MakØk3, Fall 2010 (blok 2)

�Business cycles and monetary stabilization policies�

Henrik JensenDepartment of EconomicsUniversity of Copenhagen

Lecture 3, November 30: The Basic New Keynesian Model (Galí, Chapter 3)

c 2010 Henrik Jensen. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personaluse or distributed free.

Introduction

� The classical �ex-price model(s) covered so far cannot account for realistic short-run dynamics fol-lowing monetary shocks

�In case there are any real e¤ects, the magnitude appears modest

�No liquidity e¤ect present (long run = short run)

Some �sand in the wheels�are necessary to explain the e¤ects of money on output seen in data

� Obvious candidate is incomplete nominal adjustment. In �ex-price models, money neutrality holdsboth in the short and long run

�If money neutrality fails in the short run, the impact of monetary policy will clearly be stronger

� Classical model is therefore amended with nominal rigidities (and explicit determination of prices by�rms)

� Result is Dynamic Stochastic General Equilibrium (DSGE) model of the New-Keynesian type

1

A basic New-Keynesian model

� Goods market

�Demand side: Households consume a basket of goods (based on utility maximization)

�Supply side: Firms produce di¤erent consumption goods (maximize pro�ts under monopolisticcompetition)

� Labor market

�Demand side: Firms hire labor (maximize pro�ts under monopolistic competition)

�Supply side: Households supply labor (based on utility maximization)

� Financial markets

�Households optimally invest in a one-period risk-less bond

�Households also hold money. We again just assume it (and generally do not make much use of ithere, as focus will be on interest-rate policies)

2

� Wages are perfectly �exible (this assumption is relaxed in Chapter 6)

� All goods prices are subject to in�exibility, i.e., stickiness:

�Every period, each �rm faces a state-independent probability � of �being stuck�with its price

�So, 0 � � � 1 is natural �index�for stickiness in the economy

� The probability is independent of when the �rm last changed its price; i.e., time-dependent pricestickiness (as opposed to state-dependent price stickiness)

� Stylistic representation of �staggering.�The independence of history facilitates aggregation:

�� is the fraction of �rms not adjusting prices in a period

�1� � is the fraction of �rms adjusting is a period

�1 + � + �2 + �3 + :::: = 1= (1� �) is average duration of a price contract

� When �allowed�to change the price, expectations about future prices become of importance

3

A view at prices changes: Belgium, 2001-2005

Source: Cornille and Dossche (2006)

4

A view at prices changes: France and Italy, 1996-2001

Source: Dhyne et al. (2005)

5

A view at prices changes: (Monthly) frequencies of price adjustments; Euro area vs. US

Source: Dhyne et al. (2005) (with US data based on Bils and Klenow, 2004)

6

A view at prices changes: France, hazard rates in clothing industry

Source: Baudry et al. (2004)

7

A view at prices changes: France: Hazard Rates in service industry

Source: Baudry et al. (2004)

8

Summary of �ndings from data:

� Aggregate prices are sticky (as also indicated by VAR analyses)

� Price setting in di¤erent sectors di¤ers;

�some sectors exhibit long periods of �xed prices (time dependent price setting)

�some sectors exhibit continuous changing prices

� Price setting is asynchronized across sectors

� In some sectors, the probability of price adjustment is (nearly) independent of the life-span of theprice

Relation to theoretical modelling of price rigidities:

� The staggered price setting scheme captures many of these features, and their aggregate implications,in a simple and handy way

� Modelling due to Calvo (1983); now just called �Calvo pricing�

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Household behavior

� The essentials of household behavior are like in the basic classical model

� Household maximize expected utility

E01Xt=0

�tU (Ct; Nt) ; 0 < � < 1:

� Main di¤erence: Ct is not one good, but a basket of the di¤erent goods produced by the di¤erentproducers (indexed by i 2 [0; 1])

Ct ��Z 1

0

Ct (i)"�1" di

� ""�1

; " > 1

� Budget constraint therefore becomes:Z 1

0

Pt (i)Ct (i) di +QtBt � Bt�1 +WtNt � Tt

(and a No-Ponzi Game constraint ruling out explosive debt)

� Household chooses optimal paths of Ct, Nt and Bt (taking prices Pt, Wt and Qt as given), but mustnow choose the composition of Ct as well

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Finding the optimal composition of Ct:

� Assume that total nominal expenditures on consumption are ZtZ 1

0

Pt (i)Ct (i) di = Zt

� Optimal demand for Ct (i) ; is found by

maxCt(i)

