Cagan and Lucas Models Presented by Carolina Silva 01/27/2005

Cagan and Lucas Models

Presented by Carolina Silva

01/27/2005

Introduction

I will present two models that determine nominal exchange rates:

•The monetary model: Cagan model•Lucas Model

Even though the first one is an ad-hoc model, many of its predictions are implied by models with solid microfoundations, and it is the basis for work in other topics. The Lucas model is one of those solid microfoundations exchange rate determination models.

I. Cagan Model of Money and Prices

In his 1956 paper, Cagan studied seven cases of hyperinflation. He defined periods of hyperinflations as those where the price level of goods in terms of money rises at a rate averaging at least 50% per month

These huge inflations are not things of the past, for example, between April 1984 and July 1985, Bolivia’s price level rose by 23,000%

This implies an annual inflation rate of almost 13,000

Cagan Model

Let M denote a country’s money supply and P its price level, Cagan’s model for the demand of real money balances M/P is:

inflation. expected respect to

withbalances realfor demand ofcity semielasti theis and log

t,period of end at the held balancesey nominalmon of log where

)( 1

Pp

m

ppEpm ttttdt

Cagan justifies the exclusion of real variables such as output and interest rate from the money demand function, arguing that during hyperinflation the expected future inflation swamps all other influences on money demand.

Solving the ModelWhich are the implication of Cagan’s demand function to the relationship between money and the price level?

Assuming an exogenous money supply m, in equilibrium:

(1) )(

:becomes demandmoney the thus,

1 ttttt

tdt

ppEpm

mm

So, we have an equation explaining price-level dynamics in terms of the money supply.

Solving the Model

(3) 11

1

:get that we

bubbles) especulativ no (ie, zero be to termsecond theAssuming

1lim

11

1

sts

ts

t

Tt

T

Ts

ts

ts

t

mp

pmp

... , 3 2 tt ppFirst, for the nonstochastic perfect foresight, ie, by successive substitution of we get that:

Is this a reasonable solution of (2)?

(2), )( 1 tttt pppm

Simple Cases

tmmt 1. Constant money supply:

mpmp

mppppm

tts

s

ts

t

ttttt

11

also, and

)( 1

Simple Cases

tmmt

2. Constant percentage growth rate:

Guessing that the price level is also growing at rate , and substituting this guess in equations (2) and (3), we get again the same answer from both:

tt mp

3. Solution (3) also covers more general money supply processes.

The Stochastic Cagan Model

Given the linearity of the Cagan equation, extending its solution to a stochastic environment is straightforward. Under the no bubble assumption, we have that:

(4) )(11

ts

st

ts

t mEp

The Cagan Model in Continuous Time

Sometimes is easier to work in continuous time. In this case, the Cagan nonstochastic demand (2) becomes:

0 implies assumption bubble no thewhere

)/exp(]/)(exp[1

:get that wemethods equations aldifferenti using

0

0

b

tbdsmtsp

ppm

t

st

ttt

Seignorage

t

tt

P

MM 1Seignorage

Definition: represents the real revenues a government acquires by using newly issued money to buy goods and nonmoney assets:

Most hyperinflations stem from the government’s need for seignorage revenues. What is the seignorage-revenue-maximizing rate of inflation? Rewriting seignorage as:

t

t

t

tt

P

M

M

MM 1Seignorage

we can see that, if higher money growth raises expected inflation, the demand for real balances M/P will fall, so that a rise in money growth does not necessarily augment seignorage revenues.

Seignorage

11

1

t

t

t

t

P

P

M

M

t

t

t

t

P

P

P

M 1

Finding the seignorage-revenue-maximizing rate of inflation is easy if we look only at constant rates of money growth:

Now, exponentiating Cagan’s perfect foresight demand, we get:

Substituting these in the second seignorage equation:

1)1()1(1

Seignorage

Seignorage

1 0)1)(1()1(

:is respect to with FOC theThus,

max21

Cagan was surprised because, at least in a portion of each hyperinflation he studied, governments seem to put the money to grow at rates higher than the optimal one.

