants 19
single ldquocompositerdquo commodity The technology is summarized by the productionfunction
yt = f (Nt K)
where K denotes the capital stock the firm inherits at time t Nt denotes the employ-ment of labor services arranged at time t for period t (from time t to time t + 1)and yt denotes the constant rate of production of the commodity for period t (fromtime t to time t + 1)1 Similarly for period i (i = t + 1 t + 2 ) which runsfrom time i to time i + 1 output produced is given by2
yi = f (Ni Ki)
Assumption 32 During each period firms sell the output produced in the outputmarket For output produced during period t (from time t to time t+1) let pt denoteits price when it is sold during the period up to and including time t + 1 Let pt+1denote the price of output produced during period t + 1 that is sold beyond timet +1 up to and including time t +2 and so on At time t the price level associatedwith the prior period is denoted by p
Assumption 33 At the start of each period households rent their labor servicesto firms for the period At the start of period t agreements to exchange the Ntlabor services during period t (from time t to t + 1) are entered into at the moneywage rate denoted by wt Similarly at the start of period t + 1 Nt+1 labor servicesare exchanged at the money wage rate wt+1 and so on3
Assumption 34 At the end of each period two types of financial assets areexchanged bonds (in the form of perpetuities) and equity shares Bonds promiseto pay a fixed money (coupon) payment z each future period in perpetuity Let pbpbt and pbt+1 denote the money price of such bonds in markets at the end of periodstminus1 t and t+1 respectively the gross (nominal) interest rates over period t (fromtime t to time t + 1) and over period t + 1 (from time t + 1 to time t + 2) are thengiven by4
1 + r = (z + pbt)pb
1 + rt = (z + pbt+1)pbt
Note that if ri = rt i = t + 1 then successive substitution for the priceof bonds in future periods will result in the following expression for the price ofbonds at time t
pb = 1
1 + r(z + pbt) = 1
1 + r
[z +
infinsumi=1
z
(1 + rt)i
]
= 1
1 + r
(z + z
rt
)
20 Firms as market participants
where pbt = zrt The number of previously issued bonds outstanding at time t isdenoted by B5
Assumption 35 Equity shares are the second type of financial asset exchangedat the end of each period Equity shares (ldquostocksrdquo) are contracts that obligate theissuer (firms) to pay to bearers at the end of each future period the income from thesale of output produced during the period net of other contractual obligations ofthe firms (eg wage payments to suppliers of labor services and interest paymentsto holders of bonds issued by firms) Let S denote the number of previously issuedequity shares outstanding at time t Holders of these equity shares are the ldquoownersof the firmsrdquo Let pe pet and pet+1 denote the money price of equity sharesexchanged in markets at the end of periods t minus 1 t and t + 1 respectively Thegross (nominal) rate of return on equity shares over period t (from time t to timet + 1) and period t + 1 (from time t + 1 to time t + 2) is then given by6
1 + re = [( ptdtS) + pet)]pe
1 + ret = [( pt+1dt+1St) + pet+1)]pet
where ptdt and pt+1dt+1 denote total nominal dividend payments made at the endof periods t and t + 1 (at time t + 1 and time t + 2) respectively St denotesthe anticipated number of equity shares outstanding after the equity market at theend of period t7 Note that the price of an equity share indicates the fact that thepurchase of an equity share entitles the holder to a portion of future (not current)dividends
Assumption 36 Households view bonds and equity shares as perfect substitutesldquoPerfect substitutesrdquo means that if equality in yields did not hold households wouldrefuse to purchase the asset with the lower yield forcing an adjustment in its pricethat would result in equivalent yields8 With bonds and equity shares as perfectsubstitutes we can speak of a single ldquofinancial asset marketrdquo that incorporatesboth bonds and equity shares and determines a single ldquointerest raterdquo
Assumption 37 There are incomplete markets Let pei i = t + 1 t + 2
denote the expectation formed in period t concerning the price of the consumptiongood over period i9 Similarly in period t we have pe
ei i = t + 1 and wei
i = t + 1 Such expectations are assumed to be held with subjective certaintyallowing us to abstract from risk considerations for the moment
Assumption 38 There are positive transaction costs to arranging exchanges ofthe consumption commodity during each period t Money holdings serve to reducethe transaction costs of arranging exchanges during a period10
Assumption 39 It is prohibitively costly for individuals to directly store theldquocompositerdquo commodity for consumption in future periods However output notconsumed during the period can be transformed (by ldquofirmsrdquo) into output in thesubsequent periods through the augmentation of the capital stock which permitshigher rates of production of output in future periods
Firms as market participants 21
Individual experiments firms
As always we start our analysis at the individual level The behavior of two typesof agents must now be considered ndash firms and households We start with firmsIn doing so we consider a ldquorepresentativerdquo agent a unit whose behavior exceptfor scale is identical to the behavior of the aggregate of such units Thus the samenotation will be used to represent both the individual unit and the aggregate of allunits In addition we consider an infinite time horizon
You should now recognize that an analysis of a representative unit neglectscertain potentially important ldquodistributionalrdquo aspects of the problem For instancefor households the ldquoreal indebtedness effectsrdquo of a price change on demand maynot be offsetting in the aggregate but that potential impact is ignored11 For firmsthe distribution of the initial capital stock can given adjustment costs affect totalemployment and output but that too is ignored12
To consider the behavior of a firm (or more specifically the manager whodirects production for the representative firm) we assume
Assumption 310 Technology is represented by the concave production function
yt = f (Nt K)
where yt denotes the firmrsquos planned (at time t) constant rate of output for the timeperiod from time t to t +1 to be sold during the period and at time t +1 Nt denotesthe firmrsquos planned (at time t) rate of employment of labor during period (t t + 1)with labor services purchased in the labor market at time t and K denotes thefirmrsquos planned capital stock for period t Recall that to simplify matters we takethe capital stock for the current period K as fixed at the individual firm levelThis would be the case given appropriate capital adjustment costs13
Assumption 311 The representative firm will choose the most preferred inputcombination and implied output given technology and prices (both current andanticipated future prices) At time t the objective of the firm is to maximize theexpected real market value of the S equity shares
Vt = peS
p
where pe is the price of equity shares at the end of period t minus 1 (at time t)14
A restatement of the firmrsquos objective
We have indicated that the objective of the firm at time t is to maximize the realmarket value of the S equity shares as given by
Vt = peS
p
22 Firms as market participants
To understand what underlies this market value of the firm we have to definethe elements underlying the price of equity shares and dividends We start byexamining what lies behind the price of equity shares
The assumption that equity shares and bonds are perfect substitutes means thatthe price of an equity share can be expressed as
pe = [ptdtS + pet](1 + r)
where dt denotes real dividends at the end of period t so that ptdt denotes nominaldividends S denotes the number of equity shares outstanding at time t pet is theprice of equity shares at the end of period t and r denotes the interest rate overperiod t (ie from time t to t + 1)
By successively substituting in a similar expression for the price of equity sharesin the next period we obtain for an infinite horizon that15
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
That is the price of an equity share at the end of period t minus1 (at time t) pe reflectsthe anticipated discounted future stream of nominal dividends per share16
Since the real value of the firm is given by Vt = peSp we can now expressthe value of the firm as
Vt = [Sp(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
which means that the objective of the firm can be stated in terms of maximizingthe discounted stream of current and future dividends Before examining whatdetermines dividends each period let us simplify the above expression for Vtby putting it in terms of real dividends each period To do so note that bydefinition
pt equiv p(1 + π)
pt+k equiv p(1 + π)
⎛⎝ kprod
j=1
(1 + rt+jminus1)
⎞⎠ k = 1 2 3
where πt+j denotes the rate of change in the price level between period t + j andt + j + 1 Thus we have
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
Firms as market participants 23
where R = (1 + r)(1 + π) and Ri = (1 + ri)(1 + πi) denotes the real gross rateof interest for period i (from time i + 1 to time i + 2 i = t t + 1 ) Our nextstep in outlining the firmrsquos problem is to obtain an expression for real dividendsin each period
Dividends and the firm distribution constraint
In general we may denote real dividends at the end of any period as the differencebetween a firmrsquos total real revenues during the period and the total costs incurredduring the period Total revenues derive from the sale of output produced duringthe period In addition one could add revenues from a change in the numberof equity shares outstanding or a change in the number of bonds outstanding atthe end of the period17 Total costs to the firm in any period include the agreedupon wage payments to the labor hired during the period payments to replacedepreciated capital plus coupon payments at the end of the period to holders ofpreviously issued bonds Payments at the end of the period for the purchase ofcapital and associated adjustment costs could be counted as well18 That is firmsare constrained to have
dividends = revenue from sale of output
minus wages
minus interest payments
+ funds from change in outstanding bonds and stocks
minus costs to replace depreciated capital add new capital
and capital adjustment costs
The above constraint is typically divided into two separate constraints Oneconstraint earmarks funds raised from the change in outstanding bonds andequity shares at the end of the period to pay for or ldquofinancerdquo the installationof new capital stock during the period plus any capital adjustment costs Thisis the ldquofirm financing constraintrdquo The remaining revenues minus expendituresthen determine the level of dividends This part is called the ldquofirm distributionconstraintrdquo
The firm distribution constraint simply states that the revenues from the saleof output that exceed expenditures to meet wage payments purchases of capitalto replace that used up in the production process and interest payments to bondholders at the end of the period are distributed at the end of the period to householdsas dividends Thus real dividends at the end of period t are given by
dt = yt minus (wtpt)Nt minus zBpt minus δK
According to this expression real dividends at the end of period t equal realrevenues derived from the sale of output yt produced during period t minus costs
24 Firms as market participants
to the firm during period t that reflect the real wage wtpt times the quantity oflabor hired Nt less real coupon payments at the end of the period on previouslyissued bonds zBpt plus purchases of capital during the prior period to replacethat used up in the production process δK 19
In similar manner real dividends for periods t + 1 and t + 2 (paid at the end ofeach period) are given by
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
We can rewrite the above definition of dividends to see more clearly why itis also termed the ldquofirm distribution constraintrdquo This constraint simply says thatrevenue from the sale of output net of that retained to replace capital used up inthe production process (ie ldquonet productrdquo) is distributed to households either aswage payments interest payments or dividends That is
dt + (wtpt)Nt + zBpt = yt minus δK
dt+1 + (wt+1pt+1)Nt+1 + zBtpt+1 = yt+1 minus δKt+1
The firm financing constraint
The second part of the general constraint that firmsrsquo total expenditures equalrevenues is that changes in the firmrsquos holdings of capital as well as any asso-ciated adjustment costs are financed by a change in outstanding equity sharesandor bonds This linking of funding for capital purchases to the issuing of equityshares and bonds is denoted the ldquofirm financing constraintrdquo For instance at theend of periods t + 1 t + 2 we have the following firm financing constraints
Kt+1 minus K + ψ(Int) = [ pet middot (St minus S) + pbt middot (Bt minus B)]pt
Kt+2 minus Kt+1 + ψ(Int+1) = [ pet+1 middot (St+1 minus St) + pbt+1 middot (Bt+1 minus Bt)]pt+1
where ψ(Ii) denotes the costs of installing new capital at rate Ii during period i(between time i and time i + 1) a cost that depends directly on the rate of netinvestment (Ini = Ki+1 minusKi) planned at time t to occur between time i and i+120
Recall that we assume that firmsrsquo plans with respect to the number of stocks andbonds that will be outstanding following the financial market at the end of period i(time i + 1) mirror householdsrsquo expectations concerning the number of bonds andstocks that will be outstanding so we do not distinguish between firmsrsquo plans andhouseholdsrsquo expectations with respect to these variables
Firms as market participants 25
Since bonds and equity shares are perfect substitutes we can rewrite the abovefirm financing constraints in the simpler form
Int + ψ(Int) = At minus At = net At
Int+1 + ψ(Int+1) = At+1 minus At+1 = net At+1
where Ini denotes net real investment (ie Ki+1 minusKi) Ai denotes the real planned(at time t) value of total equity shares and bonds to be outstanding after the financialmarket at the end of period i and Ai denotes the initial real value of equity sharesand bonds for period i reflecting financing and capital decisions in prior periodsbut period i prices For instance
At = [ petSt + pbtBt]pt and At = [ petS + pbtB]pt
At+1 = [ pet+1St+1 + pbt+1Bt+1]pt+1 and
At+1 = [ pet+1St + pbt+1Bt]pt+1
There are several aspects of interest with respect to the firm financingconstraints First note that net capital purchases planned for period i to be installedbetween time i and time i + 1 are paid for at the end of period i when completelyinstalled from the sale of financial assets at that time
Second note that firms purchase the output of the composite good to augmentthe capital stock That is we have a ldquoone-sectorrdquo model in which the same goodserves both households (for consumption) and firms (for investment) There isonly a single commodity price A typical extension is a two-sector model in whichtwo goods are produced a consumption good and a capital good In such casesa new variable the relative price of the capital good in terms of the consumptiongood is introduced
Third note that the firm financing constraint holds whether the firm financescapital with new bonds new equity shares or ldquoretained earningsrdquo Suppose forinstance that a firm plans to add 100 units to its capital stock by buying a newpiece of machinery If the firm issues a bond with real value of 100 to pay for themachinery then there is a direct 100 unit increase (the new bond) in the value of thefinancial assets issued by the firm Note that the value of the current shareholdersrsquostock is unchanged in this case of bond financed investment While it is true thatthe tangible assets of the firm have increased by the 100 addition to capital thisbenefit to shareholders is exactly offset by the fact that the firmrsquos debt has alsoincreased by 10021
If the firm finances the 100 net investment by issuing new shares of stock equalto 100 again there is a 100 unit increase (the new equity shares) in the real valueof the financial assets issued by the firm As with bond financing however thereal value of the initial shareholdersrsquo stock is unchanged when the firm finances itscapital purchases by issuing new equity shares The new shares do not dilute thevalue of the shares of the initial shareholders since the capital purchase increases
26 Firms as market participants
the firmrsquos tangible assets by 100 which is exactly the real value of the new equityshares issued
Finally if the firm finances the 100 net investment through retained earningsthere is in essence a 100 unit increase in the value of the financial assets issued bythe firm for the following reason When a firm retains earnings in order to financea capital purchase the current stockholders own the right to the income generatedfrom the additional capital As a consequence the value of their equity shares risesto reflect the value of the new capital owned by the firm We could equivalentlyview this as the firm paying out 100 units in dividends to its initial shareholderswho then use the dividends to buy additional ldquoconstant valuerdquo equity shares equalto the value of the capital purchased by the firm In other words when the firmuses retained earnings to finance its investment spending it is implicitly issuingnew financial assets ndash equity shares
The nature of capital adjustment costs
An important aspect of the above financing constraint is that it incorporatespotential adjustment costs to purchases of capital as captured by the terms ψ(middot)which depend on Int Int+1 where Ini denotes the planned (at time t) net rateof investment for period i22 The total cost of capital purchases is thus the sum of(a) the real payments (or receipts if negative) involved in the purchase (or sale ifnegative) of capital in the output market and (b) potential real payments denotedldquoinstallationrdquo or adjustment costs associated with new capital acquisitions Forperiod t adjustment costs are given by ψ(Int) where Int denotes net investmentbetween time t and t + 1 Gross investment for period t is given by Int + δK Notethat we can thus decompose gross investment over the period into the change inthe capital stock Kt+1 minus K which is termed ldquoplanned net investmentrdquo and thereplacement of capital used up in the production process δK which is termedldquodepreciationrdquo23
To understand the conversion of the above analysis to continuous time we notethat in general adjustment costs over period t of length h are given by hψ(Inth)where the limit of the term Inth defines the rate of net investment That is incontinuous time the planned rate of investment would be defined by the rate ofgross investment
it = limhrarr0
It
h= lim
hrarr0
Kt+h minus K + hδK
h= Kt + δKt
and the adjustment cost function in continuous time would be ψ(int)For the aggregate rate of gross investment (a flow) to be defined by the above
expression K must equal K To achieve this one of two approaches is typicallytaken One approach assumes zero adjustment costs in that ψ equiv 0 This situationsometimes referred to as the case of ldquoperfect malleabilityrdquo means that the rateof investment may not be defined at the level of the individual firm That is ifthe existing capital stock were higher or lower than the planned level investment
Firms as market participants 27
would be infinitely positive or negative However it can be shown that withzero adjustment costs the output market in a continuous-time model at a pointin time is simply a ldquocapital marketrdquo and the expression Kt = K emerges as anequilibrium condition with respect to the capital market at time t24 Thus wecan apply LrsquoHospitalrsquos rule to define the aggregate rate of (net) investment in thecontinuous-time model with zero adjustment costs as25
it = limhrarr0
Kt+h minus K
h= limhrarr0 d(Kt+h minus K)dh
limhrarr0 dhdh= K
where it is rate of investmentIn contrast to the case of zero adjustment cost one can assume adjustment costs
that take the following form
ψ(0) = 0
ψ(β) gt 0 if β gt 0 ψ primeprime gt 0 and limβrarrinfin ψ(β) = infin
ψ(β) lt 0 if β lt 0
The above set of assumptions reflects the presumption that adjustment costsincrease at an increasing rate with the rate of change in capital and that it isinfinitely costly to change the capital stock arbitrarily fast The result of suchadjustment costs in both discrete-time analysis and continuous-time analysis isthat at time t the firm chooses Kt = K That is at time t the firm views the inher-ited capital stock as optimal since it is prohibitively costly to change the capitalstock at a point in time given such adjustment costs26
As we will see with ldquocosts of installing a unit of new capitalrdquo there will be adifference between the market value of capital goods in place and their replacementcost In particular the ratio of these two values known as ldquoTobinrsquos Qrdquo will exceedone In addition adjustment costs will mean that the firmrsquos decision with respectto investment will not be myopic (ie plans will not be based on forecasts thatextend only one period into the future) Rather the firm will consider all futureperiods in making current investment decisions
The firm problem a general statement
One way to state the optimization problem faced by the firm is to say that attime t the firm makes plans with respect to current and future employment oflabor (Nt Nt+1 ) the future employment of capital (Kt+1 Kt+2 ) the stockof outstanding bonds (Bt Bt+1 ) and the stock of outstanding equity shares(St St+1 ) in order to maximize the real value of the previously issued equityshares outstanding at time t with that real value given in general form by
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
28 Firms as market participants
Such plans are subject to the combined distribution and financing constraints listedabove as well as to the production function That is in general the firmrsquos problemcan be stated as27
max(SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
subject to the financing constraints
minus [Kt+1 minus K + ψ(Kt+1 minus K)] + (petpt) middot [Bt minus B]+ (pbtpt) middot (St minus S)] = 0
minus [Kt+2 minus Kt+1 + ψ(Kt+2 minus Kt+1)] + (pet+1pt+1) middot [Bt+1 minus Bt]+ (pbt+1pt+1) middot (St+1 minus St)] = 0
and the distribution constraints
dt = yt minus (wtpt)Nt minus zBpt minus δK
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
and given the production functions
yt = f (Nt K)
yt+1 = f (Nt+1 Kt+1)
for i = t t + 1 The above problem has a recursive nature to it At the start of any given period
the firm inherits a stock of capital an outstanding stock of bonds and an outstand-ing stock of equity shares28 These variables are state variables Each period thefirm chooses a set of the ldquocontrolrdquo variables ndash employment investment Thesechoices in conjunction with the production function result in outcomes in termsof (a) a one-period return (dividends) at the end of the period and (b) a new setof ldquostaterdquo variables ndash capital stock and stock of financial assets (equity shares andbonds) ndash inherited in the subsequent period Further note that the objective of thefirm is additive in these one-period returns (dividends) Thus the problem is oneto which we can apply Bellmanrsquos dynamic programming technique
To reformulate the problem facing the firm as a dynamic programming prob-lem we use the conventional notation of dynamic programming problems29
Firms as market participants 29
Specifically the above problem can be viewed as involving
(a) A set of ldquocontrolrdquo variables each period
zt = Nt Int zt+1 = Nt+1 Int+1 etc
(b) A set of ldquostaterdquo variables each period
xt = K B S xt+1 = Kt+1 Bt St etc
(c) ldquoTransition functionsrdquo that link the choices specifically the choice of netinvestment during the period to the capital stock available at the start ofthe next period as well as the stock of financial assets outstanding in thesubsequent period For instance the choice of the net investment rate Int during period t dictates Kt+1 given K since
Kt+1 = Int + K
From the firm financing constraint we know that the choice of the investmentrate also determines the real stock of financial assets
( petpt)[Bt minus B] + ( pbtpt)[St minus S] = Int + ψ(Int)
(d) A set of one-period return functions (evaluated at the end of period t)30
rt(K B S) = dt rt+1(Kt+1 Bt St) = Sdt+1StRt etc
Since bonds and equity shares are perfect substitutes the above problem cannotbe solved for a unique optimal number of bonds or equity shares to have outstandingeach period Thus without any loss of generality we may restrict our focus toeither bond or equity share financing That is we can hold constant either bonds(ie Bi = B i = t t + 1 ) or equity shares (ie Si = S i = t t + 1 )Alternatively we can combine the distribution and financing constraints into asingle expression for dividends and hold constant both equity shares and bondsIn this case we have ldquoretained earningsrdquo financing of changes in the capital stockIt is this case that we consider below31
Simplifying the firm problem ldquoretained earnings financingrdquo
If we assume that capital expenditures are financed from ldquoretained earningsrdquo thereis a single state variable the stock of capital The problem facing the firm thencan be simply stated as follows The Bellman equation for period t given inheritedcapital stock K is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
30 Firms as market participants
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
Substituting the above definitions for real dividends and output into the Bellmanequation and substituting in the transition function the first-order conditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
Equation (31) is the standard condition that labor is employed up to the pointwhere the real marginal gain for an additional unit of labor in terms of the increasein output attained in the current period (ie the marginal product of labor partftpartNt)equals the real marginal cost as reflected by the real wage wtpt Equation (32)indicating the optimal choice of investment is discussed in the next section
Optimal investment (and the future capital stock) zeroadjustment costs
To express the optimal condition for investment and thus the future capital stockin a more transparent form we need to expand upon the effect of an increase inthe capital stock on the value function for period t + 1 In other words we need toclarify the nature of the term dW (Kt+1)dKt+1 in Equation (32) To do so let usconsider the Bellman equation for period t + 1 To simplify matters we initiallyfocus on the case of zero adjustment costs (ie that ψ equiv 0 implying that ψ prime
i = 0i = t t +1 ) Given the inherited capital stock Kt+1 for period t +1 and a fixedstock of equity shares the Bellman equation for period t + 1 is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1(Rt)minus1 + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and (assuming retained earnings financing of capital changes and zero adjustmentcosts) the following definitions for real dividends and output for period t + 1
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBtpt+1 minus Int+1
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 31
Thus we have32
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (33)
We can use first-order conditions for the Bellman equation for period t + 1 toclarify the nature of dW (Kt+2)dKt+2 in Equation (33) In particular we havethat the optimal choice of investment in period t + 1 satisfies
minus 1
Rt+ dW (Kt+2)
dKt+2= 0 (34)
Substituting (34) into (33) we obtain the following expression for the effect of achange in the inherited capital stock on the value function for period t + 1
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt (35)
Substituting (35) into equation (32) and recalling our assumption that ψ prime = 0we thus have that the optimal choice of investment in period t satisfies
minus1 + 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt= 0 (36)
The above expression can be rearranged to obtain
partft+1
partKt+1= mt minus δ (37)
where mt Fisherrsquos expected real rate of interest equals Rt minus1 or (rt minusπt)(1+πt)The interpretation of (37) is fairly straightforward Each period the firm chooseslabor and capital such that the marginal gain in the subsequent period in terms ofincreased output equals the real marginal cost For capital the marginal cost is therate of depreciation plus the expected real rate of interest An explanation of thisreal ldquouserrdquo or ldquorentalrdquo cost of capital follows
Over period t the firm pays for one unit of capital at price pt Since the firmcould have instead used these funds to reduce the outstanding stock of bonds by pt the cost of this capital (reduced dividends) in nominal terms is pt(1 + rt) In realterms the cost one period later is anticipated to be pt(1 + rt)pt+1 After oneperiod 1 minus δ of the capital remains so that the sale of the remaining capital afterone period of use (or the reduced purchases of new capital) reaps a nominal returnof (1 minus δ)pt+1 and real return (1 minus δ) The real rental cost of the unit of capital isthus33
pt(1 + rt)pt+1 minus (1 minus δ) = (1 + rt)(1 + πt+1) minus 1 + δ
= (rt minus πt)(1 + πt) + δ
= mt + δ
32 Firms as market participants
Summarizing our discussion in the case of zero adjustment costs the optimalbehavior of firms in periods t and t +1 is given by the following demand functionsfor period t and t + 1
N dt = N d(wtpt K)
I dnt = I d
n (mt + δ wt+1pt+1 K)
where I dnt = Kd
t+1minusK and Kdt+1 = Kd
t (mt+δ wt+1pt+1) Note that the anticipatedreal wage next period affects Id
nt since changes in the real wage affect the employ-ment of labor and thus assuming part2f partNpartK does not equal zero the marginalproduct of capital A similar statement explains why K enters as an argument inthe labor demand function
An important feature of the above is that planned investment demand when thereare zero adjustment costs simply depends on adjacent expected real user costs ofcapital and real wages This reflects the fact that with zero adjustment cost capitaldemand is a function of the expected real user cost of capital and the real wageover the next period alone
Financing choices and different debt-to-equityratios a digression
We have characterized the above choice of the capital stock under the presumptionthat the firm finances capital purchases through retained earnings Yet we can showthat the planned (at time t) choice of the optimal capital stock at time t+1 t+2 is independent of the method of financing given that (a) bonds and equity sharesare assumed to be perfect substitutes and (b) there is no cost to arranging theexchange of financial assets (otherwise the retained earnings financing method ispreferred) This result is sometimes referred to as the ldquoModiglianindashMiller theoremrdquowhich states that the total value of the firm is independent of its financial structureThat is the present value of the stream of dividends to the initial owners is inde-pendent of how liabilities are divided between bonds and equity shares The resultis that the capital structure is indeterminant
The view that the method of financing capital purchases is largely irrelevant isa very useful simplification for macroeconomic analysis However you should beaware of some complicating factors that we are ignoring factors that can causefirms to care about the method by which they finance their capital purchases
When a firm issues bonds the value of its outstanding debt rises When it issuesstocks the value of its outstanding equity shares increases Thus the method offinancing capital purchases affects what is known as the firmrsquos debt-to-equityratio34 Financing capital purchases with bonds will increase the firmrsquos debt-to-equity ratio while financing capital purchases with equity shares (eitherexplicitly or implicitly by using retained earnings) will reduce the firmrsquos debt-to-equity ratio Two factors that can influence a firmrsquos desired ldquocapital structurerdquoare tax considerations and bankruptcy costs35
Firms as market participants 33
The corporate taxes that firms pay are calculated as a percentage of earningsFor tax purposes corporate earnings are equal to total revenue net of costs wherecosts are calculated as including not only wages and payments for raw materi-als and intermediate goods but also interest payments to bondholders If a firmfinances its purchases of capital using bonds the interest it pays in the futurewill reduce its taxable earnings and thus the taxes that it has to pay This meansthat a firm can lower its future tax liability by raising its debt-to-equity ratio ndashthat is by financing new capital purchases with new bonds rather than equityshares
Raising the debt-to-equity ratio however is generally not without costs whichare typically referred to as ldquobankruptcy costsrdquo Unlike equity shares which promiseshareholders dividend payments if profits are sufficiently high bonds promise fixedpayments to their holders Greater debt thus increases the fixed obligations thatfirms must meet in the future This means that a fall in future revenues is morelikely to force the firm into bankruptcy
Bankruptcy occurs when a firmrsquos revenues do not cover its costs and it isforced to default on its obligations to bondholders Associated with bankruptcyare bankruptcy costs the most obvious being the hefty legal costs associated witheither reorganizing or undergoing a court-supervised liquidation The existence ofbankruptcy costs serves to limit the amount of borrowing a firm will undertake Itwill hesitate to increase its debt-to-equity ratio beyond some level since the gainin tax savings will be offset by the costs associated with an increased likelihoodof incurring bankruptcy costs
To summarize a firm can be viewed as having an optimal debt-to-equity ratiothat reflects a tradeoff of tax and bankruptcy cost considerations Table 31 liststhe general level of debt-to-equity for a sample of industries in US manufacturingNote that the debt-to-equity ratios vary widely among the industries in the sampleranging from significantly over one to significantly less than one The ratio ishighest in the steel industry where the value of debt is close to 17 times the
Table 31 Debt-to-equity ratios across select industries
Book value Market valueof equity of equity
Steel 1973 1665Petroleum refining 1548 1117Textiles 1405 1296Motor vehicles 0922 0594Plastics 0843 0792Machine tools 0472 0425Pharmaceuticals 0194 0079
Source Kester (1986) The book value of equity is computed fromaccounting sources while the market value of equity is obtained bymultiplying the number of outstanding shares by the current marketprice of the outstanding shares
34 Firms as market participants
market value of outstanding market shares In contrast for pharmaceuticals thedebt-to-equity ratio is only 0079 indicating that the industry uses bond financingvery little instead financing its investment activities almost exclusively throughthe issuance of equity
Adjustment costs for capital and Tobinrsquos Q
Let us now consider the choice of capital when there exist adjustment costs Tokeep the maximization problem simple we shall continue to assume ldquoretainedearningsrdquo financing of changes in the capital stock we would however obtainidentical results with bond or equity share financing As we have seen in the caseof retained earnings financing the problem facing the firm is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t36
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
As before substituting the above definitions for real dividends and output intothe Bellman equation and substituting in the transition function the first-orderconditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
The problem facing the firm in period t+1 assuming retained earnings financingof capital changes is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1Rt + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and given the following definitions for real dividends and output for period t37
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBpt+1 minus Int+1 minus ψ(Int+1)
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 35
Again we can substitute the above definitions for real dividends and output intothe Bellman equation for period t+1 and substituting in the transition function (inparticular note the fact that dKt+2dInt+1 = 1) obtain the following first-orderconditions
partft+1
partNt+1minus wt+1
pt+1= 0 (38)
minus (1 + ψ primet+1) + dW (Kt+2)
dKt+2= 0 where ψ prime
t+1 = dψ(Int+1)
dInt+1 (39)
Finally from our expression for the value function at time t + 1 W (Kt+1) wehave that
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (310)
Substituting (39) into (310) and then substituting the resulting expression fordW (Kt+1)dKt+1 into the first-order condition for Int (equation (32)) we obtain
minus(1 + ψ primet ) + 1
Rt
(partft+1
partKt+1 minus δd
)+ 1 + ψ prime
t+1
Rt= 0 (311)
Rearranging and simplifying we have that
partft+1
partKt+1= mt + δ + (mt + 1)ψ prime
t minus ψ primet+1
where ψ primet = ψ prime(I d
nt) and ψ primet+1 = ψ prime(I d
nt+1) An important feature of adjustmentcosts that is highlighted by the above equation is that the choice of investment inperiod t is now linked to the optimal choice of investment next period Since thisholds for each period in the future the choice of investment today is linked toinvestment decisions over all subsequent periods
We can simplify and rearrange the above first-order condition for investment inperiod t to obtain what is known as ldquoTobinrsquos Qrdquo To do so let us first assume anidentical real rate of return over time in particular we then have Ri = Rt = 1+mi = t + 1 t + 2 where (1 + m) equiv (1 + r)(1 + π) Let us also assume thefirm has attained its optimal capital stock so that Kd
t+1 = K and I dnt = 0 If the
production function is separable into capital and labor then the assumption of aninvariant real interest rate implies that I d
ni i = t +1 t +2 equal zero as well38
In this case ψ primet and ψ prime
t+1 can be replaced by a common ψ prime(0) gt 0 Then we maywrite the first-order condition as
partft+1partKt+1 minus m minus δ = ψ primem
Dividing by m and adding one to both sides we have
Tobinrsquos ldquomarginalrdquo Q equiv 1 + [partft+1partKt+1 minus m minus δ]m = 1 + ψ prime gt 1(312)
36 Firms as market participants
The above provides the definition for Tobinrsquos Q39 More precisely we have Tobinrsquosldquomarginalrdquo Q for it represents the ratio of the market value of an additional unitof capital to its replacement cost
Given adjustment costs the market value of an additional unit of capital exceedsits replacement cost so Tobinrsquos marginal Q is greater than one Since investmentdemand planned over the coming period determines marginal adjustment costsψ prime(I d
nt) we see that investment can be written in terms of Tobinrsquos Q40 The optimalrate of investment is that rate for which Q minus 1 is equal to the marginal cost ofinstallation Thus net investment demand is sometimes expressed as
Idnt = I d
nt(Q minus 1)dId
nt
d(Q minus 1)gt 0
The Q theory of investment is not operational as long as Q is not observableWhile marginal Q is not typically apparent with some additional assumptionswe can show that the expression known as Tobinrsquos ldquoaveragerdquo Q is identical toldquomarginalrdquo Q Tobinrsquos ldquoaveragerdquo Q is defined as the ratio of the total value of thefirmrsquos existing capital (the market value of its equity shares) to its total replacementcost and these variables are more easily measured41 In particular for a one-sectormodel in which the cost of capital and output are identical42
Tobinrsquos ldquoaveragerdquo Q equiv V
K
Hayashi (1982) has shown that if the firm is a price-taker if the production func-tion is linear homogeneous in K and N and if expectations of future real interestrates and real wages are static then the marginal and average Q are identical43 Tosee why this is the case note that if the real interest rate and real wage are invariantand the firm is at the optimal level of capital so that the capital stock is invariantover time then omitting time identifiers we have
V =infinsum
k=1
dRk
where (since ψ(Idnt) = ψ(0) = 0)
d = y minus (wp)N minus δK
Thus
V = [y minus (wp)N minus δK]m
where m equiv R minus 1 By Eulerrsquos theorem for a linear homogeneous production func-tion (ie K(partf partK) = y minus N (partf partN )) and the marginal productivity condition
Firms as market participants 37
for labor (ie wp = partf partN ) which reflects the price-taker assumption the firsttwo terms in the expression become K(partf partK)44 Dividing by K we thus obtain
Tobinrsquos ldquoaveragerdquo Q equiv V K = [partf partK minus m minus δ]m + 1
which as we can see is the same expression as that for Tobinrsquos marginal QAlternatively we can write the above as
V = [partf partK minus m minus δ]m + 1 = (Q minus 1) ([partf partK minus m minus δ]m)
Adjustment costs for labor labor as a ldquoquasi-fixed factorrdquo
Our prior characterization of the optimal choice of labor reflects an underlyingproduction process that incorporates a very simple view of labor For instancelabor markets are restricted to be only spot markets That is we rule out multiperiodlabor contracts Yet there is an extensive body of literature that investigates variousrationales for and the implications of such multiperiod (implicit) labor contractsOne reason why long-term contracts might emerge is an attempt by firms who areless risk-averse than workers to smooth out income over time45
A second reason why multiperiod labor contracts might emerge is if there areldquoadjustment costsrdquo to changes in the size of the labor force Adjustment costs couldreflect the fact that in order to hire new workers firms must incur hiring and trainingcosts Adjustment costs mean that a firm would view potential new hires andpreviously employed workers as imperfect substitutes and this would provide animpetus for multiperiod labor contracts The absence of adjustment costs simplifiesthe analysis by eliminating a rationale for multiperiod labor contracts and a choiceof labor given adjustment costs It also simplifies the analysis in two other ways
First the absence of adjustment costs suggests that we can measure labor ser-vices as the product of the fraction of the period each labor supplier works andthe number of individuals hired That is given no adjustment costs differences inldquohours workedrdquo and ldquonumber employedrdquo that leave total work hours unchanged areviewed by the firm as equivalent in terms of production In contrast with positiveadjustment costs firms would have a preference for meeting temporary changesin output by changing hours rather than by changing the number employed
A second implication of the absence of adjustment costs is that the employerviews as equivalent two workers working at ldquohalf-speedrdquo (and receiving ldquohalf-wagesrdquo) and one worker working at ldquofull-speedrdquo for ldquofull-wagesrdquo In contrastgiven adjustment costs firms would have a preference for meeting temporaryoutput changes by altering not only hours per worker but also the intensity that eachemployee was asked to work In fact output per work hour or ldquolabor productivityrdquodoes typically increase more rapidly during a recovery suggesting a more intensiveuse of labor
In contrast labor productivity growth is typically less rapid when the growth intotal output slackens This phenomenon is due in part to employersrsquo hoarding laborin slack times so as not to lose trained employees whom they will want when there
38 Firms as market participants
is an upturn in demand (That is to reduce subsequent adjustment costs given afuture upturn in production) Hoarding labor means that employers keep on moreworkers than necessary to produce the current output so that each worker has lesswork to do than normal The labor hoarding phenomenon also referred to as thelabor reserve hypothesis is the formal term for changes in the ldquointensityrdquo at whichlabor is used and explains lower output per work hour during periods of slackdemand46
Conclusion
The nature of the firm was discussed with the emphasis on profit maximizationDecisions of firm owners facing a variety of constraints and costs were analyzedwith particular attention paid to how financing constraints and adjustment costsaffected firm profits and the ability to adjust production levels A link was madebetween households and firms that will lead us into the next chapter Firms hireworkers who of course constitute households in the economy Workers are paidwages as costs to the firm but are an integral part of the production processMoreover firms may face costs of adjustment and other costly phenomena asso-ciated with decisions to alter the use of workers in the production process Takentogether then we see a link between firms and households as firm decisions havethe propensity to affect consumer income
4 Households as marketparticipants
Introduction
This chapter brings the household into our model of the macroeconomySpecifically the householdrsquos ability to obtain utility through consumption andthe labor supply decision is modeled within a choice framework The solutionis then used to formulate predictions about labor supply The concept of time iscritical to a thorough understanding of household behavior in the marketplace anda good deal of this chapter is spent analyzing intertemporal choices
The life-cycle and permanent income hypotheses are introduced and a theory ofportfolio choice is developed Finally the chapter ties many of these issues togetherand addresses the macroeconomic questions of absence of money illusion the realbalance effect and the real indebtedness effect
Individual experiments households
Two agents inhabit our expanded macroeconomic model with production firmsand households Having just discussed the nature of decisions confronting firmswe turn now to those confronting households These decisions can be broken downinto three types consumptionsaving portfolio and labor supply Consider nowthe first two decisions which should be familiar
The ldquoconsumptionsavingrdquo decision The representative household must deter-mine at time t the consumption purchases over each period at the implied ratect ct+1 We term this problem the ldquoFisherianrdquo problem
The ldquoportfoliordquo decision The individual must determine at time t the collectionof assets to hold at the end of each period In our expanded economy there areostensibly three types of assets
1 nominal money balances planned at time t to be held at the end of periodi Mi i = t t + 1
2 the nominal value of bonds planned at time t to be held at the end ofperiod i pbiBi i = t t + 1 where Bi denotes the planned number ofbonds held and pbi denotes the money price of bonds at the end of periodi and
40 Households as market participants
3 the nominal value of equity shares planned at time t to be held at the endof period i peiSi i = t t + 1 where pei denotes the money price ofan equity share at the end of period i and Si denotes the number of equityshares planned at time t to be held at the end of period i
Since bonds and equity shares are perfect substitutes we can consider them asa single entity with respect to householdsrsquo portfolio decisions We will let Aidenote the real holdings of financial assets planned at time t to be held at theend of period i i = t t + 1 That is for period t
At = [petSt + pbtBt]pt
and so on Excluding current dividend and interest payments the real value ofinherited financial assets at the end of period i reflecting portfolio decisions inthe prior period will be denoted by Ai i = t t + 1 For instance
At = [petS + pbtB]pt
At+1 = [pet+1St + pbt+1Bt]pt+1
The household problem
We start our analysis of the representative household as usual by discussing thehouseholdrsquos preferences constraints and objectives In particular we make thefollowing assumptions
Assumption 41 The representative householdrsquos preferences are described bythe utility function
u(ct ct+1 Mtpt Mt+1pt+1 1minusNt 1minusNt+1 )
where ci denotes the householdrsquos planned (at time t) rate of consumption dur-ing period i (from time i to time i + 1) Mi denotes the representative householdrsquosplanned (at time t) nominal holdings of money at the end of period i and Ni denotesthe householdrsquos planned (at time t) rate of supply of labor services during period i(from time i to time i +1) such that 1minusNi denotes the planned rate of leisure dur-ing period i It is assumed that partupartci gt 0 partupart(Mipi) gt 0 and partupart(1 minus Ni)
gt 0 i = t t + 1 Macroeconomics often assumes a time-separable utilityfunction a form of the utility function that ensures ldquotime consistencyrdquo1 In par-ticular following the tradition of macroeconomics we will assume that the totalutility for the representative household at time t with an infinite planning horizonis given by
infinsumi=t
β iminustu(ci Mipi 1 minus Ni)
Households as market participants 41
where β denotes the fixed personal or ldquoutilityrdquo discount factor with 0 lt β lt 12
In the above note that the one-period utility function for period t (time t to t + 1)is u(ct Mtpt 1minusNt) for period t +1 (time t +1 to t +2) it is u(ct+1 Mt+1pt+11 minus Nt+1) and so on Further note the infinite time horizon
Assumption 42 Individuals will choose the most preferred sequence of con-sumption money holdings and labor supply from the set of feasible alternatives(rationality) The feasible set of consumption money holdings and leisure
(ct ct+1 Mtpt Mt+1pt+1 1 minus Nt 1 minus Nt+1 )
is defined by the set of equalities
(wtpt)Nt + zBpt + At + Mpt minus [ct + Mtpt + At] = 0
(wt+1pt+1)Nt+1 + zBtpt+1 + At+1 + Mtpt+1
minus [ct+1 + Mt+1pt+1 + At+1] = 0
Note that we assume the budget constraints are met with equality Further note thatthe sum of dividends wage payments and interest payments equals total outputminus depreciation For instance for period t
(wtpt)Nt + dt + z middot Bpt = yt minus δK
and so on This is simply the firm distribution constraint for period t
Several aspects of the above problem deserve further elaboration First a wordon notation for future variables It is common in macroeconomics to derive themicroeconomic theoretical restrictions for the aggregate model under the conditionof certainty even though the analysis is then applied to situations that involvepotential stochastic elements One obvious way to eliminate considerations ofuncertainty from the analysis is to assume perfect foresight A second way is toassume that individual expectations of future events are point estimates held withsubjective certainty Note that either approach simplifies the analysis and in manycases this simplification gives us results that are not overturned if risk were to besystematically incorporated into the analysis
In the analysis of individual behavior below we will often for notationalsimplicity not distinguish between future prices and the expectations of futureprices Assuming expectations are held with subjective certainty this lack of dis-tinction will not be serious in discussing the result of the optimization problemsThat is the findings for perfect foresight can be made identical to those withoutthe assumption of perfect foresight by switching actual future prices for expectedprices Sometimes for clarity we will explicitly denote expected future prices(point estimates held with subjective certainty) by the superscript ldquoerdquo
42 Households as market participants
A second aspect of the above analysis that may initially appear odd concerningthe above one-period utility function for period i (from time i to time i+1) is that itseems that we are mixing money balances that occur at one time with consumptionand leisure that occur at an earlier time The reason for this is that money balancesare a stock variable and we are recording their value at the end of each period ithat is at time i + 13 The following scenario for the discrete-time analysis mayhelp clarify what is going on
At time t the labor market takes place and agreements are made to exchangelabor services at rate Nt over the period (t t + 1) for the money wage wt Duringthe period an output market operates in which firms sell output produced at rate yt During the period households receive money wages wt At the end of the period(time t + 1) households anticipate real interest payments zBpt from their priorpurchases of bonds (B) and real dividends dt from their prior purchase of equityshares stock (S)4 Given the above income sources as well as inherited nominalholdings of money (M ) and the anticipated value of inherited financial asset At atthe end of the period households plan an average rate of consumption ct duringthe period
At the end of period t after all income is received and final planned pur-chases of consumption goods are made the remainder reflects householdsrsquo planned(at time t) end-of-the-period change in real money balances ((Mt minus M )pt) andplanned changes in real financial assets holdings (At minus At) For financial assetsthe real price for bonds at the end of period t is pbtpt and the real price for equityshares is petpt
According to the above scenario the sale of labor services at rate Nt and rateof consumption ct over the period from time t to t + 1 tend to coincide whilereal money balances Mtpt and real financial asset holdings At can be viewed asthe planned (at time t) real stocks of such assets to be held at the end of period tIn continuous-time analysis as the length of the period h goes to zero the rate atwhich leisure is lost from supplying labor services during the period (Nt) the rateof consumption (ct) and the stocks of real money and real financial asset holdingswould coincide
A third aspect of the above analysis is that we have interpreted 1 minus Ni as theportion of the period of length 1 that the individual spends at leisure given thesupply of labor at rate Ni This is a simplification however for at the same timewe have suggested that the ldquoutility yieldrdquo of money is derived from its ability toreduce the transaction costs in arranging exchanges with such transaction costsreflecting at least in part a loss of leisure To explicitly incorporate such a viewof money leisure during a period i of length h given the sale of labor services Niand the real money balances Mipi held at the end of the period would be givenby h(1 minus Ni minus (Mipi)) where the function (Mipi) reflects transactions costsin terms of the loss of leisure The fact that prime lt 0 indicates that increased moneyholdings raise utility by reducing leisure lost in arranging transactions In this casethe one-period utility function would formally be given by u(ci 1minusNiminus(Mipi))with partupartci gt 0 and partupart(1 minus Ni minus (Mipi)) gt 0
Households as market participants 43
The general solution to the household problem
We can express the household problem in terms of a set of Bellman equationsAssuming perfect foresight (or equivalently interpreting future prices dividendetc as expectations of such variables held with subjective certainty) we thus havefor period t (time t to t + 1)
W (xt) = maxct Nt
Mtpt xt+1
[u(ct Mtpt 1 minus Nt) + W (xt+1)]
subject to the transition function
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt] Mtpt
and given xt Rmt is the real gross rate of interest on money that is the gross realrate of return on money and equals one divided by one plus the rate of inflation(1(1+πt)) The term xt is the total value in period t derived from the ldquoinheritedrdquoholdings of money bonds and stocks This total value is the sum of currentdividends and interest (received at the end of period t) on stock and bond holdingsacquired previously the real value of these financial assets at the end of period texclusive of these current interest and dividend payments and the real value ofpreviously acquired money holdings
xt equiv dt + zBpt + At + Mpt
where
At equiv [petS + pbtB]pt
The difference between the total real value derived in period t from inheritedbonds stocks and money balances plus real wage income xt + (wtpt)Nt andconsumption in period t ct reflects the acquisition of bonds equity shares andmoney holdings at the end of period t by the representative household Letting Atdenote the planned holdings of financial assets at the end of period t we thus havefrom the household budget constraint that
At + Mtpt = xt + (wtpt)Nt minus ct
where
At equiv [petSt + pbtBt]pt
Recall that Rt the gross real rate of return on financial assets equals one plus thenominal interest rate divided by one plus the rate of inflation ((1 + rt)(1 + πt))Thus in period t + 1 and given our definition of Rmt the inherited real value of
44 Households as market participants
bonds stocks and money balances including dividends and interest payments isgiven by
xt+1 = RtAt + RmtMtpt
Substituting in the expression for At derived from the household budget constraint(ie At = xt + (wtpt)Nt minus ct minus Mtpt) and rearranging we obtain the transitionfunction
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt]Mtpt
Substituting the transition function into Bellmanrsquos equation for period t wehave the following first-order conditions for ct Nt and Mtpt assuming interiorsolutions (ie ct gt 0 1 gt Nt gt 0 and Mtpt gt 0)
partutpartct minus (partW (xt+1)partxt+1)Rt = 0 (41)
partutpart(1 minus Nt) + (partW (xt+1)partxt+1)Rt(wtpt) = 0 (42)
partutpart(Mtpt) minus (partW (xt+1)partxt+1)(Rt minus Rmt) = 0 (43)
The above conditions indicate that for period t (time t to time t + 1) we havefrom equations (41) and (42) that
partutpart(1 minus Nt)
partutpartct= wt
pt (44)
In words the optimal choice of leisure is such that the marginal value of leisurein terms of consumption that is the marginal rate of substitution between leisureand consumption as given by
partutpart(1 minus Nt)
partutpartct
equals the marginal cost of leisure in terms of consumption forgone in the currentperiod as given by the real wage wtpt
From equations (41) and (43) we have for period t that
partutpart(Mtpt)
partutpartct= (Rt)
minus1(Rt minus Rmt)
In words the optimal choice of real money balances is such that the marginal rateof substitution between real money balances and consumption as given by
partutpart(Mtpt)
partutpartct
Households as market participants 45
equals the marginal cost in terms of the present value of the loss in interest incomein the subsequent period due to the holding of money balances instead of financialassets as given by the expression (Rt)
minus1(Rt minus Rmt) Recall that Rt minus Rmt equals(1 + rt)(1 + πt) minus (1(1 + πt)) which is simply rt(1 + πt) or essentially theanticipated nominal rate of interest Given that Rt = (1 + rt)(1 + πt) we thushave that (Rt)
minus1(Rt minus Rmt) = rt(1 + rt)We can expand upon the above discussion of the first-order conditions for
period t by first noting that W (xt+1) is defined by
W (xt+1) = maxct+1Nt+1
Mt+1pt+1xt+2
[βu(ct+1 Mt+1pt+1 1 minus Nt+1) + W (xt+2)]
where
xt+2 = Rt+1[xt+1 + (wt+1pt+1)Nt+1 minus ct+1] minus [Rt+1 minus Rmt+1]Mt+1pt+1
given
xt+1 equiv dt+1 + zBtpt+1 + At+1 + Mtpt+1
At+1 equiv [pet+1St + pbt+1Bt]pt+1
Again substituting the transition function into the Bellman equation for periodt + 1 we have the following first-order conditions for period t + 1
βpartut+1partct+1 minus (partW (xt+2)partxt+2)Rt+1 = 0 (41prime)minus βpartut+1part(1 minus Nt+1) + (partW (xt+2)partxt+2)Rt+1wt+1pt+1 = 0 (42prime)βpartut+1part(Mt+1pt+1) minus (partW (xt+2)partxt+2)(Rt+1 minus Rmt+1) = 0 (43prime)
Now consider the impact of the change in xt+1 on the value function W (xt+1)The above first-order conditions imply that the indirect effects of the change inxt+1 on W (xt+1) through the effect of such a change on the choice of the optimalvalues of consumption labor supply and real money balances for period t + 1 arezero5 This is simply an application of the envelope theorem which states thatthe change in the objective function adjusting the choice variables optimally isequal to the change in the objective function when one does not adjust the choicevariables This fact along with the transition function for xt+2 gives us
partW (xt+1)partxt+1 = (partW (ct+2)partxt+2)Rt+1 (45)
Substituting equation (41prime) into (45) we obtain
partW (xt+1)partxt+1 = βpartut+1partct+1 (46)
46 Households as market participants
Alternatively by substituting in (42prime) we can obtain6
partW (xt+1)partxt+1 = β[partut+1part(1 minus Nt+1)]pt+1wt+1 (47)
Combining equations (41) and (46) gives us the standard Fisherian solutionfor the optimal allocation of consumption between period t (time t to t + 1) andperiod t + 1 (time t + 1 to t + 2)
partutpartct
βpartut+1partct+1= Rt
Combining equations (43) and (46) we obtain the standard expression for theoptimal portfolio choice of money
partutpart(Mtpt)
βpartut+1partct+1= Rt minus Rmt
Combining equations (42) and (47) gives us an expression for the optimalallocation of labor supply over time
partutpart(1 minus Nt)
βpartut+1part(1 minus Nt+1)= pt+1
wt+1Rt
wt
pt (48)
Having discussed the ldquoFisherian problemrdquo and the ldquoportfolio problemrdquo con-fronting the household we turn our attention in the next section to the ldquolaborsupply problemrdquo as captured by equations (44) and (48) Before doing so how-ever a general comment should be made with respect to the discussions to followas well as the preceding discussions of the Fisherian problem and the portfoliodecision
In focusing on first-order conditions with respect to the particular variablesat issue (ie first-order conditions for consumption now and next period forthe Fisherian problem first-order conditions for real money holdings and futureconsumption for the portfolio problem and first-order conditions for labor sup-ply now and next period for the labor supply decision) one has a tendency toforget that the optimizing problem involves the simultaneous choice of consump-tion portfolio and leisure In general this means that the analysis is often notas straightforward as it may first appear For instance a change in the currentreal wage or an expected real interest rate can affect the first-order conditionconcerning the choice of labor supply through its impact on the choice of consump-tion if part2upartcpartN = 0 for then the change in consumption alters the ldquomarginalutilityrdquo of leisure One simple way to abstract from these ldquoindirectrdquo effects is toassume that the utility function is separable not only across time but also withrespect to consumption leisure and real money balances each period such thatpart2upartcpartN = part2upartcpart(Mp) = part2upartNpart(Mp) = 0
Households as market participants 47
The choice of hours within a period
Equation (44) indicates that at the optimal labor supply the household cannot bemade better off by trading consumption for leisure within periods at the expectedreal wage for the period Equation (48) indicates that at the optimum the householdcannot be made better off by trading leisure across periods given the relevant realwages and the real interest rate7 Figure 41 captures the first situation that is theoptimal choice of consumption and leisure within a period
For the moment let us hold anticipated real interest rates constant Further letus assume unit elastic expectations with respect to wages as well as prices so thata change in the current wage or price level that alters the current real wage changesfuture expected real wages as well so that there is no change in the current realwage relative to future expected real wages In addition let us hold constant for themoment the effect on current real money balances of a change in the current realwage These assumptions help us to mimic the traditional ldquostaticrdquo or single-periodanalysis (eg Patinkin) of the effect of a change in the current periodrsquos anticipatedreal wage on individualsrsquo labor supply during period t Under such circumstancesan increase in the real wage for period t can have ambiguous effects As Patinkin(1965) states ldquofor simplicity it is assumed that [labor] supply is an increasingfunction of the real wage though there are well known reservations on this scorerdquo
To understand what Patinkin is referring to consider an increase in the real wagedue to a rise in the money wage wt Consider one possible result on the householdrsquoslabor supply decision The increase in the net real wage means a steeper budgetline as the householdrsquos optimal leisurendashconsumption combination changes from1 minus N s
t and cdt (call this choice A) to (1 minus N s
t )prime and (cdt )prime (choice C)
An increase in the real wage has two conceptually distinct effects on the house-holdrsquos labor supply decision an income effect and a substitution effect The incomeeffect refers to the fact that an increase in the real wage makes the household betteroff because it leads to an increase in the householdrsquos feasible consumption set in
$
Leisure
Figure 41 Consumption and leisure
48 Households as market participants
the current period The higher real wage means that the household if it so desirescan increase both its leisure and consumption Thus the household is able to reacha higher indifference curve which has associated with it a higher utility levelThe substitution effect refers to the fact that the increase in the real wage makesan hour of leisure relatively more expensive in terms of consumption that must beforgone
To isolate the substitution and income effects of the change in the net real wagesuppose that after the real wage increases we temporarily take away just enoughof the householdrsquos nonlabor income so that it is just able to attain its originalindifference curve The householdrsquos choice of leisure and income would then be(1 minus N s
t )primeprime and (cdt )primeprime (call this choice B) Thus if we hold the householdrsquos utility
level constant the increase in the real wage leads unambiguously to a lower levelof leisure since an hour of leisure is relatively more expensive the householdsubstitutes away from leisure choosing to work more hours The movement fromchoice A to choice B constitutes the pure ldquosubstitution effectrdquo of the higher netreal wage8
Now suppose that we give the household back the nonlabor income that wetemporarily took away This causes an outward shift in the budget line and thehouseholdrsquos new choice of leisure and consumption would be (1 minus N s
t )prime and (cdt )prime
(choice C) The movement from B to C constitutes the ldquoincome effectrdquo of thechange in the net real wage In the present case the income effect on the choiceof leisure is positive reflecting the assumption that leisure is a normal good
Note that the substitution and income effects on leisure work in opposite direc-tions The substitution effect of the higher real wage causes leisure to fall andhours worked to rise while the income effect causes leisure to rise and hoursworked to fall The net effect on leisure and hours worked depends on whicheffect dominates9 If the substitution effect dominates then a higher real wageresults in a decrease in desired leisure and an increase in desired working hoursIf the income effect dominates the opposite is true and the individualrsquos laborsupply curve is ldquobackward bendingrdquo when plotted against the real wage
The available evidence suggests that for many workers the income effect tendsto dominate slightly Estimates are that for men an increase of 10 percent in the realwage results in approximately a 15 percent reduction in the hours worked Thisreduction in hours worked reflects an income effect of approximately minus25 percentand a substitution effect of about 1 percent Other evidence suggests a similarpattern for working women10
The choice of participation within a period
Thus far we have considered a household representative of those who are in thelabor force working a positive number of hours Yet this masks the unambiguouseffect of a higher real wage on the labor supply of those households not in the laborforce To show this we consider a corner solution with respect to labor supply inparticular a household denoted a that has chosen not to participate in the labormarket For such a household let us return to the Bellman equation for period t and
Households as market participants 49
introduce explicitly the nonnegativity constraint for labor supply (ie Nt ge 0)Letting micron denote the multiplier associated with this constraint we have as first-order conditions for household arsquos consumption and labor supply11
partuatpartcat minus (partW (xat+1)partxat+1)Rt = 0
minus partuatpart(1 minus Nat) + micron + (partW (xat+1)partxat+1)Rtwtpt = 0
Nat ge 0 micronNat = 0
Substituting the first equation into the second and rearranging gives
minuspartuatpart(1 minus Nat)
partuatpartcat= wt
pt+ micron
partuatpartcat
For individuals not participating in the labor force micron ge 0 The ldquocorner solutionrdquo isa case in which the marginal rate of substitution of leisure in terms of consumptionis greater than the real wage In other words the absolute value of the indifferencecurve is equal to or greater than the absolute value of the budget line at the pointwhere N s
at = 0Note that at the optimal choice the corresponding indifference curve is more
steeply sloped than the budget line Thus even when all hours are devoted toleisure and none to work the individualrsquos marginal rate of substitution of leisurein terms of consumption still exceeds the real wage rate In other words theindividualrsquos valuation of leisure exceeds the marketrsquos valuation of leisure As aresult individual a does not find it worthwhile to participate in the labor market
The greater the real wage the greater is the probability that a given individualwill choose to participate in the labor market A higher real wage rotates thebudget line outward Since the individual is not working an increase in the netreal wage does not make him better off and thus has no income effect There isonly a substitution effect Thus if the real wage rises sufficiently the individualcan be induced to enter the labor market
According to the above analysis the economy-wide labor supply response toan increase in the current real wage is a combination of an ambiguous effecton the labor supply of those currently working but an unambiguous increase inthe labor supply among those not working12 It is the net of these two effects theldquohoursrdquo decision and the ldquoparticipationrdquo decision that is captured by the aggregatelabor supply function it is commonly assumed that this net effect is such that theaggregate quantity of labor supplied is an increasing function of the current realwage
The labor supply intertemporal substitution hypothesis
Our discussion has yet to consider ldquothe labor market intertemporal substitutionhypothesis (ISH) which states that labor supply responds positively to transitoryincreases in real wages and increases in the real interest rate a central hypoth-esis of modern competitive models of the business cyclerdquo (Alogoskoufis 1987)
50 Households as market participants
To do so we simply expand our focus to the inherent intertemporal decisionconfronting the household In particular recall our expression (48) for the optimalallocation of labor supply over time The view of labor supply embedded in (48)has life-cycle as well as business-cycle implications With respect to life-cycleimplications the theory predicts that workers will concentrate their labor supplyin years of peak earnings consuming leisure in larger than average amounts duringchildhood and old age
With respect to the business cycle the above helps explain an apparent contra-diction in the static theory of labor supply ndash the observed wage inelasticity of laborsupply in the long run with short-run fluctuations in employment which requirean elastic labor supply if one takes a ldquomarket-clearingrdquo approach with respect tothe labor market13 It does so by introducing a distinction between a permanentchange in the real wage and a temporary or transitory change in the real wage
To show the intertemporal substitution effect with respect to labor supply spec-ify wlowastplowast as the permanent or ldquonormalrdquo real wage with the anticipated real wagenext period equal to this value such that
wipi = wlowastplowast i = t + 1 t + 2
Further we assume that
Ri = Rlowast i = t t + 1 t + 2
In this case equation (48) becomes
partutpart(1 minus Nt)
βpartut+1part(1 minus N lowast)= Rlowast wtpt
wlowastplowast (48prime)
where N lowast denotes the ldquolong-runrdquo supply of labor at ldquonormalrdquo wages Further letus assume that for the representative household βRlowast = 1 Then equation (48prime)becomes
partutpart(1 minus Nt)
partut+1part(1 minus N lowast)= wtpt
wlowastplowast (48primeprime)
Equation (48primeprime) indicates that if the current real wage is higher than the normal realwage then ldquomore labor is supplied than would be implied by the long-run laborsupply functionrdquo That is this theory views suppliers of labor as reacting primarilyto three variables an anticipated ldquonormalrdquo or ldquopermanentrdquo real wage rate whichcorresponds to the wage rate in the usual one-period analysis of the laborndashleisurechoice and has a negligible effect on labor supply the deviation of the current realwage from this normal rate which has a strong positive effect on labor supplyand the expected real rate of interest (Lucas and Rapping 1970 284ndash285)14
The above theory provides the underlying microtheoretical basis for the fol-lowing statement ldquomeasured unemployment (more exactly its nonfrictionalcomponent) is then viewed as consisting of persons who regard the wage ratesat which they can currently be employed as temporarily low and who therefore
Households as market participants 51
choose to wait or search for improved conditions rather than invest in moving oroccupational changerdquo (Lucas and Rapping 1970 285)
Empirical tests seem to provide some support for this intertemporal substitutionhypothesis15 Alogoskoufis (1987 950) finds that for measures of the total numberof employees the real wage and interest rate elasticities are high and relativelywell determined ldquoThe elasticity of labor supply to transitory changes in real wagesis around unity and is statistically significant at conventional significance levelswith one exception The real interest rate always has a significant independentinfluencerdquo Note that as suggested by equation (48) Alogoskoufis finds that arise in the real interest rate increases current labor supply
Note that equation (48) does not explicitly identify the initial asset holdings asa variable that affects the relative choice of labor supply across periods Similarlythe condition for the optimal choice of consumption across periods did not have theinitial value of assets affecting the relative consumption purchases across periodsThis is a property of time-separable preferences
Special topics in intertemporal choices
As we have seen the intertemporal problem confronting the individual involvessimultaneous decisions with respect to consumption versus saving and with respectto the composition of the asset portfolio To make some sense of what is involvedwe start by considering what is known as the ldquoFisherianrdquo problem which focuseson the individualrsquos choice of consumption across time
Fisherian analysis
The Fisherian problem typically deals with the allocation of consumption acrosstime when there is a single means by which income can be allocated across time16
To restrict our analysis to such a case we can simply omit real money holdingsfrom the utility function Further we consider only interior solutions with respectto consumption (ie cai gt 0 i = t t + T )17
Thus the maximization problem becomes18
maxcat cat+T
xat+1xat+T+1
t+Tsumi=t
β iminustua(cai)
subject to
minus xat+1 + Rt[xat + cat minus cat] = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1] = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2] = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ] = 0
xat+T+1 ge 0
52 Households as market participants
Recall that Ri = (1 + ri)(1 + πi) i = t t + T denotes the ldquogross real rateof returnrdquo on agent arsquos portfolio between the end of period i and the end of periodi + 1 and asset holdings are solely in the form of bonds such that
xat equiv (1 + r)pbBapt
xat+1 equiv (1 + rt)pbtBatpt+1 = [(1 + rt)(1 + πt)] pbtBatpt
xat+i equiv (1 + rt+iminus1)pbt+iminus1Bat+iminus1pt+i
= [(1 + rt+iminus1)(1 + πt+iminus1)] pbt+iminus1Bat+iminus1pt+iminus1
i = 2 T + 1
Let λi i = t t +T denote the Lagrange multipliers for the constraints linkingthe total value of real asset holdings at the end of period i with the total value ofreal asset holdings at the end of period i + 1 Let microT denote the multiplier for thenonnegativity condition that xat+T+1 ge 0 Then the first-order conditions for theimplied Lagrangian L include
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai) = 0 i = t t + T 19
partLpartmicroT = xat+T+1 ge 0
microT ge 0
where uai = ua(cai) i = t t + T 20 Note that the above set of first-order
conditions consist of 3(T + 1) + 1 equations to determine 3(T + 1) + 1 variablesThe variables to be determined are cai i = t t + T xai i = t t + T + 1and microT Assuming continuity and strict concavity in ua(middot) and given a convex setof constraints
xai+1 xai cai| minus xai+1 + Ri[xai + cai minus cai] ge 0there is a unique solution to the problem
There are several implications of the above first-order conditions First theconditions imply that the desired total real value of assets (bonds) inherited at timet + T + 1 xd
at+T+1 will equal zero if duat+T dcat+T gt 0 In particular from the
condition
partLpartcat+T = βT (duat+T dcat+T ) minus λt+T Rt+T = 0
we see that if duat+T dcat+T gt 0 then λt+T gt 0 From the condition
partLpartxat+T+1 = minusλt+T + microT = 0
Households as market participants 53
we thus have that microT gt 0 This in turn implies from the condition
microT partLpartmicroT = microT xat+T+1 = 0
that xat+T+1 = 0 This finding should not be surprising With a time horizonof t + T agent a perceives no gain (utility) from acquiring assets at time t + Tto finance consumption in period t + T + 1 and a clear loss from doing so attime t + T in terms of consumption forgone given the assumption of nonsatiation(dua
t+T dcat+T gt 0) Second from the first set of equations we know that betweenany two periods i and i + 1
β iminustduai dcai
β iminust+1duai+1dcai+1
= λiRi
λi+1Ri+1 i = t t + T minus 1
This expression can be simplified to obtain
duai dcai
βduai+1dcai+1
= Ri i = t t + T minus 1
where Ri the real gross return between the end of periods i and i + 1 is given by(1+ri)(1+πi) Ri has been called Fisherrsquos ldquo(gross) real interest raterdquo since he wasone of the first to provide a lucid account of its role in determining consumptionacross time
Fisherrsquos ldquo(net) real interest raterdquo denoted by mi is then defined by
1 + mi equiv Ri = (1 + ri)(1 + πi)
Subtracting one from both sides and rearranging we have
mi = (ri minus πi)(1 + πi)
Thus for small expected rates of inflation we have the approximation21
mi = ri minus πi
In words the real interest rate is approximately equal to the nominal interest rateminus the rate of inflation The expected real interest rate is then the nominalinterest rate minus the expected rate of inflation
There are several features of the above that should be noted First if an individ-ualrsquos discount factor (β lt 1) equals the reciprocal of the real gross rate of interest([Ri]minus1 = (1 + πi)(1 + ri)) between periods i and i + 1 so that
βRi = 1
then the above expression of the first-order conditions for cai and cai+1 indicatesthat dua
i dcai = duai+1dcai+1 Given the concavity of the single-period utility
function (ua(middot)) and our assumption of a time-invariant one-period utility function
54 Households as market participants
it then follows that the individual will choose the same rate of consumption acrossperiods i and i + 1 in such a case22 This constant path of consumption betweenthe two periods can be said to emerge if an individualrsquos ldquorate of time preferencerdquoequals the (real) interest rate
If the expected gross return between two periods were higher (or for an individualwith a higher discount factor) the fact that β gt (Ri)
minus1 or equivalently βRi gt 1means from the first-order conditions that dua
i dcai gt duai+1dcai+1 Given the
assumed concavity of the one-period utility function the implication is that agentarsquos consumption during period i would be less than during period i + 1 That isβRi gt 1 rArr cd
ai lt cdai+1 Conversely if the expected real gross return were to be
lower (or for an individual with a lower discount factor) the fact that β lt (Ri)minus1
or equivalently βRi lt 1 means that the individualrsquos consumption during period iwould be greater than during period i + 1 That is βRi lt 1 rArr cd
ai gt cdai+1
For the two-period case (say periods t and t + 1) the optimal consumption ineach of the two periods can be shown graphically by the point of tangency betweenan indifference curve with slope minus(dua
i dcai)β(duai+1dcai+1) and a budget line
with slope minusRt 23 If one were to place cat+1 on the vertical axis and cat on thehorizontal axis then it would be possible to determine whether an individual is aborrower or a lender in any period For example if disposable income were lessthan consumption in the first period t then the individual is a lender at time tNote that points on the same indifference curve are such that
d[ua(cat) + βua(cat+1)
] = (duat dcat+1)dcat + β(dua
t+1dcat+1)dcat+1
= 0
Rearranging we have the slope of an indifference curve given by
dcat+1
dcat= minus dua
t dcat
βduat+1dcat+1
Note that points on the budget line satisfy the present value constraint
cat + (Rt)minus1cat+1 = cat + xat + (Rt)
minus1cat+1
Rearranging we have
cat+1 = Rt[cat + xat minus cat] + cat+1
such that the slope of the budget line is given by
dcat+1dcat = minusRt
Thus at the point of tangency between the budget line and an indifference curve
duat dcat
βduat+1dcat+1
= Rt
Households as market participants 55
which is the expression we obtained previously concerning the optimal choice ofconsumption between periods i and i + 1
Note that the above analysis is for an individual consumer Thus aggregate con-sumption need not behave as that predicted above for the individual For instancean aging population could lead to variations in aggregate consumption that reflectthe aggregation at different times across agents with differing characteristics
Life-cycle and permanent income hypotheses
We have seen how a householdrsquos consumption in any period is not constrained bythe income it receives during that period but rather that the discounted value oflifetime consumption is constrained by the discounted stream of income accruingto the household over its lifetime plus initial asset holdings While income tendsto rise and fall during the lifetime of an individual through appropriate saving andborrowing the individual can maintain a smooth or constant rate of consumptionover his lifetime This smoothing of consumption across time plays a critical rolein Franco Modiglianirsquos ldquolife-cycle hypothesisrdquo of consumption24
A stylized pattern of income and consumption expenditures over an individ-ualrsquos lifetime is the following Prior to retirement income exceeds consumptionand saving is positive During this period saving increases household wealth Onretirement consumption is financed by dissaving During the retirement periodhousehold wealth falls as people draw on their accumulated savings to financeconsumption Implied in this discussion is an inverted U-shape wealthndashage pro-file (save during pre-retirement years and dissave in years following retirementrunning down the stock of accumulated wealth) A number of studies of aggregatehousehold consumption and saving behavior support this wealthndashage pattern25
To make clear the implications of consumption smoothing for the demand forthe consumption good at time t let us make the simplifying assumptions that
(a) for any period i = t t + T Ri = R and(b) the individualrsquos personal discount rate β equals the constant real ldquomarketrdquo
discount rate Rminus126
From the first-order conditions we thus have the result that agent a willcompletely smooth out consumption spending across time so that consumptioncd
ai = cda i = t t + T In this case we can use the prior combined budget
constraint to obtain
cdat =
xat +
t+Tsumi=t
(caiRiminust)
where which equals 1sumt+T
i=t (1Riminust) is what Modigliani has calledthe ldquoproportionality factorrdquo and indicates the proportion of householdsrsquo totalresources ndash consisting of initial assets current income and anticipated futureincome ndash devoted to consumption each year
56 Households as market participants
An important implication of Modiglianirsquos life-cycle hypothesis is that thefraction of an increase in current income (cat) that goes toward increased cur-rent consumption (the ldquomarginal propensity to consumerdquo) will vary depending onwhether the increase in current disposable income is accompanied by an equivalentincrease in anticipated future income (cat i = t + 1 t + T )27 If a change incurrent income is viewed as ldquotransitoryrdquo most of the increase in income will goto saving in order to finance increased consumption during future years
This idea that the effect of a change in current disposable income on consump-tion demand depends on the degree to which the change in income is viewed astemporary or permanent lies at the heart of Milton Friedmanrsquos permanent incomehypothesis The permanent income hypothesis is like the life-cycle hypothesis inthat it emphasizes the fact that consumption demand in period t depends not onlyon current income but also on anticipated income in the future periods Permanentincome is that income which if received each year over a householdrsquos time hori-zon would yield an income stream with present value exactly equal to the presentvalue of the householdrsquos anticipated income stream That is permanent income cpis defined by the following equation
t+Tsumi=t
(cpRiminust) =t+Tsumi=t
(caiRiminust) + xat (49)
Factoring out cp on the left-hand side of (49) and rearranging we have
cp =
xat +
t+Tsumi=t
(caiRiminust)
(410)
Equation (410) indicates that permanent income is simply a weighted average ofcurrent and future incomes but in this case income in the more distant future isweighted less heavily since it is discounted more highly
Comparing permanent income to agent arsquos consumption demand in period tcd
at if there is complete smoothing of consumption spending across time then weobtain
cdat = cp
The implication of this equation is that the marginal propensity to consume out ofa change in current income that is perceived as permanent is equal to one whilethe marginal propensity to consume out of a change in current disposable incomethat is perceived as entirely transitory (having little impact on permanent income)is small Changes in transitory components of income are almost entirely saved ifpositive or borrowed if negative28
Our discussion so far of the impact of changes in income on current consump-tion demand has been restricted to what might be referred to as the effects ofldquotransitoryrdquo versus ldquopermanentrdquo income changes In doing so we have assumed adeterministic world in which individuals have perfect foresight concerning future
Households as market participants 57
income streams But what happens if individuals do not have perfect foresight Inparticular what if we introduce stochastic elements so that realized future incomeis a random variable Then the above theories suggest a difference in the responseof consumption demand to income changes that are anticipated or expected versusunanticipated changes In particular the life-cyclepermanent income hypothesiswould predict that previously anticipated (or expected) changes in income wouldhave no effect on consumption demand since consumption plans have alreadyincorporated this income29
Portfolio choice
Now let us consider the more general case in which agent a chooses not onlyconsumption across time but the portfolio of assets (money and bonds) That isconsider the following problem
maxcat cat+T
xat+1 xat+T+1Matpt Mat+T pt+T
t+Tsumi=t
β iminustua(cai Maipi)
subject to
minus xat+1 + Rt[xat + cat minus cat] minus [Rt minus Rmt]Matpt = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1]minus [Rt+1 minus Rmt+1]Mat+1pt+1 = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2]minus [Rt+2 minus Rmt+2]Mat+2pt+2 = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ]minus [Rt+T minus Rmt+T ]Mat+T pt+T = 0
xat+T+1 ge 0
As before to simplify the problem we assume interior solutions in this case notonly with respect to the consumption good but also with respect to money holdingsAgain let λi i = t t + T denote the Lagrange multipliers for the constraintslinking the real asset holdings at the end of period i with their real value at the endof period i + 1 and let microT denote the multiplier for the nonnegativity conditionthat xat+T+1 ge 0 The first-order conditions are
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpart(Maipi) = β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0 i = t t + T
58 Households as market participants
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai)
minus [Ri minus Rmi]Maipi = 0 i = t t + T 30
partLpartmicroT = xat+T+1 ge 0
microT partLpartmicroT = microT xat+T+1 = 0
λi ge 0 i = t t + T
microT ge 0
where uai = ua(cai Maipi) i = t t + T To isolate the portfolio choice
consider the portfolio choice of money and bond holdings for a given level ofcurrent consumption That is let us look at the optimal choice of Maipi given thatcai is held constant From the second set of conditions
β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0
Substituting the condition for the optimal choice of the total value of assets in thatperiod
λi = λi+1Ri+1
we obtain
β iminustpartuai part(Maipi) minus λi+1Ri+1[Ri minus Rmi] = 0
Now substituting the condition for the optimal choice of consumption next period(i + 1) as given by
β iminust+1duai+1dcai+1 minus λi+1Ri+1 = 0
we obtain
β iminustpartuai part(Maipi) minus β iminust+1(dua
i+1dcai+1)[Ri minus Rmi] = 0
Rearranging gives
duai d(Maipi)
βduai+1dcai+1
= Ri minus Rmi
Recalling that the expected gross real return on bonds in period i Ri equals(1 + ri)(1 + πi) and that the expected gross real return on money Rmi equals1(1 + πi) the above expression can be written as
duai d(Maipi)
βduai+1dcai+1
= ri
1 minus πi
Households as market participants 59
Note that in the limit (as the length of the period goes to zero) the expected rate ofinflation term would vanish The implication is that the optimal division of assetsbetween money and bonds depends primarily on the money interest rate aloneWhat is essentially being shown is that an increase in money holdings with nochange in current consumption means a reduction in bond holdings and thus theloss of nominal interest income ri or real interest income ri(1 + πi)
Absence of money illusion real balance effects and realindebtedness effects
There are two aspects of agent arsquos demand function that should be noted Firstagent arsquos demand functions at time t can be shown to be homogeneous of degree 0 inthe current price level pt initial money balances M a and initial bond holdings BaIn this economy this is said to reflect the ldquoabsence of money illusionrdquo One criticalreason for this is the assumption of unit elastic expectations with respect to futureprices so that changes in the current price level leave unchanged the expectedrates of change in the price level in subsequent periods Also note that the currentand expected future money payments attached to bonds are being held constantso that given the fixed money payment x on maturity interest rates are unchangedAlternatively one could have money prices and the fixed future money paymentattached to one-period bonds rise by the same proportion
The above implies that individual arsquos demand for the consumption good andreal money balances at time t can be represented by
cdat = cd
at(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
M datpt = M d
atpt(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
where xat is the individualrsquos real wealth at the end of period t as given by
Wat = xat + cat +t+Tminus1sum
j=t
⎡⎣ jprod
i=t
(Ri)
⎤⎦
minus1
(caj+1)
From the budget constraint for period t we know that the above two demandconditions imply a real demand for bonds of a similar form since
pbtBdatpt = cat + xat minus
[cd
at + M datpt
]
Note that the above demand functions do not depend solely on wealth and thepattern of expected real (gross) rates of interest since given the portfolio choice agiven pattern of expected real (gross) interest rates could alter demand dependingon the underlying values of the money interest rate As before individual arsquos excessdemand function for the consumption good and money in period t are defined byzat = cd
at minus cat zam = M datpt minus M apt and zab = pbtBd
atpt 31
60 Households as market participants
The ldquoreal balance effectrdquo indicates the effect of a change in real balances (M apt)
on individual demand for goods other than money As before there is a real balanceeffect that reflects a wealth effect That is a decrease in initial real balances leadsto a reduction in real money demand M d
atpt and a reduction in the real demandfor bonds pbtBd
atpt There is also what might be referred to as a ldquoreal indebtednesseffectrdquo in that a change in prices alters not only real money balances but also realinitial debt If Ba gt 0 an increase in pt reduces wealth while if Ba lt 0 anincrease in pt increases wealth This is why the characterization of the absence ofmoney illusion has been expanded to include changes in Ba
It is typical in macroeconomics to adopt the convention of the ldquorepresentativeagentrdquo to reduce notational clutter Recall that the ldquorepresentative agentrdquo is essen-tially the average agent For instance if we let cd
at denote the demand for theconsumption good in period t by representative agent a and cd
t market demandat the time then cd
at = cdt Thus depending on the context we can interpret cd
tas demand by the representative agent or market demand Recall that in doingso we essentially ignore distribution effects such as effects on market demandof changes in the distribution of initial endowments of commodities or moneybalances or of changes in the distribution of future endowments In the context ofthe real indebtedness effect since in the aggregate B = 0 this effect is removedfrom our analysis
Intertemporal substitution the evidence
We have focused above on the behavior of an individual with respect to the plannedpath of consumption across time As Robert Hall (1988 340) indicates
The essential idea is that consumers plan to change their consumptionfrom one year to the next by an amount that depends on their expectationsof real interest rates Actual movements of consumption differ from plannedmovements by a completely unpredictable random variable that indexes allinformation available next year that was not incorporated in the planningprocess the year before If expectations of real interest rates shift then thereshould be a corresponding shift in the rate of change of consumption Themagnitude of the response of consumption to a change in real interest rateexpectations measures the intertemporal elasticity of substitution32
Hall (1988 340ndash341) goes on to state that
the basic model of the joint distribution of consumption and the return earnedby one asset that has emerged is the following The joint distribution ofthe log of consumption in period t log ct and the (real) return earned by theassets from period t minus 1 to period t mtminus1 is normal with a covariance matrixthat is unchanging over time The means obey the linear relation
E(log ct) = k + ctminus1 + σE(mtminus1) (411)
Households as market participants 61
That is the expected change in the log of consumption is a parameter σ times the expected real return plus a constant If the expected real interestrate E(mtminus1) is observed directly then the key parameter σ can be estimatedsimply by regressing the change in the log of consumption on the expectedreal rate That regression also has the property that no other variable knownin period t minus 1 belongs in the regression
Hall proceeds to estimate the parameter using aggregate data on consumptionand finds that there is ldquolittle basis for a conclusion that the behavior of aggregateconsumption in the United States in the twentieth century reveals an importantpositive value of the intertemporal elasticity of substitutionrdquo (1988 356)
Hallrsquos empirical finding is of importance to macroeconomic analysis Thework by Hall and others is also of interest as an example of how theoreticalmacroeconomic analysis specifically Fisherian analysis can be tested To seethe link between the theory developed in the prior section and the proposed test(equation (411)) assume the following
Assumption 43 The path of aggregate consumption reflects agent arsquos decisionsconcerning the optimal allocation of consumption across time That is we treatagent a as the ldquorepresentative consumerrdquo Thus cai i = t t+T which denotesconsumption in period i of the representative agent a differs only in scale from cii = t t + T which denotes the aggregate level of consumption This assump-tion that an aggregate variable can be viewed as reflecting decisions of a representa-tive agent is not innocuous For instance the actual path of aggregate consumptioncould well differ from that predicted by an analysis of individualsrsquo optimaldecisions due to changes across time in the composition of individuals in theeconomy33
Assumption 44 Individualsrsquo expectations in period t minus 1 of future real interestrates incorporate all information available as of period t minus 1 New informationoccurring in period t that alters consumption from what was planned results inthe distribution of consumption being ldquolog normal conditional on informationavailable last period that is log ct is normal with mean E(ct)rdquo (Hall 1988 342)
Assumption 45 In period t minus 1 the representative agentrsquos utility function takesthe following form
t+Tsumi=tminus1
expminusδi + ((δ minus 1)δ) log(ci)
where c gt 0 σ gt 0 and δ gt 0 This exponential utility function has the followingdesired properties
1 It is time-separable2 If consumption were equal across any two periods the individual would place
greater value on an increase in consumption in the earlier period ndash that is if
62 Households as market participants
ctminus1 = ct then
expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)gt expminusδt + ((δ minus 1)δ) log(ct)
3 Given 1 gt σ ge 0 utility increases in any period with increased consumptionbut at a decreasing rate ndash that is
du(ctminus1) = dctminus1
= [(1 minus δ)(δctminus1)] middot expminusδt(t minus 1)
+ ((δ minus 1)δ) log(ctminus1)gt 0 d2u(ctminus1)dc2
tminus1 lt 0
Note that the ldquointertemporal elasticity of substitutionrdquo will be given by σ As σ approaches 0 substitution of consumption across time in response tochanges in the real interest rate will approach zero as well
Assumption 46 Individualsrsquo forecasts of future variables are held with subjec-tive certainty This last assumption is a departure from Hallrsquos analysis that allowsus for the moment to maintain the ldquodeterministicrdquo aspect of the prior optimizationproblem That is we continue to assume that individualrsquos expectations of futurevariables such as expected rates of inflation and future interest rates are held withsubjective certainty
Given the above assumptions we know from our prior discussion that thechoice of consumption for periods t minus1 and t must satisfy the following first-ordercondition
dudctminus1
dudct= Rtminus1
Substituting in the appropriate expressions for the marginal utility of consumptionin periods t minus 1 and t we obtain
[(1 minus δ)δctminus1] middot expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)[(1 minus δ)δct] middot expδ + ((δ minus 1)δ) log(ct)
or
(ctctminus1) expδ + ((δ minus 1)δ) log(ctminus1ct) = Rtminus1
Taking the logarithm of both sides of the above expression and rearranging theabove first-order condition becomes
log(ctctminus1) + δ minus ((δ minus 1)δ) log(ctminus1ct) = log(Rtminus1)
Households as market participants 63
which simplifies to
δ + (1δ) log(ctminus1ct) = log(Rtminus1)
or
log ct = minusδσ + log ctminus1 + σ log(Rtminus1) (412)
which is similar in form to (411) Note that the ldquointertemporal elasticity ofsubstitutionrdquo is given by
σ = (dctdRtminus1)ctRtminus1
The form of equation (42) can be made closer to that of equation (411) if wenote that we can define the ldquoinstantaneous real rate of interestrdquo associated withcontinuous compounding mtminus1 by the expression
exp(mtminus1) = Rtminus1
Then (412) becomes
log ct = minusδσ + log ctminus1 + σmtminus1
Conclusion
This chapter has developed an in-depth understanding of household behaviorA good deal of the discussion has dealt with intertemporal choices and the tradeoffsinherent in consuming today versus consuming in the future Many policy-relatedissues in macroeconomics are related to decisions made today that are not indepen-dent of future states or activities This issue will arise again and again throughoutthis book and it is imperative that one comprehend the nature of decision-makingand time
5 Summarizing the behavior andconstraints of firms andhouseholds
Introduction
In this chapter we summarize our discussion of the behavior of firms andhouseholds in the simple Walrasian model with money and production In doingso we consider first the nature of constraints faced by the participants in the econ-omy with respect to decisions during period t and then their behavior in termsof demand andor supply Along the way we will try to simplify the notationand introduce various expectations and assumptions of different macroeconomicmodels We start our discussion with firms
Summarizing firmsrsquo constraints
We have seen how we can divide the general constraint facing firms that totalrevenues from all sources just exhausts expenditures each period into two sepa-rate constraints One is the ldquofirm financing constraintrdquo which states that desiredchanges in the capital stock as well as any capital adjustment costs are financedby issuing new bonds or equity shares That is for period t
I dnt + ψ(I d
nt) minus net Ast = 0 (51)
where
net Ast equiv As
t minus At
Ast equiv [
pbtBst + petS
st
]pt
At equiv [pbtB + petS
]pt
Idnt equiv Kd
t+1 minus K
Note that we implicitly assume that firmsrsquo plans for purchasing capital during theperiod correctly anticipate the price of output (capital) during the period and theprices of bonds and equity shares to be issued at the end of the period to financesuch purchases
Behavior and constraints 65
The second constraint the ldquofirm distribution constraintrdquo is that all revenuesfrom the sale of output net of that required to replace capital used up in the produc-tion process during the period be distributed to households either as wages interestpayments or dividends At time t firmsrsquo anticipated distribution constraint isgiven by
dt + (wtpt)Ndt + zBpt minus (ys
t minus δK) = 0 (52)
where z is the coupon payment and planned output supply is related to labordemand by the production function
yst = f (N d
t K) (53)
Equation (52) implicitly assumes that firms at time t have perfect foresight withrespect to the price of output during period t Alternatively the firm financingconstraint would take the above form if we presumed there were futures marketsat time t for the exchange of output during period t and financial assets at the endof period t
The labor market at time t determines employment for the period at a level N lowastt
and an associated rate of output denoted by ylowastt At the realized price of output the
firm distribution constraint for period t will turn out to be
dt + (wtpt)Nlowastt + zBpt minus ( ylowast
t minus δK) = 0 (54)
As (54) indicates actual real output during the period net of that used to replacedepreciated capital will be distributed to households1
Summarizing householdsrsquo constraints
With respect to households there is a single budget constraint for period t Like thatof the firm distribution constraint its form changes depending on what is assumedconcerning the correctness of expectations or the timing of markets As we haveseen at time t households make plans with respect to labor supply consumptiondemand and desired additions to their real holdings of financial assets and moneybalances based on a perceived constraint of the form
cdt + (M d
t minus M )pet + net Ad
t minus (wtpt)eN s
t minus (dt + zBpt) = 0 (55)
where
net Adt equiv Ad
t minus At
Adt equiv
[pbtB
dt + petS
dt
]pt
At equiv [pbtB + petS
]pt
66 Behavior and constraints
We presume that households have perfect foresight at time t with respect to thereal value of financial assets and the real value of dividends plus interest paymentsreceived from firms at the end of period t We leave open the possibility of errorsin expectations (held with subjective certainty) concerning the price level as itaffects real money balances and the real wage The term pt would replace pe
t if wepresumed perfect foresight on the part of households concerning the price level orequivalently presumed that there were futures markets at time t for the exchangeof output during period t
The labor market at time t determines employment and output for the periodAs before we let N lowast
t and ylowastt denote the actual rate of employment and production
of output during period t In such a case the actual firm distribution constraint(54) can be substituted into the household budget constraint Since prices will beknown by households at this point we may also replace the expected prices ofoutput bonds and equity shares by their actual prices Thus during period t thehousehold budget constraint for period t can then be expressed as
cdt + (M d
t minus M )pt + net Adt minus ( ylowast
t minus δK) = 0 (56)
As discussed below householdsrsquo behavior in the output and financial markets willbe based on realized income and prices and will differ from their plans made attime t based on expected prices and a labor supply decision unless they possessperfect foresight at time t concerning the prices that will prevail over period t andthe labor market clears with actual employment equal to labor supply2
Walrasrsquo law labor market and other markets at time t
Recall that Walrasrsquo law reflects the summing up of the constraints faced by individ-ual agents in the economy Since our preceding analysis concerned the constraintsof the ldquorepresentativerdquo firm and household we need only sum the constraints ofsuch representative agents to obtain Walrasrsquo law There still remains a potentialproblem however as to which of the different versions of the constraints enumer-ated above for firms and households to use The choice as one would suspectdepends on whether the market for labor effectively occurs at the same time as themarkets for output and financial assets or at different times
One version of Walrasrsquo law essentially combines the market for labor at time twith a futures market at time t for the exchange of output during period t andfinancial assets at the end of period t Equivalently this version of Walrasrsquo lawassumes limited perfect foresight at time t by both firms and households withrespect to prices for the period3 In such a case we would sum constraints (51)(52) and (55) (with pt replacing pe
t in (55)) to obtain
[cd
t + I dnt + δK + ψ(I d
nt) minus yst
]+[net Ad
t minus net Ast
]+ (wtpt)
[N d
t minus N st
]+ [M d
t minus M ]pt = 0 (57)
Behavior and constraints 67
Thus we have that the sum of excess demand across four markets ndash the laboroutput financial and money markets ndash must equal zero4 Note that one of thesefour markets the money ldquomarketrdquo reflects the equality between the demand forand supply of money The money ldquomarketrdquo is not of course like other marketsof an economy ndash which is why the word ldquomarketrdquo is in quotes That is unlike theother markets the money ldquomarketrdquo is not a place where the exchange of goods(eg labor financial assets or output) takes place5
Walrasrsquo law sequential markets and potential lackof perfect foresight
There is a modification to make with respect to the above that is required if marketsoccur sequentially and if there is not perfect foresight on the part of all agents attime t concerning prices for period t The sequential nature of markets is clearin that we have the labor market occurring at time t while the markets for outputand financial assets occur during the period In addition households in particularmay not correctly foresee at time t the price of output for period t Under suchcircumstances the prior version of Walrasrsquo law no longer holds for it would thensum constraints that are only anticipated not realized Instead given the sequentialnature of the markets we must break the analysis down into an analysis of thelabor market and an analysis of the other three markets
At time t the labor market occurs Assuming a competitive equilibrium for thelabor market we have a money wage determined at time t such that
N st = N d
t
Underlying the supply of labor at time t are householdsrsquo plans with respect toconsumption demand and saving (either in the form of financial assets or money)during the period These plans are influenced at time t by the anticipated pricelevel for commodities pe
t among other variables and as such these plans may notbe feasible given realized prices during period t
Once the labor market ends employment and output are determined for theperiod at levels N lowast
t and ylowastt respectively At that point households make plans with
respect to consumption and saving in light of the realized prices and the resultingeffective household budget constraint That realized household budget constraintis simply equation (56) which incorporates the actual firm distribution constraintAdding the firm financing constraint (51) we obtain a modified Walrasrsquo law forthe markets during period t of the form
[cd
t + I dnt + δK + ψ(I d
nt) minus ylowastt
]+[net Ad
t minus net Ast
]+ [M d
t minus M ]pt = 0
In the absence of perfect foresight the demands for consumption money balancesand financial assets during the period expressed in the above equation can differfrom the plans made at time t
68 Behavior and constraints
Summarizing firm behavior with limited perfect foresight
Consider firmsrsquo optimal plans at time t Note that at time t firms have an expectationof the price of output for the period We shall continue to assume that firms haveperfect foresight at time t with respect to the price of output over the periodWith respect to labor demand a diminishing marginal product of labor implies aninverse relationship between the real wage and labor demand
N st = N d
t (wtpt K)
It is typical to assume that the labor market is such that firms achieve employ-ment equal to that demanded In this case given the production function and thenature of the labor demand function we have an output supply function for periodt of the form6
yst = yd
t (wtpt K)
An increase in the real wage reduces labor demand and thus output supply so thatwe have
partN dt part(wtpt) lt 0 and partys
t part(wtpt) lt 0
Now consider firmsrsquo behavior with respect to investment and consequent finan-cial asset supply Given a diminishing marginal product of capital capital demandis inversely related to the expected real user cost of capital7
Assuming labor and capital are complements in the production process(part2f partKpartN gt 0) capital demand will be inversely related to the expected realwage in the subsequent period as well8 In particular in the absence of adjustmentcosts (for both capital and labor) we have the following capital demand function
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (58)
with
partKdt+1part(me
t + δ) lt 0 and partKdt+1part(we
t+1pet+1) lt 09
The above demand for capital stock at the end of period t (in place at time t +1)implies a net investment demand function for period t of the form
I dnt = I d
nt(met + δ we
t+1pet+1 K)
where net investment demand is inversely related to the expected real user cost ofcapital me
t + δ the anticipated real wage in the next period wet+1pe
t+1 and theexisting capital stock at time t K
Recall that the firm financing constraint in the absence of capital adjustmentcosts equates firmsrsquo net real financial asset supply to net investment demandThus given the nature of the net investment demand function the net real financial
Behavior and constraints 69
asset supply function for firms at the end of period t (at time t + 1) is identical tonet investment demand or
net Ast = net As
t (met + δ we
t+1pet+1 K)
where net real financial asset supply for period t like net investment demandduring period t is inversely related to the expected real user cost of capital theanticipated real wage in the next period and the existing capital stock at time t
With convex adjustment costs the optimal capital stock (as well as investmentdemand) depends on the entire future path of the expected real user cost of capitaland real wages That is with adjustment costs
Kdt+1 = Kd
t+1(met + δ me
t+1 + δ wet+1pe
t+1 wet+2pe
t+2 ) (59)
We can rewrite the above demand function for capital given adjustment costs soas to collapse future periods into essentially a single subsequent period10 To doso recall that the expected real rate of interest for period i me
i equals (ri minus πei )
(1+πei ) Let us assume static expectations concerning future interest rates so that
ri = rt i = t + 1 t + 2 Further let us assume static expectations concerningfuture expected inflation so that πe
i = πet i = t + 1 t + 2 The result is that
the expected constant real rate of interest between periods t and t + 1 is expectedto prevail in the future so that me
i = met i = t + 1 t + 2 We can then rewrite
the demand function for capital accumulated at the end of period t in the simplerform
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1 we
t+2pet+2 )
Now note that we can decompose the anticipated wage for period i and theexpected price level for period i into two components the wage or price levelin period i minus 1 and the expected rate of change in wages or prices respectivelyIn particular the anticipated money wage and price level for period t + 2 can beexpressed by
wet+2 equiv we
t+1(1 + πewt+1) and pe
t+2 equiv pet+1(1 + πe
t+1)
Let us now assume static expectations with respect to the rate of change in wagesbeyond the next period so that πe
wi = πewt+1 i = t + 2 t + 3 Recall that
we have already assumed a constant rate of price inflation in subsequent periodsIf we then add the assumption that beyond the next period the (constant) rate ofinflation in wages (πe
wt+1) equals the expected rate of inflation in prices (πet+1)
we have that wei pe
i = wet+1pe
t+1 i = t + 2 t + 3 In words the real wagein the subsequent period we
t+1pet+1 i = t + 2 t + 3 is anticipated to persist
indefinitely11 We can now rewrite the capital demand function given adjustmentcost (equation (59) as
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (510)
70 Behavior and constraints
Note that given our expectation assumptions the capital demand function withadjustment costs (equation (54)) has the same form as the capital demand functionin the absence of capital adjustment costs However the actual response to changesin the real user cost of capital or in the anticipated real wage in the subsequentperiod would typically be less with adjustment costs as such costs would leadfirms to only gradually move toward a new optimal capital stock
Equation (510) captures the idea that the demand for capital to be in placeat the end of period t (at time t + 1) depends inversely on the expected realuser cost of capital and that assuming capital and labor are complements capitaldemand depends inversely on the anticipated future real wage Similarly sincenet investment demand during period t simply reflects the difference betweencapital demand at the end of the period and the initial capital stock we haveinvestment demand being inversely related to the expected real user (or rental) costof capital and to the anticipated real wage for the subsequent period Finally fromthe firm financing constraint that relates firmsrsquo net real financial asset supply tonet investment demand we have firmsrsquo net financial asset supply being inverselyrelated to the expected real user cost of capital and the expected future real wageThus in summary we have
partKdt+1part(me
t + δ) lt 0 partI dntpart(me
t + δ) lt 0
partKdt+1part(we
t+1pet+1) lt 0 partI d
ntpart(wet+1pe
t+1) lt 0
partnet Ast part(me
t + δ) gt 0 partnet Ast part(we
t+1pet+1) lt 0
where mei equiv (rt minus πe
t )(1 + πet ) Note that gross investment demand is given by
Idt equiv Kd
t+1 minus K + δK = I dnt + δK
Summarizing household behavior with limitedperfect foresight
Householdsrsquo plans at time t concerning the labor supply choice the consump-tionsaving choice and the portfolio choice for period t are constrained by theanticipated household budget constraint as given by equation (55) From ourintertemporal analysis of these optimal choices we know that in general at time twe thus have labor supply at time t
N st = N s
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
consumption demand planned at time t
cdt = cd
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
Behavior and constraints 71
and money demand planned at time t
Ldt = Ld
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
where for notational ease we have defined planned real money demand by the termLd
t From the anticipated budget constraint as given by
cdt + (M d
t minus M )pet + net Ad
t minus (wtpet )N
st minus (dt + zBpt) = 0
we can infer net real financial asset demand planned at time t net Adt from
householdsrsquo plans concerning money demand consumption demand and laborsupply
With perfect foresight at time t concerning the prices for period t house-holdsrsquo plans made at time t concerning consumption money demand andnet real financial asset demand during period t will be fulfilled Thus lettingxt = dt + zBpt + At + M tpt we can express the above behavior conditions asfollows labor supply at time t is
N st = N s
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
consumption demand for period t is
cdt = cd
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
and money demand for period t is
Ldt = Ld
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
where Ldt equiv M d
t pt As we did with respect to firmsrsquo capital demand function given adjustment
costs we now introduce expectation assumptions that essentially collapse theentire sequence of future periods into a single future period First we assume staticexpectations concerning future interest rates so that ri = rt i = t + 1 t + 2 Next we assume static expectations with respect to future rates of inflation suchthat πe
i = πet i = t + 1 t + 2 Finally we assume that the expected rate of
wage inflation beyond the next period is constant and equal to the expected rate ofinflation (ie πe
wi = πewt+1 i = t + 2 t + 3 and πe
wt+1 = πet+1) so that the
real wage anticipated for the next period is expected to prevail indefinitely Giventhe above restrictive expectation assumptions we can rewrite the householdsrsquobehavioral functions in the following form the labor supply at time t becomes
N st = N s
t (wtpt wet+1pe
t+1 rt πet xt)
consumption demand for period t becomes
cdt = cd
t (wtpt wet+1pe
t+1 rt πet xt)
72 Behavior and constraints
and money demand for period t becomes
Ldt = Ld
t (wtpt wet+1pe
t+1 rt πet xt)
where real money demand at the end of period t is given by Ldt equiv M d
t pt Our discussion of the labor supply decision suggests that labor supply is directly
related to the real wage and inversely related to the real wage next period (whichreflects the real wage in all subsequent periods given our expectation assumptions)This response to changes in the real wages incorporates the intertemporal substitu-tion hypothesis The intertemporal substitution hypothesis also suggests that laborsupply is directly related to the interest rate and inversely related to the expectedrate of inflation in that either change implies a higher expected real rate of inter-est and thus a substitution from leisure to increased labor supply today Finallyhigher real initial holdings of financial assets real money balances or income inthe current period from bond and equity shares holdings will reduce labor supplyif leisure is a normal good In summary we thus have
partN st part(wtpt) gt 0 partN s
t part(wet+1pe
t+1) lt 0 partN st partrt gt 0
partN st partπe
t lt 0 partN st part xt lt 0
Our discussion of the consumptionsaving decision suggests that consumptiondemand is directly related to both the current and anticipated future real wagesince an increase in either implies an increase in the discounted stream of incomeFocusing on the Fisherian analysis of the allocation of consumption across timeconsumption demand would be inversely related to the money interest rate anddirectly related to the expected rate of inflation since either change implies ahigher expected real rate of interest Finally an increase in xt (reflecting sayhigher real initial holdings of financial assets increased real money balancesor higher income in the current period from bond and equity share holdings) willraise consumption demand in the current period In summary we thus have
partcdt part(wtpt) gt 0 partcd
t part(wet+1pe
t+1) gt 0 partcdt partrt lt 0
partcdt partπe
t gt 0 partcdt part xt gt 0
Our discussion of the portfolio decision suggests that real money demand isdirectly related to both the current and anticipated future real wage since an increasein either implies an increase in the discounted stream of income Focusing on theportfolio analysis money demand would be inversely related to the money interestrate as households shift from money to financial asset holdings Any effect of achange in the expected rate of inflation is indirect and will be ignored Finallyan increase in xt (reflecting say higher real initial holdings of financial assetsincreased real money balances or higher income in the current period from bondand equity share holdings) will likely raise money demand in the current period
Behavior and constraints 73
In summary we thus have
partLdt part(wtpt) gt 0 partLd
t part(wet+1pe
t+1) gt 0 partLdt partrt lt 0
partLdt part xt gt 0
So far our discussion has not included householdsrsquo real net financial assetdemand To see what determines this we can simply use the budget constraintand money labor and commodity demand functions Specifically rewriting thebudget constraint for period t
net Adt = (wtpt)N
st + dt + zBpt minus cd
t minus (M dt minus M )pt
An increase in the current real wage initial real money balances or dividendand interest payments for the current period is presumed to increase not onlycurrent consumption and money demand but also future consumption and moneydemand and thus increase net financial asset demand by households From theintertemporal substitution hypothesis an increase in the expected future real wagewill reduce current labor supply as well as increase current consumption demandboth changes imply a fall in net real financial asset demand for households
From the Fisherian analysis a higher expected rate of inflation will reducethe expected real rate of interest the above constraint indicates that the resultingincrease in current consumption demand will reduce householdsrsquo acquisition offinancial assets Similarly an increase in real initial financial asset holdings byraising both current consumption and money demand will lead to a fall in real netfinancial asset demand On the other hand a higher money interest rate due toboth the Fisherian effect on current consumption demand and the portfolio effecton money demand implies a higher net financial asset demand In summary wethus have
partnet Adt part(wtpt) lt (or gt or =) 0 partnet Ad
t part(wet+1pe
t+1) lt 0
partnet Adt partrt gt 0
partnet Adt partπe
t lt 0 partnet Adt partAt lt 0 partnet Ad
t partMpt) gt 0
partnet Adt part(dt + zBpt) lt (or gt or =) 0
where there are ambiguous effects on net financial asset demand of a change in thereal wage and of a change in anticipated dividend and interest payments becausean increase in either raises both consumption demand and money demand Notethat in the limit as the length of the period goes to zero the household budgetconstraint at time t becomes
net Adt + (M d
t minus M )pt = 0
74 Behavior and constraints
as all the flow terms go to zero at a point in time In this case assuming real moneydemand is directly related to income we obtain the unambiguous effect of
partnet Adt part(dt + zBpt) lt 0 partnet Ad
t part(wtpt) lt 0
However in the period analysis net Adt reflects net real financial asset demand at the
end of the period Thus assuming any increase in income is not fully reflected inan increased rate of consumption we have an offsetting effect and thus ambiguity
Summarizing household behavior without perfectforesight
If we presume that households learn of prices that will exist for period t aftertime t and they differ from what was expected then the actual demands for outputmoney and financial assets can differ from those reflecting plans made at time tThe key reason for this is that the actual constraint faced by households will differfrom that anticipated In particular using the actual firm distribution constraintwe replace anticipated real income from wages dividends and interest paymentsfrom firms with the actual real income net of depreciation ( ylowast
t minusδK) The resultingrealized household budget constraint after time t is then
cdt + (M d
t minus M )pet + net Ad
t minus ( ylowastt minus δK) = 0
If anticipations by households were incorrect at time t concerning prices or div-idends during the period then revisions in plans for consumption and saving willbe made in light of the actual budget constraint faced In this case the actual house-hold demand functions for output and money are written as follows consumptiondemand during period t is
cdt = cd
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
money demand during period t is
Ldt = Ld
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
and we replace partcdt part(wtpt) gt 0 and partcd
t part(dt + zBpt) gt 0 with
partcdt partylowast
t gt 0
Note that the term partcdt partylowast
t is referred to as the ldquomarginal propensity toconsumerdquo12 Similarly we replace partLd
t part(wtpt) gt 0 and partLdt part(dt + zBpt) gt 0
with
partLdt partylowast
t gt 0
Behavior and constraints 75
Money illusion and the real balance effect
Let us consider (sufficient) assumptions under which demands and supplies arehomogeneous of degree 0 in current wages prices and the nominal stock ofmoney ndash that is there is the absence of money illusion Note that in consideringwhether or not there exists money illusion we must now look at the behavior notonly of households but also of firms Consider firms first
Assuming perfect foresight on the part of firms it is clear that current labordemand is homogeneous of degree 0 in current prices (the wage rate wt and the pricelevel pt) Thus so also current output supply Assuming unit elastic expectationswith respect to wages and prices in all future periods and expectations of futureinterest rates that are invariant to changes in current prices and wages we havethat capital demand and thus also investment demand and firmsrsquo net real financialasset supply are homogeneous of degree 0 in current prices
We already assumed static expectations concerning future interest rates and unitelastic expectations with respect to prices beyond period t + 1 in terms of nextperiodrsquos price to obtain a simple form for the demand functions for investmentThus we need only add (a) the assumption of unit elastic expectations concerningthe price of output between period t and t+1 (implying an expected rate of inflationπe
t and thus an expected real rate of interest that is independent of a change in theprice level pt)
13 and (b) unit elastic expectations concerning next periodrsquos wageand price level (implying an anticipated real wage next period independent of anequiproportionate change in wages and prices in the current period) to obtain theabsence of money illusion with respect to firmsrsquo investment demand14
To obtain the absence of money illusion with respect to households we mustshow that each of the arguments in their demand and supply functions is invariantto an equiproportional change in current prices wages and the nominal stockof money Given perfect foresight the current real wage and initial real moneybalances meet this condition But what about the expected real wage next periodthe expected rate of inflation current dividends and interest payments and the realvalue of initial financial assets holdings As it turns out the assumption of unitelastic expectations with respect to all future prices and wages is again critical inshowing these to be invariant as it was in deriving a capital demand homogeneousof degree 0 in current wages and prices
First it is clear that the assumption of unit elastic expectations with respectto future wages and prices makes future real wages invariant to equiproportionalchanges in the current wages and prices But note that in so doing we have elimi-nated the ldquointertemporal substitution hypothesisrdquo effect on labor supply of a changein the current real wage Similarly the assumption of unit elastic expectations withrespect to the future price level eliminates any effect of a change in the price levelon the expected rate of inflation Finally assuming that expectations of nominalfuture interest rates are invariant to an equiproportionate change in current priceswages and the money supply the expected real rate of interest will not be affectedby equiproportionate changes in wages prices and the money supply But notethat the ldquointertemporal substitution hypothesisrdquo impact on labor supply of a change
76 Behavior and constraints
in the expected real rate of interest initiated by a change in the current price levelis now absent as well15
What is left in order to obtain the absence of money illusion for households isto show that changes in current prices leave current dividend payments and thereal value of initial bond and stock holdings unchanged Once again as we seebelow the assumption of unit elastic expectations concerning next periodrsquos wagesand prices will be invoked to achieve this What we are looking for are sufficientassumptions that will result in the terms dt + zBpt and At being homogeneous ofdegree 0 in wt and pt
From the firm distribution constraint (52) and assuming firmsrsquo labor demandis satisfied (ie N lowast
t = N dt and thus ylowast
t = yst ) we know that
dt + zBpt = yst minus δK minus (wtpt)N
dt
Since N dt and ys
t are homogeneous of degree 0 in prices it is clear that the sumof current dividends plus interest payments is not affected by equiproportionatechanges in both the current wage and price level A higher price level does alterthe composition of payments however as real dividends rise and real interestpayments fall
Now consider At To show that this can be homogeneous of degree 0 in thecurrent wage and price level note from the firm distribution constraint and fromthe assumption that firmsrsquo demand for labor is satisfied in subsequent periods that
dt + zBpet+1 = f (N d
t+1 Kdt+1) minus (we
t+1pet+1)N
dt+1 minus δKd
t+1
A similar equation holds for future periods as well At simply reflects the presentvalue of such future real payments using the appropriate expected real interest ratesfor discounting The assumption of unit elastic expectations concerning prices inall future periods coupled with expectations of future nominal interest rates thatare unaffected by an equiproportionate change in money prices and the moneysupply thus means that A is homogeneous of degree 0 with regard to a change inprices (price level and wages) and the money supply in period t16
A special case of the above is if we assume static expectations concerning futureinterest rates (ie ri = rt i = t + 1 t + 2 ) and zero adjustment costs In thiscase Kd
i i = t + 1 t + 2 would be the same in each future period Therewould be a constant labor demand (N d
i = N dt+1 i = t + 2 t + 3 ) as well given
the invariant real wage in conjunction with no change in their capital stock Nowrecall that At is defined by
At equiv [pbtB + petS]pt
where pbtBt is the present value of future interest payments and petS is the presentvalue of future dividends Since At is simply the present value of the now constantfuture stream of dividends and interest payments discounted using an invariant
Behavior and constraints 77
expected real rate of interest we thus have
At = [dt+1 + zBtpet+1]me
t
Since dt+1 +zBtpet+1 = f (N d
t+1 Kdt+1)minus (we
t+1pet+1)N
dt+1 minusδKd
t+1 from the firmdistribution constraint we can rewrite the initial real holdings of financial assetsas
At = [ f (N dt+1 Kd
t+1) minus (wet+1pe
t+1)Ndt+1 minus δKd
t+1]met
which is homogeneous of degree 0 in current wages and prices if we assume that theanticipated real wage next period is independent of an equiproportionate changein the current wages and prices In other words to obtain the absence of moneyillusion for households we assume as we did with firms that there are unit elasticexpectations concerning wages and prices in the next period
Note that an increase in the money interest rate and the expected rate of inflationthat leaves the expected real rate of interest unchanged will alter the compositionof financial asset holdings although not the total To see this recall that with staticexpectations concerning the interest rate the price of bonds at the end of period tis given by
pbt =infinsum
i=1
z(1 + rt)i = zrt
An increase in the money interest rate and expected rate of inflation such that thereal rate of interest is unchanged means a fall in the price of bonds but an offsettingrise in the price of stock as over time real dividend payments will be rising morerapidly while real interest payments will be falling more rapidly Similarly anincrease in the current level of prices although leaving At unaffected wouldreduce the real value of bond holdings and lead to an exactly offsetting increasein the real value of equity share holdings
The real balance effect is apparent from the nature of the demand and supplyfunctions In particular an increase in nominal money balances or fall in prices withno change in nominal money balances will increase initial real money holdingsand in general lead to an increase in consumption demand real money demandand real net financial asset demand In general labor supply would fall
6 The simple neoclassicalmacroeconomic model (withoutgovernment or depositoryinstitutions)
Introduction
We have now covered a substantial part of the underlying structure for a simpleaggregate model of an economy with production The specific elements of theldquomicroeconomic foundationsrdquo of this aggregate model developed so far have dealtwith the optimizing behavior of individuals (ldquorepresentativerdquo firms and house-holds) in a setting in which individuals take prices as given Implicit in thesediscussions is another part of the microeconomic foundations the way in whichindividual markets operate We have been assuming that prices adjust so that thepresumption that buyers and sellers are price-takers is justified In other wordsequilibrium within individual markets entails price adjustment to equate suppliesand demands In addition we will assume that all individuals in the economy cor-rectly foresee period trsquos output prices when input supply and production decisionsare made at the start of the period As discussed below however there are otheroptions
Static macroeconomic models the options
Grandmont (1977 542) notes that
one way to look at the evolution of an economic system is to view it as asuccession of temporary or short-run competitive equilibrium That is onepostulates that at each date prices move fast enough to match supply anddemand Although one assumes equilibrium in each period the economicsystem displays a disequilibrium feature along a sequence of temporary com-petitive equilibrium at each date the plans of the agents for the future arenot coordinated and thus will be in general incompatible this is to becontrasted with the perfect foresight approach where by definition such adisequilibrium phenomenon cannot occur
This temporary equilibrium view of the economy is characteristic of the simplestatic neoclassical model a model in which all prices adjust to maintainequilibrium1 The second key assumption of the neoclassical model is that agents
Simple neoclassical macroeconomic model 79
are informed about prices within the period In particular when making their laborsupply and demand decisions at the start of the period households and firms areassumed to correctly anticipate the prices they will have to pay to purchase outputduring the period As it turns out this element of ldquoperfect foresightrdquo with respectto markets through time t +1 is a critical feature of the analysis Before examiningthe implications of these assumptions however some history on the origin of theneoclassical model might be helpful
The phrase ldquoneoclassical macroeconomic modelrdquo is a descendant of ldquoclassicalrdquoeconomic theory as reflected in the work of Sir William Petty during the 1600sIn Das Kapital Karl Marx (1976 85) stated that ldquoby classical Political EconomyI understand that economy which since the time of W Petty has investigated thereal relations of production in bourgeois societyrdquo As Marx suggested early classi-cal economists focused on the determinants of the economyrsquos productive capacityThe neoclassical macroeconomic model shares this focus on the productive capac-ity of the economy as the determinant of total output It also turns out to be thestatic precursor to much of the current analysis in the macroeconomic literaturethat falls under the heading of ldquoreal business cycle theoryrdquo
While the simple static neoclassical model along with its dynamic and stochas-tic counterparts is one popular approach to macroeconomic analysis there areother approaches In fact even though static analysis is restricted to markets in thecurrent period there remains enough flexibility to introduce at least four ways ofcharacterizing macroeconomic analysis
1 ldquoNeoclassical modelrdquo competitive equilibria are assumed to exist in cur-rent and future markets and limited perfect foresight is assumed for allparticipants
2 ldquoIllusion modelrdquo competitive equilibria are assumed to exist in current andfuture markets but imperfect foresight is assumed on the part of some agentsThe result is like the ldquoLucas supply functionrdquo popularized by Lucas (1973)in which output can respond directly to increases in the actual output prices
3 ldquoKeynesian modelrdquo a competitive equilibrium is assumed not to exist inthe labor market as the money wage is fixed and employment is demand-determined However other prices in particular the prices of output arepresumed to reflect competitive equilibria A rational expectation version ofthis model is developed by Fisher
4 ldquoNon-market-clearing modelrdquo a competitive equilibrium is assumed not toexist in the output market as the price of output is fixed above the competitiveequilibrium level This model forms the basis of much of what appears inundergraduate macroeconomic analysis including the IS-LM model
The neoclassical model with its assumptions of flexible prices and informedagents provides a benchmark against which we can compare the predictions ofother (static) macroeconomic models2 It also provides insight into the nature of thestationary states for dynamic macroeconomic models that presume market-clearingprices and accurate forecasts of prices
80 Simple neoclassical macroeconomic model
Hicksian temporary equilibrium and Walrasrsquo law
For the market for any good ldquocompetitiverdquo equilibrium is defined by equalitybetween market demand and supply3 A temporary competitive equilibrium for theeconomy during period t will be characterized by a money wage for labor (wlowast
t )a money price for the consumption commodity ( plowast
t ) a money price for bonds( plowast
bt) a money price for equity shares ( plowastet) allocations to households in terms
of employment consumption bond holdings equity share holdings and moneybalances (N lowast
t clowastt Blowast
t Slowastt and M lowast
t ) and allocations to firms in terms of employmentoutput investment bond issues and equity share issues (N lowast
t ylowastt Ilowast
t Blowastt Slowast
t ) suchthat
bull these allocations are in the demand (supply) set of each agentbull these allocations are feasible
Together these two conditions imply prices determined in the labor output bondand equity shares markets for period t that result in zero excess aggregate demandfor labor output bonds equity shares and money Thus we may rewrite theconditions for a general equilibrium as a money wage a price of output a price ofbonds and a price of equity shares such that
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Bst = Bd
t
Sst = Sd
t
Mpt = M dt pt
where
I dnt = Kd
t+1 minus K
yst = f (N d
t K)
Given our assumption that bonds and equity shares are perfect substitutes ingeneral there will not be a unique equilibrium in terms of the number of bondsand equity shares supplied or demanded although the total value of financialassets supplied or demanded will be determinant Thus we replace the bond andequity share markets with a single market the financial market The equilibriumconditions then become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Simple neoclassical macroeconomic model 81
Ast = Ad
t
Mpt = M dt pt
where
Ast equiv [ pbtB
st + petS
st ]pt
Adt equiv [ pbtB
dt + petS
dt ]pt
We can view the financial market as simultaneously determining the price ofbonds pbt the price of equity shares pet and the interest rate rt That is once oneof these is known the other two are implied For instance from our definition ofthe interest rate
1 + rt equiv [z + pbt+1]pt
we see that given the coupon rate and expected price of bonds in the subsequentperiod the interest rate rt implies a price of bonds pbt As perfect substitutesbonds and equity shares must offer the identical expected gross return Thus wehave that
1 + rt = 1 + ret equiv [dt+1 + pet+1]pet
As you can see given expectations of future dividends and the future price ofequity shares an interest rate rt also implies a price of equity shares pet We oftentalk of the financial market in terms of an equilibrium interest rate The aboveshould make it clear that associated with such an equilibrium interest rate areprices of bonds and equity shares And a rise (fall) in the interest rate means a fall(rise) in the prices of bonds and equity shares
We will make one additional change in the characterization of the financialmarket to put it in terms of additional demands and supplies that is put it in netrather than gross terms The reason for this is that net financial asset demands andsupplies are what correspond to household saving in the form of financial assetsand firm investment Thus the equilibrium conditions become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
net Ast = net Ad
t
Mpt = M dt pt
where
net Ast equiv As
t minus At
net Adt equiv Ad
t minus At
At equiv [pbtB + petS]pt
82 Simple neoclassical macroeconomic model
According to the above we have four equilibrium conditions but only threeprices ndash the money wage rate the level of output prices and the interest rate ndashto be determined As usual Walrasrsquo law is invoked to show that only n minus 1 ofthe n equilibrium conditions are independent However the nature of Walrasrsquo lawdepends on whether we assume limited perfect foresight or not
Walrasrsquo law for limited perfect foresight sums up the constraints faced by theindividual agents in the economy at time t to obtain
[cdt + I d
nt + δK + ψ(I dnt) minus ys
t ] + [net Adt minus net As
t ]+ (wtpt)[N d
t minus N st ] + M d
t pt minus Mpt = 0
Thus the sum of excess demands for output financial assets labor and moneymust equal zero
When there is not perfect foresight at time t concerning prices for period t inthe output and financial markets we have the equilibrium condition for the labormarket
N st minus N d
t = 0
and the modified Walrasrsquo law based on the resulting employment and output ofthe form
[cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t ] + [net Adt minus net As
t ] + M dt pt minus Mpt = 0
In this case the money wage employment and thus output are determined inthe labor market and the modified Walrasrsquo law indicates that the price level andinterest rate are determined by any two of the remaining three markets
As it turns out most macroeconomic analysis takes this second approach tosolving for equilibrium That is the analysis focuses on the labor (and other input)markets and determines the effect of changes in output price (and potentially othervariables such as the interest rate) on equilibrium employment and thus outputThis generates an ldquoaggregate supply equationrdquo which is then combined with twoof the remaining three equilibrium equations ndash typically the commodity and moneymarket equilibrium conditions ndash to determine the equilibrium price level interestrate and output The modified Walrasrsquo law is invoked to ensure equilibrium inthe financial market at this point Working backward one can infer from theaggregate supply equation the equilibrium money wage and employment impliedby the analysis
The advantage of the above approach is that it can be used whether or not thereexists limited perfect foresight at time t with respect to the price level and whetheror not prices adjust to clear markets A disadvantage of the analysis is that inthe case of the neoclassical model with limited perfect foresight and competitiveequilibrium it arbitrarily breaks up the analysis of markets In doing so it requiresthat demand functions for such goods as money and commodities be specifiedwith income as an argument This form of the demand functions obscures the fact
Simple neoclassical macroeconomic model 83
that as we have seen in standard general equilibrium analysis demand functionsdepend on prices (including the real wage) and income is a variable determinedby the choice of labor supply
With these qualifications in mind we take the standard approach of macroeco-nomics and separate out the analysis of the labor (and other input) markets (theldquoaggregate supplyrdquo part) from the other markets (the ldquoaggregate demandrdquo part)The next section considers the labor market at time t
Labor market equilibrium
At the start of period t the labor market takes place and a rate of production of outputis determined From our analysis of firm behavior at time t we know that behindlabor demand is an expected price of output over the period and associated expectedreal wage an existing capital stock and existing technology as incorporated in theproduction function We shall assume that firms correctly anticipate at time t theprice of output for the period so that firms confront the actual real wage wtpt
From our analysis of household decision-making at time t we know that behindlabor supply at time t is not only the expected price of output and implied expectedreal wage but also such factors as the relationship of the anticipated current realwage to anticipated future real wages the expected real rate of interest anticipatedwealth in the form of financial asset and real money holdings and anticipated cur-rent nonlabor income Like firms we will assume households have limited perfectforesight in that at time t they correctly foresee the price level for period tAssuming a Walrasian or ldquocompetitive equilibriumrdquo view of the labor marketsthe money wage wt adjusts to achieve equilibrium in the labor market under thesecircumstances
We have already seen how static expectations concerning future rates of wageand price inflation along with the assumption of unit elastic expectations concern-ing wages and prices next period simplify the labor supply function by removingexpected future real wages as explicit arguments Patinkin (1965) among oth-ers goes several steps beyond these simplifying assumptions and assumes that allother variables excepting the real wage do not have a significant impact on laborsupply4 Thus equilibrium in the labor market is given by a level of employmentNt and money wage wt such that
Nt = N dt (wtpt K) and Nt = N s
t (wtpt)
As Patinkin (1965 264) notes
it will immediately be recognized that we have greatly oversimplified theanalysis of this market Both the demand and supply functions for labor shouldactually be presented as dependent on the real value of bond and moneyholdings as well as on the real wage rate5 Further if we were to permit thefirm to vary its input of capital its demand for labor would depend also onthe rate of interest Finally a full utility analysis of individual behavior wouldshow the supply of labor also to depend on this rate6
84 Simple neoclassical macroeconomic model
Besides a competitive labor market and the above simplified labor supplyfunction the neoclassical model assumes that suppliers correctly anticipate theaggregate price level As we will see the assumption of the neoclassical modelthat suppliers like firms have perfect foresight at time t with respect to the priceof output for the period means that changes in the price of output have no effect onoutput supply This characteristic of the neoclassical model implies an underlyingldquoblock recursiverdquo nature to the analysis as described below
At the start of the period the labor market occurs with employment and thusoutput and the money wage being determined for the period Employment andoutput are determined based on individualsrsquo expectations of subsequent variablessuch as the price of output And the level of employment and output influencethe remaining variables to be determined But in the neoclassical model the levelof employment and output are not influenced by the remaining variables to bedetermined We can thus solve the equilibrium sequentially looking first at thelabor market and the determination of employment and output then looking atthe output financial and money markets and the determination of the price leveland interest rate
Financial market equilibrium
With regard to the financial market Patinkin (1965 215) states
a decrease in the price of bonds (and equity shares) decreases the amountdemanded of consumption commodities it will also be assumed that itdecreases the amount demanded of money balances hence by the house-holdrsquos budget constraint their total expenditures on [net] bond holdings mustincrease
Thus Patinkin invokes a ldquoFisherian effectrdquo and a ldquoportfolio effectrdquo of a change inthe interest rate to obtain
part(net Adt )partpbt lt 0
From the firm financing constraint and the fact that a firmrsquos net investment demandis inversely related to the interest rate we have
part(net Ast )partpbt gt 0
The above analysis differs slightly from that found in Patinkin (1965 214) Forinstance Patinkin uses the assumption of static expectations concerning interestrates and a coupon rate z equal to 1 to express the price of bonds as the reciprocalof the interest rate that is
pbt = 1rt
Simple neoclassical macroeconomic model 85
Further Patinkin assumes no equity shares so his financial market consists only ofbonds Finally Patinkin does not graph net real bond demand and supply Ratherhe graphs the total number of bonds demanded and supplied Thus Patinkin hasdemands and supplies of bonds of the form
Bs = rtptAst equiv Bs
t
Bd = rtptAdt equiv Bd
t
As might be expected Patinkinrsquos bond demand and supply curves differ in naturefrom net real financial asset demand and supply curves For instance it is clearfrom the analysis of investment demand that a rise in the interest rate (a fall inthe price of bonds) will lead to reduced investment demand and thus reduced netreal financial asset supply This relationship between investment and firmsrsquo net realfinancial asset supply follows directly from the firm financing constraint whichgiven pbt = 1rt is of the form
net At equiv 1ptrt[Bst minus B] = I d
nt + ψ(I dbt)
Naturally if net investment were initially zero and there were zero adjustment costs(so that Bs
t = B) the fall in investment that results from a rise in the interest ratewould imply a similar decrease in the number of bonds supplied (Bs
t ) Howeverif initially Bs
t gt B then a higher interest rate even though it decreases invest-ment could at the same time increase the number of bonds supplied As Patinkin(1965 217) observes
consider the effect of an increase in the rate of interest (fall in 1rt) The inter-nal consistency of our model requires that this decrease the amount of realbonds supplied
However this need not reduce the number of bonds supplied As Patinkin(1965 217) continues
a rise in the interest rate ( pbt falls) has lowered the price received for bondsand so may increase the number of bonds necessary to finance the firmrsquosexpenditures on investment commodities (Bs
t gt B initially) even thoughthese expenditures have decreased
A similar analysis would apply to a comparison of householdsrsquo demand for bondsin terms of numbers with household net real financial asset demand
Money ldquomarketrdquo equilibrium
As we know from Walrasrsquo law having depicted equilibrium in the labor andfinancial markets we need only look at one of the other two remaining markets
86 Simple neoclassical macroeconomic model
the output market or the money ldquomarketrdquo Consider a money market in whichnominal money supply M and nominal money demand M d
t equiv ptLdt are plotted
against the reciprocal of the price level which indicates the ldquorelativerdquo price of oneunit of money in terms of output If there is no real balance effect with respect toreal money holdings then the money demand curve is invariant to a change in theprice level (elasticity of demand equals 1) If there is a real balance effect withrespect to money demand then a fall in the price level (a rise in the relative price ofmoney (1pt)) would lead to a less than proportionate decrease in nominal moneydemand (elasticity of demand less than 1)
Aggregate supply and demand an introduction
Temporary equilibrium for the economy can be characterized in several ways Aswe alluded to above one way common in journal articles and textbooks is to dividethe analysis under the headings of ldquoaggregate supplyrdquo and ldquoaggregate demandrdquo
Under the heading of ldquoaggregate supplyrdquo is an analysis of the input markets inparticular the labor market One aim is to determine input prices (in particular themoney wage) the employment of inputs and the implied production of output thatoccur at different levels of output prices (and potentially different levels of interestrates) The term ldquoaggregate supplyrdquo is applied to this analysis for the simple reasonthat it determines the ldquosupplyrdquo of total output at different prices
Under the heading of ldquoaggregate demandrdquo is an analysis of the other marketsin the economy during period t in particular the output financial and moneymarkets The aim is to determine the level of output prices and the interest ratethat occur at different levels of output The term ldquoaggregate demandrdquo attached tothis analysis reflects the fact that the analysis determines how the price level andinterest rate adjust to equate the ldquodemandrdquo for total output to different levels ofproduction
Combining the aggregate supply and demand analysis we can determine theoutput price level and interest rate associated with temporary equilibrium as wellas the underlying equilibrium money wage real wage employment consumptioninvestment and real money balances To understand more clearly what is involvedin aggregate supply and demand analysis and how they can be combined we con-sider below the specific case of the neoclassical model starting with the aggregatesupply
Equilibrium and aggregate supply
As we have said behind aggregate supply is an analysis of various input marketsto determine the response of total output to changes in such variables as theprice level7 In the neoclassical model changes in prices lead to equiproportionatechanges in money wages with no change in the equilibrium level of employmentand thus no change in aggregate supply To formally show this let us start withthe following statement of equilibrium in the labor market in terms of a money
Simple neoclassical macroeconomic model 87
wage wt and level of employment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpt) minus Nt = 0
A critical aspect of the above is the fact that suppliers in particular suppliers oflabor correctly anticipate the price level that will exist with respect to output sothat wtpt replaces wtpe
t in the labor supply functionTotally differentiating the above two equations with respect to wt Nt and pt
one obtains[(partN d
t part(wtpt))(1pt)
(partN st part(wtpt))(1pt)
minus1minus1
] [dwtdNt
]=[(partN d
t part(wtpt))(wtdpt( pt)2)
(partN st part(wtpt))(wtdpt( pt)
2)
]
Applying Cramerrsquos rule one obtains
dwtdpt = wtpt and dNtdpt = 0
Thus in the neoclassical model the real wage and employment level determinedin the labor market are independent of changes in the price level8
The ldquoaggregate supply equationrdquo combines the analysis of the labor market (andother input markets) and resulting determination of employment of various inputswith the production function to determine the resulting output supplied For theneoclassical model the aggregate supply equation is of the form
ylowastt = ylowast
t (K ) (61)
What is important about this equation is that the price level and interest rateare not arguments in the supply equation Of course changes in the capital stockchanges in technology or changes in the supply of other inputs (eg changes inthe oil supply) can affect output Similarly a change in the composition of thelabor force or government policies that affect labor supply can affect equilibriumemployment and thus output9
We may summarize the above findings graphically Consider an increase in pt Given perfect foresight both firms and households at time t would anticipate thishigher price level In the labor market the result would be an increase in theequilibrium money wage in the same proportion as the increase in the expectedprice level so that the anticipated real wage would remain the same as wouldemployment and output This outcome for the labor market is shown in Figure 61Note that the result of no change in employment or output in light of a higher outputprice simply requires that both labor demand and supply curves shift vertically bythe same amount Such equal shifts reflect the fact that the same increase in themoney wage leaves both firms and households anticipating the same real wage asbefore
In undergraduate textbooks the fact that changes in the price of output leaveemployment and real output unaffected in the neoclassical model is often shown
88 Simple neoclassical macroeconomic model
Labor
N demand
N supply
Figure 61 Labor market equilibrium
in ( pt yt) space by a vertical ldquoaggregate supply curverdquo Such a curve summarizesthe underlying behavior for the economy-wide labor market Later we will see thatunder other assumptions such as embedded in Lucasrsquos model (ldquomoney illusionrdquo)and the Keynesian model (fixed nominal wage) the aggregate supply curve willbe upward sloping
It is important to realize that the aggregate supply shown in Figure 62 is not thetypical market supply curve of microeconomics In microeconomics a higher priceof good x and consequent increase in the demand for inputs by firms producing thatgood draws inputs away from the production of other goods so that the higherprice of good x induces increased output of that good and associated increasedemployment of inputs (such as labor) in the production of the good Implicit inthis analysis is that there is reduced production of other goods in the economyHowever as the above analysis makes clear if the focus is on the aggregation ofall commodity markets a higher price level no longer induces increased aggregateoutput unless the quantity of total inputs supplies rises which will not be the caseunder neoclassical assumptions10
The natural rate of unemployment
The key feature of the above analysis of aggregate supply is the assumption thatprices adjust to continuously maintain equilibrium in the various markets and thatindividuals are perfectly informed concerning prices The result is a level of realoutput sometimes called the ldquofull employment levelrdquo So far missing from theanalysis however is any mention of unemployment If one expands the model tointroduce unemployment the rate of unemployment is called the ldquonaturalrdquo rate11
By ldquonaturalrdquo is meant that it is the rate of unemployment that the economy will
Simple neoclassical macroeconomic model 89
Output
Price
Aggregate supply
Figure 62 Aggregate supply
gravitate to as prices adjust to clear markets and individuals become fully informedconcerning prices (ie under the assumptions of the neoclassical model)
To introduce unemployment into the analysis when the labor market is in equilib-rium at full employment requires recognition of two facts First the labor marketis in a constant state of flux Not only do individuals enter and leave the laborforce continually but labor demand varies continuously across firms as they expe-rience variations in the relative demand for their output This instability of jobsthemselves has been estimated to account for roughly one-quarter of the averageunemployment rate as in an average year one in every nine jobs disappears and onein every eight is newly created (Leonard 1988) Second information is costly Ittakes time for new workers entering the labor force and for workers who have beenlaid off or have quit previous jobs to discover which employers have vacanciesand how wages vary across employers
When we take into account the continuous flows to unemployment together withworkersrsquo imperfect information about job vacancies we see that unemploymentis no longer inconsistent with the neoclassical model At any moment there existnew entrants into the labor market who are spending time searching for acceptablejobs There also exist laid-off workers who are either searching for alternativejobs or awaiting recall And there are workers who have quit their jobs and aresearching for other jobs This kind of unemployment is generally referred to asfrictional unemployment
Sometimes part of frictional unemployment is called structural unemploymentwith structural unemployment occurring because of a change in the compositionor ldquostructurerdquo of aggregate output across firms For example the replacement ofsteel with plastic in automobiles led to a shift in employment from steel factoriesto firms making plastic During this transition some steel workers experiencedstructural unemployment
90 Simple neoclassical macroeconomic model
To summarize when the labor market is in equilibrium there exists a positiveunemployment rate because workers continuously move into and out of the laborforce and between jobs12 To signify that this unemployment rate is ldquonaturalrdquo orconsistent with equilibrium in the labor market it is generally called the naturalrate of unemployment13 The corresponding level of employment is then oftenreferred to as the full employment level
Like equilibrium output and employment in the neoclassical model ldquosupplyfactorsrdquo determine the natural rate of unemployment as well14 Among these isthe demographic composition of the labor force but also unemployment insuran-ce and minimum wage legislation In recent years a large body of literature hasanalyzed labor markets and the sources of unemployment with the focus on searchand labor contracts Note that an understanding of the natural rate of unemploy-ment is important in determining the effects of government policies aimed at theunemployed
Equilibrium and aggregate demand
The aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) The ldquoISrdquo equation since it is simply the expression forequilibrium during period t with respect to the output market is given by15
cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t = 0 (62)
Note that the equation is termed the ldquoISrdquo equation because we can rewrite it toobtain
Idnt + δK + ψ(I d
nt) = ylowastt minus cd
t
indicating that equilibrium in the output market is equivalent to the equal-ity between Investment expenditures (the left-hand side of the equation) andhousehold Saving (the right-hand side of the equation)16
The assumption of unit elastic expectations concerning wages and prices givesthe following simple form for householdsrsquo consumption demand and firmsrsquoinvestment demand functions during period t17
cdt = cd
t (rt πet At Mpt ylowast
t )
and
Idnt = I d
nt(met + δ K)
The ldquoLMrdquo equation since it is simply the expression for equilibrium in period twith respect to the ldquomoneyrdquo market is given by
Ldt = Mpt (63)
Simple neoclassical macroeconomic model 91
This equation is termed the ldquoLMrdquo equation for it reflects the equality betweenLiquidity preference or demand and the Money supply We have seen that ourassumption of unit elastic expectations concerning wages and prices gives us thefollowing simple form for the real money demand function
Ldt = Ld
t (rt πet At Mpt ylowast
t )
From the modified Walrasrsquo law it follows that a price level and interest rate thatsatisfy the ldquoISrdquo equation and ldquoLMrdquo equation for a given level of output ylowast
t willalso result in equilibrium in the financial market
We thus have three equations (the aggregate supply equation (61) the ldquoISrdquoequation (62) and the ldquoLMrdquo equation (63)) that can be solved for the equilib-rium levels of output price and interest rate Looking at what underlies thesethree equations we can then infer the changes in money wages and employment(the labor market) as well as changes in the components of output demand (invest-ment and consumption demand) Equilibrium can be depicted in terms of a pricelevel and interest rate (Patinkinrsquos CCndashLLndashBB curves) or in terms of a price leveland output (ie aggregate demand and supply curves) We start with Patinkinrsquosdepiction of equilibrium
Depiction of equilibrium the Patinkin analysis
The neoclassical model aggregate supply is independent of changes in the pricelevel and interest rate18 This means that for any given level of output we canfocus on equations (62) and (63) to determine the equilibrium interest rate andprice level This is a reflection of the ldquoblock recursiverdquo nature of the solution tothe neoclassical model mentioned earlier
Patinkin (1965) suggests a graphical way of showing such an equilibrium com-bination of price level and interest rate using any two of three curves denoted theCC LL and BB curves The CC curve depicts combinations of the price level andinterest rate that satisfy the equilibrium condition for output (62) the LL curvedepicts combinations that satisfy the equilibrium condition for money (63) Wherethese two lines so constructed intersect it must then be the case that this price com-bination satisfies two of the three equilibrium conditions simultaneously It followsfrom the modified Walrasrsquo law that the curve indicating various combinations ofthe price level and interest rate that satisfy the equilibrium condition with respectto the financial market (the BB curve) goes through this point as well19
For the market for output equilibrium in period t is characterized by (62) Recallthat real output ylowast
t denotes that output reflecting the capital stock K technologyand the employment of labor determined in the labor market at time t From theneoclassical assumptions with respect to labor demand and supply equilibriumemployment and thus output are unchanged for any change in the price level orinterest rate
Let us presume that the pair ( plowastt rlowast
t ) is associated with equilibrium in the outputmarket In (price interest rate) space this combination is identified by a unique
92 Simple neoclassical macroeconomic model
point on the CC curve that identifies combinations of pt and rt associated withequilibrium in the commodity market (Figure 63)
To understand what lies behind the shape of the CC curve depicted above totallydifferentiate the market-clearing condition for the commodity market with respectto pt and rt Doing so we obtain20
[partcdt part(Mpt)] minus [Mp2
t ]dpt + [partcdt partrt + (1 + ψ prime)partI d
ntpartrt]drt = 0
Rearranging we have that the slope of the CC curve is given by
drtdpt | ydt minus ylowast
t = 0= [partcd
t part(Mpt)][Mp2t ][partcd
t partrt + (1 + ψ prime)partI dntpartrt] lt 0
The negative slope of the CC curve can be explained in the following way Thepositive numerator of the expression for the slope reflects the real balance effectwith respect to commodities this term indicates the fall in consumption demandthat would accompany a rise in the price level for such a rise reduces agentsrsquowealth in the form of initial real money balances21 The negative denominatorindicates the effect of a change in the interest rate on consumption and investmentdemand
Now let us consider the money market As before there is a unique point thatindicates an interest rate and a price of output at which there is equilibrium inthe economy and this point on the LL curve identifies combinations of pt andrt associated with equilibrium in the money market (Figure 64) Recall that theLL curve identifies combinations of pt and rt associated with equilibrium in themoney ldquomarketrdquo as given by (63)
The slope of the LL curve is given by totally differentiating the zero excessdemand condition with respect to money Doing so and rearranging one obtains
drtdpt |Ldt minus Mpt = 0 = [1 minus partLd
t part(Mpt)][Mp2t ][minuspartLd
t partrt] gt 0
r
CC
p
Figure 63 Commodity market
Simple neoclassical macroeconomic model 93
rLL
p
Figure 64 Money market
Note that the numerator of this term is positive The change in real money balanceswill be greater than the consequent change in real money demand given real balanceeffects with respect to financial assets andor commodities The denominator ispositive as well reflecting the fact that an increase in the interest causes householdsto shift their portfolio out of money holdings into bonds
Putting together the CC and LL curves we have equilibrium in the economyat the point where these two curves intersect From the modified Walrasrsquo law weknow that at this point demand for financial assets equals supply as well In factthere is a corresponding BB curve that goes through this same intersection
As with the CC curve to understand what lies behind the shape of the BB curvewe can totally differentiate the market-clearing condition for the financial marketwith respect to pt and rt That market-clearing condition is
net Adt = net As
t = 0
where in simplest form
net Ast = net As
t (met + δ K)
net Adt = net Ad
t (rt πet At Mpt ylowast
t )
Differentiating we obtain
[partnet Adt part(Mpt)] minus [Mp2
t ]dpt + [partnet Adt partrt minus partnet As
t partrt]drt = 0
Rearranging we have that the slope of the BB curve is given by
drtdpt |net Adt minus net As
t = 0= [partnet Ad
t part(Mpt)][Mp2t ][partnet Ad
t rt minus partnet Ast partrt] gt 0
94 Simple neoclassical macroeconomic model
The positive slope of the BB curve can be explained in the following way Thepositive numerator of the slope expression reflects the real balance effect withrespect to financial assets The term indicates the fall in net real financial assetdemand (and planned decrease in future consumption) that would accompany arise in the price level for such a rise reduces agentsrsquo wealth in the form of initialreal money balances The positive denominator indicates the effect of a change inthe interest rate on household lending and firm borrowing A rise in the interestrate would tend to decrease current consumption demand (ldquoFisherianrdquo effect) andmoney demand (ldquoportfoliordquo effect) and thus increase household net financial assetdemand On the other hand a rise in the interest rate would tend to reduce firminvestment demand by raising the expected real user cost of capital and thusreduce firm net real financial asset supply
While both the slope of the BB curve and the LL curve are positive it is thecase that the LL curve is steeper
A depiction of equilibrium aggregate demand andsupply curves
Using Patinkinrsquos analysis a higher output would require a lower price level tomaintain equilibrium in the output financial and money markets Alternativelyone can show the effect of a change in ylowast
t on pt and rt by totally differentiatingequations (62) and (63) with respect to pt rt and ylowast
t The aggregate demand curve summarizes this inverse relationship between the
price level and output arising from an analysis of the output financial and moneymarkets Specifically such a curve depicts combinations of output and price levelassociated with equilibrium in the output financial and money markets It isdownward sloping indicating that an increase in output requires a lower pricelevel to clear the output financial and money markets Behind a movement downthe aggregate demand curve are larger real balances that stimulate output demandeither directly (through a real balance effect on consumption demand) or indirectly(the resulting increase in real money balances leads to an increase in net realfinancial asset demand by households and thus a lower interest rate)22
It is important to realize that the phrase ldquoequilibrium in the output mar-ketrdquo in this context abstracts from supply-side considerations At each pricelevel the aggregate demand curve indicates the output that if produced wouldequal output demand (along with satisfying the equilibrium conditions withrespect to the financial and money markets) Production of output equal tothat demanded would occur if firms sought simply to produce to meet mar-ket demand and if workers were readily available for employment so thatfirms could hire to achieve production equal to what was demanded The termldquoequilibrium in the output marketrdquo does not imply equality between outputdemand and the output that our analysis of the labor market suggests would besupplied
Simple neoclassical macroeconomic model 95
Output
Price
Aggregate supply
Aggregate demand
Figure 65 Macroeconomic equilibrium
Figure 65 combines the aggregate demand and aggregate supply curves Wherethey intersect we know by construction that the resulting price level ( pt)0 andoutput ( yt)0 are such that
bull the labor market is in equilibrium (the economy is at a point on the aggregatesupply curve) and
bull the output financial and money markets are in equilibrium (the economy isat a point on the aggregate demand curve)
Conclusion
Using a very simple framework this chapter has developed a powerful macroeco-nomic model of the economy This model is grounded in the general equilibriumtheory set forth in earlier chapters Moreover this model highlights that degree ofconnectedness that exists among the output labor financial and money marketsThe microfoundations of macroeconomics have been emphasized and it has beenshown how market forces work in such a way as to lead to aggregate supply anddemand two key elements in any macroeconomic model
7 Empirical macroeconomicsTraditional approaches and timeseries models
Introduction
Quantitative approaches to analyzing economic data provide meaningful anduseful insight for understanding how variables interact and how they might beexpected to behave in a variety of circumstances including the future This chapteroutlines the traditional econometric-based method and the relatively simple butoften more elegant time series method for analyzing economic data in the timedomain The stochastic nature of economic data is discussed and the now com-mon ARIMA model found in much of the empirical macroeconomic literature isdeveloped in parts The chapter provides a solid background for understandingldquomacroeconometricsrdquo and time series analysis
Traditional approaches
Empirical macroeconomics can be roughly divided into two approaches ndash a tradi-tional approach that draws heavily on macroeconomic theory and a more recentapproach advocated for forecasting that does not rely to any great extent on the-ory To consider the former we start by presenting a simple theoretical model ofthe macroeconomy The model is used to illustrate traditional empirical analysesreflecting (a) tests of behavioral hypotheses (b) tests of reduced-form expressionsfor various economic aggregates and (c) the construction of econometric modelsfor policy simulations and forecasting The more recent empirical macroeconomicsapproach of using time series models for forecasting aggregate variables whichplaces less reliance on macroeconomic theory is then briefly considered
A simple theoretical macroeconomic model
Consider the following linear approximation of the simple static classical model(closed economy) such as that popularized by Sargent (1987a 20)1 The economyis divided into four markets a labor market (where wages w and total employ-ment n are determined) an output market (where total output y and the pricelevel p are determined) a financial market (where the interest rate r is deter-mined) and a money ldquomarketrdquo Our goal is to construct a model that will determine
Empirical macroeconomics 97
such endogenous variables as the level of employment wages output prices andthe interest rate
From Walrasrsquo law (as we will see more clearly later on) we need only explicitlyconsider three of the above four markets in the analysis For the labor market letus assume the following linear approximations for the key behavioral relations2
nd = f minus g(wp) (71)
ns = h + j(wp) (72)
These two linear equations indicate that firmsrsquo labor demand nd is inversely relatedto the real wage (wp) and householdsrsquo labor supply ns is directly related to thereal wage3
For the output market the key underlying behavioral and technological relationsare in linear form
cd = a + b( y minus T ) minus cr (73)
id = d minus er (74)
ys = f (n K) (75)
Equation (73) indicates that householdsrsquo consumption demand cd is directlyrelated to real disposable income (income y minus lump-sum taxes T ) andinversely related to the interest rate r4 Equation (74) indicates that firmsrsquo invest-ment demand id is inversely related to the interest rate r Equation (75) is theaggregate production function relating employment n and capital stock k tototal output supplied ys
For the money ldquomarketrdquo the key underlying behavioral relation is
Ld = l middot y minus m middot r (76)
Equation (76) indicates that householdsrsquo real money demand Ld is directly relatedto real income and inversely related to the interest rate Nominal money supplyM s is exogenous
General equilibrium in this economy means an equilibrium level of employmentnlowast wage wlowast price level plowast output ylowast and interest rate rlowast such that the labor marketis in equilibrium
ns minus n = 0 (77)
nd minus n = 0 (78)
the output market is in equilibrium
ys minus y = 0 (79)
cd + id + gd minus y = 0 (710)
98 Empirical macroeconomics
and the money ldquomarketrdquo is in equilibrium
Ld minus M sp = 0 (711)
Note that the component of output demand gd in (710) reflects exogenous realgovernment demand for output
The above macroeconomic model consists of 11 equations Macroeconomicmodels often can be represented by a system of equations This system of equationsis sometimes said to be a ldquostructural modelrdquo because the form is given from theunderlying theory As we will see below we can solve this system of equationsfor each of the ldquoendogenousrdquo variables as a function solely of the predeter-mined or ldquoexogenousrdquo variables5 Such solutions can be called the ldquoreduced-formrdquosolutions
By substituting the behavioral equations (71)ndash(76) into the equilibrium condi-tions we obtain the following set of five equilibrium conditions that can be solvedfor the five variables nlowast wlowast plowast ylowast and rlowast6
f minus g(wp) minus n = 0
h + j(wp) minus n = 0
f (n K) minus y = 0
a + by minus bT minus cr + d minus er + gd minus y = 0
ly minus mr minus M sp = 0
(712)
Note that the first three equations in the system (712) can be solved to obtainwlowast nlowast and ylowastas a function of the price level plowast In particular we obtain7
wlowast = [(f minus h)(g + j)]plowast (713)
nlowast = [1( j + g)][ jf + gh] (714)
ylowast = f (nlowast K) = f ([1( j + g)][ jf + gh] K) (715)
Equation (715) is an example of a reduced-form expression for output for itexpresses output solely in terms of the exogenous variables It is a special caseof what is called the ldquoaggregate supply equationrdquo in which the price level doesnot affect the level of output produced8 Note that equation (713) indicates thatchanges in the price level lead to equiproportionate changes in the equilibriummoney wage so that changes in the price level do not lead to changes in theequilibrium real wage
To obtain a reduced-form expression for the price level we can use the reduced-form expression for output in conjunction with the last two equations in the system(712) These last two equations are sometimes termed the ldquoIS equationrdquo andthe ldquoLM equationrdquo respectively9 Solving the LM equation (the last equation ofthe system (712)) for r and substituting both this expression for r and the priorexpression for equilibrium output ylowast (715) into the IS equation (the penultimate
Empirical macroeconomics 99
equation of the system (712)) we obtain the reduced-form expression for theequilibrium price level
plowast = M scprimem
[1 minus b + cprimem] f ([1( j + g)][ jf + gh] K) minus aprime (716)
where aprime = a + d + gd minus b middot T and cprime = c + eChanges in the variable aprime indicate changes in the autonomous component of
consumption (the term a in (73)) autonomous investment demand (d in (74))and government spending or taxing (gd or T respectively) By ldquoautonomousrdquo wemean that part of householdsrsquo and firmsrsquo output demand that is independent of thevariables to be determined by the analysis particularly income and the interestrate The variable cprime indicates the combined response of consumption demand (cin (73)) and investment demand (d in (74)) to a unit change in the interest rate
Note that the procedure to obtain equation (716) could be alternatively describedas follows First combine the last two equations in the system (712) to eliminatethe interest rate r The result is termed the ldquoaggregate demand equationrdquo Thisrelates the price level to the level of output at which the money and output markets(and thus by a modified version of Walrasrsquo law the financial market) are inequilibrium In this context equilibrium in the output market is defined as thelevel of production that if produced would equal output demand This aggregatedemand equation would then be combined with the aggregate supply equation(715) to determine the equilibrium price level or output10
Tests of behavioral hypotheses
Theoretical macroeconomic models embody predictions concerning factors thatinfluence the behavior of various groups in the economy For instance considerthe behavioral equations (71)ndash(76) in the prior simple macroeconomic modelWe could test the prediction that investment is inversely related to the interest rate(see (74)) or that money demand depends inversely on the interest rate (see (76))Or we could expand our theory of consumption behavior (equation (73)) to testa particular behavioral relationship between aggregate household consumptionand permanent income11 Or we could expand our view of labor supply behavior(equation (72)) as Stuart (1981) did in an examination of Swedish data to test theprediction that sufficiently high marginal tax rates will reduce the economy-widelabor supply
Tests of reduced-form hypotheses
Theoretical macroeconomic models also generate predictions concerning thereasons for fluctuations in such aggregate variables as real output unemploy-ment and the level of prices These predictions typically reflect the reduced-formsolutions of macroeconomic models Examples of reduced forms in our simplemacroeconomic model are equations (715) for output and (716) for price
100 Empirical macroeconomics
One example of an empirical test of macroeconomic theory that focuses ontesting the reduced-form predictions is Taylor (1979) Specifically Taylor teststhe reduced-form relationships between the money supply and the logarithmof income from trend and between the money supply and the rate of infla-tion using quarterly US data from 1953 to 197512 Similarly Christopher Sims(1972) uses quarterly US data on nominal output and the money supply to testwhether changes in the money supply lead to changes in the current dollar valueof national income (see Ewing 2001)
Large-scale econometric models and forecasting
Besides testing macroeconomic theories empirical macroeconomic analysis oftenseeks to forecast the future paths of economic aggregates Interest in forecast-ing stems not only from an obvious curiosity about the future path of aggregatevariables such as real output and prices but also from the fact that many ifnot all macroeconomic theories suggest that expectations of future events influ-ence current activity In this context forecasts can be used to proxy individualsrsquoexpectations of future events in tests of various aspects of macroeconomic models
One approach to making forecasts of future aggregate variables is to rely ontheoretical macroeconomics as a guide in the construction of large-scale econo-metric models Behavioral equations that are more detailed disaggregated versionsof equations (71)ndash(76) are estimated and coefficients are checked to make surethey agree with theory These models reflect attempts to produce large systems thatfaithfully represent the interrelationships in a complex national economy Givenpostulated paths of the exogenous variables the actual estimated equations arethen used to generate forecasts of the various aggregate variables such as outputand its components (eg consumption and investment) prices and interest rates
While large-scale econometric models as forecasting devices have their advo-cates (and many individuals reveal they have a positive value by willinglypaying for their forecasts) Granger and Newbold (1986 292ndash293) among oth-ers question the value of such a forecasting approach They note that ldquoteamsof macroeconomists have constructed forecasting models involving hundreds ofsimultaneous equations fitted to data that time series analysts would view as neitherplentiful nor of especially high qualityrdquo
An alternative to the large-scale macroeconomic models that is suggested areldquotime series modelsrdquo As summarized by Granger and Newbold (1986) ldquoin theiranxiety econometricians have failed to touch some very important basesrdquo thatinclude
bull the fact that there are many areas in which ldquoeconomic theory is not terriblywell developedrdquo
bull the fact that even where the theory is satisfactory it is ldquoalmost invariablyinsufficiently precise about dynamic specifications in the sense that it is clearthat one structure must be appropriaterdquo
Empirical macroeconomics 101
bull the fact that even after appeal to economic theory there will be error termssince ldquono theory provides a completely accurate description of the behaviorof economic agents so that any postulated equation necessarily includes astochastic error termrdquo
Given these problems many macroeconomists suggest that the appropriate start-ing point to forecasting future macroeconomic variables is the use of time seriesmodels One result is that such time series modelling terms as AR ARMA andARIMA now abound in the macroeconomic literature It is thus useful to brieflyreview the nature of time series models However at the outset it should be madeclear that the discussion below is not complete but rather is provided simply asan introduction to some terms and concepts Proofs of various propositions andrigorous definitions of various properties of time series models can be found intime series texts such as Enders (2004) and Mills (1999)
Time series models
Let us take as our premise the idea that the actual observed time series of somevariable yt t = 0 1 T (eg the logarithm of economy-wide output for thelast 30 years) is the realization of some theoretical process which can be called aldquostochastic processrdquo As phrased by Harvey (1993) ldquoeach observation in a stochas-tic process is a random variable and the observations evolve in time according tocertain probabilistic laws Thus a stochastic process may be defined as a collectionof random variables which are ordered in timerdquo To forecast future values of yt one needs a model that defines the mechanisms by which the observations aregenerated
A distinguishing feature of a pure univariate time series model is that move-ments in yt are ldquoexplainedrdquo solely in terms of its own past or by its position inrelation to time13 That is time series models look for patterns in the past move-ments of a particular variable and use that information to predict future movementsof the variable In general a time series modelrsquos forecast of y based on knownvalues yT+1 is given by
yT+1 = E( yT+1|y0 yT )
As Pindyck and Rubinfeld (1991) suggest ldquoin a sense a time-series model is just asophisticated method of extrapolation yet it may often provide a very effec-tive tool for forecastingrdquo In this sense time series models are more along the linesof empirical analyses that ldquolet the data speak for themselvesrdquo rather than empiricalanalyses that (strictly speaking) ldquotest economic theoriesrdquo As Harvey (1993) statesldquoan essential feature of time series models is that they do not involve behaviouralrelationshipsrdquo Time series models reflect a ldquostatisticalrdquo approach to forecastingHowever the patterns in the data discovered by time series models do influencetheoretical discussions of the macroeconomy and tests of macroeconomic theo-ries concerning behavior have incorporated time series models14 An example of
102 Empirical macroeconomics
how time series analysis has influenced theoretical macroeconomic discussions isgiven at the end of this section
Some properties of stochastic processes
There are several key properties of stochastic processes for time series that wecan introduce One is ldquostationarityrdquo For a stochastic process to be stationary thefollowing conditions must be satisfied for all t
E( yt) = micro
E[( yt minus micro)2] = σ 2y
E[( yt minus micro)( ytminusk minus micro)] = γk
Note that γ0 = σ 2y In words if a series is stationary the mean of the series is
invariant to time the variance of the series is invariant to time and the covarianceof the series is invariant to time As Pindyck and Rubinfeld (1991) note ldquoif astochastic process is stationary the probability distribution p( yt) is the same forall time t and its shape (or at least some of its properties) can be inferred by lookingat a histogram of the observations y1 yT that make up the observed seriesrdquo15
Also an estimate of the mean micro of the process can be obtained from the samplemean of the series
y = 1
T
Tsumj=0
yt
and an estimate of the variance σ 2y can be obtained from the sample variance16
σ 2y = 1
T
Tsumj=0
( yt minus y)2
As Granger and Newbold (1986 4) phrase it ldquoa stationarity assumption is equiv-alent to saying that the generating mechanism of the process is time-invariantso that neither the form nor the parameter values of the generation procedurechange through timerdquo The simplest example of a stationary stochastic process isa sequence of uncorrelated random variables with constant mean and variance
A second property of a stochastic process is the ldquoautocorrelation functionrdquo Theautocorrelation function provides us with a measure of how much correlation thereis (and by implication how much interdependency there is) between neighboringdata points in the series yt For stationary processes the autocorrelation with lagk is given by17
ρk = γkγ0
= E[( yt minus micro)( ytminusk minus micro)]σ 2y
Empirical macroeconomics 103
For any stochastic process ρ0 = 1 If the stochastic process is simple ldquowhitenoiserdquo (ie yt = εt where εt is an independently distributed random variablewith zero mean and finite variance) then the autocorrelation function for thisprocess is given by
ρ0 = 1 ρk = 0 for k gt 018
A simple example of a stochastic process is the ldquorandom walkrdquo process inwhich each successive change in yt is drawn independently from a probabilitydistribution with zero mean19 Thus yt is determined by
yt = ytminus1 + εt (717)
where εt is a sequence of uncorrelated random variables (E(εtεs) = 0 for t = s)with mean zero (E(εt) = 0) and constant variance (E(ε2
t ) = σ 2e for all t) Recall
that a sequence εt of this kind is typically called ldquowhite noiserdquo If the stochasticprocess is a random walk the one-period-ahead forecast of yt+1 is simply yt
A simple extension of the random walk process is to incorporate a trend in theseries yt We then obtain the following stochastic process known as a random walkwith drift20
yt = ytminus1 + d + εt (718)
The one-period-ahead forecast of yt+1 is now yt + d By repeatedly substitutingfor past values of yt into (718) we obtain
yt =tminus1sumj=0
εtminusj + td + y0 (719)
For the random walk process without drift (d = 0) we see from (719) that thefirst requirement for stationarity namely that the mean be constant over time issatisfied if y0 is fixed21 That is E( yt) = E( y0) Nevertheless the process is notstationary since Var( yt) = tσ 2
e The random walk process tends to meander awayfrom its starting value but exhibits no particular trend in doing so22
Autoregressive processes
A process similar to the random walk that is stationary is called the ldquofirst-orderautoregressive processrdquo or AR(1)23
The AR(1) process is given by
yt = φytminus1 + (1 minus φ)micro + εt (720)
A necessary condition for stationarity is that |φ| lt 1 in which case E( yt) equalsthe constant term micro in (720)24
104 Empirical macroeconomics
One possible example of an autoregressive process is the total numberunemployed each month (Granger and Newbold 1986) Let the total number unem-ployed in one month be yt This number might be thought to consist of a fixedproportion φ of those unemployed in the previous month (the others having foundemployment) plus a new group of workers seeking jobs If the new additionsare considered to form a white noise series with positive mean micro(1 minus φ) thenthe unemployment series is a first-order autoregressive process expressed byequation (720)
For convenience only it is often assumed that the process has a zero mean iemicro = 0 Note that we can always define a new variable yprime
t equiv yt minus micro such that yprimet
has a zero mean and is given by the AR(1) process
yprimet = φyprime
tminus1 + εt (721)
with |φ| lt 1 Setting micro = 0 thus simply means that the variable yt (egemployment real output etc) is measured in terms of deviations from its mean
By successive substitution for yt in (720) we obtain (assuming micro = 0)
yt =tminus1sumj=0
φjεtminusj + φty0 (722)
If the process is regarded as having started at some point in the remote past and|φ| lt 1 then we can write
yt =infinsum
j=0
φjεtminusj (723)
Given that εt is white noise we thus have that E( yt) = micro = 0 Assumingstationarity (|φ| lt 1) we know that the variance and covariances are constantRecall that εt is white noise such that E(εtεs) = 0 for s = t We thus have25
γ0 = σ 2y = E[( yt minus micro)2] = E
⎡⎢⎣⎡⎣ infinsum
j=0
φj(εtminusj)
⎤⎦
2⎤⎥⎦ = σ 2
e (1 minus φ2)
γ1 = E[( yt minus micro)( yt+1 minus micro)] = φσ 2e (1 minus φ2)
γ2 = E[( yt minus micro)( yt+2 minus micro)] = φ2σ 2e (1 minus φ2) etc
The autocorrelation function for AR(1) is thus particularly simple ndash it begins atρ0 = 1 and then declines geometrically ρk = φk Note that this process hasan infinite memory The current value of the process depends on all past valuesalthough the magnitude of this dependence declines with time
In general an autoregressive process of order p is generated by a weighted aver-age of past observations going back p periods together with a random disturbance
Empirical macroeconomics 105
in the current period The AR( p) process is thus given by26
yt = φ1ytminus1 + φ2ytminus2 + φ3ytminus3 + middot middot middot + φpytminusp
+ (1 minus φ1 minus φ2 minus middot middot middot minus φp)micro + εt
= micro + εt +psum
j=1
φj( ytminusj minus micro)
(724)
with a necessary condition for stationarity being that φ1 + φ2 + middot middot middot + φp lt 127
The next section discusses necessary and sufficient conditions for stationarity
A brief digression on necessary and sufficient conditionsfor stationarity
To get some idea of what are necessary and sufficient conditions for stationaritylet us consider two specific cases AR(1) and AR(2) In the AR(1) case
yprimet minus φyprime
tminus1 = εt
where yprimet equiv yt minus micro Note that εt can be termed the ldquostochasticrdquo component of the
expression and the remainder the ldquodeterministicrdquo component28
Focusing on the deterministic component and assuming for convenience thatmicro = 0 we have a homogeneous first-order difference equation of the form29
yt minus φytminus1 = 0 (725)
To solve this equation let us try the general solution
yt = Abt (726)
which naturally implies ytminus1 = Abtminus130 The problem then is to find the valuesof A and b Substituting the trial solution into the above difference equation weobtain
Abt minus φAbtminus1 = 0
which by multiplying through by b1minustA can be rewritten as
b minus φ = 0 (727)
Equation (727) is called the ldquoauxiliaryrdquo or characteristic equation of (725) Thisequation provides us with the solution for b which is simply that b = φ Thissolution value is sometimes referred to as the ldquorootrdquo of the characteristic equation
Using (727) to substitute for b in (726) we thus have as the solution to thefirst-order difference equation (725) the expression
yt = Aφt (728)
106 Empirical macroeconomics
The ldquoequilibriumrdquo solution for yt given a homogeneous difference equation suchas (725) is yt = 0 That is if yt equiv 0 then yt will not change over time Thisequilibrium solution is ldquostablerdquo if as t rarr infin yt rarr 0 That is deviations fromthe equilibrium yt will return toward that equilibrium value Obviously given oursolution (728) a necessary and sufficient condition for stability is that |φ| lt 1
In the context of an AR(1) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the first-order homogeneous difference equation (725)that is the deterministic component of yt for an AR(1) process The condition ofstationarity is thus that |φ| lt 1 The effect of a given shock will mitigate overtime if this stationarity condition is met A special nonstationary case is the ldquounitrootrdquo case of the random walk process in which φ = 1 In that case the effect ofa given shock will not dampen with the passage of time
Now let us consider the AR(2) case where
yprimet minus φ1yprime
tminus1 minus φ2yprimetminus2 = εt
in which yprimet equiv yt minus micro As before εt can be termed the ldquostochasticrdquo component of
the expression Assuming again for convenience that micro = 0 we can express thedeterministic component as a homogeneous second-order difference equation ofthe form
yt minus φ1ytminus1 minus φ2ytminus2 = 0 (729)
To solve this equation let us try yt = Abt which naturally implies ytminus1 = Abtminus1
and ytminus2 = Abtminus2 The problem then is to find the values of A and b Substitutingthe trial solution into the above difference equation we obtain
Abt minus φ1Abtminus1 minus φ2Abtminus2 = 0
which by multiplying through by b2minustA can be rewritten as
b2 minus φ1b minus φ2 = 0 (730)
The quadratic equation (730) is called the ldquoauxiliaryrdquo or ldquocharacteristic equationrdquoof the second-order difference equation (729) The roots of this equation will bethe solutions of (730)31 The two ldquocharacteristicrdquo roots m1 and m2 may be foundin the usual way from the formula32
m1 m2 = [φ1 plusmn (φ21 + 4φ2)
12]2 (731)
each of which is acceptable in the solution Abt Note that for the quadraticequation (730) the roots m1 and m2 satisfy
(b minus m1)(b minus m2) = 0
where φ1 = m1 + m2 and φ2 = minusm1m2
Empirical macroeconomics 107
Expression (731) provides us with two values for b and thus two potentialsolutions for yt
yt = A1mt1 and yt = A2mt
2
The general solution to a second-order difference equation combines these twoby taking the sum Thus the general solution of the second-order homogeneousdifference equation (729) is33
yt = A1mt1 + A2mt
2 (732)
Inspection of (732) suggests that stability for the linear second-order homo-geneous difference equation (729) requires that both roots of the characteristicequation have an absolute value less than one This reflects the fact that the gen-eral solution of the second-order homogeneous difference equation includes bothroots m1 and m2 The root with the higher absolute value is sometimes termed theldquodominant rootrdquo If both m1 and m2 are less than unity in absolute value yt willbe close to zero if t is large Equivalently if the dominant root is less than one inabsolute value convergence will occur
In the context of an AR(2) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the second-order homogeneous difference equation (729)that is the deterministic component of the process The condition of stationarityis thus that |m1| lt 1 and |m2 lt 1| That is for the AR(2) process stationarityrequires that the roots of the characteristic equation (730) are less than one inabsolute value ndash that is that they all lie inside the unit circle34
In terms of φ1 and φ2 the conditions for stationarity may thus be defined asfollows35
φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 φ2 gt minus1
As you can see the prior necessary condition that φ1 + φ2 lt 1 is augmented withtwo other conditions
We have seen that stability for the second-order homogeneous differenceequation requires that the maximum of (|m1| |m2|) ndash that is the dominant root ndash beless than 1 To see how one obtains the three necessary conditions listed above forstabilitywe assume this is the case and determine the resulting restrictions placedon the coefficients φ1 and φ236 Recall that
m1 m2 = [φ1 plusmn (φ21 plusmn 4φ2)
12]2
Suppose φ21 + 4φ2 gt 0 so that the roots are real A maximum of (|m1| |m2|) less
than one means that
2 minus φ1 gt (φ21 + 4φ2)
12 gt minus2 minus φ1
108 Empirical macroeconomics
and
2 minus φ1 gt minus(φ21 + 4φ2)
12 gt minus2 minus φ1
The sum of the roots is φ1 and since we are assuming each root is between minus1and 1 it must be the case that 2 gt φ1 gt minus2 or 0 gt minus2minusφ1 and 2minusφ1 gt 0 Thusthe second and third inequalities are always true and hence place no restrictionson φ1 and φ2
The first and fourth inequalities squared read
(2 minus φ1)2 gt φ2
1 + 4φ2 and φ21 + 4φ2 lt (minus2 minus φ1)
2
respectively These two expressions can be rewritten to obtain
φ1 + φ2 lt 1 and minus φ1 + φ2 lt 1
which are the first two conditions for stability The third condition for stability(φ2 gt minus1) is obtained from the case of complex conjugate roots37 The threenecessary conditions for the dominant root being less than one in absolute value arealso sufficient conditions This can be verified by showing using these inequalitiesthat it is possible to reverse the steps in the above calculations and arrive at themaximum of (|m1| |m2|) being less than one
In general a necessary and sufficient condition for stationarity of an AR( p)process is if the roots of the characteristic equation
b p minus φ1b pminus1 minus middot middot middot minus φp = 0 (733)
are less than one in absolute valueNote that for the AR(2) case there are three possible situations If φ2
