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Monetary Economics: Macro Aspects, 19/5 2014

Henrik JensenDepartment of EconomicsUniversity of Copenhagen

Monetary policymaking and the recent crisis

1. The case for quantitative easing

Literature: A. J. Auerbach and M. Obstfeld (2005): “The Case for Open-Market Purchases in a Liquidity

Trap” American Economic Review  95, 110–137.

Brief talk about exam: Hints and advice

c   2014 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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Introductory remarks

 Recent monetary policy developments, since the start of the world-wide crises post-2008, have had

two major consequences(A) Moved leading policy rates (US and in Euroland) close to zero:

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(B) Made central banks expand their balance sheets; “quantitative easing” (“QE”):

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 In a deep recession with nominal policy rates at zero, the usual interest-rate channel is “dead” (A)

 Will quantitative easing then work instead, i.e., is (B) the answer?

 Isn’t the situation where the Zero Lower Bound on policy rates binds called a “Liquidity Trap”?

Source: Blanchard et al. (2010): Macroeconomics. A European Perspective (Prentice Hall), Figure 5.11.

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 Why bother increasing the money supply?

 Some reasons pertain to …nancial market conditions per se:

– After Lehman Brothers collapse, most markets “froze”; much of the quantitative easing was aimed

at addressing con…dence crises in …nancial markets e.g., facilitating loans not happening in the

market; helping out …rms with “toxic assets,” etc.

 From a macroeconomic stabilization point of view, however, quantitative easing may have an e¤ect

also

 I.e., expansive monetary policy may be e¤ective in a “trap”

 The purpose of Auerbach and Obstfeld’s paper is to show under which circumstances, in a model

that more or less uses all we have learned!

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The AO model and the e¤ectiveness of QE

The model in the simple version: ‡exible prices

 Model is representative agent, in…nite-horizon model of the cash-in-advance type

 Household utility

U 0 =1X

t=0

  tY

s=0

 s

!U  (C t; Lt)

Model allows for “temporary patience”, i.e.,  s >  1  is possible, but limt!1Q

ts=0  s = 0

 Speci…c utility function

U  (C t; Lt) = log C t ktLt   (1)

 Households hold government debt,  Bt, and money,  M t. Total real wealth (beginning of period) is

de…ned as

V t = Bt + M t

P t1

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 Households’ budget constraint is therefore

V t+1 = Bt (1 + it)

P t+

 M tP t

+ tP t

+ wtLt

P t T t C t

 Nominal interest rate is   it 0. Real interest rate is de…ned as 1 + rt (1 + it) = (P t=P t1)

 Using the de…nition of real wealth and the real interest rate:

V t+1   =  Bt (1 + it) P t1

P tP t1+

 M tP t

+ tP t

+ wtLt

P t T t C t

=  Bt (1 + rt)

P t1+

 M tP t

+ tP t

+ wtLt

P t T t C t

=   V t (1 + rt) (1 + rt)  M 

tP t1 +

 M t

P t + 

tP t +

 wtL

tP t

T t C t

=   V t (1 + rt) (1 + it) M tP t

+ M tP t

+ tP t

+ wtLt

P t T t C t

=   V t (1 + rt) itM tP t

+ tP t

+ wtLt

P t T t C t

 CIA constraint:

M t (1 +  t) P tC t   (2)

A consumption tax,  t, is also paid by cash

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 We know that a CIA constraint is binding when  it >  0, and non binding when  it = 0 (cf. Chapter 3

in Walsh)

 We therefore have the “complementary slackness” condition:

itM t =  it (1 +  t) P tC t

 Inserted into budget constraint:

V t+1   =   V t (1 + rt) itM tP t

+ tP t

+ wtLt

P t T t C t

=   V t (1 + rt) it (1 +  t) C t + tP t

+ wtLt

P t T t C t

=   V t (1 + rt) + t

P t+ w

tLtP t

T t (1 + it +  tit) C t

 Total real tax payments are T t =   tC t  such that

V t+1   =   V t (1 + rt) + tP t

+ wtLt

P t  tC t (1 + it +  tit) C t

=   V t (1 + rt) +

 t

P t +

 wtLt

P t (1 + it) (1 +  t) C t

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 Optimal behavior by households:

