Ch07 Christiano et al

CHAPTER7

DSGE Models for Monetary

Policy Analysis$

Lawrence J.Christiano,*Mathias Trabandt,**and Karl Walentin{

*Department of Economics,Northwestern University

**European Central Bank,Germany and Sveriges Riksbank,Sweden

{Research Division,Sveriges Riksbank,Sweden

Contents

1.Introduction286

2.Simple Model289

2.1Private economy290

2.1.1Households290

2.1.2Firms290

2.1.3Aggregate resources and the private sector equilibrium conditions294

2.2Log-linearized equilibrium with Taylor rule296

2.3Frisch labor supply elasticity299

3.Simple Model:Some Implications for Monetary Policy302

3.1Taylor principle303

3.2Monetary policy and inefficient booms309

3.3Using unemployment to estimate the output gap311

3.3.1A measure of the information content of unemployment311

3.3.2The CTW model of unemployment312

3.3.3Limited information Bayesian inference315

3.3.4Estimating the output gap using the CTW model319

3.4Using HP-filtered output to estimate the output gap326

4.Medium-Sized DSGE Model331

4.1Goods production331

4.2Households334

4.2.1Households and the labor market335

4.2.2Wages,employment and monopoly unions338

4.2.3Capital accumulation340

4.2.4Household optimization problem343

4.3Fiscal and monetary authorities and equilibrium344

4.4Adjustment cost functions344

5.Estimation Strategy345

5.1VAR step345

$We are grateful for advice from Michael Woodford and for comments from Volker Wieland.The views expressed in

this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the

European Central Bank or of Sveriges Riksbank.We are grateful for assistance from Daisuke Ikeda and Matthias Kehrig.

Handbook of Monetary Economics,Volume3A#2011Elsevier B.V.

ISSN0169-7218,DOI:10.1016/S0169-7218(11)03007-3All rights reserved.285

286Lawrence J.Christiano et al.

5.2Impulse response matching step347

5.3Computation of V348

5.4Laplace approximation of the posterior distribution350

6.Medium-Sized DSGE Model:Results351

6.1VAR results351

6.1.1Monetary policy shocks351

6.1.2Technology shocks355

6.2Model results355

6.2.1Parameters355

6.2.2Impulse responses358

6.3Assessing VAR robustness and accuracy of the Laplace approximation360

7.Conclusion362

References364

Abstract

Monetary DSGE models are widely used because they fit the data well and they can be used to

address important monetary policy questions.We provide a selective review of these

developments.Policy analysis with DSGE models requires using data to assign numerical

values to model parameters.The chapter describes and implements Bayesian moment

matching and impulse response matching procedures for this purpose.

JEL Classification:E2,E3,E5,J6

Keywords

Frisch Labor Supply Elasticity

HP Filter

Impulse Response Function

Limited Information Bayesian Estimation

Materials Input for Production

New Keynesian DSGE Models

Output Gap

Potential Output

Taylor Principle

Unemployment

Vector Autoregression

Working Capital Channel

1.INTRODUCTION

There has been enormous progress in recent years in the development of dynamic,sto-chastic general equilibrium(DSGE)models for the purpose of monetary policy analy-sis.These models have been shown to fit aggregate data well by conventional econometric measures.For example,they have been shown to do as well or better than simple atheoretical statistical models at forecasting outside the sample of data on which they were estimated.In part because of these successes,a consensus has formed around

a particular model structure,the New Keynesian model.

Our objective is to present a selective review of these developments.We present seve-ral examples to illustrate the kind of policy questions the models can be used to address. We also convey a sense of how well the models fit the data.In all cases,our discussion takes place in the simplest version of the model required to make our point.As a result, we do not develop one single model.Instead,we work with several models.

We begin by presenting a detailed derivation of a version of the standard New Keynesian model with price-setting frictions and no capital or other complications.We then use versions of this simple model to address several important policy issues.For example,the past few decades have witnessed the emergence of a consensus that mone-tary policy ought to respond aggressively to changes in actual or expected inflation.This prescription for monetary policy is known as the“Taylor principle.”The standard version of the simple model is used to articulate why this prescription is a good one.However, alternative versions of the model can be used to identify potential pitfalls for the Taylor principle.In particular,a policy-induced rise in the nominal interest rate may destabilize the economy by perversely giving a direct boost to inflation.This can happen if the standard model is modified to incorporate a so-called working capital channel,which corresponds to the assumption that firms must borrow to finance their variable inputs.

