Advanced Macroeconomics

course: Yann Algan & Jean-Olivier Hairault

F. Ferroni - F. Pappada - P. Winant

2010 2011

Master Economics and Public Policy

ENSAE-ScPo-X, M1

Course Ecole Polytechnique

Contents

I Rules of the Game 2

II Problem Sets 3

PS#1 IS-LM Model and Economic Policies 4

PS#2 AD-AS Model and the Phillips Curve 6

PS#3 Measuring fluctuations and Economic Policies 9

PS#4 The Real Business Cycle Model 12

PS#5 Money and Fluctuations 15

PS#6 New Keynesian Economics 19

PS#7 Nominal Rigidities 21

PS#8 Monetary Policy in New Keynesian Models 24

III Documents 26

1

Part I

Rules of the Game

Organisation

This handout includes the problem sets and the texts for the lecture Advanced Macroe-conomics taught by Profs Yann Algan and Jean-Olivier Hairault. The objective of thesetutorials is to solve the problem sets and comment the texts related to each session. Inorder to promote your participation during the class, you have to read the texts beforeeach session.

Evaluation

Our evaluation on the contents of these tutorials is based on four homeworks accordingto the following schedule :

Homework 1: Session 1 (10% of the final mark).

Homework 2: Session 3 (10%).

Homework 3: Session 5 (10%).

Final Homework: Session 6 (70%).

Each homework not given in due time receives a mark 0/20.

Software

You may asked to use softwares in order to solve some problem sets. We recommendOctave1 to compute statistics and plot graphs, and Dynare2 to run simulations of themodels.

Filippo Ferronifilippo.ferroni@banque-france.fr

Francesco Pappadafrancesco.pappada@ensae.fr

Pablo Winantpablo.winant@gmail.com

1http://www.gnu.org/software/octave/2http://www.dynare.org/

2

Part II

Problem Sets

3

Problem Set 1

IS-LM Model and Economic Policies

* * *

Documents

1. Crack in the system and The Crisis: Basic Mechanisms and Appro-priate Policies, IMF working papers 2009, Olivier Blanchard

2. Barro: Government spending is no free lunch:Wall Street Journal: http://online.wsj.com/article/SB123258618204604599.html

3. Krugman: War and Non-remembrance:http://krugman.blogs.nytimes.com/2009/01/22/war-and-non-remembrance/

Questions

1. Texts 1: Explain the basic mechanisms leading to the financial crisis.

2. Texts 2 and 3: Use the Barro-Krugman arguments to discuss the assumptionsrequired for a keynesian stimulus plan to be efficient.

Exercise : IS-LM Model and Policies

Lets imagine a closed economy that obeys the short-run IS-LM model. This Keyne-sian economy is described by the following set of equations:

C = C0 + c (Y T ) (1)I = I0 + b1Y b2r (2)

where C is consumption, T is the level of taxes, I is investment, r the interest rate andb1 > 0, b2 > 0, C0 > 0, I0 > 0, and 0 < c < 1 are constants. The money demand writesdown

Md

P= d1Y d2r, (3)

where P is the price level, assumed to be fixed, and d1 > 0 and d2 > 0 are constants.The government controls the exogenous level of taxes T , the money supply MP and thegovernment spending G.

4

1. Comment the equations of consumption and investment

2. Determine the equilibrium on the product market. Explain this IS relationship andillustrate with a graph.

3. Comment the demand for money

4. Determine the equilibrium on the money market. Explain this LM relationship andillustrate with a graph.

5. Determine the equilibrium output, that is the equilibrium on the product marketcompatible with the equilibrium on the money market. Represent the relationshipin the (Y, r) diagram.

6. Suppose that the government increases its spending by G. The government fi-nances this policy by increasing public debt. Which curve will shift, if any? Calcu-late by how much it will shift (the multiplier) and draw a diagram that shows theimpact of this policy.Distinguish the situation where the adjustments on the money market are ignored (ris considered as exogenous) and when they are taken into account (r is endogenous).Explain the crowding out effect of investment when r is endogenous.

