MakØk3, Fall 2010 (blok 2)
�Business cycles and monetary stabilization policies�
Henrik JensenDepartment of EconomicsUniversity of Copenhagen
Lecture 4, December 7: Monetary Policy Design in the Basic New Keynesian Model (Galí,Chapter 4)
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.
� We have now developed a simple model for business cycle and monetary policy analysis
�E.g., we can examine the economy�s response to various shocks (including policy shocks)
� Next step is to examine the model�s normative implications: I.e., how should monetary policy beconducted?
� What should be the goals of monetary policy?
� What can and what cannot monetary policy achieve
� For this purpose we identify the ine¢ ciencies of the New Keynesian economy, and evaluate whetherand how policy can remedy these
� Importantly, a model consistent welfare criterion will be developed to assess various simple, subopti-mal policy rules
1
Properties of the New-Keynesian model
Basic equations summarized:
� �NKPC��t = �Et f�t+1g + �eyt; eyt � yt � ynt
� � (1� �) (1� ��)�
1� �1� � + �"
� (1� �) + ' + �1� � > 0
� �DIS� eyt = Et feyt+1g � ��1 (it � Et f�t+1g � rnt ) ; rnt � � + �Et f�ynt+1g
2
Properties of a (friendly) �command economy�
� Relevant benchmark, or, ideal outcome, is the allocation chosen by a benevolent social planner (itidenti�es the e¢ cient outcomes)
� The planner maximizes
E01Xt=0
�tU (Ct; Nt) = E01Xt=0
�tU
�Z 1
0
Ct (i)"�1" di
� ""�1
;
Z 1
0
Nt (i) di
!;
subject toCt (i) = AtNt (i)
1�� all i 2 [0; 1]
� By nature of the consumption basket, Ct (i) 6= Ct (j) is never optimal for a given Ct. Therefore,optimality requires
Ct (i) = Ct; Nt (i) = Nt; all i 2 [0; 1] :
� The problem then simpli�es tomaxU
�AtN
1�at ; Nt
�Optimality condition:
(1� �)AtN��t Uc:t + Un;t = 0 � Un;t
Uc:t= (1� �)AtN��
t �MPNt
3
Ine¢ ciencies in the New Keynesian model
Monopolistic competition
� Monopolistic competition implies that prices are a markup over aggregate marginal costs; even under�exible prices:
Pt =MWt
MPNt; M� "
"� 1 > 1
� The model�s labor market equilibrium:
�Un;tUc:t
=Wt
Pt=MPNtM < MPNt
Monopolistic competition results in too low employment and output
� Monetary policy is useless in addressing this market-structure ine¢ ciency
� Fiscal (tax) policy can (in theory) solve the problem. Assume a labor cost subsidy � (�nanced lumpsum from consumers):
Pt =M(1� � )Wt
MPNt
�Monopoly distortion is eliminated ifM (1� � ) = 1; this is assumed�Requires � = "�1
4
Nominal rigidities
� Price rigidities cause mark-up �uctuations with sticky prices:
Mt =Pt
(1� � )WtMPNt=
PtMWtMPNt
Wt
Pt=MPNt
MMt
6=MPNt
� Staggered price setting causes price and thus output dispersion:
Ct (i) 6= Ct (j) when Pt (i) 6= Pt (j)
� Ine¢ ciencies due to nominal rigidities can be addressed by monetary policy (at least in part)
5
What should monetary policy ideally do?
� Assume the labor subsidy � = "�1 is in place; the natural rate of output is then e¢ cient
� Eliminate markup �uctuations, i.e., secure that cmct = 0�Equivalent of securing: eyt = 0 all t
� Avoid any price dispersion
�Assuming no past relative price dispersion, Pt�1 (i) = Pt�1, all i 2 [0; 1]�No �rms will change prices when cmct = 0, mct = mc�Hence, Pt+j (i) = Pt+j = Pt�1+j, all i 2 [0; 1], j = 0; 1; 2; ; ;�Equivalent of fully stable aggregate prices:
�t = 0 all t
6
How can this be done?
� Surprisingly there is no policy trade-o¤s� the ideal policy goals are attainable
�(Special to the simple shock structure; by some denoted �a divine coincidence�.)