We then turn to the much discussed issue of the interaction between monetary pol-icy and volatility in asset prices and other aggregate economic variables.We explain how vigorous application of the Taylor principle could inadvertently trigger an ineffi-cient boom in output and asset prices.

Finally,we discuss the use of DSGE models for addressing a key policy question:How big is the gap between the level of economic activity and the best level that is achievable by policy?An estimate of the output gap not only provides an indication about how effi-ciently resources are being used,but in the New Keynesian framework,the output gap is also a signal of inflation https://www.wendangku.net/doc/6714201501.html,rmally,the unemployment rate is thought to pro-vide a direct observation on the efficiency of resource allocation.For example,a large increase in the number of people reporting to be“ready and willing to work”but not employed suggests,at least at a casual level,that resources are being wasted and that the output gap is negative.DSGE models can be used to formalize and assess these informal hunches.We do this by introducing unemployment into the standard New Keynesian model along the lines recently proposed in Christiano,Trabandt,and Walentin(2010a; CTW).We use the model to describe circumstances in which we can expect the unem-ployment rate to provide useful information about the output gap.We also report evi-dence suggesting that these conditions may be satisfied in the U.S.data.

Although the creators of the Hodrick and Prescott(1997;HP)filter never intended it to be used to estimate the New Keynesian output gap concept,it is often used for this purpose.We show that whether the HP filter is a good estimator of the gap depends sensitively on the details of the underlying model economy.This discussion involves a careful review of the intuition of how the New Keynesian model responds to shocks.Interestingly,a New Keynesian model fit to U.S.data suggests the condi-tions are satisfied for the HP filter to be a good estimator of the output gap.In our 287

DSGE Models for Monetary Policy Analysis

288Lawrence J.Christiano et al.

discussion,we explain that there are several caveats that must be taken into account before concluding that the HP filter is a good estimator of the output gap.

Policy analysis with DSGE models,even the simple analyses summarized earlier, require assigning values to model parameters.In recent years,the Bayesian approach to econometrics has taken over as the dominant one for this purpose.In conventional appli-cations,the Bayesian approach is a so-called full information procedure because the ana-lyst specifies the joint likelihood of the available observations in complete detail.As a result,many of the limited information tools in macroeconomists’econometric toolbox have been deemphasized in recent times.These tools include methods that match model and data second moments and that match model and empirical impulse response func-tions.Following the work of Chernozhukov and Hong(2003),Kim(2002),Kwan (1999)and others,we show how the Bayesian approach can be applied in limited infor-mation contexts.We apply a Bayesian moment matching approach in Section3.3.3and a Bayesian impulse response function matching approach in Section5.2.

The new monetary DSGE models are of interest not just because they represent laboratories for the analysis of important monetary policy questions.They are also of interest because they appear to resolve a classic empirical puzzle about the effects of monetary policy.It has long been thought that it is virtually impossible to explain the very slow response of inflation to a monetary disturbance without appealing to completely implausible assumptions about price frictions(see,e.g.,Mankiw,2000).

However,it turns out that modern DSGE models do provide an account of the inertia in inflation and the strong response of real variables to monetary policy disturbances, without appealing to seemingly implausible parameter values.Moreover,the models simultaneously explain the dynamic response of the economy to other shocks.We review these important findings.We explain in detail the contribution of each feature of the consensus medium-sized New Keynesian model in achieving this result.This discussion closely follows the analyses in Christiano,Eichenbaum,and Evans(2005;

CEE)and Altig,Christiano,Eichenbaum,and Linde′(2005;ACEL).