7. Suppose that the government finances this fiscal stimulus by increasing taxes. Whatis the impact of this policy on output (calculate the multiplier). Compare withthe previous policy. What would be the difference in the size of the multiplierunder Ricardian equivalence. How relevant would be the Ricardian equivalenceassumption?

8. Suppose that the government decides to cut taxes instead of increasing spending.Illustrate the effects of this expansionary fiscal policy by using a diagram. Calculatethe multiplier and compare with the multiplier of public spending. Which policy isthe most efficient?

9. Illustrate the impact of a expansionary monetary policy

10. Define policy-mix policies? What is the expected impact? Illustrate graphically.Discuss the policy of the FED and the ECB in this realm.

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5

Problem Set 2

AD-AS Model and the Phillips Curve

* * *

Documents

1. Inflation expectations, Uncertainty, the Phillips Curve and MonetaryPolicy FED DC, 2009

2. Rethinking Macroeconomic Policy, IMF 2010

Questions

Text 1: Explain the Phillips curve, the formation of inflation expectations, andtheir implication for monetary policy

Text 2:

What are the main lessons from the crisis drawn by the authors?

What are the reforms of the monetary policy advocated by the authors ? Doyou think Central Bank should have higher inflation rates targeting ?

Exercise 1: WS-PS and AD-AS models

We assume an economy where total output Y is given by

Y = N,

where N is total employment. Lets define the wage setting equation of workers by

W = P e (1 u) b,

where W is the bargained wage, P e is the price expected by the worker at the beginningof the period and which will hold during the period, u is the unemployment rate, and brepresent unemployment benefits.

Lets define the price setting equation of firms by

P = (1 + )W,

where P is the actual level of prices, is the level of the mark-up of firms over laborcosts. We assume that total population is L = 1. The unemployment rate is defined asu = (LN)/L = 1N.

6

1. Explain the wage determination WS. What are the different price expectationsyou could think at? What are the different labor market institutions which couldenter into WS?

2. Explain the price determination PS. How can you explain the presence of a mark-up? What are the factors influencing the size of the mark-up?

3. Illustrate the equilibrium on the labor market with the curves WS and PS in thediagram (u,WP ). Discuss the influence of labor and product market rigidities on

WP

and u.

4. By substituting WS into PS, give an expression of P as a function of P e. What isthe name of such a relationship? Comment the role of the different variables.

5. By using the fact that Y = 1 u, define output as a function of P , P e , and b.Why do we call this relationship Aggregate Supply?

Represent graphically the AS curve in the diagram (Y, P ). Comments the role ofthe different variables. Why is the aggregate supply an increasing function of pricesP?

6. Assume that aggregate demand is just given by the relationship

Y =M

P+G,

where MP represents the real money supply and G total output. Explain the deriva-tion of this expression from the IS-LM model. Represent the AD curve in the plan(Y, P ).

7. Represent graphically the equilibrium output in this economy.

Illustrate the impact of positive demand shocks on MP and G. Illustrate the impactof adverse supply shocks linked to market regulation and b. Discuss these supplyshocks with respect to the French market regulation.

8. Assume that workers have perfect expectations P e = P . Why is this situationreferred to the long-run?

Write down the expressions for real wages WP and the equilibrium unemploymentrate u. Comment.

9. Represent the Aggregate Supply curve when P = P e. Define the equilibrium outputin this case.

10. What is the role of demand shocks and supply shocks in the long-run?

7

Exercise 2: Phillips curve and the credibility of monetary policy

Assume that the Phillips relationship is given by :

t = et + 0.1 2ut

where t is the actual inflation rate between period t and t 1, and et is the expectedinflation rate given by:

et = t1

Assume that the actual rate of inflation in (t1) is equal to 0. In period t, the governmenttargets an unemployment rate of 4%.

