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
This implies an annual inflation rate of almost 13,000
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%
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:
mtd  pt  Et ( pt 1  pt )
where m  log of nominalmon ey balances held at the end of period t,
p  log P and  is the semielasti city of demand for real balances with
respect to expected inflation.
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 Model
Which 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:
mtd  mt , thus the money demand becomes :
mt  pt  Et ( pt 1  pt )
(1)
So, we have an equation explaining price-level dynamics in terms
of the money supply.
Solving the Model
First, for the nonstochastic perfect foresight, ie, mt  pt   ( pt 1  pt ) (2),
by successive substitution of pt  2 , pt 3 ... we get that:
s t
T
  
  
1

 ms  lim 
 pt T
pt 

T  1  
1   s t  1   



Assuming the second term to be zero (ie, no speculativ e bubbles)
we get that :
s t
  
1

 ms
pt 

1   s t  1   

Is this a reasonable solution of (2)?
(3)
Simple Cases
1. Constant money supply: mt  m t
mt  pt   ( pt 1  pt )  pt  m
and also,
s t
  


 ms  pt  m
pt 

1   s t  1   

Simple Cases
2. Constant percentage growth rate: mt  m  t
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:
pt  mt  
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:
s t
  


 Et (ms )
pt 

1   s t  1   

(4)
The Cagan Model in Continuous Time
Sometimes is easier to work in continuous time. In this case, the
Cagan nonstochastic demand (2) becomes:
mt  pt  p t
using differenti al equations methods we get that :
pt 
1

exp[ ( s  t ) /  ]m ds  b


s
0
exp( t /  )
t
where the no bubble assumption implies b0  0
Seignorage
Definition: represents the real revenues a government acquires
by using newly issued money to buy goods and nonmoney
assets:
M t  M t 1
Seignorage 
Pt
Most hyperinflations stem from the government’s need for
seignorage revenues. What is the seignorage-revenuemaximizing rate of inflation? Rewriting seignorage as:
Seignorage 
M t  M t 1 M t
Mt
Pt
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
Finding the seignorage-revenue-maximizing rate of inflation is
easy if we look only at constant rates of money growth:
1  
Mt
P
 t
M t 1 Pt 1
Now, exponentiating Cagan’s perfect foresight demand, we get:
M t  Pt 1 

 
Pt  Pt 

Substituting these in the second seignorage equation:
Seignorage 

1 
(1   )    (1   )  1
Seignorage
Thus, the FOC with respect to  is :
(1   )
 1
  (  1)(1   )
  2
0  
max

1

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
A variant of Cagan’s model: a SOE with exogenous real output
and money demand given by:
mt  pt  i t 1  yt
(1),
where i  log( 1  i) with i the nominal interest rate, p is the log of
price and y is the log of real output.
Let  t be the nominal exchange rate (foreign in terms of home),
and P * denote the world foreign-currency price of the consumption
basket with home-currency price P .
Then PPP  Pt   t Pt * or in logs pt  et  pt*
(2)
  t 1 

and UIP  1  it 1  (1  i ) Et 
 t 
(3)
*
t 1
A Simple Monetary Model of Exchange
Rates
An approximation in logs of UIP is:
i t 1  i*t 1  Et et 1  et
(4)
Substituting the log PPP and (4) in eq. (1) gives:
(mt  yt  i*t 1  pt* )  et   ( Et et 1  et )
(5)
and the solution t o the exchange rate is :
s t
  
1
*
*


et 
E
(
m


y


i

p

t
s
s
s 1
s)


1   s t  1   

(6)
A Simple Monetary Model of Exchange
Rates
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 price
In the sense that it depends on expectations of future variables,
just like other assets.
Monetary Policy to Fix the Exchange
Rate
Consider a special case of the SOE Cagan exchange rate model:
mt  et   ( Et et 1  et )
(1)
Suppose the government fixes the nominal exchange rate
permanently at e , then substituting in (1) we get that:
mt  m  e
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 countryspecific good, using no labor or capital input => fruit trees.
•Evolution of output:
xt  g t xt 1 and yt  g t* yt 1 , where g t and g t* are random and its
stochastic processes are known by agents.
•Each firm issues a perfectly divisible share of common stock
which is traded in a competitive market.
The Barter Economy
•Firms pay out all of their output as dividends to shareholders,
which are the sole source of support for individuals.
•We will let xt be the numeraire good.
•Under this framework, the wealth a domestic agent brings to
period t is:
Wt  wxt1 ( xt  et )  wyt1 (qt y y  et* )
•And the agent has to allocate this wealth between consumption
and new share purchases:
Wt  et wxt  et*wyt  cxt  qt c yt
The Barter Economy
Equating the last two equations we get the budget constraint for
domestics:
cxt  qt c yt  et wxt  et*wyt  wxt1 ( xt  et )  wyt1 (qt y y  et* ) (1)
In this way, domestic agents have to choose sequences
c
xt  j , c yt  j , wxt  j , wyt  j