� Letting it = rnt , is compatible with attaining both eyt = 0 and �t = 0�t = �Et f�t+1g + �eyt;eyt = Et feyt+1g � ��1 (it � Et f�t+1g � rnt )
� Problem. Setting it = rnt leads to dynamic the system� eyt�t
�= AO
�Et feyt+1gEt f�t+1g
�; AO �
�1 ��1
� � + ���1
�� AO has one eigenvalue above one, and one below one. Indeterminacy; i.e., in�nitely many stationaryin�ation and output gap paths
7
� One could therefore follow the previously considered Taylor rule, amended with a response to thenatural rate of interest:
it = rnt + ���t + �yeyt; ��; �y � 0;
� This leads to the familiar dynamics � eyt�t
�= AT
�Et feyt+1gEt f�t+1g
�
AT � �� 1� ����� � + �
�� + �y
� � ; � 1
� + ��� + �y
� Uniqueness requires:0 < (�� � 1)� + �y (1� �)
(the �Taylor principle�)
� The optimal allocation will be achieved in equilibrium
8
� Such a rule, however, poses a practical problem: rnt is not observed in real time
� Therefore, more simple rules can be considered; i.e., rules depending on observable variables
� But how should one to assess their performance?
� I.e., how is it possible to compare one rule to another?
� By developing a welfare criterion!
9
Welfare criterion in the NK model
� The relevant welfare criterion in the NK model is
E0
( 1Xt=0
�tU (Ct; Nt)
)� How do we use this together with the log-linearized model?
� W is approximated by a second-order Taylor expansion (a �rst-order expansion would not rankdi¤erent monetary policies, as these do not a¤ect longs-run levels; i.e., steady states)
� Important second-order approximation for a variable Z:Zt � ZZ
' bzt + 12bz2t
where bzt � log (Zt=Z)� The approximation is performed around an e¢ cient steady state� yields a simple expression
� If the approximation is around an ine¢ cient steady state, one may get �spurious�welfare resultsby using a log-linear model (the ignored second-order terms may become important, i.e., policydependent)
10
Initial Taylor expansion
Ut � U ' UcC�Ct � CC
�+ UnN
�Nt �NN
�+1
2UccC
2
�Ct � CC
�2+1
2UnnN
2
�Nt �NN
�2(hence, separability, Ucn = 0, is assumed)
� In log-deviations
Ut � U ' UcC
�bct + 12bc2t� + UnN �bnt + 12bn2t
�+1
2UccC
2
�bct + 12bc2t�2 + 12UnnN 2
�bnt + 12bn2t�2
' UcC
�bct + 12bc2t� + UnN �bnt + 12bn2t
�+1
2UccC
2bc2t + 12UnnN 2bn2tas bc3t ' bc4t ' bn3t ' bn4t ' 0 in a second-order expansion
� Rearranging:
Ut � U ' UcC�bct + 1
2bc2t + 12UccCUc bc2t
�+ UnN
�bnt + 12bn2t + 12UnnNUn bn2t�
� Simplifying:Ut � U ' UcC
�bct + 1� �2bc2t� + UnN �bnt + 1 + '2 bn2t�
where� � �UccC
Uc> 0; ' � UnnN
Un> 0
11
� Using the goods-market equilibrium condition bct = byt:Ut � U ' UcC
�byt + 1� �2by2t� + UnN �bnt + 1 + '2 bn2t�
� Now comes a �tricky�part: Rewrite bnt in terms of outputRelationship between employment, output and relative prices
� From last lecture:
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) + logZ 1
0
�Pt (i)
Pt
�� "1��di;
(1� �)nt = yt � at + dt; dt � (1� �) logZ 1
0
�Pt (i)
Pt
�� "1��di:
� Around a zero in�ation steady state (where d = 0):
(1� �) bnt = byt � at + dt12
� We need to �nd dt, the measure of price dispersion, as it is second-order term that will have welfaree¤ects.
� Start by the de�nition of the price index:
Pt =
�Z 1
0
Pt (i)1�" di
� 11�"
� Then,
1 =
Z 1
0
�Pt (i)
Ptdi�1�"
=
Z 1
0
exp [(1� ") (pt (i)� pt)]di
' 1 + (1� ")Z 1
0
(pt (i)� pt) di +(1� ")2
2
Z 1
0
(pt (i)� pt)2 di (*)
in a second-order approximation around p (i) = p.