U C  t (1 + it) (1 +  t) = 0;

U L + twtP t

= 0;

t +  

t+1

t+1 (1 + r

t+1) = 0;

where t  is the Lagrange multiplier on the budget constraint

 With the speci…c utility function:

1

C t=   t (1 + it) (1 +  t) ;

kt   =   twtP t

;

t   =    t+1t+1 (1 + rt+1) ;

 This results in labor supply and Euler equations:

C t =  wt

kt (1 + it) (1 +  t) P t(3)

1

C t (1 + i

t) (1 +  

t)

  =    t+11

C t+1 (1 + i

t+1) (1 +  

t+1)

 (1 + rt+1)

C t+1C t

=    t+1(1 + it) (1 +  t)

(1 + it+1) (1 +  t+1) (1 + rt+1)

C t+1C t

=    t+1 (1 + it)  P t (1 +  t)

P t+1 (1 +  t+1)  (4)

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Monetary policy at the zero lower bound

 Assume that some shock(s) has made the CIA constraint non-binding (say   0   "), such that the

economy starts in a “liquidity trap” with i0 = 0:

 Assume that at some  T > 0, iT  > 0, i.e., at some future date the zero lower bound will not bind (but

the CIA constraint will)

– E.g., if  it  is a short-term interest rate, then the observation of positive long-term rates, suggests

that this may not be a bad assumption

 The question is whether a change in the money stock,  M 0, can have any e¤ects on the economy?

 Assume a constant consumption tax, and combine (3) and (4):

C t+1C t

=  t+1 (1 + it)  P tP t+1

wt+1kt (1 + it) P twtkt+1 (1 + it+1) P t+1

=  t+1 (1 + it)  P tP t+1

wt+1ktwtkt+1 (1 + it+1)

 =  t+1

An “inverse Euler equation”:wt+1kt+1

= (1 + it+1)  t+1wtkt

(5)

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 So, as long as the economy is at the zero lower bound, for  t < T   1

wt+1kt+1

=  t+1wtkt

Nominal wage growth and price growth are determined (apparently) without reference to monetary

policy

 When the economy is out of the zero lower bound, for  t > T   1

C t+1C t

=  t+1 (1 + it)  P tP t+1

combined with the binding CIA constraint  M t = (1 +  t) P tC t   implies

M t+1M t

=  t+1 (1 + it)   (7)

Inserted into the “inverse Euler equation”

wtkt

=   t t+1

M t+1M t

wt1kt1

 So, out of the zero lower bound, nominal wage growth (and price growth) is determined by nominal

money growth.

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 For t  =  T   1 we haveC T 

C T 1=  T 

P T 1P T 

;

and thus by the CIA constraint, which binds in T :

M T 

C T 1 (1 +  T ) =  T P T 1

and thus by the labor supply equation at  T   1, where iT 1 = 0,

M T kT 1wT 1

=  T 

or,

wT 1=kT 1 = M T = T    (*)

 This “terminal condition” shows that wages (and prices) just prior to getting out of the zero lower

bound is determined by money!

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 This “terminal condition” (*) “unravels” back to period–0  wages and prices:

w0k0

=QT 1

s=0   1s

M T  T 

t T   1   (9a’)

 This shows that a temporary increase in the money supply at  t   = 0  will have no e¤ects unless it

a¤ects the future money supply

– Con…rms the nature of the liquidity trap; current money is neutral in every sense of the word: It

does not a¤ect ANY variable

– Shows that quantitative easing nevertheless can have e¤ects on prices, if   future  money supply is

a¤ected; i.e., a permanent increase in the money supply will permanently increase all prices and

wages in the same proportion

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 In a sticky-price version of the model (with two types of monopolistically produced goods and two-

period price contracts), such an ability to a¤ect aggregate prices, will have bene…cial output e¤ects

 Crucial here is  expectations about future policy  (and thus the ability to commit)

– If quantitative easing is expected to be reversed, it may have no e¤ects on prices and output

– Echoes Paul Krugman’s now famous statements (1998,  Brookings Papers on Economic Activity )

about the need for central banks to obtain credibility of being “irresponsible” when in a liquiditytrap

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