1. Discuss the relevance of the formation of inflation expectations.

2. Calculate the inflation rate for periods t, t+ 1, t+ 2, t+ 3.

3. Assume now that inflation expectations are perfect: et = t. Is there a potentialtrade-off between unemployment and inflation?

4. Assume now that there are some nominal rigidities related to wage contracts. Halfof wages are readjusted during the year and thus indexed on actual inflation et .The remaining wages are sticky and indexed on past inflation t1. So now, et =0.5t + 0.5. Rewrite the Phillips curve

5. Redo the math for questions [b].

6. What is the impact of wage contracts and nominal rigidities for demand policiesand the trade-off between inflation and unemployment?

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8

Problem Set 3

Measuring fluctuations and Economic Policies

* * *

Exercise 1: Demand and supply shocks (Blanchard and Quah,1989)

1. What are the economic assumptions behind Blanchard and Quahs modeling? Showthat the identifying restrictions on the VAR model stem from these assumptions.

2. Fishers model

Yt = Mt Pt + atYt = Nt + tPt = Wt tWt = W |{Et1Nt = N}Mt = Mt1 + e

dt

t = t1 + est

where Y , P , M et W are the log of output, prices, money and nominal wages. N ,N et denote current employment, full-employment and productivity. ed et es areuncorrelated demand and supply white noise shocks.

(a) In this model, which elements reflect a Keynesian conception of fluctuations?

(b) Solve the model.Hint : Express unemployment U = N N and growth Y = Yt Yt1 as afunction of supply and demand shocks.

(c) This model justifies Blanchard and Quahs assumptions and identifying re-strictions. Explain.

(d) Assume now that nominal wage is flexible and ensure full employment. Dodemand shocks affect economic activity, in the short run? in the long run?

3. Supply and demand shocks

(a) Define trend as output absent market imperfection and cycle as the dynamicsof effective output around its trend.Can the supply component of output be described as the trend and thedemand component of output described as the cycle?

9

(b) Do you think that this kind of decomposition is relevant for real business cyclestheory?

4. The articles results

(a) In the short run, how does unemployment respond to a positive supply shock?Is it in line with what is predicted by Fishers model?

(b) In Fishers model described in question (2), assume now that prices P are nolonger flexible. The equation Pt = Wt t is now replaced by Pt = (Wt t),with 0 < < 1. Assume that the only shock is a positive supply shock est indate t.How does unemployment respond to this shock? Discuss the values of a and. Explain the response of unemployment described in the article.

(c) Why does unemployment finally go back to its initial level?

(d) According to the results table 1 page 665, are the two oil shocks demand orsupply shocks? Comment.

(e) According to this article, can supply or demand shocks be excluded from theexplanation of output fluctuations in the short and in the medium run?

Exercise 2: How well does the IS-LM model fit postwar US data?(Gal, 1992)

1. AnalyticsConsider the model in the introduction to the paper:

y = + s + (i Ep+1) + is (IS equation)m p = y i+ md (LM equation)m = ms (money supply process)p = p1 + (y s) (Phillips curve)

where y denotes the log of GNP, i is the nominal interest rate, p is the log of theprice level, m is the log of the money stock, is the first difference operator, Eis the expectational operator, and s, is, md and ms are stochastic processesdescribing aggregate supply, spending, money demand, and money supply drivingforces. p+1 is exogenously given.

(a) Give the rationale behind the specification of the IS and LM equations.

(b) Assume that mt = m + ms. What are the effects of a one-period increase inms on the equilibrium in the IS-LM model?

2. Some identification issues

10

(a) Read the paper. Explain the identification strategy the author uses. Whatare the short-run and long-run restrictions?

(b) Explain in words the theoretical/empirical motivation of the identifying restric-tions Gal uses. What do you think about them? Criticize them on theoreticaland empirical grounds. Can they be tested?

3. What are the main results of the paper in terms of monetary policy?

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11

Problem Set 4

The Real Business Cycle Model

* * *

Documents

1. L.H. Summers, Some Skeptical Observations on Real Business Cycle The-ory, Federal Reserve Bank of Minneapolis Quarterly Review, Vol. 10, No. 4, Fall1986.