1 +4φ2 gt 0the square root in (731) is a real number and m1 and m2 will be real and distinctIn this case the solution to (729) is given by
yt = A1mt1 + A2mt
2
where A1 and A2 are constants which depend on the starting values y0 and yminus1The second situation is that of repeated roots where φ2
1 + 4φ2 = 0 such thatm1 = m2 = m In this case the solution to (729) takes the form
yt = A3mt + A4tmt
If |m| lt 1 the damping force of mt will dominate both termsThe third situation for the AR(2) process is when φ2
1 + 4φ2 lt 0 in which casethe roots are a pair of complex conjugates (ie m1 m2 = h plusmn vi where h = φ12v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12) Thus the solutionto (729) is given by
yt = A5(h + vi)t + A6(h minus vi)t
Empirical macroeconomics 109
Appealing to De Moivrersquos theorem this expression can be transformed intotrigonometric terms The general form of the solution is
yt = pt(A7 cos λt + A8 sin λt)
where p is the modulus of the roots = (minusφ2)12) and λ satisfies the two conditions
cos λ = hp and sin λ = vp As in the other two cases if the absolute valueof the conjugate complex roots |h plusmn vi| lt 1 is less than one the process isstable The time path followed by yt in response to a shock is cyclical but theperiodic fluctuation will mitigate as time passes (ldquosinusoidal decayrdquo) if the processis stationary
Moving average processes
In a moving average process the process yt is described completely by a weightedsum of current and lagged random variables The simplest moving average processthe moving average process of order 1 or MA(1) takes the form
yt = micro + εt + θεtminus1 (734)
The term ldquomoving averagerdquo reflects the fact that a series so characterized will besmoother than the original white noise series εt In general such a moving averageprocess of order q MA(q) is written as38
yt = micro + εt + θ1εtminus1 + θ2εtminus2 + middot middot middot + θqεtminusq
= micro + εt +qsum
j=1
θjεtminusj (735)
As before if yt has nonzero mean we can focus with no loss of generality on thetransformed series yprime
t = yt minus micro which has zero meanOne possible example of a moving average process is economy-wide output
(Granger and Newbold 1986) Output yt could be in equilibrium (at mean micro)but is potentially moved from its equilibrium position each period by a seriesof unpredictable events such as periods of exceptional weather or strikes If thesystem is such that the effects of such events are not immediately assimilated butexert an influence on output for q periods then a moving average model can arise
Note that by repeatedly substituting for lagged valued of εt into an MA(1)process (equation (734)) one obtains39
yt =infinsum
j=1
(minusθ)jytminusj + εt (736)
If yt is not to depend on a shock to the system arising at some point in the remotepast θ must be less than one in absolute value Comparing (736) to (724) we
110 Empirical macroeconomics
see that assuming what was previously denoted the stationarity condition but whatin this context is called the ldquoinvertibility conditionrdquo namely that |θ | lt 1 then anMA(1) process can be represented by an AR(infin) process40 The weights on pastvalues of yt for this AR(infin) process decline exponentially
In general if similar invertibility conditions are met then any finite-ordermoving average process has an equivalent autoregressive process of infinite orderLikewise we have already shown (see (723)) that if the AR(1) process is station-ary it is equivalent to a moving average process of infinite order MA(infin) In factfor any stationary autoregressive process of any order there exists an equivalentmoving average process of infinite order so that the autoregressive process isldquoinvertiblerdquo into a moving average process
Mixed autoregressive moving average processes
An obvious generalization of the above discussion is to combine an AR( p) process(ie (724)) and an MA(q) process (ie (735)) An autoregressive moving averageprocess of order ( p q) or ARMA( p q) can be written as41
yt = micro + εt +psum
j=1
φj( ytminusj minus micro) +qsum
j=1
θjεtminusj (737)
where as before εt is a zero-mean white noise For simplicity consider the casewhere micro = 0
Whether or not a mixed process is stationary depends solely on its autoregres-sive part If the ARMA process is stationary then there is an equivalent MA(qprime)process Similarly provided invertibility conditions hold there is an AR( pprime) pro-cess equivalent to the above ARMA( p q) process It thus follows that a stationaryARMA process can always be well approximated by a high-order MA process andthat if the process obeys the invertibility condition it can also be well approxi-mated by a high-order AR process However an ARMA process has the advantageof ldquoparsimonyrdquo in that the mixed model ARMA( p q) often can achieve as gooda fit as say an AR( pprime) but uses fewer parameters (ie p + q lt pprime)
As an example of an ARMA( p q) process we need only consider a case wherethe variable yt is the sum of a ldquotrue seriesrdquo that is AR( p) plus a white noiseobservation error Thus an ARMA( p q) series results42
Integrated processes
In series arising in economics the assumption of stationarity is often veryrestrictive That is often the characteristics of the underlying stochastic processgenerating a time series appear to change over time With economic data some-times a transformation (such as taking the logarithm of the variable) can result in astationary series In other cases we can obtain a stationary series by differencingone or more times We say that yt is a ldquohomogeneous nonstationary process of
Empirical macroeconomics 111
order drdquo if
wt = dyt
is a stationary series Here denotes differencing that is
yt = yt minus ytminus1 2yt = yt minus ytminus1 = yt minus 2ytminus1 + ytminus2
The process yt is an ldquointegratedrdquo process if after a series yt has been differencedto produce a stationary series wt the series wt can be modelled as an ARMAprocess If wt = dyt and wt is an ARMA( p q) process then we say that yt is anldquointegrated autoregressive moving average process of order ( p d q)rdquo or simplyARIMA ( p d q)
An application of time series models
To gain a better understanding of how time series models are used in macroeco-nomic empirical work let us consider the time series for the logarithm of the levelof real output to be denoted by yt As we stated above theoretical macroeconomicmodels typically postulate shocks or ldquoimpulsesrdquo to the economy (eg shocks totechnology money supply and government policy) that in conjunction with aprediction of how various markets in the economy adjust to these shocks offersan explanation of the actual fluctuations in aggregate variables One importantquestion that can arise is whether such shocks are ldquotransitoryrdquo in that their effectsdo not persist or ldquopermanentrdquo where the economy moves to a new level of realoutput
The ldquotraditionalrdquo answer to the above question of transitory versus permanenteffects of shocks has been according to Campbell and Mankiw (1987a 111) thatfluctuations in real output ldquoprimarily reflect temporary deviations of productionfrom trendrdquo That is the traditional view has been that quarterly real output couldbe represented by
yt = Tt + St + Xt (738)
where Tt is a deterministic component representing trend St is a determinis-tic seasonal component and Xt is a stationary autoregressive process with nodeterministic component43
Trend and seasonal components were typically estimated in various fashionsand then eliminated The focus of empirical work had then been on examiningvariations in the detrended seasonally adjusted real output series For exampleBlanchard (1981) estimated the following second-order autoregressive process fordeviations of the log of seasonally adjusted real output from its estimated trendylowast
t and obtained the equation
ylowastt = 134ylowast
tminus1 minus 042ylowasttminus2 + εt (739)
112 Empirical macroeconomics
Note that the necessary conditions for stationarity with respect to the series ylowastt
(namely φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 and φ2 gt minus1) are metThe traditional view is embodied in (739) A shock to output increases for
a few quarters but the effect ultimately dies out In fact only 8 percent of ashock remains after 20 quarters Assuming an ARMA(22) process Blanchardand Fischer (1989 9) obtain a similar result
ylowastt = 131ylowast
tminus1 minus 042ylowasttminus2 + εt minus 006εtminus1 + 025εtminus2 (740)
where by construction εt is that part of the deviation of current output from trendthat cannot be predicted from past output As Blanchard and Fischer note
A shock has an effect on GNP that increases initially and then decreases overtime After 10 quarters the effect is still 40 of the initial impact after 20quarters all but 3 of the effect has disappeared The view that reversiblecyclical fluctuations account for most of the short-term movements of realGNP and unemployment has been dominant for most of the last century
However as Granger and Newbold (1986 37) note ldquothe more modern view is thatas far as possible the trend seasonal and lsquoirregularrsquo components should be han-dled simultaneously in a single model aimed at depicting as faithfully as possiblethe behavior of a given time seriesrdquo Seasonality can be treated through a gener-alization of ARMA models44 More importantly for the question at hand trend isgenerally treated by differencing leading to the consideration of the ldquointegratedprocessesrdquo suggested above In fact Campbell and Mankiw (1987b) indicate thatthe answer to the question of the importance of temporary versus permanent shocksto output is biased if detrended data are used45 Campbell and Mankiw go on topoint out that examining the differences in the logarithm of real output does notprejudge the issue of whether shocks to the economy are transitory or permanentFor instance suppose that yt follows an IMA(11) process so that
yt minus ytminus1 = d + εt minus θεtminus1
Then a unit impulse in yt changes the forecast of yt+n by 1 minus θ regardless of nldquoHence depending on the value of θ news about current GNP could have a largeor small effect on onersquos forecast of GNP in ten yearsrdquo (Campbell and Mankiw1987b 860)46
Campbell and Mankiw consider ARMA( p q) processes for the difference in thelog of real output for p = 0 1 2 3 and for q = 0 1 2 347 Interestingly they finda high level of persistence in shocks One intriguing suggestion of Campbell andMankiw (1987b 868) is that ldquowhen we examine postwar annual data we cannotreject the hypothesis that the log of real GNP is a random walk with drift In thiscase the impulse response is unity at all horizonsrdquo Recall that the random walkprocess with drift (718) is an example of a nonstationary process that is first-order
Empirical macroeconomics 113
homogeneous nonstationary To see this simply consider the series wt that resultsfrom differencing the random walk ndash that is the series
wt = yt minus ytminus1 = d + εt (741)
Since εt are assumed independent over time wt is clearly a stationary process Theterm d + εt is a white noise process In this example yt is an ARIMA(010) andyt is an ARMA(00) The impulse response to a shock is unity in such a case
As we will see later one way to interpret the above finding of persistence isto place weight on ldquoaggregate supplyrdquo shocks such as technological disturbancesrather than on ldquoaggregate demandrdquo shocks such as changes in the money supplyin explaining fluctuations in real output That is the finding gives credence tothe view of Nelson and Plosser (1982) among others that real (ie permanent)shocks dominate as a source of output fluctuations48
However as pointed out by Campbell and Mankiw (1987b 877) the Nelsonand Plosser conclusion is an extreme
one can attribute a major role to supply shocks without completely abandoninga role for demand shocks For example suppose that output Y (= log y) isthe sum of two components a supply-driven ldquotrendrdquo Y T and demand-driveldquocyclerdquo Y c that are uncorrelated at all leads and lags Suppose further thatY T is a first-order autoregressive process with parameter φ and that Y c issome stationary process If trend output is approximately a random walk sothat φ is small then the finding of great persistence implies that fluctuationsin the cycle are small relative to fluctuations in the trend If the change inthe trend is highly serially correlated (φ is large) however the finding ofpersistence is consistent with a substantial cyclical component
Campbell and Mankiw (1987b 877) go on to suggest that
a second way to interpret the finding of persistence is to abandon the
natural rate hypothesis Models of multiple equilibria might explain a long-lasting effect of aggregate demand shocks if shocks to aggregate demand canmove the economy between equilibria Shocks to aggregate demand couldhave permanent effects if technological innovation is affected by the businesscycle
Note that so far we have considered only a single or univariate time series Theconcepts involved however can be extended to multivariate series For instancea simple vector autoregression model (VAR) could take the form
yt = ytminus1 + εt
where now yt is considered an N times 1 vector reflecting N variables the randomdisturbance term εt is also an N times 1 vector and is an N times N matrix of parame-ters The disturbances are uncorrelated over time but may be contemporaneously
114 Empirical macroeconomics
correlated As in the univariate case such a model is usually only fitted to vari-ables which are stationary (possibly obtained by logarithmic transformations orby taking first or second differences) As in the univariate case the objectivewould be to find a model that transforms a vector of time series into a whitenoise vector As Harvey (1993) notes ldquoalthough a model of (this) form is oftenused for forecasting in econometrics economic theory will typically place a priorirestrictions on elements of rdquo There are also questions concerning the correlationbetween innovations across different series of data for example the link betweeninnovations in money supply and output But given that our aim is to review basicmacroeconomic theory we will not pursue such topics
Conclusion
This chapter has emphasized the empirical nature of macroeconomics Inparticular time series analysis and econometric methods have been shown tobe useful tools for analyzing macroeconomic data Many of the policy conclu-sions that will be developed in the remainder of the book can be tested using thesemethods The results obtained from a thorough understanding of the time seriesproperties of important macroeconomic variables such as interest rates employ-ment numbers real output measures and the like together with the underlyingtheory may be used to provide policy recommendations Moreover more andmore businesses are relying on empirical macroeconomics when making operatingdecisions
Appendix translation of higher-order difference equationsinto lower order
Any higher-order difference equation can be interpreted in terms of an equivalentsystem of first-order difference equations To see this consider equation (729)
yt minus φ1ytminus1 minus φ2ytminus2 = 0
To facilitate matters we start by shifting the origin of this second-order homo-geneous difference equation from t = minus2 to t = minus1 such that we may rewrite(729) as
yt+1 minus φ1yt minus φ2ytminus1 = 0 (7A1)
Now let us introduce the new variable xt defined as
xt = ytminus1
which means that xt+1 = yt We may then express the second-order differenceequation (7A1) by means of two first-order simultaneous equations
yt+1 minus φ1yt minus φ2xt = 0
xt+1 minus yt = 0(7A2)
Empirical macroeconomics 115
In matrix notation (7A2) becomes[yt+1xt+1
]= A
[ytxt
]
where A is the square nonsingular matrix defined as
A =[φ1 φ21 0
]
The ldquocharacteristic equation of a square matrixrdquo is defined by the determinant
|A minus λI | = 0
where I is the unit matrix[
1 00 1
]and λ is a scalar variable
Letting b = λ the ldquocharacteristic equation of matrix Ardquo
|A minus λI | =∣∣∣∣φ1 minus b φ2
1 minusb
∣∣∣∣ = b2 minus φ1b minus φ2 = 0
is identical to equation (730) which was termed the ldquocharacteristic equation of thesecond-order homogeneous difference equationrdquo The values for the λs (or bs) arethe roots of the characteristic equation One reason for restating the higher-orderdifference equation in matrix form is that necessary and sufficient conditions forstability can be expressed in terms of conditions on the matrix A
8 The neoclassical model
Introduction
This chapter turns our attention to one of the most popular and sometimescontroversial models used by macroeconomists the neoclassical model Themodel in its purest form is often used as a benchmark or starting point for addingldquorealismrdquo to the structure of the economy However one views the neoclassicalmodel there is no denying that it is a powerful tool for generating predictionsabout movements in economic aggregates As such this chapter works throughthe comparative static exercise of a change in the money supply This exerciseprovides a great deal of insight into how the monetary authority might influencethe economy or whether it can influence the economy at all A number of importantissues are raised for example money neutrality and money illusion Additionallythe chapter formally derives the aggregate supply curve in the neoclassical contextThe model has serious implications for the existence of a natural rate and also forthe formation of expectations
Comparative statics for the neoclassical model
We can use our previous method of analysis known as ldquocomparative staticrdquoanalysis to examine the effects of various shocks on the equilibrium level ofprices As the name suggests comparative static analysis is concerned with thecomparison of different equilibrium states associated with different sets of valuesof parameters and exogenous variables In the current context of the neoclassicalmodel the labor market can be isolated from the other markets (ie there is aldquoblock recursiverdquo character to the equilibrium solution) In such a context we candistinguish two types of exogenous variables
One type ldquosupply-siderdquo variables alter the level of output at any given pricelevel Such variables could include changes in technology and the existing capitalstock in the current version of the model or we could expand the analysis toinclude changes in the supply of other inputs (such as oil) changes in governmentpolicies that affect incentives to supply labor and changes in government policiesthat affect the incentives to invest and thus affect the productive capacity of theeconomy over time
Neoclassical model 117
The second type of exogenous variables ldquodemand-siderdquo variables do not alterthe current supply of output at prevailing prices but instead impact equilibriumprices and interest rates as determined in the output financial and money marketsBelow we consider the impact of demand-side ldquoshocksrdquo such as changes in initialmoney balances and the expected inflation rate
The macroeconomic approach to general equilibriumhow it can obscure
The neoclassical model can be viewed as a special case of a Walrasian generalequilibrium system distinguished by the existence of money1 As Clower (1965)notes
income magnitudes do not appear as independent variables in demand andsupply functions of the (Walrasian) general equilibrium model for incomesare defined in terms of quantities as well as prices and quantity variablesnever appear explicitly in the market excess demand functions of traditionaltheory To be sure income variables could be introduced by taking factorsupplies as given parameters but this would preclude the formulation of ageneral equilibrium model containing supply functions of all marketable factorservices
Referring to Chapter 9 of Patinkin (1965) Clower goes on to note that this point
was apparently overlooked by Patinkin when he formulated his ldquogeneral the-oryrdquo of macroeconomics It is instructive to notice that this chapter is notsupplemented by a mathematical appendix I do not mean to suggest thatauthors may not put such variables as they please into their models My pointis that such variables that can be shown to be fundamentally dependent onothers should not then be manipulated independently
What Clower is referring to is the fact that the neoclassical model with limitedperfect foresight effectively determines at time t the money wage for the labormarket and the (futures) price of output and interest rate Given perfect foresightindividual optimizing behavior will generate planned consumption and moneydemand functions at time t for period t that include the real wage rather thanincome That is since labor supply is a choice variable at time t labor income (theproduct of the real wage times labor supply) should not appear as an argument inhouseholdsrsquo demand and supply functions (Note that labor income along withdividend and interest payments equals output minus depreciation)
Formally we can discover how a ldquosmallrdquo change in one (or more) of the exoge-nous variables affects equilibrium values by totally differentiating the equilibriumconditions with respect to the prices to be determined and the exogenous variableand then solving for the implied change in equilibrium prices required to maintainequilibrium2 The resulting changes in equilibrium prices then imply changes in
118 Neoclassical model
the equilibrium levels of employment output consumption investment and realmoney holdings
However we choose instead the standard macroeconomic approach of arbitrar-ily separating the analysis into an analysis of the labor market and the resultingemployment and output (the ldquoaggregate supply equationrdquo) and then an analysisof the other markets and the detemination of the price level and the interest rate(the ldquoISrdquo and ldquoLMrdquo equations) While Clower is right in that this obscures thetraditional ldquogeneral equilibriumrdquo nature of the neoclassical model the approach isuseful in that it allows us to more easily extend the analysis to situations in whichprices are not market-clearing or to situations in which there is not limited perfectforesight
Given the assumption of perfect foresight on the part of both firms and workersconcerning the price level pt we have seen that the aggregate supply equation isindependent of the price level or the interest rate We know from the modifiedWalrasrsquo law that any two of the three excess demand conditions with respect tomoney financial assets and output can be used in the comparative static analysisUsing the equilibrium conditions with respect to output and money (the ldquoISrdquo andldquoLMrdquo equations) we thus have the three equations
yt = yt(K ) (81)
cdt (rt πe
t At Mpt yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (82)
Ldt (rt πe
t At Mpt yt) minus Mpt = 0 (83)
to determine equilibrium output yt price level pt and interest rate rt Recallthat from the modified Walrasrsquo law we know that there is a fourth equation theequilibrium condition for the financial market that is implied by (82) and (83)This fourth equilibrium condition is
net Adt (rt πe
t At Mpt yt) minus net Ast (m
et + δ K) = 0 (84)
A change in the money supply the comparative statics
It is clear from the above that the neoclassical aggregate supply equation (81)determines output while the IS and LM equations (82) and (83) determine theprice level and interest rate given the equilibrium level of output Focusing on thelatter two equations and the determination of the price level and interest rate fora given level of output total differentiation with respect to the equilibrium prices( pt and rt) and the money supply change gives the following system of linearequations in matrix form3
[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpartrpartLdpartr
] [dpdr
]=[ minus(partcdpart(Mp))dMp(1 minus partLdpart(Mp))dMp
]
Neoclassical model 119
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm4 Solving the above linear-equationsystem for dp and dr using Cramerrsquos rule we obtain
dp
dM= minus(partcdpart(Mp))(1p)partLdpartr minus (1 minus partLdpart(Mp))(1p)partydpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpartr
= p
M
dr
dM=
minus[(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)minus(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)](1p)
minus(partcdpart(Mp))(Mp2)partLdpartrminus(1minuspartLdpart(Mp))(Mp2)partydpartr
= 0
As you can see dpp = dMM and drdM = 0 That is a change in the moneysupply leads to the same proportional change in the price of the consumption goodand no change in the interest rate With regard to the latter result the interest ratedoes not change as the only thing that changes the interest rate is the rate of changein prices Since the price level adjusts there is no change in the interest rate and theLM curve does not change Note that from the labor market equilibrium conditionthat underlies the aggregate supply equation we know that the change in the pricelevel results in an equiproportionate change in the money wage The result is thatwe have ldquoneutrality of moneyrdquo In other words if individuals correctly anticipatethe effect of a change in the money supply on the price level as is the case inthe deterministic model under the assumption of limited perfect foresight thenmonetary changes have no ldquorealrdquo effects Real output the real wage the expectedreal rate of interest real consumption real investment and the real money supplyare all unaffected by the monetary change
The basic reason why changes in the money supply are ldquoneutralrdquo is the absenceof money illusion on the part of both firms and households In particular house-hold demand and supply functions indicate that if a change in money balancesis accompanied by an equiproportionate change in the price of output then thereis no change in any demands That is household demand and supply functionsare homogeneous of degree zero in money balances and prices This absence ofldquomoney illusionrdquo occurs assuming
bull perfect foresight at time t for price during period t so that a change in priceresults in no change in the real wage or employment (when coupled with thesimilar assumption of limited perfect foresight on the part of firms)
bull ldquoneutral distribution effectsrdquo such that the shift in wealth from prior creditorsto debtors that would accompany a rise in the price level leaves aggregatedemands unchanged
bull unit elastic expectations so that changes in the current price level can beviewed as leaving unaffected the expected rates of change in the pricelevel
120 Neoclassical model
As we will see later this money neutrality result of the neoclassical model formsthe basis of what has become known as the ldquopolicy ineffectiveness propositionrdquo(see McCallum 1979)
The ldquodynamicsrdquo of the system and the neutrality of money a review
As we discussed earlier the ldquostoryrdquo often told with respect to the dynamics of theabove situation is a ldquoloanable funds theoryrdquo of interest rate determination in whichthe interest rate moves to clear the financial market5 In this case the ldquotatonnementprocessrdquo or movement toward equilibrium involves
dpt = f ( ydt minus yt) f (0) = 0 df d( yd
t minus yt) gt 0
dpbt = fb(net Adt minus As
t ) fb(0) = 0 dfbd(net Adt minus net As
t ) gt 0
Note that we put ldquostoryrdquo in quotes since the analysis itself simply identifies variousequilibrium points with adjustments in prices to reach a different equilibrium pointgiven a shock essentially occurring without any passage of time Neverthelessconsider the following story with respect to a rise in the money supply
In the Patinkin analysis described earlier if the money supply were to doublefrom M to 2M the LL BB and CC curves would shift to the right so that the newequilibrium would occur at the original interest rate but at a price level double theoriginal one Such a once-and-for-all change in the initial money balances leavesthe money interest rate and real demands unchanged (the neutrality of money)
Superneutrality an informal review
Superneutrality of money occurs when changes in the rate of growth of the moneysupply leave the paths of capital and real output unaffected Although the aboveanalysis is static in nature it does provide some insight into the issue of thesuperneutrality of money To see how suppose each period the economy can bereplicated in every way That is the money supply equilibrium price level andinterest rate are identical each period If expectations of price changes are correctexpected inflation would equal zero For the economy to replicate itself it mustalso be the case that the initial capital stock is optimal such that the capital stockand thus output does not change over time With zero adjustment costs such asituation would imply a marginal product of capital equal to the expected real usercost of capital (me
t + δ where met equiv (rt minus πe
t )(1 + πet )) net investment demand
equal to zero each period and gross investment demand equal to δK Now let us compare this situation to an alternative sequence of temporary equi-
librium in which the economy is identical in every way except one the moneysupply is increased each period by a constant percentage from its level in the priorperiod Other things being equal the above analysis would suggest that one dif-ference across periods would be a rise in prices due to the positive growth in themoney supply Let us further assume that expectations of inflation adjust to reflect
Neoclassical model 121
what Sargent and Wallace (1975) would characterize as a new ldquosystematic moneysupply rulerdquo
Without fully developing the appropriate dynamic analysis it is easy to see thatone obvious adjustment of the static analysis given a growing money supply (asopposed to one that does not change across periods) is thus a higher expectedinflation rate (positive as opposed to zero) In fact the analysis of the static modelcan mimic to some extent the effects of a higher rate of growth in the money supplyby considering the impact of an increase in exogenous inflationary expectations
Following Sargent and Wallace (1975) among others assume that consump-tion demand depends on the expected real rate of interest r minus π not separatelyon its components (the money interest rate and the expected rate of inflation)6
Further assume that changes in expected inflation do not affect real moneydemand7 The comparative static results are then[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpart(r minus π)
partLdpartr
] [dpdr
]=[minus(partydpart(r minus π))dπ
0dπ
]
where partydpart(r minusπ) = partcdpart(r minusπ)+ (1 +ψ prime)partI dpart(r minusπ) Solving the abovelinear equation system for dp and dr using Cramerrsquos rule we obtain
dp
dπ=
minus(partydpart(r minus π))partLdpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
dr
dπ=
minus(partLdpart(Mp))(Mp2)partydpart(r minus π)
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
As we have discussed before if there is no real balance effect with respect to con-sumption demand (partcdpart(Mp) = 0) or if real money demand does not respondto changes in the interest rate (partLdpartr = 0) then we can see from the above that
dr = minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)dπ
minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)= dπ
so that the change in the expected rate of inflation results in no change in theexpected real rate of interest (r minus π ) In this case money is superneutral in thata change in the growth of the money supply (although it alters inflation and thusgiven perfect foresight expected inflation) leaves the expected real rate of interestunchanged and thus does not affect investment and the future size of the capitalstock which depend inversely on the expected real interest rate
122 Neoclassical model
Recall that Sargent and Wallace (1975) have argued for superneutrality ofmoney In contrast Begg (1980) has noted that steady-state analyses of growthmodels with money like the analysis above do find that different rates of growthin the money supply have real effects (through changes in the expected real grossinterest rate)8 As the above analysis makes clear there are two conditions eitherof which is sufficient that will result in money being superneutral Begg (1980293) describes them as follows ldquoThe first condition is that the level of real moneybalances is not an argument in the consumption function and it is this conditionwhich distinguishes the rational expectations model of Sargent and Wallace fromthe analysis of growth models with money The second condition is that the demandfor money is independent of the nominal interest raterdquo (as well as π ) Otherwisea higher expected inflation will result in a fall in the expected real rate of interestand a higher price level
In the Patinkin framework with consumption demand and investment demanddepending on the expected real interest rate at a given price level and higherexpected inflation the CC curve must shift up vertically by the amount of theincrease in expected inflation so as to maintain the same expected real rate ofinterest But the LL curve does not shift with a change in expected inflationThus given a downward-sloping CC curve and an upward-sloping LL curve thenew equilibrium money interest rate does not rise by the extent of the increase inexpected inflation Superneutrality of money seems not to hold
In such a case the static analysis thus predicts a lower real stock of money eachperiod higher investment and a greater capital stock next period This suggests anew steady state in which the capital stock is greater In fact a complete dynamicanalysis confirms these predictions a higher steady-state capital stock is associatedwith an increase in the rate of growth of the money supply and consequent increasedexpected inflation
The role of the key assumptions of the neoclassical model
At this point it might be useful to review the role played by the two key assumptionsof the neoclassical model namely price flexibility and complete information onprices In most cases and this is no exception one can gain an understanding ofthe role of a particular assumption by exploring how the analysis would proceedif the assumption were not made Consider first the implications of dropping theneoclassical modelrsquos assumption that prices are perfectly flexible
Suppose that the demand for output decreases and firms cannot sell all theydesire at prevailing prices With flexible prices output prices will fall and thefalling price level restores output demand to its previous level However if outputprices are inflexible and do not adjust downward in response to a reduced demandfor output firms will respond by reducing production Reduced production willlead to a lower level of labor demand and thus a fall in employment Further labordemand will no longer depend upon the real wage but will be determined by whatcan be sold in the output market In short if output prices are inflexible then theoverall level of demand for goods and services is paramount in determining thelevel of employment
Neoclassical model 123
A second modification of the neoclassical model is to assume that output pricesare flexible but money wages are not If wages are ldquostickyrdquo relative to outputprices then changes in the price level will alter the real wage For instance inthe late 1970s high inflation rates led workers and employers in certain industriesto bargain for long-term contracts with a high rate of growth of the money wageThe high expected inflation did not materialize in the early 1980s The lower rateof inflation with no change in the rate of increase in wages meant a rise in thereal wage The resulting fall in the demand for labor led to lower employmentand output Given that wages are not perfectly flexible (eg ldquomultiperiod laborcontracts without complete indexationrdquo) output prices below that anticipated whenlabor contracts are signed lead to a fall in output and employment With inflexiblemoney wages the aggregate supply equation includes the price level of output asa determinant of output supply
Let us consider one more modification of the neoclassical model Suppose thatworkers have incomplete information on output prices and the real wage A firmdetermines its relevant real wage by dividing the money wage it pays its workersby the price it anticipates for the particular product its workers produce On theother hand workers must anticipate prices for a variety of different goods to bepurchased in order to determine their relevant real wage Thus firms may moreaccurately anticipate changes in prices and thus real wages than workers
Now suppose that there is an increase in output prices Given our currentassumption of incomplete information this will not only lead to an increase infirmsrsquo demand for labor and to higher wages as firms anticipate the fall in realwages but may also lead to an increase in the quantity of labor supplied for thefollowing reason Workers who have not anticipated the rise in output priceswill perceive the higher money wages as implying a rise in the real wage andwill increase their supply of labor accordingly As a result equilibrium employ-ment will rise with an increase in output prices Once again the aggregate supplyequation will incorporate the current price level as a potential determinant
The illusion model one modification of the neoclassical model
Our first departure from the neoclassical model is to introduce the potential forldquoimperfectrdquo foresight on the part of suppliers at time t concerning the price levelfor period t As a consequence the ldquonotionalrdquo or planned demands made at time tbased on anticipated prices for the period can differ from ldquoeffectiverdquo or realizeddemands andor supplies at the actual prevailing prices Further realized or effec-tive demands now depend on the quantity constraints experienced in other marketsas well as prices That is realized output now becomes a determinant of actualconsumption and money demand on the part of households
Real wage illusion the labor market and aggregate supply
As we have seen equilibrium employment is determined at the start of each periodin the labor market To formally show this let us start with the following statement
124 Neoclassical model
of equilibrium in the labor market in terms of a money wage wt and level ofemployment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpe
t ) minus Nt = 0
A critical aspect of the above is the fact that suppliers ndash in particular suppliers oflabor ndash may not correctly anticipate the price level that will exist with respect tooutput In particular we let pe
t denote suppliersrsquo expectation formed at the start ofperiod t of the price level for period t We assume this expectation is held withsubjective certainty so that we may express the real wage expected by supplierssimply by wtpe
t It is this anticipated real wage not the realized real wage wtpt that appears in the labor supply function
Firms on the other hand are presumed to correctly forecast the actual realwage9 Note that we retain the presumption that the money wage adjusts to clearthe labor market Similarly we implicitly have assumed that the price level adjuststo clear the output market such that firms are price-takers in the output marketand thus labor demand depends on the real wage rather than on a sales constraint
Initially let us assume that householdsrsquo anticipated level of prices is correctThat is we start with a money wage and level of employment consistent with theneoclassical model But we now assume that any change in the price level willnot be fully anticipated In particular we assume that
pet = g( pt) 1 gt gprime ge 0
The fact that gprime lt 1 implies imperfect foresight at time t on the part of householdsconcerning the price of output for period t If gprime gt 0 it indicates that householdsto some extent but not completely (gprime lt 1) anticipate changes in the equilibriumlevel of prices that will prevail for period t
Totally differentiating the above two equations representing equilibrium in thelabor market with respect to wt Nt pt and noting our prior assumption that pe
t = ptinitially one obtains[minus(partN d
t part(wtpt))pt(partN s
t part(wtpet ))pt
minus1minus1
] [dwtdNt
]=[
(partN dt part(wtpt))wtdpt( pt)
2
(partN st part(wtpe
t ))wtgprimedpt( pt)2
]
Applying Cramerrsquos rule gives
dNt
dpt= (partN d
t part(wtpt))(partN st part(wtpe
t ))(wt( pt)2)(gprime minus 1)
minus(partN dt part(wtpt))pt + (partN s
t part(wtpet ))pe
t
gt 0
dwt
dpt= minuspartN d
t part(wtpt) + (partN st part(wtpe
t ))gprime
minuspartN dt part(wtpt) + partN s
t part(wtpet )
wt
ptgt 0
Note that if gprime = 1 then we have the standard neoclassical result that dNtdpt = 0and dwtwt = dptpt so that a change in the price level results in no change in
Neoclassical model 125
either employment or the real wage However with gprime lt 1 we have that anincrease in the price level leads to a rise in employment and an increase in themoney wage less than proportional to the increase in the price level such thatthe real wage falls (ie dptpt gt dwtwt gt 0) Given the aggregate produc-tion function yt = f (Nt K) we thus have an ldquoaggregate supply functionrdquo ofthe form
yt = yt( pt minus pet K ) (81prime)
so that aggregate supply depends directly on the difference between the price levelfor period t and the price level anticipated by suppliers at time t
Consider the above findings with respect to the labor market and suppose thatthere is an increase in pt Given perfect foresight on the part of firms at time t thelabor demand curve shifts up vertically so that at the higher money wage associatedwith the same real wage demand would be the same However given 1 gt gprime ge 0the vertical shift upward in the supply curve is less than this with the result thatequilibrium employment rises as the money wage rises by proportionately lessthan the rise in prices
In undergraduate textbooks the fact that changes in the price of output can nowaffect real output is shown in ( pt yt) by an upward-sloping ldquoaggregate supplycurverdquo as in Figure 81 Recall that in the neoclassical model the aggregate supplycurve is vertical In either model such a curve summarizes the underlying eventsin the labor market
The above character of the money illusion model is sometimes said to reflect theldquonatural rate hypothesisrdquo The natural rate hypothesis posits that fully anticipatedincreases in prices have no effect on the rate of real economic activity ndash specificallyreal output employment and thus unemployment Thus we will refer to the abovemodel as a static version of a ldquonatural rate modelrdquo
p
y
Figure 81 Upward-sloping aggregate supply
126 Neoclassical model
Equilibrium aggregate supply and demand
An important feature of macroeconomic theories is that to a large extent they aredistinguished by their different treatment of labor markets What this means isthat the aggregate demand side is typical of macroeconomic models Recall thatthe aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) Thus the equilibrium output price level and interest rate aregiven by equations (81prime) (82) and (83) Equation (81prime) is the aggregate supplyequation of a natural rate model (82) is the ldquoISrdquo equation depicting equilibriumbetween output demand and production and (83) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market
At this point we will simplify equations (82) and (83) by removing the realbalance effect representing them as follows
cdt (rt πe
t At yt) + I dnt(m
et + δ K) + δK + ψ(I d
t ) minus yt = 0 (82prime)
Ldt (rt πe
t At yt) minus Mpt = 0 (83prime)
In our model this implies that
partnet Adt part(Mpt) = 1
As we will see one justification for this form is if real money balances are notpart of household wealth which can be the case when we introduce depositoryinstitutions into the analysis In the meantime the above assumption makes theanalysis not only simpler but also more in line with traditional macroeconomicanalysis
Graphically the equilibrium price level and output can be shown using theaggregate demand and supply curves In Figure 82 the equilibrium output andprice level are thus given by plowast
t and ylowastt Looking at what underlies these curves
we can then infer the changes in money wages and employment (specifically froman analysis of the labor market that underlies the aggregate supply curve) as wellas changes in the interest rate and the components of investment and consumptioncomponents of output demand (specifically from an analysis of the output andmoney markets that underlie the aggregate demand curve)
Money supply change comparative statics for a naturalrate model
Collecting the aggregate supply IS and LM equations for the natural rate modelunder consideration we have (81prime) (82prime) and (83prime)
These three equations determine the equilibrium output the price level andthe interest rate Substituting (81prime) into (82prime) and (83prime) in order to focus on the
Neoclassical model 127
p
y
p
y
Aggregate supply
Aggregate demand
Figure 82 Macroeconomic equilibrium with upward-sloping supply
determination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form10
[ minus(partcdparty minus 1)partyp(partLdparty)partypartp + Mp2
partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + c1 + ψ prime)partI dpartm11 The term partypartp gt 0 reflectsthe direct effect of the price level on output as implied by the aggregate supplyequation (81prime)12 It is important to note that the equilibrium condition with respectto the labor market is incorporated into the above analysis in the form of thisaggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp)minus 1)(1p)
(partcdparty minus 1)partypartp(partLdpartr)minus((partLdparty)(partypartp)+ Mp2)(partydpartr)
lt 0
As you can see we no longer have drdM = 0 Further letting x denote thedenominator for the expression for dp we have
dp
p= dM
M
1
x
128 Neoclassical model
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater13
Let us now consider the effect of the money supply shock on other variablesFrom our analysis of the labor market we know that
dp
pgt
dw
wgt 0
so that while the money wage rises the real wage falls We also know from thelabor market that the rise in the price level leads to higher employment and thusan increase in output From the demand functions we can derive the effects of thechange in the money supply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Graphically one can show the effect of the money supply shock in terms ofthe aggregate demand and aggregate supply curves Note that this exercise simplyinvolves a shift out of the aggregate demand curve
The natural rate hypothesis and expectation formation a preview
An important feature of the above analysis one already noted is that real activity ndashin particular employment output and unemployment ndash changes only to the extentthat price changes are not fully anticipated As we have just seen this ldquonatural ratehypothesisrdquo introduces the logical foundations for a monetary change to have realeffects
However note that monetary changes have real effects only to the extent that theresulting changes in the price level are not fully anticipated To understand whenthis might occur we first have to indicate why individuals may err in their formationof expectations concerning the price level The Lucas model suggests one way ofexplaining errors in forecasts such that suppliers only partially anticipate a changein the price level (ie 1 gt gprime ge 0) This model introduces the assumption ofrational expectations
Combining rational expectations with the natural rate hypothesis results in a verypowerful statement concerning monetary policy which has so far not been madeclear If the actions of monetary authorities are predictable under the presumptionof rational expectations individuals will correctly predict the consequences onprices The result in the context of a natural rate model is that such predictable
Neoclassical model 129
monetary changes will have no real effects In the deterministic world we are backto the neoclassical model The analysis above then refers only to an ldquounexpectedrdquoincrease in the money supply or to monetary ldquosurprisesrdquo Only such random shocksto the money supply will have real effects
Conclusion
A formal neoclassical model of the macroeconomy has been introduced and fullydeveloped The issues of money neutrality and money illusion have been discussedand it has been seen that money supply changes have no effect on real economicactivity when the assumptions of the neoclassical model hold However a numberof issues have been raised namely the existence of a natural rate and the potentialeffect of unanticipated money supply changes on economic activity
9 The ldquoKeynesian modelrdquo withfixed money wageModifying the neoclassical model
Introduction
The first modification of the neoclassical model is presented in this chapterTo begin we introduce the very realistic assumption that nominal wages are fixedat least for a period of time The ramifications of this change in the model aredeveloped in the context of the aggregate supply and demand model As with theneoclassical model we perform a comparative statics exercise in which the mone-tary authority changes the money supply and we trace out the effects of this actionon the economic aggregates in the model The model is then made slightly morecomplete and issues associated with sticky wages and the natural rate hypothesisare discussed We introduce the concept of rational expectations and the first ldquoover-lappingrdquo model and show that the Keynesian model has important implicationsfor the conduct of monetary policy
The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
In the standard neoclassical model it is assumed that prices adjust in all marketsto equate demand and supply With respect to the labor market this implies a spotmarket at the start of each period in which one-period labor contracts are enteredinto and an associated one-period wage set Yet employment contracts are likelyto be multiperiod in the presence of hiring and training costs That is to minimizehiring and training costs firms seek long-term relationships with their employees
Firms promote long-term relationships with their employees by offering higherwages to their experienced workers As a consequence long-time employeesbecome attached or ldquoloyalrdquo to their employers since the wages they receive aregreater than those that other firms would offer them In essence employers aresharing the returns to their hiring and training investment with their workers inorder to reduce the number who quit A long-term attachment of workers to par-ticular firms could also stem from the high cost to workers of finding alternativeemployment as obtaining such employment means that workers must generallyinterview various employers visit employment agencies and spend valuable timesimply waiting for decisions on job applications to be made
Keynesian model 131
Given long-term employment contracts between firms and their workers wagesare typically specified for extended periods of time These long-term wage agree-ments are sometimes explicit as with many labor union contracts1 In other casesonly an implicit understanding exists on the wages that a firm will pay its employeesover some extended period of time If these contracts or understandings specifywages in money terms and if modifying these agreements is costly then thereexists an inherent inflexibility in money wages ndash that is there are ldquostickyrdquo wages2
This assumption of ldquostickyrdquo nominal wages is often viewed as the critical aspectof what has been termed the ldquoKeynesianrdquo macroeconomic model
If money wages are ldquostickyrdquo relative to prices then changes in the price levelwill alter the real wage For instance in the late 1970s high inflation rates ledworkers and employers in certain industries to bargain for long-term contractswith a high rate of growth of the money wage The high expected inflation did notmaterialize in the early 1980s The lower rate of inflation with no change in therate of increase in wages meant a rise in the real wage The resulting fall in thedemand for labor led to lower employment and output In this section we formallydevelop these results in the context of a static neoclassical macroeconomic modelwith the additional assumption of a fixed money wage The subsequent section thendevelops a linear rational expectations version of this model in which overlappingmultiperiod employment contracts introduce an element of nominal wage rigidity
Fixed money wage the labor market and aggregate supply
As we have seen in the competitive (spot) labor market of the neoclassical model(or in the Lucas-type macroeconomic model) the money wage and employmentare determined at the start of each period in the labor market If we accept theneoclassical modelrsquos assumption of limited perfect foresight on the part of bothlabor suppliers and firms we have equilibrium in the labor market in terms of amoney wage wt and level of employment Nt determined such that
N dt (wtpt K) minus Nt = 0 (91)
N st (wtpt) minus Nt = 0 (92)
In this case a change in the price level pt leads to an equiproportionate change inthe money wage wt and no change in employment Nt
We now seek to modify this analysis by assuming a fixed money wage wt = wfor period t As Sargent (1987a 21) states
the essential difference between the classical model and the Keynesian modelis the absence from the latter of the classical labor supply curve combined withthe labor market equilibrium condition Since there is one fewer equation inthe Keynesian model it can determine only six endogenous variables insteadof the seven determined in the classical model3 To close the Keynesianmodel the money wage is regarded as an exogenous variable one that atany point in time can be regarded as being given from outside the model
132 Keynesian model
perhaps from the past behavior of itself and other endogenous or exogenousvariables It bears emphasizing that the equation that we have deletedin moving from the classical to the Keynesian model [equation (92)] is acombination of a supply schedule (and) an equilibrium condition Notethat we continue to require that employment satisfy the labor demand schedule[equation (91)]
Sargent goes on to say that
we shall think of the labor supply schedule as being satisfied and helping todetermine the unemployment rate Usually the model is assumed to reachequilibrium in a position satisfying Nt lt N s
t so that there is an excess supplyof labor
Totally differentiating the labor demand condition (91) that determines the levelof employment with respect to the price level and employment we have
minus(partN dt (partwpt))(wp2
t )dpt minus dNt = 0 (93)
which can be rearranged to give
dNtdpt = minus(partN dt part(wpt))(wp2
t ) gt 0 (94)
where the sign reflects the presumption that partN dt part(wtpt) lt 0
In the simple case of no labor adjustment costs labor demand is defined bythe equality between the marginal product of labor and the real wage that ispartf (Nt K)partNt = wpt 4 Differentiating this implies that
[part2ftpartN 2t ]dNt = minus(wp2
t )dpt
or rearranging
dNtdpt = minus(wp2t )[part2ftpartN 2
t ] gt 0 (95)
given diminishing returns to the labor input (ie part2ftpartN 2t lt 0)
Combining the above analysis with the aggregate production function yt =f (Nt K) we thus have the ldquoaggregate supply equationrdquo
yt = yt( ptw K ) with partytpart( ptw) gt 0 (96)
Thus (as in a Lucas-type model) we have an aggregate supply that can dependdirectly on the price level for period t
The above findings can be understood with respect to the labor market Considera decrease in pt Given limited perfect foresight on the part of both firms andhouseholds at time t there is a downward (vertical) shift in labor demand so thatat the lower money wage wlowast
t associated with the same real wage demand would
Keynesian model 133
be the same Similarly the labor supply curve shifts down vertically so that atthis lower money wage wlowast
t labor supply would be the same as well Howevermultiperiod labor contracts fix the money wage at w so that the lower price level(and implied higher real wage) results in a fall in employment (which is nowdemand-determined) and an excess supply of labor
In the opposite case of a rise in the price level that can lead to an excessdemand in the labor market at the fixed money wage the presumption remainsthat employment is demand-determined This presumption reflects the view thatat least temporarily firms can direct workers with whom they have long-termemployment contracts to work overtime or extra shifts which the workers wouldotherwise not volunteer for
The above story provides a rationale for an upward-sloping ldquoaggregate sup-ply curverdquo Both contrast with the neoclassical model in which the aggregatesupply curve is vertical as a fall in the price level results in an equiproportionatefall in the money wage so that the real wage and employment remain unchangedIn the fixed wage model the underlying events in the labor market summarized bythe aggregate supply curve are the change in the real wage and thus labor demandand employment that accompany a price change when the money wage is fixedSuch an aggregate supply curve is upward-sloping
Equilibrium aggregate supply and demand
As we have noted before an important feature of macroeconomic theories is that toa large extent they are distinguished by their different treatment of labor marketsWhat this means is that the aggregate demand side is similar across macroeconomicmodels Recall that the aggregate demand side of macroeconomic models typicallyconsiders the equilibrium conditions of two of the remaining three markets inparticular the output market (reflected by an ldquoISrdquo equation) and the money market(reflected by an ldquoLMrdquo or ldquoportfoliordquo equation) Thus for the Keynesian modelwith fixed money wage the equilibrium output price level and interest rate aregiven by the following three equations
yt = yt( ptw K ) (96)
cdt (rt πe
t+1 At yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (97)
Ldt (rt πe
t+1 At yt) minus Mpt = 0 (98)
Equation (96) is the aggregate supply equation of a Keynesian model with fixedmoney wage (97) is the ldquoISrdquo equation depicting equilibrium between outputdemand and production and equation (98) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market Note that we havesimplified the IS and LM equations by removing the real balance effect for con-sumption demand and money demand5 Equations (97) and (98) can be combinedto eliminate the interest rate The resulting equation is referred to as the ldquoaggregatedemand equationrdquo
134 Keynesian model
The equilibrium price level and output ( plowastt and ylowast
t ) can be shown graphicallyusing aggregate demand and supply curves Looking at what underlies thesecurves we can then infer the change in employment (specifically from an analysisof the labor market that underlies the aggregate supply curve) as well as changes inthe interest rate and the investment and consumption components of output demand(specifically from an analysis of the output and money markets that underlie theaggregate demand curve)
A change in the money supply the comparative statics for theKeynesian model
The aggregate supply IS and LM equations (96)ndash(98) for the static Keynesianmodel under consideration determine the equilibrium output the price level andthe interest rate Substituting (96) into (97) and (98) in order to focus on thedetermination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form6[ minus(partcdparty minus 1)partypartp
(partLdparty)(partypartp) + Mp2partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm7 The term partypartp gt 0 reflects thedirect effect on the price level as implied by the aggregate supply equation (96)8
Note that the equilibrium condition with respect to the labor market is incorporatedinto the analysis in the form of this aggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp) minus 1)p
(partcdparty minus 1)partypartp(partLdpartr) minus (partLdparty)(partypartp) + (Mp2)(partydpartr)
lt 0
In contrast to the neoclassical model we no longer have drdM = 0 Furtherletting x denote the denominator for the expression for dp we have
dp
p= dM
M
1
x
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Keynesian model 135
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater
Consider now the effect of the money supply shock on other variables Fromour analysis of the labor market we know that w is fixed so that the increase inthe price level means a fall in the real wage and thus increased labor demandemployment and thus an increase in output From the demand functions for con-sumption and investment we can derive the effects of the change in the moneysupply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
and
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Note that the effect of the money supply shock is to increase the aggregate demandwhile not affecting aggregate supply
Sticky wages and the natural rate hypothesis
Expectations play an important role in the two modifications of the neoclassicalmodel In the modification with fixed wages the level at which negotiators fixfuture money wages depends on the expectation formed when wages were set con-cerning future prices The higher the expectation of future prices the higher thelevel of wages set in the labor agreements between workers and firms The pre-sumption is that workers and firms attempt to set future wages at their anticipatedmarket-clearing levels Associated with these anticipated market-clearing wagesis a particular real wage a natural rate of unemployment and a full employmentor natural rate of output
If price expectations turn out to be incorrect then output will vary from itsnatural rate For instance a shock that causes actual output prices to fall belowthose expected means that the money wage is fixed at a level that is too highfor full employment Consequently employment and output fall below the fullemployment level
In the typical Lucas-type model firms and workers set wages for the currentperiod based on incomplete information as well As we saw if suppliersrsquo expec-tations are incorrect then output will deviate from the full employment level Forexample a shock that causes actual output prices to fall below those expectedmeans lower employment and output as workers mistake lower money wages forlower real wages In fact higher real wages accompany the lower price level andthis is the source of the reduced demand for labor and employment
The two modifications of the neoclassical model have a second common ele-ment Both predict that a macroeconomic demand shock ultimately affects onlythe level of prices Even though money wages in the Keynesian model are fixed
136 Keynesian model
for the current period we know that money wages are not fixed forever Over timelabor agreements are renegotiated and money wages change to once again equatethe ldquoexpectedrdquo future demand for and supply of labor Over time in the absenceof further shocks the economy would thus tend to behave as neoclassical analysispredicts money wages and output prices would adjust to restore equilibrium tothe various markets in the economy
While the Keynesian model with fixed money wage admits the tendency foroutput to approach its natural level over time it does introduce a potential rolefor monetary policy to play in dampening fluctuations in output In so doing itchallenges the policy ineffectiveness view of Sargent and Wallace The best-knownexamples employing the Keynesian model to demonstrate the potential stabilizingpowers of monetary policy under rational expectations are the dual papers byFischer (1977) and Phelps and Taylor (1977)
A linear rational expectations version of the Keynesian model
Counting the above discussion we have so far considered three different modelsthat can be used to assess monetary policy One model is along the lines of theneoclassical model with limited perfect foresight Since this view of the economypredicts the neutrality of money a role for monetary policy either as an instigatorof output fluctuations or as an instrument to dampen output fluctuations is missingAs Mankiw (1987) suggests people who adopt this model ldquoview economic fluctu-ations through the lens of real business cycle theoryrdquo in which output fluctuationsare traced to ldquosupply-siderdquo disturbances
As Mankiw goes on to note however ldquothere are surely readers who believe thatmonetary policy has real short-run effects because of temporary misperceptions ornominal rigiditiesrdquo Mankiw is referring to individuals who adopt either the Lucas-type model or the ldquoKeynesianrdquo fixed money wage model9 Either one as we haveseen introduces a role for monetary policy as an instigator of output fluctuationsHowever these two models do differ as to whether monetary policy can be aninstrument to dampen output fluctuations in the context of rational expectations
A Lucas-type model built on ldquotemporary misperceptionsrdquo when coupled withrational expectations leaves little if any room for countercyclical monetary policyIn fact following the analysis of Sargent and Wallace it can be shown that whilerandom monetary shocks can impact output deterministic monetary policy basedon a set of policy rules is ineffective in counteracting fluctuations in output givenrational expectation10 Further attempts at discretionary monetary policy in thiscontext only result in a suboptimal (ldquotoo highrdquo) rate of inflation (Barro and Gordon1983) Thus in this model there remains a ldquostochasticrdquo neutrality of money
As the analysis in the previous section suggests however a ldquoKeynesian-typerdquomodel built on ldquonominal rigiditiesrdquo might introduce a role for monetary policyin stabilizing output even in the context of rational expectations The reasoningfor this is that wages (or as we will see later prices) can be set prior to thereceipt of information by the monetary authority that enters into the money supplyrule In this context as Phelps and Taylor (1977) state ldquoeven systematic and
Keynesian model 137
correctly anticipated policy can make a difference for the stability of output in arational expectations model with sticky prices and wagesrdquo11 Below we considerone example of such a model that counters the Sargent and Wallace ineffectivenessproposition a model proposed by Fischer that assumes ldquostickyrdquo wages
The supply equation with overlapping two-period labor contracts
The Lucas aggregate supply equation with adjustment costs can be expressed as
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (99)
where Yt = ln yt minus ln yn denotes difference between the logarithm of output forperiod t and the logarithm of the natural rate of output (which we have normalizedto equal zero) Pt = ln Pt is the logarithm of the price level for period t Etminus1Pt isthe expectation of the logarithm of the price level for period t using all informationavailable up to the end of period t minus 1 (at time t) and γ θ is a positive constant12
The supply of output as expressed by equation (99) satisfies the conditionthat employment equals labor demand (92) The fact that a higher price level Ptinduces firms to increase employment and thus output reflects the underlying lowerequilibrium real wage that accompanies the higher price level when suppliers donot anticipate the higher price level Thus we could express (99) in the form13
Yt = (Pt minus Wt + φ) + λYtminus1 (99prime)
where Wt is the logarithm of the equilibrium nominal wage for period t Theterm φ in (99prime) is defined such that if Etminus1Pt = Pt then the resulting log ofthe equilibrium real wage (ie ln(wtpt)) equals φ Ignoring the lagged outputterm in equation (99prime) this equilibrium real wage is the one associated with thenatural level of output and employment14 We will follow others and assume forconvenience that φ = 0 implying an equilibrium real wage with no surprisesequal to one
According to (99prime) if an increase in the price level is accompanied by a lessthan proportionate increase in the equilibrium money wage Pt minusWt rises (the realwage falls) and employment and output increase In the Lucas-type model such anevent occurs if suppliers do not forecast the price increase However Sargent andWallacersquos ineffectiveness proposition eliminates deterministic monetary policyrules as a source of such a price rise not matched by a similar rise in wages if(a) wages are set each period to equate labor demand and supply and (b) rationalexpectations are assumed In this case individuals and the monetary authoritiesare assumed to have a common set of information based on events up to the endof period t minus 1 (at time t) Individuals are also privy to the monetary policy ruleand they know the structure of the economy Thus they have knowledge of thedeterministic monetary policy to be followed during period t and its effect on theprice level for period t as predicted by the model Assuming flexible wages thispredicted effect of monetary policy on prices will be factored into the setting of themoney wage for period t As a consequence such deterministic monetary policy
138 Keynesian model
cannot change the real wage and thus leaves employment and output unaffectedas well
Obviously this chain of reasoning breaks down if money wages for period twere set prior to time t For example this policy ineffectiveness doctrine disap-pears if some wages are set at the end of period t minus 2 for period t In this casenew information that arrives during period t minus 1 can be incorporated by the mon-etary authorities into their money supply rule Even with rational expectationsthe implications of this cannot be used by individuals to adjust the money wagefor period t since by assumption the money wage is fixed Thus deterministicmonetary policy based on information revealed during period t minus 1 can alter thereal wage employment and output
To formally develop this potential stabilizing role of monetary policy in a moreldquoelegantrdquo fashion let us consider Fischerrsquos model The model disaggregates theeconomy into two sectors and assumes that the sectors alternate in setting multi-period employment contracts that fix nominal wages In particular ldquosuppose thatall labor contracts run for two periods and that the contract drawn up at the endof period t minus 2 specifies nominal wages for periods t minus 1 and t [Assume] thatcontracts are drawn up to maintain constancy of the real wagerdquo (Fischer 1977198) In other words
tminusiWt = EtminusiPt i = 1 2 (910)
where tminusiWt is the logarithm of the wage set at the end of period tminus i for period t15
The idea embodied in (910) that wages are set for more than one period iscritical to Fischerrsquos finding It essentially means that in any period half of thelabor contracts have fixed money wages16 Given that the wage is predeterminedfor each firm the aggregate supply equation is given by
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911)
where ut is a stochastic ldquorealrdquo disturbance or ldquosupply shockrdquo that impinges onproduction in each period17 Substituting (910) into (911) we can rewrite theaggregate supply equation as
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911prime)
A complete model except for specifying the source of expectations
Equation (911prime) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To close the model we require LM and ISequations The explicit derivation of these is left to the next chapter but let usassume for the time being that the LM equation is given by
mt minus Pt = α1Yt minus α2 middot rt minus εt
Keynesian model 139
and the IS equation by
Yt = Xt minus β1(rt minus πet+1) + ut
Here mt = ln Mt mt = mt + εt such that the deterministic component mt is setby government authorities (ie the monetary authority) according to a monetaryrule Further rt minus πe
t+1 represents the expected real rate of interest Xt denotesa vector of exogenous variables that affect output demand εt and ut are randomterms associated with output demand and money supply respectively and assumedindependent (ie E(εtut) = 0) and well behaved
Combining the LM and IS equations to eliminate the interest rate rt we obtain
Yt = Xt + ut minus (β1α2)(minusmt minus εt + Pt minus α1Yt) + β1πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = [α2(α2 + α1β1)][Xt + ut + (β1α2)(mt + εt minus Pt) + β1πet+1] (912)
Note that if α2 = 0 (changes in the interest rate do not affect money demand) andα1 = 1 (the income elasticity of real money demand is one) then this simplifies towhat Fischer refers to as a ldquovelocity equationrdquo
Yt = mt minus Pt + vt (913)
where vt = εt is now to be interpreted as a money demand disturbance termaffecting the ldquovelocityrdquo of money18
To see why (913) is called a ldquovelocity equationrdquo note that the assumption ofmoney demand being independent of the interest rate allows us to capture therelationship between income and the price level summarized by the aggregatedemand equation by looking solely at the LM equation (ie neglecting the ISequation) In particular if we assume that real money demand can be expressedby the equation
Ldt = yt(exp(minusvt))
then equating real money demand to real money supply (Mtpt) gives us
yt(exp(minusvt)) = Mtpt (914)
Taking the logarithm of the equilibrium condition with respect to the money market(914) we obtain
ln yt minus vt = ln Mt minus ln pt
Given Pt = ln pt Yt = ln yt and mt = ln(Mt) this is simply equation (913)
140 Keynesian model
Note that equilibrium velocity is defined as the ratio of nominal output to themoney supply Thus rearranging (914) we have that
Equilibrium velocity equiv ptyt
Mt= exp(vt)
which explains the interpretation of vt as a ldquovelocityrdquo disturbance Fischer assumesthat vt has a zero mean so that expected velocity is one A vt above zero meansa decrease in real money demand relative to real output and thus an increase inequilibrium velocity As (914) makes clear for a given Mt a higher vt implies ahigher pt andor a higher yt to maintain equilibrium with respect to the demandfor and supply of money
As Fischer states (913) is ldquothe simplest way of taking demand consid-erations into accountrdquo In sum then the macroeconomics model considered byFischer is given by the aggregate supply equation (911prime) and the aggregate demandequation (913)
Combining (911prime) and (913) to eliminate Yt we have
12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut = mt minus Pt + vt
This can be solved to give the reduced-form equation for the price level Pt
Pt = 12
[(12 Etminus1Pt + 1
2 Etminus2Pt
)minus ut + mt + vt
] (915)
Combining (911prime) and (913) to eliminate Pt we have
Yt = 12 (mt minus Yt + vt minus Etminus1Pt) + 1
2 (mt minus Yt + vt minus Etminus2Pt) + ut
This can be solved for the reduced-form equation for output Yt
Yt = 12
[mt + vt + ut minus 1
2 Etminus1Pt minus 12 Etminus2Pt
] (916)
According to equations (915) and (916) an increase in the ldquorealrdquo disturbanceterm ut leads to higher equilibrium output and a reduced price level Intuitivelythis corresponds to a shift to the right in the ldquoaggregate supply curverdquo On the otherhand an increase in the ldquovelocityrdquo disturbance term vt corresponds to a shift to theright in the ldquoaggregate demand curverdquo and thus leads to a higher output and pricelevel given an upward-sloping aggregate supply curve Note that an increase in vtmeans a lower real money demand at each level of income The shift in the aggre-gate demand curve reflects the fact that a higher price level andor higher output isrequired to restore equilibrium in the money commodity and financial markets
If expectations can be taken as exogenous with respect to money supply changesthen (916) indicates that money supply changes can affect output But this wasalso the case for a Lucas-type model The next step is thus to see what happens
Keynesian model 141
when we assume rational expectations As will become clear even with rationalexpectations expectations formed at the end of period t minus 2 can be viewed asexogenous with respect to monetary changes planned for period t based on infor-mation obtained during period t minus 1 Thus monetary policy has the potential tooffset the persistent effect of disturbances that originate during period t minus 1
Introducing rational expectations
Let us now assume that individuals form their expectations Etminus1Pt and Etminus2Ptldquorationallyrdquo in that
Etminus2Pt = E(Pt |tminus2) (917)
Etminus1Pt = E(Pt |tminus1) (918)
indicating that EtminusiPt is the mathematical expectation of Pt conditional on theinformation set tminusi which is all information available at the end of period t minus ii = 1 2 Taking the expectation of (915) at the end of period t minus 2 we thus have
Etminus2Pt = Etminus2
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (919)
Note that Etminus2Etminus1Pt = Etminus2Pt Thus (919) becomes
Etminus2Pt = Etminus2minusut + vt + mt (920)
Taking the expectation of (915) at the end of period t minus1 (at time t) we then have
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (921)
Substituting (920) into (921) gives
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2minusut + vt + mt minus ut + vt + mt
]
(922)
Rearranging
34 Etminus1Pt = 1
4 Etminus2minusut + vt + mt + 12 Etminus1minusut + vt + mt
or
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + mt (923)
which is Fischerrsquos (1977) equation (16)19
142 Keynesian model
Let the money supply be determined by the simple linear rule
mt = a1utminus1 + b1vtminus1
Since mt is a function only of information available up to the end of period t minus 1(at time t) Etminus1mt = mt Accordingly (923) can be written as
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + 23 mt (924)
Substituting (920) and (924) into the reduced-form equation for output (916)we obtain
Yt = 12 [ut + vt + mt] minus 1
4
[13 Etminus2minusut + vt + mt
+ 23 Etminus1minusut + vt + 2
3 mt
]minus 1
4 [Etminus2minusut + vt + mt](925)
Equation (925) simplifies to
Yt = 13 (mt minus Etminus2mt) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt
(926)
which is Fischerrsquos (1977) equation (18)As Fischer (1977 196) notes
disturbances aside this very simple macro model would be assumed in equi-librium to have the real wage set at its full employment level would imply theneutrality of money and would obviously have no role for monetary policyin affecting the level of output A potential role for monetary policy is createdby the presence of the disturbances ut and vt that are assumed to affect thelevel of output each period Each of the disturbances is assumed to follow afirst-order autoregressive scheme
ut = ρ1 middot utminus1 + εt where |ρ1| lt 1 (927)
vt = ρ2 middot vtminus1 + ηt where |ρ2| lt 1 (928)
where εt and ηt are mutually and serially uncorrelated stochastic terms withexpectation zero and finite variances σ 2
e and σ 2n respectively
Given equations (927) and (928) and the money supply rule
mt = a1utminus1 + b1vtminus1 (929)
Etminus2mt = a1ρ1utminus2 + b1ρ2vtminus2 (930)
Keynesian model 143
so that
mt minus Etminus2mt = a1utminus1 + b1vtminus1 minus [a1ρ1utminus2 + b1ρ2vtminus2]= a1εtminus1 + b1ηtminus1
(931)
According to (931)
the difference between the actual money stock in period t and that stockas predicted two periods earlier arises from the reactions of the monetaryauthority to the disturbances εtminus1 and ηtminus1 occurring in the interim It isprecisely these disturbances that cannot influence the nominal wage for thesecond period of wage contracts entered into at time t minus 2
(Fischer 1977 199)
Substituting (931) into (926)
Yt = 13 (a1εtminus1 + b1ηtminus1) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt (932)
From (927) and (928) we know that
ut + vt = (ρ1utminus1 + εt) + (ρ2vtminus1 + ηt)
= (ρ1utminus1 + εt) + (ρ22vtminus2 + ρ2ηtminus1 + ηt)
since by substitution vt = ρ22vtminus2 + ρ2ηtminus1 + ηt we also have that
Etminus1ut minus vt = ρ1utminus1 minus ρ2vtminus1 = ρ1utminus1 minus ρ22vtminus2 minus ρ2ηtminus1
and
Etminus2ut minus vt = ρ21utminus2 minus ρ2
2vtminus2
Thus we can rewrite (932) as
Yt = 12 (εtminus1 + ηtminus1) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
(933)
which is Fischerrsquos (1977) equation (21)20
Fischer (1977 199) notes that
before we examine the variance of output as a function of the parameters a1and b1 it is worth explaining why the values of those parameters affect thebehavior of output even when the parameters are fully known The essential
144 Keynesian model
reason is that between the time the two-year contract is drawn up and the lastyear of operation of that contract there is time for the monetary authorityto react to new information about recent economic disturbances Given thenegotiated second-period nominal wage the way the monetary authority reactsto disturbances will affect the real wage for the second period of the contractand thus output
Optimal monetary policy rules the effectiveness of policy
As in our discussion of Sargent and Wallace let us presume that the goal of themonetary authority focuses solely on output In particular suppose that the mone-tary authority desires to set the money supply in order to minimize the fluctuationin the log of output around some desired level Then the objective can be expressedas to
min Etminus1(Yt minus Y lowast)2
Let us assume that Y lowast = Yn = 0 so that the objective becomes to
min Etminus1(Yt)2 (934)
From (933) we have that
(Yt)2 =
[12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21 utminus2
]times[
12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
]
(935)
Note that E(εi) = E(ηi) = 0 E(ε2i ) = σ 2
e E(η2i ) = σ 2
n and that our independenceassumptions imply that E(εiηi) = 0 and for i = s E(εiεs) = 0 and E(εiηs) = 0Thus substituting (935) into (934) we have the following explicit form for theobjective
min Etminus1(Yt)2 = σ 2
e
[14 + 1
9 (a1 + 2ρ1)2]
+ (ρ21utminus2)
2
+ σ 2n
[14 + 1
9 (b1 + ρ2)2]
(936)
Given (936) the optimal monetary rule is to choose values for a1 and b1 suchthat
a1 = minus2ρ1 b1 = minusρ2 (937)
The above findings correspond to Fischerrsquos (1977) equation (23)21
Keynesian model 145
As Fischer (1977 200) states
to interpret the monetary rule examine [equation (933)] It can be seen therethat the level of output is affected by current disturbances (εt + ηt) that can-not be offset by monetary policy by disturbances (εtminus1 and ηtminus1) that haveoccurred since the signing of the older of the existing labor contracts andby a lagged real disturbance utminus2 The disturbances εtminus1 and ηtminus1 can bewholly offset by monetary policy and that is precisely what equation [(937)]indicates The utminus2 disturbance on the other hand was known when the olderlabor contract was drawn up and cannot be offset by monetary policy becauseit is taken into account in wage setting Note however that the stabilizationis achieved by affecting the real wage of those in the second year of laborcontracts and thus should not be expected to be available to attain arbitrarylevels of output ndash the use of too active a policy would lead to a change in thestructure of contracts
[A] more general interpretation of the monetary rule is to accommodatereal disturbances that tend to increase the price level and to counteract nominaldisturbances that tend to increase the price level
Fischer concludes by noting that
given a structure of contracts there is some room for maneuver by the mon-etary authorities ndash which is to say that their policies can though will notnecessarily be stabilizing
Conclusion
This chapter presented the sticky money wage or Keynesian model of the macro-economy We find that in contrast to the neoclassical model changes in the pricelevel affect real variables and the amount of labor employed in the economy Thusmoney has real effects The development of an upward-sloping aggregate supplycurve has dramatic implications for the conduct of monetary policy However it isshown that the expectations of agents in the economy also play an important rolein whether or not monetary policy is effective
10 The Lucas model
Introduction
As we have seen anticipated changes in prices have no impact on real vari-ables in the neoclassical model A key element of this model is the ldquoessentialpresumption that nominal output is determined on the aggregate demand sideof the economy with the division into real output and the price level largely depen-dent on the behavior of suppliers of labor and goodsrdquo (Lucas 1973) As such thismodel implies no link between price changes and real output
We have also seen how the natural rate model allows one to introduce a linkbetween unanticipated price changes and real output The seminal paper by Lucas(1973) formally develops a more complete model of the potential for ldquoshort-runsupply behavior (resulting) from suppliersrsquo lack of information on some of theprices relevant to their decisionsrdquo Lucasrsquos explanation of a tradeoff between unem-ployment and inflation ldquois that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relative pricechangesrdquo
As with the ldquoillusion modelrdquo Lucas postulates ldquorational agents whose deci-sions depend on relative prices only placed in an economic setting where theycannot distinguish relative from general price movementsrdquo That is we retain thehypothesis that prices adjust to clear markets
Lucas adds to the simple (static) natural rate model so far discussed by explicitlymodeling the source of forecast errors In doing so he assumes that ldquoinferenceson these relevant unobserved prices are made optimally (or lsquorationallyrsquo) in lightof the stochastic nature of the economyrdquo Below we outline Lucasrsquos model1
The ldquoislandrdquo paradigm
Lucasrsquos model begins by disaggregating the economy into a number of what havebeen called ldquosectorsrdquo ldquomarketsrdquo or ldquoislandsrdquo As Lucas says ldquowe imagine sup-pliers as located in a large number of scattered competitive markets Demand forgoods in each period is distributed unevenly over markets leading to relative aswell as general price movementsrdquo In terms of our previous analysis one couldthink of each of the n sectors in the economy as inhabited by firms producing the
Lucas model 147
ith commodity (i = 1 n) Associated with each sector or ldquoislandrdquo is a set ofworkers and thus a labor market
For firms producing commodity i in period t the key relative price is the priceof their output relative to the wage paid in sector i or pitwit where pit is themoney price of commodity i (produced in sector i) and wit is the money wage forlabor in sector i It is assumed that individuals (firms and workers) in sector irsquoslabor market know the money wage and price of commodity i For labor suppliersin sector i however the key relative price is witpt where pt is the economy-wideprice level reflecting the fact that suppliers plan to use money wages to purchasea bundle of goods consisting of all n commodities2
It is this setup of ldquodispersed marketsrdquo and ldquoinformational discrepanciesrdquo thatLucas uses to generate a correlation between price changes and output ndash the famousLucas supply equation
The supply function for a particular sector
The Lucas model assumes a competitive labor market for sector or ldquoislandrdquo i suchthat the equilibrium level of employment and money wage equate market demandand supply With respect to the labor demand function let us start by assumingthe simple CobbndashDouglas production function such that the marginal product oflabor is given by
a(Nit)minusα where a gt 03
The profit-maximizing condition for the representative firm producingcommodity i is to equate the money wage to the marginal product of labor multi-plied by the money price of output The resulting optimal labor demand can thusbe defined by the equation4
wit = pita(N dit )
minusα
Taking the natural log of this equation and rearranging we have
ln N dit = 1
α[ln pit minus ln wit + ln a] (101)
With respect to labor supply let us assume for the moment that the real wage isknown Further let us assume that the labor supply function takes the followinglogarithmic form
ln N sit = 1
βln
wit
pit
where β is a positive constantEmployment contracts entered into at time t in sector i specify the money
wage wit so that element of the real wage is known However if the price level is
148 Lucas model
unknown then the expected real wage based on information available at time t isequal to witEt(1pt) The associated expected logarithm of the real wage is then5
Et ln(witpt) = ln wit minus Et(ln pt)
Given the assumed log-linear labor supply function and ignoring the implicationsof uncertainty for labor supply we thus have
ln N sit =
(1
β
)(ln wit minus Et(ln pt)) (102)
Equilibrium in the labor market for sector i entails a level of employment Nitand money wage wit such that the demand for labor equals the supply In loga-rithmic form and using the specific labor demand and supply functions given byequations (101) and (102) equilibrium requires that the log of the money wageln wit and the log of employment ln Nit be such that
ln Nit = 1
α(ln pit minus ln wit + ln a)
and
ln Nit = 1
β(ln wit minus Et(ln pt))
Substituting the first expression into the second to eliminate the logarithm of themoney wage we have
ln Nit = 1
α(ln pit + ln a) minus 1
α(β ln Nit + Et(ln pt))
which upon rearranging becomes
ln Nit = ln Nni + [1 + (α + β)][ln pit minus Et(ln pt)] (103)
where ln Nni = (ln a)(α + β)Equation (103) indicates that the logarithm of equilibrium employment in the ith
sector and thus the production of commodity i depends directly on the expectationof logarithm of the ratio of the price of commodity i(pit) to the general level ofprices pt The term Nni can be viewed as the ldquonormalrdquo level of employment Notethat we have abstracted from population growth and other factors that would resultin this ldquonormalrdquo level of employment varying across time
Given the assumption of a simple CobbndashDouglas production function of theform yit = (Nit)
1minusα(Ki)α we thus have
ln yit = ln yni + γ [ln pit minus Et(ln pt)] (104)
where γ = (1 minus α)(α + β) and ln yni = (1 minus α) ln Nni + α ln Ki The term yni isdenoted by Lucas as the ldquonormalrdquo level of output in the particular sector or market i
Lucas model 149
under consideration As Lucas (1973 327) states the ldquoquantity supplied in eachmarket will be viewed as the product of a normal (or secular) component commonto all markets and a cyclical component which varies from market to market
The cyclical component varies with perceived relative prices and with its ownlagged valuerdquo Note that for the moment we do not include the lagged value ofoutput in the above supply function One can justify the inclusion of such byassuming adjustment costs
The source of forecasting errors
According to (104) output of commodity i depends critically on suppliersrsquo fore-cast of the log of the general level of prices Et(ln pt) Now consider how sucha forecast may be obtained in a stochastic environment First it is assumed thatagents in sector i ndash in particular suppliers of labor involved in the production ofcommodity i ndash know commodity irsquos price pit However the exact extent to whichany change in the money price of commodity i reflects a change in the overalllevel of money prices as opposed to a change in commodity irsquos price relative toother prices is unknown It is this uncertainty that leads suppliers in sector i tomisinterpret a change in the general price level in terms of a change in a relativeprice
To be concrete suppose Etminus1(ln pt) incorporates all information available atthe end of period t minus 1 The logarithm of the actual price level will vary from thelogarithm of this expected price level to the extent that there are ldquosurprisesrdquo withrespect to the aggregate price level Letting ξt denote this ldquosurpriserdquo for period twe have
ln pt = Etminus1(ln pt) + ξt (105)
We assume that ξt which is that part of the price level that cannot be predictedfrom past data is a normally distributed random variable with zero expectationand variance σ 26
At the start of period t suppliers in market i receive one additional piece ofinformation the logarithm of the price of commodity i ln pit This signal isassumed to contain some information about the logarithm of the overall pricelevel in that
ln pit = ln pt + zit (106)
where zit is a normally distributed random variable with zero mean and varianceσ 2
z Thus using equation (105) to substitute for ln pt
ln pit = Etminus1(ln pt) + ξt + zit (107)
In words the logarithm of the nominal price for the sector ln pit is assumed toinform the supplier of the sum of the current ldquowhite noiserdquo innovations to the
150 Lucas model
relative price process in that sector (zit) and the innovations to aggregate demandand thus the economy-wide level of prices (ξt)7
We can then express the expectation of the logarithm of the price level at time tfor suppliers in sector i given the observed logarithm of the price of commodity iln pit by
Et(ln pt) equiv Eln pt |Etminus1(ln pt) ln pit (108)
Formally we have the joint distribution of two random variables f (ln pit ln pt)where one of them ln pit is known to take a particular value The problem is thebasic one of ldquobivariate regressionrdquo in that we have to determine the conditionalexpectation Eln pt | ln pit namely the ldquoaveragerdquo value of ln pt for the givenvalue of ln pit 8 As we shall see in the next section the resulting expression for theexpected general price level (in logs) given ln pit is observed can be expressedin linear form as
Et(ln pt) = Etminus1(ln pt) + (1 minus θ)(ln pit minus Etminus1(ln pt)) where 0 le θ le 1(109)
A digression on linear regression analysis
Let us assume a linear regression equation that links the observed logarithm of theprice of commodity i to the logarithm of the general level of prices of the form9
Etln pt | ln pit = a0 + a1(ln pit) (1010)
We can express the regression coefficients a0 and a1 in terms of some of the lowermoments of the joint distribution of ln pt and ln pit namely in terms of10
Eln pit = Etminus1(ln pt) (from (107))
Eln pt = Etminus1(ln pt) (from (105))11
Varln pit = σ 2 + σ 2z (from (107))
Cov(ln pit ln pt) = E(minusξt minus zit)(minusξt) = σ 2 + Ezit middot ξtIn general Ezit middot ξt = Cov(zit ξt) + EzitEξt However given that Ezit =Eξt = 0 and given the assumption that zit and ut are independent variables sothat Cov(zit ξt) = 0 we have12
Cov(ln pit ln pt) = σ 2
From (1010) we have that13
E(ln pt | ln pit) equivint
(ln pt)φ(ln pt | ln pit)d ln pt = a0 + a1(ln pit) (1011)
Lucas model 151
where φ(middot) is the conditional density function of ln pt given ln pit If we thenmultiply the expression on both sides of (1011) by the marginal density functionof ln pit denoted by g(ln pit) and integrate on ln pit we obtainintint
(ln pt)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0g(ln pit)d ln pit +int
a1(ln pit)g(ln pit)d ln pit
or
Eln pt = a0 + a1Eln pit (1012)
since φ(ln pt | ln pit)g ln(pit) = f (ln pit ln pt) Had we multiplied the expressionon both sides of (1011) also by ln pit before integrating on ln pit we would haveobtainedintint
(ln pt)(ln pit)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0(ln pit)g(ln pit)d ln pit +int
a1(ln pit)2g(ln pit)d ln pit
or
E(ln pit)(ln pt) = a0Eln pit + a1E(ln pit)2 (1013)
Solving (1012) and (1013) for a0 and a1 and making use of the fact that
E(ln pit)(ln pt) = Cov(ln pit ln pt) + E(ln pit)Eln ptand
E(ln pit)2 = Var(ln pit) + [Eln pit]2
we find that
a0 = Eln pt minus [Cov(ln pit ln pt)Eln pit](Var(ln pit))
a1 = (Cov(ln pit ln pt))(Var(ln pit))
Hence we can write equation (1010) as
Et(ln pt) equiv E(ln pt | ln pit)
= E(ln pt) + [Cov(ln pit ln pt)Var(ln pit)](ln pt minus Eln pit)Substituting in the above expressions for means variance and covariance we havethus derived (109) with
1 minus θ = σ 2(σ 2 + σ 2z )
152 Lucas model
Equation (109) indicates that agentsrsquo rational expectation of the current pricelevel is a ldquolinear least-squares projectionrdquo That is one could rewrite (109) as
Et(ln pt) = θEtminus1(ln pt) + (1 minus θ) ln pit (1014)
where θ = σ 2z (σ 2 + σ 2
z ) To see why this is called ldquolinear least-squaresrdquo notethat we could start with (1014) (the ldquolinearrdquo part of the projection) and thenpick θ to minimize the variance in this forecast or projection of ln pt (the ldquoleast-squaresrdquo part of the projection) In particular substituting in (105) for Etminus1(ln pt)
(ie Etminus1(ln pt) = ln pt minus ξt) and (106) for ln pit (ie ln pit = ln pt + zit)(1014) becomes
Et(ln pt) = ln pt minus θξt + (1 minus θ)zit (1014prime)
The problem of picking θ to minimize the variance of this projection can then beexpressed as14
minθ
Eln pt minus θξt + (1 minus θ)zit minus ln pt2 = θσ 2 + (1 minus θ)σ 2z
Taking the derivative of the above expression with respect to θ setting it equal tozero (ldquoleast squaresrdquo) and solving for θ we verify that θ = σ 2
z (σ 2 + σ 2z )
As Sargent (1987a 442) points out
the parameter θ is the fraction of the conditional variance in ln pit due torelative price variation The larger is this fraction the smaller is the weightplaced on ln pit in revising Etminus1(ln pt) to form Et(ln pt) This makes sensesince the larger is θ the more likely it is that a change in ln pit reflects arelative rather than a general price change15
Equation (1014) can be substituted into the supply function for commodity i(104) to obtain
ln yit = ln yni + γ θ(ln pit minus Etminus1(ln pt)) (1015)
where as before ln yni = (1 minus α)(ln Nni) + α(ln Ki) As noted above if weassumed adjustment costs then a lagged output term could be added to (1015)In this case we would have16
ln yni = ln yni + γ θ [ln pit minus Etminus1(ln pt)] + λ(ln yitminus1 minus ln yni) (1015prime)
If suppliers were able to observe the actual value of the price level so thatEt(ln pt) = ln pt then going back to (104) one could express the resulting ldquofullinformationrdquo output produced in sector i by
ln ylowastit = ln yni + γ [ln pit minus ln pt]
Lucas model 153
which given ln pit = ln pt + zit simply becomes
ln ylowastit = ln yni + γ zit
As you can see since the expectation of the random shock to relative prices zit iszero ln yni has the natural interpretation as the expected output of sector i givenfull information
The Lucas aggregate supply function
Equation (1015) is close to what is known as the ldquoLucas aggregate supplyfunctionrdquo Without adjustment costs the Lucas supply function takes the form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) (1016)
With adjustment costs the Lucas supply function takes the general form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) + λ(ln ytminus1 minus ln yn) (1016prime)
where yn denotes the natural rate of output17 For simplicity we have assumed thatthe natural rate of total output is constant across periods
The term ln yn can be interpreted either as the logarithm of output for theldquorepresentativerdquo sector or as the logarithm of total output across the n sectorsLet us assume the former interpretation The average level of real output can bedefined by
yt equiv 1
pt
⎡⎣ nprod
i=1
pityit
⎤⎦
1n
where [prodni=1 pityit]1n is the geometric mean of nominal output across the n markets
or sectors Taking logs we have the following definition for the logarithm ofaverage output
ln yt equiv minus ln pt + 1
n
nsumi=1
(ln pit + ln yit) (1017)
We will assume that the overall price level is constructed as a geometric mean ofindividual prices such that
pt equiv⎡⎣ nprod
i=1
pit
⎤⎦
1n
Taking logs
ln pt equiv 1
n
nsumi=1
ln pit
154 Lucas model
Substituting the above into (1017) we have the following definition for thelogarithm of average output
ln yt equiv 1
n
nsumi=1
ln yit (1018)
Substituting into (1018) the supply functions for the individual sectors as givenby (1015) we thus have18
ln yt equiv 1
n
nsumi=1
[ln yni + γ θ(ln pit minus Etminus1(ln pt))] (1019)
Recall that ln pit = ln pt + zit where zit is a normal random variable indepen-dently distributed across markets with a mean of zero and variance σ 2
z Substitutingthis into (1015) and rearranging we have
ln yt = 1
n
nsumi=1
[ln yni] + γ θ(ln pt minus Etminus1(ln pt)) + 1
n
nsumi=1
γ θzit (1020)
As the number of markets n approaches infinity from the law of large numberswe know that the sum of the zit divided by n approaches zero19 Thus for a largenumber of markets we may approximate (1020) by
ln yt = 1
n
⎡⎣ nsum
i=1
ln yni + γ θ(ln pt minus Etminus1(ln pt))
⎤⎦ (1021)
By definition the logarithm of the geometric average of ldquonormalrdquo output acrossmarkets ln yn is given by nminus1 sumn
i=1 ln yni Thus we can rewrite (1021) as (1016)in which ln yn is the logarithm of ldquonormalrdquo output that would occur if there were nosurprises with respect to the aggregate price level that is when ln pt = Etminus1(ln pt)Note that for simplicity we assume the natural rate is constant over time
Equation (1016) is the Lucas aggregate supply equation with the last termmissing As noted above if we include adjustment costs then we obtain (1016prime)indicating that the deviation of real output from its ldquonaturalrdquo level or trend isassociated with a deviation in the price level from that expected and past deviationsof output from the natural rate This last term makes output serially correlatedover time
The Lucas supply function and the Phillips curve
The Lucas supply function predicts a direct correlation between unanticipatedprice changes and output and thus a potential tradeoff between price changesand unemployment if one assumes that unemployment and output are inverselyrelated This potential inverse relationship between unemployment and inflation
Lucas model 155
is sometimes referred to as the ldquoPhillips curverdquo after AW Phillips who noted theempirical relationship between wage inflation and unemployment for the Britisheconomy for the 100 years up to 1957 (see Phillips 1958) Later depictions of thePhillips curve replaced the rate of change in wages with the inflation rate
To see this Phillips relationship more clearly rearrange the aggregate Lucassupply function without adjustment costs (1016) to obtain
ln pt = (ln yt minus ln yn)γ θ + Etminus1(ln pt)
Subtracting ln ptminus1 from both sides of this aggregate supply equation we have
ln(ptptminus1) = (ln yt minus ln yn)γ θ + Etminus1(ln(ptptminus1)) (1022)
Let the term πt denote the rate of inflation between periods t minus 1 and t20
πt equiv (pt minus ptminus1)ptminus1 = (ptptminus1) minus 1
It is common in macroeconomics to approximate the above rate of change inprices by the log of the ratio of the two prices If the ratio equals one then thelog equals zero which is the rate of inflation If the ratio is 1 + x and x is a smallproportion then the log of this ratio approximately equals the actual inflation rateFor instance if ptptminus1 = 105 so that inflation is 005 or 5 percent then the logof 105 is 00488 which approximates this 005 rate of inflation Thus we have
πt asymp ln(ptptminus1) = ln pt minus ln ptminus1
Using the above approximation for the rate of inflation we can rewrite(1022) as
πt = (ln yt minus ln yn)γ θ + Etminus1πt (1023)
which as Sargent (1987a 443) states
is in the form of a standard natural rate Phillips curve relating inflation (πt)
directly to output (ln yt) and to expected inflation (Etminus1πt) According to[(1023)] the Phillips curve shifts up in the (πt yt) plane by the exact amountof any increase in expected inflation This characteristic of equation [(1023)]is often taken as the hallmark of the natural unemployment rate hypothesisIt seems to offer an explanation for why the Phillips curve tradeoff worsenedas average inflation rates increased over the 1970s in many western countries
If we assumed that due to adjustment cost the lagged deviation in output fromthe natural level affects the current deviation as the Lucas aggregate supplyequation (1016prime) suggests then in terms of rates of change in prices we wouldhave
πt = (ln yt minus ln yn)γ θ + Etminus1πt minus (λγ θ)(ln ytminus1 minus ln yn) (1023prime)
156 Lucas model
We could instead express (1023) in terms of unemployment by assuming thereis a linear inverse relationship between deviations in output from the natural rateand deviations in the actual level of unemployment Ut from its natural rate Unsuch that
ln ytminus1 minus ln yn = minus(Ut minus Un)
where is a positive constant Substituting the above into (1023) we thus have
πt = minus(γ θ)(Ut minus Un) + Etminus1(πt) (1023primeprime)
indicating the inverse relationship between unanticipated price changes and theactual level of unemployment Rearranging (1023primeprime) we have that
Ut = Un minus (γ θ)[πt minus Etminus1(πt)] (1023primeprimeprime)
where γ θ gt 0 Equation (1023primeprimeprime) is the typical expression of the Phillips curvefound in the literature It indicates that deviations in the unemployment rate belowits natural level must be accompanied by deviations in the actual rate of inflationabove that expected It reflects the ldquonatural rate of unemployment hypothesisrdquo asoriginally coined by Friedman (1968 11)
There is always a temporary trade-off between inflation and unemploymentthere is no permanent trade-off The temporary trade-off comes not frominflation per se but from unanticipated inflation which generally means froma rising rate of inflation
Recall that as Barro and Gordon (1983 592) observed the term Etminus1(πt) in(1023primeprimeprime) is the
prior expectation of inflation for period t [which is] distinguished from theexpectation that is conditional on partial information about current pricesThis distinction arises in models (eg Lucas 1972 1973 Barro 1976) inwhich people operate in localized markets with incomplete information aboutcontemporaneous nominal aggregates In this setting the Phillips curve slopecoefficient (γ θ) turns out to depend on the relative variances for generaland market-specific shocks
Variability in prices and the tradeoff
As Lucas (1973 333) states
demand policies [can] tend to move inflation rates and output (relative totrend) in the same direction or alternatively unemployment and inflation inopposite directions The conventional Phillips curve account of this observedco-movement says that the terms of the tradeoff arise from the relatively stable
Lucas model 157
structural features of the economy and are thus independent of the nature ofthe aggregate demand policy pursued The alternative explanation of the sameobserved tradeoff is that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relativeprice changes
Taking Lucasrsquos alternative viewpoint two aspects concerning the tradeoff aresuggested First as Lucas states ldquochanges in average inflation rates will notincrease average outputrdquo As we have seen if we compare the expected pricelevel for period t with the price level for the prior period the difference wouldincorporate individualsrsquo expectation of this average rate of inflation (along with anumber of other potentially relevant variables) Second ldquothe higher the variancein average prices the less lsquofavorablersquo will be the observed tradeoffrdquo We considerthis second point below by referring back to the simple Lucas supply functionwithout lagged output (1016)
Recall that the term ξt given by (105) denotes that part of the price level thatcannot be predicted from past data We have assumed that this ldquosurpriserdquo term ξtis a normally distributed random variable with zero expectation and variance σ 2Substituting this into (1016) we have that
ln yt minus ln yn = γ θξt (1024)
where θ = σ 2z (σ 2 + σ 2
z ) and γ = (1 minus α)(α + β) Recall that θ is the weightattached to the expected price level prior to observing pit
Equation (1024) indicates that deviations in output from the natural level dependsolely on surprises In his statement concerning the variance of prices Lucas ispointing out that the impact of ldquosurprisesrdquo on output relative to its natural leveldepends on the ldquosloperdquo term λθ which is given by
λθ = 1 minus α
α + β
σ 2z
σ 2 + σ 2z
As Sargent (1987a 444) notes
a ldquofavorablerdquo tradeoff between output and unexpected inflation (that is a largevalue of γ θ ) will exist only when σ 2 is small relative to σ 2
z An attempt byauthorities to exploit the tradeoff between output and unexpected inflationmore fully by changing aggregate demand regimes might increase the vari-ance σ 2 relative to σ 2
z and thus change the slope γ θ This is yet anotherexample of how agentsrsquo optimal decision rules change in response to changesin the random processes governing the exogenous variables they base theirdecisions on
Sargentrsquos last point is another example of the ldquoLucas critiquerdquo in this contextwith respect to the validity of using past econometric estimates of a tradeoff inpredicting future tradeoffs
158 Lucas model
Note that although the tradeoff worsens with higher variability in prices theeffect of higher variability in prices on the variance of output about the natural rateis unclear In particular from (1016) we know that the variance in the differencebetween output and the natural rate is simply (γ θ)2σ 2 Given our definition ofγ θ the variance of the logarithm of output becomes
Var(ln yt) = γ 2(
σ 2z
σ 2 + σ 2z
)2
σ 2
Differentiating with respect to σ 2 we have
partVar(ln yt)
partσ 2 = γ 2(
σ 2z
σ 2 + σ 2z
)2
minus 2γ 2σ 2 σ 2z
(σ 2 + σ 2z )
σ 2z
(σ 2 + σ 2z )2
=(
γ 2σ 2z
σ 2 + σ 2z
)2 (1 minus 2σ 2
σ 2 + σ 2z
)
As the above expression indicates by itself an increase in the variation in theaverage price level (σ 2) will increase the variation in the logarithm of output for agiven ldquosloperdquo (γ θ) On the other hand as Lucas pointed out such an increase inthe variation in price level will result in a reduction in the effect of any given pricechange on output which by itself would decrease the variation in the logarithm ofoutput
Substituting (105) into the expanded Lucas supply function with lagged outputwe have the Lucas supply function of the form
ln yt minus ln yn = γ θξt + λ(ln ytminus1 minus ln yn)
where as before θ = σ 2z (σ 2 + σ 2
z ) Substituting for prior differences in outputfrom its natural level we thus have
ln yt minus ln yn = γ θ
infinsumi=0
λiξtminusi (1025)
which shows that the deviation of output from its natural rate depends on thecurrent and all previous values of the ldquoaggregate demand shockrdquo that affects theequilibrium price level
A complete model except for specifying thesource of expectations
Equation (1016) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To simplify the analysis we will normalize outputso that the natural level of real output is equal to 1 We have already assumed that
Lucas model 159
the natural level of real output is constant over time These two assumptions allowus to write (1016) in the more compact form
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (1026)
where Yt denotes log of real output supply for period t (or equivalently the deviationin output from its natural level for period t) Pt denotes the log of the price leveland Etminus1Pt denotes the expectation of log of the price level What we now requireis a characterization of the aggregate demand side of the economy as typicallysummarized by the LM and IS equations
To obtain an explicit form for the portfolio or LM equation we start by assuminga real money demand function for the end of period t of the form
Ldt = yα1
t exp[minus(α2rt)] (1027)
where yt is real output α1 gt 0 and α2 gt 0 indicating that real money demand isdirectly related to real output but inversely related to the nominal interest rate21
Let Mt denote the nominal money supply at the end of period t (previously thishas been denoted by M ) and let mt denote the logarithm of this money supplyfor period t such that mt = ln Mt Further let us assume a logarithmic supply ofmoney function of the form
mt = mt + εt (1028)
Equation (1028) separates the logarithm of the money supply into two compo-nents a deterministic component mt set by government authorities according toa rule tying money supply changes to past variables and a random component εt which is assumed to be normally distributed with zero mean This random term isalso assumed to be serially independent (ie E(εtεs) = 0 for s = t)
The LM equation is simply the money market equilibrium condition equatingthe real supply of money to real money demand and thus is given by
Mtpt = Ldt (1029)
Taking logs of the equilibrium condition (1029) and substituting the logarithm ofthe money demand function (1027) and the money supply function (1028) wehave
mt minus Pt = α1Yt minus α2rt minus εt (1030)
where Pt = ln pt Equation (1030) is the standard log-linear form of the portfolioor LM equation22
To obtain an explicit form for the IS equation which is the equilibrium conditionin terms of equating output production to the demand for output we must postulatea specific form for output demand One common assumption is to include the
160 Lucas model
expected real rate of interest rt minus πet+1 as a determinant of output demand To do
so let
πet+1 equiv (pe
t+1 minus pt)pt = (pet+1pt) minus 1
As before we can approximate the expected rate of change in prices by the expecta-tion of the log of the ratio of the future to current price level (ie πe
t+1 asymp Pet+1minusPt
where Pet+1 is the expected log of the price level for period t+1) Thus the expected
real rate of interest becomes rt minus πet+1
Letting the term Xt denote a vector of exogenous variables that also affectsoutput demand we have in log-linear form the following equilibrium conditionfor the output market
Yt = Xt minus β1(rt minus πet+1) + ut (1031)
where ut is a serially independent stationary random process with mean zeroand finite variance equal to σ 2
u The random terms for output demand and moneysupply εt and ut respectively are assumed independent (ie E(εtut) = 0)
Summarizing we have a model consisting of the aggregate supply equation(1026) the LM equation (1030) and the IS equation (1031) which can be solvedfor the equilibrium output price and interest rate
In particular combining the LM and IS equations to eliminate the interest ratert we obtain
Yt = Xt + ut minus (β1α2) middot (minusmt minus εt + Pt minus α1Yt) + β1 middot πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
] (1032)
Or in terms of the price level we have an ldquoaggregate demand equationrdquo of theform
Pt = mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1) (1033)
Equations (1032) and (1033) indicate the inverse relationship between the pricelevel and output that is shown graphically by a downward-sloping aggregatedemand curve
With respect to (1033) note that the expected rate of inflation for the nextperiod πe
t+1 is viewed as a distinct entity Our prior assumption of unit elasticexpectations concerning the expected log of the future price level Pe
t+1 wouldimply that this term is in fact independent of changes in the current price levelNote also that if output were unchanged then an x percent change in the moneysupply would result in an x percent change in the price level This is the standardresult of the ldquoneoclassicalrdquo model23
Lucas model 161
Combining the above aggregate demand equation (1032) with the aggregatesupply equation to eliminate the output term Yt we have24
α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
]
= γ θ(Pt minus Etminus1Pt) + λYtminus1
Solving the above for the equilibrium price level one obtains
Pt = 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1)
+ J0
(Etminus1Pt minus λYtminus1
γ θ
)] (1034)
where J0 = γ θ(α2 + β1α1) Expression (1034) is sometimes called the reduced-form equation for the price level
Rearranging (1034) we have
Pt minus J0
J0 + β1Etminus1Pt
= 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1) minus J0
λYtminus1
γ θ
] (1034prime)
Let us assume perfect foresight meaning that Etminus1Pt = Pt Noting that1 minus J0(J0 + β1) = β1(J0 + β1) we can solve (1034prime) for the equilibriumprice level under this hypothesis of perfect foresight obtaining
Pt = mt + εtα2
β1(Xt + ut + β1π
et+1) minus J0
λ
β1γ θYtminus1 (1034primeprime)
As equation (1034primeprime) makes clear under the presumption of limited perfectforesight the predictions are those of the neoclassical model
(a) a change in the money supply (in log form given by mt + εt) results in anequiproportionate change in the price level (in log form given by Pt)
(b) an increase in expected inflation πet+1 raises the price level
(c) a higher level of lagged output (Ytminus1) lowers the price level(d) an increase in output demand (Xt + ut) raises prices
Following a procedure similar to that used to derive (1034) if we combine theaggregate demand equation (1033) and aggregate supply equation to eliminatethe price level we have
Yt = λYtminus1 minus γ θEtminus1Pt
+ γ θ
mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1)
162 Lucas model
Solving for the equilibrium real output we have
Yt = β1
J0 + β1
[λYtminus1 + γ θ
[mt + εt minus Etminus1Pt + α2
β1(Xt + ut + β1π
et+1)
]]
(1035)
The above expression is sometimes call the reduced-form equation for real outputAccording to (1035) changes in the money supply (in logs given by mt = mt +εt)will affect real output to the extent that the impact of such changes on prices is notfully anticipated
Note that with perfect foresight we have Etminus1Pt = Pt In this case substituting(1034primeprime) into (1035) for Etminus1Pt we obtain
Yt = β1
J0 + β1
(λYtminus1 + J0
λ
β1Ytminus1
)= λYtminus1 (1035primeprime)
Thus we have the standard neoclassical result that output in the current period isindependent of demand-side changes such as changes in expected inflation or themoney supply
One source of expectations autoregressive expectations
As Shiller (1978) has noted
one of the most difficult problems which confronts builders of macroeco-nomic models is the need to model the mechanism by which the public formsits expectations of future economic variables Many of the most importanttheoretical macroeconomic behavioral relations (eg the supply equationinvestment saving) depend critically on public expectations of future eco-nomic variables yet we often do not even have any data on what theseexpectations are
This and the next section suggest two approaches that have been taken to modelexpectations in particular the expected price level that enters into the aggre-gate supply equations These two approaches to expectation formation are thedistributed lag (or adaptive) scheme and the rational expectations scheme
To understand the ideas behind distributed lag schemes as the source of expec-tations we start by noting that the price level pt can be broken down into acombination of the price level for the previous period ptminus1 multiplied by theratio of the price level this period to last period
pt equiv ptminus1(ptptminus1)
Taking logs and recalling that ln(ptptminus1) is approximately equal to the rate ofinflation πt we thus have
Pt equiv ln pt = ln ptminus1 + πt
Lucas model 163
Taking expectations at time t assuming that at a minimum the price level for theprior period is known we have
Etminus1Pt equiv ln ptminus1 + Etminus1πt (1036)
Until the 1970s the approach to modeling the source of the expected rate of infla-tion embedded in (1036) was to assume individuals forecast the rate of inflationby looking at past inflation rates A common quantitative representation of thishypothesis originated by Fisher (1930) was to have individualsrsquo expectation ofthe inflation rate behave like a weighted average or ldquodistributed lagrdquo of recent pastinflation rates That is
Etminus1πt =qsum
i=1
ηiπtminusi (1037)
where the ηi are fixed numbers A typical idea behind this distributed lag approachto anticipated inflation was that individuals have ldquoadaptive expectationsrdquo whichmeant that individuals adjusted or ldquoadaptedrdquo their expectations of the rate of infla-tion in light of the actual forecast error made concerning the prior periodrsquos inflationrate Specifically adaptive expectations can be expressed as
Etminus1πt = Etminus2πtminus1 + δ(πtminus1 minus Etminus2πtminus1)
= δπtminus1 + (1 minus δ)Etminus1πtminus1(1038)
where 1 gt δ gt 0 Successive substitution allows us to rewrite (1038) as
Etminus1πt = δπtminus1 + (1 minus δ)δπtminus2 + (1 minus δ)2δπtminus3 + middot middot middotor
Etminus1πt =infinsum
i=1
δ(1 minus δ)iminus1πtminusi (1039)
As you can see (1039) is simply a specific form of equation (1037) in whichthe ηi place declining weight on past inflation rates the more distant they are andq = infin Given declining weights we can obtain a reasonable approximation of(1039) even if we truncate the distributed lag on past inflation after q periods aslong as q is reasonably large andor δ is reasonably large
Now let us place the above discussion not in terms of past rates of inflation butinstead in terms of past price levels Recalling the approximation
πtminusi asymp ln ptminusi minus ln ptminusiminus1
we can rewrite (1037) in the form
Etminus1πt =qsum
i=1
ηi(ln ptminusi minus ln ptminusiminus1) (1040)
164 Lucas model
Writing this out we have
Etminus1πt = η1(ln ptminus1 minus ln ptminus2) + η2(ln ptminus2 minus ln ptminus3)
+ η3(ln ptminus3 minus ln ptminus4) + middot middot middot + ηq(ln ptminusq minus ln ptminusqminus1)
Thus we may rewrite (1040) as
Etminus1πt = η1 ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
(1041)
Combining (1041) with equation (1036) we obtain the following expression forthe expectation formed at time t concerning the log of the price level
Etminus1Pt = (1 + η1) ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
or
Etminus1Pt =q+1sumi=1
vi ln ptminusi (1042)
Equation (1042) is what Sargent and Wallace (1975) refer to as ldquoautoregressiveexpectationsrdquo
A second source of expectations rational expectations
The papers by Lucas (1972 1973) and Sargent and Wallace (1975) suggestedthat in macroeconomic model building a different approach to specifying thesource of expectation is preferred As Cukierman (1986) summarizes this ldquorationalexpectationsrdquo approach to the modeling of inflationary expectations is
based on the maintained hypothesis that individuals know the structure of theeconomy and of governmentrsquos decision rule and that they use this structurein conjunction with the available information in order to form an optimalpredictor of future inflation [this approach] requires a precise specificationof the model of the economy as well as of the information sets of individualsEmpirical tests of this hypothesis are therefore joint tests of the validity of theexpectational hypothesis as well as of the postulated structure of the economyand of the particular assumptions made about the information possessed byindividuals
A ldquostructure of the economyrdquo was derived based on the Lucas aggregate sup-ply equation that included the assumption of market-clearing wages and pricesSuppose that individuals know this model and accept it as reflecting the structureof the economy As we saw above this model was solved to obtain (1034) the
Lucas model 165
reduced form for the equilibrium price level in period t We assume that at time tindividuals form their expectations Etminus1Pt ldquorationallyrdquo in that
Etminus1Pt = E(Pt |tminus1) (1043)
indicating that Etminus1Pt is the mathematical expectation of Pt conditional on theinformation set tminus1 which is all information available at period t minus1 As Sargentnotes (1987a 440)
Lucas assumed that tminus1 included information on all lagged values of ln pitand lagged values of real output in all markets One could equally well con-ceive of less comprehensive definitions of tminus1 For now along with Lucaswe suppose that tminus1 includes a comprehensive list of variables includinglagged outputs and prices in all markets
At the end of period t minus 1 individuals of course know Etminus1Pt as well as Ytminus1and πe
t+1 Thus taking the expectation of (1034) and subtracting it from Pt wehave
Pt minus Etminus1Pt = β1
J0 + β1(mt + εt minus Etminus1mt) + α2
J0 + β1(Xt minus Etminus1Xt + ut)
(1044)
In Sargent and Wallace (1975 244) the deterministic part of the money supplymt is assumed to reflect a ldquolinear feedback rulerdquo of the form
mt = Gθlowastt (1045)
where ldquoθlowastt represents the set of current and past values of all of the endogenous and
exogenous variables in the system as of the end of period t minus 1 and G is a vectorof parameters conformable to θlowast
t rdquo A simple example of a monetary feedback rulewould be
mt = a0 + a1Ytminus1 (1046)
where a0 and a1 are positive constantsLet us assume that individualsrsquo information set tminus1 includes not only the
structure of the economy (as summarized by the above linear macroeconomicmodel) but also the money supply rule In the particular example of (1046) theyknow a0 a1 and Ytminus1 and thus mt Then the assumption of rational expectationsimplies that Etminus1mt = mt Furthermore since Xt represents the deterministicpart of the vector of exogenous variables affecting output demand we have thatEtminus1Xt = Xt Thus (1044) becomes
Pt minus Etminus1Pt = 1
J0 + β1(β1εt + α2ut) (1047)
166 Lucas model
Substituting (1047) into the aggregate supply equation one obtains
Yt = γ θ1
J0 + β1(β1εt + α2ut) + λYtminus1 (1048)
An important feature of (1048) is that a deviation in output from its natural levelwhich is represented by the term Yt different from zero given our assumption thatthe natural output level is normalized to equal one is determined only by pastdeviations and ldquosurprisesrdquo with respect to the money supply (the term εt) andoutput demand (the term ut) The ldquodeterministicrdquo or predictable component of anymoney supply change has no real effects Equation (1048) should look familiarGiven λ lt 1 it suggests that the times series for deviations of the logarithmof output from its natural rate is a stationary autoregressive process of order 1or AR(1)
The above analysis is an example of a ldquolinear rational expectations modelrdquoThe result that ldquopredictablerdquo monetary policy has no real effects reflects the twinassumptions of the natural rate hypothesis and rational expectations It should notbe surprising that deterministic monetary policy has no effect for the model ishomogeneous of degree 0 in mt Pt and Etminus1Pt This is a critical characteristic ofa ldquonatural rate modelrdquo
Conclusion
The main focus of this chapter has been on the development of the Lucas supplyfunction The model is often discussed in the context of the ldquoislandrdquo paradigmin which we specify the supply function for a particular sector in the economyThe role of forecasting errors is introduced and from that the Lucas aggregatesupply function is constructed and the relationship of this function to the Phillipscurve is discussed A number of other issues were discussed involving variabilityin prices and the corresponding economic tradeoff and then a complete modelwas introduced except for specifying the source of expectations Two sources ofexpectations were then described autoregressive expectations and rational expec-tations The implications of expectations were then discussed in their historicalcontext
11 Policy
Introduction
This chapter extends the earlier discussions about the actions of the monetaryauthority and how these actions affect the macroeconomy Perhaps the most inter-esting issue of monetary economics is addressed here that is the optimal roleof monetary policy The chapter highlights the differences in model results thatdepend on what type of expectations are assumed Particular attention is givento the Sargent and Wallace ldquoineffectiveness propositionsrdquo and the Phillips curveOther issues are then introduced including the ldquorule versus discretionrdquo debatetime inconsistency and the role of credibility and enforcement
Optimal monetary policy
In the 1970s the articles by Sargent and Wallace (1975 1976) and Lucas (19721973) altered the view of how one should assess the impact of monetary policy onthe economy and by implication what is optimal monetary policy The discussionstarts with the premise that monetary policy should be conducted according to arule or set of rules As Sargent and Wallace (1976 169) state
It is widely agreed that monetary policy should obey a rule that is a scheduleexpressing the setting of the monetary authorityrsquos instrument (eg the moneysupply) as a function of all the information it has received up through thecurrent moment Such a rule has the happy characteristic that in any givenset of circumstances the optimal setting for policy is unique If by remotechance the same circumstances should prevail at two different dates theappropriate settings for monetary policy would be identical1
The SargentndashWallace premise that monetary rules are preferred leads them toexplore the form of the optimal rule But as we will discover in going over thepaper by Barro and Gordon (1983) there is a question of whether monetary rulescan be enforced over time If not then what is typically left in these models isa ldquosecond bestrdquo solution involving the determination of optimal ldquodiscretionaryrdquo
168 Policy
policy Note that we use the term ldquosecond bestrdquo because enforceable rules tend todominate discretion in these models2
Accepting the premise that monetary policy can adopt enforceable rules stillleaves open the specification of the optimal set of rules The simplest rule sug-gested by Friedman (1959) would increase the money supply at a constant rateeach year perhaps 3 percent3 More complex rules known as ldquoreactive rulesrdquowould specify in advance how the growth of the money supply will change basedon new information on the state of the economy One such rule suggested is thatthe growth in the monetary base and thus the money supply automatically adjustwhenever the growth of nominal GNP deviates from its trend (McCallum 1985)Another reactive rule suggests that the government commit itself to holding theCPI to a preannounced target and adjust the monetary base and thus the moneysupply accordingly (Hall 1982)4
Below we begin our discussion of optimal monetary policy by reviewing theanalysis of optimal enforceable rules as suggested by the Sargent and Wallace(1975 1976) papers and reviewed in Sargent (1987a Chapter 17) This discussionis in the context of a natural rate model without rational expectations and then inthe context of a model which assumes rational expectations
Optimal monetary policy exogenous expectations
As we saw previously the reduced form for the log of the output in period t canbe expressed as
Yt = H0
[λYtminus1 + γ θ
[minusEtminus1Pt + mt + εt + α2
β1(Xt + ut + β1π
et+1)
]]
(111)
where H0 = β1(J0 +β1) and J0 = γ θ(α2 +β1α1) Recall that Yt is the differencein period t between the logarithm of output and the logarithm of the natural levelof output Normalizing so that the natural level of output equals one we canequivalently interpret Yt in equation (111) as total output As Sargent and Wallace(1976) state ldquoYt can be thought of as the unemployment rate or the deviation ofreal GNP from lsquopotentialrsquo GNP This equation should be thought of as the reducedform of a simple econometric modelrdquo
Recall that the log of the money supply in period t mt is the sum of adeterministic component mt and the random component εt with variance σ 2
e 5
To understand the impact of monetary changes on real GNP we must firstconsider how the expected log of the price level Etminus1Pt varies with changes inmonetary policy One approach in the spirit of ldquoautoregressive expectationsrdquo isto assume that the expected price level is independent of the current monetarypolicy This essentially means viewing the expectation of the log of the price level(Etminus1Pt) as exogenous6 Given this assumption we may rewrite the reduced-form
Policy 169
equation for output (111) as
Yt = a0 + a1Ytminus1 + a2mt + vt (112)
where vt = H0γ θ(α2β1)ut is a serially independent normally distributed randomvariable with variance σ 2
v and mean zero mt is the log of the money supply forperiod t where mt = mt + εt Ytminus1 is lagged output and a0 a1 and a2 areparameters
Suppose that the monetary authority desires to set the money supply in orderto minimize the fluctuation in the log of output around some desired level Let usassume that the log of this desired level denoted Y lowast is above the log of the naturallevel of output Yn7 Then the objective can be expressed as
min Etminus1(Yt minus Y lowast)2
We can break this expression into two terms in that the objective can beequivalently expressed as
min Etminus1(Yt minus Etminus1Yt)2 + (Etminus1Yt minus Y lowast)2 (113)
The second way of expressing the objective allows us to see the objective asminimizing the sum of two terms the variance of Yt conditional on informationup to the end of period t minus 1 and the ldquobias squaredrdquo around Y lowast The secondterm the bias squared around Y lowast is the reason for an ldquoactivistrdquo monetary policyEquation (113) indicates that the optimal monetary policy entails
1 minimizing the variance in the random component of the money supply Thisfollows since the first term in equation (113) the variance of Yt is given bya2
2σ2e +σ 2
v 8 If feasible complete elimination of the random component to themoney supply (ie a purely ldquodeterministicrdquo money supply) is optimal suchthat εt = 0 for all t and thus σ 2
e = 02 setting Etminus1Yt = Y lowast so as to make the second term in equation (113) equal
to zero From equation (112) this means a monetary policy such that
Etminus1(a0 + a1 middot Ytminus1 + a2 middot mt + vt) = Y lowast
Noting that Etminus1Yt = 0 and that Etminus1mt = mt we see that this deterministic partof the optimal monetary policy is defined by the equation
a0 + a1Ytminus1 + a2mt = Y lowast
which can be solved for the optimal deterministic monetary policy rule
mt = g0 minus g1Ytminus1 (114)
where g0 = (Y lowast minus a0)a2 and g1 = a1a2
170 Policy
An equivalent expression for the optimal monetary rule (114) is derived inSargent (1987a Chapter 17) under the presumption of exogenous expectations Inparticular Sargent uses equation (112) to substitute out for Ytminus1 in (114) so thatthe optimal (deterministic) monetary rule (114) becomes
mt = g0 minus g1[a0 + a1Ytminus2 + a2mtminus1 + vtminus1] (115)
Now substituting into (115) the expression for Ytminus2 suggested by equation (112)we have
mt = g0 minus g1[a0 + a2mtminus1 + vtminus1] minus g1a1[a0 + a1Ytminus3 + a2mtminus2 + vtminus2](116)
Continuing to successively substitute for Ytminusi i = 3 4 we have Sargentrsquosequivalent expression for the optimal (deterministic) policy rule as given by9
mt = g0 minus g1a0 minus⎛⎝g1a2
infinsumiminus1
aiminus11 mtminusi
⎞⎠ minus
(g1
infinsumi=1
aiminus11 + vtminusi
) (117)
Following the above optimal monetary policy (ie reducing any random compo-nent to the money supply to its minimum level and establishing the rule for thedeterministic component of the money supply as specified by (114) or (117)) wehave by construction that
Etminus1Yt = Y lowast and Yt = Y lowast + a2εt + vt
where vt = H0γ θ(α2β1)ut and the variance of the random component of themoney supply εt is set at its lowest feasible level Thus optimal monetary policyin essence sets output each period equal to Y lowast plus irreducible noise As Sargentand Wallace (1976 171) note
the application of the rule eliminates all serial correlation in output since thisis the way to minimize the variance in output The basic idea is that wherethe effects of shocks to a goal variable (like GNP) display a stable pattern ofpersistence (serial correlation) and hence are predictable the authority canimprove the behavior of the goal variable by inducing offsetting movementsin its instruments
Note that without the lag term for output g1 in (114) equals zero
Adaptive expectations and the accelerationist result
The well-known ldquoaccelerationist outcomerdquo concerning the path of inflation isimplied by the above analysis if expectations are adaptive and if the aim of mone-tary policy is to keep output above its natural level To see this let us go back to the
Policy 171
aggregate supply equation that underlies the reduced form for output To simplifylet us abstract from the stochastic elements in demand εt and ut (as well as anysupply-side disturbances) and also from adjustment costs (ie omit the laggedoutput term) Then the aggregate supply equation
Yt = γ θ [πt minus Etminus1πt] (118)
reflects the actual path that output will take10
To derive the accelerationist result assume that Y lowast gt Yn By normalizationYn = 0 so that to have Etminus1Yt = Y lowast gt Yn we must have Etminus1Yt gt 0Equation (118) suggests that to achieve an expected level of output greater thanits natural level the government must pursue a monetary policy that results inthe actual rate of inflation typically being above that expected In particular thedesire to keep output above its natural level means that monetary policy in period tresults in the actual inflation rate πt such that
πt minus Etminus1πt = Y lowastγ θ gt 0
If expectations are adaptive then we have that
Etminus1πt minus Etminus2πtminus1 = δ(πtminus1 minus Etminus2πtminus1)
The assumption of adaptive expectations coupled with Y lowast gt Yn = 0 thus impliesthat
Etπt+1 = Etminus1πt + δ(πt minus Etminus1πt) = Etminus1πt + δY lowastγ θ
In words the fact that individuals underestimate inflation this period (by theamount Y lowastγ θ ) leads them to adjust (ldquoadaptrdquo) their expectations of inflationupward (by the amount δY lowastγ θ ) The result is that to keep expected output atthe level Y lowast gt Yn next period means an increase in the inflation rate by theamount δY lowastγ θ each period As Blanchard and Fischer (1989 572) note
this is the famous accelerationist result derived by Friedman (1968) andPhelps (1968) using their Phillips curve together with the adaptive expec-tations assumption The explanation is simple if the government is trying tokeep output above the natural rate it has to produce inflation at a higher ratethan expected each period Since the expected inflation is a weighted averageof past inflation rates the actual rate must be increasing
Now let us assume instead that Y lowast = 0 To provide a role for an activist monetarypolicy let us reintroduce the lagged output term λYtminus1 into the right-hand sideof (118) Thus monetary policy can eliminate the effect of past deviations inoutput from the natural rate on current output In other words a policy aimed
172 Policy
at setting Etminus1Yt = Y lowast would imply altering inflation relative to expected onlywhen lagged output deviated from the natural level Rather than the prior result ofan ever-increasing inflation with Y lowast gt Yn with Y lowast = Yn we have that inflationsimply wanders11 For instance if logged lagged output Ytminus1 fell below the log ofthe natural level of output Yn = 0 then the difference Ytminus1 would be negativeOther things being equal this would imply a lower output in the current periodTo counteract this the government would pursue a monetary policy that results inthe actual rate of inflation being above that expected
The above discussion helps us understand the comment of Hall (1976) that
the benefits of inflation derive from the use of expansionary power to trickeconomic agents into behaving in socially preferable ways even thoughtheir behavior is not in their own interests The gap between actual andexpected inflation measures the extent of the trickery the optimal pol-icy is not nearly as expansionary when expectations adjust rapidly andmost of the effect of an inflationary policy is dissipated in costly anticipatedinflation
The above extract raises the following question Can the monetary authoritiessystematically trick the public in order to exploit the link between inflation andoutput For Sargent and Wallace and others the answer is no due to the existenceof rational expectations
Rational expectations and the SargentndashWallace ineffectivenessproposition
As Sargent (1987a) notes a critical aspect of the simple example of an optimalmonetary rule as given by (114) (or equivalently (117)) ldquois the implicit assumptionthat agentrsquos decision rules remain unchanged in the face of alternative stochas-tic processes for the control variable that different feedback rules implyrdquo WhatSargent means in this context is that the optimal monetary rule has been derivedunder the presumption that private agents do not take this rule into account informing their expectation of the price level Under this assumption one couldestimate the parameters of the reduced-form equation output (a0 a1 and a2 in(112)) independently of the feedback rule (114)
However Sargent and Wallace (1976) criticize this view In particular theyargue that ldquoin the reduced forms are embedded the responses of expectations tothe way policy is formed Changes in the way policy is made then ought not toleave the parameters of estimated reduced forms unchangedrdquo12 In other wordsrational individuals would clearly seek out and use information on how monetaryauthorities act as well as on the structure of the economy in forming expectationsof prices
Let us now consider the following version of the reduced-form equation foroutput (112) that explicitly includes the potential role of expected monetary policy
Policy 173
when individuals form expectations on prices13
Yt = a0 + a1Ytminus1 + a2(mt minus Etminus1) + vt (119)
For a given anticipated log of the money supply Etminus1mt we have as before theoptimal (deterministic) monetary rule of the form
mt = g0 minus g1Ytminus1 (1110)
so that
mt = g0 minus g1Ytminus1 + εt (1111)
where εt is the irreducible random element in the money supply determinationprocess Now assume that the public knows the monetary authoritiesrsquo feedbackrule Then our assumption of rational expectation (ie individuals use all availableinformation in forming expectations) implies that
Etminus1(mt) = g0 minus g1Ytminus1 (1112)
Combining (119) (1111) and (1112) the reduced form for output is now given by
Yt = a0 + a1Ytminus1 + a2εt + vt (1113)
so that the biased squared term in the objective of the monetary authorities(Etminus1Yt minus Y lowast)2 equals (a0 + a1Ytminus1 minus Y lowast)2
As is clear from (1113) there is no role for systematic monetary policy to affectreal output As Sargent (1987a 459) notes ldquothe bias squared is independent ofthe parameters of the money supply rulerdquo The optimal policy is then to makemonetary policy deterministic if feasible for then the variance of output (given bya2
2σ2e + σ 2
v ) is minimized by setting σ 2e = 0 Until we add an inflation objective
any deterministic rule will be equally as good for none will have any impact onoutput This is once again an example of the neutrality of money
As Sargent (1987a 458) notes ldquopolicy rules should be deterministic and involveno surprisesrdquo He goes on to argue that we
have therefore established the following stochastic neutrality theorem thatcharacterizes our model one deterministic feedback rule on the basis of theinformation set tminus1 which is common to the public and to the authority is asgood as any other deterministic feedback rule Via deterministic feedbackrules the monetary authority is powerless to combat the business cycle (theserial correlation in Yt)
Naturally if one abandons rational expectations or the natural rate hypothesis thenthis ldquostochastic neutralityrdquo result need not hold
174 Policy
The SargentndashWallace ineffectiveness proposition in thecontext of the Phillips curve
The above finding of the ldquoneutrality of moneyrdquo in the context of a stochasticlinear natural rate model with rational expectations is viewed by Sargent (1987a459) as
the antithesis of our earlier result rationalizing the activist Keynesian policyrules The reader is invited to verify that the truth of the neutrality theoremis not dependent on the particular information set assumed It will continueto hold for any specification of tminus1 so long as the public and the authorityshare the same information set
He concludes
The preceding results provide a [weak] defense for following rules with-out feedback Simple x-percent growth rules do as well as any deterministicfeedback rule and dominate rules with a stochastic component
Below we recast the SargentndashWallace ineffectiveness proposition in terms ofthe expectational Phillips curve This makes the discussion more in line with thenext sectionrsquos review of some implications of non-enforceable monetary rules Inaddition we add to the government goalrsquos an inflation objective In particular wemodify our analysis in the following four ways
1 we alter the objective to be in terms of unemployment rather than output2 we expand our objective function to include inflation3 we link inflation to unemployment via a modified Lucas supply equation4 we incorporate rational expectations
Our first task is to convert the objective of the government into unemploymentterms Before we assumed that the government simply sought to minimize thefluctuations in output about a particular level In particular if we let Zt denote thecost incurred in period t we assumed the objective was to
min Etminus1Zt where Zt equiv (Yt minus Y lowast)2 (1114)
We have previously assumed that the deviation of unemployment from its naturalrate is linearly related to the deviation of the log of output from the log of its naturallevel In particular we assumed
minus(Ut minus Un) = Yt
Policy 175
given the normalization of the natural level of output such that Yn equiv ln yn = 0Substituting the above into (1114) the problem facing the government policy-maker becomes
min Etminus1Zt where Zt = a(Ut minus kUn)2 a = 2 gt 0
k = 1 minus Y lowastUn (1115)
Note in (1115) that we assume k lt 1 which is equivalent to assuming thatthe log of optimal level of output Y lowast is greater than the log of the natural levelof output which by normalization has been set equal to zero As Blanchard andFischer (1989 596ndash597) suggest
The most plausible justification [for k lt 1] is the presence of distortions orimperfections that causes the natural rate of unemployment to be too high Thisjustification allows the loss function to be consistent with the single-periodutility function of private agents Another is that the governmentrsquos objectivefunction as shaped by the electoral process leads the government to seek toraise output above the natural level
Having converted the objective function into unemployment terms the next stepis to expand the objective function to include an inflation goal In particular let usassume that the cost in period t includes a term reflecting differences between theactual inflation rate πt and an optimal rate of inflation πlowast Assuming a simplequadratic form we have14
Zt = a(Ut minus kUn)2 + b(πt minus πlowast)2 (1116)
The problem of the government policy-maker is then
min Etminus1Zt where Zt equiv a(Ut minus kUn)2 + b(πt minus πlowast)2
As before we can decompose this objective to obtain the following equivalentexpression for the object of the government policy-maker
min a[Etminus1(Ut minus kUn)2 + (Etminus1Ut minus kUn)
2]+ b[Etminus1(πt minus πlowast)2 + (Etπt minus πlowast)2] (1117)
Our third task is to link inflation to unemployment Recall that if we ignore thelagged term with respect to output in the Lucas supply equation assume unem-ployment is linearly related to real output and approximate inflation by the log ofthe ratio of the price level this period to the price level last period then we canmanipulate the Lucas supply equation to obtain the expression
Ut = Un minus α(πt minus Etminus1πt) (1118)
176 Policy
where α = (γ θ) gt 015 Substituting (1118) into (1116) the governmentrsquosobjective becomes
min Etminus1a(1 minus k)Un minus α(πt minus Etminus1πt)2 + b middot (πt minus πlowast)2
Our fourth and final task is to introduce rational expectations As we saw earlierwe obtain the reduced form for the deviation of the price level from that expectedin period t
Pt minus Etminus1Pt = 1
J0 + β0(β1εt + α2ut) (1119)
if (a) the government follows a particular rule in determining monetary policy(b) that rule is known to the public and (c) there exist rational expectations
An algebraic manipulation ndash simultaneously subtracting and adding the log ofthe price level for period t minus 1 to the left-hand side of equation (1119) ndash bringsus closer to having an expression that may be interpreted in terms of inflation
ln( ptptminus1) minus Etminus1(ln( ptptminus1) = 1
J0 + β1(β1εt + α2ut) (1120)
Using the log of the price ratio as an approximation for inflation we thus canapproximate (1120) as
πt minus Etminus1πt = 1
J0 + β1(β1εt + α2ut) (1121)
Substituting the above which reflects the rational expectations approach tomodeling expectations into the new government objective we have
min Etminus1
a
[(1 minus k)Un minus α
J0 + β1(β1εt + α2ut)
]2
+ b(πt minus πlowast)2
or
min a[(1 minus k)Un]2 +(
α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1 b(πt minus πlowast)2 (1122)
since Etminus1εt = Etminus1ut = Etminus1εtut = 0It is clear from (1122) that with rational expectations monetary policy can play
no role in helping the government meet its objective concerning unemploymentIn other words in the context of the Lucas model the assumption of rationalexpectations means that the expected loss from deviations in unemployment fromits desired level and thus production from its desired level is independent ofthe deterministic monetary rule As a consequence the minimization problem asgiven by the first half of equation (1117) becomes simply one of specifying any
Policy 177
deterministic monetary policy rule and eliminating any random changes in themoney supply (ie setting σ 2
e = 0 if feasible)But the objective of the government now also includes an objective concerning
the rate of inflation so we need an expression for the equilibrium rate of changein prices πt that incorporates rational expectations To do so we start with thereduced form for the log of the price level obtained previously which is given by
Pt = 1
J0 + β1
[β1mt + εt + α2(Xt + ut + β1π
et+1) + J0
(Etminus1Pt minus λYtminus1
γ θ
)]
(1123)
Taking the difference between the reduced form for the log of the price level forperiod t and for period tminus1 we can obtain an expression for πt the rate of inflationbetween period t and t minus 1 of the form
πt = 1
J0 + β1[β1mt + α2(Xt minus Xtminus1 + ut minus utminus1) + α2β1(π
et+1 minus πw
t )
+ J0(Etminus1Pt minus Etminus2Ptminus1) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1124)
where πmt approximates the rate of change in the money supply (ie πmt =mt minus mtminus1 = ln(MtMtminus1)) Assuming rational expectations we have fromequation (1119) that
Etminus1Pt = Pt minus 1
J0 + β1(β1εt + α2ut)
and similarly
Etminus2Ptminus1 = Ptminus1 minus 1
J0 + β1(β1εtminus1 + α2utminus1)
so that
Etminus1Pt minus Etminus2Ptminus1 = πt + 1
J0 + β1[β1(εtminus1 minus εt) + α2(utminus1 minus ut)]
(1125)
Substituting this expression into equation (1124) one obtains
πtβ1
J0 + β1= 1
J0 + β1[β1πmt + α2(Xt minus Xtminus1 + ut minus utminus1)
+ α2β1 middot (πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1126)
178 Policy
where we use the fact that 1 minus J0(J0 + β1) = β1(J0 + β1) Solving for πt we have
πt = πmt + α2
β1(Xt minus Xtminus1 + ut minus utminus1)
+ α2(πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)
(1127)
which given πmt = mt minus mtminus1 = mt + εt minus (mtminus1 + εtminus1) can be rearranged toobtain
πt = πmt + εt minus εtminus1 + α2
β1(Xt minus Xtminus1 + ut minus utminus1) + α2(π
et+1 minus πe
t )
minus J0λ
β1γ θ(Ytminus1 minus Ytminus2) (1128)
where πmt = mt minus mtminus1If we assume no change in exogenous (nonrandom) demand factors this period
compared to the last period (ie Xt = Xtminus1) no change in the expected future infla-tion between last period and this period (ie πe
t+1 = πet ) and ignore adjustment
costs (ie λ = 0) we can simplify equation (1128) to obtain
πt = πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) (1129)
The three assumptions we made to derive equation (1129) from equation (1128)largely limit any differences between period t and t minus 1 to differences in the sizeof the money supply In fact the two periods differ only by the deterministiccomponent of the money supply and by random factors where these randomfactors ndash the term ut minus utminus1 with respect to output demand and the term εt minus εtminus1with respect to the irreducible random component of the money supply ndash havemean zero In other words all potential changes in aggregate demand or productionexcept money supply changes that would lead to different expected price levels inthe two periods have been removed
Substituting equation (1129) into (1122) we thus obtain the following completegovernment objective function under rational expectations
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1b[πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) minus πlowast]2
]
from which we see that constant monetary growth will not achieve a constant rateof inflation unless we neglect the lagged disturbance terms16 To obtain the resultof an optimal monetary rule in the form of a constant rate of growth in the moneysupply we must further simplify and neglect the lagged disturbance terms If we
Policy 179
for the moment ignore the lagged stochastic terms εtminus1 and utminus1 we have theobjective
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ b(πmt minus πlowast)2 + σ 2e +
(α2
β1
)2
σ 2u
]
As before to minimize the loss requires that one reduces the random variation inthe money supply to zero (if feasible) so that σ 2
e = 0 That is the optimal monetaryrule is ldquodeterministicrdquo Further given an inflation objective and no reason otherthan monetary policy for prices to change the obvious optimal rule is simply toset the determinant rate of change in the money supply equal to the desired rate ofinflation That is the optimal policy is
πmt = πlowast
Note that this rule assumes no shocks to the economy If there were a steady rateof increase in output (and thus the real demand for output) then that would raisethe optimal constant rate of change in the money supply to achieve a given rate ofchange in prices Thus Friedmanrsquos ldquo3 percentrdquo rule for monetary growth presumeda 3 percent growth in real output so as to be consistent with a zero rate of inflation
Rules versus discretion monetary policy and timeinconsistency
The SargentndashWallace ineffectiveness proposition has been used by many to supportarguments for the government not adopting activist policy rules to offset fluctua-tions ndash particularly downturns ndash in an economyrsquos real output reflecting demand-sidedisturbances The reason as we have seen is that such policies have no real effectsonce one assumes rational expectations although the attempt can lead to higherinflation
Yet as Barro and Gordon (1983) point out
empirical studies indicate the presence of countercyclical monetary policyat least for the post-World War II United States ndash rises in the unemploymentrate appear to generate subsequent expansions in monetary growth Within thenatural rate framework it is difficult to reconcile this countercyclical behaviorwith rationality of the policymaker
As Barro and Gordon go on to say ldquoa principal object of our analysis is to achievethis reconciliationrdquo That is rather than saying what policy rules governmentshould follow Barro and Gordon want to explain why government acts the way itdoes ndash thus the term ldquopositive theoryrdquo in the title of their paper
180 Policy
The BarrondashGordon approach combines a number of topics that we haveconsidered before First they utilize a natural rate model like the Lucas modelSecond they assume rational expectations And third and most interestingly theyprovide us with a nice example of the phenomenon of ldquotime inconsistencyrdquo in thecontext of optimal monetary policy when there exists the potential Phillips curvetradeoff
Contrasting the BarrondashGordon and SargentndashWallace policyenvironments
In the SargentndashWallace view of the optimal monetary policy choice it is assumedthat the policy-maker makes a once-and-for-all decision with respect to the partic-ular monetary policy rule (reactive or not) Under certain assumptions as we haveseen this leads to the natural conclusion that optimal monetary policy entails thesimple rule of a constant rate of growth in the money supply so that the resultingaverage rate of inflation equals the desired level If the desired level were zerothen inflation would be set equal to zero through appropriate monetary policy
Barro and Gordon (1983 598) have questioned this result on the basis that ldquotheremay be no mechanism in place to constrain the policymaker to stick to the rule astime evolvesrdquo The result is that the policy-maker decides each period the optimalmonetary policy to follow In other words ldquothough the objective function anddecision rules of private agents are identicalrdquo Barro and Gordon obtain differentresults from Sargent and Wallace because ldquothe problems differ in the opportunitysets of the policymakerrdquo Below we illustrate the exact nature of this differenceby first showing how Barro and Gordonrsquos setup provides an example of the ldquotimeinconsistency problemrdquo not present in the Sargent and Wallace problem and thenderive the equilibrium for an economy characterized by the policy-makers whoare allowed each period to pick a potentially new optimal monetary policy
Time inconsistency an example in the context of optimalmonetary policy
In an important paper on the ldquotime inconsistencyrdquo problem of optimal policyKydland and Prescott (1977) point out situations in which the optimal policiesdecided at time t would be changed at time t + 117 In the context of monetarypolicy as Kydland and Prescott note
the reason that such policies are suboptimal is not due to myopia The effect of(monetary policy) upon the entire future is taken into consideration Rather thesuboptimality arises because there is no mechanism to induce future policy-makers to take into consideration the effect of their policy via the expectationsmechanism upon current decisions of agents
Let us see what this means by way of a concrete example
Policy 181
A simple example of ldquotime inconsistencyrdquo following Kydland and Prescott isconstructed below To simplify the discussion somewhat for the moment we ignoreuncertainty and restrict our attention to two periods In this setting the govern-ment through monetary policy can determine each period the actual inflation raterather than the expected rate of inflation For periods 0 and 1 the choice variablesfacing the monetary policy-maker are thus π0 the actual inflation in period 0 andπ1 the actual inflation in period 1 The inflation choice impacts the unemploymentrate in that U0 and U1 the unemployment rates in periods 0 and 1 are given bythe Phillips curve formulation (1118)
U0 = Un minus α(π0 minus Eminus1π0)
U1 = Un minus α(π1 minus E0π1)
where the state variables Eminus1π0 and E0π1 denote the expected rate of inflation inperiods 0 and 1 respectively In periods 0 and 1 the objective for periods 0 and 1is to minimize the simple quadratic forms
Z0 = a(U0 minus kUn)2 + b(π0 minus πlowast)2
Z1 = a(U1 minus kUn)2 + b(π1 minus πlowast)2
where the constants a and b are positive It is assumed that k lt 1 to capture theidea that distortions exist in the economy that an activist policy can address18
Substituting in the ldquoPhillips curverdquo relations in period 0 the present value of theobjective function is
Z = Z0 + β middot Z1 = a[(1 minus k)Un minus α(π0 minus Eminus1π0)]2 + b[π0 minus πlowast]2
+ βa[(1 minus k)Un minus α(π1 minus E0π1)]2 + βb[π1 minus πlowast]2
where β is the constant discount factorLet us presume that the expectations of inflation the ldquostaterdquo variables are
exogenous and equal to the desired level each period (ie Eminus1π0 = E0π1 = πlowast)In period 0 the optimal inflation rates for periods 0 and 1 (as determined bymonetary policy) are then
partZpartπ0 = minus2aαq0 + 2b(π0 minus πlowast) = 0
partZpartπ1 = β[minus2aαq1 + 2b(π1 minus πlowast)] = 0
where qi = (1 minus k)Un minus α(πi minus πlowast) i = 0 1 In period 1 the objective is
Z1 = a[(1 minus k)Un minus α(π1 minus E0π1)]2 + b[π0 minus πlowast]2
and the optimal solution given π0 and E0π1 = πlowast is thus
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
182 Policy
Comparing this first-order condition to partZ0partπ1 it is clear that in the case ofexogenous expectations the optimal solution is time-consistent Further it impliesa rate of inflation greater than the expected (optimal) rate of πlowast each period toachieve an unemployment rate below the natural This follows given that one ofthe objectives is to approach a level of unemployment equal to kUn with k lt 1
What happens however if individualsrsquo expectations of inflation for period 1 isa forecast that correctly anticipates the optimal future policy decision That is wehave ldquorational expectationsrdquo such that in a deterministic world
E0π1 = h(0) = π1
where 0 is the information set at the end of period zeroWhat the optimal inflation policy is now depends on whether policy-makers
can ldquocommitrdquo to future policy actions Let us start by assuming they can placeconstraints on future policy We know that in period 0 the optimal solution for π1is then given by
βpartZ1
partπ1+ partZ
partE0π1
dh(middot)dπ1
= β[2b(π1 minus πlowast)] = 0
Since dh(middot)partπ1 = 1 the optimal planned (at time 0) rate of inflation for period 1is equal to the desired level πlowast This is the SargentndashWallace ineffectivenessproposition
However once period 1 occurs E0π1 is set (let us assume it equals πlowast) If thepolicy-maker was not then constrained by the prior specification of a monetarypolicy to achieve a rate of inflation equal to πlowast they would choose π1 such that
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
which implies π1 gt πlowast The ldquotime inconsistencyrdquo arises as you can see becauseparth(middot)partπ1 = 0 As Barro and Gordon state ldquothe term lsquotime inconsistencyrsquo refersto the policymakerrsquos incentives to deviate from the rule when private agents expectit to be followedrdquo
Equilibrium when monetary rules are not enforceable
As Barro and Gordon note in the time inconsistency problem ldquoconstraints onfuture policy actions are infeasible by assumptionrdquo In contrast in the SargentndashWallace view
rules are enforceable so that the policymaker can commit the course of futurepolicy (and thus of expectations) In the former case the time-inconsistentsolution is not an equilibrium given the problem facing the policymaker Inthe latter case the incentives to deviate from the rule are irrelevant sincecommitments are assumed to be binding
(Barro and Gordon 1983 599)
Policy 183
As the above extract suggests in the ldquotime inconsistency caserdquo we have not yetcharacterized a rational expectationsrsquo equilibrium since in our example individu-alsrsquo expectations of inflation for period 1 were incorrect According to Barro andGordon there are three features of an equilibrium The first is ldquoa decision rulefor private agents which determines their actions as a function of their currentinformationrdquo These actions of private agents based on current information aresummarized by the Phillips curve
Ut = Un + ζt minus α middot (πt minus Etminus1πt) (1130)
where ζt a random variable with zero mean and variance σ 2z has been added
to denote a real shock that affects the natural unemployment rate for the currentperiod only19
The second feature of an equilibrium is ldquoa policy rule which specifies the behav-ior of policy instruments as a function of the policymakerrsquos current informationsetrdquo This policy rule is given by the choice of inflation rates πt+i i = 0 1 2 by the monetary authorities with the following objective
min Et
⎡⎣ infinsum
i=0
β iZt+i|It
⎤⎦
where Zt+i equiv a(Ut+i minus kUn)2 + b(πt+i minus πlowast)2 a gt 0 b gt 0 and 0 le k le 1
The term It denotes the initial state of information and 1 gt β gt 0 is the constantdiscount factor (β = 1(1+rreal) where rreal is the exogenous real rate of interest)For simplicity we will assume that the optimal level of inflation is zero so thatπlowast = 0
The third feature is an expectations function which determines the expectationsof private agents as a function of their current information Assuming that ldquothepublic understands the nature of the policymakerrsquos optimization problem in eachperiodrdquo then Etminus1πt = πt where πt is the optimal choice of inflation by thepolicy-maker for period t
Combining the information contained in our discussion of the first and secondfeatures of equilibrium the policy-makerrsquos optimal choice of πt minimizes
EtZt = Et[a(Ut minus kUn)2 + b(πt)
2]= Et[a((1 minus k)Un minus α(πt minus Etminus1πt))
2 + b(πt)2]
Given that Eξt = 0 the expression to be minimized by the policy-maker can bewritten as
EtZt = [a((1 minus k)Un minus α(πt minus Etminus1πt))2 + b(πt)
2]A critical point to note is that each period the policy-maker inherits Etminus1πt andtakes that expected rate of inflation as given in the above optimal choice of πt
184 Policy
The first-order condition for period t is thus
partEtZtpartπt = 2a((1 minus k)Un minus α(πt minus Etminus1πt))(minusα) + 2b(πt) = 0
which can be simplified and rearranged to obtain the following expression for theoptimal inflation rate
πt = aα
b[(1 minus k)Un minus α(πt minus Etminus1πt)] (1131)
From the third feature of equilibrium we know that the public realizes the problemfaced by the policy-maker in terms of choosing the optimal rate of inflation forperiod t as defined by (1131) and thus Etminus1πt = πt so that the second term dropsout and we have
πt = aα
b(1 minus k)Un = Etminus1πt gt πlowast = 0 (1132)
as long as k lt 1 We thus have that expectations are rational and individualsoptimize subject to these expectations Since Etminus1πt = πt we have that EtUt = UnThus ldquothe equilibrium solution delivers the same unemployment rate and a higherrate of inflation at each daterdquo than is the case in the SargentndashWallace problem wherethere is a ldquorules-type equilibriumrdquo Given the optimal rate of inflation πlowast = 0a rules-type equilibrium with rational expectations would have the actual rate ofinflation equal to zero
As Barro and Gordon (1983 608) conclude
under a discretionary regime the policymaker performs optimally subject toan assumed inability to commit future actions The framework assumes ratio-nality within the given institutional mode Excessive inflation apparentlyunrewarding countercyclical policy responses and reactions of monetarygrowth and inflation to other exogenous influences can be viewed as prod-ucts of rational calculation under a regime where long-term commitments areprecluded
The model stresses the importance of monetary institutions which deter-mine the underlying rules of the game A purely discretionary environmentcontrasts with regimes such as a gold standard or paper-money constitutionin which monetary growth and inflation are determined via choices amongalternative rules (the SargentWallace approach) The rule of law or equivalentcommitments about future governmental behavior are important for inflationjust as they are for other areas that are influenced by possibly shifting publicpolicies
An alternative to the ldquorule of lawrdquo is reputation As Blanchard and Fischer(1989 599) state
reputation is the most interesting and persuasive explanation of how gov-ernments avoid dynamic inconsistency Governments know that they can do
Policy 185
better than the shortsighted solution over the long run They hope by actingconsistently over long periods to build a reputation that will cause the privatesector to believe their announcements The key to the answer [to the ques-tion of whether reputation can sustain the optimal policy] is the specificationof private sector expectations of how the public reacts to broken promises
Conclusion
This chapter has presented an overview of many of the major themes and issuesfaced in macroeconomics in terms of the conduct of monetary policy The roles ofexpectations the ldquoineffectiveness propositionrdquo the modified Phillips curve rulesversus discretion time inconsistency credibility and enforcement are all dealt within this chapter Perhaps the major contribution of this chapter is that it highlightsmany of the issues and problems that real-world monetary authorities face whendeciding what course of action to take
12 Open economy
Introduction
The field of international economics can be roughly categorized as concernedwith either the real side or the finance side of international issues The ldquorealrdquoside focuses on such basic questions as why trade occurs between countries whatdetermines the terms of trade and how government policies such as tariffs do orquotas affect trade The ldquofinancerdquo side makes explicit the fact that countries differin currencies in order to focus on such questions as what determines exchangerates and how macroeconomic shocks in one country (eg a change in the supplyof money) affect its economy and the economies of the countries with which ittrades
In the discussion below we examine questions more like those considered by theldquofinancerdquo side Namely we extend simple macroeconomic analysis to an ldquoopenrdquoeconomy that is an economy that incorporates a foreign sector This analysisdiffers from traditional macroeconomic analysis of a ldquoclosedrdquo economy in thefollowing respects
1 There is trade of composite commodities and financial assets between twocountries For the moment we assume not only that the two countries producedifferentiated output but also that the financial assets issued by the firms andgovernment of one country are not perfect substitutes for the financial assetsissued by firms and government of the second country
2 The two economies are isolated in that individuals in each country can onlypurchase or sell labor services in their own labor markets
3 The two economies are differentiated in that each has its own media ofexchange This means that an individual in the domestic economy whoseeks to purchase foreign goods must exchange domestic money for foreignmoney1 Similarly an individual in the foreign country must exchange hisforeign money for the domestic money to purchase the domestic goods Suchexchanges take place in the foreign exchange market at the prevailing ldquoforeignexchange raterdquo or the price of one currency in terms of the second currencyWe will let et denote the exchange rate for domestic money For instanceif the domestic country is the USA and the foreign country is Japan then
Open economy 187
et on December 1 1989 was 1434 yen in that 1434 yen = 1 dollar Thisimplies that 1et the price of dollars in terms of yen was 0006973 dollarson December 1 1989
4 There is distinct government policy (fiscal and monetary) in each country inthat each countryrsquos government determines spending taxation and monetarypolicy for its economy
To understand how macroeconomic analysis is altered in the above ldquoopen econ-omyrdquo setting it is instructive to first consider the nature of the constraints faced bythe various participants in an open economy We then examine how the behaviorof these participants is affected by changes in such variables as exchange ratesWith that background we are ready to consider some simple examples of openmacroeconomic analysis such as the effect of a change in the money supply in thecontext of an open economy neoclassical model
Open economy participants constraints and Walrasrsquo law
To begin our task of modeling an open economy recall that it is typical ofmacroeconomic models to simplify the economy by grouping markets into broadcategories In a closed economy there were three important markets the outputfinancial and labor markets One could add to this the money ldquomarketrdquo sinceequilibrium required that the demand for money equaled supply In an open econ-omy we add another market the foreign exchange market where the currencyof one country is traded for that of the other country In a closed economy theparticipants in the various markets in the economy could be placed in one of fivecategories households firms government (fiscal side) the central bank and pri-vate depository institutions In an open economy we add one more participantforeigners
As we have seen a common theme of macroeconomic models is their emphasison the interdependencies among markets Macroeconomics recognizes that eventsin one market imply changes in other markets as well This ldquogeneral equilibriumrdquoapproach contrasts with the ldquopartial equilibriumrdquo approach of microeconomicswhich is less concerned with how changes in one market affect all other marketsTo fully understand the links across markets it is useful to specify the ldquofinancingconstraintsrdquo faced by the participants in the various markets Our discussion ofopen economy macroeconomics thus begins by introducing these financing con-straints for firms households government (fiscal side) the central bank privatedepository institutions and foreigners We then sum these constraints to obtain amodified Walrasrsquo law
The financing constraints in an open economy
We start our discussion of the constraints faced by the participants in an openeconomy by considering the financing constraint faced by the new participant for-eigners Foreigners purchase domestic output and financial assets Let X d
t denote
188 Open economy
foreignersrsquo demand for domestic output (exports) and let net Adft denote foreignersrsquo
desired real change in their holdings of domestic financial assetsTo purchase domestic output and financial assets foreigners must first acquire
domestic money That is foreigners must finance these purchases either fromincome generated from their previously acquired holdings of domestic financialassets or by supplying foreign currency in exchange for the domestic currencyin the foreign exchange markets In particular we have the following foreignerfinancing constraint
X dt + net Ad
ft minus αf (zBf pt + dt) minus FCst etpt = 0 (121)
Note that for simplicity we limit foreignersrsquo holdings of domestic financial assetsto private financial assets (ie we do not have them holding government bonds)Further we assume foreignersrsquo portfolio of holdings of domestic financial assetsis identical to2 that of domestic households and that they own αf 1 gt αf ge 0 ofthe total value of financial assets issued by domestic firms Thus αf (zBf pt + dt)
is the real income foreigners gain from their holdings of domestic financial assets3
In equation (121) the term FCst etpt denotes foreignersrsquo real supply of foreign
currency in the foreign exchange market FCst is foreignersrsquo supply in units of the
foreign currency in period t Multiplying by 1et the price of the foreign currencyin terms of the domestic currency puts this in terms of the domestic currencyDividing by the price level pt then puts it in real terms (ie in terms of the domesticcomposite commodity) Embedded in the desired change in foreignersrsquo holdingsof domestic financial assets are changes in the desired holdings by foreign centralbanks (ie changes in foreign central banks ldquointernational reservesrdquo)
In addition to the above new constraint that accompanies the introduction ofa new participant to the economy foreigners we have constraints faced by thedomestic households the central bank private depository institutions government(fiscal side) and firms In the case of households and the central bank we modifythe constraints introduced in previous chapters to incorporate the exchanges ofcommodities and financial assets with foreigners In particular for householdsthe budget constraint in an open economy can be expressed by
bdt + cd
t + zdt + (M d
t minus M )pt + net Adht + net AFd
ht minus [yt minus αf (zBf pt + dt)
+ α(zBff pft + pftdft)etpt minus δK minus Tnt] = 0 (122)
where the new term zdt denotes real imports net Ad
ht denotes householdsrsquo desiredchange in their real holdings of foreign assets and α(zBff pft + pftdft) denotes theincome (in terms of the foreign currency) gained from holding the proportion aof the financial assets issued by foreign firms4 This income (in foreign currency)times the price of foreign currency in terms of domestic currency (1et) gives thedomestic currency value of income from foreign asset holdings Dividing by theprice level pt puts the income in terms of the composite commodity (ie in realterms)
Open economy 189
Total consumption of commodities in period t is now the sum of cdt purchase
of output and zdt imports of commodities produced abroad5 Similarly the total
desired change in financial assets is the sum of net Adht the desired change in hold-
ings of domestic financial assets and net AFdht the desired change in holdings of
foreign financial assets Note that in incorporating the firm distribution constraintinto the household budget constraint we have taken into account the fact that notall domestic output yt is income to domestic households since foreigners own theshare αf of domestic firms On the other hand households have an additionalsource of income from the ownership of foreign financial assets
For the central bank the (stock) financing constraint equates the sum of thechange in real (domestic) financial asset holdings and international assets to thereal change in the monetary base or
net Adct + net AFd
ct minus (MBst minus MB)pt = 0 (123)
where net Adct = pbt(Bd
gct minus Bgc)pt MB = R + C and MBst = Rs
t + Cst 6 We
have added the term net AFdct to denote the real demand for additional international
(foreign currency denominated) assets by the central bank In particular
net AFdct = pfbt(B
dfct minus Bfc)etpt
The quantity pfbt(Bdfct minusBfc) is the change in the amount of foreign assets demanded
by the central bank in terms of the foreign currency pfbt denotes the price of foreignbonds in terms of foreign currency and the numbers of such bonds demanded andinitially held by the domestic central bank are denoted respectively by Bd
fct and
Bfc Multiplying this quantity by the price of foreign currency in domestic currencyterms (1et) gives the domestic currency value of foreign assets demanded by thecentral bank Then dividing by the price level pt puts the net demand for interna-tional assets by the central bank net AFd
ct in terms of the composite commodity(ie in real terms)
For private depository institutions the (stock) financing constraint indicates thatprivate depository institutions can be viewed as financing additions to reserves andto financial assets holdings by creating deposits or
net Atpb + (Rd
t minus R)pt minus (Dst minus D)pt = 0 (124)
where net reserves demanded or initially held are denoted by Rdt and R respec-
tively and checkable deposits supplied or initially outstanding are denoted byDs
t and D respectively For simplicity we assume that all private demands forforeign financial assets as well as for foreign commodities are captured in thehousehold budget constraint To the extent that private banks purchase foreignfinancial assets they can be viewed as acting as financial intermediaries forhouseholds
190 Open economy
For the domestic government (fiscal side) the financing constraint is
gdct minus T lowast
nt minus net Asgt = 0 (125)
where T lowastnt = Tt minustrt minusz(Bgh+Bgp)pt tr denotes transfer payments and net As
gt =Pbt(Bs
gt minusBg)pt Equation (125) incorporates the ldquoflowrdquo financing constraint forthe central bank In doing so it is assumed that the central bank claims on realresources just exhaust its interest payments on government debt holdings plus anyincome associated with central bank holdings of foreign assets Finally for firmsthe financing constraint on capital purchases is given by
Idnt + ψ(I d
nt) minus net Asft = 0 (126)
Note that we assume for simplicity that neither firms nor the government purchaseforeign commodities
Walrasrsquo law and the balance of payments
We now sum the constraints faced by the six participants in an open economy(foreigners households the central bank private banks the government andfirms) as given by equations (121)ndash(126) In doing so assume equilibrium withrespect to the demand and supply of bank reserves (ie the Rs
t part of MBst in the
central bank constraint equals Rdt in the private banks constraint) and note that
the initial monetary base MB equals R + C that the money supply is defined byM s
t = Dst + Cs
t and that the initial money supply is given by M = D + C Weobtain
[Bdt + Cd
t + X dt + gd
ct + δK + I dnt + ψ(I d
nt) minus yt] + [M dt pt minus Mpt]
+ [net Adht + net Ad
ft + net Adct + net Ad
pt minus net Asgt minus net As
ft]+ [zdlowast
t + net AFdht + net AFd
ct minus FCst etpt] = 0 (127)
where zdlowastt + net AFd
ht + net AFdct minus FCs
t etpt denotes householdsrsquo imports notfinanced out of income generated from foreign asset holdings Equation (127) isan example of the modified Walrasrsquo law for an open economy7 As we discussbelow (127) can be viewed as stating that the sum of excess demands in fourmarkets must equal zero
The first term in (127) reflects excess demand in the output market where thedemand now includes foreignersrsquo demand for domestic output (exports) Note thatby adding and subtracting import demand we could express the excess demandfor output in the form
bdt + cdlowast
t + (xdt minus zd
t ) + gdct + δK + I d
nt + ψ(I dnt) minus yt (128)
where the term cdlowastt denotes householdsrsquo total consumption in terms of both domes-
tic and foreign output (ie cdlowastt = cd
t + zdt ) Excess demand in the output market
Open economy 191
is often expressed this way with the consumption term denoting total householdconsumption such that the ldquonetrdquo export demand term (xd
t minus zdt ) appears
Setting the excess demand for output term in (127) to zero gives us the ISequation in an open economy The second term in (127) reflects the excess demandfor money Note that we assume that only domestic households desire to hold thedomestic money (foreigners seek the domestic money in the foreign exchangemarket not to hold but as a means to purchase the domestic output or financialassets) Setting this second term in (127) to zero gives us the standard LM equation
The third term in (127) reflects excess demand in the financial market In goingto an open economy we add to the demand side of the financial market a demandfor domestic financial assets by foreigners (which could include private foreignagents as well as foreign central banks) In addition note that householdsrsquo demandfor domestic financial assets (net Ad
ht) no longer reflects householdsrsquo total demandfor financial assets since they now have the option of purchasing foreign financialassets (net AFd
t )The fourth and final term in (127) reflects excess real demand for foreign
currency in the foreign exchange market8 As before we can view output ( yt)or the price of output ( pt) as determined in the output market and the domesticinterest rate (rt) as determined in the financial market Now we can view the foreignexchange rate (et) as determined in the foreign exchange market The demand forforeign currency reflects zdlowast
t the demand associated with householdsrsquo purchases offoreign commodities that could not be financed from the foreign currency earningsof their holdings of foreign financial assets net AFd
ht the demand for foreigncurrency associated with householdsrsquo purchases of additional foreign financialassets and net AFd
ct the demand associated with the central bankrsquos desired changein international reserves Note that the real demand for the foreign currency couldinstead be stated as the real supply of the domestic currency in the foreign exchangemarket
FCst etpt denotes the real supply of foreign currency or real demand for the
domestic currency in the foreign exchange market which from (121) reflectsforeignersrsquo purchases of financial assets and exports net of those financed throughdomestic currency earnings on financial assets held by foreigners Foreignersrsquopurchases of financial assets include both foreign private (ie foreign householdsrsquo)and foreign public (ie foreign central bank) purchases
The balance of payments accounts
Let us assume for now that the domestic economy discussed above is the USA TheUS Department of Commerce actually measures the various sources of the demandfor and supply of dollars in the foreign exchange markets cited above These data ofinternational transactions are presented as the US balance of payments accountsTable 121 summarizes its major components Transactions are categorized aseither sources of the real demand for or the real supply of dollars in the foreignexchange market for the US dollar9 Equivalently they reflect the real supply ofor real demand for foreign currency in the foreign exchange market
192 Open economy
Table 121 The US balance of payments accounts (in ldquorealrdquo terms)
Real demand for dollars Real supply of dollars
1 US exports of goods and services xt US imports of goods and services zt2 Transfers (interest and dividends to US
holders of foreign financial assetsgovernment grants and gifts)
Transfers (interest and dividends toforeign holders of US financial assetsgovernment grants and gifts)
α(zBff pft + pftdft)etpt αf (zBf pt + dt)
3 Foreign purchases of US financialassets (capital inflow) net Ad
ft
US purchases of foreign financial assets(capital outflow) net AFd
h + net AFdct
The net demand for dollars associated with the first component of the balanceof payments accounts (exports minus imports) is called the balance of trade Ifthe balance of trade is negative as it has been recently for the USA then the USAis said to experience a balance of trade deficit If it is positive as was the case for106 consecutive years from the end of the Civil War to 1971 as well as throughmost of the 1970s then a balance of trade surplus is said to exist
The third component of the demand for and supply of dollars in the balance ofpayments accounts is the capital account This capital account can be divided intoa private part and a public part On the demand side the private part measures thereal dollars demanded by foreigners other than foreign central banks to financepurchases of US financial assets On the supply side the private part measuresthe real dollars supplied by US households to buy foreign financial assets Thesecurrency exchanges are called private international capital flows Private inter-national capital flows associated with the demand for dollars are referred to asUS private international capital inflows since they reflect the inflow of foreigncurrency due to private foreignersrsquo purchases of US financial assets Private inter-national capital flows associated with the supply of dollars are referred to as USprivate international capital outflows since they reflect the outflow of dollars dueto US householdsrsquo purchases of foreign financial assets
Summing the net demand (demand minus supply) for dollars associated withthe first two components plus the private international capital flows and adjustingfor measurement errors (the discrepancy term) we obtain what is called the USbalance of payments10 In years in which the balance of payments is negative it isreferred to as a balance of payments deficit On the other hand a positive balanceof payments is called a balance of payments surplus
When there is a surplus or deficit in the balance of payments accounts thenequality between the real demand for and real supply of dollars is brought about byan offsetting deficit or surplus on what is known as ldquothe official reserve transactionbalancerdquo The official reserve transaction balance reflects the intervention into theforeign exchange market by the US central bank (the Fed) andor by foreign centralbanks This is the ldquopublicrdquo part of international capital flows Whenever there existsa balance of payments deficit in the USA then (on net) central banks demand USdollars in the foreign exchange markets If this were solely the US central bank
Open economy 193
intervening this means that net AFdct wasis negative (the Fed wasis a net supplier
of dollars in the foreign exchange market) and the US central bank would havelostlose international reserves
Behavior in an open economy
With an understanding of the constraints faced by the various participants in anopen economy we now consider the behavior of these participants in particularthe determinants of imports (zd
t ) exports (xdt ) private international capital inflows
(a part of net Adft) and private international capital outflows (net AFd
ht) In the firstpart of this section we examine how a change in the foreign exchange rate for thedomestic currency (to be concrete the US dollar) can alter the relative price offoreign goods and thus lead to a change in the division of household consumptionbetween purchases of domestic goods and purchases of foreign goods11 We alsoexamine other factors that influence US imports of goods and services and thusthe real demand for foreign currency (real supply of US dollars) in the foreignexchange markets
In the second part of this section we consider the factors that influence howhouseholds divide their real accumulation of financial assets between US stocksand bonds and the financial assets of foreign countries As we saw above house-hold purchases of foreign financial assets constitute capital outflows since inorder to make these purchases households must demand foreign currency (supplydollars) in the foreign exchange market We will see how differences in foreignand domestic interest rates and the expected appreciation or depreciation of theUS dollar determine the relative returns on foreign and domestic financial assetsThese relative returns in turn affect householdsrsquo portfolio choices between foreignand domestic financial assets and the real demand for foreign currency (real supplyof dollars) in the foreign exchange market
In the third and fourth parts of this section we turn to the other side of theforeign exchange market to examine determinants of foreignersrsquo purchases of USgoods and services and of US financial assets In particular we consider how suchfactors as exchange rates affect foreignersrsquo demand for US goods (US exports)and how such factors as relative interest rates and the expected rate of changein the exchange rate affect foreignersrsquo demand for US financial assets We finishthis section by illustrating graphically the behavior discussed in terms of the realdemand for and supply of the domestic currency (the US dollar)
Householdsrsquo demand for imports
When deciding whether to purchase foreign or domestic goods households look attheir relative prices To be concrete consider two countries The domestic countryis the USA and the foreign country is Japan The relative price of Japanese goods isthen the real quantity of US goods that must be sacrificed to purchase the foreigngood For example if the price of a Japanese car is $6000 and the price of aUS computer is $1500 then the relative price of a Japanese car in terms of US
194 Open economy
computers is 4 computers If the relative price of Japanese cars rises the USAwill import fewer Japanese cars The relative price of foreign goods sometimesreferred to as the terms of trade is an important determinant of the quantity ofimports The relative prices of foreign goods depend on the dollar prices of USgoods the prices of foreign goods in their own currency and foreign exchangerates (the price of one currency in terms of a second currency) The simple examplethat follows illustrates this point Suppose that the price of a US computer is $1500and that in Japan the price of a Japanese car is 600000 yen The third ldquopricerdquo thatwe need to know in order to compute the relative price of a Japanese car in terms ofUS computers is the foreign exchange rate in particular the price of a yen in termsof dollars Suppose that it takes et yen to buy one dollar in the foreign exchangemarkets Then it takes 1et dollars to buy one yen Returning to our example ofJapanese cars and US computers if et = 100 yen per dollar and the Japanese carhad a yen price of 600000 then its dollar price would be 6000
In general the calculation of the relative price of Japanese goods is thus
Relative price of Japanese goods (in terms of US goods)
= Yen price of Japanese goods ( pft)
times Price of yen in terms of dollars (1et)Dollar price of US goods ( pt)
= pft
etpt
According to this expression a rise in the yen price of Japanese goods ( pft)
raises their relative price Similarly a fall in the dollar price of US commodities( pt) raises the relative price of Japanese goods Finally a rise in the price of yen interms of dollars (1et) also increases the relative price of Japanese goods12 Sinceall three changes mean a higher relative price of Japanese goods and thus anincrease in the cost to US buyers of Japanese goods in terms of US goods forgoneall three changes reduce US imports of Japanese goods The above relative priceexpression for a basket of foreign goods is sometimes termed the real exchangerate for foreign goods ndash that is
Real exchange rates (foreign goods)
= Dollar price of foreign goods
Dollar price of US goods= pft(1et)
pt= pft
ptet
While a rise in the relative price of foreign goods will lead to a reduction in thequantity of foreign goods bought we have to be careful not to infer from this thatUS import demand will necessarily be inversely related to the relative or real priceof imports The reason for this is that our measure of US real import demand is interms of the US good and services not in terms of the foreign good An examplewill highlight this distinction
Open economy 195
Suppose that the dollar depreciates (et falls) With the implied appreciation offoreign currency (1et rises) the price of the foreign good in terms of US goodsbecomes greater as our expression for the real exchange rate for foreign goods( pftpt et) indicates With a higher price fewer foreign goods will be purchased ndashthis is clear This by itself would suggest a fall in the value of imports into theUSA But the higher price also means that each foreign good purchased will costmore in terms of US goods that must be sacrificed This by itself would suggest arise in the value of US imports (measured in terms of US goods that must be paidto obtain the imports) The net impact of a change in the relative price of foreigngoods on US imports measured in terms of US goods depends on which of thesetwo effects is stronger
It is typically assumed that over time the effect of a change in the relative pricesof imports on the quantity of imports purchased dominates so that the value ofimports in terms of US goods will fall with a rise in the relative price of importsThis is what we will assume13 Formally this condition requires that the priceelasticity of demand for foreign goods be greater than one That is a 1 percentincrease in the price of foreign goods must cause a greater than 1 percent reductionin the amount of foreign goods that US households demand14
Besides the relative prices of imports real disposable income affects US importdemand An increase in disposable income can lead to a rise in household con-sumption demand In an open economy with foreign trade a rise in consumptiondemand means an increase in purchases not only of domestically produced goodsbut also of foreign goods Thus householdsrsquo import demand is directly related totheir disposable income To summarize household real import demand zd
t dependsinversely on the relative price of foreign goods pftetpt and directly on disposableincome yt minus δK minus Tnt
zdt = zd
t ( pftetpt yt minus δK minus Tnt ) (129)
Capital outflows householdsrsquo demand for foreign financial assets
When deciding whether to purchase US or foreign financial assets householdscompare domestic and foreign rates of return The nominal rate of return on USfinancial assets is simply the money interest rate rt The comparable nominal rateof return on foreign financial assets is not so simple to identify To explain howto compute this return which we will denote rlowast
t let us suppose that a householdlends one dollar in the foreign financial market
If the price of a dollar is et units of the foreign currency say yen then in termsof the foreign currency the household lends et yen If foreign financial assets offerthe interest rate rft then one period from now the household will have et(1 + rft)
yen At that time the household can convert these yen holdings back to dollars atthe exchange rate then existing At the time the money is lent this future exchangerate may be uncertain15 Let householdsrsquo expectation of this future exchange ratebe denoted by ee
t+116 Then the household expects to convert its et(1 + rft) yennext year into et(1 + rft)ee
t+1 dollars Subtracting the one dollar with which the
196 Open economy
household started the rate of return to lending in foreign financial markets rlowast isgiven by
rlowastt = (et(1 + rft)ee
t+1) minus 1 = [et(1 + rft) minus eet+1]ee
t+1
We can simplify the above equation by noting that the expected future dollarexchange rate ee
t+1 yen per dollar equals the current exchange rate of et yen times(1 + θe
t+1) where θet+1 is the expected rate of change in the price of a dollar in
terms of yen Substituting the expression et(1+θet+1) for the expected future dollar
exchange rate eet+1 into the above expression for rlowast
t we have
rlowastt = [et(1 + rft) minus et(1 + θe
t+1)]et(1 + θet+1) = (rft minus θe
t+1)(1 + θet+1)
Since the expected rate of appreciation of a dollar is typically small we canapproximate the above expression by
rlowastt = rft minus θe
t+1 (1210)
Equation (1210) has a straightforward interpretation The return to lending in theforeign financial market equals the difference between the foreign interest rateand the expected rate of change in the price of the dollar The return to lendingin foreign financial markets increases with a higher foreign interest rate rft anddecreases with a higher expected rate of increase in the price of the dollar (θe
t+1)The higher the expected rate of increase in the price of the dollar the lower theexpected return to lending in foreign financial markets since for a given numberof dollars sold for foreign currency at the start of the period fewer dollars can bebought back at the end of the period
When households choose between purchasing domestic and foreign financialassets they compare the domestic interest rate rt with the rate of return to lendingin the foreign financial markets rlowast
t We can thus express household real demandfor additional foreign financial assets as
net AFdht = net AFd
ht(rt rft minus θet+1 ) (1211)
In (1211) an increase in the US interest rate or a fall in the rate of return onforeign financial assets implies a reduction in householdsrsquo real demand for foreignfinancial assets The three dots in (1211) reflect other factors that have been leftunspecified For instance changes in the political stability of foreign governmentsare one unspecified factor that would likely impact on US householdsrsquo demandfor foreign financial assets Equation (1211) suggests that we should add anotherfacet to our previous discussion on householdsrsquo demand for US financial assetsnet Ad
ht In addition to such factors as real income taxes the US money interestrate and the expected rate of inflation household demand for US financial assetsdepends on the expected return to lending abroad rft minus θe
t+1
Open economy 197
Foreignersrsquo demand for exports
Just as US demand for foreign goods depends on relative prices so too isforeignersrsquo demand for US goods based on the relative prices of those goodsThe relative prices of US goods to foreigners measure what foreigners have togive up of their own goods in order to purchase US goods As we have seenthese relative prices depend on the money prices of US goods the money pricesof foreign goods and the exchange rate Considering ldquocompositerdquo goods for eachcountry the relative price of US goods to foreigners or the real exchange rate forforeign goods is given by
Real exchange rates (US goods)
= Dollar price of US goods
Dollar price of foreign goods= pt
pft(1et)= ptet
pft
Note that the real exchange rate for US goods is simply the inverse of the previouslyobtained real exchange rate for foreign goods The price of a dollar in terms ofan index of foreign currencies rose by 70 percent in 1984 and 1985 The resultingrise in the relative prices of US goods contributed significantly to a reduction inforeign demand for US goods and the large US trade deficit of the mid-1980sSimilarly the dramatic fall in the price of a dollar in the subsequent period fromlate 1985 to 1988 led to an increase in US exports Thus we have
xdt = xd
t ( ptetpft ) (1212)
Equation (1212) indicates that export demand falls with a rise in the relative priceof US goods to foreigners
We know from our previous discussion that an increase in US disposable incomeleads to a rise in household purchases of both domestically produced output andforeign goods and services By the same token an increase in foreignersrsquo dispos-able income leads to a rise in their purchases of US goods Thus among the itemsmissing in (1212) that determine foreignersrsquo real export demand xd
t is foreigndisposable income
Capital inflows foreignersrsquo demand for financial assets
In 1960 purchases of US financial assets by foreigners were approximately one-half the amount of purchases of foreign financial assets by US citizens In the USfinancial markets foreign purchases of new US financial assets were less than5 percent of household and depository institution purchases Twenty-five yearslater foreigners were purchasing four times as many US financial assets than theUSA was purchasing abroad In the US financial market close to 30 percent ofnew US financial assets were being purchased by foreigners This dramatic changein capital inflows to the USA is one indication of the growing importance to theUS economy of international trade not only in goods but also in financial assets
198 Open economy
As with US households we will assume that foreigners decide to purchase eitherUS assets or financial assets of their own country by comparing the rates of returnon the two types of financial assets For foreigners the nominal rate of return ondomestic assets is the money interest rate in their own country (rft) The expectedrate of return to foreigners on US financial assets equals the US interest rate plusthe expected change in the price of the dollar in the foreign exchange market (iert + θe
t+1)Not surprisingly the expected return to foreigners lending in US financial mar-
kets increases when the US interest rate rt increases Not as obvious is that thereturn also increases with an increase in the expected rate of change in the price ofthe dollar θe
t+1 This is because foreigners lending in US financial markets converttheir currency to dollars to make the loans When the loans are repaid they thenconvert dollars back to their own currency If the dollar is anticipated to appreciateduring the course of the year then part of their expected return to lending in theUSA is the increase in the value of the dollars (in terms of their own currency)
Summarizing we can express the foreign demand for US financial assetsnet Ad
ft as
net Adft = net Ad
ft(rft rt θet+1 ) (1213)
Equation (1213) indicates that foreignersrsquo demand for US financial assetsincreases if the US interest rate (rt) rises or the expected rate of change in theprice of the dollar (θe
t+1) increases or the foreign interest rate (rft) falls
The foreign exchange market the real demand and supply of dollars
The change in the price of a dollar (in terms of a second currency) affects the realquantity of dollars supplied and demanded in the foreign exchange market Letus start with the price of a dollar set at the equilibrium level of (et)0 let us say100 yen If the price of a dollar now falls to (et)1 say 50 yen then this depreciationof the dollar (appreciation of the yen) leads to a reduction in the real quantity ofdollars supplied from Q0 to Q1
A fall in the price of a dollar from 100 to 50 yen means a rise in the dollar priceof a yen from 001 to 002 dollars or from 1 cent to 2 cents Even though there isno increase in the yen price of Japanese goods the dollar price of Japanese goodsrises For example a 600000 yen Japanese car that formerly cost 6000 dollars(600000 times 001) now costs 12000 dollars (600000 times 002) If the dollar prices ofUS goods have not changed then the relative or real prices of Japanese cars haverisen In our example this means that US households must give up an increasedamount of US goods to obtain one more Japanese car
