macroeconomics - hec unil excess supply in the good market ... topic 1: rational expectations ... (a...

Macroeconomics

Luigi Iovino

Sep. 17, 2013

Iovino () Macroeconomics Sep. 17, 2013 1 / 56

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)

Offi ce hour: Tuesday 17-18 (Veronica)


Rules of the game

Evaluation:

1 mid-term: 40%

1 final exam (exam session): 60%

Final grade = max {40%.mid-term + 60%.final exam, final exam}


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 fixed prices

Price adjustment is given by the Phillips curve

Large-scale macro macroeconometric models used to forecasteconomic time series


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...) basedon empirical relationships

Not derived from rigorous microeconomic foundations

5 Static relationships


IS: Equilibrium in the good market.

Y︸︷︷︸Supply of goods

= C(Y − T ) + I (i) + G︸︷︷︸Aggregate demand for goods

LM: Equilibrium in the market for money.

M︸︷︷︸Supply of money

= P ∗ L(i ,Y )︸︷︷︸Demand for money


Advantages → simplicity

Limits →1 The simple behavioral equations neglect important aspects(ex., consumption depends only on current income)

Most importantly:2 No Dynamic aspects3 How do people react to changes in policies?

Behavior depends on policy

⇒ We cannot rely on the behavioral equations to makepolicy recommendations (Lucas critique)


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.


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)


Topic 1: Rational expectations


Phillips curve before 1970


Phillips curve after 1970


In the 1970s, expectations are absent from models or taken asgiven

Problems start to emerge when the Phillips curve was criticizedon two grounds

1 Empirically, the Phillips curve seemed not to hold any more

2 Theoretically, the Phillips curve appeared to be shaky, preciselybecause of how expectations were incorporated


Phillips curve is about Inflation/Unemployment trade-off:

=⇒ Price inflation → real wages are lower → firms produce more and hiremore

Policymakers (eg, the ECB) have a menu of choices between Inflationand Unemployment → need to pick their favourite point

This is based on a very simple “reduced form”economy


Friedman, Phelps, Lucas, Sargent, and Wallace modified the PC

They consider an “expectation-augmented”Phillips curve:

πt = µ+ πet − αut

They also provide a theory of the formation of expectations:

Adaptive expectations (Friedman, Phelps, 1968)

Rational expectations (Lucas, 1976)


Important implications of new PC

In the long-run expectations are right: “natural rate ofunemployment”

The long-run level of unemployment depends only on the structuralcharacteristics of the economy

There is no long-run output/inflation trade-off


Let’s derive the PC more carefully!

Wage-setting equation:

w = pe (1− αu)

w is the nominal wage, pe is the expected price and u is unemployment.Interpretation?

Price-setting equation:p = (1+ µ)w

µ is the mark-up set by firms over their cost (w)


This yields:p = (1+ µ)pe (1− αu)

We use this condition to infer:

1+ πt = (1+ µ)(1+ πet )(1− αut )

where πt is the inflation rate at time t

Expand and omit negligible products to obtain:

πt = µ+ πet − αut


There is a negative relation between inflation and unemployment

→ A “menu choice”of policy. Intuition?

But it depends on expected inflation. Intuition?


Unemployment depends on the error in expectations:

ut =µ− (πt − πet )

α

The natural rate of unemployment u∗ is the rate such that agentsmake no error (πt − πet = 0):

u∗ =µ

α


Given expectations, there is indeed a negative relationshipbetween inflation and unemployment:

πt = µ+ π − αut

This makes sense of the empirical PC (before the 1970s)

When inflation expectations are stable → high inflation translatesinto low unemployment (agents expect high real wages)

But πet cannot be assumed constant when there are changes incurrent and past inflation!


Thus, we cannot take πet to be constant → need to model expectations

First, assume agents use their observations of past inflation: adaptiveexpectations (Milton Friedman, Edmund Phelps, 1968)

I Agents observe past inflation πt−1 and adjust their expectations inaccordance with their previous errors:

πet = πet−1 + λ(πt−1 − πet−1︸︷︷︸Past error

)

0 < λ < 1 is the adaptation speed


How do expectations evolve in time?

