Money in Hart's model of imperfect competition

European Journal of POLITICAL

European Journal of Political Economy ECONOMY Vol. 11 (1995) 557-575 ELSEVIER

Money in Hart's model of imperfect competition

Neil Rankin Department of Economics, University of Warwick, Coventry, CV4 7AL, UK

Accepted for publication April 1994

Abstract

Hart's macroeconomic model with imperfect competition is recast in a temporary monetary equilibrium setting, first with a single representative consumer, then with overlapping generations of consumers. Production technology is generalised to permit unemployment for a wider range of parameter values. The key result is that an increase in the money supply raises output provided that the elasticity of expectations of future with respect to current prices is not unity: whether above or below unity is immaterial. It is later argued that a unit elasticity is not a necessary requirement for 'rational' forecasting behaviour.

JEL classification: E52

Keywords: Imperfect competition; Money non-neutrality; Expectations

I. Introduction

Hart 's (1982) is perhaps the most intriguing and least understood of the papers which in the last decade and a half have derived results on macroeconomic policy effectiveness from imperfectly competitive microfoundations. 1 It is very tempting to see Hart 's model as demonstrating the non-neutrality of money, but - formally at least - this interpretation is ruled out by the absence of anything called 'money ' from the model. Instead there is a 'non-produced good ' in fixed supply, and it is increases in the stock of this good which are shown to raise output. In this paper,

1 For a survey of this literature, see Dixon and Rankin (1994).

0176-2680/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 7 6 - 2 6 8 0 ( 9 4 ) 0 0 0 6 5 - 4

558 N. Rankin / European Journal of Political Economy 11 (1995) 557-575

we set out to explore the role for money and monetary policy in Hart 's model. 2 In common with some other developers of Hart 's contribution, 3 we argue that it is possible to view Hart 's non-produced good as 'money ' , if we accept the notion of ' temporary monetary equilibrium' (TME) as an appropriate framework for short- run analysis. The key feature of the TME concept, which has a distinguished history going back to Hicks (1946) and Patinkin (1956), is its use of 'backward- looking' expectations. Although it may appear to be a major step backwards to employ, in the 1990's, anything other than pure ' forward-looking' or 'rational' expectations, we will defend its use in the present context, and show that the scope for monetary policy which results differs in important ways from that which results from the use of similar expectations hypotheses in more familiar, perfectly competitive, models of the supply side. In doing so we explain more intuitively, and also generalise substantially, Hart 's original results.

In Section 2 we examine an economy with a single representative consumer. This is close to Hart 's original model. We first show that by introducing a CES production technology as an alternative to the Cobb-Douglas one originally used, it becomes possible to sustain an equilibrium with unemployment under a much wider range of assumptions about consumer behaviour. With the original technology, many simple parameterisations of consumers' preferences result in full employment, in which case monetary policy is automatically ineffective, so that Hart was obliged t9 add a number of restrictions on preferences to avoid this. Second, and the main result of the paper, we show that whenever the expectations elasticity embedded in the rule used by consumers to forecast future prices differs from unity, below or above, a monetary expansion raises output. Since Hart 's model may be interpreted as containing an elasticity of expectations equal to zero, this is a substantial generalisation of his finding. Most significantly, because it does not matter whether the elasticity is above or below unity, the fact that we have no a priori views about its likely value does not weaken the qualitative conclusion that monetary policy will be effective. By contrast, in conventional theories of aggregate supply, the same expectations hypothesis yields opposite effects of money on output depending on which side of unity the expectations elasticity lies.

In Section 3 we replace the representative consumer by overlapping generations of consumers. This enables us to consider the 'rationality' of different forecasting rules, since we can then compare expectations with outcomes. Although, as would

2 A closely-related contribution which was circulated as a discussion paper (Rankin, 1988) looks at fiscal policy in the same model, but without the overlapping-generations extension. (This has been scheduled for publication for some considerable time, but at the time of writing has yet to appear.)

3 In particular d'Aspremont et al. (1989, 1990). These papers focus on imperfect competition in the goods market and show how it can lead to bounded labour demand, such that with a competitive labour market there would be unemployment even at a zero wage. Here we focus instead on imperfect competition in the labour market.

N. Rankin / European Journal of Political Economy 11 (1995) 557-575 559

be expected, pure ' forward-looking' expectations cause money to be neutral, backward-looking expectations still result in a positive effect of money on output if the expectations elasticity is not unity, subject to some minor qualifications. 4 We defend backward-looking expectations by pointing out that under imperfect competition the assumption of forward-looking expectations is abnormally demanding, in that it requires consumers to have much more information than in a perfectly competitive economy. The only unquestionable requirement of 'rationality' in any expectations formation mechanism is that no forecasting errors should persist indefinitely. We show that a large number of backward-looking forecasting rules exist which satisfy this, and in particular it places no constraint on the value of the expectations elasticity.