The depreciation of the dollar and resulting higher relative price for Japanesegoods leads US households to reduce the quantity of Japanese goods demandedHowever as we discussed above the fact that the quantity of Japanese goodspurchased falls does not necessarily mean that the quantity of dollars suppliedin the foreign exchange market also falls There are two countervailing forces
Open economy 199
at work here While the purchase of fewer Japanese goods would reduce thequantity of dollars supplied the fact that each Japanese good has a higher pricewould increase the quantity of dollars supplied By assuming that the first effectoutweighs the second effect we conclude that a depreciation of the dollar causesthe real quantity of dollars supplied in the foreign exchange market to fall Thusthere is an upward-sloping supply of dollars curve17
Naturally behind the supply of dollars in the foreign exchange market is notonly US real import demand but also the real demand by households and the UScentral bank for additional foreign financial assets (net AFd
ht + net AFdct) As we
saw above this sum of the US import demand and demand for foreign financialassets can be interpreted as representing not only a supply of dollars but also ademand for foreign currency18
The above discussion also highlights the effect of a change in the price ofa dollar (in terms of a second currency) on the quantity of dollars demandedin the foreign exchange market The depreciation of the dollar (or appreciationof the yen) from (et)0 to (et)1 leads to an increase in the quantity of dol-lars demanded US in real terms The fall in the price increases the quantity of dollarsdemanded because it lowers the relative price of US goods to foreigners A fall inthe price of a dollar from 100 yen to 50 yen causes a rise in the dollar price of theyen from 1 cent to 2 cents Even though there has been no increase in the dollarprice of US goods the yen price of US goods falls and the Japanese increase theirdemand for US goods As a consequence the real quantity of dollars demandedin the foreign exchange market increases
Naturally behind the demand for dollars in the foreign exchange market isnot only foreignersrsquo export demand but also their demand for US financial assets(net AFd
ht) As we have seen this sum of the export demand and foreignersrsquo demandfor US financial assets can be interpreted as representing not only a demand fordollars but also a supply of foreign currency
Simple examples of open economy (static) macroeconomicanalysis
The statement of Walrasrsquo law for an open economy indicates that for static macro-economic analysis of an open economy we need look at only three of the fourexcess demand conditions reflecting the output financial foreign exchange andmoney markets Standard practice is to look at the output money and foreignexchange markets In the neoclassical model these three equations would besolved to obtain the equilibrium price level interest rate and exchange rate ( pt rt and et respectively) In the BarrondashGrossman disequilibrium analysis thesethree equilibrium conditions could be solved to obtain the equilibrium outputinterest rate and exchange rate ( yt rt and et respectively) In the Lucas modelor the Keynesian fixed money wage model these three equilibrium conditionsalong with the appropriate aggregate supply equation could be solved for theequilibrium price level output interest rate and exchange rate ( pt yt rt and et respectively)
200 Open economy
With flexible exchange rates (ie exchange rates determined without theintervention of central banks) the basic change in the macroeconomic analysisis to recognize that the net export component of output demand (seeequation (128)) is sensitive to interest rate changes which implies the IS curve(in (interest rate output) space) is flatter The reason why a higher US interest ratereduces net export demand is that a higher interest rate increases capital inflows(net Ad
ft) and reduces capital outflows (net AFdht) The first change increases the
demand for dollars in the foreign exchange market while the second reduces thesupply of dollars in the foreign exchange market The result is an appreciationof the dollar that reduces exports and increases imports Note that with flexibleexchange rates a change in either the price level or income does not affect netexport demand
Money supply changes in the neoclassical model purchasingpower parity
In general when considering two countries the country with the lower inflationrate will tend to have an exchange rate that is appreciating at a rate approximatelyequal to the difference in inflation rates between the two countries This patternof changes in foreign exchange rates is sometimes said to reflect the purchasingpower parity condition The condition of purchasing power parity means that thepurchasing power of each countryrsquos currency is the same whether the currency isused to purchase domestic goods or foreign goods Purchasing power parity existsfor a monetary shock in the neoclassical model since a money supply change willlead to changes not only in domestic prices but also in foreign exchange rates suchthat relative prices remain constant
Interest-rate parity
There are of course a number of variations to the above analysis For instancewe could assume that domestic and foreign financial assets are perfect substitutesand that the domestic country is sufficiently small in its interactions with foreignfinancial markets that it takes the foreign interest rate rft as a given In this caseof ldquointerest rate parityrdquo since the return to lending at home rt must equal that oflending abroad rlowast
t = rft minus θet+1 we have that
rft = rt + θet+1 (a constant) (1214)
One use of (1214) is in the well-known Dornbusch model (see Dornbusch 1976)Assume that output and the price of output are fixed initially Assume θe
t+1 = 0initially such that rt = rft Now consider an increase in the money supply Tomaintain equilibrium with respect to the demand and supply of money the interestrate rt must decrease According to (1214) the resulting increase in internationalcapital outflows and reduction in international capital inflows must reduce theexchange rate such that it is expected to appreciate at a rate equal to the difference
Open economy 201
between xdt and the new lower rt This is Dornbuschrsquos famous ldquoexchange rate
overshootingrdquo That is the analysis implies a fall in the exchange rate below whatit will ultimately be after income or prices adjust
A second use of (1214) is to combine it with the assumption that exchangerates are fixed (through appropriate central bank intervention in the foreignexchange markets) An example taking this approach is the MundellndashFlemingmodel (Fleming 1962 Mundell 1968)
Conclusion
This chapter has brought together many of the issues associated with analyzinga macroeconomy that participates in the open or global economy In keepingwith the basic model framework of this book the household is now viewed ashaving a demand for imported products as well as domestically produced goodsand services Additionally firms are allowed to sell to agents in foreign coun-tries This flow of goods and services across borders logically leads to a flow offunds Moreover this interconnectedness among trading partners implies that theactions of one country in particular those of the monetary authority may influenceconditions in the other country
Notes
1 Introduction
1 Empirically an economyrsquos total output is measured by real gross domestic product (orindustrial production) employment is estimated from company records or householdsurveys estimates of unemployment are compiled from household surveys or statisticson recipients of unemployment benefits and price indexes are computed to measurechanges in the overall level of prices
2 Leon Walras first derived the result in his book Elements drsquoEconomie Politique Purefirst published in 1874ndash77 (see Walras 1954)
3 An endogenous variable is one that is determined by the model An exogenous variableis one that is taken as given by the model
4 An element that distinguishes both static and dynamic macroeconomic analysis fromArrowndashDebreu analysis is the incorporation of money The exceptions to this in macro-economic analysis are real business cycle theories which for the most part are purelyldquorealrdquo in the sense that money is not an intrinsic part of the analysis
5 Note that an alternative to exogenous expectations is to specify an ldquoexpectation func-tionrdquo If this function relates expectations in certain specific ways to all currentinformation such that the expectation function reflects ldquorational expectationsrdquo thenstatic analysis is converted to dynamic analysis
6 In dynamic models this would be termed ldquoperfect foresightrdquo in a deterministic settingand ldquorational expectationsrdquo in a stochastic setting
7 One could say that the effect of such a variable is implicit in the exogenous level ofexpected future prices of the static analysis
8 A similar example of an incomplete listing of effects is a change in current governmentpolicies A change in current government actions may require changes in governmentactions in subsequent periods that will impact future markets Yet the construction ofstatic analysis does not require such effects to be spelled out
9 Forward markets are markets in which agreements are made that specify the prices atwhich goods will be exchanged in the future Goods are thus in essence indexed by thedate of trade
10 This is the ArrowndashDebreu contingent-claim interpretation of a competition equilibriummodel (see Arrow 1964 Debreu 1959)
11 In some cases one can attain the stationary state under the less restrictive requirementthat certain exogenous variables simply grow at a steady rate over time
12 While classical economists did not fully articulate their model Patinkin (1965) is widelycited as providing a comprehensive review and formalization of many of the key ideasunderlying the classical economistsrsquo views The result may be denoted the prototypeof the neoclassical (static) model Dynamic neoclassical macroeconomic models arebroadly based on neoclassical growth models with the addition of shocks of variouskinds Among the classic works developing neoclassical growth models are Solow(1956) Cass (1965) Koopmans (1965) and Sidrauski (1967a 1967b)
Notes 203
13 Two of the most widely known proponents of new classical economics are Robert Lucasand Thomas Sargent It has been suggested by some however that this line of analysisis less the extension of classical analysis than it is the antithesis of Keynesian analysisas discussed below See Niehans (1987) for this interpretation
14 This is changing as indicated by Howitt (1985) Shapiro and Stiglitz (1984) Weitzman(1985) and by the papers cited in the May 1988 American Economic Review
15 Howitt goes on to say that there is the ldquoreciprocal Keynesian question of how exactlythe economic system manages to overcome all the obvious coordination problems thatstand in the way of attaining the state of equilibrium common to new classical economicsmodelsrdquo
2 Walrasian economy
1 A brief description of the theory is given in Patinkin (1965 note B)2 In reality of course there is no auctioneer In fact some think that Walras would have
been the first to question such a description of the economy as realistically capturingthe true dynamic process by which the economy reaches the equilibrium described bysupply and demand curves See Walker (1987) who points out that Walras promoted aldquodisequilibrium-production model of tatonnementrdquo as more representative of Walrasrsquothought than the tatonnement model with an ldquoauctioneerrdquo
3 We use the letter T to denote the number of commodities because later we distinguishthe T commodities according to time of availability (ie from period 1 to period T )
4 One could replace the assumption of known prices by the assumption that individualsform expectations at time t about prices at time t and that such expectations are correctThis has been referred to as a situation in which expectations satisfy the assumption ofldquoweak consistencyrdquo
5 With zero transactions costs equality of purchase and sale prices could be viewed asforced by arbitrage conditions
6 Walras was one of the first to note this point Note that πjj = 17 See Debreu (1959 Chapter 2) for a discussion of such accounting prices8 A distinctive feature of macroeconomics is to alter traditional ArrowndashDebreu general
equilibrium analysis in such a way that we can reinterpret accounting prices as moneyprices and determine the level of money prices
9 Others characterize transactions costs in similar fashion For instance Alchian and Dem-setz (1972) cite the costs of ldquoformingrdquo ldquonegotiatingrdquo and ldquoenforcing contractsrdquo whileDahlman (1979) writes of ldquosearch and information costsrdquo ldquobargaining and decisioncostsrdquo and ldquopolicing and enforcingrdquo costs
10 See Varian (1992) for a discussion on these points By ldquowell behavedrdquo we mean thatpartuapartcai gt 0 and ua is strictly quasi-concave
11 In this case the Lagrangian is written ignoring the non-negativity constraints onconsumption of commodity i i = 1 T Below we provide an equivalent character-ization of the constrained maximization problem that incorporates these constraints inthe Lagrangian
12 A function y = f (x1 xn xn+1 xm) is said to be homogeneous of degree k inthe arguments x1 through xn if
f (λx1 λxn xn+1 xm) = λk f (x1 xn xn+1 xm)
3 Firms as market participants
1 With zero ldquoadjustment costsrdquo there could exist a market for capital at time t such thatthe capital employed during the initial period Kt is conceptually distinct from capitalinherited K In this case however equilibrium in the capital market at time t wouldthen imply that Kt = K
204 Notes
2 Note that output is a ldquoflow variablerdquo That is for a period i of length h the outputproduced is given by hyi The term yi is thus the ldquorate of outputrdquo over period i Sinceoutput is a flow variable at a point in time the rate of production is not defined (Asyou can see for any rate of output the limit of production as the length of the periodgoes to zero is zero)
3 In general for a period i of length h labor services are denoted by hNi such that totalwage payments at the end of the period are wihNi Like the labor input one shouldthink of the capital input in flow terms with the stock of capital determining the rateor flow of capital services
4 For simplicity of notation these expressions presume perfect foresight5 In general for a period i of length h the nominal interest rate from the end of that period
to the end of the next period is given by hri = hzpbi + (pbi+h minus pbi)pbi 6 For simplicity of notation these expressions assume perfect foresight with respect to
future prices of equity shares In addition the expressions presume future dividends areknown
7 To minimize notational clutter we have chosen to not denote such anticipations withthe expectation operator We do presume that agents (firms and households) have com-mon expectations (assumed to be held with subjective certainty) concerning plans withrespect to the issuing of equity shares
8 For instance if ri lt rei for period i there would be zero demand for bonds Theresulting excess supply of bonds would lead to a fall in the price of bonds and thus arise in the return on bonds until equality across rates of return held
9 We assume for the moment perfect foresight at time t with respect to prices at the endof the period (at time t + 1)
10 As in the prior models including money balances in the utility function reflectshow money can save (leisure) time required to make exchanges within a period Forsimplicity we limit money holdings to agents labelled ldquohouseholdsrdquo
11 For the representative household holdings of bonds issued by other households mustbe zero so that there is no real indebtedness effect with respect to the bonds exchangedamong representative households
12 In general if we assume there are ni firms producing commodity i and that there are mdifferent commodities then
yt =msum
i=1
⎡⎣ nisum
j=1
( pip)fij(Nijt Kijt)
⎤⎦
The above is a stylized view of how the empirical counterpart to total output real grossdomestic product is actually computed by the Commerce Department
13 Note that it is assumed that the capital stock K generates a fixed rate of capital servicesThat is we do not consider variation in the ldquoutilization of capitalrdquo If we did so thenvariation in the services flowing from the capital stock could be considered an additionalchoice variable with capital utilization presumably affecting the extent of depreciationin the capital stock over time
14 Fama and Miller (1972) cite conditions under which the ldquoowners of the firmrdquo iethe holders of the S equity shares will direct the managers of the firm at time t tomaximize Vt
15 In continuous time
petminus1 =int infin
t[psdsSs]eminusr(sminust)ds
where the interest rate r in the above expression is formally rs
Notes 205
16 Note that Stminus1 equiv S Assuming ri = rt i = t + 1 we have
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[pt+k dt+kSt+kminus1](1 + rt)k
⎤⎦
17 If the change is negative net revenues available for distribution as dividends would bereduced
18 We will explain more fully below the nature of these ldquoadjustment costsrdquo19 As suggested earlier if the utilization of capital were viewed as a choice variable then
it would be natural to have δ directly related to utilization of capital during the period20 Later we will say more about investment and the nature of adjustment costs21 Note that this discussion assumes for simplicity zero adjustment costs22 We follow Sargent and implicitly assume adjustment costs are related to net not gross
investment Net investment measures the change in the capital stock As we will seelater the result is a slightly different expression for Tobinrsquos Q than found elsewherewhen adjustment costs depend on gross investment We could also expand the natureof ψ so that adjustment costs depend on the size of the capital stock as is done in suchpapers as Lucas (1967) Uzawa (1969) and Gould (1968)
23 In the National Income and Product Accounts of the USA that report various measuresof the activity in the economy depreciation is measured by what is called the ldquocapitalconsumption allowancerdquo
24 The presence of such markets reflects the existence of ldquoperfect capital marketsrdquo in themacroeconomics literature Remember that with respect to the choice of investment forthe individual firm zero adjustment costs imply a potential discrete jump in the capitalstock at a point in time so that investment for the individual firm may not be defined
25 LrsquoHospitalrsquos rule states that if a is a number if f (x) and g(x) are differentiable and g(x)does not equal zero for all x on some interval 0 lt |x minusa| lt ε if the limit of f (x) equalszero as x approaches a and if the limit of g(x) is zero as x approaches zero then whenthe limit of the ratio f (x)gprime(x) as x approaches a exists or is infinite it equals the limitof f (x)g(x) as x approaches a
26 In continuous-time or discrete-time models such adjustment costs mean that the ldquocapitalmarketrdquo at time t is eliminated
27 Note that since bonds and equity shares are perfect substitutes the optimization problemwill not provide a breakdown into the optimal number of bonds versus equity shares
28 Recall that we are holding the stock of equity shares outstanding constant29 Note that we ignore the potential choice of capital at time t Kt and associated choice
of bonds setting Kt = K This in fact would be the case with capital adjustment costs30 If we evaluated returns from the start of a period each of the return functions would be
multiplied by 1R31 The equivalence of bond equity share and retained earnings financing of changes in
the capital stock can be shown32 This expression reflects the envelope theorem In particular we have that
dW (Kt+1)
dInt+1
dInt+1
dKt+1= dW (Kt+1)
dNt+1
dNt+1
dKt+1= 0
33 If the price of capital differed from the price of output but both prices were expected tochange at the same rate so that the relative price of capital was assumed to be constantover time then the expected real user or rental cost of capital would be (pkp)(mt + δ)where pkp denotes the relative price of capital
206 Notes
34 Formally at time t the inherited debt-to-equity ratio is given by pbBpeS35 Naturally there are other factors not discussed36 Note that during period t when the length of the period between planned purchases of
capital (at time t) and final installation (at time t + 1) is 1 then net investment Int isgiven by Int = Kt+1 minus K and adjustment costs are given by ψ(Int)
37 During period t+1 net investment is defined by Int+1 = Kt+2 minusKt+1 and adjustmentcosts are given by ψ(Int+1)
38 We can see from the general nature of the investment demand functions that if theproduction function were not separable then the assumption of a constant real wageover time as well as a constant expected real rate of return over time would obtain thisresult
39 See Sargent (1987a 11) for a similar expression in continuous time Note that thetwo expressions would be exact if we take the limit as the length of the period goesto zero The equality between ψ prime
t+h and ψ primet+2h that typically would be an approx-
imation in discrete time holds exactly in the limit In addition the definition ofthe real interest rate for a period of length h 1 + hm = (1 + hr)(1 + hπ) orm = (r minus π)(1 + hπ) indicates that in the limit (as h goes to zero) m = r minus π If we had assumed that adjustment costs were based on gross not net investmentthen the fraction on the left-hand side of (312) would include the term minusδψ in thedenominator
40 The Q theory of investment demand was suggested by Tobin (1969) Sargent is oneauthor who expresses investment demand in this way
41 In fact empirical measures of Tobinrsquos average Q have been constructed although suchmeasures are more complex than those discussed here since they must incorporateinfluences of the tax system (such as investment tax credits accelerated depreciationallowances and the like) on the optimal choice of the capital stock
42 In a two-sector model Tobinrsquos average Q is given by pV p1K where pV is the nominalvalue of the firm and p1 denotes the price of capital which differs from p the price ofoutput
43 In our case there is a single argument of the adjustment function ndash net investment Ifone follows Hayashi (1982) and assumes as he does that adjustment costs depend ongross investment then one must add the assumption that there is a linear homogeneousadjustment function such that ψ(δk) = ψ primeδk over the appropriate range Hayashirsquosadjustment cost function (like others) includes the stock of capital as well
44 Eulerrsquos theorem (or law) is that if the function f (middot) is a differentiable functionhomogeneous of degree 1 (linear homogeneous) with f Rn rarr R then
f (x) equivnsum
i=1
[partf (x)partxi]xi
Replacing the real wage with the marginal product of labor reflects the assumption thatthe firm is a price-taker in both the output and labor markets
45 See for example Azariadis (1976) A second reason is that new workers and previouslyemployed workers may not be considered perfect substitutes (eg see Oi 1962)
46 Taylor (1972) and Hall (1980) are among those who have examined thisphenomenon
4 Households as market participants
1 Time consistency means in this context that households will follow through on priorplans as the starting date advances Strotz (1955ndash56) discussed this point in terms of autility maximizing problem
Notes 207
2 An example of nonseparable preferences is a ldquohabit persistence modelrdquo in which utilityis given by
infinsumi=t
βiminustu(ci ciminus1 )
so that consumption at time t depends on prior consumption Alternatively one couldhave utility given by
infinsumi=t
βiminustu(ci 1 minus Ni 1 minus Niminus1)
such that past work becomes a pertinent state variable for the current period Kydlandand Prescott (1982) is one important exception to the macroeconomic literature inassuming nonseparable preferences
3 This is sometimes referred to as an ldquoend-of-periodrdquo equilibrium specification in themarket for assets Alternative asset specifications for discrete-time analysis have beendiscussed by among others Foley (1975) As Edi Karni (1978) pointed out Patinkinrsquosmodel is an end-of-period model
4 For simplicity we ignore the financial asset markets at time t If we assumed portfolioadjustment costs then it would be the case that at time t desired bond and moneyholdings would be B and S respectively
5 That is (partW (xt+1)partcdt+1)(partcd
t+1partxt+1) = 0 since partW (xt+1)partcdt+1 = 0 similarly
the indirect effects of a change in xt+1 on partW (xt+1) through its impact on optimallabor supply and real money balance holdings are zero
6 For completeness note that by substituting equation (43prime) into (45) we could have theequivalent expression
partW (xt+1)partxt+1 = β[partut+1part(Mt+1pt+1)]Rt+1[Rt+1 minus Rmt+1]7 These conditions appear in various forms throughout the macroeconomics literature
see for instance Mankiw et al (1985) or Barro and King (1984)8 The resulting demand for leisure function is termed a ldquoHicksian or compensated demand
functionrdquo as it is constructed by varying the price of leisure (the real wage) and incomeso as to keep the individual at a fixed level of utility
9 The resulting demand function for leisure that incorporates both the income and sub-stitution effects of changes in the real wage is an example of a ldquoMarshallianrdquo demandfunction
10 Borjas and Heckman (1978) See also Pencavel (1985) for a survey of estimates Forevidence on the effect of wage increases on the labor supply of working women seeNakamura and Nakamura (1981) and Robinson and Tomes (1985)
11 Note that we continue to assume nonsatiation such that partutpartct gt 012 The second statement presumes sufficient dispersion in preferences so that at each real
wage there are some individuals who are just indifferent between a zero and positivelabor supply Thus any rise in the real wage will increase labor force participation
13 This point was first formally developed in the classic paper by Lucas and Rapping(1970) The role of ldquomarket clearingrdquo in macroeconomic analysis will be clearer laterwhen we consider alternative characterizations of the labor market
14 To be exact this result assumes that utility is separable and concave in leisure That isit is assumed that part2upartcpartN = part2upartNpart(Mp) = 0 and part2upartN 2 lt 0
15 Alogoskoufis (1987) provides a good review of the empirical analysis in this area
208 Notes
16 Fisherian ndash or in Patinkinrsquos (1965) terminology ldquoFisherinerdquo ndash analysis takes itsname from the classic ldquotime-preferencerdquo analysis of Fisher see in particular Fisher(1930)
17 To assure this one could let duadcai gt 0 with limcairarr0(duadcai) rarr infin andlimcairarrinfin(duadcai) rarr 0
18 Equivalently one could assume partuapart(Maip) equiv 0 for all i Given the nonnegativityconstraint on Maipi i = t t + T and the presumption of a positive nominalinterest rate ri i = t t +T minus1 the optimal solution would be M d
aipi = 0 for all iThat is bonds will dominate money as an asset and only bonds will be held if assetholdings are positive The reason is simple ndash the unique attribute of money holdingsas a way to reduce ldquotransaction costsrdquo has not been introduced by providing a ldquoutilityyieldrdquo to holding money
19 Note that the equality sign here rather than the ge sign reflects the fact that the first-order conditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
20 Note that we assume a time-invariant one-period utility function The notation uai
reflects the fact that utility of agent a in period i depends on consumption inperiod i cai
21 In general for a period of length h we have that 1 + hm = (1 + hri)(1 + hπi)Solving for hmi and then dividing through by h we have that mi = (ri minusπi)(1+hπi)Thus in the limit as the length of each period goes to zero mi = ri minus πi Thus incontinuous-time analysis the real rate of interest is exactly defined by ri minus πi
22 In the discussion to follow we maintain the assumption of a time-invariant utilityfunction such that ua
i (cai) = ua(cai) for i = t t + T 23 At the point where cat = cat+1 the slope of the indifference curve is minus1β Assuming
cat = cat+1 and no initial assets or debt (ie zBat = 0) if 1β = Rt then the resultof the same consumption in each period would imply the individual would be neither alender nor a borrower
24 As Modigliani (1996) has stated ldquothe consumption and saving decisions of householdsat each point of time reflect a more or less conscious attempt at achieving the preferreddistribution of consumption over the life cycle subject to the constraint imposed bythe resources accruing to the household over the lifetimerdquo Modigliani summarizes hiscontribution to the analysis of consumption behavior in his Nobel lecture of December1985 (see Modigliani 1986)
25 Examples of such discussions are Diamond and Hausman (1984) and Hurd (1987)However what happens in the aggregate does mask different behavior among subgroupsof the populations For instance Burbidge and Robb (1985) find for Canadian data thatwhile an inverted U-shaped profile exists for the ldquoaveragerdquo Canadian household ldquowhitecollarrdquo households do appear to continue to accumulate wealth years after both husbandand wife have left the labor force
26 If individual a were the representative agent then consumption smoothing would implythat the aggregate endowment of the consumption good is identical across periods ieci = ci+1 i = t t + T minus 1
27 Note that in the case of multiperiod bonds agent arsquos future income could includepayments derived from the initial holdings of assets Since the present value of suchfuture ldquoincomerdquo is incorporated in the current value of the assets operationally futureincome cat i = t+1 t+T is defined as income other than derived from initial assetholdings In a production context the source of such income would be compensationfor labor services sold
28 In an economy with production this transitory component of current income can reflectsuch events as a temporary layoff a short-run opportunity to work overtime or atemporary tax rebate
29 This discussion ignores the effects of uncertainty on optimal consumption plans
Notes 209
30 Note that the equality sign here rather than the ge sign reflects the fact that the first-orderconditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
31 Note that with one-period bonds individual a has a zero initial endowment of bondsthat continue into the future at time t
32 Other papers on this topic include Hansen and Singleton (1983) and Mankiw et al(1985)
33 An example of a compositional change that could affect aggregate consumption butwould not be accounted for in the analysis to follow is a change in the proportionof retired individuals in the economy Thus on this ground at least Hallrsquos empiricalfindings cannot be taken as the last word on consumption behavior at the level of theindividual
5 Summarizing the behavior and constraints of firms and households
1 Caballero and Engel (1999) provide an empirical study of investment dynamics in thecontext of manufacturing firms
2 Weber (1998) provides empirical evidence on the link between the financial marketsand consumption spending
3 We refer to this perfect foresight as ldquolimited perfect foresightrdquo since it concerns only thecurrent period This focus on current markets alone typical of static analysis impliesldquoexpectations functions of the agentsrdquo with respect to prices in subsequent periodsthat do not reflect the underlying analysis of markets beyond the current period Thusbeyond the current period expectations do not have the property of perfect foresight (orrational expectations in a nondeterministic setting)
4 In an open economy ndash that is one which admits a foreign sector ndash we would add afourth market the market for foreign exchange
5 Note that in a fully monetized economy money enters on one side of every exchange ndashpurchase or sale ndash in these three markets
6 Note that money holdings and money demand arise only for ldquohouseholdsrdquo To the extentfirms do hold money and make choices with respect to the size of such holdings wepresume their behavior would be similar to that of households and so lump firms withhouseholds with respect to such activity Thus the behavior of the firm is restricted tolabor demand output supply investment demand and financial asset supply
7 Recall that the expected real user cost of capital is mt + δ where δ is the rate ofdepreciation of capital and mt is the expected real rate of interest (ie mt equiv (rt minusπe
t )(1+πet ) where rt is the money interest rate and πe
t is the expected rate of inflationbetween periods t and t + 1)
8 As previously for simplicity we continue to assume that expectations of future pricesare held with subjective certainty
9 If part2f partKpartN = 0 then the real wage would not enter as an argument in the capitaldemand function nor would the existing capital stock affect labor demand
10 That is we can express a demand function in a form similar to one that would beobtained if the analysis were to consider only two periods
11 Note that we do allow the expected wage inflation between period t and t + 1 πewt
to differ from subsequent wage inflation so that the expected real wage next periodwe
t+1pet+1 equiv wt(1 + πe
wt)pt(1 + πet ) can differ from the current real wage This
introduces the possibility of intertemporal substitution of labor supply in response to achange in the current real wage
12 Assuming less than unit elastic expectations with respect to future income streams wouldassure from the standard Fisherian problem that the marginal propensity to consumewould be less than one
13 Recall that πet equiv ( pe
t+1 minus pt)pt
210 Notes
14 Recall that firms are presumed not to hold money balances so there are no real balanceeffects to concern us with respect to firmsrsquo demands or supplies
15 As Lucas and Rapping (1970) point out introducing other expectation assumptionscan retain the ldquointertemporal substitution hypothesisrdquo in the context of changes in thecurrent price level but in doing so money illusion is introduced In their own wordsthe
labor-supply equation is not homogeneous in current wages and current prices(such that) there is ldquomoney illusionrdquo in the supply of labor ldquomoney illusionrdquoresults not from a myopic concentration on money values but from our assumptionthat the suppliers of labor are adaptive on the level of prices expecting a returnto normal price levels regardless of current prices and from the empirical factthat the nominal interest rate does not change in proportion to the actual rate ofinflation With these expectations it is to a supplierrsquos advantage to increase hiscurrent supply of labor and his current money savings when prices rise
(Lucas and Rapping 1970 268ndash269)
Lucas and Rapping are particularly looking at the effect of a change in the current pricelevel on the expected real rate of interest Their assumption of ldquoadaptiverdquo expectationsimplies that an increase in pt results in a fall in πe
t equiv (pet+1 minus pt)pt a rise in the
expected real rate of interest and thus a rise in labor supply16 That is future technology capital demand labor demand the real wage the rate of
depreciation and future real interest rates are all unchanged by such a change in pricesand the money supply
6 The simple neoclassical macroeconomic model (without governmentor depository institutions)
1 In particular it is assumed that the money wage rate adjusts to continuously maintainequality between the demand for and supply of labor the price of output adjusts tomaintain equality between the demand for and supply of output and the price of financialassets (and thus the interest rate) adjusts to maintain equality between the demand for andsupply of financial assets Patinkin (1965) provides one of the first complete accountingsof this model
2 That is we would expect prices in the various markets to eventually adjust to eliminateany possible excess demands or supplies in the economy We would also expect agentsultimately to correctly anticipate the price level The neoclassical model can be modifiedto explain the workings of the economy in the face of incomplete information and priceinflexibility
3 As before the ldquolaws of motionrdquo dictating how prices change to reach equilibrium aregiven by Walrasrsquo excess demand hypothesis and we maintain the assumption that noexchange occurs until an equilibrium is reached (the recontracting assumption)
4 Alternatively one could assume that expectations at time t concerning these futurevariables are constant
5 Note that Patinkin has firms as well as households managing a portfolio of financialassets and money balances which is why he includes the demand function for labor inthe above statement In our analysis this statement applies to the labor supply functionalone
6 This last sentence anticipates the intertemporal substitution hypothesis7 We ignore the potential effect of changes in the interest rate on labor supply and thus
employment8 This reflects the assumption that households and firms share common expectations
concerning the price level (in fact for both pet = pt)
Notes 211
9 Reasons such as these for changes in output form the basis of much of the currentanalysis in the literature with respect to ldquoreal business cyclesrdquo
10 This is perhaps too extreme a statement To the extent that a higher price level isanticipated the resulting lower real money balances could lead to an increase in laborsupply at any given real wage and consequently increased employment and output Alsothere is a potential effect of changes in the price level on aggregate labor supply throughthe impact of such changes on the expected real rate of interest if unit elastic expecta-tions concerning future prices are not assumed Recall that we follow macroeconomictradition and abstract from these possibilities
11 In a recent study Ewing et al (2002) develop a model of the equilibrium unemploymentrate and examine how it responds to unanticipated changes in real output
12 Fairlie and Kletzer (1998) discuss the issues revolving around job displacement13 Unemployment may also result if prices in the economy do not adjust quickly enough to
ensure that all markets (particularly the labor market) are continuously in equilibriumUnemployment associated with labor market disequilibrium is sometimes referred toas ldquoinvoluntaryrdquo unemployment We analyze such unemployment later
14 See for instance the paper by Evans (1989) that examines the relationship betweenoutput and unemployment in the United States
15 The ldquoISrdquo equation is sometimes referred to as the aggregate demand equation indicatingthat it reflects equality between total or aggregate demand for output and productionNote that equilibrium in the output market is being described in terms of demand equalto what is produced ylowast
t The aggregate supply equation indicates what will be produced16 This is the case only for this simple aggregate model without government17 Such an assumption removes the anticipated real wage next period as an argument
in these demand functions Recall that earlier ldquostaticrdquo assumptions concerning futureinterest rates and rates of inflation have already simplified the form of these functions
18 However ldquorealrdquo or ldquosupplyrdquo shocks such as the above-mentioned changes in technologycapital stock supply of other inputs such as oil or in labor supply at prevailing realwages can affect real output
19 This analysis should be familiar since we performed a similar analysis for the economywithout production
20 Note that unit elastic expectations imply that partπet partpt = 0
21 Note that without a real balance effect the CC curve would be horizontal22 The lower interest rate abstracts from a real balance effect in the output market so that
the CC curve is horizontal
7 Empirical macroeconomics traditional approaches and time series models
1 To reduce notational clutter we suppress time subscripts All variables are period tvariables
2 Note that in Sargent (1987a 20) equation (71) is replaced by the ldquorepresentativerdquofirmrsquos first-order condition for the optimal use of the labor input given a competitivelabor market that is the condition that the real wage equals the marginal product oflabor wp = Fn(n K) Note that if the production function F(n k) is separable in thelabor and capital inputs such that f (n K) = v(n)+u(K) and v(n) = (1g)(fnminusn22)then equation (71) is identical to Sargentrsquos equation since fn(n K) = (1g)(f minus n)
3 Unless otherwise noted all parameters in this model such as f g h and j in equations(71) and (72) are assumed to be positive
4 In this context ldquolump-sumrdquo taxes are taxes independent of income Equation (73) isan example of the ldquoconsumption functionrdquo
5 Endogenous variables are variables whose values are determined by the analysis6 Sargent adds equations (73) and (74) to the system (712) in order to determine
consumption and investment demand as well
212 Notes
7 To derive wlowast start with the equilibrium condition nd = ns Substituting the first twoequations of 712 the equilibrium money wage satisfies
f minus g middot (wlowastp) = h + j middot (wlowastp)
Then one can simply solve this equation for the equilibrium money wage wlowast We canthen obtain the equilibrium employment level nlowast by substituting the expression for wlowastinto either of the first two equations of 712
8 This is a property of ldquoclassicalrdquo macroeconomic models in that monetary changesthat alter the price level do not affect real variables such as the real wage output andemployment
9 The IS equation indicates combinations of the interest rate r and output y at which ifthe output were produced and that interest rate prevailed output demand would equaloutput produced The LM equation indicates combinations of r and y that will equatethe demand for and supply of money
10 For the model under consideration the ldquoaggregate demand curverdquo (a plot in ( p y) spaceof the aggregate demand equation) slopes downward and the ldquoaggregate supply curverdquo(a plot in ( p y) space of the aggregate supply curve) is vertical The intersection of theaggregate demand and supply curves graphically determines the equilibrium output andprice level
11 See Altonji and Siow (1987) Ewing and Payne (1998) examined the relation-ship between the personal savings rate and consumer sentiment in the context of aconsumption model
12 Note that Taylor assumes that certain demands for instance consumption demand maydepend on past as well as current values of output and the money supply so that thereduced-form expression estimated includes lagged values of income and the real moneysupply
13 Time series analysis can be viewed as primarily the art of specifying the most likelystochastic process that could have generated an observed time series
14 That is forecasts generated by time series models have been used to proxy individualsrsquoexpectations of future events in tests of various theoretical macroeconomic models
15 A histogram is a plot of the frequency distribution of a set of observations16 If the process is also ldquoergodicrdquo these statistics give consistent estimates of the mean and
variance Ergodicity basically requires that observations sufficiently far apart should bealmost uncorrelated Then by averaging a series through time one is continually addingnew and useful information to the average For a rigorous explanation of this conceptsee Hannan (1970 201)
17 The variable γk k = 0 1 2 is termed the autocovariance function18 Or more generally k = 0 Note that ρk = ρminusk 19 Some have suggested that stock market prices follow a random walk See Campbell
et al (1997)20 Note that if yt is taken to be the logarithm of real output then the trend reflects a constant
rate of growth of output equal to d in the absence of shocks21 This assumes y0 is the initial value of the function yt 22 A random walk is an example of a class of nonstationary processes known as ldquointe-
gratedrdquo processes that can be made stationary by the application of a time-invariantldquofilterrdquo As defined by Granger and Newbold (1986) ldquoif a series wt is formed by a linearcombination of terms of a series yt so that wt = summ
j=minuss cjytminusj then wt is called a
lsquofilteredrsquo version of yt If only past terms of yt are involved so that wt = summj=0 cjytminusj
then wt might be called a one-sided or backward-looking filterrdquo23 This is a special case of a class of stochastic processes known as Markov processes24 Recall that for the random walk process φ = 1 in which case the process was not
stationary as the variance of the process becomes larger and larger with time
Notes 213
25 To obtain this result note that
E
⎡⎣ infinsum
j=0
(φ2)jε2tminusj
⎤⎦ =
infinsumj=0
(φ2)jσ 2e = σ 2
e (1 minus φ2)
26 Often the ldquolag operatorrdquo L or equivalently the ldquobackward shift operatorrdquo B is usedto express this equation Lτ yt (or Bτ yt) = ytminusτ τ = 1 2 3 There are associatedpolynomials in the lag operator such that
d(L) = d0 + d1L + d2L2 + d3L3 + middot middot middot + dpLp
Letting
φ(L) = 1 minus φ1L minus φ2L2 minus φ3L3 minus middot middot middot minus φpLp
we can thus express an AR( p) process for yt as φ(L)yt = εt 27 If yt is an AR( p) process it may be described in the following way ldquoan appropriate
finite backward-looking filter applied to yt will produce a white noise seriesrdquo (Grangerand Newbold 1986 32)
28 If there were a single shock to yt at time 0 (ie εt = 0 for all t gt 0) then the deterministiccomponent would signify the deviation of the time path from its equilibrium level Aswe will see in this case stationarity would imply ldquostability of equilibriumrdquo in that ytwould converge to its equilibrium value over time
29 A linear difference equation of order s is of the form
yt =ssum
j=1
ajytminusj + c
30 Successive substitution (the ldquoiterative method of solutionrdquo) reveals this essential natureof the solution In particular substituting for past values of yt in (725) results inyt = φty0
31 The appendix to this chapter shows how one can reinterpret higher-order differenceequations as a system of first-order difference equations
32 Note that in general a quadratic equation of the form
ax2 + bx + c = 0
can be solved using the quadratic formula
x1 x2 = [minusb plusmn (b2 minus 4ac)12]2a
In our case a = 1 b = minus(m1 + m2) = minusφ1 and c = m1m2 = minusφ233 This assumes m1 = m2The solution for repeated roots is discussed briefly below34 A variable x is said to be inside the unit circle if x lt |1| An alternative way of
expressing this condition is in terms of the associated polynomial 1 minusφ1L minusφ2L2 = 0This polynomial equation is similar to (730) except that b is replaced by 1L and thewhole equation is multiplied through by L2 The stationarity condition is that the rootsof this polynomial equation should lie outside the unit circle
35 Recall that φ1 = m1 + m2 and φ2 = minusm1m236 The following discussion follows Goldberg (1958 171ndash172)
214 Notes
37 As discussed below in this case the two roots are m1 = h + vi and m2 = h minus vi whereh = φ12 v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12 The product
of these two roots is (φ21 minus 4φ2 minus φ2
1)4 = minusφ2 which is the square of the modulus
of the roots We are assuming (minusφ2)12 lt 1 so we have the condition that φ2 lt 1 orφ2 gt minus1
38 Note that if yt is an MA process then it is a ldquobackward looking filterrdquo applied to a whitenoise process As before we can use a lag operator L (or backward shift operator B) toexpress an MA(q) process for yt as yt = micro + θ(L)εt where the polynomial in the lagoperator θ(L) is given by
θ(L) = 1 + θ1L + θ2L2 + middot middot middot + θqLq
39 Recall that as discussed above we assume for simplicity zero mean for yt that ismicro = 0
40 For a more detailed description of invertibility see Granger and Newbold (1986) orBox and Jenkins (1970)
41 Note that ARMA( p 0) equiv AR( p) and ARMA(0 q) equiv MA(q) Using the lag operatorthe ARMA model (in the case of a zero mean) can be simply expressed as φ(L)Yt =θ(L)εt
42 In general if yt is ARMA( p1 q1) and xt is ARMA( p2 q2) the sum zt = yt + xt isARMA( p3 q3) where p3 le p1 + p2 and q3 le max(p1 + q2 p2 + q1) A proof of thisis found in Granger and Morris (1976)
43 A time sequence T (t) is called ldquodeterministicrdquo if there exists a function of past andpresent values gt = g(T (t minus j)) j = 0 1 such that E[(Tt+1 minus gt)
2] = 0 If thefunction gt is a linear function of Ttminusj j ge 0 then Tt is called ldquolinear deterministicrdquo
44 For instance for a stationary series of quarterly data one could postulate a simplefourth-order seasonal AR process of the form
yt = φ4ytminus4 + εt
This is a special case of AR(4) with φ1 = φ2 = φ3 = 0 The model could be extendedto include both AR and MA terms at other seasonal lags for instance the followingARMA(21)
yt = φ4ytminus4 + φ8ytminus8 + θ4εtminus4 + εt
To add other than seasonal components one could simply fill in the gaps (eg addterms such as φ1ytminus1 θ1εtminus1 and the like to the above) Other options are discussedby Harvey (1993) and others
45 Campbell and Mankiw argue that even if the log of output followed a random walk withdrift indicating that the effect of any shock persists indefinitely into the future estimatesusing the detrended series would be biased and erroneously conclude otherwise
46 Note that if the log of real output is an ARMA( p q) process then the differencedprocess will be an ARMA( p q + 1) process This means that to allow for stationaritywith respect to the level of real output requires at least one moving average process forthe differenced series
47 Equivalently for the logarithm of real output they are considering ARIMA(p 1 q)processes for p = 0 1 2 3 and for q = 0 1 2 3
48 Such a finding is often termed as supportive of real business cycle theories
Notes 215
8 The neoclassical model
1 The presence of money reflects the introduction of ldquoimperfectrdquo or costly informationA medium of exchange can arise to minimize costs incurred by participants in theeconomy when there exists imperfect information on potential exchange partners
2 In so doing we assume that the new equilibrium like the initial one exists Further wedisregard the process of adjustment of the variables to the new equilibrium Alterna-tively we could introduce ldquolaws of motionrdquo for the equilibrium values (eg the excessdemand hypothesis for price changes) and examine whether the equilibriums are stable
3 For notational simplicity we let Ld = Ldt cd = cd
t Id = Idnt p = pt r = rt
yd = cdt + Id
nt + δK + ψ(Idt ) and M = M
4 For simplicity we let the expected real rate of interest component of the expected realuser cost of capital ((rt minus πe
t )(1 + πet )) be approximated by rt minus πe
t as representedby r minus π
5 One alternative is for the interest rate to adjust to equate the demand for and supply ofmoney
6 That is we assume that consumption demand is a function of the expected gross realrate of interest Rt not its components (rt and πe
t ) This would be the case if one couldview the consumption decision from the point of view of the pure ldquoFisherian problemrdquoin essence separating the allocation of consumption across time decisions from theldquoportfolio problemrdquo Note that for notational ease we not only let rt denote the moneyinterest rate rt but also let π denote the expected rate of inflation πe
t 7 This would be the case if our focus was solely on the portfolio choice problem8 Examples of growth models with money include Tobin (1965) and Sidrauski (1967b) As
Begg (1980 293) notes ldquoin a steady state any expectations generating mechanism willyield correct predictions thus the steady state analysis of growth models with moneymay be viewed as a special case of the rational expectations model with systematicmonetary policyrdquo
9 This asymmetry in information can reflect an aggregation across labor markets in whicheach firm determines labor demand based on its correct anticipation of the price of theparticular commodity it produces while suppliers of labor determine labor supply basedon their potentially incorrect anticipation of the overall level of prices reflecting theidea that suppliers are concerned with the purchasing power of wages in terms ofcommodities not restricted to the particular commodity that they produce
10 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt r = rt y = yt and M = M
11 Recall that for simplicity we let the expected real rate of interest component of theexpected real user cost of capital (rt minus πe
t )(1 + πet ) be approximated by rt minus πe
t asrepresented by r minus π
12 Note that partypartp = 0 for the neoclassical model given limited perfect foresight13 This is obvious from the graph of aggregate demand and supply curves if one compares
the vertical aggregate supply curve of the neoclassical model dpp = dMM with theupward-sloping aggregate supply curve of the model with real wage illusion Note thatthe shift in the aggregate demand curve for a given change in the money supply isidentical in either case
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
1 Typically a union contract runs for three years Often however there are provisions thatpermit parts of the agreement to be renegotiated at specific times during the three-yearcontract period
2 As a general rule labor agreements are specified in money terms An exception to thisis the cost of living agreements (COLAs) as part of union wage contracts in the United
216 Notes
States which became popular during the 1960s and 1970s COLAs adjust money wagesautomatically to changes in prices typically using changes in the Consumer Price Indexto measure price changes However the percentage of all workers who have contractswith COLAs is fairly small as less than 20 percent of the total labor force is covered bycollective bargaining agreements Further not all COLA clauses offer full protectionagainst general price increases as wages may rise by only some fraction of the increaseof the CPI Considering these qualifications for the time being we simplify by assumingthat all labor agreements are specified in money terms
3 These seven variables for the classical model are made up of three ldquopricesrdquo along withfour variables implied from the behavioral equations The three ldquopricesrdquo are moneywage price level and interest rate determined by the equilibrium conditions for the labormarket and two of the three other markets (output financial andor money markets)The variable implied from the behavioral equations are employment (labor demandfunction) output (production function) consumption (consumption demand function)and investment (investment demand function)
4 Note that this analysis is inconsistent with the idea that long-term employment contractsthat fix the money wage for several periods arise due to adjustment costs with respectto the labor input
5 As discussed before one justification for this form is if real money balances are not partof household wealth which can be the case when we introduce depository institutionsinto the analysis
6 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt Idnt = Id r = rt
y = yt and M = M 7 Recall that for simplicity we let the expected real rate of interest component of the
expected real user cost of capital (rt minus πet )(1 + πe
t ) be approximated by rt minus πet as
represented by r minus π 8 Note that partypartp = 0 for the neoclassical model given limited perfect foresight9 The Lucas model was introduced in Chapter 8 and is covered in more depth in
Chapter 1010 An exception to this statement occurs if monetary authorities can react to period t
disturbances that is if monetary authoritiesrsquo information set includes the values of therandom shocks in period t which are not known to private agents
11 Phelps and Taylor (1977) go on to state that they ldquodo not pretend to have a rigorousunderstanding of [why prices andor wages are set in advance] In the ancient andhonorable tradition of Keynesians past we take it for granted that there are disadvantagesfrom too-frequent or too-precipitate revisions of price lists and wage schedulesrdquo
12 Note that θ is affected by the variability of relative price shocks in relation to generalprice shocks
13 We assume for simplicity that the coefficient on Pt minus Wt + φ is one14 Note that if Pt minus Wt + φ = 0 then equation (99prime) is a first-order linear difference
equation of the form YtminusλYtminus1 = 0 with solution Yt = c0λt In the limit as t approachesinfinity Yt equals zero Recall that the logarithm of the natural level of output is zero
15 Recall our assumption that the scale factor in the determination of the real wage is equalto zero for convenience
16 As Fischer shows if wages were set only for the current period t that is tminusiWt = φ +EtminusiPt then his results would be like those obtained by the Sargent and Wallace modelwith rational expectations for similar reasons This can be seen clearly on substituting(910) into (99prime) in which case one obtains the standard Lucas-like aggregate supplyequation
17 Later it will be assumed that ut and utminus1 are correlated so that information obtainedduring period t minus 1 will help predict variations in output for period t Like the laggedoutput term in the Lucas equation this introduction of serial correlation in output isnecessary (but not sufficient) if monetary policy rules that dictate the money supply
Notes 217
for period t based on information obtained up to the end of period t minus 1 are to serve astabilizing role
18 Note that in Fischerrsquos paper minusvt rather than vt is used in (913) Our vt is more in linewith other models
19 With the one exception noted before that Fischer has minusvt replacing vt 20 Again note that Fischer has minusηt and minusρ2 in the above equation since our vt is his minusvt 21 Fischer has b1 = minusρ2 since our vt is his minusvt
10 The Lucas model
1 The discussion follows that found in Lucas (1973) and Sargent (1987a 438ndash446)2 Or as Sargent (1987a 483) states ldquoan employee cares about his prospective wage
measured not in terms of own-market goods but in terms of an economy-wide averagebundle of goods the assumption is that the labor supplier works in one market butshops in many other marketsrdquo
3 The CobbndashDouglas production function would be given by yit = (Nit)1minusα(Ki)
α whereKi denotes the inherited capital stock for sector or ldquoislandrdquo i Nit is the employment oflabor for sector i and yit is the production of commodity i by sector i during period t Inthis case the marginal product of labor is given by partyitpartNit = (1 minus α)(Nit)
minusα(Ki)α
which implies that a = (1 minus α)(Ki)α
4 Solving this expression for the demand for labor and differentiating with respect to thereal wage relevant for a firm producing commodity i we have
partN dit part(witpit) = (1 minus α)minus1α(minus1α)(witpit)
(minus1minusα)α lt 0
Note that our particular form for the labor demand function implies a constant elasticityIn particular the elasticity of demand for labor is given by
partN dit
part(witpit)
witpit
N dit
= minus 1
α
5 Up to this point we have assumed that expectations are held with ldquosubjective certaintyrdquoso that Et(1pt) = 1Etpt and the expected real wage can be represented as witEtpt However we now assume that individuals view the price level as a random variableIn this setting the expression for the expected real wage is Et(witpt) or witEt(1pt)which is not the same as witEtpt In fact the relationship between witEt(1pt) andwitEtpt is shown by Jensenrsquos inequality witEt(1pt) ge wit(1Etpt) This inequalityreflects the fact that the function f ( pt) = 1pt is convex rather than linear in pt Forinstance if pt = p0 with probability t and p1 with probability 1 minus t then we haveEt(1pt) = tf ( p0) + (1 minus t)f ( p1) gt f (tp0 + (1 minus t)p1) = 1Etpt where 1 gt t gt 0On the other hand ln(1pt) is linear in ln pt so the log of the real wage is linear in thelog of the price level Thus we express the labor supply function in logarithmic formand consider the expectation of the log of the price level
6 The term ξt is assumed to be serially independent which means that E(ξtξs) = 0 fort = s
7 We assume that zit and ξt are statistically independent8 In problems involving more than two random variables ndash that is a ldquomultivariate regres-
sionrdquo ndash we are correspondingly concerned with the term E(z|x y) the expected valueof z for given values of x and y and so on
9 The discussion that follows is standard statistical theory Note that if the joint distributionof the two variables is a bivariate normal density function then the regression of ln pton ln pit is linear
218 Notes
10 In general if x1 x2 xn are random variables a1 a2 an are constants andq = a1x1 + a2x2 + middot middot middot + anxn then
E(q) =nsum
i=1
aiE(xi)
Var(q) =nsum
i=1
a2i Var(xi) + 2
sumiltj
aiajCov(xixj)
wheresum
iltj means that the summation extends over all values of i and j from 1 to nfor which i lt j This expression is derived from the definition of the variance
Var(q) equiv E([q minus E(q)]2) = E
⎛⎜⎝⎡⎣ nsum
i=1
ai(xi minus microi)
⎤⎦
2⎞⎟⎠
which can be expanded by means of the multinomial theorem according to which forexample (a+b+ c +d)2 = a2 +b2 + c2 +d2 +2ab+2ac +2ad +2bc +2bd +2cdNote that
Cov(xixj) = E[(xi minus microi)(xj minus microj)]
If the xi i = 1 n are independent then
Var(q) =nsum
i=1
a2i middot Var(xi)
11 Note that these first two expectations are taken without information on ln pit 12 While it is true that if two random variables are independent they are also uncorre-
lated the converse does not necessarily hold That is two random variables that areuncorrelated are not necessarily independent However two random variables havingthe bivariate normal distribution are independent if and only if they are uncorrelated
13 In general for any two random variables x1 and x2 with joint density function f (x1 x2)the marginal density of x2 g(x2) is obtained by integrating out from minusinfin to +infinthe other variable Thus g(x2) = int
f (x1 x2)dx1 Similarly we can obtain the marginaldensity of x1 by integrating out x2 The conditional probability density function of therandom variable x1 given that the random variable x2 takes on the value x2 is definedby φ(x1|x2) = f (x1 x2)g(x2) assuming that g(x2) does not equal zero
14 To obtain the following expression we use the fact that the random variables ξt and zitare independent with zero means and variances σ 2 and σ 2
z respectively15 Note that this ldquosignal extractionrdquo problem appears in a number of different contexts
such as in statistical theories of discrimination in labor economics (Aigner and Cain1977 Lundberg and Startz 1983) and in the industrial organization literature
16 If there were a trend in the natural level of output then ln ynit would replace the firstln yni and in the lagged term ln ynitminus1 would replace the second ln yni
17 The last term in (1016prime) indicates the deviation of output in the prior period from itsldquonormalrdquo level It is presumed that λ lt 1
18 Recall that this supply function assumes no adjustment costs and thus does not have alagged output term in it
Notes 219
19 In general if z1 z2 zn are independent random variables having the same distribu-tion with mean micro and variance σ 2
z and if z = (z1 + z2 + + zn)n then E(z) = micro
and Var(z) = σ 2z n Note that as n goes to infinity the variance of z goes to zero
20 Recall that we have let π(πtminus1) denote ( pt minus ptminus1)ptminus1 This was done so thatthe terms making up the expected real interest rate rt minus πe
t would have the sametime subscript πe
t = ( pet+1 minus pt)pt In the discussions to follow however we shift
time subscripts so that the previously denoted π = π and the previously denoted πet
becomes πet+1
21 Note that the natural log of real output is ln( yt) = Yt 22 Sometimes the money demand function is simplified by assuming α2 = 0 so that there
are no effects of interest rate changes on real money demand23 This follows given mt = mt +εt so that dPtdmt = 1 Recall that Pt and mt are logs of
the price level and the money supply respectively so that dPt = d(ln pt) = (1pt)dptand dmt = d(ln Mt) = (1Mt)dMt
24 Note that we are somewhat imprecise in our statement that we are eliminating the termfor output from the equation Recall that the exact interpretation of Yt would be as thedifference between the logarithm of output and the logarithm of the natural level ofoutput Equivalently we may call Yt the log of the ratio of output to its natural level
11 Policy
1 The alternative to a rule is called ldquopurely discretionary monetary policyrdquo in whichmoney supply changes are made purely at the discretion of the government dependingon its current reading of the economy and current set of objectives
2 This view is not universally accepted For instance according to the Nobel prize winnerJames Tobin (1985) the government should be free to do as it sees fit and he sees manyreasons ldquofor the Fedrsquos reluctance to tie its own hands as much as lsquorulesrsquo advocates wishrdquo Sargent and Wallace (1976) identify as ldquoKeynesiansrdquo individuals who believegovernment monetary policy should attempt to ldquolean against the windrdquo in an effort toattenuate the business cycle In the view of Sargent and Wallace and others the monetaryauthority has no scope to conduct countercyclical policy Not surprisingly those whodisagree with this view suggest alternative models of the economy such as those withldquostickyrdquo prices more favorable to their alternative views
3 Those who argue for a rule such as this are sometimes called ldquomonetaristsrdquo Someadvocates of a constant growth rate in the money supply would restrict the length oftime during which a particular rate of growth was fixed For instance William Pooleformer member of the Presidentrsquos Council of Economic Advisers (1982ndash1985) andnow with the Fed suggests that monetary rules should be adopted but that the ruleshould be ldquosubject to change at any time upon presentation of a convincing case withsupporting evidencerdquo (Poole 1985)
4 Hallrsquos suggestion echoes Simonsrsquo (1936) proposal for ldquoa monetary rule of maintainingthe constancy of some price indexrdquo Wayne Angell a 1986 appointee to the Board ofGovernors of the Federal Reserve (the monetary authorities for the USA) has arguedfor a monetary policy that will stabilize a price index constructed from a basket of basiccommodities (perhaps including gold)
5 Recall also that we assume the random variables εt and ut (the random variable asso-ciated with output demand) are independent with zero mean and respective variancesσ 2
e and σ 2u
6 An example of such ldquoexogenousrdquo price expectations would be the autoregressiveexpectations
7 Recall that the natural level of output was normalized to equal one so that Yn equiv ln yn = 0Thus if the optimal level of output is the natural level then the objective would be tominimize Etminus1(Yt)
2
220 Notes
8 Recall that we are assuming that εt and vt are independent random variables From(112)
Etminus1Yt = a0 + a1Ytminus1 + a2mt
so that Yt minus Etminus1Yt = a2εt + vt Squaring this expression and noting that E(εt) = 0E(vt) = 0 and given independence E(εt middot vt) = 0 we obtain
Etminus1(Yt minus Etminus1Yt)2 = a2
2σ 2e + σ 2
v
9 See Sargentrsquos (1987a 453) equation (13) for the corresponding expression Recall thatg0 = (Y lowast minus a0)a2
10 Note that ln yn = 0 due to normalization11 To tie down the inflation rate we would have to add an inflation objective to the goals
of the government12 Note that this is another example of the Lucas critique in which econometric estimates
of a specific set of parameters based on past policies cannot be used to project the impactof new different monetary rules to be followed in the future for with these new policiesthe parameters will change
13 This equation is derived from combining the price error expression with the aggregatesupply equation given earlier
14 The assumptions leading up to (1116) have been chosen so that (1116) matches thecriterion found in Barro and Gordon (1983) We will contrast the results of that paperwith our findings here shortly
15 The form of this equation mirrors that in Barro and Gordon16 Consider what would happen if we did not neglect the lagged disturbance terms For
instance let us say one of the disturbance terms (εtminus1 or utminus1) is positive rather thanzero while the other is held equal to zero According to the reduced-form equation forthe price level for period t minus 1 the result would be a higher price level in period t minus 1with the positive lagged disturbance term If the rate of growth in the money supplybetween period t minus 1 and t was the same in both cases then the inflation rate wouldbe higher in the case when both lagged disturbance terms are zero Thus to maintain aconstant inflation rate a lower money supply growth is implied if lagged disturbanceterms are zero instead of positive
17 Kydland and Prescott were awarded the 2004 Nobel prize in economics for their work18 This form of the objective function introduced above is discussed in Barro and Gordon
(1983) Without the assumption of k lt 1 a zero average rate of inflation would beoptimal regardless of the nature of expectation formation
19 For simplicity we assume there is no persistence in real shocks Thus we assume λ = 0in the original Barro and Gordon model
12 Open economy
1 For discussion purposes the terms ldquodomesticrdquo and ldquoforeignrdquo are used with respect tothe domestic perspective
2 For simplicity we assume foreigners do not desire to hold domestic money3 We assume for simplicity that foreigners are not taxed by the domestic government4 That is the domestic households own the remaining share of bonds issued by foreign
firms Bff and α of the equity share issues by foreign firms The total dividends (interms of the foreign currency) paid by foreign firms in period t are denoted by pftdftwhere pft is the price level of the foreign country and dft are real dividends of foreignfirms
Notes 221
5 One might also include bdt as purchase of output by private depository institutions is
included in the household budget constraint This reflects the replacement of dividendspaid by private depository institutions to their shareholders (households) by the differ-ence between banksrsquo interest income and their purchases of the composite commodityThis definition of dividends for private banks has been termed the ldquoflowrdquo constraint forprivate depository institutions
6 Recall that MB denotes the monetary base R denotes reserves and C denotes currencyin the hands of the nonbank public
7 Recall that we use the term ldquomodifiedrdquo as we have substituted the firm distribution con-straint into the household budget constraint for labor income We thus have effectivelysuppressed the labor market from the markets under consideration This modified lawis useful in understanding how the aggregate demand equation is derived
8 By ldquoreal demandrdquo we mean in units of the domestic commodity9 The term ldquorealrdquo means in terms of the domestic composite commodity
10 In measuring the exports imports and international capital flows a number of itemsare often missed For instance the clandestine transfer of funds from the Philippines toUS bank accounts would generate a demand for dollars On the other hand secretiveimports of heroin from Turkey result in a supply of dollars in international marketsThe net of such unmeasured transactions are lumped under the heading of ldquostatisticaldiscrepancyrdquo in the balance of payments accounts We have omitted this item fromTable 121
11 For simplicity we assume that the exchange rate affects only the division of totalconsumption between imports and purchases of the domestic output total consumptionis assumed to be unaffected by such exchange rate changes
12 Note that an increase in the price of the yen or an ldquoappreciationrdquo of the yen means afall in the price of a dollar in terms of yen or a ldquodepreciationrdquo of the dollar
13 If the price elasticity with respect to imports was less than one in short run a rise in therelative price of imports due to a depreciation of the dollar could in fact lead to a risein the value of imports as well This short-run phenomenon when applied to the path ofnet exports over time is referred to as the ldquoJ-curve effectrdquo
14 Actually for much of the analysis to follow we need only assume the weaker ldquoMarshallndashLernerrdquo condition that the sum of the price elasticity of demand for imports and theprice elasticity of demand for exports exceeds one This assures that a price of the dollarbelow its equilibrium level will be associated with an excess demand for dollars in theforeign exchange markets while a price of the dollar above its equilibrium level willbe associated with an excess supply of dollars
15 Given future markets for foreign currency this is not always the case16 We assume that such expectation is held with subjective certainty17 There are several reasons why in the short run the supply curve may not be upward-
sloping First it often takes time for US purchasers to adjust purchases in light of achange in relative prices Second the prices of domestic goods that are close substitutesto the imported goods can rise significantly in the short run as domestic producers hitshort-run production constraints The third complication that has the effect of makingthe supply of dollars curve less likely to be upward-sloping is that foreign producersat least in the short run often adjust the foreign currency prices of goods they exportto partially offset the impact of exchange rate changes on the prices of their goodsin foreign markets For instance Knetter (1987) found that with a depreciation of thedollar (appreciation of the West German Mark) West German exporters often reducedthe Mark price of their exports so as to minimize the rise in the dollar price of Germangoods that would result from the appreciation of the Mark For the time being weabstract from such short-run considerations although this is not to lessen the importanceof this phenomenon as the experience during the 1985ndash1987 period indicates With adepreciation of the dollar the dollar value of imports grew as the USA had to supply
222 Notes
more dollars for each unit of imported goods and there was initially little reduction inthe quantity of goods imported
18 It is important to remember that while we talk of households as being the only privatedemanders of foreign goods and financial assets this is purely a simplifying deviceFirms also demand foreign goods and private depository institutions demand foreignfinancial assets The analysis would be more complex if we explicitly recognized thesedemands but our conclusions would be unchanged since we can subsume in householdactions the actions of firms and depository institutions in foreign markets
References
Aigner DJ and Cain GC (1977) Statistical theories of discrimination in labor marketsIndustrial and Labor Relations Review 30(2) 175ndash187
Alchian A and Demsetz H (1972) Production information costs and economicorganization American Economic Review 62 777ndash795
Alogoskoufis GS (1987) On intertemporal substitution and aggregate labor supplyJournal of Political Economy 95 938ndash960
Altonji J and Siow A (1987) Testing the response of consumption to income changeswith (noisy) panel data Quarterly Journal of Economics 102 293ndash328
Arrow K (1964) The role of securities in the optimal allocation of risk-bearing Review ofEconomic Studies 31 (April) 91ndash96
Arrow K and Hahn FH (1971) General Competitive Analysis San Francisco CAHolden-Day
Azariadis C (1976) On the incidence of unemployment Review of Economic Studies 43115ndash125
Barro RJ (1976) Rational expectations and the role of monetary policy Journal ofMonetary Economics 2 1ndash32
Barro RJ and Gordon D (1983) A positive theory of monetary policy in a natural ratemodel Journal of Political Economy 91 589ndash610
Barro RJ and Grossman HI (1971) A general disequilibrium model of income andemployment American Economic Review 61 82ndash93
Barro RJ and King RG (1984) Time-separable preferences and intertemporal substitu-tions models of business cycles Quarterly Journal of Economics 99 817ndash839
Begg DKH (1980) Rational expectations and the non-neutrality of systematic monetarypolicy Review of Economic Studies 47 293ndash303
Blanchard OJ (1981) What is left of the multiplier accelerator American EconomicReview 71 150ndash154
Blanchard OJ and Fischer S (1989) Lectures on Macroeconomics Cambridge MA MITPress
Borjas GJ and Heckman JJ (1978) Labor supply estimates for public policy evaluationNBER Working Paper No W0299 November
Box GEP and Jenkins GM (1970) Time Series Analysis San Francisco CAHolden-Day
Burbidge JB and Robb AL (1985) Evidence on wealthndashage profiles in Canadian cross-section data Canadian Journal of Economics 18 854ndash875
Caballero R and Engel E (1999) Explaining investment dynamics in US manufacturingA generalized (S s) approach Econometrica 67 783ndash826
224 References
Campbell JY and Mankiw NG (1987a) Permanent and transitory components inmacroeconomic fluctuations American Economic Review 77 111ndash117
Campbell JY and Mankiw NG (1987b) Are output fluctuations transitory QuarterlyJournal of Economics 102 857ndash880
Campbell JY Lo AW and MacKinlay AC (1997) The Econometrics of FinancialMarkets Princeton NJ Princeton University Press
Cass D (1965) Optimal growth in an aggregate model of capital accumulation Review ofEconomic Studies 32 233ndash240
Clower RW (1965) The Keynesian Counter-revolution A theoretical appraisal InFH Hahn and FPR Brechling (eds) The Theory of Interest Rates London Macmillan
Coase RH (1960) The problem of social cost Journal of Law and Economics 31ndash44
Cukierman A (1986) Measuring inflationary expectations Journal of Monetary Eco-nomics 17 315ndash324
Dahlman C (1979) The problem of externality Journal of Law and Economics 22141ndash162
Debreu G (1959) The Theory of Value New York WileyDiamond PA and Hausman JA (1984) Individual retirement and savings behaviour
Journal of Public Economics 23 81ndash114Dornbusch R (1976) Expectations and exchange rate dynamics Journal of Political
Economy 84 1161ndash1176Enders W (2004) Applied Econometric Time Series 2nd edn Hoboken NJ WileyEvans G (1989) Output and unemployment dynamics in the United States 1950ndash1985
Journal of Applied Econometrics 4 213ndash237Ewing BT (2001) Cross-effects of fundamental state variables Journal of
Macroeconomics 23 633ndash645Ewing BT and Payne JE (1998) The long-run relation between the personal savings rate
and consumer sentiment Financial Counseling and Planning 9(1) 89ndash96Ewing BT Levernier W and Malik F (2002) Differential effects of output shocks on
unemployment rates by race and gender Southern Economic Journal 68 584ndash599Fama EF and Miller MH (1972) The Theory of Finance New York Holt Rinehart and
WinstonFischer S (1977) Long-term contracts rational expectations and the optimal money supply
rule Journal of Political Economy 85 191ndash205Fisher I (1930) The Theory of Interest New York MacmillanFleming MJ (1962) Domestic financial policies under fixed and under floating exchange
rates IMF Staff Papers 9 369ndash379Foley KD (1975) On two specifications of asset equilibrium in macroeconomic models
Journal of Political Economy 83 303ndash324Friedman M (1959) A Program for Monetary Stability New York Fordham University
PressFriedman M (1968) The role of monetary policy American Economic Review 58 1ndash17Goldberg S (1958) Difference Equations New York WileyGould JP (1968) Adjustment cost in the theory of investment of the firm Review of
Economic Studies 35 47ndash56Grandmont JM (1977) Temporary general equilibrium theory Econometrica 45
535ndash572Granger CWJ and Morris M (1976) Time series modelling and interpretation Journal
of the Royal Statistical Society Series A 139 246ndash257
References 225
Granger CWJ and Newbold P (1986) Forecasting Economic Time Series 2nd ednOrlando FL Academic Press
Hall R (1976) The Phillips curve and macroeconomic policy In K Brunner andAH Meltzer (eds) The Phillips Curve and Labor Markets Carnegie-RochesterConference Series on Public Policy Amsterdam North-Holland
Hall R (1980) Employment fluctuation and wage rigidity In GL Perry (ed) BrookingsPapers on Economic Activity pp 91ndash123 Washington DC Brookings Institution
Hall RE (1982) Explorations in the Gold Standard and related policies for stabilizingthe dollar In RE Hall (ed) Inflation Causes and Effects Chicago IL University ofChicago Press
Hall RE (1988) Intertemporal substitution in consumption Journal of Political Economy96 339ndash357
Hannan EJ (1970) Multiple Time Series New York WileyHansen B (1970) A Survey of General Equilibrium Systems New York McGraw-HillHansen LP and Singleton KJ (1983) Stochastic consumption risk aversion and the
temporal behavior of asset returns Journal of Political Economy 91 249ndash265Harvey AC (1993) Time Series Models Cambridge MA MIT PressHayashi F (1982) Tobinrsquos marginal q and average q A neoclassical interpretation
Econometrica 50(1) 213ndash224Hicks J (1939) Value and Capital Oxford Clarendon PressHowitt P (1985) Transaction costs in the theory of unemployment American Economic
Review 75 88ndash100Howitt P (1986) Conversations with economists A review essay Journal of Monetary
Economics 18 103ndash118Hurd M (1987) Savings of the elderly and desired bequests American Economic Review
77 298ndash312Karni E (1978) Period analysis and continuous analysis in Patinkinrsquos macroeconomic
model Journal of Economic Theory 17 134ndash140Kester WC (1986) Capital ownership structure A comparison of the United States and
Japanese manufacturing corporations Financial Management 15(1) 5ndash16Keynes JM (1936) General Theory of Employment Interest and Money London
MacmillanKletzer LG and Fairlie R (1998) Jobs lost jobs regained An analysis of blackwhite
differences in job displacement in the 1980s Industrial Relations 37 460ndash477Knetter M (1987) Export prices and exchange rates Theory and evidence Working paper
Stanford University NovemberKoopmans T (1965) On the concept of optimal economic growth In Proceedings ndash
Study Week on the Econometric Approach to Development Planning Chicago ILRand-McNally
Kydland FE and Prescott EC (1977) Rules rather than discretion The inconsistency ofoptimal plans Journal of Political Economy 85 473ndash491
Kydland FE and Prescott EC (1982) Time to build and aggregate fluctuationsEconometrica 50 1345ndash1370
Leonard JS (1988) In the wrong place at the wrong time The extent of frictional andstructural unemployment NBER Working Paper No 1979
Lucas RE (1967) Adjustment costs and the theory of supply Journal of Political Economy75 321ndash334
Lucas RE (1972) Expectations and the neutrality of money Journal of Economic Theory4 103ndash124
226 References
Lucas RE (1973) Some international evidence on outputndashinflation tradeoffs AmericanEconomic Review 63 326ndash334
Lucas R Jr (1981) Methods and problems in business cycle theory In Studies in Business-Cycle Theory Cambridge MA MIT Press First published in Journal of Money Creditand Banking 12 November 1980
Lucas RE and Rapping LA (1970) Real wages employment and inflation InES Phelps (ed) Microeconomic Foundations of Employment and Inflation TheoryNew York WW Norton
Lundberg S and Startz R (1983) Private discrimination and social intervention incompetitive labor markets American Economic Review 73(3) 340ndash347
McCallum B (1979) The current state of the policy-ineffectiveness debate AmericanEconomic Review 69 240ndash245
McCallum B (1985) On consequences and criticisms of monetary targeting Journal ofMoney Credit and Banking 17 570ndash597
Mankiw NG (1987) The optimal collection of seigniorage Theory and evidence Journalof Monetary Economics 20 327ndash341
Mankiw NG Rotemberg JJ and Summers LH (1985) Intertemporal substitution inmacroeconomics Quarterly Journal of Economics 100 225ndash251
Marx K (1976) Capital Vol 1 A Critique of Political Economy HarmondsworthPenguin
Mills TC (1999) The Econometric Modelling of Financial Time Series 2nd ednCambridge Cambridge University Press
Modigliani F (1966) Life cycle hypothesis of saving the demand for wealth and the supplyof capital Social Research 33 160ndash217
Modigliani F (1986) Life cycle individual thrift and the wealth of nations AmericanEconomic Review 76 297ndash313
Mundell RA (1968) International Economics New York MacmillanNakamura A and Nakamura M (1981) A comparison of the labor force behavior of
married women in the US and Canada with special attention to the impact of incometaxes Econometrica 49 451ndash489
Nelson CR and Plosser CI (1982) Trends and random walks in macroeconomictime series Some evidence and implications Journal of Monetary Economics 10139ndash162
Niehans J (1987) Classical monetary theory new and old Journal of Money Credit andBanking 19 409ndash424
Oi WY (1962) Labor as a quasi-fixed factor Journal of Political Economy 70 538ndash555Patinkin D (1965) Money Interest and Prices New York Harper amp RowPencavel J (1985) Labor supply of men A survey In Orley Ashenfelter (ed) Handbook
of Labor Economics Amsterdam North-HollandPhelps ES (1968) Money-wage dynamics and labor market equilibrium Journal of
Political Economy 76 678ndash711Phelps ES and Taylor JB (1977) Stabilizing powers of monetary policy under rational
expectations Journal of Political Economy 85 163ndash190Phillips AW (1958) The relationship between unemployment and the rate of change in
money wage rates in the United Kingdom 1861ndash1957 Economica 25 283ndash299Pindyck RS and Rubinfeld DL (1991) Econometric Models and Economic Forecasts
3rd edn New York McGraw-HillPoole W (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo
Journal of Money Credit and Banking 17 602ndash605
References 227
Radford RA (1945) The economic organization of a prisoner of war camp Economica12 189ndash201
Robinson C and Tomes N (1985) More on the labour supply of Canadian womenCanadian Journal of Economics 18 156ndash163
Sargent T (1987a) Macroeconomic Theory 2nd edn Boston MA Academic PressSargent T (1987b) Dynamic Macroeconomic Analysis Cambridge MA Harvard
University PressSargent TJ and Wallace N (1975) ldquoRationalrdquo expectations the optimal monetary
instrument and the optimal money supply rule Journal of Political Economy 83241ndash254
Sargent TJ and Wallace N (1976) Rational expectations and the theory of economicpolicy Journal of Monetary Economics 2 169ndash183
Shapiro C and Stiglitz JE (1984) Equilibrium unemployment as a worker disciplinedevice American Economic Review 74 433ndash444
Shiller RJ (1978) Rational expectations and the dynamic structure of macroeconomicmodels Journal of Monetary Economics 4 1ndash44
Sidrauski M (1967a) Rational choice and patterns of growth in a monetary economyAmerican Economic Review 57 534ndash544
Sidrauski M (1967b) Inflation and economic growth Journal of Political Economy 75797ndash810
Simons HC (1936) Rules versus authorities in monetary policy Journal of PoliticalEconomy 44 1ndash30
Sims C (1972) Money income and causality American Economic Review 62 540ndash552Solow R (1956) A contribution to the theory of economic growth Quarterly Journal of
Economics 70 65ndash94Strotz RH (1955ndash1956) Myopia and inconsistency in dynamic utility maximization
Review of Economic Studies 23 165ndash180Stuart CE (1981) Swedish tax rates labor supply and tax revenues Journal of Political
Economy 89 1020ndash1038Taylor J (1972) The behaviour of unemployment and unfilled vacancies Great Britain
1958ndash71 An alternative view Economic Journal 82 1352ndash1365Taylor JB (1979) Estimation and control of a macroeconomic model with rational
expectations Econometrica 47 1267ndash1286Tobin J (1965) Money and economic growth Econometrica 33 671ndash684Tobin J (1969) A general equilibrium approach to monetary theory Journal of Money
Credit and Banking 1(1) 15ndash29Tobin J (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo or
Monetary targeting Dead at last Journal of Money Credit and Banking 17 605ndash610Uzawa H (1969) Time preference and the Penrose effect in a two-class model of economic
growth Journal of Political Economy 77 628ndash652Varian H (1992) Microeconomic Analysis 3rd edn New York WW NortonWalker DA (1987) Walrasrsquo theories of tatonnement Journal of Political Economy 95
758ndash774Walras L (1954) Elements of Pure Economics trans W Jaffeacute London George Allen amp
UnwinWeber CE (1998) Consumption spending and the paper-bill spread Theory and evidence
Economic Inquiry 36 575ndash589Weitzman M (1985) The simple macroeconomics of profit sharing American Economic
Review 75 937ndash953
Index
accelerationist outcome 170ndash2aggregate demand 7 86 90ndash1 94ndash5 99
Keynesian model 133ndash4 135 139 140Lucas model 160ndash1 neoclassical model126 127 128 146 shocks 158 seealso demand
aggregate supply 82 86ndash90 91 94ndash5 98113 Keynesian model 132 133ndash4 135138 140 145 Lucas model 137153ndash4 155 158 160ndash1 164 166monetary policy 170ndash1 neoclassicalmodel 118 123 125ndash6 127 128 133see also supply
aggregation issues 5ndash6 8 14ndash15Alchian A 203n9Alogoskoufis GS 51Angell Wayne 219n4Arrow-Debreu theory 2 202n4 n10
203n8assets 51 52 53 portfolio decision
39ndash40 57ndash9 tangible 25ndash6 see alsofinancial assets
autocorrelation function 102ndash3autoregressive expectations 162ndash4 168autoregressive processes 103ndash5 110 111
113 166
balance of payments 190ndash3bankruptcy 32 33Barro RJ 156 167 179ndash80 182ndash4Begg DKH 122 215n8behavioral hypotheses 96 98 99 100Bellman equation firms 29 30 31 34
35 households 43 44 45 48ndash9Blanchard OJ 111 112 171 175 184ndash5bonds 19ndash20 23 27ndash9 32ndash3 financial
market equilibrium 84ndash5 firmfinancing constraint 24ndash6 64 Fisherian
problem 52 foreign 189 households39 40 43ndash4 72 204n11 money illusion59 portfolio choice 57 58 59 93 realvalue 77 temporary equilibrium 80 81
budget constraint 7 16 household 43 4455 65ndash6 67 70ndash1 73ndash4 open economy188 189
Burbidge JB 208n25business cycle 5 6 49 50 173 real
business cycle theory 79 136 202n4technological innovation 113
Caballero R 209n1Campbell JY 111 112 113 214n45capital 23 24 31 68 69ndash70 cost of
31ndash2 36 69 70 94 120 209n7 firmfinancing constraint 24 25 26 64international flows 192 193 195ndash6197ndash8 200 Tobinrsquos Q 36 see alsocapital stock
capital account 192capital adjustment costs 21 26ndash7 34ndash7
69ndash70capital markets 27 203n1 205n24capital stock 3 18ndash19 20ndash1 23 25 27ndash9
optimal investment 30 31 69 70retained earnings financing 29ndash30 32superneutrality of money 120 121 122Tobinrsquos Q 35 36
central banks 187 188 189 190191 192ndash3
classical economics 4 5 79 212n8Clower RW 117 118Coase RH 11CobbndashDouglas production function 147
148 217n3commodities 8 9ndash10 14 16 18ndash19 20
individual experiments 11 13ndash14
Index 229
Lucas model 146ndash7 market equilibrium92 open economy 188ndash9
comparative static analysis 116ndash23consumer behavior 11 12ndash13consumption 20 39 40ndash6 47ndash8 67
70ndash4 aggregate 55 61 demand 97 99195 215n6 financial asset demand 94Fisherian problem 51ndash5 individualexperiments 11 12ndash13 intertemporalsubstitution hypothesis 51 60ndash3Keynesian model 134 135 laborparticipation 49 life-cycle hypothesis50 55ndash6 57 money supply shocks 128neoclassical model 117ndash18 123 126neutrality of money 119 open economy190ndash1 193 permanent incomehypothesis 56ndash7 portfolio choice 5758 59 real balance effect 77 92superneutrality of money 121 122
continuous-time analysis 4cost of capital 31ndash2 36 69 70 94 120
209n7cost of living agreements (COLAs) 215n2costs bankruptcy 33 capital adjustment
21 26ndash7 34ndash7 69ndash70 dividends 23ndash4firm financing constraint 24
Cramerrsquos rule 87 119 121 127 134Cukierman A 164currency depreciation 193 195 198ndash9
221n17
Dahlman C 203n9De Moivrersquos theorem 109Debreu G 8debt 32 60debt-to-equity ratio 32ndash4decision-making behavior 5ndash6 38 51 see
also portfolio decisiondemand excess 13ndash14 15ndash16 67 118
133 190ndash1 199 individual demandfunctions 13 14 labor marketequilibrium 83 Lucas model 146156ndash7 market demand functions 1415 money 86 neoclassical model82ndash3 119 122 123 new classicaleconomics 5 price elasticity of 195real balance effect 60 77 Sayrsquos law 716 temporary equilibrium 80Walrasian model 7 15 see alsoaggregate demand
Demsetz H 203n9depository institutions 126 187 188 189
221n5 222n18
depreciation 26 31 74 205n23deterministic feedback rules 172 173 174deterministic models 6disequilibrium 78 199distributed lag schemes 162ndash4dividends depository institutions 221n5
firms 20 22 23ndash4 29 30 32 34ndash5future 76ndash7 households 43 44 66
Dornbusch model 200ndash1dynamic analysis 2 3ndash4 6 79 122dynamic programming 28ndash9
econometric models 96 100ndash1employment 1 19 128 148 full
employment model 88 142 Keynesianmodel 130ndash1 132ndash3 134 135 137neoclassical model 82 83 84 91 122123ndash5 126 see also labor labor marketlabor supply
Engel E 209n1equilibrium 1 5 7 15 199 aggregate
demand 90ndash1 94ndash5 99 aggregatesupply 86ndash8 94ndash5 employment123ndash5 financial markets 80ndash1 84ndash6illusion model 79 Keynesian model79 133ndash4 137 139ndash40 labor market83ndash4 88 90 119 123ndash5 127 131 148loanable funds theory 120 Lucas model159 160 162 monetary policy 182ndash4money market 85ndash6 92ndash3 126 159multiple equilibria models 113 naturalrate of unemployment 89 90neoclassical model 78 79 117ndash18 126non- market-clearing model 79 partial187 Patinkin analysis 91ndash4 stationarity106 stochastic models 6 temporary 278 80ndash3 86 120 velocity 139ndash40 seealso general equilibrium
equity shares 18 19ndash20 21ndash2 23 27ndash932ndash3 firm financing constraint 24ndash664 households 40 43 72 real value77 temporary equilibrium 80 81
ergodicity 212n16Eulerrsquos theorem 206n44Ewing BT 211n11 212n11exchange rate 186ndash7 191 193ndash6 197
199 200ndash1expectations 41 59 84 119 202n5
adaptive 163 170ndash2 210n15 aggregatedemand 90ndash1 autoregressive 162ndash4168 inflation 62 69 71 77 120ndash2163 164 interest rate 60 61 62 6971 75ndash7 Keynesian model 135 136ndash8
230 Index
expectations (Continued)140 141ndash4 145 labor marketequilibrium 83 Lucas model 157 160optimal monetary policy 168ndash70 180182 183ndash4 output 68 rational 3128ndash9 136ndash8 141ndash4 164ndash6 172ndash4176ndash8 182ndash4 suppliers 124 135 weakconsistency 203n4 see also forecasting
exports 190ndash1 193 197 200
financial assets 77 85 93ndash4 118 firms19ndash20 25 26 28 29 68ndash9 70households 40 42 43ndash4 66 71 72ndash4labor market equilibrium 83 openeconomy 186 187ndash8 189 191 192193 195ndash6 197ndash9 temporaryequilibrium 81ndash2 see also bondsequity shares
financial market 1 36 67 96ndash7equilibrium 80ndash1 84ndash6 93 94 95 118open economy 191 195ndash6 197 199
firm distribution constraint 23ndash4 28 4165 67 74 76ndash7 189
firm financing constraint 23 24ndash6 28ndash964 67 68ndash70 84ndash5
firms 18ndash38 64ndash5 68ndash70 130 135aggregate supply 87 capital adjustmentcosts 26ndash7 34ndash7 dividends 23ndash4investment demand 97 laboradjustment costs 37ndash8 money illusion75 119 neoclassical model 79objectives 21ndash3 open economy 187188 189 190 222n18 optimalinvestment 30ndash2 real wage illusion124 125 retained earnings financing29ndash30 34
Fischer S 112 136 137 138ndash45 171175 184ndash5
Fisher I 79 163 208n16Fisherian problem 39 46 51ndash5 209n12
215n6forecasting 96 100ndash1 101ndash14 128
149ndash50 see also expectationsforeign exchange market 186ndash7 188 191
193ndash5 198ndash9 200foreigners 187ndash90 191 197ndash8Friedman Milton 56 156 168 171 179futures market 3 7 8 10 65 66
general equilibrium 1 2 11 95 97 187neoclassical model 117 118 prices 710 15ndash16 Walrasian model 8 see alsoequilibrium
GNP see gross national productGordon D 156 167 179ndash80 182ndash4government 187 190Grandmont JM 78Granger CWJ 100 102 112 212n22gross national product (GNP) 112 168
habit persistence model 207n2Hall Robert 60ndash1 62 172Hansen B 8Harvey AC 101 114Hayashi F 36 206n43Hicks John 2 8households 18 19 39ndash63 65ndash6 70ndash4
aggregate supply 87 bonds and equityshares 20 24 consumption demand97 dividends 23 24 Fisherian problem39 46 51ndash5 imperfect foresight 124intertemporal substitution hypothesis49ndash51 60ndash3 labor supply problem46ndash51 life-cycle hypothesis 55ndash6money illusion 75ndash6 77 119neoclassical model 79 open economy187 188 189 191 193ndash6 222n18permanent income hypothesis 56ndash7portfolio decision 39ndash40 46 57ndash9price changes 21
Howitt P 5 203n15
imperfect foresight 123 124imports 193ndash5 199 200income 59 74 97 208n27 foreign goods
195 197 life-cycle hypothesis 55ndash657 neoclassical model 117 permanentincome hypothesis 56ndash7 see also wages
income effects 47ndash8 49individual experiments 11ndash14 21 39inflation 53 59 72 196 200
expectations 62 69 71 77 120ndash2 163164 Lucas model 146 154ndash6 157160 161 monetary policy 136 170ndash2174ndash9 180 181ndash2 183ndash4superneutrality of money 121 122wages 123 131 see also prices
integrated processes 110ndash11 112interest rate 1 4 20 31 36 59
consumption demand 97 equilibrium91ndash4 equity shares 22 23 expectations60 61 62 69 71 75ndash7 financialmarket equilibrium 84 85 94Fisherian problem 53 54 foreignfinancial assets 195 196 198household problem 43 45 46 47
Index 231
intertemporal substitution hypothesis49 50 51 63 72 Keynesian model133 134 labor demand 83 loanablefunds theory 120 Lucas model159ndash60 neoclassical model 118 119126ndash7 neutrality of money 119 parity200ndash1 superneutrality of money 121ndash2temporary equilibrium 81 82
intertemporal substitution hypothesis (ISH)49ndash51 60ndash3 72 73 75ndash6
invertibility condition 110investment 3 29 84 85 128 capital
adjustment costs 26ndash7 35 36 demand32 68ndash70 75 90 94 97 99 120206n40 Keynesian model 134 135neoclassical model 118 126 neutralityof money 119 optimal 30ndash2superneutrality of money 121 122supply-side variables 116
IS equation 79 90ndash1 98ndash9 118 126 191export demand 200 Keynesian model133 134 138ndash9 Lucas model 159 160
ISH see intertemporal substitutionhypothesis
lsquoislandrsquo paradigm 146ndash9
J-curve effect 221n13
Karni Edi 207n3Keynes John Maynard 4Keynesian model 5 79 88 130ndash45 199Knetter M 221n17Kydland FE 180ndash1 207n2 220n17
labor 19 21 27 42 66 adjustment costs37ndash8 demand 68 76 97 132ndash3 147215n9 217n4 Keynesian models 5marginal product of 30 optimalinvestment 31 participation 48ndash9temporary equilibrium 80 see alsoemployment labor supply wages
labor market 1 18 37 65 96ndash7 186aggregate supply 86ndash8 94 95 119intertemporal substitution hypothesis49ndash51 Keynesian model 131ndash3 134Lucas model 147 148 natural rate ofunemployment 89 90 neoclassicalmodel 83ndash4 118 124ndash5 127 Walrasrsquolaw 66ndash7 82 see also employment
labor reserve hypothesis 37ndash8labor supply 39 40 41 45 46ndash9 215n9
intertemporal substitution hypothesis49ndash51 72 73 75ndash6 210n15 Keynesian
model 132 133 Lucas model 147ndash8neoclassical model 79 83ndash4 117 123124 perfect foresight 66 70 71 realbalance effect 77 supply-side variables116 tax rates 99 see also employmentlabor
leisure 40ndash1 42 44 46ndash8 49 50 72life-cycle hypothesis 50 55ndash6 57linear regression analysis 150ndash3LM equation 79 90ndash1 98ndash9 118 126
191 Keynesian model 133 134 138ndash9Lucas model 159 160
loanable funds theory 120Lucas RE 6 50ndash1 146 149 156ndash7
164 167 203n13 210n15Lucas model 88 128 135ndash6 140 146ndash66
176 199Lucas supply function 79 137 153ndash6
158 174 175
Mankiw NG 111 112 113 136 214n45market clearing 4ndash5 6 7 15 50 79
commodity market 92 financial market93 Lucas model 146 164 wages 135
market experiments 11 14ndash16markets 1 2 20 78 187 aggregate
demand 86 90ndash1 capital 27 203n1205n24 foreign exchange 186ndash7 188191 193ndash5 198ndash9 200 futures 3 7 810 65 66 Lucas model 146 149 154sequential 67 see also financial marketlabor market money market
Marx Karl 79microeconomics 5 6 78 88 187mixed autoregressive moving average
processes 110Modigliani Franco 55ndash6 208n24ModiglianindashMiller theorem 32monetary policy 166 167ndash85 187
activist 169 171 174 discretionary136 167ndash8 184 219n1 Keynesianmodel 136ndash8 141 142 144ndash5 policyineffectiveness proposition 120 136137ndash8 172ndash9 182
money classical analysis 4 comparativestatic analysis 118 demand 71ndash4 7797 121 122 139ndash40 159 financialmarket equilibrium 84 householdproblem 41 42 43ndash4 45 46 66neoclassical model 117ndash18 neutralityof 119ndash20 136 142 173 174portfolio decision 39 57 58 59 72superneutrality of 120ndash2
232 Index
money (Continued)temporary equilibrium 80 82transaction costs 11 utility yield of 42see also money market money supply
money illusion 59 75ndash6 77 88 119 125210n15
money market 7 16 67 96ndash9 199aggregate demand 90 94 95equilibrium 85ndash6 92ndash3 126 159Keynesian model 133 134
money supply 3 76 100 119 168 190Keynesian model 134ndash5 142 Lucasmodel 159ndash60 161 162 monetarypolicy 144 169 170 173 178ndash9 180money market equilibrium 86 91neoclassical model 116 118ndash20 126ndash9purchasing power parity 200superneutrality of money 120ndash2 seealso money
moving average processes 109ndash10multiple equilibria models 113MundellndashFleming model 201
natural rate hypothesis 125 128ndash9 135166
Nelson CR 113neoclassical model 4ndash5 116ndash29 146 199
202n12 Keynesian model comparison133 136 Lucas model comparison 160162 modification of 130ndash1 purchasingpower parity 200 simple 78ndash95
new classical economics 4ndash5 6 7Newbold P 100 102 112 212n22numeraire 7 8 9ndash10
official reserve transaction balance 192open economy 186ndash201output 1 18 19 20 66ndash7 202n1
aggregate demand 86 90 94 95 139aggregate supply 86 87ndash8 classicalanalysis 4 comparative static analysis118 Dornbusch model 200 equilibrium80 82 91ndash2 94 126 expectations 6875 firm distribution constraint 65Keynesian model 5 133ndash4 135 137ndash8143ndash4 145 labor market equilibrium83 84 Lucas model 79 146 147148ndash9 153ndash5 157ndash62 165ndash6 monetarypolicy 144 168 170ndash2 173 174ndash5179 money supply shocks 128 movingaverage processes 109 neoclassicalmodel 117ndash18 122 123 125 126neutrality of money 119 open economy
187ndash8 189 190ndash1 199 simpletheoretical model 96ndash9 supply-sidevariables 116 136 time series model111ndash14
overlapping generations models 6
Patinkin D 8 11 120 202n12 critiqueof 117 equilibrium 83 84ndash5 91ndash4firms 210n5 labor supply 47
Payne JE 212n11perfect competition 7 11perfect foresight 3 87 firms 65 68ndash70
75 households 41 43 56ndash7 66 70ndash475 labor market equilibrium 83 125Lucas model 161 162 neoclassicalmodel 78 79 84 117 118 119 131Walrasrsquo law 66 67 82
perfect substitution 20 22 29 32period (discrete) analysis 4permanent income hypothesis 56ndash7Petty William 79Phelps ES 136ndash7 171 216n11Phillips AW 155Phillips curve 154ndash5 156ndash7 171 174ndash9
180 181 183Pindyck RS 101 102Plosser CI 113policy see monetary policypolicy ineffectiveness proposition 120
136 137ndash8 172ndash9 182Poole William 219n3portfolio decision 39ndash40 46 51 57ndash9 72
93 193preferences 11 40Prescott EC 180ndash1 207n2 220n17price elasticity of demand 195prices 1 2 3 19 68 accounting 7 10
13 15 aggregate demand 90 91 9495 aggregate supply 86 98autoregressive expectations 162ndash4classical analysis 4 comparative staticanalysis 116 demand-side variables117 equity shares 20 22 forecastingerrors 149ndash50 foreign goods 193ndash5197 future 41 59 75 76 householdbudget constraint 66 74 individualexperiments 11 Keynesian model 133134 135 137 145 labor marketequilibrium 83 84 Lucas model 146147 149 153 154 156ndash8 159ndash64market experiments 11 microeconomicfoundations 6 monetary policy 172ndash3money market equilibrium 86 natural
Index 233
rate hypothesis 125 natural rate ofunemployment 88ndash9 neoclassicalmodel 78ndash9 117ndash20 122ndash3 124ndash5126ndash8 130 136 146 new classicaleconomics 5 7 perfect foresight 71purchasing power parity 200 realindebtedness effect 21 relative 7 89ndash10 11 13 15ndash16 sequential markets67 superneutrality of money 120tatonnement process 15 temporaryequilibrium 80 82 Walrasian model 79ndash10 15 see also inflation
private international capital flows 192193 195ndash6 197ndash8 200
production 18ndash19productivity of labor 37purchasing power parity 200
Radford RA 8lsquorandom walkrsquo process 103 106 112ndash13Rapping LA 50ndash1 210n15rational expectations 3 128ndash9 136ndash8
141ndash4 164ndash6 172ndash4 176ndash8 182ndash4 seealso expectations
real balance effect 60 77 86 92ndash3 94121
real indebtedness effect 60real user cost of capital 31ndash2 36 69 70
94 120 209n7recontracting assumption 15reduced-form expressions 96 98 99ndash100
Lucas model 161 162 monetary policy168ndash9 172ndash3 177
reputation 184ndash5retained earnings financing 29ndash30 32 34risk 41Robb AL 208n25Rubinfeld DL 101 102
Sargent T 8 96 131ndash2 211n2 n6autoregressive expectations 164employees 217n2 investment demand206n40 Keynesians 219n2 linearregression analysis 152 monetarypolicy 136ndash7 167 168 170 172ndash4180 182 money supply 165 newclassical economics 203n13outputinflation tradeoff 157 Phillipscurve 155 superneutrality of money121 122
Say Jean-Baptiste 7Sayrsquos law 8 16seasonality 111 112
shareholders 25ndash6shares see equity sharesShiller RJ 162shocks 1 6 153 158 186 demand-side
117 monetary policy 170 moneysupply 128 129 135 Phillips curve156 time-series models 111ndash13 wages135ndash6
Simons HC 219n4Sims Christopher 100spot markets 2 37 130 131spot prices 10static analysis 2ndash4 47 50 79 116ndash23stationarity 102 103 104 105ndash9 110ndash11
113 114stationary analysis 2 3ndash4stochastic processes 6 41 101 102ndash3
110 172 173Stuart CE 99substitution effects 47ndash8 49 see also
intertemporal substitution hypothesissupply labor market equilibrium 83
Lucas supply function 79 137 153ndash6158 174 175 neoclassical model 119new classical economics 5 real balanceeffect 77 Sayrsquos law 7 16 temporaryequilibrium 80 86 Walrasian model 715 see also aggregate supply laborsupply money supply
tatonnement process 7 15 120tax issues 3 32 33 99Taylor JB 100 136ndash7 212n12 216n11technology 18ndash19 21terms of trade 194lsquotime inconsistency problemrsquo 180ndash2time series models 96 100 101ndash14Tobin James 206n40 219n2Tobinrsquos Q 27 35ndash7transaction costs 5 7 10ndash11 20
42 208n18
uncertainty 6 41unemployment 1 50ndash1 104 125
frictional 89 involuntary 211n13Lucas model 146 154ndash6 monetarypolicy 174ndash5 176 179 181ndash2 183184 natural rate of 88ndash90 135 156174ndash5
utility 13 17 40ndash1 42 46 48 61ndash2
velocity equations 139ndash40
234 Index
wages 18 19 23 24 32 36 aggregatedemand 90 91 aggregate supply 86ndash7anticipated 69 70 71 75ndash6 77 124household problem 42 43 44 47ndash866 intertemporal substitution hypothesis50 51 72 73 Keynesian model 130ndash3135ndash6 137ndash8 142ndash4 145 labordemand 68 97 labor marketequilibrium 83 84 labor participation48ndash9 Lucas model 147ndash8 moneysupply shocks 128 neoclassical model117 122ndash3 126 neutrality ofmoney 119 real wage illusion 123ndash5sticky 123 131 135ndash6 137temporary equilibrium 80 82 see alsoincome
Wallace N autoregressive expectations164 Keynesians 219n2 monetarypolicy 136ndash7 167 168 170 172 180182 money supply 165 superneutralityof money 121 122
Walrasrsquo law 1 8 14 16 85 91 balance ofpayments 190ndash1 excess demand 118financial assets 93 labor market 66ndash7market equilibrium 97 99 openeconomy 187 199 perfect foresight82 sequential markets 67
Walras Leon 1 16 202n2 203n2Walrasian models 7ndash11 15 18ndash20wealth 55 59 60 83Weber CE 209n2working hours 47ndash8
Routledge Advanced Texts in Economics and Finance
Financial EconometricsPeijie Wang
Macroeconomics for Developing Countries 2nd editionRaghbendra Jha
Advanced Mathematical EconomicsRakesh V Vohra
Advanced Econometric TheoryJohn S Chipman
Understanding Macroeconomic TheoryJohn M Barron Bradley T Ewing and Gerald J Lynch
Understanding MacroeconomicTheory
John M Barron Bradley T Ewingand Gerald J Lynch
First published 2006by Routledge270 Madison Ave New York NY 10016
Simultaneously published in the UKby Routledge2 Park Square Milton Park Abingdon Oxon OX14 4RN
Routledge is an imprint of the Taylor amp Francis Group an informa business
copy 2006 John M Barron Bradley T Ewing and Gerald J Lynch
All rights reserved No part of this book may be reprinted orreproduced or utilized in any form or by any electronicmechanical or other means now known or hereafterinvented including photocopying and recording or in anyinformation storage or retrieval system without permission inwriting from the publishers
Library of Congress Cataloging in Publication DataBarron John M
Understanding macroeconomic theory John M BarronBradley T Ewing and Gerald J Lynch
p cmIncludes bibliographical references and index1 Macroeconomics I Ewing Bradley T II Lynch Gerald J III Title
HB1725B3753 2006339ndashdc22 2005026372
British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library
ISBN10 0ndash415ndash70195ndash3 ISBN13 978ndash0ndash415ndash70195ndash2 (hbk)ISBN10 0ndash415ndash70196ndash1 ISBN13 978ndash0ndash415ndash70196ndash9 (pbk)ISBN10 0ndash203ndash08822ndash0 ISBN13 978ndash0ndash203ndash08822ndash7 (ebk)
This edition published in the Taylor amp Francis e-Library 2006
ldquoTo purchase your own copy of this or any of Taylor amp Francis or Routledgersquoscollection of thousands of eBooks please go to wwweBookstoretandfcoukrdquo
Contents
1 Introduction 1
2 Walrasian economy 7
3 Firms as market participants 18
4 Households as market participants 39
5 Summarizing the behavior and constraints of firms andhouseholds 64
6 The simple neoclassical macroeconomic model (withoutgovernment or depository institutions) 78
7 Empirical macroeconomics traditional approaches and timeseries models 96
8 The neoclassical model 116
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model 130
10 The Lucas model 146
11 Policy 167
12 Open economy 186
Notes 202References 223Index 228
1 Introduction
The topics of macroeconomics
At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activity canbe summarized by aggregate variables such as an economyrsquos total production ofvarious goods and services the aggregate level of employment and unemploymentthe general level of interest rates and the overall level of prices1 Macroeconomicsis the study of movements in such economy-wide variables as output employmentand prices
The focus of this book will be on developing simple theoretical models thatprovide insight into the reasons for fluctuations in such aggregate variables Thesemodels explore how shocks or ldquoimpulsesrdquo to the economy (eg changes to tech-nology the money supply or government policy) impact individualsrsquo behavior inspecific markets and the resulting implications in terms of changes in aggregatevariables
An overview of some facets of theoreticalmacroeconomic analysis
Given the breadth of economic activity in an economy the study of macroeco-nomics must involve an examination of a variety of different markets For instanceit is common for macroeconomic analysis to consider exchanges of labor servicesin the labor markets of consumption and capital goods in the output marketsand of financial assets in the financial markets The fact that macroeconomicssimultaneously analyses exchanges of different goods in different markets meansthat macroeconomic theory is a general equilibrium theory That is macro-economic theory must by necessity incorporate the links across markets that arefundamental to general equilibrium analysis As we will see throughout this booka key reflection of the links across markets is Walrasrsquo law named in honor ofthe nineteenth-century French economist Leon Walras2 Simply put Walrasrsquo lawnotes that the budget constraints faced by individual agents in the economy sug-gest that if n minus 1 of the n markets in the economy are in equilibrium then the nthmarket must be in equilibrium We will repeatedly rely on Walrasrsquo law or variantsof it to simplify macroeconomic analysis
2 Introduction
While macroeconomic theories have in common (a) an attempt to explainfluctuations in aggregate variables and (b) a general equilibrium character thereremain wide differences among macroeconomic models Below we break downthese differences across macroeconomic models in several ways in order to makesome sense of what passes for simple theoretical macroeconomic analysis
Static dynamic and stationary analysis
One way of breaking down macroeconomic analyses is into static models dynamicmodels and stationary analysis of dynamic models Static macroeconomic modelsanalyze the economy at a point in time They consider the determination of pro-duction exchange and prices of various goods only for the markets that currentlyexist John Hicks (1939) sketched out an analysis of ldquospotrdquo or ldquotemporaryrdquo equi-librium The advantage to such an approach is that it provides for rather simpleldquocomparative staticrdquo analysis of the effects of changes in a variety of exogenousvariables on the endogenous variables3 Such static analysis is useful in providinginsight into a variety of questions of interest
Static macroeconomic analysis can be viewed as a modification of a Walrasiangeneral equilibrium analysis or what is commonly referred to as ldquoArrowndashDebreutheoryrdquo (Arrow and Hahn 1971 Debreu 1959) In ArrowndashDebreu theory eachcommodity is described by its physical characteristics its location and its dateof availability It is assumed there are a complete set of spot and forwardmarkets Prices adjust to clear all markets However if one restricts attentionto just spot markets then one moves from traditional Walrasian generalequilibrium to an analysis of ldquotemporary equilibriumrdquo a phrase coined by Hicks(1939) This restriction to spot markets is one element of static macroeconomicanalysis4
A second element of static models is that if there is a future then staticmacroeconomic analysis simply assumes given expectations of future prices andenvironment How expectations of future events are formed is left unspecified sothat expectations of future prices become simply an element in the set of exogenousvariables5
While static analysis provides insights there are several disadvantages of staticanalysis severe enough that it alone does not provide an adequate grounding inmacroeconomic analysis The key disadvantage of static analysis is that it breaksties between current events and future events To show the limitations of staticanalysis let us suppose that underlying a simple static macroeconomic analy-sis of current markets is a microeconomic analysis of individualsrsquo decisions thatidentifies the anticipated future level of prices as one of the exogenous variablesaffecting current behavior As we have seen static analysis takes expectations ofsuch variables as future prices as exogenous variables Doing so however resultsin (a) an incomplete enumeration of exogenous variables that can impact currenteconomic activity and (b) a potentially incomplete accounting of the effects of theimpact on current economic activity of a change in those exogenous variables thatare identified by the analysis
Introduction 3
To illustrate the first point of an incomplete listing of exogenous variables letus suppose that the static model identifies changes in the current money supplyas one factor that influences current prices This suggests that if we replicate thestatic analysis in future periods changes in the money supply in the future would beshown to affect prices at that time It seems natural to then presume that individualsrsquoanticipation of future prices would incorporate this link between changes in thefuture money supply and future prices in forming their expectations of futureprice levels so that the anticipated future money supply becomes a determinantof current activity6 Yet static analysis since it does not analyze markets beyondthe current period will not identify the potential impact of future changes in themoney supply on current activity7
To illustrate the second point of an incomplete accounting of the effects of achange in an exogenous variable let us suppose that underlying the static macro-economic analysis of current markets is a microeconomic analysis of firmsrsquo currentinvestment behavior that identifies the anticipated future tax levels as well as futureprices as two exogenous variables affecting investment decisions Thus staticanalysis would suggest that a change in future tax levels will impact current activitythrough the direct effect on current investment It is not hard to see however that(a) the current change in investment means a different future capital stock and(b) the change in future tax levels could affect future as well as current investmentEither or both of these changes would likely impact future prices and if such animpact were anticipated be a second way that future tax changes impact currentactivity8
An obvious way to avoid the above problems is to introduce forward marketsso that the macroeconomic analysis determines the prices of goods to be tradedin the future along with the prices of goods traded at the current time9 In doingso we have moved from static to dynamic analysis That is the macroeconomicmodels now determine the paths of variables (such as prices) over time rather thanprices (and other variables) at only one point in time
In a deterministic setting this expansion of dynamic analysis incorporates thenotion of ldquoperfect foresightrdquo in which individuals correctly anticipate all futureprices If there were uncertainty the analysis indexes goods by both the date oftrade and the ldquostate of naturerdquo with trades contingent on the realized state ofnature10 The result is that at each date there is a distribution of potential prices atwhich trade for a good could occur and given common knowledge of likelihood ofthe states of nature expectations of future prices would be defined by the analysis(ldquorational expectationsrdquo)
Once dynamic analysis is introduced we can consider a special limiting form ofdynamic analysis termed stationary analysis The aim of stationary analysis is toidentify in the context of a dynamic model the limiting tendencies of endogenousvariables such as the capital stock or the rate of growth in prices given that theexogenous variables remain constant or stationary over time11
While stationary analysis is distinct from static analysis in some cases one canthink of static analysis as a form of stationary analysis That is static analysis insome cases can be viewed as the outcome that would emerge each period given
4 Introduction
that the exogenous variables remain constant (or in some cases grow at a steadyrate over time) and given that one picks the correct fixed level of certain keyexogenous variables (eg the capital stock and the rate of change in the moneysupply) Note however that this implies that for static analysis to perfectly mimicstationary analysis one must to all intents and purposes have first executed theunderlying dynamic analysis
Period (discrete) versus continuous-time analysis
Macroeconomic analysis can be broken down into period or discrete-time macro-economic models and continuous-time macroeconomic models Substantivedifferences in terms of theoretical predictions do not exist between these two typesof analyses if one is careful to assure identical underlying assumptions Yet the twoanalyses do differ in the analytical techniques used For instance while discretemacroeconomics relies on the techniques of dynamic programming and differenceequations to characterize elements of the model in similar circumstances con-tinuous macroeconomic analysis turns to the techniques of optimal control anddifferential equations
Although substantive issues are not raised by the discrete- versus continuous-time dichotomy it is sometimes argued that one is preferred to the other Forinstance an attractive feature of the continuous-time analysis is that it highlightsquite clearly the distinctions between stocks and flows something that is not soclearly discernable in discrete analysis On the other hand an attractive feature ofdiscrete analysis is that it makes more transparent the link between the theoreticalanalysis and empirical testing since such analysis coincides with the obvious factthat empirical data on macroeconomic variables is discrete
New classical economics versus non-market-clearing
Classical analysis refers to the widely adopted view of how the macroeconomyshould be modeled that existed prior to the experience of the Great Depressionand John Maynard Keynesrsquo General Theory of Employment Interest and Money(1936) In classical theory the real side of the economy was separate from themoney side Classical analysis of the ldquorealrdquo side of the economy is aimed atdetermining such variables as total production relative prices the real rate ofinterest and the distribution of output Classical analysis of the money side ofthe economy meant analysis is aimed at determining money prices and nominalinterest rates
The separation of the monetary side from the real side in classical or neoclassicalanalysis led to the prediction that monetary changes do not have any effect onreal variables such as total output12 A similar prediction is often obtained bymore recent macroeconomic analysis and this is one reason why this more recentanalysis is referred to as the new classical economics13 Alternative labels ofthese new classical models include rational expectations models with market
Introduction 5
clearing neoclassical models of business fluctuations and equilibrium businesscycle models
A common feature of the analyses of new classical economics besides thefact that it suggests a divorcement of monetary changes from the real side ofthe economy is that prices are determined in the analysis so as to clear marketsThis view that prices serve to equate demands and supplies a view common tomicroeconomics is taken as an important strength of the analysis for it meansthat the models have consistent ldquomicroeconomic foundationsrdquo One implicationof the market-clearing assumption is the same as in microeconomics ndash the analysissuggests that all gains to exchange have been extracted
Contrasting the new classical economics with what preceded it helps one putthis rebirth of classical analysis into perspective Following the Great Depressionmacroeconomic analysis took as its main premise the idea that markets did notclear ndash in particular that prices did not adjust In this context the business cyclewas defined by ldquomarket failurerdquo and the role of government to stabilize the eco-nomy was clear There are a number of different types of non-market-clearing orKeynesian models One version of such Keynesian models that popularized byPatinkin (1965 chapters 13ndash14) Clower (1965) and Barro and Grossman (1971)takes as given output prices such that the output market fails to clear A secondmodel popularized by Fischer (1977) Phelps and Taylor (1977) and Sargent(1987a) as an alternative formalization of the Keynesian model takes as given theprice of labor such that the labor market fails to clear
The common theme of these non-market-clearing analyses is that for variousreasons prices do not clear markets and concepts such as excess demand and supplyplay a role in the analysis Yet no concise reason is given as to why there is marketfailure other than suggesting such items as ldquocoordination problemsrdquo and ldquotrans-action costsrdquo14 The result is that such analysis is challenged by the new classicaleconomics as lacking the microeconomic foundations for price determination AsHowitt (1986 108) suggests such a view ldquoforces the proponent of active stabi-lization policy to explain the precise nature of the impediments of transacting andcommunicating that prevent private arrangements from exhausting all gains fromtraderdquo15 This is not an easy task according to Howitt since ldquoimpediments to com-munication in a model simple enough for an economist to understand will typicallyalso be simple enough that the economist can think of institutional changes thatwould overcome themrdquo (1989 108)
Microeconomic foundations and aggregation issues
An important feature of macroeconomic analysis is that it reflects the aggre-gation of individual decisions A common approach to such aggregation is toassume ldquorepresentativerdquo agents characterize their optimal behavior then use suchbehavioral specifications in building the macroeconomic model Thus much ofmacroeconomic analysis entails looking at individualsrsquo decisions such as house-holdsrsquo decisions to work consume and save or firmsrsquo decisions to produceborrow and invest in capital
6 Introduction
These characterizations of optimizing individual behavior make up part of thebuilding blocks or ldquomicroeconomic foundationsrdquo of macroeconomic analysisYet microeconomic foundations of macroeconomics are not restricted to suchanalysis For instance such foundations also include a characterization of howprices in individual markets are determined as we saw in our discussion of newclassical economics
In developing the microeconomic foundations of macroeconomic models wewill often be struck by the extent to which the analysis restricts any role forheterogeneity or diversity among the individual agents in the economy Yet suchdiversity can in certain instances be critical to the analysis One attempt to introducediverse or heterogeneous agents into macroeconomic analysis is represented by theoverlapping generations models These models also have the advantage of beinggenuinely dynamic in nature and as such represent one area of macroeconomicsthat has recently received significant attention
Deterministic versus stochastic
In recent years an important element to macroeconomic models has been to intro-duce stochastic elements The rationale is clear the presence of uncertainty as tofuture events is real As noted by Lucas (1981 286)
the idea that speculative elements play a key role in business cycles that theseevents seem to involve agents reacting to imperfect signals in a way whichafter the fact appears inappropriate has been commonplace in the verbaltradition of business cycle theory at least since Mitchell It is now entirelypractical to view price and quantity paths that follow complicated stochasticprocesses as equilibrium ldquopointsrdquo in an appropriately specified space
As the quote suggests in dynamic models especially for new classical economicswhere market clearing is presumed stochastic elements are incorporated into theanalysis so that the role played by shocks to an economy in a dynamic setting canbe well defined
2 Walrasian economy
Introduction
This chapter develops a competitive model of the economy The key assumptionsneeded for this model are spelled out in detail One important aspect of this modelis the ldquonumerairerdquo or the commodity price that is used as a reference in the modelA distinction is made between accounting prices and relative prices and it isseen that traditional general equilibrium analysis does not determine the levelof accounting prices but rather simply relative prices A number of modificationsto the model are mentioned and add a sense of ldquorealismrdquo to the framework Thesemodifications include the introduction of futures markets quantity constraintsand the costs associated with carrying out a transaction
The chapter continues by considering individual decision-making and the theoryof the consumer and how this relates to the determination of market demand Thegeneral equilibrium conditions are stated in terms of relative prices and allocationsA theme carried throughout this book is emphasized and that is the use of theaggregate budget constraint Finally it is shown that explicitly excluding moneyas a ldquomarketrdquo allows one to understand how Sayrsquos conclusion that ldquosupply createsits own demandrdquo is arrived at in an economy composed of a single aggregatecommodity
A simple Walrasian model
As discussed previously the idea that prices adjust to clear markets is commonto much of new classical macroeconomics Thus we begin our examination ofmacroeconomic analysis by considering an economy consisting of perfectly com-petitive markets This means that individuals take prices at which exchanges canbe made as parametric and prices adjust to eliminate excess demands so that indi-vidualsrsquo plans at given prices are feasible As the title to this section suggests sucha characterization is sometimes referred to as being indicative of a ldquoWalrasianrdquoeconomy Walras described the process by which prices adjust to excess demand orsupply as a groping or tatonnement process (see Walras 1954)1 A fictitious auc-tioneer calls out different prices for the various markets and no exchange occursuntil equilibrium prices are reached2
8 Walrasian economy
Our analysis also begins at a very simple level The term ldquosimplerdquo reflects atleast the following three characteristics of the economy that we consider
1 It is a barter economy That is any commodity can be freely traded for anyother commodity There is no role for a medium of exchange (money) inreducing the costs of arranging exchanges
2 It is an exchange economy That is there is no production Instead individualshave initial fixed endowments of various commodities
3 It is a timeless economy Goods are indexed by physical characteristics andlocation but not dated according to availability This rules out futures mar-kets or the formation of expectations of future events and planning Such aneconomy was suggested by Hicks (1939) Patinkin (1965 Chapter 1) andHansen (1970 Chapter 4) provide a more detailed view of such an economyAn actual example of a pure barter exchange economy is offered by Radford(1945) The simple model of the economy developed below is useful in high-lighting such concepts as relative prices the numeraire individual versusmarket experiments aggregation issues conditions for general equilibriumand Walrasrsquo and Sayrsquos laws
The first model developed below also takes an approach to modeling the econ-omy that is in vogue in current theoretical macroeconomics As Sargent (1987b)states the ldquoattraction of (such) general equilibrium models is their internal con-sistency one is assured the agentsrsquo choices are derived from a common set ofassumptionsrdquo
Yet this advantage of general equilibrium analysis is not fully exploited until theelements of time money and production are introduced and so we will expand thediscussion in subsequent sections by introducing such features Below we intro-duce in more detail some of the key assumptions underlying the simple Walrasianmodel we start with
Key assumptions underlying a simple Walrasian model
As noted by Debreu (1959 74) ldquoan economy is defined by m consumers (charac-terized by their consumption sets and their preferences) n producers (characterizedby their production sets) and the total resources (the available quantities of thevarious commodities which are a priori given)rdquo As discussed above we considera special case of Debreursquos ldquoconcept of an economyrdquo one in which productionis absent As the following set of assumptions makes clear we also restrict ouranalysis to private ownership economies with a price system In particular assume(partial listing)
Assumption 21 There are m individuals (agents) in the economy indexed bya = 1 m There are T commodities indexed by i = 1 T 3 Agent arsquos initial
Walrasian economy 9
endowment of commodity i is denoted by cai cai ge 0 i = 1 T Naturally
msuma=1
cai = ci gt 0
Note that this assumption reflects an exchange economy in which private propertyrights exist ldquoPrivate property rightsrdquo means that for each unit of each good theexclusive right to determine use has been assigned to a particular individual
Assumption 22 All exchanges occur at a single point in time
Assumption 23 Each individual confronts the same known set of prices at whichexchange can occur4 A relative (purchase) price of commodity i indicates the unitsof commodity j required to purchase one unit of commodity i A relative (sale)price of commodity i indicates units of commodity j received when one unit ofcommodity i is sold
Assumption 24 Purchase and sale prices are identical for each commodity Thismeans that there are no ldquoprice spreadsrdquo which would suggest either a gain to anindividual buying and selling the same commodity or the presence of costs tomaking an exchange5 Thus for the T commodities there are T 2 exchange ratesor relative prices taking two commodities at a time
The numeraire
While there are T 2 exchange rates the complete set of exchange rates can bededuced directly or indirectly by the set of T minus 1 relative prices
(π1j πjminus1 j πj+1 j πTj)
where the πij denotes the price of commodity i in terms of commodity j6 In thelisting of relative prices (π1j πjminus1 j πj+1 j πTj) commodity j is referredto as the ldquonumerairerdquo
To see how the set of relative prices reduces to T minus 1 we rely on the factthat πhh = πhjπhj for all h j and k Let us see what this means for a simpleexample of three commodities h j and k There are then T 2 or nine differentrelative prices which are πhh πjj πkk πhj πjh πhk πkh πjk and πkj But we canuse the relationship πhh = πhjπhj to reduce this to T minus 1 = 2 relative prices withinformational content In particular we know that
1 πhh πhkπhj and similarly for πjj and πkk when h = j = k In other wordsthe exchange rate of a commodity with itself is unity This reduces from nineto six the number of relative prices for which information is required
2 πjh = 1πhj when k = j (such that πkj = πkk = 1) Similarly πhk = 1πkhand πjk = 1πkj For example if the jth commodity is pears and the hth
10 Walrasian economy
commodity is oranges then if πjh = 3 (3 oranges = 1 pear) πhj = 13(13 pear = 1 orange) This reduces from six to three the number of relativeprices for which information is required to reconstruct the complete set ofrelative prices
3 Finally πkh = πkjπhj (for k = j = h) For example if the jth commodity ispears the kth commodity is apples and the hth commodity is oranges thenif πkj = 3 (3 pears = 1 apple) and πhj = 13 (13 pears = 1 orange) thenπkh = 9 (9 oranges = 1 apple) That is with 9 oranges you can get 3 pearswhich in turn will purchase 1 apple This drops us from three to two relativeprices required to reconstruct the complete set of relative prices Since thenumber of commodities T = 3 we have shown how the T 2 relative pricescan be constructed from T minus 1 relative prices
In subsequent discussions we will arbitrarily let commodity T be the numerairesuch that the set of relative prices can be summarized by
(π1T πTminus1T )
For simplicity let us change notation such that πiT = πi i = 1 T Thus theset of relative prices can be rewritten as
(π1 πTminus1)
Note that πT = 1 since we are assuming the T th good is the numeraireIn traditional general equilibrium theory there is a concept of ldquoaccounting pricesrdquo
as well as the concept of relative prices Accounting prices can be represented bya set of real numbers (say pi i = 1 T ) attached to the T commodities7 Therelationship between these accounting prices and the set of relative prices that doimpinge on behavior is that πi = pipT i = 1 T (for Pi = 0) As we will seetraditional general equilibrium analysis does not determine the level of accountingprices but rather simply relative prices8
Anticipating future modifications
Before continuing it might be useful to anticipate some of the subsequent changeswe will make in the characterization of the economy Besides the introduction ofproduction we will
bull introduce time implying either forward (futures) markets or an important rolefor expectations of future spot prices
bull introduce quantity constraints that can arise if prices are fixed at non-market-clearing levels (ie depart from a Walrasian framework) and
bull introduce the cost of carrying out an exchange
Walrasian economy 11
With respect to point (c) such costs have been characterized by Coase (1960) asldquotransaction costsrdquo arising from the costs ldquonecessary to discover who it is that onewishes to deal with and to inform people that one wishes to dealrdquo the costs ofldquoconduct[ing] negotiations leading up to a bargain and to draw up a contractrdquoand the costs ldquoto undertake the inspection needed to make sure that the terms ofthe contract are being observedrdquo9 Such costs will alter the nature of contracts(exchange agreements) formed and may provide the reason for ldquoprice rigiditiesrdquoSuch costs also suggest a role for money
Transaction costs are assumed to be zero in the simple Walrasian sys-tem outlined above Sometimes this is referred to as a situation wherethere is ldquoperfect informationrdquo or where markets are complete and ldquoperfectlycompetitiverdquo
Individual experiments
General equilibrium analysis can be divided into what Patinkin (1965) refers to asldquoindividual experimentsrdquo and ldquomarket experimentsrdquo In the context of the simpleWalrasian barter exchange economy individual experiments consider the behaviorof individual agents given an initial endowment and preferences when confrontedwith a set of prices Market experiments consider the resulting determination ofprices
In the simple Walrasian model under consideration individual experimentsreplicate standard microeconomic analysis of consumer behavior In particularassume
Assumption 25 Individual arsquos preferences are described by his utility functionua(ca1 caT ) where ca1 caT denote agent arsquos consumption bundle cai ge 0i = 1 T ua maps the set of all T -tuples of non-negative numbers into the setof all real numbers (ua RT+ rarr R) We make the appropriate assumptions withrespect to individualsrsquo preferences such that a utility function exists and is wellbehaved10
Assumption 26 Individual a will choose the most preferred consumption bundlefrom the set of feasible alternatives (rationality) Given the possibility of cost-less exchange at the set of relative prices represented by (π1 πTminus1) feasibleconsumption bundles or sets (ca1 caT ) are defined by
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0
wheresumT
i=1 πi cai denotes the initial endowment of individual a in terms ofcommodity T The above expression defines the budget set
12 Walrasian economy
The consumer problem
From Assumptions 25 and 26 individual arsquos optimum consumption bundle is thesolution to the problem
maxCa1CaT
ua(ca1 caT )
subject to
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0 cai ge 0 i = 1 T
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λ
L(ca1 caT λ) = ua(ca1 caT ) + λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
with first-order (necessary) conditions being11
partL
partcaile 0 i = 1 T
partL
partcaicai = 0 i = 1 T
cai ge 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
λ ge 0
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λmicro1microT
L(ca1 caT λ micro1 microT ) = ua(ca1 caT )
+ λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
Walrasian economy 13
with the necessary conditions being
partL
partcai= 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
partL
partmicroige 0 i = 1 T
microipartL
partmicroi= 0 i = 1 T
λ ge 0
microi ge 0 i = 1 T
Individual demands and excess demands
The optimal consumption bundle for agent a is defined by the above first-orderconditions and will be denoted by the (demand) set
(ca1 caT ) cai ge 0 i = 1 T
Individual arsquos demand functions will be of the form
cdai
(π1 πTminus1
Tsumi=1
πi cai
) i = 1 T
That is individual arsquos demand (consumption) of commodity i depends on the T minus1relative prices of commodities and the initial endowment Note that the form ofthe utility function implies the utility-maximizing consumption bundle meets thebudget constraint with equality Thus at the optimal bundle we have
partL
partλ=
Tsumi=1
πi cai minusTsum
i=1
πicdai = 0
An important point to note about demand functions is that they are homogeneousof degree zero in what might be called accounting prices12 Accounting prices aredefined such that pi = πi middot pT i = 1 T so it is clear that if all prices increaseby the multiple θ relative prices and the initial endowment are unchanged
Individual arsquos excess demand function for commodity i is defined byzai = cd
ai minus cai If zai is positive agent a is a net buyer of commodity i whileif zai is negative the agent is a net seller of commodity i The market value
14 Walrasian economy
(in terms of the numeraire) of the quantity of the ith commodity that individuala seeks to exchange (buy or sell) is then given by πizai From the budget con-straint we know that
sumTi=1 πi(cd
ai minus cai) = 0 orsumT
i=1 πizai = 0 In other words foreach individual the market value (in terms of commodity T ) of individual excessdemands must sum to zero This rather obvious finding generates what is referredto as Walrasrsquo law as we will see
Market experiments
In the previous section we reviewed the nature of individual demand functions andindividual excess demand functions Now consider the collection of m individualsAggregating or summing individual demand functions we obtain an aggregate orldquomarketrdquo demand function for commodity i of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci1 Tsum
i=1
πi
)equiv
msuma=1
cdai(middot)
Similarly summing agentsrsquo excess demand functions for commodity i gives usthe aggregate or ldquomarketrdquo excess demand function for commodity i of the form
zi equivmsum
a=1
(zai) equivmsum
a=1
(cdai minus cai)
Note that a zero aggregate excess demand for commodity i does not imply thatno exchange of commodity i occurs among the m agents However as we haveseen a zero individual excess demand for commodity i does imply no exchangeof commodity i by that particular individual
Aggregation issues
So far our aggregations have remained true to the underlying microeconomicanalysis Yet this is rarely the case in macroeconomic analysis which typicallyabstracts from what might be termed ldquodistributionalrdquo effects An example of thisin the above context as we will see later is to ignore the effects of the distributionof initial endowments across individuals on market demands such that the marketdemand function for commodity i is assumed to be of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci
)
where
Tsumi=1
πicai equivmsum
a=1
(Tsum
i=1
πi cai
)
Walrasian economy 15
As you can see with heterogeneity (either in initial endowments or preferences)such a posited aggregate market demand function is unlikely to follow exactlyfrom the underlying microeconomic analysis One should keep in mind suchapproximations when interpreting macroeconomic analysis
Equilibrium an isolated market
With respect to a single market equilibrium is characterized by an accounting pricepi and implied relative price πi = pipT such that cd
i = ci (demand equals fixedendowment) or equivalently zi = 0 (excess demand equals zero) The tatonnementprocess is the description of how prices change to clear the market In the Walrasianmodel movement toward equilibrium the tatonnement process involves twofacets
1 The Walrasian excess demand hypothesis which indicates that the accountingprice of commodity i rises if there is excess demand and falls if there is excesssupply That is
dpi = fi(zi) i = 1 T fi(0) = 0dfidzi
gt 0
In terms of relative prices
dπi = dpi
pT= f (zi)
pT i = 1 T minus 1
Note that the change in price is not across time since each market is assumedto clear instantaneously at the same point in time
2 The recontracting assumption which states that offers to buy or sell at var-ious relative prices are not binding unless market(s) clear Only when theequilibrium price (or price vector) is obtained are contracts then made final
General equilibrium (conditions)
A general equilibrium will be characterized by a set of T minus 1 relative prices(πlowast
1 πlowasttminus1) and allocations (clowast
a1 clowastaT ) for individual a a = 1 m such
that
clowastai = cd
ai
(πlowast
1 πlowasttminus1
Tsumi=1
πi cai
) i = 1 T a = 1 m (21)
rArr clowasti equiv
msuma=1
clowastai =
msuma=1
cdai equiv cd
i
clowasti = ci i = 1 T (22)
16 Walrasian economy
Equation (21) indicates that the equilibrium allocation must be optimal in thatit must satisfy all demands for commodities at the specified set of pricesEquation (22) indicates that such an allocation must be feasible that is sumto total resource endowment Together these two conditions imply a set of relativeprices such that excess demands are zero or
cdi = ci i = 1 T
For questions concerning the existence uniqueness and stability of generalequilibrium consult Varian (1992) Debreu (1959) and Arrow and Hahn (1971)
Walrasrsquo law and Sayrsquos law
Note that the above statement of general equilibrium involves setting T excessdemand equations equal to zero but there are only T minus 1 unknowns (relativeprices) Walras solved this problem by showing that one of the equations arbitrarilychosen can be deduced from the other T minus 1 equations In other words there areonly T minus 1 independent equations To show the dependency remember that thebudget constraint for each individual is given by
Tsumi=1
πicdai minus
Tsumi=1
πi cai = 0
Summing across all individuals it must then be the case that
msuma=1
(Tsum
i=1
πi cai minusTsum
i=1
πicdai
)= 0
Rearranging and substituting in cdi for
summaminus1 cd
ai and ci forsumm
aminus1 cai we obtain
Tsumi=1
πi(cdai minus cai) = 0 or
Tsumi=1
πizi = 0
The above is an explicit statement of Walrasrsquo law Walrasrsquo law states that the sumof the excess demands across all markets must be zero Note that in summing theexcess demand of each commodity is weighted by its relative price so that we aresumming common units (ie all excess demands are in units of the numeraire)The above aggregate budget constraint is sometimes referred to as Sayrsquos law orSayrsquos identity If there is a distinction between the two it is that Sayrsquos law explicitlyexcluded money as a ldquomarketrdquo In this setting one can understand how Sayrsquosconclusion that ldquosupply creates its own demandrdquo is arrived at in an economycomposed of a single aggregate commodity
Walrasian economy 17
Conclusion
This chapter has developed a general equilibrium framework that sets the stagefor a thorough understanding of how the macroeconomy works Particular atten-tion has been paid to the development of relative prices and the development ofaggregate demand through a process of many individual consumers operating in anenvironment in which they set out to maximize their own utility This frameworkand the tool of constrained optimization is used throughout this book
3 Firms as market participants
Introduction
In this chapter the simple Walrasian model is discussed in the context of moneyfinancial assets and production The chapter clearly illustrates the firmrsquos objectivethat is to maximize profits However the firm is constrained in that it must financepurchases of capital and equipment as well as pay its workers Moreover attentionis paid to all the costs faced by the firm not just the obvious ones The investmentand financing decisions of firms are discussed and issues related to Tobinrsquos Q anddebt-to-equity are explored This chapter provides a detailed examination of therole that firms play in the macroeconomy
A simple Walrasian model with money financialassets and production
In the exchange economy we just considered endowments of the commodity goodwere magically bestowed on individuals each period We now introduce productionas the source of commodities At the start of each period there now exists a ldquolaborrdquomarket in which labor services are exchanged New agents denoted ldquofirmsrdquo hirethe labor services provided by households During the period firms combine thelabor services with an existing capital stock to produce output (commodities)which is sold in the output market net of output retained to replace capital usedup during production Revenues from the sale of output are distributed to ldquohouse-holdsrdquo in the form of wages during the period At the end of the period interestpayments and dividends are made to households out of revenues Each period firmsalso enter the output market to augment their capital stock with such purchasesfinanced by the issue of bonds and a new financial asset denoted ldquoequity sharesrdquoIn particular we assume
Assumption 31 There are new agents in the economy denoted as ldquofirmsrdquo Theseagents are initially endowed at time t with a capital stock K and a technologyfor transforming capital services from a capital stock and labor services into a
Firms as market participants 19
single ldquocompositerdquo commodity The technology is summarized by the productionfunction
yt = f (Nt K)
where K denotes the capital stock the firm inherits at time t Nt denotes the employ-ment of labor services arranged at time t for period t (from time t to time t + 1)and yt denotes the constant rate of production of the commodity for period t (fromtime t to time t + 1)1 Similarly for period i (i = t + 1 t + 2 ) which runsfrom time i to time i + 1 output produced is given by2
yi = f (Ni Ki)
Assumption 32 During each period firms sell the output produced in the outputmarket For output produced during period t (from time t to time t+1) let pt denoteits price when it is sold during the period up to and including time t + 1 Let pt+1denote the price of output produced during period t + 1 that is sold beyond timet +1 up to and including time t +2 and so on At time t the price level associatedwith the prior period is denoted by p
Assumption 33 At the start of each period households rent their labor servicesto firms for the period At the start of period t agreements to exchange the Ntlabor services during period t (from time t to t + 1) are entered into at the moneywage rate denoted by wt Similarly at the start of period t + 1 Nt+1 labor servicesare exchanged at the money wage rate wt+1 and so on3
Assumption 34 At the end of each period two types of financial assets areexchanged bonds (in the form of perpetuities) and equity shares Bonds promiseto pay a fixed money (coupon) payment z each future period in perpetuity Let pbpbt and pbt+1 denote the money price of such bonds in markets at the end of periodstminus1 t and t+1 respectively the gross (nominal) interest rates over period t (fromtime t to time t + 1) and over period t + 1 (from time t + 1 to time t + 2) are thengiven by4
1 + r = (z + pbt)pb
1 + rt = (z + pbt+1)pbt
Note that if ri = rt i = t + 1 then successive substitution for the priceof bonds in future periods will result in the following expression for the price ofbonds at time t
pb = 1
1 + r(z + pbt) = 1
1 + r
[z +
infinsumi=1
z
(1 + rt)i
]
= 1
1 + r
(z + z
rt
)
20 Firms as market participants
where pbt = zrt The number of previously issued bonds outstanding at time t isdenoted by B5
Assumption 35 Equity shares are the second type of financial asset exchangedat the end of each period Equity shares (ldquostocksrdquo) are contracts that obligate theissuer (firms) to pay to bearers at the end of each future period the income from thesale of output produced during the period net of other contractual obligations ofthe firms (eg wage payments to suppliers of labor services and interest paymentsto holders of bonds issued by firms) Let S denote the number of previously issuedequity shares outstanding at time t Holders of these equity shares are the ldquoownersof the firmsrdquo Let pe pet and pet+1 denote the money price of equity sharesexchanged in markets at the end of periods t minus 1 t and t + 1 respectively Thegross (nominal) rate of return on equity shares over period t (from time t to timet + 1) and period t + 1 (from time t + 1 to time t + 2) is then given by6
1 + re = [( ptdtS) + pet)]pe
1 + ret = [( pt+1dt+1St) + pet+1)]pet
where ptdt and pt+1dt+1 denote total nominal dividend payments made at the endof periods t and t + 1 (at time t + 1 and time t + 2) respectively St denotesthe anticipated number of equity shares outstanding after the equity market at theend of period t7 Note that the price of an equity share indicates the fact that thepurchase of an equity share entitles the holder to a portion of future (not current)dividends
Assumption 36 Households view bonds and equity shares as perfect substitutesldquoPerfect substitutesrdquo means that if equality in yields did not hold households wouldrefuse to purchase the asset with the lower yield forcing an adjustment in its pricethat would result in equivalent yields8 With bonds and equity shares as perfectsubstitutes we can speak of a single ldquofinancial asset marketrdquo that incorporatesboth bonds and equity shares and determines a single ldquointerest raterdquo
Assumption 37 There are incomplete markets Let pei i = t + 1 t + 2
denote the expectation formed in period t concerning the price of the consumptiongood over period i9 Similarly in period t we have pe
ei i = t + 1 and wei
i = t + 1 Such expectations are assumed to be held with subjective certaintyallowing us to abstract from risk considerations for the moment
Assumption 38 There are positive transaction costs to arranging exchanges ofthe consumption commodity during each period t Money holdings serve to reducethe transaction costs of arranging exchanges during a period10
Assumption 39 It is prohibitively costly for individuals to directly store theldquocompositerdquo commodity for consumption in future periods However output notconsumed during the period can be transformed (by ldquofirmsrdquo) into output in thesubsequent periods through the augmentation of the capital stock which permitshigher rates of production of output in future periods
Firms as market participants 21
Individual experiments firms
As always we start our analysis at the individual level The behavior of two typesof agents must now be considered ndash firms and households We start with firmsIn doing so we consider a ldquorepresentativerdquo agent a unit whose behavior exceptfor scale is identical to the behavior of the aggregate of such units Thus the samenotation will be used to represent both the individual unit and the aggregate of allunits In addition we consider an infinite time horizon
You should now recognize that an analysis of a representative unit neglectscertain potentially important ldquodistributionalrdquo aspects of the problem For instancefor households the ldquoreal indebtedness effectsrdquo of a price change on demand maynot be offsetting in the aggregate but that potential impact is ignored11 For firmsthe distribution of the initial capital stock can given adjustment costs affect totalemployment and output but that too is ignored12
To consider the behavior of a firm (or more specifically the manager whodirects production for the representative firm) we assume
Assumption 310 Technology is represented by the concave production function
yt = f (Nt K)
where yt denotes the firmrsquos planned (at time t) constant rate of output for the timeperiod from time t to t +1 to be sold during the period and at time t +1 Nt denotesthe firmrsquos planned (at time t) rate of employment of labor during period (t t + 1)with labor services purchased in the labor market at time t and K denotes thefirmrsquos planned capital stock for period t Recall that to simplify matters we takethe capital stock for the current period K as fixed at the individual firm levelThis would be the case given appropriate capital adjustment costs13
Assumption 311 The representative firm will choose the most preferred inputcombination and implied output given technology and prices (both current andanticipated future prices) At time t the objective of the firm is to maximize theexpected real market value of the S equity shares
Vt = peS
p
where pe is the price of equity shares at the end of period t minus 1 (at time t)14
A restatement of the firmrsquos objective
We have indicated that the objective of the firm at time t is to maximize the realmarket value of the S equity shares as given by
Vt = peS
p
22 Firms as market participants
To understand what underlies this market value of the firm we have to definethe elements underlying the price of equity shares and dividends We start byexamining what lies behind the price of equity shares
The assumption that equity shares and bonds are perfect substitutes means thatthe price of an equity share can be expressed as
pe = [ptdtS + pet](1 + r)
where dt denotes real dividends at the end of period t so that ptdt denotes nominaldividends S denotes the number of equity shares outstanding at time t pet is theprice of equity shares at the end of period t and r denotes the interest rate overperiod t (ie from time t to t + 1)
By successively substituting in a similar expression for the price of equity sharesin the next period we obtain for an infinite horizon that15
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
That is the price of an equity share at the end of period t minus1 (at time t) pe reflectsthe anticipated discounted future stream of nominal dividends per share16
Since the real value of the firm is given by Vt = peSp we can now expressthe value of the firm as
Vt = [Sp(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
which means that the objective of the firm can be stated in terms of maximizingthe discounted stream of current and future dividends Before examining whatdetermines dividends each period let us simplify the above expression for Vtby putting it in terms of real dividends each period To do so note that bydefinition
pt equiv p(1 + π)
pt+k equiv p(1 + π)
⎛⎝ kprod
j=1
(1 + rt+jminus1)
⎞⎠ k = 1 2 3
where πt+j denotes the rate of change in the price level between period t + j andt + j + 1 Thus we have
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
Firms as market participants 23
where R = (1 + r)(1 + π) and Ri = (1 + ri)(1 + πi) denotes the real gross rateof interest for period i (from time i + 1 to time i + 2 i = t t + 1 ) Our nextstep in outlining the firmrsquos problem is to obtain an expression for real dividendsin each period
Dividends and the firm distribution constraint
In general we may denote real dividends at the end of any period as the differencebetween a firmrsquos total real revenues during the period and the total costs incurredduring the period Total revenues derive from the sale of output produced duringthe period In addition one could add revenues from a change in the numberof equity shares outstanding or a change in the number of bonds outstanding atthe end of the period17 Total costs to the firm in any period include the agreedupon wage payments to the labor hired during the period payments to replacedepreciated capital plus coupon payments at the end of the period to holders ofpreviously issued bonds Payments at the end of the period for the purchase ofcapital and associated adjustment costs could be counted as well18 That is firmsare constrained to have
dividends = revenue from sale of output
minus wages
minus interest payments
+ funds from change in outstanding bonds and stocks
minus costs to replace depreciated capital add new capital
and capital adjustment costs
The above constraint is typically divided into two separate constraints Oneconstraint earmarks funds raised from the change in outstanding bonds andequity shares at the end of the period to pay for or ldquofinancerdquo the installationof new capital stock during the period plus any capital adjustment costs Thisis the ldquofirm financing constraintrdquo The remaining revenues minus expendituresthen determine the level of dividends This part is called the ldquofirm distributionconstraintrdquo
The firm distribution constraint simply states that the revenues from the saleof output that exceed expenditures to meet wage payments purchases of capitalto replace that used up in the production process and interest payments to bondholders at the end of the period are distributed at the end of the period to householdsas dividends Thus real dividends at the end of period t are given by
dt = yt minus (wtpt)Nt minus zBpt minus δK
According to this expression real dividends at the end of period t equal realrevenues derived from the sale of output yt produced during period t minus costs
24 Firms as market participants
to the firm during period t that reflect the real wage wtpt times the quantity oflabor hired Nt less real coupon payments at the end of the period on previouslyissued bonds zBpt plus purchases of capital during the prior period to replacethat used up in the production process δK 19
In similar manner real dividends for periods t + 1 and t + 2 (paid at the end ofeach period) are given by
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
We can rewrite the above definition of dividends to see more clearly why itis also termed the ldquofirm distribution constraintrdquo This constraint simply says thatrevenue from the sale of output net of that retained to replace capital used up inthe production process (ie ldquonet productrdquo) is distributed to households either aswage payments interest payments or dividends That is
dt + (wtpt)Nt + zBpt = yt minus δK
dt+1 + (wt+1pt+1)Nt+1 + zBtpt+1 = yt+1 minus δKt+1
The firm financing constraint
The second part of the general constraint that firmsrsquo total expenditures equalrevenues is that changes in the firmrsquos holdings of capital as well as any asso-ciated adjustment costs are financed by a change in outstanding equity sharesandor bonds This linking of funding for capital purchases to the issuing of equityshares and bonds is denoted the ldquofirm financing constraintrdquo For instance at theend of periods t + 1 t + 2 we have the following firm financing constraints
Kt+1 minus K + ψ(Int) = [ pet middot (St minus S) + pbt middot (Bt minus B)]pt
Kt+2 minus Kt+1 + ψ(Int+1) = [ pet+1 middot (St+1 minus St) + pbt+1 middot (Bt+1 minus Bt)]pt+1
where ψ(Ii) denotes the costs of installing new capital at rate Ii during period i(between time i and time i + 1) a cost that depends directly on the rate of netinvestment (Ini = Ki+1 minusKi) planned at time t to occur between time i and i+120
Recall that we assume that firmsrsquo plans with respect to the number of stocks andbonds that will be outstanding following the financial market at the end of period i(time i + 1) mirror householdsrsquo expectations concerning the number of bonds andstocks that will be outstanding so we do not distinguish between firmsrsquo plans andhouseholdsrsquo expectations with respect to these variables
Firms as market participants 25
Since bonds and equity shares are perfect substitutes we can rewrite the abovefirm financing constraints in the simpler form
Int + ψ(Int) = At minus At = net At
Int+1 + ψ(Int+1) = At+1 minus At+1 = net At+1
where Ini denotes net real investment (ie Ki+1 minusKi) Ai denotes the real planned(at time t) value of total equity shares and bonds to be outstanding after the financialmarket at the end of period i and Ai denotes the initial real value of equity sharesand bonds for period i reflecting financing and capital decisions in prior periodsbut period i prices For instance
At = [ petSt + pbtBt]pt and At = [ petS + pbtB]pt
At+1 = [ pet+1St+1 + pbt+1Bt+1]pt+1 and
At+1 = [ pet+1St + pbt+1Bt]pt+1
There are several aspects of interest with respect to the firm financingconstraints First note that net capital purchases planned for period i to be installedbetween time i and time i + 1 are paid for at the end of period i when completelyinstalled from the sale of financial assets at that time
Second note that firms purchase the output of the composite good to augmentthe capital stock That is we have a ldquoone-sectorrdquo model in which the same goodserves both households (for consumption) and firms (for investment) There isonly a single commodity price A typical extension is a two-sector model in whichtwo goods are produced a consumption good and a capital good In such casesa new variable the relative price of the capital good in terms of the consumptiongood is introduced
Third note that the firm financing constraint holds whether the firm financescapital with new bonds new equity shares or ldquoretained earningsrdquo Suppose forinstance that a firm plans to add 100 units to its capital stock by buying a newpiece of machinery If the firm issues a bond with real value of 100 to pay for themachinery then there is a direct 100 unit increase (the new bond) in the value of thefinancial assets issued by the firm Note that the value of the current shareholdersrsquostock is unchanged in this case of bond financed investment While it is true thatthe tangible assets of the firm have increased by the 100 addition to capital thisbenefit to shareholders is exactly offset by the fact that the firmrsquos debt has alsoincreased by 10021
If the firm finances the 100 net investment by issuing new shares of stock equalto 100 again there is a 100 unit increase (the new equity shares) in the real valueof the financial assets issued by the firm As with bond financing however thereal value of the initial shareholdersrsquo stock is unchanged when the firm finances itscapital purchases by issuing new equity shares The new shares do not dilute thevalue of the shares of the initial shareholders since the capital purchase increases
26 Firms as market participants
the firmrsquos tangible assets by 100 which is exactly the real value of the new equityshares issued
Finally if the firm finances the 100 net investment through retained earningsthere is in essence a 100 unit increase in the value of the financial assets issued bythe firm for the following reason When a firm retains earnings in order to financea capital purchase the current stockholders own the right to the income generatedfrom the additional capital As a consequence the value of their equity shares risesto reflect the value of the new capital owned by the firm We could equivalentlyview this as the firm paying out 100 units in dividends to its initial shareholderswho then use the dividends to buy additional ldquoconstant valuerdquo equity shares equalto the value of the capital purchased by the firm In other words when the firmuses retained earnings to finance its investment spending it is implicitly issuingnew financial assets ndash equity shares
The nature of capital adjustment costs
An important aspect of the above financing constraint is that it incorporatespotential adjustment costs to purchases of capital as captured by the terms ψ(middot)which depend on Int Int+1 where Ini denotes the planned (at time t) net rateof investment for period i22 The total cost of capital purchases is thus the sum of(a) the real payments (or receipts if negative) involved in the purchase (or sale ifnegative) of capital in the output market and (b) potential real payments denotedldquoinstallationrdquo or adjustment costs associated with new capital acquisitions Forperiod t adjustment costs are given by ψ(Int) where Int denotes net investmentbetween time t and t + 1 Gross investment for period t is given by Int + δK Notethat we can thus decompose gross investment over the period into the change inthe capital stock Kt+1 minus K which is termed ldquoplanned net investmentrdquo and thereplacement of capital used up in the production process δK which is termedldquodepreciationrdquo23
To understand the conversion of the above analysis to continuous time we notethat in general adjustment costs over period t of length h are given by hψ(Inth)where the limit of the term Inth defines the rate of net investment That is incontinuous time the planned rate of investment would be defined by the rate ofgross investment
it = limhrarr0
It
h= lim
hrarr0
Kt+h minus K + hδK
h= Kt + δKt
and the adjustment cost function in continuous time would be ψ(int)For the aggregate rate of gross investment (a flow) to be defined by the above
expression K must equal K To achieve this one of two approaches is typicallytaken One approach assumes zero adjustment costs in that ψ equiv 0 This situationsometimes referred to as the case of ldquoperfect malleabilityrdquo means that the rateof investment may not be defined at the level of the individual firm That is ifthe existing capital stock were higher or lower than the planned level investment
Firms as market participants 27
would be infinitely positive or negative However it can be shown that withzero adjustment costs the output market in a continuous-time model at a pointin time is simply a ldquocapital marketrdquo and the expression Kt = K emerges as anequilibrium condition with respect to the capital market at time t24 Thus wecan apply LrsquoHospitalrsquos rule to define the aggregate rate of (net) investment in thecontinuous-time model with zero adjustment costs as25
it = limhrarr0
Kt+h minus K
h= limhrarr0 d(Kt+h minus K)dh
limhrarr0 dhdh= K
where it is rate of investmentIn contrast to the case of zero adjustment cost one can assume adjustment costs
that take the following form
ψ(0) = 0
ψ(β) gt 0 if β gt 0 ψ primeprime gt 0 and limβrarrinfin ψ(β) = infin
ψ(β) lt 0 if β lt 0
The above set of assumptions reflects the presumption that adjustment costsincrease at an increasing rate with the rate of change in capital and that it isinfinitely costly to change the capital stock arbitrarily fast The result of suchadjustment costs in both discrete-time analysis and continuous-time analysis isthat at time t the firm chooses Kt = K That is at time t the firm views the inher-ited capital stock as optimal since it is prohibitively costly to change the capitalstock at a point in time given such adjustment costs26
As we will see with ldquocosts of installing a unit of new capitalrdquo there will be adifference between the market value of capital goods in place and their replacementcost In particular the ratio of these two values known as ldquoTobinrsquos Qrdquo will exceedone In addition adjustment costs will mean that the firmrsquos decision with respectto investment will not be myopic (ie plans will not be based on forecasts thatextend only one period into the future) Rather the firm will consider all futureperiods in making current investment decisions
The firm problem a general statement
One way to state the optimization problem faced by the firm is to say that attime t the firm makes plans with respect to current and future employment oflabor (Nt Nt+1 ) the future employment of capital (Kt+1 Kt+2 ) the stockof outstanding bonds (Bt Bt+1 ) and the stock of outstanding equity shares(St St+1 ) in order to maximize the real value of the previously issued equityshares outstanding at time t with that real value given in general form by
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
28 Firms as market participants
Such plans are subject to the combined distribution and financing constraints listedabove as well as to the production function That is in general the firmrsquos problemcan be stated as27
max(SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
subject to the financing constraints
minus [Kt+1 minus K + ψ(Kt+1 minus K)] + (petpt) middot [Bt minus B]+ (pbtpt) middot (St minus S)] = 0
minus [Kt+2 minus Kt+1 + ψ(Kt+2 minus Kt+1)] + (pet+1pt+1) middot [Bt+1 minus Bt]+ (pbt+1pt+1) middot (St+1 minus St)] = 0
and the distribution constraints
dt = yt minus (wtpt)Nt minus zBpt minus δK
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
and given the production functions
yt = f (Nt K)
yt+1 = f (Nt+1 Kt+1)
for i = t t + 1 The above problem has a recursive nature to it At the start of any given period
the firm inherits a stock of capital an outstanding stock of bonds and an outstand-ing stock of equity shares28 These variables are state variables Each period thefirm chooses a set of the ldquocontrolrdquo variables ndash employment investment Thesechoices in conjunction with the production function result in outcomes in termsof (a) a one-period return (dividends) at the end of the period and (b) a new setof ldquostaterdquo variables ndash capital stock and stock of financial assets (equity shares andbonds) ndash inherited in the subsequent period Further note that the objective of thefirm is additive in these one-period returns (dividends) Thus the problem is oneto which we can apply Bellmanrsquos dynamic programming technique
To reformulate the problem facing the firm as a dynamic programming prob-lem we use the conventional notation of dynamic programming problems29
Firms as market participants 29
Specifically the above problem can be viewed as involving
(a) A set of ldquocontrolrdquo variables each period
zt = Nt Int zt+1 = Nt+1 Int+1 etc
(b) A set of ldquostaterdquo variables each period
xt = K B S xt+1 = Kt+1 Bt St etc
(c) ldquoTransition functionsrdquo that link the choices specifically the choice of netinvestment during the period to the capital stock available at the start ofthe next period as well as the stock of financial assets outstanding in thesubsequent period For instance the choice of the net investment rate Int during period t dictates Kt+1 given K since
Kt+1 = Int + K
From the firm financing constraint we know that the choice of the investmentrate also determines the real stock of financial assets
( petpt)[Bt minus B] + ( pbtpt)[St minus S] = Int + ψ(Int)
(d) A set of one-period return functions (evaluated at the end of period t)30
rt(K B S) = dt rt+1(Kt+1 Bt St) = Sdt+1StRt etc
Since bonds and equity shares are perfect substitutes the above problem cannotbe solved for a unique optimal number of bonds or equity shares to have outstandingeach period Thus without any loss of generality we may restrict our focus toeither bond or equity share financing That is we can hold constant either bonds(ie Bi = B i = t t + 1 ) or equity shares (ie Si = S i = t t + 1 )Alternatively we can combine the distribution and financing constraints into asingle expression for dividends and hold constant both equity shares and bondsIn this case we have ldquoretained earningsrdquo financing of changes in the capital stockIt is this case that we consider below31
Simplifying the firm problem ldquoretained earnings financingrdquo
If we assume that capital expenditures are financed from ldquoretained earningsrdquo thereis a single state variable the stock of capital The problem facing the firm thencan be simply stated as follows The Bellman equation for period t given inheritedcapital stock K is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
30 Firms as market participants
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
Substituting the above definitions for real dividends and output into the Bellmanequation and substituting in the transition function the first-order conditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
Equation (31) is the standard condition that labor is employed up to the pointwhere the real marginal gain for an additional unit of labor in terms of the increasein output attained in the current period (ie the marginal product of labor partftpartNt)equals the real marginal cost as reflected by the real wage wtpt Equation (32)indicating the optimal choice of investment is discussed in the next section
Optimal investment (and the future capital stock) zeroadjustment costs
To express the optimal condition for investment and thus the future capital stockin a more transparent form we need to expand upon the effect of an increase inthe capital stock on the value function for period t + 1 In other words we need toclarify the nature of the term dW (Kt+1)dKt+1 in Equation (32) To do so let usconsider the Bellman equation for period t + 1 To simplify matters we initiallyfocus on the case of zero adjustment costs (ie that ψ equiv 0 implying that ψ prime
i = 0i = t t +1 ) Given the inherited capital stock Kt+1 for period t +1 and a fixedstock of equity shares the Bellman equation for period t + 1 is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1(Rt)minus1 + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and (assuming retained earnings financing of capital changes and zero adjustmentcosts) the following definitions for real dividends and output for period t + 1
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBtpt+1 minus Int+1
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 31
Thus we have32
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (33)
We can use first-order conditions for the Bellman equation for period t + 1 toclarify the nature of dW (Kt+2)dKt+2 in Equation (33) In particular we havethat the optimal choice of investment in period t + 1 satisfies
minus 1
Rt+ dW (Kt+2)
dKt+2= 0 (34)
Substituting (34) into (33) we obtain the following expression for the effect of achange in the inherited capital stock on the value function for period t + 1
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt (35)
Substituting (35) into equation (32) and recalling our assumption that ψ prime = 0we thus have that the optimal choice of investment in period t satisfies
minus1 + 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt= 0 (36)
The above expression can be rearranged to obtain
partft+1
partKt+1= mt minus δ (37)
where mt Fisherrsquos expected real rate of interest equals Rt minus1 or (rt minusπt)(1+πt)The interpretation of (37) is fairly straightforward Each period the firm chooseslabor and capital such that the marginal gain in the subsequent period in terms ofincreased output equals the real marginal cost For capital the marginal cost is therate of depreciation plus the expected real rate of interest An explanation of thisreal ldquouserrdquo or ldquorentalrdquo cost of capital follows
Over period t the firm pays for one unit of capital at price pt Since the firmcould have instead used these funds to reduce the outstanding stock of bonds by pt the cost of this capital (reduced dividends) in nominal terms is pt(1 + rt) In realterms the cost one period later is anticipated to be pt(1 + rt)pt+1 After oneperiod 1 minus δ of the capital remains so that the sale of the remaining capital afterone period of use (or the reduced purchases of new capital) reaps a nominal returnof (1 minus δ)pt+1 and real return (1 minus δ) The real rental cost of the unit of capital isthus33
pt(1 + rt)pt+1 minus (1 minus δ) = (1 + rt)(1 + πt+1) minus 1 + δ
= (rt minus πt)(1 + πt) + δ
= mt + δ
32 Firms as market participants
Summarizing our discussion in the case of zero adjustment costs the optimalbehavior of firms in periods t and t +1 is given by the following demand functionsfor period t and t + 1
N dt = N d(wtpt K)
I dnt = I d
n (mt + δ wt+1pt+1 K)
where I dnt = Kd
t+1minusK and Kdt+1 = Kd
t (mt+δ wt+1pt+1) Note that the anticipatedreal wage next period affects Id
nt since changes in the real wage affect the employ-ment of labor and thus assuming part2f partNpartK does not equal zero the marginalproduct of capital A similar statement explains why K enters as an argument inthe labor demand function
An important feature of the above is that planned investment demand when thereare zero adjustment costs simply depends on adjacent expected real user costs ofcapital and real wages This reflects the fact that with zero adjustment cost capitaldemand is a function of the expected real user cost of capital and the real wageover the next period alone
Financing choices and different debt-to-equityratios a digression
We have characterized the above choice of the capital stock under the presumptionthat the firm finances capital purchases through retained earnings Yet we can showthat the planned (at time t) choice of the optimal capital stock at time t+1 t+2 is independent of the method of financing given that (a) bonds and equity sharesare assumed to be perfect substitutes and (b) there is no cost to arranging theexchange of financial assets (otherwise the retained earnings financing method ispreferred) This result is sometimes referred to as the ldquoModiglianindashMiller theoremrdquowhich states that the total value of the firm is independent of its financial structureThat is the present value of the stream of dividends to the initial owners is inde-pendent of how liabilities are divided between bonds and equity shares The resultis that the capital structure is indeterminant
The view that the method of financing capital purchases is largely irrelevant isa very useful simplification for macroeconomic analysis However you should beaware of some complicating factors that we are ignoring factors that can causefirms to care about the method by which they finance their capital purchases
When a firm issues bonds the value of its outstanding debt rises When it issuesstocks the value of its outstanding equity shares increases Thus the method offinancing capital purchases affects what is known as the firmrsquos debt-to-equityratio34 Financing capital purchases with bonds will increase the firmrsquos debt-to-equity ratio while financing capital purchases with equity shares (eitherexplicitly or implicitly by using retained earnings) will reduce the firmrsquos debt-to-equity ratio Two factors that can influence a firmrsquos desired ldquocapital structurerdquoare tax considerations and bankruptcy costs35
Firms as market participants 33
The corporate taxes that firms pay are calculated as a percentage of earningsFor tax purposes corporate earnings are equal to total revenue net of costs wherecosts are calculated as including not only wages and payments for raw materi-als and intermediate goods but also interest payments to bondholders If a firmfinances its purchases of capital using bonds the interest it pays in the futurewill reduce its taxable earnings and thus the taxes that it has to pay This meansthat a firm can lower its future tax liability by raising its debt-to-equity ratio ndashthat is by financing new capital purchases with new bonds rather than equityshares
Raising the debt-to-equity ratio however is generally not without costs whichare typically referred to as ldquobankruptcy costsrdquo Unlike equity shares which promiseshareholders dividend payments if profits are sufficiently high bonds promise fixedpayments to their holders Greater debt thus increases the fixed obligations thatfirms must meet in the future This means that a fall in future revenues is morelikely to force the firm into bankruptcy
Bankruptcy occurs when a firmrsquos revenues do not cover its costs and it isforced to default on its obligations to bondholders Associated with bankruptcyare bankruptcy costs the most obvious being the hefty legal costs associated witheither reorganizing or undergoing a court-supervised liquidation The existence ofbankruptcy costs serves to limit the amount of borrowing a firm will undertake Itwill hesitate to increase its debt-to-equity ratio beyond some level since the gainin tax savings will be offset by the costs associated with an increased likelihoodof incurring bankruptcy costs
To summarize a firm can be viewed as having an optimal debt-to-equity ratiothat reflects a tradeoff of tax and bankruptcy cost considerations Table 31 liststhe general level of debt-to-equity for a sample of industries in US manufacturingNote that the debt-to-equity ratios vary widely among the industries in the sampleranging from significantly over one to significantly less than one The ratio ishighest in the steel industry where the value of debt is close to 17 times the
Table 31 Debt-to-equity ratios across select industries
Book value Market valueof equity of equity
Steel 1973 1665Petroleum refining 1548 1117Textiles 1405 1296Motor vehicles 0922 0594Plastics 0843 0792Machine tools 0472 0425Pharmaceuticals 0194 0079
Source Kester (1986) The book value of equity is computed fromaccounting sources while the market value of equity is obtained bymultiplying the number of outstanding shares by the current marketprice of the outstanding shares
34 Firms as market participants
market value of outstanding market shares In contrast for pharmaceuticals thedebt-to-equity ratio is only 0079 indicating that the industry uses bond financingvery little instead financing its investment activities almost exclusively throughthe issuance of equity
Adjustment costs for capital and Tobinrsquos Q
Let us now consider the choice of capital when there exist adjustment costs Tokeep the maximization problem simple we shall continue to assume ldquoretainedearningsrdquo financing of changes in the capital stock we would however obtainidentical results with bond or equity share financing As we have seen in the caseof retained earnings financing the problem facing the firm is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t36
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
As before substituting the above definitions for real dividends and output intothe Bellman equation and substituting in the transition function the first-orderconditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
The problem facing the firm in period t+1 assuming retained earnings financingof capital changes is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1Rt + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and given the following definitions for real dividends and output for period t37
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBpt+1 minus Int+1 minus ψ(Int+1)
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 35
Again we can substitute the above definitions for real dividends and output intothe Bellman equation for period t+1 and substituting in the transition function (inparticular note the fact that dKt+2dInt+1 = 1) obtain the following first-orderconditions
partft+1
partNt+1minus wt+1
pt+1= 0 (38)
minus (1 + ψ primet+1) + dW (Kt+2)
dKt+2= 0 where ψ prime
t+1 = dψ(Int+1)
dInt+1 (39)
Finally from our expression for the value function at time t + 1 W (Kt+1) wehave that
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (310)
Substituting (39) into (310) and then substituting the resulting expression fordW (Kt+1)dKt+1 into the first-order condition for Int (equation (32)) we obtain
minus(1 + ψ primet ) + 1
Rt
(partft+1
partKt+1 minus δd
)+ 1 + ψ prime
t+1
Rt= 0 (311)
Rearranging and simplifying we have that
partft+1
partKt+1= mt + δ + (mt + 1)ψ prime
t minus ψ primet+1
where ψ primet = ψ prime(I d
nt) and ψ primet+1 = ψ prime(I d
nt+1) An important feature of adjustmentcosts that is highlighted by the above equation is that the choice of investment inperiod t is now linked to the optimal choice of investment next period Since thisholds for each period in the future the choice of investment today is linked toinvestment decisions over all subsequent periods
We can simplify and rearrange the above first-order condition for investment inperiod t to obtain what is known as ldquoTobinrsquos Qrdquo To do so let us first assume anidentical real rate of return over time in particular we then have Ri = Rt = 1+mi = t + 1 t + 2 where (1 + m) equiv (1 + r)(1 + π) Let us also assume thefirm has attained its optimal capital stock so that Kd
t+1 = K and I dnt = 0 If the
production function is separable into capital and labor then the assumption of aninvariant real interest rate implies that I d
ni i = t +1 t +2 equal zero as well38
In this case ψ primet and ψ prime
t+1 can be replaced by a common ψ prime(0) gt 0 Then we maywrite the first-order condition as
partft+1partKt+1 minus m minus δ = ψ primem
Dividing by m and adding one to both sides we have
Tobinrsquos ldquomarginalrdquo Q equiv 1 + [partft+1partKt+1 minus m minus δ]m = 1 + ψ prime gt 1(312)
36 Firms as market participants
The above provides the definition for Tobinrsquos Q39 More precisely we have Tobinrsquosldquomarginalrdquo Q for it represents the ratio of the market value of an additional unitof capital to its replacement cost
Given adjustment costs the market value of an additional unit of capital exceedsits replacement cost so Tobinrsquos marginal Q is greater than one Since investmentdemand planned over the coming period determines marginal adjustment costsψ prime(I d
nt) we see that investment can be written in terms of Tobinrsquos Q40 The optimalrate of investment is that rate for which Q minus 1 is equal to the marginal cost ofinstallation Thus net investment demand is sometimes expressed as
Idnt = I d
nt(Q minus 1)dId
nt
d(Q minus 1)gt 0
The Q theory of investment is not operational as long as Q is not observableWhile marginal Q is not typically apparent with some additional assumptionswe can show that the expression known as Tobinrsquos ldquoaveragerdquo Q is identical toldquomarginalrdquo Q Tobinrsquos ldquoaveragerdquo Q is defined as the ratio of the total value of thefirmrsquos existing capital (the market value of its equity shares) to its total replacementcost and these variables are more easily measured41 In particular for a one-sectormodel in which the cost of capital and output are identical42
Tobinrsquos ldquoaveragerdquo Q equiv V
K
Hayashi (1982) has shown that if the firm is a price-taker if the production func-tion is linear homogeneous in K and N and if expectations of future real interestrates and real wages are static then the marginal and average Q are identical43 Tosee why this is the case note that if the real interest rate and real wage are invariantand the firm is at the optimal level of capital so that the capital stock is invariantover time then omitting time identifiers we have
V =infinsum
k=1
dRk
where (since ψ(Idnt) = ψ(0) = 0)
d = y minus (wp)N minus δK
Thus
V = [y minus (wp)N minus δK]m
where m equiv R minus 1 By Eulerrsquos theorem for a linear homogeneous production func-tion (ie K(partf partK) = y minus N (partf partN )) and the marginal productivity condition
Firms as market participants 37
for labor (ie wp = partf partN ) which reflects the price-taker assumption the firsttwo terms in the expression become K(partf partK)44 Dividing by K we thus obtain
Tobinrsquos ldquoaveragerdquo Q equiv V K = [partf partK minus m minus δ]m + 1
which as we can see is the same expression as that for Tobinrsquos marginal QAlternatively we can write the above as
V = [partf partK minus m minus δ]m + 1 = (Q minus 1) ([partf partK minus m minus δ]m)
Adjustment costs for labor labor as a ldquoquasi-fixed factorrdquo
Our prior characterization of the optimal choice of labor reflects an underlyingproduction process that incorporates a very simple view of labor For instancelabor markets are restricted to be only spot markets That is we rule out multiperiodlabor contracts Yet there is an extensive body of literature that investigates variousrationales for and the implications of such multiperiod (implicit) labor contractsOne reason why long-term contracts might emerge is an attempt by firms who areless risk-averse than workers to smooth out income over time45
A second reason why multiperiod labor contracts might emerge is if there areldquoadjustment costsrdquo to changes in the size of the labor force Adjustment costs couldreflect the fact that in order to hire new workers firms must incur hiring and trainingcosts Adjustment costs mean that a firm would view potential new hires andpreviously employed workers as imperfect substitutes and this would provide animpetus for multiperiod labor contracts The absence of adjustment costs simplifiesthe analysis by eliminating a rationale for multiperiod labor contracts and a choiceof labor given adjustment costs It also simplifies the analysis in two other ways
First the absence of adjustment costs suggests that we can measure labor ser-vices as the product of the fraction of the period each labor supplier works andthe number of individuals hired That is given no adjustment costs differences inldquohours workedrdquo and ldquonumber employedrdquo that leave total work hours unchanged areviewed by the firm as equivalent in terms of production In contrast with positiveadjustment costs firms would have a preference for meeting temporary changesin output by changing hours rather than by changing the number employed
A second implication of the absence of adjustment costs is that the employerviews as equivalent two workers working at ldquohalf-speedrdquo (and receiving ldquohalf-wagesrdquo) and one worker working at ldquofull-speedrdquo for ldquofull-wagesrdquo In contrastgiven adjustment costs firms would have a preference for meeting temporaryoutput changes by altering not only hours per worker but also the intensity that eachemployee was asked to work In fact output per work hour or ldquolabor productivityrdquodoes typically increase more rapidly during a recovery suggesting a more intensiveuse of labor
In contrast labor productivity growth is typically less rapid when the growth intotal output slackens This phenomenon is due in part to employersrsquo hoarding laborin slack times so as not to lose trained employees whom they will want when there
38 Firms as market participants
is an upturn in demand (That is to reduce subsequent adjustment costs given afuture upturn in production) Hoarding labor means that employers keep on moreworkers than necessary to produce the current output so that each worker has lesswork to do than normal The labor hoarding phenomenon also referred to as thelabor reserve hypothesis is the formal term for changes in the ldquointensityrdquo at whichlabor is used and explains lower output per work hour during periods of slackdemand46
Conclusion
The nature of the firm was discussed with the emphasis on profit maximizationDecisions of firm owners facing a variety of constraints and costs were analyzedwith particular attention paid to how financing constraints and adjustment costsaffected firm profits and the ability to adjust production levels A link was madebetween households and firms that will lead us into the next chapter Firms hireworkers who of course constitute households in the economy Workers are paidwages as costs to the firm but are an integral part of the production processMoreover firms may face costs of adjustment and other costly phenomena asso-ciated with decisions to alter the use of workers in the production process Takentogether then we see a link between firms and households as firm decisions havethe propensity to affect consumer income
4 Households as marketparticipants
Introduction
This chapter brings the household into our model of the macroeconomySpecifically the householdrsquos ability to obtain utility through consumption andthe labor supply decision is modeled within a choice framework The solutionis then used to formulate predictions about labor supply The concept of time iscritical to a thorough understanding of household behavior in the marketplace anda good deal of this chapter is spent analyzing intertemporal choices
The life-cycle and permanent income hypotheses are introduced and a theory ofportfolio choice is developed Finally the chapter ties many of these issues togetherand addresses the macroeconomic questions of absence of money illusion the realbalance effect and the real indebtedness effect
Individual experiments households
Two agents inhabit our expanded macroeconomic model with production firmsand households Having just discussed the nature of decisions confronting firmswe turn now to those confronting households These decisions can be broken downinto three types consumptionsaving portfolio and labor supply Consider nowthe first two decisions which should be familiar
The ldquoconsumptionsavingrdquo decision The representative household must deter-mine at time t the consumption purchases over each period at the implied ratect ct+1 We term this problem the ldquoFisherianrdquo problem
The ldquoportfoliordquo decision The individual must determine at time t the collectionof assets to hold at the end of each period In our expanded economy there areostensibly three types of assets
1 nominal money balances planned at time t to be held at the end of periodi Mi i = t t + 1
2 the nominal value of bonds planned at time t to be held at the end ofperiod i pbiBi i = t t + 1 where Bi denotes the planned number ofbonds held and pbi denotes the money price of bonds at the end of periodi and
40 Households as market participants
3 the nominal value of equity shares planned at time t to be held at the endof period i peiSi i = t t + 1 where pei denotes the money price ofan equity share at the end of period i and Si denotes the number of equityshares planned at time t to be held at the end of period i
Since bonds and equity shares are perfect substitutes we can consider them asa single entity with respect to householdsrsquo portfolio decisions We will let Aidenote the real holdings of financial assets planned at time t to be held at theend of period i i = t t + 1 That is for period t
At = [petSt + pbtBt]pt
and so on Excluding current dividend and interest payments the real value ofinherited financial assets at the end of period i reflecting portfolio decisions inthe prior period will be denoted by Ai i = t t + 1 For instance
At = [petS + pbtB]pt
At+1 = [pet+1St + pbt+1Bt]pt+1
The household problem
We start our analysis of the representative household as usual by discussing thehouseholdrsquos preferences constraints and objectives In particular we make thefollowing assumptions
Assumption 41 The representative householdrsquos preferences are described bythe utility function
u(ct ct+1 Mtpt Mt+1pt+1 1minusNt 1minusNt+1 )
where ci denotes the householdrsquos planned (at time t) rate of consumption dur-ing period i (from time i to time i + 1) Mi denotes the representative householdrsquosplanned (at time t) nominal holdings of money at the end of period i and Ni denotesthe householdrsquos planned (at time t) rate of supply of labor services during period i(from time i to time i +1) such that 1minusNi denotes the planned rate of leisure dur-ing period i It is assumed that partupartci gt 0 partupart(Mipi) gt 0 and partupart(1 minus Ni)
gt 0 i = t t + 1 Macroeconomics often assumes a time-separable utilityfunction a form of the utility function that ensures ldquotime consistencyrdquo1 In par-ticular following the tradition of macroeconomics we will assume that the totalutility for the representative household at time t with an infinite planning horizonis given by
infinsumi=t
β iminustu(ci Mipi 1 minus Ni)
Households as market participants 41
where β denotes the fixed personal or ldquoutilityrdquo discount factor with 0 lt β lt 12
In the above note that the one-period utility function for period t (time t to t + 1)is u(ct Mtpt 1minusNt) for period t +1 (time t +1 to t +2) it is u(ct+1 Mt+1pt+11 minus Nt+1) and so on Further note the infinite time horizon
Assumption 42 Individuals will choose the most preferred sequence of con-sumption money holdings and labor supply from the set of feasible alternatives(rationality) The feasible set of consumption money holdings and leisure
(ct ct+1 Mtpt Mt+1pt+1 1 minus Nt 1 minus Nt+1 )
is defined by the set of equalities
(wtpt)Nt + zBpt + At + Mpt minus [ct + Mtpt + At] = 0
(wt+1pt+1)Nt+1 + zBtpt+1 + At+1 + Mtpt+1
minus [ct+1 + Mt+1pt+1 + At+1] = 0
Note that we assume the budget constraints are met with equality Further note thatthe sum of dividends wage payments and interest payments equals total outputminus depreciation For instance for period t
(wtpt)Nt + dt + z middot Bpt = yt minus δK
and so on This is simply the firm distribution constraint for period t
Several aspects of the above problem deserve further elaboration First a wordon notation for future variables It is common in macroeconomics to derive themicroeconomic theoretical restrictions for the aggregate model under the conditionof certainty even though the analysis is then applied to situations that involvepotential stochastic elements One obvious way to eliminate considerations ofuncertainty from the analysis is to assume perfect foresight A second way is toassume that individual expectations of future events are point estimates held withsubjective certainty Note that either approach simplifies the analysis and in manycases this simplification gives us results that are not overturned if risk were to besystematically incorporated into the analysis
In the analysis of individual behavior below we will often for notationalsimplicity not distinguish between future prices and the expectations of futureprices Assuming expectations are held with subjective certainty this lack of dis-tinction will not be serious in discussing the result of the optimization problemsThat is the findings for perfect foresight can be made identical to those withoutthe assumption of perfect foresight by switching actual future prices for expectedprices Sometimes for clarity we will explicitly denote expected future prices(point estimates held with subjective certainty) by the superscript ldquoerdquo
42 Households as market participants
A second aspect of the above analysis that may initially appear odd concerningthe above one-period utility function for period i (from time i to time i+1) is that itseems that we are mixing money balances that occur at one time with consumptionand leisure that occur at an earlier time The reason for this is that money balancesare a stock variable and we are recording their value at the end of each period ithat is at time i + 13 The following scenario for the discrete-time analysis mayhelp clarify what is going on
At time t the labor market takes place and agreements are made to exchangelabor services at rate Nt over the period (t t + 1) for the money wage wt Duringthe period an output market operates in which firms sell output produced at rate yt During the period households receive money wages wt At the end of the period(time t + 1) households anticipate real interest payments zBpt from their priorpurchases of bonds (B) and real dividends dt from their prior purchase of equityshares stock (S)4 Given the above income sources as well as inherited nominalholdings of money (M ) and the anticipated value of inherited financial asset At atthe end of the period households plan an average rate of consumption ct duringthe period
At the end of period t after all income is received and final planned pur-chases of consumption goods are made the remainder reflects householdsrsquo planned(at time t) end-of-the-period change in real money balances ((Mt minus M )pt) andplanned changes in real financial assets holdings (At minus At) For financial assetsthe real price for bonds at the end of period t is pbtpt and the real price for equityshares is petpt
According to the above scenario the sale of labor services at rate Nt and rateof consumption ct over the period from time t to t + 1 tend to coincide whilereal money balances Mtpt and real financial asset holdings At can be viewed asthe planned (at time t) real stocks of such assets to be held at the end of period tIn continuous-time analysis as the length of the period h goes to zero the rate atwhich leisure is lost from supplying labor services during the period (Nt) the rateof consumption (ct) and the stocks of real money and real financial asset holdingswould coincide
A third aspect of the above analysis is that we have interpreted 1 minus Ni as theportion of the period of length 1 that the individual spends at leisure given thesupply of labor at rate Ni This is a simplification however for at the same timewe have suggested that the ldquoutility yieldrdquo of money is derived from its ability toreduce the transaction costs in arranging exchanges with such transaction costsreflecting at least in part a loss of leisure To explicitly incorporate such a viewof money leisure during a period i of length h given the sale of labor services Niand the real money balances Mipi held at the end of the period would be givenby h(1 minus Ni minus (Mipi)) where the function (Mipi) reflects transactions costsin terms of the loss of leisure The fact that prime lt 0 indicates that increased moneyholdings raise utility by reducing leisure lost in arranging transactions In this casethe one-period utility function would formally be given by u(ci 1minusNiminus(Mipi))with partupartci gt 0 and partupart(1 minus Ni minus (Mipi)) gt 0