There is an econometric technique that is particularly well-suited to the shock-based analysis described in the previous paragraph.It is the one that matches impulse response functions estimated by vector autoregressions(VARs)with the corresponding objects in a https://www.wendangku.net/doc/6714201501.html,ing U.S.macroeconomic data,we show how the parameters of the consen-sus DSGE model are estimated by this impulse response matching procedure.The advan-tage of this econometric approach is transparency and focus.The transparency reflects that the estimation strategy has a simple graphical representation,involving objects—impulse response functions—about which economists have strong intuition.The advantage of focus comes from the possibility of studying the empirical properties of a model without having to specify a full set of shocks.As noted previously,we show how to implement the impulse response matching strategy using Bayesian methods.In particular,we are able to implement all the machinery of priors and posteriors,as well as the marginal likelihood as

a measure of model fit in our impulse response function matching exercise.

This chapter is organized as follows.Section2describes the simple New Keynesian model without capital.The following section reviews some policy implications of that model.The medium-sized version of the model,designed to econometrically address a rich set of macroeconomic data,is described in Section4.Section5reviews our Bayesian impulse response matching strategy.Section6reviews the results,and conclusions are offered in Sec-tion7.Many algebraic derivations are relegated to a separate technical appendix.1

2.SIMPLE MODEL

This section analyzes versions of the standard Calvo-sticky price New Keynesian model without capital.In practice,the analysis of the standard New Keynesian model often begins with the familiar three equations:the linearized“Phillips curve,”“IS curve,”and monetary policy rule.We cannot simply begin with these three equations here because we also study departures from the standard model.For this reason,we must derive the equilibrium conditions from their foundations.

The version of the New Keynesian model studied in this section is the one considered in Clarida,Gali,and Gertler(1999)and Woodford(2003),modified in two ways.First, we introduce the working capital channel emphasized by CEE and Barth and Ramey (2002).2The working capital channel results from the assumption that firms’variable inputs must be financed by short-term loans.With this assumption,changes in the interest rate affect the economy by changing firms’variable production costs,in addition to operating through the usual spending mechanism.There are several reasons to take the working capital channel https://www.wendangku.net/doc/6714201501.html,ing U.S.Flow of Funds data,Barth and Ramey (2002)argued that a substantial fraction of firms’variable input costs are borrowed in advance.Christiano,Eichenbaum,and Evans(1996)provided VAR evidence suggesting the presence of a working capital channel.Chowdhury,Hoffmann,and Schabert(2006) and Ravenna and Walsh(2006)provided additional evidence supporting the working capital channel,based on instrumental variables estimates of a suitably modified Phillips curve.Finally,Section4shows that incorporating the working capital channel helps to explain the“price puzzle”in the VAR literature and provides a response to Ball’s (1994)“dis-inflationary boom”critique of sticky price models.

We explore a second modification to the classic New Keynesian model by incor-porating the assumption about materials inputs proposed in Basu(1995).Basu argued that a large part—as much as half—of a firm’s output is used as inputs by other firms. The working capital channel introduces the interest rate into costs while the materials assumption makes those costs big.In the next section we show that these two factors have potentially far-reaching consequences for monetary policy.

1The technical appendix can be found at https://www.wendangku.net/doc/6714201501.html,/faculty/christiano/research/ Handbook/technical_appendix.pdf.

2The first monetary DSGE model we are aware of that incorporates a working capital channel is Fuerst(1992).Other early examples include Christiano(1991)and Christiano and Eichenbaum(1992b).289

DSGE Models for Monetary Policy Analysis

This section is organized as follows.We begin in subsection 2.1by describing the private sector of the economy,and deriving equilibrium conditions associated with optimization and market clearing.In subsection 2.2,we specify the monetary policy rule and define the Taylor rule equilibrium.Subsection 2.3discusses the interpretation of a key parameter in our utility function.The parameter controls the elasticity with which the labor input in our model economy adjusts in response to a change in the real wage.Traditionally,this parameter has been viewed as being restricted by microeco-nomic evidence on the Frisch labor supply elasticity.We summarize recent thinking stimulated by the seminal work of Rogerson (1988)and Hansen (1985),according to which this parameter is not restricted by evidence on the Frisch elasticity.