2. E.C. Prescott, Response to a Skeptic, Federal Reserve Bank of MinneapolisQuarterly Review, Vol. 10, No. 4, Fall 1986.

3. R.E. Manuelli, Modern Business Cycle Analysis: a Guide to the Prescott-Summers Debate, Federal Reserve Bank of Minneapolis Quarterly Review, Vol.10, No. 4, Fall 1986.

Questions

Text 1: Explain why the RBC theory leads to question some conclusions of the Key-nesian economics. What are the main objections formulated by Summers towardthe RBC theory?

Text 2: How does Prescott respond to the Summers claims?

Exercise 1: Analytics

Consider a perfectly competitive economy with two representative agent, a firm anda household, and with one good, produced with the following technology:

Yt = AtKt N

1t

where Kt denotes the capital stock, Nt the labour and At a stochastic technological shock.The capital fully depreciates after one period, i.e. :

Kt+1 = It

where It is investment at date t. The household can work up to N and consume Ct atdate t.

12

At each date t, he chooses to work Nt, to consume Ct and to invest It and maximizeshis intertemporal utility function:

maxEt

=t

t logC + (N N )

s.t. t, C + I WR N + rI1

where r is the real interest rate, WR the real wage and the psychological discountfactor.The lagrangian associated to this problem is:

Lt = Et

=t

t [logC + log(N N )] + [WR N + rI1 C I ]

1. Find the first order conditions for the firm and the household.

2. Assume that households have naive expectations on their output-consumption ratio(i.e. Et[Yt+1/Ct+1] = Yt/Ct). Show that consumption, investment, and employmentare defined by:

Ct = (1 )Yt, It = Yt, Nt =(1 )N

1 + (1 )

3. Show that the log of output is:

yt =at

1 L + n+ ln()

1 where L is the lag operator.

Exercise 2: The Real Business Cycle Model

In this exercise, we investigate the role of the productivity shocks persistence and itseffects on BC properties of the canonical model. Lets consider an economy populatedby individuals that maximize utility, i.e. solve the following problem:

max{ct,ht,kt+1}t=0 E0

t=0 t[ln(ct) + ln(1 ht)]

subject to exp(zt)kt h1t + (1 )kt ct + kt+1 , t 0

k0 given

where ct, ht, kt represent respectively the consumption, labor effort, and capital stock.Parameters (0, 1), > 0, (0, 1), (0, 1) are respectively the subjective dis-count factor, Frish elasticity, the share of capital in value added, and the depreciationrate. Finally, the stochastic process zt follows the law of motion zt+1 = zt + t, where|| < 1 and t N (0, 2 ) and iid.

1. Derive the first order conditions for an optimal path {ct, ht, kt+1}t=0.

13

2. Introducing the following variables: production, yt, investment, it, real interest rate,rt, and wage wt, so that

yt = exp(zt)kt h

1t

it = kt+1 (1 )ktrt = yt/ktwt = (1 )yt/ht.

Give the value of these variables at the steady state.

3. Let define the log deviation s of variable s around its steady state value s, that isst = ln(st) ln(s). Compute the log-linearization of the model, i.e. write the modelin terms of hatted variables at the first order.

4. Define the vector of endogenous variables xt = (ct, ht, kt+1, yt, it, rt, wt). Show thatxt satisfies

Et[F xt+1 +Gxt +Hxt1 + Lzt+1 +Mzt] = 0

zt+1 = Nzt + t+1, Et[t+1] = 0,

and give expression for the matrices F,G,H, L,M , and N .

5. Solving the linear dynamic model consists in finding the two matrices P and Q sothat

xt = P xt1 +Qzt.

Show that P must solve a quadratic equation.

6. Use the providedmatlab routines to solve the model and plot the Impulse ResponseFunctions for each variables. We will use the following calibration

0.984 3.48 0.33 0.025 0.98 0.0072

7. Simulate the model with 50 draws of 115 periods long. After logging and detrend-ing, compute the standard deviation and correlation with output for the endogenousvariables.