Households as market participants 43
The general solution to the household problem
We can express the household problem in terms of a set of Bellman equationsAssuming perfect foresight (or equivalently interpreting future prices dividendetc as expectations of such variables held with subjective certainty) we thus havefor period t (time t to t + 1)
W (xt) = maxct Nt
Mtpt xt+1
[u(ct Mtpt 1 minus Nt) + W (xt+1)]
subject to the transition function
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt] Mtpt
and given xt Rmt is the real gross rate of interest on money that is the gross realrate of return on money and equals one divided by one plus the rate of inflation(1(1+πt)) The term xt is the total value in period t derived from the ldquoinheritedrdquoholdings of money bonds and stocks This total value is the sum of currentdividends and interest (received at the end of period t) on stock and bond holdingsacquired previously the real value of these financial assets at the end of period texclusive of these current interest and dividend payments and the real value ofpreviously acquired money holdings
xt equiv dt + zBpt + At + Mpt
where
At equiv [petS + pbtB]pt
The difference between the total real value derived in period t from inheritedbonds stocks and money balances plus real wage income xt + (wtpt)Nt andconsumption in period t ct reflects the acquisition of bonds equity shares andmoney holdings at the end of period t by the representative household Letting Atdenote the planned holdings of financial assets at the end of period t we thus havefrom the household budget constraint that
At + Mtpt = xt + (wtpt)Nt minus ct
where
At equiv [petSt + pbtBt]pt
Recall that Rt the gross real rate of return on financial assets equals one plus thenominal interest rate divided by one plus the rate of inflation ((1 + rt)(1 + πt))Thus in period t + 1 and given our definition of Rmt the inherited real value of
44 Households as market participants
bonds stocks and money balances including dividends and interest payments isgiven by
xt+1 = RtAt + RmtMtpt
Substituting in the expression for At derived from the household budget constraint(ie At = xt + (wtpt)Nt minus ct minus Mtpt) and rearranging we obtain the transitionfunction
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt]Mtpt
Substituting the transition function into Bellmanrsquos equation for period t wehave the following first-order conditions for ct Nt and Mtpt assuming interiorsolutions (ie ct gt 0 1 gt Nt gt 0 and Mtpt gt 0)
partutpartct minus (partW (xt+1)partxt+1)Rt = 0 (41)
partutpart(1 minus Nt) + (partW (xt+1)partxt+1)Rt(wtpt) = 0 (42)
partutpart(Mtpt) minus (partW (xt+1)partxt+1)(Rt minus Rmt) = 0 (43)
The above conditions indicate that for period t (time t to time t + 1) we havefrom equations (41) and (42) that
partutpart(1 minus Nt)
partutpartct= wt
pt (44)
In words the optimal choice of leisure is such that the marginal value of leisurein terms of consumption that is the marginal rate of substitution between leisureand consumption as given by
partutpart(1 minus Nt)
partutpartct
equals the marginal cost of leisure in terms of consumption forgone in the currentperiod as given by the real wage wtpt
From equations (41) and (43) we have for period t that
partutpart(Mtpt)
partutpartct= (Rt)
minus1(Rt minus Rmt)
In words the optimal choice of real money balances is such that the marginal rateof substitution between real money balances and consumption as given by
partutpart(Mtpt)
partutpartct
Households as market participants 45
equals the marginal cost in terms of the present value of the loss in interest incomein the subsequent period due to the holding of money balances instead of financialassets as given by the expression (Rt)
minus1(Rt minus Rmt) Recall that Rt minus Rmt equals(1 + rt)(1 + πt) minus (1(1 + πt)) which is simply rt(1 + πt) or essentially theanticipated nominal rate of interest Given that Rt = (1 + rt)(1 + πt) we thushave that (Rt)
minus1(Rt minus Rmt) = rt(1 + rt)We can expand upon the above discussion of the first-order conditions for
period t by first noting that W (xt+1) is defined by
W (xt+1) = maxct+1Nt+1
Mt+1pt+1xt+2
[βu(ct+1 Mt+1pt+1 1 minus Nt+1) + W (xt+2)]
where
xt+2 = Rt+1[xt+1 + (wt+1pt+1)Nt+1 minus ct+1] minus [Rt+1 minus Rmt+1]Mt+1pt+1
given
xt+1 equiv dt+1 + zBtpt+1 + At+1 + Mtpt+1
At+1 equiv [pet+1St + pbt+1Bt]pt+1
Again substituting the transition function into the Bellman equation for periodt + 1 we have the following first-order conditions for period t + 1
βpartut+1partct+1 minus (partW (xt+2)partxt+2)Rt+1 = 0 (41prime)minus βpartut+1part(1 minus Nt+1) + (partW (xt+2)partxt+2)Rt+1wt+1pt+1 = 0 (42prime)βpartut+1part(Mt+1pt+1) minus (partW (xt+2)partxt+2)(Rt+1 minus Rmt+1) = 0 (43prime)
Now consider the impact of the change in xt+1 on the value function W (xt+1)The above first-order conditions imply that the indirect effects of the change inxt+1 on W (xt+1) through the effect of such a change on the choice of the optimalvalues of consumption labor supply and real money balances for period t + 1 arezero5 This is simply an application of the envelope theorem which states thatthe change in the objective function adjusting the choice variables optimally isequal to the change in the objective function when one does not adjust the choicevariables This fact along with the transition function for xt+2 gives us
partW (xt+1)partxt+1 = (partW (ct+2)partxt+2)Rt+1 (45)
Substituting equation (41prime) into (45) we obtain
partW (xt+1)partxt+1 = βpartut+1partct+1 (46)
46 Households as market participants
Alternatively by substituting in (42prime) we can obtain6
partW (xt+1)partxt+1 = β[partut+1part(1 minus Nt+1)]pt+1wt+1 (47)
Combining equations (41) and (46) gives us the standard Fisherian solutionfor the optimal allocation of consumption between period t (time t to t + 1) andperiod t + 1 (time t + 1 to t + 2)
partutpartct
βpartut+1partct+1= Rt
Combining equations (43) and (46) we obtain the standard expression for theoptimal portfolio choice of money
partutpart(Mtpt)
βpartut+1partct+1= Rt minus Rmt
Combining equations (42) and (47) gives us an expression for the optimalallocation of labor supply over time
partutpart(1 minus Nt)
βpartut+1part(1 minus Nt+1)= pt+1
wt+1Rt
wt
pt (48)
Having discussed the ldquoFisherian problemrdquo and the ldquoportfolio problemrdquo con-fronting the household we turn our attention in the next section to the ldquolaborsupply problemrdquo as captured by equations (44) and (48) Before doing so how-ever a general comment should be made with respect to the discussions to followas well as the preceding discussions of the Fisherian problem and the portfoliodecision
In focusing on first-order conditions with respect to the particular variablesat issue (ie first-order conditions for consumption now and next period forthe Fisherian problem first-order conditions for real money holdings and futureconsumption for the portfolio problem and first-order conditions for labor sup-ply now and next period for the labor supply decision) one has a tendency toforget that the optimizing problem involves the simultaneous choice of consump-tion portfolio and leisure In general this means that the analysis is often notas straightforward as it may first appear For instance a change in the currentreal wage or an expected real interest rate can affect the first-order conditionconcerning the choice of labor supply through its impact on the choice of consump-tion if part2upartcpartN = 0 for then the change in consumption alters the ldquomarginalutilityrdquo of leisure One simple way to abstract from these ldquoindirectrdquo effects is toassume that the utility function is separable not only across time but also withrespect to consumption leisure and real money balances each period such thatpart2upartcpartN = part2upartcpart(Mp) = part2upartNpart(Mp) = 0
Households as market participants 47
The choice of hours within a period
Equation (44) indicates that at the optimal labor supply the household cannot bemade better off by trading consumption for leisure within periods at the expectedreal wage for the period Equation (48) indicates that at the optimum the householdcannot be made better off by trading leisure across periods given the relevant realwages and the real interest rate7 Figure 41 captures the first situation that is theoptimal choice of consumption and leisure within a period
For the moment let us hold anticipated real interest rates constant Further letus assume unit elastic expectations with respect to wages as well as prices so thata change in the current wage or price level that alters the current real wage changesfuture expected real wages as well so that there is no change in the current realwage relative to future expected real wages In addition let us hold constant for themoment the effect on current real money balances of a change in the current realwage These assumptions help us to mimic the traditional ldquostaticrdquo or single-periodanalysis (eg Patinkin) of the effect of a change in the current periodrsquos anticipatedreal wage on individualsrsquo labor supply during period t Under such circumstancesan increase in the real wage for period t can have ambiguous effects As Patinkin(1965) states ldquofor simplicity it is assumed that [labor] supply is an increasingfunction of the real wage though there are well known reservations on this scorerdquo
To understand what Patinkin is referring to consider an increase in the real wagedue to a rise in the money wage wt Consider one possible result on the householdrsquoslabor supply decision The increase in the net real wage means a steeper budgetline as the householdrsquos optimal leisurendashconsumption combination changes from1 minus N s
t and cdt (call this choice A) to (1 minus N s
t )prime and (cdt )prime (choice C)
An increase in the real wage has two conceptually distinct effects on the house-holdrsquos labor supply decision an income effect and a substitution effect The incomeeffect refers to the fact that an increase in the real wage makes the household betteroff because it leads to an increase in the householdrsquos feasible consumption set in
$
Leisure
Figure 41 Consumption and leisure
48 Households as market participants
the current period The higher real wage means that the household if it so desirescan increase both its leisure and consumption Thus the household is able to reacha higher indifference curve which has associated with it a higher utility levelThe substitution effect refers to the fact that the increase in the real wage makesan hour of leisure relatively more expensive in terms of consumption that must beforgone
To isolate the substitution and income effects of the change in the net real wagesuppose that after the real wage increases we temporarily take away just enoughof the householdrsquos nonlabor income so that it is just able to attain its originalindifference curve The householdrsquos choice of leisure and income would then be(1 minus N s
t )primeprime and (cdt )primeprime (call this choice B) Thus if we hold the householdrsquos utility
level constant the increase in the real wage leads unambiguously to a lower levelof leisure since an hour of leisure is relatively more expensive the householdsubstitutes away from leisure choosing to work more hours The movement fromchoice A to choice B constitutes the pure ldquosubstitution effectrdquo of the higher netreal wage8
Now suppose that we give the household back the nonlabor income that wetemporarily took away This causes an outward shift in the budget line and thehouseholdrsquos new choice of leisure and consumption would be (1 minus N s
t )prime and (cdt )prime
(choice C) The movement from B to C constitutes the ldquoincome effectrdquo of thechange in the net real wage In the present case the income effect on the choiceof leisure is positive reflecting the assumption that leisure is a normal good
Note that the substitution and income effects on leisure work in opposite direc-tions The substitution effect of the higher real wage causes leisure to fall andhours worked to rise while the income effect causes leisure to rise and hoursworked to fall The net effect on leisure and hours worked depends on whicheffect dominates9 If the substitution effect dominates then a higher real wageresults in a decrease in desired leisure and an increase in desired working hoursIf the income effect dominates the opposite is true and the individualrsquos laborsupply curve is ldquobackward bendingrdquo when plotted against the real wage
The available evidence suggests that for many workers the income effect tendsto dominate slightly Estimates are that for men an increase of 10 percent in the realwage results in approximately a 15 percent reduction in the hours worked Thisreduction in hours worked reflects an income effect of approximately minus25 percentand a substitution effect of about 1 percent Other evidence suggests a similarpattern for working women10
The choice of participation within a period
Thus far we have considered a household representative of those who are in thelabor force working a positive number of hours Yet this masks the unambiguouseffect of a higher real wage on the labor supply of those households not in the laborforce To show this we consider a corner solution with respect to labor supply inparticular a household denoted a that has chosen not to participate in the labormarket For such a household let us return to the Bellman equation for period t and
Households as market participants 49
introduce explicitly the nonnegativity constraint for labor supply (ie Nt ge 0)Letting micron denote the multiplier associated with this constraint we have as first-order conditions for household arsquos consumption and labor supply11
partuatpartcat minus (partW (xat+1)partxat+1)Rt = 0
minus partuatpart(1 minus Nat) + micron + (partW (xat+1)partxat+1)Rtwtpt = 0
Nat ge 0 micronNat = 0
Substituting the first equation into the second and rearranging gives
minuspartuatpart(1 minus Nat)
partuatpartcat= wt
pt+ micron
partuatpartcat
For individuals not participating in the labor force micron ge 0 The ldquocorner solutionrdquo isa case in which the marginal rate of substitution of leisure in terms of consumptionis greater than the real wage In other words the absolute value of the indifferencecurve is equal to or greater than the absolute value of the budget line at the pointwhere N s
at = 0Note that at the optimal choice the corresponding indifference curve is more
steeply sloped than the budget line Thus even when all hours are devoted toleisure and none to work the individualrsquos marginal rate of substitution of leisurein terms of consumption still exceeds the real wage rate In other words theindividualrsquos valuation of leisure exceeds the marketrsquos valuation of leisure As aresult individual a does not find it worthwhile to participate in the labor market
The greater the real wage the greater is the probability that a given individualwill choose to participate in the labor market A higher real wage rotates thebudget line outward Since the individual is not working an increase in the netreal wage does not make him better off and thus has no income effect There isonly a substitution effect Thus if the real wage rises sufficiently the individualcan be induced to enter the labor market
According to the above analysis the economy-wide labor supply response toan increase in the current real wage is a combination of an ambiguous effecton the labor supply of those currently working but an unambiguous increase inthe labor supply among those not working12 It is the net of these two effects theldquohoursrdquo decision and the ldquoparticipationrdquo decision that is captured by the aggregatelabor supply function it is commonly assumed that this net effect is such that theaggregate quantity of labor supplied is an increasing function of the current realwage
The labor supply intertemporal substitution hypothesis
Our discussion has yet to consider ldquothe labor market intertemporal substitutionhypothesis (ISH) which states that labor supply responds positively to transitoryincreases in real wages and increases in the real interest rate a central hypoth-esis of modern competitive models of the business cyclerdquo (Alogoskoufis 1987)
50 Households as market participants
To do so we simply expand our focus to the inherent intertemporal decisionconfronting the household In particular recall our expression (48) for the optimalallocation of labor supply over time The view of labor supply embedded in (48)has life-cycle as well as business-cycle implications With respect to life-cycleimplications the theory predicts that workers will concentrate their labor supplyin years of peak earnings consuming leisure in larger than average amounts duringchildhood and old age
With respect to the business cycle the above helps explain an apparent contra-diction in the static theory of labor supply ndash the observed wage inelasticity of laborsupply in the long run with short-run fluctuations in employment which requirean elastic labor supply if one takes a ldquomarket-clearingrdquo approach with respect tothe labor market13 It does so by introducing a distinction between a permanentchange in the real wage and a temporary or transitory change in the real wage
To show the intertemporal substitution effect with respect to labor supply spec-ify wlowastplowast as the permanent or ldquonormalrdquo real wage with the anticipated real wagenext period equal to this value such that
wipi = wlowastplowast i = t + 1 t + 2
Further we assume that
Ri = Rlowast i = t t + 1 t + 2
In this case equation (48) becomes
partutpart(1 minus Nt)
βpartut+1part(1 minus N lowast)= Rlowast wtpt
wlowastplowast (48prime)
where N lowast denotes the ldquolong-runrdquo supply of labor at ldquonormalrdquo wages Further letus assume that for the representative household βRlowast = 1 Then equation (48prime)becomes
partutpart(1 minus Nt)
partut+1part(1 minus N lowast)= wtpt
wlowastplowast (48primeprime)
Equation (48primeprime) indicates that if the current real wage is higher than the normal realwage then ldquomore labor is supplied than would be implied by the long-run laborsupply functionrdquo That is this theory views suppliers of labor as reacting primarilyto three variables an anticipated ldquonormalrdquo or ldquopermanentrdquo real wage rate whichcorresponds to the wage rate in the usual one-period analysis of the laborndashleisurechoice and has a negligible effect on labor supply the deviation of the current realwage from this normal rate which has a strong positive effect on labor supplyand the expected real rate of interest (Lucas and Rapping 1970 284ndash285)14
The above theory provides the underlying microtheoretical basis for the fol-lowing statement ldquomeasured unemployment (more exactly its nonfrictionalcomponent) is then viewed as consisting of persons who regard the wage ratesat which they can currently be employed as temporarily low and who therefore
Households as market participants 51
choose to wait or search for improved conditions rather than invest in moving oroccupational changerdquo (Lucas and Rapping 1970 285)
Empirical tests seem to provide some support for this intertemporal substitutionhypothesis15 Alogoskoufis (1987 950) finds that for measures of the total numberof employees the real wage and interest rate elasticities are high and relativelywell determined ldquoThe elasticity of labor supply to transitory changes in real wagesis around unity and is statistically significant at conventional significance levelswith one exception The real interest rate always has a significant independentinfluencerdquo Note that as suggested by equation (48) Alogoskoufis finds that arise in the real interest rate increases current labor supply
Note that equation (48) does not explicitly identify the initial asset holdings asa variable that affects the relative choice of labor supply across periods Similarlythe condition for the optimal choice of consumption across periods did not have theinitial value of assets affecting the relative consumption purchases across periodsThis is a property of time-separable preferences
Special topics in intertemporal choices
As we have seen the intertemporal problem confronting the individual involvessimultaneous decisions with respect to consumption versus saving and with respectto the composition of the asset portfolio To make some sense of what is involvedwe start by considering what is known as the ldquoFisherianrdquo problem which focuseson the individualrsquos choice of consumption across time
Fisherian analysis
The Fisherian problem typically deals with the allocation of consumption acrosstime when there is a single means by which income can be allocated across time16
To restrict our analysis to such a case we can simply omit real money holdingsfrom the utility function Further we consider only interior solutions with respectto consumption (ie cai gt 0 i = t t + T )17
Thus the maximization problem becomes18
maxcat cat+T
xat+1xat+T+1
t+Tsumi=t
β iminustua(cai)
subject to
minus xat+1 + Rt[xat + cat minus cat] = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1] = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2] = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ] = 0
xat+T+1 ge 0
52 Households as market participants
Recall that Ri = (1 + ri)(1 + πi) i = t t + T denotes the ldquogross real rateof returnrdquo on agent arsquos portfolio between the end of period i and the end of periodi + 1 and asset holdings are solely in the form of bonds such that
xat equiv (1 + r)pbBapt
xat+1 equiv (1 + rt)pbtBatpt+1 = [(1 + rt)(1 + πt)] pbtBatpt
xat+i equiv (1 + rt+iminus1)pbt+iminus1Bat+iminus1pt+i
= [(1 + rt+iminus1)(1 + πt+iminus1)] pbt+iminus1Bat+iminus1pt+iminus1
i = 2 T + 1
Let λi i = t t +T denote the Lagrange multipliers for the constraints linkingthe total value of real asset holdings at the end of period i with the total value ofreal asset holdings at the end of period i + 1 Let microT denote the multiplier for thenonnegativity condition that xat+T+1 ge 0 Then the first-order conditions for theimplied Lagrangian L include
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai) = 0 i = t t + T 19
partLpartmicroT = xat+T+1 ge 0
microT ge 0
where uai = ua(cai) i = t t + T 20 Note that the above set of first-order
conditions consist of 3(T + 1) + 1 equations to determine 3(T + 1) + 1 variablesThe variables to be determined are cai i = t t + T xai i = t t + T + 1and microT Assuming continuity and strict concavity in ua(middot) and given a convex setof constraints
xai+1 xai cai| minus xai+1 + Ri[xai + cai minus cai] ge 0there is a unique solution to the problem
There are several implications of the above first-order conditions First theconditions imply that the desired total real value of assets (bonds) inherited at timet + T + 1 xd
at+T+1 will equal zero if duat+T dcat+T gt 0 In particular from the
condition
partLpartcat+T = βT (duat+T dcat+T ) minus λt+T Rt+T = 0
we see that if duat+T dcat+T gt 0 then λt+T gt 0 From the condition
partLpartxat+T+1 = minusλt+T + microT = 0
Households as market participants 53
we thus have that microT gt 0 This in turn implies from the condition
microT partLpartmicroT = microT xat+T+1 = 0
that xat+T+1 = 0 This finding should not be surprising With a time horizonof t + T agent a perceives no gain (utility) from acquiring assets at time t + Tto finance consumption in period t + T + 1 and a clear loss from doing so attime t + T in terms of consumption forgone given the assumption of nonsatiation(dua
t+T dcat+T gt 0) Second from the first set of equations we know that betweenany two periods i and i + 1
β iminustduai dcai
β iminust+1duai+1dcai+1
= λiRi
λi+1Ri+1 i = t t + T minus 1
This expression can be simplified to obtain
duai dcai
βduai+1dcai+1
= Ri i = t t + T minus 1
where Ri the real gross return between the end of periods i and i + 1 is given by(1+ri)(1+πi) Ri has been called Fisherrsquos ldquo(gross) real interest raterdquo since he wasone of the first to provide a lucid account of its role in determining consumptionacross time
Fisherrsquos ldquo(net) real interest raterdquo denoted by mi is then defined by
1 + mi equiv Ri = (1 + ri)(1 + πi)
Subtracting one from both sides and rearranging we have
mi = (ri minus πi)(1 + πi)
Thus for small expected rates of inflation we have the approximation21
mi = ri minus πi
In words the real interest rate is approximately equal to the nominal interest rateminus the rate of inflation The expected real interest rate is then the nominalinterest rate minus the expected rate of inflation
There are several features of the above that should be noted First if an individ-ualrsquos discount factor (β lt 1) equals the reciprocal of the real gross rate of interest([Ri]minus1 = (1 + πi)(1 + ri)) between periods i and i + 1 so that
βRi = 1
then the above expression of the first-order conditions for cai and cai+1 indicatesthat dua
i dcai = duai+1dcai+1 Given the concavity of the single-period utility
function (ua(middot)) and our assumption of a time-invariant one-period utility function
54 Households as market participants
it then follows that the individual will choose the same rate of consumption acrossperiods i and i + 1 in such a case22 This constant path of consumption betweenthe two periods can be said to emerge if an individualrsquos ldquorate of time preferencerdquoequals the (real) interest rate
If the expected gross return between two periods were higher (or for an individualwith a higher discount factor) the fact that β gt (Ri)
minus1 or equivalently βRi gt 1means from the first-order conditions that dua
i dcai gt duai+1dcai+1 Given the
assumed concavity of the one-period utility function the implication is that agentarsquos consumption during period i would be less than during period i + 1 That isβRi gt 1 rArr cd
ai lt cdai+1 Conversely if the expected real gross return were to be
lower (or for an individual with a lower discount factor) the fact that β lt (Ri)minus1
or equivalently βRi lt 1 means that the individualrsquos consumption during period iwould be greater than during period i + 1 That is βRi lt 1 rArr cd
ai gt cdai+1
For the two-period case (say periods t and t + 1) the optimal consumption ineach of the two periods can be shown graphically by the point of tangency betweenan indifference curve with slope minus(dua
i dcai)β(duai+1dcai+1) and a budget line
with slope minusRt 23 If one were to place cat+1 on the vertical axis and cat on thehorizontal axis then it would be possible to determine whether an individual is aborrower or a lender in any period For example if disposable income were lessthan consumption in the first period t then the individual is a lender at time tNote that points on the same indifference curve are such that
d[ua(cat) + βua(cat+1)
] = (duat dcat+1)dcat + β(dua
t+1dcat+1)dcat+1
= 0
Rearranging we have the slope of an indifference curve given by
dcat+1
dcat= minus dua
t dcat
βduat+1dcat+1
Note that points on the budget line satisfy the present value constraint
cat + (Rt)minus1cat+1 = cat + xat + (Rt)
minus1cat+1
Rearranging we have
cat+1 = Rt[cat + xat minus cat] + cat+1
such that the slope of the budget line is given by
dcat+1dcat = minusRt
Thus at the point of tangency between the budget line and an indifference curve
duat dcat
βduat+1dcat+1
= Rt
Households as market participants 55
which is the expression we obtained previously concerning the optimal choice ofconsumption between periods i and i + 1
Note that the above analysis is for an individual consumer Thus aggregate con-sumption need not behave as that predicted above for the individual For instancean aging population could lead to variations in aggregate consumption that reflectthe aggregation at different times across agents with differing characteristics
Life-cycle and permanent income hypotheses
We have seen how a householdrsquos consumption in any period is not constrained bythe income it receives during that period but rather that the discounted value oflifetime consumption is constrained by the discounted stream of income accruingto the household over its lifetime plus initial asset holdings While income tendsto rise and fall during the lifetime of an individual through appropriate saving andborrowing the individual can maintain a smooth or constant rate of consumptionover his lifetime This smoothing of consumption across time plays a critical rolein Franco Modiglianirsquos ldquolife-cycle hypothesisrdquo of consumption24
A stylized pattern of income and consumption expenditures over an individ-ualrsquos lifetime is the following Prior to retirement income exceeds consumptionand saving is positive During this period saving increases household wealth Onretirement consumption is financed by dissaving During the retirement periodhousehold wealth falls as people draw on their accumulated savings to financeconsumption Implied in this discussion is an inverted U-shape wealthndashage pro-file (save during pre-retirement years and dissave in years following retirementrunning down the stock of accumulated wealth) A number of studies of aggregatehousehold consumption and saving behavior support this wealthndashage pattern25
To make clear the implications of consumption smoothing for the demand forthe consumption good at time t let us make the simplifying assumptions that
(a) for any period i = t t + T Ri = R and(b) the individualrsquos personal discount rate β equals the constant real ldquomarketrdquo
discount rate Rminus126
From the first-order conditions we thus have the result that agent a willcompletely smooth out consumption spending across time so that consumptioncd
ai = cda i = t t + T In this case we can use the prior combined budget
constraint to obtain
cdat =
xat +
t+Tsumi=t
(caiRiminust)
where which equals 1sumt+T
i=t (1Riminust) is what Modigliani has calledthe ldquoproportionality factorrdquo and indicates the proportion of householdsrsquo totalresources ndash consisting of initial assets current income and anticipated futureincome ndash devoted to consumption each year
56 Households as market participants
An important implication of Modiglianirsquos life-cycle hypothesis is that thefraction of an increase in current income (cat) that goes toward increased cur-rent consumption (the ldquomarginal propensity to consumerdquo) will vary depending onwhether the increase in current disposable income is accompanied by an equivalentincrease in anticipated future income (cat i = t + 1 t + T )27 If a change incurrent income is viewed as ldquotransitoryrdquo most of the increase in income will goto saving in order to finance increased consumption during future years
This idea that the effect of a change in current disposable income on consump-tion demand depends on the degree to which the change in income is viewed astemporary or permanent lies at the heart of Milton Friedmanrsquos permanent incomehypothesis The permanent income hypothesis is like the life-cycle hypothesis inthat it emphasizes the fact that consumption demand in period t depends not onlyon current income but also on anticipated income in the future periods Permanentincome is that income which if received each year over a householdrsquos time hori-zon would yield an income stream with present value exactly equal to the presentvalue of the householdrsquos anticipated income stream That is permanent income cpis defined by the following equation
t+Tsumi=t
(cpRiminust) =t+Tsumi=t
(caiRiminust) + xat (49)
Factoring out cp on the left-hand side of (49) and rearranging we have
cp =
xat +
t+Tsumi=t
(caiRiminust)
(410)
Equation (410) indicates that permanent income is simply a weighted average ofcurrent and future incomes but in this case income in the more distant future isweighted less heavily since it is discounted more highly
Comparing permanent income to agent arsquos consumption demand in period tcd
at if there is complete smoothing of consumption spending across time then weobtain
cdat = cp
The implication of this equation is that the marginal propensity to consume out ofa change in current income that is perceived as permanent is equal to one whilethe marginal propensity to consume out of a change in current disposable incomethat is perceived as entirely transitory (having little impact on permanent income)is small Changes in transitory components of income are almost entirely saved ifpositive or borrowed if negative28
Our discussion so far of the impact of changes in income on current consump-tion demand has been restricted to what might be referred to as the effects ofldquotransitoryrdquo versus ldquopermanentrdquo income changes In doing so we have assumed adeterministic world in which individuals have perfect foresight concerning future
Households as market participants 57
income streams But what happens if individuals do not have perfect foresight Inparticular what if we introduce stochastic elements so that realized future incomeis a random variable Then the above theories suggest a difference in the responseof consumption demand to income changes that are anticipated or expected versusunanticipated changes In particular the life-cyclepermanent income hypothesiswould predict that previously anticipated (or expected) changes in income wouldhave no effect on consumption demand since consumption plans have alreadyincorporated this income29
Portfolio choice
Now let us consider the more general case in which agent a chooses not onlyconsumption across time but the portfolio of assets (money and bonds) That isconsider the following problem
maxcat cat+T
xat+1 xat+T+1Matpt Mat+T pt+T
t+Tsumi=t
β iminustua(cai Maipi)
subject to
minus xat+1 + Rt[xat + cat minus cat] minus [Rt minus Rmt]Matpt = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1]minus [Rt+1 minus Rmt+1]Mat+1pt+1 = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2]minus [Rt+2 minus Rmt+2]Mat+2pt+2 = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ]minus [Rt+T minus Rmt+T ]Mat+T pt+T = 0
xat+T+1 ge 0
As before to simplify the problem we assume interior solutions in this case notonly with respect to the consumption good but also with respect to money holdingsAgain let λi i = t t + T denote the Lagrange multipliers for the constraintslinking the real asset holdings at the end of period i with their real value at the endof period i + 1 and let microT denote the multiplier for the nonnegativity conditionthat xat+T+1 ge 0 The first-order conditions are
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpart(Maipi) = β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0 i = t t + T
58 Households as market participants
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai)
minus [Ri minus Rmi]Maipi = 0 i = t t + T 30
partLpartmicroT = xat+T+1 ge 0
microT partLpartmicroT = microT xat+T+1 = 0
λi ge 0 i = t t + T
microT ge 0
where uai = ua(cai Maipi) i = t t + T To isolate the portfolio choice
consider the portfolio choice of money and bond holdings for a given level ofcurrent consumption That is let us look at the optimal choice of Maipi given thatcai is held constant From the second set of conditions
β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0
Substituting the condition for the optimal choice of the total value of assets in thatperiod
λi = λi+1Ri+1
we obtain
β iminustpartuai part(Maipi) minus λi+1Ri+1[Ri minus Rmi] = 0
Now substituting the condition for the optimal choice of consumption next period(i + 1) as given by
β iminust+1duai+1dcai+1 minus λi+1Ri+1 = 0
we obtain
β iminustpartuai part(Maipi) minus β iminust+1(dua
i+1dcai+1)[Ri minus Rmi] = 0
Rearranging gives
duai d(Maipi)
βduai+1dcai+1
= Ri minus Rmi
Recalling that the expected gross real return on bonds in period i Ri equals(1 + ri)(1 + πi) and that the expected gross real return on money Rmi equals1(1 + πi) the above expression can be written as
duai d(Maipi)
βduai+1dcai+1
= ri
1 minus πi
Households as market participants 59
Note that in the limit (as the length of the period goes to zero) the expected rate ofinflation term would vanish The implication is that the optimal division of assetsbetween money and bonds depends primarily on the money interest rate aloneWhat is essentially being shown is that an increase in money holdings with nochange in current consumption means a reduction in bond holdings and thus theloss of nominal interest income ri or real interest income ri(1 + πi)
Absence of money illusion real balance effects and realindebtedness effects
There are two aspects of agent arsquos demand function that should be noted Firstagent arsquos demand functions at time t can be shown to be homogeneous of degree 0 inthe current price level pt initial money balances M a and initial bond holdings BaIn this economy this is said to reflect the ldquoabsence of money illusionrdquo One criticalreason for this is the assumption of unit elastic expectations with respect to futureprices so that changes in the current price level leave unchanged the expectedrates of change in the price level in subsequent periods Also note that the currentand expected future money payments attached to bonds are being held constantso that given the fixed money payment x on maturity interest rates are unchangedAlternatively one could have money prices and the fixed future money paymentattached to one-period bonds rise by the same proportion
The above implies that individual arsquos demand for the consumption good andreal money balances at time t can be represented by
cdat = cd
at(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
M datpt = M d
atpt(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
where xat is the individualrsquos real wealth at the end of period t as given by
Wat = xat + cat +t+Tminus1sum
j=t
⎡⎣ jprod
i=t
(Ri)
⎤⎦
minus1
(caj+1)
From the budget constraint for period t we know that the above two demandconditions imply a real demand for bonds of a similar form since
pbtBdatpt = cat + xat minus
[cd
at + M datpt
]
Note that the above demand functions do not depend solely on wealth and thepattern of expected real (gross) rates of interest since given the portfolio choice agiven pattern of expected real (gross) interest rates could alter demand dependingon the underlying values of the money interest rate As before individual arsquos excessdemand function for the consumption good and money in period t are defined byzat = cd
at minus cat zam = M datpt minus M apt and zab = pbtBd
atpt 31
60 Households as market participants
The ldquoreal balance effectrdquo indicates the effect of a change in real balances (M apt)
on individual demand for goods other than money As before there is a real balanceeffect that reflects a wealth effect That is a decrease in initial real balances leadsto a reduction in real money demand M d
atpt and a reduction in the real demandfor bonds pbtBd
atpt There is also what might be referred to as a ldquoreal indebtednesseffectrdquo in that a change in prices alters not only real money balances but also realinitial debt If Ba gt 0 an increase in pt reduces wealth while if Ba lt 0 anincrease in pt increases wealth This is why the characterization of the absence ofmoney illusion has been expanded to include changes in Ba
It is typical in macroeconomics to adopt the convention of the ldquorepresentativeagentrdquo to reduce notational clutter Recall that the ldquorepresentative agentrdquo is essen-tially the average agent For instance if we let cd
at denote the demand for theconsumption good in period t by representative agent a and cd
t market demandat the time then cd
at = cdt Thus depending on the context we can interpret cd
tas demand by the representative agent or market demand Recall that in doingso we essentially ignore distribution effects such as effects on market demandof changes in the distribution of initial endowments of commodities or moneybalances or of changes in the distribution of future endowments In the context ofthe real indebtedness effect since in the aggregate B = 0 this effect is removedfrom our analysis
Intertemporal substitution the evidence
We have focused above on the behavior of an individual with respect to the plannedpath of consumption across time As Robert Hall (1988 340) indicates
The essential idea is that consumers plan to change their consumptionfrom one year to the next by an amount that depends on their expectationsof real interest rates Actual movements of consumption differ from plannedmovements by a completely unpredictable random variable that indexes allinformation available next year that was not incorporated in the planningprocess the year before If expectations of real interest rates shift then thereshould be a corresponding shift in the rate of change of consumption Themagnitude of the response of consumption to a change in real interest rateexpectations measures the intertemporal elasticity of substitution32
Hall (1988 340ndash341) goes on to state that
the basic model of the joint distribution of consumption and the return earnedby one asset that has emerged is the following The joint distribution ofthe log of consumption in period t log ct and the (real) return earned by theassets from period t minus 1 to period t mtminus1 is normal with a covariance matrixthat is unchanging over time The means obey the linear relation
E(log ct) = k + ctminus1 + σE(mtminus1) (411)
Households as market participants 61
That is the expected change in the log of consumption is a parameter σ times the expected real return plus a constant If the expected real interestrate E(mtminus1) is observed directly then the key parameter σ can be estimatedsimply by regressing the change in the log of consumption on the expectedreal rate That regression also has the property that no other variable knownin period t minus 1 belongs in the regression
Hall proceeds to estimate the parameter using aggregate data on consumptionand finds that there is ldquolittle basis for a conclusion that the behavior of aggregateconsumption in the United States in the twentieth century reveals an importantpositive value of the intertemporal elasticity of substitutionrdquo (1988 356)
Hallrsquos empirical finding is of importance to macroeconomic analysis Thework by Hall and others is also of interest as an example of how theoreticalmacroeconomic analysis specifically Fisherian analysis can be tested To seethe link between the theory developed in the prior section and the proposed test(equation (411)) assume the following
Assumption 43 The path of aggregate consumption reflects agent arsquos decisionsconcerning the optimal allocation of consumption across time That is we treatagent a as the ldquorepresentative consumerrdquo Thus cai i = t t+T which denotesconsumption in period i of the representative agent a differs only in scale from cii = t t + T which denotes the aggregate level of consumption This assump-tion that an aggregate variable can be viewed as reflecting decisions of a representa-tive agent is not innocuous For instance the actual path of aggregate consumptioncould well differ from that predicted by an analysis of individualsrsquo optimaldecisions due to changes across time in the composition of individuals in theeconomy33
Assumption 44 Individualsrsquo expectations in period t minus 1 of future real interestrates incorporate all information available as of period t minus 1 New informationoccurring in period t that alters consumption from what was planned results inthe distribution of consumption being ldquolog normal conditional on informationavailable last period that is log ct is normal with mean E(ct)rdquo (Hall 1988 342)
Assumption 45 In period t minus 1 the representative agentrsquos utility function takesthe following form
t+Tsumi=tminus1
expminusδi + ((δ minus 1)δ) log(ci)
where c gt 0 σ gt 0 and δ gt 0 This exponential utility function has the followingdesired properties
1 It is time-separable2 If consumption were equal across any two periods the individual would place
greater value on an increase in consumption in the earlier period ndash that is if
62 Households as market participants
ctminus1 = ct then
expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)gt expminusδt + ((δ minus 1)δ) log(ct)
3 Given 1 gt σ ge 0 utility increases in any period with increased consumptionbut at a decreasing rate ndash that is
du(ctminus1) = dctminus1
= [(1 minus δ)(δctminus1)] middot expminusδt(t minus 1)
+ ((δ minus 1)δ) log(ctminus1)gt 0 d2u(ctminus1)dc2
tminus1 lt 0
Note that the ldquointertemporal elasticity of substitutionrdquo will be given by σ As σ approaches 0 substitution of consumption across time in response tochanges in the real interest rate will approach zero as well
Assumption 46 Individualsrsquo forecasts of future variables are held with subjec-tive certainty This last assumption is a departure from Hallrsquos analysis that allowsus for the moment to maintain the ldquodeterministicrdquo aspect of the prior optimizationproblem That is we continue to assume that individualrsquos expectations of futurevariables such as expected rates of inflation and future interest rates are held withsubjective certainty
Given the above assumptions we know from our prior discussion that thechoice of consumption for periods t minus1 and t must satisfy the following first-ordercondition
dudctminus1
dudct= Rtminus1
Substituting in the appropriate expressions for the marginal utility of consumptionin periods t minus 1 and t we obtain
[(1 minus δ)δctminus1] middot expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)[(1 minus δ)δct] middot expδ + ((δ minus 1)δ) log(ct)
or
(ctctminus1) expδ + ((δ minus 1)δ) log(ctminus1ct) = Rtminus1
Taking the logarithm of both sides of the above expression and rearranging theabove first-order condition becomes
log(ctctminus1) + δ minus ((δ minus 1)δ) log(ctminus1ct) = log(Rtminus1)
Households as market participants 63
which simplifies to
δ + (1δ) log(ctminus1ct) = log(Rtminus1)
or
log ct = minusδσ + log ctminus1 + σ log(Rtminus1) (412)
which is similar in form to (411) Note that the ldquointertemporal elasticity ofsubstitutionrdquo is given by
σ = (dctdRtminus1)ctRtminus1
The form of equation (42) can be made closer to that of equation (411) if wenote that we can define the ldquoinstantaneous real rate of interestrdquo associated withcontinuous compounding mtminus1 by the expression
exp(mtminus1) = Rtminus1
Then (412) becomes
log ct = minusδσ + log ctminus1 + σmtminus1
Conclusion
This chapter has developed an in-depth understanding of household behaviorA good deal of the discussion has dealt with intertemporal choices and the tradeoffsinherent in consuming today versus consuming in the future Many policy-relatedissues in macroeconomics are related to decisions made today that are not indepen-dent of future states or activities This issue will arise again and again throughoutthis book and it is imperative that one comprehend the nature of decision-makingand time
5 Summarizing the behavior andconstraints of firms andhouseholds
Introduction
In this chapter we summarize our discussion of the behavior of firms andhouseholds in the simple Walrasian model with money and production In doingso we consider first the nature of constraints faced by the participants in the econ-omy with respect to decisions during period t and then their behavior in termsof demand andor supply Along the way we will try to simplify the notationand introduce various expectations and assumptions of different macroeconomicmodels We start our discussion with firms
Summarizing firmsrsquo constraints
We have seen how we can divide the general constraint facing firms that totalrevenues from all sources just exhausts expenditures each period into two sepa-rate constraints One is the ldquofirm financing constraintrdquo which states that desiredchanges in the capital stock as well as any capital adjustment costs are financedby issuing new bonds or equity shares That is for period t
I dnt + ψ(I d
nt) minus net Ast = 0 (51)
where
net Ast equiv As
t minus At
Ast equiv [
pbtBst + petS
st
]pt
At equiv [pbtB + petS
]pt
Idnt equiv Kd
t+1 minus K
Note that we implicitly assume that firmsrsquo plans for purchasing capital during theperiod correctly anticipate the price of output (capital) during the period and theprices of bonds and equity shares to be issued at the end of the period to financesuch purchases
Behavior and constraints 65
The second constraint the ldquofirm distribution constraintrdquo is that all revenuesfrom the sale of output net of that required to replace capital used up in the produc-tion process during the period be distributed to households either as wages interestpayments or dividends At time t firmsrsquo anticipated distribution constraint isgiven by
dt + (wtpt)Ndt + zBpt minus (ys
t minus δK) = 0 (52)
where z is the coupon payment and planned output supply is related to labordemand by the production function
yst = f (N d
t K) (53)
Equation (52) implicitly assumes that firms at time t have perfect foresight withrespect to the price of output during period t Alternatively the firm financingconstraint would take the above form if we presumed there were futures marketsat time t for the exchange of output during period t and financial assets at the endof period t
The labor market at time t determines employment for the period at a level N lowastt
and an associated rate of output denoted by ylowastt At the realized price of output the
firm distribution constraint for period t will turn out to be
dt + (wtpt)Nlowastt + zBpt minus ( ylowast
t minus δK) = 0 (54)
As (54) indicates actual real output during the period net of that used to replacedepreciated capital will be distributed to households1
Summarizing householdsrsquo constraints
With respect to households there is a single budget constraint for period t Like thatof the firm distribution constraint its form changes depending on what is assumedconcerning the correctness of expectations or the timing of markets As we haveseen at time t households make plans with respect to labor supply consumptiondemand and desired additions to their real holdings of financial assets and moneybalances based on a perceived constraint of the form
cdt + (M d
t minus M )pet + net Ad
t minus (wtpt)eN s
t minus (dt + zBpt) = 0 (55)
where
net Adt equiv Ad
t minus At
Adt equiv
[pbtB
dt + petS
dt
]pt
At equiv [pbtB + petS
]pt
66 Behavior and constraints
We presume that households have perfect foresight at time t with respect to thereal value of financial assets and the real value of dividends plus interest paymentsreceived from firms at the end of period t We leave open the possibility of errorsin expectations (held with subjective certainty) concerning the price level as itaffects real money balances and the real wage The term pt would replace pe
t if wepresumed perfect foresight on the part of households concerning the price level orequivalently presumed that there were futures markets at time t for the exchangeof output during period t
The labor market at time t determines employment and output for the periodAs before we let N lowast
t and ylowastt denote the actual rate of employment and production
of output during period t In such a case the actual firm distribution constraint(54) can be substituted into the household budget constraint Since prices will beknown by households at this point we may also replace the expected prices ofoutput bonds and equity shares by their actual prices Thus during period t thehousehold budget constraint for period t can then be expressed as
cdt + (M d
t minus M )pt + net Adt minus ( ylowast
t minus δK) = 0 (56)
As discussed below householdsrsquo behavior in the output and financial markets willbe based on realized income and prices and will differ from their plans made attime t based on expected prices and a labor supply decision unless they possessperfect foresight at time t concerning the prices that will prevail over period t andthe labor market clears with actual employment equal to labor supply2
Walrasrsquo law labor market and other markets at time t
Recall that Walrasrsquo law reflects the summing up of the constraints faced by individ-ual agents in the economy Since our preceding analysis concerned the constraintsof the ldquorepresentativerdquo firm and household we need only sum the constraints ofsuch representative agents to obtain Walrasrsquo law There still remains a potentialproblem however as to which of the different versions of the constraints enumer-ated above for firms and households to use The choice as one would suspectdepends on whether the market for labor effectively occurs at the same time as themarkets for output and financial assets or at different times
One version of Walrasrsquo law essentially combines the market for labor at time twith a futures market at time t for the exchange of output during period t andfinancial assets at the end of period t Equivalently this version of Walrasrsquo lawassumes limited perfect foresight at time t by both firms and households withrespect to prices for the period3 In such a case we would sum constraints (51)(52) and (55) (with pt replacing pe
t in (55)) to obtain
[cd
t + I dnt + δK + ψ(I d
nt) minus yst
]+[net Ad
t minus net Ast
]+ (wtpt)
[N d
t minus N st
]+ [M d
t minus M ]pt = 0 (57)
Behavior and constraints 67
Thus we have that the sum of excess demand across four markets ndash the laboroutput financial and money markets ndash must equal zero4 Note that one of thesefour markets the money ldquomarketrdquo reflects the equality between the demand forand supply of money The money ldquomarketrdquo is not of course like other marketsof an economy ndash which is why the word ldquomarketrdquo is in quotes That is unlike theother markets the money ldquomarketrdquo is not a place where the exchange of goods(eg labor financial assets or output) takes place5
Walrasrsquo law sequential markets and potential lackof perfect foresight
There is a modification to make with respect to the above that is required if marketsoccur sequentially and if there is not perfect foresight on the part of all agents attime t concerning prices for period t The sequential nature of markets is clearin that we have the labor market occurring at time t while the markets for outputand financial assets occur during the period In addition households in particularmay not correctly foresee at time t the price of output for period t Under suchcircumstances the prior version of Walrasrsquo law no longer holds for it would thensum constraints that are only anticipated not realized Instead given the sequentialnature of the markets we must break the analysis down into an analysis of thelabor market and an analysis of the other three markets
At time t the labor market occurs Assuming a competitive equilibrium for thelabor market we have a money wage determined at time t such that
N st = N d
t
Underlying the supply of labor at time t are householdsrsquo plans with respect toconsumption demand and saving (either in the form of financial assets or money)during the period These plans are influenced at time t by the anticipated pricelevel for commodities pe
t among other variables and as such these plans may notbe feasible given realized prices during period t
Once the labor market ends employment and output are determined for theperiod at levels N lowast
t and ylowastt respectively At that point households make plans with
respect to consumption and saving in light of the realized prices and the resultingeffective household budget constraint That realized household budget constraintis simply equation (56) which incorporates the actual firm distribution constraintAdding the firm financing constraint (51) we obtain a modified Walrasrsquo law forthe markets during period t of the form
[cd
t + I dnt + δK + ψ(I d
nt) minus ylowastt
]+[net Ad
t minus net Ast
]+ [M d
t minus M ]pt = 0
In the absence of perfect foresight the demands for consumption money balancesand financial assets during the period expressed in the above equation can differfrom the plans made at time t
68 Behavior and constraints
Summarizing firm behavior with limited perfect foresight
Consider firmsrsquo optimal plans at time t Note that at time t firms have an expectationof the price of output for the period We shall continue to assume that firms haveperfect foresight at time t with respect to the price of output over the periodWith respect to labor demand a diminishing marginal product of labor implies aninverse relationship between the real wage and labor demand
N st = N d
t (wtpt K)
It is typical to assume that the labor market is such that firms achieve employ-ment equal to that demanded In this case given the production function and thenature of the labor demand function we have an output supply function for periodt of the form6
yst = yd
t (wtpt K)
An increase in the real wage reduces labor demand and thus output supply so thatwe have
partN dt part(wtpt) lt 0 and partys
t part(wtpt) lt 0
Now consider firmsrsquo behavior with respect to investment and consequent finan-cial asset supply Given a diminishing marginal product of capital capital demandis inversely related to the expected real user cost of capital7
Assuming labor and capital are complements in the production process(part2f partKpartN gt 0) capital demand will be inversely related to the expected realwage in the subsequent period as well8 In particular in the absence of adjustmentcosts (for both capital and labor) we have the following capital demand function
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (58)
with
partKdt+1part(me
t + δ) lt 0 and partKdt+1part(we
t+1pet+1) lt 09
The above demand for capital stock at the end of period t (in place at time t +1)implies a net investment demand function for period t of the form
I dnt = I d
nt(met + δ we
t+1pet+1 K)
where net investment demand is inversely related to the expected real user cost ofcapital me
t + δ the anticipated real wage in the next period wet+1pe
t+1 and theexisting capital stock at time t K
Recall that the firm financing constraint in the absence of capital adjustmentcosts equates firmsrsquo net real financial asset supply to net investment demandThus given the nature of the net investment demand function the net real financial
Behavior and constraints 69
asset supply function for firms at the end of period t (at time t + 1) is identical tonet investment demand or
net Ast = net As
t (met + δ we
t+1pet+1 K)
where net real financial asset supply for period t like net investment demandduring period t is inversely related to the expected real user cost of capital theanticipated real wage in the next period and the existing capital stock at time t
With convex adjustment costs the optimal capital stock (as well as investmentdemand) depends on the entire future path of the expected real user cost of capitaland real wages That is with adjustment costs
Kdt+1 = Kd
t+1(met + δ me
t+1 + δ wet+1pe
t+1 wet+2pe
t+2 ) (59)
We can rewrite the above demand function for capital given adjustment costs soas to collapse future periods into essentially a single subsequent period10 To doso recall that the expected real rate of interest for period i me
i equals (ri minus πei )
(1+πei ) Let us assume static expectations concerning future interest rates so that
ri = rt i = t + 1 t + 2 Further let us assume static expectations concerningfuture expected inflation so that πe
i = πet i = t + 1 t + 2 The result is that
the expected constant real rate of interest between periods t and t + 1 is expectedto prevail in the future so that me
i = met i = t + 1 t + 2 We can then rewrite
the demand function for capital accumulated at the end of period t in the simplerform
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1 we
t+2pet+2 )
Now note that we can decompose the anticipated wage for period i and theexpected price level for period i into two components the wage or price levelin period i minus 1 and the expected rate of change in wages or prices respectivelyIn particular the anticipated money wage and price level for period t + 2 can beexpressed by
wet+2 equiv we
t+1(1 + πewt+1) and pe
t+2 equiv pet+1(1 + πe
t+1)
Let us now assume static expectations with respect to the rate of change in wagesbeyond the next period so that πe
wi = πewt+1 i = t + 2 t + 3 Recall that
we have already assumed a constant rate of price inflation in subsequent periodsIf we then add the assumption that beyond the next period the (constant) rate ofinflation in wages (πe
wt+1) equals the expected rate of inflation in prices (πet+1)
we have that wei pe
i = wet+1pe
t+1 i = t + 2 t + 3 In words the real wagein the subsequent period we
t+1pet+1 i = t + 2 t + 3 is anticipated to persist
indefinitely11 We can now rewrite the capital demand function given adjustmentcost (equation (59) as
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (510)
70 Behavior and constraints
Note that given our expectation assumptions the capital demand function withadjustment costs (equation (54)) has the same form as the capital demand functionin the absence of capital adjustment costs However the actual response to changesin the real user cost of capital or in the anticipated real wage in the subsequentperiod would typically be less with adjustment costs as such costs would leadfirms to only gradually move toward a new optimal capital stock
Equation (510) captures the idea that the demand for capital to be in placeat the end of period t (at time t + 1) depends inversely on the expected realuser cost of capital and that assuming capital and labor are complements capitaldemand depends inversely on the anticipated future real wage Similarly sincenet investment demand during period t simply reflects the difference betweencapital demand at the end of the period and the initial capital stock we haveinvestment demand being inversely related to the expected real user (or rental) costof capital and to the anticipated real wage for the subsequent period Finally fromthe firm financing constraint that relates firmsrsquo net real financial asset supply tonet investment demand we have firmsrsquo net financial asset supply being inverselyrelated to the expected real user cost of capital and the expected future real wageThus in summary we have
partKdt+1part(me
t + δ) lt 0 partI dntpart(me
t + δ) lt 0
partKdt+1part(we
t+1pet+1) lt 0 partI d
ntpart(wet+1pe
t+1) lt 0
partnet Ast part(me
t + δ) gt 0 partnet Ast part(we
t+1pet+1) lt 0
where mei equiv (rt minus πe
t )(1 + πet ) Note that gross investment demand is given by
Idt equiv Kd
t+1 minus K + δK = I dnt + δK
Summarizing household behavior with limitedperfect foresight
Householdsrsquo plans at time t concerning the labor supply choice the consump-tionsaving choice and the portfolio choice for period t are constrained by theanticipated household budget constraint as given by equation (55) From ourintertemporal analysis of these optimal choices we know that in general at time twe thus have labor supply at time t
N st = N s
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
consumption demand planned at time t
cdt = cd
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
Behavior and constraints 71
and money demand planned at time t
Ldt = Ld
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
where for notational ease we have defined planned real money demand by the termLd
t From the anticipated budget constraint as given by
cdt + (M d
t minus M )pet + net Ad
t minus (wtpet )N
st minus (dt + zBpt) = 0
we can infer net real financial asset demand planned at time t net Adt from
householdsrsquo plans concerning money demand consumption demand and laborsupply
With perfect foresight at time t concerning the prices for period t house-holdsrsquo plans made at time t concerning consumption money demand andnet real financial asset demand during period t will be fulfilled Thus lettingxt = dt + zBpt + At + M tpt we can express the above behavior conditions asfollows labor supply at time t is
N st = N s
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
consumption demand for period t is
cdt = cd
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
and money demand for period t is
Ldt = Ld
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
where Ldt equiv M d
t pt As we did with respect to firmsrsquo capital demand function given adjustment
costs we now introduce expectation assumptions that essentially collapse theentire sequence of future periods into a single future period First we assume staticexpectations concerning future interest rates so that ri = rt i = t + 1 t + 2 Next we assume static expectations with respect to future rates of inflation suchthat πe
i = πet i = t + 1 t + 2 Finally we assume that the expected rate of
wage inflation beyond the next period is constant and equal to the expected rate ofinflation (ie πe
wi = πewt+1 i = t + 2 t + 3 and πe
wt+1 = πet+1) so that the
real wage anticipated for the next period is expected to prevail indefinitely Giventhe above restrictive expectation assumptions we can rewrite the householdsrsquobehavioral functions in the following form the labor supply at time t becomes
N st = N s
t (wtpt wet+1pe
t+1 rt πet xt)
consumption demand for period t becomes
cdt = cd
t (wtpt wet+1pe
t+1 rt πet xt)
72 Behavior and constraints
and money demand for period t becomes
Ldt = Ld
t (wtpt wet+1pe
t+1 rt πet xt)
where real money demand at the end of period t is given by Ldt equiv M d
t pt Our discussion of the labor supply decision suggests that labor supply is directly
related to the real wage and inversely related to the real wage next period (whichreflects the real wage in all subsequent periods given our expectation assumptions)This response to changes in the real wages incorporates the intertemporal substitu-tion hypothesis The intertemporal substitution hypothesis also suggests that laborsupply is directly related to the interest rate and inversely related to the expectedrate of inflation in that either change implies a higher expected real rate of inter-est and thus a substitution from leisure to increased labor supply today Finallyhigher real initial holdings of financial assets real money balances or income inthe current period from bond and equity shares holdings will reduce labor supplyif leisure is a normal good In summary we thus have
partN st part(wtpt) gt 0 partN s
t part(wet+1pe
t+1) lt 0 partN st partrt gt 0
partN st partπe
t lt 0 partN st part xt lt 0
Our discussion of the consumptionsaving decision suggests that consumptiondemand is directly related to both the current and anticipated future real wagesince an increase in either implies an increase in the discounted stream of incomeFocusing on the Fisherian analysis of the allocation of consumption across timeconsumption demand would be inversely related to the money interest rate anddirectly related to the expected rate of inflation since either change implies ahigher expected real rate of interest Finally an increase in xt (reflecting sayhigher real initial holdings of financial assets increased real money balancesor higher income in the current period from bond and equity share holdings) willraise consumption demand in the current period In summary we thus have
partcdt part(wtpt) gt 0 partcd
t part(wet+1pe
t+1) gt 0 partcdt partrt lt 0
partcdt partπe
t gt 0 partcdt part xt gt 0
Our discussion of the portfolio decision suggests that real money demand isdirectly related to both the current and anticipated future real wage since an increasein either implies an increase in the discounted stream of income Focusing on theportfolio analysis money demand would be inversely related to the money interestrate as households shift from money to financial asset holdings Any effect of achange in the expected rate of inflation is indirect and will be ignored Finallyan increase in xt (reflecting say higher real initial holdings of financial assetsincreased real money balances or higher income in the current period from bondand equity share holdings) will likely raise money demand in the current period
Behavior and constraints 73
In summary we thus have
partLdt part(wtpt) gt 0 partLd
t part(wet+1pe
t+1) gt 0 partLdt partrt lt 0
partLdt part xt gt 0
So far our discussion has not included householdsrsquo real net financial assetdemand To see what determines this we can simply use the budget constraintand money labor and commodity demand functions Specifically rewriting thebudget constraint for period t
net Adt = (wtpt)N
st + dt + zBpt minus cd
t minus (M dt minus M )pt
An increase in the current real wage initial real money balances or dividendand interest payments for the current period is presumed to increase not onlycurrent consumption and money demand but also future consumption and moneydemand and thus increase net financial asset demand by households From theintertemporal substitution hypothesis an increase in the expected future real wagewill reduce current labor supply as well as increase current consumption demandboth changes imply a fall in net real financial asset demand for households
From the Fisherian analysis a higher expected rate of inflation will reducethe expected real rate of interest the above constraint indicates that the resultingincrease in current consumption demand will reduce householdsrsquo acquisition offinancial assets Similarly an increase in real initial financial asset holdings byraising both current consumption and money demand will lead to a fall in real netfinancial asset demand On the other hand a higher money interest rate due toboth the Fisherian effect on current consumption demand and the portfolio effecton money demand implies a higher net financial asset demand In summary wethus have
partnet Adt part(wtpt) lt (or gt or =) 0 partnet Ad
t part(wet+1pe
t+1) lt 0
partnet Adt partrt gt 0
partnet Adt partπe
t lt 0 partnet Adt partAt lt 0 partnet Ad
t partMpt) gt 0
partnet Adt part(dt + zBpt) lt (or gt or =) 0
where there are ambiguous effects on net financial asset demand of a change in thereal wage and of a change in anticipated dividend and interest payments becausean increase in either raises both consumption demand and money demand Notethat in the limit as the length of the period goes to zero the household budgetconstraint at time t becomes
net Adt + (M d
t minus M )pt = 0
74 Behavior and constraints
as all the flow terms go to zero at a point in time In this case assuming real moneydemand is directly related to income we obtain the unambiguous effect of
partnet Adt part(dt + zBpt) lt 0 partnet Ad
t part(wtpt) lt 0
However in the period analysis net Adt reflects net real financial asset demand at the
end of the period Thus assuming any increase in income is not fully reflected inan increased rate of consumption we have an offsetting effect and thus ambiguity
Summarizing household behavior without perfectforesight
If we presume that households learn of prices that will exist for period t aftertime t and they differ from what was expected then the actual demands for outputmoney and financial assets can differ from those reflecting plans made at time tThe key reason for this is that the actual constraint faced by households will differfrom that anticipated In particular using the actual firm distribution constraintwe replace anticipated real income from wages dividends and interest paymentsfrom firms with the actual real income net of depreciation ( ylowast
t minusδK) The resultingrealized household budget constraint after time t is then
cdt + (M d
t minus M )pet + net Ad
t minus ( ylowastt minus δK) = 0
If anticipations by households were incorrect at time t concerning prices or div-idends during the period then revisions in plans for consumption and saving willbe made in light of the actual budget constraint faced In this case the actual house-hold demand functions for output and money are written as follows consumptiondemand during period t is
cdt = cd
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
money demand during period t is
Ldt = Ld
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
and we replace partcdt part(wtpt) gt 0 and partcd
t part(dt + zBpt) gt 0 with
partcdt partylowast
t gt 0
Note that the term partcdt partylowast
t is referred to as the ldquomarginal propensity toconsumerdquo12 Similarly we replace partLd
t part(wtpt) gt 0 and partLdt part(dt + zBpt) gt 0
with
partLdt partylowast
t gt 0
Behavior and constraints 75
Money illusion and the real balance effect
Let us consider (sufficient) assumptions under which demands and supplies arehomogeneous of degree 0 in current wages prices and the nominal stock ofmoney ndash that is there is the absence of money illusion Note that in consideringwhether or not there exists money illusion we must now look at the behavior notonly of households but also of firms Consider firms first
Assuming perfect foresight on the part of firms it is clear that current labordemand is homogeneous of degree 0 in current prices (the wage rate wt and the pricelevel pt) Thus so also current output supply Assuming unit elastic expectationswith respect to wages and prices in all future periods and expectations of futureinterest rates that are invariant to changes in current prices and wages we havethat capital demand and thus also investment demand and firmsrsquo net real financialasset supply are homogeneous of degree 0 in current prices
We already assumed static expectations concerning future interest rates and unitelastic expectations with respect to prices beyond period t + 1 in terms of nextperiodrsquos price to obtain a simple form for the demand functions for investmentThus we need only add (a) the assumption of unit elastic expectations concerningthe price of output between period t and t+1 (implying an expected rate of inflationπe
t and thus an expected real rate of interest that is independent of a change in theprice level pt)
13 and (b) unit elastic expectations concerning next periodrsquos wageand price level (implying an anticipated real wage next period independent of anequiproportionate change in wages and prices in the current period) to obtain theabsence of money illusion with respect to firmsrsquo investment demand14
To obtain the absence of money illusion with respect to households we mustshow that each of the arguments in their demand and supply functions is invariantto an equiproportional change in current prices wages and the nominal stockof money Given perfect foresight the current real wage and initial real moneybalances meet this condition But what about the expected real wage next periodthe expected rate of inflation current dividends and interest payments and the realvalue of initial financial assets holdings As it turns out the assumption of unitelastic expectations with respect to all future prices and wages is again critical inshowing these to be invariant as it was in deriving a capital demand homogeneousof degree 0 in current wages and prices
First it is clear that the assumption of unit elastic expectations with respectto future wages and prices makes future real wages invariant to equiproportionalchanges in the current wages and prices But note that in so doing we have elimi-nated the ldquointertemporal substitution hypothesisrdquo effect on labor supply of a changein the current real wage Similarly the assumption of unit elastic expectations withrespect to the future price level eliminates any effect of a change in the price levelon the expected rate of inflation Finally assuming that expectations of nominalfuture interest rates are invariant to an equiproportionate change in current priceswages and the money supply the expected real rate of interest will not be affectedby equiproportionate changes in wages prices and the money supply But notethat the ldquointertemporal substitution hypothesisrdquo impact on labor supply of a change
76 Behavior and constraints
in the expected real rate of interest initiated by a change in the current price levelis now absent as well15
What is left in order to obtain the absence of money illusion for households isto show that changes in current prices leave current dividend payments and thereal value of initial bond and stock holdings unchanged Once again as we seebelow the assumption of unit elastic expectations concerning next periodrsquos wagesand prices will be invoked to achieve this What we are looking for are sufficientassumptions that will result in the terms dt + zBpt and At being homogeneous ofdegree 0 in wt and pt
From the firm distribution constraint (52) and assuming firmsrsquo labor demandis satisfied (ie N lowast
t = N dt and thus ylowast
t = yst ) we know that
dt + zBpt = yst minus δK minus (wtpt)N
dt
Since N dt and ys
t are homogeneous of degree 0 in prices it is clear that the sumof current dividends plus interest payments is not affected by equiproportionatechanges in both the current wage and price level A higher price level does alterthe composition of payments however as real dividends rise and real interestpayments fall
Now consider At To show that this can be homogeneous of degree 0 in thecurrent wage and price level note from the firm distribution constraint and fromthe assumption that firmsrsquo demand for labor is satisfied in subsequent periods that
dt + zBpet+1 = f (N d
t+1 Kdt+1) minus (we
t+1pet+1)N
dt+1 minus δKd
t+1
A similar equation holds for future periods as well At simply reflects the presentvalue of such future real payments using the appropriate expected real interest ratesfor discounting The assumption of unit elastic expectations concerning prices inall future periods coupled with expectations of future nominal interest rates thatare unaffected by an equiproportionate change in money prices and the moneysupply thus means that A is homogeneous of degree 0 with regard to a change inprices (price level and wages) and the money supply in period t16
A special case of the above is if we assume static expectations concerning futureinterest rates (ie ri = rt i = t + 1 t + 2 ) and zero adjustment costs In thiscase Kd
i i = t + 1 t + 2 would be the same in each future period Therewould be a constant labor demand (N d
i = N dt+1 i = t + 2 t + 3 ) as well given
the invariant real wage in conjunction with no change in their capital stock Nowrecall that At is defined by
At equiv [pbtB + petS]pt
where pbtBt is the present value of future interest payments and petS is the presentvalue of future dividends Since At is simply the present value of the now constantfuture stream of dividends and interest payments discounted using an invariant
Behavior and constraints 77
expected real rate of interest we thus have
At = [dt+1 + zBtpet+1]me
t
Since dt+1 +zBtpet+1 = f (N d
t+1 Kdt+1)minus (we
t+1pet+1)N
dt+1 minusδKd
t+1 from the firmdistribution constraint we can rewrite the initial real holdings of financial assetsas
At = [ f (N dt+1 Kd
t+1) minus (wet+1pe
t+1)Ndt+1 minus δKd
t+1]met
which is homogeneous of degree 0 in current wages and prices if we assume that theanticipated real wage next period is independent of an equiproportionate changein the current wages and prices In other words to obtain the absence of moneyillusion for households we assume as we did with firms that there are unit elasticexpectations concerning wages and prices in the next period
Note that an increase in the money interest rate and the expected rate of inflationthat leaves the expected real rate of interest unchanged will alter the compositionof financial asset holdings although not the total To see this recall that with staticexpectations concerning the interest rate the price of bonds at the end of period tis given by
pbt =infinsum
i=1
z(1 + rt)i = zrt
An increase in the money interest rate and expected rate of inflation such that thereal rate of interest is unchanged means a fall in the price of bonds but an offsettingrise in the price of stock as over time real dividend payments will be rising morerapidly while real interest payments will be falling more rapidly Similarly anincrease in the current level of prices although leaving At unaffected wouldreduce the real value of bond holdings and lead to an exactly offsetting increasein the real value of equity share holdings
The real balance effect is apparent from the nature of the demand and supplyfunctions In particular an increase in nominal money balances or fall in prices withno change in nominal money balances will increase initial real money holdingsand in general lead to an increase in consumption demand real money demandand real net financial asset demand In general labor supply would fall
6 The simple neoclassicalmacroeconomic model (withoutgovernment or depositoryinstitutions)
Introduction
We have now covered a substantial part of the underlying structure for a simpleaggregate model of an economy with production The specific elements of theldquomicroeconomic foundationsrdquo of this aggregate model developed so far have dealtwith the optimizing behavior of individuals (ldquorepresentativerdquo firms and house-holds) in a setting in which individuals take prices as given Implicit in thesediscussions is another part of the microeconomic foundations the way in whichindividual markets operate We have been assuming that prices adjust so that thepresumption that buyers and sellers are price-takers is justified In other wordsequilibrium within individual markets entails price adjustment to equate suppliesand demands In addition we will assume that all individuals in the economy cor-rectly foresee period trsquos output prices when input supply and production decisionsare made at the start of the period As discussed below however there are otheroptions
Static macroeconomic models the options
Grandmont (1977 542) notes that
one way to look at the evolution of an economic system is to view it as asuccession of temporary or short-run competitive equilibrium That is onepostulates that at each date prices move fast enough to match supply anddemand Although one assumes equilibrium in each period the economicsystem displays a disequilibrium feature along a sequence of temporary com-petitive equilibrium at each date the plans of the agents for the future arenot coordinated and thus will be in general incompatible this is to becontrasted with the perfect foresight approach where by definition such adisequilibrium phenomenon cannot occur
This temporary equilibrium view of the economy is characteristic of the simplestatic neoclassical model a model in which all prices adjust to maintainequilibrium1 The second key assumption of the neoclassical model is that agents
Simple neoclassical macroeconomic model 79
are informed about prices within the period In particular when making their laborsupply and demand decisions at the start of the period households and firms areassumed to correctly anticipate the prices they will have to pay to purchase outputduring the period As it turns out this element of ldquoperfect foresightrdquo with respectto markets through time t +1 is a critical feature of the analysis Before examiningthe implications of these assumptions however some history on the origin of theneoclassical model might be helpful
The phrase ldquoneoclassical macroeconomic modelrdquo is a descendant of ldquoclassicalrdquoeconomic theory as reflected in the work of Sir William Petty during the 1600sIn Das Kapital Karl Marx (1976 85) stated that ldquoby classical Political EconomyI understand that economy which since the time of W Petty has investigated thereal relations of production in bourgeois societyrdquo As Marx suggested early classi-cal economists focused on the determinants of the economyrsquos productive capacityThe neoclassical macroeconomic model shares this focus on the productive capac-ity of the economy as the determinant of total output It also turns out to be thestatic precursor to much of the current analysis in the macroeconomic literaturethat falls under the heading of ldquoreal business cycle theoryrdquo
While the simple static neoclassical model along with its dynamic and stochas-tic counterparts is one popular approach to macroeconomic analysis there areother approaches In fact even though static analysis is restricted to markets in thecurrent period there remains enough flexibility to introduce at least four ways ofcharacterizing macroeconomic analysis
1 ldquoNeoclassical modelrdquo competitive equilibria are assumed to exist in cur-rent and future markets and limited perfect foresight is assumed for allparticipants
2 ldquoIllusion modelrdquo competitive equilibria are assumed to exist in current andfuture markets but imperfect foresight is assumed on the part of some agentsThe result is like the ldquoLucas supply functionrdquo popularized by Lucas (1973)in which output can respond directly to increases in the actual output prices
3 ldquoKeynesian modelrdquo a competitive equilibrium is assumed not to exist inthe labor market as the money wage is fixed and employment is demand-determined However other prices in particular the prices of output arepresumed to reflect competitive equilibria A rational expectation version ofthis model is developed by Fisher
4 ldquoNon-market-clearing modelrdquo a competitive equilibrium is assumed not toexist in the output market as the price of output is fixed above the competitiveequilibrium level This model forms the basis of much of what appears inundergraduate macroeconomic analysis including the IS-LM model
The neoclassical model with its assumptions of flexible prices and informedagents provides a benchmark against which we can compare the predictions ofother (static) macroeconomic models2 It also provides insight into the nature of thestationary states for dynamic macroeconomic models that presume market-clearingprices and accurate forecasts of prices
80 Simple neoclassical macroeconomic model
Hicksian temporary equilibrium and Walrasrsquo law
For the market for any good ldquocompetitiverdquo equilibrium is defined by equalitybetween market demand and supply3 A temporary competitive equilibrium for theeconomy during period t will be characterized by a money wage for labor (wlowast
t )a money price for the consumption commodity ( plowast
t ) a money price for bonds( plowast
bt) a money price for equity shares ( plowastet) allocations to households in terms
of employment consumption bond holdings equity share holdings and moneybalances (N lowast
t clowastt Blowast
t Slowastt and M lowast
t ) and allocations to firms in terms of employmentoutput investment bond issues and equity share issues (N lowast
t ylowastt Ilowast
t Blowastt Slowast
t ) suchthat
bull these allocations are in the demand (supply) set of each agentbull these allocations are feasible
Together these two conditions imply prices determined in the labor output bondand equity shares markets for period t that result in zero excess aggregate demandfor labor output bonds equity shares and money Thus we may rewrite theconditions for a general equilibrium as a money wage a price of output a price ofbonds and a price of equity shares such that
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Bst = Bd
t
Sst = Sd
t
Mpt = M dt pt
where
I dnt = Kd
t+1 minus K
yst = f (N d
t K)
Given our assumption that bonds and equity shares are perfect substitutes ingeneral there will not be a unique equilibrium in terms of the number of bondsand equity shares supplied or demanded although the total value of financialassets supplied or demanded will be determinant Thus we replace the bond andequity share markets with a single market the financial market The equilibriumconditions then become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Simple neoclassical macroeconomic model 81
Ast = Ad
t
Mpt = M dt pt
where
Ast equiv [ pbtB
st + petS
st ]pt
Adt equiv [ pbtB
dt + petS
dt ]pt
We can view the financial market as simultaneously determining the price ofbonds pbt the price of equity shares pet and the interest rate rt That is once oneof these is known the other two are implied For instance from our definition ofthe interest rate
1 + rt equiv [z + pbt+1]pt
we see that given the coupon rate and expected price of bonds in the subsequentperiod the interest rate rt implies a price of bonds pbt As perfect substitutesbonds and equity shares must offer the identical expected gross return Thus wehave that
1 + rt = 1 + ret equiv [dt+1 + pet+1]pet
As you can see given expectations of future dividends and the future price ofequity shares an interest rate rt also implies a price of equity shares pet We oftentalk of the financial market in terms of an equilibrium interest rate The aboveshould make it clear that associated with such an equilibrium interest rate areprices of bonds and equity shares And a rise (fall) in the interest rate means a fall(rise) in the prices of bonds and equity shares
We will make one additional change in the characterization of the financialmarket to put it in terms of additional demands and supplies that is put it in netrather than gross terms The reason for this is that net financial asset demands andsupplies are what correspond to household saving in the form of financial assetsand firm investment Thus the equilibrium conditions become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
net Ast = net Ad
t
Mpt = M dt pt
where
net Ast equiv As
t minus At
net Adt equiv Ad
t minus At
At equiv [pbtB + petS]pt
82 Simple neoclassical macroeconomic model
According to the above we have four equilibrium conditions but only threeprices ndash the money wage rate the level of output prices and the interest rate ndashto be determined As usual Walrasrsquo law is invoked to show that only n minus 1 ofthe n equilibrium conditions are independent However the nature of Walrasrsquo lawdepends on whether we assume limited perfect foresight or not
Walrasrsquo law for limited perfect foresight sums up the constraints faced by theindividual agents in the economy at time t to obtain
[cdt + I d
nt + δK + ψ(I dnt) minus ys
t ] + [net Adt minus net As
t ]+ (wtpt)[N d
t minus N st ] + M d
t pt minus Mpt = 0
Thus the sum of excess demands for output financial assets labor and moneymust equal zero
When there is not perfect foresight at time t concerning prices for period t inthe output and financial markets we have the equilibrium condition for the labormarket
N st minus N d
t = 0
and the modified Walrasrsquo law based on the resulting employment and output ofthe form
[cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t ] + [net Adt minus net As
t ] + M dt pt minus Mpt = 0
In this case the money wage employment and thus output are determined inthe labor market and the modified Walrasrsquo law indicates that the price level andinterest rate are determined by any two of the remaining three markets
As it turns out most macroeconomic analysis takes this second approach tosolving for equilibrium That is the analysis focuses on the labor (and other input)markets and determines the effect of changes in output price (and potentially othervariables such as the interest rate) on equilibrium employment and thus outputThis generates an ldquoaggregate supply equationrdquo which is then combined with twoof the remaining three equilibrium equations ndash typically the commodity and moneymarket equilibrium conditions ndash to determine the equilibrium price level interestrate and output The modified Walrasrsquo law is invoked to ensure equilibrium inthe financial market at this point Working backward one can infer from theaggregate supply equation the equilibrium money wage and employment impliedby the analysis
The advantage of the above approach is that it can be used whether or not thereexists limited perfect foresight at time t with respect to the price level and whetheror not prices adjust to clear markets A disadvantage of the analysis is that inthe case of the neoclassical model with limited perfect foresight and competitiveequilibrium it arbitrarily breaks up the analysis of markets In doing so it requiresthat demand functions for such goods as money and commodities be specifiedwith income as an argument This form of the demand functions obscures the fact
Simple neoclassical macroeconomic model 83
that as we have seen in standard general equilibrium analysis demand functionsdepend on prices (including the real wage) and income is a variable determinedby the choice of labor supply
With these qualifications in mind we take the standard approach of macroeco-nomics and separate out the analysis of the labor (and other input) markets (theldquoaggregate supplyrdquo part) from the other markets (the ldquoaggregate demandrdquo part)The next section considers the labor market at time t
Labor market equilibrium
At the start of period t the labor market takes place and a rate of production of outputis determined From our analysis of firm behavior at time t we know that behindlabor demand is an expected price of output over the period and associated expectedreal wage an existing capital stock and existing technology as incorporated in theproduction function We shall assume that firms correctly anticipate at time t theprice of output for the period so that firms confront the actual real wage wtpt
From our analysis of household decision-making at time t we know that behindlabor supply at time t is not only the expected price of output and implied expectedreal wage but also such factors as the relationship of the anticipated current realwage to anticipated future real wages the expected real rate of interest anticipatedwealth in the form of financial asset and real money holdings and anticipated cur-rent nonlabor income Like firms we will assume households have limited perfectforesight in that at time t they correctly foresee the price level for period tAssuming a Walrasian or ldquocompetitive equilibriumrdquo view of the labor marketsthe money wage wt adjusts to achieve equilibrium in the labor market under thesecircumstances
We have already seen how static expectations concerning future rates of wageand price inflation along with the assumption of unit elastic expectations concern-ing wages and prices next period simplify the labor supply function by removingexpected future real wages as explicit arguments Patinkin (1965) among oth-ers goes several steps beyond these simplifying assumptions and assumes that allother variables excepting the real wage do not have a significant impact on laborsupply4 Thus equilibrium in the labor market is given by a level of employmentNt and money wage wt such that
Nt = N dt (wtpt K) and Nt = N s
t (wtpt)
As Patinkin (1965 264) notes
it will immediately be recognized that we have greatly oversimplified theanalysis of this market Both the demand and supply functions for labor shouldactually be presented as dependent on the real value of bond and moneyholdings as well as on the real wage rate5 Further if we were to permit thefirm to vary its input of capital its demand for labor would depend also onthe rate of interest Finally a full utility analysis of individual behavior wouldshow the supply of labor also to depend on this rate6
84 Simple neoclassical macroeconomic model
Besides a competitive labor market and the above simplified labor supplyfunction the neoclassical model assumes that suppliers correctly anticipate theaggregate price level As we will see the assumption of the neoclassical modelthat suppliers like firms have perfect foresight at time t with respect to the priceof output for the period means that changes in the price of output have no effect onoutput supply This characteristic of the neoclassical model implies an underlyingldquoblock recursiverdquo nature to the analysis as described below
At the start of the period the labor market occurs with employment and thusoutput and the money wage being determined for the period Employment andoutput are determined based on individualsrsquo expectations of subsequent variablessuch as the price of output And the level of employment and output influencethe remaining variables to be determined But in the neoclassical model the levelof employment and output are not influenced by the remaining variables to bedetermined We can thus solve the equilibrium sequentially looking first at thelabor market and the determination of employment and output then looking atthe output financial and money markets and the determination of the price leveland interest rate
Financial market equilibrium
With regard to the financial market Patinkin (1965 215) states
a decrease in the price of bonds (and equity shares) decreases the amountdemanded of consumption commodities it will also be assumed that itdecreases the amount demanded of money balances hence by the house-holdrsquos budget constraint their total expenditures on [net] bond holdings mustincrease
Thus Patinkin invokes a ldquoFisherian effectrdquo and a ldquoportfolio effectrdquo of a change inthe interest rate to obtain
part(net Adt )partpbt lt 0
From the firm financing constraint and the fact that a firmrsquos net investment demandis inversely related to the interest rate we have
part(net Ast )partpbt gt 0
The above analysis differs slightly from that found in Patinkin (1965 214) Forinstance Patinkin uses the assumption of static expectations concerning interestrates and a coupon rate z equal to 1 to express the price of bonds as the reciprocalof the interest rate that is
pbt = 1rt
Simple neoclassical macroeconomic model 85
Further Patinkin assumes no equity shares so his financial market consists only ofbonds Finally Patinkin does not graph net real bond demand and supply Ratherhe graphs the total number of bonds demanded and supplied Thus Patinkin hasdemands and supplies of bonds of the form
Bs = rtptAst equiv Bs
t
Bd = rtptAdt equiv Bd
t
As might be expected Patinkinrsquos bond demand and supply curves differ in naturefrom net real financial asset demand and supply curves For instance it is clearfrom the analysis of investment demand that a rise in the interest rate (a fall inthe price of bonds) will lead to reduced investment demand and thus reduced netreal financial asset supply This relationship between investment and firmsrsquo net realfinancial asset supply follows directly from the firm financing constraint whichgiven pbt = 1rt is of the form
net At equiv 1ptrt[Bst minus B] = I d
nt + ψ(I dbt)
Naturally if net investment were initially zero and there were zero adjustment costs(so that Bs
t = B) the fall in investment that results from a rise in the interest ratewould imply a similar decrease in the number of bonds supplied (Bs
t ) Howeverif initially Bs
t gt B then a higher interest rate even though it decreases invest-ment could at the same time increase the number of bonds supplied As Patinkin(1965 217) observes
consider the effect of an increase in the rate of interest (fall in 1rt) The inter-nal consistency of our model requires that this decrease the amount of realbonds supplied
However this need not reduce the number of bonds supplied As Patinkin(1965 217) continues
a rise in the interest rate ( pbt falls) has lowered the price received for bondsand so may increase the number of bonds necessary to finance the firmrsquosexpenditures on investment commodities (Bs
t gt B initially) even thoughthese expenditures have decreased
A similar analysis would apply to a comparison of householdsrsquo demand for bondsin terms of numbers with household net real financial asset demand
Money ldquomarketrdquo equilibrium
As we know from Walrasrsquo law having depicted equilibrium in the labor andfinancial markets we need only look at one of the other two remaining markets
86 Simple neoclassical macroeconomic model
the output market or the money ldquomarketrdquo Consider a money market in whichnominal money supply M and nominal money demand M d
t equiv ptLdt are plotted
against the reciprocal of the price level which indicates the ldquorelativerdquo price of oneunit of money in terms of output If there is no real balance effect with respect toreal money holdings then the money demand curve is invariant to a change in theprice level (elasticity of demand equals 1) If there is a real balance effect withrespect to money demand then a fall in the price level (a rise in the relative price ofmoney (1pt)) would lead to a less than proportionate decrease in nominal moneydemand (elasticity of demand less than 1)
Aggregate supply and demand an introduction
Temporary equilibrium for the economy can be characterized in several ways Aswe alluded to above one way common in journal articles and textbooks is to dividethe analysis under the headings of ldquoaggregate supplyrdquo and ldquoaggregate demandrdquo
Under the heading of ldquoaggregate supplyrdquo is an analysis of the input markets inparticular the labor market One aim is to determine input prices (in particular themoney wage) the employment of inputs and the implied production of output thatoccur at different levels of output prices (and potentially different levels of interestrates) The term ldquoaggregate supplyrdquo is applied to this analysis for the simple reasonthat it determines the ldquosupplyrdquo of total output at different prices
Under the heading of ldquoaggregate demandrdquo is an analysis of the other marketsin the economy during period t in particular the output financial and moneymarkets The aim is to determine the level of output prices and the interest ratethat occur at different levels of output The term ldquoaggregate demandrdquo attached tothis analysis reflects the fact that the analysis determines how the price level andinterest rate adjust to equate the ldquodemandrdquo for total output to different levels ofproduction
Combining the aggregate supply and demand analysis we can determine theoutput price level and interest rate associated with temporary equilibrium as wellas the underlying equilibrium money wage real wage employment consumptioninvestment and real money balances To understand more clearly what is involvedin aggregate supply and demand analysis and how they can be combined we con-sider below the specific case of the neoclassical model starting with the aggregatesupply
Equilibrium and aggregate supply
As we have said behind aggregate supply is an analysis of various input marketsto determine the response of total output to changes in such variables as theprice level7 In the neoclassical model changes in prices lead to equiproportionatechanges in money wages with no change in the equilibrium level of employmentand thus no change in aggregate supply To formally show this let us start withthe following statement of equilibrium in the labor market in terms of a money
Simple neoclassical macroeconomic model 87
wage wt and level of employment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpt) minus Nt = 0
A critical aspect of the above is the fact that suppliers in particular suppliers oflabor correctly anticipate the price level that will exist with respect to output sothat wtpt replaces wtpe
t in the labor supply functionTotally differentiating the above two equations with respect to wt Nt and pt
one obtains[(partN d
t part(wtpt))(1pt)
(partN st part(wtpt))(1pt)
minus1minus1
] [dwtdNt
]=[(partN d
t part(wtpt))(wtdpt( pt)2)
(partN st part(wtpt))(wtdpt( pt)
2)
]
Applying Cramerrsquos rule one obtains
dwtdpt = wtpt and dNtdpt = 0
Thus in the neoclassical model the real wage and employment level determinedin the labor market are independent of changes in the price level8
The ldquoaggregate supply equationrdquo combines the analysis of the labor market (andother input markets) and resulting determination of employment of various inputswith the production function to determine the resulting output supplied For theneoclassical model the aggregate supply equation is of the form
ylowastt = ylowast
t (K ) (61)
What is important about this equation is that the price level and interest rateare not arguments in the supply equation Of course changes in the capital stockchanges in technology or changes in the supply of other inputs (eg changes inthe oil supply) can affect output Similarly a change in the composition of thelabor force or government policies that affect labor supply can affect equilibriumemployment and thus output9
We may summarize the above findings graphically Consider an increase in pt Given perfect foresight both firms and households at time t would anticipate thishigher price level In the labor market the result would be an increase in theequilibrium money wage in the same proportion as the increase in the expectedprice level so that the anticipated real wage would remain the same as wouldemployment and output This outcome for the labor market is shown in Figure 61Note that the result of no change in employment or output in light of a higher outputprice simply requires that both labor demand and supply curves shift vertically bythe same amount Such equal shifts reflect the fact that the same increase in themoney wage leaves both firms and households anticipating the same real wage asbefore
In undergraduate textbooks the fact that changes in the price of output leaveemployment and real output unaffected in the neoclassical model is often shown
88 Simple neoclassical macroeconomic model
Labor
N demand
N supply
Figure 61 Labor market equilibrium
in ( pt yt) space by a vertical ldquoaggregate supply curverdquo Such a curve summarizesthe underlying behavior for the economy-wide labor market Later we will see thatunder other assumptions such as embedded in Lucasrsquos model (ldquomoney illusionrdquo)and the Keynesian model (fixed nominal wage) the aggregate supply curve willbe upward sloping
It is important to realize that the aggregate supply shown in Figure 62 is not thetypical market supply curve of microeconomics In microeconomics a higher priceof good x and consequent increase in the demand for inputs by firms producing thatgood draws inputs away from the production of other goods so that the higherprice of good x induces increased output of that good and associated increasedemployment of inputs (such as labor) in the production of the good Implicit inthis analysis is that there is reduced production of other goods in the economyHowever as the above analysis makes clear if the focus is on the aggregation ofall commodity markets a higher price level no longer induces increased aggregateoutput unless the quantity of total inputs supplies rises which will not be the caseunder neoclassical assumptions10
The natural rate of unemployment
The key feature of the above analysis of aggregate supply is the assumption thatprices adjust to continuously maintain equilibrium in the various markets and thatindividuals are perfectly informed concerning prices The result is a level of realoutput sometimes called the ldquofull employment levelrdquo So far missing from theanalysis however is any mention of unemployment If one expands the model tointroduce unemployment the rate of unemployment is called the ldquonaturalrdquo rate11
By ldquonaturalrdquo is meant that it is the rate of unemployment that the economy will
Simple neoclassical macroeconomic model 89
Output
Price
Aggregate supply
Figure 62 Aggregate supply
gravitate to as prices adjust to clear markets and individuals become fully informedconcerning prices (ie under the assumptions of the neoclassical model)
To introduce unemployment into the analysis when the labor market is in equilib-rium at full employment requires recognition of two facts First the labor marketis in a constant state of flux Not only do individuals enter and leave the laborforce continually but labor demand varies continuously across firms as they expe-rience variations in the relative demand for their output This instability of jobsthemselves has been estimated to account for roughly one-quarter of the averageunemployment rate as in an average year one in every nine jobs disappears and onein every eight is newly created (Leonard 1988) Second information is costly Ittakes time for new workers entering the labor force and for workers who have beenlaid off or have quit previous jobs to discover which employers have vacanciesand how wages vary across employers
When we take into account the continuous flows to unemployment together withworkersrsquo imperfect information about job vacancies we see that unemploymentis no longer inconsistent with the neoclassical model At any moment there existnew entrants into the labor market who are spending time searching for acceptablejobs There also exist laid-off workers who are either searching for alternativejobs or awaiting recall And there are workers who have quit their jobs and aresearching for other jobs This kind of unemployment is generally referred to asfrictional unemployment
Sometimes part of frictional unemployment is called structural unemploymentwith structural unemployment occurring because of a change in the compositionor ldquostructurerdquo of aggregate output across firms For example the replacement ofsteel with plastic in automobiles led to a shift in employment from steel factoriesto firms making plastic During this transition some steel workers experiencedstructural unemployment
90 Simple neoclassical macroeconomic model
To summarize when the labor market is in equilibrium there exists a positiveunemployment rate because workers continuously move into and out of the laborforce and between jobs12 To signify that this unemployment rate is ldquonaturalrdquo orconsistent with equilibrium in the labor market it is generally called the naturalrate of unemployment13 The corresponding level of employment is then oftenreferred to as the full employment level
Like equilibrium output and employment in the neoclassical model ldquosupplyfactorsrdquo determine the natural rate of unemployment as well14 Among these isthe demographic composition of the labor force but also unemployment insuran-ce and minimum wage legislation In recent years a large body of literature hasanalyzed labor markets and the sources of unemployment with the focus on searchand labor contracts Note that an understanding of the natural rate of unemploy-ment is important in determining the effects of government policies aimed at theunemployed
Equilibrium and aggregate demand
The aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) The ldquoISrdquo equation since it is simply the expression forequilibrium during period t with respect to the output market is given by15
cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t = 0 (62)
Note that the equation is termed the ldquoISrdquo equation because we can rewrite it toobtain
Idnt + δK + ψ(I d
nt) = ylowastt minus cd
t
indicating that equilibrium in the output market is equivalent to the equal-ity between Investment expenditures (the left-hand side of the equation) andhousehold Saving (the right-hand side of the equation)16
The assumption of unit elastic expectations concerning wages and prices givesthe following simple form for householdsrsquo consumption demand and firmsrsquoinvestment demand functions during period t17
cdt = cd
t (rt πet At Mpt ylowast
t )
and
Idnt = I d
nt(met + δ K)
The ldquoLMrdquo equation since it is simply the expression for equilibrium in period twith respect to the ldquomoneyrdquo market is given by
Ldt = Mpt (63)
Simple neoclassical macroeconomic model 91
This equation is termed the ldquoLMrdquo equation for it reflects the equality betweenLiquidity preference or demand and the Money supply We have seen that ourassumption of unit elastic expectations concerning wages and prices gives us thefollowing simple form for the real money demand function
Ldt = Ld
t (rt πet At Mpt ylowast
t )
From the modified Walrasrsquo law it follows that a price level and interest rate thatsatisfy the ldquoISrdquo equation and ldquoLMrdquo equation for a given level of output ylowast
t willalso result in equilibrium in the financial market
We thus have three equations (the aggregate supply equation (61) the ldquoISrdquoequation (62) and the ldquoLMrdquo equation (63)) that can be solved for the equilib-rium levels of output price and interest rate Looking at what underlies thesethree equations we can then infer the changes in money wages and employment(the labor market) as well as changes in the components of output demand (invest-ment and consumption demand) Equilibrium can be depicted in terms of a pricelevel and interest rate (Patinkinrsquos CCndashLLndashBB curves) or in terms of a price leveland output (ie aggregate demand and supply curves) We start with Patinkinrsquosdepiction of equilibrium
Depiction of equilibrium the Patinkin analysis
The neoclassical model aggregate supply is independent of changes in the pricelevel and interest rate18 This means that for any given level of output we canfocus on equations (62) and (63) to determine the equilibrium interest rate andprice level This is a reflection of the ldquoblock recursiverdquo nature of the solution tothe neoclassical model mentioned earlier
Patinkin (1965) suggests a graphical way of showing such an equilibrium com-bination of price level and interest rate using any two of three curves denoted theCC LL and BB curves The CC curve depicts combinations of the price level andinterest rate that satisfy the equilibrium condition for output (62) the LL curvedepicts combinations that satisfy the equilibrium condition for money (63) Wherethese two lines so constructed intersect it must then be the case that this price com-bination satisfies two of the three equilibrium conditions simultaneously It followsfrom the modified Walrasrsquo law that the curve indicating various combinations ofthe price level and interest rate that satisfy the equilibrium condition with respectto the financial market (the BB curve) goes through this point as well19
For the market for output equilibrium in period t is characterized by (62) Recallthat real output ylowast
t denotes that output reflecting the capital stock K technologyand the employment of labor determined in the labor market at time t From theneoclassical assumptions with respect to labor demand and supply equilibriumemployment and thus output are unchanged for any change in the price level orinterest rate
Let us presume that the pair ( plowastt rlowast
t ) is associated with equilibrium in the outputmarket In (price interest rate) space this combination is identified by a unique
92 Simple neoclassical macroeconomic model
point on the CC curve that identifies combinations of pt and rt associated withequilibrium in the commodity market (Figure 63)
To understand what lies behind the shape of the CC curve depicted above totallydifferentiate the market-clearing condition for the commodity market with respectto pt and rt Doing so we obtain20
[partcdt part(Mpt)] minus [Mp2
t ]dpt + [partcdt partrt + (1 + ψ prime)partI d
ntpartrt]drt = 0
Rearranging we have that the slope of the CC curve is given by
drtdpt | ydt minus ylowast
t = 0= [partcd
t part(Mpt)][Mp2t ][partcd
t partrt + (1 + ψ prime)partI dntpartrt] lt 0
The negative slope of the CC curve can be explained in the following way Thepositive numerator of the expression for the slope reflects the real balance effectwith respect to commodities this term indicates the fall in consumption demandthat would accompany a rise in the price level for such a rise reduces agentsrsquowealth in the form of initial real money balances21 The negative denominatorindicates the effect of a change in the interest rate on consumption and investmentdemand
Now let us consider the money market As before there is a unique point thatindicates an interest rate and a price of output at which there is equilibrium inthe economy and this point on the LL curve identifies combinations of pt andrt associated with equilibrium in the money market (Figure 64) Recall that theLL curve identifies combinations of pt and rt associated with equilibrium in themoney ldquomarketrdquo as given by (63)
The slope of the LL curve is given by totally differentiating the zero excessdemand condition with respect to money Doing so and rearranging one obtains
drtdpt |Ldt minus Mpt = 0 = [1 minus partLd
t part(Mpt)][Mp2t ][minuspartLd
t partrt] gt 0
r
CC
p
Figure 63 Commodity market
Simple neoclassical macroeconomic model 93
rLL
p
Figure 64 Money market
Note that the numerator of this term is positive The change in real money balanceswill be greater than the consequent change in real money demand given real balanceeffects with respect to financial assets andor commodities The denominator ispositive as well reflecting the fact that an increase in the interest causes householdsto shift their portfolio out of money holdings into bonds
Putting together the CC and LL curves we have equilibrium in the economyat the point where these two curves intersect From the modified Walrasrsquo law weknow that at this point demand for financial assets equals supply as well In factthere is a corresponding BB curve that goes through this same intersection
As with the CC curve to understand what lies behind the shape of the BB curvewe can totally differentiate the market-clearing condition for the financial marketwith respect to pt and rt That market-clearing condition is
net Adt = net As
t = 0
where in simplest form
net Ast = net As
t (met + δ K)
net Adt = net Ad
t (rt πet At Mpt ylowast
t )
Differentiating we obtain
[partnet Adt part(Mpt)] minus [Mp2
t ]dpt + [partnet Adt partrt minus partnet As
t partrt]drt = 0
Rearranging we have that the slope of the BB curve is given by
drtdpt |net Adt minus net As
t = 0= [partnet Ad
t part(Mpt)][Mp2t ][partnet Ad
t rt minus partnet Ast partrt] gt 0
94 Simple neoclassical macroeconomic model
The positive slope of the BB curve can be explained in the following way Thepositive numerator of the slope expression reflects the real balance effect withrespect to financial assets The term indicates the fall in net real financial assetdemand (and planned decrease in future consumption) that would accompany arise in the price level for such a rise reduces agentsrsquo wealth in the form of initialreal money balances The positive denominator indicates the effect of a change inthe interest rate on household lending and firm borrowing A rise in the interestrate would tend to decrease current consumption demand (ldquoFisherianrdquo effect) andmoney demand (ldquoportfoliordquo effect) and thus increase household net financial assetdemand On the other hand a rise in the interest rate would tend to reduce firminvestment demand by raising the expected real user cost of capital and thusreduce firm net real financial asset supply
While both the slope of the BB curve and the LL curve are positive it is thecase that the LL curve is steeper
A depiction of equilibrium aggregate demand andsupply curves
Using Patinkinrsquos analysis a higher output would require a lower price level tomaintain equilibrium in the output financial and money markets Alternativelyone can show the effect of a change in ylowast
t on pt and rt by totally differentiatingequations (62) and (63) with respect to pt rt and ylowast
t The aggregate demand curve summarizes this inverse relationship between the
price level and output arising from an analysis of the output financial and moneymarkets Specifically such a curve depicts combinations of output and price levelassociated with equilibrium in the output financial and money markets It isdownward sloping indicating that an increase in output requires a lower pricelevel to clear the output financial and money markets Behind a movement downthe aggregate demand curve are larger real balances that stimulate output demandeither directly (through a real balance effect on consumption demand) or indirectly(the resulting increase in real money balances leads to an increase in net realfinancial asset demand by households and thus a lower interest rate)22
It is important to realize that the phrase ldquoequilibrium in the output mar-ketrdquo in this context abstracts from supply-side considerations At each pricelevel the aggregate demand curve indicates the output that if produced wouldequal output demand (along with satisfying the equilibrium conditions withrespect to the financial and money markets) Production of output equal tothat demanded would occur if firms sought simply to produce to meet mar-ket demand and if workers were readily available for employment so thatfirms could hire to achieve production equal to what was demanded The termldquoequilibrium in the output marketrdquo does not imply equality between outputdemand and the output that our analysis of the labor market suggests would besupplied
Simple neoclassical macroeconomic model 95
Output
Price
Aggregate supply
Aggregate demand
Figure 65 Macroeconomic equilibrium
Figure 65 combines the aggregate demand and aggregate supply curves Wherethey intersect we know by construction that the resulting price level ( pt)0 andoutput ( yt)0 are such that
bull the labor market is in equilibrium (the economy is at a point on the aggregatesupply curve) and
bull the output financial and money markets are in equilibrium (the economy isat a point on the aggregate demand curve)
Conclusion
Using a very simple framework this chapter has developed a powerful macroeco-nomic model of the economy This model is grounded in the general equilibriumtheory set forth in earlier chapters Moreover this model highlights that degree ofconnectedness that exists among the output labor financial and money marketsThe microfoundations of macroeconomics have been emphasized and it has beenshown how market forces work in such a way as to lead to aggregate supply anddemand two key elements in any macroeconomic model
7 Empirical macroeconomicsTraditional approaches and timeseries models
Introduction
Quantitative approaches to analyzing economic data provide meaningful anduseful insight for understanding how variables interact and how they might beexpected to behave in a variety of circumstances including the future This chapteroutlines the traditional econometric-based method and the relatively simple butoften more elegant time series method for analyzing economic data in the timedomain The stochastic nature of economic data is discussed and the now com-mon ARIMA model found in much of the empirical macroeconomic literature isdeveloped in parts The chapter provides a solid background for understandingldquomacroeconometricsrdquo and time series analysis
Traditional approaches
Empirical macroeconomics can be roughly divided into two approaches ndash a tradi-tional approach that draws heavily on macroeconomic theory and a more recentapproach advocated for forecasting that does not rely to any great extent on the-ory To consider the former we start by presenting a simple theoretical model ofthe macroeconomy The model is used to illustrate traditional empirical analysesreflecting (a) tests of behavioral hypotheses (b) tests of reduced-form expressionsfor various economic aggregates and (c) the construction of econometric modelsfor policy simulations and forecasting The more recent empirical macroeconomicsapproach of using time series models for forecasting aggregate variables whichplaces less reliance on macroeconomic theory is then briefly considered
A simple theoretical macroeconomic model
Consider the following linear approximation of the simple static classical model(closed economy) such as that popularized by Sargent (1987a 20)1 The economyis divided into four markets a labor market (where wages w and total employ-ment n are determined) an output market (where total output y and the pricelevel p are determined) a financial market (where the interest rate r is deter-mined) and a money ldquomarketrdquo Our goal is to construct a model that will determine
Empirical macroeconomics 97
such endogenous variables as the level of employment wages output prices andthe interest rate
From Walrasrsquo law (as we will see more clearly later on) we need only explicitlyconsider three of the above four markets in the analysis For the labor market letus assume the following linear approximations for the key behavioral relations2
nd = f minus g(wp) (71)
ns = h + j(wp) (72)
These two linear equations indicate that firmsrsquo labor demand nd is inversely relatedto the real wage (wp) and householdsrsquo labor supply ns is directly related to thereal wage3
For the output market the key underlying behavioral and technological relationsare in linear form
cd = a + b( y minus T ) minus cr (73)
id = d minus er (74)
ys = f (n K) (75)
Equation (73) indicates that householdsrsquo consumption demand cd is directlyrelated to real disposable income (income y minus lump-sum taxes T ) andinversely related to the interest rate r4 Equation (74) indicates that firmsrsquo invest-ment demand id is inversely related to the interest rate r Equation (75) is theaggregate production function relating employment n and capital stock k tototal output supplied ys
For the money ldquomarketrdquo the key underlying behavioral relation is
Ld = l middot y minus m middot r (76)
Equation (76) indicates that householdsrsquo real money demand Ld is directly relatedto real income and inversely related to the interest rate Nominal money supplyM s is exogenous
General equilibrium in this economy means an equilibrium level of employmentnlowast wage wlowast price level plowast output ylowast and interest rate rlowast such that the labor marketis in equilibrium
ns minus n = 0 (77)
nd minus n = 0 (78)
the output market is in equilibrium
ys minus y = 0 (79)
cd + id + gd minus y = 0 (710)
98 Empirical macroeconomics
and the money ldquomarketrdquo is in equilibrium
Ld minus M sp = 0 (711)
Note that the component of output demand gd in (710) reflects exogenous realgovernment demand for output
The above macroeconomic model consists of 11 equations Macroeconomicmodels often can be represented by a system of equations This system of equationsis sometimes said to be a ldquostructural modelrdquo because the form is given from theunderlying theory As we will see below we can solve this system of equationsfor each of the ldquoendogenousrdquo variables as a function solely of the predeter-mined or ldquoexogenousrdquo variables5 Such solutions can be called the ldquoreduced-formrdquosolutions
By substituting the behavioral equations (71)ndash(76) into the equilibrium condi-tions we obtain the following set of five equilibrium conditions that can be solvedfor the five variables nlowast wlowast plowast ylowast and rlowast6
f minus g(wp) minus n = 0
h + j(wp) minus n = 0
f (n K) minus y = 0
a + by minus bT minus cr + d minus er + gd minus y = 0
ly minus mr minus M sp = 0
(712)
Note that the first three equations in the system (712) can be solved to obtainwlowast nlowast and ylowastas a function of the price level plowast In particular we obtain7
wlowast = [(f minus h)(g + j)]plowast (713)
nlowast = [1( j + g)][ jf + gh] (714)
ylowast = f (nlowast K) = f ([1( j + g)][ jf + gh] K) (715)
Equation (715) is an example of a reduced-form expression for output for itexpresses output solely in terms of the exogenous variables It is a special caseof what is called the ldquoaggregate supply equationrdquo in which the price level doesnot affect the level of output produced8 Note that equation (713) indicates thatchanges in the price level lead to equiproportionate changes in the equilibriummoney wage so that changes in the price level do not lead to changes in theequilibrium real wage
To obtain a reduced-form expression for the price level we can use the reduced-form expression for output in conjunction with the last two equations in the system(712) These last two equations are sometimes termed the ldquoIS equationrdquo andthe ldquoLM equationrdquo respectively9 Solving the LM equation (the last equation ofthe system (712)) for r and substituting both this expression for r and the priorexpression for equilibrium output ylowast (715) into the IS equation (the penultimate
Empirical macroeconomics 99
equation of the system (712)) we obtain the reduced-form expression for theequilibrium price level
plowast = M scprimem
[1 minus b + cprimem] f ([1( j + g)][ jf + gh] K) minus aprime (716)
where aprime = a + d + gd minus b middot T and cprime = c + eChanges in the variable aprime indicate changes in the autonomous component of
consumption (the term a in (73)) autonomous investment demand (d in (74))and government spending or taxing (gd or T respectively) By ldquoautonomousrdquo wemean that part of householdsrsquo and firmsrsquo output demand that is independent of thevariables to be determined by the analysis particularly income and the interestrate The variable cprime indicates the combined response of consumption demand (cin (73)) and investment demand (d in (74)) to a unit change in the interest rate
Note that the procedure to obtain equation (716) could be alternatively describedas follows First combine the last two equations in the system (712) to eliminatethe interest rate r The result is termed the ldquoaggregate demand equationrdquo Thisrelates the price level to the level of output at which the money and output markets(and thus by a modified version of Walrasrsquo law the financial market) are inequilibrium In this context equilibrium in the output market is defined as thelevel of production that if produced would equal output demand This aggregatedemand equation would then be combined with the aggregate supply equation(715) to determine the equilibrium price level or output10
Tests of behavioral hypotheses
Theoretical macroeconomic models embody predictions concerning factors thatinfluence the behavior of various groups in the economy For instance considerthe behavioral equations (71)ndash(76) in the prior simple macroeconomic modelWe could test the prediction that investment is inversely related to the interest rate(see (74)) or that money demand depends inversely on the interest rate (see (76))Or we could expand our theory of consumption behavior (equation (73)) to testa particular behavioral relationship between aggregate household consumptionand permanent income11 Or we could expand our view of labor supply behavior(equation (72)) as Stuart (1981) did in an examination of Swedish data to test theprediction that sufficiently high marginal tax rates will reduce the economy-widelabor supply
Tests of reduced-form hypotheses
Theoretical macroeconomic models also generate predictions concerning thereasons for fluctuations in such aggregate variables as real output unemploy-ment and the level of prices These predictions typically reflect the reduced-formsolutions of macroeconomic models Examples of reduced forms in our simplemacroeconomic model are equations (715) for output and (716) for price
100 Empirical macroeconomics
One example of an empirical test of macroeconomic theory that focuses ontesting the reduced-form predictions is Taylor (1979) Specifically Taylor teststhe reduced-form relationships between the money supply and the logarithmof income from trend and between the money supply and the rate of infla-tion using quarterly US data from 1953 to 197512 Similarly Christopher Sims(1972) uses quarterly US data on nominal output and the money supply to testwhether changes in the money supply lead to changes in the current dollar valueof national income (see Ewing 2001)
Large-scale econometric models and forecasting
Besides testing macroeconomic theories empirical macroeconomic analysis oftenseeks to forecast the future paths of economic aggregates Interest in forecast-ing stems not only from an obvious curiosity about the future path of aggregatevariables such as real output and prices but also from the fact that many ifnot all macroeconomic theories suggest that expectations of future events influ-ence current activity In this context forecasts can be used to proxy individualsrsquoexpectations of future events in tests of various aspects of macroeconomic models
One approach to making forecasts of future aggregate variables is to rely ontheoretical macroeconomics as a guide in the construction of large-scale econo-metric models Behavioral equations that are more detailed disaggregated versionsof equations (71)ndash(76) are estimated and coefficients are checked to make surethey agree with theory These models reflect attempts to produce large systems thatfaithfully represent the interrelationships in a complex national economy Givenpostulated paths of the exogenous variables the actual estimated equations arethen used to generate forecasts of the various aggregate variables such as outputand its components (eg consumption and investment) prices and interest rates
While large-scale econometric models as forecasting devices have their advo-cates (and many individuals reveal they have a positive value by willinglypaying for their forecasts) Granger and Newbold (1986 292ndash293) among oth-ers question the value of such a forecasting approach They note that ldquoteamsof macroeconomists have constructed forecasting models involving hundreds ofsimultaneous equations fitted to data that time series analysts would view as neitherplentiful nor of especially high qualityrdquo
An alternative to the large-scale macroeconomic models that is suggested areldquotime series modelsrdquo As summarized by Granger and Newbold (1986) ldquoin theiranxiety econometricians have failed to touch some very important basesrdquo thatinclude
bull the fact that there are many areas in which ldquoeconomic theory is not terriblywell developedrdquo
bull the fact that even where the theory is satisfactory it is ldquoalmost invariablyinsufficiently precise about dynamic specifications in the sense that it is clearthat one structure must be appropriaterdquo
Empirical macroeconomics 101
bull the fact that even after appeal to economic theory there will be error termssince ldquono theory provides a completely accurate description of the behaviorof economic agents so that any postulated equation necessarily includes astochastic error termrdquo
Given these problems many macroeconomists suggest that the appropriate start-ing point to forecasting future macroeconomic variables is the use of time seriesmodels One result is that such time series modelling terms as AR ARMA andARIMA now abound in the macroeconomic literature It is thus useful to brieflyreview the nature of time series models However at the outset it should be madeclear that the discussion below is not complete but rather is provided simply asan introduction to some terms and concepts Proofs of various propositions andrigorous definitions of various properties of time series models can be found intime series texts such as Enders (2004) and Mills (1999)
Time series models
Let us take as our premise the idea that the actual observed time series of somevariable yt t = 0 1 T (eg the logarithm of economy-wide output for thelast 30 years) is the realization of some theoretical process which can be called aldquostochastic processrdquo As phrased by Harvey (1993) ldquoeach observation in a stochas-tic process is a random variable and the observations evolve in time according tocertain probabilistic laws Thus a stochastic process may be defined as a collectionof random variables which are ordered in timerdquo To forecast future values of yt one needs a model that defines the mechanisms by which the observations aregenerated
A distinguishing feature of a pure univariate time series model is that move-ments in yt are ldquoexplainedrdquo solely in terms of its own past or by its position inrelation to time13 That is time series models look for patterns in the past move-ments of a particular variable and use that information to predict future movementsof the variable In general a time series modelrsquos forecast of y based on knownvalues yT+1 is given by
yT+1 = E( yT+1|y0 yT )
As Pindyck and Rubinfeld (1991) suggest ldquoin a sense a time-series model is just asophisticated method of extrapolation yet it may often provide a very effec-tive tool for forecastingrdquo In this sense time series models are more along the linesof empirical analyses that ldquolet the data speak for themselvesrdquo rather than empiricalanalyses that (strictly speaking) ldquotest economic theoriesrdquo As Harvey (1993) statesldquoan essential feature of time series models is that they do not involve behaviouralrelationshipsrdquo Time series models reflect a ldquostatisticalrdquo approach to forecastingHowever the patterns in the data discovered by time series models do influencetheoretical discussions of the macroeconomy and tests of macroeconomic theo-ries concerning behavior have incorporated time series models14 An example of
102 Empirical macroeconomics
how time series analysis has influenced theoretical macroeconomic discussions isgiven at the end of this section
Some properties of stochastic processes
There are several key properties of stochastic processes for time series that wecan introduce One is ldquostationarityrdquo For a stochastic process to be stationary thefollowing conditions must be satisfied for all t
E( yt) = micro
E[( yt minus micro)2] = σ 2y
E[( yt minus micro)( ytminusk minus micro)] = γk
Note that γ0 = σ 2y In words if a series is stationary the mean of the series is
invariant to time the variance of the series is invariant to time and the covarianceof the series is invariant to time As Pindyck and Rubinfeld (1991) note ldquoif astochastic process is stationary the probability distribution p( yt) is the same forall time t and its shape (or at least some of its properties) can be inferred by lookingat a histogram of the observations y1 yT that make up the observed seriesrdquo15
Also an estimate of the mean micro of the process can be obtained from the samplemean of the series
y = 1
T
Tsumj=0
yt
and an estimate of the variance σ 2y can be obtained from the sample variance16
σ 2y = 1
T
Tsumj=0
( yt minus y)2
As Granger and Newbold (1986 4) phrase it ldquoa stationarity assumption is equiv-alent to saying that the generating mechanism of the process is time-invariantso that neither the form nor the parameter values of the generation procedurechange through timerdquo The simplest example of a stationary stochastic process isa sequence of uncorrelated random variables with constant mean and variance
A second property of a stochastic process is the ldquoautocorrelation functionrdquo Theautocorrelation function provides us with a measure of how much correlation thereis (and by implication how much interdependency there is) between neighboringdata points in the series yt For stationary processes the autocorrelation with lagk is given by17
ρk = γkγ0
= E[( yt minus micro)( ytminusk minus micro)]σ 2y
Empirical macroeconomics 103
For any stochastic process ρ0 = 1 If the stochastic process is simple ldquowhitenoiserdquo (ie yt = εt where εt is an independently distributed random variablewith zero mean and finite variance) then the autocorrelation function for thisprocess is given by
ρ0 = 1 ρk = 0 for k gt 018
A simple example of a stochastic process is the ldquorandom walkrdquo process inwhich each successive change in yt is drawn independently from a probabilitydistribution with zero mean19 Thus yt is determined by
yt = ytminus1 + εt (717)
where εt is a sequence of uncorrelated random variables (E(εtεs) = 0 for t = s)with mean zero (E(εt) = 0) and constant variance (E(ε2
t ) = σ 2e for all t) Recall
that a sequence εt of this kind is typically called ldquowhite noiserdquo If the stochasticprocess is a random walk the one-period-ahead forecast of yt+1 is simply yt
A simple extension of the random walk process is to incorporate a trend in theseries yt We then obtain the following stochastic process known as a random walkwith drift20
yt = ytminus1 + d + εt (718)
The one-period-ahead forecast of yt+1 is now yt + d By repeatedly substitutingfor past values of yt into (718) we obtain
yt =tminus1sumj=0
εtminusj + td + y0 (719)
For the random walk process without drift (d = 0) we see from (719) that thefirst requirement for stationarity namely that the mean be constant over time issatisfied if y0 is fixed21 That is E( yt) = E( y0) Nevertheless the process is notstationary since Var( yt) = tσ 2
e The random walk process tends to meander awayfrom its starting value but exhibits no particular trend in doing so22
Autoregressive processes
A process similar to the random walk that is stationary is called the ldquofirst-orderautoregressive processrdquo or AR(1)23
The AR(1) process is given by
yt = φytminus1 + (1 minus φ)micro + εt (720)
A necessary condition for stationarity is that |φ| lt 1 in which case E( yt) equalsthe constant term micro in (720)24
104 Empirical macroeconomics
One possible example of an autoregressive process is the total numberunemployed each month (Granger and Newbold 1986) Let the total number unem-ployed in one month be yt This number might be thought to consist of a fixedproportion φ of those unemployed in the previous month (the others having foundemployment) plus a new group of workers seeking jobs If the new additionsare considered to form a white noise series with positive mean micro(1 minus φ) thenthe unemployment series is a first-order autoregressive process expressed byequation (720)
For convenience only it is often assumed that the process has a zero mean iemicro = 0 Note that we can always define a new variable yprime
t equiv yt minus micro such that yprimet
has a zero mean and is given by the AR(1) process
yprimet = φyprime
tminus1 + εt (721)
with |φ| lt 1 Setting micro = 0 thus simply means that the variable yt (egemployment real output etc) is measured in terms of deviations from its mean
By successive substitution for yt in (720) we obtain (assuming micro = 0)
yt =tminus1sumj=0
φjεtminusj + φty0 (722)
If the process is regarded as having started at some point in the remote past and|φ| lt 1 then we can write
yt =infinsum
j=0
φjεtminusj (723)
Given that εt is white noise we thus have that E( yt) = micro = 0 Assumingstationarity (|φ| lt 1) we know that the variance and covariances are constantRecall that εt is white noise such that E(εtεs) = 0 for s = t We thus have25
γ0 = σ 2y = E[( yt minus micro)2] = E
⎡⎢⎣⎡⎣ infinsum
j=0
φj(εtminusj)
⎤⎦
2⎤⎥⎦ = σ 2
e (1 minus φ2)
γ1 = E[( yt minus micro)( yt+1 minus micro)] = φσ 2e (1 minus φ2)
γ2 = E[( yt minus micro)( yt+2 minus micro)] = φ2σ 2e (1 minus φ2) etc
The autocorrelation function for AR(1) is thus particularly simple ndash it begins atρ0 = 1 and then declines geometrically ρk = φk Note that this process hasan infinite memory The current value of the process depends on all past valuesalthough the magnitude of this dependence declines with time
In general an autoregressive process of order p is generated by a weighted aver-age of past observations going back p periods together with a random disturbance
Empirical macroeconomics 105
in the current period The AR( p) process is thus given by26
yt = φ1ytminus1 + φ2ytminus2 + φ3ytminus3 + middot middot middot + φpytminusp
+ (1 minus φ1 minus φ2 minus middot middot middot minus φp)micro + εt
= micro + εt +psum
j=1
φj( ytminusj minus micro)
(724)
with a necessary condition for stationarity being that φ1 + φ2 + middot middot middot + φp lt 127
The next section discusses necessary and sufficient conditions for stationarity
A brief digression on necessary and sufficient conditionsfor stationarity
To get some idea of what are necessary and sufficient conditions for stationaritylet us consider two specific cases AR(1) and AR(2) In the AR(1) case
yprimet minus φyprime
tminus1 = εt
where yprimet equiv yt minus micro Note that εt can be termed the ldquostochasticrdquo component of the
expression and the remainder the ldquodeterministicrdquo component28
Focusing on the deterministic component and assuming for convenience thatmicro = 0 we have a homogeneous first-order difference equation of the form29
yt minus φytminus1 = 0 (725)
To solve this equation let us try the general solution
yt = Abt (726)
which naturally implies ytminus1 = Abtminus130 The problem then is to find the valuesof A and b Substituting the trial solution into the above difference equation weobtain
Abt minus φAbtminus1 = 0
which by multiplying through by b1minustA can be rewritten as
b minus φ = 0 (727)
Equation (727) is called the ldquoauxiliaryrdquo or characteristic equation of (725) Thisequation provides us with the solution for b which is simply that b = φ Thissolution value is sometimes referred to as the ldquorootrdquo of the characteristic equation
Using (727) to substitute for b in (726) we thus have as the solution to thefirst-order difference equation (725) the expression
yt = Aφt (728)
106 Empirical macroeconomics
The ldquoequilibriumrdquo solution for yt given a homogeneous difference equation suchas (725) is yt = 0 That is if yt equiv 0 then yt will not change over time Thisequilibrium solution is ldquostablerdquo if as t rarr infin yt rarr 0 That is deviations fromthe equilibrium yt will return toward that equilibrium value Obviously given oursolution (728) a necessary and sufficient condition for stability is that |φ| lt 1
In the context of an AR(1) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the first-order homogeneous difference equation (725)that is the deterministic component of yt for an AR(1) process The condition ofstationarity is thus that |φ| lt 1 The effect of a given shock will mitigate overtime if this stationarity condition is met A special nonstationary case is the ldquounitrootrdquo case of the random walk process in which φ = 1 In that case the effect ofa given shock will not dampen with the passage of time
Now let us consider the AR(2) case where
yprimet minus φ1yprime
tminus1 minus φ2yprimetminus2 = εt
in which yprimet equiv yt minus micro As before εt can be termed the ldquostochasticrdquo component of
the expression Assuming again for convenience that micro = 0 we can express thedeterministic component as a homogeneous second-order difference equation ofthe form
yt minus φ1ytminus1 minus φ2ytminus2 = 0 (729)
To solve this equation let us try yt = Abt which naturally implies ytminus1 = Abtminus1
and ytminus2 = Abtminus2 The problem then is to find the values of A and b Substitutingthe trial solution into the above difference equation we obtain
Abt minus φ1Abtminus1 minus φ2Abtminus2 = 0
which by multiplying through by b2minustA can be rewritten as
b2 minus φ1b minus φ2 = 0 (730)
The quadratic equation (730) is called the ldquoauxiliaryrdquo or ldquocharacteristic equationrdquoof the second-order difference equation (729) The roots of this equation will bethe solutions of (730)31 The two ldquocharacteristicrdquo roots m1 and m2 may be foundin the usual way from the formula32
m1 m2 = [φ1 plusmn (φ21 + 4φ2)
12]2 (731)
each of which is acceptable in the solution Abt Note that for the quadraticequation (730) the roots m1 and m2 satisfy
(b minus m1)(b minus m2) = 0
where φ1 = m1 + m2 and φ2 = minusm1m2
Empirical macroeconomics 107
Expression (731) provides us with two values for b and thus two potentialsolutions for yt
yt = A1mt1 and yt = A2mt
2
The general solution to a second-order difference equation combines these twoby taking the sum Thus the general solution of the second-order homogeneousdifference equation (729) is33
yt = A1mt1 + A2mt
2 (732)
Inspection of (732) suggests that stability for the linear second-order homo-geneous difference equation (729) requires that both roots of the characteristicequation have an absolute value less than one This reflects the fact that the gen-eral solution of the second-order homogeneous difference equation includes bothroots m1 and m2 The root with the higher absolute value is sometimes termed theldquodominant rootrdquo If both m1 and m2 are less than unity in absolute value yt willbe close to zero if t is large Equivalently if the dominant root is less than one inabsolute value convergence will occur
In the context of an AR(2) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the second-order homogeneous difference equation (729)that is the deterministic component of the process The condition of stationarityis thus that |m1| lt 1 and |m2 lt 1| That is for the AR(2) process stationarityrequires that the roots of the characteristic equation (730) are less than one inabsolute value ndash that is that they all lie inside the unit circle34
In terms of φ1 and φ2 the conditions for stationarity may thus be defined asfollows35
φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 φ2 gt minus1
As you can see the prior necessary condition that φ1 + φ2 lt 1 is augmented withtwo other conditions
We have seen that stability for the second-order homogeneous differenceequation requires that the maximum of (|m1| |m2|) ndash that is the dominant root ndash beless than 1 To see how one obtains the three necessary conditions listed above forstabilitywe assume this is the case and determine the resulting restrictions placedon the coefficients φ1 and φ236 Recall that
m1 m2 = [φ1 plusmn (φ21 plusmn 4φ2)
12]2
Suppose φ21 + 4φ2 gt 0 so that the roots are real A maximum of (|m1| |m2|) less
than one means that
2 minus φ1 gt (φ21 + 4φ2)
12 gt minus2 minus φ1
108 Empirical macroeconomics
and
2 minus φ1 gt minus(φ21 + 4φ2)
12 gt minus2 minus φ1
The sum of the roots is φ1 and since we are assuming each root is between minus1and 1 it must be the case that 2 gt φ1 gt minus2 or 0 gt minus2minusφ1 and 2minusφ1 gt 0 Thusthe second and third inequalities are always true and hence place no restrictionson φ1 and φ2
The first and fourth inequalities squared read
(2 minus φ1)2 gt φ2
1 + 4φ2 and φ21 + 4φ2 lt (minus2 minus φ1)
2
respectively These two expressions can be rewritten to obtain
φ1 + φ2 lt 1 and minus φ1 + φ2 lt 1
which are the first two conditions for stability The third condition for stability(φ2 gt minus1) is obtained from the case of complex conjugate roots37 The threenecessary conditions for the dominant root being less than one in absolute value arealso sufficient conditions This can be verified by showing using these inequalitiesthat it is possible to reverse the steps in the above calculations and arrive at themaximum of (|m1| |m2|) being less than one
In general a necessary and sufficient condition for stationarity of an AR( p)process is if the roots of the characteristic equation
b p minus φ1b pminus1 minus middot middot middot minus φp = 0 (733)
are less than one in absolute valueNote that for the AR(2) case there are three possible situations If φ2
1 +4φ2 gt 0the square root in (731) is a real number and m1 and m2 will be real and distinctIn this case the solution to (729) is given by
yt = A1mt1 + A2mt
2
where A1 and A2 are constants which depend on the starting values y0 and yminus1The second situation is that of repeated roots where φ2
1 + 4φ2 = 0 such thatm1 = m2 = m In this case the solution to (729) takes the form
yt = A3mt + A4tmt
If |m| lt 1 the damping force of mt will dominate both termsThe third situation for the AR(2) process is when φ2
1 + 4φ2 lt 0 in which casethe roots are a pair of complex conjugates (ie m1 m2 = h plusmn vi where h = φ12v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12) Thus the solutionto (729) is given by
yt = A5(h + vi)t + A6(h minus vi)t
Empirical macroeconomics 109
Appealing to De Moivrersquos theorem this expression can be transformed intotrigonometric terms The general form of the solution is
yt = pt(A7 cos λt + A8 sin λt)
where p is the modulus of the roots = (minusφ2)12) and λ satisfies the two conditions
cos λ = hp and sin λ = vp As in the other two cases if the absolute valueof the conjugate complex roots |h plusmn vi| lt 1 is less than one the process isstable The time path followed by yt in response to a shock is cyclical but theperiodic fluctuation will mitigate as time passes (ldquosinusoidal decayrdquo) if the processis stationary
Moving average processes
In a moving average process the process yt is described completely by a weightedsum of current and lagged random variables The simplest moving average processthe moving average process of order 1 or MA(1) takes the form
yt = micro + εt + θεtminus1 (734)
The term ldquomoving averagerdquo reflects the fact that a series so characterized will besmoother than the original white noise series εt In general such a moving averageprocess of order q MA(q) is written as38
yt = micro + εt + θ1εtminus1 + θ2εtminus2 + middot middot middot + θqεtminusq
= micro + εt +qsum
j=1
θjεtminusj (735)
As before if yt has nonzero mean we can focus with no loss of generality on thetransformed series yprime
t = yt minus micro which has zero meanOne possible example of a moving average process is economy-wide output
(Granger and Newbold 1986) Output yt could be in equilibrium (at mean micro)but is potentially moved from its equilibrium position each period by a seriesof unpredictable events such as periods of exceptional weather or strikes If thesystem is such that the effects of such events are not immediately assimilated butexert an influence on output for q periods then a moving average model can arise
Note that by repeatedly substituting for lagged valued of εt into an MA(1)process (equation (734)) one obtains39
yt =infinsum
j=1
(minusθ)jytminusj + εt (736)
If yt is not to depend on a shock to the system arising at some point in the remotepast θ must be less than one in absolute value Comparing (736) to (724) we
110 Empirical macroeconomics
see that assuming what was previously denoted the stationarity condition but whatin this context is called the ldquoinvertibility conditionrdquo namely that |θ | lt 1 then anMA(1) process can be represented by an AR(infin) process40 The weights on pastvalues of yt for this AR(infin) process decline exponentially
In general if similar invertibility conditions are met then any finite-ordermoving average process has an equivalent autoregressive process of infinite orderLikewise we have already shown (see (723)) that if the AR(1) process is station-ary it is equivalent to a moving average process of infinite order MA(infin) In factfor any stationary autoregressive process of any order there exists an equivalentmoving average process of infinite order so that the autoregressive process isldquoinvertiblerdquo into a moving average process
Mixed autoregressive moving average processes
An obvious generalization of the above discussion is to combine an AR( p) process(ie (724)) and an MA(q) process (ie (735)) An autoregressive moving averageprocess of order ( p q) or ARMA( p q) can be written as41
yt = micro + εt +psum
j=1
φj( ytminusj minus micro) +qsum
j=1
θjεtminusj (737)
where as before εt is a zero-mean white noise For simplicity consider the casewhere micro = 0
Whether or not a mixed process is stationary depends solely on its autoregres-sive part If the ARMA process is stationary then there is an equivalent MA(qprime)process Similarly provided invertibility conditions hold there is an AR( pprime) pro-cess equivalent to the above ARMA( p q) process It thus follows that a stationaryARMA process can always be well approximated by a high-order MA process andthat if the process obeys the invertibility condition it can also be well approxi-mated by a high-order AR process However an ARMA process has the advantageof ldquoparsimonyrdquo in that the mixed model ARMA( p q) often can achieve as gooda fit as say an AR( pprime) but uses fewer parameters (ie p + q lt pprime)
As an example of an ARMA( p q) process we need only consider a case wherethe variable yt is the sum of a ldquotrue seriesrdquo that is AR( p) plus a white noiseobservation error Thus an ARMA( p q) series results42
Integrated processes
In series arising in economics the assumption of stationarity is often veryrestrictive That is often the characteristics of the underlying stochastic processgenerating a time series appear to change over time With economic data some-times a transformation (such as taking the logarithm of the variable) can result in astationary series In other cases we can obtain a stationary series by differencingone or more times We say that yt is a ldquohomogeneous nonstationary process of
Empirical macroeconomics 111
order drdquo if
wt = dyt
is a stationary series Here denotes differencing that is
yt = yt minus ytminus1 2yt = yt minus ytminus1 = yt minus 2ytminus1 + ytminus2
The process yt is an ldquointegratedrdquo process if after a series yt has been differencedto produce a stationary series wt the series wt can be modelled as an ARMAprocess If wt = dyt and wt is an ARMA( p q) process then we say that yt is anldquointegrated autoregressive moving average process of order ( p d q)rdquo or simplyARIMA ( p d q)
An application of time series models
To gain a better understanding of how time series models are used in macroeco-nomic empirical work let us consider the time series for the logarithm of the levelof real output to be denoted by yt As we stated above theoretical macroeconomicmodels typically postulate shocks or ldquoimpulsesrdquo to the economy (eg shocks totechnology money supply and government policy) that in conjunction with aprediction of how various markets in the economy adjust to these shocks offersan explanation of the actual fluctuations in aggregate variables One importantquestion that can arise is whether such shocks are ldquotransitoryrdquo in that their effectsdo not persist or ldquopermanentrdquo where the economy moves to a new level of realoutput
The ldquotraditionalrdquo answer to the above question of transitory versus permanenteffects of shocks has been according to Campbell and Mankiw (1987a 111) thatfluctuations in real output ldquoprimarily reflect temporary deviations of productionfrom trendrdquo That is the traditional view has been that quarterly real output couldbe represented by
yt = Tt + St + Xt (738)
where Tt is a deterministic component representing trend St is a determinis-tic seasonal component and Xt is a stationary autoregressive process with nodeterministic component43
Trend and seasonal components were typically estimated in various fashionsand then eliminated The focus of empirical work had then been on examiningvariations in the detrended seasonally adjusted real output series For exampleBlanchard (1981) estimated the following second-order autoregressive process fordeviations of the log of seasonally adjusted real output from its estimated trendylowast
t and obtained the equation
ylowastt = 134ylowast
tminus1 minus 042ylowasttminus2 + εt (739)
112 Empirical macroeconomics
Note that the necessary conditions for stationarity with respect to the series ylowastt
(namely φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 and φ2 gt minus1) are metThe traditional view is embodied in (739) A shock to output increases for
a few quarters but the effect ultimately dies out In fact only 8 percent of ashock remains after 20 quarters Assuming an ARMA(22) process Blanchardand Fischer (1989 9) obtain a similar result
ylowastt = 131ylowast
tminus1 minus 042ylowasttminus2 + εt minus 006εtminus1 + 025εtminus2 (740)
where by construction εt is that part of the deviation of current output from trendthat cannot be predicted from past output As Blanchard and Fischer note
A shock has an effect on GNP that increases initially and then decreases overtime After 10 quarters the effect is still 40 of the initial impact after 20quarters all but 3 of the effect has disappeared The view that reversiblecyclical fluctuations account for most of the short-term movements of realGNP and unemployment has been dominant for most of the last century
However as Granger and Newbold (1986 37) note ldquothe more modern view is thatas far as possible the trend seasonal and lsquoirregularrsquo components should be han-dled simultaneously in a single model aimed at depicting as faithfully as possiblethe behavior of a given time seriesrdquo Seasonality can be treated through a gener-alization of ARMA models44 More importantly for the question at hand trend isgenerally treated by differencing leading to the consideration of the ldquointegratedprocessesrdquo suggested above In fact Campbell and Mankiw (1987b) indicate thatthe answer to the question of the importance of temporary versus permanent shocksto output is biased if detrended data are used45 Campbell and Mankiw go on topoint out that examining the differences in the logarithm of real output does notprejudge the issue of whether shocks to the economy are transitory or permanentFor instance suppose that yt follows an IMA(11) process so that
yt minus ytminus1 = d + εt minus θεtminus1
Then a unit impulse in yt changes the forecast of yt+n by 1 minus θ regardless of nldquoHence depending on the value of θ news about current GNP could have a largeor small effect on onersquos forecast of GNP in ten yearsrdquo (Campbell and Mankiw1987b 860)46
Campbell and Mankiw consider ARMA( p q) processes for the difference in thelog of real output for p = 0 1 2 3 and for q = 0 1 2 347 Interestingly they finda high level of persistence in shocks One intriguing suggestion of Campbell andMankiw (1987b 868) is that ldquowhen we examine postwar annual data we cannotreject the hypothesis that the log of real GNP is a random walk with drift In thiscase the impulse response is unity at all horizonsrdquo Recall that the random walkprocess with drift (718) is an example of a nonstationary process that is first-order
Empirical macroeconomics 113
homogeneous nonstationary To see this simply consider the series wt that resultsfrom differencing the random walk ndash that is the series
wt = yt minus ytminus1 = d + εt (741)
Since εt are assumed independent over time wt is clearly a stationary process Theterm d + εt is a white noise process In this example yt is an ARIMA(010) andyt is an ARMA(00) The impulse response to a shock is unity in such a case
As we will see later one way to interpret the above finding of persistence isto place weight on ldquoaggregate supplyrdquo shocks such as technological disturbancesrather than on ldquoaggregate demandrdquo shocks such as changes in the money supplyin explaining fluctuations in real output That is the finding gives credence tothe view of Nelson and Plosser (1982) among others that real (ie permanent)shocks dominate as a source of output fluctuations48
However as pointed out by Campbell and Mankiw (1987b 877) the Nelsonand Plosser conclusion is an extreme
one can attribute a major role to supply shocks without completely abandoninga role for demand shocks For example suppose that output Y (= log y) isthe sum of two components a supply-driven ldquotrendrdquo Y T and demand-driveldquocyclerdquo Y c that are uncorrelated at all leads and lags Suppose further thatY T is a first-order autoregressive process with parameter φ and that Y c issome stationary process If trend output is approximately a random walk sothat φ is small then the finding of great persistence implies that fluctuationsin the cycle are small relative to fluctuations in the trend If the change inthe trend is highly serially correlated (φ is large) however the finding ofpersistence is consistent with a substantial cyclical component
Campbell and Mankiw (1987b 877) go on to suggest that
a second way to interpret the finding of persistence is to abandon the
natural rate hypothesis Models of multiple equilibria might explain a long-lasting effect of aggregate demand shocks if shocks to aggregate demand canmove the economy between equilibria Shocks to aggregate demand couldhave permanent effects if technological innovation is affected by the businesscycle
Note that so far we have considered only a single or univariate time series Theconcepts involved however can be extended to multivariate series For instancea simple vector autoregression model (VAR) could take the form
yt = ytminus1 + εt
where now yt is considered an N times 1 vector reflecting N variables the randomdisturbance term εt is also an N times 1 vector and is an N times N matrix of parame-ters The disturbances are uncorrelated over time but may be contemporaneously
114 Empirical macroeconomics
correlated As in the univariate case such a model is usually only fitted to vari-ables which are stationary (possibly obtained by logarithmic transformations orby taking first or second differences) As in the univariate case the objectivewould be to find a model that transforms a vector of time series into a whitenoise vector As Harvey (1993) notes ldquoalthough a model of (this) form is oftenused for forecasting in econometrics economic theory will typically place a priorirestrictions on elements of rdquo There are also questions concerning the correlationbetween innovations across different series of data for example the link betweeninnovations in money supply and output But given that our aim is to review basicmacroeconomic theory we will not pursue such topics
Conclusion
This chapter has emphasized the empirical nature of macroeconomics Inparticular time series analysis and econometric methods have been shown tobe useful tools for analyzing macroeconomic data Many of the policy conclu-sions that will be developed in the remainder of the book can be tested using thesemethods The results obtained from a thorough understanding of the time seriesproperties of important macroeconomic variables such as interest rates employ-ment numbers real output measures and the like together with the underlyingtheory may be used to provide policy recommendations Moreover more andmore businesses are relying on empirical macroeconomics when making operatingdecisions
Appendix translation of higher-order difference equationsinto lower order
Any higher-order difference equation can be interpreted in terms of an equivalentsystem of first-order difference equations To see this consider equation (729)
yt minus φ1ytminus1 minus φ2ytminus2 = 0
To facilitate matters we start by shifting the origin of this second-order homo-geneous difference equation from t = minus2 to t = minus1 such that we may rewrite(729) as
yt+1 minus φ1yt minus φ2ytminus1 = 0 (7A1)
Now let us introduce the new variable xt defined as
xt = ytminus1
which means that xt+1 = yt We may then express the second-order differenceequation (7A1) by means of two first-order simultaneous equations
yt+1 minus φ1yt minus φ2xt = 0
xt+1 minus yt = 0(7A2)
Empirical macroeconomics 115
In matrix notation (7A2) becomes[yt+1xt+1
]= A
[ytxt
]
where A is the square nonsingular matrix defined as
A =[φ1 φ21 0
]
The ldquocharacteristic equation of a square matrixrdquo is defined by the determinant
|A minus λI | = 0
where I is the unit matrix[
1 00 1
]and λ is a scalar variable
Letting b = λ the ldquocharacteristic equation of matrix Ardquo
|A minus λI | =∣∣∣∣φ1 minus b φ2
1 minusb
∣∣∣∣ = b2 minus φ1b minus φ2 = 0
is identical to equation (730) which was termed the ldquocharacteristic equation of thesecond-order homogeneous difference equationrdquo The values for the λs (or bs) arethe roots of the characteristic equation One reason for restating the higher-orderdifference equation in matrix form is that necessary and sufficient conditions forstability can be expressed in terms of conditions on the matrix A
8 The neoclassical model
Introduction
This chapter turns our attention to one of the most popular and sometimescontroversial models used by macroeconomists the neoclassical model Themodel in its purest form is often used as a benchmark or starting point for addingldquorealismrdquo to the structure of the economy However one views the neoclassicalmodel there is no denying that it is a powerful tool for generating predictionsabout movements in economic aggregates As such this chapter works throughthe comparative static exercise of a change in the money supply This exerciseprovides a great deal of insight into how the monetary authority might influencethe economy or whether it can influence the economy at all A number of importantissues are raised for example money neutrality and money illusion Additionallythe chapter formally derives the aggregate supply curve in the neoclassical contextThe model has serious implications for the existence of a natural rate and also forthe formation of expectations
Comparative statics for the neoclassical model
We can use our previous method of analysis known as ldquocomparative staticrdquoanalysis to examine the effects of various shocks on the equilibrium level ofprices As the name suggests comparative static analysis is concerned with thecomparison of different equilibrium states associated with different sets of valuesof parameters and exogenous variables In the current context of the neoclassicalmodel the labor market can be isolated from the other markets (ie there is aldquoblock recursiverdquo character to the equilibrium solution) In such a context we candistinguish two types of exogenous variables
One type ldquosupply-siderdquo variables alter the level of output at any given pricelevel Such variables could include changes in technology and the existing capitalstock in the current version of the model or we could expand the analysis toinclude changes in the supply of other inputs (such as oil) changes in governmentpolicies that affect incentives to supply labor and changes in government policiesthat affect the incentives to invest and thus affect the productive capacity of theeconomy over time
Neoclassical model 117
The second type of exogenous variables ldquodemand-siderdquo variables do not alterthe current supply of output at prevailing prices but instead impact equilibriumprices and interest rates as determined in the output financial and money marketsBelow we consider the impact of demand-side ldquoshocksrdquo such as changes in initialmoney balances and the expected inflation rate
The macroeconomic approach to general equilibriumhow it can obscure
The neoclassical model can be viewed as a special case of a Walrasian generalequilibrium system distinguished by the existence of money1 As Clower (1965)notes
income magnitudes do not appear as independent variables in demand andsupply functions of the (Walrasian) general equilibrium model for incomesare defined in terms of quantities as well as prices and quantity variablesnever appear explicitly in the market excess demand functions of traditionaltheory To be sure income variables could be introduced by taking factorsupplies as given parameters but this would preclude the formulation of ageneral equilibrium model containing supply functions of all marketable factorservices
Referring to Chapter 9 of Patinkin (1965) Clower goes on to note that this point
was apparently overlooked by Patinkin when he formulated his ldquogeneral the-oryrdquo of macroeconomics It is instructive to notice that this chapter is notsupplemented by a mathematical appendix I do not mean to suggest thatauthors may not put such variables as they please into their models My pointis that such variables that can be shown to be fundamentally dependent onothers should not then be manipulated independently
What Clower is referring to is the fact that the neoclassical model with limitedperfect foresight effectively determines at time t the money wage for the labormarket and the (futures) price of output and interest rate Given perfect foresightindividual optimizing behavior will generate planned consumption and moneydemand functions at time t for period t that include the real wage rather thanincome That is since labor supply is a choice variable at time t labor income (theproduct of the real wage times labor supply) should not appear as an argument inhouseholdsrsquo demand and supply functions (Note that labor income along withdividend and interest payments equals output minus depreciation)
Formally we can discover how a ldquosmallrdquo change in one (or more) of the exoge-nous variables affects equilibrium values by totally differentiating the equilibriumconditions with respect to the prices to be determined and the exogenous variableand then solving for the implied change in equilibrium prices required to maintainequilibrium2 The resulting changes in equilibrium prices then imply changes in
118 Neoclassical model
the equilibrium levels of employment output consumption investment and realmoney holdings
However we choose instead the standard macroeconomic approach of arbitrar-ily separating the analysis into an analysis of the labor market and the resultingemployment and output (the ldquoaggregate supply equationrdquo) and then an analysisof the other markets and the detemination of the price level and the interest rate(the ldquoISrdquo and ldquoLMrdquo equations) While Clower is right in that this obscures thetraditional ldquogeneral equilibriumrdquo nature of the neoclassical model the approach isuseful in that it allows us to more easily extend the analysis to situations in whichprices are not market-clearing or to situations in which there is not limited perfectforesight
Given the assumption of perfect foresight on the part of both firms and workersconcerning the price level pt we have seen that the aggregate supply equation isindependent of the price level or the interest rate We know from the modifiedWalrasrsquo law that any two of the three excess demand conditions with respect tomoney financial assets and output can be used in the comparative static analysisUsing the equilibrium conditions with respect to output and money (the ldquoISrdquo andldquoLMrdquo equations) we thus have the three equations
yt = yt(K ) (81)
cdt (rt πe
t At Mpt yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (82)
Ldt (rt πe
t At Mpt yt) minus Mpt = 0 (83)
to determine equilibrium output yt price level pt and interest rate rt Recallthat from the modified Walrasrsquo law we know that there is a fourth equation theequilibrium condition for the financial market that is implied by (82) and (83)This fourth equilibrium condition is
net Adt (rt πe
t At Mpt yt) minus net Ast (m
et + δ K) = 0 (84)
A change in the money supply the comparative statics
It is clear from the above that the neoclassical aggregate supply equation (81)determines output while the IS and LM equations (82) and (83) determine theprice level and interest rate given the equilibrium level of output Focusing on thelatter two equations and the determination of the price level and interest rate fora given level of output total differentiation with respect to the equilibrium prices( pt and rt) and the money supply change gives the following system of linearequations in matrix form3
[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpartrpartLdpartr
] [dpdr
]=[ minus(partcdpart(Mp))dMp(1 minus partLdpart(Mp))dMp
]
Neoclassical model 119
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm4 Solving the above linear-equationsystem for dp and dr using Cramerrsquos rule we obtain
dp
dM= minus(partcdpart(Mp))(1p)partLdpartr minus (1 minus partLdpart(Mp))(1p)partydpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpartr
= p
M
dr
dM=
minus[(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)minus(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)](1p)
minus(partcdpart(Mp))(Mp2)partLdpartrminus(1minuspartLdpart(Mp))(Mp2)partydpartr
= 0
As you can see dpp = dMM and drdM = 0 That is a change in the moneysupply leads to the same proportional change in the price of the consumption goodand no change in the interest rate With regard to the latter result the interest ratedoes not change as the only thing that changes the interest rate is the rate of changein prices Since the price level adjusts there is no change in the interest rate and theLM curve does not change Note that from the labor market equilibrium conditionthat underlies the aggregate supply equation we know that the change in the pricelevel results in an equiproportionate change in the money wage The result is thatwe have ldquoneutrality of moneyrdquo In other words if individuals correctly anticipatethe effect of a change in the money supply on the price level as is the case inthe deterministic model under the assumption of limited perfect foresight thenmonetary changes have no ldquorealrdquo effects Real output the real wage the expectedreal rate of interest real consumption real investment and the real money supplyare all unaffected by the monetary change
The basic reason why changes in the money supply are ldquoneutralrdquo is the absenceof money illusion on the part of both firms and households In particular house-hold demand and supply functions indicate that if a change in money balancesis accompanied by an equiproportionate change in the price of output then thereis no change in any demands That is household demand and supply functionsare homogeneous of degree zero in money balances and prices This absence ofldquomoney illusionrdquo occurs assuming
bull perfect foresight at time t for price during period t so that a change in priceresults in no change in the real wage or employment (when coupled with thesimilar assumption of limited perfect foresight on the part of firms)
bull ldquoneutral distribution effectsrdquo such that the shift in wealth from prior creditorsto debtors that would accompany a rise in the price level leaves aggregatedemands unchanged
bull unit elastic expectations so that changes in the current price level can beviewed as leaving unaffected the expected rates of change in the pricelevel
120 Neoclassical model
As we will see later this money neutrality result of the neoclassical model formsthe basis of what has become known as the ldquopolicy ineffectiveness propositionrdquo(see McCallum 1979)
The ldquodynamicsrdquo of the system and the neutrality of money a review
As we discussed earlier the ldquostoryrdquo often told with respect to the dynamics of theabove situation is a ldquoloanable funds theoryrdquo of interest rate determination in whichthe interest rate moves to clear the financial market5 In this case the ldquotatonnementprocessrdquo or movement toward equilibrium involves
dpt = f ( ydt minus yt) f (0) = 0 df d( yd
t minus yt) gt 0
dpbt = fb(net Adt minus As
t ) fb(0) = 0 dfbd(net Adt minus net As
t ) gt 0
Note that we put ldquostoryrdquo in quotes since the analysis itself simply identifies variousequilibrium points with adjustments in prices to reach a different equilibrium pointgiven a shock essentially occurring without any passage of time Neverthelessconsider the following story with respect to a rise in the money supply
In the Patinkin analysis described earlier if the money supply were to doublefrom M to 2M the LL BB and CC curves would shift to the right so that the newequilibrium would occur at the original interest rate but at a price level double theoriginal one Such a once-and-for-all change in the initial money balances leavesthe money interest rate and real demands unchanged (the neutrality of money)
Superneutrality an informal review
Superneutrality of money occurs when changes in the rate of growth of the moneysupply leave the paths of capital and real output unaffected Although the aboveanalysis is static in nature it does provide some insight into the issue of thesuperneutrality of money To see how suppose each period the economy can bereplicated in every way That is the money supply equilibrium price level andinterest rate are identical each period If expectations of price changes are correctexpected inflation would equal zero For the economy to replicate itself it mustalso be the case that the initial capital stock is optimal such that the capital stockand thus output does not change over time With zero adjustment costs such asituation would imply a marginal product of capital equal to the expected real usercost of capital (me
t + δ where met equiv (rt minus πe
t )(1 + πet )) net investment demand
equal to zero each period and gross investment demand equal to δK Now let us compare this situation to an alternative sequence of temporary equi-
librium in which the economy is identical in every way except one the moneysupply is increased each period by a constant percentage from its level in the priorperiod Other things being equal the above analysis would suggest that one dif-ference across periods would be a rise in prices due to the positive growth in themoney supply Let us further assume that expectations of inflation adjust to reflect
Neoclassical model 121
what Sargent and Wallace (1975) would characterize as a new ldquosystematic moneysupply rulerdquo
Without fully developing the appropriate dynamic analysis it is easy to see thatone obvious adjustment of the static analysis given a growing money supply (asopposed to one that does not change across periods) is thus a higher expectedinflation rate (positive as opposed to zero) In fact the analysis of the static modelcan mimic to some extent the effects of a higher rate of growth in the money supplyby considering the impact of an increase in exogenous inflationary expectations
Following Sargent and Wallace (1975) among others assume that consump-tion demand depends on the expected real rate of interest r minus π not separatelyon its components (the money interest rate and the expected rate of inflation)6
Further assume that changes in expected inflation do not affect real moneydemand7 The comparative static results are then[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpart(r minus π)
partLdpartr
] [dpdr
]=[minus(partydpart(r minus π))dπ
0dπ
]
where partydpart(r minusπ) = partcdpart(r minusπ)+ (1 +ψ prime)partI dpart(r minusπ) Solving the abovelinear equation system for dp and dr using Cramerrsquos rule we obtain
dp
dπ=
minus(partydpart(r minus π))partLdpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
dr
dπ=
minus(partLdpart(Mp))(Mp2)partydpart(r minus π)
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
As we have discussed before if there is no real balance effect with respect to con-sumption demand (partcdpart(Mp) = 0) or if real money demand does not respondto changes in the interest rate (partLdpartr = 0) then we can see from the above that
dr = minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)dπ
minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)= dπ
so that the change in the expected rate of inflation results in no change in theexpected real rate of interest (r minus π ) In this case money is superneutral in thata change in the growth of the money supply (although it alters inflation and thusgiven perfect foresight expected inflation) leaves the expected real rate of interestunchanged and thus does not affect investment and the future size of the capitalstock which depend inversely on the expected real interest rate
122 Neoclassical model
Recall that Sargent and Wallace (1975) have argued for superneutrality ofmoney In contrast Begg (1980) has noted that steady-state analyses of growthmodels with money like the analysis above do find that different rates of growthin the money supply have real effects (through changes in the expected real grossinterest rate)8 As the above analysis makes clear there are two conditions eitherof which is sufficient that will result in money being superneutral Begg (1980293) describes them as follows ldquoThe first condition is that the level of real moneybalances is not an argument in the consumption function and it is this conditionwhich distinguishes the rational expectations model of Sargent and Wallace fromthe analysis of growth models with money The second condition is that the demandfor money is independent of the nominal interest raterdquo (as well as π ) Otherwisea higher expected inflation will result in a fall in the expected real rate of interestand a higher price level
In the Patinkin framework with consumption demand and investment demanddepending on the expected real interest rate at a given price level and higherexpected inflation the CC curve must shift up vertically by the amount of theincrease in expected inflation so as to maintain the same expected real rate ofinterest But the LL curve does not shift with a change in expected inflationThus given a downward-sloping CC curve and an upward-sloping LL curve thenew equilibrium money interest rate does not rise by the extent of the increase inexpected inflation Superneutrality of money seems not to hold
In such a case the static analysis thus predicts a lower real stock of money eachperiod higher investment and a greater capital stock next period This suggests anew steady state in which the capital stock is greater In fact a complete dynamicanalysis confirms these predictions a higher steady-state capital stock is associatedwith an increase in the rate of growth of the money supply and consequent increasedexpected inflation
The role of the key assumptions of the neoclassical model
At this point it might be useful to review the role played by the two key assumptionsof the neoclassical model namely price flexibility and complete information onprices In most cases and this is no exception one can gain an understanding ofthe role of a particular assumption by exploring how the analysis would proceedif the assumption were not made Consider first the implications of dropping theneoclassical modelrsquos assumption that prices are perfectly flexible
Suppose that the demand for output decreases and firms cannot sell all theydesire at prevailing prices With flexible prices output prices will fall and thefalling price level restores output demand to its previous level However if outputprices are inflexible and do not adjust downward in response to a reduced demandfor output firms will respond by reducing production Reduced production willlead to a lower level of labor demand and thus a fall in employment Further labordemand will no longer depend upon the real wage but will be determined by whatcan be sold in the output market In short if output prices are inflexible then theoverall level of demand for goods and services is paramount in determining thelevel of employment
Neoclassical model 123
A second modification of the neoclassical model is to assume that output pricesare flexible but money wages are not If wages are ldquostickyrdquo relative to outputprices then changes in the price level will alter the real wage For instance inthe late 1970s high inflation rates led workers and employers in certain industriesto bargain for long-term contracts with a high rate of growth of the money wageThe high expected inflation did not materialize in the early 1980s The lower rateof inflation with no change in the rate of increase in wages meant a rise in thereal wage The resulting fall in the demand for labor led to lower employmentand output Given that wages are not perfectly flexible (eg ldquomultiperiod laborcontracts without complete indexationrdquo) output prices below that anticipated whenlabor contracts are signed lead to a fall in output and employment With inflexiblemoney wages the aggregate supply equation includes the price level of output asa determinant of output supply
Let us consider one more modification of the neoclassical model Suppose thatworkers have incomplete information on output prices and the real wage A firmdetermines its relevant real wage by dividing the money wage it pays its workersby the price it anticipates for the particular product its workers produce On theother hand workers must anticipate prices for a variety of different goods to bepurchased in order to determine their relevant real wage Thus firms may moreaccurately anticipate changes in prices and thus real wages than workers
Now suppose that there is an increase in output prices Given our currentassumption of incomplete information this will not only lead to an increase infirmsrsquo demand for labor and to higher wages as firms anticipate the fall in realwages but may also lead to an increase in the quantity of labor supplied for thefollowing reason Workers who have not anticipated the rise in output priceswill perceive the higher money wages as implying a rise in the real wage andwill increase their supply of labor accordingly As a result equilibrium employ-ment will rise with an increase in output prices Once again the aggregate supplyequation will incorporate the current price level as a potential determinant
The illusion model one modification of the neoclassical model
Our first departure from the neoclassical model is to introduce the potential forldquoimperfectrdquo foresight on the part of suppliers at time t concerning the price levelfor period t As a consequence the ldquonotionalrdquo or planned demands made at time tbased on anticipated prices for the period can differ from ldquoeffectiverdquo or realizeddemands andor supplies at the actual prevailing prices Further realized or effec-tive demands now depend on the quantity constraints experienced in other marketsas well as prices That is realized output now becomes a determinant of actualconsumption and money demand on the part of households
Real wage illusion the labor market and aggregate supply
As we have seen equilibrium employment is determined at the start of each periodin the labor market To formally show this let us start with the following statement
124 Neoclassical model
of equilibrium in the labor market in terms of a money wage wt and level ofemployment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpe
t ) minus Nt = 0
A critical aspect of the above is the fact that suppliers ndash in particular suppliers oflabor ndash may not correctly anticipate the price level that will exist with respect tooutput In particular we let pe
t denote suppliersrsquo expectation formed at the start ofperiod t of the price level for period t We assume this expectation is held withsubjective certainty so that we may express the real wage expected by supplierssimply by wtpe
t It is this anticipated real wage not the realized real wage wtpt that appears in the labor supply function
Firms on the other hand are presumed to correctly forecast the actual realwage9 Note that we retain the presumption that the money wage adjusts to clearthe labor market Similarly we implicitly have assumed that the price level adjuststo clear the output market such that firms are price-takers in the output marketand thus labor demand depends on the real wage rather than on a sales constraint
Initially let us assume that householdsrsquo anticipated level of prices is correctThat is we start with a money wage and level of employment consistent with theneoclassical model But we now assume that any change in the price level willnot be fully anticipated In particular we assume that
pet = g( pt) 1 gt gprime ge 0
The fact that gprime lt 1 implies imperfect foresight at time t on the part of householdsconcerning the price of output for period t If gprime gt 0 it indicates that householdsto some extent but not completely (gprime lt 1) anticipate changes in the equilibriumlevel of prices that will prevail for period t
Totally differentiating the above two equations representing equilibrium in thelabor market with respect to wt Nt pt and noting our prior assumption that pe
t = ptinitially one obtains[minus(partN d
t part(wtpt))pt(partN s
t part(wtpet ))pt
minus1minus1
] [dwtdNt
]=[
(partN dt part(wtpt))wtdpt( pt)
2
(partN st part(wtpe
t ))wtgprimedpt( pt)2
]
Applying Cramerrsquos rule gives
dNt
dpt= (partN d
t part(wtpt))(partN st part(wtpe
t ))(wt( pt)2)(gprime minus 1)
minus(partN dt part(wtpt))pt + (partN s
t part(wtpet ))pe
t
gt 0
dwt
dpt= minuspartN d
t part(wtpt) + (partN st part(wtpe
t ))gprime
minuspartN dt part(wtpt) + partN s
t part(wtpet )
wt
ptgt 0
Note that if gprime = 1 then we have the standard neoclassical result that dNtdpt = 0and dwtwt = dptpt so that a change in the price level results in no change in
Neoclassical model 125
either employment or the real wage However with gprime lt 1 we have that anincrease in the price level leads to a rise in employment and an increase in themoney wage less than proportional to the increase in the price level such thatthe real wage falls (ie dptpt gt dwtwt gt 0) Given the aggregate produc-tion function yt = f (Nt K) we thus have an ldquoaggregate supply functionrdquo ofthe form
yt = yt( pt minus pet K ) (81prime)
so that aggregate supply depends directly on the difference between the price levelfor period t and the price level anticipated by suppliers at time t
Consider the above findings with respect to the labor market and suppose thatthere is an increase in pt Given perfect foresight on the part of firms at time t thelabor demand curve shifts up vertically so that at the higher money wage associatedwith the same real wage demand would be the same However given 1 gt gprime ge 0the vertical shift upward in the supply curve is less than this with the result thatequilibrium employment rises as the money wage rises by proportionately lessthan the rise in prices
In undergraduate textbooks the fact that changes in the price of output can nowaffect real output is shown in ( pt yt) by an upward-sloping ldquoaggregate supplycurverdquo as in Figure 81 Recall that in the neoclassical model the aggregate supplycurve is vertical In either model such a curve summarizes the underlying eventsin the labor market
The above character of the money illusion model is sometimes said to reflect theldquonatural rate hypothesisrdquo The natural rate hypothesis posits that fully anticipatedincreases in prices have no effect on the rate of real economic activity ndash specificallyreal output employment and thus unemployment Thus we will refer to the abovemodel as a static version of a ldquonatural rate modelrdquo
p
y
Figure 81 Upward-sloping aggregate supply
126 Neoclassical model
Equilibrium aggregate supply and demand
An important feature of macroeconomic theories is that to a large extent they aredistinguished by their different treatment of labor markets What this means isthat the aggregate demand side is typical of macroeconomic models Recall thatthe aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) Thus the equilibrium output price level and interest rate aregiven by equations (81prime) (82) and (83) Equation (81prime) is the aggregate supplyequation of a natural rate model (82) is the ldquoISrdquo equation depicting equilibriumbetween output demand and production and (83) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market
At this point we will simplify equations (82) and (83) by removing the realbalance effect representing them as follows
cdt (rt πe
t At yt) + I dnt(m
et + δ K) + δK + ψ(I d
t ) minus yt = 0 (82prime)
Ldt (rt πe
t At yt) minus Mpt = 0 (83prime)
In our model this implies that
partnet Adt part(Mpt) = 1
As we will see one justification for this form is if real money balances are notpart of household wealth which can be the case when we introduce depositoryinstitutions into the analysis In the meantime the above assumption makes theanalysis not only simpler but also more in line with traditional macroeconomicanalysis
Graphically the equilibrium price level and output can be shown using theaggregate demand and supply curves In Figure 82 the equilibrium output andprice level are thus given by plowast
t and ylowastt Looking at what underlies these curves
we can then infer the changes in money wages and employment (specifically froman analysis of the labor market that underlies the aggregate supply curve) as wellas changes in the interest rate and the components of investment and consumptioncomponents of output demand (specifically from an analysis of the output andmoney markets that underlie the aggregate demand curve)
Money supply change comparative statics for a naturalrate model
Collecting the aggregate supply IS and LM equations for the natural rate modelunder consideration we have (81prime) (82prime) and (83prime)
These three equations determine the equilibrium output the price level andthe interest rate Substituting (81prime) into (82prime) and (83prime) in order to focus on the
Neoclassical model 127
p
y
p
y
Aggregate supply
Aggregate demand
Figure 82 Macroeconomic equilibrium with upward-sloping supply
determination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form10
[ minus(partcdparty minus 1)partyp(partLdparty)partypartp + Mp2
partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + c1 + ψ prime)partI dpartm11 The term partypartp gt 0 reflectsthe direct effect of the price level on output as implied by the aggregate supplyequation (81prime)12 It is important to note that the equilibrium condition with respectto the labor market is incorporated into the above analysis in the form of thisaggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp)minus 1)(1p)
(partcdparty minus 1)partypartp(partLdpartr)minus((partLdparty)(partypartp)+ Mp2)(partydpartr)
lt 0
As you can see we no longer have drdM = 0 Further letting x denote thedenominator for the expression for dp we have
dp
p= dM
M
1
x
128 Neoclassical model
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater13
Let us now consider the effect of the money supply shock on other variablesFrom our analysis of the labor market we know that
dp
pgt
dw
wgt 0
so that while the money wage rises the real wage falls We also know from thelabor market that the rise in the price level leads to higher employment and thusan increase in output From the demand functions we can derive the effects of thechange in the money supply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Graphically one can show the effect of the money supply shock in terms ofthe aggregate demand and aggregate supply curves Note that this exercise simplyinvolves a shift out of the aggregate demand curve
The natural rate hypothesis and expectation formation a preview
An important feature of the above analysis one already noted is that real activity ndashin particular employment output and unemployment ndash changes only to the extentthat price changes are not fully anticipated As we have just seen this ldquonatural ratehypothesisrdquo introduces the logical foundations for a monetary change to have realeffects
However note that monetary changes have real effects only to the extent that theresulting changes in the price level are not fully anticipated To understand whenthis might occur we first have to indicate why individuals may err in their formationof expectations concerning the price level The Lucas model suggests one way ofexplaining errors in forecasts such that suppliers only partially anticipate a changein the price level (ie 1 gt gprime ge 0) This model introduces the assumption ofrational expectations
Combining rational expectations with the natural rate hypothesis results in a verypowerful statement concerning monetary policy which has so far not been madeclear If the actions of monetary authorities are predictable under the presumptionof rational expectations individuals will correctly predict the consequences onprices The result in the context of a natural rate model is that such predictable
Neoclassical model 129
monetary changes will have no real effects In the deterministic world we are backto the neoclassical model The analysis above then refers only to an ldquounexpectedrdquoincrease in the money supply or to monetary ldquosurprisesrdquo Only such random shocksto the money supply will have real effects
Conclusion
A formal neoclassical model of the macroeconomy has been introduced and fullydeveloped The issues of money neutrality and money illusion have been discussedand it has been seen that money supply changes have no effect on real economicactivity when the assumptions of the neoclassical model hold However a numberof issues have been raised namely the existence of a natural rate and the potentialeffect of unanticipated money supply changes on economic activity
9 The ldquoKeynesian modelrdquo withfixed money wageModifying the neoclassical model
Introduction
The first modification of the neoclassical model is presented in this chapterTo begin we introduce the very realistic assumption that nominal wages are fixedat least for a period of time The ramifications of this change in the model aredeveloped in the context of the aggregate supply and demand model As with theneoclassical model we perform a comparative statics exercise in which the mone-tary authority changes the money supply and we trace out the effects of this actionon the economic aggregates in the model The model is then made slightly morecomplete and issues associated with sticky wages and the natural rate hypothesisare discussed We introduce the concept of rational expectations and the first ldquoover-lappingrdquo model and show that the Keynesian model has important implicationsfor the conduct of monetary policy
The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
In the standard neoclassical model it is assumed that prices adjust in all marketsto equate demand and supply With respect to the labor market this implies a spotmarket at the start of each period in which one-period labor contracts are enteredinto and an associated one-period wage set Yet employment contracts are likelyto be multiperiod in the presence of hiring and training costs That is to minimizehiring and training costs firms seek long-term relationships with their employees
Firms promote long-term relationships with their employees by offering higherwages to their experienced workers As a consequence long-time employeesbecome attached or ldquoloyalrdquo to their employers since the wages they receive aregreater than those that other firms would offer them In essence employers aresharing the returns to their hiring and training investment with their workers inorder to reduce the number who quit A long-term attachment of workers to par-ticular firms could also stem from the high cost to workers of finding alternativeemployment as obtaining such employment means that workers must generallyinterview various employers visit employment agencies and spend valuable timesimply waiting for decisions on job applications to be made
Keynesian model 131
Given long-term employment contracts between firms and their workers wagesare typically specified for extended periods of time These long-term wage agree-ments are sometimes explicit as with many labor union contracts1 In other casesonly an implicit understanding exists on the wages that a firm will pay its employeesover some extended period of time If these contracts or understandings specifywages in money terms and if modifying these agreements is costly then thereexists an inherent inflexibility in money wages ndash that is there are ldquostickyrdquo wages2
This assumption of ldquostickyrdquo nominal wages is often viewed as the critical aspectof what has been termed the ldquoKeynesianrdquo macroeconomic model
If money wages are ldquostickyrdquo relative to prices then changes in the price levelwill alter the real wage For instance in the late 1970s high inflation rates ledworkers and employers in certain industries to bargain for long-term contractswith a high rate of growth of the money wage The high expected inflation did notmaterialize in the early 1980s The lower rate of inflation with no change in therate of increase in wages meant a rise in the real wage The resulting fall in thedemand for labor led to lower employment and output In this section we formallydevelop these results in the context of a static neoclassical macroeconomic modelwith the additional assumption of a fixed money wage The subsequent section thendevelops a linear rational expectations version of this model in which overlappingmultiperiod employment contracts introduce an element of nominal wage rigidity
Fixed money wage the labor market and aggregate supply
As we have seen in the competitive (spot) labor market of the neoclassical model(or in the Lucas-type macroeconomic model) the money wage and employmentare determined at the start of each period in the labor market If we accept theneoclassical modelrsquos assumption of limited perfect foresight on the part of bothlabor suppliers and firms we have equilibrium in the labor market in terms of amoney wage wt and level of employment Nt determined such that
N dt (wtpt K) minus Nt = 0 (91)
N st (wtpt) minus Nt = 0 (92)
In this case a change in the price level pt leads to an equiproportionate change inthe money wage wt and no change in employment Nt
We now seek to modify this analysis by assuming a fixed money wage wt = wfor period t As Sargent (1987a 21) states
the essential difference between the classical model and the Keynesian modelis the absence from the latter of the classical labor supply curve combined withthe labor market equilibrium condition Since there is one fewer equation inthe Keynesian model it can determine only six endogenous variables insteadof the seven determined in the classical model3 To close the Keynesianmodel the money wage is regarded as an exogenous variable one that atany point in time can be regarded as being given from outside the model
132 Keynesian model
perhaps from the past behavior of itself and other endogenous or exogenousvariables It bears emphasizing that the equation that we have deletedin moving from the classical to the Keynesian model [equation (92)] is acombination of a supply schedule (and) an equilibrium condition Notethat we continue to require that employment satisfy the labor demand schedule[equation (91)]
Sargent goes on to say that
we shall think of the labor supply schedule as being satisfied and helping todetermine the unemployment rate Usually the model is assumed to reachequilibrium in a position satisfying Nt lt N s
t so that there is an excess supplyof labor
Totally differentiating the labor demand condition (91) that determines the levelof employment with respect to the price level and employment we have
minus(partN dt (partwpt))(wp2
t )dpt minus dNt = 0 (93)
which can be rearranged to give
dNtdpt = minus(partN dt part(wpt))(wp2
t ) gt 0 (94)
