Macroeconomics
Luigi Iovino
Sep. 17, 2013
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Rules of the game
2 sessions every other week:
Monday 15-19 (lectures)
Tuesday 13-17 (lectures)
Some sessions will cover exercises with the TA
Assistant: Veronica Preotu (Extranef 122)
O¢ ce hour: Tuesday 17-18 (Veronica)
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Rules of the game
Evaluation:
1 mid-term: 40%
1 …nal exam (exam session): 60%
Final grade = max {40%.mid-term + 60%.…nal exam, …nal exam}
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Short Review
What you learnt in Macro 101:
The IS-LM model (Hicks, 1937), an interpretation of Keynes
(1936), gives the mechanics of the economy with …xed prices
Price adjustment is given by the Phillips curve
Large-scale macro macroeconometric models used to forecast
economic time series
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Assumptions?
1
3 markets
2
Prices are rigid (short-run)
3
Excess supply in the good market
4
Behavioral assumptions (good demand, demand for money...) based
on empirical relationships
Not derived from rigorous microeconomic foundations
5
Static relationships
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IS: Equilibrium in the good market.
Y
|{z}
Supply of goods
= C (Y
|
T ) + I (i ) + G
{z
}
Aggregate demand for goods
LM: Equilibrium in the market for money.
M
|{z}
=
Supply of money
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P
|
L(i, Y )
{z }
Demand for money
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Advantages ! simplicity
Limits !
1 The simple behavioral equations neglect important aspects
(ex., consumption depends only on current income)
Most importantly:
2
No Dynamic aspects
3
How do people react to changes in policies?
Behavior depends on policy
) We cannot rely on the behavioral equations to make
policy recommendations (Lucas critique)
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1
First revolution: Rational Expectations
2
Second revolution: Dynamics
General change in perspective: individual behaviors are
“micro-founded"
) derived from the utility maximization by rational agents.
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Topics
1
Rational expectations
2
Consumption
3
Investment
4
Money
5
Nominal rigidities
6
Fiscal policy
Also, you will learn to do:
Solve a simple model with rational expectations
Dynamic optimization (Lagrangian, Bellman Equation)
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Topic 1: Rational expectations
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Phillips curve before 1970
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Phillips curve after 1970
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In the 1970s, expectations are absent from models or taken as
given
Problems start to emerge when the Phillips curve was criticized
on two grounds
1
Empirically, the Phillips curve seemed not to hold any more
2
Theoretically, the Phillips curve appeared to be shaky, precisely
because of how expectations were incorporated
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Phillips curve is about In‡ation/Unemployment trade-o¤:
=) Price in‡ation ! real wages are lower ! …rms produce more and hire
more
Policymakers (eg, the ECB) have a menu of choices between In‡ation
and Unemployment ! need to pick their favourite point
This is based on a very simple “reduced form” economy
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Friedman, Phelps, Lucas, Sargent, and Wallace modi…ed the PC
They consider an “expectation-augmented” Phillips curve:
π t = µ + π et
αut
They also provide a theory of the formation of expectations:
Adaptive expectations (Friedman, Phelps, 1968)
Rational expectations (Lucas, 1976)
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Important implications of new PC
In the long-run expectations are right: “natural rate of
unemployment”
The long-run level of unemployment depends only on the structural
characteristics of the economy
There is no long-run output/in‡ation trade-o¤
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Let’s derive the PC more carefully!
Wage-setting equation:
w = p e (1
αu )
w is the nominal wage, p e is the expected price and u is unemployment.
Interpretation?
Price-setting equation:
p = (1 + µ )w
µ is the mark-up set by …rms over their cost (w )
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This yields:
p = (1 + µ )p e (1
αu )
We use this condition to infer:
1 + π t = (1 + µ)(1 + π et )(1
αu t )
where π t is the in‡ation rate at time t
Expand and omit negligible products to obtain:
π t = µ + π et
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αut
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There is a negative relation between in‡ation and unemployment
! A “menu choice” of policy. Intuition?
But it depends on expected in‡ation. Intuition?
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Unemployment depends on the error in expectations:
ut =
µ
(π t
α
π et )
The natural rate of unemployment u is the rate such that agents
make no error (π t π et = 0):
u =
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µ
α
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Given expectations, there is indeed a negative relationship
between in‡ation and unemployment:
π t = µ + π̄
αu t
This makes sense of the empirical PC (before the 1970s)
When in‡ation expectations are stable ! high in‡ation translates
into low unemployment (agents expect high real wages)
But π et cannot be assumed constant when there are changes in
current and past in‡ation!