8. Redo questions 6-7 with = 0.2 and compare the results.

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14

Problem Set 5

Money and Fluctuations

* * *

Exercise 1: Money in the utility function

Consider an economy composed of Nt households with period utility function:

Ut = u(ct,mt)

where ct is the household consumption, mt is the stock of money per capita, in realterms: mt =

MtPtNt

where Mt is money demand, Pt the level of prices. The utility functionis increasing in c and in m, strictly concave and continually differentiable. The householdchooses {ct,mt, t 0} that maximizes its intertemporal utility:

W =

t=0

tu(ct,mt)

with 0 < < 1 the subjective discount rate.The aggregate resource constraint implies3 :

Yt + Tt + (1 )Kt1 +(1 + it1)Bt1

Pt+

Mt1

Pt= Ct +Kt +

Mt

Pt+

Bt

Pt

Yt = F (Kt1, Nt) is the aggregate output (and F is a production function supposedhomogeneous and with constant returns to scale, satisfying Inada conditions) and Ttdenotes real public transfers, financed by money emissions.

1. The households program

(a) Denote bt =Bt

PtNtbonds held by each household at date t, expressed as a

function of consumption goods, t = Tt/Nt per capita transfers and t =PtPt1Pt1

the inflation rate. Suppose that population grows at a constant rate n.Derive the household budget constraint from the aggregate resource constraint.

(b) Let t be the household resources at date t. Write down the householdsprogram as a Bellman equation. Derive the first order condtions regarding c,m and b. Write down the transversality conditions. Comment.

First order conditions, equilibrium in the money market (Mt = M st with Mst the

exogenous money supply) and the budget constraint characterize the model equi-librium. As households are identical, bond holdings are null in equilibrium.

3Capital letters are used for aggregate variables and small letters for per capita variables: xt =XtNt

.

15

2. Stationary equilibrium

To keep things simple, suppose that the population growth rate is null and thatmoney grows at rate . Consider the stationary equilibrium in which per capitamoney holdings are constant(mt = m, where x denotes the stationary value of x.)

(a) What is the stationary inflation rate?

(b) Use the government budget constraint to show that real transfers per capitaare: = m1+ .

(c) Derive the stationary value of the capital labour ratio, of consumption and ofoutput. Is money neutral in this model? superneutral?

(d) Suppose the utility function is of the form:

u =ac

1bt + (1 a)m1bt

11b

Find the stationary real money demand:

m

c=

1 aa

1/b 1 + 1 +

1/b

Comment.

(e) The government choose in order to reach the optimal value of m in thestationary equilibrium. What is the optimal behaviour of the government?

Exercise 2: a Cash-in-advance Model

In this problem set, we will try to answer some questional issues on money and businesscycle fluctuations. More precisely, we will follow Cooley and Hansen [T.F. Cooley andG.D. Hansen (1989) The Inflation Tax in a Real Business Cycle Model, The AmericanEconomic Review, 79(4), pp.73348] and answer the following questions:

Does money and the form of money supply rules affect the nature and amplitudeof the business cycle?

How does anticipated inflation affect the nature and amplitude of the businesscycle?

Lets consider an economy populated by a representative household seeking to maxi-mize an intertemporal utility function

E0

t=0

t(ln ct Bht), 0 < < 1

depending on consumption ct and labour effort ht, and where E0 is the expectationoperator with respect to inforation available at date 0. This household faces a budgetconstraint:

ct + xt +mt/pt wtht + rtkt + (mt1 + (gt 1)Mt1)/pt

16

where xt is the investment, defined by kt+1 = (1 )kt + xt, 0 1, mt the nominalmoney balances, pt a price index, wt the real wage, rt the real interest rate and kt thecapital stock (k0 given.) Per capita money supply, Mt, evolves according to Mt = gtMt1,with gt+1 = gt g

1 exp(t+1), t iid with variance 2 . The household faces anotherconstraint on cash good

ptct mt1 + (gt 1)Mt11. Give a rationale for a cash-in-advance constraint. What is the basic difference with

the money-in-utility-function paradigm?