where the sign reflects the presumption that partN dt part(wtpt) lt 0
In the simple case of no labor adjustment costs labor demand is defined bythe equality between the marginal product of labor and the real wage that ispartf (Nt K)partNt = wpt 4 Differentiating this implies that
[part2ftpartN 2t ]dNt = minus(wp2
t )dpt
or rearranging
dNtdpt = minus(wp2t )[part2ftpartN 2
t ] gt 0 (95)
given diminishing returns to the labor input (ie part2ftpartN 2t lt 0)
Combining the above analysis with the aggregate production function yt =f (Nt K) we thus have the ldquoaggregate supply equationrdquo
yt = yt( ptw K ) with partytpart( ptw) gt 0 (96)
Thus (as in a Lucas-type model) we have an aggregate supply that can dependdirectly on the price level for period t
The above findings can be understood with respect to the labor market Considera decrease in pt Given limited perfect foresight on the part of both firms andhouseholds at time t there is a downward (vertical) shift in labor demand so thatat the lower money wage wlowast
t associated with the same real wage demand would
Keynesian model 133
be the same Similarly the labor supply curve shifts down vertically so that atthis lower money wage wlowast
t labor supply would be the same as well Howevermultiperiod labor contracts fix the money wage at w so that the lower price level(and implied higher real wage) results in a fall in employment (which is nowdemand-determined) and an excess supply of labor
In the opposite case of a rise in the price level that can lead to an excessdemand in the labor market at the fixed money wage the presumption remainsthat employment is demand-determined This presumption reflects the view thatat least temporarily firms can direct workers with whom they have long-termemployment contracts to work overtime or extra shifts which the workers wouldotherwise not volunteer for
The above story provides a rationale for an upward-sloping ldquoaggregate sup-ply curverdquo Both contrast with the neoclassical model in which the aggregatesupply curve is vertical as a fall in the price level results in an equiproportionatefall in the money wage so that the real wage and employment remain unchangedIn the fixed wage model the underlying events in the labor market summarized bythe aggregate supply curve are the change in the real wage and thus labor demandand employment that accompany a price change when the money wage is fixedSuch an aggregate supply curve is upward-sloping
Equilibrium aggregate supply and demand
As we have noted before an important feature of macroeconomic theories is that toa large extent they are distinguished by their different treatment of labor marketsWhat this means is that the aggregate demand side is similar across macroeconomicmodels Recall that the aggregate demand side of macroeconomic models typicallyconsiders the equilibrium conditions of two of the remaining three markets inparticular the output market (reflected by an ldquoISrdquo equation) and the money market(reflected by an ldquoLMrdquo or ldquoportfoliordquo equation) Thus for the Keynesian modelwith fixed money wage the equilibrium output price level and interest rate aregiven by the following three equations
yt = yt( ptw K ) (96)
cdt (rt πe
t+1 At yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (97)
Ldt (rt πe
t+1 At yt) minus Mpt = 0 (98)
Equation (96) is the aggregate supply equation of a Keynesian model with fixedmoney wage (97) is the ldquoISrdquo equation depicting equilibrium between outputdemand and production and equation (98) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market Note that we havesimplified the IS and LM equations by removing the real balance effect for con-sumption demand and money demand5 Equations (97) and (98) can be combinedto eliminate the interest rate The resulting equation is referred to as the ldquoaggregatedemand equationrdquo
134 Keynesian model
The equilibrium price level and output ( plowastt and ylowast
t ) can be shown graphicallyusing aggregate demand and supply curves Looking at what underlies thesecurves we can then infer the change in employment (specifically from an analysisof the labor market that underlies the aggregate supply curve) as well as changes inthe interest rate and the investment and consumption components of output demand(specifically from an analysis of the output and money markets that underlie theaggregate demand curve)
A change in the money supply the comparative statics for theKeynesian model
The aggregate supply IS and LM equations (96)ndash(98) for the static Keynesianmodel under consideration determine the equilibrium output the price level andthe interest rate Substituting (96) into (97) and (98) in order to focus on thedetermination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form6[ minus(partcdparty minus 1)partypartp
(partLdparty)(partypartp) + Mp2partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm7 The term partypartp gt 0 reflects thedirect effect on the price level as implied by the aggregate supply equation (96)8
Note that the equilibrium condition with respect to the labor market is incorporatedinto the analysis in the form of this aggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp) minus 1)p
(partcdparty minus 1)partypartp(partLdpartr) minus (partLdparty)(partypartp) + (Mp2)(partydpartr)
lt 0
In contrast to the neoclassical model we no longer have drdM = 0 Furtherletting x denote the denominator for the expression for dp we have
dp
p= dM
M
1
x
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Keynesian model 135
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater
Consider now the effect of the money supply shock on other variables Fromour analysis of the labor market we know that w is fixed so that the increase inthe price level means a fall in the real wage and thus increased labor demandemployment and thus an increase in output From the demand functions for con-sumption and investment we can derive the effects of the change in the moneysupply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
and
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Note that the effect of the money supply shock is to increase the aggregate demandwhile not affecting aggregate supply
Sticky wages and the natural rate hypothesis
Expectations play an important role in the two modifications of the neoclassicalmodel In the modification with fixed wages the level at which negotiators fixfuture money wages depends on the expectation formed when wages were set con-cerning future prices The higher the expectation of future prices the higher thelevel of wages set in the labor agreements between workers and firms The pre-sumption is that workers and firms attempt to set future wages at their anticipatedmarket-clearing levels Associated with these anticipated market-clearing wagesis a particular real wage a natural rate of unemployment and a full employmentor natural rate of output
If price expectations turn out to be incorrect then output will vary from itsnatural rate For instance a shock that causes actual output prices to fall belowthose expected means that the money wage is fixed at a level that is too highfor full employment Consequently employment and output fall below the fullemployment level
In the typical Lucas-type model firms and workers set wages for the currentperiod based on incomplete information as well As we saw if suppliersrsquo expec-tations are incorrect then output will deviate from the full employment level Forexample a shock that causes actual output prices to fall below those expectedmeans lower employment and output as workers mistake lower money wages forlower real wages In fact higher real wages accompany the lower price level andthis is the source of the reduced demand for labor and employment
The two modifications of the neoclassical model have a second common ele-ment Both predict that a macroeconomic demand shock ultimately affects onlythe level of prices Even though money wages in the Keynesian model are fixed
136 Keynesian model
for the current period we know that money wages are not fixed forever Over timelabor agreements are renegotiated and money wages change to once again equatethe ldquoexpectedrdquo future demand for and supply of labor Over time in the absenceof further shocks the economy would thus tend to behave as neoclassical analysispredicts money wages and output prices would adjust to restore equilibrium tothe various markets in the economy
While the Keynesian model with fixed money wage admits the tendency foroutput to approach its natural level over time it does introduce a potential rolefor monetary policy to play in dampening fluctuations in output In so doing itchallenges the policy ineffectiveness view of Sargent and Wallace The best-knownexamples employing the Keynesian model to demonstrate the potential stabilizingpowers of monetary policy under rational expectations are the dual papers byFischer (1977) and Phelps and Taylor (1977)
A linear rational expectations version of the Keynesian model
Counting the above discussion we have so far considered three different modelsthat can be used to assess monetary policy One model is along the lines of theneoclassical model with limited perfect foresight Since this view of the economypredicts the neutrality of money a role for monetary policy either as an instigatorof output fluctuations or as an instrument to dampen output fluctuations is missingAs Mankiw (1987) suggests people who adopt this model ldquoview economic fluctu-ations through the lens of real business cycle theoryrdquo in which output fluctuationsare traced to ldquosupply-siderdquo disturbances
As Mankiw goes on to note however ldquothere are surely readers who believe thatmonetary policy has real short-run effects because of temporary misperceptions ornominal rigiditiesrdquo Mankiw is referring to individuals who adopt either the Lucas-type model or the ldquoKeynesianrdquo fixed money wage model9 Either one as we haveseen introduces a role for monetary policy as an instigator of output fluctuationsHowever these two models do differ as to whether monetary policy can be aninstrument to dampen output fluctuations in the context of rational expectations
A Lucas-type model built on ldquotemporary misperceptionsrdquo when coupled withrational expectations leaves little if any room for countercyclical monetary policyIn fact following the analysis of Sargent and Wallace it can be shown that whilerandom monetary shocks can impact output deterministic monetary policy basedon a set of policy rules is ineffective in counteracting fluctuations in output givenrational expectation10 Further attempts at discretionary monetary policy in thiscontext only result in a suboptimal (ldquotoo highrdquo) rate of inflation (Barro and Gordon1983) Thus in this model there remains a ldquostochasticrdquo neutrality of money
As the analysis in the previous section suggests however a ldquoKeynesian-typerdquomodel built on ldquonominal rigiditiesrdquo might introduce a role for monetary policyin stabilizing output even in the context of rational expectations The reasoningfor this is that wages (or as we will see later prices) can be set prior to thereceipt of information by the monetary authority that enters into the money supplyrule In this context as Phelps and Taylor (1977) state ldquoeven systematic and
Keynesian model 137
correctly anticipated policy can make a difference for the stability of output in arational expectations model with sticky prices and wagesrdquo11 Below we considerone example of such a model that counters the Sargent and Wallace ineffectivenessproposition a model proposed by Fischer that assumes ldquostickyrdquo wages
The supply equation with overlapping two-period labor contracts
The Lucas aggregate supply equation with adjustment costs can be expressed as
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (99)
where Yt = ln yt minus ln yn denotes difference between the logarithm of output forperiod t and the logarithm of the natural rate of output (which we have normalizedto equal zero) Pt = ln Pt is the logarithm of the price level for period t Etminus1Pt isthe expectation of the logarithm of the price level for period t using all informationavailable up to the end of period t minus 1 (at time t) and γ θ is a positive constant12
The supply of output as expressed by equation (99) satisfies the conditionthat employment equals labor demand (92) The fact that a higher price level Ptinduces firms to increase employment and thus output reflects the underlying lowerequilibrium real wage that accompanies the higher price level when suppliers donot anticipate the higher price level Thus we could express (99) in the form13
Yt = (Pt minus Wt + φ) + λYtminus1 (99prime)
where Wt is the logarithm of the equilibrium nominal wage for period t Theterm φ in (99prime) is defined such that if Etminus1Pt = Pt then the resulting log ofthe equilibrium real wage (ie ln(wtpt)) equals φ Ignoring the lagged outputterm in equation (99prime) this equilibrium real wage is the one associated with thenatural level of output and employment14 We will follow others and assume forconvenience that φ = 0 implying an equilibrium real wage with no surprisesequal to one
According to (99prime) if an increase in the price level is accompanied by a lessthan proportionate increase in the equilibrium money wage Pt minusWt rises (the realwage falls) and employment and output increase In the Lucas-type model such anevent occurs if suppliers do not forecast the price increase However Sargent andWallacersquos ineffectiveness proposition eliminates deterministic monetary policyrules as a source of such a price rise not matched by a similar rise in wages if(a) wages are set each period to equate labor demand and supply and (b) rationalexpectations are assumed In this case individuals and the monetary authoritiesare assumed to have a common set of information based on events up to the endof period t minus 1 (at time t) Individuals are also privy to the monetary policy ruleand they know the structure of the economy Thus they have knowledge of thedeterministic monetary policy to be followed during period t and its effect on theprice level for period t as predicted by the model Assuming flexible wages thispredicted effect of monetary policy on prices will be factored into the setting of themoney wage for period t As a consequence such deterministic monetary policy
138 Keynesian model
cannot change the real wage and thus leaves employment and output unaffectedas well
Obviously this chain of reasoning breaks down if money wages for period twere set prior to time t For example this policy ineffectiveness doctrine disap-pears if some wages are set at the end of period t minus 2 for period t In this casenew information that arrives during period t minus 1 can be incorporated by the mon-etary authorities into their money supply rule Even with rational expectationsthe implications of this cannot be used by individuals to adjust the money wagefor period t since by assumption the money wage is fixed Thus deterministicmonetary policy based on information revealed during period t minus 1 can alter thereal wage employment and output
To formally develop this potential stabilizing role of monetary policy in a moreldquoelegantrdquo fashion let us consider Fischerrsquos model The model disaggregates theeconomy into two sectors and assumes that the sectors alternate in setting multi-period employment contracts that fix nominal wages In particular ldquosuppose thatall labor contracts run for two periods and that the contract drawn up at the endof period t minus 2 specifies nominal wages for periods t minus 1 and t [Assume] thatcontracts are drawn up to maintain constancy of the real wagerdquo (Fischer 1977198) In other words
tminusiWt = EtminusiPt i = 1 2 (910)
where tminusiWt is the logarithm of the wage set at the end of period tminus i for period t15
The idea embodied in (910) that wages are set for more than one period iscritical to Fischerrsquos finding It essentially means that in any period half of thelabor contracts have fixed money wages16 Given that the wage is predeterminedfor each firm the aggregate supply equation is given by
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911)
where ut is a stochastic ldquorealrdquo disturbance or ldquosupply shockrdquo that impinges onproduction in each period17 Substituting (910) into (911) we can rewrite theaggregate supply equation as
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911prime)
A complete model except for specifying the source of expectations
Equation (911prime) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To close the model we require LM and ISequations The explicit derivation of these is left to the next chapter but let usassume for the time being that the LM equation is given by
mt minus Pt = α1Yt minus α2 middot rt minus εt
Keynesian model 139
and the IS equation by
Yt = Xt minus β1(rt minus πet+1) + ut
Here mt = ln Mt mt = mt + εt such that the deterministic component mt is setby government authorities (ie the monetary authority) according to a monetaryrule Further rt minus πe
t+1 represents the expected real rate of interest Xt denotesa vector of exogenous variables that affect output demand εt and ut are randomterms associated with output demand and money supply respectively and assumedindependent (ie E(εtut) = 0) and well behaved
Combining the LM and IS equations to eliminate the interest rate rt we obtain
Yt = Xt + ut minus (β1α2)(minusmt minus εt + Pt minus α1Yt) + β1πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = [α2(α2 + α1β1)][Xt + ut + (β1α2)(mt + εt minus Pt) + β1πet+1] (912)
Note that if α2 = 0 (changes in the interest rate do not affect money demand) andα1 = 1 (the income elasticity of real money demand is one) then this simplifies towhat Fischer refers to as a ldquovelocity equationrdquo
Yt = mt minus Pt + vt (913)
where vt = εt is now to be interpreted as a money demand disturbance termaffecting the ldquovelocityrdquo of money18
To see why (913) is called a ldquovelocity equationrdquo note that the assumption ofmoney demand being independent of the interest rate allows us to capture therelationship between income and the price level summarized by the aggregatedemand equation by looking solely at the LM equation (ie neglecting the ISequation) In particular if we assume that real money demand can be expressedby the equation
Ldt = yt(exp(minusvt))
then equating real money demand to real money supply (Mtpt) gives us
yt(exp(minusvt)) = Mtpt (914)
Taking the logarithm of the equilibrium condition with respect to the money market(914) we obtain
ln yt minus vt = ln Mt minus ln pt
Given Pt = ln pt Yt = ln yt and mt = ln(Mt) this is simply equation (913)
140 Keynesian model
Note that equilibrium velocity is defined as the ratio of nominal output to themoney supply Thus rearranging (914) we have that
Equilibrium velocity equiv ptyt
Mt= exp(vt)
which explains the interpretation of vt as a ldquovelocityrdquo disturbance Fischer assumesthat vt has a zero mean so that expected velocity is one A vt above zero meansa decrease in real money demand relative to real output and thus an increase inequilibrium velocity As (914) makes clear for a given Mt a higher vt implies ahigher pt andor a higher yt to maintain equilibrium with respect to the demandfor and supply of money
As Fischer states (913) is ldquothe simplest way of taking demand consid-erations into accountrdquo In sum then the macroeconomics model considered byFischer is given by the aggregate supply equation (911prime) and the aggregate demandequation (913)
Combining (911prime) and (913) to eliminate Yt we have
12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut = mt minus Pt + vt
This can be solved to give the reduced-form equation for the price level Pt
Pt = 12
[(12 Etminus1Pt + 1
2 Etminus2Pt
)minus ut + mt + vt
] (915)
Combining (911prime) and (913) to eliminate Pt we have
Yt = 12 (mt minus Yt + vt minus Etminus1Pt) + 1
2 (mt minus Yt + vt minus Etminus2Pt) + ut
This can be solved for the reduced-form equation for output Yt
Yt = 12
[mt + vt + ut minus 1
2 Etminus1Pt minus 12 Etminus2Pt
] (916)
According to equations (915) and (916) an increase in the ldquorealrdquo disturbanceterm ut leads to higher equilibrium output and a reduced price level Intuitivelythis corresponds to a shift to the right in the ldquoaggregate supply curverdquo On the otherhand an increase in the ldquovelocityrdquo disturbance term vt corresponds to a shift to theright in the ldquoaggregate demand curverdquo and thus leads to a higher output and pricelevel given an upward-sloping aggregate supply curve Note that an increase in vtmeans a lower real money demand at each level of income The shift in the aggre-gate demand curve reflects the fact that a higher price level andor higher output isrequired to restore equilibrium in the money commodity and financial markets
If expectations can be taken as exogenous with respect to money supply changesthen (916) indicates that money supply changes can affect output But this wasalso the case for a Lucas-type model The next step is thus to see what happens
Keynesian model 141
when we assume rational expectations As will become clear even with rationalexpectations expectations formed at the end of period t minus 2 can be viewed asexogenous with respect to monetary changes planned for period t based on infor-mation obtained during period t minus 1 Thus monetary policy has the potential tooffset the persistent effect of disturbances that originate during period t minus 1
Introducing rational expectations
Let us now assume that individuals form their expectations Etminus1Pt and Etminus2Ptldquorationallyrdquo in that
Etminus2Pt = E(Pt |tminus2) (917)
Etminus1Pt = E(Pt |tminus1) (918)
indicating that EtminusiPt is the mathematical expectation of Pt conditional on theinformation set tminusi which is all information available at the end of period t minus ii = 1 2 Taking the expectation of (915) at the end of period t minus 2 we thus have
Etminus2Pt = Etminus2
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (919)
Note that Etminus2Etminus1Pt = Etminus2Pt Thus (919) becomes
Etminus2Pt = Etminus2minusut + vt + mt (920)
Taking the expectation of (915) at the end of period t minus1 (at time t) we then have
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (921)
Substituting (920) into (921) gives
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2minusut + vt + mt minus ut + vt + mt
]
(922)
Rearranging
34 Etminus1Pt = 1
4 Etminus2minusut + vt + mt + 12 Etminus1minusut + vt + mt
or
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + mt (923)
which is Fischerrsquos (1977) equation (16)19
142 Keynesian model
Let the money supply be determined by the simple linear rule
mt = a1utminus1 + b1vtminus1
Since mt is a function only of information available up to the end of period t minus 1(at time t) Etminus1mt = mt Accordingly (923) can be written as
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + 23 mt (924)
Substituting (920) and (924) into the reduced-form equation for output (916)we obtain
Yt = 12 [ut + vt + mt] minus 1
4
[13 Etminus2minusut + vt + mt
+ 23 Etminus1minusut + vt + 2
3 mt
]minus 1
4 [Etminus2minusut + vt + mt](925)
Equation (925) simplifies to
Yt = 13 (mt minus Etminus2mt) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt
(926)
which is Fischerrsquos (1977) equation (18)As Fischer (1977 196) notes
disturbances aside this very simple macro model would be assumed in equi-librium to have the real wage set at its full employment level would imply theneutrality of money and would obviously have no role for monetary policyin affecting the level of output A potential role for monetary policy is createdby the presence of the disturbances ut and vt that are assumed to affect thelevel of output each period Each of the disturbances is assumed to follow afirst-order autoregressive scheme
ut = ρ1 middot utminus1 + εt where |ρ1| lt 1 (927)
vt = ρ2 middot vtminus1 + ηt where |ρ2| lt 1 (928)
where εt and ηt are mutually and serially uncorrelated stochastic terms withexpectation zero and finite variances σ 2
e and σ 2n respectively
Given equations (927) and (928) and the money supply rule
mt = a1utminus1 + b1vtminus1 (929)
Etminus2mt = a1ρ1utminus2 + b1ρ2vtminus2 (930)
Keynesian model 143
so that
mt minus Etminus2mt = a1utminus1 + b1vtminus1 minus [a1ρ1utminus2 + b1ρ2vtminus2]= a1εtminus1 + b1ηtminus1
(931)
According to (931)
the difference between the actual money stock in period t and that stockas predicted two periods earlier arises from the reactions of the monetaryauthority to the disturbances εtminus1 and ηtminus1 occurring in the interim It isprecisely these disturbances that cannot influence the nominal wage for thesecond period of wage contracts entered into at time t minus 2
(Fischer 1977 199)
Substituting (931) into (926)
Yt = 13 (a1εtminus1 + b1ηtminus1) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt (932)
From (927) and (928) we know that
ut + vt = (ρ1utminus1 + εt) + (ρ2vtminus1 + ηt)
= (ρ1utminus1 + εt) + (ρ22vtminus2 + ρ2ηtminus1 + ηt)
since by substitution vt = ρ22vtminus2 + ρ2ηtminus1 + ηt we also have that
Etminus1ut minus vt = ρ1utminus1 minus ρ2vtminus1 = ρ1utminus1 minus ρ22vtminus2 minus ρ2ηtminus1
and
Etminus2ut minus vt = ρ21utminus2 minus ρ2
2vtminus2
Thus we can rewrite (932) as
Yt = 12 (εtminus1 + ηtminus1) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
(933)
which is Fischerrsquos (1977) equation (21)20
Fischer (1977 199) notes that
before we examine the variance of output as a function of the parameters a1and b1 it is worth explaining why the values of those parameters affect thebehavior of output even when the parameters are fully known The essential
144 Keynesian model
reason is that between the time the two-year contract is drawn up and the lastyear of operation of that contract there is time for the monetary authorityto react to new information about recent economic disturbances Given thenegotiated second-period nominal wage the way the monetary authority reactsto disturbances will affect the real wage for the second period of the contractand thus output
Optimal monetary policy rules the effectiveness of policy
As in our discussion of Sargent and Wallace let us presume that the goal of themonetary authority focuses solely on output In particular suppose that the mone-tary authority desires to set the money supply in order to minimize the fluctuationin the log of output around some desired level Then the objective can be expressedas to
min Etminus1(Yt minus Y lowast)2
Let us assume that Y lowast = Yn = 0 so that the objective becomes to
min Etminus1(Yt)2 (934)
From (933) we have that
(Yt)2 =
[12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21 utminus2
]times[
12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
]
(935)
Note that E(εi) = E(ηi) = 0 E(ε2i ) = σ 2
e E(η2i ) = σ 2
n and that our independenceassumptions imply that E(εiηi) = 0 and for i = s E(εiεs) = 0 and E(εiηs) = 0Thus substituting (935) into (934) we have the following explicit form for theobjective
min Etminus1(Yt)2 = σ 2
e
[14 + 1
9 (a1 + 2ρ1)2]
+ (ρ21utminus2)
2
+ σ 2n
[14 + 1
9 (b1 + ρ2)2]
(936)
Given (936) the optimal monetary rule is to choose values for a1 and b1 suchthat
a1 = minus2ρ1 b1 = minusρ2 (937)
The above findings correspond to Fischerrsquos (1977) equation (23)21
Keynesian model 145
As Fischer (1977 200) states
to interpret the monetary rule examine [equation (933)] It can be seen therethat the level of output is affected by current disturbances (εt + ηt) that can-not be offset by monetary policy by disturbances (εtminus1 and ηtminus1) that haveoccurred since the signing of the older of the existing labor contracts andby a lagged real disturbance utminus2 The disturbances εtminus1 and ηtminus1 can bewholly offset by monetary policy and that is precisely what equation [(937)]indicates The utminus2 disturbance on the other hand was known when the olderlabor contract was drawn up and cannot be offset by monetary policy becauseit is taken into account in wage setting Note however that the stabilizationis achieved by affecting the real wage of those in the second year of laborcontracts and thus should not be expected to be available to attain arbitrarylevels of output ndash the use of too active a policy would lead to a change in thestructure of contracts
[A] more general interpretation of the monetary rule is to accommodatereal disturbances that tend to increase the price level and to counteract nominaldisturbances that tend to increase the price level
Fischer concludes by noting that
given a structure of contracts there is some room for maneuver by the mon-etary authorities ndash which is to say that their policies can though will notnecessarily be stabilizing
Conclusion
This chapter presented the sticky money wage or Keynesian model of the macro-economy We find that in contrast to the neoclassical model changes in the pricelevel affect real variables and the amount of labor employed in the economy Thusmoney has real effects The development of an upward-sloping aggregate supplycurve has dramatic implications for the conduct of monetary policy However it isshown that the expectations of agents in the economy also play an important rolein whether or not monetary policy is effective
10 The Lucas model
Introduction
As we have seen anticipated changes in prices have no impact on real vari-ables in the neoclassical model A key element of this model is the ldquoessentialpresumption that nominal output is determined on the aggregate demand sideof the economy with the division into real output and the price level largely depen-dent on the behavior of suppliers of labor and goodsrdquo (Lucas 1973) As such thismodel implies no link between price changes and real output
We have also seen how the natural rate model allows one to introduce a linkbetween unanticipated price changes and real output The seminal paper by Lucas(1973) formally develops a more complete model of the potential for ldquoshort-runsupply behavior (resulting) from suppliersrsquo lack of information on some of theprices relevant to their decisionsrdquo Lucasrsquos explanation of a tradeoff between unem-ployment and inflation ldquois that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relative pricechangesrdquo
As with the ldquoillusion modelrdquo Lucas postulates ldquorational agents whose deci-sions depend on relative prices only placed in an economic setting where theycannot distinguish relative from general price movementsrdquo That is we retain thehypothesis that prices adjust to clear markets
Lucas adds to the simple (static) natural rate model so far discussed by explicitlymodeling the source of forecast errors In doing so he assumes that ldquoinferenceson these relevant unobserved prices are made optimally (or lsquorationallyrsquo) in lightof the stochastic nature of the economyrdquo Below we outline Lucasrsquos model1
The ldquoislandrdquo paradigm
Lucasrsquos model begins by disaggregating the economy into a number of what havebeen called ldquosectorsrdquo ldquomarketsrdquo or ldquoislandsrdquo As Lucas says ldquowe imagine sup-pliers as located in a large number of scattered competitive markets Demand forgoods in each period is distributed unevenly over markets leading to relative aswell as general price movementsrdquo In terms of our previous analysis one couldthink of each of the n sectors in the economy as inhabited by firms producing the
Lucas model 147
ith commodity (i = 1 n) Associated with each sector or ldquoislandrdquo is a set ofworkers and thus a labor market
For firms producing commodity i in period t the key relative price is the priceof their output relative to the wage paid in sector i or pitwit where pit is themoney price of commodity i (produced in sector i) and wit is the money wage forlabor in sector i It is assumed that individuals (firms and workers) in sector irsquoslabor market know the money wage and price of commodity i For labor suppliersin sector i however the key relative price is witpt where pt is the economy-wideprice level reflecting the fact that suppliers plan to use money wages to purchasea bundle of goods consisting of all n commodities2
It is this setup of ldquodispersed marketsrdquo and ldquoinformational discrepanciesrdquo thatLucas uses to generate a correlation between price changes and output ndash the famousLucas supply equation
The supply function for a particular sector
The Lucas model assumes a competitive labor market for sector or ldquoislandrdquo i suchthat the equilibrium level of employment and money wage equate market demandand supply With respect to the labor demand function let us start by assumingthe simple CobbndashDouglas production function such that the marginal product oflabor is given by
a(Nit)minusα where a gt 03
The profit-maximizing condition for the representative firm producingcommodity i is to equate the money wage to the marginal product of labor multi-plied by the money price of output The resulting optimal labor demand can thusbe defined by the equation4
wit = pita(N dit )
minusα
Taking the natural log of this equation and rearranging we have
ln N dit = 1
α[ln pit minus ln wit + ln a] (101)
With respect to labor supply let us assume for the moment that the real wage isknown Further let us assume that the labor supply function takes the followinglogarithmic form
ln N sit = 1
βln
wit
pit
where β is a positive constantEmployment contracts entered into at time t in sector i specify the money
wage wit so that element of the real wage is known However if the price level is
148 Lucas model
unknown then the expected real wage based on information available at time t isequal to witEt(1pt) The associated expected logarithm of the real wage is then5
Et ln(witpt) = ln wit minus Et(ln pt)
Given the assumed log-linear labor supply function and ignoring the implicationsof uncertainty for labor supply we thus have
ln N sit =
(1
β
)(ln wit minus Et(ln pt)) (102)
Equilibrium in the labor market for sector i entails a level of employment Nitand money wage wit such that the demand for labor equals the supply In loga-rithmic form and using the specific labor demand and supply functions given byequations (101) and (102) equilibrium requires that the log of the money wageln wit and the log of employment ln Nit be such that
ln Nit = 1
α(ln pit minus ln wit + ln a)
and
ln Nit = 1
β(ln wit minus Et(ln pt))
Substituting the first expression into the second to eliminate the logarithm of themoney wage we have
ln Nit = 1
α(ln pit + ln a) minus 1
α(β ln Nit + Et(ln pt))
which upon rearranging becomes
ln Nit = ln Nni + [1 + (α + β)][ln pit minus Et(ln pt)] (103)
where ln Nni = (ln a)(α + β)Equation (103) indicates that the logarithm of equilibrium employment in the ith
sector and thus the production of commodity i depends directly on the expectationof logarithm of the ratio of the price of commodity i(pit) to the general level ofprices pt The term Nni can be viewed as the ldquonormalrdquo level of employment Notethat we have abstracted from population growth and other factors that would resultin this ldquonormalrdquo level of employment varying across time
Given the assumption of a simple CobbndashDouglas production function of theform yit = (Nit)
1minusα(Ki)α we thus have
ln yit = ln yni + γ [ln pit minus Et(ln pt)] (104)
where γ = (1 minus α)(α + β) and ln yni = (1 minus α) ln Nni + α ln Ki The term yni isdenoted by Lucas as the ldquonormalrdquo level of output in the particular sector or market i
Lucas model 149
under consideration As Lucas (1973 327) states the ldquoquantity supplied in eachmarket will be viewed as the product of a normal (or secular) component commonto all markets and a cyclical component which varies from market to market
The cyclical component varies with perceived relative prices and with its ownlagged valuerdquo Note that for the moment we do not include the lagged value ofoutput in the above supply function One can justify the inclusion of such byassuming adjustment costs
The source of forecasting errors
According to (104) output of commodity i depends critically on suppliersrsquo fore-cast of the log of the general level of prices Et(ln pt) Now consider how sucha forecast may be obtained in a stochastic environment First it is assumed thatagents in sector i ndash in particular suppliers of labor involved in the production ofcommodity i ndash know commodity irsquos price pit However the exact extent to whichany change in the money price of commodity i reflects a change in the overalllevel of money prices as opposed to a change in commodity irsquos price relative toother prices is unknown It is this uncertainty that leads suppliers in sector i tomisinterpret a change in the general price level in terms of a change in a relativeprice
To be concrete suppose Etminus1(ln pt) incorporates all information available atthe end of period t minus 1 The logarithm of the actual price level will vary from thelogarithm of this expected price level to the extent that there are ldquosurprisesrdquo withrespect to the aggregate price level Letting ξt denote this ldquosurpriserdquo for period twe have
ln pt = Etminus1(ln pt) + ξt (105)
We assume that ξt which is that part of the price level that cannot be predictedfrom past data is a normally distributed random variable with zero expectationand variance σ 26
At the start of period t suppliers in market i receive one additional piece ofinformation the logarithm of the price of commodity i ln pit This signal isassumed to contain some information about the logarithm of the overall pricelevel in that
ln pit = ln pt + zit (106)
where zit is a normally distributed random variable with zero mean and varianceσ 2
z Thus using equation (105) to substitute for ln pt
ln pit = Etminus1(ln pt) + ξt + zit (107)
In words the logarithm of the nominal price for the sector ln pit is assumed toinform the supplier of the sum of the current ldquowhite noiserdquo innovations to the
150 Lucas model
relative price process in that sector (zit) and the innovations to aggregate demandand thus the economy-wide level of prices (ξt)7
We can then express the expectation of the logarithm of the price level at time tfor suppliers in sector i given the observed logarithm of the price of commodity iln pit by
Et(ln pt) equiv Eln pt |Etminus1(ln pt) ln pit (108)
Formally we have the joint distribution of two random variables f (ln pit ln pt)where one of them ln pit is known to take a particular value The problem is thebasic one of ldquobivariate regressionrdquo in that we have to determine the conditionalexpectation Eln pt | ln pit namely the ldquoaveragerdquo value of ln pt for the givenvalue of ln pit 8 As we shall see in the next section the resulting expression for theexpected general price level (in logs) given ln pit is observed can be expressedin linear form as
Et(ln pt) = Etminus1(ln pt) + (1 minus θ)(ln pit minus Etminus1(ln pt)) where 0 le θ le 1(109)
A digression on linear regression analysis
Let us assume a linear regression equation that links the observed logarithm of theprice of commodity i to the logarithm of the general level of prices of the form9
Etln pt | ln pit = a0 + a1(ln pit) (1010)
We can express the regression coefficients a0 and a1 in terms of some of the lowermoments of the joint distribution of ln pt and ln pit namely in terms of10
Eln pit = Etminus1(ln pt) (from (107))
Eln pt = Etminus1(ln pt) (from (105))11
Varln pit = σ 2 + σ 2z (from (107))
Cov(ln pit ln pt) = E(minusξt minus zit)(minusξt) = σ 2 + Ezit middot ξtIn general Ezit middot ξt = Cov(zit ξt) + EzitEξt However given that Ezit =Eξt = 0 and given the assumption that zit and ut are independent variables sothat Cov(zit ξt) = 0 we have12
Cov(ln pit ln pt) = σ 2
From (1010) we have that13
E(ln pt | ln pit) equivint
(ln pt)φ(ln pt | ln pit)d ln pt = a0 + a1(ln pit) (1011)
Lucas model 151
where φ(middot) is the conditional density function of ln pt given ln pit If we thenmultiply the expression on both sides of (1011) by the marginal density functionof ln pit denoted by g(ln pit) and integrate on ln pit we obtainintint
(ln pt)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0g(ln pit)d ln pit +int
a1(ln pit)g(ln pit)d ln pit
or
Eln pt = a0 + a1Eln pit (1012)
since φ(ln pt | ln pit)g ln(pit) = f (ln pit ln pt) Had we multiplied the expressionon both sides of (1011) also by ln pit before integrating on ln pit we would haveobtainedintint
(ln pt)(ln pit)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0(ln pit)g(ln pit)d ln pit +int
a1(ln pit)2g(ln pit)d ln pit
or
E(ln pit)(ln pt) = a0Eln pit + a1E(ln pit)2 (1013)
Solving (1012) and (1013) for a0 and a1 and making use of the fact that
E(ln pit)(ln pt) = Cov(ln pit ln pt) + E(ln pit)Eln ptand
E(ln pit)2 = Var(ln pit) + [Eln pit]2
we find that
a0 = Eln pt minus [Cov(ln pit ln pt)Eln pit](Var(ln pit))
a1 = (Cov(ln pit ln pt))(Var(ln pit))
Hence we can write equation (1010) as
Et(ln pt) equiv E(ln pt | ln pit)
= E(ln pt) + [Cov(ln pit ln pt)Var(ln pit)](ln pt minus Eln pit)Substituting in the above expressions for means variance and covariance we havethus derived (109) with
1 minus θ = σ 2(σ 2 + σ 2z )
152 Lucas model
Equation (109) indicates that agentsrsquo rational expectation of the current pricelevel is a ldquolinear least-squares projectionrdquo That is one could rewrite (109) as
Et(ln pt) = θEtminus1(ln pt) + (1 minus θ) ln pit (1014)
where θ = σ 2z (σ 2 + σ 2
z ) To see why this is called ldquolinear least-squaresrdquo notethat we could start with (1014) (the ldquolinearrdquo part of the projection) and thenpick θ to minimize the variance in this forecast or projection of ln pt (the ldquoleast-squaresrdquo part of the projection) In particular substituting in (105) for Etminus1(ln pt)
(ie Etminus1(ln pt) = ln pt minus ξt) and (106) for ln pit (ie ln pit = ln pt + zit)(1014) becomes
Et(ln pt) = ln pt minus θξt + (1 minus θ)zit (1014prime)
The problem of picking θ to minimize the variance of this projection can then beexpressed as14
minθ
Eln pt minus θξt + (1 minus θ)zit minus ln pt2 = θσ 2 + (1 minus θ)σ 2z
Taking the derivative of the above expression with respect to θ setting it equal tozero (ldquoleast squaresrdquo) and solving for θ we verify that θ = σ 2
z (σ 2 + σ 2z )
As Sargent (1987a 442) points out
the parameter θ is the fraction of the conditional variance in ln pit due torelative price variation The larger is this fraction the smaller is the weightplaced on ln pit in revising Etminus1(ln pt) to form Et(ln pt) This makes sensesince the larger is θ the more likely it is that a change in ln pit reflects arelative rather than a general price change15
Equation (1014) can be substituted into the supply function for commodity i(104) to obtain
ln yit = ln yni + γ θ(ln pit minus Etminus1(ln pt)) (1015)
where as before ln yni = (1 minus α)(ln Nni) + α(ln Ki) As noted above if weassumed adjustment costs then a lagged output term could be added to (1015)In this case we would have16
ln yni = ln yni + γ θ [ln pit minus Etminus1(ln pt)] + λ(ln yitminus1 minus ln yni) (1015prime)
If suppliers were able to observe the actual value of the price level so thatEt(ln pt) = ln pt then going back to (104) one could express the resulting ldquofullinformationrdquo output produced in sector i by
ln ylowastit = ln yni + γ [ln pit minus ln pt]
Lucas model 153
which given ln pit = ln pt + zit simply becomes
ln ylowastit = ln yni + γ zit
As you can see since the expectation of the random shock to relative prices zit iszero ln yni has the natural interpretation as the expected output of sector i givenfull information
The Lucas aggregate supply function
Equation (1015) is close to what is known as the ldquoLucas aggregate supplyfunctionrdquo Without adjustment costs the Lucas supply function takes the form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) (1016)
With adjustment costs the Lucas supply function takes the general form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) + λ(ln ytminus1 minus ln yn) (1016prime)
where yn denotes the natural rate of output17 For simplicity we have assumed thatthe natural rate of total output is constant across periods
The term ln yn can be interpreted either as the logarithm of output for theldquorepresentativerdquo sector or as the logarithm of total output across the n sectorsLet us assume the former interpretation The average level of real output can bedefined by
yt equiv 1
pt
⎡⎣ nprod
i=1
pityit
⎤⎦
1n
where [prodni=1 pityit]1n is the geometric mean of nominal output across the n markets
or sectors Taking logs we have the following definition for the logarithm ofaverage output
ln yt equiv minus ln pt + 1
n
nsumi=1
(ln pit + ln yit) (1017)
We will assume that the overall price level is constructed as a geometric mean ofindividual prices such that
pt equiv⎡⎣ nprod
i=1
pit
⎤⎦
1n
Taking logs
ln pt equiv 1
n
nsumi=1
ln pit
154 Lucas model
Substituting the above into (1017) we have the following definition for thelogarithm of average output
ln yt equiv 1
n
nsumi=1
ln yit (1018)
Substituting into (1018) the supply functions for the individual sectors as givenby (1015) we thus have18
ln yt equiv 1
n
nsumi=1
[ln yni + γ θ(ln pit minus Etminus1(ln pt))] (1019)
Recall that ln pit = ln pt + zit where zit is a normal random variable indepen-dently distributed across markets with a mean of zero and variance σ 2
z Substitutingthis into (1015) and rearranging we have
ln yt = 1
n
nsumi=1
[ln yni] + γ θ(ln pt minus Etminus1(ln pt)) + 1
n
nsumi=1
γ θzit (1020)
As the number of markets n approaches infinity from the law of large numberswe know that the sum of the zit divided by n approaches zero19 Thus for a largenumber of markets we may approximate (1020) by
ln yt = 1
n
⎡⎣ nsum
i=1
ln yni + γ θ(ln pt minus Etminus1(ln pt))
⎤⎦ (1021)
By definition the logarithm of the geometric average of ldquonormalrdquo output acrossmarkets ln yn is given by nminus1 sumn
i=1 ln yni Thus we can rewrite (1021) as (1016)in which ln yn is the logarithm of ldquonormalrdquo output that would occur if there were nosurprises with respect to the aggregate price level that is when ln pt = Etminus1(ln pt)Note that for simplicity we assume the natural rate is constant over time
Equation (1016) is the Lucas aggregate supply equation with the last termmissing As noted above if we include adjustment costs then we obtain (1016prime)indicating that the deviation of real output from its ldquonaturalrdquo level or trend isassociated with a deviation in the price level from that expected and past deviationsof output from the natural rate This last term makes output serially correlatedover time
The Lucas supply function and the Phillips curve
The Lucas supply function predicts a direct correlation between unanticipatedprice changes and output and thus a potential tradeoff between price changesand unemployment if one assumes that unemployment and output are inverselyrelated This potential inverse relationship between unemployment and inflation
Lucas model 155
is sometimes referred to as the ldquoPhillips curverdquo after AW Phillips who noted theempirical relationship between wage inflation and unemployment for the Britisheconomy for the 100 years up to 1957 (see Phillips 1958) Later depictions of thePhillips curve replaced the rate of change in wages with the inflation rate
To see this Phillips relationship more clearly rearrange the aggregate Lucassupply function without adjustment costs (1016) to obtain
ln pt = (ln yt minus ln yn)γ θ + Etminus1(ln pt)
Subtracting ln ptminus1 from both sides of this aggregate supply equation we have
ln(ptptminus1) = (ln yt minus ln yn)γ θ + Etminus1(ln(ptptminus1)) (1022)
Let the term πt denote the rate of inflation between periods t minus 1 and t20
πt equiv (pt minus ptminus1)ptminus1 = (ptptminus1) minus 1
It is common in macroeconomics to approximate the above rate of change inprices by the log of the ratio of the two prices If the ratio equals one then thelog equals zero which is the rate of inflation If the ratio is 1 + x and x is a smallproportion then the log of this ratio approximately equals the actual inflation rateFor instance if ptptminus1 = 105 so that inflation is 005 or 5 percent then the logof 105 is 00488 which approximates this 005 rate of inflation Thus we have
πt asymp ln(ptptminus1) = ln pt minus ln ptminus1
Using the above approximation for the rate of inflation we can rewrite(1022) as
πt = (ln yt minus ln yn)γ θ + Etminus1πt (1023)
which as Sargent (1987a 443) states
is in the form of a standard natural rate Phillips curve relating inflation (πt)
directly to output (ln yt) and to expected inflation (Etminus1πt) According to[(1023)] the Phillips curve shifts up in the (πt yt) plane by the exact amountof any increase in expected inflation This characteristic of equation [(1023)]is often taken as the hallmark of the natural unemployment rate hypothesisIt seems to offer an explanation for why the Phillips curve tradeoff worsenedas average inflation rates increased over the 1970s in many western countries
If we assumed that due to adjustment cost the lagged deviation in output fromthe natural level affects the current deviation as the Lucas aggregate supplyequation (1016prime) suggests then in terms of rates of change in prices we wouldhave
πt = (ln yt minus ln yn)γ θ + Etminus1πt minus (λγ θ)(ln ytminus1 minus ln yn) (1023prime)
156 Lucas model
We could instead express (1023) in terms of unemployment by assuming thereis a linear inverse relationship between deviations in output from the natural rateand deviations in the actual level of unemployment Ut from its natural rate Unsuch that
ln ytminus1 minus ln yn = minus(Ut minus Un)
where is a positive constant Substituting the above into (1023) we thus have
πt = minus(γ θ)(Ut minus Un) + Etminus1(πt) (1023primeprime)
indicating the inverse relationship between unanticipated price changes and theactual level of unemployment Rearranging (1023primeprime) we have that
Ut = Un minus (γ θ)[πt minus Etminus1(πt)] (1023primeprimeprime)
where γ θ gt 0 Equation (1023primeprimeprime) is the typical expression of the Phillips curvefound in the literature It indicates that deviations in the unemployment rate belowits natural level must be accompanied by deviations in the actual rate of inflationabove that expected It reflects the ldquonatural rate of unemployment hypothesisrdquo asoriginally coined by Friedman (1968 11)
There is always a temporary trade-off between inflation and unemploymentthere is no permanent trade-off The temporary trade-off comes not frominflation per se but from unanticipated inflation which generally means froma rising rate of inflation
Recall that as Barro and Gordon (1983 592) observed the term Etminus1(πt) in(1023primeprimeprime) is the
prior expectation of inflation for period t [which is] distinguished from theexpectation that is conditional on partial information about current pricesThis distinction arises in models (eg Lucas 1972 1973 Barro 1976) inwhich people operate in localized markets with incomplete information aboutcontemporaneous nominal aggregates In this setting the Phillips curve slopecoefficient (γ θ) turns out to depend on the relative variances for generaland market-specific shocks
Variability in prices and the tradeoff
As Lucas (1973 333) states
demand policies [can] tend to move inflation rates and output (relative totrend) in the same direction or alternatively unemployment and inflation inopposite directions The conventional Phillips curve account of this observedco-movement says that the terms of the tradeoff arise from the relatively stable
Lucas model 157
structural features of the economy and are thus independent of the nature ofthe aggregate demand policy pursued The alternative explanation of the sameobserved tradeoff is that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relativeprice changes
Taking Lucasrsquos alternative viewpoint two aspects concerning the tradeoff aresuggested First as Lucas states ldquochanges in average inflation rates will notincrease average outputrdquo As we have seen if we compare the expected pricelevel for period t with the price level for the prior period the difference wouldincorporate individualsrsquo expectation of this average rate of inflation (along with anumber of other potentially relevant variables) Second ldquothe higher the variancein average prices the less lsquofavorablersquo will be the observed tradeoffrdquo We considerthis second point below by referring back to the simple Lucas supply functionwithout lagged output (1016)
Recall that the term ξt given by (105) denotes that part of the price level thatcannot be predicted from past data We have assumed that this ldquosurpriserdquo term ξtis a normally distributed random variable with zero expectation and variance σ 2Substituting this into (1016) we have that
ln yt minus ln yn = γ θξt (1024)
where θ = σ 2z (σ 2 + σ 2
z ) and γ = (1 minus α)(α + β) Recall that θ is the weightattached to the expected price level prior to observing pit
Equation (1024) indicates that deviations in output from the natural level dependsolely on surprises In his statement concerning the variance of prices Lucas ispointing out that the impact of ldquosurprisesrdquo on output relative to its natural leveldepends on the ldquosloperdquo term λθ which is given by
λθ = 1 minus α
α + β
σ 2z
σ 2 + σ 2z
As Sargent (1987a 444) notes
a ldquofavorablerdquo tradeoff between output and unexpected inflation (that is a largevalue of γ θ ) will exist only when σ 2 is small relative to σ 2
z An attempt byauthorities to exploit the tradeoff between output and unexpected inflationmore fully by changing aggregate demand regimes might increase the vari-ance σ 2 relative to σ 2
z and thus change the slope γ θ This is yet anotherexample of how agentsrsquo optimal decision rules change in response to changesin the random processes governing the exogenous variables they base theirdecisions on
Sargentrsquos last point is another example of the ldquoLucas critiquerdquo in this contextwith respect to the validity of using past econometric estimates of a tradeoff inpredicting future tradeoffs
158 Lucas model
Note that although the tradeoff worsens with higher variability in prices theeffect of higher variability in prices on the variance of output about the natural rateis unclear In particular from (1016) we know that the variance in the differencebetween output and the natural rate is simply (γ θ)2σ 2 Given our definition ofγ θ the variance of the logarithm of output becomes
Var(ln yt) = γ 2(
σ 2z
σ 2 + σ 2z
)2
σ 2
Differentiating with respect to σ 2 we have
partVar(ln yt)
partσ 2 = γ 2(
σ 2z
σ 2 + σ 2z
)2
minus 2γ 2σ 2 σ 2z
(σ 2 + σ 2z )
σ 2z
(σ 2 + σ 2z )2
=(
γ 2σ 2z
σ 2 + σ 2z
)2 (1 minus 2σ 2
σ 2 + σ 2z
)
As the above expression indicates by itself an increase in the variation in theaverage price level (σ 2) will increase the variation in the logarithm of output for agiven ldquosloperdquo (γ θ) On the other hand as Lucas pointed out such an increase inthe variation in price level will result in a reduction in the effect of any given pricechange on output which by itself would decrease the variation in the logarithm ofoutput
Substituting (105) into the expanded Lucas supply function with lagged outputwe have the Lucas supply function of the form
ln yt minus ln yn = γ θξt + λ(ln ytminus1 minus ln yn)
where as before θ = σ 2z (σ 2 + σ 2
z ) Substituting for prior differences in outputfrom its natural level we thus have
ln yt minus ln yn = γ θ
infinsumi=0
λiξtminusi (1025)
which shows that the deviation of output from its natural rate depends on thecurrent and all previous values of the ldquoaggregate demand shockrdquo that affects theequilibrium price level
A complete model except for specifying thesource of expectations
Equation (1016) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To simplify the analysis we will normalize outputso that the natural level of real output is equal to 1 We have already assumed that
Lucas model 159
the natural level of real output is constant over time These two assumptions allowus to write (1016) in the more compact form
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (1026)
where Yt denotes log of real output supply for period t (or equivalently the deviationin output from its natural level for period t) Pt denotes the log of the price leveland Etminus1Pt denotes the expectation of log of the price level What we now requireis a characterization of the aggregate demand side of the economy as typicallysummarized by the LM and IS equations
To obtain an explicit form for the portfolio or LM equation we start by assuminga real money demand function for the end of period t of the form
Ldt = yα1
t exp[minus(α2rt)] (1027)
where yt is real output α1 gt 0 and α2 gt 0 indicating that real money demand isdirectly related to real output but inversely related to the nominal interest rate21
Let Mt denote the nominal money supply at the end of period t (previously thishas been denoted by M ) and let mt denote the logarithm of this money supplyfor period t such that mt = ln Mt Further let us assume a logarithmic supply ofmoney function of the form
mt = mt + εt (1028)
Equation (1028) separates the logarithm of the money supply into two compo-nents a deterministic component mt set by government authorities according toa rule tying money supply changes to past variables and a random component εt which is assumed to be normally distributed with zero mean This random term isalso assumed to be serially independent (ie E(εtεs) = 0 for s = t)
The LM equation is simply the money market equilibrium condition equatingthe real supply of money to real money demand and thus is given by
Mtpt = Ldt (1029)
Taking logs of the equilibrium condition (1029) and substituting the logarithm ofthe money demand function (1027) and the money supply function (1028) wehave
mt minus Pt = α1Yt minus α2rt minus εt (1030)
where Pt = ln pt Equation (1030) is the standard log-linear form of the portfolioor LM equation22
To obtain an explicit form for the IS equation which is the equilibrium conditionin terms of equating output production to the demand for output we must postulatea specific form for output demand One common assumption is to include the
160 Lucas model
expected real rate of interest rt minus πet+1 as a determinant of output demand To do
so let
πet+1 equiv (pe
t+1 minus pt)pt = (pet+1pt) minus 1
As before we can approximate the expected rate of change in prices by the expecta-tion of the log of the ratio of the future to current price level (ie πe
t+1 asymp Pet+1minusPt
where Pet+1 is the expected log of the price level for period t+1) Thus the expected
real rate of interest becomes rt minus πet+1
Letting the term Xt denote a vector of exogenous variables that also affectsoutput demand we have in log-linear form the following equilibrium conditionfor the output market
Yt = Xt minus β1(rt minus πet+1) + ut (1031)
where ut is a serially independent stationary random process with mean zeroand finite variance equal to σ 2
u The random terms for output demand and moneysupply εt and ut respectively are assumed independent (ie E(εtut) = 0)
Summarizing we have a model consisting of the aggregate supply equation(1026) the LM equation (1030) and the IS equation (1031) which can be solvedfor the equilibrium output price and interest rate
In particular combining the LM and IS equations to eliminate the interest ratert we obtain
Yt = Xt + ut minus (β1α2) middot (minusmt minus εt + Pt minus α1Yt) + β1 middot πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
] (1032)
Or in terms of the price level we have an ldquoaggregate demand equationrdquo of theform
Pt = mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1) (1033)
Equations (1032) and (1033) indicate the inverse relationship between the pricelevel and output that is shown graphically by a downward-sloping aggregatedemand curve
With respect to (1033) note that the expected rate of inflation for the nextperiod πe
t+1 is viewed as a distinct entity Our prior assumption of unit elasticexpectations concerning the expected log of the future price level Pe
t+1 wouldimply that this term is in fact independent of changes in the current price levelNote also that if output were unchanged then an x percent change in the moneysupply would result in an x percent change in the price level This is the standardresult of the ldquoneoclassicalrdquo model23
Lucas model 161
Combining the above aggregate demand equation (1032) with the aggregatesupply equation to eliminate the output term Yt we have24
α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
]
= γ θ(Pt minus Etminus1Pt) + λYtminus1
Solving the above for the equilibrium price level one obtains
Pt = 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1)
+ J0
(Etminus1Pt minus λYtminus1
γ θ
)] (1034)
where J0 = γ θ(α2 + β1α1) Expression (1034) is sometimes called the reduced-form equation for the price level
Rearranging (1034) we have
Pt minus J0
J0 + β1Etminus1Pt
= 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1) minus J0
λYtminus1
γ θ
] (1034prime)
Let us assume perfect foresight meaning that Etminus1Pt = Pt Noting that1 minus J0(J0 + β1) = β1(J0 + β1) we can solve (1034prime) for the equilibriumprice level under this hypothesis of perfect foresight obtaining
Pt = mt + εtα2
β1(Xt + ut + β1π
et+1) minus J0
λ
β1γ θYtminus1 (1034primeprime)
As equation (1034primeprime) makes clear under the presumption of limited perfectforesight the predictions are those of the neoclassical model
(a) a change in the money supply (in log form given by mt + εt) results in anequiproportionate change in the price level (in log form given by Pt)
(b) an increase in expected inflation πet+1 raises the price level
(c) a higher level of lagged output (Ytminus1) lowers the price level(d) an increase in output demand (Xt + ut) raises prices
Following a procedure similar to that used to derive (1034) if we combine theaggregate demand equation (1033) and aggregate supply equation to eliminatethe price level we have
Yt = λYtminus1 minus γ θEtminus1Pt
+ γ θ
mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1)
162 Lucas model
Solving for the equilibrium real output we have
Yt = β1
J0 + β1
[λYtminus1 + γ θ
[mt + εt minus Etminus1Pt + α2
β1(Xt + ut + β1π
et+1)
]]
(1035)
The above expression is sometimes call the reduced-form equation for real outputAccording to (1035) changes in the money supply (in logs given by mt = mt +εt)will affect real output to the extent that the impact of such changes on prices is notfully anticipated
Note that with perfect foresight we have Etminus1Pt = Pt In this case substituting(1034primeprime) into (1035) for Etminus1Pt we obtain
Yt = β1
J0 + β1
(λYtminus1 + J0
λ
β1Ytminus1
)= λYtminus1 (1035primeprime)
Thus we have the standard neoclassical result that output in the current period isindependent of demand-side changes such as changes in expected inflation or themoney supply
One source of expectations autoregressive expectations
As Shiller (1978) has noted
one of the most difficult problems which confronts builders of macroeco-nomic models is the need to model the mechanism by which the public formsits expectations of future economic variables Many of the most importanttheoretical macroeconomic behavioral relations (eg the supply equationinvestment saving) depend critically on public expectations of future eco-nomic variables yet we often do not even have any data on what theseexpectations are
This and the next section suggest two approaches that have been taken to modelexpectations in particular the expected price level that enters into the aggre-gate supply equations These two approaches to expectation formation are thedistributed lag (or adaptive) scheme and the rational expectations scheme
To understand the ideas behind distributed lag schemes as the source of expec-tations we start by noting that the price level pt can be broken down into acombination of the price level for the previous period ptminus1 multiplied by theratio of the price level this period to last period
pt equiv ptminus1(ptptminus1)
Taking logs and recalling that ln(ptptminus1) is approximately equal to the rate ofinflation πt we thus have
Pt equiv ln pt = ln ptminus1 + πt
Lucas model 163
Taking expectations at time t assuming that at a minimum the price level for theprior period is known we have
Etminus1Pt equiv ln ptminus1 + Etminus1πt (1036)
Until the 1970s the approach to modeling the source of the expected rate of infla-tion embedded in (1036) was to assume individuals forecast the rate of inflationby looking at past inflation rates A common quantitative representation of thishypothesis originated by Fisher (1930) was to have individualsrsquo expectation ofthe inflation rate behave like a weighted average or ldquodistributed lagrdquo of recent pastinflation rates That is
Etminus1πt =qsum
i=1
ηiπtminusi (1037)
where the ηi are fixed numbers A typical idea behind this distributed lag approachto anticipated inflation was that individuals have ldquoadaptive expectationsrdquo whichmeant that individuals adjusted or ldquoadaptedrdquo their expectations of the rate of infla-tion in light of the actual forecast error made concerning the prior periodrsquos inflationrate Specifically adaptive expectations can be expressed as
Etminus1πt = Etminus2πtminus1 + δ(πtminus1 minus Etminus2πtminus1)
= δπtminus1 + (1 minus δ)Etminus1πtminus1(1038)
where 1 gt δ gt 0 Successive substitution allows us to rewrite (1038) as
Etminus1πt = δπtminus1 + (1 minus δ)δπtminus2 + (1 minus δ)2δπtminus3 + middot middot middotor
Etminus1πt =infinsum
i=1
δ(1 minus δ)iminus1πtminusi (1039)
As you can see (1039) is simply a specific form of equation (1037) in whichthe ηi place declining weight on past inflation rates the more distant they are andq = infin Given declining weights we can obtain a reasonable approximation of(1039) even if we truncate the distributed lag on past inflation after q periods aslong as q is reasonably large andor δ is reasonably large
Now let us place the above discussion not in terms of past rates of inflation butinstead in terms of past price levels Recalling the approximation
πtminusi asymp ln ptminusi minus ln ptminusiminus1
we can rewrite (1037) in the form
Etminus1πt =qsum
i=1
ηi(ln ptminusi minus ln ptminusiminus1) (1040)
164 Lucas model
Writing this out we have
Etminus1πt = η1(ln ptminus1 minus ln ptminus2) + η2(ln ptminus2 minus ln ptminus3)
+ η3(ln ptminus3 minus ln ptminus4) + middot middot middot + ηq(ln ptminusq minus ln ptminusqminus1)
Thus we may rewrite (1040) as
Etminus1πt = η1 ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
(1041)
Combining (1041) with equation (1036) we obtain the following expression forthe expectation formed at time t concerning the log of the price level
Etminus1Pt = (1 + η1) ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
or
Etminus1Pt =q+1sumi=1
vi ln ptminusi (1042)
Equation (1042) is what Sargent and Wallace (1975) refer to as ldquoautoregressiveexpectationsrdquo
A second source of expectations rational expectations
The papers by Lucas (1972 1973) and Sargent and Wallace (1975) suggestedthat in macroeconomic model building a different approach to specifying thesource of expectation is preferred As Cukierman (1986) summarizes this ldquorationalexpectationsrdquo approach to the modeling of inflationary expectations is
based on the maintained hypothesis that individuals know the structure of theeconomy and of governmentrsquos decision rule and that they use this structurein conjunction with the available information in order to form an optimalpredictor of future inflation [this approach] requires a precise specificationof the model of the economy as well as of the information sets of individualsEmpirical tests of this hypothesis are therefore joint tests of the validity of theexpectational hypothesis as well as of the postulated structure of the economyand of the particular assumptions made about the information possessed byindividuals
A ldquostructure of the economyrdquo was derived based on the Lucas aggregate sup-ply equation that included the assumption of market-clearing wages and pricesSuppose that individuals know this model and accept it as reflecting the structureof the economy As we saw above this model was solved to obtain (1034) the
Lucas model 165
reduced form for the equilibrium price level in period t We assume that at time tindividuals form their expectations Etminus1Pt ldquorationallyrdquo in that
Etminus1Pt = E(Pt |tminus1) (1043)
indicating that Etminus1Pt is the mathematical expectation of Pt conditional on theinformation set tminus1 which is all information available at period t minus1 As Sargentnotes (1987a 440)
Lucas assumed that tminus1 included information on all lagged values of ln pitand lagged values of real output in all markets One could equally well con-ceive of less comprehensive definitions of tminus1 For now along with Lucaswe suppose that tminus1 includes a comprehensive list of variables includinglagged outputs and prices in all markets
At the end of period t minus 1 individuals of course know Etminus1Pt as well as Ytminus1and πe
t+1 Thus taking the expectation of (1034) and subtracting it from Pt wehave
Pt minus Etminus1Pt = β1
J0 + β1(mt + εt minus Etminus1mt) + α2
J0 + β1(Xt minus Etminus1Xt + ut)
(1044)
In Sargent and Wallace (1975 244) the deterministic part of the money supplymt is assumed to reflect a ldquolinear feedback rulerdquo of the form
mt = Gθlowastt (1045)
where ldquoθlowastt represents the set of current and past values of all of the endogenous and
exogenous variables in the system as of the end of period t minus 1 and G is a vectorof parameters conformable to θlowast
t rdquo A simple example of a monetary feedback rulewould be
mt = a0 + a1Ytminus1 (1046)
where a0 and a1 are positive constantsLet us assume that individualsrsquo information set tminus1 includes not only the
structure of the economy (as summarized by the above linear macroeconomicmodel) but also the money supply rule In the particular example of (1046) theyknow a0 a1 and Ytminus1 and thus mt Then the assumption of rational expectationsimplies that Etminus1mt = mt Furthermore since Xt represents the deterministicpart of the vector of exogenous variables affecting output demand we have thatEtminus1Xt = Xt Thus (1044) becomes
Pt minus Etminus1Pt = 1
J0 + β1(β1εt + α2ut) (1047)
166 Lucas model
Substituting (1047) into the aggregate supply equation one obtains
Yt = γ θ1
J0 + β1(β1εt + α2ut) + λYtminus1 (1048)
An important feature of (1048) is that a deviation in output from its natural levelwhich is represented by the term Yt different from zero given our assumption thatthe natural output level is normalized to equal one is determined only by pastdeviations and ldquosurprisesrdquo with respect to the money supply (the term εt) andoutput demand (the term ut) The ldquodeterministicrdquo or predictable component of anymoney supply change has no real effects Equation (1048) should look familiarGiven λ lt 1 it suggests that the times series for deviations of the logarithmof output from its natural rate is a stationary autoregressive process of order 1or AR(1)
The above analysis is an example of a ldquolinear rational expectations modelrdquoThe result that ldquopredictablerdquo monetary policy has no real effects reflects the twinassumptions of the natural rate hypothesis and rational expectations It should notbe surprising that deterministic monetary policy has no effect for the model ishomogeneous of degree 0 in mt Pt and Etminus1Pt This is a critical characteristic ofa ldquonatural rate modelrdquo
Conclusion
The main focus of this chapter has been on the development of the Lucas supplyfunction The model is often discussed in the context of the ldquoislandrdquo paradigmin which we specify the supply function for a particular sector in the economyThe role of forecasting errors is introduced and from that the Lucas aggregatesupply function is constructed and the relationship of this function to the Phillipscurve is discussed A number of other issues were discussed involving variabilityin prices and the corresponding economic tradeoff and then a complete modelwas introduced except for specifying the source of expectations Two sources ofexpectations were then described autoregressive expectations and rational expec-tations The implications of expectations were then discussed in their historicalcontext
11 Policy
Introduction
This chapter extends the earlier discussions about the actions of the monetaryauthority and how these actions affect the macroeconomy Perhaps the most inter-esting issue of monetary economics is addressed here that is the optimal roleof monetary policy The chapter highlights the differences in model results thatdepend on what type of expectations are assumed Particular attention is givento the Sargent and Wallace ldquoineffectiveness propositionsrdquo and the Phillips curveOther issues are then introduced including the ldquorule versus discretionrdquo debatetime inconsistency and the role of credibility and enforcement
Optimal monetary policy
In the 1970s the articles by Sargent and Wallace (1975 1976) and Lucas (19721973) altered the view of how one should assess the impact of monetary policy onthe economy and by implication what is optimal monetary policy The discussionstarts with the premise that monetary policy should be conducted according to arule or set of rules As Sargent and Wallace (1976 169) state
It is widely agreed that monetary policy should obey a rule that is a scheduleexpressing the setting of the monetary authorityrsquos instrument (eg the moneysupply) as a function of all the information it has received up through thecurrent moment Such a rule has the happy characteristic that in any givenset of circumstances the optimal setting for policy is unique If by remotechance the same circumstances should prevail at two different dates theappropriate settings for monetary policy would be identical1
The SargentndashWallace premise that monetary rules are preferred leads them toexplore the form of the optimal rule But as we will discover in going over thepaper by Barro and Gordon (1983) there is a question of whether monetary rulescan be enforced over time If not then what is typically left in these models isa ldquosecond bestrdquo solution involving the determination of optimal ldquodiscretionaryrdquo
168 Policy
policy Note that we use the term ldquosecond bestrdquo because enforceable rules tend todominate discretion in these models2
Accepting the premise that monetary policy can adopt enforceable rules stillleaves open the specification of the optimal set of rules The simplest rule sug-gested by Friedman (1959) would increase the money supply at a constant rateeach year perhaps 3 percent3 More complex rules known as ldquoreactive rulesrdquowould specify in advance how the growth of the money supply will change basedon new information on the state of the economy One such rule suggested is thatthe growth in the monetary base and thus the money supply automatically adjustwhenever the growth of nominal GNP deviates from its trend (McCallum 1985)Another reactive rule suggests that the government commit itself to holding theCPI to a preannounced target and adjust the monetary base and thus the moneysupply accordingly (Hall 1982)4
Below we begin our discussion of optimal monetary policy by reviewing theanalysis of optimal enforceable rules as suggested by the Sargent and Wallace(1975 1976) papers and reviewed in Sargent (1987a Chapter 17) This discussionis in the context of a natural rate model without rational expectations and then inthe context of a model which assumes rational expectations
Optimal monetary policy exogenous expectations
As we saw previously the reduced form for the log of the output in period t canbe expressed as
Yt = H0
[λYtminus1 + γ θ
[minusEtminus1Pt + mt + εt + α2
β1(Xt + ut + β1π
et+1)
]]
(111)
where H0 = β1(J0 +β1) and J0 = γ θ(α2 +β1α1) Recall that Yt is the differencein period t between the logarithm of output and the logarithm of the natural levelof output Normalizing so that the natural level of output equals one we canequivalently interpret Yt in equation (111) as total output As Sargent and Wallace(1976) state ldquoYt can be thought of as the unemployment rate or the deviation ofreal GNP from lsquopotentialrsquo GNP This equation should be thought of as the reducedform of a simple econometric modelrdquo
Recall that the log of the money supply in period t mt is the sum of adeterministic component mt and the random component εt with variance σ 2
e 5
To understand the impact of monetary changes on real GNP we must firstconsider how the expected log of the price level Etminus1Pt varies with changes inmonetary policy One approach in the spirit of ldquoautoregressive expectationsrdquo isto assume that the expected price level is independent of the current monetarypolicy This essentially means viewing the expectation of the log of the price level(Etminus1Pt) as exogenous6 Given this assumption we may rewrite the reduced-form
Policy 169
equation for output (111) as
Yt = a0 + a1Ytminus1 + a2mt + vt (112)
where vt = H0γ θ(α2β1)ut is a serially independent normally distributed randomvariable with variance σ 2
v and mean zero mt is the log of the money supply forperiod t where mt = mt + εt Ytminus1 is lagged output and a0 a1 and a2 areparameters
Suppose that the monetary authority desires to set the money supply in orderto minimize the fluctuation in the log of output around some desired level Let usassume that the log of this desired level denoted Y lowast is above the log of the naturallevel of output Yn7 Then the objective can be expressed as
min Etminus1(Yt minus Y lowast)2
We can break this expression into two terms in that the objective can beequivalently expressed as
min Etminus1(Yt minus Etminus1Yt)2 + (Etminus1Yt minus Y lowast)2 (113)
The second way of expressing the objective allows us to see the objective asminimizing the sum of two terms the variance of Yt conditional on informationup to the end of period t minus 1 and the ldquobias squaredrdquo around Y lowast The secondterm the bias squared around Y lowast is the reason for an ldquoactivistrdquo monetary policyEquation (113) indicates that the optimal monetary policy entails
1 minimizing the variance in the random component of the money supply Thisfollows since the first term in equation (113) the variance of Yt is given bya2
2σ2e +σ 2
v 8 If feasible complete elimination of the random component to themoney supply (ie a purely ldquodeterministicrdquo money supply) is optimal suchthat εt = 0 for all t and thus σ 2
e = 02 setting Etminus1Yt = Y lowast so as to make the second term in equation (113) equal
to zero From equation (112) this means a monetary policy such that
Etminus1(a0 + a1 middot Ytminus1 + a2 middot mt + vt) = Y lowast
Noting that Etminus1Yt = 0 and that Etminus1mt = mt we see that this deterministic partof the optimal monetary policy is defined by the equation
a0 + a1Ytminus1 + a2mt = Y lowast
which can be solved for the optimal deterministic monetary policy rule
mt = g0 minus g1Ytminus1 (114)
where g0 = (Y lowast minus a0)a2 and g1 = a1a2
170 Policy
An equivalent expression for the optimal monetary rule (114) is derived inSargent (1987a Chapter 17) under the presumption of exogenous expectations Inparticular Sargent uses equation (112) to substitute out for Ytminus1 in (114) so thatthe optimal (deterministic) monetary rule (114) becomes
mt = g0 minus g1[a0 + a1Ytminus2 + a2mtminus1 + vtminus1] (115)
Now substituting into (115) the expression for Ytminus2 suggested by equation (112)we have
mt = g0 minus g1[a0 + a2mtminus1 + vtminus1] minus g1a1[a0 + a1Ytminus3 + a2mtminus2 + vtminus2](116)
Continuing to successively substitute for Ytminusi i = 3 4 we have Sargentrsquosequivalent expression for the optimal (deterministic) policy rule as given by9
mt = g0 minus g1a0 minus⎛⎝g1a2
infinsumiminus1
aiminus11 mtminusi
⎞⎠ minus
(g1
infinsumi=1
aiminus11 + vtminusi
) (117)
Following the above optimal monetary policy (ie reducing any random compo-nent to the money supply to its minimum level and establishing the rule for thedeterministic component of the money supply as specified by (114) or (117)) wehave by construction that
Etminus1Yt = Y lowast and Yt = Y lowast + a2εt + vt
where vt = H0γ θ(α2β1)ut and the variance of the random component of themoney supply εt is set at its lowest feasible level Thus optimal monetary policyin essence sets output each period equal to Y lowast plus irreducible noise As Sargentand Wallace (1976 171) note
the application of the rule eliminates all serial correlation in output since thisis the way to minimize the variance in output The basic idea is that wherethe effects of shocks to a goal variable (like GNP) display a stable pattern ofpersistence (serial correlation) and hence are predictable the authority canimprove the behavior of the goal variable by inducing offsetting movementsin its instruments
Note that without the lag term for output g1 in (114) equals zero
Adaptive expectations and the accelerationist result
The well-known ldquoaccelerationist outcomerdquo concerning the path of inflation isimplied by the above analysis if expectations are adaptive and if the aim of mone-tary policy is to keep output above its natural level To see this let us go back to the
Policy 171
aggregate supply equation that underlies the reduced form for output To simplifylet us abstract from the stochastic elements in demand εt and ut (as well as anysupply-side disturbances) and also from adjustment costs (ie omit the laggedoutput term) Then the aggregate supply equation
Yt = γ θ [πt minus Etminus1πt] (118)
reflects the actual path that output will take10
To derive the accelerationist result assume that Y lowast gt Yn By normalizationYn = 0 so that to have Etminus1Yt = Y lowast gt Yn we must have Etminus1Yt gt 0Equation (118) suggests that to achieve an expected level of output greater thanits natural level the government must pursue a monetary policy that results inthe actual rate of inflation typically being above that expected In particular thedesire to keep output above its natural level means that monetary policy in period tresults in the actual inflation rate πt such that
πt minus Etminus1πt = Y lowastγ θ gt 0
If expectations are adaptive then we have that
Etminus1πt minus Etminus2πtminus1 = δ(πtminus1 minus Etminus2πtminus1)
The assumption of adaptive expectations coupled with Y lowast gt Yn = 0 thus impliesthat
Etπt+1 = Etminus1πt + δ(πt minus Etminus1πt) = Etminus1πt + δY lowastγ θ
In words the fact that individuals underestimate inflation this period (by theamount Y lowastγ θ ) leads them to adjust (ldquoadaptrdquo) their expectations of inflationupward (by the amount δY lowastγ θ ) The result is that to keep expected output atthe level Y lowast gt Yn next period means an increase in the inflation rate by theamount δY lowastγ θ each period As Blanchard and Fischer (1989 572) note
this is the famous accelerationist result derived by Friedman (1968) andPhelps (1968) using their Phillips curve together with the adaptive expec-tations assumption The explanation is simple if the government is trying tokeep output above the natural rate it has to produce inflation at a higher ratethan expected each period Since the expected inflation is a weighted averageof past inflation rates the actual rate must be increasing
Now let us assume instead that Y lowast = 0 To provide a role for an activist monetarypolicy let us reintroduce the lagged output term λYtminus1 into the right-hand sideof (118) Thus monetary policy can eliminate the effect of past deviations inoutput from the natural rate on current output In other words a policy aimed
172 Policy
at setting Etminus1Yt = Y lowast would imply altering inflation relative to expected onlywhen lagged output deviated from the natural level Rather than the prior result ofan ever-increasing inflation with Y lowast gt Yn with Y lowast = Yn we have that inflationsimply wanders11 For instance if logged lagged output Ytminus1 fell below the log ofthe natural level of output Yn = 0 then the difference Ytminus1 would be negativeOther things being equal this would imply a lower output in the current periodTo counteract this the government would pursue a monetary policy that results inthe actual rate of inflation being above that expected
The above discussion helps us understand the comment of Hall (1976) that
the benefits of inflation derive from the use of expansionary power to trickeconomic agents into behaving in socially preferable ways even thoughtheir behavior is not in their own interests The gap between actual andexpected inflation measures the extent of the trickery the optimal pol-icy is not nearly as expansionary when expectations adjust rapidly andmost of the effect of an inflationary policy is dissipated in costly anticipatedinflation
The above extract raises the following question Can the monetary authoritiessystematically trick the public in order to exploit the link between inflation andoutput For Sargent and Wallace and others the answer is no due to the existenceof rational expectations
Rational expectations and the SargentndashWallace ineffectivenessproposition
As Sargent (1987a) notes a critical aspect of the simple example of an optimalmonetary rule as given by (114) (or equivalently (117)) ldquois the implicit assumptionthat agentrsquos decision rules remain unchanged in the face of alternative stochas-tic processes for the control variable that different feedback rules implyrdquo WhatSargent means in this context is that the optimal monetary rule has been derivedunder the presumption that private agents do not take this rule into account informing their expectation of the price level Under this assumption one couldestimate the parameters of the reduced-form equation output (a0 a1 and a2 in(112)) independently of the feedback rule (114)
However Sargent and Wallace (1976) criticize this view In particular theyargue that ldquoin the reduced forms are embedded the responses of expectations tothe way policy is formed Changes in the way policy is made then ought not toleave the parameters of estimated reduced forms unchangedrdquo12 In other wordsrational individuals would clearly seek out and use information on how monetaryauthorities act as well as on the structure of the economy in forming expectationsof prices
Let us now consider the following version of the reduced-form equation foroutput (112) that explicitly includes the potential role of expected monetary policy
Policy 173
when individuals form expectations on prices13
Yt = a0 + a1Ytminus1 + a2(mt minus Etminus1) + vt (119)
For a given anticipated log of the money supply Etminus1mt we have as before theoptimal (deterministic) monetary rule of the form
mt = g0 minus g1Ytminus1 (1110)
so that
mt = g0 minus g1Ytminus1 + εt (1111)
where εt is the irreducible random element in the money supply determinationprocess Now assume that the public knows the monetary authoritiesrsquo feedbackrule Then our assumption of rational expectation (ie individuals use all availableinformation in forming expectations) implies that
Etminus1(mt) = g0 minus g1Ytminus1 (1112)
Combining (119) (1111) and (1112) the reduced form for output is now given by
Yt = a0 + a1Ytminus1 + a2εt + vt (1113)
so that the biased squared term in the objective of the monetary authorities(Etminus1Yt minus Y lowast)2 equals (a0 + a1Ytminus1 minus Y lowast)2
As is clear from (1113) there is no role for systematic monetary policy to affectreal output As Sargent (1987a 459) notes ldquothe bias squared is independent ofthe parameters of the money supply rulerdquo The optimal policy is then to makemonetary policy deterministic if feasible for then the variance of output (given bya2
2σ2e + σ 2
v ) is minimized by setting σ 2e = 0 Until we add an inflation objective
any deterministic rule will be equally as good for none will have any impact onoutput This is once again an example of the neutrality of money
As Sargent (1987a 458) notes ldquopolicy rules should be deterministic and involveno surprisesrdquo He goes on to argue that we
have therefore established the following stochastic neutrality theorem thatcharacterizes our model one deterministic feedback rule on the basis of theinformation set tminus1 which is common to the public and to the authority is asgood as any other deterministic feedback rule Via deterministic feedbackrules the monetary authority is powerless to combat the business cycle (theserial correlation in Yt)
Naturally if one abandons rational expectations or the natural rate hypothesis thenthis ldquostochastic neutralityrdquo result need not hold
174 Policy
The SargentndashWallace ineffectiveness proposition in thecontext of the Phillips curve
The above finding of the ldquoneutrality of moneyrdquo in the context of a stochasticlinear natural rate model with rational expectations is viewed by Sargent (1987a459) as
the antithesis of our earlier result rationalizing the activist Keynesian policyrules The reader is invited to verify that the truth of the neutrality theoremis not dependent on the particular information set assumed It will continueto hold for any specification of tminus1 so long as the public and the authorityshare the same information set
He concludes
The preceding results provide a [weak] defense for following rules with-out feedback Simple x-percent growth rules do as well as any deterministicfeedback rule and dominate rules with a stochastic component
Below we recast the SargentndashWallace ineffectiveness proposition in terms ofthe expectational Phillips curve This makes the discussion more in line with thenext sectionrsquos review of some implications of non-enforceable monetary rules Inaddition we add to the government goalrsquos an inflation objective In particular wemodify our analysis in the following four ways
1 we alter the objective to be in terms of unemployment rather than output2 we expand our objective function to include inflation3 we link inflation to unemployment via a modified Lucas supply equation4 we incorporate rational expectations
Our first task is to convert the objective of the government into unemploymentterms Before we assumed that the government simply sought to minimize thefluctuations in output about a particular level In particular if we let Zt denote thecost incurred in period t we assumed the objective was to
min Etminus1Zt where Zt equiv (Yt minus Y lowast)2 (1114)
We have previously assumed that the deviation of unemployment from its naturalrate is linearly related to the deviation of the log of output from the log of its naturallevel In particular we assumed
minus(Ut minus Un) = Yt
Policy 175
given the normalization of the natural level of output such that Yn equiv ln yn = 0Substituting the above into (1114) the problem facing the government policy-maker becomes
min Etminus1Zt where Zt = a(Ut minus kUn)2 a = 2 gt 0
k = 1 minus Y lowastUn (1115)
Note in (1115) that we assume k lt 1 which is equivalent to assuming thatthe log of optimal level of output Y lowast is greater than the log of the natural levelof output which by normalization has been set equal to zero As Blanchard andFischer (1989 596ndash597) suggest
The most plausible justification [for k lt 1] is the presence of distortions orimperfections that causes the natural rate of unemployment to be too high Thisjustification allows the loss function to be consistent with the single-periodutility function of private agents Another is that the governmentrsquos objectivefunction as shaped by the electoral process leads the government to seek toraise output above the natural level
Having converted the objective function into unemployment terms the next stepis to expand the objective function to include an inflation goal In particular let usassume that the cost in period t includes a term reflecting differences between theactual inflation rate πt and an optimal rate of inflation πlowast Assuming a simplequadratic form we have14
Zt = a(Ut minus kUn)2 + b(πt minus πlowast)2 (1116)
The problem of the government policy-maker is then
min Etminus1Zt where Zt equiv a(Ut minus kUn)2 + b(πt minus πlowast)2
As before we can decompose this objective to obtain the following equivalentexpression for the object of the government policy-maker
min a[Etminus1(Ut minus kUn)2 + (Etminus1Ut minus kUn)
2]+ b[Etminus1(πt minus πlowast)2 + (Etπt minus πlowast)2] (1117)
Our third task is to link inflation to unemployment Recall that if we ignore thelagged term with respect to output in the Lucas supply equation assume unem-ployment is linearly related to real output and approximate inflation by the log ofthe ratio of the price level this period to the price level last period then we canmanipulate the Lucas supply equation to obtain the expression
Ut = Un minus α(πt minus Etminus1πt) (1118)
176 Policy
where α = (γ θ) gt 015 Substituting (1118) into (1116) the governmentrsquosobjective becomes
min Etminus1a(1 minus k)Un minus α(πt minus Etminus1πt)2 + b middot (πt minus πlowast)2
Our fourth and final task is to introduce rational expectations As we saw earlierwe obtain the reduced form for the deviation of the price level from that expectedin period t
Pt minus Etminus1Pt = 1
J0 + β0(β1εt + α2ut) (1119)
if (a) the government follows a particular rule in determining monetary policy(b) that rule is known to the public and (c) there exist rational expectations
An algebraic manipulation ndash simultaneously subtracting and adding the log ofthe price level for period t minus 1 to the left-hand side of equation (1119) ndash bringsus closer to having an expression that may be interpreted in terms of inflation
ln( ptptminus1) minus Etminus1(ln( ptptminus1) = 1
J0 + β1(β1εt + α2ut) (1120)
Using the log of the price ratio as an approximation for inflation we thus canapproximate (1120) as
πt minus Etminus1πt = 1
J0 + β1(β1εt + α2ut) (1121)
Substituting the above which reflects the rational expectations approach tomodeling expectations into the new government objective we have
min Etminus1
a
[(1 minus k)Un minus α
J0 + β1(β1εt + α2ut)
]2
+ b(πt minus πlowast)2
or
min a[(1 minus k)Un]2 +(
α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1 b(πt minus πlowast)2 (1122)
since Etminus1εt = Etminus1ut = Etminus1εtut = 0It is clear from (1122) that with rational expectations monetary policy can play
no role in helping the government meet its objective concerning unemploymentIn other words in the context of the Lucas model the assumption of rationalexpectations means that the expected loss from deviations in unemployment fromits desired level and thus production from its desired level is independent ofthe deterministic monetary rule As a consequence the minimization problem asgiven by the first half of equation (1117) becomes simply one of specifying any
Policy 177
deterministic monetary policy rule and eliminating any random changes in themoney supply (ie setting σ 2
e = 0 if feasible)But the objective of the government now also includes an objective concerning
the rate of inflation so we need an expression for the equilibrium rate of changein prices πt that incorporates rational expectations To do so we start with thereduced form for the log of the price level obtained previously which is given by
Pt = 1
J0 + β1
[β1mt + εt + α2(Xt + ut + β1π
et+1) + J0
(Etminus1Pt minus λYtminus1
γ θ
)]
(1123)
Taking the difference between the reduced form for the log of the price level forperiod t and for period tminus1 we can obtain an expression for πt the rate of inflationbetween period t and t minus 1 of the form
πt = 1
J0 + β1[β1mt + α2(Xt minus Xtminus1 + ut minus utminus1) + α2β1(π
et+1 minus πw
t )
+ J0(Etminus1Pt minus Etminus2Ptminus1) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1124)
where πmt approximates the rate of change in the money supply (ie πmt =mt minus mtminus1 = ln(MtMtminus1)) Assuming rational expectations we have fromequation (1119) that
Etminus1Pt = Pt minus 1
J0 + β1(β1εt + α2ut)
and similarly
Etminus2Ptminus1 = Ptminus1 minus 1
J0 + β1(β1εtminus1 + α2utminus1)
so that
Etminus1Pt minus Etminus2Ptminus1 = πt + 1
J0 + β1[β1(εtminus1 minus εt) + α2(utminus1 minus ut)]
(1125)
Substituting this expression into equation (1124) one obtains
πtβ1
J0 + β1= 1
J0 + β1[β1πmt + α2(Xt minus Xtminus1 + ut minus utminus1)
+ α2β1 middot (πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1126)
178 Policy
where we use the fact that 1 minus J0(J0 + β1) = β1(J0 + β1) Solving for πt we have
πt = πmt + α2
β1(Xt minus Xtminus1 + ut minus utminus1)
+ α2(πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)
(1127)
which given πmt = mt minus mtminus1 = mt + εt minus (mtminus1 + εtminus1) can be rearranged toobtain
πt = πmt + εt minus εtminus1 + α2
β1(Xt minus Xtminus1 + ut minus utminus1) + α2(π
et+1 minus πe
t )
minus J0λ
β1γ θ(Ytminus1 minus Ytminus2) (1128)
where πmt = mt minus mtminus1If we assume no change in exogenous (nonrandom) demand factors this period
compared to the last period (ie Xt = Xtminus1) no change in the expected future infla-tion between last period and this period (ie πe
t+1 = πet ) and ignore adjustment
costs (ie λ = 0) we can simplify equation (1128) to obtain
πt = πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) (1129)
The three assumptions we made to derive equation (1129) from equation (1128)largely limit any differences between period t and t minus 1 to differences in the sizeof the money supply In fact the two periods differ only by the deterministiccomponent of the money supply and by random factors where these randomfactors ndash the term ut minus utminus1 with respect to output demand and the term εt minus εtminus1with respect to the irreducible random component of the money supply ndash havemean zero In other words all potential changes in aggregate demand or productionexcept money supply changes that would lead to different expected price levels inthe two periods have been removed
Substituting equation (1129) into (1122) we thus obtain the following completegovernment objective function under rational expectations
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1b[πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) minus πlowast]2
]
from which we see that constant monetary growth will not achieve a constant rateof inflation unless we neglect the lagged disturbance terms16 To obtain the resultof an optimal monetary rule in the form of a constant rate of growth in the moneysupply we must further simplify and neglect the lagged disturbance terms If we
Policy 179
for the moment ignore the lagged stochastic terms εtminus1 and utminus1 we have theobjective
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ b(πmt minus πlowast)2 + σ 2e +
(α2
β1
)2
σ 2u
]
As before to minimize the loss requires that one reduces the random variation inthe money supply to zero (if feasible) so that σ 2
e = 0 That is the optimal monetaryrule is ldquodeterministicrdquo Further given an inflation objective and no reason otherthan monetary policy for prices to change the obvious optimal rule is simply toset the determinant rate of change in the money supply equal to the desired rate ofinflation That is the optimal policy is
πmt = πlowast
Note that this rule assumes no shocks to the economy If there were a steady rateof increase in output (and thus the real demand for output) then that would raisethe optimal constant rate of change in the money supply to achieve a given rate ofchange in prices Thus Friedmanrsquos ldquo3 percentrdquo rule for monetary growth presumeda 3 percent growth in real output so as to be consistent with a zero rate of inflation
Rules versus discretion monetary policy and timeinconsistency
The SargentndashWallace ineffectiveness proposition has been used by many to supportarguments for the government not adopting activist policy rules to offset fluctua-tions ndash particularly downturns ndash in an economyrsquos real output reflecting demand-sidedisturbances The reason as we have seen is that such policies have no real effectsonce one assumes rational expectations although the attempt can lead to higherinflation
Yet as Barro and Gordon (1983) point out
empirical studies indicate the presence of countercyclical monetary policyat least for the post-World War II United States ndash rises in the unemploymentrate appear to generate subsequent expansions in monetary growth Within thenatural rate framework it is difficult to reconcile this countercyclical behaviorwith rationality of the policymaker
As Barro and Gordon go on to say ldquoa principal object of our analysis is to achievethis reconciliationrdquo That is rather than saying what policy rules governmentshould follow Barro and Gordon want to explain why government acts the way itdoes ndash thus the term ldquopositive theoryrdquo in the title of their paper
180 Policy
The BarrondashGordon approach combines a number of topics that we haveconsidered before First they utilize a natural rate model like the Lucas modelSecond they assume rational expectations And third and most interestingly theyprovide us with a nice example of the phenomenon of ldquotime inconsistencyrdquo in thecontext of optimal monetary policy when there exists the potential Phillips curvetradeoff
Contrasting the BarrondashGordon and SargentndashWallace policyenvironments
In the SargentndashWallace view of the optimal monetary policy choice it is assumedthat the policy-maker makes a once-and-for-all decision with respect to the partic-ular monetary policy rule (reactive or not) Under certain assumptions as we haveseen this leads to the natural conclusion that optimal monetary policy entails thesimple rule of a constant rate of growth in the money supply so that the resultingaverage rate of inflation equals the desired level If the desired level were zerothen inflation would be set equal to zero through appropriate monetary policy
Barro and Gordon (1983 598) have questioned this result on the basis that ldquotheremay be no mechanism in place to constrain the policymaker to stick to the rule astime evolvesrdquo The result is that the policy-maker decides each period the optimalmonetary policy to follow In other words ldquothough the objective function anddecision rules of private agents are identicalrdquo Barro and Gordon obtain differentresults from Sargent and Wallace because ldquothe problems differ in the opportunitysets of the policymakerrdquo Below we illustrate the exact nature of this differenceby first showing how Barro and Gordonrsquos setup provides an example of the ldquotimeinconsistency problemrdquo not present in the Sargent and Wallace problem and thenderive the equilibrium for an economy characterized by the policy-makers whoare allowed each period to pick a potentially new optimal monetary policy
Time inconsistency an example in the context of optimalmonetary policy
In an important paper on the ldquotime inconsistencyrdquo problem of optimal policyKydland and Prescott (1977) point out situations in which the optimal policiesdecided at time t would be changed at time t + 117 In the context of monetarypolicy as Kydland and Prescott note
the reason that such policies are suboptimal is not due to myopia The effect of(monetary policy) upon the entire future is taken into consideration Rather thesuboptimality arises because there is no mechanism to induce future policy-makers to take into consideration the effect of their policy via the expectationsmechanism upon current decisions of agents
Let us see what this means by way of a concrete example
Policy 181
A simple example of ldquotime inconsistencyrdquo following Kydland and Prescott isconstructed below To simplify the discussion somewhat for the moment we ignoreuncertainty and restrict our attention to two periods In this setting the govern-ment through monetary policy can determine each period the actual inflation raterather than the expected rate of inflation For periods 0 and 1 the choice variablesfacing the monetary policy-maker are thus π0 the actual inflation in period 0 andπ1 the actual inflation in period 1 The inflation choice impacts the unemploymentrate in that U0 and U1 the unemployment rates in periods 0 and 1 are given bythe Phillips curve formulation (1118)
U0 = Un minus α(π0 minus Eminus1π0)
U1 = Un minus α(π1 minus E0π1)
where the state variables Eminus1π0 and E0π1 denote the expected rate of inflation inperiods 0 and 1 respectively In periods 0 and 1 the objective for periods 0 and 1is to minimize the simple quadratic forms
Z0 = a(U0 minus kUn)2 + b(π0 minus πlowast)2
Z1 = a(U1 minus kUn)2 + b(π1 minus πlowast)2
where the constants a and b are positive It is assumed that k lt 1 to capture theidea that distortions exist in the economy that an activist policy can address18
Substituting in the ldquoPhillips curverdquo relations in period 0 the present value of theobjective function is
Z = Z0 + β middot Z1 = a[(1 minus k)Un minus α(π0 minus Eminus1π0)]2 + b[π0 minus πlowast]2
+ βa[(1 minus k)Un minus α(π1 minus E0π1)]2 + βb[π1 minus πlowast]2
where β is the constant discount factorLet us presume that the expectations of inflation the ldquostaterdquo variables are
exogenous and equal to the desired level each period (ie Eminus1π0 = E0π1 = πlowast)In period 0 the optimal inflation rates for periods 0 and 1 (as determined bymonetary policy) are then
partZpartπ0 = minus2aαq0 + 2b(π0 minus πlowast) = 0
partZpartπ1 = β[minus2aαq1 + 2b(π1 minus πlowast)] = 0
where qi = (1 minus k)Un minus α(πi minus πlowast) i = 0 1 In period 1 the objective is
Z1 = a[(1 minus k)Un minus α(π1 minus E0π1)]2 + b[π0 minus πlowast]2
and the optimal solution given π0 and E0π1 = πlowast is thus
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
182 Policy
Comparing this first-order condition to partZ0partπ1 it is clear that in the case ofexogenous expectations the optimal solution is time-consistent Further it impliesa rate of inflation greater than the expected (optimal) rate of πlowast each period toachieve an unemployment rate below the natural This follows given that one ofthe objectives is to approach a level of unemployment equal to kUn with k lt 1
What happens however if individualsrsquo expectations of inflation for period 1 isa forecast that correctly anticipates the optimal future policy decision That is wehave ldquorational expectationsrdquo such that in a deterministic world
E0π1 = h(0) = π1
where 0 is the information set at the end of period zeroWhat the optimal inflation policy is now depends on whether policy-makers
can ldquocommitrdquo to future policy actions Let us start by assuming they can placeconstraints on future policy We know that in period 0 the optimal solution for π1is then given by
βpartZ1
partπ1+ partZ
partE0π1
dh(middot)dπ1
= β[2b(π1 minus πlowast)] = 0
Since dh(middot)partπ1 = 1 the optimal planned (at time 0) rate of inflation for period 1is equal to the desired level πlowast This is the SargentndashWallace ineffectivenessproposition
However once period 1 occurs E0π1 is set (let us assume it equals πlowast) If thepolicy-maker was not then constrained by the prior specification of a monetarypolicy to achieve a rate of inflation equal to πlowast they would choose π1 such that
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
which implies π1 gt πlowast The ldquotime inconsistencyrdquo arises as you can see becauseparth(middot)partπ1 = 0 As Barro and Gordon state ldquothe term lsquotime inconsistencyrsquo refersto the policymakerrsquos incentives to deviate from the rule when private agents expectit to be followedrdquo
Equilibrium when monetary rules are not enforceable
As Barro and Gordon note in the time inconsistency problem ldquoconstraints onfuture policy actions are infeasible by assumptionrdquo In contrast in the SargentndashWallace view
rules are enforceable so that the policymaker can commit the course of futurepolicy (and thus of expectations) In the former case the time-inconsistentsolution is not an equilibrium given the problem facing the policymaker Inthe latter case the incentives to deviate from the rule are irrelevant sincecommitments are assumed to be binding
(Barro and Gordon 1983 599)
Policy 183
As the above extract suggests in the ldquotime inconsistency caserdquo we have not yetcharacterized a rational expectationsrsquo equilibrium since in our example individu-alsrsquo expectations of inflation for period 1 were incorrect According to Barro andGordon there are three features of an equilibrium The first is ldquoa decision rulefor private agents which determines their actions as a function of their currentinformationrdquo These actions of private agents based on current information aresummarized by the Phillips curve
Ut = Un + ζt minus α middot (πt minus Etminus1πt) (1130)
where ζt a random variable with zero mean and variance σ 2z has been added
to denote a real shock that affects the natural unemployment rate for the currentperiod only19
The second feature of an equilibrium is ldquoa policy rule which specifies the behav-ior of policy instruments as a function of the policymakerrsquos current informationsetrdquo This policy rule is given by the choice of inflation rates πt+i i = 0 1 2 by the monetary authorities with the following objective
min Et
⎡⎣ infinsum
i=0
β iZt+i|It
⎤⎦
where Zt+i equiv a(Ut+i minus kUn)2 + b(πt+i minus πlowast)2 a gt 0 b gt 0 and 0 le k le 1
The term It denotes the initial state of information and 1 gt β gt 0 is the constantdiscount factor (β = 1(1+rreal) where rreal is the exogenous real rate of interest)For simplicity we will assume that the optimal level of inflation is zero so thatπlowast = 0
The third feature is an expectations function which determines the expectationsof private agents as a function of their current information Assuming that ldquothepublic understands the nature of the policymakerrsquos optimization problem in eachperiodrdquo then Etminus1πt = πt where πt is the optimal choice of inflation by thepolicy-maker for period t
Combining the information contained in our discussion of the first and secondfeatures of equilibrium the policy-makerrsquos optimal choice of πt minimizes
EtZt = Et[a(Ut minus kUn)2 + b(πt)
2]= Et[a((1 minus k)Un minus α(πt minus Etminus1πt))
2 + b(πt)2]
Given that Eξt = 0 the expression to be minimized by the policy-maker can bewritten as
EtZt = [a((1 minus k)Un minus α(πt minus Etminus1πt))2 + b(πt)
2]A critical point to note is that each period the policy-maker inherits Etminus1πt andtakes that expected rate of inflation as given in the above optimal choice of πt
184 Policy
The first-order condition for period t is thus
partEtZtpartπt = 2a((1 minus k)Un minus α(πt minus Etminus1πt))(minusα) + 2b(πt) = 0
which can be simplified and rearranged to obtain the following expression for theoptimal inflation rate
πt = aα
b[(1 minus k)Un minus α(πt minus Etminus1πt)] (1131)
From the third feature of equilibrium we know that the public realizes the problemfaced by the policy-maker in terms of choosing the optimal rate of inflation forperiod t as defined by (1131) and thus Etminus1πt = πt so that the second term dropsout and we have
πt = aα
b(1 minus k)Un = Etminus1πt gt πlowast = 0 (1132)
as long as k lt 1 We thus have that expectations are rational and individualsoptimize subject to these expectations Since Etminus1πt = πt we have that EtUt = UnThus ldquothe equilibrium solution delivers the same unemployment rate and a higherrate of inflation at each daterdquo than is the case in the SargentndashWallace problem wherethere is a ldquorules-type equilibriumrdquo Given the optimal rate of inflation πlowast = 0a rules-type equilibrium with rational expectations would have the actual rate ofinflation equal to zero
As Barro and Gordon (1983 608) conclude
under a discretionary regime the policymaker performs optimally subject toan assumed inability to commit future actions The framework assumes ratio-nality within the given institutional mode Excessive inflation apparentlyunrewarding countercyclical policy responses and reactions of monetarygrowth and inflation to other exogenous influences can be viewed as prod-ucts of rational calculation under a regime where long-term commitments areprecluded
The model stresses the importance of monetary institutions which deter-mine the underlying rules of the game A purely discretionary environmentcontrasts with regimes such as a gold standard or paper-money constitutionin which monetary growth and inflation are determined via choices amongalternative rules (the SargentWallace approach) The rule of law or equivalentcommitments about future governmental behavior are important for inflationjust as they are for other areas that are influenced by possibly shifting publicpolicies
An alternative to the ldquorule of lawrdquo is reputation As Blanchard and Fischer(1989 599) state
reputation is the most interesting and persuasive explanation of how gov-ernments avoid dynamic inconsistency Governments know that they can do
Policy 185
better than the shortsighted solution over the long run They hope by actingconsistently over long periods to build a reputation that will cause the privatesector to believe their announcements The key to the answer [to the ques-tion of whether reputation can sustain the optimal policy] is the specificationof private sector expectations of how the public reacts to broken promises
Conclusion
This chapter has presented an overview of many of the major themes and issuesfaced in macroeconomics in terms of the conduct of monetary policy The roles ofexpectations the ldquoineffectiveness propositionrdquo the modified Phillips curve rulesversus discretion time inconsistency credibility and enforcement are all dealt within this chapter Perhaps the major contribution of this chapter is that it highlightsmany of the issues and problems that real-world monetary authorities face whendeciding what course of action to take
12 Open economy
Introduction
The field of international economics can be roughly categorized as concernedwith either the real side or the finance side of international issues The ldquorealrdquoside focuses on such basic questions as why trade occurs between countries whatdetermines the terms of trade and how government policies such as tariffs do orquotas affect trade The ldquofinancerdquo side makes explicit the fact that countries differin currencies in order to focus on such questions as what determines exchangerates and how macroeconomic shocks in one country (eg a change in the supplyof money) affect its economy and the economies of the countries with which ittrades
In the discussion below we examine questions more like those considered by theldquofinancerdquo side Namely we extend simple macroeconomic analysis to an ldquoopenrdquoeconomy that is an economy that incorporates a foreign sector This analysisdiffers from traditional macroeconomic analysis of a ldquoclosedrdquo economy in thefollowing respects
1 There is trade of composite commodities and financial assets between twocountries For the moment we assume not only that the two countries producedifferentiated output but also that the financial assets issued by the firms andgovernment of one country are not perfect substitutes for the financial assetsissued by firms and government of the second country
2 The two economies are isolated in that individuals in each country can onlypurchase or sell labor services in their own labor markets
3 The two economies are differentiated in that each has its own media ofexchange This means that an individual in the domestic economy whoseeks to purchase foreign goods must exchange domestic money for foreignmoney1 Similarly an individual in the foreign country must exchange hisforeign money for the domestic money to purchase the domestic goods Suchexchanges take place in the foreign exchange market at the prevailing ldquoforeignexchange raterdquo or the price of one currency in terms of the second currencyWe will let et denote the exchange rate for domestic money For instanceif the domestic country is the USA and the foreign country is Japan then
Open economy 187
et on December 1 1989 was 1434 yen in that 1434 yen = 1 dollar Thisimplies that 1et the price of dollars in terms of yen was 0006973 dollarson December 1 1989
4 There is distinct government policy (fiscal and monetary) in each country inthat each countryrsquos government determines spending taxation and monetarypolicy for its economy
To understand how macroeconomic analysis is altered in the above ldquoopen econ-omyrdquo setting it is instructive to first consider the nature of the constraints faced bythe various participants in an open economy We then examine how the behaviorof these participants is affected by changes in such variables as exchange ratesWith that background we are ready to consider some simple examples of openmacroeconomic analysis such as the effect of a change in the money supply in thecontext of an open economy neoclassical model
Open economy participants constraints and Walrasrsquo law
To begin our task of modeling an open economy recall that it is typical ofmacroeconomic models to simplify the economy by grouping markets into broadcategories In a closed economy there were three important markets the outputfinancial and labor markets One could add to this the money ldquomarketrdquo sinceequilibrium required that the demand for money equaled supply In an open econ-omy we add another market the foreign exchange market where the currencyof one country is traded for that of the other country In a closed economy theparticipants in the various markets in the economy could be placed in one of fivecategories households firms government (fiscal side) the central bank and pri-vate depository institutions In an open economy we add one more participantforeigners
As we have seen a common theme of macroeconomic models is their emphasison the interdependencies among markets Macroeconomics recognizes that eventsin one market imply changes in other markets as well This ldquogeneral equilibriumrdquoapproach contrasts with the ldquopartial equilibriumrdquo approach of microeconomicswhich is less concerned with how changes in one market affect all other marketsTo fully understand the links across markets it is useful to specify the ldquofinancingconstraintsrdquo faced by the participants in the various markets Our discussion ofopen economy macroeconomics thus begins by introducing these financing con-straints for firms households government (fiscal side) the central bank privatedepository institutions and foreigners We then sum these constraints to obtain amodified Walrasrsquo law
The financing constraints in an open economy
We start our discussion of the constraints faced by the participants in an openeconomy by considering the financing constraint faced by the new participant for-eigners Foreigners purchase domestic output and financial assets Let X d
t denote
188 Open economy
foreignersrsquo demand for domestic output (exports) and let net Adft denote foreignersrsquo
desired real change in their holdings of domestic financial assetsTo purchase domestic output and financial assets foreigners must first acquire
domestic money That is foreigners must finance these purchases either fromincome generated from their previously acquired holdings of domestic financialassets or by supplying foreign currency in exchange for the domestic currencyin the foreign exchange markets In particular we have the following foreignerfinancing constraint
X dt + net Ad
ft minus αf (zBf pt + dt) minus FCst etpt = 0 (121)
Note that for simplicity we limit foreignersrsquo holdings of domestic financial assetsto private financial assets (ie we do not have them holding government bonds)Further we assume foreignersrsquo portfolio of holdings of domestic financial assetsis identical to2 that of domestic households and that they own αf 1 gt αf ge 0 ofthe total value of financial assets issued by domestic firms Thus αf (zBf pt + dt)
is the real income foreigners gain from their holdings of domestic financial assets3
In equation (121) the term FCst etpt denotes foreignersrsquo real supply of foreign
currency in the foreign exchange market FCst is foreignersrsquo supply in units of the
foreign currency in period t Multiplying by 1et the price of the foreign currencyin terms of the domestic currency puts this in terms of the domestic currencyDividing by the price level pt then puts it in real terms (ie in terms of the domesticcomposite commodity) Embedded in the desired change in foreignersrsquo holdingsof domestic financial assets are changes in the desired holdings by foreign centralbanks (ie changes in foreign central banks ldquointernational reservesrdquo)
In addition to the above new constraint that accompanies the introduction ofa new participant to the economy foreigners we have constraints faced by thedomestic households the central bank private depository institutions government(fiscal side) and firms In the case of households and the central bank we modifythe constraints introduced in previous chapters to incorporate the exchanges ofcommodities and financial assets with foreigners In particular for householdsthe budget constraint in an open economy can be expressed by
bdt + cd
t + zdt + (M d
t minus M )pt + net Adht + net AFd
ht minus [yt minus αf (zBf pt + dt)
+ α(zBff pft + pftdft)etpt minus δK minus Tnt] = 0 (122)
where the new term zdt denotes real imports net Ad
ht denotes householdsrsquo desiredchange in their real holdings of foreign assets and α(zBff pft + pftdft) denotes theincome (in terms of the foreign currency) gained from holding the proportion aof the financial assets issued by foreign firms4 This income (in foreign currency)times the price of foreign currency in terms of domestic currency (1et) gives thedomestic currency value of income from foreign asset holdings Dividing by theprice level pt puts the income in terms of the composite commodity (ie in realterms)
Open economy 189
Total consumption of commodities in period t is now the sum of cdt purchase
of output and zdt imports of commodities produced abroad5 Similarly the total
desired change in financial assets is the sum of net Adht the desired change in hold-
ings of domestic financial assets and net AFdht the desired change in holdings of
foreign financial assets Note that in incorporating the firm distribution constraintinto the household budget constraint we have taken into account the fact that notall domestic output yt is income to domestic households since foreigners own theshare αf of domestic firms On the other hand households have an additionalsource of income from the ownership of foreign financial assets
For the central bank the (stock) financing constraint equates the sum of thechange in real (domestic) financial asset holdings and international assets to thereal change in the monetary base or
net Adct + net AFd
ct minus (MBst minus MB)pt = 0 (123)
where net Adct = pbt(Bd
gct minus Bgc)pt MB = R + C and MBst = Rs
t + Cst 6 We
have added the term net AFdct to denote the real demand for additional international
(foreign currency denominated) assets by the central bank In particular
net AFdct = pfbt(B
dfct minus Bfc)etpt
The quantity pfbt(Bdfct minusBfc) is the change in the amount of foreign assets demanded
by the central bank in terms of the foreign currency pfbt denotes the price of foreignbonds in terms of foreign currency and the numbers of such bonds demanded andinitially held by the domestic central bank are denoted respectively by Bd
fct and
Bfc Multiplying this quantity by the price of foreign currency in domestic currencyterms (1et) gives the domestic currency value of foreign assets demanded by thecentral bank Then dividing by the price level pt puts the net demand for interna-tional assets by the central bank net AFd
ct in terms of the composite commodity(ie in real terms)
For private depository institutions the (stock) financing constraint indicates thatprivate depository institutions can be viewed as financing additions to reserves andto financial assets holdings by creating deposits or
net Atpb + (Rd
t minus R)pt minus (Dst minus D)pt = 0 (124)
where net reserves demanded or initially held are denoted by Rdt and R respec-
tively and checkable deposits supplied or initially outstanding are denoted byDs
t and D respectively For simplicity we assume that all private demands forforeign financial assets as well as for foreign commodities are captured in thehousehold budget constraint To the extent that private banks purchase foreignfinancial assets they can be viewed as acting as financial intermediaries forhouseholds
190 Open economy
For the domestic government (fiscal side) the financing constraint is
gdct minus T lowast
nt minus net Asgt = 0 (125)
where T lowastnt = Tt minustrt minusz(Bgh+Bgp)pt tr denotes transfer payments and net As
gt =Pbt(Bs
gt minusBg)pt Equation (125) incorporates the ldquoflowrdquo financing constraint forthe central bank In doing so it is assumed that the central bank claims on realresources just exhaust its interest payments on government debt holdings plus anyincome associated with central bank holdings of foreign assets Finally for firmsthe financing constraint on capital purchases is given by
Idnt + ψ(I d
nt) minus net Asft = 0 (126)
Note that we assume for simplicity that neither firms nor the government purchaseforeign commodities
Walrasrsquo law and the balance of payments
We now sum the constraints faced by the six participants in an open economy(foreigners households the central bank private banks the government andfirms) as given by equations (121)ndash(126) In doing so assume equilibrium withrespect to the demand and supply of bank reserves (ie the Rs
t part of MBst in the
central bank constraint equals Rdt in the private banks constraint) and note that
the initial monetary base MB equals R + C that the money supply is defined byM s
t = Dst + Cs
t and that the initial money supply is given by M = D + C Weobtain
[Bdt + Cd
t + X dt + gd
ct + δK + I dnt + ψ(I d
nt) minus yt] + [M dt pt minus Mpt]
+ [net Adht + net Ad
ft + net Adct + net Ad
pt minus net Asgt minus net As
ft]+ [zdlowast
t + net AFdht + net AFd
ct minus FCst etpt] = 0 (127)
where zdlowastt + net AFd
ht + net AFdct minus FCs
t etpt denotes householdsrsquo imports notfinanced out of income generated from foreign asset holdings Equation (127) isan example of the modified Walrasrsquo law for an open economy7 As we discussbelow (127) can be viewed as stating that the sum of excess demands in fourmarkets must equal zero
The first term in (127) reflects excess demand in the output market where thedemand now includes foreignersrsquo demand for domestic output (exports) Note thatby adding and subtracting import demand we could express the excess demandfor output in the form
bdt + cdlowast
t + (xdt minus zd
t ) + gdct + δK + I d
nt + ψ(I dnt) minus yt (128)
where the term cdlowastt denotes householdsrsquo total consumption in terms of both domes-
tic and foreign output (ie cdlowastt = cd
t + zdt ) Excess demand in the output market
Open economy 191
is often expressed this way with the consumption term denoting total householdconsumption such that the ldquonetrdquo export demand term (xd
t minus zdt ) appears
Setting the excess demand for output term in (127) to zero gives us the ISequation in an open economy The second term in (127) reflects the excess demandfor money Note that we assume that only domestic households desire to hold thedomestic money (foreigners seek the domestic money in the foreign exchangemarket not to hold but as a means to purchase the domestic output or financialassets) Setting this second term in (127) to zero gives us the standard LM equation
The third term in (127) reflects excess demand in the financial market In goingto an open economy we add to the demand side of the financial market a demandfor domestic financial assets by foreigners (which could include private foreignagents as well as foreign central banks) In addition note that householdsrsquo demandfor domestic financial assets (net Ad
ht) no longer reflects householdsrsquo total demandfor financial assets since they now have the option of purchasing foreign financialassets (net AFd
t )The fourth and final term in (127) reflects excess real demand for foreign
currency in the foreign exchange market8 As before we can view output ( yt)or the price of output ( pt) as determined in the output market and the domesticinterest rate (rt) as determined in the financial market Now we can view the foreignexchange rate (et) as determined in the foreign exchange market The demand forforeign currency reflects zdlowast
t the demand associated with householdsrsquo purchases offoreign commodities that could not be financed from the foreign currency earningsof their holdings of foreign financial assets net AFd
ht the demand for foreigncurrency associated with householdsrsquo purchases of additional foreign financialassets and net AFd
ct the demand associated with the central bankrsquos desired changein international reserves Note that the real demand for the foreign currency couldinstead be stated as the real supply of the domestic currency in the foreign exchangemarket
FCst etpt denotes the real supply of foreign currency or real demand for the
domestic currency in the foreign exchange market which from (121) reflectsforeignersrsquo purchases of financial assets and exports net of those financed throughdomestic currency earnings on financial assets held by foreigners Foreignersrsquopurchases of financial assets include both foreign private (ie foreign householdsrsquo)and foreign public (ie foreign central bank) purchases
The balance of payments accounts
Let us assume for now that the domestic economy discussed above is the USA TheUS Department of Commerce actually measures the various sources of the demandfor and supply of dollars in the foreign exchange markets cited above These data ofinternational transactions are presented as the US balance of payments accountsTable 121 summarizes its major components Transactions are categorized aseither sources of the real demand for or the real supply of dollars in the foreignexchange market for the US dollar9 Equivalently they reflect the real supply ofor real demand for foreign currency in the foreign exchange market
192 Open economy
Table 121 The US balance of payments accounts (in ldquorealrdquo terms)
Real demand for dollars Real supply of dollars
1 US exports of goods and services xt US imports of goods and services zt2 Transfers (interest and dividends to US
holders of foreign financial assetsgovernment grants and gifts)
Transfers (interest and dividends toforeign holders of US financial assetsgovernment grants and gifts)
α(zBff pft + pftdft)etpt αf (zBf pt + dt)
3 Foreign purchases of US financialassets (capital inflow) net Ad
ft
US purchases of foreign financial assets(capital outflow) net AFd
h + net AFdct
The net demand for dollars associated with the first component of the balanceof payments accounts (exports minus imports) is called the balance of trade Ifthe balance of trade is negative as it has been recently for the USA then the USAis said to experience a balance of trade deficit If it is positive as was the case for106 consecutive years from the end of the Civil War to 1971 as well as throughmost of the 1970s then a balance of trade surplus is said to exist
The third component of the demand for and supply of dollars in the balance ofpayments accounts is the capital account This capital account can be divided intoa private part and a public part On the demand side the private part measures thereal dollars demanded by foreigners other than foreign central banks to financepurchases of US financial assets On the supply side the private part measuresthe real dollars supplied by US households to buy foreign financial assets Thesecurrency exchanges are called private international capital flows Private inter-national capital flows associated with the demand for dollars are referred to asUS private international capital inflows since they reflect the inflow of foreigncurrency due to private foreignersrsquo purchases of US financial assets Private inter-national capital flows associated with the supply of dollars are referred to as USprivate international capital outflows since they reflect the outflow of dollars dueto US householdsrsquo purchases of foreign financial assets
Summing the net demand (demand minus supply) for dollars associated withthe first two components plus the private international capital flows and adjustingfor measurement errors (the discrepancy term) we obtain what is called the USbalance of payments10 In years in which the balance of payments is negative it isreferred to as a balance of payments deficit On the other hand a positive balanceof payments is called a balance of payments surplus
When there is a surplus or deficit in the balance of payments accounts thenequality between the real demand for and real supply of dollars is brought about byan offsetting deficit or surplus on what is known as ldquothe official reserve transactionbalancerdquo The official reserve transaction balance reflects the intervention into theforeign exchange market by the US central bank (the Fed) andor by foreign centralbanks This is the ldquopublicrdquo part of international capital flows Whenever there existsa balance of payments deficit in the USA then (on net) central banks demand USdollars in the foreign exchange markets If this were solely the US central bank
Open economy 193
intervening this means that net AFdct wasis negative (the Fed wasis a net supplier
of dollars in the foreign exchange market) and the US central bank would havelostlose international reserves
Behavior in an open economy
With an understanding of the constraints faced by the various participants in anopen economy we now consider the behavior of these participants in particularthe determinants of imports (zd
t ) exports (xdt ) private international capital inflows
(a part of net Adft) and private international capital outflows (net AFd
ht) In the firstpart of this section we examine how a change in the foreign exchange rate for thedomestic currency (to be concrete the US dollar) can alter the relative price offoreign goods and thus lead to a change in the division of household consumptionbetween purchases of domestic goods and purchases of foreign goods11 We alsoexamine other factors that influence US imports of goods and services and thusthe real demand for foreign currency (real supply of US dollars) in the foreignexchange markets
In the second part of this section we consider the factors that influence howhouseholds divide their real accumulation of financial assets between US stocksand bonds and the financial assets of foreign countries As we saw above house-hold purchases of foreign financial assets constitute capital outflows since inorder to make these purchases households must demand foreign currency (supplydollars) in the foreign exchange market We will see how differences in foreignand domestic interest rates and the expected appreciation or depreciation of theUS dollar determine the relative returns on foreign and domestic financial assetsThese relative returns in turn affect householdsrsquo portfolio choices between foreignand domestic financial assets and the real demand for foreign currency (real supplyof dollars) in the foreign exchange market
In the third and fourth parts of this section we turn to the other side of theforeign exchange market to examine determinants of foreignersrsquo purchases of USgoods and services and of US financial assets In particular we consider how suchfactors as exchange rates affect foreignersrsquo demand for US goods (US exports)and how such factors as relative interest rates and the expected rate of changein the exchange rate affect foreignersrsquo demand for US financial assets We finishthis section by illustrating graphically the behavior discussed in terms of the realdemand for and supply of the domestic currency (the US dollar)
Householdsrsquo demand for imports
When deciding whether to purchase foreign or domestic goods households look attheir relative prices To be concrete consider two countries The domestic countryis the USA and the foreign country is Japan The relative price of Japanese goods isthen the real quantity of US goods that must be sacrificed to purchase the foreigngood For example if the price of a Japanese car is $6000 and the price of aUS computer is $1500 then the relative price of a Japanese car in terms of US
194 Open economy
computers is 4 computers If the relative price of Japanese cars rises the USAwill import fewer Japanese cars The relative price of foreign goods sometimesreferred to as the terms of trade is an important determinant of the quantity ofimports The relative prices of foreign goods depend on the dollar prices of USgoods the prices of foreign goods in their own currency and foreign exchangerates (the price of one currency in terms of a second currency) The simple examplethat follows illustrates this point Suppose that the price of a US computer is $1500and that in Japan the price of a Japanese car is 600000 yen The third ldquopricerdquo thatwe need to know in order to compute the relative price of a Japanese car in terms ofUS computers is the foreign exchange rate in particular the price of a yen in termsof dollars Suppose that it takes et yen to buy one dollar in the foreign exchangemarkets Then it takes 1et dollars to buy one yen Returning to our example ofJapanese cars and US computers if et = 100 yen per dollar and the Japanese carhad a yen price of 600000 then its dollar price would be 6000
In general the calculation of the relative price of Japanese goods is thus
Relative price of Japanese goods (in terms of US goods)
= Yen price of Japanese goods ( pft)
times Price of yen in terms of dollars (1et)Dollar price of US goods ( pt)
= pft
etpt
According to this expression a rise in the yen price of Japanese goods ( pft)
raises their relative price Similarly a fall in the dollar price of US commodities( pt) raises the relative price of Japanese goods Finally a rise in the price of yen interms of dollars (1et) also increases the relative price of Japanese goods12 Sinceall three changes mean a higher relative price of Japanese goods and thus anincrease in the cost to US buyers of Japanese goods in terms of US goods forgoneall three changes reduce US imports of Japanese goods The above relative priceexpression for a basket of foreign goods is sometimes termed the real exchangerate for foreign goods ndash that is
Real exchange rates (foreign goods)
= Dollar price of foreign goods
Dollar price of US goods= pft(1et)
pt= pft
ptet
While a rise in the relative price of foreign goods will lead to a reduction in thequantity of foreign goods bought we have to be careful not to infer from this thatUS import demand will necessarily be inversely related to the relative or real priceof imports The reason for this is that our measure of US real import demand is interms of the US good and services not in terms of the foreign good An examplewill highlight this distinction
Open economy 195
Suppose that the dollar depreciates (et falls) With the implied appreciation offoreign currency (1et rises) the price of the foreign good in terms of US goodsbecomes greater as our expression for the real exchange rate for foreign goods( pftpt et) indicates With a higher price fewer foreign goods will be purchased ndashthis is clear This by itself would suggest a fall in the value of imports into theUSA But the higher price also means that each foreign good purchased will costmore in terms of US goods that must be sacrificed This by itself would suggest arise in the value of US imports (measured in terms of US goods that must be paidto obtain the imports) The net impact of a change in the relative price of foreigngoods on US imports measured in terms of US goods depends on which of thesetwo effects is stronger
It is typically assumed that over time the effect of a change in the relative pricesof imports on the quantity of imports purchased dominates so that the value ofimports in terms of US goods will fall with a rise in the relative price of importsThis is what we will assume13 Formally this condition requires that the priceelasticity of demand for foreign goods be greater than one That is a 1 percentincrease in the price of foreign goods must cause a greater than 1 percent reductionin the amount of foreign goods that US households demand14
Besides the relative prices of imports real disposable income affects US importdemand An increase in disposable income can lead to a rise in household con-sumption demand In an open economy with foreign trade a rise in consumptiondemand means an increase in purchases not only of domestically produced goodsbut also of foreign goods Thus householdsrsquo import demand is directly related totheir disposable income To summarize household real import demand zd
t dependsinversely on the relative price of foreign goods pftetpt and directly on disposableincome yt minus δK minus Tnt
zdt = zd
t ( pftetpt yt minus δK minus Tnt ) (129)
Capital outflows householdsrsquo demand for foreign financial assets
When deciding whether to purchase US or foreign financial assets householdscompare domestic and foreign rates of return The nominal rate of return on USfinancial assets is simply the money interest rate rt The comparable nominal rateof return on foreign financial assets is not so simple to identify To explain howto compute this return which we will denote rlowast
t let us suppose that a householdlends one dollar in the foreign financial market
If the price of a dollar is et units of the foreign currency say yen then in termsof the foreign currency the household lends et yen If foreign financial assets offerthe interest rate rft then one period from now the household will have et(1 + rft)
yen At that time the household can convert these yen holdings back to dollars atthe exchange rate then existing At the time the money is lent this future exchangerate may be uncertain15 Let householdsrsquo expectation of this future exchange ratebe denoted by ee
t+116 Then the household expects to convert its et(1 + rft) yennext year into et(1 + rft)ee
t+1 dollars Subtracting the one dollar with which the
196 Open economy
household started the rate of return to lending in foreign financial markets rlowast isgiven by
rlowastt = (et(1 + rft)ee
t+1) minus 1 = [et(1 + rft) minus eet+1]ee
t+1
We can simplify the above equation by noting that the expected future dollarexchange rate ee
t+1 yen per dollar equals the current exchange rate of et yen times(1 + θe
t+1) where θet+1 is the expected rate of change in the price of a dollar in
terms of yen Substituting the expression et(1+θet+1) for the expected future dollar
exchange rate eet+1 into the above expression for rlowast
t we have
rlowastt = [et(1 + rft) minus et(1 + θe
t+1)]et(1 + θet+1) = (rft minus θe
t+1)(1 + θet+1)
Since the expected rate of appreciation of a dollar is typically small we canapproximate the above expression by
rlowastt = rft minus θe
t+1 (1210)
Equation (1210) has a straightforward interpretation The return to lending in theforeign financial market equals the difference between the foreign interest rateand the expected rate of change in the price of the dollar The return to lendingin foreign financial markets increases with a higher foreign interest rate rft anddecreases with a higher expected rate of increase in the price of the dollar (θe
t+1)The higher the expected rate of increase in the price of the dollar the lower theexpected return to lending in foreign financial markets since for a given numberof dollars sold for foreign currency at the start of the period fewer dollars can bebought back at the end of the period
When households choose between purchasing domestic and foreign financialassets they compare the domestic interest rate rt with the rate of return to lendingin the foreign financial markets rlowast
t We can thus express household real demandfor additional foreign financial assets as
net AFdht = net AFd
ht(rt rft minus θet+1 ) (1211)
In (1211) an increase in the US interest rate or a fall in the rate of return onforeign financial assets implies a reduction in householdsrsquo real demand for foreignfinancial assets The three dots in (1211) reflect other factors that have been leftunspecified For instance changes in the political stability of foreign governmentsare one unspecified factor that would likely impact on US householdsrsquo demandfor foreign financial assets Equation (1211) suggests that we should add anotherfacet to our previous discussion on householdsrsquo demand for US financial assetsnet Ad
ht In addition to such factors as real income taxes the US money interestrate and the expected rate of inflation household demand for US financial assetsdepends on the expected return to lending abroad rft minus θe
t+1
Open economy 197
Foreignersrsquo demand for exports
Just as US demand for foreign goods depends on relative prices so too isforeignersrsquo demand for US goods based on the relative prices of those goodsThe relative prices of US goods to foreigners measure what foreigners have togive up of their own goods in order to purchase US goods As we have seenthese relative prices depend on the money prices of US goods the money pricesof foreign goods and the exchange rate Considering ldquocompositerdquo goods for eachcountry the relative price of US goods to foreigners or the real exchange rate forforeign goods is given by
Real exchange rates (US goods)
= Dollar price of US goods
Dollar price of foreign goods= pt
pft(1et)= ptet
pft
Note that the real exchange rate for US goods is simply the inverse of the previouslyobtained real exchange rate for foreign goods The price of a dollar in terms ofan index of foreign currencies rose by 70 percent in 1984 and 1985 The resultingrise in the relative prices of US goods contributed significantly to a reduction inforeign demand for US goods and the large US trade deficit of the mid-1980sSimilarly the dramatic fall in the price of a dollar in the subsequent period fromlate 1985 to 1988 led to an increase in US exports Thus we have
xdt = xd
t ( ptetpft ) (1212)
Equation (1212) indicates that export demand falls with a rise in the relative priceof US goods to foreigners
We know from our previous discussion that an increase in US disposable incomeleads to a rise in household purchases of both domestically produced output andforeign goods and services By the same token an increase in foreignersrsquo dispos-able income leads to a rise in their purchases of US goods Thus among the itemsmissing in (1212) that determine foreignersrsquo real export demand xd
t is foreigndisposable income
Capital inflows foreignersrsquo demand for financial assets
In 1960 purchases of US financial assets by foreigners were approximately one-half the amount of purchases of foreign financial assets by US citizens In the USfinancial markets foreign purchases of new US financial assets were less than5 percent of household and depository institution purchases Twenty-five yearslater foreigners were purchasing four times as many US financial assets than theUSA was purchasing abroad In the US financial market close to 30 percent ofnew US financial assets were being purchased by foreigners This dramatic changein capital inflows to the USA is one indication of the growing importance to theUS economy of international trade not only in goods but also in financial assets
198 Open economy
As with US households we will assume that foreigners decide to purchase eitherUS assets or financial assets of their own country by comparing the rates of returnon the two types of financial assets For foreigners the nominal rate of return ondomestic assets is the money interest rate in their own country (rft) The expectedrate of return to foreigners on US financial assets equals the US interest rate plusthe expected change in the price of the dollar in the foreign exchange market (iert + θe
t+1)Not surprisingly the expected return to foreigners lending in US financial mar-
kets increases when the US interest rate rt increases Not as obvious is that thereturn also increases with an increase in the expected rate of change in the price ofthe dollar θe
t+1 This is because foreigners lending in US financial markets converttheir currency to dollars to make the loans When the loans are repaid they thenconvert dollars back to their own currency If the dollar is anticipated to appreciateduring the course of the year then part of their expected return to lending in theUSA is the increase in the value of the dollars (in terms of their own currency)
Summarizing we can express the foreign demand for US financial assetsnet Ad
ft as
net Adft = net Ad
ft(rft rt θet+1 ) (1213)
Equation (1213) indicates that foreignersrsquo demand for US financial assetsincreases if the US interest rate (rt) rises or the expected rate of change in theprice of the dollar (θe
t+1) increases or the foreign interest rate (rft) falls
The foreign exchange market the real demand and supply of dollars
The change in the price of a dollar (in terms of a second currency) affects the realquantity of dollars supplied and demanded in the foreign exchange market Letus start with the price of a dollar set at the equilibrium level of (et)0 let us say100 yen If the price of a dollar now falls to (et)1 say 50 yen then this depreciationof the dollar (appreciation of the yen) leads to a reduction in the real quantity ofdollars supplied from Q0 to Q1
A fall in the price of a dollar from 100 to 50 yen means a rise in the dollar priceof a yen from 001 to 002 dollars or from 1 cent to 2 cents Even though there isno increase in the yen price of Japanese goods the dollar price of Japanese goodsrises For example a 600000 yen Japanese car that formerly cost 6000 dollars(600000 times 001) now costs 12000 dollars (600000 times 002) If the dollar prices ofUS goods have not changed then the relative or real prices of Japanese cars haverisen In our example this means that US households must give up an increasedamount of US goods to obtain one more Japanese car
The depreciation of the dollar and resulting higher relative price for Japanesegoods leads US households to reduce the quantity of Japanese goods demandedHowever as we discussed above the fact that the quantity of Japanese goodspurchased falls does not necessarily mean that the quantity of dollars suppliedin the foreign exchange market also falls There are two countervailing forces
Open economy 199
at work here While the purchase of fewer Japanese goods would reduce thequantity of dollars supplied the fact that each Japanese good has a higher pricewould increase the quantity of dollars supplied By assuming that the first effectoutweighs the second effect we conclude that a depreciation of the dollar causesthe real quantity of dollars supplied in the foreign exchange market to fall Thusthere is an upward-sloping supply of dollars curve17
Naturally behind the supply of dollars in the foreign exchange market is notonly US real import demand but also the real demand by households and the UScentral bank for additional foreign financial assets (net AFd
ht + net AFdct) As we
saw above this sum of the US import demand and demand for foreign financialassets can be interpreted as representing not only a supply of dollars but also ademand for foreign currency18
The above discussion also highlights the effect of a change in the price ofa dollar (in terms of a second currency) on the quantity of dollars demandedin the foreign exchange market The depreciation of the dollar (or appreciationof the yen) from (et)0 to (et)1 leads to an increase in the quantity of dol-lars demanded US in real terms The fall in the price increases the quantity of dollarsdemanded because it lowers the relative price of US goods to foreigners A fall inthe price of a dollar from 100 yen to 50 yen causes a rise in the dollar price of theyen from 1 cent to 2 cents Even though there has been no increase in the dollarprice of US goods the yen price of US goods falls and the Japanese increase theirdemand for US goods As a consequence the real quantity of dollars demandedin the foreign exchange market increases
Naturally behind the demand for dollars in the foreign exchange market isnot only foreignersrsquo export demand but also their demand for US financial assets(net AFd
ht) As we have seen this sum of the export demand and foreignersrsquo demandfor US financial assets can be interpreted as representing not only a demand fordollars but also a supply of foreign currency
Simple examples of open economy (static) macroeconomicanalysis
The statement of Walrasrsquo law for an open economy indicates that for static macro-economic analysis of an open economy we need look at only three of the fourexcess demand conditions reflecting the output financial foreign exchange andmoney markets Standard practice is to look at the output money and foreignexchange markets In the neoclassical model these three equations would besolved to obtain the equilibrium price level interest rate and exchange rate ( pt rt and et respectively) In the BarrondashGrossman disequilibrium analysis thesethree equilibrium conditions could be solved to obtain the equilibrium outputinterest rate and exchange rate ( yt rt and et respectively) In the Lucas modelor the Keynesian fixed money wage model these three equilibrium conditionsalong with the appropriate aggregate supply equation could be solved for theequilibrium price level output interest rate and exchange rate ( pt yt rt and et respectively)
200 Open economy
With flexible exchange rates (ie exchange rates determined without theintervention of central banks) the basic change in the macroeconomic analysisis to recognize that the net export component of output demand (seeequation (128)) is sensitive to interest rate changes which implies the IS curve(in (interest rate output) space) is flatter The reason why a higher US interest ratereduces net export demand is that a higher interest rate increases capital inflows(net Ad
ft) and reduces capital outflows (net AFdht) The first change increases the
demand for dollars in the foreign exchange market while the second reduces thesupply of dollars in the foreign exchange market The result is an appreciationof the dollar that reduces exports and increases imports Note that with flexibleexchange rates a change in either the price level or income does not affect netexport demand
Money supply changes in the neoclassical model purchasingpower parity
In general when considering two countries the country with the lower inflationrate will tend to have an exchange rate that is appreciating at a rate approximatelyequal to the difference in inflation rates between the two countries This patternof changes in foreign exchange rates is sometimes said to reflect the purchasingpower parity condition The condition of purchasing power parity means that thepurchasing power of each countryrsquos currency is the same whether the currency isused to purchase domestic goods or foreign goods Purchasing power parity existsfor a monetary shock in the neoclassical model since a money supply change willlead to changes not only in domestic prices but also in foreign exchange rates suchthat relative prices remain constant
Interest-rate parity
There are of course a number of variations to the above analysis For instancewe could assume that domestic and foreign financial assets are perfect substitutesand that the domestic country is sufficiently small in its interactions with foreignfinancial markets that it takes the foreign interest rate rft as a given In this caseof ldquointerest rate parityrdquo since the return to lending at home rt must equal that oflending abroad rlowast
t = rft minus θet+1 we have that
rft = rt + θet+1 (a constant) (1214)
One use of (1214) is in the well-known Dornbusch model (see Dornbusch 1976)Assume that output and the price of output are fixed initially Assume θe
t+1 = 0initially such that rt = rft Now consider an increase in the money supply Tomaintain equilibrium with respect to the demand and supply of money the interestrate rt must decrease According to (1214) the resulting increase in internationalcapital outflows and reduction in international capital inflows must reduce theexchange rate such that it is expected to appreciate at a rate equal to the difference
Open economy 201
between xdt and the new lower rt This is Dornbuschrsquos famous ldquoexchange rate
overshootingrdquo That is the analysis implies a fall in the exchange rate below whatit will ultimately be after income or prices adjust
A second use of (1214) is to combine it with the assumption that exchangerates are fixed (through appropriate central bank intervention in the foreignexchange markets) An example taking this approach is the MundellndashFlemingmodel (Fleming 1962 Mundell 1968)
Conclusion
This chapter has brought together many of the issues associated with analyzinga macroeconomy that participates in the open or global economy In keepingwith the basic model framework of this book the household is now viewed ashaving a demand for imported products as well as domestically produced goodsand services Additionally firms are allowed to sell to agents in foreign coun-tries This flow of goods and services across borders logically leads to a flow offunds Moreover this interconnectedness among trading partners implies that theactions of one country in particular those of the monetary authority may influenceconditions in the other country
Notes
1 Introduction
1 Empirically an economyrsquos total output is measured by real gross domestic product (orindustrial production) employment is estimated from company records or householdsurveys estimates of unemployment are compiled from household surveys or statisticson recipients of unemployment benefits and price indexes are computed to measurechanges in the overall level of prices
2 Leon Walras first derived the result in his book Elements drsquoEconomie Politique Purefirst published in 1874ndash77 (see Walras 1954)
3 An endogenous variable is one that is determined by the model An exogenous variableis one that is taken as given by the model
4 An element that distinguishes both static and dynamic macroeconomic analysis fromArrowndashDebreu analysis is the incorporation of money The exceptions to this in macro-economic analysis are real business cycle theories which for the most part are purelyldquorealrdquo in the sense that money is not an intrinsic part of the analysis
5 Note that an alternative to exogenous expectations is to specify an ldquoexpectation func-tionrdquo If this function relates expectations in certain specific ways to all currentinformation such that the expectation function reflects ldquorational expectationsrdquo thenstatic analysis is converted to dynamic analysis
6 In dynamic models this would be termed ldquoperfect foresightrdquo in a deterministic settingand ldquorational expectationsrdquo in a stochastic setting
7 One could say that the effect of such a variable is implicit in the exogenous level ofexpected future prices of the static analysis
8 A similar example of an incomplete listing of effects is a change in current governmentpolicies A change in current government actions may require changes in governmentactions in subsequent periods that will impact future markets Yet the construction ofstatic analysis does not require such effects to be spelled out
9 Forward markets are markets in which agreements are made that specify the prices atwhich goods will be exchanged in the future Goods are thus in essence indexed by thedate of trade
10 This is the ArrowndashDebreu contingent-claim interpretation of a competition equilibriummodel (see Arrow 1964 Debreu 1959)
11 In some cases one can attain the stationary state under the less restrictive requirementthat certain exogenous variables simply grow at a steady rate over time
12 While classical economists did not fully articulate their model Patinkin (1965) is widelycited as providing a comprehensive review and formalization of many of the key ideasunderlying the classical economistsrsquo views The result may be denoted the prototypeof the neoclassical (static) model Dynamic neoclassical macroeconomic models arebroadly based on neoclassical growth models with the addition of shocks of variouskinds Among the classic works developing neoclassical growth models are Solow(1956) Cass (1965) Koopmans (1965) and Sidrauski (1967a 1967b)
Notes 203
13 Two of the most widely known proponents of new classical economics are Robert Lucasand Thomas Sargent It has been suggested by some however that this line of analysisis less the extension of classical analysis than it is the antithesis of Keynesian analysisas discussed below See Niehans (1987) for this interpretation
14 This is changing as indicated by Howitt (1985) Shapiro and Stiglitz (1984) Weitzman(1985) and by the papers cited in the May 1988 American Economic Review
15 Howitt goes on to say that there is the ldquoreciprocal Keynesian question of how exactlythe economic system manages to overcome all the obvious coordination problems thatstand in the way of attaining the state of equilibrium common to new classical economicsmodelsrdquo
2 Walrasian economy
1 A brief description of the theory is given in Patinkin (1965 note B)2 In reality of course there is no auctioneer In fact some think that Walras would have
been the first to question such a description of the economy as realistically capturingthe true dynamic process by which the economy reaches the equilibrium described bysupply and demand curves See Walker (1987) who points out that Walras promoted aldquodisequilibrium-production model of tatonnementrdquo as more representative of Walrasrsquothought than the tatonnement model with an ldquoauctioneerrdquo
3 We use the letter T to denote the number of commodities because later we distinguishthe T commodities according to time of availability (ie from period 1 to period T )
4 One could replace the assumption of known prices by the assumption that individualsform expectations at time t about prices at time t and that such expectations are correctThis has been referred to as a situation in which expectations satisfy the assumption ofldquoweak consistencyrdquo
5 With zero transactions costs equality of purchase and sale prices could be viewed asforced by arbitrage conditions
6 Walras was one of the first to note this point Note that πjj = 17 See Debreu (1959 Chapter 2) for a discussion of such accounting prices8 A distinctive feature of macroeconomics is to alter traditional ArrowndashDebreu general
equilibrium analysis in such a way that we can reinterpret accounting prices as moneyprices and determine the level of money prices
9 Others characterize transactions costs in similar fashion For instance Alchian and Dem-setz (1972) cite the costs of ldquoformingrdquo ldquonegotiatingrdquo and ldquoenforcing contractsrdquo whileDahlman (1979) writes of ldquosearch and information costsrdquo ldquobargaining and decisioncostsrdquo and ldquopolicing and enforcingrdquo costs
10 See Varian (1992) for a discussion on these points By ldquowell behavedrdquo we mean thatpartuapartcai gt 0 and ua is strictly quasi-concave
11 In this case the Lagrangian is written ignoring the non-negativity constraints onconsumption of commodity i i = 1 T Below we provide an equivalent character-ization of the constrained maximization problem that incorporates these constraints inthe Lagrangian
12 A function y = f (x1 xn xn+1 xm) is said to be homogeneous of degree k inthe arguments x1 through xn if
f (λx1 λxn xn+1 xm) = λk f (x1 xn xn+1 xm)
3 Firms as market participants
1 With zero ldquoadjustment costsrdquo there could exist a market for capital at time t such thatthe capital employed during the initial period Kt is conceptually distinct from capitalinherited K In this case however equilibrium in the capital market at time t wouldthen imply that Kt = K
204 Notes
2 Note that output is a ldquoflow variablerdquo That is for a period i of length h the outputproduced is given by hyi The term yi is thus the ldquorate of outputrdquo over period i Sinceoutput is a flow variable at a point in time the rate of production is not defined (Asyou can see for any rate of output the limit of production as the length of the periodgoes to zero is zero)
3 In general for a period i of length h labor services are denoted by hNi such that totalwage payments at the end of the period are wihNi Like the labor input one shouldthink of the capital input in flow terms with the stock of capital determining the rateor flow of capital services
4 For simplicity of notation these expressions presume perfect foresight5 In general for a period i of length h the nominal interest rate from the end of that period
to the end of the next period is given by hri = hzpbi + (pbi+h minus pbi)pbi 6 For simplicity of notation these expressions assume perfect foresight with respect to
future prices of equity shares In addition the expressions presume future dividends areknown
7 To minimize notational clutter we have chosen to not denote such anticipations withthe expectation operator We do presume that agents (firms and households) have com-mon expectations (assumed to be held with subjective certainty) concerning plans withrespect to the issuing of equity shares
8 For instance if ri lt rei for period i there would be zero demand for bonds Theresulting excess supply of bonds would lead to a fall in the price of bonds and thus arise in the return on bonds until equality across rates of return held
9 We assume for the moment perfect foresight at time t with respect to prices at the endof the period (at time t + 1)
10 As in the prior models including money balances in the utility function reflectshow money can save (leisure) time required to make exchanges within a period Forsimplicity we limit money holdings to agents labelled ldquohouseholdsrdquo
11 For the representative household holdings of bonds issued by other households mustbe zero so that there is no real indebtedness effect with respect to the bonds exchangedamong representative households
12 In general if we assume there are ni firms producing commodity i and that there are mdifferent commodities then
yt =msum
i=1
⎡⎣ nisum
j=1
( pip)fij(Nijt Kijt)
⎤⎦
The above is a stylized view of how the empirical counterpart to total output real grossdomestic product is actually computed by the Commerce Department
13 Note that it is assumed that the capital stock K generates a fixed rate of capital servicesThat is we do not consider variation in the ldquoutilization of capitalrdquo If we did so thenvariation in the services flowing from the capital stock could be considered an additionalchoice variable with capital utilization presumably affecting the extent of depreciationin the capital stock over time
14 Fama and Miller (1972) cite conditions under which the ldquoowners of the firmrdquo iethe holders of the S equity shares will direct the managers of the firm at time t tomaximize Vt
15 In continuous time
petminus1 =int infin
t[psdsSs]eminusr(sminust)ds
where the interest rate r in the above expression is formally rs
Notes 205
16 Note that Stminus1 equiv S Assuming ri = rt i = t + 1 we have
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[pt+k dt+kSt+kminus1](1 + rt)k
⎤⎦
17 If the change is negative net revenues available for distribution as dividends would bereduced
18 We will explain more fully below the nature of these ldquoadjustment costsrdquo19 As suggested earlier if the utilization of capital were viewed as a choice variable then
it would be natural to have δ directly related to utilization of capital during the period20 Later we will say more about investment and the nature of adjustment costs21 Note that this discussion assumes for simplicity zero adjustment costs22 We follow Sargent and implicitly assume adjustment costs are related to net not gross
investment Net investment measures the change in the capital stock As we will seelater the result is a slightly different expression for Tobinrsquos Q than found elsewherewhen adjustment costs depend on gross investment We could also expand the natureof ψ so that adjustment costs depend on the size of the capital stock as is done in suchpapers as Lucas (1967) Uzawa (1969) and Gould (1968)
23 In the National Income and Product Accounts of the USA that report various measuresof the activity in the economy depreciation is measured by what is called the ldquocapitalconsumption allowancerdquo
24 The presence of such markets reflects the existence of ldquoperfect capital marketsrdquo in themacroeconomics literature Remember that with respect to the choice of investment forthe individual firm zero adjustment costs imply a potential discrete jump in the capitalstock at a point in time so that investment for the individual firm may not be defined
25 LrsquoHospitalrsquos rule states that if a is a number if f (x) and g(x) are differentiable and g(x)does not equal zero for all x on some interval 0 lt |x minusa| lt ε if the limit of f (x) equalszero as x approaches a and if the limit of g(x) is zero as x approaches zero then whenthe limit of the ratio f (x)gprime(x) as x approaches a exists or is infinite it equals the limitof f (x)g(x) as x approaches a
26 In continuous-time or discrete-time models such adjustment costs mean that the ldquocapitalmarketrdquo at time t is eliminated
27 Note that since bonds and equity shares are perfect substitutes the optimization problemwill not provide a breakdown into the optimal number of bonds versus equity shares
28 Recall that we are holding the stock of equity shares outstanding constant29 Note that we ignore the potential choice of capital at time t Kt and associated choice
of bonds setting Kt = K This in fact would be the case with capital adjustment costs30 If we evaluated returns from the start of a period each of the return functions would be
multiplied by 1R31 The equivalence of bond equity share and retained earnings financing of changes in
the capital stock can be shown32 This expression reflects the envelope theorem In particular we have that
dW (Kt+1)
dInt+1
dInt+1
dKt+1= dW (Kt+1)
dNt+1
dNt+1
dKt+1= 0
33 If the price of capital differed from the price of output but both prices were expected tochange at the same rate so that the relative price of capital was assumed to be constantover time then the expected real user or rental cost of capital would be (pkp)(mt + δ)where pkp denotes the relative price of capital
206 Notes
34 Formally at time t the inherited debt-to-equity ratio is given by pbBpeS35 Naturally there are other factors not discussed36 Note that during period t when the length of the period between planned purchases of
capital (at time t) and final installation (at time t + 1) is 1 then net investment Int isgiven by Int = Kt+1 minus K and adjustment costs are given by ψ(Int)
37 During period t+1 net investment is defined by Int+1 = Kt+2 minusKt+1 and adjustmentcosts are given by ψ(Int+1)
38 We can see from the general nature of the investment demand functions that if theproduction function were not separable then the assumption of a constant real wageover time as well as a constant expected real rate of return over time would obtain thisresult
39 See Sargent (1987a 11) for a similar expression in continuous time Note that thetwo expressions would be exact if we take the limit as the length of the period goesto zero The equality between ψ prime
t+h and ψ primet+2h that typically would be an approx-
imation in discrete time holds exactly in the limit In addition the definition ofthe real interest rate for a period of length h 1 + hm = (1 + hr)(1 + hπ) orm = (r minus π)(1 + hπ) indicates that in the limit (as h goes to zero) m = r minus π If we had assumed that adjustment costs were based on gross not net investmentthen the fraction on the left-hand side of (312) would include the term minusδψ in thedenominator
40 The Q theory of investment demand was suggested by Tobin (1969) Sargent is oneauthor who expresses investment demand in this way
41 In fact empirical measures of Tobinrsquos average Q have been constructed although suchmeasures are more complex than those discussed here since they must incorporateinfluences of the tax system (such as investment tax credits accelerated depreciationallowances and the like) on the optimal choice of the capital stock
42 In a two-sector model Tobinrsquos average Q is given by pV p1K where pV is the nominalvalue of the firm and p1 denotes the price of capital which differs from p the price ofoutput
43 In our case there is a single argument of the adjustment function ndash net investment Ifone follows Hayashi (1982) and assumes as he does that adjustment costs depend ongross investment then one must add the assumption that there is a linear homogeneousadjustment function such that ψ(δk) = ψ primeδk over the appropriate range Hayashirsquosadjustment cost function (like others) includes the stock of capital as well
44 Eulerrsquos theorem (or law) is that if the function f (middot) is a differentiable functionhomogeneous of degree 1 (linear homogeneous) with f Rn rarr R then
f (x) equivnsum
i=1
[partf (x)partxi]xi
Replacing the real wage with the marginal product of labor reflects the assumption thatthe firm is a price-taker in both the output and labor markets
45 See for example Azariadis (1976) A second reason is that new workers and previouslyemployed workers may not be considered perfect substitutes (eg see Oi 1962)
46 Taylor (1972) and Hall (1980) are among those who have examined thisphenomenon
4 Households as market participants
1 Time consistency means in this context that households will follow through on priorplans as the starting date advances Strotz (1955ndash56) discussed this point in terms of autility maximizing problem
Notes 207
2 An example of nonseparable preferences is a ldquohabit persistence modelrdquo in which utilityis given by
infinsumi=t
βiminustu(ci ciminus1 )
so that consumption at time t depends on prior consumption Alternatively one couldhave utility given by
infinsumi=t
βiminustu(ci 1 minus Ni 1 minus Niminus1)
such that past work becomes a pertinent state variable for the current period Kydlandand Prescott (1982) is one important exception to the macroeconomic literature inassuming nonseparable preferences
3 This is sometimes referred to as an ldquoend-of-periodrdquo equilibrium specification in themarket for assets Alternative asset specifications for discrete-time analysis have beendiscussed by among others Foley (1975) As Edi Karni (1978) pointed out Patinkinrsquosmodel is an end-of-period model
4 For simplicity we ignore the financial asset markets at time t If we assumed portfolioadjustment costs then it would be the case that at time t desired bond and moneyholdings would be B and S respectively
5 That is (partW (xt+1)partcdt+1)(partcd
t+1partxt+1) = 0 since partW (xt+1)partcdt+1 = 0 similarly
the indirect effects of a change in xt+1 on partW (xt+1) through its impact on optimallabor supply and real money balance holdings are zero
6 For completeness note that by substituting equation (43prime) into (45) we could have theequivalent expression
partW (xt+1)partxt+1 = β[partut+1part(Mt+1pt+1)]Rt+1[Rt+1 minus Rmt+1]7 These conditions appear in various forms throughout the macroeconomics literature
see for instance Mankiw et al (1985) or Barro and King (1984)8 The resulting demand for leisure function is termed a ldquoHicksian or compensated demand
functionrdquo as it is constructed by varying the price of leisure (the real wage) and incomeso as to keep the individual at a fixed level of utility
9 The resulting demand function for leisure that incorporates both the income and sub-stitution effects of changes in the real wage is an example of a ldquoMarshallianrdquo demandfunction
10 Borjas and Heckman (1978) See also Pencavel (1985) for a survey of estimates Forevidence on the effect of wage increases on the labor supply of working women seeNakamura and Nakamura (1981) and Robinson and Tomes (1985)
11 Note that we continue to assume nonsatiation such that partutpartct gt 012 The second statement presumes sufficient dispersion in preferences so that at each real
wage there are some individuals who are just indifferent between a zero and positivelabor supply Thus any rise in the real wage will increase labor force participation
13 This point was first formally developed in the classic paper by Lucas and Rapping(1970) The role of ldquomarket clearingrdquo in macroeconomic analysis will be clearer laterwhen we consider alternative characterizations of the labor market
14 To be exact this result assumes that utility is separable and concave in leisure That isit is assumed that part2upartcpartN = part2upartNpart(Mp) = 0 and part2upartN 2 lt 0
15 Alogoskoufis (1987) provides a good review of the empirical analysis in this area
208 Notes
16 Fisherian ndash or in Patinkinrsquos (1965) terminology ldquoFisherinerdquo ndash analysis takes itsname from the classic ldquotime-preferencerdquo analysis of Fisher see in particular Fisher(1930)
17 To assure this one could let duadcai gt 0 with limcairarr0(duadcai) rarr infin andlimcairarrinfin(duadcai) rarr 0
18 Equivalently one could assume partuapart(Maip) equiv 0 for all i Given the nonnegativityconstraint on Maipi i = t t + T and the presumption of a positive nominalinterest rate ri i = t t +T minus1 the optimal solution would be M d
aipi = 0 for all iThat is bonds will dominate money as an asset and only bonds will be held if assetholdings are positive The reason is simple ndash the unique attribute of money holdingsas a way to reduce ldquotransaction costsrdquo has not been introduced by providing a ldquoutilityyieldrdquo to holding money
19 Note that the equality sign here rather than the ge sign reflects the fact that the first-order conditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
20 Note that we assume a time-invariant one-period utility function The notation uai
reflects the fact that utility of agent a in period i depends on consumption inperiod i cai
21 In general for a period of length h we have that 1 + hm = (1 + hri)(1 + hπi)Solving for hmi and then dividing through by h we have that mi = (ri minusπi)(1+hπi)Thus in the limit as the length of each period goes to zero mi = ri minus πi Thus incontinuous-time analysis the real rate of interest is exactly defined by ri minus πi
22 In the discussion to follow we maintain the assumption of a time-invariant utilityfunction such that ua
i (cai) = ua(cai) for i = t t + T 23 At the point where cat = cat+1 the slope of the indifference curve is minus1β Assuming
cat = cat+1 and no initial assets or debt (ie zBat = 0) if 1β = Rt then the resultof the same consumption in each period would imply the individual would be neither alender nor a borrower
24 As Modigliani (1996) has stated ldquothe consumption and saving decisions of householdsat each point of time reflect a more or less conscious attempt at achieving the preferreddistribution of consumption over the life cycle subject to the constraint imposed bythe resources accruing to the household over the lifetimerdquo Modigliani summarizes hiscontribution to the analysis of consumption behavior in his Nobel lecture of December1985 (see Modigliani 1986)
25 Examples of such discussions are Diamond and Hausman (1984) and Hurd (1987)However what happens in the aggregate does mask different behavior among subgroupsof the populations For instance Burbidge and Robb (1985) find for Canadian data thatwhile an inverted U-shaped profile exists for the ldquoaveragerdquo Canadian household ldquowhitecollarrdquo households do appear to continue to accumulate wealth years after both husbandand wife have left the labor force
26 If individual a were the representative agent then consumption smoothing would implythat the aggregate endowment of the consumption good is identical across periods ieci = ci+1 i = t t + T minus 1
27 Note that in the case of multiperiod bonds agent arsquos future income could includepayments derived from the initial holdings of assets Since the present value of suchfuture ldquoincomerdquo is incorporated in the current value of the assets operationally futureincome cat i = t+1 t+T is defined as income other than derived from initial assetholdings In a production context the source of such income would be compensationfor labor services sold
28 In an economy with production this transitory component of current income can reflectsuch events as a temporary layoff a short-run opportunity to work overtime or atemporary tax rebate
29 This discussion ignores the effects of uncertainty on optimal consumption plans
Notes 209
30 Note that the equality sign here rather than the ge sign reflects the fact that the first-orderconditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
31 Note that with one-period bonds individual a has a zero initial endowment of bondsthat continue into the future at time t
32 Other papers on this topic include Hansen and Singleton (1983) and Mankiw et al(1985)
33 An example of a compositional change that could affect aggregate consumption butwould not be accounted for in the analysis to follow is a change in the proportionof retired individuals in the economy Thus on this ground at least Hallrsquos empiricalfindings cannot be taken as the last word on consumption behavior at the level of theindividual
5 Summarizing the behavior and constraints of firms and households
1 Caballero and Engel (1999) provide an empirical study of investment dynamics in thecontext of manufacturing firms
2 Weber (1998) provides empirical evidence on the link between the financial marketsand consumption spending
3 We refer to this perfect foresight as ldquolimited perfect foresightrdquo since it concerns only thecurrent period This focus on current markets alone typical of static analysis impliesldquoexpectations functions of the agentsrdquo with respect to prices in subsequent periodsthat do not reflect the underlying analysis of markets beyond the current period Thusbeyond the current period expectations do not have the property of perfect foresight (orrational expectations in a nondeterministic setting)
4 In an open economy ndash that is one which admits a foreign sector ndash we would add afourth market the market for foreign exchange
5 Note that in a fully monetized economy money enters on one side of every exchange ndashpurchase or sale ndash in these three markets
6 Note that money holdings and money demand arise only for ldquohouseholdsrdquo To the extentfirms do hold money and make choices with respect to the size of such holdings wepresume their behavior would be similar to that of households and so lump firms withhouseholds with respect to such activity Thus the behavior of the firm is restricted tolabor demand output supply investment demand and financial asset supply
7 Recall that the expected real user cost of capital is mt + δ where δ is the rate ofdepreciation of capital and mt is the expected real rate of interest (ie mt equiv (rt minusπe
t )(1+πet ) where rt is the money interest rate and πe
t is the expected rate of inflationbetween periods t and t + 1)
8 As previously for simplicity we continue to assume that expectations of future pricesare held with subjective certainty
9 If part2f partKpartN = 0 then the real wage would not enter as an argument in the capitaldemand function nor would the existing capital stock affect labor demand
10 That is we can express a demand function in a form similar to one that would beobtained if the analysis were to consider only two periods
11 Note that we do allow the expected wage inflation between period t and t + 1 πewt
to differ from subsequent wage inflation so that the expected real wage next periodwe
t+1pet+1 equiv wt(1 + πe
wt)pt(1 + πet ) can differ from the current real wage This
introduces the possibility of intertemporal substitution of labor supply in response to achange in the current real wage
12 Assuming less than unit elastic expectations with respect to future income streams wouldassure from the standard Fisherian problem that the marginal propensity to consumewould be less than one
13 Recall that πet equiv ( pe
t+1 minus pt)pt
210 Notes
14 Recall that firms are presumed not to hold money balances so there are no real balanceeffects to concern us with respect to firmsrsquo demands or supplies
15 As Lucas and Rapping (1970) point out introducing other expectation assumptionscan retain the ldquointertemporal substitution hypothesisrdquo in the context of changes in thecurrent price level but in doing so money illusion is introduced In their own wordsthe
labor-supply equation is not homogeneous in current wages and current prices(such that) there is ldquomoney illusionrdquo in the supply of labor ldquomoney illusionrdquoresults not from a myopic concentration on money values but from our assumptionthat the suppliers of labor are adaptive on the level of prices expecting a returnto normal price levels regardless of current prices and from the empirical factthat the nominal interest rate does not change in proportion to the actual rate ofinflation With these expectations it is to a supplierrsquos advantage to increase hiscurrent supply of labor and his current money savings when prices rise
(Lucas and Rapping 1970 268ndash269)
Lucas and Rapping are particularly looking at the effect of a change in the current pricelevel on the expected real rate of interest Their assumption of ldquoadaptiverdquo expectationsimplies that an increase in pt results in a fall in πe
t equiv (pet+1 minus pt)pt a rise in the
expected real rate of interest and thus a rise in labor supply16 That is future technology capital demand labor demand the real wage the rate of
depreciation and future real interest rates are all unchanged by such a change in pricesand the money supply
6 The simple neoclassical macroeconomic model (without governmentor depository institutions)
1 In particular it is assumed that the money wage rate adjusts to continuously maintainequality between the demand for and supply of labor the price of output adjusts tomaintain equality between the demand for and supply of output and the price of financialassets (and thus the interest rate) adjusts to maintain equality between the demand for andsupply of financial assets Patinkin (1965) provides one of the first complete accountingsof this model
2 That is we would expect prices in the various markets to eventually adjust to eliminateany possible excess demands or supplies in the economy We would also expect agentsultimately to correctly anticipate the price level The neoclassical model can be modifiedto explain the workings of the economy in the face of incomplete information and priceinflexibility
3 As before the ldquolaws of motionrdquo dictating how prices change to reach equilibrium aregiven by Walrasrsquo excess demand hypothesis and we maintain the assumption that noexchange occurs until an equilibrium is reached (the recontracting assumption)
4 Alternatively one could assume that expectations at time t concerning these futurevariables are constant
5 Note that Patinkin has firms as well as households managing a portfolio of financialassets and money balances which is why he includes the demand function for labor inthe above statement In our analysis this statement applies to the labor supply functionalone
6 This last sentence anticipates the intertemporal substitution hypothesis7 We ignore the potential effect of changes in the interest rate on labor supply and thus
employment8 This reflects the assumption that households and firms share common expectations
concerning the price level (in fact for both pet = pt)
Notes 211
9 Reasons such as these for changes in output form the basis of much of the currentanalysis in the literature with respect to ldquoreal business cyclesrdquo
10 This is perhaps too extreme a statement To the extent that a higher price level isanticipated the resulting lower real money balances could lead to an increase in laborsupply at any given real wage and consequently increased employment and output Alsothere is a potential effect of changes in the price level on aggregate labor supply throughthe impact of such changes on the expected real rate of interest if unit elastic expecta-tions concerning future prices are not assumed Recall that we follow macroeconomictradition and abstract from these possibilities
11 In a recent study Ewing et al (2002) develop a model of the equilibrium unemploymentrate and examine how it responds to unanticipated changes in real output
12 Fairlie and Kletzer (1998) discuss the issues revolving around job displacement13 Unemployment may also result if prices in the economy do not adjust quickly enough to
ensure that all markets (particularly the labor market) are continuously in equilibriumUnemployment associated with labor market disequilibrium is sometimes referred toas ldquoinvoluntaryrdquo unemployment We analyze such unemployment later
14 See for instance the paper by Evans (1989) that examines the relationship betweenoutput and unemployment in the United States
15 The ldquoISrdquo equation is sometimes referred to as the aggregate demand equation indicatingthat it reflects equality between total or aggregate demand for output and productionNote that equilibrium in the output market is being described in terms of demand equalto what is produced ylowast
t The aggregate supply equation indicates what will be produced16 This is the case only for this simple aggregate model without government17 Such an assumption removes the anticipated real wage next period as an argument
in these demand functions Recall that earlier ldquostaticrdquo assumptions concerning futureinterest rates and rates of inflation have already simplified the form of these functions
18 However ldquorealrdquo or ldquosupplyrdquo shocks such as the above-mentioned changes in technologycapital stock supply of other inputs such as oil or in labor supply at prevailing realwages can affect real output
19 This analysis should be familiar since we performed a similar analysis for the economywithout production
20 Note that unit elastic expectations imply that partπet partpt = 0
21 Note that without a real balance effect the CC curve would be horizontal22 The lower interest rate abstracts from a real balance effect in the output market so that
the CC curve is horizontal
7 Empirical macroeconomics traditional approaches and time series models
1 To reduce notational clutter we suppress time subscripts All variables are period tvariables
2 Note that in Sargent (1987a 20) equation (71) is replaced by the ldquorepresentativerdquofirmrsquos first-order condition for the optimal use of the labor input given a competitivelabor market that is the condition that the real wage equals the marginal product oflabor wp = Fn(n K) Note that if the production function F(n k) is separable in thelabor and capital inputs such that f (n K) = v(n)+u(K) and v(n) = (1g)(fnminusn22)then equation (71) is identical to Sargentrsquos equation since fn(n K) = (1g)(f minus n)
3 Unless otherwise noted all parameters in this model such as f g h and j in equations(71) and (72) are assumed to be positive
4 In this context ldquolump-sumrdquo taxes are taxes independent of income Equation (73) isan example of the ldquoconsumption functionrdquo
5 Endogenous variables are variables whose values are determined by the analysis6 Sargent adds equations (73) and (74) to the system (712) in order to determine
consumption and investment demand as well
212 Notes
7 To derive wlowast start with the equilibrium condition nd = ns Substituting the first twoequations of 712 the equilibrium money wage satisfies
f minus g middot (wlowastp) = h + j middot (wlowastp)
Then one can simply solve this equation for the equilibrium money wage wlowast We canthen obtain the equilibrium employment level nlowast by substituting the expression for wlowastinto either of the first two equations of 712
8 This is a property of ldquoclassicalrdquo macroeconomic models in that monetary changesthat alter the price level do not affect real variables such as the real wage output andemployment
9 The IS equation indicates combinations of the interest rate r and output y at which ifthe output were produced and that interest rate prevailed output demand would equaloutput produced The LM equation indicates combinations of r and y that will equatethe demand for and supply of money
10 For the model under consideration the ldquoaggregate demand curverdquo (a plot in ( p y) spaceof the aggregate demand equation) slopes downward and the ldquoaggregate supply curverdquo(a plot in ( p y) space of the aggregate supply curve) is vertical The intersection of theaggregate demand and supply curves graphically determines the equilibrium output andprice level
11 See Altonji and Siow (1987) Ewing and Payne (1998) examined the relation-ship between the personal savings rate and consumer sentiment in the context of aconsumption model
12 Note that Taylor assumes that certain demands for instance consumption demand maydepend on past as well as current values of output and the money supply so that thereduced-form expression estimated includes lagged values of income and the real moneysupply
13 Time series analysis can be viewed as primarily the art of specifying the most likelystochastic process that could have generated an observed time series
14 That is forecasts generated by time series models have been used to proxy individualsrsquoexpectations of future events in tests of various theoretical macroeconomic models
15 A histogram is a plot of the frequency distribution of a set of observations16 If the process is also ldquoergodicrdquo these statistics give consistent estimates of the mean and
variance Ergodicity basically requires that observations sufficiently far apart should bealmost uncorrelated Then by averaging a series through time one is continually addingnew and useful information to the average For a rigorous explanation of this conceptsee Hannan (1970 201)
17 The variable γk k = 0 1 2 is termed the autocovariance function18 Or more generally k = 0 Note that ρk = ρminusk 19 Some have suggested that stock market prices follow a random walk See Campbell
et al (1997)20 Note that if yt is taken to be the logarithm of real output then the trend reflects a constant
rate of growth of output equal to d in the absence of shocks21 This assumes y0 is the initial value of the function yt 22 A random walk is an example of a class of nonstationary processes known as ldquointe-
gratedrdquo processes that can be made stationary by the application of a time-invariantldquofilterrdquo As defined by Granger and Newbold (1986) ldquoif a series wt is formed by a linearcombination of terms of a series yt so that wt = summ
j=minuss cjytminusj then wt is called a
lsquofilteredrsquo version of yt If only past terms of yt are involved so that wt = summj=0 cjytminusj
then wt might be called a one-sided or backward-looking filterrdquo23 This is a special case of a class of stochastic processes known as Markov processes24 Recall that for the random walk process φ = 1 in which case the process was not
stationary as the variance of the process becomes larger and larger with time
Notes 213
25 To obtain this result note that
E
⎡⎣ infinsum
j=0
(φ2)jε2tminusj
⎤⎦ =
infinsumj=0
(φ2)jσ 2e = σ 2
e (1 minus φ2)
26 Often the ldquolag operatorrdquo L or equivalently the ldquobackward shift operatorrdquo B is usedto express this equation Lτ yt (or Bτ yt) = ytminusτ τ = 1 2 3 There are associatedpolynomials in the lag operator such that
d(L) = d0 + d1L + d2L2 + d3L3 + middot middot middot + dpLp
Letting
φ(L) = 1 minus φ1L minus φ2L2 minus φ3L3 minus middot middot middot minus φpLp
we can thus express an AR( p) process for yt as φ(L)yt = εt 27 If yt is an AR( p) process it may be described in the following way ldquoan appropriate
finite backward-looking filter applied to yt will produce a white noise seriesrdquo (Grangerand Newbold 1986 32)
28 If there were a single shock to yt at time 0 (ie εt = 0 for all t gt 0) then the deterministiccomponent would signify the deviation of the time path from its equilibrium level Aswe will see in this case stationarity would imply ldquostability of equilibriumrdquo in that ytwould converge to its equilibrium value over time
29 A linear difference equation of order s is of the form
yt =ssum
j=1
ajytminusj + c
30 Successive substitution (the ldquoiterative method of solutionrdquo) reveals this essential natureof the solution In particular substituting for past values of yt in (725) results inyt = φty0
31 The appendix to this chapter shows how one can reinterpret higher-order differenceequations as a system of first-order difference equations
32 Note that in general a quadratic equation of the form
ax2 + bx + c = 0
can be solved using the quadratic formula
x1 x2 = [minusb plusmn (b2 minus 4ac)12]2a
In our case a = 1 b = minus(m1 + m2) = minusφ1 and c = m1m2 = minusφ233 This assumes m1 = m2The solution for repeated roots is discussed briefly below34 A variable x is said to be inside the unit circle if x lt |1| An alternative way of
expressing this condition is in terms of the associated polynomial 1 minusφ1L minusφ2L2 = 0This polynomial equation is similar to (730) except that b is replaced by 1L and thewhole equation is multiplied through by L2 The stationarity condition is that the rootsof this polynomial equation should lie outside the unit circle
35 Recall that φ1 = m1 + m2 and φ2 = minusm1m236 The following discussion follows Goldberg (1958 171ndash172)
214 Notes
37 As discussed below in this case the two roots are m1 = h + vi and m2 = h minus vi whereh = φ12 v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12 The product
of these two roots is (φ21 minus 4φ2 minus φ2
1)4 = minusφ2 which is the square of the modulus
of the roots We are assuming (minusφ2)12 lt 1 so we have the condition that φ2 lt 1 orφ2 gt minus1
38 Note that if yt is an MA process then it is a ldquobackward looking filterrdquo applied to a whitenoise process As before we can use a lag operator L (or backward shift operator B) toexpress an MA(q) process for yt as yt = micro + θ(L)εt where the polynomial in the lagoperator θ(L) is given by
θ(L) = 1 + θ1L + θ2L2 + middot middot middot + θqLq
39 Recall that as discussed above we assume for simplicity zero mean for yt that ismicro = 0
40 For a more detailed description of invertibility see Granger and Newbold (1986) orBox and Jenkins (1970)
41 Note that ARMA( p 0) equiv AR( p) and ARMA(0 q) equiv MA(q) Using the lag operatorthe ARMA model (in the case of a zero mean) can be simply expressed as φ(L)Yt =θ(L)εt
42 In general if yt is ARMA( p1 q1) and xt is ARMA( p2 q2) the sum zt = yt + xt isARMA( p3 q3) where p3 le p1 + p2 and q3 le max(p1 + q2 p2 + q1) A proof of thisis found in Granger and Morris (1976)
43 A time sequence T (t) is called ldquodeterministicrdquo if there exists a function of past andpresent values gt = g(T (t minus j)) j = 0 1 such that E[(Tt+1 minus gt)
2] = 0 If thefunction gt is a linear function of Ttminusj j ge 0 then Tt is called ldquolinear deterministicrdquo
44 For instance for a stationary series of quarterly data one could postulate a simplefourth-order seasonal AR process of the form
yt = φ4ytminus4 + εt
This is a special case of AR(4) with φ1 = φ2 = φ3 = 0 The model could be extendedto include both AR and MA terms at other seasonal lags for instance the followingARMA(21)
yt = φ4ytminus4 + φ8ytminus8 + θ4εtminus4 + εt
To add other than seasonal components one could simply fill in the gaps (eg addterms such as φ1ytminus1 θ1εtminus1 and the like to the above) Other options are discussedby Harvey (1993) and others
45 Campbell and Mankiw argue that even if the log of output followed a random walk withdrift indicating that the effect of any shock persists indefinitely into the future estimatesusing the detrended series would be biased and erroneously conclude otherwise
46 Note that if the log of real output is an ARMA( p q) process then the differencedprocess will be an ARMA( p q + 1) process This means that to allow for stationaritywith respect to the level of real output requires at least one moving average process forthe differenced series
47 Equivalently for the logarithm of real output they are considering ARIMA(p 1 q)processes for p = 0 1 2 3 and for q = 0 1 2 3
48 Such a finding is often termed as supportive of real business cycle theories
Notes 215
8 The neoclassical model
1 The presence of money reflects the introduction of ldquoimperfectrdquo or costly informationA medium of exchange can arise to minimize costs incurred by participants in theeconomy when there exists imperfect information on potential exchange partners
2 In so doing we assume that the new equilibrium like the initial one exists Further wedisregard the process of adjustment of the variables to the new equilibrium Alterna-tively we could introduce ldquolaws of motionrdquo for the equilibrium values (eg the excessdemand hypothesis for price changes) and examine whether the equilibriums are stable
3 For notational simplicity we let Ld = Ldt cd = cd
t Id = Idnt p = pt r = rt
yd = cdt + Id
nt + δK + ψ(Idt ) and M = M
4 For simplicity we let the expected real rate of interest component of the expected realuser cost of capital ((rt minus πe
t )(1 + πet )) be approximated by rt minus πe
t as representedby r minus π
5 One alternative is for the interest rate to adjust to equate the demand for and supply ofmoney
6 That is we assume that consumption demand is a function of the expected gross realrate of interest Rt not its components (rt and πe
t ) This would be the case if one couldview the consumption decision from the point of view of the pure ldquoFisherian problemrdquoin essence separating the allocation of consumption across time decisions from theldquoportfolio problemrdquo Note that for notational ease we not only let rt denote the moneyinterest rate rt but also let π denote the expected rate of inflation πe
t 7 This would be the case if our focus was solely on the portfolio choice problem8 Examples of growth models with money include Tobin (1965) and Sidrauski (1967b) As
Begg (1980 293) notes ldquoin a steady state any expectations generating mechanism willyield correct predictions thus the steady state analysis of growth models with moneymay be viewed as a special case of the rational expectations model with systematicmonetary policyrdquo
9 This asymmetry in information can reflect an aggregation across labor markets in whicheach firm determines labor demand based on its correct anticipation of the price of theparticular commodity it produces while suppliers of labor determine labor supply basedon their potentially incorrect anticipation of the overall level of prices reflecting theidea that suppliers are concerned with the purchasing power of wages in terms ofcommodities not restricted to the particular commodity that they produce
10 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt r = rt y = yt and M = M
11 Recall that for simplicity we let the expected real rate of interest component of theexpected real user cost of capital (rt minus πe
t )(1 + πet ) be approximated by rt minus πe
t asrepresented by r minus π
12 Note that partypartp = 0 for the neoclassical model given limited perfect foresight13 This is obvious from the graph of aggregate demand and supply curves if one compares
the vertical aggregate supply curve of the neoclassical model dpp = dMM with theupward-sloping aggregate supply curve of the model with real wage illusion Note thatthe shift in the aggregate demand curve for a given change in the money supply isidentical in either case
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
1 Typically a union contract runs for three years Often however there are provisions thatpermit parts of the agreement to be renegotiated at specific times during the three-yearcontract period
2 As a general rule labor agreements are specified in money terms An exception to thisis the cost of living agreements (COLAs) as part of union wage contracts in the United
216 Notes
States which became popular during the 1960s and 1970s COLAs adjust money wagesautomatically to changes in prices typically using changes in the Consumer Price Indexto measure price changes However the percentage of all workers who have contractswith COLAs is fairly small as less than 20 percent of the total labor force is covered bycollective bargaining agreements Further not all COLA clauses offer full protectionagainst general price increases as wages may rise by only some fraction of the increaseof the CPI Considering these qualifications for the time being we simplify by assumingthat all labor agreements are specified in money terms