�Z 1

0

Ct (i)"�1" di

� ""�1

s.t.Z 1

0

Pt (i)Ct (i) di = Zt

� The relevant �rst-order condition is�Z 1

0

Ct (i)"�1" di

� 1"�1

Ct (i)�1" = �Pt (i) ;�Z 1

0

Ct (i)"�1" di

� 1"�1

Ct (i)"�1" = �Pt (i)Ct (i)

where � is the Lagrange multiplier associated with the constraint

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� Integrating over all goods:�Z 1

0

Ct (i)"�1" di

� 1"�1 Z 1

0

Ct (i)"�1" di = �

Z 1

0

Pt (i)Ct (i) di

Ct = �Zt

� Let Pt be the price index associated with Ct, then PtCt = Zt and

� =1

Pt

� Inserted into �rst-order condition:�Z 1

0

Ct (i)"�1" di

� 1"�1

Ct (i)�1" =

Pt (i)

Pt;

C1"t Ct (i)

�1" =Pt (i)

Pt

� The relative demand for good i is therefore

Ct (i) =

�Pt (i)

Pt

��"Ct (1)

� Note that " is the elasticity of substitution between goods. (Hence, the limiting case of perfectcompetition applies when "!1.)

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The price index

� The precise form of the price index Pt follows from

PtCt =

Z 1

0

Pt (i)Ct (i) di

and the expression for relative demand

� I.e., we get

PtCt =

Z 1

0

Pt (i)

�Pt (i)

Pt

��"Ct di

Pt =

�Z 1

0

Pt (i)1�" di

� 11�"

13

Remaining household optimality conditions

� Precisely as in the simple classical model:

�Un;tUc;t

=Wt

Pt

Qt = �Et

�Uc;t+1Uc;t

PtPt+1

�

� With U (Ct; Nt) � C1��t1�� �

N1+'t1+' , �; ' > 0, we have the same log-linear expressions:

�Labor supply schedule:�ct + 'nt = wt � pt (2)

�Consumption-Euler equation:

ct = Et fct+1g � ��1 (it � Et f�t+1g � �) (3)

with it � � logQt and � � � log �

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Firms�behavior and aggregate in�ation dynamics� Generally, optimal price-setting of a �rm that can change its price in period t:

maxP �t

1Xk=0

�kEt�Qt;t+k

�P �t Yt+kjt � t+k

�Yt+kjt

��s.t. Yt+kjt =

�P �tPt+k

��"Ct+k (8)

� I.e., the maximal expected discounted stream of pro�ts of choosing P �t taking into account theprobabilities that it is �xed in the future

� t+k is total costs of sales at price P �t in period t + k (will depend on labor costs and productivity)

� Qt;t+k is the �stochastic discount factor�(or present-value operator), equal to the one facing consumersbetween periods t and t + k:

Qt;t+k � �k�Ct+kCt

����PtPt+k

�Note

Qt;t+1 = �

�Ct+1Ct

����PtPt+1

�Qt;t+2 = �2

�Ct+2Ct

����PtPt+2

�= �

�Ct+1Ct

����PtPt+1

��

�Ct+2Ct+1

����Pt+1Pt+2

�= Qt;t+1Qt+1;t+2

Qt;t+k = Qt;t+1Qt+1;t+2:::::Qt+k�2;t+k�1Qt+k�1;t+k

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� First-order condition for P �t :1Xk=0

�kEt

�Qt;t+k

�Yt+kjt + P

�t

@Yt+kjt@P �t

� t+kjt@Yt+kjt@P �t

��= 0;

where t+kjt � 0t+k�Yt+kjt

�is nominal marginal costs

1Xk=0

�kEt

�Qt;t+kYt+kjt

�1 +

P �tYt+kjt

@Yt+kjt@P �t

� t+kjt1

Yt+kjt

@Yt+kjt@P �t

��= 0;

1Xk=0

�kEt

�Qt;t+kYt+kjt

�1� " + t+kjt

"

P �t

��= 0:

� We then get the central pricing equation:1Xk=0

�kEt�Qt;t+kYt+kjt

�P �t �M t+kjt

�= 0; (9)

withM� "= ("� 1) > 1 being the mark-up; a measure of the monopoly distortion

16

� This is rewritten as1Xk=0

�kEt

�Qt;t+kYt+kjt

�P �tPt�1

�MMCt+kjt�t�1;t+k

��= 0; (10)

where MCt+kjt � t+kjt=Pt+k is real marginal costs and �t�1;t+k � Pt+k=Pt�1 is gross in�ationbetween t� 1 and t + k