2.1Private economy

2.1.1Households

We suppose there is a large number of identical households.The representative house-hold solves the following problem:max C t ;H t ;B t t1f g E 0X 1t ?0

b t log C t àH 1tf t 1tf !;0

P t C t tB t t1 B t R t à1tW t H t tTransfers and profits t :e2T

Here,C t and H t denote household consumption and market work,respectively.In Eq.(2),B t t1denotes the quantity of a nominal bond purchased by the household in period t and R t denotes the one-period gross nominal rate of interest on a bond pur-chased in period t .Finally,W t denotes the competitively determined nominal wage rate.The parameter,f ,is discussed in Section 2.3.

The representative household equates the marginal cost of working,in consump-tion units,with the marginal benefit,the real wage:

C t H f t ?W t P t :e3T

The representative household also equates the utility cost of the consumption foregone in acquiring a bond with the corresponding benefit:

1C t ?b E t 1C t t1R t p t t1

:e4T

Here,p t t1denotes the gross rate of inflation from t to t t1.

2.1.2Firms

A key feature of the New Keynesian model is its assumption that there are price-setting frictions.These frictions are introduced to accommodate the evidence of inertia in 290Lawrence J.Christiano et al.

aggregate inflation.Obviously,the presence of price-setting frictions requires that firms have the power to set prices,and this in turn requires the presence of monopoly power.A challenge is to create an environment in which there is monopoly power,without contradicting the obvious fact that actual economies have a very large number of firms.The Dixit-Stiglitz framework of production handles this challenge very nicely,because it has a very large number of price-setting monopolist firms.In particular,gross output is produced using a representative,competitive firm using the following technology:

Y t?

e1

Y

1

l f

i;t

di

l

f

;l f>1;e5T

where l f governs the degree of substitution between the different inputs.The repre-sentative firm takes the price of gross output,P t,and the price of intermediate inputs, P i,t,as given.Profit maximization leads to the following first-order condition:

Y i;t?Y t

P i;t

P t

àl f

l fà1

:e6T

Substituting Eq.(6)into Eq.(5)yields the following relation between the aggregate price level and the prices of intermediate goods:

P t?

e1

P

à1

l fà1

i;t

di

àel

f

à1T

:e7T

The i th intermediate good is produced by a single monopolist,who takes Eq.(6)as its demand curve.The value of l f determines how much monopoly power the i th pro-ducer has.If l f is large,then intermediate goods are poor substitutes for each other, and the monopoly supplier of good i has a lot of market power.Consistent with this, note that if l f is large,then the demand for Y i,t is relatively price inelastic(see Eq.6). If l f is close to unity,so that each Y i,t is almost a perfect substitute for Y j,t,j?i,then the i th firm faces a demand curve that is almost perfectly elastic.In this case,the firm has virtually no market power.

The production function of the i th monopolist is:

Y i;t?z t H g i;t I1àg

i;t

;0

The nominal marginal cost of the intermediate good producer is the following Cobb-Douglas function of the price of its two inputs:291

DSGE Models for Monetary Policy Analysis

marginal cost t?

P

t

1àg

1àg

W t

g

g

1

z t

:

Here, W and P are the effective prices of H i,t,and I i,t,respectively:

W t ?e1àv tTe1àctc R tTW t

P

t

?e1àv tTe1àctc R tTP t:e9TIn this expression,n t denotes a subsidy to intermediate good firms and the term involv-ing the interest rate reflects the presence of a“working capital channel.”For example, c?1corresponds to the case where the full amount of the cost of labor and materials must be financed at the beginning of the period.When c?0,no advanced financing is required.A key variable in the model is the ratio of nominal marginal cost to the price of gross output,P t:

s t?e1àv tT

1

1àg

1àg

w t

g

g

e1àctc R tT;e10T

where w t denotes the scaled real wage rate:

w t W t

z1g t Pt

:e11T

If intermediate good firms faced no price-setting frictions,they would all set their price as a fixed markup over nominal marginal cost:

l f P t s t:e12TIn fact,we assume there are price-setting frictions along the lines proposed by Calvo (1983).An intermediate firm can set its price optimally with probability1àx p,and with probability x p it must keep its price unchanged relative to what it was in the pre-vious period:

P i;t?P i;tà1:

