cagan and lucas models presented by carolina silva 01/27/2005
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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
A Simple Monetary Model of Exchange Rates
(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
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 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.
The One-Money Monetary Economy
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
The One-Money Monetary Economy
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 One-Money Monetary Economy
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).
The One-Money Monetary Economy
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”.
The One-Money Monetary Economy
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 **
The Two-Money Monetary Economy
: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.
The Two-Money Monetary Economy
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
The Two-Money Monetary Economy
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
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