Cagan and Lucas Models


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Cagan and Lucas Models

• Presented by Carolina Silva
• 01/27/2005

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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.

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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
13000
These huge inflations are not things of the past
for example between April 1984 and July 1985
Bolivias price level rose by 23000
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Cagan Model
Let M denote a countrys money supply and P its
price level Cagans model for the demand of real
money balances M/P is
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.
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Solving the Model
Which are the implication of Cagans demand
function to the relationship between money and
the price level Assuming an exogenous money
supply m in equilibrium
So we have an equation explaining price-level
dynamics in terms of the money supply.
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Solving the Model
First for the nonstochastic perfect foresight
ie by successive
substitution of we get that
Is this a reasonable solution of (2)
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Simple Cases

• Constant money supply

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Simple Cases
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
3. Solution (3) also covers more general money
supply processes.
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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
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The Cagan Model in Continuous Time
Sometimes is easier to work in continuous time.
In this case the Cagan nonstochastic demand (2)
becomes
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Seignorage
Definition represents the real revenues a
government acquires by using newly issued money
to buy goods and nonmoney assets
Most hyperinflations stem from the governments
need for seignorage revenues. What is the
seignorage-revenue-maximizing rate of inflation
Rewriting seignorage as
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.
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Seignorage
Finding the seignorage-revenue-maximizing rate of
inflation is easy if we look only at constant
rates of money growth
Now exponentiating Cagans perfect foresight
demand we get
Substituting these in the second seignorage
equation
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Seignorage

• 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.

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A Simple Monetary Model of Exchange Rates
A variant of Cagans 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 .
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A Simple Monetary Model of Exchange Rates
An approximation in logs of UIP is
Substituting the log PPP and (4) in eq. (1) gives
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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.
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Monetary Policy to Fix the Exchange Rate
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
Thus money supply becomes an endogenous
variable implying that exchange rate targets
implicitly entail decisions about monetary policy.
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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.

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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 Lucass
  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.

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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

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

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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 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

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The Barter Economy
Equating the last two equations we get the budget
constraint for domestics
In this way domestic agents have to choose
sequences
to solve

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The Barter Economy
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.
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The Barter Economy
We need to add four more constraints to clear the
markets
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The Barter Economy
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
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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

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

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The Barter Economy
Under all what we have seen and assumed the
Euler equations imply
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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
• Output levels are revealed.
• Money evolves according to
  is known. The
  economy wide increment is distributed evenly
  across H and F individuals as lump sum transfers.
  Each receive

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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 wont do it if the nominal interest rate is
positive.
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The One-Money Monetary Economy
Given these assumptions domestic agents 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
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The One-Money Monetary Economy
Using the last three equations we get that the
domestic agent problem is
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The One-Money Monetary Economy
The domestic agent problem implies the following
Euler equations
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).
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The One-Money Monetary Economy
To clear the markets we need to add the
constraints
The equilibrium of the barter economy is still
the perfect risk-pooling equilibrium
and
The only thing that has changed is the equity
pricing formulae which now include the
inflation premium.
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The One-Money Monetary Economy
Using the same constant relative risk aversion
utility function we used in the barter economy
we have that
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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
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.
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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 xs dividends are paid in dollars and
ys in euros. Agents can get the foreign currency
during security market trading. Currencies evolve
according to
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.
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The Two-Money Monetary Economy
Then we have that the home agent current-period
wealth is
And this wealth will be allocated according to
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The Two-Money Monetary Economy
And again the foreign agent have a symmetric set
of Euler eqs.
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The Two-Money Monetary Economy
Together with the Euler eqs. We have the clear
market conditions
With these eqs. We have the following equilibrium
and
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The Two-Money Monetary Economy
From the first Euler equation we get that the
nominal exchange rate is

• 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