2. The representative agent model

2.1. Overview

Three commodities are traded - goods, labour and money - by agents who are either consumers, trade unions or firms. Imperfect competition takes the form of a Cournot oligopoly between trade unions. Hart also assumed Coumot oligopoly amongst firms, but since this is irrelevant for the effects of money, we make the simpler supposition that the goods market is competitive. Imperfect competition in the labour market is essential, on the other hand, since we assume, following Hart, that the underlying competitive supply of labour is completely inelastic (households obtain no utility from leisure). With perfect competition, this supply would be fully employed, 5 thus fixing employment and output exogenously. Although exogeneity of employment with a competitive labour market can be avoided if utility of leisure is introduced, 6 we take the view, in common with critics of 'new classical' theory, that the real-wage elasticity of competitive labour supply is too small in reality to explain significant employment fluctuations. It is worth noting that even the models of monopolistic competition with menu costs, such as that of

4 The analysis here complements that in Rankin (1992), which uses the same model. Here we look at the 'neutrality' question, i.e. the effect of an increase in the level of the money supply, while Rankin (1992) looks at 'superneutrality', i.e. the effect of an increase in money's growth rate. Interestingly, here we find robustness to the forecasting rule, whereas the other paper finds strong sensitivity. Our analysis also complements Jacobsen and Schultz (1994), who look at fiscal policy in a very similar framework. Related overlapping-generations models which look at the bounded labour demand phenomenon are by Schultz (1992) and d'Aspremont et al. (1991)

5 Except in the extreme case where the market-clearing wage is zero: this possibility is examined in more depth by d'Aspremont et al. in their already-cited papers, and by Silvestre (1990).

6 Hart considers this as an extension in section 5A of his paper


Blanchard and Kiyotaki (1987), need to assume a high real-wage elasticity of the underlying competitive labour supply in order to sustain large non-neutralities of money - a problem acknowledged by Blanchard and Fischer (1989, Chapter 9). To get round this, Ball and Romer (1990) advocate the introduction of 'real rigidities' such as efficiency wages or customer markets. A nice feature of Hart ' s framework is that such devices are unnecessary: policy effectiveness is obtained despite a completely inelastic competitive labour supply.

The economy should be thought of as divided into a large number of identical locations, with agents allocated amongst them such that no worker is a customer of, or shareholder in, his own firm; and that no shareholder is a customer of her own firm. This permits us to assume that unions take their members ' consumption goods prices, and the incomes of customers at their employers ' firms, as given. Hart makes a similar assumption in order to rule out such general equilibrium ' feedback effects' . 7 To avoid cluttering the exposition with extra notation, we shall not, however, formally index variables by location.

2.2. Union behaviour

In each local labour market there are n identical unions. Households are exogenously allocated to a union. Given the local labour demand function / = g ( W; . . . ) , a labour endowment L, and the labour sales f ' of the other n - 1 unions, union i faces the problem

max Wff/ subject to ffi + f ' = g ( W ; . . . ) and if/-<< L.

Note that maximising wage revenue is consistent with maximising the utility of the union's typical member, since workers have no utility of leisure. 8 The first-order condition for solving this problem is

6 + W[~f /OW] <~ O, 6 <~ L, with complementary slack.

When the labour market is in equilibrium, symmetry amongst unions implies = f / n for all i. This gives the fundamental equation of the model:

- [ O f / O W ] [ W / f ] =- - e ( W ; . . . ) > / 1 / n . (1)

Below we shall focus on unemployment equilibria where f < nL, for which (1) must hold with equality. In general, whether the equilibrium of the model is one with unempl9yment or full employment depends on the configuration of parame- ters and policy variables.

7 See d'Aspremont et al. (1989, 1990, 1991) for analyses which take such effects into account. 8 To make this true we assume that when / / < L workers are rationed equally in hours of work.

Alternatively, if hours were a [0,1] decision, the same objective function would be implied if workers were risk-neutral and the employed were randomly chosen.


2.3. Firm behaviour

Firms are perfect competitors, and so given a production function y - - f ( f ) they maximise profits where W / p = - w = f ' ( f ) . Inverting gives the decreasing labour demand and output supply functions f = f d ( w ) , y = yS(w). For later use, we define here the elasticities of these functions as e L =- [ a f d / a w ] [ w J ] , ~s - [OyS/Ow][w/y]. Note that in general they are functions of w, or equivalently of f or y.

To obtain his specific results, Hart assumed a Cobb-Douglas form for f ( f ) . We shall work with the following more general CES form:

y = A { [ 1 - f l ] k l - 1 / ~ r + f l f l - 1 / ~ } ~ ° ' / [ ~ - l l o r > O , 0 < fl,Tr~< 1. (2)

Here k is a fixed parameter which could be interpreted as the capital stock. The limit of this function as the elasticity of substitution tr tends to unity is Ak'~II-~l f ~ , which is the form of Hart's function.