j 0
to solve:
 j

Max Et   u (cxt j , c yt j )
 j 0

st . (1)
The Barter Economy
Thus, the domestic Euler equations are:
c yt :
qt u1 (c xt , c yt )  u2 (c xt , c y y )
(2)
wxt : et u1 (c xt , c y y )  Et [u1 (c xt 1 , c yt 1 )( xt 1  et 1 )]
(3)
wyt : et*u1 (c xt , c y y )  Et [u1 (c xt 1 , c yt 1 )( qt 1 yt 1  et*1 )]
(4)



If we put an * over the variables cx , c y , wx , wy j 0 in the
domestic agent problem and in the domestic Euler equations, we
get the foreign agent problem and foreign Euler equations.
t j
t j
t j
t j
The Barter Economy
We need to add four more constraints to clear the markets:
wxt  w*xt  1
(5)
wyt  w*yt  1
(6)
c xt  c*xt  xt
(7)
c yt  c*yt  yt
(8)
The Barter Economy
Given that we have complete and competitive markets, we can
apply the welfare theorem and solve the social planner problem:
1
1
Max u (cxt , c yt )  u (c*xt , c*yt )
2
2
st . (7), (8)
and the solution will be an competitive equilibrium:
1
1

*
*
u1 (cxt , c yt )  u1 (cxt , c yt ) 
xt
yt

2
2
*
*
FOC :
 c yt  c yt 
  cxt  cxt 
1
1
2
2
*
* 
u2 (cxt , c yt )  u2 (cxt , c yt )
2
2

The Barter Economy
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,
1
wxt  w  wyt  w 
2
*
xt
*
yt
•Prices: to get an explicit solution we need to give a function
form to the utility, let
Ct  cxt c1yt 
Ct1
and u (cxt , c yt ) 
1 
The Barter Economy
Under all what we have seen and assumed, the Euler equations
imply:
1   xt
qt 
 yt
1

et
 Ct 1   et 1 
 1 

 Et 
xt
 Ct   xt 1 
1
*






e
C
e
 t 1
 1  t 1 
 Et 
qt yt
 Ct   qt yt 1 
*
t
B. The One-Money Monetary Economy
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: M t  t M t 1 where t
is known. The economy wide increment is distributed evenly
across H and F individuals as lump sum transfers. Each
receive:
M t
 (t  1)( M t 1 / 2)
2
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: 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.
The One-Money Monetary Economy
Given these assumptions, domestic agent’s period t wealth is:
Pt 1 ( wxt 1 xt 1  wyt 1 qt 1 yt 1 )
M t
Wt 
 wxt 1 et  wyt 1 e 


P
2 Pt

t  ex  dividend sharevalue
dividends
*
t
money transfer
And in the security market, the agent allocate his wealth between:
Wt 
mt
 wxt et  wyt et*
Pt
Assuming a positive nominal interest rate, the cash in advance
constraint binds:
mt  Pt (cxt  qt c yt )
The One-Money Monetary Economy
Using the last three equations, we get that the domestic agent
problem is:
 j