•Adaptative expectations may imply short run benefits from temporarily increasing the money growth rate.

•Problem: even under rational expectations, if the government can not commit to maintain the optimal rate, its revenues could be lower.

A Simple Monetary Model of Exchange Rates

output. real of log theis and price

of log theis rate,interest nominal the with )1log(i where

,(1) i 1t

y

pii

ypm ttt

(3) )1(1 UIPand

(2) logsin or PPPThen

1*11

**

t

tttt

tttttt

Eii

pepPP

t

A variant of Cagan’s model: a SOE with exogenous real output and money demand given by:

Let be the nominal exchange rate (foreign in terms of home), and denote the world foreign-currency price of the consumption basket with home-currency price . P

*P


(4) ii 1*

1t1t ttt eeE

An approximation in logs of UIP is:

Substituting the log PPP and (4) in eq. (1) gives:

(6) )i(11

1

:is rate exchange theosolution t theand

(5) )()i(

**1s

1**

1t

ssstts

ts

t

ttttttt

pymEe

eeEepym


Even though data do not support generally this model in non hyperinflation environment, this simple model yields one important insight that is preserved in more general frameworks:

The nominal exchange rate must be viewed as an asset priceThe nominal exchange rate must be viewed as an asset price

In the sense that it depends on expectations of future variables, just like other assets.

Monetary Policy to Fix the Exchange Rate

e

(1) )( 1 ttttt eeEem

Consider a special case of the SOE Cagan exchange rate model:

Suppose the government fixes the nominal exchange rate permanently at , then substituting in (1) we get that:

emmt

Thus, money supply becomes an endogenous variable, implying that exchange rate targets implicitly entail decisions

about monetary policy.

Some observations

Can the exchange rate be fixed and the government still have some monetary independence?

•Adjusting government spending can relieve monetary policy of some of the burden of fixing the exchange rate. But in practice, fiscal policy is not a useful tool for exchange rate management, because it takes too long to be implemented.

•Financial policies can help also through sterilized interventions: to keep the exchange rate fix, the government may have to buy foreign currency denominated bonds with domestic currency. To “sterilize” this, the government reverses its expansive impact by selling home currency denominated bonds for home cash.

II. Lucas Model

One of the problems of Cagan model is that the money demand function upon which it rest has no microfoundations. On the other hand, Lucas’s neoclassical model of exchange rate determination gives a rigorous theoretical framework for pricing foreign exchange and other assets.

We will see three models:

•The barter economy

•The one money monetary economy

•The two money monetary economy

In all these, markets have no imperfections and exhibit no nominal rigidities. Agents have rational expectations and complete information.

A. The Barter Economy

Here we will study the real part of the economy:

•Two countries, each inhabited by a representative agent.

•There is one “firm” in each country, which are pure endowment streams that generate a homogeneous nonstorable country-specific good, using no labor or capital input => fruit trees.

•Evolution of output:

agents.by known are processes stochastic

its and random are and where, and *1

*1 tttttttt ggygyxgx

•Each firm issues a perfectly divisible share of common stock which is traded in a competitive market.

The Barter Economy

tx

)()( *

11 tytyttxt eyqwexwWtt

•Firms pay out all of their output as dividends to shareholders, which are the sole source of support for individuals.

•We will let be the numeraire good.

•Under this framework, the wealth a domestic agent brings to period t is:

•And the agent has to allocate this wealth between consumption and new share purchases:

tttt ytxytxtt cqcweweW *

The Barter Economy

(1) )()( **

11 tytyttxytxtytx eyqwexwwewecqctttttt

(1) .

),( 0

st

ccuEMaxj

yxj

t jtjt

Equating the last two equations we get the budget constraint for domestics:

In this way, domestic agents have to choose sequences

to solve:

0,,,

jyxyx jtjtjtjtwwcc

The Barter Economy

(4) )])(,([),( :

(3) )])(,([),( :

(2) ),(),( :

*11111

*

1111

21

11

11

tttyxtyxty

ttyxtyxtx

yxyxty

eyqccuEccuew

exccuEccuew

ccuccuqc

ttytt

ttytt

ytttt

Thus, the domestic Euler equations are:

If we put an * over the variables in the domestic agent problem and in the domestic Euler equations, we get the foreign agent problem and foreign Euler equations.