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Thus, we cannot take π et to be constant ! need to model expectations
First, assume agents use their observations of past in‡ation: adaptive
expectations (Milton Friedman, Edmund Phelps, 1968)
I Agents observe past in‡ation π t 1 and adjust their expectations in
accordance with their previous errors:
π et = π et
1
+ λ(π t
|
1
π et
Past error
0 < λ < 1 is the adaptation speed
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{z
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}1
)
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How do expectations evolve in time?
π eT +1 = π eT + λ(π T
π eT )
= (1
= (1
λ)π eT + λπ T
= (1
λ)2 π eT + λ[π T +1 + (1
λ)π T ]
π eT +3 = (1
..
.
λ)3 π eT + λ[π T +2 + (1
λ ) π T +1 + ( 1
π eT +k = (1
λ)k π eT + λ
π eT +2
λ)[(1
λ)π eT + λπ T ] + λπ T +1
k 1
∑ (1
λ )k
i 1
λ )2 π T ]
π T +i
i =0
The in‡uence of the initial expectation π eT vanishes over time!
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Example
Suppose in‡ation at time T suddenly rises to 3%:
t < T : π t = π et = 0%
t
T : Actual in‡ation raises to π t = 3%
Before date T
Phillips curve:
πt = µ + 0
αut
The equilibrium unemployment rate is equal to its natural rate:
0 = µ+0
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αut ) ut = u =
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µ
α
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Example
After date T
We …rst need to compute expectations:
π eT +k = (1
λ)k π eT + λ
k 1
∑ (1
λ )k
i 1
π T +i
i =0
= [1 (1
e
π T = 0%
e
π T +1 = λ3%
λ )k ] π T
π eT +2 = [1
..
.
λ)2 ]3%
(1
Agents learn over time and expectations π eT +k converge towards 3%
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Thus, the Phillips curve evolves over time:
π T +k = µ + [ 1
(1
λ)k ]3%
αuT +k
This explains the instability of the Phillips curve!
What about unemployment rate?
(1 λ)k ]3% αuT +k
µ (1 λ)k 3%
=
α
3% = µ + [1
) uT +k
Thus, it is temporarily low and then converges to u
There is a short-run output-in‡ation trade-o¤ but (not a
long-run one)!
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So far, adaptive expectations
But there is something anappealing about this:
Agents make smaller and smaller errors. In the example,
π T +k π eT +k = (1 λ)k π T
But the errors are systematic: they depend on π T , which is observed
by agents. And yet, agents still make errors
When agents form their expectations, they should not only use all
available information but should also use it optimally
In particular, they should respond endogenously to policy and to the
structure of the economy
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Rational Expectations
Rational Expectations (Introduced by Muth (1961), used by Lucas
(1972) and Sargent and Wallace (1976))
1
Agents use all available information in order to make their best guess
of the future:
π et = E (π t jIt )
It is the information available at date t. We denote E (.) = E (.jIt )
Therefore, predictions of future economic variables are not
systematically wrong (zero error in expectation)
E (π t
2
π et ) = E (π t )
E (E (π t )) = E (π t )
E (π t ) = 0
Agents know the model of the economy
Note: rational expectations do not imply that agents are perfectly informed
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Rational Expectations
In the 1970s, there were two main reconstructive e¤orts in macro:
Imperfect information models (Lucas “Island” model, 1972)
Models with nominal rigidities (Fisher model, 1977)
Common feature of these models: expectations are rational
Di¤erences: they assume di¤erent types of frictions
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In the Lucas “Island” model, prices are ‡exible but agents
(entrepreneurs) have only partial information about monetary
policy
In the Fisher model, agents are informed of current monetary
policy but wages are predetermined
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Lucas island model
Agents do not know what happens to the overall price level because
they only observe a partial signal. Agents live in isolated islands and
have information only about the price of the good on their island
Thus, they cannot form accurate expectations as regards overall
in‡ation, even though they are rational
There are n islands indexed by i, each subject to an idiosyncratic
shock zti (a demand shock)
There is also an aggregate (common) shock et (a monetary shock)
Both shocks have zero mean and variances σ2z and σ2e , respectively.
