Lecture 3, November 30: The Basic New Keynesian Model...
Transcript of Lecture 3, November 30: The Basic New Keynesian Model...
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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.
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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
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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)
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� 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
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A view at prices changes: Belgium, 2001-2005
Source: Cornille and Dossche (2006)
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A view at prices changes: France and Italy, 1996-2001
Source: Dhyne et al. (2005)
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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)
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A view at prices changes: France, hazard rates in clothing industry
Source: Baudry et al. (2004)
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A view at prices changes: France: Hazard Rates in service industry
Source: Baudry et al. (2004)
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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�"
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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
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� 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
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� 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
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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
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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)
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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)
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� 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
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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� �)
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� 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)
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� 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!
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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)
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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
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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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