0

,,,jyxyx jtjtjtjt

wwcc

The Barter Economy

We need to add four more constraints to clear the markets:

(8)

(7)

(6) 1

(5) 1

*

*

*

*

tyy

txx

yy

xx

ycc

xcc

ww

ww

tt

tt

tt

tt

The Barter Economy

)8( ),7( .

),(2

1),(

2

1 **

st

ccuccuMaxtttt yxyx

Given that we have complete and competitive markets, we can apply the welfare theorem and solve the social planner problem:

and the solution will be an competitive equilibrium:

2

2),(

2

1),(

2

1

),(2

1),(

2

1

: **

**22

**11

tyy

txx

yxyx

yxyx ycc

xcc

ccuccu

ccuccuFOC

tttt

tttt

tttt

The Barter Economy

2

1** tttt yyxx wwww

Now we have to look for the prices and shares that support this equilibrium.

•Shares: a stock portfolio that achieves complete insurance of idiosyncratic risk is,

•Prices: to get an explicit solution we need to give a function form to the utility, let

1),( and

11 t

yxyxt

CccuccC

tttt

The Barter Economy

Under all what we have seen and assumed, the Euler equations imply:

1

*1

1

1*

1

1

1

1

1

1

1

tt

t

t

tt

tt

t

t

t

t

tt

t

t

t

tt

yq

e

C

CE

yq

e

x

e

C

CE

x

e

y

xq

B. The One-Money Monetary Economy

tttt MM where1

Here we introduce a single world currency and the idea is to do it without changing the real equilibrium reached above.

For the money to have some value at equilibrium, Lucas introduces a “cash-in-advance” constraint. As we enter period t:

1. Output levels are revealed.

2. Money evolves according to: is known. The economy wide increment is distributed evenly across H and F individuals as lump sum transfers. Each receive:

)2/)(1(2 1

tt

t MM

The One-Money Monetary Economy

3. A centralized securities market opens, where agents allocate their wealth toward stock purchases and the cash they will need for consumption.

4. Decentralized goods trading now takes place in the “shopping mall”.

5.The cash value of goods sales is distributed to stockholders as dividends, who carry these nominal payments into the next period.

Observation:Observation: the state of the world is revealed before trading, thus agents know exactly how much cash they need to finance the current period consumption plan. So, it is no necessary to carry cash from one period to the next, and they won’t do it if the nominal interest rate is positive.


transfermoney

t

t

valuesharedividendex

tytx

dividends

t

ttytxtt P

Mewew

P

yqwxwPW

tt

tt

*1111

2

)(11

11

*tytx

t

tt ewew

P

mW

tt

Given these assumptions, domestic agent’s period t wealth is:

And in the security market, the agent allocate his wealth between:

Assuming a positive nominal interest rate, the cash in advance constraint binds:

)(tt ytxtt cqcPm


Using the last three equations, we get that the domestic agent problem is:

c

2)( .

),(Max

*y

*111

1

0

t

1111

tytxtx

tytxt

tttytx

t

t

jyx

jt

ewewqc

ewewP

Myqwxw

P

Pst

ccuE

t

tttt

jtjt


The domestic agent problem implies the following Euler equations:

(4) )])(,([),( :

(3) )])(,([),( :

(1) ),(),( :

*111

111

*

111

11

21

11

11

tttt

tyxtyxty

ttt

tyxtyxtx

yxyxty

eyqP

PccuEccuew

exP

PccuEccuew

ccuccuqc

ttytt

ttytt

ytttt

The foreign agent has the same problem and Euler equations but with an * over the variables that he chooses (consumption, shares w and money holdings m).


and

1 1

*

**

**

ttt

tyytxx

yyxx

mmM

yccxcc

wwww

tttt

tttt

2

2** tyy

txx

ycc

xcc

tttt

To clear the markets we need to add the constraints:

The equilibrium of the barter economy is still the perfect risk-pooling equilibrium:

2

1** tttt yyxx wwwwand

The only thing that has changed is the equity pricing formulae, which now include the “inflation premium”.


