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)

20