� Letting Ei fpt (i)g �R 10 pt (i)di be the mean of log prices across sectors,
pt ' Ei fpt (i)g +1� "2
Z 1
0
(pt (i)� pt)2 di (*́*)
13
� Then assess the speci�c relative price expression of dt:Z 1
0
�Pt (i)
Pt
�� "1��di =
Z 1
0
exp
�� "
1� � (pt (i)� pt)�di
' 1� "
1� �
Z 1
0
(pt (i)� pt) di +1
2
�"
1� �
�2 Z 1
0
(pt (i)� pt)2 di
� From (*) we have that Z 1
0
(pt (i)� pt) di ' �1� "2
Z 1
0
(pt (i)� pt)2 di
� Hence,Z 1
0
�Pt (i)
Pt
�� "1��di ' 1 +
" (1� ")2 (1� �)
Z 1
0
(pt (i)� pt) di +1
2
�"
1� �
�2 Z 1
0
(pt (i)� pt)2 di
' 1 +1
2
"
1� �1
�
Z 1
0
(pt (i)� pt)2 di � � 1� �1� � + �"
� Using (**) we getZ 1
0
�Pt (i)
Pt
�� "1��di ' 1 + 1
2
"
1� �1
�
Z 1
0
(pt (i)� Ei fpt (i)g)2 di = 1 +1
2
"
1� �1
�vari fpt (i)g
where vari fpt (i)g is price variance across sectors
14
� Sincedt � (1� �) log
Z 1
0
�Pt (i)
Pt
�� "1��di
we get
dt '1
2
"
�vari fpt (i)g
(which also proves that we rightfully ignored it when looking at the linear dynamics, as dt is asecond-order term)
� We then substitute bnt = (1� �)�1 byt � (1� �)�1 at + (1� �)�1 dt intoUt � U ' UcC
�byt + 1� �2by2t� + UnN �bnt + 1 + '2 bn2t�
and get
Ut � U ' UcC
�byt + 1� �2by2t�
+UnN
1� �
�byt + dt + 1 + '
2 (1� �) (byt � at)2�+ t.i.p.
where t.i.p. is �terms independent of policy�and the third-order e¤ects and higher are ignored
� Rewrite so we get utility change measured as percentage change in steady-state consumption:Ut � UUcC
' byt + 1� �2by2t + UnN
UcC (1� �)
�byt + dt + 1 + '
2 (1� �) (byt � at)2�+ t.i.p.
15
� Now remember that we are approximating around an e¢ cient steady state. Hence,
�UnUc=MPN = (1� �)AN�� � (1� �) Y
N
� Therefore,�UnUc= (1� �) C
Nor,
�UnUc
N
C (1� �) = �1
� The utility approximation therefore simpli�es toUt � UUcC
' 1� �2by2t � 12 "�vari fpt (i)g � 1 + '
2 (1� �) (byt � at)2 + t.i.p.= �1
2
�"
�vari fpt (i)g + (� � 1) by2t + 1 + '1� � (byt � at)2
�+ t.i.p.
= �12
�"
�vari fpt (i)g +
�� +
� + '
1� �
� by2t � 21 + '1� �bytat�+ t.i.p.
= �12
�"
�vari fpt (i)g +
�� +
� + '
1� �
��by2t � 21 + '1� �bytynt��+ t.i.p.
asynt =
1 + '
� (1� �) + ' + �at:
16
� The welfare measure is therefore approximately
W = E01Xt=0
�tUt � UUcC
= �12E0
1Xt=0
�t�"
�vari fpt (i)g +
�� +
� + '
1� �
� ey2t � + t.i.p.� We �nally use Lemma 2 from Woodford (2003):
1Xt=0
�tvari fpt (i)g =�
(1� �) (1� ��)
1Xt=0
�t�2t
� We then get
W = E01Xt=0
�tUt � UUcC
=
�12
�� +
� + '
1� �
�E0
1Xt=0
�th"��2t + ey2t i
as
� � (1� �) (1� ��)�
1� �1� � + �"
� (1� �) + ' + �1� �
=(1� �) (1� ��)
��� (1� �) + ' + �
1� �
17
The performance of various policy rules
� With this utility-based welfare loss one can assess the performance of various policy rules
� One can perform optimal policy exercises as linear-quadratic optimization problems (next time)
� Galí exempli�es the importance of price stability in the New-Keynesian model by assessing theperformance of the policy rule
it = � + ���t + �ybyt(note: a function of byt, not ey) for various policy parameters:
18
Concluding remarks
� The New-Keynesian model o¤ers a simple framework for welfare-based policy analysis
� Models is (in principle) immune to the Lucas critique, and the welfare criterion is consistent with theone used to derive the economy�s behavioral equations
� One can rank various policy rules as well as meaningfully compare their quantitative welfare di¤er-ences
� The simple model is obviously too simple to represent the real world, but its basic features �survive�in large-scale versions used in many in�ation-targeting central banks
19
Next time(s)
Monday, December 13: Exercises:
� Prove Lemma 2 on page 89 (this will yield a prize!)
� Exercise 4.1 in Galí (2008)
� Exercise 4.2 in Galí (2008)
Tuesday, December 14Lecture: Monetary policy trade-o¤s, optimal policy and credibility issues (Galí, Chapter 5)
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