Using the same constant relative risk aversion utility function we used in the barter economy, we have that:

1

*1

1

1

1*

1

1

1

1

1

1

11

1

tt

t

t

t

t

tt

tt

t

t

t

t

t

t

tt

t

t

t

t

t

t

t

t

t

tt

yq

e

M

M

C

CE

yq

e

x

e

M

M

C

CE

x

e

x

x

M

M

P

P

y

xq

The One-Money Monetary Economy, pricing other assets

payoff ofutility marginal

11

bond thebuying ofcost utility

1 )/),((/),(11

tyxtttyx PccuEPbccutttt

1)1( tt ib

At equilibrium, the price b of a nominal bond that pays 1 dollar at the end of the period must satisfy:

If is the nominal interest rate, then

Thus, using the usual utility function, nominal interest rate will be positive in all states if the endowment growth rate and monetary growth rates are positive.

ti

C. The Two-Money Monetary Economy

Let the home currency be the “dollar”, and the foreign, the “euro”. Now, the home good x can only be purchased with dollars, and y with euros. Besides, x’s dividends are paid in dollars and y’s in euros. Agents can get the foreign currency during security market trading.

Currencies evolve according to:

1_*

1

:

:

ttt

ttt

NNeuro

MMdollar

Now we will have a new product: claims to future dollar and euro transfers. It will be assumed that initially the home agent is endowed with the whole stream of dollars and the foreign, with the hole stream of euros. Then they can trade.

The Two-Money Monetary Economy

uritiesofvaluemarket

tNtMtytx

transfersmoney

t

ttN

t

tM

dividends

tyt

tttx

t

tt

rrewew

P

NS

P

Myw

P

PSxw

P

PW

tttt

tt

tt

sec

**

1

*1

11

1111

11

11

Then, we have that the home agent current-period wealth is:

And this wealth will be allocated according to:

t

tt

t

ttNtMtytxt P

Sn

P

mrrewewW

tttt **


:eqsEuler following imply the and sconstraint

advancein cash theand equations last two by the implied BC thebefore, As*

tt yttxtt cPncPm

)])(,([),( :

)])(,([),( :

)])(,([),( :

)])(,([),( :

),(),( :

*1

1

1111

*

11

11M

*11

1

*1

11*

111

11

21

*

11

11t

11

11

tt

ttyxtyxtN

tt

tyxtyxt

ttt

ttyxtyxty

ttt

tyxtyxtx

yxyxt

tty

rP

SNccuEccur

rP

MccuEccur

eyP

PSccuEccuew

exP

PccuEccuew

ccuccuP

PSc

ttytt

tttt

ttytt

ttytt

ytttt

And again the foreign agent have a symmetric set of Euler eqs.


1 1

**

**

**

tttttt

tyytxx

yyxx

nnNmmM

yccxcc

wwww

tttt

tttt

2

2** tyy

txx

ycc

xcc

tttt

Together with the Euler eqs. We have the clear market conditions:

With these eqs. We have the following equilibrium:

2

1**** tttttttt NNMMyyxx wwww

and


From the first Euler equation, we get that the nominal exchange rate is:

t

t

t

t

yx

yxt x

y

N

M

ccu

ccuS

tt

tt

),(

),(

1

2

ConclusionConclusion: as in the monetary approach, the determinants of the : as in the monetary approach, the determinants of the nominal exchange rate are relative money supply and relative nominal exchange rate are relative money supply and relative GDPs. Two major differences are that in the Lucas model:GDPs. Two major differences are that in the Lucas model:

•S depends on preferencesS depends on preferences

•S does not depend explicitly on expectationsS does not depend explicitly on expectations

Documents

Cagan and Lucas Models Presented by Carolina Silva 01/27/2005