Max Et   u (c xt  j , c yt  j )
 j 0

Pt 1
M t
st .
( wxt 1 xt 1  wyt 1 qt 1 yt 1 ) 
 wxt 1 et  wyt 1 et*
Pt
2 Pt
 c xt  qt c y t  wx et  wy et*
The One-Money Monetary Economy
The domestic agent problem implies the following Euler
equations:
c yt :
qt u1 (c xt , c yt )  u2 (cxt , c y y )
Pt
wxt : et u1 (cxt , c y y )  Et [u1 (cxt 1 , c yt 1 )(
xt 1  et 1 )]
Pt 1
(1)
(3)
Pt
wyt : e u (cxt , c y y )  Et [u1 (cxt 1 , c yt 1 )(
qt 1 yt 1  et*1 )] (4)
Pt 1
*
t 1
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).
The One-Money Monetary Economy
To clear the markets we need to add the constraints:
wxt  w*xt  1

wyt  w*yt  1
cxt  c*xt  xt

c yt  c*yt  yt
and
M t  mt  mt*
The equilibrium of the barter economy is still the perfect riskpooling equilibrium:
1
xt
yt
*
*
*
*
wxt  wxt  wyt  wyt 
and
cxt  cxt 
 c yt  c yt 
2
2
2
The only thing that has changed is the equity pricing formulae,
which now include the “inflation premium”.
The One-Money Monetary Economy
Using the same constant relative risk aversion utility function we
used in the barter economy, we have that:
1   xt
qt 
 yt
Pt
M t xt 1

Pt 1 M t 1 xt
1

et
et 1 
 Ct 1   M t
 

  Et 

xt
 Ct   M t 1 xt 1 
1
*






e
C
M
e
 t 1
  t  t 1 
  Et 
qt yt
 Ct   M t 1 qt yt 1 
*
t
The One-Money Monetary Economy,
pricing other assets
At equilibrium, the price b of a nominal bond that pays 1 dollar at
the end of the period must satisfy:
u1 (cxt , c yt )bt / Pt  Et (u1 (cxt 1 , c yt 1 ) / Pt 1 )




utilitycost of buyingthe bond
If
it
marginal utilityof payoff
is the nominal interest rate, then
bt  (1  it ) 1
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.
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:
dollar : M t  t M t 1
euro : N t  *t N t _1
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
Then, we have that the home agent current-period wealth is:
 M t1 M t  N t 1 St N t
Pt 1
St Pt *1
Wt 
wxt 1 xt 1 
wyt 1 yt 1 

P
P
Pt
Pt
t
t
 




dividends
money transfers
 wxt 1 et  wyt 1 et*   M t 1 rt   N t 1 rt*

market value of sec urities
And this wealth will be allocated according to:
mt nt St
Wt  wxt et  wyt e  M t rt  N t rt 

Pt
Pt
*
t
*
The Two-Money Monetary Economy
As before, the BC implied by the last two equations and the cash in advance
constraint s mt  Pt cxt and nt  Pt*c yt imply the following Euler eqs :
c yt :
S t Pt *
u1 (c xt , c yt )  u 2 (c xt , c y y )
Pt
wxt :
et u1 (c xt , c y y )   Et [u1 (c xt 1 , c yt 1 )(
w yt :
S t 1 Pt *
e u (c xt , c y y )   Et [u1 (c xt 1 , c yt 1 )(
yt 1  et*1 )]
Pt 1
M
t
*
t 1
M t
: rt u1 (c xt , c yt )   Et [u1 (c xt 1 , c yt 1 )(
 rt 1 )]
Pt 1
N :
t
Pt
xt 1  et 1 )]
Pt 1
N t 1S t 1
rt u1 (c xt , c y y )   Et [u1 (c xt 1 , c yt 1 )(
 rt*1 )]
Pt 1
*
And again the foreign agent have a symmetric set of Euler eqs.
The Two-Money Monetary Economy
Together with the Euler eqs. We have the clear market conditions:
wxt  w*xt  1

wyt  w*yt  1
cxt  c*xt  xt

c yt  c*yt  yt
M t  mt  mt*

N t  nt  nt*
With these eqs. We have the following equilibrium:
wxt  w  wyt  w   M t  
*
xt
and
*
yt
*
Mt
  Nt  
xt
yt
*
cxt  c 
 c yt  c yt 
2
2
*
xt
*
Nt
1

2
The Two-Money Monetary Economy
From the first Euler equation, we get that the nominal exchange
rate is:
u2 (cxt , c yt ) M t yt
St 
u1 (cxt , c yt ) Nt xt
Conclusion: as in the monetary approach, the determinants of the
nominal exchange rate are relative money supply and relative
GDPs. Two major differences are that in the Lucas model:
•S depends on preferences
•S does not depend explicitly on expectations