2. Write the lagrangian for the household problem and derive the first order conditionsfor an optimal path.

3. In which variables one can found nominal trend? Propose a method to make theproblem stationary.

4. The firm in the economy produces output according to a Cobb-Douglas productionfunction Yt = exp(zt)KtH

1t , 0 1, with zt+1 = zt + t+1, 0 1 and

t iid with mean 0 and variance 2 . Give the first order conditions for the firmsproblem.

5. Compute the steady state for this economy.

6. We use the following calibration

B g

0.99 0.36 0.025 2.86 0.95 0.00721 1.015 0.48 0.009

Simulate the model with 50 draws of 115 periods long. After logging and detrend-ing, compute the standard deviation and correlation with output for the followingvariables: output, consumption, investment, vapital stock, hours, productivity, andprice level.

7. Redo the simulations with = 0. Compare these results with the Hansen(1985)sindivisible labor model (see below) and conclude on the initial questions.

Standard deviations in percent(a) and correlations with output(b) for an economy with indivis-ble labor

Series (a) (b)Output 1.76(0.21) 1.00(0.00)Consumption 0.51(0.08) 0.87(0.04)Investment 5.71(0.70) 0.99(0.00)Capital stock 0.47(0.10) 0.05(0.07)Hours 1.35(0.16) 0.98(0.01)Productivity 0.50(0.07) 0.87(0.03)

17

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18

Problem Set 6

New Keynesian Economics

* * *

Documents

Text: N. Gregory Mankiw, A Quick Refresher Course in Macroeconomics, Jour-nal of Economic Literature, 28, pp.16451660, Dec. 1990.

Questions

1. Try to summarize the main developments in Macroeconomic literature in 1970sand 1980s. What does the breakdown consensus refer to?

2. Which are the three big directions of research that departed from the consensusbreakdown? To what extent the introduction of rational expectations has been arevolutionary approach in Macroeconomics?

3. Discuss the time inconsistency of monetary policy argument and the rules versusdiscretion dylemma.

Exercise: Imperfect Competition and Strategic Complementari-ties

We consider an economy populated by a large number of identical agents. Eachagent draws utility from the consumption of a Constant Elasticity of Substitution (CES)aggregate, C, which correponds to a combination of all goods Ci available in the economy.Agent offers a quantity L of labor.

The representative agents utiliy function is given by

U = C 1

L

with

C =

1

0

C

1

i di

1

where > 1 and > 1.Let P denotes the price level index

P =

1

0

P1i di

11

19

1. Write the problem of an agent that wants to achieve a level C of aggregate con-sumption by optimaly choosing the intermediate goods and taking their price Pi asgiven.

2. Solve this problem and comment the role of the parameter related to elasticity ofsubstitution.

3. Express the relative demand of intermediate good Ci with respect to the aggregateconsumption C.

4. Give the budget constraint of the representative agent and determine the level oflabor supply for a given wage rate w. Compute the equilibrium condition on theproduct market.

5. Money, M , is held for transaction purpose on the product market. How writes theaggregate demand curve?

6. Each intermediate good is yield by a unique, monopolistically competitive firm.Production technology is linear in labor. Write the problem of firm i. Comment.

7. Give the first order condition. Express the relationship between the optimal relativeprice and the real marginal cost mc.

8. Give the relationship between the optimal price, aggregate price index, and moneyholdings. Comment on the possible strategic interactions.

9. How this situation compare to the competitive case ?

10. Show that the game between all intermediate firms reveals externalities and strate-gic complementarities.

11. Does this framework support a Keynesian view on unemployment? Discuss theclassical feature of this model.

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20

Problem Set 7

Nominal Rigidities

* * *

Documents

Text: Laurence Ball, N. Gregory Mankiw, David Romer, The New Keynesian Eco-nomics and the Output-Inflation Trade-Off, Brookings Papers on Economic Ac-tivity, 1988(1), pp.182, 1988.