They are independent
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Lucas island model
The key is that nobody can distinguish between zti and et
De…ne the price of the good being produced and sold on island i as
pti = pt + zti
where pt is the aggregate price level
The price level on island i deviates from the aggregate, but each
agent only observe pti , NOT its separate components
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Lucas island model
Aggregate Price is endogenous
Aggregate shocks to prices ! aggregate price di¤ers from average
E (p )
Remember
pti = pt + zti
Agents do not observe separately pti and pt , only pti , the price at
home!
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Lucas island model
When I see the price of the good I produce, pti , I do not know whether
changes come from the overall price level or from the price level on my
island zti
Why should that matter?
Producers care about the relative price of their goods:
If my good becomes relatively more expensive (a shock in zti ) ! I
produce more of it
But if my good is becoming more expensive along with everything
else (a shock in pt ) ! I leave production unchanged
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Lucas island model
Production on island i:
yti = ȳ + a[pti
E (pt jpti )]
Producers increase output only when they expect relative price change
=) All we need is to solve for is E (pt jpti )
In general, this problem can be complex...
But when zti and pt are normal, it can be solved easily
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Lucas island model
Conjecture pt
N E (p ) , σ2p (we solve for E (p ) and σ2p later)
When random variables are Normal, the relationship between E (pt jpti ) and
pti is linear:
E (pt jpti ) = α + βpti
More precisely, we have that:
E (pt jpti ) = E (pt ) +
Cov (pt , pti ) i
[ pt
V (pti )
E (pti )]
σ2p
[p i
σ2p + σ2z t
E (p )]
) E (pt jpti ) = E (p ) +
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Lucas island model
Back to the production function:
yti = ȳ + a[pti
E (p )
= ȳ + a
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σ2p
pi
σ2p + σ2z t
σ2z
pi
σ2p + σ2z t
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E (p ) ]
E (p )
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Lucas island model
Note the following:
When there are no island speci…c shock, σ2z = 0, producers simply do
not alter their production plans, leaving them at ȳ . Why?
Observed deviations between the price on their island and what they
expect it to be (E (p )) comes from an aggregate shock!
In other words, they know all prices on all islands have shifted
identically ! No reason to produce more
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Lucas island model
Now sum across islands:
yt =
1
N
N
i =1
= ȳ + a
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σ2
1
N
∑ yti = ȳ + N a σ2 +z σ2 ∑
p
σ2z
( pt
σ2p + σ2z
Macroeconomics
pti
E (p )
z i =1
E (p ))
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Lucas island model
Across the MACROeconomy, production only responds to
unexpected changes to aggregate prices
And the sensitivity of this response increases with the variance of
idiosyncratic shocks
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Lucas island model
Thus, changes in prices (engineered for instance via monetary policy)
can have an e¤ect on output only if they are unexpected!!
The expectations-augmented Phillips curve told us a similar story:
unemployment responds only to deviation of π et from π t
Now, however, the agents’rational expectations are solved within the
model
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Lucas island model
Let’s now embed this “Lucas” supply curve in general equilibrium
models of the macroeconomy (i.e. with demand and supply)
Aggregate Supply (AS) was just derived:
yt = ȳ + α(pt
Ept )
(AS )
where α = aσ2z /(σ2p + σ2z )
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Lucas island model
Demand arises from an equation of exchange, MV = PY , which in logs
rewrites
yt = mt
pt + v
mt = E ( m ) + e t , e t
(AD )
N 0, σ2e is set by the Central Bank
v is constant (for simplicity, assume v = ȳ )
We want to use this framework to think about the e¤ect of (monetary)
policy on real activity in a rational expectations equilibrium
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Lucas island model
Use (AD) to solve for prices:
pt = m t
yt + v
In equilibrium AD = AS:
ȳ + α (pt
Thus,
pt =
and
yt = ȳ
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E (p )) = mt
pt + v
1
α
E (p ) +
mt
1+α
1+α
α
α
E (p ) +
mt
1+α
1+α
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Lucas island model
Need to solve for E (p ) and σ2p :
1
From pt =
2
We get
α
1 +α E
(p ) +
Iovino ()
2
1
1 +α
=) E (p ) = E (m)
E (p ) =
pt
Thus, σ2p =
1
1 + α mt
1
( mt
1+α
E (m ))
σ2m
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Lucas island model
Substitute this back into supply:
yt = ȳ +
α
(mt
1+α
Emt )
(AS )
This is a so-called reduced form, i.e. it expresses endogenous variables
(output) in function of the model parameters (ȳ , α) and policy choices
(mt )
Economically, this means that monetary policy can a¤ect economic
activity only if it is not expected (Sargent and Wallace, 1976)
Intuition of the mechanism?