2.4. Consumer behaviour

Hart's consumer is atemporal, and obtains utility from consumption of the produced, and of a non-produced, good. It is natural to want to think of the non-produced good as 'money'. However, some justification is required for including money in the utility function. If money is the only asset 9 then the standard justification is to view the utility function as ' partially indirect', obtained from an intertemporal utility maximisation problem where money is held to transfer purchasing power from period t to t + 1. This is the approach underlying the concept of temporary monetary equilibrium (TME) as developed by Patinkin (1956), which has been given a modem restatement by Grandmont (1983). To introduce money into Hart's model, like d'Aspremont et al. (1989, 1990) we reinterpret it as a simple form of TME.

Suppose the consumer expects to live two periods but to have no income in the second. To consume in period + 1 he must accumulate money balances M, which will yield expected consumption c e - e e +1 - M/p+ 1, where p+ 1 is the expected future price. His problem is thus

max u(c, C~l ) subject to [ M o + Y ] / p = c + [p~+l/p]c~+l,

where M 0 is the money endowment and Y is the first-period money income. If preferences are homothetic as we shall assume, following Hart, then the demand function for current consumption will clearly take the form

c = or( p + l / p ) [ M o + Y ] / p . (3)

9 'Money' should therefore be interpreted broadly, to include all nominal outside assets, in this context.


a( . ) is a function which is increasing if current and future consumption are gross substitutes, decreasing if they are gross complements. Thus it captures ' intertemporal substitution effects' (positive or negative) on consumption. These effects have a central role in the model: if a ' = 0 non-neutrality will not operate, as we will see.

Since TME is a concept designed to apply to the short run, it is taken as read that consumers need to learn and that they forecast P+I using backward-looking forecasting rules whose arguments potentially include all current and past variables. A simple class of such rules which we shall use in what follows is P~-I = / z P z', where y = [Opt_ l/Op][p/p~+l] is the (constant) elasticity of expectations. In these terms, HaWs consumer is the special case where y = O, so that p~_ t is an exogenous subjective parameter independent of current variables. P~-1 can then be subsumed into u(.) and utility can be written as u(c, M). Another special case is that of unit-elastic expectations where y = 1. In this case utility can be written as u(c, M/p), subsuming /z into the function. Hart argues in the conclusion to his paper that to interpret the non-produced g o o d as 'money ' the utility function would have to take the latter form, and that this would make money neutral (as indeed we show). We believe this is too pessimistic: the TME concept does not suggest that one value of the expectations elasticity is more reasonable than another, and in Section 3, we support this by directly examining the 'rationality' of different forecasting rules.

Using the constant-elasticity forecasting rule in (3), we may compute the price-elasticity of goods demand as

[OclOp][p/c ] - ED(p) = -- 1 + [ y -- 1] tzp z'- 'a'( Ixp~'-l)/a ( /zpr- 1).

(4)

This is a generalisation of Hart ' s formula (see r / (p) , p. 117 in his paper). From it we can see that unit-elastic expectations are special in that they make the demand elasticity equal to - 1 . More generally E o is a function of the price level. To determine the nature of this function we now appeal to a particular utility function, the CES:

= p > 0 , 0 < ~ < l . ( s )

Depending on whether the elasticity of intertemporal substitution p is greater or less than one, there is (respectively) gross substitutability or complementarity between current and future consumption. Using this the expressions for a ( . ) and ~D turn out to be:

a ( p + , / p ) = 1 / [ 1 + 6°[p+i /p] ' - ° ] , (6)

-1 + t ,-,1I p -11 / [1 + "-'].


1

l-['/-llIP-1]

l - ty - l ]~ - l ]

P

[~:-llIo-U > 0

P

['y-1l[p-l] < 0

Fig. 1.

- eo as a function of p is plotted in Fig. 1. It varies in a neighbourhood of unity whose size is determined by the magnitude of [ y - 1][ p - 1]. A key feature for our purposes is that - ~ o is increasing in p irrespective of the sign of ~ / - 1. Although this is a consequence of the CES functional form, this form is probably the most widely-used one for problems of intertemporal choice, and has a number of well-known and desirable features. The only case where - e o is not increasing for "y ~ 1 is the 'Cobb-Douglas ' special case which occurs as p -~ 1 implying there is no intertemporal substitution effect. As we will see, this has identical implications to ~/= 1.

2.5. Imperfectly competitive equilibrium

We can think of the equilibrium as determined in stages. Consider first a particular location, and begin with the local goods market. Local consumers have a total nominal budget of M 0 + Y, where Y is by assumption earned at other locations. Contingent on any money wage W determined in the location's labour market, the price which clears the goods market is determined by

yS(W/p) = c~(/xpV-1)[M0 + Yl /p . (8)

This defines p as an implicit function of (W, M 0 + Y): call it p(W, M o + Y). Now consider the location's labour market. Each of the n trade unions recognises that when it restricts its labour sales, pushing up W, this raises the local goods price via (8) (but does not affect consumers' income Y). Thus the demand function which unions face (earlier denoted g(W;...)) can be defined by using p(.) in the labour demand function of the representative firm:

g(W, M o + Y) =-/d(W/p(W, M o + Y)). (9)

Partially differentiating with respect to W we obtain ~, the wage elasticity of labour demand,

