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Lucas (1972) Imperfect-Information Model
Romer (2012) sections 6.9 and 6.10
Rational expectations
Lucas supply curve with price flexibility
Signal extraction problem
The Lucas critique
Basic idea of the model “when a producer observes a change
in the price of her product, she does not know whether it
reflects in the relative price of the good or a change in the
price level”
Story: Inflation and the price of oil in the 1980.
Individual produces goods with their labour only and sell their
output in competitive markets.
Two types of shocks: 1) random changes in preferences that
change relative price of goods, and 2) money supply
(aggregate demand) disturbances that affect the price level.
6.1 The Case of Perfect Information
Representative producer of a typical good i
and the production function is:
Qi  Li
i i
And individual consumption Ci  PQ
The utility function:
U i  Ci 
1

P
Where P, the price level
is an index number 6.78
Li ,   1,
So increasing marginal disutility of work. In this first simple case,
the aggregate price level is known P. In that case, the individual
maximizes:
PL
1 
i i
Ui 
 Li ,   1,
P 
The individual chooses Li to maximize U given Pi (price taker) and P
First-order condition leads to:
 Pi 
Li   
P
1
 1
Or in log form:
i
1

( pi  p) (6.6)
 1
This is the supply of labour and the supply of good since Qi  Li
Demand
The (log-linear) demand for good i
qi  y  zi   ( pi  p),   0,
(6.76)
y is log real aggregate income
zi is the demand shock to good i (mean zero across goods)
 is the elasticity of demand
y and p are the mean across goods of:
y  qi
6.79
p  pi
6.78
y m p
And the aggregate demand is simply:
Where m is an aggregate demand shock, various interpretations
Equilibrium for the market of good i comes from 6.6 and 6.76 (qi=ℓi)
1
( pi  p)  y  zi   ( pi  p)
 1
Isolating pi give:
 1
pi 
( y  zi )  p,
1    
Taking the average on both side (z drops):
 1
p 
y  p,
1    
And the equilibrium y is zero and p = m
and, not surprisingly, money is neutral
6.2 The case of imperfect information
Producers observe the price of their good but not the price level.
We define the relative price of good i as ri=pi-p:
pi  p  ( pi  p )
pi  p  ri
6.80
The producer would like to plan its production on ri only but,
unfortunately, he does not observe ri. He just observe pi and he
has to guess ri given pi.
As discussed in Romer (pp.270-271), Lucas at this point will make
two simplifying assumptions. The second one will change the
development of macroeconomics. Let’s discuss the first to start
with.
First assumption: certainty-equivalent: the producer find the
expectation of ri given pi and then produce as much as he can.
Consequently, the level of uncertainty (risk) does not change the
producer behaviour. In this framework, 6.6 becomes:
i
1

E ( ri pi )
 1
6.81
This is not the same as maximizing utility (problem 6.1). For example,
the E(U(W)) is not the same that U(E(W)) where W is a gamble:
W has a probability of 0.5 to get 100 and 0.5 to get 0, the E(W)=50
The U(E(W))=E(U(W)) only if the Utility function is linear in W (risk
neutral). If U is concave in W, by Jensen inequality we know that
U(E(W))>E(U(W)). The certainty equivalent solution (CE) is the
maximum amount a person is willing to pay to participate to the gamble.
So CE is the value of E(U(W)). In the actual case, the U function is
concave (log-linear) on Pi/P) (to follow)
i
 1   Pi
 1 