3 These seven variables for the classical model are made up of three ldquopricesrdquo along withfour variables implied from the behavioral equations The three ldquopricesrdquo are moneywage price level and interest rate determined by the equilibrium conditions for the labormarket and two of the three other markets (output financial andor money markets)The variable implied from the behavioral equations are employment (labor demandfunction) output (production function) consumption (consumption demand function)and investment (investment demand function)
4 Note that this analysis is inconsistent with the idea that long-term employment contractsthat fix the money wage for several periods arise due to adjustment costs with respectto the labor input
5 As discussed before one justification for this form is if real money balances are not partof household wealth which can be the case when we introduce depository institutionsinto the analysis
6 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt Idnt = Id r = rt
y = yt and M = M 7 Recall that for simplicity we let the expected real rate of interest component of the
expected real user cost of capital (rt minus πet )(1 + πe
t ) be approximated by rt minus πet as
represented by r minus π 8 Note that partypartp = 0 for the neoclassical model given limited perfect foresight9 The Lucas model was introduced in Chapter 8 and is covered in more depth in
Chapter 1010 An exception to this statement occurs if monetary authorities can react to period t
disturbances that is if monetary authoritiesrsquo information set includes the values of therandom shocks in period t which are not known to private agents
11 Phelps and Taylor (1977) go on to state that they ldquodo not pretend to have a rigorousunderstanding of [why prices andor wages are set in advance] In the ancient andhonorable tradition of Keynesians past we take it for granted that there are disadvantagesfrom too-frequent or too-precipitate revisions of price lists and wage schedulesrdquo
12 Note that θ is affected by the variability of relative price shocks in relation to generalprice shocks
13 We assume for simplicity that the coefficient on Pt minus Wt + φ is one14 Note that if Pt minus Wt + φ = 0 then equation (99prime) is a first-order linear difference
equation of the form YtminusλYtminus1 = 0 with solution Yt = c0λt In the limit as t approachesinfinity Yt equals zero Recall that the logarithm of the natural level of output is zero
15 Recall our assumption that the scale factor in the determination of the real wage is equalto zero for convenience
16 As Fischer shows if wages were set only for the current period t that is tminusiWt = φ +EtminusiPt then his results would be like those obtained by the Sargent and Wallace modelwith rational expectations for similar reasons This can be seen clearly on substituting(910) into (99prime) in which case one obtains the standard Lucas-like aggregate supplyequation
17 Later it will be assumed that ut and utminus1 are correlated so that information obtainedduring period t minus 1 will help predict variations in output for period t Like the laggedoutput term in the Lucas equation this introduction of serial correlation in output isnecessary (but not sufficient) if monetary policy rules that dictate the money supply
Notes 217
for period t based on information obtained up to the end of period t minus 1 are to serve astabilizing role
18 Note that in Fischerrsquos paper minusvt rather than vt is used in (913) Our vt is more in linewith other models
19 With the one exception noted before that Fischer has minusvt replacing vt 20 Again note that Fischer has minusηt and minusρ2 in the above equation since our vt is his minusvt 21 Fischer has b1 = minusρ2 since our vt is his minusvt
10 The Lucas model
1 The discussion follows that found in Lucas (1973) and Sargent (1987a 438ndash446)2 Or as Sargent (1987a 483) states ldquoan employee cares about his prospective wage
measured not in terms of own-market goods but in terms of an economy-wide averagebundle of goods the assumption is that the labor supplier works in one market butshops in many other marketsrdquo
3 The CobbndashDouglas production function would be given by yit = (Nit)1minusα(Ki)
α whereKi denotes the inherited capital stock for sector or ldquoislandrdquo i Nit is the employment oflabor for sector i and yit is the production of commodity i by sector i during period t Inthis case the marginal product of labor is given by partyitpartNit = (1 minus α)(Nit)
minusα(Ki)α
which implies that a = (1 minus α)(Ki)α
4 Solving this expression for the demand for labor and differentiating with respect to thereal wage relevant for a firm producing commodity i we have
partN dit part(witpit) = (1 minus α)minus1α(minus1α)(witpit)
(minus1minusα)α lt 0
Note that our particular form for the labor demand function implies a constant elasticityIn particular the elasticity of demand for labor is given by
partN dit
part(witpit)
witpit
N dit
= minus 1
α
5 Up to this point we have assumed that expectations are held with ldquosubjective certaintyrdquoso that Et(1pt) = 1Etpt and the expected real wage can be represented as witEtpt However we now assume that individuals view the price level as a random variableIn this setting the expression for the expected real wage is Et(witpt) or witEt(1pt)which is not the same as witEtpt In fact the relationship between witEt(1pt) andwitEtpt is shown by Jensenrsquos inequality witEt(1pt) ge wit(1Etpt) This inequalityreflects the fact that the function f ( pt) = 1pt is convex rather than linear in pt Forinstance if pt = p0 with probability t and p1 with probability 1 minus t then we haveEt(1pt) = tf ( p0) + (1 minus t)f ( p1) gt f (tp0 + (1 minus t)p1) = 1Etpt where 1 gt t gt 0On the other hand ln(1pt) is linear in ln pt so the log of the real wage is linear in thelog of the price level Thus we express the labor supply function in logarithmic formand consider the expectation of the log of the price level
6 The term ξt is assumed to be serially independent which means that E(ξtξs) = 0 fort = s
7 We assume that zit and ξt are statistically independent8 In problems involving more than two random variables ndash that is a ldquomultivariate regres-
sionrdquo ndash we are correspondingly concerned with the term E(z|x y) the expected valueof z for given values of x and y and so on
9 The discussion that follows is standard statistical theory Note that if the joint distributionof the two variables is a bivariate normal density function then the regression of ln pton ln pit is linear
218 Notes
10 In general if x1 x2 xn are random variables a1 a2 an are constants andq = a1x1 + a2x2 + middot middot middot + anxn then
E(q) =nsum
i=1
aiE(xi)
Var(q) =nsum
i=1
a2i Var(xi) + 2
sumiltj
aiajCov(xixj)
wheresum
iltj means that the summation extends over all values of i and j from 1 to nfor which i lt j This expression is derived from the definition of the variance
Var(q) equiv E([q minus E(q)]2) = E
⎛⎜⎝⎡⎣ nsum
i=1
ai(xi minus microi)
⎤⎦
2⎞⎟⎠
which can be expanded by means of the multinomial theorem according to which forexample (a+b+ c +d)2 = a2 +b2 + c2 +d2 +2ab+2ac +2ad +2bc +2bd +2cdNote that
Cov(xixj) = E[(xi minus microi)(xj minus microj)]
If the xi i = 1 n are independent then
Var(q) =nsum
i=1
a2i middot Var(xi)
11 Note that these first two expectations are taken without information on ln pit 12 While it is true that if two random variables are independent they are also uncorre-
lated the converse does not necessarily hold That is two random variables that areuncorrelated are not necessarily independent However two random variables havingthe bivariate normal distribution are independent if and only if they are uncorrelated
13 In general for any two random variables x1 and x2 with joint density function f (x1 x2)the marginal density of x2 g(x2) is obtained by integrating out from minusinfin to +infinthe other variable Thus g(x2) = int
f (x1 x2)dx1 Similarly we can obtain the marginaldensity of x1 by integrating out x2 The conditional probability density function of therandom variable x1 given that the random variable x2 takes on the value x2 is definedby φ(x1|x2) = f (x1 x2)g(x2) assuming that g(x2) does not equal zero
14 To obtain the following expression we use the fact that the random variables ξt and zitare independent with zero means and variances σ 2 and σ 2
z respectively15 Note that this ldquosignal extractionrdquo problem appears in a number of different contexts
such as in statistical theories of discrimination in labor economics (Aigner and Cain1977 Lundberg and Startz 1983) and in the industrial organization literature
16 If there were a trend in the natural level of output then ln ynit would replace the firstln yni and in the lagged term ln ynitminus1 would replace the second ln yni
17 The last term in (1016prime) indicates the deviation of output in the prior period from itsldquonormalrdquo level It is presumed that λ lt 1
18 Recall that this supply function assumes no adjustment costs and thus does not have alagged output term in it
Notes 219
19 In general if z1 z2 zn are independent random variables having the same distribu-tion with mean micro and variance σ 2
z and if z = (z1 + z2 + + zn)n then E(z) = micro
and Var(z) = σ 2z n Note that as n goes to infinity the variance of z goes to zero
20 Recall that we have let π(πtminus1) denote ( pt minus ptminus1)ptminus1 This was done so thatthe terms making up the expected real interest rate rt minus πe
t would have the sametime subscript πe
t = ( pet+1 minus pt)pt In the discussions to follow however we shift
time subscripts so that the previously denoted π = π and the previously denoted πet
becomes πet+1
21 Note that the natural log of real output is ln( yt) = Yt 22 Sometimes the money demand function is simplified by assuming α2 = 0 so that there
are no effects of interest rate changes on real money demand23 This follows given mt = mt +εt so that dPtdmt = 1 Recall that Pt and mt are logs of
the price level and the money supply respectively so that dPt = d(ln pt) = (1pt)dptand dmt = d(ln Mt) = (1Mt)dMt
24 Note that we are somewhat imprecise in our statement that we are eliminating the termfor output from the equation Recall that the exact interpretation of Yt would be as thedifference between the logarithm of output and the logarithm of the natural level ofoutput Equivalently we may call Yt the log of the ratio of output to its natural level
11 Policy
1 The alternative to a rule is called ldquopurely discretionary monetary policyrdquo in whichmoney supply changes are made purely at the discretion of the government dependingon its current reading of the economy and current set of objectives
2 This view is not universally accepted For instance according to the Nobel prize winnerJames Tobin (1985) the government should be free to do as it sees fit and he sees manyreasons ldquofor the Fedrsquos reluctance to tie its own hands as much as lsquorulesrsquo advocates wishrdquo Sargent and Wallace (1976) identify as ldquoKeynesiansrdquo individuals who believegovernment monetary policy should attempt to ldquolean against the windrdquo in an effort toattenuate the business cycle In the view of Sargent and Wallace and others the monetaryauthority has no scope to conduct countercyclical policy Not surprisingly those whodisagree with this view suggest alternative models of the economy such as those withldquostickyrdquo prices more favorable to their alternative views
3 Those who argue for a rule such as this are sometimes called ldquomonetaristsrdquo Someadvocates of a constant growth rate in the money supply would restrict the length oftime during which a particular rate of growth was fixed For instance William Pooleformer member of the Presidentrsquos Council of Economic Advisers (1982ndash1985) andnow with the Fed suggests that monetary rules should be adopted but that the ruleshould be ldquosubject to change at any time upon presentation of a convincing case withsupporting evidencerdquo (Poole 1985)
4 Hallrsquos suggestion echoes Simonsrsquo (1936) proposal for ldquoa monetary rule of maintainingthe constancy of some price indexrdquo Wayne Angell a 1986 appointee to the Board ofGovernors of the Federal Reserve (the monetary authorities for the USA) has arguedfor a monetary policy that will stabilize a price index constructed from a basket of basiccommodities (perhaps including gold)
5 Recall also that we assume the random variables εt and ut (the random variable asso-ciated with output demand) are independent with zero mean and respective variancesσ 2
e and σ 2u
6 An example of such ldquoexogenousrdquo price expectations would be the autoregressiveexpectations
7 Recall that the natural level of output was normalized to equal one so that Yn equiv ln yn = 0Thus if the optimal level of output is the natural level then the objective would be tominimize Etminus1(Yt)
2
220 Notes
8 Recall that we are assuming that εt and vt are independent random variables From(112)
Etminus1Yt = a0 + a1Ytminus1 + a2mt
so that Yt minus Etminus1Yt = a2εt + vt Squaring this expression and noting that E(εt) = 0E(vt) = 0 and given independence E(εt middot vt) = 0 we obtain
Etminus1(Yt minus Etminus1Yt)2 = a2
2σ 2e + σ 2
v
9 See Sargentrsquos (1987a 453) equation (13) for the corresponding expression Recall thatg0 = (Y lowast minus a0)a2
10 Note that ln yn = 0 due to normalization11 To tie down the inflation rate we would have to add an inflation objective to the goals
of the government12 Note that this is another example of the Lucas critique in which econometric estimates
of a specific set of parameters based on past policies cannot be used to project the impactof new different monetary rules to be followed in the future for with these new policiesthe parameters will change
13 This equation is derived from combining the price error expression with the aggregatesupply equation given earlier
14 The assumptions leading up to (1116) have been chosen so that (1116) matches thecriterion found in Barro and Gordon (1983) We will contrast the results of that paperwith our findings here shortly
15 The form of this equation mirrors that in Barro and Gordon16 Consider what would happen if we did not neglect the lagged disturbance terms For
instance let us say one of the disturbance terms (εtminus1 or utminus1) is positive rather thanzero while the other is held equal to zero According to the reduced-form equation forthe price level for period t minus 1 the result would be a higher price level in period t minus 1with the positive lagged disturbance term If the rate of growth in the money supplybetween period t minus 1 and t was the same in both cases then the inflation rate wouldbe higher in the case when both lagged disturbance terms are zero Thus to maintain aconstant inflation rate a lower money supply growth is implied if lagged disturbanceterms are zero instead of positive
17 Kydland and Prescott were awarded the 2004 Nobel prize in economics for their work18 This form of the objective function introduced above is discussed in Barro and Gordon
(1983) Without the assumption of k lt 1 a zero average rate of inflation would beoptimal regardless of the nature of expectation formation
19 For simplicity we assume there is no persistence in real shocks Thus we assume λ = 0in the original Barro and Gordon model
12 Open economy
1 For discussion purposes the terms ldquodomesticrdquo and ldquoforeignrdquo are used with respect tothe domestic perspective
2 For simplicity we assume foreigners do not desire to hold domestic money3 We assume for simplicity that foreigners are not taxed by the domestic government4 That is the domestic households own the remaining share of bonds issued by foreign
firms Bff and α of the equity share issues by foreign firms The total dividends (interms of the foreign currency) paid by foreign firms in period t are denoted by pftdftwhere pft is the price level of the foreign country and dft are real dividends of foreignfirms
Notes 221
5 One might also include bdt as purchase of output by private depository institutions is
included in the household budget constraint This reflects the replacement of dividendspaid by private depository institutions to their shareholders (households) by the differ-ence between banksrsquo interest income and their purchases of the composite commodityThis definition of dividends for private banks has been termed the ldquoflowrdquo constraint forprivate depository institutions
6 Recall that MB denotes the monetary base R denotes reserves and C denotes currencyin the hands of the nonbank public
7 Recall that we use the term ldquomodifiedrdquo as we have substituted the firm distribution con-straint into the household budget constraint for labor income We thus have effectivelysuppressed the labor market from the markets under consideration This modified lawis useful in understanding how the aggregate demand equation is derived
8 By ldquoreal demandrdquo we mean in units of the domestic commodity9 The term ldquorealrdquo means in terms of the domestic composite commodity
10 In measuring the exports imports and international capital flows a number of itemsare often missed For instance the clandestine transfer of funds from the Philippines toUS bank accounts would generate a demand for dollars On the other hand secretiveimports of heroin from Turkey result in a supply of dollars in international marketsThe net of such unmeasured transactions are lumped under the heading of ldquostatisticaldiscrepancyrdquo in the balance of payments accounts We have omitted this item fromTable 121
11 For simplicity we assume that the exchange rate affects only the division of totalconsumption between imports and purchases of the domestic output total consumptionis assumed to be unaffected by such exchange rate changes
12 Note that an increase in the price of the yen or an ldquoappreciationrdquo of the yen means afall in the price of a dollar in terms of yen or a ldquodepreciationrdquo of the dollar
13 If the price elasticity with respect to imports was less than one in short run a rise in therelative price of imports due to a depreciation of the dollar could in fact lead to a risein the value of imports as well This short-run phenomenon when applied to the path ofnet exports over time is referred to as the ldquoJ-curve effectrdquo
14 Actually for much of the analysis to follow we need only assume the weaker ldquoMarshallndashLernerrdquo condition that the sum of the price elasticity of demand for imports and theprice elasticity of demand for exports exceeds one This assures that a price of the dollarbelow its equilibrium level will be associated with an excess demand for dollars in theforeign exchange markets while a price of the dollar above its equilibrium level willbe associated with an excess supply of dollars
15 Given future markets for foreign currency this is not always the case16 We assume that such expectation is held with subjective certainty17 There are several reasons why in the short run the supply curve may not be upward-
sloping First it often takes time for US purchasers to adjust purchases in light of achange in relative prices Second the prices of domestic goods that are close substitutesto the imported goods can rise significantly in the short run as domestic producers hitshort-run production constraints The third complication that has the effect of makingthe supply of dollars curve less likely to be upward-sloping is that foreign producersat least in the short run often adjust the foreign currency prices of goods they exportto partially offset the impact of exchange rate changes on the prices of their goodsin foreign markets For instance Knetter (1987) found that with a depreciation of thedollar (appreciation of the West German Mark) West German exporters often reducedthe Mark price of their exports so as to minimize the rise in the dollar price of Germangoods that would result from the appreciation of the Mark For the time being weabstract from such short-run considerations although this is not to lessen the importanceof this phenomenon as the experience during the 1985ndash1987 period indicates With adepreciation of the dollar the dollar value of imports grew as the USA had to supply
222 Notes
more dollars for each unit of imported goods and there was initially little reduction inthe quantity of goods imported
18 It is important to remember that while we talk of households as being the only privatedemanders of foreign goods and financial assets this is purely a simplifying deviceFirms also demand foreign goods and private depository institutions demand foreignfinancial assets The analysis would be more complex if we explicitly recognized thesedemands but our conclusions would be unchanged since we can subsume in householdactions the actions of firms and depository institutions in foreign markets
References
Aigner DJ and Cain GC (1977) Statistical theories of discrimination in labor marketsIndustrial and Labor Relations Review 30(2) 175ndash187
Alchian A and Demsetz H (1972) Production information costs and economicorganization American Economic Review 62 777ndash795
Alogoskoufis GS (1987) On intertemporal substitution and aggregate labor supplyJournal of Political Economy 95 938ndash960
Altonji J and Siow A (1987) Testing the response of consumption to income changeswith (noisy) panel data Quarterly Journal of Economics 102 293ndash328
Arrow K (1964) The role of securities in the optimal allocation of risk-bearing Review ofEconomic Studies 31 (April) 91ndash96
Arrow K and Hahn FH (1971) General Competitive Analysis San Francisco CAHolden-Day
Azariadis C (1976) On the incidence of unemployment Review of Economic Studies 43115ndash125
Barro RJ (1976) Rational expectations and the role of monetary policy Journal ofMonetary Economics 2 1ndash32
Barro RJ and Gordon D (1983) A positive theory of monetary policy in a natural ratemodel Journal of Political Economy 91 589ndash610
Barro RJ and Grossman HI (1971) A general disequilibrium model of income andemployment American Economic Review 61 82ndash93
Barro RJ and King RG (1984) Time-separable preferences and intertemporal substitu-tions models of business cycles Quarterly Journal of Economics 99 817ndash839
Begg DKH (1980) Rational expectations and the non-neutrality of systematic monetarypolicy Review of Economic Studies 47 293ndash303
Blanchard OJ (1981) What is left of the multiplier accelerator American EconomicReview 71 150ndash154
Blanchard OJ and Fischer S (1989) Lectures on Macroeconomics Cambridge MA MITPress
Borjas GJ and Heckman JJ (1978) Labor supply estimates for public policy evaluationNBER Working Paper No W0299 November
Box GEP and Jenkins GM (1970) Time Series Analysis San Francisco CAHolden-Day
Burbidge JB and Robb AL (1985) Evidence on wealthndashage profiles in Canadian cross-section data Canadian Journal of Economics 18 854ndash875
Caballero R and Engel E (1999) Explaining investment dynamics in US manufacturingA generalized (S s) approach Econometrica 67 783ndash826
224 References
Campbell JY and Mankiw NG (1987a) Permanent and transitory components inmacroeconomic fluctuations American Economic Review 77 111ndash117
Campbell JY and Mankiw NG (1987b) Are output fluctuations transitory QuarterlyJournal of Economics 102 857ndash880
Campbell JY Lo AW and MacKinlay AC (1997) The Econometrics of FinancialMarkets Princeton NJ Princeton University Press
Cass D (1965) Optimal growth in an aggregate model of capital accumulation Review ofEconomic Studies 32 233ndash240
Clower RW (1965) The Keynesian Counter-revolution A theoretical appraisal InFH Hahn and FPR Brechling (eds) The Theory of Interest Rates London Macmillan
Coase RH (1960) The problem of social cost Journal of Law and Economics 31ndash44
Cukierman A (1986) Measuring inflationary expectations Journal of Monetary Eco-nomics 17 315ndash324
Dahlman C (1979) The problem of externality Journal of Law and Economics 22141ndash162
Debreu G (1959) The Theory of Value New York WileyDiamond PA and Hausman JA (1984) Individual retirement and savings behaviour
Journal of Public Economics 23 81ndash114Dornbusch R (1976) Expectations and exchange rate dynamics Journal of Political
Economy 84 1161ndash1176Enders W (2004) Applied Econometric Time Series 2nd edn Hoboken NJ WileyEvans G (1989) Output and unemployment dynamics in the United States 1950ndash1985
Journal of Applied Econometrics 4 213ndash237Ewing BT (2001) Cross-effects of fundamental state variables Journal of
Macroeconomics 23 633ndash645Ewing BT and Payne JE (1998) The long-run relation between the personal savings rate
and consumer sentiment Financial Counseling and Planning 9(1) 89ndash96Ewing BT Levernier W and Malik F (2002) Differential effects of output shocks on
unemployment rates by race and gender Southern Economic Journal 68 584ndash599Fama EF and Miller MH (1972) The Theory of Finance New York Holt Rinehart and
WinstonFischer S (1977) Long-term contracts rational expectations and the optimal money supply
rule Journal of Political Economy 85 191ndash205Fisher I (1930) The Theory of Interest New York MacmillanFleming MJ (1962) Domestic financial policies under fixed and under floating exchange
rates IMF Staff Papers 9 369ndash379Foley KD (1975) On two specifications of asset equilibrium in macroeconomic models
Journal of Political Economy 83 303ndash324Friedman M (1959) A Program for Monetary Stability New York Fordham University
PressFriedman M (1968) The role of monetary policy American Economic Review 58 1ndash17Goldberg S (1958) Difference Equations New York WileyGould JP (1968) Adjustment cost in the theory of investment of the firm Review of
Economic Studies 35 47ndash56Grandmont JM (1977) Temporary general equilibrium theory Econometrica 45
535ndash572Granger CWJ and Morris M (1976) Time series modelling and interpretation Journal
of the Royal Statistical Society Series A 139 246ndash257
References 225
Granger CWJ and Newbold P (1986) Forecasting Economic Time Series 2nd ednOrlando FL Academic Press
Hall R (1976) The Phillips curve and macroeconomic policy In K Brunner andAH Meltzer (eds) The Phillips Curve and Labor Markets Carnegie-RochesterConference Series on Public Policy Amsterdam North-Holland
Hall R (1980) Employment fluctuation and wage rigidity In GL Perry (ed) BrookingsPapers on Economic Activity pp 91ndash123 Washington DC Brookings Institution
Hall RE (1982) Explorations in the Gold Standard and related policies for stabilizingthe dollar In RE Hall (ed) Inflation Causes and Effects Chicago IL University ofChicago Press
Hall RE (1988) Intertemporal substitution in consumption Journal of Political Economy96 339ndash357
Hannan EJ (1970) Multiple Time Series New York WileyHansen B (1970) A Survey of General Equilibrium Systems New York McGraw-HillHansen LP and Singleton KJ (1983) Stochastic consumption risk aversion and the
temporal behavior of asset returns Journal of Political Economy 91 249ndash265Harvey AC (1993) Time Series Models Cambridge MA MIT PressHayashi F (1982) Tobinrsquos marginal q and average q A neoclassical interpretation
Econometrica 50(1) 213ndash224Hicks J (1939) Value and Capital Oxford Clarendon PressHowitt P (1985) Transaction costs in the theory of unemployment American Economic
Review 75 88ndash100Howitt P (1986) Conversations with economists A review essay Journal of Monetary
Economics 18 103ndash118Hurd M (1987) Savings of the elderly and desired bequests American Economic Review
77 298ndash312Karni E (1978) Period analysis and continuous analysis in Patinkinrsquos macroeconomic
model Journal of Economic Theory 17 134ndash140Kester WC (1986) Capital ownership structure A comparison of the United States and
Japanese manufacturing corporations Financial Management 15(1) 5ndash16Keynes JM (1936) General Theory of Employment Interest and Money London
MacmillanKletzer LG and Fairlie R (1998) Jobs lost jobs regained An analysis of blackwhite
differences in job displacement in the 1980s Industrial Relations 37 460ndash477Knetter M (1987) Export prices and exchange rates Theory and evidence Working paper
Stanford University NovemberKoopmans T (1965) On the concept of optimal economic growth In Proceedings ndash
Study Week on the Econometric Approach to Development Planning Chicago ILRand-McNally
Kydland FE and Prescott EC (1977) Rules rather than discretion The inconsistency ofoptimal plans Journal of Political Economy 85 473ndash491
Kydland FE and Prescott EC (1982) Time to build and aggregate fluctuationsEconometrica 50 1345ndash1370
Leonard JS (1988) In the wrong place at the wrong time The extent of frictional andstructural unemployment NBER Working Paper No 1979
Lucas RE (1967) Adjustment costs and the theory of supply Journal of Political Economy75 321ndash334
Lucas RE (1972) Expectations and the neutrality of money Journal of Economic Theory4 103ndash124
226 References
Lucas RE (1973) Some international evidence on outputndashinflation tradeoffs AmericanEconomic Review 63 326ndash334
Lucas R Jr (1981) Methods and problems in business cycle theory In Studies in Business-Cycle Theory Cambridge MA MIT Press First published in Journal of Money Creditand Banking 12 November 1980
Lucas RE and Rapping LA (1970) Real wages employment and inflation InES Phelps (ed) Microeconomic Foundations of Employment and Inflation TheoryNew York WW Norton
Lundberg S and Startz R (1983) Private discrimination and social intervention incompetitive labor markets American Economic Review 73(3) 340ndash347
McCallum B (1979) The current state of the policy-ineffectiveness debate AmericanEconomic Review 69 240ndash245
McCallum B (1985) On consequences and criticisms of monetary targeting Journal ofMoney Credit and Banking 17 570ndash597
Mankiw NG (1987) The optimal collection of seigniorage Theory and evidence Journalof Monetary Economics 20 327ndash341
Mankiw NG Rotemberg JJ and Summers LH (1985) Intertemporal substitution inmacroeconomics Quarterly Journal of Economics 100 225ndash251
Marx K (1976) Capital Vol 1 A Critique of Political Economy HarmondsworthPenguin
Mills TC (1999) The Econometric Modelling of Financial Time Series 2nd ednCambridge Cambridge University Press
Modigliani F (1966) Life cycle hypothesis of saving the demand for wealth and the supplyof capital Social Research 33 160ndash217
Modigliani F (1986) Life cycle individual thrift and the wealth of nations AmericanEconomic Review 76 297ndash313
Mundell RA (1968) International Economics New York MacmillanNakamura A and Nakamura M (1981) A comparison of the labor force behavior of
married women in the US and Canada with special attention to the impact of incometaxes Econometrica 49 451ndash489
Nelson CR and Plosser CI (1982) Trends and random walks in macroeconomictime series Some evidence and implications Journal of Monetary Economics 10139ndash162
Niehans J (1987) Classical monetary theory new and old Journal of Money Credit andBanking 19 409ndash424
Oi WY (1962) Labor as a quasi-fixed factor Journal of Political Economy 70 538ndash555Patinkin D (1965) Money Interest and Prices New York Harper amp RowPencavel J (1985) Labor supply of men A survey In Orley Ashenfelter (ed) Handbook
of Labor Economics Amsterdam North-HollandPhelps ES (1968) Money-wage dynamics and labor market equilibrium Journal of
Political Economy 76 678ndash711Phelps ES and Taylor JB (1977) Stabilizing powers of monetary policy under rational
expectations Journal of Political Economy 85 163ndash190Phillips AW (1958) The relationship between unemployment and the rate of change in
money wage rates in the United Kingdom 1861ndash1957 Economica 25 283ndash299Pindyck RS and Rubinfeld DL (1991) Econometric Models and Economic Forecasts
3rd edn New York McGraw-HillPoole W (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo
Journal of Money Credit and Banking 17 602ndash605
References 227
Radford RA (1945) The economic organization of a prisoner of war camp Economica12 189ndash201
Robinson C and Tomes N (1985) More on the labour supply of Canadian womenCanadian Journal of Economics 18 156ndash163
Sargent T (1987a) Macroeconomic Theory 2nd edn Boston MA Academic PressSargent T (1987b) Dynamic Macroeconomic Analysis Cambridge MA Harvard
University PressSargent TJ and Wallace N (1975) ldquoRationalrdquo expectations the optimal monetary
instrument and the optimal money supply rule Journal of Political Economy 83241ndash254
Sargent TJ and Wallace N (1976) Rational expectations and the theory of economicpolicy Journal of Monetary Economics 2 169ndash183
Shapiro C and Stiglitz JE (1984) Equilibrium unemployment as a worker disciplinedevice American Economic Review 74 433ndash444
Shiller RJ (1978) Rational expectations and the dynamic structure of macroeconomicmodels Journal of Monetary Economics 4 1ndash44
Sidrauski M (1967a) Rational choice and patterns of growth in a monetary economyAmerican Economic Review 57 534ndash544
Sidrauski M (1967b) Inflation and economic growth Journal of Political Economy 75797ndash810
Simons HC (1936) Rules versus authorities in monetary policy Journal of PoliticalEconomy 44 1ndash30
Sims C (1972) Money income and causality American Economic Review 62 540ndash552Solow R (1956) A contribution to the theory of economic growth Quarterly Journal of
Economics 70 65ndash94Strotz RH (1955ndash1956) Myopia and inconsistency in dynamic utility maximization
Review of Economic Studies 23 165ndash180Stuart CE (1981) Swedish tax rates labor supply and tax revenues Journal of Political
Economy 89 1020ndash1038Taylor J (1972) The behaviour of unemployment and unfilled vacancies Great Britain
1958ndash71 An alternative view Economic Journal 82 1352ndash1365Taylor JB (1979) Estimation and control of a macroeconomic model with rational
expectations Econometrica 47 1267ndash1286Tobin J (1965) Money and economic growth Econometrica 33 671ndash684Tobin J (1969) A general equilibrium approach to monetary theory Journal of Money
Credit and Banking 1(1) 15ndash29Tobin J (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo or
Monetary targeting Dead at last Journal of Money Credit and Banking 17 605ndash610Uzawa H (1969) Time preference and the Penrose effect in a two-class model of economic
growth Journal of Political Economy 77 628ndash652Varian H (1992) Microeconomic Analysis 3rd edn New York WW NortonWalker DA (1987) Walrasrsquo theories of tatonnement Journal of Political Economy 95
758ndash774Walras L (1954) Elements of Pure Economics trans W Jaffeacute London George Allen amp
UnwinWeber CE (1998) Consumption spending and the paper-bill spread Theory and evidence
Economic Inquiry 36 575ndash589Weitzman M (1985) The simple macroeconomics of profit sharing American Economic
Review 75 937ndash953
Index
accelerationist outcome 170ndash2aggregate demand 7 86 90ndash1 94ndash5 99
Keynesian model 133ndash4 135 139 140Lucas model 160ndash1 neoclassical model126 127 128 146 shocks 158 seealso demand
aggregate supply 82 86ndash90 91 94ndash5 98113 Keynesian model 132 133ndash4 135138 140 145 Lucas model 137153ndash4 155 158 160ndash1 164 166monetary policy 170ndash1 neoclassicalmodel 118 123 125ndash6 127 128 133see also supply
aggregation issues 5ndash6 8 14ndash15Alchian A 203n9Alogoskoufis GS 51Angell Wayne 219n4Arrow-Debreu theory 2 202n4 n10
203n8assets 51 52 53 portfolio decision
39ndash40 57ndash9 tangible 25ndash6 see alsofinancial assets
autocorrelation function 102ndash3autoregressive expectations 162ndash4 168autoregressive processes 103ndash5 110 111
113 166
balance of payments 190ndash3bankruptcy 32 33Barro RJ 156 167 179ndash80 182ndash4Begg DKH 122 215n8behavioral hypotheses 96 98 99 100Bellman equation firms 29 30 31 34
35 households 43 44 45 48ndash9Blanchard OJ 111 112 171 175 184ndash5bonds 19ndash20 23 27ndash9 32ndash3 financial
market equilibrium 84ndash5 firmfinancing constraint 24ndash6 64 Fisherian
problem 52 foreign 189 households39 40 43ndash4 72 204n11 money illusion59 portfolio choice 57 58 59 93 realvalue 77 temporary equilibrium 80 81
budget constraint 7 16 household 43 4455 65ndash6 67 70ndash1 73ndash4 open economy188 189
Burbidge JB 208n25business cycle 5 6 49 50 173 real
business cycle theory 79 136 202n4technological innovation 113
Caballero R 209n1Campbell JY 111 112 113 214n45capital 23 24 31 68 69ndash70 cost of
31ndash2 36 69 70 94 120 209n7 firmfinancing constraint 24 25 26 64international flows 192 193 195ndash6197ndash8 200 Tobinrsquos Q 36 see alsocapital stock
capital account 192capital adjustment costs 21 26ndash7 34ndash7
69ndash70capital markets 27 203n1 205n24capital stock 3 18ndash19 20ndash1 23 25 27ndash9
optimal investment 30 31 69 70retained earnings financing 29ndash30 32superneutrality of money 120 121 122Tobinrsquos Q 35 36
central banks 187 188 189 190191 192ndash3
classical economics 4 5 79 212n8Clower RW 117 118Coase RH 11CobbndashDouglas production function 147
148 217n3commodities 8 9ndash10 14 16 18ndash19 20
individual experiments 11 13ndash14
Index 229
Lucas model 146ndash7 market equilibrium92 open economy 188ndash9
comparative static analysis 116ndash23consumer behavior 11 12ndash13consumption 20 39 40ndash6 47ndash8 67
70ndash4 aggregate 55 61 demand 97 99195 215n6 financial asset demand 94Fisherian problem 51ndash5 individualexperiments 11 12ndash13 intertemporalsubstitution hypothesis 51 60ndash3Keynesian model 134 135 laborparticipation 49 life-cycle hypothesis50 55ndash6 57 money supply shocks 128neoclassical model 117ndash18 123 126neutrality of money 119 open economy190ndash1 193 permanent incomehypothesis 56ndash7 portfolio choice 5758 59 real balance effect 77 92superneutrality of money 121 122
continuous-time analysis 4cost of capital 31ndash2 36 69 70 94 120
209n7cost of living agreements (COLAs) 215n2costs bankruptcy 33 capital adjustment
21 26ndash7 34ndash7 69ndash70 dividends 23ndash4firm financing constraint 24
Cramerrsquos rule 87 119 121 127 134Cukierman A 164currency depreciation 193 195 198ndash9
221n17
Dahlman C 203n9De Moivrersquos theorem 109Debreu G 8debt 32 60debt-to-equity ratio 32ndash4decision-making behavior 5ndash6 38 51 see
also portfolio decisiondemand excess 13ndash14 15ndash16 67 118
133 190ndash1 199 individual demandfunctions 13 14 labor marketequilibrium 83 Lucas model 146156ndash7 market demand functions 1415 money 86 neoclassical model82ndash3 119 122 123 new classicaleconomics 5 price elasticity of 195real balance effect 60 77 Sayrsquos law 716 temporary equilibrium 80Walrasian model 7 15 see alsoaggregate demand
Demsetz H 203n9depository institutions 126 187 188 189
221n5 222n18
depreciation 26 31 74 205n23deterministic feedback rules 172 173 174deterministic models 6disequilibrium 78 199distributed lag schemes 162ndash4dividends depository institutions 221n5
firms 20 22 23ndash4 29 30 32 34ndash5future 76ndash7 households 43 44 66
Dornbusch model 200ndash1dynamic analysis 2 3ndash4 6 79 122dynamic programming 28ndash9
econometric models 96 100ndash1employment 1 19 128 148 full
employment model 88 142 Keynesianmodel 130ndash1 132ndash3 134 135 137neoclassical model 82 83 84 91 122123ndash5 126 see also labor labor marketlabor supply
Engel E 209n1equilibrium 1 5 7 15 199 aggregate
demand 90ndash1 94ndash5 99 aggregatesupply 86ndash8 94ndash5 employment123ndash5 financial markets 80ndash1 84ndash6illusion model 79 Keynesian model79 133ndash4 137 139ndash40 labor market83ndash4 88 90 119 123ndash5 127 131 148loanable funds theory 120 Lucas model159 160 162 monetary policy 182ndash4money market 85ndash6 92ndash3 126 159multiple equilibria models 113 naturalrate of unemployment 89 90neoclassical model 78 79 117ndash18 126non- market-clearing model 79 partial187 Patinkin analysis 91ndash4 stationarity106 stochastic models 6 temporary 278 80ndash3 86 120 velocity 139ndash40 seealso general equilibrium
equity shares 18 19ndash20 21ndash2 23 27ndash932ndash3 firm financing constraint 24ndash664 households 40 43 72 real value77 temporary equilibrium 80 81
ergodicity 212n16Eulerrsquos theorem 206n44Ewing BT 211n11 212n11exchange rate 186ndash7 191 193ndash6 197
199 200ndash1expectations 41 59 84 119 202n5
adaptive 163 170ndash2 210n15 aggregatedemand 90ndash1 autoregressive 162ndash4168 inflation 62 69 71 77 120ndash2163 164 interest rate 60 61 62 6971 75ndash7 Keynesian model 135 136ndash8
230 Index
expectations (Continued)140 141ndash4 145 labor marketequilibrium 83 Lucas model 157 160optimal monetary policy 168ndash70 180182 183ndash4 output 68 rational 3128ndash9 136ndash8 141ndash4 164ndash6 172ndash4176ndash8 182ndash4 suppliers 124 135 weakconsistency 203n4 see also forecasting
exports 190ndash1 193 197 200
financial assets 77 85 93ndash4 118 firms19ndash20 25 26 28 29 68ndash9 70households 40 42 43ndash4 66 71 72ndash4labor market equilibrium 83 openeconomy 186 187ndash8 189 191 192193 195ndash6 197ndash9 temporaryequilibrium 81ndash2 see also bondsequity shares
financial market 1 36 67 96ndash7equilibrium 80ndash1 84ndash6 93 94 95 118open economy 191 195ndash6 197 199
firm distribution constraint 23ndash4 28 4165 67 74 76ndash7 189
firm financing constraint 23 24ndash6 28ndash964 67 68ndash70 84ndash5
firms 18ndash38 64ndash5 68ndash70 130 135aggregate supply 87 capital adjustmentcosts 26ndash7 34ndash7 dividends 23ndash4investment demand 97 laboradjustment costs 37ndash8 money illusion75 119 neoclassical model 79objectives 21ndash3 open economy 187188 189 190 222n18 optimalinvestment 30ndash2 real wage illusion124 125 retained earnings financing29ndash30 34
Fischer S 112 136 137 138ndash45 171175 184ndash5
Fisher I 79 163 208n16Fisherian problem 39 46 51ndash5 209n12
215n6forecasting 96 100ndash1 101ndash14 128
149ndash50 see also expectationsforeign exchange market 186ndash7 188 191
193ndash5 198ndash9 200foreigners 187ndash90 191 197ndash8Friedman Milton 56 156 168 171 179futures market 3 7 8 10 65 66
general equilibrium 1 2 11 95 97 187neoclassical model 117 118 prices 710 15ndash16 Walrasian model 8 see alsoequilibrium
GNP see gross national productGordon D 156 167 179ndash80 182ndash4government 187 190Grandmont JM 78Granger CWJ 100 102 112 212n22gross national product (GNP) 112 168
habit persistence model 207n2Hall Robert 60ndash1 62 172Hansen B 8Harvey AC 101 114Hayashi F 36 206n43Hicks John 2 8households 18 19 39ndash63 65ndash6 70ndash4
aggregate supply 87 bonds and equityshares 20 24 consumption demand97 dividends 23 24 Fisherian problem39 46 51ndash5 imperfect foresight 124intertemporal substitution hypothesis49ndash51 60ndash3 labor supply problem46ndash51 life-cycle hypothesis 55ndash6money illusion 75ndash6 77 119neoclassical model 79 open economy187 188 189 191 193ndash6 222n18permanent income hypothesis 56ndash7portfolio decision 39ndash40 46 57ndash9price changes 21
Howitt P 5 203n15
imperfect foresight 123 124imports 193ndash5 199 200income 59 74 97 208n27 foreign goods
195 197 life-cycle hypothesis 55ndash657 neoclassical model 117 permanentincome hypothesis 56ndash7 see also wages
income effects 47ndash8 49individual experiments 11ndash14 21 39inflation 53 59 72 196 200
expectations 62 69 71 77 120ndash2 163164 Lucas model 146 154ndash6 157160 161 monetary policy 136 170ndash2174ndash9 180 181ndash2 183ndash4superneutrality of money 121 122wages 123 131 see also prices
integrated processes 110ndash11 112interest rate 1 4 20 31 36 59
consumption demand 97 equilibrium91ndash4 equity shares 22 23 expectations60 61 62 69 71 75ndash7 financialmarket equilibrium 84 85 94Fisherian problem 53 54 foreignfinancial assets 195 196 198household problem 43 45 46 47
Index 231
intertemporal substitution hypothesis49 50 51 63 72 Keynesian model133 134 labor demand 83 loanablefunds theory 120 Lucas model159ndash60 neoclassical model 118 119126ndash7 neutrality of money 119 parity200ndash1 superneutrality of money 121ndash2temporary equilibrium 81 82
intertemporal substitution hypothesis (ISH)49ndash51 60ndash3 72 73 75ndash6
invertibility condition 110investment 3 29 84 85 128 capital
adjustment costs 26ndash7 35 36 demand32 68ndash70 75 90 94 97 99 120206n40 Keynesian model 134 135neoclassical model 118 126 neutralityof money 119 optimal 30ndash2superneutrality of money 121 122supply-side variables 116
IS equation 79 90ndash1 98ndash9 118 126 191export demand 200 Keynesian model133 134 138ndash9 Lucas model 159 160
ISH see intertemporal substitutionhypothesis
lsquoislandrsquo paradigm 146ndash9
J-curve effect 221n13
Karni Edi 207n3Keynes John Maynard 4Keynesian model 5 79 88 130ndash45 199Knetter M 221n17Kydland FE 180ndash1 207n2 220n17
labor 19 21 27 42 66 adjustment costs37ndash8 demand 68 76 97 132ndash3 147215n9 217n4 Keynesian models 5marginal product of 30 optimalinvestment 31 participation 48ndash9temporary equilibrium 80 see alsoemployment labor supply wages
labor market 1 18 37 65 96ndash7 186aggregate supply 86ndash8 94 95 119intertemporal substitution hypothesis49ndash51 Keynesian model 131ndash3 134Lucas model 147 148 natural rate ofunemployment 89 90 neoclassicalmodel 83ndash4 118 124ndash5 127 Walrasrsquolaw 66ndash7 82 see also employment
labor reserve hypothesis 37ndash8labor supply 39 40 41 45 46ndash9 215n9
intertemporal substitution hypothesis49ndash51 72 73 75ndash6 210n15 Keynesian
model 132 133 Lucas model 147ndash8neoclassical model 79 83ndash4 117 123124 perfect foresight 66 70 71 realbalance effect 77 supply-side variables116 tax rates 99 see also employmentlabor
leisure 40ndash1 42 44 46ndash8 49 50 72life-cycle hypothesis 50 55ndash6 57linear regression analysis 150ndash3LM equation 79 90ndash1 98ndash9 118 126
191 Keynesian model 133 134 138ndash9Lucas model 159 160
loanable funds theory 120Lucas RE 6 50ndash1 146 149 156ndash7
164 167 203n13 210n15Lucas model 88 128 135ndash6 140 146ndash66
176 199Lucas supply function 79 137 153ndash6
158 174 175
Mankiw NG 111 112 113 136 214n45market clearing 4ndash5 6 7 15 50 79
commodity market 92 financial market93 Lucas model 146 164 wages 135
market experiments 11 14ndash16markets 1 2 20 78 187 aggregate
demand 86 90ndash1 capital 27 203n1205n24 foreign exchange 186ndash7 188191 193ndash5 198ndash9 200 futures 3 7 810 65 66 Lucas model 146 149 154sequential 67 see also financial marketlabor market money market
Marx Karl 79microeconomics 5 6 78 88 187mixed autoregressive moving average
processes 110Modigliani Franco 55ndash6 208n24ModiglianindashMiller theorem 32monetary policy 166 167ndash85 187
activist 169 171 174 discretionary136 167ndash8 184 219n1 Keynesianmodel 136ndash8 141 142 144ndash5 policyineffectiveness proposition 120 136137ndash8 172ndash9 182
money classical analysis 4 comparativestatic analysis 118 demand 71ndash4 7797 121 122 139ndash40 159 financialmarket equilibrium 84 householdproblem 41 42 43ndash4 45 46 66neoclassical model 117ndash18 neutralityof 119ndash20 136 142 173 174portfolio decision 39 57 58 59 72superneutrality of 120ndash2
232 Index
money (Continued)temporary equilibrium 80 82transaction costs 11 utility yield of 42see also money market money supply
money illusion 59 75ndash6 77 88 119 125210n15
money market 7 16 67 96ndash9 199aggregate demand 90 94 95equilibrium 85ndash6 92ndash3 126 159Keynesian model 133 134
money supply 3 76 100 119 168 190Keynesian model 134ndash5 142 Lucasmodel 159ndash60 161 162 monetarypolicy 144 169 170 173 178ndash9 180money market equilibrium 86 91neoclassical model 116 118ndash20 126ndash9purchasing power parity 200superneutrality of money 120ndash2 seealso money
moving average processes 109ndash10multiple equilibria models 113MundellndashFleming model 201
natural rate hypothesis 125 128ndash9 135166
Nelson CR 113neoclassical model 4ndash5 116ndash29 146 199
202n12 Keynesian model comparison133 136 Lucas model comparison 160162 modification of 130ndash1 purchasingpower parity 200 simple 78ndash95
new classical economics 4ndash5 6 7Newbold P 100 102 112 212n22numeraire 7 8 9ndash10
official reserve transaction balance 192open economy 186ndash201output 1 18 19 20 66ndash7 202n1
aggregate demand 86 90 94 95 139aggregate supply 86 87ndash8 classicalanalysis 4 comparative static analysis118 Dornbusch model 200 equilibrium80 82 91ndash2 94 126 expectations 6875 firm distribution constraint 65Keynesian model 5 133ndash4 135 137ndash8143ndash4 145 labor market equilibrium83 84 Lucas model 79 146 147148ndash9 153ndash5 157ndash62 165ndash6 monetarypolicy 144 168 170ndash2 173 174ndash5179 money supply shocks 128 movingaverage processes 109 neoclassicalmodel 117ndash18 122 123 125 126neutrality of money 119 open economy
187ndash8 189 190ndash1 199 simpletheoretical model 96ndash9 supply-sidevariables 116 136 time series model111ndash14
overlapping generations models 6
Patinkin D 8 11 120 202n12 critiqueof 117 equilibrium 83 84ndash5 91ndash4firms 210n5 labor supply 47
Payne JE 212n11perfect competition 7 11perfect foresight 3 87 firms 65 68ndash70
75 households 41 43 56ndash7 66 70ndash475 labor market equilibrium 83 125Lucas model 161 162 neoclassicalmodel 78 79 84 117 118 119 131Walrasrsquo law 66 67 82
perfect substitution 20 22 29 32period (discrete) analysis 4permanent income hypothesis 56ndash7Petty William 79Phelps ES 136ndash7 171 216n11Phillips AW 155Phillips curve 154ndash5 156ndash7 171 174ndash9
180 181 183Pindyck RS 101 102Plosser CI 113policy see monetary policypolicy ineffectiveness proposition 120
136 137ndash8 172ndash9 182Poole William 219n3portfolio decision 39ndash40 46 51 57ndash9 72
93 193preferences 11 40Prescott EC 180ndash1 207n2 220n17price elasticity of demand 195prices 1 2 3 19 68 accounting 7 10
13 15 aggregate demand 90 91 9495 aggregate supply 86 98autoregressive expectations 162ndash4classical analysis 4 comparative staticanalysis 116 demand-side variables117 equity shares 20 22 forecastingerrors 149ndash50 foreign goods 193ndash5197 future 41 59 75 76 householdbudget constraint 66 74 individualexperiments 11 Keynesian model 133134 135 137 145 labor marketequilibrium 83 84 Lucas model 146147 149 153 154 156ndash8 159ndash64market experiments 11 microeconomicfoundations 6 monetary policy 172ndash3money market equilibrium 86 natural
Index 233
rate hypothesis 125 natural rate ofunemployment 88ndash9 neoclassicalmodel 78ndash9 117ndash20 122ndash3 124ndash5126ndash8 130 136 146 new classicaleconomics 5 7 perfect foresight 71purchasing power parity 200 realindebtedness effect 21 relative 7 89ndash10 11 13 15ndash16 sequential markets67 superneutrality of money 120tatonnement process 15 temporaryequilibrium 80 82 Walrasian model 79ndash10 15 see also inflation
private international capital flows 192193 195ndash6 197ndash8 200
production 18ndash19productivity of labor 37purchasing power parity 200
Radford RA 8lsquorandom walkrsquo process 103 106 112ndash13Rapping LA 50ndash1 210n15rational expectations 3 128ndash9 136ndash8
141ndash4 164ndash6 172ndash4 176ndash8 182ndash4 seealso expectations
real balance effect 60 77 86 92ndash3 94121
real indebtedness effect 60real user cost of capital 31ndash2 36 69 70
94 120 209n7recontracting assumption 15reduced-form expressions 96 98 99ndash100
Lucas model 161 162 monetary policy168ndash9 172ndash3 177
reputation 184ndash5retained earnings financing 29ndash30 32 34risk 41Robb AL 208n25Rubinfeld DL 101 102
Sargent T 8 96 131ndash2 211n2 n6autoregressive expectations 164employees 217n2 investment demand206n40 Keynesians 219n2 linearregression analysis 152 monetarypolicy 136ndash7 167 168 170 172ndash4180 182 money supply 165 newclassical economics 203n13outputinflation tradeoff 157 Phillipscurve 155 superneutrality of money121 122
Say Jean-Baptiste 7Sayrsquos law 8 16seasonality 111 112
shareholders 25ndash6shares see equity sharesShiller RJ 162shocks 1 6 153 158 186 demand-side
117 monetary policy 170 moneysupply 128 129 135 Phillips curve156 time-series models 111ndash13 wages135ndash6
Simons HC 219n4Sims Christopher 100spot markets 2 37 130 131spot prices 10static analysis 2ndash4 47 50 79 116ndash23stationarity 102 103 104 105ndash9 110ndash11
113 114stationary analysis 2 3ndash4stochastic processes 6 41 101 102ndash3
110 172 173Stuart CE 99substitution effects 47ndash8 49 see also
intertemporal substitution hypothesissupply labor market equilibrium 83
Lucas supply function 79 137 153ndash6158 174 175 neoclassical model 119new classical economics 5 real balanceeffect 77 Sayrsquos law 7 16 temporaryequilibrium 80 86 Walrasian model 715 see also aggregate supply laborsupply money supply
tatonnement process 7 15 120tax issues 3 32 33 99Taylor JB 100 136ndash7 212n12 216n11technology 18ndash19 21terms of trade 194lsquotime inconsistency problemrsquo 180ndash2time series models 96 100 101ndash14Tobin James 206n40 219n2Tobinrsquos Q 27 35ndash7transaction costs 5 7 10ndash11 20
42 208n18
uncertainty 6 41unemployment 1 50ndash1 104 125
frictional 89 involuntary 211n13Lucas model 146 154ndash6 monetarypolicy 174ndash5 176 179 181ndash2 183184 natural rate of 88ndash90 135 156174ndash5
utility 13 17 40ndash1 42 46 48 61ndash2
velocity equations 139ndash40
234 Index
wages 18 19 23 24 32 36 aggregatedemand 90 91 aggregate supply 86ndash7anticipated 69 70 71 75ndash6 77 124household problem 42 43 44 47ndash866 intertemporal substitution hypothesis50 51 72 73 Keynesian model 130ndash3135ndash6 137ndash8 142ndash4 145 labordemand 68 97 labor marketequilibrium 83 84 labor participation48ndash9 Lucas model 147ndash8 moneysupply shocks 128 neoclassical model117 122ndash3 126 neutrality ofmoney 119 real wage illusion 123ndash5sticky 123 131 135ndash6 137temporary equilibrium 80 82 see alsoincome
Wallace N autoregressive expectations164 Keynesians 219n2 monetarypolicy 136ndash7 167 168 170 172 180182 money supply 165 superneutralityof money 121 122
Walrasrsquo law 1 8 14 16 85 91 balance ofpayments 190ndash1 excess demand 118financial assets 93 labor market 66ndash7market equilibrium 97 99 openeconomy 187 199 perfect foresight82 sequential markets 67
Walras Leon 1 16 202n2 203n2Walrasian models 7ndash11 15 18ndash20wealth 55 59 60 83Weber CE 209n2working hours 47ndash8
Understanding MacroeconomicTheory
John M Barron Bradley T Ewingand Gerald J Lynch
First published 2006by Routledge270 Madison Ave New York NY 10016
Simultaneously published in the UKby Routledge2 Park Square Milton Park Abingdon Oxon OX14 4RN
Routledge is an imprint of the Taylor amp Francis Group an informa business
copy 2006 John M Barron Bradley T Ewing and Gerald J Lynch
All rights reserved No part of this book may be reprinted orreproduced or utilized in any form or by any electronicmechanical or other means now known or hereafterinvented including photocopying and recording or in anyinformation storage or retrieval system without permission inwriting from the publishers
Library of Congress Cataloging in Publication DataBarron John M
Understanding macroeconomic theory John M BarronBradley T Ewing and Gerald J Lynch
p cmIncludes bibliographical references and index1 Macroeconomics I Ewing Bradley T II Lynch Gerald J III Title
HB1725B3753 2006339ndashdc22 2005026372
British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library
ISBN10 0ndash415ndash70195ndash3 ISBN13 978ndash0ndash415ndash70195ndash2 (hbk)ISBN10 0ndash415ndash70196ndash1 ISBN13 978ndash0ndash415ndash70196ndash9 (pbk)ISBN10 0ndash203ndash08822ndash0 ISBN13 978ndash0ndash203ndash08822ndash7 (ebk)
This edition published in the Taylor amp Francis e-Library 2006
ldquoTo purchase your own copy of this or any of Taylor amp Francis or Routledgersquoscollection of thousands of eBooks please go to wwweBookstoretandfcoukrdquo
Contents
1 Introduction 1
2 Walrasian economy 7
3 Firms as market participants 18
4 Households as market participants 39
5 Summarizing the behavior and constraints of firms andhouseholds 64
6 The simple neoclassical macroeconomic model (withoutgovernment or depository institutions) 78
7 Empirical macroeconomics traditional approaches and timeseries models 96
8 The neoclassical model 116
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model 130
10 The Lucas model 146
11 Policy 167
12 Open economy 186
Notes 202References 223Index 228
1 Introduction
The topics of macroeconomics
At each point in time individuals in an economy are making choices with respectto the acquisition sale andor use of a variety of different goods Such activity canbe summarized by aggregate variables such as an economyrsquos total production ofvarious goods and services the aggregate level of employment and unemploymentthe general level of interest rates and the overall level of prices1 Macroeconomicsis the study of movements in such economy-wide variables as output employmentand prices
The focus of this book will be on developing simple theoretical models thatprovide insight into the reasons for fluctuations in such aggregate variables Thesemodels explore how shocks or ldquoimpulsesrdquo to the economy (eg changes to tech-nology the money supply or government policy) impact individualsrsquo behavior inspecific markets and the resulting implications in terms of changes in aggregatevariables
An overview of some facets of theoreticalmacroeconomic analysis
Given the breadth of economic activity in an economy the study of macroeco-nomics must involve an examination of a variety of different markets For instanceit is common for macroeconomic analysis to consider exchanges of labor servicesin the labor markets of consumption and capital goods in the output marketsand of financial assets in the financial markets The fact that macroeconomicssimultaneously analyses exchanges of different goods in different markets meansthat macroeconomic theory is a general equilibrium theory That is macro-economic theory must by necessity incorporate the links across markets that arefundamental to general equilibrium analysis As we will see throughout this booka key reflection of the links across markets is Walrasrsquo law named in honor ofthe nineteenth-century French economist Leon Walras2 Simply put Walrasrsquo lawnotes that the budget constraints faced by individual agents in the economy sug-gest that if n minus 1 of the n markets in the economy are in equilibrium then the nthmarket must be in equilibrium We will repeatedly rely on Walrasrsquo law or variantsof it to simplify macroeconomic analysis
2 Introduction
While macroeconomic theories have in common (a) an attempt to explainfluctuations in aggregate variables and (b) a general equilibrium character thereremain wide differences among macroeconomic models Below we break downthese differences across macroeconomic models in several ways in order to makesome sense of what passes for simple theoretical macroeconomic analysis
Static dynamic and stationary analysis
One way of breaking down macroeconomic analyses is into static models dynamicmodels and stationary analysis of dynamic models Static macroeconomic modelsanalyze the economy at a point in time They consider the determination of pro-duction exchange and prices of various goods only for the markets that currentlyexist John Hicks (1939) sketched out an analysis of ldquospotrdquo or ldquotemporaryrdquo equi-librium The advantage to such an approach is that it provides for rather simpleldquocomparative staticrdquo analysis of the effects of changes in a variety of exogenousvariables on the endogenous variables3 Such static analysis is useful in providinginsight into a variety of questions of interest
Static macroeconomic analysis can be viewed as a modification of a Walrasiangeneral equilibrium analysis or what is commonly referred to as ldquoArrowndashDebreutheoryrdquo (Arrow and Hahn 1971 Debreu 1959) In ArrowndashDebreu theory eachcommodity is described by its physical characteristics its location and its dateof availability It is assumed there are a complete set of spot and forwardmarkets Prices adjust to clear all markets However if one restricts attentionto just spot markets then one moves from traditional Walrasian generalequilibrium to an analysis of ldquotemporary equilibriumrdquo a phrase coined by Hicks(1939) This restriction to spot markets is one element of static macroeconomicanalysis4
A second element of static models is that if there is a future then staticmacroeconomic analysis simply assumes given expectations of future prices andenvironment How expectations of future events are formed is left unspecified sothat expectations of future prices become simply an element in the set of exogenousvariables5
While static analysis provides insights there are several disadvantages of staticanalysis severe enough that it alone does not provide an adequate grounding inmacroeconomic analysis The key disadvantage of static analysis is that it breaksties between current events and future events To show the limitations of staticanalysis let us suppose that underlying a simple static macroeconomic analy-sis of current markets is a microeconomic analysis of individualsrsquo decisions thatidentifies the anticipated future level of prices as one of the exogenous variablesaffecting current behavior As we have seen static analysis takes expectations ofsuch variables as future prices as exogenous variables Doing so however resultsin (a) an incomplete enumeration of exogenous variables that can impact currenteconomic activity and (b) a potentially incomplete accounting of the effects of theimpact on current economic activity of a change in those exogenous variables thatare identified by the analysis
Introduction 3
To illustrate the first point of an incomplete listing of exogenous variables letus suppose that the static model identifies changes in the current money supplyas one factor that influences current prices This suggests that if we replicate thestatic analysis in future periods changes in the money supply in the future would beshown to affect prices at that time It seems natural to then presume that individualsrsquoanticipation of future prices would incorporate this link between changes in thefuture money supply and future prices in forming their expectations of futureprice levels so that the anticipated future money supply becomes a determinantof current activity6 Yet static analysis since it does not analyze markets beyondthe current period will not identify the potential impact of future changes in themoney supply on current activity7
To illustrate the second point of an incomplete accounting of the effects of achange in an exogenous variable let us suppose that underlying the static macro-economic analysis of current markets is a microeconomic analysis of firmsrsquo currentinvestment behavior that identifies the anticipated future tax levels as well as futureprices as two exogenous variables affecting investment decisions Thus staticanalysis would suggest that a change in future tax levels will impact current activitythrough the direct effect on current investment It is not hard to see however that(a) the current change in investment means a different future capital stock and(b) the change in future tax levels could affect future as well as current investmentEither or both of these changes would likely impact future prices and if such animpact were anticipated be a second way that future tax changes impact currentactivity8
An obvious way to avoid the above problems is to introduce forward marketsso that the macroeconomic analysis determines the prices of goods to be tradedin the future along with the prices of goods traded at the current time9 In doingso we have moved from static to dynamic analysis That is the macroeconomicmodels now determine the paths of variables (such as prices) over time rather thanprices (and other variables) at only one point in time
In a deterministic setting this expansion of dynamic analysis incorporates thenotion of ldquoperfect foresightrdquo in which individuals correctly anticipate all futureprices If there were uncertainty the analysis indexes goods by both the date oftrade and the ldquostate of naturerdquo with trades contingent on the realized state ofnature10 The result is that at each date there is a distribution of potential prices atwhich trade for a good could occur and given common knowledge of likelihood ofthe states of nature expectations of future prices would be defined by the analysis(ldquorational expectationsrdquo)
Once dynamic analysis is introduced we can consider a special limiting form ofdynamic analysis termed stationary analysis The aim of stationary analysis is toidentify in the context of a dynamic model the limiting tendencies of endogenousvariables such as the capital stock or the rate of growth in prices given that theexogenous variables remain constant or stationary over time11
While stationary analysis is distinct from static analysis in some cases one canthink of static analysis as a form of stationary analysis That is static analysis insome cases can be viewed as the outcome that would emerge each period given
4 Introduction
that the exogenous variables remain constant (or in some cases grow at a steadyrate over time) and given that one picks the correct fixed level of certain keyexogenous variables (eg the capital stock and the rate of change in the moneysupply) Note however that this implies that for static analysis to perfectly mimicstationary analysis one must to all intents and purposes have first executed theunderlying dynamic analysis
Period (discrete) versus continuous-time analysis
Macroeconomic analysis can be broken down into period or discrete-time macro-economic models and continuous-time macroeconomic models Substantivedifferences in terms of theoretical predictions do not exist between these two typesof analyses if one is careful to assure identical underlying assumptions Yet the twoanalyses do differ in the analytical techniques used For instance while discretemacroeconomics relies on the techniques of dynamic programming and differenceequations to characterize elements of the model in similar circumstances con-tinuous macroeconomic analysis turns to the techniques of optimal control anddifferential equations
Although substantive issues are not raised by the discrete- versus continuous-time dichotomy it is sometimes argued that one is preferred to the other Forinstance an attractive feature of the continuous-time analysis is that it highlightsquite clearly the distinctions between stocks and flows something that is not soclearly discernable in discrete analysis On the other hand an attractive feature ofdiscrete analysis is that it makes more transparent the link between the theoreticalanalysis and empirical testing since such analysis coincides with the obvious factthat empirical data on macroeconomic variables is discrete
New classical economics versus non-market-clearing
Classical analysis refers to the widely adopted view of how the macroeconomyshould be modeled that existed prior to the experience of the Great Depressionand John Maynard Keynesrsquo General Theory of Employment Interest and Money(1936) In classical theory the real side of the economy was separate from themoney side Classical analysis of the ldquorealrdquo side of the economy is aimed atdetermining such variables as total production relative prices the real rate ofinterest and the distribution of output Classical analysis of the money side ofthe economy meant analysis is aimed at determining money prices and nominalinterest rates
The separation of the monetary side from the real side in classical or neoclassicalanalysis led to the prediction that monetary changes do not have any effect onreal variables such as total output12 A similar prediction is often obtained bymore recent macroeconomic analysis and this is one reason why this more recentanalysis is referred to as the new classical economics13 Alternative labels ofthese new classical models include rational expectations models with market
Introduction 5
clearing neoclassical models of business fluctuations and equilibrium businesscycle models
A common feature of the analyses of new classical economics besides thefact that it suggests a divorcement of monetary changes from the real side ofthe economy is that prices are determined in the analysis so as to clear marketsThis view that prices serve to equate demands and supplies a view common tomicroeconomics is taken as an important strength of the analysis for it meansthat the models have consistent ldquomicroeconomic foundationsrdquo One implicationof the market-clearing assumption is the same as in microeconomics ndash the analysissuggests that all gains to exchange have been extracted
Contrasting the new classical economics with what preceded it helps one putthis rebirth of classical analysis into perspective Following the Great Depressionmacroeconomic analysis took as its main premise the idea that markets did notclear ndash in particular that prices did not adjust In this context the business cyclewas defined by ldquomarket failurerdquo and the role of government to stabilize the eco-nomy was clear There are a number of different types of non-market-clearing orKeynesian models One version of such Keynesian models that popularized byPatinkin (1965 chapters 13ndash14) Clower (1965) and Barro and Grossman (1971)takes as given output prices such that the output market fails to clear A secondmodel popularized by Fischer (1977) Phelps and Taylor (1977) and Sargent(1987a) as an alternative formalization of the Keynesian model takes as given theprice of labor such that the labor market fails to clear
The common theme of these non-market-clearing analyses is that for variousreasons prices do not clear markets and concepts such as excess demand and supplyplay a role in the analysis Yet no concise reason is given as to why there is marketfailure other than suggesting such items as ldquocoordination problemsrdquo and ldquotrans-action costsrdquo14 The result is that such analysis is challenged by the new classicaleconomics as lacking the microeconomic foundations for price determination AsHowitt (1986 108) suggests such a view ldquoforces the proponent of active stabi-lization policy to explain the precise nature of the impediments of transacting andcommunicating that prevent private arrangements from exhausting all gains fromtraderdquo15 This is not an easy task according to Howitt since ldquoimpediments to com-munication in a model simple enough for an economist to understand will typicallyalso be simple enough that the economist can think of institutional changes thatwould overcome themrdquo (1989 108)
Microeconomic foundations and aggregation issues
An important feature of macroeconomic analysis is that it reflects the aggre-gation of individual decisions A common approach to such aggregation is toassume ldquorepresentativerdquo agents characterize their optimal behavior then use suchbehavioral specifications in building the macroeconomic model Thus much ofmacroeconomic analysis entails looking at individualsrsquo decisions such as house-holdsrsquo decisions to work consume and save or firmsrsquo decisions to produceborrow and invest in capital
6 Introduction
These characterizations of optimizing individual behavior make up part of thebuilding blocks or ldquomicroeconomic foundationsrdquo of macroeconomic analysisYet microeconomic foundations of macroeconomics are not restricted to suchanalysis For instance such foundations also include a characterization of howprices in individual markets are determined as we saw in our discussion of newclassical economics
In developing the microeconomic foundations of macroeconomic models wewill often be struck by the extent to which the analysis restricts any role forheterogeneity or diversity among the individual agents in the economy Yet suchdiversity can in certain instances be critical to the analysis One attempt to introducediverse or heterogeneous agents into macroeconomic analysis is represented by theoverlapping generations models These models also have the advantage of beinggenuinely dynamic in nature and as such represent one area of macroeconomicsthat has recently received significant attention
Deterministic versus stochastic
In recent years an important element to macroeconomic models has been to intro-duce stochastic elements The rationale is clear the presence of uncertainty as tofuture events is real As noted by Lucas (1981 286)
the idea that speculative elements play a key role in business cycles that theseevents seem to involve agents reacting to imperfect signals in a way whichafter the fact appears inappropriate has been commonplace in the verbaltradition of business cycle theory at least since Mitchell It is now entirelypractical to view price and quantity paths that follow complicated stochasticprocesses as equilibrium ldquopointsrdquo in an appropriately specified space
As the quote suggests in dynamic models especially for new classical economicswhere market clearing is presumed stochastic elements are incorporated into theanalysis so that the role played by shocks to an economy in a dynamic setting canbe well defined
2 Walrasian economy
Introduction
This chapter develops a competitive model of the economy The key assumptionsneeded for this model are spelled out in detail One important aspect of this modelis the ldquonumerairerdquo or the commodity price that is used as a reference in the modelA distinction is made between accounting prices and relative prices and it isseen that traditional general equilibrium analysis does not determine the levelof accounting prices but rather simply relative prices A number of modificationsto the model are mentioned and add a sense of ldquorealismrdquo to the framework Thesemodifications include the introduction of futures markets quantity constraintsand the costs associated with carrying out a transaction
The chapter continues by considering individual decision-making and the theoryof the consumer and how this relates to the determination of market demand Thegeneral equilibrium conditions are stated in terms of relative prices and allocationsA theme carried throughout this book is emphasized and that is the use of theaggregate budget constraint Finally it is shown that explicitly excluding moneyas a ldquomarketrdquo allows one to understand how Sayrsquos conclusion that ldquosupply createsits own demandrdquo is arrived at in an economy composed of a single aggregatecommodity
A simple Walrasian model
As discussed previously the idea that prices adjust to clear markets is commonto much of new classical macroeconomics Thus we begin our examination ofmacroeconomic analysis by considering an economy consisting of perfectly com-petitive markets This means that individuals take prices at which exchanges canbe made as parametric and prices adjust to eliminate excess demands so that indi-vidualsrsquo plans at given prices are feasible As the title to this section suggests sucha characterization is sometimes referred to as being indicative of a ldquoWalrasianrdquoeconomy Walras described the process by which prices adjust to excess demand orsupply as a groping or tatonnement process (see Walras 1954)1 A fictitious auc-tioneer calls out different prices for the various markets and no exchange occursuntil equilibrium prices are reached2
8 Walrasian economy
Our analysis also begins at a very simple level The term ldquosimplerdquo reflects atleast the following three characteristics of the economy that we consider
1 It is a barter economy That is any commodity can be freely traded for anyother commodity There is no role for a medium of exchange (money) inreducing the costs of arranging exchanges
2 It is an exchange economy That is there is no production Instead individualshave initial fixed endowments of various commodities
3 It is a timeless economy Goods are indexed by physical characteristics andlocation but not dated according to availability This rules out futures mar-kets or the formation of expectations of future events and planning Such aneconomy was suggested by Hicks (1939) Patinkin (1965 Chapter 1) andHansen (1970 Chapter 4) provide a more detailed view of such an economyAn actual example of a pure barter exchange economy is offered by Radford(1945) The simple model of the economy developed below is useful in high-lighting such concepts as relative prices the numeraire individual versusmarket experiments aggregation issues conditions for general equilibriumand Walrasrsquo and Sayrsquos laws
The first model developed below also takes an approach to modeling the econ-omy that is in vogue in current theoretical macroeconomics As Sargent (1987b)states the ldquoattraction of (such) general equilibrium models is their internal con-sistency one is assured the agentsrsquo choices are derived from a common set ofassumptionsrdquo
Yet this advantage of general equilibrium analysis is not fully exploited until theelements of time money and production are introduced and so we will expand thediscussion in subsequent sections by introducing such features Below we intro-duce in more detail some of the key assumptions underlying the simple Walrasianmodel we start with
Key assumptions underlying a simple Walrasian model
As noted by Debreu (1959 74) ldquoan economy is defined by m consumers (charac-terized by their consumption sets and their preferences) n producers (characterizedby their production sets) and the total resources (the available quantities of thevarious commodities which are a priori given)rdquo As discussed above we considera special case of Debreursquos ldquoconcept of an economyrdquo one in which productionis absent As the following set of assumptions makes clear we also restrict ouranalysis to private ownership economies with a price system In particular assume(partial listing)
Assumption 21 There are m individuals (agents) in the economy indexed bya = 1 m There are T commodities indexed by i = 1 T 3 Agent arsquos initial
Walrasian economy 9
endowment of commodity i is denoted by cai cai ge 0 i = 1 T Naturally
msuma=1
cai = ci gt 0
Note that this assumption reflects an exchange economy in which private propertyrights exist ldquoPrivate property rightsrdquo means that for each unit of each good theexclusive right to determine use has been assigned to a particular individual
Assumption 22 All exchanges occur at a single point in time
Assumption 23 Each individual confronts the same known set of prices at whichexchange can occur4 A relative (purchase) price of commodity i indicates the unitsof commodity j required to purchase one unit of commodity i A relative (sale)price of commodity i indicates units of commodity j received when one unit ofcommodity i is sold
Assumption 24 Purchase and sale prices are identical for each commodity Thismeans that there are no ldquoprice spreadsrdquo which would suggest either a gain to anindividual buying and selling the same commodity or the presence of costs tomaking an exchange5 Thus for the T commodities there are T 2 exchange ratesor relative prices taking two commodities at a time
The numeraire
While there are T 2 exchange rates the complete set of exchange rates can bededuced directly or indirectly by the set of T minus 1 relative prices
(π1j πjminus1 j πj+1 j πTj)
where the πij denotes the price of commodity i in terms of commodity j6 In thelisting of relative prices (π1j πjminus1 j πj+1 j πTj) commodity j is referredto as the ldquonumerairerdquo
To see how the set of relative prices reduces to T minus 1 we rely on the factthat πhh = πhjπhj for all h j and k Let us see what this means for a simpleexample of three commodities h j and k There are then T 2 or nine differentrelative prices which are πhh πjj πkk πhj πjh πhk πkh πjk and πkj But we canuse the relationship πhh = πhjπhj to reduce this to T minus 1 = 2 relative prices withinformational content In particular we know that
1 πhh πhkπhj and similarly for πjj and πkk when h = j = k In other wordsthe exchange rate of a commodity with itself is unity This reduces from nineto six the number of relative prices for which information is required
2 πjh = 1πhj when k = j (such that πkj = πkk = 1) Similarly πhk = 1πkhand πjk = 1πkj For example if the jth commodity is pears and the hth
10 Walrasian economy
commodity is oranges then if πjh = 3 (3 oranges = 1 pear) πhj = 13(13 pear = 1 orange) This reduces from six to three the number of relativeprices for which information is required to reconstruct the complete set ofrelative prices
3 Finally πkh = πkjπhj (for k = j = h) For example if the jth commodity ispears the kth commodity is apples and the hth commodity is oranges thenif πkj = 3 (3 pears = 1 apple) and πhj = 13 (13 pears = 1 orange) thenπkh = 9 (9 oranges = 1 apple) That is with 9 oranges you can get 3 pearswhich in turn will purchase 1 apple This drops us from three to two relativeprices required to reconstruct the complete set of relative prices Since thenumber of commodities T = 3 we have shown how the T 2 relative pricescan be constructed from T minus 1 relative prices
In subsequent discussions we will arbitrarily let commodity T be the numerairesuch that the set of relative prices can be summarized by
(π1T πTminus1T )
For simplicity let us change notation such that πiT = πi i = 1 T Thus theset of relative prices can be rewritten as
(π1 πTminus1)
Note that πT = 1 since we are assuming the T th good is the numeraireIn traditional general equilibrium theory there is a concept of ldquoaccounting pricesrdquo
as well as the concept of relative prices Accounting prices can be represented bya set of real numbers (say pi i = 1 T ) attached to the T commodities7 Therelationship between these accounting prices and the set of relative prices that doimpinge on behavior is that πi = pipT i = 1 T (for Pi = 0) As we will seetraditional general equilibrium analysis does not determine the level of accountingprices but rather simply relative prices8
Anticipating future modifications
Before continuing it might be useful to anticipate some of the subsequent changeswe will make in the characterization of the economy Besides the introduction ofproduction we will
bull introduce time implying either forward (futures) markets or an important rolefor expectations of future spot prices
bull introduce quantity constraints that can arise if prices are fixed at non-market-clearing levels (ie depart from a Walrasian framework) and
bull introduce the cost of carrying out an exchange
Walrasian economy 11
With respect to point (c) such costs have been characterized by Coase (1960) asldquotransaction costsrdquo arising from the costs ldquonecessary to discover who it is that onewishes to deal with and to inform people that one wishes to dealrdquo the costs ofldquoconduct[ing] negotiations leading up to a bargain and to draw up a contractrdquoand the costs ldquoto undertake the inspection needed to make sure that the terms ofthe contract are being observedrdquo9 Such costs will alter the nature of contracts(exchange agreements) formed and may provide the reason for ldquoprice rigiditiesrdquoSuch costs also suggest a role for money
Transaction costs are assumed to be zero in the simple Walrasian sys-tem outlined above Sometimes this is referred to as a situation wherethere is ldquoperfect informationrdquo or where markets are complete and ldquoperfectlycompetitiverdquo
Individual experiments
General equilibrium analysis can be divided into what Patinkin (1965) refers to asldquoindividual experimentsrdquo and ldquomarket experimentsrdquo In the context of the simpleWalrasian barter exchange economy individual experiments consider the behaviorof individual agents given an initial endowment and preferences when confrontedwith a set of prices Market experiments consider the resulting determination ofprices
In the simple Walrasian model under consideration individual experimentsreplicate standard microeconomic analysis of consumer behavior In particularassume
Assumption 25 Individual arsquos preferences are described by his utility functionua(ca1 caT ) where ca1 caT denote agent arsquos consumption bundle cai ge 0i = 1 T ua maps the set of all T -tuples of non-negative numbers into the setof all real numbers (ua RT+ rarr R) We make the appropriate assumptions withrespect to individualsrsquo preferences such that a utility function exists and is wellbehaved10
Assumption 26 Individual a will choose the most preferred consumption bundlefrom the set of feasible alternatives (rationality) Given the possibility of cost-less exchange at the set of relative prices represented by (π1 πTminus1) feasibleconsumption bundles or sets (ca1 caT ) are defined by
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0
wheresumT
i=1 πi cai denotes the initial endowment of individual a in terms ofcommodity T The above expression defines the budget set
12 Walrasian economy
The consumer problem
From Assumptions 25 and 26 individual arsquos optimum consumption bundle is thesolution to the problem
maxCa1CaT
ua(ca1 caT )
subject to
Tsumi=1
πi cai minusTsum
i=1
πicai ge 0 cai ge 0 i = 1 T
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λ
L(ca1 caT λ) = ua(ca1 caT ) + λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
with first-order (necessary) conditions being11
partL
partcaile 0 i = 1 T
partL
partcaicai = 0 i = 1 T
cai ge 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
λ ge 0
The constrained maximization problem can be translated into the unconstrainedLagrangian expression
maxca1caT λmicro1microT
L(ca1 caT λ micro1 microT ) = ua(ca1 caT )
+ λ
(Tsum
i=1
πi cai minusTsum
i=1
πicai
)
Walrasian economy 13
with the necessary conditions being
partL
partcai= 0 i = 1 T
partL
partλge 0
λpartL
partλ= 0
partL
partmicroige 0 i = 1 T
microipartL
partmicroi= 0 i = 1 T
λ ge 0
microi ge 0 i = 1 T
Individual demands and excess demands
The optimal consumption bundle for agent a is defined by the above first-orderconditions and will be denoted by the (demand) set
(ca1 caT ) cai ge 0 i = 1 T
Individual arsquos demand functions will be of the form
cdai
(π1 πTminus1
Tsumi=1
πi cai
) i = 1 T
That is individual arsquos demand (consumption) of commodity i depends on the T minus1relative prices of commodities and the initial endowment Note that the form ofthe utility function implies the utility-maximizing consumption bundle meets thebudget constraint with equality Thus at the optimal bundle we have
partL
partλ=
Tsumi=1
πi cai minusTsum
i=1
πicdai = 0
An important point to note about demand functions is that they are homogeneousof degree zero in what might be called accounting prices12 Accounting prices aredefined such that pi = πi middot pT i = 1 T so it is clear that if all prices increaseby the multiple θ relative prices and the initial endowment are unchanged
Individual arsquos excess demand function for commodity i is defined byzai = cd
ai minus cai If zai is positive agent a is a net buyer of commodity i whileif zai is negative the agent is a net seller of commodity i The market value
14 Walrasian economy
(in terms of the numeraire) of the quantity of the ith commodity that individuala seeks to exchange (buy or sell) is then given by πizai From the budget con-straint we know that
sumTi=1 πi(cd
ai minus cai) = 0 orsumT
i=1 πizai = 0 In other words foreach individual the market value (in terms of commodity T ) of individual excessdemands must sum to zero This rather obvious finding generates what is referredto as Walrasrsquo law as we will see
Market experiments
In the previous section we reviewed the nature of individual demand functions andindividual excess demand functions Now consider the collection of m individualsAggregating or summing individual demand functions we obtain an aggregate orldquomarketrdquo demand function for commodity i of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci1 Tsum
i=1
πi
)equiv
msuma=1
cdai(middot)
Similarly summing agentsrsquo excess demand functions for commodity i gives usthe aggregate or ldquomarketrdquo excess demand function for commodity i of the form
zi equivmsum
a=1
(zai) equivmsum
a=1
(cdai minus cai)
Note that a zero aggregate excess demand for commodity i does not imply thatno exchange of commodity i occurs among the m agents However as we haveseen a zero individual excess demand for commodity i does imply no exchangeof commodity i by that particular individual
Aggregation issues
So far our aggregations have remained true to the underlying microeconomicanalysis Yet this is rarely the case in macroeconomic analysis which typicallyabstracts from what might be termed ldquodistributionalrdquo effects An example of thisin the above context as we will see later is to ignore the effects of the distributionof initial endowments across individuals on market demands such that the marketdemand function for commodity i is assumed to be of the form
cdi
(π1 πTminus1
Tsumi=1
πi ci
)
where
Tsumi=1
πicai equivmsum
a=1
(Tsum
i=1
πi cai
)
Walrasian economy 15
As you can see with heterogeneity (either in initial endowments or preferences)such a posited aggregate market demand function is unlikely to follow exactlyfrom the underlying microeconomic analysis One should keep in mind suchapproximations when interpreting macroeconomic analysis
Equilibrium an isolated market
With respect to a single market equilibrium is characterized by an accounting pricepi and implied relative price πi = pipT such that cd
i = ci (demand equals fixedendowment) or equivalently zi = 0 (excess demand equals zero) The tatonnementprocess is the description of how prices change to clear the market In the Walrasianmodel movement toward equilibrium the tatonnement process involves twofacets
1 The Walrasian excess demand hypothesis which indicates that the accountingprice of commodity i rises if there is excess demand and falls if there is excesssupply That is
dpi = fi(zi) i = 1 T fi(0) = 0dfidzi
gt 0
In terms of relative prices
dπi = dpi
pT= f (zi)
pT i = 1 T minus 1
Note that the change in price is not across time since each market is assumedto clear instantaneously at the same point in time
2 The recontracting assumption which states that offers to buy or sell at var-ious relative prices are not binding unless market(s) clear Only when theequilibrium price (or price vector) is obtained are contracts then made final
General equilibrium (conditions)
A general equilibrium will be characterized by a set of T minus 1 relative prices(πlowast
1 πlowasttminus1) and allocations (clowast
a1 clowastaT ) for individual a a = 1 m such
that
clowastai = cd
ai
(πlowast
1 πlowasttminus1
Tsumi=1
πi cai
) i = 1 T a = 1 m (21)
rArr clowasti equiv
msuma=1
clowastai =
msuma=1
cdai equiv cd
i
clowasti = ci i = 1 T (22)
16 Walrasian economy
Equation (21) indicates that the equilibrium allocation must be optimal in thatit must satisfy all demands for commodities at the specified set of pricesEquation (22) indicates that such an allocation must be feasible that is sumto total resource endowment Together these two conditions imply a set of relativeprices such that excess demands are zero or
cdi = ci i = 1 T
For questions concerning the existence uniqueness and stability of generalequilibrium consult Varian (1992) Debreu (1959) and Arrow and Hahn (1971)
Walrasrsquo law and Sayrsquos law
Note that the above statement of general equilibrium involves setting T excessdemand equations equal to zero but there are only T minus 1 unknowns (relativeprices) Walras solved this problem by showing that one of the equations arbitrarilychosen can be deduced from the other T minus 1 equations In other words there areonly T minus 1 independent equations To show the dependency remember that thebudget constraint for each individual is given by
Tsumi=1
πicdai minus
Tsumi=1
πi cai = 0
Summing across all individuals it must then be the case that
msuma=1
(Tsum
i=1
πi cai minusTsum
i=1
πicdai
)= 0
Rearranging and substituting in cdi for
summaminus1 cd
ai and ci forsumm
aminus1 cai we obtain
Tsumi=1
πi(cdai minus cai) = 0 or
Tsumi=1
πizi = 0
The above is an explicit statement of Walrasrsquo law Walrasrsquo law states that the sumof the excess demands across all markets must be zero Note that in summing theexcess demand of each commodity is weighted by its relative price so that we aresumming common units (ie all excess demands are in units of the numeraire)The above aggregate budget constraint is sometimes referred to as Sayrsquos law orSayrsquos identity If there is a distinction between the two it is that Sayrsquos law explicitlyexcluded money as a ldquomarketrdquo In this setting one can understand how Sayrsquosconclusion that ldquosupply creates its own demandrdquo is arrived at in an economycomposed of a single aggregate commodity
Walrasian economy 17
Conclusion
This chapter has developed a general equilibrium framework that sets the stagefor a thorough understanding of how the macroeconomy works Particular atten-tion has been paid to the development of relative prices and the development ofaggregate demand through a process of many individual consumers operating in anenvironment in which they set out to maximize their own utility This frameworkand the tool of constrained optimization is used throughout this book
3 Firms as market participants
Introduction
In this chapter the simple Walrasian model is discussed in the context of moneyfinancial assets and production The chapter clearly illustrates the firmrsquos objectivethat is to maximize profits However the firm is constrained in that it must financepurchases of capital and equipment as well as pay its workers Moreover attentionis paid to all the costs faced by the firm not just the obvious ones The investmentand financing decisions of firms are discussed and issues related to Tobinrsquos Q anddebt-to-equity are explored This chapter provides a detailed examination of therole that firms play in the macroeconomy
A simple Walrasian model with money financialassets and production
In the exchange economy we just considered endowments of the commodity goodwere magically bestowed on individuals each period We now introduce productionas the source of commodities At the start of each period there now exists a ldquolaborrdquomarket in which labor services are exchanged New agents denoted ldquofirmsrdquo hirethe labor services provided by households During the period firms combine thelabor services with an existing capital stock to produce output (commodities)which is sold in the output market net of output retained to replace capital usedup during production Revenues from the sale of output are distributed to ldquohouse-holdsrdquo in the form of wages during the period At the end of the period interestpayments and dividends are made to households out of revenues Each period firmsalso enter the output market to augment their capital stock with such purchasesfinanced by the issue of bonds and a new financial asset denoted ldquoequity sharesrdquoIn particular we assume
Assumption 31 There are new agents in the economy denoted as ldquofirmsrdquo Theseagents are initially endowed at time t with a capital stock K and a technologyfor transforming capital services from a capital stock and labor services into a
Firms as market participants 19
single ldquocompositerdquo commodity The technology is summarized by the productionfunction
yt = f (Nt K)
where K denotes the capital stock the firm inherits at time t Nt denotes the employ-ment of labor services arranged at time t for period t (from time t to time t + 1)and yt denotes the constant rate of production of the commodity for period t (fromtime t to time t + 1)1 Similarly for period i (i = t + 1 t + 2 ) which runsfrom time i to time i + 1 output produced is given by2
yi = f (Ni Ki)
Assumption 32 During each period firms sell the output produced in the outputmarket For output produced during period t (from time t to time t+1) let pt denoteits price when it is sold during the period up to and including time t + 1 Let pt+1denote the price of output produced during period t + 1 that is sold beyond timet +1 up to and including time t +2 and so on At time t the price level associatedwith the prior period is denoted by p
Assumption 33 At the start of each period households rent their labor servicesto firms for the period At the start of period t agreements to exchange the Ntlabor services during period t (from time t to t + 1) are entered into at the moneywage rate denoted by wt Similarly at the start of period t + 1 Nt+1 labor servicesare exchanged at the money wage rate wt+1 and so on3
Assumption 34 At the end of each period two types of financial assets areexchanged bonds (in the form of perpetuities) and equity shares Bonds promiseto pay a fixed money (coupon) payment z each future period in perpetuity Let pbpbt and pbt+1 denote the money price of such bonds in markets at the end of periodstminus1 t and t+1 respectively the gross (nominal) interest rates over period t (fromtime t to time t + 1) and over period t + 1 (from time t + 1 to time t + 2) are thengiven by4
1 + r = (z + pbt)pb
1 + rt = (z + pbt+1)pbt
Note that if ri = rt i = t + 1 then successive substitution for the priceof bonds in future periods will result in the following expression for the price ofbonds at time t
pb = 1
1 + r(z + pbt) = 1
1 + r
[z +
infinsumi=1
z
(1 + rt)i
]
= 1
1 + r
(z + z
rt
)
20 Firms as market participants
where pbt = zrt The number of previously issued bonds outstanding at time t isdenoted by B5
Assumption 35 Equity shares are the second type of financial asset exchangedat the end of each period Equity shares (ldquostocksrdquo) are contracts that obligate theissuer (firms) to pay to bearers at the end of each future period the income from thesale of output produced during the period net of other contractual obligations ofthe firms (eg wage payments to suppliers of labor services and interest paymentsto holders of bonds issued by firms) Let S denote the number of previously issuedequity shares outstanding at time t Holders of these equity shares are the ldquoownersof the firmsrdquo Let pe pet and pet+1 denote the money price of equity sharesexchanged in markets at the end of periods t minus 1 t and t + 1 respectively Thegross (nominal) rate of return on equity shares over period t (from time t to timet + 1) and period t + 1 (from time t + 1 to time t + 2) is then given by6
1 + re = [( ptdtS) + pet)]pe
1 + ret = [( pt+1dt+1St) + pet+1)]pet
where ptdt and pt+1dt+1 denote total nominal dividend payments made at the endof periods t and t + 1 (at time t + 1 and time t + 2) respectively St denotesthe anticipated number of equity shares outstanding after the equity market at theend of period t7 Note that the price of an equity share indicates the fact that thepurchase of an equity share entitles the holder to a portion of future (not current)dividends
Assumption 36 Households view bonds and equity shares as perfect substitutesldquoPerfect substitutesrdquo means that if equality in yields did not hold households wouldrefuse to purchase the asset with the lower yield forcing an adjustment in its pricethat would result in equivalent yields8 With bonds and equity shares as perfectsubstitutes we can speak of a single ldquofinancial asset marketrdquo that incorporatesboth bonds and equity shares and determines a single ldquointerest raterdquo
Assumption 37 There are incomplete markets Let pei i = t + 1 t + 2
denote the expectation formed in period t concerning the price of the consumptiongood over period i9 Similarly in period t we have pe
ei i = t + 1 and wei
i = t + 1 Such expectations are assumed to be held with subjective certaintyallowing us to abstract from risk considerations for the moment
Assumption 38 There are positive transaction costs to arranging exchanges ofthe consumption commodity during each period t Money holdings serve to reducethe transaction costs of arranging exchanges during a period10
Assumption 39 It is prohibitively costly for individuals to directly store theldquocompositerdquo commodity for consumption in future periods However output notconsumed during the period can be transformed (by ldquofirmsrdquo) into output in thesubsequent periods through the augmentation of the capital stock which permitshigher rates of production of output in future periods
Firms as market participants 21
Individual experiments firms
As always we start our analysis at the individual level The behavior of two typesof agents must now be considered ndash firms and households We start with firmsIn doing so we consider a ldquorepresentativerdquo agent a unit whose behavior exceptfor scale is identical to the behavior of the aggregate of such units Thus the samenotation will be used to represent both the individual unit and the aggregate of allunits In addition we consider an infinite time horizon
You should now recognize that an analysis of a representative unit neglectscertain potentially important ldquodistributionalrdquo aspects of the problem For instancefor households the ldquoreal indebtedness effectsrdquo of a price change on demand maynot be offsetting in the aggregate but that potential impact is ignored11 For firmsthe distribution of the initial capital stock can given adjustment costs affect totalemployment and output but that too is ignored12
To consider the behavior of a firm (or more specifically the manager whodirects production for the representative firm) we assume
Assumption 310 Technology is represented by the concave production function
yt = f (Nt K)
where yt denotes the firmrsquos planned (at time t) constant rate of output for the timeperiod from time t to t +1 to be sold during the period and at time t +1 Nt denotesthe firmrsquos planned (at time t) rate of employment of labor during period (t t + 1)with labor services purchased in the labor market at time t and K denotes thefirmrsquos planned capital stock for period t Recall that to simplify matters we takethe capital stock for the current period K as fixed at the individual firm levelThis would be the case given appropriate capital adjustment costs13
Assumption 311 The representative firm will choose the most preferred inputcombination and implied output given technology and prices (both current andanticipated future prices) At time t the objective of the firm is to maximize theexpected real market value of the S equity shares
Vt = peS
p
where pe is the price of equity shares at the end of period t minus 1 (at time t)14
A restatement of the firmrsquos objective
We have indicated that the objective of the firm at time t is to maximize the realmarket value of the S equity shares as given by
Vt = peS
p
22 Firms as market participants
To understand what underlies this market value of the firm we have to definethe elements underlying the price of equity shares and dividends We start byexamining what lies behind the price of equity shares
The assumption that equity shares and bonds are perfect substitutes means thatthe price of an equity share can be expressed as
pe = [ptdtS + pet](1 + r)
where dt denotes real dividends at the end of period t so that ptdt denotes nominaldividends S denotes the number of equity shares outstanding at time t pet is theprice of equity shares at the end of period t and r denotes the interest rate overperiod t (ie from time t to t + 1)
By successively substituting in a similar expression for the price of equity sharesin the next period we obtain for an infinite horizon that15
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
That is the price of an equity share at the end of period t minus1 (at time t) pe reflectsthe anticipated discounted future stream of nominal dividends per share16
Since the real value of the firm is given by Vt = peSp we can now expressthe value of the firm as
Vt = [Sp(1 + r)]⎡⎣ptdtS +
infinsumk=1
[(pt+kdt+k)St+kminus1]kprod
j=1
(1 + rt+jminus1)
⎤⎦
which means that the objective of the firm can be stated in terms of maximizingthe discounted stream of current and future dividends Before examining whatdetermines dividends each period let us simplify the above expression for Vtby putting it in terms of real dividends each period To do so note that bydefinition
pt equiv p(1 + π)
pt+k equiv p(1 + π)
⎛⎝ kprod
j=1
(1 + rt+jminus1)
⎞⎠ k = 1 2 3
where πt+j denotes the rate of change in the price level between period t + j andt + j + 1 Thus we have
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
Firms as market participants 23
where R = (1 + r)(1 + π) and Ri = (1 + ri)(1 + πi) denotes the real gross rateof interest for period i (from time i + 1 to time i + 2 i = t t + 1 ) Our nextstep in outlining the firmrsquos problem is to obtain an expression for real dividendsin each period
Dividends and the firm distribution constraint
In general we may denote real dividends at the end of any period as the differencebetween a firmrsquos total real revenues during the period and the total costs incurredduring the period Total revenues derive from the sale of output produced duringthe period In addition one could add revenues from a change in the numberof equity shares outstanding or a change in the number of bonds outstanding atthe end of the period17 Total costs to the firm in any period include the agreedupon wage payments to the labor hired during the period payments to replacedepreciated capital plus coupon payments at the end of the period to holders ofpreviously issued bonds Payments at the end of the period for the purchase ofcapital and associated adjustment costs could be counted as well18 That is firmsare constrained to have
dividends = revenue from sale of output
minus wages
minus interest payments
+ funds from change in outstanding bonds and stocks
minus costs to replace depreciated capital add new capital
and capital adjustment costs
The above constraint is typically divided into two separate constraints Oneconstraint earmarks funds raised from the change in outstanding bonds andequity shares at the end of the period to pay for or ldquofinancerdquo the installationof new capital stock during the period plus any capital adjustment costs Thisis the ldquofirm financing constraintrdquo The remaining revenues minus expendituresthen determine the level of dividends This part is called the ldquofirm distributionconstraintrdquo
The firm distribution constraint simply states that the revenues from the saleof output that exceed expenditures to meet wage payments purchases of capitalto replace that used up in the production process and interest payments to bondholders at the end of the period are distributed at the end of the period to householdsas dividends Thus real dividends at the end of period t are given by
dt = yt minus (wtpt)Nt minus zBpt minus δK
According to this expression real dividends at the end of period t equal realrevenues derived from the sale of output yt produced during period t minus costs
24 Firms as market participants
to the firm during period t that reflect the real wage wtpt times the quantity oflabor hired Nt less real coupon payments at the end of the period on previouslyissued bonds zBpt plus purchases of capital during the prior period to replacethat used up in the production process δK 19
In similar manner real dividends for periods t + 1 and t + 2 (paid at the end ofeach period) are given by
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
We can rewrite the above definition of dividends to see more clearly why itis also termed the ldquofirm distribution constraintrdquo This constraint simply says thatrevenue from the sale of output net of that retained to replace capital used up inthe production process (ie ldquonet productrdquo) is distributed to households either aswage payments interest payments or dividends That is
dt + (wtpt)Nt + zBpt = yt minus δK
dt+1 + (wt+1pt+1)Nt+1 + zBtpt+1 = yt+1 minus δKt+1
The firm financing constraint
The second part of the general constraint that firmsrsquo total expenditures equalrevenues is that changes in the firmrsquos holdings of capital as well as any asso-ciated adjustment costs are financed by a change in outstanding equity sharesandor bonds This linking of funding for capital purchases to the issuing of equityshares and bonds is denoted the ldquofirm financing constraintrdquo For instance at theend of periods t + 1 t + 2 we have the following firm financing constraints
Kt+1 minus K + ψ(Int) = [ pet middot (St minus S) + pbt middot (Bt minus B)]pt
Kt+2 minus Kt+1 + ψ(Int+1) = [ pet+1 middot (St+1 minus St) + pbt+1 middot (Bt+1 minus Bt)]pt+1
where ψ(Ii) denotes the costs of installing new capital at rate Ii during period i(between time i and time i + 1) a cost that depends directly on the rate of netinvestment (Ini = Ki+1 minusKi) planned at time t to occur between time i and i+120
Recall that we assume that firmsrsquo plans with respect to the number of stocks andbonds that will be outstanding following the financial market at the end of period i(time i + 1) mirror householdsrsquo expectations concerning the number of bonds andstocks that will be outstanding so we do not distinguish between firmsrsquo plans andhouseholdsrsquo expectations with respect to these variables
Firms as market participants 25
Since bonds and equity shares are perfect substitutes we can rewrite the abovefirm financing constraints in the simpler form
Int + ψ(Int) = At minus At = net At
Int+1 + ψ(Int+1) = At+1 minus At+1 = net At+1
where Ini denotes net real investment (ie Ki+1 minusKi) Ai denotes the real planned(at time t) value of total equity shares and bonds to be outstanding after the financialmarket at the end of period i and Ai denotes the initial real value of equity sharesand bonds for period i reflecting financing and capital decisions in prior periodsbut period i prices For instance
At = [ petSt + pbtBt]pt and At = [ petS + pbtB]pt
At+1 = [ pet+1St+1 + pbt+1Bt+1]pt+1 and
At+1 = [ pet+1St + pbt+1Bt]pt+1
There are several aspects of interest with respect to the firm financingconstraints First note that net capital purchases planned for period i to be installedbetween time i and time i + 1 are paid for at the end of period i when completelyinstalled from the sale of financial assets at that time
Second note that firms purchase the output of the composite good to augmentthe capital stock That is we have a ldquoone-sectorrdquo model in which the same goodserves both households (for consumption) and firms (for investment) There isonly a single commodity price A typical extension is a two-sector model in whichtwo goods are produced a consumption good and a capital good In such casesa new variable the relative price of the capital good in terms of the consumptiongood is introduced
Third note that the firm financing constraint holds whether the firm financescapital with new bonds new equity shares or ldquoretained earningsrdquo Suppose forinstance that a firm plans to add 100 units to its capital stock by buying a newpiece of machinery If the firm issues a bond with real value of 100 to pay for themachinery then there is a direct 100 unit increase (the new bond) in the value of thefinancial assets issued by the firm Note that the value of the current shareholdersrsquostock is unchanged in this case of bond financed investment While it is true thatthe tangible assets of the firm have increased by the 100 addition to capital thisbenefit to shareholders is exactly offset by the fact that the firmrsquos debt has alsoincreased by 10021
If the firm finances the 100 net investment by issuing new shares of stock equalto 100 again there is a 100 unit increase (the new equity shares) in the real valueof the financial assets issued by the firm As with bond financing however thereal value of the initial shareholdersrsquo stock is unchanged when the firm finances itscapital purchases by issuing new equity shares The new shares do not dilute thevalue of the shares of the initial shareholders since the capital purchase increases
26 Firms as market participants
the firmrsquos tangible assets by 100 which is exactly the real value of the new equityshares issued
Finally if the firm finances the 100 net investment through retained earningsthere is in essence a 100 unit increase in the value of the financial assets issued bythe firm for the following reason When a firm retains earnings in order to financea capital purchase the current stockholders own the right to the income generatedfrom the additional capital As a consequence the value of their equity shares risesto reflect the value of the new capital owned by the firm We could equivalentlyview this as the firm paying out 100 units in dividends to its initial shareholderswho then use the dividends to buy additional ldquoconstant valuerdquo equity shares equalto the value of the capital purchased by the firm In other words when the firmuses retained earnings to finance its investment spending it is implicitly issuingnew financial assets ndash equity shares
The nature of capital adjustment costs
An important aspect of the above financing constraint is that it incorporatespotential adjustment costs to purchases of capital as captured by the terms ψ(middot)which depend on Int Int+1 where Ini denotes the planned (at time t) net rateof investment for period i22 The total cost of capital purchases is thus the sum of(a) the real payments (or receipts if negative) involved in the purchase (or sale ifnegative) of capital in the output market and (b) potential real payments denotedldquoinstallationrdquo or adjustment costs associated with new capital acquisitions Forperiod t adjustment costs are given by ψ(Int) where Int denotes net investmentbetween time t and t + 1 Gross investment for period t is given by Int + δK Notethat we can thus decompose gross investment over the period into the change inthe capital stock Kt+1 minus K which is termed ldquoplanned net investmentrdquo and thereplacement of capital used up in the production process δK which is termedldquodepreciationrdquo23
To understand the conversion of the above analysis to continuous time we notethat in general adjustment costs over period t of length h are given by hψ(Inth)where the limit of the term Inth defines the rate of net investment That is incontinuous time the planned rate of investment would be defined by the rate ofgross investment
it = limhrarr0
It
h= lim
hrarr0
Kt+h minus K + hδK
h= Kt + δKt
and the adjustment cost function in continuous time would be ψ(int)For the aggregate rate of gross investment (a flow) to be defined by the above
expression K must equal K To achieve this one of two approaches is typicallytaken One approach assumes zero adjustment costs in that ψ equiv 0 This situationsometimes referred to as the case of ldquoperfect malleabilityrdquo means that the rateof investment may not be defined at the level of the individual firm That is ifthe existing capital stock were higher or lower than the planned level investment
Firms as market participants 27
would be infinitely positive or negative However it can be shown that withzero adjustment costs the output market in a continuous-time model at a pointin time is simply a ldquocapital marketrdquo and the expression Kt = K emerges as anequilibrium condition with respect to the capital market at time t24 Thus wecan apply LrsquoHospitalrsquos rule to define the aggregate rate of (net) investment in thecontinuous-time model with zero adjustment costs as25
it = limhrarr0
Kt+h minus K
h= limhrarr0 d(Kt+h minus K)dh
limhrarr0 dhdh= K
where it is rate of investmentIn contrast to the case of zero adjustment cost one can assume adjustment costs
that take the following form
ψ(0) = 0
ψ(β) gt 0 if β gt 0 ψ primeprime gt 0 and limβrarrinfin ψ(β) = infin
ψ(β) lt 0 if β lt 0
The above set of assumptions reflects the presumption that adjustment costsincrease at an increasing rate with the rate of change in capital and that it isinfinitely costly to change the capital stock arbitrarily fast The result of suchadjustment costs in both discrete-time analysis and continuous-time analysis isthat at time t the firm chooses Kt = K That is at time t the firm views the inher-ited capital stock as optimal since it is prohibitively costly to change the capitalstock at a point in time given such adjustment costs26
As we will see with ldquocosts of installing a unit of new capitalrdquo there will be adifference between the market value of capital goods in place and their replacementcost In particular the ratio of these two values known as ldquoTobinrsquos Qrdquo will exceedone In addition adjustment costs will mean that the firmrsquos decision with respectto investment will not be myopic (ie plans will not be based on forecasts thatextend only one period into the future) Rather the firm will consider all futureperiods in making current investment decisions
The firm problem a general statement
One way to state the optimization problem faced by the firm is to say that attime t the firm makes plans with respect to current and future employment oflabor (Nt Nt+1 ) the future employment of capital (Kt+1 Kt+2 ) the stockof outstanding bonds (Bt Bt+1 ) and the stock of outstanding equity shares(St St+1 ) in order to maximize the real value of the previously issued equityshares outstanding at time t with that real value given in general form by
Vt = (SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
28 Firms as market participants
Such plans are subject to the combined distribution and financing constraints listedabove as well as to the production function That is in general the firmrsquos problemcan be stated as27
max(SR)
⎡⎣dtS +
infinsumk=1
[dt+kSt+kminus1]kprod
j=1
Rt+jminus1
⎤⎦
subject to the financing constraints
minus [Kt+1 minus K + ψ(Kt+1 minus K)] + (petpt) middot [Bt minus B]+ (pbtpt) middot (St minus S)] = 0
minus [Kt+2 minus Kt+1 + ψ(Kt+2 minus Kt+1)] + (pet+1pt+1) middot [Bt+1 minus Bt]+ (pbt+1pt+1) middot (St+1 minus St)] = 0
and the distribution constraints
dt = yt minus (wtpt)Nt minus zBpt minus δK
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus zBtpt+1 minus δKt+1
dt+2 = yt+2 minus (wt+2pt+2)Nt+2 minus zBt+1pt+2 minus δKt+2
and given the production functions
yt = f (Nt K)
yt+1 = f (Nt+1 Kt+1)
for i = t t + 1 The above problem has a recursive nature to it At the start of any given period
the firm inherits a stock of capital an outstanding stock of bonds and an outstand-ing stock of equity shares28 These variables are state variables Each period thefirm chooses a set of the ldquocontrolrdquo variables ndash employment investment Thesechoices in conjunction with the production function result in outcomes in termsof (a) a one-period return (dividends) at the end of the period and (b) a new setof ldquostaterdquo variables ndash capital stock and stock of financial assets (equity shares andbonds) ndash inherited in the subsequent period Further note that the objective of thefirm is additive in these one-period returns (dividends) Thus the problem is oneto which we can apply Bellmanrsquos dynamic programming technique
To reformulate the problem facing the firm as a dynamic programming prob-lem we use the conventional notation of dynamic programming problems29
Firms as market participants 29
Specifically the above problem can be viewed as involving
(a) A set of ldquocontrolrdquo variables each period
zt = Nt Int zt+1 = Nt+1 Int+1 etc
(b) A set of ldquostaterdquo variables each period
xt = K B S xt+1 = Kt+1 Bt St etc
(c) ldquoTransition functionsrdquo that link the choices specifically the choice of netinvestment during the period to the capital stock available at the start ofthe next period as well as the stock of financial assets outstanding in thesubsequent period For instance the choice of the net investment rate Int during period t dictates Kt+1 given K since
Kt+1 = Int + K
From the firm financing constraint we know that the choice of the investmentrate also determines the real stock of financial assets
( petpt)[Bt minus B] + ( pbtpt)[St minus S] = Int + ψ(Int)
(d) A set of one-period return functions (evaluated at the end of period t)30
rt(K B S) = dt rt+1(Kt+1 Bt St) = Sdt+1StRt etc
Since bonds and equity shares are perfect substitutes the above problem cannotbe solved for a unique optimal number of bonds or equity shares to have outstandingeach period Thus without any loss of generality we may restrict our focus toeither bond or equity share financing That is we can hold constant either bonds(ie Bi = B i = t t + 1 ) or equity shares (ie Si = S i = t t + 1 )Alternatively we can combine the distribution and financing constraints into asingle expression for dividends and hold constant both equity shares and bondsIn this case we have ldquoretained earningsrdquo financing of changes in the capital stockIt is this case that we consider below31
Simplifying the firm problem ldquoretained earnings financingrdquo
If we assume that capital expenditures are financed from ldquoretained earningsrdquo thereis a single state variable the stock of capital The problem facing the firm thencan be simply stated as follows The Bellman equation for period t given inheritedcapital stock K is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
30 Firms as market participants
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
Substituting the above definitions for real dividends and output into the Bellmanequation and substituting in the transition function the first-order conditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
Equation (31) is the standard condition that labor is employed up to the pointwhere the real marginal gain for an additional unit of labor in terms of the increasein output attained in the current period (ie the marginal product of labor partftpartNt)equals the real marginal cost as reflected by the real wage wtpt Equation (32)indicating the optimal choice of investment is discussed in the next section
Optimal investment (and the future capital stock) zeroadjustment costs
To express the optimal condition for investment and thus the future capital stockin a more transparent form we need to expand upon the effect of an increase inthe capital stock on the value function for period t + 1 In other words we need toclarify the nature of the term dW (Kt+1)dKt+1 in Equation (32) To do so let usconsider the Bellman equation for period t + 1 To simplify matters we initiallyfocus on the case of zero adjustment costs (ie that ψ equiv 0 implying that ψ prime
i = 0i = t t +1 ) Given the inherited capital stock Kt+1 for period t +1 and a fixedstock of equity shares the Bellman equation for period t + 1 is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1(Rt)minus1 + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and (assuming retained earnings financing of capital changes and zero adjustmentcosts) the following definitions for real dividends and output for period t + 1
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBtpt+1 minus Int+1
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 31
Thus we have32
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (33)
We can use first-order conditions for the Bellman equation for period t + 1 toclarify the nature of dW (Kt+2)dKt+2 in Equation (33) In particular we havethat the optimal choice of investment in period t + 1 satisfies
minus 1
Rt+ dW (Kt+2)
dKt+2= 0 (34)
Substituting (34) into (33) we obtain the following expression for the effect of achange in the inherited capital stock on the value function for period t + 1
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt (35)
Substituting (35) into equation (32) and recalling our assumption that ψ prime = 0we thus have that the optimal choice of investment in period t satisfies
minus1 + 1
Rt
(partft+1
partKt+1 minus δ
)+ 1
Rt= 0 (36)
The above expression can be rearranged to obtain
partft+1
partKt+1= mt minus δ (37)
where mt Fisherrsquos expected real rate of interest equals Rt minus1 or (rt minusπt)(1+πt)The interpretation of (37) is fairly straightforward Each period the firm chooseslabor and capital such that the marginal gain in the subsequent period in terms ofincreased output equals the real marginal cost For capital the marginal cost is therate of depreciation plus the expected real rate of interest An explanation of thisreal ldquouserrdquo or ldquorentalrdquo cost of capital follows
Over period t the firm pays for one unit of capital at price pt Since the firmcould have instead used these funds to reduce the outstanding stock of bonds by pt the cost of this capital (reduced dividends) in nominal terms is pt(1 + rt) In realterms the cost one period later is anticipated to be pt(1 + rt)pt+1 After oneperiod 1 minus δ of the capital remains so that the sale of the remaining capital afterone period of use (or the reduced purchases of new capital) reaps a nominal returnof (1 minus δ)pt+1 and real return (1 minus δ) The real rental cost of the unit of capital isthus33
pt(1 + rt)pt+1 minus (1 minus δ) = (1 + rt)(1 + πt+1) minus 1 + δ
= (rt minus πt)(1 + πt) + δ
= mt + δ
32 Firms as market participants
Summarizing our discussion in the case of zero adjustment costs the optimalbehavior of firms in periods t and t +1 is given by the following demand functionsfor period t and t + 1
N dt = N d(wtpt K)
I dnt = I d
n (mt + δ wt+1pt+1 K)
where I dnt = Kd
t+1minusK and Kdt+1 = Kd
t (mt+δ wt+1pt+1) Note that the anticipatedreal wage next period affects Id
nt since changes in the real wage affect the employ-ment of labor and thus assuming part2f partNpartK does not equal zero the marginalproduct of capital A similar statement explains why K enters as an argument inthe labor demand function
An important feature of the above is that planned investment demand when thereare zero adjustment costs simply depends on adjacent expected real user costs ofcapital and real wages This reflects the fact that with zero adjustment cost capitaldemand is a function of the expected real user cost of capital and the real wageover the next period alone
Financing choices and different debt-to-equityratios a digression
We have characterized the above choice of the capital stock under the presumptionthat the firm finances capital purchases through retained earnings Yet we can showthat the planned (at time t) choice of the optimal capital stock at time t+1 t+2 is independent of the method of financing given that (a) bonds and equity sharesare assumed to be perfect substitutes and (b) there is no cost to arranging theexchange of financial assets (otherwise the retained earnings financing method ispreferred) This result is sometimes referred to as the ldquoModiglianindashMiller theoremrdquowhich states that the total value of the firm is independent of its financial structureThat is the present value of the stream of dividends to the initial owners is inde-pendent of how liabilities are divided between bonds and equity shares The resultis that the capital structure is indeterminant
The view that the method of financing capital purchases is largely irrelevant isa very useful simplification for macroeconomic analysis However you should beaware of some complicating factors that we are ignoring factors that can causefirms to care about the method by which they finance their capital purchases
When a firm issues bonds the value of its outstanding debt rises When it issuesstocks the value of its outstanding equity shares increases Thus the method offinancing capital purchases affects what is known as the firmrsquos debt-to-equityratio34 Financing capital purchases with bonds will increase the firmrsquos debt-to-equity ratio while financing capital purchases with equity shares (eitherexplicitly or implicitly by using retained earnings) will reduce the firmrsquos debt-to-equity ratio Two factors that can influence a firmrsquos desired ldquocapital structurerdquoare tax considerations and bankruptcy costs35
Firms as market participants 33
The corporate taxes that firms pay are calculated as a percentage of earningsFor tax purposes corporate earnings are equal to total revenue net of costs wherecosts are calculated as including not only wages and payments for raw materi-als and intermediate goods but also interest payments to bondholders If a firmfinances its purchases of capital using bonds the interest it pays in the futurewill reduce its taxable earnings and thus the taxes that it has to pay This meansthat a firm can lower its future tax liability by raising its debt-to-equity ratio ndashthat is by financing new capital purchases with new bonds rather than equityshares
Raising the debt-to-equity ratio however is generally not without costs whichare typically referred to as ldquobankruptcy costsrdquo Unlike equity shares which promiseshareholders dividend payments if profits are sufficiently high bonds promise fixedpayments to their holders Greater debt thus increases the fixed obligations thatfirms must meet in the future This means that a fall in future revenues is morelikely to force the firm into bankruptcy
Bankruptcy occurs when a firmrsquos revenues do not cover its costs and it isforced to default on its obligations to bondholders Associated with bankruptcyare bankruptcy costs the most obvious being the hefty legal costs associated witheither reorganizing or undergoing a court-supervised liquidation The existence ofbankruptcy costs serves to limit the amount of borrowing a firm will undertake Itwill hesitate to increase its debt-to-equity ratio beyond some level since the gainin tax savings will be offset by the costs associated with an increased likelihoodof incurring bankruptcy costs
To summarize a firm can be viewed as having an optimal debt-to-equity ratiothat reflects a tradeoff of tax and bankruptcy cost considerations Table 31 liststhe general level of debt-to-equity for a sample of industries in US manufacturingNote that the debt-to-equity ratios vary widely among the industries in the sampleranging from significantly over one to significantly less than one The ratio ishighest in the steel industry where the value of debt is close to 17 times the
Table 31 Debt-to-equity ratios across select industries
Book value Market valueof equity of equity
Steel 1973 1665Petroleum refining 1548 1117Textiles 1405 1296Motor vehicles 0922 0594Plastics 0843 0792Machine tools 0472 0425Pharmaceuticals 0194 0079
Source Kester (1986) The book value of equity is computed fromaccounting sources while the market value of equity is obtained bymultiplying the number of outstanding shares by the current marketprice of the outstanding shares
34 Firms as market participants
market value of outstanding market shares In contrast for pharmaceuticals thedebt-to-equity ratio is only 0079 indicating that the industry uses bond financingvery little instead financing its investment activities almost exclusively throughthe issuance of equity
Adjustment costs for capital and Tobinrsquos Q
Let us now consider the choice of capital when there exist adjustment costs Tokeep the maximization problem simple we shall continue to assume ldquoretainedearningsrdquo financing of changes in the capital stock we would however obtainidentical results with bond or equity share financing As we have seen in the caseof retained earnings financing the problem facing the firm is
W (K) = maxNt Int Kt+1
dt + W (Kt+1)
subject to the transition function
Kt+1 = Int + K
and given the following definitions for real dividends and output for period t36
dt = yt minus (wtpt)Nt minus δK minus zBpt minus Int minus ψ(Int)
yt = f (Nt K)
As before substituting the above definitions for real dividends and output intothe Bellman equation and substituting in the transition function the first-orderconditions are
partftpartNt
minus wt
pt= 0 (31)
minus (1 + ψ primet ) + dW (Kt+1)
dKt+1= 0 where ψ prime
t = dψ(Int)
dInt (32)
The problem facing the firm in period t+1 assuming retained earnings financingof capital changes is
W (Kt+1) = maxNt+1 Int+1 Kt+2
dt+1Rt + W (Kt+2)
subject to the transition function
Kt+2 = Int+1 + Kt+1
and given the following definitions for real dividends and output for period t37
dt+1 = yt+1 minus (wt+1pt+1)Nt+1 minus δKt+1 minus zBpt+1 minus Int+1 minus ψ(Int+1)
yt+1 = f (Nt+1 Kt+1)
Firms as market participants 35
Again we can substitute the above definitions for real dividends and output intothe Bellman equation for period t+1 and substituting in the transition function (inparticular note the fact that dKt+2dInt+1 = 1) obtain the following first-orderconditions
partft+1
partNt+1minus wt+1
pt+1= 0 (38)
minus (1 + ψ primet+1) + dW (Kt+2)
dKt+2= 0 where ψ prime
t+1 = dψ(Int+1)
dInt+1 (39)
Finally from our expression for the value function at time t + 1 W (Kt+1) wehave that
dW (Kt+1)
dKt+1= 1
Rt
(partft+1
partKt+1 minus δ
)+ dW (Kt+2)
dKt+2 (310)
Substituting (39) into (310) and then substituting the resulting expression fordW (Kt+1)dKt+1 into the first-order condition for Int (equation (32)) we obtain
minus(1 + ψ primet ) + 1
Rt
(partft+1
partKt+1 minus δd
)+ 1 + ψ prime
t+1
Rt= 0 (311)
Rearranging and simplifying we have that
partft+1
partKt+1= mt + δ + (mt + 1)ψ prime
t minus ψ primet+1
where ψ primet = ψ prime(I d
nt) and ψ primet+1 = ψ prime(I d
nt+1) An important feature of adjustmentcosts that is highlighted by the above equation is that the choice of investment inperiod t is now linked to the optimal choice of investment next period Since thisholds for each period in the future the choice of investment today is linked toinvestment decisions over all subsequent periods
We can simplify and rearrange the above first-order condition for investment inperiod t to obtain what is known as ldquoTobinrsquos Qrdquo To do so let us first assume anidentical real rate of return over time in particular we then have Ri = Rt = 1+mi = t + 1 t + 2 where (1 + m) equiv (1 + r)(1 + π) Let us also assume thefirm has attained its optimal capital stock so that Kd
t+1 = K and I dnt = 0 If the
production function is separable into capital and labor then the assumption of aninvariant real interest rate implies that I d
ni i = t +1 t +2 equal zero as well38
In this case ψ primet and ψ prime
t+1 can be replaced by a common ψ prime(0) gt 0 Then we maywrite the first-order condition as
partft+1partKt+1 minus m minus δ = ψ primem
Dividing by m and adding one to both sides we have
Tobinrsquos ldquomarginalrdquo Q equiv 1 + [partft+1partKt+1 minus m minus δ]m = 1 + ψ prime gt 1(312)
36 Firms as market participants
The above provides the definition for Tobinrsquos Q39 More precisely we have Tobinrsquosldquomarginalrdquo Q for it represents the ratio of the market value of an additional unitof capital to its replacement cost
Given adjustment costs the market value of an additional unit of capital exceedsits replacement cost so Tobinrsquos marginal Q is greater than one Since investmentdemand planned over the coming period determines marginal adjustment costsψ prime(I d
nt) we see that investment can be written in terms of Tobinrsquos Q40 The optimalrate of investment is that rate for which Q minus 1 is equal to the marginal cost ofinstallation Thus net investment demand is sometimes expressed as
Idnt = I d
nt(Q minus 1)dId
nt
d(Q minus 1)gt 0
The Q theory of investment is not operational as long as Q is not observableWhile marginal Q is not typically apparent with some additional assumptionswe can show that the expression known as Tobinrsquos ldquoaveragerdquo Q is identical toldquomarginalrdquo Q Tobinrsquos ldquoaveragerdquo Q is defined as the ratio of the total value of thefirmrsquos existing capital (the market value of its equity shares) to its total replacementcost and these variables are more easily measured41 In particular for a one-sectormodel in which the cost of capital and output are identical42
Tobinrsquos ldquoaveragerdquo Q equiv V
K
Hayashi (1982) has shown that if the firm is a price-taker if the production func-tion is linear homogeneous in K and N and if expectations of future real interestrates and real wages are static then the marginal and average Q are identical43 Tosee why this is the case note that if the real interest rate and real wage are invariantand the firm is at the optimal level of capital so that the capital stock is invariantover time then omitting time identifiers we have
V =infinsum
k=1
dRk
where (since ψ(Idnt) = ψ(0) = 0)
d = y minus (wp)N minus δK
Thus
V = [y minus (wp)N minus δK]m
where m equiv R minus 1 By Eulerrsquos theorem for a linear homogeneous production func-tion (ie K(partf partK) = y minus N (partf partN )) and the marginal productivity condition
Firms as market participants 37
for labor (ie wp = partf partN ) which reflects the price-taker assumption the firsttwo terms in the expression become K(partf partK)44 Dividing by K we thus obtain
Tobinrsquos ldquoaveragerdquo Q equiv V K = [partf partK minus m minus δ]m + 1
which as we can see is the same expression as that for Tobinrsquos marginal QAlternatively we can write the above as
V = [partf partK minus m minus δ]m + 1 = (Q minus 1) ([partf partK minus m minus δ]m)
Adjustment costs for labor labor as a ldquoquasi-fixed factorrdquo
Our prior characterization of the optimal choice of labor reflects an underlyingproduction process that incorporates a very simple view of labor For instancelabor markets are restricted to be only spot markets That is we rule out multiperiodlabor contracts Yet there is an extensive body of literature that investigates variousrationales for and the implications of such multiperiod (implicit) labor contractsOne reason why long-term contracts might emerge is an attempt by firms who areless risk-averse than workers to smooth out income over time45
A second reason why multiperiod labor contracts might emerge is if there areldquoadjustment costsrdquo to changes in the size of the labor force Adjustment costs couldreflect the fact that in order to hire new workers firms must incur hiring and trainingcosts Adjustment costs mean that a firm would view potential new hires andpreviously employed workers as imperfect substitutes and this would provide animpetus for multiperiod labor contracts The absence of adjustment costs simplifiesthe analysis by eliminating a rationale for multiperiod labor contracts and a choiceof labor given adjustment costs It also simplifies the analysis in two other ways
First the absence of adjustment costs suggests that we can measure labor ser-vices as the product of the fraction of the period each labor supplier works andthe number of individuals hired That is given no adjustment costs differences inldquohours workedrdquo and ldquonumber employedrdquo that leave total work hours unchanged areviewed by the firm as equivalent in terms of production In contrast with positiveadjustment costs firms would have a preference for meeting temporary changesin output by changing hours rather than by changing the number employed
A second implication of the absence of adjustment costs is that the employerviews as equivalent two workers working at ldquohalf-speedrdquo (and receiving ldquohalf-wagesrdquo) and one worker working at ldquofull-speedrdquo for ldquofull-wagesrdquo In contrastgiven adjustment costs firms would have a preference for meeting temporaryoutput changes by altering not only hours per worker but also the intensity that eachemployee was asked to work In fact output per work hour or ldquolabor productivityrdquodoes typically increase more rapidly during a recovery suggesting a more intensiveuse of labor
In contrast labor productivity growth is typically less rapid when the growth intotal output slackens This phenomenon is due in part to employersrsquo hoarding laborin slack times so as not to lose trained employees whom they will want when there
38 Firms as market participants
is an upturn in demand (That is to reduce subsequent adjustment costs given afuture upturn in production) Hoarding labor means that employers keep on moreworkers than necessary to produce the current output so that each worker has lesswork to do than normal The labor hoarding phenomenon also referred to as thelabor reserve hypothesis is the formal term for changes in the ldquointensityrdquo at whichlabor is used and explains lower output per work hour during periods of slackdemand46
Conclusion
The nature of the firm was discussed with the emphasis on profit maximizationDecisions of firm owners facing a variety of constraints and costs were analyzedwith particular attention paid to how financing constraints and adjustment costsaffected firm profits and the ability to adjust production levels A link was madebetween households and firms that will lead us into the next chapter Firms hireworkers who of course constitute households in the economy Workers are paidwages as costs to the firm but are an integral part of the production processMoreover firms may face costs of adjustment and other costly phenomena asso-ciated with decisions to alter the use of workers in the production process Takentogether then we see a link between firms and households as firm decisions havethe propensity to affect consumer income
4 Households as marketparticipants
Introduction
This chapter brings the household into our model of the macroeconomySpecifically the householdrsquos ability to obtain utility through consumption andthe labor supply decision is modeled within a choice framework The solutionis then used to formulate predictions about labor supply The concept of time iscritical to a thorough understanding of household behavior in the marketplace anda good deal of this chapter is spent analyzing intertemporal choices
The life-cycle and permanent income hypotheses are introduced and a theory ofportfolio choice is developed Finally the chapter ties many of these issues togetherand addresses the macroeconomic questions of absence of money illusion the realbalance effect and the real indebtedness effect
Individual experiments households
Two agents inhabit our expanded macroeconomic model with production firmsand households Having just discussed the nature of decisions confronting firmswe turn now to those confronting households These decisions can be broken downinto three types consumptionsaving portfolio and labor supply Consider nowthe first two decisions which should be familiar
The ldquoconsumptionsavingrdquo decision The representative household must deter-mine at time t the consumption purchases over each period at the implied ratect ct+1 We term this problem the ldquoFisherianrdquo problem
The ldquoportfoliordquo decision The individual must determine at time t the collectionof assets to hold at the end of each period In our expanded economy there areostensibly three types of assets
1 nominal money balances planned at time t to be held at the end of periodi Mi i = t t + 1
2 the nominal value of bonds planned at time t to be held at the end ofperiod i pbiBi i = t t + 1 where Bi denotes the planned number ofbonds held and pbi denotes the money price of bonds at the end of periodi and
40 Households as market participants
3 the nominal value of equity shares planned at time t to be held at the endof period i peiSi i = t t + 1 where pei denotes the money price ofan equity share at the end of period i and Si denotes the number of equityshares planned at time t to be held at the end of period i
Since bonds and equity shares are perfect substitutes we can consider them asa single entity with respect to householdsrsquo portfolio decisions We will let Aidenote the real holdings of financial assets planned at time t to be held at theend of period i i = t t + 1 That is for period t
At = [petSt + pbtBt]pt
and so on Excluding current dividend and interest payments the real value ofinherited financial assets at the end of period i reflecting portfolio decisions inthe prior period will be denoted by Ai i = t t + 1 For instance
At = [petS + pbtB]pt
At+1 = [pet+1St + pbt+1Bt]pt+1
The household problem
We start our analysis of the representative household as usual by discussing thehouseholdrsquos preferences constraints and objectives In particular we make thefollowing assumptions
Assumption 41 The representative householdrsquos preferences are described bythe utility function
u(ct ct+1 Mtpt Mt+1pt+1 1minusNt 1minusNt+1 )
where ci denotes the householdrsquos planned (at time t) rate of consumption dur-ing period i (from time i to time i + 1) Mi denotes the representative householdrsquosplanned (at time t) nominal holdings of money at the end of period i and Ni denotesthe householdrsquos planned (at time t) rate of supply of labor services during period i(from time i to time i +1) such that 1minusNi denotes the planned rate of leisure dur-ing period i It is assumed that partupartci gt 0 partupart(Mipi) gt 0 and partupart(1 minus Ni)
gt 0 i = t t + 1 Macroeconomics often assumes a time-separable utilityfunction a form of the utility function that ensures ldquotime consistencyrdquo1 In par-ticular following the tradition of macroeconomics we will assume that the totalutility for the representative household at time t with an infinite planning horizonis given by
infinsumi=t
β iminustu(ci Mipi 1 minus Ni)
Households as market participants 41
where β denotes the fixed personal or ldquoutilityrdquo discount factor with 0 lt β lt 12
In the above note that the one-period utility function for period t (time t to t + 1)is u(ct Mtpt 1minusNt) for period t +1 (time t +1 to t +2) it is u(ct+1 Mt+1pt+11 minus Nt+1) and so on Further note the infinite time horizon
Assumption 42 Individuals will choose the most preferred sequence of con-sumption money holdings and labor supply from the set of feasible alternatives(rationality) The feasible set of consumption money holdings and leisure
(ct ct+1 Mtpt Mt+1pt+1 1 minus Nt 1 minus Nt+1 )
is defined by the set of equalities
(wtpt)Nt + zBpt + At + Mpt minus [ct + Mtpt + At] = 0
(wt+1pt+1)Nt+1 + zBtpt+1 + At+1 + Mtpt+1
minus [ct+1 + Mt+1pt+1 + At+1] = 0
Note that we assume the budget constraints are met with equality Further note thatthe sum of dividends wage payments and interest payments equals total outputminus depreciation For instance for period t
(wtpt)Nt + dt + z middot Bpt = yt minus δK
and so on This is simply the firm distribution constraint for period t
Several aspects of the above problem deserve further elaboration First a wordon notation for future variables It is common in macroeconomics to derive themicroeconomic theoretical restrictions for the aggregate model under the conditionof certainty even though the analysis is then applied to situations that involvepotential stochastic elements One obvious way to eliminate considerations ofuncertainty from the analysis is to assume perfect foresight A second way is toassume that individual expectations of future events are point estimates held withsubjective certainty Note that either approach simplifies the analysis and in manycases this simplification gives us results that are not overturned if risk were to besystematically incorporated into the analysis
In the analysis of individual behavior below we will often for notationalsimplicity not distinguish between future prices and the expectations of futureprices Assuming expectations are held with subjective certainty this lack of dis-tinction will not be serious in discussing the result of the optimization problemsThat is the findings for perfect foresight can be made identical to those withoutthe assumption of perfect foresight by switching actual future prices for expectedprices Sometimes for clarity we will explicitly denote expected future prices(point estimates held with subjective certainty) by the superscript ldquoerdquo
42 Households as market participants
A second aspect of the above analysis that may initially appear odd concerningthe above one-period utility function for period i (from time i to time i+1) is that itseems that we are mixing money balances that occur at one time with consumptionand leisure that occur at an earlier time The reason for this is that money balancesare a stock variable and we are recording their value at the end of each period ithat is at time i + 13 The following scenario for the discrete-time analysis mayhelp clarify what is going on
At time t the labor market takes place and agreements are made to exchangelabor services at rate Nt over the period (t t + 1) for the money wage wt Duringthe period an output market operates in which firms sell output produced at rate yt During the period households receive money wages wt At the end of the period(time t + 1) households anticipate real interest payments zBpt from their priorpurchases of bonds (B) and real dividends dt from their prior purchase of equityshares stock (S)4 Given the above income sources as well as inherited nominalholdings of money (M ) and the anticipated value of inherited financial asset At atthe end of the period households plan an average rate of consumption ct duringthe period
At the end of period t after all income is received and final planned pur-chases of consumption goods are made the remainder reflects householdsrsquo planned(at time t) end-of-the-period change in real money balances ((Mt minus M )pt) andplanned changes in real financial assets holdings (At minus At) For financial assetsthe real price for bonds at the end of period t is pbtpt and the real price for equityshares is petpt
According to the above scenario the sale of labor services at rate Nt and rateof consumption ct over the period from time t to t + 1 tend to coincide whilereal money balances Mtpt and real financial asset holdings At can be viewed asthe planned (at time t) real stocks of such assets to be held at the end of period tIn continuous-time analysis as the length of the period h goes to zero the rate atwhich leisure is lost from supplying labor services during the period (Nt) the rateof consumption (ct) and the stocks of real money and real financial asset holdingswould coincide
A third aspect of the above analysis is that we have interpreted 1 minus Ni as theportion of the period of length 1 that the individual spends at leisure given thesupply of labor at rate Ni This is a simplification however for at the same timewe have suggested that the ldquoutility yieldrdquo of money is derived from its ability toreduce the transaction costs in arranging exchanges with such transaction costsreflecting at least in part a loss of leisure To explicitly incorporate such a viewof money leisure during a period i of length h given the sale of labor services Niand the real money balances Mipi held at the end of the period would be givenby h(1 minus Ni minus (Mipi)) where the function (Mipi) reflects transactions costsin terms of the loss of leisure The fact that prime lt 0 indicates that increased moneyholdings raise utility by reducing leisure lost in arranging transactions In this casethe one-period utility function would formally be given by u(ci 1minusNiminus(Mipi))with partupartci gt 0 and partupart(1 minus Ni minus (Mipi)) gt 0
Households as market participants 43
The general solution to the household problem
We can express the household problem in terms of a set of Bellman equationsAssuming perfect foresight (or equivalently interpreting future prices dividendetc as expectations of such variables held with subjective certainty) we thus havefor period t (time t to t + 1)
W (xt) = maxct Nt
Mtpt xt+1
[u(ct Mtpt 1 minus Nt) + W (xt+1)]
subject to the transition function
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt] Mtpt
and given xt Rmt is the real gross rate of interest on money that is the gross realrate of return on money and equals one divided by one plus the rate of inflation(1(1+πt)) The term xt is the total value in period t derived from the ldquoinheritedrdquoholdings of money bonds and stocks This total value is the sum of currentdividends and interest (received at the end of period t) on stock and bond holdingsacquired previously the real value of these financial assets at the end of period texclusive of these current interest and dividend payments and the real value ofpreviously acquired money holdings
xt equiv dt + zBpt + At + Mpt
where
At equiv [petS + pbtB]pt
The difference between the total real value derived in period t from inheritedbonds stocks and money balances plus real wage income xt + (wtpt)Nt andconsumption in period t ct reflects the acquisition of bonds equity shares andmoney holdings at the end of period t by the representative household Letting Atdenote the planned holdings of financial assets at the end of period t we thus havefrom the household budget constraint that
At + Mtpt = xt + (wtpt)Nt minus ct
where
At equiv [petSt + pbtBt]pt
Recall that Rt the gross real rate of return on financial assets equals one plus thenominal interest rate divided by one plus the rate of inflation ((1 + rt)(1 + πt))Thus in period t + 1 and given our definition of Rmt the inherited real value of
44 Households as market participants
bonds stocks and money balances including dividends and interest payments isgiven by
xt+1 = RtAt + RmtMtpt
Substituting in the expression for At derived from the household budget constraint(ie At = xt + (wtpt)Nt minus ct minus Mtpt) and rearranging we obtain the transitionfunction
xt+1 = Rt[xt + (wtpt)Nt minus ct] minus [Rt minus Rmt]Mtpt
Substituting the transition function into Bellmanrsquos equation for period t wehave the following first-order conditions for ct Nt and Mtpt assuming interiorsolutions (ie ct gt 0 1 gt Nt gt 0 and Mtpt gt 0)
partutpartct minus (partW (xt+1)partxt+1)Rt = 0 (41)
partutpart(1 minus Nt) + (partW (xt+1)partxt+1)Rt(wtpt) = 0 (42)
partutpart(Mtpt) minus (partW (xt+1)partxt+1)(Rt minus Rmt) = 0 (43)
The above conditions indicate that for period t (time t to time t + 1) we havefrom equations (41) and (42) that
partutpart(1 minus Nt)
partutpartct= wt
pt (44)
In words the optimal choice of leisure is such that the marginal value of leisurein terms of consumption that is the marginal rate of substitution between leisureand consumption as given by
partutpart(1 minus Nt)
partutpartct
equals the marginal cost of leisure in terms of consumption forgone in the currentperiod as given by the real wage wtpt
From equations (41) and (43) we have for period t that
partutpart(Mtpt)
partutpartct= (Rt)
minus1(Rt minus Rmt)
In words the optimal choice of real money balances is such that the marginal rateof substitution between real money balances and consumption as given by
partutpart(Mtpt)
partutpartct
Households as market participants 45
equals the marginal cost in terms of the present value of the loss in interest incomein the subsequent period due to the holding of money balances instead of financialassets as given by the expression (Rt)
minus1(Rt minus Rmt) Recall that Rt minus Rmt equals(1 + rt)(1 + πt) minus (1(1 + πt)) which is simply rt(1 + πt) or essentially theanticipated nominal rate of interest Given that Rt = (1 + rt)(1 + πt) we thushave that (Rt)
minus1(Rt minus Rmt) = rt(1 + rt)We can expand upon the above discussion of the first-order conditions for
period t by first noting that W (xt+1) is defined by
W (xt+1) = maxct+1Nt+1
Mt+1pt+1xt+2
[βu(ct+1 Mt+1pt+1 1 minus Nt+1) + W (xt+2)]
where
xt+2 = Rt+1[xt+1 + (wt+1pt+1)Nt+1 minus ct+1] minus [Rt+1 minus Rmt+1]Mt+1pt+1
given
xt+1 equiv dt+1 + zBtpt+1 + At+1 + Mtpt+1
At+1 equiv [pet+1St + pbt+1Bt]pt+1
Again substituting the transition function into the Bellman equation for periodt + 1 we have the following first-order conditions for period t + 1
βpartut+1partct+1 minus (partW (xt+2)partxt+2)Rt+1 = 0 (41prime)minus βpartut+1part(1 minus Nt+1) + (partW (xt+2)partxt+2)Rt+1wt+1pt+1 = 0 (42prime)βpartut+1part(Mt+1pt+1) minus (partW (xt+2)partxt+2)(Rt+1 minus Rmt+1) = 0 (43prime)
Now consider the impact of the change in xt+1 on the value function W (xt+1)The above first-order conditions imply that the indirect effects of the change inxt+1 on W (xt+1) through the effect of such a change on the choice of the optimalvalues of consumption labor supply and real money balances for period t + 1 arezero5 This is simply an application of the envelope theorem which states thatthe change in the objective function adjusting the choice variables optimally isequal to the change in the objective function when one does not adjust the choicevariables This fact along with the transition function for xt+2 gives us
partW (xt+1)partxt+1 = (partW (ct+2)partxt+2)Rt+1 (45)
Substituting equation (41prime) into (45) we obtain
partW (xt+1)partxt+1 = βpartut+1partct+1 (46)
46 Households as market participants
Alternatively by substituting in (42prime) we can obtain6
partW (xt+1)partxt+1 = β[partut+1part(1 minus Nt+1)]pt+1wt+1 (47)
Combining equations (41) and (46) gives us the standard Fisherian solutionfor the optimal allocation of consumption between period t (time t to t + 1) andperiod t + 1 (time t + 1 to t + 2)
partutpartct
βpartut+1partct+1= Rt
Combining equations (43) and (46) we obtain the standard expression for theoptimal portfolio choice of money
partutpart(Mtpt)
βpartut+1partct+1= Rt minus Rmt
Combining equations (42) and (47) gives us an expression for the optimalallocation of labor supply over time
partutpart(1 minus Nt)
βpartut+1part(1 minus Nt+1)= pt+1
wt+1Rt
wt
pt (48)
Having discussed the ldquoFisherian problemrdquo and the ldquoportfolio problemrdquo con-fronting the household we turn our attention in the next section to the ldquolaborsupply problemrdquo as captured by equations (44) and (48) Before doing so how-ever a general comment should be made with respect to the discussions to followas well as the preceding discussions of the Fisherian problem and the portfoliodecision
In focusing on first-order conditions with respect to the particular variablesat issue (ie first-order conditions for consumption now and next period forthe Fisherian problem first-order conditions for real money holdings and futureconsumption for the portfolio problem and first-order conditions for labor sup-ply now and next period for the labor supply decision) one has a tendency toforget that the optimizing problem involves the simultaneous choice of consump-tion portfolio and leisure In general this means that the analysis is often notas straightforward as it may first appear For instance a change in the currentreal wage or an expected real interest rate can affect the first-order conditionconcerning the choice of labor supply through its impact on the choice of consump-tion if part2upartcpartN = 0 for then the change in consumption alters the ldquomarginalutilityrdquo of leisure One simple way to abstract from these ldquoindirectrdquo effects is toassume that the utility function is separable not only across time but also withrespect to consumption leisure and real money balances each period such thatpart2upartcpartN = part2upartcpart(Mp) = part2upartNpart(Mp) = 0
Households as market participants 47
The choice of hours within a period
Equation (44) indicates that at the optimal labor supply the household cannot bemade better off by trading consumption for leisure within periods at the expectedreal wage for the period Equation (48) indicates that at the optimum the householdcannot be made better off by trading leisure across periods given the relevant realwages and the real interest rate7 Figure 41 captures the first situation that is theoptimal choice of consumption and leisure within a period
For the moment let us hold anticipated real interest rates constant Further letus assume unit elastic expectations with respect to wages as well as prices so thata change in the current wage or price level that alters the current real wage changesfuture expected real wages as well so that there is no change in the current realwage relative to future expected real wages In addition let us hold constant for themoment the effect on current real money balances of a change in the current realwage These assumptions help us to mimic the traditional ldquostaticrdquo or single-periodanalysis (eg Patinkin) of the effect of a change in the current periodrsquos anticipatedreal wage on individualsrsquo labor supply during period t Under such circumstancesan increase in the real wage for period t can have ambiguous effects As Patinkin(1965) states ldquofor simplicity it is assumed that [labor] supply is an increasingfunction of the real wage though there are well known reservations on this scorerdquo
To understand what Patinkin is referring to consider an increase in the real wagedue to a rise in the money wage wt Consider one possible result on the householdrsquoslabor supply decision The increase in the net real wage means a steeper budgetline as the householdrsquos optimal leisurendashconsumption combination changes from1 minus N s
t and cdt (call this choice A) to (1 minus N s
t )prime and (cdt )prime (choice C)
An increase in the real wage has two conceptually distinct effects on the house-holdrsquos labor supply decision an income effect and a substitution effect The incomeeffect refers to the fact that an increase in the real wage makes the household betteroff because it leads to an increase in the householdrsquos feasible consumption set in
$
Leisure
Figure 41 Consumption and leisure
48 Households as market participants
the current period The higher real wage means that the household if it so desirescan increase both its leisure and consumption Thus the household is able to reacha higher indifference curve which has associated with it a higher utility levelThe substitution effect refers to the fact that the increase in the real wage makesan hour of leisure relatively more expensive in terms of consumption that must beforgone
To isolate the substitution and income effects of the change in the net real wagesuppose that after the real wage increases we temporarily take away just enoughof the householdrsquos nonlabor income so that it is just able to attain its originalindifference curve The householdrsquos choice of leisure and income would then be(1 minus N s
t )primeprime and (cdt )primeprime (call this choice B) Thus if we hold the householdrsquos utility
level constant the increase in the real wage leads unambiguously to a lower levelof leisure since an hour of leisure is relatively more expensive the householdsubstitutes away from leisure choosing to work more hours The movement fromchoice A to choice B constitutes the pure ldquosubstitution effectrdquo of the higher netreal wage8
Now suppose that we give the household back the nonlabor income that wetemporarily took away This causes an outward shift in the budget line and thehouseholdrsquos new choice of leisure and consumption would be (1 minus N s
t )prime and (cdt )prime
(choice C) The movement from B to C constitutes the ldquoincome effectrdquo of thechange in the net real wage In the present case the income effect on the choiceof leisure is positive reflecting the assumption that leisure is a normal good
Note that the substitution and income effects on leisure work in opposite direc-tions The substitution effect of the higher real wage causes leisure to fall andhours worked to rise while the income effect causes leisure to rise and hoursworked to fall The net effect on leisure and hours worked depends on whicheffect dominates9 If the substitution effect dominates then a higher real wageresults in a decrease in desired leisure and an increase in desired working hoursIf the income effect dominates the opposite is true and the individualrsquos laborsupply curve is ldquobackward bendingrdquo when plotted against the real wage
The available evidence suggests that for many workers the income effect tendsto dominate slightly Estimates are that for men an increase of 10 percent in the realwage results in approximately a 15 percent reduction in the hours worked Thisreduction in hours worked reflects an income effect of approximately minus25 percentand a substitution effect of about 1 percent Other evidence suggests a similarpattern for working women10
The choice of participation within a period
Thus far we have considered a household representative of those who are in thelabor force working a positive number of hours Yet this masks the unambiguouseffect of a higher real wage on the labor supply of those households not in the laborforce To show this we consider a corner solution with respect to labor supply inparticular a household denoted a that has chosen not to participate in the labormarket For such a household let us return to the Bellman equation for period t and
Households as market participants 49
introduce explicitly the nonnegativity constraint for labor supply (ie Nt ge 0)Letting micron denote the multiplier associated with this constraint we have as first-order conditions for household arsquos consumption and labor supply11
partuatpartcat minus (partW (xat+1)partxat+1)Rt = 0
minus partuatpart(1 minus Nat) + micron + (partW (xat+1)partxat+1)Rtwtpt = 0
Nat ge 0 micronNat = 0
Substituting the first equation into the second and rearranging gives
minuspartuatpart(1 minus Nat)
partuatpartcat= wt
pt+ micron
partuatpartcat
For individuals not participating in the labor force micron ge 0 The ldquocorner solutionrdquo isa case in which the marginal rate of substitution of leisure in terms of consumptionis greater than the real wage In other words the absolute value of the indifferencecurve is equal to or greater than the absolute value of the budget line at the pointwhere N s
at = 0Note that at the optimal choice the corresponding indifference curve is more
steeply sloped than the budget line Thus even when all hours are devoted toleisure and none to work the individualrsquos marginal rate of substitution of leisurein terms of consumption still exceeds the real wage rate In other words theindividualrsquos valuation of leisure exceeds the marketrsquos valuation of leisure As aresult individual a does not find it worthwhile to participate in the labor market
The greater the real wage the greater is the probability that a given individualwill choose to participate in the labor market A higher real wage rotates thebudget line outward Since the individual is not working an increase in the netreal wage does not make him better off and thus has no income effect There isonly a substitution effect Thus if the real wage rises sufficiently the individualcan be induced to enter the labor market
According to the above analysis the economy-wide labor supply response toan increase in the current real wage is a combination of an ambiguous effecton the labor supply of those currently working but an unambiguous increase inthe labor supply among those not working12 It is the net of these two effects theldquohoursrdquo decision and the ldquoparticipationrdquo decision that is captured by the aggregatelabor supply function it is commonly assumed that this net effect is such that theaggregate quantity of labor supplied is an increasing function of the current realwage
The labor supply intertemporal substitution hypothesis
Our discussion has yet to consider ldquothe labor market intertemporal substitutionhypothesis (ISH) which states that labor supply responds positively to transitoryincreases in real wages and increases in the real interest rate a central hypoth-esis of modern competitive models of the business cyclerdquo (Alogoskoufis 1987)
50 Households as market participants
To do so we simply expand our focus to the inherent intertemporal decisionconfronting the household In particular recall our expression (48) for the optimalallocation of labor supply over time The view of labor supply embedded in (48)has life-cycle as well as business-cycle implications With respect to life-cycleimplications the theory predicts that workers will concentrate their labor supplyin years of peak earnings consuming leisure in larger than average amounts duringchildhood and old age
With respect to the business cycle the above helps explain an apparent contra-diction in the static theory of labor supply ndash the observed wage inelasticity of laborsupply in the long run with short-run fluctuations in employment which requirean elastic labor supply if one takes a ldquomarket-clearingrdquo approach with respect tothe labor market13 It does so by introducing a distinction between a permanentchange in the real wage and a temporary or transitory change in the real wage
To show the intertemporal substitution effect with respect to labor supply spec-ify wlowastplowast as the permanent or ldquonormalrdquo real wage with the anticipated real wagenext period equal to this value such that
wipi = wlowastplowast i = t + 1 t + 2
Further we assume that
Ri = Rlowast i = t t + 1 t + 2
In this case equation (48) becomes
partutpart(1 minus Nt)
βpartut+1part(1 minus N lowast)= Rlowast wtpt
wlowastplowast (48prime)
where N lowast denotes the ldquolong-runrdquo supply of labor at ldquonormalrdquo wages Further letus assume that for the representative household βRlowast = 1 Then equation (48prime)becomes
partutpart(1 minus Nt)
partut+1part(1 minus N lowast)= wtpt
wlowastplowast (48primeprime)
Equation (48primeprime) indicates that if the current real wage is higher than the normal realwage then ldquomore labor is supplied than would be implied by the long-run laborsupply functionrdquo That is this theory views suppliers of labor as reacting primarilyto three variables an anticipated ldquonormalrdquo or ldquopermanentrdquo real wage rate whichcorresponds to the wage rate in the usual one-period analysis of the laborndashleisurechoice and has a negligible effect on labor supply the deviation of the current realwage from this normal rate which has a strong positive effect on labor supplyand the expected real rate of interest (Lucas and Rapping 1970 284ndash285)14
The above theory provides the underlying microtheoretical basis for the fol-lowing statement ldquomeasured unemployment (more exactly its nonfrictionalcomponent) is then viewed as consisting of persons who regard the wage ratesat which they can currently be employed as temporarily low and who therefore
Households as market participants 51
choose to wait or search for improved conditions rather than invest in moving oroccupational changerdquo (Lucas and Rapping 1970 285)
Empirical tests seem to provide some support for this intertemporal substitutionhypothesis15 Alogoskoufis (1987 950) finds that for measures of the total numberof employees the real wage and interest rate elasticities are high and relativelywell determined ldquoThe elasticity of labor supply to transitory changes in real wagesis around unity and is statistically significant at conventional significance levelswith one exception The real interest rate always has a significant independentinfluencerdquo Note that as suggested by equation (48) Alogoskoufis finds that arise in the real interest rate increases current labor supply
Note that equation (48) does not explicitly identify the initial asset holdings asa variable that affects the relative choice of labor supply across periods Similarlythe condition for the optimal choice of consumption across periods did not have theinitial value of assets affecting the relative consumption purchases across periodsThis is a property of time-separable preferences
Special topics in intertemporal choices
As we have seen the intertemporal problem confronting the individual involvessimultaneous decisions with respect to consumption versus saving and with respectto the composition of the asset portfolio To make some sense of what is involvedwe start by considering what is known as the ldquoFisherianrdquo problem which focuseson the individualrsquos choice of consumption across time
Fisherian analysis
The Fisherian problem typically deals with the allocation of consumption acrosstime when there is a single means by which income can be allocated across time16
To restrict our analysis to such a case we can simply omit real money holdingsfrom the utility function Further we consider only interior solutions with respectto consumption (ie cai gt 0 i = t t + T )17
Thus the maximization problem becomes18
maxcat cat+T
xat+1xat+T+1
t+Tsumi=t
β iminustua(cai)
subject to
minus xat+1 + Rt[xat + cat minus cat] = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1] = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2] = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ] = 0
xat+T+1 ge 0
52 Households as market participants
Recall that Ri = (1 + ri)(1 + πi) i = t t + T denotes the ldquogross real rateof returnrdquo on agent arsquos portfolio between the end of period i and the end of periodi + 1 and asset holdings are solely in the form of bonds such that
xat equiv (1 + r)pbBapt
xat+1 equiv (1 + rt)pbtBatpt+1 = [(1 + rt)(1 + πt)] pbtBatpt
xat+i equiv (1 + rt+iminus1)pbt+iminus1Bat+iminus1pt+i
= [(1 + rt+iminus1)(1 + πt+iminus1)] pbt+iminus1Bat+iminus1pt+iminus1
i = 2 T + 1
Let λi i = t t +T denote the Lagrange multipliers for the constraints linkingthe total value of real asset holdings at the end of period i with the total value ofreal asset holdings at the end of period i + 1 Let microT denote the multiplier for thenonnegativity condition that xat+T+1 ge 0 Then the first-order conditions for theimplied Lagrangian L include
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai) = 0 i = t t + T 19
partLpartmicroT = xat+T+1 ge 0
microT ge 0
where uai = ua(cai) i = t t + T 20 Note that the above set of first-order
conditions consist of 3(T + 1) + 1 equations to determine 3(T + 1) + 1 variablesThe variables to be determined are cai i = t t + T xai i = t t + T + 1and microT Assuming continuity and strict concavity in ua(middot) and given a convex setof constraints
xai+1 xai cai| minus xai+1 + Ri[xai + cai minus cai] ge 0there is a unique solution to the problem
There are several implications of the above first-order conditions First theconditions imply that the desired total real value of assets (bonds) inherited at timet + T + 1 xd
at+T+1 will equal zero if duat+T dcat+T gt 0 In particular from the
condition
partLpartcat+T = βT (duat+T dcat+T ) minus λt+T Rt+T = 0
we see that if duat+T dcat+T gt 0 then λt+T gt 0 From the condition
partLpartxat+T+1 = minusλt+T + microT = 0
Households as market participants 53
we thus have that microT gt 0 This in turn implies from the condition
microT partLpartmicroT = microT xat+T+1 = 0
that xat+T+1 = 0 This finding should not be surprising With a time horizonof t + T agent a perceives no gain (utility) from acquiring assets at time t + Tto finance consumption in period t + T + 1 and a clear loss from doing so attime t + T in terms of consumption forgone given the assumption of nonsatiation(dua
t+T dcat+T gt 0) Second from the first set of equations we know that betweenany two periods i and i + 1
β iminustduai dcai
β iminust+1duai+1dcai+1
= λiRi
λi+1Ri+1 i = t t + T minus 1
This expression can be simplified to obtain
duai dcai
βduai+1dcai+1
= Ri i = t t + T minus 1
where Ri the real gross return between the end of periods i and i + 1 is given by(1+ri)(1+πi) Ri has been called Fisherrsquos ldquo(gross) real interest raterdquo since he wasone of the first to provide a lucid account of its role in determining consumptionacross time
Fisherrsquos ldquo(net) real interest raterdquo denoted by mi is then defined by
1 + mi equiv Ri = (1 + ri)(1 + πi)
Subtracting one from both sides and rearranging we have
mi = (ri minus πi)(1 + πi)
Thus for small expected rates of inflation we have the approximation21
mi = ri minus πi
In words the real interest rate is approximately equal to the nominal interest rateminus the rate of inflation The expected real interest rate is then the nominalinterest rate minus the expected rate of inflation
There are several features of the above that should be noted First if an individ-ualrsquos discount factor (β lt 1) equals the reciprocal of the real gross rate of interest([Ri]minus1 = (1 + πi)(1 + ri)) between periods i and i + 1 so that
βRi = 1
then the above expression of the first-order conditions for cai and cai+1 indicatesthat dua
i dcai = duai+1dcai+1 Given the concavity of the single-period utility
function (ua(middot)) and our assumption of a time-invariant one-period utility function
54 Households as market participants
it then follows that the individual will choose the same rate of consumption acrossperiods i and i + 1 in such a case22 This constant path of consumption betweenthe two periods can be said to emerge if an individualrsquos ldquorate of time preferencerdquoequals the (real) interest rate
If the expected gross return between two periods were higher (or for an individualwith a higher discount factor) the fact that β gt (Ri)
minus1 or equivalently βRi gt 1means from the first-order conditions that dua
i dcai gt duai+1dcai+1 Given the
assumed concavity of the one-period utility function the implication is that agentarsquos consumption during period i would be less than during period i + 1 That isβRi gt 1 rArr cd
ai lt cdai+1 Conversely if the expected real gross return were to be
lower (or for an individual with a lower discount factor) the fact that β lt (Ri)minus1
or equivalently βRi lt 1 means that the individualrsquos consumption during period iwould be greater than during period i + 1 That is βRi lt 1 rArr cd
ai gt cdai+1
For the two-period case (say periods t and t + 1) the optimal consumption ineach of the two periods can be shown graphically by the point of tangency betweenan indifference curve with slope minus(dua
i dcai)β(duai+1dcai+1) and a budget line
with slope minusRt 23 If one were to place cat+1 on the vertical axis and cat on thehorizontal axis then it would be possible to determine whether an individual is aborrower or a lender in any period For example if disposable income were lessthan consumption in the first period t then the individual is a lender at time tNote that points on the same indifference curve are such that
d[ua(cat) + βua(cat+1)
] = (duat dcat+1)dcat + β(dua
t+1dcat+1)dcat+1
= 0
Rearranging we have the slope of an indifference curve given by
dcat+1
dcat= minus dua
t dcat
βduat+1dcat+1
Note that points on the budget line satisfy the present value constraint
cat + (Rt)minus1cat+1 = cat + xat + (Rt)
minus1cat+1
Rearranging we have
cat+1 = Rt[cat + xat minus cat] + cat+1
such that the slope of the budget line is given by
dcat+1dcat = minusRt
Thus at the point of tangency between the budget line and an indifference curve
duat dcat
βduat+1dcat+1
= Rt
Households as market participants 55
which is the expression we obtained previously concerning the optimal choice ofconsumption between periods i and i + 1
Note that the above analysis is for an individual consumer Thus aggregate con-sumption need not behave as that predicted above for the individual For instancean aging population could lead to variations in aggregate consumption that reflectthe aggregation at different times across agents with differing characteristics
Life-cycle and permanent income hypotheses
We have seen how a householdrsquos consumption in any period is not constrained bythe income it receives during that period but rather that the discounted value oflifetime consumption is constrained by the discounted stream of income accruingto the household over its lifetime plus initial asset holdings While income tendsto rise and fall during the lifetime of an individual through appropriate saving andborrowing the individual can maintain a smooth or constant rate of consumptionover his lifetime This smoothing of consumption across time plays a critical rolein Franco Modiglianirsquos ldquolife-cycle hypothesisrdquo of consumption24
A stylized pattern of income and consumption expenditures over an individ-ualrsquos lifetime is the following Prior to retirement income exceeds consumptionand saving is positive During this period saving increases household wealth Onretirement consumption is financed by dissaving During the retirement periodhousehold wealth falls as people draw on their accumulated savings to financeconsumption Implied in this discussion is an inverted U-shape wealthndashage pro-file (save during pre-retirement years and dissave in years following retirementrunning down the stock of accumulated wealth) A number of studies of aggregatehousehold consumption and saving behavior support this wealthndashage pattern25
To make clear the implications of consumption smoothing for the demand forthe consumption good at time t let us make the simplifying assumptions that
(a) for any period i = t t + T Ri = R and(b) the individualrsquos personal discount rate β equals the constant real ldquomarketrdquo
discount rate Rminus126
From the first-order conditions we thus have the result that agent a willcompletely smooth out consumption spending across time so that consumptioncd
ai = cda i = t t + T In this case we can use the prior combined budget
constraint to obtain
cdat =
xat +
t+Tsumi=t
(caiRiminust)
where which equals 1sumt+T
i=t (1Riminust) is what Modigliani has calledthe ldquoproportionality factorrdquo and indicates the proportion of householdsrsquo totalresources ndash consisting of initial assets current income and anticipated futureincome ndash devoted to consumption each year
56 Households as market participants
An important implication of Modiglianirsquos life-cycle hypothesis is that thefraction of an increase in current income (cat) that goes toward increased cur-rent consumption (the ldquomarginal propensity to consumerdquo) will vary depending onwhether the increase in current disposable income is accompanied by an equivalentincrease in anticipated future income (cat i = t + 1 t + T )27 If a change incurrent income is viewed as ldquotransitoryrdquo most of the increase in income will goto saving in order to finance increased consumption during future years
This idea that the effect of a change in current disposable income on consump-tion demand depends on the degree to which the change in income is viewed astemporary or permanent lies at the heart of Milton Friedmanrsquos permanent incomehypothesis The permanent income hypothesis is like the life-cycle hypothesis inthat it emphasizes the fact that consumption demand in period t depends not onlyon current income but also on anticipated income in the future periods Permanentincome is that income which if received each year over a householdrsquos time hori-zon would yield an income stream with present value exactly equal to the presentvalue of the householdrsquos anticipated income stream That is permanent income cpis defined by the following equation
t+Tsumi=t
(cpRiminust) =t+Tsumi=t
(caiRiminust) + xat (49)
Factoring out cp on the left-hand side of (49) and rearranging we have
cp =
xat +
t+Tsumi=t
(caiRiminust)
(410)
Equation (410) indicates that permanent income is simply a weighted average ofcurrent and future incomes but in this case income in the more distant future isweighted less heavily since it is discounted more highly
Comparing permanent income to agent arsquos consumption demand in period tcd
at if there is complete smoothing of consumption spending across time then weobtain
cdat = cp
The implication of this equation is that the marginal propensity to consume out ofa change in current income that is perceived as permanent is equal to one whilethe marginal propensity to consume out of a change in current disposable incomethat is perceived as entirely transitory (having little impact on permanent income)is small Changes in transitory components of income are almost entirely saved ifpositive or borrowed if negative28
Our discussion so far of the impact of changes in income on current consump-tion demand has been restricted to what might be referred to as the effects ofldquotransitoryrdquo versus ldquopermanentrdquo income changes In doing so we have assumed adeterministic world in which individuals have perfect foresight concerning future
Households as market participants 57
income streams But what happens if individuals do not have perfect foresight Inparticular what if we introduce stochastic elements so that realized future incomeis a random variable Then the above theories suggest a difference in the responseof consumption demand to income changes that are anticipated or expected versusunanticipated changes In particular the life-cyclepermanent income hypothesiswould predict that previously anticipated (or expected) changes in income wouldhave no effect on consumption demand since consumption plans have alreadyincorporated this income29
Portfolio choice
Now let us consider the more general case in which agent a chooses not onlyconsumption across time but the portfolio of assets (money and bonds) That isconsider the following problem
maxcat cat+T
xat+1 xat+T+1Matpt Mat+T pt+T
t+Tsumi=t
β iminustua(cai Maipi)
subject to
minus xat+1 + Rt[xat + cat minus cat] minus [Rt minus Rmt]Matpt = 0
minus xat+2 + Rt+1[xat+1 + cat+1 minus cat+1]minus [Rt+1 minus Rmt+1]Mat+1pt+1 = 0
minus xat+3 + Rt+2[xat+2 + cat+2 minus cat+2]minus [Rt+2 minus Rmt+2]Mat+2pt+2 = 0
minus xat+T+1 + Rt+T [xat+T + cat+T minus cat+T ]minus [Rt+T minus Rmt+T ]Mat+T pt+T = 0
xat+T+1 ge 0
As before to simplify the problem we assume interior solutions in this case notonly with respect to the consumption good but also with respect to money holdingsAgain let λi i = t t + T denote the Lagrange multipliers for the constraintslinking the real asset holdings at the end of period i with their real value at the endof period i + 1 and let microT denote the multiplier for the nonnegativity conditionthat xat+T+1 ge 0 The first-order conditions are
partLpartcai = β iminustduai dcai minus λiRi = 0 i = t t + T
partLpart(Maipi) = β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0 i = t t + T
58 Households as market participants
partLpartxai+1 = minusλi + λi+1Ri+1 = 0 i = t t + T minus 1
partLpartxat+T+1 = minusλt+T + microT = 0
partLpartλi = minusxai+1 + Ri(xai + cai minus cai)
minus [Ri minus Rmi]Maipi = 0 i = t t + T 30
partLpartmicroT = xat+T+1 ge 0
microT partLpartmicroT = microT xat+T+1 = 0
λi ge 0 i = t t + T
microT ge 0
where uai = ua(cai Maipi) i = t t + T To isolate the portfolio choice
consider the portfolio choice of money and bond holdings for a given level ofcurrent consumption That is let us look at the optimal choice of Maipi given thatcai is held constant From the second set of conditions
β iminustpartuai part(Maipi) minus λi[Ri minus Rmi] = 0
Substituting the condition for the optimal choice of the total value of assets in thatperiod
λi = λi+1Ri+1
we obtain
β iminustpartuai part(Maipi) minus λi+1Ri+1[Ri minus Rmi] = 0
Now substituting the condition for the optimal choice of consumption next period(i + 1) as given by
β iminust+1duai+1dcai+1 minus λi+1Ri+1 = 0
we obtain
β iminustpartuai part(Maipi) minus β iminust+1(dua
i+1dcai+1)[Ri minus Rmi] = 0
Rearranging gives
duai d(Maipi)
βduai+1dcai+1
= Ri minus Rmi
Recalling that the expected gross real return on bonds in period i Ri equals(1 + ri)(1 + πi) and that the expected gross real return on money Rmi equals1(1 + πi) the above expression can be written as
duai d(Maipi)
βduai+1dcai+1
= ri
1 minus πi
Households as market participants 59
Note that in the limit (as the length of the period goes to zero) the expected rate ofinflation term would vanish The implication is that the optimal division of assetsbetween money and bonds depends primarily on the money interest rate aloneWhat is essentially being shown is that an increase in money holdings with nochange in current consumption means a reduction in bond holdings and thus theloss of nominal interest income ri or real interest income ri(1 + πi)
Absence of money illusion real balance effects and realindebtedness effects
There are two aspects of agent arsquos demand function that should be noted Firstagent arsquos demand functions at time t can be shown to be homogeneous of degree 0 inthe current price level pt initial money balances M a and initial bond holdings BaIn this economy this is said to reflect the ldquoabsence of money illusionrdquo One criticalreason for this is the assumption of unit elastic expectations with respect to futureprices so that changes in the current price level leave unchanged the expectedrates of change in the price level in subsequent periods Also note that the currentand expected future money payments attached to bonds are being held constantso that given the fixed money payment x on maturity interest rates are unchangedAlternatively one could have money prices and the fixed future money paymentattached to one-period bonds rise by the same proportion
The above implies that individual arsquos demand for the consumption good andreal money balances at time t can be represented by
cdat = cd
at(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
M datpt = M d
atpt(rt rt+1 rt+Tminus1 πt πt+Tminus1 Wat)
where xat is the individualrsquos real wealth at the end of period t as given by
Wat = xat + cat +t+Tminus1sum
j=t
⎡⎣ jprod
i=t
(Ri)
⎤⎦
minus1
(caj+1)
From the budget constraint for period t we know that the above two demandconditions imply a real demand for bonds of a similar form since
pbtBdatpt = cat + xat minus
[cd
at + M datpt
]
Note that the above demand functions do not depend solely on wealth and thepattern of expected real (gross) rates of interest since given the portfolio choice agiven pattern of expected real (gross) interest rates could alter demand dependingon the underlying values of the money interest rate As before individual arsquos excessdemand function for the consumption good and money in period t are defined byzat = cd
at minus cat zam = M datpt minus M apt and zab = pbtBd
atpt 31
60 Households as market participants
The ldquoreal balance effectrdquo indicates the effect of a change in real balances (M apt)
on individual demand for goods other than money As before there is a real balanceeffect that reflects a wealth effect That is a decrease in initial real balances leadsto a reduction in real money demand M d
atpt and a reduction in the real demandfor bonds pbtBd
atpt There is also what might be referred to as a ldquoreal indebtednesseffectrdquo in that a change in prices alters not only real money balances but also realinitial debt If Ba gt 0 an increase in pt reduces wealth while if Ba lt 0 anincrease in pt increases wealth This is why the characterization of the absence ofmoney illusion has been expanded to include changes in Ba
It is typical in macroeconomics to adopt the convention of the ldquorepresentativeagentrdquo to reduce notational clutter Recall that the ldquorepresentative agentrdquo is essen-tially the average agent For instance if we let cd
at denote the demand for theconsumption good in period t by representative agent a and cd
t market demandat the time then cd
at = cdt Thus depending on the context we can interpret cd
tas demand by the representative agent or market demand Recall that in doingso we essentially ignore distribution effects such as effects on market demandof changes in the distribution of initial endowments of commodities or moneybalances or of changes in the distribution of future endowments In the context ofthe real indebtedness effect since in the aggregate B = 0 this effect is removedfrom our analysis
Intertemporal substitution the evidence
We have focused above on the behavior of an individual with respect to the plannedpath of consumption across time As Robert Hall (1988 340) indicates
The essential idea is that consumers plan to change their consumptionfrom one year to the next by an amount that depends on their expectationsof real interest rates Actual movements of consumption differ from plannedmovements by a completely unpredictable random variable that indexes allinformation available next year that was not incorporated in the planningprocess the year before If expectations of real interest rates shift then thereshould be a corresponding shift in the rate of change of consumption Themagnitude of the response of consumption to a change in real interest rateexpectations measures the intertemporal elasticity of substitution32
Hall (1988 340ndash341) goes on to state that
the basic model of the joint distribution of consumption and the return earnedby one asset that has emerged is the following The joint distribution ofthe log of consumption in period t log ct and the (real) return earned by theassets from period t minus 1 to period t mtminus1 is normal with a covariance matrixthat is unchanging over time The means obey the linear relation
E(log ct) = k + ctminus1 + σE(mtminus1) (411)
Households as market participants 61
That is the expected change in the log of consumption is a parameter σ times the expected real return plus a constant If the expected real interestrate E(mtminus1) is observed directly then the key parameter σ can be estimatedsimply by regressing the change in the log of consumption on the expectedreal rate That regression also has the property that no other variable knownin period t minus 1 belongs in the regression
Hall proceeds to estimate the parameter using aggregate data on consumptionand finds that there is ldquolittle basis for a conclusion that the behavior of aggregateconsumption in the United States in the twentieth century reveals an importantpositive value of the intertemporal elasticity of substitutionrdquo (1988 356)
Hallrsquos empirical finding is of importance to macroeconomic analysis Thework by Hall and others is also of interest as an example of how theoreticalmacroeconomic analysis specifically Fisherian analysis can be tested To seethe link between the theory developed in the prior section and the proposed test(equation (411)) assume the following
Assumption 43 The path of aggregate consumption reflects agent arsquos decisionsconcerning the optimal allocation of consumption across time That is we treatagent a as the ldquorepresentative consumerrdquo Thus cai i = t t+T which denotesconsumption in period i of the representative agent a differs only in scale from cii = t t + T which denotes the aggregate level of consumption This assump-tion that an aggregate variable can be viewed as reflecting decisions of a representa-tive agent is not innocuous For instance the actual path of aggregate consumptioncould well differ from that predicted by an analysis of individualsrsquo optimaldecisions due to changes across time in the composition of individuals in theeconomy33
Assumption 44 Individualsrsquo expectations in period t minus 1 of future real interestrates incorporate all information available as of period t minus 1 New informationoccurring in period t that alters consumption from what was planned results inthe distribution of consumption being ldquolog normal conditional on informationavailable last period that is log ct is normal with mean E(ct)rdquo (Hall 1988 342)
Assumption 45 In period t minus 1 the representative agentrsquos utility function takesthe following form
t+Tsumi=tminus1
expminusδi + ((δ minus 1)δ) log(ci)
where c gt 0 σ gt 0 and δ gt 0 This exponential utility function has the followingdesired properties
1 It is time-separable2 If consumption were equal across any two periods the individual would place
greater value on an increase in consumption in the earlier period ndash that is if
62 Households as market participants
ctminus1 = ct then
expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)gt expminusδt + ((δ minus 1)δ) log(ct)
3 Given 1 gt σ ge 0 utility increases in any period with increased consumptionbut at a decreasing rate ndash that is
du(ctminus1) = dctminus1
= [(1 minus δ)(δctminus1)] middot expminusδt(t minus 1)
+ ((δ minus 1)δ) log(ctminus1)gt 0 d2u(ctminus1)dc2
tminus1 lt 0
Note that the ldquointertemporal elasticity of substitutionrdquo will be given by σ As σ approaches 0 substitution of consumption across time in response tochanges in the real interest rate will approach zero as well
Assumption 46 Individualsrsquo forecasts of future variables are held with subjec-tive certainty This last assumption is a departure from Hallrsquos analysis that allowsus for the moment to maintain the ldquodeterministicrdquo aspect of the prior optimizationproblem That is we continue to assume that individualrsquos expectations of futurevariables such as expected rates of inflation and future interest rates are held withsubjective certainty
Given the above assumptions we know from our prior discussion that thechoice of consumption for periods t minus1 and t must satisfy the following first-ordercondition
dudctminus1
dudct= Rtminus1
Substituting in the appropriate expressions for the marginal utility of consumptionin periods t minus 1 and t we obtain
[(1 minus δ)δctminus1] middot expminusδ(t minus 1) + ((δ minus 1)δ) log(ctminus1)[(1 minus δ)δct] middot expδ + ((δ minus 1)δ) log(ct)
or
(ctctminus1) expδ + ((δ minus 1)δ) log(ctminus1ct) = Rtminus1
Taking the logarithm of both sides of the above expression and rearranging theabove first-order condition becomes
log(ctctminus1) + δ minus ((δ minus 1)δ) log(ctminus1ct) = log(Rtminus1)
Households as market participants 63
which simplifies to
δ + (1δ) log(ctminus1ct) = log(Rtminus1)
or
log ct = minusδσ + log ctminus1 + σ log(Rtminus1) (412)
which is similar in form to (411) Note that the ldquointertemporal elasticity ofsubstitutionrdquo is given by
σ = (dctdRtminus1)ctRtminus1
The form of equation (42) can be made closer to that of equation (411) if wenote that we can define the ldquoinstantaneous real rate of interestrdquo associated withcontinuous compounding mtminus1 by the expression
exp(mtminus1) = Rtminus1
Then (412) becomes
log ct = minusδσ + log ctminus1 + σmtminus1
Conclusion
This chapter has developed an in-depth understanding of household behaviorA good deal of the discussion has dealt with intertemporal choices and the tradeoffsinherent in consuming today versus consuming in the future Many policy-relatedissues in macroeconomics are related to decisions made today that are not indepen-dent of future states or activities This issue will arise again and again throughoutthis book and it is imperative that one comprehend the nature of decision-makingand time
5 Summarizing the behavior andconstraints of firms andhouseholds
Introduction
In this chapter we summarize our discussion of the behavior of firms andhouseholds in the simple Walrasian model with money and production In doingso we consider first the nature of constraints faced by the participants in the econ-omy with respect to decisions during period t and then their behavior in termsof demand andor supply Along the way we will try to simplify the notationand introduce various expectations and assumptions of different macroeconomicmodels We start our discussion with firms
Summarizing firmsrsquo constraints
We have seen how we can divide the general constraint facing firms that totalrevenues from all sources just exhausts expenditures each period into two sepa-rate constraints One is the ldquofirm financing constraintrdquo which states that desiredchanges in the capital stock as well as any capital adjustment costs are financedby issuing new bonds or equity shares That is for period t
I dnt + ψ(I d
nt) minus net Ast = 0 (51)
where
net Ast equiv As
t minus At
Ast equiv [
pbtBst + petS
st
]pt
At equiv [pbtB + petS
]pt
Idnt equiv Kd
t+1 minus K
Note that we implicitly assume that firmsrsquo plans for purchasing capital during theperiod correctly anticipate the price of output (capital) during the period and theprices of bonds and equity shares to be issued at the end of the period to financesuch purchases
Behavior and constraints 65
The second constraint the ldquofirm distribution constraintrdquo is that all revenuesfrom the sale of output net of that required to replace capital used up in the produc-tion process during the period be distributed to households either as wages interestpayments or dividends At time t firmsrsquo anticipated distribution constraint isgiven by
dt + (wtpt)Ndt + zBpt minus (ys
t minus δK) = 0 (52)
where z is the coupon payment and planned output supply is related to labordemand by the production function
yst = f (N d
t K) (53)
Equation (52) implicitly assumes that firms at time t have perfect foresight withrespect to the price of output during period t Alternatively the firm financingconstraint would take the above form if we presumed there were futures marketsat time t for the exchange of output during period t and financial assets at the endof period t
The labor market at time t determines employment for the period at a level N lowastt
and an associated rate of output denoted by ylowastt At the realized price of output the
firm distribution constraint for period t will turn out to be
dt + (wtpt)Nlowastt + zBpt minus ( ylowast
t minus δK) = 0 (54)
As (54) indicates actual real output during the period net of that used to replacedepreciated capital will be distributed to households1
Summarizing householdsrsquo constraints
With respect to households there is a single budget constraint for period t Like thatof the firm distribution constraint its form changes depending on what is assumedconcerning the correctness of expectations or the timing of markets As we haveseen at time t households make plans with respect to labor supply consumptiondemand and desired additions to their real holdings of financial assets and moneybalances based on a perceived constraint of the form
cdt + (M d
t minus M )pet + net Ad
t minus (wtpt)eN s
t minus (dt + zBpt) = 0 (55)
where
net Adt equiv Ad
t minus At
Adt equiv
[pbtB
dt + petS
dt
]pt
At equiv [pbtB + petS
]pt
66 Behavior and constraints
We presume that households have perfect foresight at time t with respect to thereal value of financial assets and the real value of dividends plus interest paymentsreceived from firms at the end of period t We leave open the possibility of errorsin expectations (held with subjective certainty) concerning the price level as itaffects real money balances and the real wage The term pt would replace pe
t if wepresumed perfect foresight on the part of households concerning the price level orequivalently presumed that there were futures markets at time t for the exchangeof output during period t
The labor market at time t determines employment and output for the periodAs before we let N lowast
t and ylowastt denote the actual rate of employment and production
of output during period t In such a case the actual firm distribution constraint(54) can be substituted into the household budget constraint Since prices will beknown by households at this point we may also replace the expected prices ofoutput bonds and equity shares by their actual prices Thus during period t thehousehold budget constraint for period t can then be expressed as
cdt + (M d
t minus M )pt + net Adt minus ( ylowast
t minus δK) = 0 (56)
As discussed below householdsrsquo behavior in the output and financial markets willbe based on realized income and prices and will differ from their plans made attime t based on expected prices and a labor supply decision unless they possessperfect foresight at time t concerning the prices that will prevail over period t andthe labor market clears with actual employment equal to labor supply2
Walrasrsquo law labor market and other markets at time t
Recall that Walrasrsquo law reflects the summing up of the constraints faced by individ-ual agents in the economy Since our preceding analysis concerned the constraintsof the ldquorepresentativerdquo firm and household we need only sum the constraints ofsuch representative agents to obtain Walrasrsquo law There still remains a potentialproblem however as to which of the different versions of the constraints enumer-ated above for firms and households to use The choice as one would suspectdepends on whether the market for labor effectively occurs at the same time as themarkets for output and financial assets or at different times
One version of Walrasrsquo law essentially combines the market for labor at time twith a futures market at time t for the exchange of output during period t andfinancial assets at the end of period t Equivalently this version of Walrasrsquo lawassumes limited perfect foresight at time t by both firms and households withrespect to prices for the period3 In such a case we would sum constraints (51)(52) and (55) (with pt replacing pe
t in (55)) to obtain
[cd
t + I dnt + δK + ψ(I d
nt) minus yst
]+[net Ad
t minus net Ast
]+ (wtpt)
[N d
t minus N st
]+ [M d
t minus M ]pt = 0 (57)
Behavior and constraints 67
Thus we have that the sum of excess demand across four markets ndash the laboroutput financial and money markets ndash must equal zero4 Note that one of thesefour markets the money ldquomarketrdquo reflects the equality between the demand forand supply of money The money ldquomarketrdquo is not of course like other marketsof an economy ndash which is why the word ldquomarketrdquo is in quotes That is unlike theother markets the money ldquomarketrdquo is not a place where the exchange of goods(eg labor financial assets or output) takes place5
Walrasrsquo law sequential markets and potential lackof perfect foresight
There is a modification to make with respect to the above that is required if marketsoccur sequentially and if there is not perfect foresight on the part of all agents attime t concerning prices for period t The sequential nature of markets is clearin that we have the labor market occurring at time t while the markets for outputand financial assets occur during the period In addition households in particularmay not correctly foresee at time t the price of output for period t Under suchcircumstances the prior version of Walrasrsquo law no longer holds for it would thensum constraints that are only anticipated not realized Instead given the sequentialnature of the markets we must break the analysis down into an analysis of thelabor market and an analysis of the other three markets
At time t the labor market occurs Assuming a competitive equilibrium for thelabor market we have a money wage determined at time t such that
N st = N d
t
Underlying the supply of labor at time t are householdsrsquo plans with respect toconsumption demand and saving (either in the form of financial assets or money)during the period These plans are influenced at time t by the anticipated pricelevel for commodities pe