" = "L "o/ [ "s + "o], (10)


where E L, E s, E D are the elasticities defined earlier. This expression for E can now be used in the condition (1) for a Cournot equilibrium. We assume (1) to hold with equality, so that unemployment exists, and rearrange to get

- E o ( p ) = Es(y) /[1 +net(y)] - z ( y ) . (11)

The right-hand side of (11) may conveniently be written as a function z(y) whose form depends only on that of f ( f ) (and on n). Eq. (11) - which is the same for

10 each location - provides an 'aggregate supply' relationship between p and y. Consider secondly the economy-wide equilibrium. If there is symmetry across

the different locations, then the money income disposed of by consumers at any location must equal the money value of income generated there: Y = p c =py. Using this in (3) and solving for y gives us an 'aggregate demand' relationship:

y = or( /xp ~'- 1)[1 - or(/.~pV- 1)] - 1 Mo/p" (12)

To ensure that this has the usual negative slope under CES preferences we shall need to assume that [7 - 1][ p - 1] < 1.

The equilibrium can be depicted by plotting (11) and (12) in (y , p)-space to obtain an 'AD-AS ' diagram. Alternative versions are given in Fig. 3. An equilibrium with unemployment may not exist for certain configurations of parameter values, signifying that equilibrium is at full employment. Examples of this will be seen in the next sub-section. Where an unemployment equilibrium does exist, we will study the effect of monetary policy on output by considering the comparative statics with respect to M 0. A rise in M 0 clearly shifts the AD curve to the right. The effect on y therefore depends, exactly as in macroeconomics textbooks, on the shape of the AS, which we now discuss.

2.6. The neutrality of money: Unit-elastic expectations

Suppose, first, that 7 = 1. By (4) or (7) this forces - E o to equal unity. Eq. (11) then becomes z (y )= 1. Thus (11) determines y independently of p. The ' A S ' is vertical in this case, and therefore - as Hart predicted in his conclusion - money cannot affect output.

However, this assumes that a solution to z ( y ) = 1 exists. To examine the existence question we need to use the CES production function (2) to find the form of the function z(y). Mechanical calculations yield the following expression for z (y) :

7r{1 - [1 - 13 ]k 1 - 1 / ~ [ y / a ] [ ' -"] /~ '~} z ( y ) =

n -- 1 / o ' + [Tr-- 1 + 1/o,]{1 - [1 - f l ] k l -1 /~[y /A] I i -~] /~r}"

(13)

10 This is a slight misuse of language since (11) also depends on demand, via the form of En(p). However, we will see that it plays the same role as a conventional AS function.

7t/[x+n-1]

N. Rankin / European Journal of Political Economy I1 (1995) 557-575 565

J (a) o < l/n

Z

; f y y

1

n/lit+n-l]

D --% (b) l / n < o < 1

1

n[~/[Tt[~+n-I

1

7t/[rt+n-l]

0 Y 0 Y (c) o = 1 (d) 1 < o

Fig. 2.

This function is graphed in Fig. 2. The shape depends critically on the value of the elasticity of substitution between labour and capital, ~r. We may distinguish four cases: ~r < 1/n, 1/n < o- < 1, o- = 1, and cr > 1. It is clear from the diagrams that z(y) = 1 only has a solution, i.e. an equilibrium with unemployment only exists, if cr < 1/n. In the other three cases, equilibrium with unit-elastic expectations must be a full employment one. 11 Note, however, that at full employment the exogenous labour supply nL determines output directly, from y =f(nL) , so that money is clearly still unable to affect output.

It is interesting to recall that Hart's chosen production function is the Cobb- Douglas case where o" = 1. This implies that if we were to allow ~/= 1 in Hart's original model, full employment would be the outcome. Therefore a benefit of generalising the production technology is that (focusing on the case o-< 1/n) whenever 7 deviates from unity, permitting demand elasticity to vary in the neighbourhood of unity, output will respond to demand elasticity. The level of activity is thereby freed from the restriction imposed by the exogenous competitive labour supply.

11 Or else no equilibrium exists at all: this second possibility can be ruled out, but we omit the proof since it is uninteresting.


2. 7. The neutrality of money: Non-unit-elastic expectations

Now suppose 3' ~ 1: how does this affect the AS? We look first at the case or < 1/n, which ensures that an unemployment equilibrium exists. The slope of the AS curve defined by (11) depends on the signs of the relationships between --eD and p (see Fig. 1), and between z and y (see Fig. 2). The former is always positive, and so is the latter when tr < 1/n. Therefore the AS curve is upward- sloping for any non-unitary value of 7. This is illustrated in panel (a) of Fig. 3. It follows that an increase in M0, which shifts the AD curve out, raises price and output as in the familiar textbook diagram.

This provides an example of a positive multiplier for a policy which can be unambiguously interpreted as monetary policy. Hart's original demonstration, by comparison, was strictly speaking only a proof that a discovery of some valued natural resource would have a positive multiplier. To this extent it was dealing with a 'real' shock, and it is less surprising that such a shock should have real effects. Our extension also shows that if the original model is interpreted as a monetary model, then the implicit assumption that 3' = 0 which it embodies is not necessary for Keynesian effects: any value of the expectations elasticity different from unity will suffice. This is the key point for the monetary interpretation of the model. The concept of TME is agnostic as to the value of 3': given this, for the

AD(M~) / ~AD(M0)

Y (a) a < l/n

AS

(c) o" = 1 Y

Fig. 3.