Consider the1àx p intermediate good firms that are able to set their prices optimally in period t.There are no state variables in the intermediate good firm problem and all the firms face the same demand curve.As a result,all firms able to optimize their prices in period t choose the same price,which we denote by e P t.It is clear that optimizing firms do not set e P t equal to Eq.(12).Setting e P t to Eq.(12)would be optimal from the perspective of the current period,but it does not take into account the possibility that the firm may be stuck with e P t for several periods into the future.Instead,the intermediate good firms that have an opportunity to reoptimize their price in the current period,do so to solve:

max e P t E t

X1

j?0

ex p bTj u ttj e P t Y i;ttjàP ttj s ttj Y i;ttj

àá

;e13T

292Lawrence J.Christiano et al.

subject to the demand curve,Eq.(6),and the definition of marginal cost,Eq.(10).In Eq.(13),b j u t tj is the multiplier on the household’s nominal period t tj budget constraint.Because they are the owners of the intermediate good firms,households are the recipients of firm profits.In this way,it is natural that the firm should weigh profits in different dates and states of nature using b j u t tj .Intermediate good firms take u t tj as given.The nature of the family’s preferences,Eq.(1),implies:

u t tj ?1P t tj C t tj

:In Eq.(13)the presence of x p reflects that intermediate good firms are only concerned with future scenarios in which they are not able to reoptimize the price chosen in period t .

The first-order condition associated with Eq.(13)is

e p t ?

E t P 1j ?0ebx p Tj eX t ;j Tàl f l f à1l f s t tj E t P 1j ?0ebx p Tj eX i ;j Tà1l f à1?K f t

F f t ;e14Twhere K f t and F f t denote the numerator and denominator of the ratio after the first

equality,respectively.Also,e p t e P t P t ;X t ;j 1p t tj áááp t tj j >01j ?0

:8<:Not surprisingly,Eq.(14)implies e P

t is set to Eq.(12)when x p ?0.When x p >0,optimizing firms set their prices so that Eq.(12)is satisfied on average.It is useful to write the numerator and denominator in Eq.(14)in recursive form.Thus,

K f t ?l f s t tbx p E t p l f l f à1t t1K f t t1;

e15TF f t ?1tbx p E t p 1l f à1t t1F f

t t1:e16TExpression (7)simplifies when we take into account that (i)the 1àx p intermediate good firms that set their price optimally all set it to e P

t and (ii)the x p firms that cannot reset their price are selected at random from the set of all firms.Doing so,

e p t ?1àx p p 1l

f à1t 1àx p

2435àel f à1T:e17TIt is convenient to use Eq.(17)to eliminate e p t in Eq.(14):

293

DSGE Models for Monetary Policy Analysis

K f t?F f t

1àx p p

1

l fà1

t

1àx p

@

1

A

àel fà1T

:e18T

When g<1,cost minimization by the i th intermediate good producer leads it to equate the relative price of its labor and materials inputs to the corresponding relative marginal productivities:

W t P t ?W t

P t

?g

1àg

I i;t

H i;t

?g

1àg

I t

H t

:e19T

Evidently,each firm uses the same ratio of inputs,regardless of its output price,P i,t.

2.1.3Aggregate resources and the private sector equilibrium conditions

A notable feature of the New Keynesian model is the absence of an aggregate production function.That is,given information about aggregate inputs and technology,it is not pos-sible to say what aggregate output,Y t,is.This is because Y t also depends on how inputs are distributed among the various intermediate good producers.For a given amount of aggre-gate inputs,Y t is maximized by distributing the inputs equally across producers.An unequal distribution of inputs results in a lower level of Y t.In the New Keynesian model with Calvo price frictions,resources are unequally allocated across intermediate good firms if,and only if,P i,t differs across i.Price dispersion in the model is caused by the inter-action of inflation with price-setting frictions.With price dispersion,the price mecha-nism ceases to allocate resources efficiently,as too much production is done in firms with low prices and too little in the firms with high prices.Yun(1996)derived a very sim-ple formula that characterizes the loss of output due to price dispersion.We re-derive the analog of Yun’s(1996)formula that is relevant for our setting.