t among other variables and as such these plans may notbe feasible given realized prices during period t
Once the labor market ends employment and output are determined for theperiod at levels N lowast
t and ylowastt respectively At that point households make plans with
respect to consumption and saving in light of the realized prices and the resultingeffective household budget constraint That realized household budget constraintis simply equation (56) which incorporates the actual firm distribution constraintAdding the firm financing constraint (51) we obtain a modified Walrasrsquo law forthe markets during period t of the form
[cd
t + I dnt + δK + ψ(I d
nt) minus ylowastt
]+[net Ad
t minus net Ast
]+ [M d
t minus M ]pt = 0
In the absence of perfect foresight the demands for consumption money balancesand financial assets during the period expressed in the above equation can differfrom the plans made at time t
68 Behavior and constraints
Summarizing firm behavior with limited perfect foresight
Consider firmsrsquo optimal plans at time t Note that at time t firms have an expectationof the price of output for the period We shall continue to assume that firms haveperfect foresight at time t with respect to the price of output over the periodWith respect to labor demand a diminishing marginal product of labor implies aninverse relationship between the real wage and labor demand
N st = N d
t (wtpt K)
It is typical to assume that the labor market is such that firms achieve employ-ment equal to that demanded In this case given the production function and thenature of the labor demand function we have an output supply function for periodt of the form6
yst = yd
t (wtpt K)
An increase in the real wage reduces labor demand and thus output supply so thatwe have
partN dt part(wtpt) lt 0 and partys
t part(wtpt) lt 0
Now consider firmsrsquo behavior with respect to investment and consequent finan-cial asset supply Given a diminishing marginal product of capital capital demandis inversely related to the expected real user cost of capital7
Assuming labor and capital are complements in the production process(part2f partKpartN gt 0) capital demand will be inversely related to the expected realwage in the subsequent period as well8 In particular in the absence of adjustmentcosts (for both capital and labor) we have the following capital demand function
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (58)
with
partKdt+1part(me
t + δ) lt 0 and partKdt+1part(we
t+1pet+1) lt 09
The above demand for capital stock at the end of period t (in place at time t +1)implies a net investment demand function for period t of the form
I dnt = I d
nt(met + δ we
t+1pet+1 K)
where net investment demand is inversely related to the expected real user cost ofcapital me
t + δ the anticipated real wage in the next period wet+1pe
t+1 and theexisting capital stock at time t K
Recall that the firm financing constraint in the absence of capital adjustmentcosts equates firmsrsquo net real financial asset supply to net investment demandThus given the nature of the net investment demand function the net real financial
Behavior and constraints 69
asset supply function for firms at the end of period t (at time t + 1) is identical tonet investment demand or
net Ast = net As
t (met + δ we
t+1pet+1 K)
where net real financial asset supply for period t like net investment demandduring period t is inversely related to the expected real user cost of capital theanticipated real wage in the next period and the existing capital stock at time t
With convex adjustment costs the optimal capital stock (as well as investmentdemand) depends on the entire future path of the expected real user cost of capitaland real wages That is with adjustment costs
Kdt+1 = Kd
t+1(met + δ me
t+1 + δ wet+1pe
t+1 wet+2pe
t+2 ) (59)
We can rewrite the above demand function for capital given adjustment costs soas to collapse future periods into essentially a single subsequent period10 To doso recall that the expected real rate of interest for period i me
i equals (ri minus πei )
(1+πei ) Let us assume static expectations concerning future interest rates so that
ri = rt i = t + 1 t + 2 Further let us assume static expectations concerningfuture expected inflation so that πe
i = πet i = t + 1 t + 2 The result is that
the expected constant real rate of interest between periods t and t + 1 is expectedto prevail in the future so that me
i = met i = t + 1 t + 2 We can then rewrite
the demand function for capital accumulated at the end of period t in the simplerform
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1 we
t+2pet+2 )
Now note that we can decompose the anticipated wage for period i and theexpected price level for period i into two components the wage or price levelin period i minus 1 and the expected rate of change in wages or prices respectivelyIn particular the anticipated money wage and price level for period t + 2 can beexpressed by
wet+2 equiv we
t+1(1 + πewt+1) and pe
t+2 equiv pet+1(1 + πe
t+1)
Let us now assume static expectations with respect to the rate of change in wagesbeyond the next period so that πe
wi = πewt+1 i = t + 2 t + 3 Recall that
we have already assumed a constant rate of price inflation in subsequent periodsIf we then add the assumption that beyond the next period the (constant) rate ofinflation in wages (πe
wt+1) equals the expected rate of inflation in prices (πet+1)
we have that wei pe
i = wet+1pe
t+1 i = t + 2 t + 3 In words the real wagein the subsequent period we
t+1pet+1 i = t + 2 t + 3 is anticipated to persist
indefinitely11 We can now rewrite the capital demand function given adjustmentcost (equation (59) as
Kdt+1 = Kd
t+1(met + δ we
t+1pet+1) (510)
70 Behavior and constraints
Note that given our expectation assumptions the capital demand function withadjustment costs (equation (54)) has the same form as the capital demand functionin the absence of capital adjustment costs However the actual response to changesin the real user cost of capital or in the anticipated real wage in the subsequentperiod would typically be less with adjustment costs as such costs would leadfirms to only gradually move toward a new optimal capital stock
Equation (510) captures the idea that the demand for capital to be in placeat the end of period t (at time t + 1) depends inversely on the expected realuser cost of capital and that assuming capital and labor are complements capitaldemand depends inversely on the anticipated future real wage Similarly sincenet investment demand during period t simply reflects the difference betweencapital demand at the end of the period and the initial capital stock we haveinvestment demand being inversely related to the expected real user (or rental) costof capital and to the anticipated real wage for the subsequent period Finally fromthe firm financing constraint that relates firmsrsquo net real financial asset supply tonet investment demand we have firmsrsquo net financial asset supply being inverselyrelated to the expected real user cost of capital and the expected future real wageThus in summary we have
partKdt+1part(me
t + δ) lt 0 partI dntpart(me
t + δ) lt 0
partKdt+1part(we
t+1pet+1) lt 0 partI d
ntpart(wet+1pe
t+1) lt 0
partnet Ast part(me
t + δ) gt 0 partnet Ast part(we
t+1pet+1) lt 0
where mei equiv (rt minus πe
t )(1 + πet ) Note that gross investment demand is given by
Idt equiv Kd
t+1 minus K + δK = I dnt + δK
Summarizing household behavior with limitedperfect foresight
Householdsrsquo plans at time t concerning the labor supply choice the consump-tionsaving choice and the portfolio choice for period t are constrained by theanticipated household budget constraint as given by equation (55) From ourintertemporal analysis of these optimal choices we know that in general at time twe thus have labor supply at time t
N st = N s
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
consumption demand planned at time t
cdt = cd
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
Behavior and constraints 71
and money demand planned at time t
Ldt = Ld
t (wtpet we
t+1pet+1 rt rt+1 πe
t πet+1 At Mpe
t dt
+ zBpt)
where for notational ease we have defined planned real money demand by the termLd
t From the anticipated budget constraint as given by
cdt + (M d
t minus M )pet + net Ad
t minus (wtpet )N
st minus (dt + zBpt) = 0
we can infer net real financial asset demand planned at time t net Adt from
householdsrsquo plans concerning money demand consumption demand and laborsupply
With perfect foresight at time t concerning the prices for period t house-holdsrsquo plans made at time t concerning consumption money demand andnet real financial asset demand during period t will be fulfilled Thus lettingxt = dt + zBpt + At + M tpt we can express the above behavior conditions asfollows labor supply at time t is
N st = N s
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
consumption demand for period t is
cdt = cd
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
and money demand for period t is
Ldt = Ld
t (wtpt wet+1pe
t+1 rt rt+1 πet πe
t+1 xt)
where Ldt equiv M d
t pt As we did with respect to firmsrsquo capital demand function given adjustment
costs we now introduce expectation assumptions that essentially collapse theentire sequence of future periods into a single future period First we assume staticexpectations concerning future interest rates so that ri = rt i = t + 1 t + 2 Next we assume static expectations with respect to future rates of inflation suchthat πe
i = πet i = t + 1 t + 2 Finally we assume that the expected rate of
wage inflation beyond the next period is constant and equal to the expected rate ofinflation (ie πe
wi = πewt+1 i = t + 2 t + 3 and πe
wt+1 = πet+1) so that the
real wage anticipated for the next period is expected to prevail indefinitely Giventhe above restrictive expectation assumptions we can rewrite the householdsrsquobehavioral functions in the following form the labor supply at time t becomes
N st = N s
t (wtpt wet+1pe
t+1 rt πet xt)
consumption demand for period t becomes
cdt = cd
t (wtpt wet+1pe
t+1 rt πet xt)
72 Behavior and constraints
and money demand for period t becomes
Ldt = Ld
t (wtpt wet+1pe
t+1 rt πet xt)
where real money demand at the end of period t is given by Ldt equiv M d
t pt Our discussion of the labor supply decision suggests that labor supply is directly
related to the real wage and inversely related to the real wage next period (whichreflects the real wage in all subsequent periods given our expectation assumptions)This response to changes in the real wages incorporates the intertemporal substitu-tion hypothesis The intertemporal substitution hypothesis also suggests that laborsupply is directly related to the interest rate and inversely related to the expectedrate of inflation in that either change implies a higher expected real rate of inter-est and thus a substitution from leisure to increased labor supply today Finallyhigher real initial holdings of financial assets real money balances or income inthe current period from bond and equity shares holdings will reduce labor supplyif leisure is a normal good In summary we thus have
partN st part(wtpt) gt 0 partN s
t part(wet+1pe
t+1) lt 0 partN st partrt gt 0
partN st partπe
t lt 0 partN st part xt lt 0
Our discussion of the consumptionsaving decision suggests that consumptiondemand is directly related to both the current and anticipated future real wagesince an increase in either implies an increase in the discounted stream of incomeFocusing on the Fisherian analysis of the allocation of consumption across timeconsumption demand would be inversely related to the money interest rate anddirectly related to the expected rate of inflation since either change implies ahigher expected real rate of interest Finally an increase in xt (reflecting sayhigher real initial holdings of financial assets increased real money balancesor higher income in the current period from bond and equity share holdings) willraise consumption demand in the current period In summary we thus have
partcdt part(wtpt) gt 0 partcd
t part(wet+1pe
t+1) gt 0 partcdt partrt lt 0
partcdt partπe
t gt 0 partcdt part xt gt 0
Our discussion of the portfolio decision suggests that real money demand isdirectly related to both the current and anticipated future real wage since an increasein either implies an increase in the discounted stream of income Focusing on theportfolio analysis money demand would be inversely related to the money interestrate as households shift from money to financial asset holdings Any effect of achange in the expected rate of inflation is indirect and will be ignored Finallyan increase in xt (reflecting say higher real initial holdings of financial assetsincreased real money balances or higher income in the current period from bondand equity share holdings) will likely raise money demand in the current period
Behavior and constraints 73
In summary we thus have
partLdt part(wtpt) gt 0 partLd
t part(wet+1pe
t+1) gt 0 partLdt partrt lt 0
partLdt part xt gt 0
So far our discussion has not included householdsrsquo real net financial assetdemand To see what determines this we can simply use the budget constraintand money labor and commodity demand functions Specifically rewriting thebudget constraint for period t
net Adt = (wtpt)N
st + dt + zBpt minus cd
t minus (M dt minus M )pt
An increase in the current real wage initial real money balances or dividendand interest payments for the current period is presumed to increase not onlycurrent consumption and money demand but also future consumption and moneydemand and thus increase net financial asset demand by households From theintertemporal substitution hypothesis an increase in the expected future real wagewill reduce current labor supply as well as increase current consumption demandboth changes imply a fall in net real financial asset demand for households
From the Fisherian analysis a higher expected rate of inflation will reducethe expected real rate of interest the above constraint indicates that the resultingincrease in current consumption demand will reduce householdsrsquo acquisition offinancial assets Similarly an increase in real initial financial asset holdings byraising both current consumption and money demand will lead to a fall in real netfinancial asset demand On the other hand a higher money interest rate due toboth the Fisherian effect on current consumption demand and the portfolio effecton money demand implies a higher net financial asset demand In summary wethus have
partnet Adt part(wtpt) lt (or gt or =) 0 partnet Ad
t part(wet+1pe
t+1) lt 0
partnet Adt partrt gt 0
partnet Adt partπe
t lt 0 partnet Adt partAt lt 0 partnet Ad
t partMpt) gt 0
partnet Adt part(dt + zBpt) lt (or gt or =) 0
where there are ambiguous effects on net financial asset demand of a change in thereal wage and of a change in anticipated dividend and interest payments becausean increase in either raises both consumption demand and money demand Notethat in the limit as the length of the period goes to zero the household budgetconstraint at time t becomes
net Adt + (M d
t minus M )pt = 0
74 Behavior and constraints
as all the flow terms go to zero at a point in time In this case assuming real moneydemand is directly related to income we obtain the unambiguous effect of
partnet Adt part(dt + zBpt) lt 0 partnet Ad
t part(wtpt) lt 0
However in the period analysis net Adt reflects net real financial asset demand at the
end of the period Thus assuming any increase in income is not fully reflected inan increased rate of consumption we have an offsetting effect and thus ambiguity
Summarizing household behavior without perfectforesight
If we presume that households learn of prices that will exist for period t aftertime t and they differ from what was expected then the actual demands for outputmoney and financial assets can differ from those reflecting plans made at time tThe key reason for this is that the actual constraint faced by households will differfrom that anticipated In particular using the actual firm distribution constraintwe replace anticipated real income from wages dividends and interest paymentsfrom firms with the actual real income net of depreciation ( ylowast
t minusδK) The resultingrealized household budget constraint after time t is then
cdt + (M d
t minus M )pet + net Ad
t minus ( ylowastt minus δK) = 0
If anticipations by households were incorrect at time t concerning prices or div-idends during the period then revisions in plans for consumption and saving willbe made in light of the actual budget constraint faced In this case the actual house-hold demand functions for output and money are written as follows consumptiondemand during period t is
cdt = cd
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
money demand during period t is
Ldt = Ld
t (wet+1pe
t+1 rt πet At Mpt ylowast
t )
and we replace partcdt part(wtpt) gt 0 and partcd
t part(dt + zBpt) gt 0 with
partcdt partylowast
t gt 0
Note that the term partcdt partylowast
t is referred to as the ldquomarginal propensity toconsumerdquo12 Similarly we replace partLd
t part(wtpt) gt 0 and partLdt part(dt + zBpt) gt 0
with
partLdt partylowast
t gt 0
Behavior and constraints 75
Money illusion and the real balance effect
Let us consider (sufficient) assumptions under which demands and supplies arehomogeneous of degree 0 in current wages prices and the nominal stock ofmoney ndash that is there is the absence of money illusion Note that in consideringwhether or not there exists money illusion we must now look at the behavior notonly of households but also of firms Consider firms first
Assuming perfect foresight on the part of firms it is clear that current labordemand is homogeneous of degree 0 in current prices (the wage rate wt and the pricelevel pt) Thus so also current output supply Assuming unit elastic expectationswith respect to wages and prices in all future periods and expectations of futureinterest rates that are invariant to changes in current prices and wages we havethat capital demand and thus also investment demand and firmsrsquo net real financialasset supply are homogeneous of degree 0 in current prices
We already assumed static expectations concerning future interest rates and unitelastic expectations with respect to prices beyond period t + 1 in terms of nextperiodrsquos price to obtain a simple form for the demand functions for investmentThus we need only add (a) the assumption of unit elastic expectations concerningthe price of output between period t and t+1 (implying an expected rate of inflationπe
t and thus an expected real rate of interest that is independent of a change in theprice level pt)
13 and (b) unit elastic expectations concerning next periodrsquos wageand price level (implying an anticipated real wage next period independent of anequiproportionate change in wages and prices in the current period) to obtain theabsence of money illusion with respect to firmsrsquo investment demand14
To obtain the absence of money illusion with respect to households we mustshow that each of the arguments in their demand and supply functions is invariantto an equiproportional change in current prices wages and the nominal stockof money Given perfect foresight the current real wage and initial real moneybalances meet this condition But what about the expected real wage next periodthe expected rate of inflation current dividends and interest payments and the realvalue of initial financial assets holdings As it turns out the assumption of unitelastic expectations with respect to all future prices and wages is again critical inshowing these to be invariant as it was in deriving a capital demand homogeneousof degree 0 in current wages and prices
First it is clear that the assumption of unit elastic expectations with respectto future wages and prices makes future real wages invariant to equiproportionalchanges in the current wages and prices But note that in so doing we have elimi-nated the ldquointertemporal substitution hypothesisrdquo effect on labor supply of a changein the current real wage Similarly the assumption of unit elastic expectations withrespect to the future price level eliminates any effect of a change in the price levelon the expected rate of inflation Finally assuming that expectations of nominalfuture interest rates are invariant to an equiproportionate change in current priceswages and the money supply the expected real rate of interest will not be affectedby equiproportionate changes in wages prices and the money supply But notethat the ldquointertemporal substitution hypothesisrdquo impact on labor supply of a change
76 Behavior and constraints
in the expected real rate of interest initiated by a change in the current price levelis now absent as well15
What is left in order to obtain the absence of money illusion for households isto show that changes in current prices leave current dividend payments and thereal value of initial bond and stock holdings unchanged Once again as we seebelow the assumption of unit elastic expectations concerning next periodrsquos wagesand prices will be invoked to achieve this What we are looking for are sufficientassumptions that will result in the terms dt + zBpt and At being homogeneous ofdegree 0 in wt and pt
From the firm distribution constraint (52) and assuming firmsrsquo labor demandis satisfied (ie N lowast
t = N dt and thus ylowast
t = yst ) we know that
dt + zBpt = yst minus δK minus (wtpt)N
dt
Since N dt and ys
t are homogeneous of degree 0 in prices it is clear that the sumof current dividends plus interest payments is not affected by equiproportionatechanges in both the current wage and price level A higher price level does alterthe composition of payments however as real dividends rise and real interestpayments fall
Now consider At To show that this can be homogeneous of degree 0 in thecurrent wage and price level note from the firm distribution constraint and fromthe assumption that firmsrsquo demand for labor is satisfied in subsequent periods that
dt + zBpet+1 = f (N d
t+1 Kdt+1) minus (we
t+1pet+1)N
dt+1 minus δKd
t+1
A similar equation holds for future periods as well At simply reflects the presentvalue of such future real payments using the appropriate expected real interest ratesfor discounting The assumption of unit elastic expectations concerning prices inall future periods coupled with expectations of future nominal interest rates thatare unaffected by an equiproportionate change in money prices and the moneysupply thus means that A is homogeneous of degree 0 with regard to a change inprices (price level and wages) and the money supply in period t16
A special case of the above is if we assume static expectations concerning futureinterest rates (ie ri = rt i = t + 1 t + 2 ) and zero adjustment costs In thiscase Kd
i i = t + 1 t + 2 would be the same in each future period Therewould be a constant labor demand (N d
i = N dt+1 i = t + 2 t + 3 ) as well given
the invariant real wage in conjunction with no change in their capital stock Nowrecall that At is defined by
At equiv [pbtB + petS]pt
where pbtBt is the present value of future interest payments and petS is the presentvalue of future dividends Since At is simply the present value of the now constantfuture stream of dividends and interest payments discounted using an invariant
Behavior and constraints 77
expected real rate of interest we thus have
At = [dt+1 + zBtpet+1]me
t
Since dt+1 +zBtpet+1 = f (N d
t+1 Kdt+1)minus (we
t+1pet+1)N
dt+1 minusδKd
t+1 from the firmdistribution constraint we can rewrite the initial real holdings of financial assetsas
At = [ f (N dt+1 Kd
t+1) minus (wet+1pe
t+1)Ndt+1 minus δKd
t+1]met
which is homogeneous of degree 0 in current wages and prices if we assume that theanticipated real wage next period is independent of an equiproportionate changein the current wages and prices In other words to obtain the absence of moneyillusion for households we assume as we did with firms that there are unit elasticexpectations concerning wages and prices in the next period
Note that an increase in the money interest rate and the expected rate of inflationthat leaves the expected real rate of interest unchanged will alter the compositionof financial asset holdings although not the total To see this recall that with staticexpectations concerning the interest rate the price of bonds at the end of period tis given by
pbt =infinsum
i=1
z(1 + rt)i = zrt
An increase in the money interest rate and expected rate of inflation such that thereal rate of interest is unchanged means a fall in the price of bonds but an offsettingrise in the price of stock as over time real dividend payments will be rising morerapidly while real interest payments will be falling more rapidly Similarly anincrease in the current level of prices although leaving At unaffected wouldreduce the real value of bond holdings and lead to an exactly offsetting increasein the real value of equity share holdings
The real balance effect is apparent from the nature of the demand and supplyfunctions In particular an increase in nominal money balances or fall in prices withno change in nominal money balances will increase initial real money holdingsand in general lead to an increase in consumption demand real money demandand real net financial asset demand In general labor supply would fall
6 The simple neoclassicalmacroeconomic model (withoutgovernment or depositoryinstitutions)
Introduction
We have now covered a substantial part of the underlying structure for a simpleaggregate model of an economy with production The specific elements of theldquomicroeconomic foundationsrdquo of this aggregate model developed so far have dealtwith the optimizing behavior of individuals (ldquorepresentativerdquo firms and house-holds) in a setting in which individuals take prices as given Implicit in thesediscussions is another part of the microeconomic foundations the way in whichindividual markets operate We have been assuming that prices adjust so that thepresumption that buyers and sellers are price-takers is justified In other wordsequilibrium within individual markets entails price adjustment to equate suppliesand demands In addition we will assume that all individuals in the economy cor-rectly foresee period trsquos output prices when input supply and production decisionsare made at the start of the period As discussed below however there are otheroptions
Static macroeconomic models the options
Grandmont (1977 542) notes that
one way to look at the evolution of an economic system is to view it as asuccession of temporary or short-run competitive equilibrium That is onepostulates that at each date prices move fast enough to match supply anddemand Although one assumes equilibrium in each period the economicsystem displays a disequilibrium feature along a sequence of temporary com-petitive equilibrium at each date the plans of the agents for the future arenot coordinated and thus will be in general incompatible this is to becontrasted with the perfect foresight approach where by definition such adisequilibrium phenomenon cannot occur
This temporary equilibrium view of the economy is characteristic of the simplestatic neoclassical model a model in which all prices adjust to maintainequilibrium1 The second key assumption of the neoclassical model is that agents
Simple neoclassical macroeconomic model 79
are informed about prices within the period In particular when making their laborsupply and demand decisions at the start of the period households and firms areassumed to correctly anticipate the prices they will have to pay to purchase outputduring the period As it turns out this element of ldquoperfect foresightrdquo with respectto markets through time t +1 is a critical feature of the analysis Before examiningthe implications of these assumptions however some history on the origin of theneoclassical model might be helpful
The phrase ldquoneoclassical macroeconomic modelrdquo is a descendant of ldquoclassicalrdquoeconomic theory as reflected in the work of Sir William Petty during the 1600sIn Das Kapital Karl Marx (1976 85) stated that ldquoby classical Political EconomyI understand that economy which since the time of W Petty has investigated thereal relations of production in bourgeois societyrdquo As Marx suggested early classi-cal economists focused on the determinants of the economyrsquos productive capacityThe neoclassical macroeconomic model shares this focus on the productive capac-ity of the economy as the determinant of total output It also turns out to be thestatic precursor to much of the current analysis in the macroeconomic literaturethat falls under the heading of ldquoreal business cycle theoryrdquo
While the simple static neoclassical model along with its dynamic and stochas-tic counterparts is one popular approach to macroeconomic analysis there areother approaches In fact even though static analysis is restricted to markets in thecurrent period there remains enough flexibility to introduce at least four ways ofcharacterizing macroeconomic analysis
1 ldquoNeoclassical modelrdquo competitive equilibria are assumed to exist in cur-rent and future markets and limited perfect foresight is assumed for allparticipants
2 ldquoIllusion modelrdquo competitive equilibria are assumed to exist in current andfuture markets but imperfect foresight is assumed on the part of some agentsThe result is like the ldquoLucas supply functionrdquo popularized by Lucas (1973)in which output can respond directly to increases in the actual output prices
3 ldquoKeynesian modelrdquo a competitive equilibrium is assumed not to exist inthe labor market as the money wage is fixed and employment is demand-determined However other prices in particular the prices of output arepresumed to reflect competitive equilibria A rational expectation version ofthis model is developed by Fisher
4 ldquoNon-market-clearing modelrdquo a competitive equilibrium is assumed not toexist in the output market as the price of output is fixed above the competitiveequilibrium level This model forms the basis of much of what appears inundergraduate macroeconomic analysis including the IS-LM model
The neoclassical model with its assumptions of flexible prices and informedagents provides a benchmark against which we can compare the predictions ofother (static) macroeconomic models2 It also provides insight into the nature of thestationary states for dynamic macroeconomic models that presume market-clearingprices and accurate forecasts of prices
80 Simple neoclassical macroeconomic model
Hicksian temporary equilibrium and Walrasrsquo law
For the market for any good ldquocompetitiverdquo equilibrium is defined by equalitybetween market demand and supply3 A temporary competitive equilibrium for theeconomy during period t will be characterized by a money wage for labor (wlowast
t )a money price for the consumption commodity ( plowast
t ) a money price for bonds( plowast
bt) a money price for equity shares ( plowastet) allocations to households in terms
of employment consumption bond holdings equity share holdings and moneybalances (N lowast
t clowastt Blowast
t Slowastt and M lowast
t ) and allocations to firms in terms of employmentoutput investment bond issues and equity share issues (N lowast
t ylowastt Ilowast
t Blowastt Slowast
t ) suchthat
bull these allocations are in the demand (supply) set of each agentbull these allocations are feasible
Together these two conditions imply prices determined in the labor output bondand equity shares markets for period t that result in zero excess aggregate demandfor labor output bonds equity shares and money Thus we may rewrite theconditions for a general equilibrium as a money wage a price of output a price ofbonds and a price of equity shares such that
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Bst = Bd
t
Sst = Sd
t
Mpt = M dt pt
where
I dnt = Kd
t+1 minus K
yst = f (N d
t K)
Given our assumption that bonds and equity shares are perfect substitutes ingeneral there will not be a unique equilibrium in terms of the number of bondsand equity shares supplied or demanded although the total value of financialassets supplied or demanded will be determinant Thus we replace the bond andequity share markets with a single market the financial market The equilibriumconditions then become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
Simple neoclassical macroeconomic model 81
Ast = Ad
t
Mpt = M dt pt
where
Ast equiv [ pbtB
st + petS
st ]pt
Adt equiv [ pbtB
dt + petS
dt ]pt
We can view the financial market as simultaneously determining the price ofbonds pbt the price of equity shares pet and the interest rate rt That is once oneof these is known the other two are implied For instance from our definition ofthe interest rate
1 + rt equiv [z + pbt+1]pt
we see that given the coupon rate and expected price of bonds in the subsequentperiod the interest rate rt implies a price of bonds pbt As perfect substitutesbonds and equity shares must offer the identical expected gross return Thus wehave that
1 + rt = 1 + ret equiv [dt+1 + pet+1]pet
As you can see given expectations of future dividends and the future price ofequity shares an interest rate rt also implies a price of equity shares pet We oftentalk of the financial market in terms of an equilibrium interest rate The aboveshould make it clear that associated with such an equilibrium interest rate areprices of bonds and equity shares And a rise (fall) in the interest rate means a fall(rise) in the prices of bonds and equity shares
We will make one additional change in the characterization of the financialmarket to put it in terms of additional demands and supplies that is put it in netrather than gross terms The reason for this is that net financial asset demands andsupplies are what correspond to household saving in the form of financial assetsand firm investment Thus the equilibrium conditions become
N st = N d
t
yst = cd
t + I dnt + δK + ψ(I d
nt)
net Ast = net Ad
t
Mpt = M dt pt
where
net Ast equiv As
t minus At
net Adt equiv Ad
t minus At
At equiv [pbtB + petS]pt
82 Simple neoclassical macroeconomic model
According to the above we have four equilibrium conditions but only threeprices ndash the money wage rate the level of output prices and the interest rate ndashto be determined As usual Walrasrsquo law is invoked to show that only n minus 1 ofthe n equilibrium conditions are independent However the nature of Walrasrsquo lawdepends on whether we assume limited perfect foresight or not
Walrasrsquo law for limited perfect foresight sums up the constraints faced by theindividual agents in the economy at time t to obtain
[cdt + I d
nt + δK + ψ(I dnt) minus ys
t ] + [net Adt minus net As
t ]+ (wtpt)[N d
t minus N st ] + M d
t pt minus Mpt = 0
Thus the sum of excess demands for output financial assets labor and moneymust equal zero
When there is not perfect foresight at time t concerning prices for period t inthe output and financial markets we have the equilibrium condition for the labormarket
N st minus N d
t = 0
and the modified Walrasrsquo law based on the resulting employment and output ofthe form
[cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t ] + [net Adt minus net As
t ] + M dt pt minus Mpt = 0
In this case the money wage employment and thus output are determined inthe labor market and the modified Walrasrsquo law indicates that the price level andinterest rate are determined by any two of the remaining three markets
As it turns out most macroeconomic analysis takes this second approach tosolving for equilibrium That is the analysis focuses on the labor (and other input)markets and determines the effect of changes in output price (and potentially othervariables such as the interest rate) on equilibrium employment and thus outputThis generates an ldquoaggregate supply equationrdquo which is then combined with twoof the remaining three equilibrium equations ndash typically the commodity and moneymarket equilibrium conditions ndash to determine the equilibrium price level interestrate and output The modified Walrasrsquo law is invoked to ensure equilibrium inthe financial market at this point Working backward one can infer from theaggregate supply equation the equilibrium money wage and employment impliedby the analysis
The advantage of the above approach is that it can be used whether or not thereexists limited perfect foresight at time t with respect to the price level and whetheror not prices adjust to clear markets A disadvantage of the analysis is that inthe case of the neoclassical model with limited perfect foresight and competitiveequilibrium it arbitrarily breaks up the analysis of markets In doing so it requiresthat demand functions for such goods as money and commodities be specifiedwith income as an argument This form of the demand functions obscures the fact
Simple neoclassical macroeconomic model 83
that as we have seen in standard general equilibrium analysis demand functionsdepend on prices (including the real wage) and income is a variable determinedby the choice of labor supply
With these qualifications in mind we take the standard approach of macroeco-nomics and separate out the analysis of the labor (and other input) markets (theldquoaggregate supplyrdquo part) from the other markets (the ldquoaggregate demandrdquo part)The next section considers the labor market at time t
Labor market equilibrium
At the start of period t the labor market takes place and a rate of production of outputis determined From our analysis of firm behavior at time t we know that behindlabor demand is an expected price of output over the period and associated expectedreal wage an existing capital stock and existing technology as incorporated in theproduction function We shall assume that firms correctly anticipate at time t theprice of output for the period so that firms confront the actual real wage wtpt
From our analysis of household decision-making at time t we know that behindlabor supply at time t is not only the expected price of output and implied expectedreal wage but also such factors as the relationship of the anticipated current realwage to anticipated future real wages the expected real rate of interest anticipatedwealth in the form of financial asset and real money holdings and anticipated cur-rent nonlabor income Like firms we will assume households have limited perfectforesight in that at time t they correctly foresee the price level for period tAssuming a Walrasian or ldquocompetitive equilibriumrdquo view of the labor marketsthe money wage wt adjusts to achieve equilibrium in the labor market under thesecircumstances
We have already seen how static expectations concerning future rates of wageand price inflation along with the assumption of unit elastic expectations concern-ing wages and prices next period simplify the labor supply function by removingexpected future real wages as explicit arguments Patinkin (1965) among oth-ers goes several steps beyond these simplifying assumptions and assumes that allother variables excepting the real wage do not have a significant impact on laborsupply4 Thus equilibrium in the labor market is given by a level of employmentNt and money wage wt such that
Nt = N dt (wtpt K) and Nt = N s
t (wtpt)
As Patinkin (1965 264) notes
it will immediately be recognized that we have greatly oversimplified theanalysis of this market Both the demand and supply functions for labor shouldactually be presented as dependent on the real value of bond and moneyholdings as well as on the real wage rate5 Further if we were to permit thefirm to vary its input of capital its demand for labor would depend also onthe rate of interest Finally a full utility analysis of individual behavior wouldshow the supply of labor also to depend on this rate6
84 Simple neoclassical macroeconomic model
Besides a competitive labor market and the above simplified labor supplyfunction the neoclassical model assumes that suppliers correctly anticipate theaggregate price level As we will see the assumption of the neoclassical modelthat suppliers like firms have perfect foresight at time t with respect to the priceof output for the period means that changes in the price of output have no effect onoutput supply This characteristic of the neoclassical model implies an underlyingldquoblock recursiverdquo nature to the analysis as described below
At the start of the period the labor market occurs with employment and thusoutput and the money wage being determined for the period Employment andoutput are determined based on individualsrsquo expectations of subsequent variablessuch as the price of output And the level of employment and output influencethe remaining variables to be determined But in the neoclassical model the levelof employment and output are not influenced by the remaining variables to bedetermined We can thus solve the equilibrium sequentially looking first at thelabor market and the determination of employment and output then looking atthe output financial and money markets and the determination of the price leveland interest rate
Financial market equilibrium
With regard to the financial market Patinkin (1965 215) states
a decrease in the price of bonds (and equity shares) decreases the amountdemanded of consumption commodities it will also be assumed that itdecreases the amount demanded of money balances hence by the house-holdrsquos budget constraint their total expenditures on [net] bond holdings mustincrease
Thus Patinkin invokes a ldquoFisherian effectrdquo and a ldquoportfolio effectrdquo of a change inthe interest rate to obtain
part(net Adt )partpbt lt 0
From the firm financing constraint and the fact that a firmrsquos net investment demandis inversely related to the interest rate we have
part(net Ast )partpbt gt 0
The above analysis differs slightly from that found in Patinkin (1965 214) Forinstance Patinkin uses the assumption of static expectations concerning interestrates and a coupon rate z equal to 1 to express the price of bonds as the reciprocalof the interest rate that is
pbt = 1rt
Simple neoclassical macroeconomic model 85
Further Patinkin assumes no equity shares so his financial market consists only ofbonds Finally Patinkin does not graph net real bond demand and supply Ratherhe graphs the total number of bonds demanded and supplied Thus Patinkin hasdemands and supplies of bonds of the form
Bs = rtptAst equiv Bs
t
Bd = rtptAdt equiv Bd
t
As might be expected Patinkinrsquos bond demand and supply curves differ in naturefrom net real financial asset demand and supply curves For instance it is clearfrom the analysis of investment demand that a rise in the interest rate (a fall inthe price of bonds) will lead to reduced investment demand and thus reduced netreal financial asset supply This relationship between investment and firmsrsquo net realfinancial asset supply follows directly from the firm financing constraint whichgiven pbt = 1rt is of the form
net At equiv 1ptrt[Bst minus B] = I d
nt + ψ(I dbt)
Naturally if net investment were initially zero and there were zero adjustment costs(so that Bs
t = B) the fall in investment that results from a rise in the interest ratewould imply a similar decrease in the number of bonds supplied (Bs
t ) Howeverif initially Bs
t gt B then a higher interest rate even though it decreases invest-ment could at the same time increase the number of bonds supplied As Patinkin(1965 217) observes
consider the effect of an increase in the rate of interest (fall in 1rt) The inter-nal consistency of our model requires that this decrease the amount of realbonds supplied
However this need not reduce the number of bonds supplied As Patinkin(1965 217) continues
a rise in the interest rate ( pbt falls) has lowered the price received for bondsand so may increase the number of bonds necessary to finance the firmrsquosexpenditures on investment commodities (Bs
t gt B initially) even thoughthese expenditures have decreased
A similar analysis would apply to a comparison of householdsrsquo demand for bondsin terms of numbers with household net real financial asset demand
Money ldquomarketrdquo equilibrium
As we know from Walrasrsquo law having depicted equilibrium in the labor andfinancial markets we need only look at one of the other two remaining markets
86 Simple neoclassical macroeconomic model
the output market or the money ldquomarketrdquo Consider a money market in whichnominal money supply M and nominal money demand M d
t equiv ptLdt are plotted
against the reciprocal of the price level which indicates the ldquorelativerdquo price of oneunit of money in terms of output If there is no real balance effect with respect toreal money holdings then the money demand curve is invariant to a change in theprice level (elasticity of demand equals 1) If there is a real balance effect withrespect to money demand then a fall in the price level (a rise in the relative price ofmoney (1pt)) would lead to a less than proportionate decrease in nominal moneydemand (elasticity of demand less than 1)
Aggregate supply and demand an introduction
Temporary equilibrium for the economy can be characterized in several ways Aswe alluded to above one way common in journal articles and textbooks is to dividethe analysis under the headings of ldquoaggregate supplyrdquo and ldquoaggregate demandrdquo
Under the heading of ldquoaggregate supplyrdquo is an analysis of the input markets inparticular the labor market One aim is to determine input prices (in particular themoney wage) the employment of inputs and the implied production of output thatoccur at different levels of output prices (and potentially different levels of interestrates) The term ldquoaggregate supplyrdquo is applied to this analysis for the simple reasonthat it determines the ldquosupplyrdquo of total output at different prices
Under the heading of ldquoaggregate demandrdquo is an analysis of the other marketsin the economy during period t in particular the output financial and moneymarkets The aim is to determine the level of output prices and the interest ratethat occur at different levels of output The term ldquoaggregate demandrdquo attached tothis analysis reflects the fact that the analysis determines how the price level andinterest rate adjust to equate the ldquodemandrdquo for total output to different levels ofproduction
Combining the aggregate supply and demand analysis we can determine theoutput price level and interest rate associated with temporary equilibrium as wellas the underlying equilibrium money wage real wage employment consumptioninvestment and real money balances To understand more clearly what is involvedin aggregate supply and demand analysis and how they can be combined we con-sider below the specific case of the neoclassical model starting with the aggregatesupply
Equilibrium and aggregate supply
As we have said behind aggregate supply is an analysis of various input marketsto determine the response of total output to changes in such variables as theprice level7 In the neoclassical model changes in prices lead to equiproportionatechanges in money wages with no change in the equilibrium level of employmentand thus no change in aggregate supply To formally show this let us start withthe following statement of equilibrium in the labor market in terms of a money
Simple neoclassical macroeconomic model 87
wage wt and level of employment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpt) minus Nt = 0
A critical aspect of the above is the fact that suppliers in particular suppliers oflabor correctly anticipate the price level that will exist with respect to output sothat wtpt replaces wtpe
t in the labor supply functionTotally differentiating the above two equations with respect to wt Nt and pt
one obtains[(partN d
t part(wtpt))(1pt)
(partN st part(wtpt))(1pt)
minus1minus1
] [dwtdNt
]=[(partN d
t part(wtpt))(wtdpt( pt)2)
(partN st part(wtpt))(wtdpt( pt)
2)
]
Applying Cramerrsquos rule one obtains
dwtdpt = wtpt and dNtdpt = 0
Thus in the neoclassical model the real wage and employment level determinedin the labor market are independent of changes in the price level8
The ldquoaggregate supply equationrdquo combines the analysis of the labor market (andother input markets) and resulting determination of employment of various inputswith the production function to determine the resulting output supplied For theneoclassical model the aggregate supply equation is of the form
ylowastt = ylowast
t (K ) (61)
What is important about this equation is that the price level and interest rateare not arguments in the supply equation Of course changes in the capital stockchanges in technology or changes in the supply of other inputs (eg changes inthe oil supply) can affect output Similarly a change in the composition of thelabor force or government policies that affect labor supply can affect equilibriumemployment and thus output9
We may summarize the above findings graphically Consider an increase in pt Given perfect foresight both firms and households at time t would anticipate thishigher price level In the labor market the result would be an increase in theequilibrium money wage in the same proportion as the increase in the expectedprice level so that the anticipated real wage would remain the same as wouldemployment and output This outcome for the labor market is shown in Figure 61Note that the result of no change in employment or output in light of a higher outputprice simply requires that both labor demand and supply curves shift vertically bythe same amount Such equal shifts reflect the fact that the same increase in themoney wage leaves both firms and households anticipating the same real wage asbefore
In undergraduate textbooks the fact that changes in the price of output leaveemployment and real output unaffected in the neoclassical model is often shown
88 Simple neoclassical macroeconomic model
Labor
N demand
N supply
Figure 61 Labor market equilibrium
in ( pt yt) space by a vertical ldquoaggregate supply curverdquo Such a curve summarizesthe underlying behavior for the economy-wide labor market Later we will see thatunder other assumptions such as embedded in Lucasrsquos model (ldquomoney illusionrdquo)and the Keynesian model (fixed nominal wage) the aggregate supply curve willbe upward sloping
It is important to realize that the aggregate supply shown in Figure 62 is not thetypical market supply curve of microeconomics In microeconomics a higher priceof good x and consequent increase in the demand for inputs by firms producing thatgood draws inputs away from the production of other goods so that the higherprice of good x induces increased output of that good and associated increasedemployment of inputs (such as labor) in the production of the good Implicit inthis analysis is that there is reduced production of other goods in the economyHowever as the above analysis makes clear if the focus is on the aggregation ofall commodity markets a higher price level no longer induces increased aggregateoutput unless the quantity of total inputs supplies rises which will not be the caseunder neoclassical assumptions10
The natural rate of unemployment
The key feature of the above analysis of aggregate supply is the assumption thatprices adjust to continuously maintain equilibrium in the various markets and thatindividuals are perfectly informed concerning prices The result is a level of realoutput sometimes called the ldquofull employment levelrdquo So far missing from theanalysis however is any mention of unemployment If one expands the model tointroduce unemployment the rate of unemployment is called the ldquonaturalrdquo rate11
By ldquonaturalrdquo is meant that it is the rate of unemployment that the economy will
Simple neoclassical macroeconomic model 89
Output
Price
Aggregate supply
Figure 62 Aggregate supply
gravitate to as prices adjust to clear markets and individuals become fully informedconcerning prices (ie under the assumptions of the neoclassical model)
To introduce unemployment into the analysis when the labor market is in equilib-rium at full employment requires recognition of two facts First the labor marketis in a constant state of flux Not only do individuals enter and leave the laborforce continually but labor demand varies continuously across firms as they expe-rience variations in the relative demand for their output This instability of jobsthemselves has been estimated to account for roughly one-quarter of the averageunemployment rate as in an average year one in every nine jobs disappears and onein every eight is newly created (Leonard 1988) Second information is costly Ittakes time for new workers entering the labor force and for workers who have beenlaid off or have quit previous jobs to discover which employers have vacanciesand how wages vary across employers
When we take into account the continuous flows to unemployment together withworkersrsquo imperfect information about job vacancies we see that unemploymentis no longer inconsistent with the neoclassical model At any moment there existnew entrants into the labor market who are spending time searching for acceptablejobs There also exist laid-off workers who are either searching for alternativejobs or awaiting recall And there are workers who have quit their jobs and aresearching for other jobs This kind of unemployment is generally referred to asfrictional unemployment
Sometimes part of frictional unemployment is called structural unemploymentwith structural unemployment occurring because of a change in the compositionor ldquostructurerdquo of aggregate output across firms For example the replacement ofsteel with plastic in automobiles led to a shift in employment from steel factoriesto firms making plastic During this transition some steel workers experiencedstructural unemployment
90 Simple neoclassical macroeconomic model
To summarize when the labor market is in equilibrium there exists a positiveunemployment rate because workers continuously move into and out of the laborforce and between jobs12 To signify that this unemployment rate is ldquonaturalrdquo orconsistent with equilibrium in the labor market it is generally called the naturalrate of unemployment13 The corresponding level of employment is then oftenreferred to as the full employment level
Like equilibrium output and employment in the neoclassical model ldquosupplyfactorsrdquo determine the natural rate of unemployment as well14 Among these isthe demographic composition of the labor force but also unemployment insuran-ce and minimum wage legislation In recent years a large body of literature hasanalyzed labor markets and the sources of unemployment with the focus on searchand labor contracts Note that an understanding of the natural rate of unemploy-ment is important in determining the effects of government policies aimed at theunemployed
Equilibrium and aggregate demand
The aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) The ldquoISrdquo equation since it is simply the expression forequilibrium during period t with respect to the output market is given by15
cdt + I d
nt + δK + ψ(I dnt) minus ylowast
t = 0 (62)
Note that the equation is termed the ldquoISrdquo equation because we can rewrite it toobtain
Idnt + δK + ψ(I d
nt) = ylowastt minus cd
t
indicating that equilibrium in the output market is equivalent to the equal-ity between Investment expenditures (the left-hand side of the equation) andhousehold Saving (the right-hand side of the equation)16
The assumption of unit elastic expectations concerning wages and prices givesthe following simple form for householdsrsquo consumption demand and firmsrsquoinvestment demand functions during period t17
cdt = cd
t (rt πet At Mpt ylowast
t )
and
Idnt = I d
nt(met + δ K)
The ldquoLMrdquo equation since it is simply the expression for equilibrium in period twith respect to the ldquomoneyrdquo market is given by
Ldt = Mpt (63)
Simple neoclassical macroeconomic model 91
This equation is termed the ldquoLMrdquo equation for it reflects the equality betweenLiquidity preference or demand and the Money supply We have seen that ourassumption of unit elastic expectations concerning wages and prices gives us thefollowing simple form for the real money demand function
Ldt = Ld
t (rt πet At Mpt ylowast
t )
From the modified Walrasrsquo law it follows that a price level and interest rate thatsatisfy the ldquoISrdquo equation and ldquoLMrdquo equation for a given level of output ylowast
t willalso result in equilibrium in the financial market
We thus have three equations (the aggregate supply equation (61) the ldquoISrdquoequation (62) and the ldquoLMrdquo equation (63)) that can be solved for the equilib-rium levels of output price and interest rate Looking at what underlies thesethree equations we can then infer the changes in money wages and employment(the labor market) as well as changes in the components of output demand (invest-ment and consumption demand) Equilibrium can be depicted in terms of a pricelevel and interest rate (Patinkinrsquos CCndashLLndashBB curves) or in terms of a price leveland output (ie aggregate demand and supply curves) We start with Patinkinrsquosdepiction of equilibrium
Depiction of equilibrium the Patinkin analysis
The neoclassical model aggregate supply is independent of changes in the pricelevel and interest rate18 This means that for any given level of output we canfocus on equations (62) and (63) to determine the equilibrium interest rate andprice level This is a reflection of the ldquoblock recursiverdquo nature of the solution tothe neoclassical model mentioned earlier
Patinkin (1965) suggests a graphical way of showing such an equilibrium com-bination of price level and interest rate using any two of three curves denoted theCC LL and BB curves The CC curve depicts combinations of the price level andinterest rate that satisfy the equilibrium condition for output (62) the LL curvedepicts combinations that satisfy the equilibrium condition for money (63) Wherethese two lines so constructed intersect it must then be the case that this price com-bination satisfies two of the three equilibrium conditions simultaneously It followsfrom the modified Walrasrsquo law that the curve indicating various combinations ofthe price level and interest rate that satisfy the equilibrium condition with respectto the financial market (the BB curve) goes through this point as well19
For the market for output equilibrium in period t is characterized by (62) Recallthat real output ylowast
t denotes that output reflecting the capital stock K technologyand the employment of labor determined in the labor market at time t From theneoclassical assumptions with respect to labor demand and supply equilibriumemployment and thus output are unchanged for any change in the price level orinterest rate
Let us presume that the pair ( plowastt rlowast
t ) is associated with equilibrium in the outputmarket In (price interest rate) space this combination is identified by a unique
92 Simple neoclassical macroeconomic model
point on the CC curve that identifies combinations of pt and rt associated withequilibrium in the commodity market (Figure 63)
To understand what lies behind the shape of the CC curve depicted above totallydifferentiate the market-clearing condition for the commodity market with respectto pt and rt Doing so we obtain20
[partcdt part(Mpt)] minus [Mp2
t ]dpt + [partcdt partrt + (1 + ψ prime)partI d
ntpartrt]drt = 0
Rearranging we have that the slope of the CC curve is given by
drtdpt | ydt minus ylowast
t = 0= [partcd
t part(Mpt)][Mp2t ][partcd
t partrt + (1 + ψ prime)partI dntpartrt] lt 0
The negative slope of the CC curve can be explained in the following way Thepositive numerator of the expression for the slope reflects the real balance effectwith respect to commodities this term indicates the fall in consumption demandthat would accompany a rise in the price level for such a rise reduces agentsrsquowealth in the form of initial real money balances21 The negative denominatorindicates the effect of a change in the interest rate on consumption and investmentdemand
Now let us consider the money market As before there is a unique point thatindicates an interest rate and a price of output at which there is equilibrium inthe economy and this point on the LL curve identifies combinations of pt andrt associated with equilibrium in the money market (Figure 64) Recall that theLL curve identifies combinations of pt and rt associated with equilibrium in themoney ldquomarketrdquo as given by (63)
The slope of the LL curve is given by totally differentiating the zero excessdemand condition with respect to money Doing so and rearranging one obtains
drtdpt |Ldt minus Mpt = 0 = [1 minus partLd
t part(Mpt)][Mp2t ][minuspartLd
t partrt] gt 0
r
CC
p
Figure 63 Commodity market
Simple neoclassical macroeconomic model 93
rLL
p
Figure 64 Money market
Note that the numerator of this term is positive The change in real money balanceswill be greater than the consequent change in real money demand given real balanceeffects with respect to financial assets andor commodities The denominator ispositive as well reflecting the fact that an increase in the interest causes householdsto shift their portfolio out of money holdings into bonds
Putting together the CC and LL curves we have equilibrium in the economyat the point where these two curves intersect From the modified Walrasrsquo law weknow that at this point demand for financial assets equals supply as well In factthere is a corresponding BB curve that goes through this same intersection
As with the CC curve to understand what lies behind the shape of the BB curvewe can totally differentiate the market-clearing condition for the financial marketwith respect to pt and rt That market-clearing condition is
net Adt = net As
t = 0
where in simplest form
net Ast = net As
t (met + δ K)
net Adt = net Ad
t (rt πet At Mpt ylowast
t )
Differentiating we obtain
[partnet Adt part(Mpt)] minus [Mp2
t ]dpt + [partnet Adt partrt minus partnet As
t partrt]drt = 0
Rearranging we have that the slope of the BB curve is given by
drtdpt |net Adt minus net As
t = 0= [partnet Ad
t part(Mpt)][Mp2t ][partnet Ad
t rt minus partnet Ast partrt] gt 0
94 Simple neoclassical macroeconomic model
The positive slope of the BB curve can be explained in the following way Thepositive numerator of the slope expression reflects the real balance effect withrespect to financial assets The term indicates the fall in net real financial assetdemand (and planned decrease in future consumption) that would accompany arise in the price level for such a rise reduces agentsrsquo wealth in the form of initialreal money balances The positive denominator indicates the effect of a change inthe interest rate on household lending and firm borrowing A rise in the interestrate would tend to decrease current consumption demand (ldquoFisherianrdquo effect) andmoney demand (ldquoportfoliordquo effect) and thus increase household net financial assetdemand On the other hand a rise in the interest rate would tend to reduce firminvestment demand by raising the expected real user cost of capital and thusreduce firm net real financial asset supply
While both the slope of the BB curve and the LL curve are positive it is thecase that the LL curve is steeper
A depiction of equilibrium aggregate demand andsupply curves
Using Patinkinrsquos analysis a higher output would require a lower price level tomaintain equilibrium in the output financial and money markets Alternativelyone can show the effect of a change in ylowast
t on pt and rt by totally differentiatingequations (62) and (63) with respect to pt rt and ylowast
t The aggregate demand curve summarizes this inverse relationship between the
price level and output arising from an analysis of the output financial and moneymarkets Specifically such a curve depicts combinations of output and price levelassociated with equilibrium in the output financial and money markets It isdownward sloping indicating that an increase in output requires a lower pricelevel to clear the output financial and money markets Behind a movement downthe aggregate demand curve are larger real balances that stimulate output demandeither directly (through a real balance effect on consumption demand) or indirectly(the resulting increase in real money balances leads to an increase in net realfinancial asset demand by households and thus a lower interest rate)22
It is important to realize that the phrase ldquoequilibrium in the output mar-ketrdquo in this context abstracts from supply-side considerations At each pricelevel the aggregate demand curve indicates the output that if produced wouldequal output demand (along with satisfying the equilibrium conditions withrespect to the financial and money markets) Production of output equal tothat demanded would occur if firms sought simply to produce to meet mar-ket demand and if workers were readily available for employment so thatfirms could hire to achieve production equal to what was demanded The termldquoequilibrium in the output marketrdquo does not imply equality between outputdemand and the output that our analysis of the labor market suggests would besupplied
Simple neoclassical macroeconomic model 95
Output
Price
Aggregate supply
Aggregate demand
Figure 65 Macroeconomic equilibrium
Figure 65 combines the aggregate demand and aggregate supply curves Wherethey intersect we know by construction that the resulting price level ( pt)0 andoutput ( yt)0 are such that
bull the labor market is in equilibrium (the economy is at a point on the aggregatesupply curve) and
bull the output financial and money markets are in equilibrium (the economy isat a point on the aggregate demand curve)
Conclusion
Using a very simple framework this chapter has developed a powerful macroeco-nomic model of the economy This model is grounded in the general equilibriumtheory set forth in earlier chapters Moreover this model highlights that degree ofconnectedness that exists among the output labor financial and money marketsThe microfoundations of macroeconomics have been emphasized and it has beenshown how market forces work in such a way as to lead to aggregate supply anddemand two key elements in any macroeconomic model
7 Empirical macroeconomicsTraditional approaches and timeseries models
Introduction
Quantitative approaches to analyzing economic data provide meaningful anduseful insight for understanding how variables interact and how they might beexpected to behave in a variety of circumstances including the future This chapteroutlines the traditional econometric-based method and the relatively simple butoften more elegant time series method for analyzing economic data in the timedomain The stochastic nature of economic data is discussed and the now com-mon ARIMA model found in much of the empirical macroeconomic literature isdeveloped in parts The chapter provides a solid background for understandingldquomacroeconometricsrdquo and time series analysis
Traditional approaches
Empirical macroeconomics can be roughly divided into two approaches ndash a tradi-tional approach that draws heavily on macroeconomic theory and a more recentapproach advocated for forecasting that does not rely to any great extent on the-ory To consider the former we start by presenting a simple theoretical model ofthe macroeconomy The model is used to illustrate traditional empirical analysesreflecting (a) tests of behavioral hypotheses (b) tests of reduced-form expressionsfor various economic aggregates and (c) the construction of econometric modelsfor policy simulations and forecasting The more recent empirical macroeconomicsapproach of using time series models for forecasting aggregate variables whichplaces less reliance on macroeconomic theory is then briefly considered
A simple theoretical macroeconomic model
Consider the following linear approximation of the simple static classical model(closed economy) such as that popularized by Sargent (1987a 20)1 The economyis divided into four markets a labor market (where wages w and total employ-ment n are determined) an output market (where total output y and the pricelevel p are determined) a financial market (where the interest rate r is deter-mined) and a money ldquomarketrdquo Our goal is to construct a model that will determine
Empirical macroeconomics 97
such endogenous variables as the level of employment wages output prices andthe interest rate
From Walrasrsquo law (as we will see more clearly later on) we need only explicitlyconsider three of the above four markets in the analysis For the labor market letus assume the following linear approximations for the key behavioral relations2
nd = f minus g(wp) (71)
ns = h + j(wp) (72)
These two linear equations indicate that firmsrsquo labor demand nd is inversely relatedto the real wage (wp) and householdsrsquo labor supply ns is directly related to thereal wage3
For the output market the key underlying behavioral and technological relationsare in linear form
cd = a + b( y minus T ) minus cr (73)
id = d minus er (74)
ys = f (n K) (75)
Equation (73) indicates that householdsrsquo consumption demand cd is directlyrelated to real disposable income (income y minus lump-sum taxes T ) andinversely related to the interest rate r4 Equation (74) indicates that firmsrsquo invest-ment demand id is inversely related to the interest rate r Equation (75) is theaggregate production function relating employment n and capital stock k tototal output supplied ys
For the money ldquomarketrdquo the key underlying behavioral relation is
Ld = l middot y minus m middot r (76)
Equation (76) indicates that householdsrsquo real money demand Ld is directly relatedto real income and inversely related to the interest rate Nominal money supplyM s is exogenous
General equilibrium in this economy means an equilibrium level of employmentnlowast wage wlowast price level plowast output ylowast and interest rate rlowast such that the labor marketis in equilibrium
ns minus n = 0 (77)
nd minus n = 0 (78)
the output market is in equilibrium
ys minus y = 0 (79)
cd + id + gd minus y = 0 (710)
98 Empirical macroeconomics
and the money ldquomarketrdquo is in equilibrium
Ld minus M sp = 0 (711)
Note that the component of output demand gd in (710) reflects exogenous realgovernment demand for output
The above macroeconomic model consists of 11 equations Macroeconomicmodels often can be represented by a system of equations This system of equationsis sometimes said to be a ldquostructural modelrdquo because the form is given from theunderlying theory As we will see below we can solve this system of equationsfor each of the ldquoendogenousrdquo variables as a function solely of the predeter-mined or ldquoexogenousrdquo variables5 Such solutions can be called the ldquoreduced-formrdquosolutions
By substituting the behavioral equations (71)ndash(76) into the equilibrium condi-tions we obtain the following set of five equilibrium conditions that can be solvedfor the five variables nlowast wlowast plowast ylowast and rlowast6
f minus g(wp) minus n = 0
h + j(wp) minus n = 0
f (n K) minus y = 0
a + by minus bT minus cr + d minus er + gd minus y = 0
ly minus mr minus M sp = 0
(712)
Note that the first three equations in the system (712) can be solved to obtainwlowast nlowast and ylowastas a function of the price level plowast In particular we obtain7
wlowast = [(f minus h)(g + j)]plowast (713)
nlowast = [1( j + g)][ jf + gh] (714)
ylowast = f (nlowast K) = f ([1( j + g)][ jf + gh] K) (715)
Equation (715) is an example of a reduced-form expression for output for itexpresses output solely in terms of the exogenous variables It is a special caseof what is called the ldquoaggregate supply equationrdquo in which the price level doesnot affect the level of output produced8 Note that equation (713) indicates thatchanges in the price level lead to equiproportionate changes in the equilibriummoney wage so that changes in the price level do not lead to changes in theequilibrium real wage
To obtain a reduced-form expression for the price level we can use the reduced-form expression for output in conjunction with the last two equations in the system(712) These last two equations are sometimes termed the ldquoIS equationrdquo andthe ldquoLM equationrdquo respectively9 Solving the LM equation (the last equation ofthe system (712)) for r and substituting both this expression for r and the priorexpression for equilibrium output ylowast (715) into the IS equation (the penultimate
Empirical macroeconomics 99
equation of the system (712)) we obtain the reduced-form expression for theequilibrium price level
plowast = M scprimem
[1 minus b + cprimem] f ([1( j + g)][ jf + gh] K) minus aprime (716)
where aprime = a + d + gd minus b middot T and cprime = c + eChanges in the variable aprime indicate changes in the autonomous component of
consumption (the term a in (73)) autonomous investment demand (d in (74))and government spending or taxing (gd or T respectively) By ldquoautonomousrdquo wemean that part of householdsrsquo and firmsrsquo output demand that is independent of thevariables to be determined by the analysis particularly income and the interestrate The variable cprime indicates the combined response of consumption demand (cin (73)) and investment demand (d in (74)) to a unit change in the interest rate
Note that the procedure to obtain equation (716) could be alternatively describedas follows First combine the last two equations in the system (712) to eliminatethe interest rate r The result is termed the ldquoaggregate demand equationrdquo Thisrelates the price level to the level of output at which the money and output markets(and thus by a modified version of Walrasrsquo law the financial market) are inequilibrium In this context equilibrium in the output market is defined as thelevel of production that if produced would equal output demand This aggregatedemand equation would then be combined with the aggregate supply equation(715) to determine the equilibrium price level or output10
Tests of behavioral hypotheses
Theoretical macroeconomic models embody predictions concerning factors thatinfluence the behavior of various groups in the economy For instance considerthe behavioral equations (71)ndash(76) in the prior simple macroeconomic modelWe could test the prediction that investment is inversely related to the interest rate(see (74)) or that money demand depends inversely on the interest rate (see (76))Or we could expand our theory of consumption behavior (equation (73)) to testa particular behavioral relationship between aggregate household consumptionand permanent income11 Or we could expand our view of labor supply behavior(equation (72)) as Stuart (1981) did in an examination of Swedish data to test theprediction that sufficiently high marginal tax rates will reduce the economy-widelabor supply
Tests of reduced-form hypotheses
Theoretical macroeconomic models also generate predictions concerning thereasons for fluctuations in such aggregate variables as real output unemploy-ment and the level of prices These predictions typically reflect the reduced-formsolutions of macroeconomic models Examples of reduced forms in our simplemacroeconomic model are equations (715) for output and (716) for price
100 Empirical macroeconomics
One example of an empirical test of macroeconomic theory that focuses ontesting the reduced-form predictions is Taylor (1979) Specifically Taylor teststhe reduced-form relationships between the money supply and the logarithmof income from trend and between the money supply and the rate of infla-tion using quarterly US data from 1953 to 197512 Similarly Christopher Sims(1972) uses quarterly US data on nominal output and the money supply to testwhether changes in the money supply lead to changes in the current dollar valueof national income (see Ewing 2001)
Large-scale econometric models and forecasting
Besides testing macroeconomic theories empirical macroeconomic analysis oftenseeks to forecast the future paths of economic aggregates Interest in forecast-ing stems not only from an obvious curiosity about the future path of aggregatevariables such as real output and prices but also from the fact that many ifnot all macroeconomic theories suggest that expectations of future events influ-ence current activity In this context forecasts can be used to proxy individualsrsquoexpectations of future events in tests of various aspects of macroeconomic models
One approach to making forecasts of future aggregate variables is to rely ontheoretical macroeconomics as a guide in the construction of large-scale econo-metric models Behavioral equations that are more detailed disaggregated versionsof equations (71)ndash(76) are estimated and coefficients are checked to make surethey agree with theory These models reflect attempts to produce large systems thatfaithfully represent the interrelationships in a complex national economy Givenpostulated paths of the exogenous variables the actual estimated equations arethen used to generate forecasts of the various aggregate variables such as outputand its components (eg consumption and investment) prices and interest rates
While large-scale econometric models as forecasting devices have their advo-cates (and many individuals reveal they have a positive value by willinglypaying for their forecasts) Granger and Newbold (1986 292ndash293) among oth-ers question the value of such a forecasting approach They note that ldquoteamsof macroeconomists have constructed forecasting models involving hundreds ofsimultaneous equations fitted to data that time series analysts would view as neitherplentiful nor of especially high qualityrdquo
An alternative to the large-scale macroeconomic models that is suggested areldquotime series modelsrdquo As summarized by Granger and Newbold (1986) ldquoin theiranxiety econometricians have failed to touch some very important basesrdquo thatinclude
bull the fact that there are many areas in which ldquoeconomic theory is not terriblywell developedrdquo
bull the fact that even where the theory is satisfactory it is ldquoalmost invariablyinsufficiently precise about dynamic specifications in the sense that it is clearthat one structure must be appropriaterdquo
Empirical macroeconomics 101
bull the fact that even after appeal to economic theory there will be error termssince ldquono theory provides a completely accurate description of the behaviorof economic agents so that any postulated equation necessarily includes astochastic error termrdquo
Given these problems many macroeconomists suggest that the appropriate start-ing point to forecasting future macroeconomic variables is the use of time seriesmodels One result is that such time series modelling terms as AR ARMA andARIMA now abound in the macroeconomic literature It is thus useful to brieflyreview the nature of time series models However at the outset it should be madeclear that the discussion below is not complete but rather is provided simply asan introduction to some terms and concepts Proofs of various propositions andrigorous definitions of various properties of time series models can be found intime series texts such as Enders (2004) and Mills (1999)
Time series models
Let us take as our premise the idea that the actual observed time series of somevariable yt t = 0 1 T (eg the logarithm of economy-wide output for thelast 30 years) is the realization of some theoretical process which can be called aldquostochastic processrdquo As phrased by Harvey (1993) ldquoeach observation in a stochas-tic process is a random variable and the observations evolve in time according tocertain probabilistic laws Thus a stochastic process may be defined as a collectionof random variables which are ordered in timerdquo To forecast future values of yt one needs a model that defines the mechanisms by which the observations aregenerated
A distinguishing feature of a pure univariate time series model is that move-ments in yt are ldquoexplainedrdquo solely in terms of its own past or by its position inrelation to time13 That is time series models look for patterns in the past move-ments of a particular variable and use that information to predict future movementsof the variable In general a time series modelrsquos forecast of y based on knownvalues yT+1 is given by
yT+1 = E( yT+1|y0 yT )
As Pindyck and Rubinfeld (1991) suggest ldquoin a sense a time-series model is just asophisticated method of extrapolation yet it may often provide a very effec-tive tool for forecastingrdquo In this sense time series models are more along the linesof empirical analyses that ldquolet the data speak for themselvesrdquo rather than empiricalanalyses that (strictly speaking) ldquotest economic theoriesrdquo As Harvey (1993) statesldquoan essential feature of time series models is that they do not involve behaviouralrelationshipsrdquo Time series models reflect a ldquostatisticalrdquo approach to forecastingHowever the patterns in the data discovered by time series models do influencetheoretical discussions of the macroeconomy and tests of macroeconomic theo-ries concerning behavior have incorporated time series models14 An example of
102 Empirical macroeconomics
how time series analysis has influenced theoretical macroeconomic discussions isgiven at the end of this section
Some properties of stochastic processes
There are several key properties of stochastic processes for time series that wecan introduce One is ldquostationarityrdquo For a stochastic process to be stationary thefollowing conditions must be satisfied for all t
E( yt) = micro
E[( yt minus micro)2] = σ 2y
E[( yt minus micro)( ytminusk minus micro)] = γk
Note that γ0 = σ 2y In words if a series is stationary the mean of the series is
invariant to time the variance of the series is invariant to time and the covarianceof the series is invariant to time As Pindyck and Rubinfeld (1991) note ldquoif astochastic process is stationary the probability distribution p( yt) is the same forall time t and its shape (or at least some of its properties) can be inferred by lookingat a histogram of the observations y1 yT that make up the observed seriesrdquo15
Also an estimate of the mean micro of the process can be obtained from the samplemean of the series
y = 1
T
Tsumj=0
yt
and an estimate of the variance σ 2y can be obtained from the sample variance16
σ 2y = 1
T
Tsumj=0
( yt minus y)2
As Granger and Newbold (1986 4) phrase it ldquoa stationarity assumption is equiv-alent to saying that the generating mechanism of the process is time-invariantso that neither the form nor the parameter values of the generation procedurechange through timerdquo The simplest example of a stationary stochastic process isa sequence of uncorrelated random variables with constant mean and variance
A second property of a stochastic process is the ldquoautocorrelation functionrdquo Theautocorrelation function provides us with a measure of how much correlation thereis (and by implication how much interdependency there is) between neighboringdata points in the series yt For stationary processes the autocorrelation with lagk is given by17
ρk = γkγ0
= E[( yt minus micro)( ytminusk minus micro)]σ 2y
Empirical macroeconomics 103
For any stochastic process ρ0 = 1 If the stochastic process is simple ldquowhitenoiserdquo (ie yt = εt where εt is an independently distributed random variablewith zero mean and finite variance) then the autocorrelation function for thisprocess is given by
ρ0 = 1 ρk = 0 for k gt 018
A simple example of a stochastic process is the ldquorandom walkrdquo process inwhich each successive change in yt is drawn independently from a probabilitydistribution with zero mean19 Thus yt is determined by
yt = ytminus1 + εt (717)
where εt is a sequence of uncorrelated random variables (E(εtεs) = 0 for t = s)with mean zero (E(εt) = 0) and constant variance (E(ε2
t ) = σ 2e for all t) Recall
that a sequence εt of this kind is typically called ldquowhite noiserdquo If the stochasticprocess is a random walk the one-period-ahead forecast of yt+1 is simply yt
A simple extension of the random walk process is to incorporate a trend in theseries yt We then obtain the following stochastic process known as a random walkwith drift20
yt = ytminus1 + d + εt (718)
The one-period-ahead forecast of yt+1 is now yt + d By repeatedly substitutingfor past values of yt into (718) we obtain
yt =tminus1sumj=0
εtminusj + td + y0 (719)
For the random walk process without drift (d = 0) we see from (719) that thefirst requirement for stationarity namely that the mean be constant over time issatisfied if y0 is fixed21 That is E( yt) = E( y0) Nevertheless the process is notstationary since Var( yt) = tσ 2
e The random walk process tends to meander awayfrom its starting value but exhibits no particular trend in doing so22
Autoregressive processes
A process similar to the random walk that is stationary is called the ldquofirst-orderautoregressive processrdquo or AR(1)23
The AR(1) process is given by
yt = φytminus1 + (1 minus φ)micro + εt (720)
A necessary condition for stationarity is that |φ| lt 1 in which case E( yt) equalsthe constant term micro in (720)24
104 Empirical macroeconomics
One possible example of an autoregressive process is the total numberunemployed each month (Granger and Newbold 1986) Let the total number unem-ployed in one month be yt This number might be thought to consist of a fixedproportion φ of those unemployed in the previous month (the others having foundemployment) plus a new group of workers seeking jobs If the new additionsare considered to form a white noise series with positive mean micro(1 minus φ) thenthe unemployment series is a first-order autoregressive process expressed byequation (720)
For convenience only it is often assumed that the process has a zero mean iemicro = 0 Note that we can always define a new variable yprime
t equiv yt minus micro such that yprimet
has a zero mean and is given by the AR(1) process
yprimet = φyprime
tminus1 + εt (721)
with |φ| lt 1 Setting micro = 0 thus simply means that the variable yt (egemployment real output etc) is measured in terms of deviations from its mean
By successive substitution for yt in (720) we obtain (assuming micro = 0)
yt =tminus1sumj=0
φjεtminusj + φty0 (722)
If the process is regarded as having started at some point in the remote past and|φ| lt 1 then we can write
yt =infinsum
j=0
φjεtminusj (723)
Given that εt is white noise we thus have that E( yt) = micro = 0 Assumingstationarity (|φ| lt 1) we know that the variance and covariances are constantRecall that εt is white noise such that E(εtεs) = 0 for s = t We thus have25
γ0 = σ 2y = E[( yt minus micro)2] = E
⎡⎢⎣⎡⎣ infinsum
j=0
φj(εtminusj)
⎤⎦
2⎤⎥⎦ = σ 2
e (1 minus φ2)
γ1 = E[( yt minus micro)( yt+1 minus micro)] = φσ 2e (1 minus φ2)
γ2 = E[( yt minus micro)( yt+2 minus micro)] = φ2σ 2e (1 minus φ2) etc
The autocorrelation function for AR(1) is thus particularly simple ndash it begins atρ0 = 1 and then declines geometrically ρk = φk Note that this process hasan infinite memory The current value of the process depends on all past valuesalthough the magnitude of this dependence declines with time
In general an autoregressive process of order p is generated by a weighted aver-age of past observations going back p periods together with a random disturbance
Empirical macroeconomics 105
in the current period The AR( p) process is thus given by26
yt = φ1ytminus1 + φ2ytminus2 + φ3ytminus3 + middot middot middot + φpytminusp
+ (1 minus φ1 minus φ2 minus middot middot middot minus φp)micro + εt
= micro + εt +psum
j=1
φj( ytminusj minus micro)
(724)
with a necessary condition for stationarity being that φ1 + φ2 + middot middot middot + φp lt 127
The next section discusses necessary and sufficient conditions for stationarity
A brief digression on necessary and sufficient conditionsfor stationarity
To get some idea of what are necessary and sufficient conditions for stationaritylet us consider two specific cases AR(1) and AR(2) In the AR(1) case
yprimet minus φyprime
tminus1 = εt
where yprimet equiv yt minus micro Note that εt can be termed the ldquostochasticrdquo component of the
expression and the remainder the ldquodeterministicrdquo component28
Focusing on the deterministic component and assuming for convenience thatmicro = 0 we have a homogeneous first-order difference equation of the form29
yt minus φytminus1 = 0 (725)
To solve this equation let us try the general solution
yt = Abt (726)
which naturally implies ytminus1 = Abtminus130 The problem then is to find the valuesof A and b Substituting the trial solution into the above difference equation weobtain
Abt minus φAbtminus1 = 0
which by multiplying through by b1minustA can be rewritten as
b minus φ = 0 (727)
Equation (727) is called the ldquoauxiliaryrdquo or characteristic equation of (725) Thisequation provides us with the solution for b which is simply that b = φ Thissolution value is sometimes referred to as the ldquorootrdquo of the characteristic equation
Using (727) to substitute for b in (726) we thus have as the solution to thefirst-order difference equation (725) the expression
yt = Aφt (728)
106 Empirical macroeconomics
The ldquoequilibriumrdquo solution for yt given a homogeneous difference equation suchas (725) is yt = 0 That is if yt equiv 0 then yt will not change over time Thisequilibrium solution is ldquostablerdquo if as t rarr infin yt rarr 0 That is deviations fromthe equilibrium yt will return toward that equilibrium value Obviously given oursolution (728) a necessary and sufficient condition for stability is that |φ| lt 1
In the context of an AR(1) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the first-order homogeneous difference equation (725)that is the deterministic component of yt for an AR(1) process The condition ofstationarity is thus that |φ| lt 1 The effect of a given shock will mitigate overtime if this stationarity condition is met A special nonstationary case is the ldquounitrootrdquo case of the random walk process in which φ = 1 In that case the effect ofa given shock will not dampen with the passage of time
Now let us consider the AR(2) case where
yprimet minus φ1yprime
tminus1 minus φ2yprimetminus2 = εt
in which yprimet equiv yt minus micro As before εt can be termed the ldquostochasticrdquo component of
the expression Assuming again for convenience that micro = 0 we can express thedeterministic component as a homogeneous second-order difference equation ofthe form
yt minus φ1ytminus1 minus φ2ytminus2 = 0 (729)
To solve this equation let us try yt = Abt which naturally implies ytminus1 = Abtminus1
and ytminus2 = Abtminus2 The problem then is to find the values of A and b Substitutingthe trial solution into the above difference equation we obtain
Abt minus φ1Abtminus1 minus φ2Abtminus2 = 0
which by multiplying through by b2minustA can be rewritten as
b2 minus φ1b minus φ2 = 0 (730)
The quadratic equation (730) is called the ldquoauxiliaryrdquo or ldquocharacteristic equationrdquoof the second-order difference equation (729) The roots of this equation will bethe solutions of (730)31 The two ldquocharacteristicrdquo roots m1 and m2 may be foundin the usual way from the formula32
m1 m2 = [φ1 plusmn (φ21 + 4φ2)
12]2 (731)
each of which is acceptable in the solution Abt Note that for the quadraticequation (730) the roots m1 and m2 satisfy
(b minus m1)(b minus m2) = 0
where φ1 = m1 + m2 and φ2 = minusm1m2
Empirical macroeconomics 107
Expression (731) provides us with two values for b and thus two potentialsolutions for yt
yt = A1mt1 and yt = A2mt
2
The general solution to a second-order difference equation combines these twoby taking the sum Thus the general solution of the second-order homogeneousdifference equation (729) is33
yt = A1mt1 + A2mt
2 (732)
Inspection of (732) suggests that stability for the linear second-order homo-geneous difference equation (729) requires that both roots of the characteristicequation have an absolute value less than one This reflects the fact that the gen-eral solution of the second-order homogeneous difference equation includes bothroots m1 and m2 The root with the higher absolute value is sometimes termed theldquodominant rootrdquo If both m1 and m2 are less than unity in absolute value yt willbe close to zero if t is large Equivalently if the dominant root is less than one inabsolute value convergence will occur
In the context of an AR(2) process the notion of ldquostationarityrdquo corresponds tothe notion of stability for the second-order homogeneous difference equation (729)that is the deterministic component of the process The condition of stationarityis thus that |m1| lt 1 and |m2 lt 1| That is for the AR(2) process stationarityrequires that the roots of the characteristic equation (730) are less than one inabsolute value ndash that is that they all lie inside the unit circle34
In terms of φ1 and φ2 the conditions for stationarity may thus be defined asfollows35
φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 φ2 gt minus1
As you can see the prior necessary condition that φ1 + φ2 lt 1 is augmented withtwo other conditions
We have seen that stability for the second-order homogeneous differenceequation requires that the maximum of (|m1| |m2|) ndash that is the dominant root ndash beless than 1 To see how one obtains the three necessary conditions listed above forstabilitywe assume this is the case and determine the resulting restrictions placedon the coefficients φ1 and φ236 Recall that
m1 m2 = [φ1 plusmn (φ21 plusmn 4φ2)
12]2
Suppose φ21 + 4φ2 gt 0 so that the roots are real A maximum of (|m1| |m2|) less
than one means that
2 minus φ1 gt (φ21 + 4φ2)
12 gt minus2 minus φ1
108 Empirical macroeconomics
and
2 minus φ1 gt minus(φ21 + 4φ2)
12 gt minus2 minus φ1
The sum of the roots is φ1 and since we are assuming each root is between minus1and 1 it must be the case that 2 gt φ1 gt minus2 or 0 gt minus2minusφ1 and 2minusφ1 gt 0 Thusthe second and third inequalities are always true and hence place no restrictionson φ1 and φ2
The first and fourth inequalities squared read
(2 minus φ1)2 gt φ2
1 + 4φ2 and φ21 + 4φ2 lt (minus2 minus φ1)
2
respectively These two expressions can be rewritten to obtain
φ1 + φ2 lt 1 and minus φ1 + φ2 lt 1
which are the first two conditions for stability The third condition for stability(φ2 gt minus1) is obtained from the case of complex conjugate roots37 The threenecessary conditions for the dominant root being less than one in absolute value arealso sufficient conditions This can be verified by showing using these inequalitiesthat it is possible to reverse the steps in the above calculations and arrive at themaximum of (|m1| |m2|) being less than one
In general a necessary and sufficient condition for stationarity of an AR( p)process is if the roots of the characteristic equation
b p minus φ1b pminus1 minus middot middot middot minus φp = 0 (733)
are less than one in absolute valueNote that for the AR(2) case there are three possible situations If φ2
1 +4φ2 gt 0the square root in (731) is a real number and m1 and m2 will be real and distinctIn this case the solution to (729) is given by
yt = A1mt1 + A2mt
2
where A1 and A2 are constants which depend on the starting values y0 and yminus1The second situation is that of repeated roots where φ2
1 + 4φ2 = 0 such thatm1 = m2 = m In this case the solution to (729) takes the form
yt = A3mt + A4tmt
If |m| lt 1 the damping force of mt will dominate both termsThe third situation for the AR(2) process is when φ2
1 + 4φ2 lt 0 in which casethe roots are a pair of complex conjugates (ie m1 m2 = h plusmn vi where h = φ12v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12) Thus the solutionto (729) is given by
yt = A5(h + vi)t + A6(h minus vi)t
Empirical macroeconomics 109
Appealing to De Moivrersquos theorem this expression can be transformed intotrigonometric terms The general form of the solution is
yt = pt(A7 cos λt + A8 sin λt)
where p is the modulus of the roots = (minusφ2)12) and λ satisfies the two conditions
cos λ = hp and sin λ = vp As in the other two cases if the absolute valueof the conjugate complex roots |h plusmn vi| lt 1 is less than one the process isstable The time path followed by yt in response to a shock is cyclical but theperiodic fluctuation will mitigate as time passes (ldquosinusoidal decayrdquo) if the processis stationary
Moving average processes
In a moving average process the process yt is described completely by a weightedsum of current and lagged random variables The simplest moving average processthe moving average process of order 1 or MA(1) takes the form
yt = micro + εt + θεtminus1 (734)
The term ldquomoving averagerdquo reflects the fact that a series so characterized will besmoother than the original white noise series εt In general such a moving averageprocess of order q MA(q) is written as38
yt = micro + εt + θ1εtminus1 + θ2εtminus2 + middot middot middot + θqεtminusq
= micro + εt +qsum
j=1
θjεtminusj (735)
As before if yt has nonzero mean we can focus with no loss of generality on thetransformed series yprime
t = yt minus micro which has zero meanOne possible example of a moving average process is economy-wide output
(Granger and Newbold 1986) Output yt could be in equilibrium (at mean micro)but is potentially moved from its equilibrium position each period by a seriesof unpredictable events such as periods of exceptional weather or strikes If thesystem is such that the effects of such events are not immediately assimilated butexert an influence on output for q periods then a moving average model can arise
Note that by repeatedly substituting for lagged valued of εt into an MA(1)process (equation (734)) one obtains39
yt =infinsum
j=1
(minusθ)jytminusj + εt (736)
If yt is not to depend on a shock to the system arising at some point in the remotepast θ must be less than one in absolute value Comparing (736) to (724) we
110 Empirical macroeconomics
see that assuming what was previously denoted the stationarity condition but whatin this context is called the ldquoinvertibility conditionrdquo namely that |θ | lt 1 then anMA(1) process can be represented by an AR(infin) process40 The weights on pastvalues of yt for this AR(infin) process decline exponentially
In general if similar invertibility conditions are met then any finite-ordermoving average process has an equivalent autoregressive process of infinite orderLikewise we have already shown (see (723)) that if the AR(1) process is station-ary it is equivalent to a moving average process of infinite order MA(infin) In factfor any stationary autoregressive process of any order there exists an equivalentmoving average process of infinite order so that the autoregressive process isldquoinvertiblerdquo into a moving average process
Mixed autoregressive moving average processes
An obvious generalization of the above discussion is to combine an AR( p) process(ie (724)) and an MA(q) process (ie (735)) An autoregressive moving averageprocess of order ( p q) or ARMA( p q) can be written as41
yt = micro + εt +psum
j=1
φj( ytminusj minus micro) +qsum
j=1
θjεtminusj (737)
where as before εt is a zero-mean white noise For simplicity consider the casewhere micro = 0
Whether or not a mixed process is stationary depends solely on its autoregres-sive part If the ARMA process is stationary then there is an equivalent MA(qprime)process Similarly provided invertibility conditions hold there is an AR( pprime) pro-cess equivalent to the above ARMA( p q) process It thus follows that a stationaryARMA process can always be well approximated by a high-order MA process andthat if the process obeys the invertibility condition it can also be well approxi-mated by a high-order AR process However an ARMA process has the advantageof ldquoparsimonyrdquo in that the mixed model ARMA( p q) often can achieve as gooda fit as say an AR( pprime) but uses fewer parameters (ie p + q lt pprime)
As an example of an ARMA( p q) process we need only consider a case wherethe variable yt is the sum of a ldquotrue seriesrdquo that is AR( p) plus a white noiseobservation error Thus an ARMA( p q) series results42
Integrated processes
In series arising in economics the assumption of stationarity is often veryrestrictive That is often the characteristics of the underlying stochastic processgenerating a time series appear to change over time With economic data some-times a transformation (such as taking the logarithm of the variable) can result in astationary series In other cases we can obtain a stationary series by differencingone or more times We say that yt is a ldquohomogeneous nonstationary process of
Empirical macroeconomics 111
order drdquo if
wt = dyt
is a stationary series Here denotes differencing that is
yt = yt minus ytminus1 2yt = yt minus ytminus1 = yt minus 2ytminus1 + ytminus2
The process yt is an ldquointegratedrdquo process if after a series yt has been differencedto produce a stationary series wt the series wt can be modelled as an ARMAprocess If wt = dyt and wt is an ARMA( p q) process then we say that yt is anldquointegrated autoregressive moving average process of order ( p d q)rdquo or simplyARIMA ( p d q)
An application of time series models
To gain a better understanding of how time series models are used in macroeco-nomic empirical work let us consider the time series for the logarithm of the levelof real output to be denoted by yt As we stated above theoretical macroeconomicmodels typically postulate shocks or ldquoimpulsesrdquo to the economy (eg shocks totechnology money supply and government policy) that in conjunction with aprediction of how various markets in the economy adjust to these shocks offersan explanation of the actual fluctuations in aggregate variables One importantquestion that can arise is whether such shocks are ldquotransitoryrdquo in that their effectsdo not persist or ldquopermanentrdquo where the economy moves to a new level of realoutput
The ldquotraditionalrdquo answer to the above question of transitory versus permanenteffects of shocks has been according to Campbell and Mankiw (1987a 111) thatfluctuations in real output ldquoprimarily reflect temporary deviations of productionfrom trendrdquo That is the traditional view has been that quarterly real output couldbe represented by
yt = Tt + St + Xt (738)
where Tt is a deterministic component representing trend St is a determinis-tic seasonal component and Xt is a stationary autoregressive process with nodeterministic component43
Trend and seasonal components were typically estimated in various fashionsand then eliminated The focus of empirical work had then been on examiningvariations in the detrended seasonally adjusted real output series For exampleBlanchard (1981) estimated the following second-order autoregressive process fordeviations of the log of seasonally adjusted real output from its estimated trendylowast
t and obtained the equation
ylowastt = 134ylowast
tminus1 minus 042ylowasttminus2 + εt (739)
112 Empirical macroeconomics
Note that the necessary conditions for stationarity with respect to the series ylowastt
(namely φ1 + φ2 lt 1 minusφ1 + φ2 lt 1 and φ2 gt minus1) are metThe traditional view is embodied in (739) A shock to output increases for
a few quarters but the effect ultimately dies out In fact only 8 percent of ashock remains after 20 quarters Assuming an ARMA(22) process Blanchardand Fischer (1989 9) obtain a similar result
ylowastt = 131ylowast
tminus1 minus 042ylowasttminus2 + εt minus 006εtminus1 + 025εtminus2 (740)
where by construction εt is that part of the deviation of current output from trendthat cannot be predicted from past output As Blanchard and Fischer note
A shock has an effect on GNP that increases initially and then decreases overtime After 10 quarters the effect is still 40 of the initial impact after 20quarters all but 3 of the effect has disappeared The view that reversiblecyclical fluctuations account for most of the short-term movements of realGNP and unemployment has been dominant for most of the last century
However as Granger and Newbold (1986 37) note ldquothe more modern view is thatas far as possible the trend seasonal and lsquoirregularrsquo components should be han-dled simultaneously in a single model aimed at depicting as faithfully as possiblethe behavior of a given time seriesrdquo Seasonality can be treated through a gener-alization of ARMA models44 More importantly for the question at hand trend isgenerally treated by differencing leading to the consideration of the ldquointegratedprocessesrdquo suggested above In fact Campbell and Mankiw (1987b) indicate thatthe answer to the question of the importance of temporary versus permanent shocksto output is biased if detrended data are used45 Campbell and Mankiw go on topoint out that examining the differences in the logarithm of real output does notprejudge the issue of whether shocks to the economy are transitory or permanentFor instance suppose that yt follows an IMA(11) process so that
yt minus ytminus1 = d + εt minus θεtminus1
Then a unit impulse in yt changes the forecast of yt+n by 1 minus θ regardless of nldquoHence depending on the value of θ news about current GNP could have a largeor small effect on onersquos forecast of GNP in ten yearsrdquo (Campbell and Mankiw1987b 860)46
Campbell and Mankiw consider ARMA( p q) processes for the difference in thelog of real output for p = 0 1 2 3 and for q = 0 1 2 347 Interestingly they finda high level of persistence in shocks One intriguing suggestion of Campbell andMankiw (1987b 868) is that ldquowhen we examine postwar annual data we cannotreject the hypothesis that the log of real GNP is a random walk with drift In thiscase the impulse response is unity at all horizonsrdquo Recall that the random walkprocess with drift (718) is an example of a nonstationary process that is first-order
Empirical macroeconomics 113
homogeneous nonstationary To see this simply consider the series wt that resultsfrom differencing the random walk ndash that is the series
wt = yt minus ytminus1 = d + εt (741)
Since εt are assumed independent over time wt is clearly a stationary process Theterm d + εt is a white noise process In this example yt is an ARIMA(010) andyt is an ARMA(00) The impulse response to a shock is unity in such a case
As we will see later one way to interpret the above finding of persistence isto place weight on ldquoaggregate supplyrdquo shocks such as technological disturbancesrather than on ldquoaggregate demandrdquo shocks such as changes in the money supplyin explaining fluctuations in real output That is the finding gives credence tothe view of Nelson and Plosser (1982) among others that real (ie permanent)shocks dominate as a source of output fluctuations48
However as pointed out by Campbell and Mankiw (1987b 877) the Nelsonand Plosser conclusion is an extreme
one can attribute a major role to supply shocks without completely abandoninga role for demand shocks For example suppose that output Y (= log y) isthe sum of two components a supply-driven ldquotrendrdquo Y T and demand-driveldquocyclerdquo Y c that are uncorrelated at all leads and lags Suppose further thatY T is a first-order autoregressive process with parameter φ and that Y c issome stationary process If trend output is approximately a random walk sothat φ is small then the finding of great persistence implies that fluctuationsin the cycle are small relative to fluctuations in the trend If the change inthe trend is highly serially correlated (φ is large) however the finding ofpersistence is consistent with a substantial cyclical component
Campbell and Mankiw (1987b 877) go on to suggest that
a second way to interpret the finding of persistence is to abandon the
natural rate hypothesis Models of multiple equilibria might explain a long-lasting effect of aggregate demand shocks if shocks to aggregate demand canmove the economy between equilibria Shocks to aggregate demand couldhave permanent effects if technological innovation is affected by the businesscycle
Note that so far we have considered only a single or univariate time series Theconcepts involved however can be extended to multivariate series For instancea simple vector autoregression model (VAR) could take the form
yt = ytminus1 + εt
where now yt is considered an N times 1 vector reflecting N variables the randomdisturbance term εt is also an N times 1 vector and is an N times N matrix of parame-ters The disturbances are uncorrelated over time but may be contemporaneously
114 Empirical macroeconomics
correlated As in the univariate case such a model is usually only fitted to vari-ables which are stationary (possibly obtained by logarithmic transformations orby taking first or second differences) As in the univariate case the objectivewould be to find a model that transforms a vector of time series into a whitenoise vector As Harvey (1993) notes ldquoalthough a model of (this) form is oftenused for forecasting in econometrics economic theory will typically place a priorirestrictions on elements of rdquo There are also questions concerning the correlationbetween innovations across different series of data for example the link betweeninnovations in money supply and output But given that our aim is to review basicmacroeconomic theory we will not pursue such topics
Conclusion
This chapter has emphasized the empirical nature of macroeconomics Inparticular time series analysis and econometric methods have been shown tobe useful tools for analyzing macroeconomic data Many of the policy conclu-sions that will be developed in the remainder of the book can be tested using thesemethods The results obtained from a thorough understanding of the time seriesproperties of important macroeconomic variables such as interest rates employ-ment numbers real output measures and the like together with the underlyingtheory may be used to provide policy recommendations Moreover more andmore businesses are relying on empirical macroeconomics when making operatingdecisions
Appendix translation of higher-order difference equationsinto lower order
Any higher-order difference equation can be interpreted in terms of an equivalentsystem of first-order difference equations To see this consider equation (729)
yt minus φ1ytminus1 minus φ2ytminus2 = 0
To facilitate matters we start by shifting the origin of this second-order homo-geneous difference equation from t = minus2 to t = minus1 such that we may rewrite(729) as
yt+1 minus φ1yt minus φ2ytminus1 = 0 (7A1)
Now let us introduce the new variable xt defined as
xt = ytminus1
which means that xt+1 = yt We may then express the second-order differenceequation (7A1) by means of two first-order simultaneous equations
yt+1 minus φ1yt minus φ2xt = 0
xt+1 minus yt = 0(7A2)
Empirical macroeconomics 115
In matrix notation (7A2) becomes[yt+1xt+1
]= A
[ytxt
]
where A is the square nonsingular matrix defined as
A =[φ1 φ21 0
]
The ldquocharacteristic equation of a square matrixrdquo is defined by the determinant
|A minus λI | = 0
where I is the unit matrix[
1 00 1
]and λ is a scalar variable
Letting b = λ the ldquocharacteristic equation of matrix Ardquo
|A minus λI | =∣∣∣∣φ1 minus b φ2
1 minusb
∣∣∣∣ = b2 minus φ1b minus φ2 = 0
is identical to equation (730) which was termed the ldquocharacteristic equation of thesecond-order homogeneous difference equationrdquo The values for the λs (or bs) arethe roots of the characteristic equation One reason for restating the higher-orderdifference equation in matrix form is that necessary and sufficient conditions forstability can be expressed in terms of conditions on the matrix A
8 The neoclassical model
Introduction
This chapter turns our attention to one of the most popular and sometimescontroversial models used by macroeconomists the neoclassical model Themodel in its purest form is often used as a benchmark or starting point for addingldquorealismrdquo to the structure of the economy However one views the neoclassicalmodel there is no denying that it is a powerful tool for generating predictionsabout movements in economic aggregates As such this chapter works throughthe comparative static exercise of a change in the money supply This exerciseprovides a great deal of insight into how the monetary authority might influencethe economy or whether it can influence the economy at all A number of importantissues are raised for example money neutrality and money illusion Additionallythe chapter formally derives the aggregate supply curve in the neoclassical contextThe model has serious implications for the existence of a natural rate and also forthe formation of expectations
Comparative statics for the neoclassical model
We can use our previous method of analysis known as ldquocomparative staticrdquoanalysis to examine the effects of various shocks on the equilibrium level ofprices As the name suggests comparative static analysis is concerned with thecomparison of different equilibrium states associated with different sets of valuesof parameters and exogenous variables In the current context of the neoclassicalmodel the labor market can be isolated from the other markets (ie there is aldquoblock recursiverdquo character to the equilibrium solution) In such a context we candistinguish two types of exogenous variables
One type ldquosupply-siderdquo variables alter the level of output at any given pricelevel Such variables could include changes in technology and the existing capitalstock in the current version of the model or we could expand the analysis toinclude changes in the supply of other inputs (such as oil) changes in governmentpolicies that affect incentives to supply labor and changes in government policiesthat affect the incentives to invest and thus affect the productive capacity of theeconomy over time
Neoclassical model 117
The second type of exogenous variables ldquodemand-siderdquo variables do not alterthe current supply of output at prevailing prices but instead impact equilibriumprices and interest rates as determined in the output financial and money marketsBelow we consider the impact of demand-side ldquoshocksrdquo such as changes in initialmoney balances and the expected inflation rate
The macroeconomic approach to general equilibriumhow it can obscure
The neoclassical model can be viewed as a special case of a Walrasian generalequilibrium system distinguished by the existence of money1 As Clower (1965)notes
income magnitudes do not appear as independent variables in demand andsupply functions of the (Walrasian) general equilibrium model for incomesare defined in terms of quantities as well as prices and quantity variablesnever appear explicitly in the market excess demand functions of traditionaltheory To be sure income variables could be introduced by taking factorsupplies as given parameters but this would preclude the formulation of ageneral equilibrium model containing supply functions of all marketable factorservices
Referring to Chapter 9 of Patinkin (1965) Clower goes on to note that this point
was apparently overlooked by Patinkin when he formulated his ldquogeneral the-oryrdquo of macroeconomics It is instructive to notice that this chapter is notsupplemented by a mathematical appendix I do not mean to suggest thatauthors may not put such variables as they please into their models My pointis that such variables that can be shown to be fundamentally dependent onothers should not then be manipulated independently
What Clower is referring to is the fact that the neoclassical model with limitedperfect foresight effectively determines at time t the money wage for the labormarket and the (futures) price of output and interest rate Given perfect foresightindividual optimizing behavior will generate planned consumption and moneydemand functions at time t for period t that include the real wage rather thanincome That is since labor supply is a choice variable at time t labor income (theproduct of the real wage times labor supply) should not appear as an argument inhouseholdsrsquo demand and supply functions (Note that labor income along withdividend and interest payments equals output minus depreciation)
Formally we can discover how a ldquosmallrdquo change in one (or more) of the exoge-nous variables affects equilibrium values by totally differentiating the equilibriumconditions with respect to the prices to be determined and the exogenous variableand then solving for the implied change in equilibrium prices required to maintainequilibrium2 The resulting changes in equilibrium prices then imply changes in
118 Neoclassical model
the equilibrium levels of employment output consumption investment and realmoney holdings
However we choose instead the standard macroeconomic approach of arbitrar-ily separating the analysis into an analysis of the labor market and the resultingemployment and output (the ldquoaggregate supply equationrdquo) and then an analysisof the other markets and the detemination of the price level and the interest rate(the ldquoISrdquo and ldquoLMrdquo equations) While Clower is right in that this obscures thetraditional ldquogeneral equilibriumrdquo nature of the neoclassical model the approach isuseful in that it allows us to more easily extend the analysis to situations in whichprices are not market-clearing or to situations in which there is not limited perfectforesight
Given the assumption of perfect foresight on the part of both firms and workersconcerning the price level pt we have seen that the aggregate supply equation isindependent of the price level or the interest rate We know from the modifiedWalrasrsquo law that any two of the three excess demand conditions with respect tomoney financial assets and output can be used in the comparative static analysisUsing the equilibrium conditions with respect to output and money (the ldquoISrdquo andldquoLMrdquo equations) we thus have the three equations
yt = yt(K ) (81)
cdt (rt πe
t At Mpt yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (82)
Ldt (rt πe
t At Mpt yt) minus Mpt = 0 (83)
to determine equilibrium output yt price level pt and interest rate rt Recallthat from the modified Walrasrsquo law we know that there is a fourth equation theequilibrium condition for the financial market that is implied by (82) and (83)This fourth equilibrium condition is
net Adt (rt πe
t At Mpt yt) minus net Ast (m
et + δ K) = 0 (84)
A change in the money supply the comparative statics
It is clear from the above that the neoclassical aggregate supply equation (81)determines output while the IS and LM equations (82) and (83) determine theprice level and interest rate given the equilibrium level of output Focusing on thelatter two equations and the determination of the price level and interest rate fora given level of output total differentiation with respect to the equilibrium prices( pt and rt) and the money supply change gives the following system of linearequations in matrix form3
[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpartrpartLdpartr
] [dpdr
]=[ minus(partcdpart(Mp))dMp(1 minus partLdpart(Mp))dMp
]
Neoclassical model 119
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm4 Solving the above linear-equationsystem for dp and dr using Cramerrsquos rule we obtain
dp
dM= minus(partcdpart(Mp))(1p)partLdpartr minus (1 minus partLdpart(Mp))(1p)partydpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpartr
= p
M
dr
dM=
minus[(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)minus(partcdpart(Mp))(Mp2)(partLdpart(Mp)minus1)](1p)
minus(partcdpart(Mp))(Mp2)partLdpartrminus(1minuspartLdpart(Mp))(Mp2)partydpartr
= 0
As you can see dpp = dMM and drdM = 0 That is a change in the moneysupply leads to the same proportional change in the price of the consumption goodand no change in the interest rate With regard to the latter result the interest ratedoes not change as the only thing that changes the interest rate is the rate of changein prices Since the price level adjusts there is no change in the interest rate and theLM curve does not change Note that from the labor market equilibrium conditionthat underlies the aggregate supply equation we know that the change in the pricelevel results in an equiproportionate change in the money wage The result is thatwe have ldquoneutrality of moneyrdquo In other words if individuals correctly anticipatethe effect of a change in the money supply on the price level as is the case inthe deterministic model under the assumption of limited perfect foresight thenmonetary changes have no ldquorealrdquo effects Real output the real wage the expectedreal rate of interest real consumption real investment and the real money supplyare all unaffected by the monetary change
The basic reason why changes in the money supply are ldquoneutralrdquo is the absenceof money illusion on the part of both firms and households In particular house-hold demand and supply functions indicate that if a change in money balancesis accompanied by an equiproportionate change in the price of output then thereis no change in any demands That is household demand and supply functionsare homogeneous of degree zero in money balances and prices This absence ofldquomoney illusionrdquo occurs assuming
bull perfect foresight at time t for price during period t so that a change in priceresults in no change in the real wage or employment (when coupled with thesimilar assumption of limited perfect foresight on the part of firms)
bull ldquoneutral distribution effectsrdquo such that the shift in wealth from prior creditorsto debtors that would accompany a rise in the price level leaves aggregatedemands unchanged
bull unit elastic expectations so that changes in the current price level can beviewed as leaving unaffected the expected rates of change in the pricelevel
120 Neoclassical model
As we will see later this money neutrality result of the neoclassical model formsthe basis of what has become known as the ldquopolicy ineffectiveness propositionrdquo(see McCallum 1979)
The ldquodynamicsrdquo of the system and the neutrality of money a review
As we discussed earlier the ldquostoryrdquo often told with respect to the dynamics of theabove situation is a ldquoloanable funds theoryrdquo of interest rate determination in whichthe interest rate moves to clear the financial market5 In this case the ldquotatonnementprocessrdquo or movement toward equilibrium involves
dpt = f ( ydt minus yt) f (0) = 0 df d( yd
t minus yt) gt 0
dpbt = fb(net Adt minus As
t ) fb(0) = 0 dfbd(net Adt minus net As
t ) gt 0
Note that we put ldquostoryrdquo in quotes since the analysis itself simply identifies variousequilibrium points with adjustments in prices to reach a different equilibrium pointgiven a shock essentially occurring without any passage of time Neverthelessconsider the following story with respect to a rise in the money supply
In the Patinkin analysis described earlier if the money supply were to doublefrom M to 2M the LL BB and CC curves would shift to the right so that the newequilibrium would occur at the original interest rate but at a price level double theoriginal one Such a once-and-for-all change in the initial money balances leavesthe money interest rate and real demands unchanged (the neutrality of money)
Superneutrality an informal review
Superneutrality of money occurs when changes in the rate of growth of the moneysupply leave the paths of capital and real output unaffected Although the aboveanalysis is static in nature it does provide some insight into the issue of thesuperneutrality of money To see how suppose each period the economy can bereplicated in every way That is the money supply equilibrium price level andinterest rate are identical each period If expectations of price changes are correctexpected inflation would equal zero For the economy to replicate itself it mustalso be the case that the initial capital stock is optimal such that the capital stockand thus output does not change over time With zero adjustment costs such asituation would imply a marginal product of capital equal to the expected real usercost of capital (me
t + δ where met equiv (rt minus πe
t )(1 + πet )) net investment demand
equal to zero each period and gross investment demand equal to δK Now let us compare this situation to an alternative sequence of temporary equi-
librium in which the economy is identical in every way except one the moneysupply is increased each period by a constant percentage from its level in the priorperiod Other things being equal the above analysis would suggest that one dif-ference across periods would be a rise in prices due to the positive growth in themoney supply Let us further assume that expectations of inflation adjust to reflect
Neoclassical model 121
what Sargent and Wallace (1975) would characterize as a new ldquosystematic moneysupply rulerdquo
Without fully developing the appropriate dynamic analysis it is easy to see thatone obvious adjustment of the static analysis given a growing money supply (asopposed to one that does not change across periods) is thus a higher expectedinflation rate (positive as opposed to zero) In fact the analysis of the static modelcan mimic to some extent the effects of a higher rate of growth in the money supplyby considering the impact of an increase in exogenous inflationary expectations
Following Sargent and Wallace (1975) among others assume that consump-tion demand depends on the expected real rate of interest r minus π not separatelyon its components (the money interest rate and the expected rate of inflation)6
Further assume that changes in expected inflation do not affect real moneydemand7 The comparative static results are then[ minus(partcdpart(MP))Mp2
(1 minus partLdpart(Mp))Mp2partydpart(r minus π)
partLdpartr
] [dpdr
]=[minus(partydpart(r minus π))dπ
0dπ
]
where partydpart(r minusπ) = partcdpart(r minusπ)+ (1 +ψ prime)partI dpart(r minusπ) Solving the abovelinear equation system for dp and dr using Cramerrsquos rule we obtain
dp
dπ=
minus(partydpart(r minus π))partLdpartr
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
dr
dπ=
minus(partLdpart(Mp))(Mp2)partydpart(r minus π)
minus(partcdpart(Mp))(Mp2)partLdpartr minus (1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)
gt 0
As we have discussed before if there is no real balance effect with respect to con-sumption demand (partcdpart(Mp) = 0) or if real money demand does not respondto changes in the interest rate (partLdpartr = 0) then we can see from the above that
dr = minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)dπ
minus(1 minus partLdpart(Mp))(Mp2)partydpart(r minus π)= dπ
so that the change in the expected rate of inflation results in no change in theexpected real rate of interest (r minus π ) In this case money is superneutral in thata change in the growth of the money supply (although it alters inflation and thusgiven perfect foresight expected inflation) leaves the expected real rate of interestunchanged and thus does not affect investment and the future size of the capitalstock which depend inversely on the expected real interest rate
122 Neoclassical model
Recall that Sargent and Wallace (1975) have argued for superneutrality ofmoney In contrast Begg (1980) has noted that steady-state analyses of growthmodels with money like the analysis above do find that different rates of growthin the money supply have real effects (through changes in the expected real grossinterest rate)8 As the above analysis makes clear there are two conditions eitherof which is sufficient that will result in money being superneutral Begg (1980293) describes them as follows ldquoThe first condition is that the level of real moneybalances is not an argument in the consumption function and it is this conditionwhich distinguishes the rational expectations model of Sargent and Wallace fromthe analysis of growth models with money The second condition is that the demandfor money is independent of the nominal interest raterdquo (as well as π ) Otherwisea higher expected inflation will result in a fall in the expected real rate of interestand a higher price level
In the Patinkin framework with consumption demand and investment demanddepending on the expected real interest rate at a given price level and higherexpected inflation the CC curve must shift up vertically by the amount of theincrease in expected inflation so as to maintain the same expected real rate ofinterest But the LL curve does not shift with a change in expected inflationThus given a downward-sloping CC curve and an upward-sloping LL curve thenew equilibrium money interest rate does not rise by the extent of the increase inexpected inflation Superneutrality of money seems not to hold
In such a case the static analysis thus predicts a lower real stock of money eachperiod higher investment and a greater capital stock next period This suggests anew steady state in which the capital stock is greater In fact a complete dynamicanalysis confirms these predictions a higher steady-state capital stock is associatedwith an increase in the rate of growth of the money supply and consequent increasedexpected inflation
The role of the key assumptions of the neoclassical model
At this point it might be useful to review the role played by the two key assumptionsof the neoclassical model namely price flexibility and complete information onprices In most cases and this is no exception one can gain an understanding ofthe role of a particular assumption by exploring how the analysis would proceedif the assumption were not made Consider first the implications of dropping theneoclassical modelrsquos assumption that prices are perfectly flexible
Suppose that the demand for output decreases and firms cannot sell all theydesire at prevailing prices With flexible prices output prices will fall and thefalling price level restores output demand to its previous level However if outputprices are inflexible and do not adjust downward in response to a reduced demandfor output firms will respond by reducing production Reduced production willlead to a lower level of labor demand and thus a fall in employment Further labordemand will no longer depend upon the real wage but will be determined by whatcan be sold in the output market In short if output prices are inflexible then theoverall level of demand for goods and services is paramount in determining thelevel of employment
Neoclassical model 123
A second modification of the neoclassical model is to assume that output pricesare flexible but money wages are not If wages are ldquostickyrdquo relative to outputprices then changes in the price level will alter the real wage For instance inthe late 1970s high inflation rates led workers and employers in certain industriesto bargain for long-term contracts with a high rate of growth of the money wageThe high expected inflation did not materialize in the early 1980s The lower rateof inflation with no change in the rate of increase in wages meant a rise in thereal wage The resulting fall in the demand for labor led to lower employmentand output Given that wages are not perfectly flexible (eg ldquomultiperiod laborcontracts without complete indexationrdquo) output prices below that anticipated whenlabor contracts are signed lead to a fall in output and employment With inflexiblemoney wages the aggregate supply equation includes the price level of output asa determinant of output supply
Let us consider one more modification of the neoclassical model Suppose thatworkers have incomplete information on output prices and the real wage A firmdetermines its relevant real wage by dividing the money wage it pays its workersby the price it anticipates for the particular product its workers produce On theother hand workers must anticipate prices for a variety of different goods to bepurchased in order to determine their relevant real wage Thus firms may moreaccurately anticipate changes in prices and thus real wages than workers
Now suppose that there is an increase in output prices Given our currentassumption of incomplete information this will not only lead to an increase infirmsrsquo demand for labor and to higher wages as firms anticipate the fall in realwages but may also lead to an increase in the quantity of labor supplied for thefollowing reason Workers who have not anticipated the rise in output priceswill perceive the higher money wages as implying a rise in the real wage andwill increase their supply of labor accordingly As a result equilibrium employ-ment will rise with an increase in output prices Once again the aggregate supplyequation will incorporate the current price level as a potential determinant
The illusion model one modification of the neoclassical model
Our first departure from the neoclassical model is to introduce the potential forldquoimperfectrdquo foresight on the part of suppliers at time t concerning the price levelfor period t As a consequence the ldquonotionalrdquo or planned demands made at time tbased on anticipated prices for the period can differ from ldquoeffectiverdquo or realizeddemands andor supplies at the actual prevailing prices Further realized or effec-tive demands now depend on the quantity constraints experienced in other marketsas well as prices That is realized output now becomes a determinant of actualconsumption and money demand on the part of households
Real wage illusion the labor market and aggregate supply
As we have seen equilibrium employment is determined at the start of each periodin the labor market To formally show this let us start with the following statement
124 Neoclassical model
of equilibrium in the labor market in terms of a money wage wt and level ofemployment Nt such that
N dt (wtpt K) minus Nt = 0
N st (wtpe
t ) minus Nt = 0
A critical aspect of the above is the fact that suppliers ndash in particular suppliers oflabor ndash may not correctly anticipate the price level that will exist with respect tooutput In particular we let pe
t denote suppliersrsquo expectation formed at the start ofperiod t of the price level for period t We assume this expectation is held withsubjective certainty so that we may express the real wage expected by supplierssimply by wtpe
t It is this anticipated real wage not the realized real wage wtpt that appears in the labor supply function
Firms on the other hand are presumed to correctly forecast the actual realwage9 Note that we retain the presumption that the money wage adjusts to clearthe labor market Similarly we implicitly have assumed that the price level adjuststo clear the output market such that firms are price-takers in the output marketand thus labor demand depends on the real wage rather than on a sales constraint
Initially let us assume that householdsrsquo anticipated level of prices is correctThat is we start with a money wage and level of employment consistent with theneoclassical model But we now assume that any change in the price level willnot be fully anticipated In particular we assume that
pet = g( pt) 1 gt gprime ge 0
The fact that gprime lt 1 implies imperfect foresight at time t on the part of householdsconcerning the price of output for period t If gprime gt 0 it indicates that householdsto some extent but not completely (gprime lt 1) anticipate changes in the equilibriumlevel of prices that will prevail for period t
Totally differentiating the above two equations representing equilibrium in thelabor market with respect to wt Nt pt and noting our prior assumption that pe
t = ptinitially one obtains[minus(partN d
t part(wtpt))pt(partN s
t part(wtpet ))pt
minus1minus1
] [dwtdNt
]=[
(partN dt part(wtpt))wtdpt( pt)
2
(partN st part(wtpe
t ))wtgprimedpt( pt)2
]
Applying Cramerrsquos rule gives
dNt
dpt= (partN d
t part(wtpt))(partN st part(wtpe
t ))(wt( pt)2)(gprime minus 1)
minus(partN dt part(wtpt))pt + (partN s
t part(wtpet ))pe
t
gt 0
dwt
dpt= minuspartN d
t part(wtpt) + (partN st part(wtpe
t ))gprime
minuspartN dt part(wtpt) + partN s
t part(wtpet )
wt
ptgt 0
Note that if gprime = 1 then we have the standard neoclassical result that dNtdpt = 0and dwtwt = dptpt so that a change in the price level results in no change in
Neoclassical model 125
either employment or the real wage However with gprime lt 1 we have that anincrease in the price level leads to a rise in employment and an increase in themoney wage less than proportional to the increase in the price level such thatthe real wage falls (ie dptpt gt dwtwt gt 0) Given the aggregate produc-tion function yt = f (Nt K) we thus have an ldquoaggregate supply functionrdquo ofthe form
yt = yt( pt minus pet K ) (81prime)
so that aggregate supply depends directly on the difference between the price levelfor period t and the price level anticipated by suppliers at time t
Consider the above findings with respect to the labor market and suppose thatthere is an increase in pt Given perfect foresight on the part of firms at time t thelabor demand curve shifts up vertically so that at the higher money wage associatedwith the same real wage demand would be the same However given 1 gt gprime ge 0the vertical shift upward in the supply curve is less than this with the result thatequilibrium employment rises as the money wage rises by proportionately lessthan the rise in prices
In undergraduate textbooks the fact that changes in the price of output can nowaffect real output is shown in ( pt yt) by an upward-sloping ldquoaggregate supplycurverdquo as in Figure 81 Recall that in the neoclassical model the aggregate supplycurve is vertical In either model such a curve summarizes the underlying eventsin the labor market
The above character of the money illusion model is sometimes said to reflect theldquonatural rate hypothesisrdquo The natural rate hypothesis posits that fully anticipatedincreases in prices have no effect on the rate of real economic activity ndash specificallyreal output employment and thus unemployment Thus we will refer to the abovemodel as a static version of a ldquonatural rate modelrdquo
p
y
Figure 81 Upward-sloping aggregate supply
126 Neoclassical model
Equilibrium aggregate supply and demand
An important feature of macroeconomic theories is that to a large extent they aredistinguished by their different treatment of labor markets What this means isthat the aggregate demand side is typical of macroeconomic models Recall thatthe aggregate demand side of macroeconomic models considers the equilibriumconditions of two of the remaining three markets in particular the output market(reflected by an ldquoISrdquo equation) and the money market (reflected by an ldquoLMrdquo orldquoportfoliordquo equation) Thus the equilibrium output price level and interest rate aregiven by equations (81prime) (82) and (83) Equation (81prime) is the aggregate supplyequation of a natural rate model (82) is the ldquoISrdquo equation depicting equilibriumbetween output demand and production and (83) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market
At this point we will simplify equations (82) and (83) by removing the realbalance effect representing them as follows
cdt (rt πe
t At yt) + I dnt(m
et + δ K) + δK + ψ(I d
t ) minus yt = 0 (82prime)
Ldt (rt πe
t At yt) minus Mpt = 0 (83prime)
In our model this implies that
partnet Adt part(Mpt) = 1
As we will see one justification for this form is if real money balances are notpart of household wealth which can be the case when we introduce depositoryinstitutions into the analysis In the meantime the above assumption makes theanalysis not only simpler but also more in line with traditional macroeconomicanalysis
Graphically the equilibrium price level and output can be shown using theaggregate demand and supply curves In Figure 82 the equilibrium output andprice level are thus given by plowast
t and ylowastt Looking at what underlies these curves
we can then infer the changes in money wages and employment (specifically froman analysis of the labor market that underlies the aggregate supply curve) as wellas changes in the interest rate and the components of investment and consumptioncomponents of output demand (specifically from an analysis of the output andmoney markets that underlie the aggregate demand curve)
Money supply change comparative statics for a naturalrate model
Collecting the aggregate supply IS and LM equations for the natural rate modelunder consideration we have (81prime) (82prime) and (83prime)
These three equations determine the equilibrium output the price level andthe interest rate Substituting (81prime) into (82prime) and (83prime) in order to focus on the
Neoclassical model 127
p
y
p
y
Aggregate supply
Aggregate demand
Figure 82 Macroeconomic equilibrium with upward-sloping supply
determination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form10
[ minus(partcdparty minus 1)partyp(partLdparty)partypartp + Mp2
partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + c1 + ψ prime)partI dpartm11 The term partypartp gt 0 reflectsthe direct effect of the price level on output as implied by the aggregate supplyequation (81prime)12 It is important to note that the equilibrium condition with respectto the labor market is incorporated into the above analysis in the form of thisaggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp)minus 1)(1p)
(partcdparty minus 1)partypartp(partLdpartr)minus((partLdparty)(partypartp)+ Mp2)(partydpartr)
lt 0
As you can see we no longer have drdM = 0 Further letting x denote thedenominator for the expression for dp we have
dp
p= dM
M
1
x
128 Neoclassical model
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater13
Let us now consider the effect of the money supply shock on other variablesFrom our analysis of the labor market we know that
dp
pgt
dw
wgt 0
so that while the money wage rises the real wage falls We also know from thelabor market that the rise in the price level leads to higher employment and thusan increase in output From the demand functions we can derive the effects of thechange in the money supply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Graphically one can show the effect of the money supply shock in terms ofthe aggregate demand and aggregate supply curves Note that this exercise simplyinvolves a shift out of the aggregate demand curve
The natural rate hypothesis and expectation formation a preview
An important feature of the above analysis one already noted is that real activity ndashin particular employment output and unemployment ndash changes only to the extentthat price changes are not fully anticipated As we have just seen this ldquonatural ratehypothesisrdquo introduces the logical foundations for a monetary change to have realeffects
However note that monetary changes have real effects only to the extent that theresulting changes in the price level are not fully anticipated To understand whenthis might occur we first have to indicate why individuals may err in their formationof expectations concerning the price level The Lucas model suggests one way ofexplaining errors in forecasts such that suppliers only partially anticipate a changein the price level (ie 1 gt gprime ge 0) This model introduces the assumption ofrational expectations
Combining rational expectations with the natural rate hypothesis results in a verypowerful statement concerning monetary policy which has so far not been madeclear If the actions of monetary authorities are predictable under the presumptionof rational expectations individuals will correctly predict the consequences onprices The result in the context of a natural rate model is that such predictable
Neoclassical model 129
monetary changes will have no real effects In the deterministic world we are backto the neoclassical model The analysis above then refers only to an ldquounexpectedrdquoincrease in the money supply or to monetary ldquosurprisesrdquo Only such random shocksto the money supply will have real effects
Conclusion
A formal neoclassical model of the macroeconomy has been introduced and fullydeveloped The issues of money neutrality and money illusion have been discussedand it has been seen that money supply changes have no effect on real economicactivity when the assumptions of the neoclassical model hold However a numberof issues have been raised namely the existence of a natural rate and the potentialeffect of unanticipated money supply changes on economic activity
9 The ldquoKeynesian modelrdquo withfixed money wageModifying the neoclassical model
Introduction
The first modification of the neoclassical model is presented in this chapterTo begin we introduce the very realistic assumption that nominal wages are fixedat least for a period of time The ramifications of this change in the model aredeveloped in the context of the aggregate supply and demand model As with theneoclassical model we perform a comparative statics exercise in which the mone-tary authority changes the money supply and we trace out the effects of this actionon the economic aggregates in the model The model is then made slightly morecomplete and issues associated with sticky wages and the natural rate hypothesisare discussed We introduce the concept of rational expectations and the first ldquoover-lappingrdquo model and show that the Keynesian model has important implicationsfor the conduct of monetary policy
The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
In the standard neoclassical model it is assumed that prices adjust in all marketsto equate demand and supply With respect to the labor market this implies a spotmarket at the start of each period in which one-period labor contracts are enteredinto and an associated one-period wage set Yet employment contracts are likelyto be multiperiod in the presence of hiring and training costs That is to minimizehiring and training costs firms seek long-term relationships with their employees
Firms promote long-term relationships with their employees by offering higherwages to their experienced workers As a consequence long-time employeesbecome attached or ldquoloyalrdquo to their employers since the wages they receive aregreater than those that other firms would offer them In essence employers aresharing the returns to their hiring and training investment with their workers inorder to reduce the number who quit A long-term attachment of workers to par-ticular firms could also stem from the high cost to workers of finding alternativeemployment as obtaining such employment means that workers must generallyinterview various employers visit employment agencies and spend valuable timesimply waiting for decisions on job applications to be made
Keynesian model 131
Given long-term employment contracts between firms and their workers wagesare typically specified for extended periods of time These long-term wage agree-ments are sometimes explicit as with many labor union contracts1 In other casesonly an implicit understanding exists on the wages that a firm will pay its employeesover some extended period of time If these contracts or understandings specifywages in money terms and if modifying these agreements is costly then thereexists an inherent inflexibility in money wages ndash that is there are ldquostickyrdquo wages2
This assumption of ldquostickyrdquo nominal wages is often viewed as the critical aspectof what has been termed the ldquoKeynesianrdquo macroeconomic model
If money wages are ldquostickyrdquo relative to prices then changes in the price levelwill alter the real wage For instance in the late 1970s high inflation rates ledworkers and employers in certain industries to bargain for long-term contractswith a high rate of growth of the money wage The high expected inflation did notmaterialize in the early 1980s The lower rate of inflation with no change in therate of increase in wages meant a rise in the real wage The resulting fall in thedemand for labor led to lower employment and output In this section we formallydevelop these results in the context of a static neoclassical macroeconomic modelwith the additional assumption of a fixed money wage The subsequent section thendevelops a linear rational expectations version of this model in which overlappingmultiperiod employment contracts introduce an element of nominal wage rigidity
Fixed money wage the labor market and aggregate supply
As we have seen in the competitive (spot) labor market of the neoclassical model(or in the Lucas-type macroeconomic model) the money wage and employmentare determined at the start of each period in the labor market If we accept theneoclassical modelrsquos assumption of limited perfect foresight on the part of bothlabor suppliers and firms we have equilibrium in the labor market in terms of amoney wage wt and level of employment Nt determined such that
N dt (wtpt K) minus Nt = 0 (91)
N st (wtpt) minus Nt = 0 (92)
In this case a change in the price level pt leads to an equiproportionate change inthe money wage wt and no change in employment Nt
We now seek to modify this analysis by assuming a fixed money wage wt = wfor period t As Sargent (1987a 21) states
the essential difference between the classical model and the Keynesian modelis the absence from the latter of the classical labor supply curve combined withthe labor market equilibrium condition Since there is one fewer equation inthe Keynesian model it can determine only six endogenous variables insteadof the seven determined in the classical model3 To close the Keynesianmodel the money wage is regarded as an exogenous variable one that atany point in time can be regarded as being given from outside the model
132 Keynesian model
perhaps from the past behavior of itself and other endogenous or exogenousvariables It bears emphasizing that the equation that we have deletedin moving from the classical to the Keynesian model [equation (92)] is acombination of a supply schedule (and) an equilibrium condition Notethat we continue to require that employment satisfy the labor demand schedule[equation (91)]
Sargent goes on to say that
we shall think of the labor supply schedule as being satisfied and helping todetermine the unemployment rate Usually the model is assumed to reachequilibrium in a position satisfying Nt lt N s
t so that there is an excess supplyof labor
Totally differentiating the labor demand condition (91) that determines the levelof employment with respect to the price level and employment we have
minus(partN dt (partwpt))(wp2
t )dpt minus dNt = 0 (93)
which can be rearranged to give
dNtdpt = minus(partN dt part(wpt))(wp2
t ) gt 0 (94)
where the sign reflects the presumption that partN dt part(wtpt) lt 0
In the simple case of no labor adjustment costs labor demand is defined bythe equality between the marginal product of labor and the real wage that ispartf (Nt K)partNt = wpt 4 Differentiating this implies that
[part2ftpartN 2t ]dNt = minus(wp2
t )dpt
or rearranging
dNtdpt = minus(wp2t )[part2ftpartN 2
t ] gt 0 (95)
given diminishing returns to the labor input (ie part2ftpartN 2t lt 0)
Combining the above analysis with the aggregate production function yt =f (Nt K) we thus have the ldquoaggregate supply equationrdquo
yt = yt( ptw K ) with partytpart( ptw) gt 0 (96)
Thus (as in a Lucas-type model) we have an aggregate supply that can dependdirectly on the price level for period t
The above findings can be understood with respect to the labor market Considera decrease in pt Given limited perfect foresight on the part of both firms andhouseholds at time t there is a downward (vertical) shift in labor demand so thatat the lower money wage wlowast
t associated with the same real wage demand would
Keynesian model 133
be the same Similarly the labor supply curve shifts down vertically so that atthis lower money wage wlowast
t labor supply would be the same as well Howevermultiperiod labor contracts fix the money wage at w so that the lower price level(and implied higher real wage) results in a fall in employment (which is nowdemand-determined) and an excess supply of labor
In the opposite case of a rise in the price level that can lead to an excessdemand in the labor market at the fixed money wage the presumption remainsthat employment is demand-determined This presumption reflects the view thatat least temporarily firms can direct workers with whom they have long-termemployment contracts to work overtime or extra shifts which the workers wouldotherwise not volunteer for
The above story provides a rationale for an upward-sloping ldquoaggregate sup-ply curverdquo Both contrast with the neoclassical model in which the aggregatesupply curve is vertical as a fall in the price level results in an equiproportionatefall in the money wage so that the real wage and employment remain unchangedIn the fixed wage model the underlying events in the labor market summarized bythe aggregate supply curve are the change in the real wage and thus labor demandand employment that accompany a price change when the money wage is fixedSuch an aggregate supply curve is upward-sloping
Equilibrium aggregate supply and demand
As we have noted before an important feature of macroeconomic theories is that toa large extent they are distinguished by their different treatment of labor marketsWhat this means is that the aggregate demand side is similar across macroeconomicmodels Recall that the aggregate demand side of macroeconomic models typicallyconsiders the equilibrium conditions of two of the remaining three markets inparticular the output market (reflected by an ldquoISrdquo equation) and the money market(reflected by an ldquoLMrdquo or ldquoportfoliordquo equation) Thus for the Keynesian modelwith fixed money wage the equilibrium output price level and interest rate aregiven by the following three equations
yt = yt( ptw K ) (96)
cdt (rt πe
t+1 At yt) + I dnt(m
et + δ K) + δK + ψ(I d
nt) minus yt = 0 (97)
Ldt (rt πe
t+1 At yt) minus Mpt = 0 (98)
Equation (96) is the aggregate supply equation of a Keynesian model with fixedmoney wage (97) is the ldquoISrdquo equation depicting equilibrium between outputdemand and production and equation (98) is the portfolio or ldquoLMrdquo equationexpressing equilibrium with respect to the money market Note that we havesimplified the IS and LM equations by removing the real balance effect for con-sumption demand and money demand5 Equations (97) and (98) can be combinedto eliminate the interest rate The resulting equation is referred to as the ldquoaggregatedemand equationrdquo
134 Keynesian model
The equilibrium price level and output ( plowastt and ylowast
t ) can be shown graphicallyusing aggregate demand and supply curves Looking at what underlies thesecurves we can then infer the change in employment (specifically from an analysisof the labor market that underlies the aggregate supply curve) as well as changes inthe interest rate and the investment and consumption components of output demand(specifically from an analysis of the output and money markets that underlie theaggregate demand curve)
A change in the money supply the comparative statics for theKeynesian model
The aggregate supply IS and LM equations (96)ndash(98) for the static Keynesianmodel under consideration determine the equilibrium output the price level andthe interest rate Substituting (96) into (97) and (98) in order to focus on thedetermination of the price level and interest rate and totally differentiating withrespect to the equilibrium prices ( pt and rt) and the money supply change givesus the following system of linear equations in matrix form6[ minus(partcdparty minus 1)partypartp
(partLdparty)(partypartp) + Mp2partydpartrpartLdpartr
] [dpdr
]=[
0dMp
]
where partydpartr = partcdpartr + (1 + ψ prime)partI dpartm7 The term partypartp gt 0 reflects thedirect effect on the price level as implied by the aggregate supply equation (96)8
Note that the equilibrium condition with respect to the labor market is incorporatedinto the analysis in the form of this aggregate supply equation
Solving the above linear equation system for dp and dr using Cramerrsquos rule weobtain
dp
dM= pM
1 + (p2M )(partypartp)[(1 minus partcdparty)(partLdpartr)(partydpartr) + partLdparty]gt 0
dr
dM= ((partcdparty minus 1)(partypartp) minus 1)p
(partcdparty minus 1)partypartp(partLdpartr) minus (partLdparty)(partypartp) + (Mp2)(partydpartr)
lt 0
In contrast to the neoclassical model we no longer have drdM = 0 Furtherletting x denote the denominator for the expression for dp we have
dp
p= dM
M
1
x
Since x gt 1 1x lt 1 and we now have that
dp
plt
dM
M
Keynesian model 135
Thus the increase in the money supply leads to a less than proportionate increasein the price level so that the real money supply is greater
Consider now the effect of the money supply shock on other variables Fromour analysis of the labor market we know that w is fixed so that the increase inthe price level means a fall in the real wage and thus increased labor demandemployment and thus an increase in output From the demand functions for con-sumption and investment we can derive the effects of the change in the moneysupply on consumption and investment as equal to
dcd
dM= dcd
dy
dy
dp
dp
dM+ dcd
dr
dr
dMgt 0
and
dId
dM= dId
dm
dm
dr
dr
dMgt 0
Note that the effect of the money supply shock is to increase the aggregate demandwhile not affecting aggregate supply
Sticky wages and the natural rate hypothesis
Expectations play an important role in the two modifications of the neoclassicalmodel In the modification with fixed wages the level at which negotiators fixfuture money wages depends on the expectation formed when wages were set con-cerning future prices The higher the expectation of future prices the higher thelevel of wages set in the labor agreements between workers and firms The pre-sumption is that workers and firms attempt to set future wages at their anticipatedmarket-clearing levels Associated with these anticipated market-clearing wagesis a particular real wage a natural rate of unemployment and a full employmentor natural rate of output
If price expectations turn out to be incorrect then output will vary from itsnatural rate For instance a shock that causes actual output prices to fall belowthose expected means that the money wage is fixed at a level that is too highfor full employment Consequently employment and output fall below the fullemployment level
In the typical Lucas-type model firms and workers set wages for the currentperiod based on incomplete information as well As we saw if suppliersrsquo expec-tations are incorrect then output will deviate from the full employment level Forexample a shock that causes actual output prices to fall below those expectedmeans lower employment and output as workers mistake lower money wages forlower real wages In fact higher real wages accompany the lower price level andthis is the source of the reduced demand for labor and employment
The two modifications of the neoclassical model have a second common ele-ment Both predict that a macroeconomic demand shock ultimately affects onlythe level of prices Even though money wages in the Keynesian model are fixed
136 Keynesian model
for the current period we know that money wages are not fixed forever Over timelabor agreements are renegotiated and money wages change to once again equatethe ldquoexpectedrdquo future demand for and supply of labor Over time in the absenceof further shocks the economy would thus tend to behave as neoclassical analysispredicts money wages and output prices would adjust to restore equilibrium tothe various markets in the economy
While the Keynesian model with fixed money wage admits the tendency foroutput to approach its natural level over time it does introduce a potential rolefor monetary policy to play in dampening fluctuations in output In so doing itchallenges the policy ineffectiveness view of Sargent and Wallace The best-knownexamples employing the Keynesian model to demonstrate the potential stabilizingpowers of monetary policy under rational expectations are the dual papers byFischer (1977) and Phelps and Taylor (1977)
A linear rational expectations version of the Keynesian model
Counting the above discussion we have so far considered three different modelsthat can be used to assess monetary policy One model is along the lines of theneoclassical model with limited perfect foresight Since this view of the economypredicts the neutrality of money a role for monetary policy either as an instigatorof output fluctuations or as an instrument to dampen output fluctuations is missingAs Mankiw (1987) suggests people who adopt this model ldquoview economic fluctu-ations through the lens of real business cycle theoryrdquo in which output fluctuationsare traced to ldquosupply-siderdquo disturbances
As Mankiw goes on to note however ldquothere are surely readers who believe thatmonetary policy has real short-run effects because of temporary misperceptions ornominal rigiditiesrdquo Mankiw is referring to individuals who adopt either the Lucas-type model or the ldquoKeynesianrdquo fixed money wage model9 Either one as we haveseen introduces a role for monetary policy as an instigator of output fluctuationsHowever these two models do differ as to whether monetary policy can be aninstrument to dampen output fluctuations in the context of rational expectations
A Lucas-type model built on ldquotemporary misperceptionsrdquo when coupled withrational expectations leaves little if any room for countercyclical monetary policyIn fact following the analysis of Sargent and Wallace it can be shown that whilerandom monetary shocks can impact output deterministic monetary policy basedon a set of policy rules is ineffective in counteracting fluctuations in output givenrational expectation10 Further attempts at discretionary monetary policy in thiscontext only result in a suboptimal (ldquotoo highrdquo) rate of inflation (Barro and Gordon1983) Thus in this model there remains a ldquostochasticrdquo neutrality of money
As the analysis in the previous section suggests however a ldquoKeynesian-typerdquomodel built on ldquonominal rigiditiesrdquo might introduce a role for monetary policyin stabilizing output even in the context of rational expectations The reasoningfor this is that wages (or as we will see later prices) can be set prior to thereceipt of information by the monetary authority that enters into the money supplyrule In this context as Phelps and Taylor (1977) state ldquoeven systematic and
Keynesian model 137
correctly anticipated policy can make a difference for the stability of output in arational expectations model with sticky prices and wagesrdquo11 Below we considerone example of such a model that counters the Sargent and Wallace ineffectivenessproposition a model proposed by Fischer that assumes ldquostickyrdquo wages
The supply equation with overlapping two-period labor contracts
The Lucas aggregate supply equation with adjustment costs can be expressed as
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (99)
where Yt = ln yt minus ln yn denotes difference between the logarithm of output forperiod t and the logarithm of the natural rate of output (which we have normalizedto equal zero) Pt = ln Pt is the logarithm of the price level for period t Etminus1Pt isthe expectation of the logarithm of the price level for period t using all informationavailable up to the end of period t minus 1 (at time t) and γ θ is a positive constant12
The supply of output as expressed by equation (99) satisfies the conditionthat employment equals labor demand (92) The fact that a higher price level Ptinduces firms to increase employment and thus output reflects the underlying lowerequilibrium real wage that accompanies the higher price level when suppliers donot anticipate the higher price level Thus we could express (99) in the form13
Yt = (Pt minus Wt + φ) + λYtminus1 (99prime)
where Wt is the logarithm of the equilibrium nominal wage for period t Theterm φ in (99prime) is defined such that if Etminus1Pt = Pt then the resulting log ofthe equilibrium real wage (ie ln(wtpt)) equals φ Ignoring the lagged outputterm in equation (99prime) this equilibrium real wage is the one associated with thenatural level of output and employment14 We will follow others and assume forconvenience that φ = 0 implying an equilibrium real wage with no surprisesequal to one
According to (99prime) if an increase in the price level is accompanied by a lessthan proportionate increase in the equilibrium money wage Pt minusWt rises (the realwage falls) and employment and output increase In the Lucas-type model such anevent occurs if suppliers do not forecast the price increase However Sargent andWallacersquos ineffectiveness proposition eliminates deterministic monetary policyrules as a source of such a price rise not matched by a similar rise in wages if(a) wages are set each period to equate labor demand and supply and (b) rationalexpectations are assumed In this case individuals and the monetary authoritiesare assumed to have a common set of information based on events up to the endof period t minus 1 (at time t) Individuals are also privy to the monetary policy ruleand they know the structure of the economy Thus they have knowledge of thedeterministic monetary policy to be followed during period t and its effect on theprice level for period t as predicted by the model Assuming flexible wages thispredicted effect of monetary policy on prices will be factored into the setting of themoney wage for period t As a consequence such deterministic monetary policy
138 Keynesian model
cannot change the real wage and thus leaves employment and output unaffectedas well
Obviously this chain of reasoning breaks down if money wages for period twere set prior to time t For example this policy ineffectiveness doctrine disap-pears if some wages are set at the end of period t minus 2 for period t In this casenew information that arrives during period t minus 1 can be incorporated by the mon-etary authorities into their money supply rule Even with rational expectationsthe implications of this cannot be used by individuals to adjust the money wagefor period t since by assumption the money wage is fixed Thus deterministicmonetary policy based on information revealed during period t minus 1 can alter thereal wage employment and output
To formally develop this potential stabilizing role of monetary policy in a moreldquoelegantrdquo fashion let us consider Fischerrsquos model The model disaggregates theeconomy into two sectors and assumes that the sectors alternate in setting multi-period employment contracts that fix nominal wages In particular ldquosuppose thatall labor contracts run for two periods and that the contract drawn up at the endof period t minus 2 specifies nominal wages for periods t minus 1 and t [Assume] thatcontracts are drawn up to maintain constancy of the real wagerdquo (Fischer 1977198) In other words
tminusiWt = EtminusiPt i = 1 2 (910)
where tminusiWt is the logarithm of the wage set at the end of period tminus i for period t15
The idea embodied in (910) that wages are set for more than one period iscritical to Fischerrsquos finding It essentially means that in any period half of thelabor contracts have fixed money wages16 Given that the wage is predeterminedfor each firm the aggregate supply equation is given by
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911)
where ut is a stochastic ldquorealrdquo disturbance or ldquosupply shockrdquo that impinges onproduction in each period17 Substituting (910) into (911) we can rewrite theaggregate supply equation as
Yt = 12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut (911prime)
A complete model except for specifying the source of expectations
Equation (911prime) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To close the model we require LM and ISequations The explicit derivation of these is left to the next chapter but let usassume for the time being that the LM equation is given by
mt minus Pt = α1Yt minus α2 middot rt minus εt
Keynesian model 139
and the IS equation by
Yt = Xt minus β1(rt minus πet+1) + ut
Here mt = ln Mt mt = mt + εt such that the deterministic component mt is setby government authorities (ie the monetary authority) according to a monetaryrule Further rt minus πe
t+1 represents the expected real rate of interest Xt denotesa vector of exogenous variables that affect output demand εt and ut are randomterms associated with output demand and money supply respectively and assumedindependent (ie E(εtut) = 0) and well behaved
Combining the LM and IS equations to eliminate the interest rate rt we obtain
Yt = Xt + ut minus (β1α2)(minusmt minus εt + Pt minus α1Yt) + β1πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = [α2(α2 + α1β1)][Xt + ut + (β1α2)(mt + εt minus Pt) + β1πet+1] (912)
Note that if α2 = 0 (changes in the interest rate do not affect money demand) andα1 = 1 (the income elasticity of real money demand is one) then this simplifies towhat Fischer refers to as a ldquovelocity equationrdquo
Yt = mt minus Pt + vt (913)
where vt = εt is now to be interpreted as a money demand disturbance termaffecting the ldquovelocityrdquo of money18
To see why (913) is called a ldquovelocity equationrdquo note that the assumption ofmoney demand being independent of the interest rate allows us to capture therelationship between income and the price level summarized by the aggregatedemand equation by looking solely at the LM equation (ie neglecting the ISequation) In particular if we assume that real money demand can be expressedby the equation
Ldt = yt(exp(minusvt))
then equating real money demand to real money supply (Mtpt) gives us
yt(exp(minusvt)) = Mtpt (914)
Taking the logarithm of the equilibrium condition with respect to the money market(914) we obtain
ln yt minus vt = ln Mt minus ln pt
Given Pt = ln pt Yt = ln yt and mt = ln(Mt) this is simply equation (913)
140 Keynesian model
Note that equilibrium velocity is defined as the ratio of nominal output to themoney supply Thus rearranging (914) we have that
Equilibrium velocity equiv ptyt
Mt= exp(vt)
which explains the interpretation of vt as a ldquovelocityrdquo disturbance Fischer assumesthat vt has a zero mean so that expected velocity is one A vt above zero meansa decrease in real money demand relative to real output and thus an increase inequilibrium velocity As (914) makes clear for a given Mt a higher vt implies ahigher pt andor a higher yt to maintain equilibrium with respect to the demandfor and supply of money
As Fischer states (913) is ldquothe simplest way of taking demand consid-erations into accountrdquo In sum then the macroeconomics model considered byFischer is given by the aggregate supply equation (911prime) and the aggregate demandequation (913)
Combining (911prime) and (913) to eliminate Yt we have
12 (Pt minus Etminus1Pt) + 1
2 (Pt minus Etminus2Pt) + ut = mt minus Pt + vt
This can be solved to give the reduced-form equation for the price level Pt
Pt = 12
[(12 Etminus1Pt + 1
2 Etminus2Pt
)minus ut + mt + vt
] (915)
Combining (911prime) and (913) to eliminate Pt we have
Yt = 12 (mt minus Yt + vt minus Etminus1Pt) + 1
2 (mt minus Yt + vt minus Etminus2Pt) + ut
This can be solved for the reduced-form equation for output Yt
Yt = 12
[mt + vt + ut minus 1
2 Etminus1Pt minus 12 Etminus2Pt
] (916)
According to equations (915) and (916) an increase in the ldquorealrdquo disturbanceterm ut leads to higher equilibrium output and a reduced price level Intuitivelythis corresponds to a shift to the right in the ldquoaggregate supply curverdquo On the otherhand an increase in the ldquovelocityrdquo disturbance term vt corresponds to a shift to theright in the ldquoaggregate demand curverdquo and thus leads to a higher output and pricelevel given an upward-sloping aggregate supply curve Note that an increase in vtmeans a lower real money demand at each level of income The shift in the aggre-gate demand curve reflects the fact that a higher price level andor higher output isrequired to restore equilibrium in the money commodity and financial markets
If expectations can be taken as exogenous with respect to money supply changesthen (916) indicates that money supply changes can affect output But this wasalso the case for a Lucas-type model The next step is thus to see what happens
Keynesian model 141
when we assume rational expectations As will become clear even with rationalexpectations expectations formed at the end of period t minus 2 can be viewed asexogenous with respect to monetary changes planned for period t based on infor-mation obtained during period t minus 1 Thus monetary policy has the potential tooffset the persistent effect of disturbances that originate during period t minus 1
Introducing rational expectations
Let us now assume that individuals form their expectations Etminus1Pt and Etminus2Ptldquorationallyrdquo in that
Etminus2Pt = E(Pt |tminus2) (917)
Etminus1Pt = E(Pt |tminus1) (918)
indicating that EtminusiPt is the mathematical expectation of Pt conditional on theinformation set tminusi which is all information available at the end of period t minus ii = 1 2 Taking the expectation of (915) at the end of period t minus 2 we thus have
Etminus2Pt = Etminus2
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (919)
Note that Etminus2Etminus1Pt = Etminus2Pt Thus (919) becomes
Etminus2Pt = Etminus2minusut + vt + mt (920)
Taking the expectation of (915) at the end of period t minus1 (at time t) we then have
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2Pt minus ut + vt + mt
] (921)
Substituting (920) into (921) gives
Etminus1Pt = Etminus1
12
[12 Etminus1Pt + 1
2 Etminus2minusut + vt + mt minus ut + vt + mt
]
(922)
Rearranging
34 Etminus1Pt = 1
4 Etminus2minusut + vt + mt + 12 Etminus1minusut + vt + mt
or
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + mt (923)
which is Fischerrsquos (1977) equation (16)19
142 Keynesian model
Let the money supply be determined by the simple linear rule
mt = a1utminus1 + b1vtminus1
Since mt is a function only of information available up to the end of period t minus 1(at time t) Etminus1mt = mt Accordingly (923) can be written as
Etminus1Pt = 13 Etminus2minusut + vt + mt + 2
3 Etminus1minusut + vt + 23 mt (924)
Substituting (920) and (924) into the reduced-form equation for output (916)we obtain
Yt = 12 [ut + vt + mt] minus 1
4
[13 Etminus2minusut + vt + mt
+ 23 Etminus1minusut + vt + 2
3 mt
]minus 1
4 [Etminus2minusut + vt + mt](925)
Equation (925) simplifies to
Yt = 13 (mt minus Etminus2mt) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt
(926)
which is Fischerrsquos (1977) equation (18)As Fischer (1977 196) notes
disturbances aside this very simple macro model would be assumed in equi-librium to have the real wage set at its full employment level would imply theneutrality of money and would obviously have no role for monetary policyin affecting the level of output A potential role for monetary policy is createdby the presence of the disturbances ut and vt that are assumed to affect thelevel of output each period Each of the disturbances is assumed to follow afirst-order autoregressive scheme
ut = ρ1 middot utminus1 + εt where |ρ1| lt 1 (927)
vt = ρ2 middot vtminus1 + ηt where |ρ2| lt 1 (928)
where εt and ηt are mutually and serially uncorrelated stochastic terms withexpectation zero and finite variances σ 2
e and σ 2n respectively
Given equations (927) and (928) and the money supply rule
mt = a1utminus1 + b1vtminus1 (929)
Etminus2mt = a1ρ1utminus2 + b1ρ2vtminus2 (930)
Keynesian model 143
so that
mt minus Etminus2mt = a1utminus1 + b1vtminus1 minus [a1ρ1utminus2 + b1ρ2vtminus2]= a1εtminus1 + b1ηtminus1
(931)
According to (931)
the difference between the actual money stock in period t and that stockas predicted two periods earlier arises from the reactions of the monetaryauthority to the disturbances εtminus1 and ηtminus1 occurring in the interim It isprecisely these disturbances that cannot influence the nominal wage for thesecond period of wage contracts entered into at time t minus 2
(Fischer 1977 199)
Substituting (931) into (926)
Yt = 13 (a1εtminus1 + b1ηtminus1) + 1
2 (ut + vt) + 16 Etminus1ut minus vt
+ 13 Etminus2ut minus vt (932)
From (927) and (928) we know that
ut + vt = (ρ1utminus1 + εt) + (ρ2vtminus1 + ηt)
= (ρ1utminus1 + εt) + (ρ22vtminus2 + ρ2ηtminus1 + ηt)
since by substitution vt = ρ22vtminus2 + ρ2ηtminus1 + ηt we also have that
Etminus1ut minus vt = ρ1utminus1 minus ρ2vtminus1 = ρ1utminus1 minus ρ22vtminus2 minus ρ2ηtminus1
and
Etminus2ut minus vt = ρ21utminus2 minus ρ2
2vtminus2
Thus we can rewrite (932) as
Yt = 12 (εtminus1 + ηtminus1) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
(933)
which is Fischerrsquos (1977) equation (21)20
Fischer (1977 199) notes that
before we examine the variance of output as a function of the parameters a1and b1 it is worth explaining why the values of those parameters affect thebehavior of output even when the parameters are fully known The essential
144 Keynesian model
reason is that between the time the two-year contract is drawn up and the lastyear of operation of that contract there is time for the monetary authorityto react to new information about recent economic disturbances Given thenegotiated second-period nominal wage the way the monetary authority reactsto disturbances will affect the real wage for the second period of the contractand thus output
Optimal monetary policy rules the effectiveness of policy
As in our discussion of Sargent and Wallace let us presume that the goal of themonetary authority focuses solely on output In particular suppose that the mone-tary authority desires to set the money supply in order to minimize the fluctuationin the log of output around some desired level Then the objective can be expressedas to
min Etminus1(Yt minus Y lowast)2
Let us assume that Y lowast = Yn = 0 so that the objective becomes to
min Etminus1(Yt)2 (934)
From (933) we have that
(Yt)2 =
[12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21 utminus2
]times[
12 (εt + ηt) + 1
3 [εtminus1(a1 + 2ρ1) + ηtminus1(b1 + ρ2)] + ρ21utminus2
]
(935)
Note that E(εi) = E(ηi) = 0 E(ε2i ) = σ 2
e E(η2i ) = σ 2
n and that our independenceassumptions imply that E(εiηi) = 0 and for i = s E(εiεs) = 0 and E(εiηs) = 0Thus substituting (935) into (934) we have the following explicit form for theobjective
min Etminus1(Yt)2 = σ 2
e
[14 + 1
9 (a1 + 2ρ1)2]
+ (ρ21utminus2)
2
+ σ 2n
[14 + 1
9 (b1 + ρ2)2]
(936)
Given (936) the optimal monetary rule is to choose values for a1 and b1 suchthat
a1 = minus2ρ1 b1 = minusρ2 (937)
The above findings correspond to Fischerrsquos (1977) equation (23)21
Keynesian model 145
As Fischer (1977 200) states
to interpret the monetary rule examine [equation (933)] It can be seen therethat the level of output is affected by current disturbances (εt + ηt) that can-not be offset by monetary policy by disturbances (εtminus1 and ηtminus1) that haveoccurred since the signing of the older of the existing labor contracts andby a lagged real disturbance utminus2 The disturbances εtminus1 and ηtminus1 can bewholly offset by monetary policy and that is precisely what equation [(937)]indicates The utminus2 disturbance on the other hand was known when the olderlabor contract was drawn up and cannot be offset by monetary policy becauseit is taken into account in wage setting Note however that the stabilizationis achieved by affecting the real wage of those in the second year of laborcontracts and thus should not be expected to be available to attain arbitrarylevels of output ndash the use of too active a policy would lead to a change in thestructure of contracts
[A] more general interpretation of the monetary rule is to accommodatereal disturbances that tend to increase the price level and to counteract nominaldisturbances that tend to increase the price level
Fischer concludes by noting that
given a structure of contracts there is some room for maneuver by the mon-etary authorities ndash which is to say that their policies can though will notnecessarily be stabilizing
Conclusion
This chapter presented the sticky money wage or Keynesian model of the macro-economy We find that in contrast to the neoclassical model changes in the pricelevel affect real variables and the amount of labor employed in the economy Thusmoney has real effects The development of an upward-sloping aggregate supplycurve has dramatic implications for the conduct of monetary policy However it isshown that the expectations of agents in the economy also play an important rolein whether or not monetary policy is effective
10 The Lucas model
Introduction
As we have seen anticipated changes in prices have no impact on real vari-ables in the neoclassical model A key element of this model is the ldquoessentialpresumption that nominal output is determined on the aggregate demand sideof the economy with the division into real output and the price level largely depen-dent on the behavior of suppliers of labor and goodsrdquo (Lucas 1973) As such thismodel implies no link between price changes and real output
We have also seen how the natural rate model allows one to introduce a linkbetween unanticipated price changes and real output The seminal paper by Lucas(1973) formally develops a more complete model of the potential for ldquoshort-runsupply behavior (resulting) from suppliersrsquo lack of information on some of theprices relevant to their decisionsrdquo Lucasrsquos explanation of a tradeoff between unem-ployment and inflation ldquois that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relative pricechangesrdquo
As with the ldquoillusion modelrdquo Lucas postulates ldquorational agents whose deci-sions depend on relative prices only placed in an economic setting where theycannot distinguish relative from general price movementsrdquo That is we retain thehypothesis that prices adjust to clear markets
Lucas adds to the simple (static) natural rate model so far discussed by explicitlymodeling the source of forecast errors In doing so he assumes that ldquoinferenceson these relevant unobserved prices are made optimally (or lsquorationallyrsquo) in lightof the stochastic nature of the economyrdquo Below we outline Lucasrsquos model1
The ldquoislandrdquo paradigm
Lucasrsquos model begins by disaggregating the economy into a number of what havebeen called ldquosectorsrdquo ldquomarketsrdquo or ldquoislandsrdquo As Lucas says ldquowe imagine sup-pliers as located in a large number of scattered competitive markets Demand forgoods in each period is distributed unevenly over markets leading to relative aswell as general price movementsrdquo In terms of our previous analysis one couldthink of each of the n sectors in the economy as inhabited by firms producing the
Lucas model 147
ith commodity (i = 1 n) Associated with each sector or ldquoislandrdquo is a set ofworkers and thus a labor market
For firms producing commodity i in period t the key relative price is the priceof their output relative to the wage paid in sector i or pitwit where pit is themoney price of commodity i (produced in sector i) and wit is the money wage forlabor in sector i It is assumed that individuals (firms and workers) in sector irsquoslabor market know the money wage and price of commodity i For labor suppliersin sector i however the key relative price is witpt where pt is the economy-wideprice level reflecting the fact that suppliers plan to use money wages to purchasea bundle of goods consisting of all n commodities2
It is this setup of ldquodispersed marketsrdquo and ldquoinformational discrepanciesrdquo thatLucas uses to generate a correlation between price changes and output ndash the famousLucas supply equation
The supply function for a particular sector
The Lucas model assumes a competitive labor market for sector or ldquoislandrdquo i suchthat the equilibrium level of employment and money wage equate market demandand supply With respect to the labor demand function let us start by assumingthe simple CobbndashDouglas production function such that the marginal product oflabor is given by
a(Nit)minusα where a gt 03
The profit-maximizing condition for the representative firm producingcommodity i is to equate the money wage to the marginal product of labor multi-plied by the money price of output The resulting optimal labor demand can thusbe defined by the equation4
wit = pita(N dit )
minusα
Taking the natural log of this equation and rearranging we have
ln N dit = 1
α[ln pit minus ln wit + ln a] (101)
With respect to labor supply let us assume for the moment that the real wage isknown Further let us assume that the labor supply function takes the followinglogarithmic form
ln N sit = 1
βln
wit
pit
where β is a positive constantEmployment contracts entered into at time t in sector i specify the money
wage wit so that element of the real wage is known However if the price level is
148 Lucas model
unknown then the expected real wage based on information available at time t isequal to witEt(1pt) The associated expected logarithm of the real wage is then5
Et ln(witpt) = ln wit minus Et(ln pt)
Given the assumed log-linear labor supply function and ignoring the implicationsof uncertainty for labor supply we thus have
ln N sit =
(1
β
)(ln wit minus Et(ln pt)) (102)
Equilibrium in the labor market for sector i entails a level of employment Nitand money wage wit such that the demand for labor equals the supply In loga-rithmic form and using the specific labor demand and supply functions given byequations (101) and (102) equilibrium requires that the log of the money wageln wit and the log of employment ln Nit be such that
ln Nit = 1
α(ln pit minus ln wit + ln a)
and
ln Nit = 1
β(ln wit minus Et(ln pt))
Substituting the first expression into the second to eliminate the logarithm of themoney wage we have
ln Nit = 1
α(ln pit + ln a) minus 1
α(β ln Nit + Et(ln pt))
which upon rearranging becomes
ln Nit = ln Nni + [1 + (α + β)][ln pit minus Et(ln pt)] (103)
where ln Nni = (ln a)(α + β)Equation (103) indicates that the logarithm of equilibrium employment in the ith
sector and thus the production of commodity i depends directly on the expectationof logarithm of the ratio of the price of commodity i(pit) to the general level ofprices pt The term Nni can be viewed as the ldquonormalrdquo level of employment Notethat we have abstracted from population growth and other factors that would resultin this ldquonormalrdquo level of employment varying across time
Given the assumption of a simple CobbndashDouglas production function of theform yit = (Nit)
1minusα(Ki)α we thus have
ln yit = ln yni + γ [ln pit minus Et(ln pt)] (104)
where γ = (1 minus α)(α + β) and ln yni = (1 minus α) ln Nni + α ln Ki The term yni isdenoted by Lucas as the ldquonormalrdquo level of output in the particular sector or market i
Lucas model 149
under consideration As Lucas (1973 327) states the ldquoquantity supplied in eachmarket will be viewed as the product of a normal (or secular) component commonto all markets and a cyclical component which varies from market to market
The cyclical component varies with perceived relative prices and with its ownlagged valuerdquo Note that for the moment we do not include the lagged value ofoutput in the above supply function One can justify the inclusion of such byassuming adjustment costs
The source of forecasting errors
According to (104) output of commodity i depends critically on suppliersrsquo fore-cast of the log of the general level of prices Et(ln pt) Now consider how sucha forecast may be obtained in a stochastic environment First it is assumed thatagents in sector i ndash in particular suppliers of labor involved in the production ofcommodity i ndash know commodity irsquos price pit However the exact extent to whichany change in the money price of commodity i reflects a change in the overalllevel of money prices as opposed to a change in commodity irsquos price relative toother prices is unknown It is this uncertainty that leads suppliers in sector i tomisinterpret a change in the general price level in terms of a change in a relativeprice
To be concrete suppose Etminus1(ln pt) incorporates all information available atthe end of period t minus 1 The logarithm of the actual price level will vary from thelogarithm of this expected price level to the extent that there are ldquosurprisesrdquo withrespect to the aggregate price level Letting ξt denote this ldquosurpriserdquo for period twe have
ln pt = Etminus1(ln pt) + ξt (105)
We assume that ξt which is that part of the price level that cannot be predictedfrom past data is a normally distributed random variable with zero expectationand variance σ 26
At the start of period t suppliers in market i receive one additional piece ofinformation the logarithm of the price of commodity i ln pit This signal isassumed to contain some information about the logarithm of the overall pricelevel in that
ln pit = ln pt + zit (106)
where zit is a normally distributed random variable with zero mean and varianceσ 2
z Thus using equation (105) to substitute for ln pt
ln pit = Etminus1(ln pt) + ξt + zit (107)
In words the logarithm of the nominal price for the sector ln pit is assumed toinform the supplier of the sum of the current ldquowhite noiserdquo innovations to the
150 Lucas model
relative price process in that sector (zit) and the innovations to aggregate demandand thus the economy-wide level of prices (ξt)7
We can then express the expectation of the logarithm of the price level at time tfor suppliers in sector i given the observed logarithm of the price of commodity iln pit by
Et(ln pt) equiv Eln pt |Etminus1(ln pt) ln pit (108)
Formally we have the joint distribution of two random variables f (ln pit ln pt)where one of them ln pit is known to take a particular value The problem is thebasic one of ldquobivariate regressionrdquo in that we have to determine the conditionalexpectation Eln pt | ln pit namely the ldquoaveragerdquo value of ln pt for the givenvalue of ln pit 8 As we shall see in the next section the resulting expression for theexpected general price level (in logs) given ln pit is observed can be expressedin linear form as
Et(ln pt) = Etminus1(ln pt) + (1 minus θ)(ln pit minus Etminus1(ln pt)) where 0 le θ le 1(109)
A digression on linear regression analysis
Let us assume a linear regression equation that links the observed logarithm of theprice of commodity i to the logarithm of the general level of prices of the form9
Etln pt | ln pit = a0 + a1(ln pit) (1010)
We can express the regression coefficients a0 and a1 in terms of some of the lowermoments of the joint distribution of ln pt and ln pit namely in terms of10
Eln pit = Etminus1(ln pt) (from (107))
Eln pt = Etminus1(ln pt) (from (105))11
Varln pit = σ 2 + σ 2z (from (107))
Cov(ln pit ln pt) = E(minusξt minus zit)(minusξt) = σ 2 + Ezit middot ξtIn general Ezit middot ξt = Cov(zit ξt) + EzitEξt However given that Ezit =Eξt = 0 and given the assumption that zit and ut are independent variables sothat Cov(zit ξt) = 0 we have12
Cov(ln pit ln pt) = σ 2
From (1010) we have that13
E(ln pt | ln pit) equivint
(ln pt)φ(ln pt | ln pit)d ln pt = a0 + a1(ln pit) (1011)
Lucas model 151
where φ(middot) is the conditional density function of ln pt given ln pit If we thenmultiply the expression on both sides of (1011) by the marginal density functionof ln pit denoted by g(ln pit) and integrate on ln pit we obtainintint
(ln pt)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0g(ln pit)d ln pit +int
a1(ln pit)g(ln pit)d ln pit
or
Eln pt = a0 + a1Eln pit (1012)
since φ(ln pt | ln pit)g ln(pit) = f (ln pit ln pt) Had we multiplied the expressionon both sides of (1011) also by ln pit before integrating on ln pit we would haveobtainedintint
(ln pt)(ln pit)φ(ln pt | ln pit)g ln(pit)d ln ptd ln pit
=int
a0(ln pit)g(ln pit)d ln pit +int
a1(ln pit)2g(ln pit)d ln pit
or
E(ln pit)(ln pt) = a0Eln pit + a1E(ln pit)2 (1013)
Solving (1012) and (1013) for a0 and a1 and making use of the fact that
E(ln pit)(ln pt) = Cov(ln pit ln pt) + E(ln pit)Eln ptand
E(ln pit)2 = Var(ln pit) + [Eln pit]2
we find that
a0 = Eln pt minus [Cov(ln pit ln pt)Eln pit](Var(ln pit))
a1 = (Cov(ln pit ln pt))(Var(ln pit))
Hence we can write equation (1010) as
Et(ln pt) equiv E(ln pt | ln pit)
= E(ln pt) + [Cov(ln pit ln pt)Var(ln pit)](ln pt minus Eln pit)Substituting in the above expressions for means variance and covariance we havethus derived (109) with
1 minus θ = σ 2(σ 2 + σ 2z )
152 Lucas model
Equation (109) indicates that agentsrsquo rational expectation of the current pricelevel is a ldquolinear least-squares projectionrdquo That is one could rewrite (109) as
Et(ln pt) = θEtminus1(ln pt) + (1 minus θ) ln pit (1014)
where θ = σ 2z (σ 2 + σ 2
z ) To see why this is called ldquolinear least-squaresrdquo notethat we could start with (1014) (the ldquolinearrdquo part of the projection) and thenpick θ to minimize the variance in this forecast or projection of ln pt (the ldquoleast-squaresrdquo part of the projection) In particular substituting in (105) for Etminus1(ln pt)
(ie Etminus1(ln pt) = ln pt minus ξt) and (106) for ln pit (ie ln pit = ln pt + zit)(1014) becomes
Et(ln pt) = ln pt minus θξt + (1 minus θ)zit (1014prime)
The problem of picking θ to minimize the variance of this projection can then beexpressed as14
minθ
Eln pt minus θξt + (1 minus θ)zit minus ln pt2 = θσ 2 + (1 minus θ)σ 2z
Taking the derivative of the above expression with respect to θ setting it equal tozero (ldquoleast squaresrdquo) and solving for θ we verify that θ = σ 2
z (σ 2 + σ 2z )
As Sargent (1987a 442) points out
the parameter θ is the fraction of the conditional variance in ln pit due torelative price variation The larger is this fraction the smaller is the weightplaced on ln pit in revising Etminus1(ln pt) to form Et(ln pt) This makes sensesince the larger is θ the more likely it is that a change in ln pit reflects arelative rather than a general price change15
Equation (1014) can be substituted into the supply function for commodity i(104) to obtain
ln yit = ln yni + γ θ(ln pit minus Etminus1(ln pt)) (1015)
where as before ln yni = (1 minus α)(ln Nni) + α(ln Ki) As noted above if weassumed adjustment costs then a lagged output term could be added to (1015)In this case we would have16
ln yni = ln yni + γ θ [ln pit minus Etminus1(ln pt)] + λ(ln yitminus1 minus ln yni) (1015prime)
If suppliers were able to observe the actual value of the price level so thatEt(ln pt) = ln pt then going back to (104) one could express the resulting ldquofullinformationrdquo output produced in sector i by
ln ylowastit = ln yni + γ [ln pit minus ln pt]
Lucas model 153
which given ln pit = ln pt + zit simply becomes
ln ylowastit = ln yni + γ zit
As you can see since the expectation of the random shock to relative prices zit iszero ln yni has the natural interpretation as the expected output of sector i givenfull information
The Lucas aggregate supply function
Equation (1015) is close to what is known as the ldquoLucas aggregate supplyfunctionrdquo Without adjustment costs the Lucas supply function takes the form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) (1016)
With adjustment costs the Lucas supply function takes the general form
ln yt = ln yn + γ θ(ln pit minus Etminus1(ln pt)) + λ(ln ytminus1 minus ln yn) (1016prime)
where yn denotes the natural rate of output17 For simplicity we have assumed thatthe natural rate of total output is constant across periods
The term ln yn can be interpreted either as the logarithm of output for theldquorepresentativerdquo sector or as the logarithm of total output across the n sectorsLet us assume the former interpretation The average level of real output can bedefined by
yt equiv 1
pt
⎡⎣ nprod
i=1
pityit
⎤⎦
1n
where [prodni=1 pityit]1n is the geometric mean of nominal output across the n markets
or sectors Taking logs we have the following definition for the logarithm ofaverage output
ln yt equiv minus ln pt + 1
n
nsumi=1
(ln pit + ln yit) (1017)
We will assume that the overall price level is constructed as a geometric mean ofindividual prices such that
pt equiv⎡⎣ nprod
i=1
pit
⎤⎦
1n
Taking logs
ln pt equiv 1
n
nsumi=1
ln pit
154 Lucas model
Substituting the above into (1017) we have the following definition for thelogarithm of average output
ln yt equiv 1
n
nsumi=1
ln yit (1018)
Substituting into (1018) the supply functions for the individual sectors as givenby (1015) we thus have18
ln yt equiv 1
n
nsumi=1
[ln yni + γ θ(ln pit minus Etminus1(ln pt))] (1019)
Recall that ln pit = ln pt + zit where zit is a normal random variable indepen-dently distributed across markets with a mean of zero and variance σ 2
z Substitutingthis into (1015) and rearranging we have
ln yt = 1
n
nsumi=1
[ln yni] + γ θ(ln pt minus Etminus1(ln pt)) + 1
n
nsumi=1
γ θzit (1020)
As the number of markets n approaches infinity from the law of large numberswe know that the sum of the zit divided by n approaches zero19 Thus for a largenumber of markets we may approximate (1020) by
ln yt = 1
n
⎡⎣ nsum
i=1
ln yni + γ θ(ln pt minus Etminus1(ln pt))
⎤⎦ (1021)
By definition the logarithm of the geometric average of ldquonormalrdquo output acrossmarkets ln yn is given by nminus1 sumn
i=1 ln yni Thus we can rewrite (1021) as (1016)in which ln yn is the logarithm of ldquonormalrdquo output that would occur if there were nosurprises with respect to the aggregate price level that is when ln pt = Etminus1(ln pt)Note that for simplicity we assume the natural rate is constant over time
Equation (1016) is the Lucas aggregate supply equation with the last termmissing As noted above if we include adjustment costs then we obtain (1016prime)indicating that the deviation of real output from its ldquonaturalrdquo level or trend isassociated with a deviation in the price level from that expected and past deviationsof output from the natural rate This last term makes output serially correlatedover time
The Lucas supply function and the Phillips curve
The Lucas supply function predicts a direct correlation between unanticipatedprice changes and output and thus a potential tradeoff between price changesand unemployment if one assumes that unemployment and output are inverselyrelated This potential inverse relationship between unemployment and inflation
Lucas model 155
is sometimes referred to as the ldquoPhillips curverdquo after AW Phillips who noted theempirical relationship between wage inflation and unemployment for the Britisheconomy for the 100 years up to 1957 (see Phillips 1958) Later depictions of thePhillips curve replaced the rate of change in wages with the inflation rate
To see this Phillips relationship more clearly rearrange the aggregate Lucassupply function without adjustment costs (1016) to obtain
ln pt = (ln yt minus ln yn)γ θ + Etminus1(ln pt)
Subtracting ln ptminus1 from both sides of this aggregate supply equation we have
ln(ptptminus1) = (ln yt minus ln yn)γ θ + Etminus1(ln(ptptminus1)) (1022)
Let the term πt denote the rate of inflation between periods t minus 1 and t20
πt equiv (pt minus ptminus1)ptminus1 = (ptptminus1) minus 1
It is common in macroeconomics to approximate the above rate of change inprices by the log of the ratio of the two prices If the ratio equals one then thelog equals zero which is the rate of inflation If the ratio is 1 + x and x is a smallproportion then the log of this ratio approximately equals the actual inflation rateFor instance if ptptminus1 = 105 so that inflation is 005 or 5 percent then the logof 105 is 00488 which approximates this 005 rate of inflation Thus we have
πt asymp ln(ptptminus1) = ln pt minus ln ptminus1
Using the above approximation for the rate of inflation we can rewrite(1022) as
πt = (ln yt minus ln yn)γ θ + Etminus1πt (1023)
which as Sargent (1987a 443) states
is in the form of a standard natural rate Phillips curve relating inflation (πt)
directly to output (ln yt) and to expected inflation (Etminus1πt) According to[(1023)] the Phillips curve shifts up in the (πt yt) plane by the exact amountof any increase in expected inflation This characteristic of equation [(1023)]is often taken as the hallmark of the natural unemployment rate hypothesisIt seems to offer an explanation for why the Phillips curve tradeoff worsenedas average inflation rates increased over the 1970s in many western countries
If we assumed that due to adjustment cost the lagged deviation in output fromthe natural level affects the current deviation as the Lucas aggregate supplyequation (1016prime) suggests then in terms of rates of change in prices we wouldhave
πt = (ln yt minus ln yn)γ θ + Etminus1πt minus (λγ θ)(ln ytminus1 minus ln yn) (1023prime)
156 Lucas model
We could instead express (1023) in terms of unemployment by assuming thereis a linear inverse relationship between deviations in output from the natural rateand deviations in the actual level of unemployment Ut from its natural rate Unsuch that
ln ytminus1 minus ln yn = minus(Ut minus Un)
where is a positive constant Substituting the above into (1023) we thus have
πt = minus(γ θ)(Ut minus Un) + Etminus1(πt) (1023primeprime)
indicating the inverse relationship between unanticipated price changes and theactual level of unemployment Rearranging (1023primeprime) we have that
Ut = Un minus (γ θ)[πt minus Etminus1(πt)] (1023primeprimeprime)
where γ θ gt 0 Equation (1023primeprimeprime) is the typical expression of the Phillips curvefound in the literature It indicates that deviations in the unemployment rate belowits natural level must be accompanied by deviations in the actual rate of inflationabove that expected It reflects the ldquonatural rate of unemployment hypothesisrdquo asoriginally coined by Friedman (1968 11)
There is always a temporary trade-off between inflation and unemploymentthere is no permanent trade-off The temporary trade-off comes not frominflation per se but from unanticipated inflation which generally means froma rising rate of inflation
Recall that as Barro and Gordon (1983 592) observed the term Etminus1(πt) in(1023primeprimeprime) is the
prior expectation of inflation for period t [which is] distinguished from theexpectation that is conditional on partial information about current pricesThis distinction arises in models (eg Lucas 1972 1973 Barro 1976) inwhich people operate in localized markets with incomplete information aboutcontemporaneous nominal aggregates In this setting the Phillips curve slopecoefficient (γ θ) turns out to depend on the relative variances for generaland market-specific shocks
Variability in prices and the tradeoff
As Lucas (1973 333) states
demand policies [can] tend to move inflation rates and output (relative totrend) in the same direction or alternatively unemployment and inflation inopposite directions The conventional Phillips curve account of this observedco-movement says that the terms of the tradeoff arise from the relatively stable
Lucas model 157
structural features of the economy and are thus independent of the nature ofthe aggregate demand policy pursued The alternative explanation of the sameobserved tradeoff is that the positive association of price changes and outputarises because suppliers misinterpret general price movements for relativeprice changes
Taking Lucasrsquos alternative viewpoint two aspects concerning the tradeoff aresuggested First as Lucas states ldquochanges in average inflation rates will notincrease average outputrdquo As we have seen if we compare the expected pricelevel for period t with the price level for the prior period the difference wouldincorporate individualsrsquo expectation of this average rate of inflation (along with anumber of other potentially relevant variables) Second ldquothe higher the variancein average prices the less lsquofavorablersquo will be the observed tradeoffrdquo We considerthis second point below by referring back to the simple Lucas supply functionwithout lagged output (1016)
Recall that the term ξt given by (105) denotes that part of the price level thatcannot be predicted from past data We have assumed that this ldquosurpriserdquo term ξtis a normally distributed random variable with zero expectation and variance σ 2Substituting this into (1016) we have that
ln yt minus ln yn = γ θξt (1024)
where θ = σ 2z (σ 2 + σ 2
z ) and γ = (1 minus α)(α + β) Recall that θ is the weightattached to the expected price level prior to observing pit
Equation (1024) indicates that deviations in output from the natural level dependsolely on surprises In his statement concerning the variance of prices Lucas ispointing out that the impact of ldquosurprisesrdquo on output relative to its natural leveldepends on the ldquosloperdquo term λθ which is given by
λθ = 1 minus α
α + β
σ 2z
σ 2 + σ 2z
As Sargent (1987a 444) notes
a ldquofavorablerdquo tradeoff between output and unexpected inflation (that is a largevalue of γ θ ) will exist only when σ 2 is small relative to σ 2
z An attempt byauthorities to exploit the tradeoff between output and unexpected inflationmore fully by changing aggregate demand regimes might increase the vari-ance σ 2 relative to σ 2
z and thus change the slope γ θ This is yet anotherexample of how agentsrsquo optimal decision rules change in response to changesin the random processes governing the exogenous variables they base theirdecisions on
Sargentrsquos last point is another example of the ldquoLucas critiquerdquo in this contextwith respect to the validity of using past econometric estimates of a tradeoff inpredicting future tradeoffs