~ A D

(b) l/n< ~ < I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ h S y Y

(d) 1 < c~

N. Rankin // European Journal of Political Economy 11 (1995) 557-575 567

model to provide any general support for positive effects of money on output, it needs to generate them for almost all values of T. This is indeed what it does. By comparison in competitive TME models, although it is well-known that non-unit- elastic expectations lead to non-neutrality (see Grandmont, 1983), this non-neutrality typically either does not involve output, or the effect on output is not robust to the sign of T - 1. The former is true of our model in its competitive version: real balances are affected by M 0 but y is not, since it is determined by the exogenous labour supply. The latter is true, for example, of the new classical model of aggregate supply based on intertemporal substitution of leisure, if combined with backward-looking expectations. 12

Secondly we look at what happens when or lies in one of its other three possible ranges. We note immediately from Figs. 1 and 2 that if [T - 1][ p - 1] is negative then an unemployment equilibrium cannot exist, since - c o is greater than unity while z is less than it, for all p. Equilibrium in this case must be at full employment, implying that monetary policy is ineffective. If an unemployment equilibrium is to exist we therefore need [ 7 - 1 ] [ p - 1 ] to be positive, and moreover sufficiently large. Under this assumption, the AS curves corresponding to cases (b)-(d) of the z(y) function in Fig. 2 are depicted in panels (b)-(d) of Fig. 3. In panel (b) we see that 1/n < or < 1 leads to a negatively sloped AS. This must have two intersections with the AD, if it has any. Here we get a very surprising 'ultra-Keynesian' result, in which the price in the low-output equilibrium falls when M 0 is raised, resulting in a rise in y which is more than proportional to the rise in M 0. 13 Panel (c) depicts the AS for ~ = 1, corresponding to the special case of a Cobb-Douglas production function. This was the technology used by Hart, and we see that it is responsible for the particularly strong result of complete price rigidity which Hart obtained. Lastly panel (d) shows that when tr > 1 the AS reverts to a more conventional upward slope. However its curvature is unconventional, suggesting that monetary policy is more, rather than less, effective at high levels of output.

In summary, in all cases where a (stable) unemployment equilibrium exists, a monetary expansion is effective in reducing unemployment given only that expectations are not unit-elastic. What is the intuitive explanation for this Keyne- sian result? We can best answer this by focusing on the case tr < I/n, where z (y) is upward-sloping. A monetary expansion will in general tend to raise the price level. With CES utility and non-unit-elastic expectations, we have seen that this raises the price elasticity of demand for goods. However, a more elastic goods demand implies a more elastic labour demand facing unions, as (10) shows. This

12 AS a trivial illustration, suppose the AS function is lny = lnp- lnp~_ 1 (see, e.g., Barro (1976)). Using our constant-elasticity forecasting rule in this we get lny = [ 1 - y ] l n p - i n / x , an AS curve whose slope depends on the sign of 3' - 1.

13 The high-output equilibrium may be shown to be unstable under a plausible adjustment process: a

proof of this is in Rankin (1988).


therefore tends to weaken unions' monopoly power. Since the general consequence of monopoly power is restriction of employment, such a weakening relaxes the restriction on employment which unions find it optimal to exercise. This mechanism is essentially what is embodied in the upward slope of the z(y) function: as - G o increases, the monopoly power of labour is reduced, and employment and output rise.

3. An overlapping-generations extension

The representative agent model has the merit of simplicity and of being very close to Hart's original, but it does not provide a consistent intertemporal structure which enables us to compare expectations with outcomes, and so to consider the 'rationality' of the different forecasting rules. Hence in this section we replace the representative consumer by overlapping generations of consumers. We show, first, that the effectiveness of monetary policy under backward-looking expectations is robust to this extension. Second, we examine whether considerations of 'rationality', short of full forward-looking expectations, eliminate forecasting rules which embody non-unit-elastic expectations, and conclude that they do not. Thirdly, for comparison, we solve the model under forward-looking expectations. Not surpris- ingly, the scope for non-neutrality here proves to be much smaller. However this exercise also highlights the extremely demanding informational assumptions which imperfect competition makes necessary for full forward-looking expectations, so that our final conclusion is that backward-looking expectations is the more reasonable hypothesis.

3.1. Overlapping generations of consumers

Suppose a constant population of consumers, who each live for two periods. In any period t, there are 'young' and 'old' individuals, denoted Y and O. Only the young have a positive labour endowment so the consumer must save for retire- ment, which she does by accumulating money when young. Her lifetime optimisa- tion problem is

c ° maxu(cV, ct°+l) subjectto Yt+St v =pt cY +Mr, Mt+S°~_el =pte+l t + l "

We assume that firms' profits are distributed to the young, so that Yt is the combined wage and profit income, or equivalently national income. S Y, S°~:~1 are current and expected lump-sum subsidies by means of which the government increases the money supply. With homothetic utility, the demand functions for

14 consumption which result are:

cVt =a(p~+a/pt)[Yt+Stv +S°~_el]/pt, c °=[Mt_ l +S°] /p t . (14)

14 The implicit non-negativity constraint on M t is assumed never to bind.


The young's demand has an almost identical form to that in Section 2, while the old's demand is just a rearrangement of her budget constraint.