Questions

1. What are the main assumptions of new Keynesian economic models?

2. Confront the new classical and new Keynesian predictions. In what respect do theyresemble? How to distinguish between them?

Exercise 1: Multiple equilibria with menu costs

We consider an economy inhabited by a large number of firms operating in an imper-fect competitive market. All variables are in log. The firm i incurs a loss when its priceis pi instead of the optial flexible one p, given by

Li = k (pi p)2 , k > 0

withp = (1 )p+ m

where m denotes the money supply, p the aggregate price index and [0, 1]. Each firmhas to pay a menu cost z > 0 if it decides to adjust its price. Production is given by

y = m p

We assume that initially m = 0 and the economy is at equilibrium with flexible prices,y = 0 and p = 0. Then m increase from 0 to m. We assume that only a fraction f offirms, 0 < f < 1, is allowed to change its prices.

1. Give the level of aggregate price p as a function of f and p. Work out expressionsfor p, y, and p with respect to f and m.

2. Plot the firms incentive to adjust its price as a function of f , measured by theprofit loss if the firm does not adjust.

21

3. A firm adjusts its price if its incentive to do so is higher than the menu cost z,otherwise the firm has no motivation to adjust or is indifferent at equality. Underwhat conditions all firms or no firm adjusting are both equilibria?

Exercise 2: Price setting a la Calvo

We consider a closed economy populated by a continuum of firms i [0, 1]. Eachfirm produces an imperfect substitute intermediate good, according to the productionfunction

Yi,t = Li,t

where Li,t denotes employment, paid on a perfectly competitive mlabor market at nominalwage Wt.

A representative household draws utility from consumption as stated by the function

Ut =

1

0

C

1

i,t di

1

with Ci,t the consumption of intermediate good i at date t, and > 0 the elasticity ofsubstitution between two intermediate goods. The representative household is assumed tospend its entire earnings on consumption, i.e. there is no intertemporal optimization. LetRt and Pt denote the households nominal revenues and price level at date t respectively,with

Pt =

1

0

P1i,t di

11

1. Write the households optimization problem and give the related first order condi-tions. Work Ci,t out as a function of Pi,t, Pt, Rt. Comment.

2. Assume that firms can freely set their prices at each period t. What is the optimalprice Pi,t for the firm i?

3. Assume at each period, only a fraction 1 of firms can set optimally its price anda cannot. Show that the optimizing firms price level decision Pi,t solve

maxPi,t

i,t = Et

+

s=t

1

1 + r

stst(Pi,t Ws)Ci,t,s

where r is the real interest rate, held constant, and Et the expectation operatorbased on available information at date t

Ci,t,s =

Pi,t

Ps

Rs

Ps

Work out the first order conditions and solve for the optimal price Pi,t = Pt.

22

4. At each date t, the price index Pt is a function of Pt1 and Pt

Pt =P

1t1 + (1 )P 1t

11

Write the ratio Pt/Pt1 as a function of Pt/Pt1. Compute the inflation rate for = 0, = 1, and 0 < < 1.

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23

Problem Set 8

Monetary Policy in New Keynesian Models

* * *

Documents

Richard Clarida, Jordi Gal and Mark Gertler, Monetary Policy Rules AndMacroe-conomic Stability: Evidence And Some Theory, The Quarterly Journal of Eco-nomics, MIT Press, vol. 115(1), pages 147-180, February 2000.

Questions

1. What is the conventional wisdom about the differences in the pre-Volcker and theVolcker-Greenspan eras of monetary policy? How do the authors identify the twoeras of monetary policy?

2. To what extent the introduction of a forward-looking rule by the central bank is keyfor the papers result? What are the main differences with respect to a standardTaylor rule?

3. In light of the authors estimates, explain why the monetary policy in the pre-Volcker era was not able to stabilize inflation neither to mitigate the impact offundamental shocks to the economy. Discuss.