 Pi


E ln
Pi   
ln E 
Pi 


P 
 P 
   1 
   1
The first ℓ term is the CE and the second ℓ is the supply of labour that
maximizes utility.
Here, CE is a simplifying assumption made by Lucas.
1
E ( ri pi )
6.81
i 
1
The second assumption is that the expectation of ri given pi is
rational à la Muth’s (1960, 1961).
“The concept of rational expectations asserts that outcomes do not
differ systematically (i.e., regularly or predictably) from what
people expected them to be. The concept is motivated by the same
thinking that led Abraham Lincolnr to assert, "You can fool some of the
people all of the time, and all of the people some of the time, but you
cannot fool all of the people all of the time." From the viewpoint of the
rational expectations doctrine, Lincoln's statement gets things right. It
does not deny that people often make forecasting errors, but it does
suggest that errors will not persistently occur on one side or the other.”
Thomas Sargent, The Concise Encyclopedia of Economics,
http://www.econlib.org/library/Enc/RationalExpectations.html
i
Following Romer (p. 295), To make the computation of E tractable,
Lucas assumes that both shocks m and zi are normally distributed
with mean E(m) and zero respectively and variance Vm and Vz.
Furthermore, the zi are independent of m.
Consequently, p and ri are normal and independent. pi which is the
sum of both is also normally distributed, its mean is the sum
of their mean and its variance is also the sum of their variance since
they are independent. In this case, using statistics, Lucas get:
Vr
E  ri pi  
 pi  E ( p)
Vr  Vp
Explanation on following slide
6.82
Vr
E  ri pi  
 pi  E ( p)
Vr  Vp
6.82
E(p) is the mean of pi
-If pi exceeds its mean, the expectation of ri exceeds also its mean (0)
-If pi equals its mean, the expectation of ri is zero
-The fraction of the departure of pi from of its mean that is estimated
to be due to a change in relative price ri equals Vr/(Vr+Vp)
Example: 1) Vp = 0, 2) Vr=Vp, Vr=0
This is the signal extraction problem: the observed variable pi
equals the signal ri plus noise p. The agent’s problem is to extract
the signal from noise.
Combining 6.81 with 6.82 we get:
i
1
Vr

 pi  E ( p)
  1 Vr  Vp
6.83
 b  pi  E ( p )
And taking the average across producers we get:
y  b( p  E  p)
6.84
Which is the Lucas Supply curve (explanation, the same as the
expectations-augmented Phillips curve).
Equilibrium
Combining 6.84 with y = m-p we get:
1
b
p
m
E ( p)
1 b
1 b
b
b
y
m
E ( p ).
1 b
1 b
6.85
6.86
We take E( ) on both side of 6.85 with E(E( ))=E( ) and we get:
E ( p )  E ( m)
6.88
Using the fact that m = E(m) + (m-E(m)) we rewrite 6.85 and 6.86:
1
p  E (m) 
( m  E (m))
1 b
b
y
(m  E (m))
1 b
6.89
6.90
The component of AD which is anticipated (observed) affects
only prices, the non-anticipated component (un-observed)
has real effect on output.
Part of an increase in m which is not anticipated is associated
to a change in relative prices and has a real effect
Monetary policy affects output only when it is not anticipated
Discussion.
The Phillips curve and the Lucas Critique
From equations 6.93 to 6.96, we see that the Lucas supply curve
implies a positive relationship between the inflation rate and
output growth.
But monetary policy cannot exploit the tradeoff. The statistical
relationship between output and inflation will change Policy Makers
tend to exploit it. This a general proposition known as the
Lucas Critique (discussion)
Empirical Application:
International Evidence on the
Output-Inflation Tradeoff
• According to Lucas model, the slope of the AS curve is
determined by the (relative) variability of AD. If the variance
of AD shocks is larger, agents attribute most of the changes in
prices to changes in price level. In this case, b in 6.84 is low,
and in the P – y plan, the AS curve is more classical, steaper.
• Lucas (1973) test this prediction. He assumes that shocks to
AD are measured changes in nominal GDP. For this to be
reasonable, 1) AD must be unit-elastic such as an increase in
AS generate a decrease in P and an increase in y but the
product between the two does not change. 2) Changes to
nominal GDP might not be predictable. (Remember it is only
the non-predictable disturbances that affect real output).
Lucas estimate the following equation for various
countries
Where y is log of real GDP and x is log of nominal gdp.
He then regresses the τ on the SD of nominal GDP:
Ball, Mankiw, and Romer (1988)
• BMR (1988) interpret the results of Lucas (1973) at the light
of the menu-cost or other Keynesian friction. When average
inflation is higher (the π bar), firms revised their prices more
often and prices are less sticky. In this case, the AS curve is
steeper. They regress:
Countries with larger mean inflation tend to have more variable
AD. So we have two theories. To test the two theories against
each other, let do a horse race:
Menu-cost win the race