158 Lucas model
Note that although the tradeoff worsens with higher variability in prices theeffect of higher variability in prices on the variance of output about the natural rateis unclear In particular from (1016) we know that the variance in the differencebetween output and the natural rate is simply (γ θ)2σ 2 Given our definition ofγ θ the variance of the logarithm of output becomes
Var(ln yt) = γ 2(
σ 2z
σ 2 + σ 2z
)2
σ 2
Differentiating with respect to σ 2 we have
partVar(ln yt)
partσ 2 = γ 2(
σ 2z
σ 2 + σ 2z
)2
minus 2γ 2σ 2 σ 2z
(σ 2 + σ 2z )
σ 2z
(σ 2 + σ 2z )2
=(
γ 2σ 2z
σ 2 + σ 2z
)2 (1 minus 2σ 2
σ 2 + σ 2z
)
As the above expression indicates by itself an increase in the variation in theaverage price level (σ 2) will increase the variation in the logarithm of output for agiven ldquosloperdquo (γ θ) On the other hand as Lucas pointed out such an increase inthe variation in price level will result in a reduction in the effect of any given pricechange on output which by itself would decrease the variation in the logarithm ofoutput
Substituting (105) into the expanded Lucas supply function with lagged outputwe have the Lucas supply function of the form
ln yt minus ln yn = γ θξt + λ(ln ytminus1 minus ln yn)
where as before θ = σ 2z (σ 2 + σ 2
z ) Substituting for prior differences in outputfrom its natural level we thus have
ln yt minus ln yn = γ θ
infinsumi=0
λiξtminusi (1025)
which shows that the deviation of output from its natural rate depends on thecurrent and all previous values of the ldquoaggregate demand shockrdquo that affects theequilibrium price level
A complete model except for specifying thesource of expectations
Equation (1016) provides us with one part of the standard macroeconomic modelthe ldquoaggregate supply equationrdquo To simplify the analysis we will normalize outputso that the natural level of real output is equal to 1 We have already assumed that
Lucas model 159
the natural level of real output is constant over time These two assumptions allowus to write (1016) in the more compact form
Yt = γ θ(Pt minus Etminus1Pt) + λYtminus1 (1026)
where Yt denotes log of real output supply for period t (or equivalently the deviationin output from its natural level for period t) Pt denotes the log of the price leveland Etminus1Pt denotes the expectation of log of the price level What we now requireis a characterization of the aggregate demand side of the economy as typicallysummarized by the LM and IS equations
To obtain an explicit form for the portfolio or LM equation we start by assuminga real money demand function for the end of period t of the form
Ldt = yα1
t exp[minus(α2rt)] (1027)
where yt is real output α1 gt 0 and α2 gt 0 indicating that real money demand isdirectly related to real output but inversely related to the nominal interest rate21
Let Mt denote the nominal money supply at the end of period t (previously thishas been denoted by M ) and let mt denote the logarithm of this money supplyfor period t such that mt = ln Mt Further let us assume a logarithmic supply ofmoney function of the form
mt = mt + εt (1028)
Equation (1028) separates the logarithm of the money supply into two compo-nents a deterministic component mt set by government authorities according toa rule tying money supply changes to past variables and a random component εt which is assumed to be normally distributed with zero mean This random term isalso assumed to be serially independent (ie E(εtεs) = 0 for s = t)
The LM equation is simply the money market equilibrium condition equatingthe real supply of money to real money demand and thus is given by
Mtpt = Ldt (1029)
Taking logs of the equilibrium condition (1029) and substituting the logarithm ofthe money demand function (1027) and the money supply function (1028) wehave
mt minus Pt = α1Yt minus α2rt minus εt (1030)
where Pt = ln pt Equation (1030) is the standard log-linear form of the portfolioor LM equation22
To obtain an explicit form for the IS equation which is the equilibrium conditionin terms of equating output production to the demand for output we must postulatea specific form for output demand One common assumption is to include the
160 Lucas model
expected real rate of interest rt minus πet+1 as a determinant of output demand To do
so let
πet+1 equiv (pe
t+1 minus pt)pt = (pet+1pt) minus 1
As before we can approximate the expected rate of change in prices by the expecta-tion of the log of the ratio of the future to current price level (ie πe
t+1 asymp Pet+1minusPt
where Pet+1 is the expected log of the price level for period t+1) Thus the expected
real rate of interest becomes rt minus πet+1
Letting the term Xt denote a vector of exogenous variables that also affectsoutput demand we have in log-linear form the following equilibrium conditionfor the output market
Yt = Xt minus β1(rt minus πet+1) + ut (1031)
where ut is a serially independent stationary random process with mean zeroand finite variance equal to σ 2
u The random terms for output demand and moneysupply εt and ut respectively are assumed independent (ie E(εtut) = 0)
Summarizing we have a model consisting of the aggregate supply equation(1026) the LM equation (1030) and the IS equation (1031) which can be solvedfor the equilibrium output price and interest rate
In particular combining the LM and IS equations to eliminate the interest ratert we obtain
Yt = Xt + ut minus (β1α2) middot (minusmt minus εt + Pt minus α1Yt) + β1 middot πet+1
which on rearranging becomes an ldquoaggregate demand equationrdquo of the form
Yt = α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
] (1032)
Or in terms of the price level we have an ldquoaggregate demand equationrdquo of theform
Pt = mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1) (1033)
Equations (1032) and (1033) indicate the inverse relationship between the pricelevel and output that is shown graphically by a downward-sloping aggregatedemand curve
With respect to (1033) note that the expected rate of inflation for the nextperiod πe
t+1 is viewed as a distinct entity Our prior assumption of unit elasticexpectations concerning the expected log of the future price level Pe
t+1 wouldimply that this term is in fact independent of changes in the current price levelNote also that if output were unchanged then an x percent change in the moneysupply would result in an x percent change in the price level This is the standardresult of the ldquoneoclassicalrdquo model23
Lucas model 161
Combining the above aggregate demand equation (1032) with the aggregatesupply equation to eliminate the output term Yt we have24
α2
α2 + α1β1
[Xt + ut + β1π
et+1 + β1
α2(mt + εt minus Pt)
]
= γ θ(Pt minus Etminus1Pt) + λYtminus1
Solving the above for the equilibrium price level one obtains
Pt = 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1)
+ J0
(Etminus1Pt minus λYtminus1
γ θ
)] (1034)
where J0 = γ θ(α2 + β1α1) Expression (1034) is sometimes called the reduced-form equation for the price level
Rearranging (1034) we have
Pt minus J0
J0 + β1Etminus1Pt
= 1
J0 + β1
[β1(mt + εt) + α2(Xt + ut + β1π
et+1) minus J0
λYtminus1
γ θ
] (1034prime)
Let us assume perfect foresight meaning that Etminus1Pt = Pt Noting that1 minus J0(J0 + β1) = β1(J0 + β1) we can solve (1034prime) for the equilibriumprice level under this hypothesis of perfect foresight obtaining
Pt = mt + εtα2
β1(Xt + ut + β1π
et+1) minus J0
λ
β1γ θYtminus1 (1034primeprime)
As equation (1034primeprime) makes clear under the presumption of limited perfectforesight the predictions are those of the neoclassical model
(a) a change in the money supply (in log form given by mt + εt) results in anequiproportionate change in the price level (in log form given by Pt)
(b) an increase in expected inflation πet+1 raises the price level
(c) a higher level of lagged output (Ytminus1) lowers the price level(d) an increase in output demand (Xt + ut) raises prices
Following a procedure similar to that used to derive (1034) if we combine theaggregate demand equation (1033) and aggregate supply equation to eliminatethe price level we have
Yt = λYtminus1 minus γ θEtminus1Pt
+ γ θ
mt + εt minus α2 + α1β1
β1Yt + α2
β1(Xt + ut + β1π
et+1)
162 Lucas model
Solving for the equilibrium real output we have
Yt = β1
J0 + β1
[λYtminus1 + γ θ
[mt + εt minus Etminus1Pt + α2
β1(Xt + ut + β1π
et+1)
]]
(1035)
The above expression is sometimes call the reduced-form equation for real outputAccording to (1035) changes in the money supply (in logs given by mt = mt +εt)will affect real output to the extent that the impact of such changes on prices is notfully anticipated
Note that with perfect foresight we have Etminus1Pt = Pt In this case substituting(1034primeprime) into (1035) for Etminus1Pt we obtain
Yt = β1
J0 + β1
(λYtminus1 + J0
λ
β1Ytminus1
)= λYtminus1 (1035primeprime)
Thus we have the standard neoclassical result that output in the current period isindependent of demand-side changes such as changes in expected inflation or themoney supply
One source of expectations autoregressive expectations
As Shiller (1978) has noted
one of the most difficult problems which confronts builders of macroeco-nomic models is the need to model the mechanism by which the public formsits expectations of future economic variables Many of the most importanttheoretical macroeconomic behavioral relations (eg the supply equationinvestment saving) depend critically on public expectations of future eco-nomic variables yet we often do not even have any data on what theseexpectations are
This and the next section suggest two approaches that have been taken to modelexpectations in particular the expected price level that enters into the aggre-gate supply equations These two approaches to expectation formation are thedistributed lag (or adaptive) scheme and the rational expectations scheme
To understand the ideas behind distributed lag schemes as the source of expec-tations we start by noting that the price level pt can be broken down into acombination of the price level for the previous period ptminus1 multiplied by theratio of the price level this period to last period
pt equiv ptminus1(ptptminus1)
Taking logs and recalling that ln(ptptminus1) is approximately equal to the rate ofinflation πt we thus have
Pt equiv ln pt = ln ptminus1 + πt
Lucas model 163
Taking expectations at time t assuming that at a minimum the price level for theprior period is known we have
Etminus1Pt equiv ln ptminus1 + Etminus1πt (1036)
Until the 1970s the approach to modeling the source of the expected rate of infla-tion embedded in (1036) was to assume individuals forecast the rate of inflationby looking at past inflation rates A common quantitative representation of thishypothesis originated by Fisher (1930) was to have individualsrsquo expectation ofthe inflation rate behave like a weighted average or ldquodistributed lagrdquo of recent pastinflation rates That is
Etminus1πt =qsum
i=1
ηiπtminusi (1037)
where the ηi are fixed numbers A typical idea behind this distributed lag approachto anticipated inflation was that individuals have ldquoadaptive expectationsrdquo whichmeant that individuals adjusted or ldquoadaptedrdquo their expectations of the rate of infla-tion in light of the actual forecast error made concerning the prior periodrsquos inflationrate Specifically adaptive expectations can be expressed as
Etminus1πt = Etminus2πtminus1 + δ(πtminus1 minus Etminus2πtminus1)
= δπtminus1 + (1 minus δ)Etminus1πtminus1(1038)
where 1 gt δ gt 0 Successive substitution allows us to rewrite (1038) as
Etminus1πt = δπtminus1 + (1 minus δ)δπtminus2 + (1 minus δ)2δπtminus3 + middot middot middotor
Etminus1πt =infinsum
i=1
δ(1 minus δ)iminus1πtminusi (1039)
As you can see (1039) is simply a specific form of equation (1037) in whichthe ηi place declining weight on past inflation rates the more distant they are andq = infin Given declining weights we can obtain a reasonable approximation of(1039) even if we truncate the distributed lag on past inflation after q periods aslong as q is reasonably large andor δ is reasonably large
Now let us place the above discussion not in terms of past rates of inflation butinstead in terms of past price levels Recalling the approximation
πtminusi asymp ln ptminusi minus ln ptminusiminus1
we can rewrite (1037) in the form
Etminus1πt =qsum
i=1
ηi(ln ptminusi minus ln ptminusiminus1) (1040)
164 Lucas model
Writing this out we have
Etminus1πt = η1(ln ptminus1 minus ln ptminus2) + η2(ln ptminus2 minus ln ptminus3)
+ η3(ln ptminus3 minus ln ptminus4) + middot middot middot + ηq(ln ptminusq minus ln ptminusqminus1)
Thus we may rewrite (1040) as
Etminus1πt = η1 ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
(1041)
Combining (1041) with equation (1036) we obtain the following expression forthe expectation formed at time t concerning the log of the price level
Etminus1Pt = (1 + η1) ln ptminus1 + (η2 minus η1) ln ptminus2 + (η3 minus η2) ln ptminus3 + middot middot middot+ (ηq minus ηqminus1) ln ptminus4 minus ηq ln ptminusqminus1
or
Etminus1Pt =q+1sumi=1
vi ln ptminusi (1042)
Equation (1042) is what Sargent and Wallace (1975) refer to as ldquoautoregressiveexpectationsrdquo
A second source of expectations rational expectations
The papers by Lucas (1972 1973) and Sargent and Wallace (1975) suggestedthat in macroeconomic model building a different approach to specifying thesource of expectation is preferred As Cukierman (1986) summarizes this ldquorationalexpectationsrdquo approach to the modeling of inflationary expectations is
based on the maintained hypothesis that individuals know the structure of theeconomy and of governmentrsquos decision rule and that they use this structurein conjunction with the available information in order to form an optimalpredictor of future inflation [this approach] requires a precise specificationof the model of the economy as well as of the information sets of individualsEmpirical tests of this hypothesis are therefore joint tests of the validity of theexpectational hypothesis as well as of the postulated structure of the economyand of the particular assumptions made about the information possessed byindividuals
A ldquostructure of the economyrdquo was derived based on the Lucas aggregate sup-ply equation that included the assumption of market-clearing wages and pricesSuppose that individuals know this model and accept it as reflecting the structureof the economy As we saw above this model was solved to obtain (1034) the
Lucas model 165
reduced form for the equilibrium price level in period t We assume that at time tindividuals form their expectations Etminus1Pt ldquorationallyrdquo in that
Etminus1Pt = E(Pt |tminus1) (1043)
indicating that Etminus1Pt is the mathematical expectation of Pt conditional on theinformation set tminus1 which is all information available at period t minus1 As Sargentnotes (1987a 440)
Lucas assumed that tminus1 included information on all lagged values of ln pitand lagged values of real output in all markets One could equally well con-ceive of less comprehensive definitions of tminus1 For now along with Lucaswe suppose that tminus1 includes a comprehensive list of variables includinglagged outputs and prices in all markets
At the end of period t minus 1 individuals of course know Etminus1Pt as well as Ytminus1and πe
t+1 Thus taking the expectation of (1034) and subtracting it from Pt wehave
Pt minus Etminus1Pt = β1
J0 + β1(mt + εt minus Etminus1mt) + α2
J0 + β1(Xt minus Etminus1Xt + ut)
(1044)
In Sargent and Wallace (1975 244) the deterministic part of the money supplymt is assumed to reflect a ldquolinear feedback rulerdquo of the form
mt = Gθlowastt (1045)
where ldquoθlowastt represents the set of current and past values of all of the endogenous and
exogenous variables in the system as of the end of period t minus 1 and G is a vectorof parameters conformable to θlowast
t rdquo A simple example of a monetary feedback rulewould be
mt = a0 + a1Ytminus1 (1046)
where a0 and a1 are positive constantsLet us assume that individualsrsquo information set tminus1 includes not only the
structure of the economy (as summarized by the above linear macroeconomicmodel) but also the money supply rule In the particular example of (1046) theyknow a0 a1 and Ytminus1 and thus mt Then the assumption of rational expectationsimplies that Etminus1mt = mt Furthermore since Xt represents the deterministicpart of the vector of exogenous variables affecting output demand we have thatEtminus1Xt = Xt Thus (1044) becomes
Pt minus Etminus1Pt = 1
J0 + β1(β1εt + α2ut) (1047)
166 Lucas model
Substituting (1047) into the aggregate supply equation one obtains
Yt = γ θ1
J0 + β1(β1εt + α2ut) + λYtminus1 (1048)
An important feature of (1048) is that a deviation in output from its natural levelwhich is represented by the term Yt different from zero given our assumption thatthe natural output level is normalized to equal one is determined only by pastdeviations and ldquosurprisesrdquo with respect to the money supply (the term εt) andoutput demand (the term ut) The ldquodeterministicrdquo or predictable component of anymoney supply change has no real effects Equation (1048) should look familiarGiven λ lt 1 it suggests that the times series for deviations of the logarithmof output from its natural rate is a stationary autoregressive process of order 1or AR(1)
The above analysis is an example of a ldquolinear rational expectations modelrdquoThe result that ldquopredictablerdquo monetary policy has no real effects reflects the twinassumptions of the natural rate hypothesis and rational expectations It should notbe surprising that deterministic monetary policy has no effect for the model ishomogeneous of degree 0 in mt Pt and Etminus1Pt This is a critical characteristic ofa ldquonatural rate modelrdquo
Conclusion
The main focus of this chapter has been on the development of the Lucas supplyfunction The model is often discussed in the context of the ldquoislandrdquo paradigmin which we specify the supply function for a particular sector in the economyThe role of forecasting errors is introduced and from that the Lucas aggregatesupply function is constructed and the relationship of this function to the Phillipscurve is discussed A number of other issues were discussed involving variabilityin prices and the corresponding economic tradeoff and then a complete modelwas introduced except for specifying the source of expectations Two sources ofexpectations were then described autoregressive expectations and rational expec-tations The implications of expectations were then discussed in their historicalcontext
11 Policy
Introduction
This chapter extends the earlier discussions about the actions of the monetaryauthority and how these actions affect the macroeconomy Perhaps the most inter-esting issue of monetary economics is addressed here that is the optimal roleof monetary policy The chapter highlights the differences in model results thatdepend on what type of expectations are assumed Particular attention is givento the Sargent and Wallace ldquoineffectiveness propositionsrdquo and the Phillips curveOther issues are then introduced including the ldquorule versus discretionrdquo debatetime inconsistency and the role of credibility and enforcement
Optimal monetary policy
In the 1970s the articles by Sargent and Wallace (1975 1976) and Lucas (19721973) altered the view of how one should assess the impact of monetary policy onthe economy and by implication what is optimal monetary policy The discussionstarts with the premise that monetary policy should be conducted according to arule or set of rules As Sargent and Wallace (1976 169) state
It is widely agreed that monetary policy should obey a rule that is a scheduleexpressing the setting of the monetary authorityrsquos instrument (eg the moneysupply) as a function of all the information it has received up through thecurrent moment Such a rule has the happy characteristic that in any givenset of circumstances the optimal setting for policy is unique If by remotechance the same circumstances should prevail at two different dates theappropriate settings for monetary policy would be identical1
The SargentndashWallace premise that monetary rules are preferred leads them toexplore the form of the optimal rule But as we will discover in going over thepaper by Barro and Gordon (1983) there is a question of whether monetary rulescan be enforced over time If not then what is typically left in these models isa ldquosecond bestrdquo solution involving the determination of optimal ldquodiscretionaryrdquo
168 Policy
policy Note that we use the term ldquosecond bestrdquo because enforceable rules tend todominate discretion in these models2
Accepting the premise that monetary policy can adopt enforceable rules stillleaves open the specification of the optimal set of rules The simplest rule sug-gested by Friedman (1959) would increase the money supply at a constant rateeach year perhaps 3 percent3 More complex rules known as ldquoreactive rulesrdquowould specify in advance how the growth of the money supply will change basedon new information on the state of the economy One such rule suggested is thatthe growth in the monetary base and thus the money supply automatically adjustwhenever the growth of nominal GNP deviates from its trend (McCallum 1985)Another reactive rule suggests that the government commit itself to holding theCPI to a preannounced target and adjust the monetary base and thus the moneysupply accordingly (Hall 1982)4
Below we begin our discussion of optimal monetary policy by reviewing theanalysis of optimal enforceable rules as suggested by the Sargent and Wallace(1975 1976) papers and reviewed in Sargent (1987a Chapter 17) This discussionis in the context of a natural rate model without rational expectations and then inthe context of a model which assumes rational expectations
Optimal monetary policy exogenous expectations
As we saw previously the reduced form for the log of the output in period t canbe expressed as
Yt = H0
[λYtminus1 + γ θ
[minusEtminus1Pt + mt + εt + α2
β1(Xt + ut + β1π
et+1)
]]
(111)
where H0 = β1(J0 +β1) and J0 = γ θ(α2 +β1α1) Recall that Yt is the differencein period t between the logarithm of output and the logarithm of the natural levelof output Normalizing so that the natural level of output equals one we canequivalently interpret Yt in equation (111) as total output As Sargent and Wallace(1976) state ldquoYt can be thought of as the unemployment rate or the deviation ofreal GNP from lsquopotentialrsquo GNP This equation should be thought of as the reducedform of a simple econometric modelrdquo
Recall that the log of the money supply in period t mt is the sum of adeterministic component mt and the random component εt with variance σ 2
e 5
To understand the impact of monetary changes on real GNP we must firstconsider how the expected log of the price level Etminus1Pt varies with changes inmonetary policy One approach in the spirit of ldquoautoregressive expectationsrdquo isto assume that the expected price level is independent of the current monetarypolicy This essentially means viewing the expectation of the log of the price level(Etminus1Pt) as exogenous6 Given this assumption we may rewrite the reduced-form
Policy 169
equation for output (111) as
Yt = a0 + a1Ytminus1 + a2mt + vt (112)
where vt = H0γ θ(α2β1)ut is a serially independent normally distributed randomvariable with variance σ 2
v and mean zero mt is the log of the money supply forperiod t where mt = mt + εt Ytminus1 is lagged output and a0 a1 and a2 areparameters
Suppose that the monetary authority desires to set the money supply in orderto minimize the fluctuation in the log of output around some desired level Let usassume that the log of this desired level denoted Y lowast is above the log of the naturallevel of output Yn7 Then the objective can be expressed as
min Etminus1(Yt minus Y lowast)2
We can break this expression into two terms in that the objective can beequivalently expressed as
min Etminus1(Yt minus Etminus1Yt)2 + (Etminus1Yt minus Y lowast)2 (113)
The second way of expressing the objective allows us to see the objective asminimizing the sum of two terms the variance of Yt conditional on informationup to the end of period t minus 1 and the ldquobias squaredrdquo around Y lowast The secondterm the bias squared around Y lowast is the reason for an ldquoactivistrdquo monetary policyEquation (113) indicates that the optimal monetary policy entails
1 minimizing the variance in the random component of the money supply Thisfollows since the first term in equation (113) the variance of Yt is given bya2
2σ2e +σ 2
v 8 If feasible complete elimination of the random component to themoney supply (ie a purely ldquodeterministicrdquo money supply) is optimal suchthat εt = 0 for all t and thus σ 2
e = 02 setting Etminus1Yt = Y lowast so as to make the second term in equation (113) equal
to zero From equation (112) this means a monetary policy such that
Etminus1(a0 + a1 middot Ytminus1 + a2 middot mt + vt) = Y lowast
Noting that Etminus1Yt = 0 and that Etminus1mt = mt we see that this deterministic partof the optimal monetary policy is defined by the equation
a0 + a1Ytminus1 + a2mt = Y lowast
which can be solved for the optimal deterministic monetary policy rule
mt = g0 minus g1Ytminus1 (114)
where g0 = (Y lowast minus a0)a2 and g1 = a1a2
170 Policy
An equivalent expression for the optimal monetary rule (114) is derived inSargent (1987a Chapter 17) under the presumption of exogenous expectations Inparticular Sargent uses equation (112) to substitute out for Ytminus1 in (114) so thatthe optimal (deterministic) monetary rule (114) becomes
mt = g0 minus g1[a0 + a1Ytminus2 + a2mtminus1 + vtminus1] (115)
Now substituting into (115) the expression for Ytminus2 suggested by equation (112)we have
mt = g0 minus g1[a0 + a2mtminus1 + vtminus1] minus g1a1[a0 + a1Ytminus3 + a2mtminus2 + vtminus2](116)
Continuing to successively substitute for Ytminusi i = 3 4 we have Sargentrsquosequivalent expression for the optimal (deterministic) policy rule as given by9
mt = g0 minus g1a0 minus⎛⎝g1a2
infinsumiminus1
aiminus11 mtminusi
⎞⎠ minus
(g1
infinsumi=1
aiminus11 + vtminusi
) (117)
Following the above optimal monetary policy (ie reducing any random compo-nent to the money supply to its minimum level and establishing the rule for thedeterministic component of the money supply as specified by (114) or (117)) wehave by construction that
Etminus1Yt = Y lowast and Yt = Y lowast + a2εt + vt
where vt = H0γ θ(α2β1)ut and the variance of the random component of themoney supply εt is set at its lowest feasible level Thus optimal monetary policyin essence sets output each period equal to Y lowast plus irreducible noise As Sargentand Wallace (1976 171) note
the application of the rule eliminates all serial correlation in output since thisis the way to minimize the variance in output The basic idea is that wherethe effects of shocks to a goal variable (like GNP) display a stable pattern ofpersistence (serial correlation) and hence are predictable the authority canimprove the behavior of the goal variable by inducing offsetting movementsin its instruments
Note that without the lag term for output g1 in (114) equals zero
Adaptive expectations and the accelerationist result
The well-known ldquoaccelerationist outcomerdquo concerning the path of inflation isimplied by the above analysis if expectations are adaptive and if the aim of mone-tary policy is to keep output above its natural level To see this let us go back to the
Policy 171
aggregate supply equation that underlies the reduced form for output To simplifylet us abstract from the stochastic elements in demand εt and ut (as well as anysupply-side disturbances) and also from adjustment costs (ie omit the laggedoutput term) Then the aggregate supply equation
Yt = γ θ [πt minus Etminus1πt] (118)
reflects the actual path that output will take10
To derive the accelerationist result assume that Y lowast gt Yn By normalizationYn = 0 so that to have Etminus1Yt = Y lowast gt Yn we must have Etminus1Yt gt 0Equation (118) suggests that to achieve an expected level of output greater thanits natural level the government must pursue a monetary policy that results inthe actual rate of inflation typically being above that expected In particular thedesire to keep output above its natural level means that monetary policy in period tresults in the actual inflation rate πt such that
πt minus Etminus1πt = Y lowastγ θ gt 0
If expectations are adaptive then we have that
Etminus1πt minus Etminus2πtminus1 = δ(πtminus1 minus Etminus2πtminus1)
The assumption of adaptive expectations coupled with Y lowast gt Yn = 0 thus impliesthat
Etπt+1 = Etminus1πt + δ(πt minus Etminus1πt) = Etminus1πt + δY lowastγ θ
In words the fact that individuals underestimate inflation this period (by theamount Y lowastγ θ ) leads them to adjust (ldquoadaptrdquo) their expectations of inflationupward (by the amount δY lowastγ θ ) The result is that to keep expected output atthe level Y lowast gt Yn next period means an increase in the inflation rate by theamount δY lowastγ θ each period As Blanchard and Fischer (1989 572) note
this is the famous accelerationist result derived by Friedman (1968) andPhelps (1968) using their Phillips curve together with the adaptive expec-tations assumption The explanation is simple if the government is trying tokeep output above the natural rate it has to produce inflation at a higher ratethan expected each period Since the expected inflation is a weighted averageof past inflation rates the actual rate must be increasing
Now let us assume instead that Y lowast = 0 To provide a role for an activist monetarypolicy let us reintroduce the lagged output term λYtminus1 into the right-hand sideof (118) Thus monetary policy can eliminate the effect of past deviations inoutput from the natural rate on current output In other words a policy aimed
172 Policy
at setting Etminus1Yt = Y lowast would imply altering inflation relative to expected onlywhen lagged output deviated from the natural level Rather than the prior result ofan ever-increasing inflation with Y lowast gt Yn with Y lowast = Yn we have that inflationsimply wanders11 For instance if logged lagged output Ytminus1 fell below the log ofthe natural level of output Yn = 0 then the difference Ytminus1 would be negativeOther things being equal this would imply a lower output in the current periodTo counteract this the government would pursue a monetary policy that results inthe actual rate of inflation being above that expected
The above discussion helps us understand the comment of Hall (1976) that
the benefits of inflation derive from the use of expansionary power to trickeconomic agents into behaving in socially preferable ways even thoughtheir behavior is not in their own interests The gap between actual andexpected inflation measures the extent of the trickery the optimal pol-icy is not nearly as expansionary when expectations adjust rapidly andmost of the effect of an inflationary policy is dissipated in costly anticipatedinflation
The above extract raises the following question Can the monetary authoritiessystematically trick the public in order to exploit the link between inflation andoutput For Sargent and Wallace and others the answer is no due to the existenceof rational expectations
Rational expectations and the SargentndashWallace ineffectivenessproposition
As Sargent (1987a) notes a critical aspect of the simple example of an optimalmonetary rule as given by (114) (or equivalently (117)) ldquois the implicit assumptionthat agentrsquos decision rules remain unchanged in the face of alternative stochas-tic processes for the control variable that different feedback rules implyrdquo WhatSargent means in this context is that the optimal monetary rule has been derivedunder the presumption that private agents do not take this rule into account informing their expectation of the price level Under this assumption one couldestimate the parameters of the reduced-form equation output (a0 a1 and a2 in(112)) independently of the feedback rule (114)
However Sargent and Wallace (1976) criticize this view In particular theyargue that ldquoin the reduced forms are embedded the responses of expectations tothe way policy is formed Changes in the way policy is made then ought not toleave the parameters of estimated reduced forms unchangedrdquo12 In other wordsrational individuals would clearly seek out and use information on how monetaryauthorities act as well as on the structure of the economy in forming expectationsof prices
Let us now consider the following version of the reduced-form equation foroutput (112) that explicitly includes the potential role of expected monetary policy
Policy 173
when individuals form expectations on prices13
Yt = a0 + a1Ytminus1 + a2(mt minus Etminus1) + vt (119)
For a given anticipated log of the money supply Etminus1mt we have as before theoptimal (deterministic) monetary rule of the form
mt = g0 minus g1Ytminus1 (1110)
so that
mt = g0 minus g1Ytminus1 + εt (1111)
where εt is the irreducible random element in the money supply determinationprocess Now assume that the public knows the monetary authoritiesrsquo feedbackrule Then our assumption of rational expectation (ie individuals use all availableinformation in forming expectations) implies that
Etminus1(mt) = g0 minus g1Ytminus1 (1112)
Combining (119) (1111) and (1112) the reduced form for output is now given by
Yt = a0 + a1Ytminus1 + a2εt + vt (1113)
so that the biased squared term in the objective of the monetary authorities(Etminus1Yt minus Y lowast)2 equals (a0 + a1Ytminus1 minus Y lowast)2
As is clear from (1113) there is no role for systematic monetary policy to affectreal output As Sargent (1987a 459) notes ldquothe bias squared is independent ofthe parameters of the money supply rulerdquo The optimal policy is then to makemonetary policy deterministic if feasible for then the variance of output (given bya2
2σ2e + σ 2
v ) is minimized by setting σ 2e = 0 Until we add an inflation objective
any deterministic rule will be equally as good for none will have any impact onoutput This is once again an example of the neutrality of money
As Sargent (1987a 458) notes ldquopolicy rules should be deterministic and involveno surprisesrdquo He goes on to argue that we
have therefore established the following stochastic neutrality theorem thatcharacterizes our model one deterministic feedback rule on the basis of theinformation set tminus1 which is common to the public and to the authority is asgood as any other deterministic feedback rule Via deterministic feedbackrules the monetary authority is powerless to combat the business cycle (theserial correlation in Yt)
Naturally if one abandons rational expectations or the natural rate hypothesis thenthis ldquostochastic neutralityrdquo result need not hold
174 Policy
The SargentndashWallace ineffectiveness proposition in thecontext of the Phillips curve
The above finding of the ldquoneutrality of moneyrdquo in the context of a stochasticlinear natural rate model with rational expectations is viewed by Sargent (1987a459) as
the antithesis of our earlier result rationalizing the activist Keynesian policyrules The reader is invited to verify that the truth of the neutrality theoremis not dependent on the particular information set assumed It will continueto hold for any specification of tminus1 so long as the public and the authorityshare the same information set
He concludes
The preceding results provide a [weak] defense for following rules with-out feedback Simple x-percent growth rules do as well as any deterministicfeedback rule and dominate rules with a stochastic component
Below we recast the SargentndashWallace ineffectiveness proposition in terms ofthe expectational Phillips curve This makes the discussion more in line with thenext sectionrsquos review of some implications of non-enforceable monetary rules Inaddition we add to the government goalrsquos an inflation objective In particular wemodify our analysis in the following four ways
1 we alter the objective to be in terms of unemployment rather than output2 we expand our objective function to include inflation3 we link inflation to unemployment via a modified Lucas supply equation4 we incorporate rational expectations
Our first task is to convert the objective of the government into unemploymentterms Before we assumed that the government simply sought to minimize thefluctuations in output about a particular level In particular if we let Zt denote thecost incurred in period t we assumed the objective was to
min Etminus1Zt where Zt equiv (Yt minus Y lowast)2 (1114)
We have previously assumed that the deviation of unemployment from its naturalrate is linearly related to the deviation of the log of output from the log of its naturallevel In particular we assumed
minus(Ut minus Un) = Yt
Policy 175
given the normalization of the natural level of output such that Yn equiv ln yn = 0Substituting the above into (1114) the problem facing the government policy-maker becomes
min Etminus1Zt where Zt = a(Ut minus kUn)2 a = 2 gt 0
k = 1 minus Y lowastUn (1115)
Note in (1115) that we assume k lt 1 which is equivalent to assuming thatthe log of optimal level of output Y lowast is greater than the log of the natural levelof output which by normalization has been set equal to zero As Blanchard andFischer (1989 596ndash597) suggest
The most plausible justification [for k lt 1] is the presence of distortions orimperfections that causes the natural rate of unemployment to be too high Thisjustification allows the loss function to be consistent with the single-periodutility function of private agents Another is that the governmentrsquos objectivefunction as shaped by the electoral process leads the government to seek toraise output above the natural level
Having converted the objective function into unemployment terms the next stepis to expand the objective function to include an inflation goal In particular let usassume that the cost in period t includes a term reflecting differences between theactual inflation rate πt and an optimal rate of inflation πlowast Assuming a simplequadratic form we have14
Zt = a(Ut minus kUn)2 + b(πt minus πlowast)2 (1116)
The problem of the government policy-maker is then
min Etminus1Zt where Zt equiv a(Ut minus kUn)2 + b(πt minus πlowast)2
As before we can decompose this objective to obtain the following equivalentexpression for the object of the government policy-maker
min a[Etminus1(Ut minus kUn)2 + (Etminus1Ut minus kUn)
2]+ b[Etminus1(πt minus πlowast)2 + (Etπt minus πlowast)2] (1117)
Our third task is to link inflation to unemployment Recall that if we ignore thelagged term with respect to output in the Lucas supply equation assume unem-ployment is linearly related to real output and approximate inflation by the log ofthe ratio of the price level this period to the price level last period then we canmanipulate the Lucas supply equation to obtain the expression
Ut = Un minus α(πt minus Etminus1πt) (1118)
176 Policy
where α = (γ θ) gt 015 Substituting (1118) into (1116) the governmentrsquosobjective becomes
min Etminus1a(1 minus k)Un minus α(πt minus Etminus1πt)2 + b middot (πt minus πlowast)2
Our fourth and final task is to introduce rational expectations As we saw earlierwe obtain the reduced form for the deviation of the price level from that expectedin period t
Pt minus Etminus1Pt = 1
J0 + β0(β1εt + α2ut) (1119)
if (a) the government follows a particular rule in determining monetary policy(b) that rule is known to the public and (c) there exist rational expectations
An algebraic manipulation ndash simultaneously subtracting and adding the log ofthe price level for period t minus 1 to the left-hand side of equation (1119) ndash bringsus closer to having an expression that may be interpreted in terms of inflation
ln( ptptminus1) minus Etminus1(ln( ptptminus1) = 1
J0 + β1(β1εt + α2ut) (1120)
Using the log of the price ratio as an approximation for inflation we thus canapproximate (1120) as
πt minus Etminus1πt = 1
J0 + β1(β1εt + α2ut) (1121)
Substituting the above which reflects the rational expectations approach tomodeling expectations into the new government objective we have
min Etminus1
a
[(1 minus k)Un minus α
J0 + β1(β1εt + α2ut)
]2
+ b(πt minus πlowast)2
or
min a[(1 minus k)Un]2 +(
α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1 b(πt minus πlowast)2 (1122)
since Etminus1εt = Etminus1ut = Etminus1εtut = 0It is clear from (1122) that with rational expectations monetary policy can play
no role in helping the government meet its objective concerning unemploymentIn other words in the context of the Lucas model the assumption of rationalexpectations means that the expected loss from deviations in unemployment fromits desired level and thus production from its desired level is independent ofthe deterministic monetary rule As a consequence the minimization problem asgiven by the first half of equation (1117) becomes simply one of specifying any
Policy 177
deterministic monetary policy rule and eliminating any random changes in themoney supply (ie setting σ 2
e = 0 if feasible)But the objective of the government now also includes an objective concerning
the rate of inflation so we need an expression for the equilibrium rate of changein prices πt that incorporates rational expectations To do so we start with thereduced form for the log of the price level obtained previously which is given by
Pt = 1
J0 + β1
[β1mt + εt + α2(Xt + ut + β1π
et+1) + J0
(Etminus1Pt minus λYtminus1
γ θ
)]
(1123)
Taking the difference between the reduced form for the log of the price level forperiod t and for period tminus1 we can obtain an expression for πt the rate of inflationbetween period t and t minus 1 of the form
πt = 1
J0 + β1[β1mt + α2(Xt minus Xtminus1 + ut minus utminus1) + α2β1(π
et+1 minus πw
t )
+ J0(Etminus1Pt minus Etminus2Ptminus1) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1124)
where πmt approximates the rate of change in the money supply (ie πmt =mt minus mtminus1 = ln(MtMtminus1)) Assuming rational expectations we have fromequation (1119) that
Etminus1Pt = Pt minus 1
J0 + β1(β1εt + α2ut)
and similarly
Etminus2Ptminus1 = Ptminus1 minus 1
J0 + β1(β1εtminus1 + α2utminus1)
so that
Etminus1Pt minus Etminus2Ptminus1 = πt + 1
J0 + β1[β1(εtminus1 minus εt) + α2(utminus1 minus ut)]
(1125)
Substituting this expression into equation (1124) one obtains
πtβ1
J0 + β1= 1
J0 + β1[β1πmt + α2(Xt minus Xtminus1 + ut minus utminus1)
+ α2β1 middot (πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)] (1126)
178 Policy
where we use the fact that 1 minus J0(J0 + β1) = β1(J0 + β1) Solving for πt we have
πt = πmt + α2
β1(Xt minus Xtminus1 + ut minus utminus1)
+ α2(πet+1 minus πe
t ) minus J0λ
γ θ(Ytminus1 minus Ytminus2)
(1127)
which given πmt = mt minus mtminus1 = mt + εt minus (mtminus1 + εtminus1) can be rearranged toobtain
πt = πmt + εt minus εtminus1 + α2
β1(Xt minus Xtminus1 + ut minus utminus1) + α2(π
et+1 minus πe
t )
minus J0λ
β1γ θ(Ytminus1 minus Ytminus2) (1128)
where πmt = mt minus mtminus1If we assume no change in exogenous (nonrandom) demand factors this period
compared to the last period (ie Xt = Xtminus1) no change in the expected future infla-tion between last period and this period (ie πe
t+1 = πet ) and ignore adjustment
costs (ie λ = 0) we can simplify equation (1128) to obtain
πt = πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) (1129)
The three assumptions we made to derive equation (1129) from equation (1128)largely limit any differences between period t and t minus 1 to differences in the sizeof the money supply In fact the two periods differ only by the deterministiccomponent of the money supply and by random factors where these randomfactors ndash the term ut minus utminus1 with respect to output demand and the term εt minus εtminus1with respect to the irreducible random component of the money supply ndash havemean zero In other words all potential changes in aggregate demand or productionexcept money supply changes that would lead to different expected price levels inthe two periods have been removed
Substituting equation (1129) into (1122) we thus obtain the following completegovernment objective function under rational expectations
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ Etminus1b[πmt + εt minus εtminus1 + α2
β1(ut minus utminus1) minus πlowast]2
]
from which we see that constant monetary growth will not achieve a constant rateof inflation unless we neglect the lagged disturbance terms16 To obtain the resultof an optimal monetary rule in the form of a constant rate of growth in the moneysupply we must further simplify and neglect the lagged disturbance terms If we
Policy 179
for the moment ignore the lagged stochastic terms εtminus1 and utminus1 we have theobjective
min a
[[(1 minus k)Un]2 +
(α
J0 + β1
)2
(β21σ 2
e + α22σ 2
u )
+ b(πmt minus πlowast)2 + σ 2e +
(α2
β1
)2
σ 2u
]
As before to minimize the loss requires that one reduces the random variation inthe money supply to zero (if feasible) so that σ 2
e = 0 That is the optimal monetaryrule is ldquodeterministicrdquo Further given an inflation objective and no reason otherthan monetary policy for prices to change the obvious optimal rule is simply toset the determinant rate of change in the money supply equal to the desired rate ofinflation That is the optimal policy is
πmt = πlowast
Note that this rule assumes no shocks to the economy If there were a steady rateof increase in output (and thus the real demand for output) then that would raisethe optimal constant rate of change in the money supply to achieve a given rate ofchange in prices Thus Friedmanrsquos ldquo3 percentrdquo rule for monetary growth presumeda 3 percent growth in real output so as to be consistent with a zero rate of inflation
Rules versus discretion monetary policy and timeinconsistency
The SargentndashWallace ineffectiveness proposition has been used by many to supportarguments for the government not adopting activist policy rules to offset fluctua-tions ndash particularly downturns ndash in an economyrsquos real output reflecting demand-sidedisturbances The reason as we have seen is that such policies have no real effectsonce one assumes rational expectations although the attempt can lead to higherinflation
Yet as Barro and Gordon (1983) point out
empirical studies indicate the presence of countercyclical monetary policyat least for the post-World War II United States ndash rises in the unemploymentrate appear to generate subsequent expansions in monetary growth Within thenatural rate framework it is difficult to reconcile this countercyclical behaviorwith rationality of the policymaker
As Barro and Gordon go on to say ldquoa principal object of our analysis is to achievethis reconciliationrdquo That is rather than saying what policy rules governmentshould follow Barro and Gordon want to explain why government acts the way itdoes ndash thus the term ldquopositive theoryrdquo in the title of their paper
180 Policy
The BarrondashGordon approach combines a number of topics that we haveconsidered before First they utilize a natural rate model like the Lucas modelSecond they assume rational expectations And third and most interestingly theyprovide us with a nice example of the phenomenon of ldquotime inconsistencyrdquo in thecontext of optimal monetary policy when there exists the potential Phillips curvetradeoff
Contrasting the BarrondashGordon and SargentndashWallace policyenvironments
In the SargentndashWallace view of the optimal monetary policy choice it is assumedthat the policy-maker makes a once-and-for-all decision with respect to the partic-ular monetary policy rule (reactive or not) Under certain assumptions as we haveseen this leads to the natural conclusion that optimal monetary policy entails thesimple rule of a constant rate of growth in the money supply so that the resultingaverage rate of inflation equals the desired level If the desired level were zerothen inflation would be set equal to zero through appropriate monetary policy
Barro and Gordon (1983 598) have questioned this result on the basis that ldquotheremay be no mechanism in place to constrain the policymaker to stick to the rule astime evolvesrdquo The result is that the policy-maker decides each period the optimalmonetary policy to follow In other words ldquothough the objective function anddecision rules of private agents are identicalrdquo Barro and Gordon obtain differentresults from Sargent and Wallace because ldquothe problems differ in the opportunitysets of the policymakerrdquo Below we illustrate the exact nature of this differenceby first showing how Barro and Gordonrsquos setup provides an example of the ldquotimeinconsistency problemrdquo not present in the Sargent and Wallace problem and thenderive the equilibrium for an economy characterized by the policy-makers whoare allowed each period to pick a potentially new optimal monetary policy
Time inconsistency an example in the context of optimalmonetary policy
In an important paper on the ldquotime inconsistencyrdquo problem of optimal policyKydland and Prescott (1977) point out situations in which the optimal policiesdecided at time t would be changed at time t + 117 In the context of monetarypolicy as Kydland and Prescott note
the reason that such policies are suboptimal is not due to myopia The effect of(monetary policy) upon the entire future is taken into consideration Rather thesuboptimality arises because there is no mechanism to induce future policy-makers to take into consideration the effect of their policy via the expectationsmechanism upon current decisions of agents
Let us see what this means by way of a concrete example
Policy 181
A simple example of ldquotime inconsistencyrdquo following Kydland and Prescott isconstructed below To simplify the discussion somewhat for the moment we ignoreuncertainty and restrict our attention to two periods In this setting the govern-ment through monetary policy can determine each period the actual inflation raterather than the expected rate of inflation For periods 0 and 1 the choice variablesfacing the monetary policy-maker are thus π0 the actual inflation in period 0 andπ1 the actual inflation in period 1 The inflation choice impacts the unemploymentrate in that U0 and U1 the unemployment rates in periods 0 and 1 are given bythe Phillips curve formulation (1118)
U0 = Un minus α(π0 minus Eminus1π0)
U1 = Un minus α(π1 minus E0π1)
where the state variables Eminus1π0 and E0π1 denote the expected rate of inflation inperiods 0 and 1 respectively In periods 0 and 1 the objective for periods 0 and 1is to minimize the simple quadratic forms
Z0 = a(U0 minus kUn)2 + b(π0 minus πlowast)2
Z1 = a(U1 minus kUn)2 + b(π1 minus πlowast)2
where the constants a and b are positive It is assumed that k lt 1 to capture theidea that distortions exist in the economy that an activist policy can address18
Substituting in the ldquoPhillips curverdquo relations in period 0 the present value of theobjective function is
Z = Z0 + β middot Z1 = a[(1 minus k)Un minus α(π0 minus Eminus1π0)]2 + b[π0 minus πlowast]2
+ βa[(1 minus k)Un minus α(π1 minus E0π1)]2 + βb[π1 minus πlowast]2
where β is the constant discount factorLet us presume that the expectations of inflation the ldquostaterdquo variables are
exogenous and equal to the desired level each period (ie Eminus1π0 = E0π1 = πlowast)In period 0 the optimal inflation rates for periods 0 and 1 (as determined bymonetary policy) are then
partZpartπ0 = minus2aαq0 + 2b(π0 minus πlowast) = 0
partZpartπ1 = β[minus2aαq1 + 2b(π1 minus πlowast)] = 0
where qi = (1 minus k)Un minus α(πi minus πlowast) i = 0 1 In period 1 the objective is
Z1 = a[(1 minus k)Un minus α(π1 minus E0π1)]2 + b[π0 minus πlowast]2
and the optimal solution given π0 and E0π1 = πlowast is thus
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
182 Policy
Comparing this first-order condition to partZ0partπ1 it is clear that in the case ofexogenous expectations the optimal solution is time-consistent Further it impliesa rate of inflation greater than the expected (optimal) rate of πlowast each period toachieve an unemployment rate below the natural This follows given that one ofthe objectives is to approach a level of unemployment equal to kUn with k lt 1
What happens however if individualsrsquo expectations of inflation for period 1 isa forecast that correctly anticipates the optimal future policy decision That is wehave ldquorational expectationsrdquo such that in a deterministic world
E0π1 = h(0) = π1
where 0 is the information set at the end of period zeroWhat the optimal inflation policy is now depends on whether policy-makers
can ldquocommitrdquo to future policy actions Let us start by assuming they can placeconstraints on future policy We know that in period 0 the optimal solution for π1is then given by
βpartZ1
partπ1+ partZ
partE0π1
dh(middot)dπ1
= β[2b(π1 minus πlowast)] = 0
Since dh(middot)partπ1 = 1 the optimal planned (at time 0) rate of inflation for period 1is equal to the desired level πlowast This is the SargentndashWallace ineffectivenessproposition
However once period 1 occurs E0π1 is set (let us assume it equals πlowast) If thepolicy-maker was not then constrained by the prior specification of a monetarypolicy to achieve a rate of inflation equal to πlowast they would choose π1 such that
partZ1partπ1 = minus2aαq1 + 2b(π1 minus πlowast) = 0
which implies π1 gt πlowast The ldquotime inconsistencyrdquo arises as you can see becauseparth(middot)partπ1 = 0 As Barro and Gordon state ldquothe term lsquotime inconsistencyrsquo refersto the policymakerrsquos incentives to deviate from the rule when private agents expectit to be followedrdquo
Equilibrium when monetary rules are not enforceable
As Barro and Gordon note in the time inconsistency problem ldquoconstraints onfuture policy actions are infeasible by assumptionrdquo In contrast in the SargentndashWallace view
rules are enforceable so that the policymaker can commit the course of futurepolicy (and thus of expectations) In the former case the time-inconsistentsolution is not an equilibrium given the problem facing the policymaker Inthe latter case the incentives to deviate from the rule are irrelevant sincecommitments are assumed to be binding
(Barro and Gordon 1983 599)
Policy 183
As the above extract suggests in the ldquotime inconsistency caserdquo we have not yetcharacterized a rational expectationsrsquo equilibrium since in our example individu-alsrsquo expectations of inflation for period 1 were incorrect According to Barro andGordon there are three features of an equilibrium The first is ldquoa decision rulefor private agents which determines their actions as a function of their currentinformationrdquo These actions of private agents based on current information aresummarized by the Phillips curve
Ut = Un + ζt minus α middot (πt minus Etminus1πt) (1130)
where ζt a random variable with zero mean and variance σ 2z has been added
to denote a real shock that affects the natural unemployment rate for the currentperiod only19
The second feature of an equilibrium is ldquoa policy rule which specifies the behav-ior of policy instruments as a function of the policymakerrsquos current informationsetrdquo This policy rule is given by the choice of inflation rates πt+i i = 0 1 2 by the monetary authorities with the following objective
min Et
⎡⎣ infinsum
i=0
β iZt+i|It
⎤⎦
where Zt+i equiv a(Ut+i minus kUn)2 + b(πt+i minus πlowast)2 a gt 0 b gt 0 and 0 le k le 1
The term It denotes the initial state of information and 1 gt β gt 0 is the constantdiscount factor (β = 1(1+rreal) where rreal is the exogenous real rate of interest)For simplicity we will assume that the optimal level of inflation is zero so thatπlowast = 0
The third feature is an expectations function which determines the expectationsof private agents as a function of their current information Assuming that ldquothepublic understands the nature of the policymakerrsquos optimization problem in eachperiodrdquo then Etminus1πt = πt where πt is the optimal choice of inflation by thepolicy-maker for period t
Combining the information contained in our discussion of the first and secondfeatures of equilibrium the policy-makerrsquos optimal choice of πt minimizes
EtZt = Et[a(Ut minus kUn)2 + b(πt)
2]= Et[a((1 minus k)Un minus α(πt minus Etminus1πt))
2 + b(πt)2]
Given that Eξt = 0 the expression to be minimized by the policy-maker can bewritten as
EtZt = [a((1 minus k)Un minus α(πt minus Etminus1πt))2 + b(πt)
2]A critical point to note is that each period the policy-maker inherits Etminus1πt andtakes that expected rate of inflation as given in the above optimal choice of πt
184 Policy
The first-order condition for period t is thus
partEtZtpartπt = 2a((1 minus k)Un minus α(πt minus Etminus1πt))(minusα) + 2b(πt) = 0
which can be simplified and rearranged to obtain the following expression for theoptimal inflation rate
πt = aα
b[(1 minus k)Un minus α(πt minus Etminus1πt)] (1131)
From the third feature of equilibrium we know that the public realizes the problemfaced by the policy-maker in terms of choosing the optimal rate of inflation forperiod t as defined by (1131) and thus Etminus1πt = πt so that the second term dropsout and we have
πt = aα
b(1 minus k)Un = Etminus1πt gt πlowast = 0 (1132)
as long as k lt 1 We thus have that expectations are rational and individualsoptimize subject to these expectations Since Etminus1πt = πt we have that EtUt = UnThus ldquothe equilibrium solution delivers the same unemployment rate and a higherrate of inflation at each daterdquo than is the case in the SargentndashWallace problem wherethere is a ldquorules-type equilibriumrdquo Given the optimal rate of inflation πlowast = 0a rules-type equilibrium with rational expectations would have the actual rate ofinflation equal to zero
As Barro and Gordon (1983 608) conclude
under a discretionary regime the policymaker performs optimally subject toan assumed inability to commit future actions The framework assumes ratio-nality within the given institutional mode Excessive inflation apparentlyunrewarding countercyclical policy responses and reactions of monetarygrowth and inflation to other exogenous influences can be viewed as prod-ucts of rational calculation under a regime where long-term commitments areprecluded
The model stresses the importance of monetary institutions which deter-mine the underlying rules of the game A purely discretionary environmentcontrasts with regimes such as a gold standard or paper-money constitutionin which monetary growth and inflation are determined via choices amongalternative rules (the SargentWallace approach) The rule of law or equivalentcommitments about future governmental behavior are important for inflationjust as they are for other areas that are influenced by possibly shifting publicpolicies
An alternative to the ldquorule of lawrdquo is reputation As Blanchard and Fischer(1989 599) state
reputation is the most interesting and persuasive explanation of how gov-ernments avoid dynamic inconsistency Governments know that they can do
Policy 185
better than the shortsighted solution over the long run They hope by actingconsistently over long periods to build a reputation that will cause the privatesector to believe their announcements The key to the answer [to the ques-tion of whether reputation can sustain the optimal policy] is the specificationof private sector expectations of how the public reacts to broken promises
Conclusion
This chapter has presented an overview of many of the major themes and issuesfaced in macroeconomics in terms of the conduct of monetary policy The roles ofexpectations the ldquoineffectiveness propositionrdquo the modified Phillips curve rulesversus discretion time inconsistency credibility and enforcement are all dealt within this chapter Perhaps the major contribution of this chapter is that it highlightsmany of the issues and problems that real-world monetary authorities face whendeciding what course of action to take
12 Open economy
Introduction
The field of international economics can be roughly categorized as concernedwith either the real side or the finance side of international issues The ldquorealrdquoside focuses on such basic questions as why trade occurs between countries whatdetermines the terms of trade and how government policies such as tariffs do orquotas affect trade The ldquofinancerdquo side makes explicit the fact that countries differin currencies in order to focus on such questions as what determines exchangerates and how macroeconomic shocks in one country (eg a change in the supplyof money) affect its economy and the economies of the countries with which ittrades
In the discussion below we examine questions more like those considered by theldquofinancerdquo side Namely we extend simple macroeconomic analysis to an ldquoopenrdquoeconomy that is an economy that incorporates a foreign sector This analysisdiffers from traditional macroeconomic analysis of a ldquoclosedrdquo economy in thefollowing respects
1 There is trade of composite commodities and financial assets between twocountries For the moment we assume not only that the two countries producedifferentiated output but also that the financial assets issued by the firms andgovernment of one country are not perfect substitutes for the financial assetsissued by firms and government of the second country
2 The two economies are isolated in that individuals in each country can onlypurchase or sell labor services in their own labor markets
3 The two economies are differentiated in that each has its own media ofexchange This means that an individual in the domestic economy whoseeks to purchase foreign goods must exchange domestic money for foreignmoney1 Similarly an individual in the foreign country must exchange hisforeign money for the domestic money to purchase the domestic goods Suchexchanges take place in the foreign exchange market at the prevailing ldquoforeignexchange raterdquo or the price of one currency in terms of the second currencyWe will let et denote the exchange rate for domestic money For instanceif the domestic country is the USA and the foreign country is Japan then
Open economy 187
et on December 1 1989 was 1434 yen in that 1434 yen = 1 dollar Thisimplies that 1et the price of dollars in terms of yen was 0006973 dollarson December 1 1989
4 There is distinct government policy (fiscal and monetary) in each country inthat each countryrsquos government determines spending taxation and monetarypolicy for its economy
To understand how macroeconomic analysis is altered in the above ldquoopen econ-omyrdquo setting it is instructive to first consider the nature of the constraints faced bythe various participants in an open economy We then examine how the behaviorof these participants is affected by changes in such variables as exchange ratesWith that background we are ready to consider some simple examples of openmacroeconomic analysis such as the effect of a change in the money supply in thecontext of an open economy neoclassical model
Open economy participants constraints and Walrasrsquo law
To begin our task of modeling an open economy recall that it is typical ofmacroeconomic models to simplify the economy by grouping markets into broadcategories In a closed economy there were three important markets the outputfinancial and labor markets One could add to this the money ldquomarketrdquo sinceequilibrium required that the demand for money equaled supply In an open econ-omy we add another market the foreign exchange market where the currencyof one country is traded for that of the other country In a closed economy theparticipants in the various markets in the economy could be placed in one of fivecategories households firms government (fiscal side) the central bank and pri-vate depository institutions In an open economy we add one more participantforeigners
As we have seen a common theme of macroeconomic models is their emphasison the interdependencies among markets Macroeconomics recognizes that eventsin one market imply changes in other markets as well This ldquogeneral equilibriumrdquoapproach contrasts with the ldquopartial equilibriumrdquo approach of microeconomicswhich is less concerned with how changes in one market affect all other marketsTo fully understand the links across markets it is useful to specify the ldquofinancingconstraintsrdquo faced by the participants in the various markets Our discussion ofopen economy macroeconomics thus begins by introducing these financing con-straints for firms households government (fiscal side) the central bank privatedepository institutions and foreigners We then sum these constraints to obtain amodified Walrasrsquo law
The financing constraints in an open economy
We start our discussion of the constraints faced by the participants in an openeconomy by considering the financing constraint faced by the new participant for-eigners Foreigners purchase domestic output and financial assets Let X d
t denote
188 Open economy
foreignersrsquo demand for domestic output (exports) and let net Adft denote foreignersrsquo
desired real change in their holdings of domestic financial assetsTo purchase domestic output and financial assets foreigners must first acquire
domestic money That is foreigners must finance these purchases either fromincome generated from their previously acquired holdings of domestic financialassets or by supplying foreign currency in exchange for the domestic currencyin the foreign exchange markets In particular we have the following foreignerfinancing constraint
X dt + net Ad
ft minus αf (zBf pt + dt) minus FCst etpt = 0 (121)
Note that for simplicity we limit foreignersrsquo holdings of domestic financial assetsto private financial assets (ie we do not have them holding government bonds)Further we assume foreignersrsquo portfolio of holdings of domestic financial assetsis identical to2 that of domestic households and that they own αf 1 gt αf ge 0 ofthe total value of financial assets issued by domestic firms Thus αf (zBf pt + dt)
is the real income foreigners gain from their holdings of domestic financial assets3
In equation (121) the term FCst etpt denotes foreignersrsquo real supply of foreign
currency in the foreign exchange market FCst is foreignersrsquo supply in units of the
foreign currency in period t Multiplying by 1et the price of the foreign currencyin terms of the domestic currency puts this in terms of the domestic currencyDividing by the price level pt then puts it in real terms (ie in terms of the domesticcomposite commodity) Embedded in the desired change in foreignersrsquo holdingsof domestic financial assets are changes in the desired holdings by foreign centralbanks (ie changes in foreign central banks ldquointernational reservesrdquo)
In addition to the above new constraint that accompanies the introduction ofa new participant to the economy foreigners we have constraints faced by thedomestic households the central bank private depository institutions government(fiscal side) and firms In the case of households and the central bank we modifythe constraints introduced in previous chapters to incorporate the exchanges ofcommodities and financial assets with foreigners In particular for householdsthe budget constraint in an open economy can be expressed by
bdt + cd
t + zdt + (M d
t minus M )pt + net Adht + net AFd
ht minus [yt minus αf (zBf pt + dt)
+ α(zBff pft + pftdft)etpt minus δK minus Tnt] = 0 (122)
where the new term zdt denotes real imports net Ad
ht denotes householdsrsquo desiredchange in their real holdings of foreign assets and α(zBff pft + pftdft) denotes theincome (in terms of the foreign currency) gained from holding the proportion aof the financial assets issued by foreign firms4 This income (in foreign currency)times the price of foreign currency in terms of domestic currency (1et) gives thedomestic currency value of income from foreign asset holdings Dividing by theprice level pt puts the income in terms of the composite commodity (ie in realterms)
Open economy 189
Total consumption of commodities in period t is now the sum of cdt purchase
of output and zdt imports of commodities produced abroad5 Similarly the total
desired change in financial assets is the sum of net Adht the desired change in hold-
ings of domestic financial assets and net AFdht the desired change in holdings of
foreign financial assets Note that in incorporating the firm distribution constraintinto the household budget constraint we have taken into account the fact that notall domestic output yt is income to domestic households since foreigners own theshare αf of domestic firms On the other hand households have an additionalsource of income from the ownership of foreign financial assets
For the central bank the (stock) financing constraint equates the sum of thechange in real (domestic) financial asset holdings and international assets to thereal change in the monetary base or
net Adct + net AFd
ct minus (MBst minus MB)pt = 0 (123)
where net Adct = pbt(Bd
gct minus Bgc)pt MB = R + C and MBst = Rs
t + Cst 6 We
have added the term net AFdct to denote the real demand for additional international
(foreign currency denominated) assets by the central bank In particular
net AFdct = pfbt(B
dfct minus Bfc)etpt
The quantity pfbt(Bdfct minusBfc) is the change in the amount of foreign assets demanded
by the central bank in terms of the foreign currency pfbt denotes the price of foreignbonds in terms of foreign currency and the numbers of such bonds demanded andinitially held by the domestic central bank are denoted respectively by Bd
fct and
Bfc Multiplying this quantity by the price of foreign currency in domestic currencyterms (1et) gives the domestic currency value of foreign assets demanded by thecentral bank Then dividing by the price level pt puts the net demand for interna-tional assets by the central bank net AFd
ct in terms of the composite commodity(ie in real terms)
For private depository institutions the (stock) financing constraint indicates thatprivate depository institutions can be viewed as financing additions to reserves andto financial assets holdings by creating deposits or
net Atpb + (Rd
t minus R)pt minus (Dst minus D)pt = 0 (124)
where net reserves demanded or initially held are denoted by Rdt and R respec-
tively and checkable deposits supplied or initially outstanding are denoted byDs
t and D respectively For simplicity we assume that all private demands forforeign financial assets as well as for foreign commodities are captured in thehousehold budget constraint To the extent that private banks purchase foreignfinancial assets they can be viewed as acting as financial intermediaries forhouseholds
190 Open economy
For the domestic government (fiscal side) the financing constraint is
gdct minus T lowast
nt minus net Asgt = 0 (125)
where T lowastnt = Tt minustrt minusz(Bgh+Bgp)pt tr denotes transfer payments and net As
gt =Pbt(Bs
gt minusBg)pt Equation (125) incorporates the ldquoflowrdquo financing constraint forthe central bank In doing so it is assumed that the central bank claims on realresources just exhaust its interest payments on government debt holdings plus anyincome associated with central bank holdings of foreign assets Finally for firmsthe financing constraint on capital purchases is given by
Idnt + ψ(I d
nt) minus net Asft = 0 (126)
Note that we assume for simplicity that neither firms nor the government purchaseforeign commodities
Walrasrsquo law and the balance of payments
We now sum the constraints faced by the six participants in an open economy(foreigners households the central bank private banks the government andfirms) as given by equations (121)ndash(126) In doing so assume equilibrium withrespect to the demand and supply of bank reserves (ie the Rs
t part of MBst in the
central bank constraint equals Rdt in the private banks constraint) and note that
the initial monetary base MB equals R + C that the money supply is defined byM s
t = Dst + Cs
t and that the initial money supply is given by M = D + C Weobtain
[Bdt + Cd
t + X dt + gd
ct + δK + I dnt + ψ(I d
nt) minus yt] + [M dt pt minus Mpt]
+ [net Adht + net Ad
ft + net Adct + net Ad
pt minus net Asgt minus net As
ft]+ [zdlowast
t + net AFdht + net AFd
ct minus FCst etpt] = 0 (127)
where zdlowastt + net AFd
ht + net AFdct minus FCs
t etpt denotes householdsrsquo imports notfinanced out of income generated from foreign asset holdings Equation (127) isan example of the modified Walrasrsquo law for an open economy7 As we discussbelow (127) can be viewed as stating that the sum of excess demands in fourmarkets must equal zero
The first term in (127) reflects excess demand in the output market where thedemand now includes foreignersrsquo demand for domestic output (exports) Note thatby adding and subtracting import demand we could express the excess demandfor output in the form
bdt + cdlowast
t + (xdt minus zd
t ) + gdct + δK + I d
nt + ψ(I dnt) minus yt (128)
where the term cdlowastt denotes householdsrsquo total consumption in terms of both domes-
tic and foreign output (ie cdlowastt = cd
t + zdt ) Excess demand in the output market
Open economy 191
is often expressed this way with the consumption term denoting total householdconsumption such that the ldquonetrdquo export demand term (xd
t minus zdt ) appears
Setting the excess demand for output term in (127) to zero gives us the ISequation in an open economy The second term in (127) reflects the excess demandfor money Note that we assume that only domestic households desire to hold thedomestic money (foreigners seek the domestic money in the foreign exchangemarket not to hold but as a means to purchase the domestic output or financialassets) Setting this second term in (127) to zero gives us the standard LM equation
The third term in (127) reflects excess demand in the financial market In goingto an open economy we add to the demand side of the financial market a demandfor domestic financial assets by foreigners (which could include private foreignagents as well as foreign central banks) In addition note that householdsrsquo demandfor domestic financial assets (net Ad
ht) no longer reflects householdsrsquo total demandfor financial assets since they now have the option of purchasing foreign financialassets (net AFd
t )The fourth and final term in (127) reflects excess real demand for foreign
currency in the foreign exchange market8 As before we can view output ( yt)or the price of output ( pt) as determined in the output market and the domesticinterest rate (rt) as determined in the financial market Now we can view the foreignexchange rate (et) as determined in the foreign exchange market The demand forforeign currency reflects zdlowast
t the demand associated with householdsrsquo purchases offoreign commodities that could not be financed from the foreign currency earningsof their holdings of foreign financial assets net AFd
ht the demand for foreigncurrency associated with householdsrsquo purchases of additional foreign financialassets and net AFd
ct the demand associated with the central bankrsquos desired changein international reserves Note that the real demand for the foreign currency couldinstead be stated as the real supply of the domestic currency in the foreign exchangemarket
FCst etpt denotes the real supply of foreign currency or real demand for the
domestic currency in the foreign exchange market which from (121) reflectsforeignersrsquo purchases of financial assets and exports net of those financed throughdomestic currency earnings on financial assets held by foreigners Foreignersrsquopurchases of financial assets include both foreign private (ie foreign householdsrsquo)and foreign public (ie foreign central bank) purchases
The balance of payments accounts
Let us assume for now that the domestic economy discussed above is the USA TheUS Department of Commerce actually measures the various sources of the demandfor and supply of dollars in the foreign exchange markets cited above These data ofinternational transactions are presented as the US balance of payments accountsTable 121 summarizes its major components Transactions are categorized aseither sources of the real demand for or the real supply of dollars in the foreignexchange market for the US dollar9 Equivalently they reflect the real supply ofor real demand for foreign currency in the foreign exchange market
192 Open economy
Table 121 The US balance of payments accounts (in ldquorealrdquo terms)
Real demand for dollars Real supply of dollars
1 US exports of goods and services xt US imports of goods and services zt2 Transfers (interest and dividends to US
holders of foreign financial assetsgovernment grants and gifts)
Transfers (interest and dividends toforeign holders of US financial assetsgovernment grants and gifts)
α(zBff pft + pftdft)etpt αf (zBf pt + dt)
3 Foreign purchases of US financialassets (capital inflow) net Ad
ft
US purchases of foreign financial assets(capital outflow) net AFd
h + net AFdct
The net demand for dollars associated with the first component of the balanceof payments accounts (exports minus imports) is called the balance of trade Ifthe balance of trade is negative as it has been recently for the USA then the USAis said to experience a balance of trade deficit If it is positive as was the case for106 consecutive years from the end of the Civil War to 1971 as well as throughmost of the 1970s then a balance of trade surplus is said to exist
The third component of the demand for and supply of dollars in the balance ofpayments accounts is the capital account This capital account can be divided intoa private part and a public part On the demand side the private part measures thereal dollars demanded by foreigners other than foreign central banks to financepurchases of US financial assets On the supply side the private part measuresthe real dollars supplied by US households to buy foreign financial assets Thesecurrency exchanges are called private international capital flows Private inter-national capital flows associated with the demand for dollars are referred to asUS private international capital inflows since they reflect the inflow of foreigncurrency due to private foreignersrsquo purchases of US financial assets Private inter-national capital flows associated with the supply of dollars are referred to as USprivate international capital outflows since they reflect the outflow of dollars dueto US householdsrsquo purchases of foreign financial assets
Summing the net demand (demand minus supply) for dollars associated withthe first two components plus the private international capital flows and adjustingfor measurement errors (the discrepancy term) we obtain what is called the USbalance of payments10 In years in which the balance of payments is negative it isreferred to as a balance of payments deficit On the other hand a positive balanceof payments is called a balance of payments surplus
When there is a surplus or deficit in the balance of payments accounts thenequality between the real demand for and real supply of dollars is brought about byan offsetting deficit or surplus on what is known as ldquothe official reserve transactionbalancerdquo The official reserve transaction balance reflects the intervention into theforeign exchange market by the US central bank (the Fed) andor by foreign centralbanks This is the ldquopublicrdquo part of international capital flows Whenever there existsa balance of payments deficit in the USA then (on net) central banks demand USdollars in the foreign exchange markets If this were solely the US central bank
Open economy 193
intervening this means that net AFdct wasis negative (the Fed wasis a net supplier
of dollars in the foreign exchange market) and the US central bank would havelostlose international reserves
Behavior in an open economy
With an understanding of the constraints faced by the various participants in anopen economy we now consider the behavior of these participants in particularthe determinants of imports (zd
t ) exports (xdt ) private international capital inflows
(a part of net Adft) and private international capital outflows (net AFd
ht) In the firstpart of this section we examine how a change in the foreign exchange rate for thedomestic currency (to be concrete the US dollar) can alter the relative price offoreign goods and thus lead to a change in the division of household consumptionbetween purchases of domestic goods and purchases of foreign goods11 We alsoexamine other factors that influence US imports of goods and services and thusthe real demand for foreign currency (real supply of US dollars) in the foreignexchange markets
In the second part of this section we consider the factors that influence howhouseholds divide their real accumulation of financial assets between US stocksand bonds and the financial assets of foreign countries As we saw above house-hold purchases of foreign financial assets constitute capital outflows since inorder to make these purchases households must demand foreign currency (supplydollars) in the foreign exchange market We will see how differences in foreignand domestic interest rates and the expected appreciation or depreciation of theUS dollar determine the relative returns on foreign and domestic financial assetsThese relative returns in turn affect householdsrsquo portfolio choices between foreignand domestic financial assets and the real demand for foreign currency (real supplyof dollars) in the foreign exchange market
In the third and fourth parts of this section we turn to the other side of theforeign exchange market to examine determinants of foreignersrsquo purchases of USgoods and services and of US financial assets In particular we consider how suchfactors as exchange rates affect foreignersrsquo demand for US goods (US exports)and how such factors as relative interest rates and the expected rate of changein the exchange rate affect foreignersrsquo demand for US financial assets We finishthis section by illustrating graphically the behavior discussed in terms of the realdemand for and supply of the domestic currency (the US dollar)
Householdsrsquo demand for imports
When deciding whether to purchase foreign or domestic goods households look attheir relative prices To be concrete consider two countries The domestic countryis the USA and the foreign country is Japan The relative price of Japanese goods isthen the real quantity of US goods that must be sacrificed to purchase the foreigngood For example if the price of a Japanese car is $6000 and the price of aUS computer is $1500 then the relative price of a Japanese car in terms of US
194 Open economy
computers is 4 computers If the relative price of Japanese cars rises the USAwill import fewer Japanese cars The relative price of foreign goods sometimesreferred to as the terms of trade is an important determinant of the quantity ofimports The relative prices of foreign goods depend on the dollar prices of USgoods the prices of foreign goods in their own currency and foreign exchangerates (the price of one currency in terms of a second currency) The simple examplethat follows illustrates this point Suppose that the price of a US computer is $1500and that in Japan the price of a Japanese car is 600000 yen The third ldquopricerdquo thatwe need to know in order to compute the relative price of a Japanese car in terms ofUS computers is the foreign exchange rate in particular the price of a yen in termsof dollars Suppose that it takes et yen to buy one dollar in the foreign exchangemarkets Then it takes 1et dollars to buy one yen Returning to our example ofJapanese cars and US computers if et = 100 yen per dollar and the Japanese carhad a yen price of 600000 then its dollar price would be 6000
In general the calculation of the relative price of Japanese goods is thus
Relative price of Japanese goods (in terms of US goods)
= Yen price of Japanese goods ( pft)
times Price of yen in terms of dollars (1et)Dollar price of US goods ( pt)
= pft
etpt
According to this expression a rise in the yen price of Japanese goods ( pft)
raises their relative price Similarly a fall in the dollar price of US commodities( pt) raises the relative price of Japanese goods Finally a rise in the price of yen interms of dollars (1et) also increases the relative price of Japanese goods12 Sinceall three changes mean a higher relative price of Japanese goods and thus anincrease in the cost to US buyers of Japanese goods in terms of US goods forgoneall three changes reduce US imports of Japanese goods The above relative priceexpression for a basket of foreign goods is sometimes termed the real exchangerate for foreign goods ndash that is
Real exchange rates (foreign goods)
= Dollar price of foreign goods
Dollar price of US goods= pft(1et)
pt= pft
ptet
While a rise in the relative price of foreign goods will lead to a reduction in thequantity of foreign goods bought we have to be careful not to infer from this thatUS import demand will necessarily be inversely related to the relative or real priceof imports The reason for this is that our measure of US real import demand is interms of the US good and services not in terms of the foreign good An examplewill highlight this distinction
Open economy 195
Suppose that the dollar depreciates (et falls) With the implied appreciation offoreign currency (1et rises) the price of the foreign good in terms of US goodsbecomes greater as our expression for the real exchange rate for foreign goods( pftpt et) indicates With a higher price fewer foreign goods will be purchased ndashthis is clear This by itself would suggest a fall in the value of imports into theUSA But the higher price also means that each foreign good purchased will costmore in terms of US goods that must be sacrificed This by itself would suggest arise in the value of US imports (measured in terms of US goods that must be paidto obtain the imports) The net impact of a change in the relative price of foreigngoods on US imports measured in terms of US goods depends on which of thesetwo effects is stronger
It is typically assumed that over time the effect of a change in the relative pricesof imports on the quantity of imports purchased dominates so that the value ofimports in terms of US goods will fall with a rise in the relative price of importsThis is what we will assume13 Formally this condition requires that the priceelasticity of demand for foreign goods be greater than one That is a 1 percentincrease in the price of foreign goods must cause a greater than 1 percent reductionin the amount of foreign goods that US households demand14
Besides the relative prices of imports real disposable income affects US importdemand An increase in disposable income can lead to a rise in household con-sumption demand In an open economy with foreign trade a rise in consumptiondemand means an increase in purchases not only of domestically produced goodsbut also of foreign goods Thus householdsrsquo import demand is directly related totheir disposable income To summarize household real import demand zd
t dependsinversely on the relative price of foreign goods pftetpt and directly on disposableincome yt minus δK minus Tnt
zdt = zd
t ( pftetpt yt minus δK minus Tnt ) (129)
Capital outflows householdsrsquo demand for foreign financial assets
When deciding whether to purchase US or foreign financial assets householdscompare domestic and foreign rates of return The nominal rate of return on USfinancial assets is simply the money interest rate rt The comparable nominal rateof return on foreign financial assets is not so simple to identify To explain howto compute this return which we will denote rlowast
t let us suppose that a householdlends one dollar in the foreign financial market
If the price of a dollar is et units of the foreign currency say yen then in termsof the foreign currency the household lends et yen If foreign financial assets offerthe interest rate rft then one period from now the household will have et(1 + rft)
yen At that time the household can convert these yen holdings back to dollars atthe exchange rate then existing At the time the money is lent this future exchangerate may be uncertain15 Let householdsrsquo expectation of this future exchange ratebe denoted by ee
t+116 Then the household expects to convert its et(1 + rft) yennext year into et(1 + rft)ee
t+1 dollars Subtracting the one dollar with which the
196 Open economy
household started the rate of return to lending in foreign financial markets rlowast isgiven by
rlowastt = (et(1 + rft)ee
t+1) minus 1 = [et(1 + rft) minus eet+1]ee
t+1
We can simplify the above equation by noting that the expected future dollarexchange rate ee
t+1 yen per dollar equals the current exchange rate of et yen times(1 + θe
t+1) where θet+1 is the expected rate of change in the price of a dollar in
terms of yen Substituting the expression et(1+θet+1) for the expected future dollar
exchange rate eet+1 into the above expression for rlowast
t we have
rlowastt = [et(1 + rft) minus et(1 + θe
t+1)]et(1 + θet+1) = (rft minus θe
t+1)(1 + θet+1)
Since the expected rate of appreciation of a dollar is typically small we canapproximate the above expression by
rlowastt = rft minus θe
t+1 (1210)
Equation (1210) has a straightforward interpretation The return to lending in theforeign financial market equals the difference between the foreign interest rateand the expected rate of change in the price of the dollar The return to lendingin foreign financial markets increases with a higher foreign interest rate rft anddecreases with a higher expected rate of increase in the price of the dollar (θe
t+1)The higher the expected rate of increase in the price of the dollar the lower theexpected return to lending in foreign financial markets since for a given numberof dollars sold for foreign currency at the start of the period fewer dollars can bebought back at the end of the period
When households choose between purchasing domestic and foreign financialassets they compare the domestic interest rate rt with the rate of return to lendingin the foreign financial markets rlowast
t We can thus express household real demandfor additional foreign financial assets as
net AFdht = net AFd
ht(rt rft minus θet+1 ) (1211)
In (1211) an increase in the US interest rate or a fall in the rate of return onforeign financial assets implies a reduction in householdsrsquo real demand for foreignfinancial assets The three dots in (1211) reflect other factors that have been leftunspecified For instance changes in the political stability of foreign governmentsare one unspecified factor that would likely impact on US householdsrsquo demandfor foreign financial assets Equation (1211) suggests that we should add anotherfacet to our previous discussion on householdsrsquo demand for US financial assetsnet Ad
ht In addition to such factors as real income taxes the US money interestrate and the expected rate of inflation household demand for US financial assetsdepends on the expected return to lending abroad rft minus θe
t+1
Open economy 197
Foreignersrsquo demand for exports
Just as US demand for foreign goods depends on relative prices so too isforeignersrsquo demand for US goods based on the relative prices of those goodsThe relative prices of US goods to foreigners measure what foreigners have togive up of their own goods in order to purchase US goods As we have seenthese relative prices depend on the money prices of US goods the money pricesof foreign goods and the exchange rate Considering ldquocompositerdquo goods for eachcountry the relative price of US goods to foreigners or the real exchange rate forforeign goods is given by
Real exchange rates (US goods)
= Dollar price of US goods
Dollar price of foreign goods= pt
pft(1et)= ptet
pft
Note that the real exchange rate for US goods is simply the inverse of the previouslyobtained real exchange rate for foreign goods The price of a dollar in terms ofan index of foreign currencies rose by 70 percent in 1984 and 1985 The resultingrise in the relative prices of US goods contributed significantly to a reduction inforeign demand for US goods and the large US trade deficit of the mid-1980sSimilarly the dramatic fall in the price of a dollar in the subsequent period fromlate 1985 to 1988 led to an increase in US exports Thus we have
xdt = xd
t ( ptetpft ) (1212)
Equation (1212) indicates that export demand falls with a rise in the relative priceof US goods to foreigners
We know from our previous discussion that an increase in US disposable incomeleads to a rise in household purchases of both domestically produced output andforeign goods and services By the same token an increase in foreignersrsquo dispos-able income leads to a rise in their purchases of US goods Thus among the itemsmissing in (1212) that determine foreignersrsquo real export demand xd
t is foreigndisposable income
Capital inflows foreignersrsquo demand for financial assets
In 1960 purchases of US financial assets by foreigners were approximately one-half the amount of purchases of foreign financial assets by US citizens In the USfinancial markets foreign purchases of new US financial assets were less than5 percent of household and depository institution purchases Twenty-five yearslater foreigners were purchasing four times as many US financial assets than theUSA was purchasing abroad In the US financial market close to 30 percent ofnew US financial assets were being purchased by foreigners This dramatic changein capital inflows to the USA is one indication of the growing importance to theUS economy of international trade not only in goods but also in financial assets
198 Open economy
As with US households we will assume that foreigners decide to purchase eitherUS assets or financial assets of their own country by comparing the rates of returnon the two types of financial assets For foreigners the nominal rate of return ondomestic assets is the money interest rate in their own country (rft) The expectedrate of return to foreigners on US financial assets equals the US interest rate plusthe expected change in the price of the dollar in the foreign exchange market (iert + θe
t+1)Not surprisingly the expected return to foreigners lending in US financial mar-
kets increases when the US interest rate rt increases Not as obvious is that thereturn also increases with an increase in the expected rate of change in the price ofthe dollar θe
t+1 This is because foreigners lending in US financial markets converttheir currency to dollars to make the loans When the loans are repaid they thenconvert dollars back to their own currency If the dollar is anticipated to appreciateduring the course of the year then part of their expected return to lending in theUSA is the increase in the value of the dollars (in terms of their own currency)
Summarizing we can express the foreign demand for US financial assetsnet Ad
ft as
net Adft = net Ad
ft(rft rt θet+1 ) (1213)
Equation (1213) indicates that foreignersrsquo demand for US financial assetsincreases if the US interest rate (rt) rises or the expected rate of change in theprice of the dollar (θe
t+1) increases or the foreign interest rate (rft) falls
The foreign exchange market the real demand and supply of dollars
The change in the price of a dollar (in terms of a second currency) affects the realquantity of dollars supplied and demanded in the foreign exchange market Letus start with the price of a dollar set at the equilibrium level of (et)0 let us say100 yen If the price of a dollar now falls to (et)1 say 50 yen then this depreciationof the dollar (appreciation of the yen) leads to a reduction in the real quantity ofdollars supplied from Q0 to Q1
A fall in the price of a dollar from 100 to 50 yen means a rise in the dollar priceof a yen from 001 to 002 dollars or from 1 cent to 2 cents Even though there isno increase in the yen price of Japanese goods the dollar price of Japanese goodsrises For example a 600000 yen Japanese car that formerly cost 6000 dollars(600000 times 001) now costs 12000 dollars (600000 times 002) If the dollar prices ofUS goods have not changed then the relative or real prices of Japanese cars haverisen In our example this means that US households must give up an increasedamount of US goods to obtain one more Japanese car
The depreciation of the dollar and resulting higher relative price for Japanesegoods leads US households to reduce the quantity of Japanese goods demandedHowever as we discussed above the fact that the quantity of Japanese goodspurchased falls does not necessarily mean that the quantity of dollars suppliedin the foreign exchange market also falls There are two countervailing forces
Open economy 199
at work here While the purchase of fewer Japanese goods would reduce thequantity of dollars supplied the fact that each Japanese good has a higher pricewould increase the quantity of dollars supplied By assuming that the first effectoutweighs the second effect we conclude that a depreciation of the dollar causesthe real quantity of dollars supplied in the foreign exchange market to fall Thusthere is an upward-sloping supply of dollars curve17
Naturally behind the supply of dollars in the foreign exchange market is notonly US real import demand but also the real demand by households and the UScentral bank for additional foreign financial assets (net AFd
ht + net AFdct) As we
saw above this sum of the US import demand and demand for foreign financialassets can be interpreted as representing not only a supply of dollars but also ademand for foreign currency18
The above discussion also highlights the effect of a change in the price ofa dollar (in terms of a second currency) on the quantity of dollars demandedin the foreign exchange market The depreciation of the dollar (or appreciationof the yen) from (et)0 to (et)1 leads to an increase in the quantity of dol-lars demanded US in real terms The fall in the price increases the quantity of dollarsdemanded because it lowers the relative price of US goods to foreigners A fall inthe price of a dollar from 100 yen to 50 yen causes a rise in the dollar price of theyen from 1 cent to 2 cents Even though there has been no increase in the dollarprice of US goods the yen price of US goods falls and the Japanese increase theirdemand for US goods As a consequence the real quantity of dollars demandedin the foreign exchange market increases
Naturally behind the demand for dollars in the foreign exchange market isnot only foreignersrsquo export demand but also their demand for US financial assets(net AFd
ht) As we have seen this sum of the export demand and foreignersrsquo demandfor US financial assets can be interpreted as representing not only a demand fordollars but also a supply of foreign currency
Simple examples of open economy (static) macroeconomicanalysis
The statement of Walrasrsquo law for an open economy indicates that for static macro-economic analysis of an open economy we need look at only three of the fourexcess demand conditions reflecting the output financial foreign exchange andmoney markets Standard practice is to look at the output money and foreignexchange markets In the neoclassical model these three equations would besolved to obtain the equilibrium price level interest rate and exchange rate ( pt rt and et respectively) In the BarrondashGrossman disequilibrium analysis thesethree equilibrium conditions could be solved to obtain the equilibrium outputinterest rate and exchange rate ( yt rt and et respectively) In the Lucas modelor the Keynesian fixed money wage model these three equilibrium conditionsalong with the appropriate aggregate supply equation could be solved for theequilibrium price level output interest rate and exchange rate ( pt yt rt and et respectively)
200 Open economy
With flexible exchange rates (ie exchange rates determined without theintervention of central banks) the basic change in the macroeconomic analysisis to recognize that the net export component of output demand (seeequation (128)) is sensitive to interest rate changes which implies the IS curve(in (interest rate output) space) is flatter The reason why a higher US interest ratereduces net export demand is that a higher interest rate increases capital inflows(net Ad
ft) and reduces capital outflows (net AFdht) The first change increases the
demand for dollars in the foreign exchange market while the second reduces thesupply of dollars in the foreign exchange market The result is an appreciationof the dollar that reduces exports and increases imports Note that with flexibleexchange rates a change in either the price level or income does not affect netexport demand
Money supply changes in the neoclassical model purchasingpower parity
In general when considering two countries the country with the lower inflationrate will tend to have an exchange rate that is appreciating at a rate approximatelyequal to the difference in inflation rates between the two countries This patternof changes in foreign exchange rates is sometimes said to reflect the purchasingpower parity condition The condition of purchasing power parity means that thepurchasing power of each countryrsquos currency is the same whether the currency isused to purchase domestic goods or foreign goods Purchasing power parity existsfor a monetary shock in the neoclassical model since a money supply change willlead to changes not only in domestic prices but also in foreign exchange rates suchthat relative prices remain constant
Interest-rate parity
There are of course a number of variations to the above analysis For instancewe could assume that domestic and foreign financial assets are perfect substitutesand that the domestic country is sufficiently small in its interactions with foreignfinancial markets that it takes the foreign interest rate rft as a given In this caseof ldquointerest rate parityrdquo since the return to lending at home rt must equal that oflending abroad rlowast
t = rft minus θet+1 we have that
rft = rt + θet+1 (a constant) (1214)
One use of (1214) is in the well-known Dornbusch model (see Dornbusch 1976)Assume that output and the price of output are fixed initially Assume θe
t+1 = 0initially such that rt = rft Now consider an increase in the money supply Tomaintain equilibrium with respect to the demand and supply of money the interestrate rt must decrease According to (1214) the resulting increase in internationalcapital outflows and reduction in international capital inflows must reduce theexchange rate such that it is expected to appreciate at a rate equal to the difference
Open economy 201
between xdt and the new lower rt This is Dornbuschrsquos famous ldquoexchange rate
overshootingrdquo That is the analysis implies a fall in the exchange rate below whatit will ultimately be after income or prices adjust
A second use of (1214) is to combine it with the assumption that exchangerates are fixed (through appropriate central bank intervention in the foreignexchange markets) An example taking this approach is the MundellndashFlemingmodel (Fleming 1962 Mundell 1968)
Conclusion
This chapter has brought together many of the issues associated with analyzinga macroeconomy that participates in the open or global economy In keepingwith the basic model framework of this book the household is now viewed ashaving a demand for imported products as well as domestically produced goodsand services Additionally firms are allowed to sell to agents in foreign coun-tries This flow of goods and services across borders logically leads to a flow offunds Moreover this interconnectedness among trading partners implies that theactions of one country in particular those of the monetary authority may influenceconditions in the other country
Notes
1 Introduction
1 Empirically an economyrsquos total output is measured by real gross domestic product (orindustrial production) employment is estimated from company records or householdsurveys estimates of unemployment are compiled from household surveys or statisticson recipients of unemployment benefits and price indexes are computed to measurechanges in the overall level of prices
2 Leon Walras first derived the result in his book Elements drsquoEconomie Politique Purefirst published in 1874ndash77 (see Walras 1954)
3 An endogenous variable is one that is determined by the model An exogenous variableis one that is taken as given by the model
4 An element that distinguishes both static and dynamic macroeconomic analysis fromArrowndashDebreu analysis is the incorporation of money The exceptions to this in macro-economic analysis are real business cycle theories which for the most part are purelyldquorealrdquo in the sense that money is not an intrinsic part of the analysis
5 Note that an alternative to exogenous expectations is to specify an ldquoexpectation func-tionrdquo If this function relates expectations in certain specific ways to all currentinformation such that the expectation function reflects ldquorational expectationsrdquo thenstatic analysis is converted to dynamic analysis
6 In dynamic models this would be termed ldquoperfect foresightrdquo in a deterministic settingand ldquorational expectationsrdquo in a stochastic setting
7 One could say that the effect of such a variable is implicit in the exogenous level ofexpected future prices of the static analysis
8 A similar example of an incomplete listing of effects is a change in current governmentpolicies A change in current government actions may require changes in governmentactions in subsequent periods that will impact future markets Yet the construction ofstatic analysis does not require such effects to be spelled out
9 Forward markets are markets in which agreements are made that specify the prices atwhich goods will be exchanged in the future Goods are thus in essence indexed by thedate of trade
10 This is the ArrowndashDebreu contingent-claim interpretation of a competition equilibriummodel (see Arrow 1964 Debreu 1959)
11 In some cases one can attain the stationary state under the less restrictive requirementthat certain exogenous variables simply grow at a steady rate over time
12 While classical economists did not fully articulate their model Patinkin (1965) is widelycited as providing a comprehensive review and formalization of many of the key ideasunderlying the classical economistsrsquo views The result may be denoted the prototypeof the neoclassical (static) model Dynamic neoclassical macroeconomic models arebroadly based on neoclassical growth models with the addition of shocks of variouskinds Among the classic works developing neoclassical growth models are Solow(1956) Cass (1965) Koopmans (1965) and Sidrauski (1967a 1967b)
Notes 203
13 Two of the most widely known proponents of new classical economics are Robert Lucasand Thomas Sargent It has been suggested by some however that this line of analysisis less the extension of classical analysis than it is the antithesis of Keynesian analysisas discussed below See Niehans (1987) for this interpretation
14 This is changing as indicated by Howitt (1985) Shapiro and Stiglitz (1984) Weitzman(1985) and by the papers cited in the May 1988 American Economic Review
15 Howitt goes on to say that there is the ldquoreciprocal Keynesian question of how exactlythe economic system manages to overcome all the obvious coordination problems thatstand in the way of attaining the state of equilibrium common to new classical economicsmodelsrdquo
2 Walrasian economy
1 A brief description of the theory is given in Patinkin (1965 note B)2 In reality of course there is no auctioneer In fact some think that Walras would have
been the first to question such a description of the economy as realistically capturingthe true dynamic process by which the economy reaches the equilibrium described bysupply and demand curves See Walker (1987) who points out that Walras promoted aldquodisequilibrium-production model of tatonnementrdquo as more representative of Walrasrsquothought than the tatonnement model with an ldquoauctioneerrdquo
3 We use the letter T to denote the number of commodities because later we distinguishthe T commodities according to time of availability (ie from period 1 to period T )
4 One could replace the assumption of known prices by the assumption that individualsform expectations at time t about prices at time t and that such expectations are correctThis has been referred to as a situation in which expectations satisfy the assumption ofldquoweak consistencyrdquo
5 With zero transactions costs equality of purchase and sale prices could be viewed asforced by arbitrage conditions
6 Walras was one of the first to note this point Note that πjj = 17 See Debreu (1959 Chapter 2) for a discussion of such accounting prices8 A distinctive feature of macroeconomics is to alter traditional ArrowndashDebreu general
equilibrium analysis in such a way that we can reinterpret accounting prices as moneyprices and determine the level of money prices
9 Others characterize transactions costs in similar fashion For instance Alchian and Dem-setz (1972) cite the costs of ldquoformingrdquo ldquonegotiatingrdquo and ldquoenforcing contractsrdquo whileDahlman (1979) writes of ldquosearch and information costsrdquo ldquobargaining and decisioncostsrdquo and ldquopolicing and enforcingrdquo costs
10 See Varian (1992) for a discussion on these points By ldquowell behavedrdquo we mean thatpartuapartcai gt 0 and ua is strictly quasi-concave
11 In this case the Lagrangian is written ignoring the non-negativity constraints onconsumption of commodity i i = 1 T Below we provide an equivalent character-ization of the constrained maximization problem that incorporates these constraints inthe Lagrangian
12 A function y = f (x1 xn xn+1 xm) is said to be homogeneous of degree k inthe arguments x1 through xn if
f (λx1 λxn xn+1 xm) = λk f (x1 xn xn+1 xm)
3 Firms as market participants
1 With zero ldquoadjustment costsrdquo there could exist a market for capital at time t such thatthe capital employed during the initial period Kt is conceptually distinct from capitalinherited K In this case however equilibrium in the capital market at time t wouldthen imply that Kt = K
204 Notes
2 Note that output is a ldquoflow variablerdquo That is for a period i of length h the outputproduced is given by hyi The term yi is thus the ldquorate of outputrdquo over period i Sinceoutput is a flow variable at a point in time the rate of production is not defined (Asyou can see for any rate of output the limit of production as the length of the periodgoes to zero is zero)
3 In general for a period i of length h labor services are denoted by hNi such that totalwage payments at the end of the period are wihNi Like the labor input one shouldthink of the capital input in flow terms with the stock of capital determining the rateor flow of capital services
4 For simplicity of notation these expressions presume perfect foresight5 In general for a period i of length h the nominal interest rate from the end of that period
to the end of the next period is given by hri = hzpbi + (pbi+h minus pbi)pbi 6 For simplicity of notation these expressions assume perfect foresight with respect to
future prices of equity shares In addition the expressions presume future dividends areknown
7 To minimize notational clutter we have chosen to not denote such anticipations withthe expectation operator We do presume that agents (firms and households) have com-mon expectations (assumed to be held with subjective certainty) concerning plans withrespect to the issuing of equity shares
8 For instance if ri lt rei for period i there would be zero demand for bonds Theresulting excess supply of bonds would lead to a fall in the price of bonds and thus arise in the return on bonds until equality across rates of return held
9 We assume for the moment perfect foresight at time t with respect to prices at the endof the period (at time t + 1)
10 As in the prior models including money balances in the utility function reflectshow money can save (leisure) time required to make exchanges within a period Forsimplicity we limit money holdings to agents labelled ldquohouseholdsrdquo
11 For the representative household holdings of bonds issued by other households mustbe zero so that there is no real indebtedness effect with respect to the bonds exchangedamong representative households
12 In general if we assume there are ni firms producing commodity i and that there are mdifferent commodities then
yt =msum
i=1
⎡⎣ nisum
j=1
( pip)fij(Nijt Kijt)
⎤⎦
The above is a stylized view of how the empirical counterpart to total output real grossdomestic product is actually computed by the Commerce Department
13 Note that it is assumed that the capital stock K generates a fixed rate of capital servicesThat is we do not consider variation in the ldquoutilization of capitalrdquo If we did so thenvariation in the services flowing from the capital stock could be considered an additionalchoice variable with capital utilization presumably affecting the extent of depreciationin the capital stock over time
14 Fama and Miller (1972) cite conditions under which the ldquoowners of the firmrdquo iethe holders of the S equity shares will direct the managers of the firm at time t tomaximize Vt
15 In continuous time
petminus1 =int infin
t[psdsSs]eminusr(sminust)ds
where the interest rate r in the above expression is formally rs
Notes 205
16 Note that Stminus1 equiv S Assuming ri = rt i = t + 1 we have
pe = [1(1 + r)]⎡⎣ptdtS +
infinsumk=1
[pt+k dt+kSt+kminus1](1 + rt)k
⎤⎦
17 If the change is negative net revenues available for distribution as dividends would bereduced
18 We will explain more fully below the nature of these ldquoadjustment costsrdquo19 As suggested earlier if the utilization of capital were viewed as a choice variable then
it would be natural to have δ directly related to utilization of capital during the period20 Later we will say more about investment and the nature of adjustment costs21 Note that this discussion assumes for simplicity zero adjustment costs22 We follow Sargent and implicitly assume adjustment costs are related to net not gross
investment Net investment measures the change in the capital stock As we will seelater the result is a slightly different expression for Tobinrsquos Q than found elsewherewhen adjustment costs depend on gross investment We could also expand the natureof ψ so that adjustment costs depend on the size of the capital stock as is done in suchpapers as Lucas (1967) Uzawa (1969) and Gould (1968)
23 In the National Income and Product Accounts of the USA that report various measuresof the activity in the economy depreciation is measured by what is called the ldquocapitalconsumption allowancerdquo
24 The presence of such markets reflects the existence of ldquoperfect capital marketsrdquo in themacroeconomics literature Remember that with respect to the choice of investment forthe individual firm zero adjustment costs imply a potential discrete jump in the capitalstock at a point in time so that investment for the individual firm may not be defined
25 LrsquoHospitalrsquos rule states that if a is a number if f (x) and g(x) are differentiable and g(x)does not equal zero for all x on some interval 0 lt |x minusa| lt ε if the limit of f (x) equalszero as x approaches a and if the limit of g(x) is zero as x approaches zero then whenthe limit of the ratio f (x)gprime(x) as x approaches a exists or is infinite it equals the limitof f (x)g(x) as x approaches a
26 In continuous-time or discrete-time models such adjustment costs mean that the ldquocapitalmarketrdquo at time t is eliminated
27 Note that since bonds and equity shares are perfect substitutes the optimization problemwill not provide a breakdown into the optimal number of bonds versus equity shares
28 Recall that we are holding the stock of equity shares outstanding constant29 Note that we ignore the potential choice of capital at time t Kt and associated choice
of bonds setting Kt = K This in fact would be the case with capital adjustment costs30 If we evaluated returns from the start of a period each of the return functions would be
multiplied by 1R31 The equivalence of bond equity share and retained earnings financing of changes in
the capital stock can be shown32 This expression reflects the envelope theorem In particular we have that
dW (Kt+1)
dInt+1
dInt+1
dKt+1= dW (Kt+1)
dNt+1
dNt+1
dKt+1= 0
33 If the price of capital differed from the price of output but both prices were expected tochange at the same rate so that the relative price of capital was assumed to be constantover time then the expected real user or rental cost of capital would be (pkp)(mt + δ)where pkp denotes the relative price of capital
206 Notes
34 Formally at time t the inherited debt-to-equity ratio is given by pbBpeS35 Naturally there are other factors not discussed36 Note that during period t when the length of the period between planned purchases of
capital (at time t) and final installation (at time t + 1) is 1 then net investment Int isgiven by Int = Kt+1 minus K and adjustment costs are given by ψ(Int)
37 During period t+1 net investment is defined by Int+1 = Kt+2 minusKt+1 and adjustmentcosts are given by ψ(Int+1)
38 We can see from the general nature of the investment demand functions that if theproduction function were not separable then the assumption of a constant real wageover time as well as a constant expected real rate of return over time would obtain thisresult
39 See Sargent (1987a 11) for a similar expression in continuous time Note that thetwo expressions would be exact if we take the limit as the length of the period goesto zero The equality between ψ prime
t+h and ψ primet+2h that typically would be an approx-
imation in discrete time holds exactly in the limit In addition the definition ofthe real interest rate for a period of length h 1 + hm = (1 + hr)(1 + hπ) orm = (r minus π)(1 + hπ) indicates that in the limit (as h goes to zero) m = r minus π If we had assumed that adjustment costs were based on gross not net investmentthen the fraction on the left-hand side of (312) would include the term minusδψ in thedenominator
40 The Q theory of investment demand was suggested by Tobin (1969) Sargent is oneauthor who expresses investment demand in this way
41 In fact empirical measures of Tobinrsquos average Q have been constructed although suchmeasures are more complex than those discussed here since they must incorporateinfluences of the tax system (such as investment tax credits accelerated depreciationallowances and the like) on the optimal choice of the capital stock
42 In a two-sector model Tobinrsquos average Q is given by pV p1K where pV is the nominalvalue of the firm and p1 denotes the price of capital which differs from p the price ofoutput
43 In our case there is a single argument of the adjustment function ndash net investment Ifone follows Hayashi (1982) and assumes as he does that adjustment costs depend ongross investment then one must add the assumption that there is a linear homogeneousadjustment function such that ψ(δk) = ψ primeδk over the appropriate range Hayashirsquosadjustment cost function (like others) includes the stock of capital as well
44 Eulerrsquos theorem (or law) is that if the function f (middot) is a differentiable functionhomogeneous of degree 1 (linear homogeneous) with f Rn rarr R then
f (x) equivnsum
i=1
[partf (x)partxi]xi
Replacing the real wage with the marginal product of labor reflects the assumption thatthe firm is a price-taker in both the output and labor markets
45 See for example Azariadis (1976) A second reason is that new workers and previouslyemployed workers may not be considered perfect substitutes (eg see Oi 1962)
46 Taylor (1972) and Hall (1980) are among those who have examined thisphenomenon
4 Households as market participants
1 Time consistency means in this context that households will follow through on priorplans as the starting date advances Strotz (1955ndash56) discussed this point in terms of autility maximizing problem
Notes 207
2 An example of nonseparable preferences is a ldquohabit persistence modelrdquo in which utilityis given by
infinsumi=t
βiminustu(ci ciminus1 )
so that consumption at time t depends on prior consumption Alternatively one couldhave utility given by
infinsumi=t
βiminustu(ci 1 minus Ni 1 minus Niminus1)
such that past work becomes a pertinent state variable for the current period Kydlandand Prescott (1982) is one important exception to the macroeconomic literature inassuming nonseparable preferences
3 This is sometimes referred to as an ldquoend-of-periodrdquo equilibrium specification in themarket for assets Alternative asset specifications for discrete-time analysis have beendiscussed by among others Foley (1975) As Edi Karni (1978) pointed out Patinkinrsquosmodel is an end-of-period model
4 For simplicity we ignore the financial asset markets at time t If we assumed portfolioadjustment costs then it would be the case that at time t desired bond and moneyholdings would be B and S respectively
5 That is (partW (xt+1)partcdt+1)(partcd
t+1partxt+1) = 0 since partW (xt+1)partcdt+1 = 0 similarly
the indirect effects of a change in xt+1 on partW (xt+1) through its impact on optimallabor supply and real money balance holdings are zero
6 For completeness note that by substituting equation (43prime) into (45) we could have theequivalent expression
partW (xt+1)partxt+1 = β[partut+1part(Mt+1pt+1)]Rt+1[Rt+1 minus Rmt+1]7 These conditions appear in various forms throughout the macroeconomics literature
see for instance Mankiw et al (1985) or Barro and King (1984)8 The resulting demand for leisure function is termed a ldquoHicksian or compensated demand
functionrdquo as it is constructed by varying the price of leisure (the real wage) and incomeso as to keep the individual at a fixed level of utility
9 The resulting demand function for leisure that incorporates both the income and sub-stitution effects of changes in the real wage is an example of a ldquoMarshallianrdquo demandfunction
10 Borjas and Heckman (1978) See also Pencavel (1985) for a survey of estimates Forevidence on the effect of wage increases on the labor supply of working women seeNakamura and Nakamura (1981) and Robinson and Tomes (1985)
11 Note that we continue to assume nonsatiation such that partutpartct gt 012 The second statement presumes sufficient dispersion in preferences so that at each real
wage there are some individuals who are just indifferent between a zero and positivelabor supply Thus any rise in the real wage will increase labor force participation
13 This point was first formally developed in the classic paper by Lucas and Rapping(1970) The role of ldquomarket clearingrdquo in macroeconomic analysis will be clearer laterwhen we consider alternative characterizations of the labor market
14 To be exact this result assumes that utility is separable and concave in leisure That isit is assumed that part2upartcpartN = part2upartNpart(Mp) = 0 and part2upartN 2 lt 0
15 Alogoskoufis (1987) provides a good review of the empirical analysis in this area
208 Notes
16 Fisherian ndash or in Patinkinrsquos (1965) terminology ldquoFisherinerdquo ndash analysis takes itsname from the classic ldquotime-preferencerdquo analysis of Fisher see in particular Fisher(1930)
17 To assure this one could let duadcai gt 0 with limcairarr0(duadcai) rarr infin andlimcairarrinfin(duadcai) rarr 0
18 Equivalently one could assume partuapart(Maip) equiv 0 for all i Given the nonnegativityconstraint on Maipi i = t t + T and the presumption of a positive nominalinterest rate ri i = t t +T minus1 the optimal solution would be M d
aipi = 0 for all iThat is bonds will dominate money as an asset and only bonds will be held if assetholdings are positive The reason is simple ndash the unique attribute of money holdingsas a way to reduce ldquotransaction costsrdquo has not been introduced by providing a ldquoutilityyieldrdquo to holding money
19 Note that the equality sign here rather than the ge sign reflects the fact that the first-order conditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
20 Note that we assume a time-invariant one-period utility function The notation uai
reflects the fact that utility of agent a in period i depends on consumption inperiod i cai
21 In general for a period of length h we have that 1 + hm = (1 + hri)(1 + hπi)Solving for hmi and then dividing through by h we have that mi = (ri minusπi)(1+hπi)Thus in the limit as the length of each period goes to zero mi = ri minus πi Thus incontinuous-time analysis the real rate of interest is exactly defined by ri minus πi
22 In the discussion to follow we maintain the assumption of a time-invariant utilityfunction such that ua
i (cai) = ua(cai) for i = t t + T 23 At the point where cat = cat+1 the slope of the indifference curve is minus1β Assuming
cat = cat+1 and no initial assets or debt (ie zBat = 0) if 1β = Rt then the resultof the same consumption in each period would imply the individual would be neither alender nor a borrower
24 As Modigliani (1996) has stated ldquothe consumption and saving decisions of householdsat each point of time reflect a more or less conscious attempt at achieving the preferreddistribution of consumption over the life cycle subject to the constraint imposed bythe resources accruing to the household over the lifetimerdquo Modigliani summarizes hiscontribution to the analysis of consumption behavior in his Nobel lecture of December1985 (see Modigliani 1986)
25 Examples of such discussions are Diamond and Hausman (1984) and Hurd (1987)However what happens in the aggregate does mask different behavior among subgroupsof the populations For instance Burbidge and Robb (1985) find for Canadian data thatwhile an inverted U-shaped profile exists for the ldquoaveragerdquo Canadian household ldquowhitecollarrdquo households do appear to continue to accumulate wealth years after both husbandand wife have left the labor force
26 If individual a were the representative agent then consumption smoothing would implythat the aggregate endowment of the consumption good is identical across periods ieci = ci+1 i = t t + T minus 1
27 Note that in the case of multiperiod bonds agent arsquos future income could includepayments derived from the initial holdings of assets Since the present value of suchfuture ldquoincomerdquo is incorporated in the current value of the assets operationally futureincome cat i = t+1 t+T is defined as income other than derived from initial assetholdings In a production context the source of such income would be compensationfor labor services sold
28 In an economy with production this transitory component of current income can reflectsuch events as a temporary layoff a short-run opportunity to work overtime or atemporary tax rebate
29 This discussion ignores the effects of uncertainty on optimal consumption plans
Notes 209
30 Note that the equality sign here rather than the ge sign reflects the fact that the first-orderconditions imply that λi gt 0 so that the condition λipartLpartλi = 0 must be met bypartLpartλi = 0
31 Note that with one-period bonds individual a has a zero initial endowment of bondsthat continue into the future at time t
32 Other papers on this topic include Hansen and Singleton (1983) and Mankiw et al(1985)
33 An example of a compositional change that could affect aggregate consumption butwould not be accounted for in the analysis to follow is a change in the proportionof retired individuals in the economy Thus on this ground at least Hallrsquos empiricalfindings cannot be taken as the last word on consumption behavior at the level of theindividual
5 Summarizing the behavior and constraints of firms and households
1 Caballero and Engel (1999) provide an empirical study of investment dynamics in thecontext of manufacturing firms
2 Weber (1998) provides empirical evidence on the link between the financial marketsand consumption spending
3 We refer to this perfect foresight as ldquolimited perfect foresightrdquo since it concerns only thecurrent period This focus on current markets alone typical of static analysis impliesldquoexpectations functions of the agentsrdquo with respect to prices in subsequent periodsthat do not reflect the underlying analysis of markets beyond the current period Thusbeyond the current period expectations do not have the property of perfect foresight (orrational expectations in a nondeterministic setting)
4 In an open economy ndash that is one which admits a foreign sector ndash we would add afourth market the market for foreign exchange
5 Note that in a fully monetized economy money enters on one side of every exchange ndashpurchase or sale ndash in these three markets
6 Note that money holdings and money demand arise only for ldquohouseholdsrdquo To the extentfirms do hold money and make choices with respect to the size of such holdings wepresume their behavior would be similar to that of households and so lump firms withhouseholds with respect to such activity Thus the behavior of the firm is restricted tolabor demand output supply investment demand and financial asset supply
7 Recall that the expected real user cost of capital is mt + δ where δ is the rate ofdepreciation of capital and mt is the expected real rate of interest (ie mt equiv (rt minusπe
t )(1+πet ) where rt is the money interest rate and πe
t is the expected rate of inflationbetween periods t and t + 1)
8 As previously for simplicity we continue to assume that expectations of future pricesare held with subjective certainty
9 If part2f partKpartN = 0 then the real wage would not enter as an argument in the capitaldemand function nor would the existing capital stock affect labor demand
10 That is we can express a demand function in a form similar to one that would beobtained if the analysis were to consider only two periods
11 Note that we do allow the expected wage inflation between period t and t + 1 πewt
to differ from subsequent wage inflation so that the expected real wage next periodwe
t+1pet+1 equiv wt(1 + πe
wt)pt(1 + πet ) can differ from the current real wage This
introduces the possibility of intertemporal substitution of labor supply in response to achange in the current real wage
12 Assuming less than unit elastic expectations with respect to future income streams wouldassure from the standard Fisherian problem that the marginal propensity to consumewould be less than one
13 Recall that πet equiv ( pe
t+1 minus pt)pt
210 Notes
14 Recall that firms are presumed not to hold money balances so there are no real balanceeffects to concern us with respect to firmsrsquo demands or supplies
15 As Lucas and Rapping (1970) point out introducing other expectation assumptionscan retain the ldquointertemporal substitution hypothesisrdquo in the context of changes in thecurrent price level but in doing so money illusion is introduced In their own wordsthe
labor-supply equation is not homogeneous in current wages and current prices(such that) there is ldquomoney illusionrdquo in the supply of labor ldquomoney illusionrdquoresults not from a myopic concentration on money values but from our assumptionthat the suppliers of labor are adaptive on the level of prices expecting a returnto normal price levels regardless of current prices and from the empirical factthat the nominal interest rate does not change in proportion to the actual rate ofinflation With these expectations it is to a supplierrsquos advantage to increase hiscurrent supply of labor and his current money savings when prices rise
(Lucas and Rapping 1970 268ndash269)
Lucas and Rapping are particularly looking at the effect of a change in the current pricelevel on the expected real rate of interest Their assumption of ldquoadaptiverdquo expectationsimplies that an increase in pt results in a fall in πe
t equiv (pet+1 minus pt)pt a rise in the
expected real rate of interest and thus a rise in labor supply16 That is future technology capital demand labor demand the real wage the rate of
depreciation and future real interest rates are all unchanged by such a change in pricesand the money supply
6 The simple neoclassical macroeconomic model (without governmentor depository institutions)
1 In particular it is assumed that the money wage rate adjusts to continuously maintainequality between the demand for and supply of labor the price of output adjusts tomaintain equality between the demand for and supply of output and the price of financialassets (and thus the interest rate) adjusts to maintain equality between the demand for andsupply of financial assets Patinkin (1965) provides one of the first complete accountingsof this model
2 That is we would expect prices in the various markets to eventually adjust to eliminateany possible excess demands or supplies in the economy We would also expect agentsultimately to correctly anticipate the price level The neoclassical model can be modifiedto explain the workings of the economy in the face of incomplete information and priceinflexibility
3 As before the ldquolaws of motionrdquo dictating how prices change to reach equilibrium aregiven by Walrasrsquo excess demand hypothesis and we maintain the assumption that noexchange occurs until an equilibrium is reached (the recontracting assumption)
4 Alternatively one could assume that expectations at time t concerning these futurevariables are constant
5 Note that Patinkin has firms as well as households managing a portfolio of financialassets and money balances which is why he includes the demand function for labor inthe above statement In our analysis this statement applies to the labor supply functionalone
6 This last sentence anticipates the intertemporal substitution hypothesis7 We ignore the potential effect of changes in the interest rate on labor supply and thus
employment8 This reflects the assumption that households and firms share common expectations
concerning the price level (in fact for both pet = pt)
Notes 211
9 Reasons such as these for changes in output form the basis of much of the currentanalysis in the literature with respect to ldquoreal business cyclesrdquo
10 This is perhaps too extreme a statement To the extent that a higher price level isanticipated the resulting lower real money balances could lead to an increase in laborsupply at any given real wage and consequently increased employment and output Alsothere is a potential effect of changes in the price level on aggregate labor supply throughthe impact of such changes on the expected real rate of interest if unit elastic expecta-tions concerning future prices are not assumed Recall that we follow macroeconomictradition and abstract from these possibilities
11 In a recent study Ewing et al (2002) develop a model of the equilibrium unemploymentrate and examine how it responds to unanticipated changes in real output
12 Fairlie and Kletzer (1998) discuss the issues revolving around job displacement13 Unemployment may also result if prices in the economy do not adjust quickly enough to
ensure that all markets (particularly the labor market) are continuously in equilibriumUnemployment associated with labor market disequilibrium is sometimes referred toas ldquoinvoluntaryrdquo unemployment We analyze such unemployment later
14 See for instance the paper by Evans (1989) that examines the relationship betweenoutput and unemployment in the United States
15 The ldquoISrdquo equation is sometimes referred to as the aggregate demand equation indicatingthat it reflects equality between total or aggregate demand for output and productionNote that equilibrium in the output market is being described in terms of demand equalto what is produced ylowast
t The aggregate supply equation indicates what will be produced16 This is the case only for this simple aggregate model without government17 Such an assumption removes the anticipated real wage next period as an argument
in these demand functions Recall that earlier ldquostaticrdquo assumptions concerning futureinterest rates and rates of inflation have already simplified the form of these functions
18 However ldquorealrdquo or ldquosupplyrdquo shocks such as the above-mentioned changes in technologycapital stock supply of other inputs such as oil or in labor supply at prevailing realwages can affect real output
19 This analysis should be familiar since we performed a similar analysis for the economywithout production
20 Note that unit elastic expectations imply that partπet partpt = 0
21 Note that without a real balance effect the CC curve would be horizontal22 The lower interest rate abstracts from a real balance effect in the output market so that
the CC curve is horizontal
7 Empirical macroeconomics traditional approaches and time series models
1 To reduce notational clutter we suppress time subscripts All variables are period tvariables
2 Note that in Sargent (1987a 20) equation (71) is replaced by the ldquorepresentativerdquofirmrsquos first-order condition for the optimal use of the labor input given a competitivelabor market that is the condition that the real wage equals the marginal product oflabor wp = Fn(n K) Note that if the production function F(n k) is separable in thelabor and capital inputs such that f (n K) = v(n)+u(K) and v(n) = (1g)(fnminusn22)then equation (71) is identical to Sargentrsquos equation since fn(n K) = (1g)(f minus n)
3 Unless otherwise noted all parameters in this model such as f g h and j in equations(71) and (72) are assumed to be positive
4 In this context ldquolump-sumrdquo taxes are taxes independent of income Equation (73) isan example of the ldquoconsumption functionrdquo
5 Endogenous variables are variables whose values are determined by the analysis6 Sargent adds equations (73) and (74) to the system (712) in order to determine
consumption and investment demand as well
212 Notes
7 To derive wlowast start with the equilibrium condition nd = ns Substituting the first twoequations of 712 the equilibrium money wage satisfies
f minus g middot (wlowastp) = h + j middot (wlowastp)
Then one can simply solve this equation for the equilibrium money wage wlowast We canthen obtain the equilibrium employment level nlowast by substituting the expression for wlowastinto either of the first two equations of 712
8 This is a property of ldquoclassicalrdquo macroeconomic models in that monetary changesthat alter the price level do not affect real variables such as the real wage output andemployment
9 The IS equation indicates combinations of the interest rate r and output y at which ifthe output were produced and that interest rate prevailed output demand would equaloutput produced The LM equation indicates combinations of r and y that will equatethe demand for and supply of money
10 For the model under consideration the ldquoaggregate demand curverdquo (a plot in ( p y) spaceof the aggregate demand equation) slopes downward and the ldquoaggregate supply curverdquo(a plot in ( p y) space of the aggregate supply curve) is vertical The intersection of theaggregate demand and supply curves graphically determines the equilibrium output andprice level
11 See Altonji and Siow (1987) Ewing and Payne (1998) examined the relation-ship between the personal savings rate and consumer sentiment in the context of aconsumption model
12 Note that Taylor assumes that certain demands for instance consumption demand maydepend on past as well as current values of output and the money supply so that thereduced-form expression estimated includes lagged values of income and the real moneysupply
13 Time series analysis can be viewed as primarily the art of specifying the most likelystochastic process that could have generated an observed time series
14 That is forecasts generated by time series models have been used to proxy individualsrsquoexpectations of future events in tests of various theoretical macroeconomic models
15 A histogram is a plot of the frequency distribution of a set of observations16 If the process is also ldquoergodicrdquo these statistics give consistent estimates of the mean and
variance Ergodicity basically requires that observations sufficiently far apart should bealmost uncorrelated Then by averaging a series through time one is continually addingnew and useful information to the average For a rigorous explanation of this conceptsee Hannan (1970 201)
17 The variable γk k = 0 1 2 is termed the autocovariance function18 Or more generally k = 0 Note that ρk = ρminusk 19 Some have suggested that stock market prices follow a random walk See Campbell
et al (1997)20 Note that if yt is taken to be the logarithm of real output then the trend reflects a constant
rate of growth of output equal to d in the absence of shocks21 This assumes y0 is the initial value of the function yt 22 A random walk is an example of a class of nonstationary processes known as ldquointe-
gratedrdquo processes that can be made stationary by the application of a time-invariantldquofilterrdquo As defined by Granger and Newbold (1986) ldquoif a series wt is formed by a linearcombination of terms of a series yt so that wt = summ
j=minuss cjytminusj then wt is called a
lsquofilteredrsquo version of yt If only past terms of yt are involved so that wt = summj=0 cjytminusj
then wt might be called a one-sided or backward-looking filterrdquo23 This is a special case of a class of stochastic processes known as Markov processes24 Recall that for the random walk process φ = 1 in which case the process was not
stationary as the variance of the process becomes larger and larger with time
Notes 213
25 To obtain this result note that
E
⎡⎣ infinsum
j=0
(φ2)jε2tminusj
⎤⎦ =
infinsumj=0
(φ2)jσ 2e = σ 2
e (1 minus φ2)
26 Often the ldquolag operatorrdquo L or equivalently the ldquobackward shift operatorrdquo B is usedto express this equation Lτ yt (or Bτ yt) = ytminusτ τ = 1 2 3 There are associatedpolynomials in the lag operator such that
d(L) = d0 + d1L + d2L2 + d3L3 + middot middot middot + dpLp
Letting
φ(L) = 1 minus φ1L minus φ2L2 minus φ3L3 minus middot middot middot minus φpLp
we can thus express an AR( p) process for yt as φ(L)yt = εt 27 If yt is an AR( p) process it may be described in the following way ldquoan appropriate
finite backward-looking filter applied to yt will produce a white noise seriesrdquo (Grangerand Newbold 1986 32)
28 If there were a single shock to yt at time 0 (ie εt = 0 for all t gt 0) then the deterministiccomponent would signify the deviation of the time path from its equilibrium level Aswe will see in this case stationarity would imply ldquostability of equilibriumrdquo in that ytwould converge to its equilibrium value over time
29 A linear difference equation of order s is of the form
yt =ssum
j=1
ajytminusj + c
30 Successive substitution (the ldquoiterative method of solutionrdquo) reveals this essential natureof the solution In particular substituting for past values of yt in (725) results inyt = φty0
31 The appendix to this chapter shows how one can reinterpret higher-order differenceequations as a system of first-order difference equations
32 Note that in general a quadratic equation of the form
ax2 + bx + c = 0
can be solved using the quadratic formula
x1 x2 = [minusb plusmn (b2 minus 4ac)12]2a
In our case a = 1 b = minus(m1 + m2) = minusφ1 and c = m1m2 = minusφ233 This assumes m1 = m2The solution for repeated roots is discussed briefly below34 A variable x is said to be inside the unit circle if x lt |1| An alternative way of
expressing this condition is in terms of the associated polynomial 1 minusφ1L minusφ2L2 = 0This polynomial equation is similar to (730) except that b is replaced by 1L and thewhole equation is multiplied through by L2 The stationarity condition is that the rootsof this polynomial equation should lie outside the unit circle
35 Recall that φ1 = m1 + m2 and φ2 = minusm1m236 The following discussion follows Goldberg (1958 171ndash172)
214 Notes
37 As discussed below in this case the two roots are m1 = h + vi and m2 = h minus vi whereh = φ12 v = (4φ2 + φ2
1)122 and i is the imaginary number (minus1)12 The product
of these two roots is (φ21 minus 4φ2 minus φ2
1)4 = minusφ2 which is the square of the modulus
of the roots We are assuming (minusφ2)12 lt 1 so we have the condition that φ2 lt 1 orφ2 gt minus1
38 Note that if yt is an MA process then it is a ldquobackward looking filterrdquo applied to a whitenoise process As before we can use a lag operator L (or backward shift operator B) toexpress an MA(q) process for yt as yt = micro + θ(L)εt where the polynomial in the lagoperator θ(L) is given by
θ(L) = 1 + θ1L + θ2L2 + middot middot middot + θqLq
39 Recall that as discussed above we assume for simplicity zero mean for yt that ismicro = 0
40 For a more detailed description of invertibility see Granger and Newbold (1986) orBox and Jenkins (1970)
41 Note that ARMA( p 0) equiv AR( p) and ARMA(0 q) equiv MA(q) Using the lag operatorthe ARMA model (in the case of a zero mean) can be simply expressed as φ(L)Yt =θ(L)εt
42 In general if yt is ARMA( p1 q1) and xt is ARMA( p2 q2) the sum zt = yt + xt isARMA( p3 q3) where p3 le p1 + p2 and q3 le max(p1 + q2 p2 + q1) A proof of thisis found in Granger and Morris (1976)
43 A time sequence T (t) is called ldquodeterministicrdquo if there exists a function of past andpresent values gt = g(T (t minus j)) j = 0 1 such that E[(Tt+1 minus gt)
2] = 0 If thefunction gt is a linear function of Ttminusj j ge 0 then Tt is called ldquolinear deterministicrdquo
44 For instance for a stationary series of quarterly data one could postulate a simplefourth-order seasonal AR process of the form
yt = φ4ytminus4 + εt
This is a special case of AR(4) with φ1 = φ2 = φ3 = 0 The model could be extendedto include both AR and MA terms at other seasonal lags for instance the followingARMA(21)
yt = φ4ytminus4 + φ8ytminus8 + θ4εtminus4 + εt
To add other than seasonal components one could simply fill in the gaps (eg addterms such as φ1ytminus1 θ1εtminus1 and the like to the above) Other options are discussedby Harvey (1993) and others
45 Campbell and Mankiw argue that even if the log of output followed a random walk withdrift indicating that the effect of any shock persists indefinitely into the future estimatesusing the detrended series would be biased and erroneously conclude otherwise
46 Note that if the log of real output is an ARMA( p q) process then the differencedprocess will be an ARMA( p q + 1) process This means that to allow for stationaritywith respect to the level of real output requires at least one moving average process forthe differenced series
47 Equivalently for the logarithm of real output they are considering ARIMA(p 1 q)processes for p = 0 1 2 3 and for q = 0 1 2 3
48 Such a finding is often termed as supportive of real business cycle theories
Notes 215
8 The neoclassical model
1 The presence of money reflects the introduction of ldquoimperfectrdquo or costly informationA medium of exchange can arise to minimize costs incurred by participants in theeconomy when there exists imperfect information on potential exchange partners
2 In so doing we assume that the new equilibrium like the initial one exists Further wedisregard the process of adjustment of the variables to the new equilibrium Alterna-tively we could introduce ldquolaws of motionrdquo for the equilibrium values (eg the excessdemand hypothesis for price changes) and examine whether the equilibriums are stable
3 For notational simplicity we let Ld = Ldt cd = cd
t Id = Idnt p = pt r = rt
yd = cdt + Id
nt + δK + ψ(Idt ) and M = M
4 For simplicity we let the expected real rate of interest component of the expected realuser cost of capital ((rt minus πe
t )(1 + πet )) be approximated by rt minus πe
t as representedby r minus π
5 One alternative is for the interest rate to adjust to equate the demand for and supply ofmoney
6 That is we assume that consumption demand is a function of the expected gross realrate of interest Rt not its components (rt and πe
t ) This would be the case if one couldview the consumption decision from the point of view of the pure ldquoFisherian problemrdquoin essence separating the allocation of consumption across time decisions from theldquoportfolio problemrdquo Note that for notational ease we not only let rt denote the moneyinterest rate rt but also let π denote the expected rate of inflation πe
t 7 This would be the case if our focus was solely on the portfolio choice problem8 Examples of growth models with money include Tobin (1965) and Sidrauski (1967b) As
Begg (1980 293) notes ldquoin a steady state any expectations generating mechanism willyield correct predictions thus the steady state analysis of growth models with moneymay be viewed as a special case of the rational expectations model with systematicmonetary policyrdquo
9 This asymmetry in information can reflect an aggregation across labor markets in whicheach firm determines labor demand based on its correct anticipation of the price of theparticular commodity it produces while suppliers of labor determine labor supply basedon their potentially incorrect anticipation of the overall level of prices reflecting theidea that suppliers are concerned with the purchasing power of wages in terms ofcommodities not restricted to the particular commodity that they produce
10 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt r = rt y = yt and M = M
11 Recall that for simplicity we let the expected real rate of interest component of theexpected real user cost of capital (rt minus πe
t )(1 + πet ) be approximated by rt minus πe
t asrepresented by r minus π
12 Note that partypartp = 0 for the neoclassical model given limited perfect foresight13 This is obvious from the graph of aggregate demand and supply curves if one compares
the vertical aggregate supply curve of the neoclassical model dpp = dMM with theupward-sloping aggregate supply curve of the model with real wage illusion Note thatthe shift in the aggregate demand curve for a given change in the money supply isidentical in either case
9 The ldquoKeynesian modelrdquo with fixed money wage modifyingthe neoclassical model
1 Typically a union contract runs for three years Often however there are provisions thatpermit parts of the agreement to be renegotiated at specific times during the three-yearcontract period
2 As a general rule labor agreements are specified in money terms An exception to thisis the cost of living agreements (COLAs) as part of union wage contracts in the United
216 Notes
States which became popular during the 1960s and 1970s COLAs adjust money wagesautomatically to changes in prices typically using changes in the Consumer Price Indexto measure price changes However the percentage of all workers who have contractswith COLAs is fairly small as less than 20 percent of the total labor force is covered bycollective bargaining agreements Further not all COLA clauses offer full protectionagainst general price increases as wages may rise by only some fraction of the increaseof the CPI Considering these qualifications for the time being we simplify by assumingthat all labor agreements are specified in money terms
3 These seven variables for the classical model are made up of three ldquopricesrdquo along withfour variables implied from the behavioral equations The three ldquopricesrdquo are moneywage price level and interest rate determined by the equilibrium conditions for the labormarket and two of the three other markets (output financial andor money markets)The variable implied from the behavioral equations are employment (labor demandfunction) output (production function) consumption (consumption demand function)and investment (investment demand function)
4 Note that this analysis is inconsistent with the idea that long-term employment contractsthat fix the money wage for several periods arise due to adjustment costs with respectto the labor input
5 As discussed before one justification for this form is if real money balances are not partof household wealth which can be the case when we introduce depository institutionsinto the analysis
6 As before for notational simplicity we let Ld = Ldt cd = cd
t p = pt Idnt = Id r = rt
y = yt and M = M 7 Recall that for simplicity we let the expected real rate of interest component of the
expected real user cost of capital (rt minus πet )(1 + πe
t ) be approximated by rt minus πet as
represented by r minus π 8 Note that partypartp = 0 for the neoclassical model given limited perfect foresight9 The Lucas model was introduced in Chapter 8 and is covered in more depth in
Chapter 1010 An exception to this statement occurs if monetary authorities can react to period t
disturbances that is if monetary authoritiesrsquo information set includes the values of therandom shocks in period t which are not known to private agents
11 Phelps and Taylor (1977) go on to state that they ldquodo not pretend to have a rigorousunderstanding of [why prices andor wages are set in advance] In the ancient andhonorable tradition of Keynesians past we take it for granted that there are disadvantagesfrom too-frequent or too-precipitate revisions of price lists and wage schedulesrdquo
12 Note that θ is affected by the variability of relative price shocks in relation to generalprice shocks
13 We assume for simplicity that the coefficient on Pt minus Wt + φ is one14 Note that if Pt minus Wt + φ = 0 then equation (99prime) is a first-order linear difference
equation of the form YtminusλYtminus1 = 0 with solution Yt = c0λt In the limit as t approachesinfinity Yt equals zero Recall that the logarithm of the natural level of output is zero
15 Recall our assumption that the scale factor in the determination of the real wage is equalto zero for convenience
16 As Fischer shows if wages were set only for the current period t that is tminusiWt = φ +EtminusiPt then his results would be like those obtained by the Sargent and Wallace modelwith rational expectations for similar reasons This can be seen clearly on substituting(910) into (99prime) in which case one obtains the standard Lucas-like aggregate supplyequation
17 Later it will be assumed that ut and utminus1 are correlated so that information obtainedduring period t minus 1 will help predict variations in output for period t Like the laggedoutput term in the Lucas equation this introduction of serial correlation in output isnecessary (but not sufficient) if monetary policy rules that dictate the money supply
Notes 217
for period t based on information obtained up to the end of period t minus 1 are to serve astabilizing role
18 Note that in Fischerrsquos paper minusvt rather than vt is used in (913) Our vt is more in linewith other models
19 With the one exception noted before that Fischer has minusvt replacing vt 20 Again note that Fischer has minusηt and minusρ2 in the above equation since our vt is his minusvt 21 Fischer has b1 = minusρ2 since our vt is his minusvt
10 The Lucas model
1 The discussion follows that found in Lucas (1973) and Sargent (1987a 438ndash446)2 Or as Sargent (1987a 483) states ldquoan employee cares about his prospective wage
measured not in terms of own-market goods but in terms of an economy-wide averagebundle of goods the assumption is that the labor supplier works in one market butshops in many other marketsrdquo
3 The CobbndashDouglas production function would be given by yit = (Nit)1minusα(Ki)
α whereKi denotes the inherited capital stock for sector or ldquoislandrdquo i Nit is the employment oflabor for sector i and yit is the production of commodity i by sector i during period t Inthis case the marginal product of labor is given by partyitpartNit = (1 minus α)(Nit)
minusα(Ki)α
which implies that a = (1 minus α)(Ki)α
4 Solving this expression for the demand for labor and differentiating with respect to thereal wage relevant for a firm producing commodity i we have
partN dit part(witpit) = (1 minus α)minus1α(minus1α)(witpit)
(minus1minusα)α lt 0
Note that our particular form for the labor demand function implies a constant elasticityIn particular the elasticity of demand for labor is given by
partN dit
part(witpit)
witpit
N dit
= minus 1
α
5 Up to this point we have assumed that expectations are held with ldquosubjective certaintyrdquoso that Et(1pt) = 1Etpt and the expected real wage can be represented as witEtpt However we now assume that individuals view the price level as a random variableIn this setting the expression for the expected real wage is Et(witpt) or witEt(1pt)which is not the same as witEtpt In fact the relationship between witEt(1pt) andwitEtpt is shown by Jensenrsquos inequality witEt(1pt) ge wit(1Etpt) This inequalityreflects the fact that the function f ( pt) = 1pt is convex rather than linear in pt Forinstance if pt = p0 with probability t and p1 with probability 1 minus t then we haveEt(1pt) = tf ( p0) + (1 minus t)f ( p1) gt f (tp0 + (1 minus t)p1) = 1Etpt where 1 gt t gt 0On the other hand ln(1pt) is linear in ln pt so the log of the real wage is linear in thelog of the price level Thus we express the labor supply function in logarithmic formand consider the expectation of the log of the price level
6 The term ξt is assumed to be serially independent which means that E(ξtξs) = 0 fort = s
7 We assume that zit and ξt are statistically independent8 In problems involving more than two random variables ndash that is a ldquomultivariate regres-
sionrdquo ndash we are correspondingly concerned with the term E(z|x y) the expected valueof z for given values of x and y and so on
9 The discussion that follows is standard statistical theory Note that if the joint distributionof the two variables is a bivariate normal density function then the regression of ln pton ln pit is linear
218 Notes
10 In general if x1 x2 xn are random variables a1 a2 an are constants andq = a1x1 + a2x2 + middot middot middot + anxn then
E(q) =nsum
i=1
aiE(xi)
Var(q) =nsum
i=1
a2i Var(xi) + 2
sumiltj
aiajCov(xixj)
wheresum
iltj means that the summation extends over all values of i and j from 1 to nfor which i lt j This expression is derived from the definition of the variance
Var(q) equiv E([q minus E(q)]2) = E
⎛⎜⎝⎡⎣ nsum
i=1
ai(xi minus microi)
⎤⎦
2⎞⎟⎠
which can be expanded by means of the multinomial theorem according to which forexample (a+b+ c +d)2 = a2 +b2 + c2 +d2 +2ab+2ac +2ad +2bc +2bd +2cdNote that
Cov(xixj) = E[(xi minus microi)(xj minus microj)]
If the xi i = 1 n are independent then
Var(q) =nsum
i=1
a2i middot Var(xi)
11 Note that these first two expectations are taken without information on ln pit 12 While it is true that if two random variables are independent they are also uncorre-
lated the converse does not necessarily hold That is two random variables that areuncorrelated are not necessarily independent However two random variables havingthe bivariate normal distribution are independent if and only if they are uncorrelated
13 In general for any two random variables x1 and x2 with joint density function f (x1 x2)the marginal density of x2 g(x2) is obtained by integrating out from minusinfin to +infinthe other variable Thus g(x2) = int
f (x1 x2)dx1 Similarly we can obtain the marginaldensity of x1 by integrating out x2 The conditional probability density function of therandom variable x1 given that the random variable x2 takes on the value x2 is definedby φ(x1|x2) = f (x1 x2)g(x2) assuming that g(x2) does not equal zero
14 To obtain the following expression we use the fact that the random variables ξt and zitare independent with zero means and variances σ 2 and σ 2
z respectively15 Note that this ldquosignal extractionrdquo problem appears in a number of different contexts
such as in statistical theories of discrimination in labor economics (Aigner and Cain1977 Lundberg and Startz 1983) and in the industrial organization literature
16 If there were a trend in the natural level of output then ln ynit would replace the firstln yni and in the lagged term ln ynitminus1 would replace the second ln yni
17 The last term in (1016prime) indicates the deviation of output in the prior period from itsldquonormalrdquo level It is presumed that λ lt 1
18 Recall that this supply function assumes no adjustment costs and thus does not have alagged output term in it
Notes 219
19 In general if z1 z2 zn are independent random variables having the same distribu-tion with mean micro and variance σ 2
z and if z = (z1 + z2 + + zn)n then E(z) = micro
and Var(z) = σ 2z n Note that as n goes to infinity the variance of z goes to zero
20 Recall that we have let π(πtminus1) denote ( pt minus ptminus1)ptminus1 This was done so thatthe terms making up the expected real interest rate rt minus πe
t would have the sametime subscript πe
t = ( pet+1 minus pt)pt In the discussions to follow however we shift
time subscripts so that the previously denoted π = π and the previously denoted πet
becomes πet+1
21 Note that the natural log of real output is ln( yt) = Yt 22 Sometimes the money demand function is simplified by assuming α2 = 0 so that there
are no effects of interest rate changes on real money demand23 This follows given mt = mt +εt so that dPtdmt = 1 Recall that Pt and mt are logs of
the price level and the money supply respectively so that dPt = d(ln pt) = (1pt)dptand dmt = d(ln Mt) = (1Mt)dMt
24 Note that we are somewhat imprecise in our statement that we are eliminating the termfor output from the equation Recall that the exact interpretation of Yt would be as thedifference between the logarithm of output and the logarithm of the natural level ofoutput Equivalently we may call Yt the log of the ratio of output to its natural level
11 Policy
1 The alternative to a rule is called ldquopurely discretionary monetary policyrdquo in whichmoney supply changes are made purely at the discretion of the government dependingon its current reading of the economy and current set of objectives
2 This view is not universally accepted For instance according to the Nobel prize winnerJames Tobin (1985) the government should be free to do as it sees fit and he sees manyreasons ldquofor the Fedrsquos reluctance to tie its own hands as much as lsquorulesrsquo advocates wishrdquo Sargent and Wallace (1976) identify as ldquoKeynesiansrdquo individuals who believegovernment monetary policy should attempt to ldquolean against the windrdquo in an effort toattenuate the business cycle In the view of Sargent and Wallace and others the monetaryauthority has no scope to conduct countercyclical policy Not surprisingly those whodisagree with this view suggest alternative models of the economy such as those withldquostickyrdquo prices more favorable to their alternative views
3 Those who argue for a rule such as this are sometimes called ldquomonetaristsrdquo Someadvocates of a constant growth rate in the money supply would restrict the length oftime during which a particular rate of growth was fixed For instance William Pooleformer member of the Presidentrsquos Council of Economic Advisers (1982ndash1985) andnow with the Fed suggests that monetary rules should be adopted but that the ruleshould be ldquosubject to change at any time upon presentation of a convincing case withsupporting evidencerdquo (Poole 1985)
4 Hallrsquos suggestion echoes Simonsrsquo (1936) proposal for ldquoa monetary rule of maintainingthe constancy of some price indexrdquo Wayne Angell a 1986 appointee to the Board ofGovernors of the Federal Reserve (the monetary authorities for the USA) has arguedfor a monetary policy that will stabilize a price index constructed from a basket of basiccommodities (perhaps including gold)
5 Recall also that we assume the random variables εt and ut (the random variable asso-ciated with output demand) are independent with zero mean and respective variancesσ 2
e and σ 2u
6 An example of such ldquoexogenousrdquo price expectations would be the autoregressiveexpectations
7 Recall that the natural level of output was normalized to equal one so that Yn equiv ln yn = 0Thus if the optimal level of output is the natural level then the objective would be tominimize Etminus1(Yt)
2
220 Notes
8 Recall that we are assuming that εt and vt are independent random variables From(112)
Etminus1Yt = a0 + a1Ytminus1 + a2mt
so that Yt minus Etminus1Yt = a2εt + vt Squaring this expression and noting that E(εt) = 0E(vt) = 0 and given independence E(εt middot vt) = 0 we obtain
Etminus1(Yt minus Etminus1Yt)2 = a2
2σ 2e + σ 2
v
9 See Sargentrsquos (1987a 453) equation (13) for the corresponding expression Recall thatg0 = (Y lowast minus a0)a2
10 Note that ln yn = 0 due to normalization11 To tie down the inflation rate we would have to add an inflation objective to the goals
of the government12 Note that this is another example of the Lucas critique in which econometric estimates
of a specific set of parameters based on past policies cannot be used to project the impactof new different monetary rules to be followed in the future for with these new policiesthe parameters will change
13 This equation is derived from combining the price error expression with the aggregatesupply equation given earlier
14 The assumptions leading up to (1116) have been chosen so that (1116) matches thecriterion found in Barro and Gordon (1983) We will contrast the results of that paperwith our findings here shortly
15 The form of this equation mirrors that in Barro and Gordon16 Consider what would happen if we did not neglect the lagged disturbance terms For
instance let us say one of the disturbance terms (εtminus1 or utminus1) is positive rather thanzero while the other is held equal to zero According to the reduced-form equation forthe price level for period t minus 1 the result would be a higher price level in period t minus 1with the positive lagged disturbance term If the rate of growth in the money supplybetween period t minus 1 and t was the same in both cases then the inflation rate wouldbe higher in the case when both lagged disturbance terms are zero Thus to maintain aconstant inflation rate a lower money supply growth is implied if lagged disturbanceterms are zero instead of positive
17 Kydland and Prescott were awarded the 2004 Nobel prize in economics for their work18 This form of the objective function introduced above is discussed in Barro and Gordon
(1983) Without the assumption of k lt 1 a zero average rate of inflation would beoptimal regardless of the nature of expectation formation
19 For simplicity we assume there is no persistence in real shocks Thus we assume λ = 0in the original Barro and Gordon model
12 Open economy
1 For discussion purposes the terms ldquodomesticrdquo and ldquoforeignrdquo are used with respect tothe domestic perspective
2 For simplicity we assume foreigners do not desire to hold domestic money3 We assume for simplicity that foreigners are not taxed by the domestic government4 That is the domestic households own the remaining share of bonds issued by foreign
firms Bff and α of the equity share issues by foreign firms The total dividends (interms of the foreign currency) paid by foreign firms in period t are denoted by pftdftwhere pft is the price level of the foreign country and dft are real dividends of foreignfirms
Notes 221
5 One might also include bdt as purchase of output by private depository institutions is
included in the household budget constraint This reflects the replacement of dividendspaid by private depository institutions to their shareholders (households) by the differ-ence between banksrsquo interest income and their purchases of the composite commodityThis definition of dividends for private banks has been termed the ldquoflowrdquo constraint forprivate depository institutions
6 Recall that MB denotes the monetary base R denotes reserves and C denotes currencyin the hands of the nonbank public
7 Recall that we use the term ldquomodifiedrdquo as we have substituted the firm distribution con-straint into the household budget constraint for labor income We thus have effectivelysuppressed the labor market from the markets under consideration This modified lawis useful in understanding how the aggregate demand equation is derived
8 By ldquoreal demandrdquo we mean in units of the domestic commodity9 The term ldquorealrdquo means in terms of the domestic composite commodity
10 In measuring the exports imports and international capital flows a number of itemsare often missed For instance the clandestine transfer of funds from the Philippines toUS bank accounts would generate a demand for dollars On the other hand secretiveimports of heroin from Turkey result in a supply of dollars in international marketsThe net of such unmeasured transactions are lumped under the heading of ldquostatisticaldiscrepancyrdquo in the balance of payments accounts We have omitted this item fromTable 121
11 For simplicity we assume that the exchange rate affects only the division of totalconsumption between imports and purchases of the domestic output total consumptionis assumed to be unaffected by such exchange rate changes
12 Note that an increase in the price of the yen or an ldquoappreciationrdquo of the yen means afall in the price of a dollar in terms of yen or a ldquodepreciationrdquo of the dollar
13 If the price elasticity with respect to imports was less than one in short run a rise in therelative price of imports due to a depreciation of the dollar could in fact lead to a risein the value of imports as well This short-run phenomenon when applied to the path ofnet exports over time is referred to as the ldquoJ-curve effectrdquo
14 Actually for much of the analysis to follow we need only assume the weaker ldquoMarshallndashLernerrdquo condition that the sum of the price elasticity of demand for imports and theprice elasticity of demand for exports exceeds one This assures that a price of the dollarbelow its equilibrium level will be associated with an excess demand for dollars in theforeign exchange markets while a price of the dollar above its equilibrium level willbe associated with an excess supply of dollars
15 Given future markets for foreign currency this is not always the case16 We assume that such expectation is held with subjective certainty17 There are several reasons why in the short run the supply curve may not be upward-
sloping First it often takes time for US purchasers to adjust purchases in light of achange in relative prices Second the prices of domestic goods that are close substitutesto the imported goods can rise significantly in the short run as domestic producers hitshort-run production constraints The third complication that has the effect of makingthe supply of dollars curve less likely to be upward-sloping is that foreign producersat least in the short run often adjust the foreign currency prices of goods they exportto partially offset the impact of exchange rate changes on the prices of their goodsin foreign markets For instance Knetter (1987) found that with a depreciation of thedollar (appreciation of the West German Mark) West German exporters often reducedthe Mark price of their exports so as to minimize the rise in the dollar price of Germangoods that would result from the appreciation of the Mark For the time being weabstract from such short-run considerations although this is not to lessen the importanceof this phenomenon as the experience during the 1985ndash1987 period indicates With adepreciation of the dollar the dollar value of imports grew as the USA had to supply
222 Notes
more dollars for each unit of imported goods and there was initially little reduction inthe quantity of goods imported
18 It is important to remember that while we talk of households as being the only privatedemanders of foreign goods and financial assets this is purely a simplifying deviceFirms also demand foreign goods and private depository institutions demand foreignfinancial assets The analysis would be more complex if we explicitly recognized thesedemands but our conclusions would be unchanged since we can subsume in householdactions the actions of firms and depository institutions in foreign markets
References
Aigner DJ and Cain GC (1977) Statistical theories of discrimination in labor marketsIndustrial and Labor Relations Review 30(2) 175ndash187
Alchian A and Demsetz H (1972) Production information costs and economicorganization American Economic Review 62 777ndash795
Alogoskoufis GS (1987) On intertemporal substitution and aggregate labor supplyJournal of Political Economy 95 938ndash960
Altonji J and Siow A (1987) Testing the response of consumption to income changeswith (noisy) panel data Quarterly Journal of Economics 102 293ndash328
Arrow K (1964) The role of securities in the optimal allocation of risk-bearing Review ofEconomic Studies 31 (April) 91ndash96
Arrow K and Hahn FH (1971) General Competitive Analysis San Francisco CAHolden-Day
Azariadis C (1976) On the incidence of unemployment Review of Economic Studies 43115ndash125
Barro RJ (1976) Rational expectations and the role of monetary policy Journal ofMonetary Economics 2 1ndash32
Barro RJ and Gordon D (1983) A positive theory of monetary policy in a natural ratemodel Journal of Political Economy 91 589ndash610
Barro RJ and Grossman HI (1971) A general disequilibrium model of income andemployment American Economic Review 61 82ndash93
Barro RJ and King RG (1984) Time-separable preferences and intertemporal substitu-tions models of business cycles Quarterly Journal of Economics 99 817ndash839
Begg DKH (1980) Rational expectations and the non-neutrality of systematic monetarypolicy Review of Economic Studies 47 293ndash303
Blanchard OJ (1981) What is left of the multiplier accelerator American EconomicReview 71 150ndash154
Blanchard OJ and Fischer S (1989) Lectures on Macroeconomics Cambridge MA MITPress
Borjas GJ and Heckman JJ (1978) Labor supply estimates for public policy evaluationNBER Working Paper No W0299 November
Box GEP and Jenkins GM (1970) Time Series Analysis San Francisco CAHolden-Day
Burbidge JB and Robb AL (1985) Evidence on wealthndashage profiles in Canadian cross-section data Canadian Journal of Economics 18 854ndash875
Caballero R and Engel E (1999) Explaining investment dynamics in US manufacturingA generalized (S s) approach Econometrica 67 783ndash826
224 References
Campbell JY and Mankiw NG (1987a) Permanent and transitory components inmacroeconomic fluctuations American Economic Review 77 111ndash117
Campbell JY and Mankiw NG (1987b) Are output fluctuations transitory QuarterlyJournal of Economics 102 857ndash880
Campbell JY Lo AW and MacKinlay AC (1997) The Econometrics of FinancialMarkets Princeton NJ Princeton University Press
Cass D (1965) Optimal growth in an aggregate model of capital accumulation Review ofEconomic Studies 32 233ndash240
Clower RW (1965) The Keynesian Counter-revolution A theoretical appraisal InFH Hahn and FPR Brechling (eds) The Theory of Interest Rates London Macmillan
Coase RH (1960) The problem of social cost Journal of Law and Economics 31ndash44
Cukierman A (1986) Measuring inflationary expectations Journal of Monetary Eco-nomics 17 315ndash324
Dahlman C (1979) The problem of externality Journal of Law and Economics 22141ndash162
Debreu G (1959) The Theory of Value New York WileyDiamond PA and Hausman JA (1984) Individual retirement and savings behaviour
Journal of Public Economics 23 81ndash114Dornbusch R (1976) Expectations and exchange rate dynamics Journal of Political
Economy 84 1161ndash1176Enders W (2004) Applied Econometric Time Series 2nd edn Hoboken NJ WileyEvans G (1989) Output and unemployment dynamics in the United States 1950ndash1985
Journal of Applied Econometrics 4 213ndash237Ewing BT (2001) Cross-effects of fundamental state variables Journal of
Macroeconomics 23 633ndash645Ewing BT and Payne JE (1998) The long-run relation between the personal savings rate
and consumer sentiment Financial Counseling and Planning 9(1) 89ndash96Ewing BT Levernier W and Malik F (2002) Differential effects of output shocks on
unemployment rates by race and gender Southern Economic Journal 68 584ndash599Fama EF and Miller MH (1972) The Theory of Finance New York Holt Rinehart and
WinstonFischer S (1977) Long-term contracts rational expectations and the optimal money supply
rule Journal of Political Economy 85 191ndash205Fisher I (1930) The Theory of Interest New York MacmillanFleming MJ (1962) Domestic financial policies under fixed and under floating exchange
rates IMF Staff Papers 9 369ndash379Foley KD (1975) On two specifications of asset equilibrium in macroeconomic models
Journal of Political Economy 83 303ndash324Friedman M (1959) A Program for Monetary Stability New York Fordham University
PressFriedman M (1968) The role of monetary policy American Economic Review 58 1ndash17Goldberg S (1958) Difference Equations New York WileyGould JP (1968) Adjustment cost in the theory of investment of the firm Review of
Economic Studies 35 47ndash56Grandmont JM (1977) Temporary general equilibrium theory Econometrica 45
535ndash572Granger CWJ and Morris M (1976) Time series modelling and interpretation Journal
of the Royal Statistical Society Series A 139 246ndash257
References 225
Granger CWJ and Newbold P (1986) Forecasting Economic Time Series 2nd ednOrlando FL Academic Press
Hall R (1976) The Phillips curve and macroeconomic policy In K Brunner andAH Meltzer (eds) The Phillips Curve and Labor Markets Carnegie-RochesterConference Series on Public Policy Amsterdam North-Holland
Hall R (1980) Employment fluctuation and wage rigidity In GL Perry (ed) BrookingsPapers on Economic Activity pp 91ndash123 Washington DC Brookings Institution
Hall RE (1982) Explorations in the Gold Standard and related policies for stabilizingthe dollar In RE Hall (ed) Inflation Causes and Effects Chicago IL University ofChicago Press
Hall RE (1988) Intertemporal substitution in consumption Journal of Political Economy96 339ndash357
Hannan EJ (1970) Multiple Time Series New York WileyHansen B (1970) A Survey of General Equilibrium Systems New York McGraw-HillHansen LP and Singleton KJ (1983) Stochastic consumption risk aversion and the
temporal behavior of asset returns Journal of Political Economy 91 249ndash265Harvey AC (1993) Time Series Models Cambridge MA MIT PressHayashi F (1982) Tobinrsquos marginal q and average q A neoclassical interpretation
Econometrica 50(1) 213ndash224Hicks J (1939) Value and Capital Oxford Clarendon PressHowitt P (1985) Transaction costs in the theory of unemployment American Economic
Review 75 88ndash100Howitt P (1986) Conversations with economists A review essay Journal of Monetary
Economics 18 103ndash118Hurd M (1987) Savings of the elderly and desired bequests American Economic Review
77 298ndash312Karni E (1978) Period analysis and continuous analysis in Patinkinrsquos macroeconomic
model Journal of Economic Theory 17 134ndash140Kester WC (1986) Capital ownership structure A comparison of the United States and
Japanese manufacturing corporations Financial Management 15(1) 5ndash16Keynes JM (1936) General Theory of Employment Interest and Money London
MacmillanKletzer LG and Fairlie R (1998) Jobs lost jobs regained An analysis of blackwhite
differences in job displacement in the 1980s Industrial Relations 37 460ndash477Knetter M (1987) Export prices and exchange rates Theory and evidence Working paper
Stanford University NovemberKoopmans T (1965) On the concept of optimal economic growth In Proceedings ndash
Study Week on the Econometric Approach to Development Planning Chicago ILRand-McNally
Kydland FE and Prescott EC (1977) Rules rather than discretion The inconsistency ofoptimal plans Journal of Political Economy 85 473ndash491
Kydland FE and Prescott EC (1982) Time to build and aggregate fluctuationsEconometrica 50 1345ndash1370
Leonard JS (1988) In the wrong place at the wrong time The extent of frictional andstructural unemployment NBER Working Paper No 1979
Lucas RE (1967) Adjustment costs and the theory of supply Journal of Political Economy75 321ndash334
Lucas RE (1972) Expectations and the neutrality of money Journal of Economic Theory4 103ndash124
226 References
Lucas RE (1973) Some international evidence on outputndashinflation tradeoffs AmericanEconomic Review 63 326ndash334
Lucas R Jr (1981) Methods and problems in business cycle theory In Studies in Business-Cycle Theory Cambridge MA MIT Press First published in Journal of Money Creditand Banking 12 November 1980
Lucas RE and Rapping LA (1970) Real wages employment and inflation InES Phelps (ed) Microeconomic Foundations of Employment and Inflation TheoryNew York WW Norton
Lundberg S and Startz R (1983) Private discrimination and social intervention incompetitive labor markets American Economic Review 73(3) 340ndash347
McCallum B (1979) The current state of the policy-ineffectiveness debate AmericanEconomic Review 69 240ndash245
McCallum B (1985) On consequences and criticisms of monetary targeting Journal ofMoney Credit and Banking 17 570ndash597
Mankiw NG (1987) The optimal collection of seigniorage Theory and evidence Journalof Monetary Economics 20 327ndash341
Mankiw NG Rotemberg JJ and Summers LH (1985) Intertemporal substitution inmacroeconomics Quarterly Journal of Economics 100 225ndash251
Marx K (1976) Capital Vol 1 A Critique of Political Economy HarmondsworthPenguin
Mills TC (1999) The Econometric Modelling of Financial Time Series 2nd ednCambridge Cambridge University Press
Modigliani F (1966) Life cycle hypothesis of saving the demand for wealth and the supplyof capital Social Research 33 160ndash217
Modigliani F (1986) Life cycle individual thrift and the wealth of nations AmericanEconomic Review 76 297ndash313
Mundell RA (1968) International Economics New York MacmillanNakamura A and Nakamura M (1981) A comparison of the labor force behavior of
married women in the US and Canada with special attention to the impact of incometaxes Econometrica 49 451ndash489
Nelson CR and Plosser CI (1982) Trends and random walks in macroeconomictime series Some evidence and implications Journal of Monetary Economics 10139ndash162
Niehans J (1987) Classical monetary theory new and old Journal of Money Credit andBanking 19 409ndash424
Oi WY (1962) Labor as a quasi-fixed factor Journal of Political Economy 70 538ndash555Patinkin D (1965) Money Interest and Prices New York Harper amp RowPencavel J (1985) Labor supply of men A survey In Orley Ashenfelter (ed) Handbook
of Labor Economics Amsterdam North-HollandPhelps ES (1968) Money-wage dynamics and labor market equilibrium Journal of
Political Economy 76 678ndash711Phelps ES and Taylor JB (1977) Stabilizing powers of monetary policy under rational
expectations Journal of Political Economy 85 163ndash190Phillips AW (1958) The relationship between unemployment and the rate of change in
money wage rates in the United Kingdom 1861ndash1957 Economica 25 283ndash299Pindyck RS and Rubinfeld DL (1991) Econometric Models and Economic Forecasts
3rd edn New York McGraw-HillPoole W (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo
Journal of Money Credit and Banking 17 602ndash605
References 227
Radford RA (1945) The economic organization of a prisoner of war camp Economica12 189ndash201
Robinson C and Tomes N (1985) More on the labour supply of Canadian womenCanadian Journal of Economics 18 156ndash163
Sargent T (1987a) Macroeconomic Theory 2nd edn Boston MA Academic PressSargent T (1987b) Dynamic Macroeconomic Analysis Cambridge MA Harvard
University PressSargent TJ and Wallace N (1975) ldquoRationalrdquo expectations the optimal monetary
instrument and the optimal money supply rule Journal of Political Economy 83241ndash254
Sargent TJ and Wallace N (1976) Rational expectations and the theory of economicpolicy Journal of Monetary Economics 2 169ndash183
Shapiro C and Stiglitz JE (1984) Equilibrium unemployment as a worker disciplinedevice American Economic Review 74 433ndash444
Shiller RJ (1978) Rational expectations and the dynamic structure of macroeconomicmodels Journal of Monetary Economics 4 1ndash44
Sidrauski M (1967a) Rational choice and patterns of growth in a monetary economyAmerican Economic Review 57 534ndash544
Sidrauski M (1967b) Inflation and economic growth Journal of Political Economy 75797ndash810
Simons HC (1936) Rules versus authorities in monetary policy Journal of PoliticalEconomy 44 1ndash30
Sims C (1972) Money income and causality American Economic Review 62 540ndash552Solow R (1956) A contribution to the theory of economic growth Quarterly Journal of
Economics 70 65ndash94Strotz RH (1955ndash1956) Myopia and inconsistency in dynamic utility maximization
Review of Economic Studies 23 165ndash180Stuart CE (1981) Swedish tax rates labor supply and tax revenues Journal of Political
Economy 89 1020ndash1038Taylor J (1972) The behaviour of unemployment and unfilled vacancies Great Britain
1958ndash71 An alternative view Economic Journal 82 1352ndash1365Taylor JB (1979) Estimation and control of a macroeconomic model with rational
expectations Econometrica 47 1267ndash1286Tobin J (1965) Money and economic growth Econometrica 33 671ndash684Tobin J (1969) A general equilibrium approach to monetary theory Journal of Money
Credit and Banking 1(1) 15ndash29Tobin J (1985) Comment on ldquoOn consequences and criticisms of monetary targetingrdquo or
Monetary targeting Dead at last Journal of Money Credit and Banking 17 605ndash610Uzawa H (1969) Time preference and the Penrose effect in a two-class model of economic
growth Journal of Political Economy 77 628ndash652Varian H (1992) Microeconomic Analysis 3rd edn New York WW NortonWalker DA (1987) Walrasrsquo theories of tatonnement Journal of Political Economy 95
758ndash774Walras L (1954) Elements of Pure Economics trans W Jaffeacute London George Allen amp
UnwinWeber CE (1998) Consumption spending and the paper-bill spread Theory and evidence
Economic Inquiry 36 575ndash589Weitzman M (1985) The simple macroeconomics of profit sharing American Economic
Review 75 937ndash953
Index
accelerationist outcome 170ndash2aggregate demand 7 86 90ndash1 94ndash5 99
Keynesian model 133ndash4 135 139 140Lucas model 160ndash1 neoclassical model126 127 128 146 shocks 158 seealso demand
aggregate supply 82 86ndash90 91 94ndash5 98113 Keynesian model 132 133ndash4 135138 140 145 Lucas model 137153ndash4 155 158 160ndash1 164 166monetary policy 170ndash1 neoclassicalmodel 118 123 125ndash6 127 128 133see also supply
aggregation issues 5ndash6 8 14ndash15Alchian A 203n9Alogoskoufis GS 51Angell Wayne 219n4Arrow-Debreu theory 2 202n4 n10
203n8assets 51 52 53 portfolio decision
39ndash40 57ndash9 tangible 25ndash6 see alsofinancial assets
autocorrelation function 102ndash3autoregressive expectations 162ndash4 168autoregressive processes 103ndash5 110 111
113 166
balance of payments 190ndash3bankruptcy 32 33Barro RJ 156 167 179ndash80 182ndash4Begg DKH 122 215n8behavioral hypotheses 96 98 99 100Bellman equation firms 29 30 31 34
35 households 43 44 45 48ndash9Blanchard OJ 111 112 171 175 184ndash5bonds 19ndash20 23 27ndash9 32ndash3 financial
market equilibrium 84ndash5 firmfinancing constraint 24ndash6 64 Fisherian
problem 52 foreign 189 households39 40 43ndash4 72 204n11 money illusion59 portfolio choice 57 58 59 93 realvalue 77 temporary equilibrium 80 81
budget constraint 7 16 household 43 4455 65ndash6 67 70ndash1 73ndash4 open economy188 189
Burbidge JB 208n25business cycle 5 6 49 50 173 real
business cycle theory 79 136 202n4technological innovation 113
Caballero R 209n1Campbell JY 111 112 113 214n45capital 23 24 31 68 69ndash70 cost of
31ndash2 36 69 70 94 120 209n7 firmfinancing constraint 24 25 26 64international flows 192 193 195ndash6197ndash8 200 Tobinrsquos Q 36 see alsocapital stock
capital account 192capital adjustment costs 21 26ndash7 34ndash7
69ndash70capital markets 27 203n1 205n24capital stock 3 18ndash19 20ndash1 23 25 27ndash9
optimal investment 30 31 69 70retained earnings financing 29ndash30 32superneutrality of money 120 121 122Tobinrsquos Q 35 36
central banks 187 188 189 190191 192ndash3
classical economics 4 5 79 212n8Clower RW 117 118Coase RH 11CobbndashDouglas production function 147
148 217n3commodities 8 9ndash10 14 16 18ndash19 20
individual experiments 11 13ndash14
Index 229
Lucas model 146ndash7 market equilibrium92 open economy 188ndash9
comparative static analysis 116ndash23consumer behavior 11 12ndash13consumption 20 39 40ndash6 47ndash8 67
70ndash4 aggregate 55 61 demand 97 99195 215n6 financial asset demand 94Fisherian problem 51ndash5 individualexperiments 11 12ndash13 intertemporalsubstitution hypothesis 51 60ndash3Keynesian model 134 135 laborparticipation 49 life-cycle hypothesis50 55ndash6 57 money supply shocks 128neoclassical model 117ndash18 123 126neutrality of money 119 open economy190ndash1 193 permanent incomehypothesis 56ndash7 portfolio choice 5758 59 real balance effect 77 92superneutrality of money 121 122
continuous-time analysis 4cost of capital 31ndash2 36 69 70 94 120
209n7cost of living agreements (COLAs) 215n2costs bankruptcy 33 capital adjustment
21 26ndash7 34ndash7 69ndash70 dividends 23ndash4firm financing constraint 24
Cramerrsquos rule 87 119 121 127 134Cukierman A 164currency depreciation 193 195 198ndash9
221n17
Dahlman C 203n9De Moivrersquos theorem 109Debreu G 8debt 32 60debt-to-equity ratio 32ndash4decision-making behavior 5ndash6 38 51 see
also portfolio decisiondemand excess 13ndash14 15ndash16 67 118
133 190ndash1 199 individual demandfunctions 13 14 labor marketequilibrium 83 Lucas model 146156ndash7 market demand functions 1415 money 86 neoclassical model82ndash3 119 122 123 new classicaleconomics 5 price elasticity of 195real balance effect 60 77 Sayrsquos law 716 temporary equilibrium 80Walrasian model 7 15 see alsoaggregate demand
Demsetz H 203n9depository institutions 126 187 188 189
221n5 222n18
depreciation 26 31 74 205n23deterministic feedback rules 172 173 174deterministic models 6disequilibrium 78 199distributed lag schemes 162ndash4dividends depository institutions 221n5
firms 20 22 23ndash4 29 30 32 34ndash5future 76ndash7 households 43 44 66
Dornbusch model 200ndash1dynamic analysis 2 3ndash4 6 79 122dynamic programming 28ndash9
econometric models 96 100ndash1employment 1 19 128 148 full
employment model 88 142 Keynesianmodel 130ndash1 132ndash3 134 135 137neoclassical model 82 83 84 91 122123ndash5 126 see also labor labor marketlabor supply
Engel E 209n1equilibrium 1 5 7 15 199 aggregate
demand 90ndash1 94ndash5 99 aggregatesupply 86ndash8 94ndash5 employment123ndash5 financial markets 80ndash1 84ndash6illusion model 79 Keynesian model79 133ndash4 137 139ndash40 labor market83ndash4 88 90 119 123ndash5 127 131 148loanable funds theory 120 Lucas model159 160 162 monetary policy 182ndash4money market 85ndash6 92ndash3 126 159multiple equilibria models 113 naturalrate of unemployment 89 90neoclassical model 78 79 117ndash18 126non- market-clearing model 79 partial187 Patinkin analysis 91ndash4 stationarity106 stochastic models 6 temporary 278 80ndash3 86 120 velocity 139ndash40 seealso general equilibrium
equity shares 18 19ndash20 21ndash2 23 27ndash932ndash3 firm financing constraint 24ndash664 households 40 43 72 real value77 temporary equilibrium 80 81
ergodicity 212n16Eulerrsquos theorem 206n44Ewing BT 211n11 212n11exchange rate 186ndash7 191 193ndash6 197
199 200ndash1expectations 41 59 84 119 202n5
adaptive 163 170ndash2 210n15 aggregatedemand 90ndash1 autoregressive 162ndash4168 inflation 62 69 71 77 120ndash2163 164 interest rate 60 61 62 6971 75ndash7 Keynesian model 135 136ndash8
230 Index
expectations (Continued)140 141ndash4 145 labor marketequilibrium 83 Lucas model 157 160optimal monetary policy 168ndash70 180182 183ndash4 output 68 rational 3128ndash9 136ndash8 141ndash4 164ndash6 172ndash4176ndash8 182ndash4 suppliers 124 135 weakconsistency 203n4 see also forecasting
exports 190ndash1 193 197 200
financial assets 77 85 93ndash4 118 firms19ndash20 25 26 28 29 68ndash9 70households 40 42 43ndash4 66 71 72ndash4labor market equilibrium 83 openeconomy 186 187ndash8 189 191 192193 195ndash6 197ndash9 temporaryequilibrium 81ndash2 see also bondsequity shares
financial market 1 36 67 96ndash7equilibrium 80ndash1 84ndash6 93 94 95 118open economy 191 195ndash6 197 199
firm distribution constraint 23ndash4 28 4165 67 74 76ndash7 189
firm financing constraint 23 24ndash6 28ndash964 67 68ndash70 84ndash5
firms 18ndash38 64ndash5 68ndash70 130 135aggregate supply 87 capital adjustmentcosts 26ndash7 34ndash7 dividends 23ndash4investment demand 97 laboradjustment costs 37ndash8 money illusion75 119 neoclassical model 79objectives 21ndash3 open economy 187188 189 190 222n18 optimalinvestment 30ndash2 real wage illusion124 125 retained earnings financing29ndash30 34
Fischer S 112 136 137 138ndash45 171175 184ndash5
Fisher I 79 163 208n16Fisherian problem 39 46 51ndash5 209n12
215n6forecasting 96 100ndash1 101ndash14 128
149ndash50 see also expectationsforeign exchange market 186ndash7 188 191
193ndash5 198ndash9 200foreigners 187ndash90 191 197ndash8Friedman Milton 56 156 168 171 179futures market 3 7 8 10 65 66
general equilibrium 1 2 11 95 97 187neoclassical model 117 118 prices 710 15ndash16 Walrasian model 8 see alsoequilibrium
GNP see gross national productGordon D 156 167 179ndash80 182ndash4government 187 190Grandmont JM 78Granger CWJ 100 102 112 212n22gross national product (GNP) 112 168
habit persistence model 207n2Hall Robert 60ndash1 62 172Hansen B 8Harvey AC 101 114Hayashi F 36 206n43Hicks John 2 8households 18 19 39ndash63 65ndash6 70ndash4
aggregate supply 87 bonds and equityshares 20 24 consumption demand97 dividends 23 24 Fisherian problem39 46 51ndash5 imperfect foresight 124intertemporal substitution hypothesis49ndash51 60ndash3 labor supply problem46ndash51 life-cycle hypothesis 55ndash6money illusion 75ndash6 77 119neoclassical model 79 open economy187 188 189 191 193ndash6 222n18permanent income hypothesis 56ndash7portfolio decision 39ndash40 46 57ndash9price changes 21
Howitt P 5 203n15
imperfect foresight 123 124imports 193ndash5 199 200income 59 74 97 208n27 foreign goods
195 197 life-cycle hypothesis 55ndash657 neoclassical model 117 permanentincome hypothesis 56ndash7 see also wages
income effects 47ndash8 49individual experiments 11ndash14 21 39inflation 53 59 72 196 200
expectations 62 69 71 77 120ndash2 163164 Lucas model 146 154ndash6 157160 161 monetary policy 136 170ndash2174ndash9 180 181ndash2 183ndash4superneutrality of money 121 122wages 123 131 see also prices
integrated processes 110ndash11 112interest rate 1 4 20 31 36 59
consumption demand 97 equilibrium91ndash4 equity shares 22 23 expectations60 61 62 69 71 75ndash7 financialmarket equilibrium 84 85 94Fisherian problem 53 54 foreignfinancial assets 195 196 198household problem 43 45 46 47
Index 231
intertemporal substitution hypothesis49 50 51 63 72 Keynesian model133 134 labor demand 83 loanablefunds theory 120 Lucas model159ndash60 neoclassical model 118 119126ndash7 neutrality of money 119 parity200ndash1 superneutrality of money 121ndash2temporary equilibrium 81 82
intertemporal substitution hypothesis (ISH)49ndash51 60ndash3 72 73 75ndash6
invertibility condition 110investment 3 29 84 85 128 capital
adjustment costs 26ndash7 35 36 demand32 68ndash70 75 90 94 97 99 120206n40 Keynesian model 134 135neoclassical model 118 126 neutralityof money 119 optimal 30ndash2superneutrality of money 121 122supply-side variables 116
IS equation 79 90ndash1 98ndash9 118 126 191export demand 200 Keynesian model133 134 138ndash9 Lucas model 159 160
ISH see intertemporal substitutionhypothesis
lsquoislandrsquo paradigm 146ndash9
J-curve effect 221n13
Karni Edi 207n3Keynes John Maynard 4Keynesian model 5 79 88 130ndash45 199Knetter M 221n17Kydland FE 180ndash1 207n2 220n17
labor 19 21 27 42 66 adjustment costs37ndash8 demand 68 76 97 132ndash3 147215n9 217n4 Keynesian models 5marginal product of 30 optimalinvestment 31 participation 48ndash9temporary equilibrium 80 see alsoemployment labor supply wages
labor market 1 18 37 65 96ndash7 186aggregate supply 86ndash8 94 95 119intertemporal substitution hypothesis49ndash51 Keynesian model 131ndash3 134Lucas model 147 148 natural rate ofunemployment 89 90 neoclassicalmodel 83ndash4 118 124ndash5 127 Walrasrsquolaw 66ndash7 82 see also employment
labor reserve hypothesis 37ndash8labor supply 39 40 41 45 46ndash9 215n9
intertemporal substitution hypothesis49ndash51 72 73 75ndash6 210n15 Keynesian
model 132 133 Lucas model 147ndash8neoclassical model 79 83ndash4 117 123124 perfect foresight 66 70 71 realbalance effect 77 supply-side variables116 tax rates 99 see also employmentlabor
leisure 40ndash1 42 44 46ndash8 49 50 72life-cycle hypothesis 50 55ndash6 57linear regression analysis 150ndash3LM equation 79 90ndash1 98ndash9 118 126
191 Keynesian model 133 134 138ndash9Lucas model 159 160
loanable funds theory 120Lucas RE 6 50ndash1 146 149 156ndash7
164 167 203n13 210n15Lucas model 88 128 135ndash6 140 146ndash66
176 199Lucas supply function 79 137 153ndash6
158 174 175
Mankiw NG 111 112 113 136 214n45market clearing 4ndash5 6 7 15 50 79
commodity market 92 financial market93 Lucas model 146 164 wages 135
market experiments 11 14ndash16markets 1 2 20 78 187 aggregate
demand 86 90ndash1 capital 27 203n1205n24 foreign exchange 186ndash7 188191 193ndash5 198ndash9 200 futures 3 7 810 65 66 Lucas model 146 149 154sequential 67 see also financial marketlabor market money market
Marx Karl 79microeconomics 5 6 78 88 187mixed autoregressive moving average
processes 110Modigliani Franco 55ndash6 208n24ModiglianindashMiller theorem 32monetary policy 166 167ndash85 187
activist 169 171 174 discretionary136 167ndash8 184 219n1 Keynesianmodel 136ndash8 141 142 144ndash5 policyineffectiveness proposition 120 136137ndash8 172ndash9 182
money classical analysis 4 comparativestatic analysis 118 demand 71ndash4 7797 121 122 139ndash40 159 financialmarket equilibrium 84 householdproblem 41 42 43ndash4 45 46 66neoclassical model 117ndash18 neutralityof 119ndash20 136 142 173 174portfolio decision 39 57 58 59 72superneutrality of 120ndash2
232 Index
money (Continued)temporary equilibrium 80 82transaction costs 11 utility yield of 42see also money market money supply
money illusion 59 75ndash6 77 88 119 125210n15
money market 7 16 67 96ndash9 199aggregate demand 90 94 95equilibrium 85ndash6 92ndash3 126 159Keynesian model 133 134
money supply 3 76 100 119 168 190Keynesian model 134ndash5 142 Lucasmodel 159ndash60 161 162 monetarypolicy 144 169 170 173 178ndash9 180money market equilibrium 86 91neoclassical model 116 118ndash20 126ndash9purchasing power parity 200superneutrality of money 120ndash2 seealso money
moving average processes 109ndash10multiple equilibria models 113MundellndashFleming model 201
natural rate hypothesis 125 128ndash9 135166
Nelson CR 113neoclassical model 4ndash5 116ndash29 146 199
202n12 Keynesian model comparison133 136 Lucas model comparison 160162 modification of 130ndash1 purchasingpower parity 200 simple 78ndash95
new classical economics 4ndash5 6 7Newbold P 100 102 112 212n22numeraire 7 8 9ndash10
official reserve transaction balance 192open economy 186ndash201output 1 18 19 20 66ndash7 202n1
aggregate demand 86 90 94 95 139aggregate supply 86 87ndash8 classicalanalysis 4 comparative static analysis118 Dornbusch model 200 equilibrium80 82 91ndash2 94 126 expectations 6875 firm distribution constraint 65Keynesian model 5 133ndash4 135 137ndash8143ndash4 145 labor market equilibrium83 84 Lucas model 79 146 147148ndash9 153ndash5 157ndash62 165ndash6 monetarypolicy 144 168 170ndash2 173 174ndash5179 money supply shocks 128 movingaverage processes 109 neoclassicalmodel 117ndash18 122 123 125 126neutrality of money 119 open economy
187ndash8 189 190ndash1 199 simpletheoretical model 96ndash9 supply-sidevariables 116 136 time series model111ndash14
overlapping generations models 6
Patinkin D 8 11 120 202n12 critiqueof 117 equilibrium 83 84ndash5 91ndash4firms 210n5 labor supply 47
Payne JE 212n11perfect competition 7 11perfect foresight 3 87 firms 65 68ndash70
75 households 41 43 56ndash7 66 70ndash475 labor market equilibrium 83 125Lucas model 161 162 neoclassicalmodel 78 79 84 117 118 119 131Walrasrsquo law 66 67 82
perfect substitution 20 22 29 32period (discrete) analysis 4permanent income hypothesis 56ndash7Petty William 79Phelps ES 136ndash7 171 216n11Phillips AW 155Phillips curve 154ndash5 156ndash7 171 174ndash9
180 181 183Pindyck RS 101 102Plosser CI 113policy see monetary policypolicy ineffectiveness proposition 120
136 137ndash8 172ndash9 182Poole William 219n3portfolio decision 39ndash40 46 51 57ndash9 72
93 193preferences 11 40Prescott EC 180ndash1 207n2 220n17price elasticity of demand 195prices 1 2 3 19 68 accounting 7 10
13 15 aggregate demand 90 91 9495 aggregate supply 86 98autoregressive expectations 162ndash4classical analysis 4 comparative staticanalysis 116 demand-side variables117 equity shares 20 22 forecastingerrors 149ndash50 foreign goods 193ndash5197 future 41 59 75 76 householdbudget constraint 66 74 individualexperiments 11 Keynesian model 133134 135 137 145 labor marketequilibrium 83 84 Lucas model 146147 149 153 154 156ndash8 159ndash64market experiments 11 microeconomicfoundations 6 monetary policy 172ndash3money market equilibrium 86 natural
Index 233
rate hypothesis 125 natural rate ofunemployment 88ndash9 neoclassicalmodel 78ndash9 117ndash20 122ndash3 124ndash5126ndash8 130 136 146 new classicaleconomics 5 7 perfect foresight 71purchasing power parity 200 realindebtedness effect 21 relative 7 89ndash10 11 13 15ndash16 sequential markets67 superneutrality of money 120tatonnement process 15 temporaryequilibrium 80 82 Walrasian model 79ndash10 15 see also inflation
private international capital flows 192193 195ndash6 197ndash8 200
production 18ndash19productivity of labor 37purchasing power parity 200
Radford RA 8lsquorandom walkrsquo process 103 106 112ndash13Rapping LA 50ndash1 210n15rational expectations 3 128ndash9 136ndash8
141ndash4 164ndash6 172ndash4 176ndash8 182ndash4 seealso expectations
real balance effect 60 77 86 92ndash3 94121
real indebtedness effect 60real user cost of capital 31ndash2 36 69 70
94 120 209n7recontracting assumption 15reduced-form expressions 96 98 99ndash100
Lucas model 161 162 monetary policy168ndash9 172ndash3 177
reputation 184ndash5retained earnings financing 29ndash30 32 34risk 41Robb AL 208n25Rubinfeld DL 101 102
Sargent T 8 96 131ndash2 211n2 n6autoregressive expectations 164employees 217n2 investment demand206n40 Keynesians 219n2 linearregression analysis 152 monetarypolicy 136ndash7 167 168 170 172ndash4180 182 money supply 165 newclassical economics 203n13outputinflation tradeoff 157 Phillipscurve 155 superneutrality of money121 122
Say Jean-Baptiste 7Sayrsquos law 8 16seasonality 111 112
shareholders 25ndash6shares see equity sharesShiller RJ 162shocks 1 6 153 158 186 demand-side
117 monetary policy 170 moneysupply 128 129 135 Phillips curve156 time-series models 111ndash13 wages135ndash6
Simons HC 219n4Sims Christopher 100spot markets 2 37 130 131spot prices 10static analysis 2ndash4 47 50 79 116ndash23stationarity 102 103 104 105ndash9 110ndash11
113 114stationary analysis 2 3ndash4stochastic processes 6 41 101 102ndash3
110 172 173Stuart CE 99substitution effects 47ndash8 49 see also
intertemporal substitution hypothesissupply labor market equilibrium 83
Lucas supply function 79 137 153ndash6158 174 175 neoclassical model 119new classical economics 5 real balanceeffect 77 Sayrsquos law 7 16 temporaryequilibrium 80 86 Walrasian model 715 see also aggregate supply laborsupply money supply
tatonnement process 7 15 120tax issues 3 32 33 99Taylor JB 100 136ndash7 212n12 216n11technology 18ndash19 21terms of trade 194lsquotime inconsistency problemrsquo 180ndash2time series models 96 100 101ndash14Tobin James 206n40 219n2Tobinrsquos Q 27 35ndash7transaction costs 5 7 10ndash11 20
42 208n18
uncertainty 6 41unemployment 1 50ndash1 104 125
frictional 89 involuntary 211n13Lucas model 146 154ndash6 monetarypolicy 174ndash5 176 179 181ndash2 183184 natural rate of 88ndash90 135 156174ndash5
utility 13 17 40ndash1 42 46 48 61ndash2
velocity equations 139ndash40
234 Index
wages 18 19 23 24 32 36 aggregatedemand 90 91 aggregate supply 86ndash7anticipated 69 70 71 75ndash6 77 124household problem 42 43 44 47ndash866 intertemporal substitution hypothesis50 51 72 73 Keynesian model 130ndash3135ndash6 137ndash8 142ndash4 145 labordemand 68 97 labor marketequilibrium 83 84 labor participation48ndash9 Lucas model 147ndash8 moneysupply shocks 128 neoclassical model117 122ndash3 126 neutrality ofmoney 119 real wage illusion 123ndash5sticky 123 131 135ndash6 137temporary equilibrium 80 82 see alsoincome
Wallace N autoregressive expectations164 Keynesians 219n2 monetarypolicy 136ndash7 167 168 170 172 180182 money supply 165 superneutralityof money 121 122
Walrasrsquo law 1 8 14 16 85 91 balance ofpayments 190ndash1 excess demand 118financial assets 93 labor market 66ndash7market equilibrium 97 99 openeconomy 187 199 perfect foresight82 sequential markets 67
Walras Leon 1 16 202n2 203n2Walrasian models 7ndash11 15 18ndash20wealth 55 59 60 83Weber CE 209n2working hours 47ndash8