πeT+1 = πeT + λ(πT − πeT )

= (1− λ)πeT + λπT

πeT+2 = (1− λ)[(1− λ)πeT + λπT ] + λπT+1

= (1− λ)2πeT + λ[πT+1 + (1− λ)πT ]

πeT+3 = (1− λ)3πeT + λ[πT+2 + (1− λ)πT+1 + (1− λ)2πT ]

...

πeT+k = (1− λ)kπeT + λk−1∑i=0(1− λ)k−i−1πT+i

The influence of the initial expectation πeT vanishes over time!


Example

Suppose inflation at time T suddenly rises to 3%:

t < T : πt = πet = 0%

t ≥ T : Actual inflation raises to πt = 3%

Before date T

Phillips curve:πt = µ+ 0− αut

The equilibrium unemployment rate is equal to its natural rate:

0 = µ+ 0− αut ⇒ ut = u∗ =µ

α


Example

After date T

We first need to compute expectations:

πeT+k = (1− λ)kπeT + λk−1∑i=0(1− λ)k−i−1πT+i

= [1− (1− λ)k ]πT

πeT = 0%

πeT+1 = λ3%

πeT+2 = [1− (1− λ)2]3%...

Agents learn over time and expectations πeT+k converge towards 3%


Thus, the Phillips curve evolves over time:

πT+k = µ+ [1− (1− λ)k ]3%− αuT+k

This explains the instability of the Phillips curve!

What about unemployment rate?

3% = µ+ [1− (1− λ)k ]3%− αuT+k

⇒ uT+k =µ− (1− λ)k3%

α

Thus, it is temporarily low and then converges to u∗

There is a short-run output-inflation trade-off but (not along-run one)!


So far, adaptive expectations

But there is something anappealing about this:

Agents make smaller and smaller errors. In the example,πT+k − πeT+k = (1− λ)kπT

But the errors are systematic: they depend on πT , which is observedby agents. And yet, agents still make errors

When agents form their expectations, they should not only use allavailable information but should also use it optimally

In particular, they should respond endogenously to policy and to thestructure of the economy


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 guessof the future:

πet = E (πt |It )It is the information available at date t. We denote E (.) = E (.|It )Therefore, predictions of future economic variables are notsystematically wrong (zero error in expectation)

E (πt − πet ) = E (πt )− E (E (πt )) = E (πt )− E (πt ) = 0

2 Agents know the model of the economy

Note: rational expectations do not imply that agents are perfectly informed


Rational Expectations

In the 1970s, there were two main reconstructive efforts in macro:

Imperfect information models (Lucas “Island”model, 1972)

Models with nominal rigidities (Fisher model, 1977)

Common feature of these models: expectations are rational

Differences: they assume different types of frictions


In the Lucas “Island”model, prices are flexible but agents(entrepreneurs) have only partial information about monetarypolicy

In the Fisher model, agents are informed of current monetarypolicy but wages are predetermined


Lucas island model

Agents do not know what happens to the overall price level becausethey only observe a partial signal. Agents live in isolated islands andhave information only about the price of the good on their island

Thus, they cannot form accurate expectations as regards overallinflation, even though they are rational

There are n islands indexed by i , each subject to an idiosyncraticshock z it (a demand shock)

There is also an aggregate (common) shock εt (a monetary shock)

Both shocks have zero mean and variances σ2z and σ2ε , respectively.They are independent


Lucas island model

The key is that nobody can distinguish between z it and εt

Define the price of the good being produced and sold on island i as

pit = pt + zit

where pt is the aggregate price level

The price level on island i deviates from the aggregate, but eachagent only observe pit , NOT its separate components


Lucas island model

Aggregate Price is endogenous

Aggregate shocks to prices → aggregate price differs from averageE (p)

Rememberpit = pt + z

it

Agents do not observe separately pit and pt , only pit , the price at

home!