Aggregate goods demand is now the sum of two components, c t --- + c v + c°t. Its price elasticity is easily shown to be a weighted average of the price elasticities of the components, the weights being their shares in total spending. From (14) we see that using the forecasting rule p~+ 1 = I~tPt v the young's elasticity will have the same form as in Section 2; while the old's elasticity is clearly just the constant - 1. The weighted average of these is

eo= -- l + [cV/ct][T--1]txtPrt-~Ol'(Ixtprt-l)/Ot(lxtP~/-l). (15)

Relative to the expression for % in Section 2, the new element is the presence of the weight cV/c,. It is this which complicates somewhat the following analysis.

3.2. The neutrality of money under backward-looking expectations

The new AD equation may be obtained by summing the two demands in (14), imposing Yt --- PtYt = PtCt as before, and solving for Yt:

yt = [ 1 - a ( tz tpT-i)]- l{ot( txtpT- ')[S Y + S°~_~] + Mt_ 1 + S?} /p t .

(16)

The AS equation is still given by (11), except that the function - e n ( p t ) now takes a different form. To obtain it we need to use (15) and solve for the young's consumption share. This may be done with the help of (14) and (16), which yields the following expression for the AS:

-~n (P t ) = 1 - [ 3 1 - 1]lxtpt~-lot'( txtpt r - l )

{ Svt + S?£-~ + M,- , + s° } × ot(tztpT_l)[sVt +so:el] + M t _ l + S o = z ( y t ) . (17)

We will assume that ¢r < 1/n holds, so that (17) has a solution when - e n = 1. (16) and (17) thus determine the current values of (Pt, Yt). The dynamic path of the economy depends, inter alia, on how/z t is related to lagged variables such as Pt-1: this is taken up in the next sub-section.

Now consider a 'step' increase in the money supply, which is correctly perceived as such. This means that either StY or S ° goes from zero to positive while S °'e remains at zero. It is clear, first of all, that the elasticity of expectations t+l remains critical to the effects of such a change. If T = 1 in (71), Yt is determined as before by 1 = z(yt), independently of either St v or S °. Unit&lastic expectations therefore still result in a vertical AS and monetary policy ineffectiveness.

Consider then the case where T 4= 1. Relative to before the analysis is compli- cated by two factors, as a result of which we relegate the details to the appendix.

5 7 0 N. Rankin / European Journal of Political Economy 11 (1995) 557-575

First, under CES utility it turns out that - eD is no longer monotonic in Pt, which implies that the AS curve is forward- or backward-bending. Second, monetary shocks now shift the AS curve as well as the AD curve, as follows from the presence of S Y and S ° in (17). In the appendix we show that, despite these complications, for an economy which is initially in a stationary state a monetary injection via a subsidy to the old will always raise output, while a monetary injection via a subsidy to the young will probably do so, the condition for the latter being that the parameter product [ y - 1][ p - 1] should lie outside the interval (0, 8P[1 + ~ P]). The mechanism through which these effects operate is by raising the price-elasticity of goods demand, as in Section 2. However, since this elasticity is a weighted average of the old 's and the young 's elasticities, a new way in which it can be (for example) increased is through shifting the share of consumption in favour of the generation with the higher elasticity. This effect sometimes rein- forces, and sometimes conflicts with, the basic positive effect via the young's elasticity. Only in a small proportion of cases does it actually overwhelm it.

3.3. The 'rationality' of backward-looking expectations

Backward-looking expectations are predicated on the idea that agents may repeat their mistakes in the medium run. Nevertheless the argument that any rational learning process should eliminate mistakes in the long run, in the absence of exogenous shocks, is one which it is hard to reject. So we now ask, does this requirement of ' rationality ' restrict the elasticity of expectations to equal unity? If this were to prove the case, the explanation which we have been advocating for monetary policy effectiveness would be undermined.