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Lucas island model
Note further that
aσ2z
α
= 2
1+α
σp + (1 + a)σ2z
The response of production to surprises in money is increasing in the
relative magnitude of σ2z , i.e. the importance of idiosyncratic shocks
Agents living in a country with a history of high aggregate in‡ation will
have relatively large values to σ2p , which means policy is increasingly
ine¤ective there
The more policy tries to use monetary policy to boost output, the harder it
becomes as the volatility of σ2p increases
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Fisher model
Other authors tried to make sense of the e¤ect of monetary policy on
output by focusing on imperfect wage and price setting, instead of
imperfect information (Phelps and Taylor (1977), Fisher (1977)
The idea is that nominal prices/wages are preset at the beginning of
the period on the basis of available information
This approach is not fundamentally di¤erent from the imperfect
information approach: some information is unavailable when
prices/wages are set. The di¤erence is that the mismatch comes in
one case from “sticky information”, in the other from “sticky
prices/wages”
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Fisher model
Fisher introduced the following model in logs:
yt = mt
yt = ȳ
pt + v
α(wt
(AD )
pt )
(AS )
wt = pte
(WS )
(AD) is the same as before
(AS) depends now on the actual real wage w p. Firms observe the
real wage. As labor comes as a cost for them, they produce more if
the real wage is low
The last equation (WS) is the wage-setting equation. Wages are
preset at the beginning of period so as to achieve yt = ȳ in
expectations
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Fisher model
We are looking for the rational-expectation solution to this model,
that is, with pte = Et 1 (pt )
We replace the wage in (AS) by its predetermined value:
yt = ȳ + α[pt
Et
1 (pt )]
(AS )
This is reminiscent of the Lucas supply curve
We have the same model as before:
yt = mt
pt + v
yt = ȳ + α[pt
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(AD )
Et
1 (pt )]
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(AS )
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Fisher model
We thus obtain the same reduced forms:
pt
Et
1 pt
=
1
( mt
1+α
Et
1 mt )
α
(mt Et 1 mt )
1+α
Similarly, money shocks a¤ect output only to the extent that they are
unanticipated
yt = ȳ +
But the channel is Keynesian: since prices are ‡exible but nominal
wages are preset, money shocks increase prices, decrease real wages
and increase output
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Fisher model
Finally, go back to the initial expectations-augmented Phillips curve to
illustrate the importance of rational expectations for monetary policy
Consider the expectations augmented Phillips curve:
yt = ȳ + b (π t
π et )
Output is above its average, long run level whenever in‡ation happens to
be above its expected level
Once again, only surprises in in‡ation matter for production
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Fisher model
Now suppose the Central Bank has a loss function that re‡ects both a
concern for output AND for in‡ation. For instance:
1
L = (yt
2
1
y )2 + θ ( π t
2
π )2
where y denote the TARGET level of output the Central Bank would
like to reach, with y > y . In other words, the Central Bank would
like to push economic activity ABOVE its long run level ȳ
Similarly π denotes the target in‡ation level - e.g. 2 or 3%, and θ
captures the relative importance of in‡ation vs. output
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Fisher model
Suppose now the Central Bank makes an announcement that it will
achieve in‡ation at π . Crucially, suppose the announcement is
believed, so that π et = π
Then the Central Bank is faced with the following minimization
problem:
1
Min (ȳ + b (π t
2
π )
1
y )2 + θ ( π t
2
π )2
Note that we have made use of the fact that, the moment the Central
Bank’s announcement is believed, a Phillips Curve arises at the
corresponding level of π et . Here at π
The Central Bank will choose ACTUAL in‡ation π t to solve this
problem
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Fisher model
Solve for π t : π t = π +
b
(y
b 2 +θ
ȳ )
Since the Central Bank wants to push output yt above its long run
level ȳ , this means it will want to in‡ate the economy ABOVE its
initial announcement of π . Why?
So the initial announcement of π cannot possibly be credible.
RATIONAL agents anticipate the Central Bank will renege on its
initial commitment, and do not believe it when it claims it will
implement π t = π
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Fisher model
Clearly, the only credible in‡ation veri…es π t = π et
Substitute that condition to get:
b
π t = π + (y
θ
ȳ )
In‡ation has to be above the very target the Central Bank has!
That so-called "in‡ation bias" increases with b, but decreases in θ.
Why?
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