Lucas island model

When I see the price of the good I produce, pit , I do not know whetherchanges come from the overall price level or from the price level on myisland z itWhy should that matter?

Producers care about the relative price of their goods:

If my good becomes relatively more expensive (a shock in z it ) → Iproduce more of it

But if my good is becoming more expensive along with everythingelse (a shock in pt) → I leave production unchanged


Lucas island model

Production on island i :

y it = y + a[pit − E (pt |pit )]

Producers increase output only when they expect relative price change

=⇒ All we need is to solve for is E (pt |pit )

In general, this problem can be complex...

But when z it and pt are normal, it can be solved easily


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 |pit ) andpit is linear:

E (pt |pit ) = α+ βpit

More precisely, we have that:

E (pt |pit ) = E (pt ) +Cov(pt , pit )V (pit )

[pit − E (pit )]

⇒ E (pt |pit ) = E (p) +σ2p

σ2p + σ2z[pit − E (p)]


Lucas island model

Back to the production function:

y it = y + a[pit − E (p)−

σ2pσ2p + σ2z

(pit − E (p)

)]

= y + aσ2z

σ2p + σ2z

(pit − E (p)

)


Lucas island model

Note the following:

When there are no island specific shock, σ2z = 0, producers simply donot alter their production plans, leaving them at y . Why?

Observed deviations between the price on their island and what theyexpect it to be (E (p)) comes from an aggregate shock!

In other words, they know all prices on all islands have shiftedidentically → No reason to produce more


Lucas island model

Now sum across islands:

yt =1N

N

∑i=1y it = y +

1Na

σ2zσ2p + σ2z

N

∑i=1

(pit − E (p)

)= y + a

σ2zσ2p + σ2z

(pt − E (p))


Lucas island model

Across the MACROeconomy, production only responds tounexpected changes to aggregate prices

And the sensitivity of this response increases with the variance ofidiosyncratic shocks


Lucas island model

Thus, changes in prices (engineered for instance via monetary policy)can have an effect on output only if they are unexpected!!

The expectations-augmented Phillips curve told us a similar story:unemployment responds only to deviation of πet from πt

Now, however, the agents’rational expectations are solved within themodel


Lucas island model

Let’s now embed this “Lucas” supply curve in general equilibriummodels of the macroeconomy (i.e. with demand and supply)

Aggregate Supply (AS) was just derived:

yt = y + α(pt − Ept ) (AS)

where α = aσ2z/(σ2p + σ2z )


Lucas island model

Demand arises from an equation of exchange, MV = PY , which in logsrewrites

yt = mt − pt + v (AD)

mt = E (m) + εt , εt ∼ N(0, σ2ε

)is set by the Central Bank

v is constant (for simplicity, assume v = y)

We want to use this framework to think about the effect of (monetary)policy on real activity in a rational expectations equilibrium


Lucas island model

Use (AD) to solve for prices:

pt = mt − yt + v

In equilibrium AD = AS:

y + α (pt − E (p)) = mt − pt + v

Thus,

pt =α

1+ αE (p) +

11+ α

mt

andyt = y −

α

1+ αE (p) +

α

1+ αmt


Lucas island model

Need to solve for E (p) and σ2p :

1 From pt = α1+αE (p) +

11+αmt =⇒ E (p) = E (m)

2 We get

pt − E (p) =1

1+ α(mt − E (m))

Thus, σ2p =( 11+α

)2σ2m


Lucas island model

Substitute this back into supply:

yt = y +α

1+ α(mt − Emt ) (AS)

This is a so-called reduced form, i.e. it expresses endogenous variables(output) in function of the model parameters (y , α) and policy choices(mt)

Economically, this means that monetary policy can affect economicactivity only if it is not expected (Sargent and Wallace, 1976)

Intuition of the mechanism?