We will confine attention to long-run states in which the money supply is held constant. 15 If the economy converges to a stationary equilibrium in which forecasts are correct, then the actual and expected price levels must be constant and equal. To see what this implies for the forecasting rule, consider a broad class of such rules, the loglinear autoregressive class:

In Pt+I = E Aslnpt-s • (18) s = 0

When Pt-s = P, say, for all s t> 0, for p~+ 1 also to equal p we obviously need the rule to satisfy ]~=0 As = 1. However, this does not constrain y : y = A o here, and we can choose h o arbitrarily and still ensure that the coefficients sum to unity. Hence what is perhaps the most convincing requirement for ' rat ional ' expectations

15 It should be clear that money is neutral in the long run. If expectations are correct in a stationary state then setting I~tPt y - 1= 1 in (17) gives 1 - - [ y - 1]a'(1)= z(y), which determines y independently of M.


is not sufficient to upset the model 's conclusions about the positive short-run effects of money on output. 16

3.4. The neutrality of money under forward-looking expectations

For comparison, we now ask what happens if agents are able perfectly to foresee the future price level. Hence we treat consumers as if they can directly observe Pt+ 1 in period t, just as they can directly observe Pt. 17 There are several ways in which this modifies, or may modify, the existing AD and AS equations. Most obviously, Pt+ 1 is replaced by its true future value Pt+ 1. Second, consumers no longer have any need to link Pt to their perception of Pt+ 1 through a subjective parameter 3', which means that 3/should be set to zero in the expression for - E o. A third, and more subtle, point is that the formula (10) for the labour demand elasticity may no longer be correct. When unions push up the wage at their location, this could in principle affect the true future price faced by consumers currently at the location. This would occur if the young consumers at the location were to remain there next period, in which case their current savings decisions would impact significantly on the future demand at the location and hence on the future price which they would face. In this case, with perfect anticipation by both consumers and unions, there should be an additional term in (10) to reflect the effect via the induced change in the true future price. For simplicity we will rule out this 'future price effect' by assuming that young consumers at the location are dispersed amongst new locations next period. This implies that their current behaviour will not affect the future price they face, and no further modifications to the AD and AS equations are then needed.

The equilibrium for (Pt , Yt) contingent on Pt+l is thus given by (16) and (17) with y = 0 and /z~pt v-1 replaced by Pt+l//Pt . This system provides an implicit difference equation linking Pt and Pt+ 1. As with any model with forward-looking expectations, equilibrium in the current period depends on the complete future time path of the economy, and this gives rise to the possibility of multiple equilibrium time paths. Following convention, we will focus on the stationary state path. Thus, if there is an unanticipated step increase in M in period t, the economy is assumed to have attained its new stationary state in t + 1. This ties down Pt+ 1, and from this we can solve recursively for Pt. Y and p in a stationary

16A further requirement of 'rationality' might be that the forecasting rule should not cause the economy to diverge. To examine this, we checked the dynamic stability of the model under the familiar 'adaptive' expectations rule. This is the special case of (18) where A s = [ 1 - ~b]~b s, 1 - ~b being the adaptation coefficient. We found that stability is assured for all tk ~ [0,1]. Thus 4' = 0, and so y = 1, is not necessary for convergence.

17 This is the expectations assumption used by other authors of overlapping-generations models with imperfect competition, such as those cited in footnote 4.


state can be found by taking (16) and (17) and setting y and the subsidy variables to zero, and lxtpt r - 1 to unity:

y = [1 - a ( 1 ) ] - a M / p , (19)

1 + a ' (1) = z ( y ) . (20)

Eq. (20) determines y independently of M and so shows that even if money should have an impact effect on output, this cannot persist beyond the period of the shock.

Consider now the impact effect itself. First suppose the monetary injection is via a subsidy to the old generation, so that in (17) S v + S°~:~1 remains at zero while S ° goes from zero to positive. We assert that this has no effect on Yt, so that this type of monetary injection is neutral on impact. To show this, one simply needs to verify that if Yt = Y as defined in (20), and if Pt increases in proportion to Pt+l (which from (19) we know increases in proportion to M t, or M t_ 1 + sO), then (16) and (17) will be satisfied. This is not a surprising result, since it is standard in monetary economics that monetary injections proportional to initial endowments are neutral when there is perfect foresight. If, second, the injection is via a subsidy to the young, then by contrast Yt will be affected. This may be shown by computing the multiplier d y t / d S t v, which is presented in the appendix. Intuitively a real effect arises be.cause the injection is not proportional to initial endowments (the young having zero initial balances), so that there is a redistribution of real wealth from old to young. The appendix shows that its sign depends on preferences: gross substitutability causes output to rise, complementarity causes it to fall.

In summary, under forward-looking expectations money is quite likely to be neutral, and when it is not it could just as well be negative as positive. However, we would argue that the forward-looking expectations hypothesis makes implausi- bly strong informational assumptions. These are already strong in the competitive context, where they still attract controversy. In an imperfectly competitive context, they are stronger still. This is because we assume that agents can correctly predict not only the values of variables along the equilibrium path of the economy, but also their values off the equilibrium path. We saw this when discussing the 'future price effect' above: if a union deviates from its optimal choice of labour sales, pushing up the local wage and price, it is assumed that consumers at the location can correctly predict the impact (zero under our assumptions, but non-zero under less convenient assumptions) on the future price which they will face. Yet, by definition, such out-of-equilibrium behaviour will never actually occur in the model, so that consumers have no opportunity to learn from experience. In a perfectly competitive world this need to forecast correctly off the equilibrium path does not arise: a small agent cannot affect any future price. The extra information which agents need in order to have forward-looking expectations in an imperfectly competitive economy therefore adds further to the already stringent requirements


of this hypothesis in a competitive economy, and makes the case for considering backward-looking hypotheses substantially stronger. 18