Exercise: Monetary Policy in the basic New Keynesian Model

Consider the basic New Keynesian model described by the equilibrium conditions :

Phillips Curve:

t = Et{t+1}+ kytIS equation:

yt = Et{yt+1}1

(it Et{t+1} rnt )

Policy Rule:

it = + t

Natural rate:

rnt = rr

nt1 +

rt

where t denotes inflation, yt is the output gap, rt is the nominal rate and rnt rnt isthe natural rate in deviations from its steady state value.

24

1. Show that the equilibrium behaviour of inflation and the output gap is given by:

t = rt1 + rt

yt = yyt1 + yrt

where and y are coefficients to be determined.

2. Determine the optimal value of coefficient if the central banks loss function isgiven by:

var(yt) + var(t)

3. Assume we now augment the Philips curve with an i.i.d. cost push shock:

t = Et{t+1}+ kyt + ut

and assume further that r = 0. How would your answer to question 2 change?

* **

25

Part III

DocumentsProblem Set 1

1. Crack in the system and The Crisis: Basic Mechanisms and Appro-priate Policies, IMF working papers 2009, Olivier Blanchard

2. Barro: Government spending is no free lunch:Wall Street Journal: http://online.wsj.com/article/SB123258618204604599.html

3. Krugman: War and Non-remembrance:http://krugman.blogs.nytimes.com/2009/01/22/war-and-non-remembrance/

Problem Set 2

1. Inflation expectations, Uncertainty, the Phillips Curve and MonetaryPolicy FED DC, 2009

2. Rethinking Macroeconomic Policy, IMF 2010

Problem Set 3

1. Blanchard, Olivier Jean and Quah, Danny, 1989. The Dynamic Effects of Aggre-gate Demand and Supply Disturbances, American Economic Review, AmericanEconomic Association, vol. 79(4), pages 655-73, September.

2. Gali, Jordi, 1992. How Well Does the IS-LM Model Fit Postwar U.S. Data, TheQuarterly Journal of Economics, MIT Press, vol. 107(2), pages 709-38, May.

Problem Set 4

1. L.H. Summers, Some Skeptical Observations on Real Business Cycle The-ory, Federal Reserve Bank of Minneapolis Quarterly Review, Vol. 10, No. 4, Fall1986.

2. E.C. Prescott, Response to a Skeptic, Federal Reserve Bank of MinneapolisQuarterly Review, Vol. 10, No. 4, Fall 1986.

3. R.E. Manuelli, Modern Business Cycle Analysis: a Guide to the Prescott-Summers Debate, Federal Reserve Bank of Minneapolis Quarterly Review, Vol.10, No. 4, Fall 1986.

Problem Set 5

1. T.F. Cooley and G.D. Hansen (1989) The Inflation Tax in a Real Business CycleModel, The American Economic Review, 79(4), pp.73348

Problem Set 6

26

1. N. Gregory Mankiw, A Quick Refresher Course in Macroeconomics, Jour-nal of Economic Literature, 28, pp.16451660, Dec. 1990.

Problem Set 7

1. Laurence Ball, N. Gregory Mankiw, David Romer, The New Keynesian Eco-nomics and the Output-Inflation Trade-Off, Brookings Papers on EconomicActivity, 1988(1), pp.182, 1988.

Problem Set 8

1. Richard Clarida, Jordi Gal and Mark Gertler, Monetary Policy Rules AndMacroeconomic Stability: Evidence And Some Theory, The QuarterlyJournal of Economics, MIT Press, vol. 115(1), pages 147-180, February 2000.

27

I Rules of the GameII Problem SetsPS#1 IS-LM Model and Economic PoliciesPS#2 AD-AS Model and the Phillips CurvePS#3 Measuring fluctuations and Economic PoliciesPS#4 The Real Business Cycle ModelPS#5 Money and FluctuationsPS#6 New Keynesian EconomicsPS#7 Nominal RigiditiesPS#8 Monetary Policy in New Keynesian Models

III Documents