� Optimality condition is log-linearized around a zero-in�ation steady state:P �tPt�1

= 1; �t�1;t+k = 1; Yt+kjt = Y; MC =M�1; Qt;t+k = �k

� One gets1Xk=0

(��)k Et�p�t � pt�1 � cmct+kjt � pt+k + pt�1

= 0

where cmct+kjt � mct+kjt �mc = mct+kjt + logM = mct+kjt + log""�1 = mct+kjt + �

� Hence,1Xk=0

(��)k p�t =

1Xk=0

(��)k Et�cmct+kjt + pt+k

p�t = (1� ��)

1Xk=0

(��)k Et�cmct+kjt + pt+k

Optimal price is a function of expected current and future real marginal costs and aggregate prices

17

� Marginal costs depend on production function and factor payments

� Each �rm has the following production function (i.e., identical technology):

Yt (i) = AtNt (i)1�� ; 0 � � < 1 (5)

� Total costsTCt = WtNt (i)

Real marginal costs

MCt (i) =Wt

PtAt (1� �)Nt (i)��

0@= Wt

Pt MPNt=

Wt

PtAt (1� �) (Yt (i) =At)� �1��

=Wt

Pt (1� �)A11��t Yt (i)

� �1��

1A� Special case of constant returns to scale, � = 0

MCt+kjt =MCt+k

�=

Wt+k

Pt+kAt+k

�

p�t = (1� ��)

1Xk=0

(��)k Et fcmct+k + pt+kg� This is the unique stationary solution to the �rst-order rational expectations di¤erence equation:

p�t = ��Et fp�t+1g + (1� ��) (cmct + pt)18

Aggregate price dynamics

� Remember, � is fraction of price setters that keep past period�s price; � is fraction of price settersthat set new prices

� By de�nition of the price index:

Pt =h� (Pt�1)

1�" + (1� �) (P �t )1�"i 11�"

=) �1�"t = � + (1� �)

�P �tPt�1

�1�"� Log-linearized around the zero-in�ation steady state:

(1� ") �t = (1� ") (1� �) (p�t � pt�1) =) �t = (1� �) (p�t � pt�1) (7)

� Rewrite the di¤erence equation for optimal price setting:

p�t � pt�1 = ��Et fp�t+1 � ptg + (1� ��) (cmct + pt)� pt�1 + ��pt

p�t � pt�1 = ��Et fp�t+1 � ptg + (1� ��) cmct + �t� Use aggregate dynamics to obtain:

(1� �)�1 �t = (1� �)�1 ��Et f�t+1g + (1� ��) cmct + �t�t = �Et f�t+1g + �cmct; � � (1� �) (1� ��)

�; lim

�!0� =1:

This is the basic in�ation-adjustment equation in New-Keynesian theory

19

General case of decreasing returns to scale, 0 < � < 1

� We de�ne aggregate real marginal costs (in logs):

mct � wt � pt �mpnt

= wt � pt �1

1� �(at � �yt)� log (1� �)

� We have for a price-changing �rm:

mct+kjt = wt+k � pt+k �1

1� �

�at+k � �yt+kjt

�� log (1� �)

= mct+k +�

1� �

�yt+kjt � yt+k

�� By the demand function in logs (here using equilibrium condition yt+k = ct+k):

mct+kjt = mct+k ��"

1� �(p�t � pt+k) (14)

� Optimal price setting:

p�t = (1� ��)

1Xk=0

(��)k Et

�cmct+k � �"

1� �(p�t � pt+k) + pt+k

�� Associated in�ation dynamics

�t = �Et f�t+1g + �cmct; � � (1� �) (1� ��)

�

1� �

1� � + �"; lim

�!0� =1: (16)

20

Equilibrium outcomes

� Goods market clearing on each goods market

Yt (i) = Ct (i) ; all 0 � i � 1

� In the aggregate:Yt = Ct;

where

Yt ��Z 1

0

Yt (i)"�1" di

� ""�1

:

� Asset market equilibrium (as in classical economy):

yt = Et fyt+1g � ��1 (it � Et f�t+1g � �) (12)

21

� Aggregate employment:

Nt =

Z 1

0

Nt (i) di =Z 1

0

�Yt (i)

At

� 11��di

Nt =

�YtAt

� 11��Z 1

0

�Yt (i)

Yt

� 11��di�

YtAt

� 11��Z 1

0

�Pt (i)

Pt

�� "1��di

� In logs:

nt =1

1� �(yt � at) + log

Z 1

0

�Pt (i)

Pt

�� "1��

;