4. Conclusions

In this paper we have conducted an explicit analysis of monetary policy in an adaptation of Hart's (1982) seminal model of imperfect competition. The results suggest grounds for optimism that his findings on policy effectiveness will carry into the monetary arena, but several new issues, in particular about the modelling of expectations in imperfectly competitive models, are raised. The feature we have emphasised is the robustness of the effect of money on output to different forecasting rules. If the positive effect is to be sustained, future work needs to test its robustness to other extensions and generalisations. One such is the inclusion of investment in the model. Investment offers another (and empirically better-founded) channel for intertemporal substitution effects, and these were noted to be central to the phenomenon investigated here. Some progress on this has been made by Rivera-Campos (1992).

Acknowledgment

I am grateful to an anonymous referee for helpful comments.

Appendix

A.1. Policy effects in the overlapping generations model with backward-looking expectations

Using CES utility, the AS equation (17) can be written more explicitly as

Xt -%(Pt ) = 1 - [ 7 - 11[ p - 11

[1 + xt] 2

Y O,e 0 [S t "I-St+l] + [i~t-l-]-St ] X [ S v + S O ~ e l ] / [ l + x t ] + [ M t _ l + S O ] ) =z(y,) , (A.1)

18 AS a referee points out, a weaker version of 'forward-looking' expectations could be postulated in which expectations are always correct on the equilibrium path, but may be incorrect off it (an example of this is given in Rankin (1992, footnote 9)). However once we have accepted the abandonment of the assumption that agents know the true model, this seems an unsatisfactory halfway house - unless, possibly, as a long-run assumption.

574 N. Rankin /European Journal of Political Economy 11 (1995) 557-575

where x t = diP[/.ttpT-1] 1-p. Note that as Pt goes from 0 to ~, x t goes from ~ to 0 if [7 - 1][ p - 1] > 0, or from 0 to o¢ if [3' - 1][ p - 1] < 0. Further, it can easily be confirmed that as x t goes from 0 to oo, xt//[l + Xt] 2 rises from 0 to a maximum where x t = 1 and then falls back to 0. Thus, if we consider - E o as a function of Pt in an initial stationary state where S v = S°--S°'~+1 = 0, we find that as Pt increases from 0 to ~, then for [3" - 1][ p - 1] > 0, --eD(Pt ) is 'valley-shaped' , falling from 1 to a minimum and returning to 1; while for [3' - 1][ p - 1] < 0, it is 'hill-shaped', rising from 1 to a maximum and returning to 1. Combined with z (y t) for the case o-< 1/n, this implies that the AS curve is forward-bending if [3' - 1][ p - 1] > 0 (i.e. downward-sloping for low, but upward-sloping for high, p~); and backward-bending if [3' - 1][ p - 1] < 0.

To determine policy effects, we need to know the economy's position on the AS curve. In a stationary state, correct expectations imply /ztp~ - 1 = 1, whence x t = di P. Since di < 1 if there is positive time preference, we have that x t < 1 initially. From this it follows that, regardless of the sign of [3' - 1][ p - 1], the AS is upward-sloping at the stationary state point. Consider first a monetary injection via a subsidy to the old. This shifts the AD to the right. Since S Y + St°if1 = 0 in an initial stationary state, we can see from Eq. (A.1) that the AS does not shift. Thus, since the AS is upward-sloping, Yt increases for all 3' 4: 1. Consider next a monetary injection via a subsidy to the young. This again shifts the AD right- wards. This time the AS also shifts: in (A.1), the term {.} increases, and thus the AS shifts left for [3' - 1][ p - 1] > 0, right for [3' - 1][ p - 1] < 0. In the latter case, AD and AS shifts are reinforcing and Yt hence increases. In the former case, there is ambiguity. To resolve this we differentiate, obtaining the following expression for the multiplier evaluated in a stationary state:

dyt dS v = [ 3 , - - 1 ] [ p - 1 ] ( [ 3 , - 1 ] [ p - 1 ] - 6 P [ l + d i P ] ) A -1, (A.2)

where

,4 = {z 'y[ dip + 1 - ( 3 ' - 1)( p - 1) ] [1 + ~p]4di_p + [ ( 3 , - 1)( p - 1 ) ] 2 d i p ( 1 - diP)}p, > O.

This is negative when [ 3 , - 1 ] [ p - 1 ] lies in the interval (0, diP[1 + diP]), and positive elsewhere.

A.2. Policy effects in the overlapping generations model with forward-looking expectations

The case of a monetary injection via a subsidy to the old was examined in the main text. For a subsidy to the young we can compute the multiplier, evaluating in a stationary state:

dyt dS v = [ p - I ]~P[1 + pdiP] A - l , (A.3)


where A is as in (A.2) but with 31-- 0. The sign of this depends on the sign of p - 1: for gross substitutability in intertemporal consumption preferences ( p > 1) it is positive, for gross complementarity ( p < 1), negative.

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Money in Hart's model of imperfect competition