Lucas island model

Note further thatα

1+ α=

aσ2zσ2p + (1+ a)σ2z

The response of production to surprises in money is increasing in therelative magnitude of σ2z , i.e. the importance of idiosyncratic shocks

Agents living in a country with a history of high aggregate inflation willhave relatively large values to σ2p , which means policy is increasinglyineffective there

The more policy tries to use monetary policy to boost output, the harder itbecomes as the volatility of σ2p increases


Fisher model

Other authors tried to make sense of the effect of monetary policy onoutput by focusing on imperfect wage and price setting, instead ofimperfect information (Phelps and Taylor (1977), Fisher (1977)

The idea is that nominal prices/wages are preset at the beginning ofthe period on the basis of available information

This approach is not fundamentally different from the imperfectinformation approach: some information is unavailable whenprices/wages are set. The difference is that the mismatch comes inone case from “sticky information”, in the other from “stickyprices/wages”


Fisher model

Fisher introduced the following model in logs:


yt = y − α(wt − pt ) (AS)

wt = pet (WS)

(AD) is the same as before

(AS) depends now on the actual real wage w − p. Firms observe thereal wage. As labor comes as a cost for them, they produce more ifthe real wage is low

The last equation (WS) is the wage-setting equation. Wages arepreset at the beginning of period so as to achieve yt = y inexpectations


Fisher model

We are looking for the rational-expectation solution to this model,that is, with pet = Et−1(pt )

We replace the wage in (AS) by its predetermined value:

yt = y + α[pt − Et−1(pt )] (AS)

This is reminiscent of the Lucas supply curve

We have the same model as before:


yt = y + α[pt − Et−1(pt )] (AS)


Fisher model

We thus obtain the same reduced forms:

pt − Et−1pt =1

1+ α(mt − Et−1mt )

yt = y +α

1+ α(mt − Et−1mt )

Similarly, money shocks affect output only to the extent that they areunanticipated

But the channel is Keynesian: since prices are flexible but nominalwages are preset, money shocks increase prices, decrease real wagesand increase output


Fisher model

Finally, go back to the initial expectations-augmented Phillips curve toillustrate the importance of rational expectations for monetary policy

Consider the expectations augmented Phillips curve:

yt = y + b(πt − πet )

Output is above its average, long run level whenever inflation happens tobe above its expected level

Once again, only surprises in inflation matter for production


Fisher model

Now suppose the Central Bank has a loss function that reflects both aconcern for output AND for inflation. For instance:

L = 12(yt − y ∗)2 +

12

θ(πt − π∗)2

where y ∗ denote the TARGET level of output the Central Bank wouldlike to reach, with y ∗ > y . In other words, the Central Bank wouldlike to push economic activity ABOVE its long run level y

Similarly π∗ denotes the target inflation level - e.g. 2 or 3%, and θcaptures the relative importance of inflation vs. output


Fisher model

Suppose now the Central Bank makes an announcement that it willachieve inflation at π∗. Crucially, suppose the announcement isbelieved, so that πet = π∗

Then the Central Bank is faced with the following minimizationproblem:

Min12(y + b(πt − π∗)− y ∗)2 + 1

2θ(πt − π∗)2

Note that we have made use of the fact that, the moment the CentralBank’s announcement is believed, a Phillips Curve arises at thecorresponding level of πet . Here at π∗

The Central Bank will choose ACTUAL inflation πt to solve thisproblem


Fisher model

Solve for πt : πt = π∗ + bb2+θ

(y ∗ − y)

Since the Central Bank wants to push output yt above its long runlevel y , this means it will want to inflate the economy ABOVE itsinitial announcement of π∗. Why?

So the initial announcement of π∗ cannot possibly be credible.RATIONAL agents anticipate the Central Bank will renege on itsinitial commitment, and do not believe it when it claims it willimplement πt = π∗


Fisher model

Clearly, the only credible inflation verifies πt = πet

Substitute that condition to get:

πt = π∗ +bθ(y ∗ − y)

Inflation has to be above the very target the Central Bank has!

That so-called "inflation bias" increases with b, but decreases in θ.Why?


macroeconomics - hec unil excess supply in the good market ... topic 1: rational expectations ... (a...

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