(1� �)nt = yt � at + dt; dt � (1� �) log

Z 1

0

�Pt (i)

Pt

�� "1��

:

� The variable dt is a natural measure of price dispersion (proportional to the variance of prices)

� When we consider log deviations from a symmetric zero-in�ation steady state, dt can be ignored (itis of second order� see Appendix 3.2 and 3.3)

� Hence, as a log-linear approximation:

yt = at + (1� �)nt; (13)

as in the classical case

22

Relationship between real marginal cost and output

� We want a dynamic system in output and in�ation; hence, marginal cost is replaced appropriatelyby output

� We havemct = wt � pt �

1

1� �(at � �yt)� log (1� �)

� Using the labor supply schedule

mct = �ct + 'nt �1

1� �(at � �yt)� log (1� �)

mct = �yt +'

1� �(yt � at)�

1

1� �(at � �yt)� log (1� �)

=� (1� �) + ' + �

1� �yt �

1 + '

1� �at � log (1� �) (17)

� Under �exible prices, � = 0, cmct = 0mct = mc

= �� (= � lnM)

� Hence, denoting ynt the �exible-price output, or, the natural rate of output,

mc = �� = � (1� �) + ' + �

1� �ynt �

1 + '

1� �at � log (1� �)

23

� The natural rate of output is given as

ynt =1 + '

� (1� �) + ' + �at �

[�� log (1� �)] (1� �)

� (1� �) + ' + �= nyaat + #

ny (19)

� Under perfect competition, � = 0, this is output as in the classical model

� We get the marginal cost in deviation from steady state:

cmct = mct �mc =� (1� �) + ' + �

1� �(yt � ynt ) (20)

� We denote eyt � yt � ynt the output gap

� The �New-Keynesian�Phillips Curve (�NKPC�):

�t = �Et f�t+1g + �eyt; � � �� (1� �) + ' + �

1� �(21)

24

� Rephrase the Euler equation in terms of the output gap:

yt = Et fyt+1g � ��1 (it � Et f�t+1g � �)eyt = Et feyt+1g � ��1 (it � Et f�t+1g � �)� ynt + Et fynt+1geyt = Et feyt+1g � ��1 (it � Et f�t+1g � rnt ) (22)

wherernt � � + �Et f�ynt+1g

is the natural rate of interest

�A dynamic �IS�curve (�DIS�)

�Let brnt � rnt � �

� New-Keynesian Phillips curve and dynamic IS curve determines output gap and in�ation conditionalon monetary policy (it).

� So, in contrast with the classical model, monetary policy matters for output determination!

25

Interest rate policy in the New Keynesian model

� The case of a simple Taylor-type interest rate rule:

it = � + ���t + �yeyt + �t; ��; �y � 0; (25)

where �t is a policy shock, vt = �vvt�1 + "vt

� The policy rule, NKPC and DIS give the dynamics� eyt�t

�= AT

�Et feyt+1gEt f�t+1g

�+BT (brt � vt) ; (26)

AT � �� 1� ���

�� � + ��� + �y

� � ; BT � �1

�

�; � 1

� + ��� + �y

� A unique non-explosive equilibrium requires the eigenvalues ofAT to be stable (within the unit circle)

� Uniqueness depends on policy-rule parameters:

0 < (�� � 1)� + �y (1� �) (27)

is necessary and su¢ cient condition (the �Taylor principle�)

�Note �� > 1 is su¢ cient condition (an �active�Taylor rule)

26

27

28

Concluding remarks

� The New-Keynesian model is (compared to the classical model) a simple but powerful tool for ana-lyzing monetary policy

� Given monetary policy works more in accordance with evidence, how can the model be used for policyevaluation?

� To come:

�Welfare e¤ects of di¤erent policies

�Optimal monetary policy and credibility issues

29

Next time(s)

Monday, December 6: Exercises:

� Derive the dynamics of the New-Keynesian model on matrix form (or, state-space form), (26)

� Derive the condition for determinacy, (27) in Galí, Chapter 3. Hint: A 2 � 2 matrix has two stableroots, when the coe¢ cients in the characteristic polynomial � 2 + a1� + a0 satisfy ja0j < 1 andja1j < 1 + a0

� Derive the explicit solutions for the output gap, in�ation, and the nominal interest rate and outputin New-Keynesian model of Galí, Chapter 3 (page 51) under an interest-rate rule (and in the absenceof technology shocks).

�Interpret the solutions economically

Tuesday, December 7:Lecture: Monetary policy design and welfare in the simple NK Model (Galí, Chapter 4)

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