Let Y?t denote the unweighted integral of gross output across intermediate good producers:

Y?t e1

Y i;t di?

e1

z t

H i;t

I i;t

g

I i;t di?z t

H t

I t

g

I t?z t H g t I1àg

t

:

Here,we have used linear homogeneity of the production function,as well as the result in Eq.(19),that all intermediate good producers use the same labor to materials ratio.An alternative representation of Y?t makes use of the demand curve,Eq.(6):

Y?t?Y t e1

P i;t

P t

àl f

fà1

di?Y t P

l f

l fà1

t

e1

eP i;tTà

l f

l fà1di?Y t P

l f

l fà1

t

eP?

t

Tà

l f

l fà1:e20T

Thus,

Y t?p?t z t H g t I1àg

t ;

294Lawrence J.Christiano et al.

where

p?t

P?t

P t

l f

l fà1

:e21T

Here,P?t1denotes Yun’s(1996)measure of the output lost due to price dispersion. From Eq.(20),

P?t?e1

eP i;tTà

l f

l fà1di

!àl fà1

l f

:e22T

According to Eq.(21),P?t is a monotone function of the ratio of two different weighted averages of intermediate good prices.The ratio of these two weighted averages can only be at its maximum of unity if all prices are the same.3 Taking into account observations(i)and(ii)after Eq.(16),Eq.(22)reduces(after dividing by P t and taking into account Eq.21)to:

p?t?e1àx pT1àx p p

1

l fà1

t

1àx p

0 @1

A

l f

tx p p

l f

l fà1

t

p?tà1

2 643

75

à1

:e23T

According to Eq.(23),there is price dispersion in the current period if there was dis-persion in the previous period and/or if there is a current shock to dispersion.Such a shock must operate through the aggregate rate of inflation.

We conclude that the relation between aggregate inputs and gross output is given by:

C ttI t?p?t z t H g t I1àg

t

:e24THere,C ttI t represents total gross output,while C t represents value added.

The private sector equilibrium conditions of the model are Eqs.(3),(4),(10),(15), (16),(18),(19),(23),and(24).This represents9equations in the following11 unknowns:

C t;H t;I t;R t;p t;p?t;K f t;F f t;W t

P t

;s t;v t:e25T

3The distortion,p?

t ,is of interest in its own right.It is a sort of“endogenous Solow residual”of the kind called for by

Prescott(1998).Whether the magnitude of fluctuations in p?t are quantitatively important given the actual price dispersion in data is something that deserves exploration.A difficulty that must be overcome,in such an exploration, is determining what the benchmark efficient dispersion of prices is in the data.In the model it is efficient for all prices to be exactly the same,but that is obviously only a convenient normalization.295

DSGE Models for Monetary Policy Analysis

As it stands,the system is underdetermined.This is not surprising,since we have said nothing about monetary policy or how n t is determined.We turn to this in the follow-ing section.

2.2Log-linearized equilibrium with Taylor rule

We log-linearize the equilibrium conditions of the model about its nonstochastic steady state.We assume that monetary policy is governed by a Taylor rule,which responds to the deviation between actual inflation and a zero inflation target.As a result,inflation is zero in the nonstochastic steady state.In addition,we suppose that the intermediate good subsidy,n t ,is set to the constant value that causes the price of goods to equal the social marginal cost of production in steady state.To see what this implies for n t ,recall that in steady state firms set their price as a markup,l f ,over mar-ginal cost.That is,they equate the object in Eq.(12)to P t ,so that:

l f s ?1:

Using Eq.(10)to substitute out for the steady state value of s ,the latter expression reduces,in steady state,to:l t e1àn Te1àc tc R T11àg 1àg w g

g "#?1:Because we assume competitive labor markets,the object in square brackets is the ratio of social marginal cost to price.As a result,it is socially efficient for this expression to equal unity.This is accomplished in the steady state by setting n as follows:

1àn ?1l f e1àc tc R T:e26T

Our treatment of policy implies that the steady-state allocations of our model economy are efficient in the sense that they coincide with the solution to a particular planning prob-lem.To define this problem,it is convenient to adopt the following scaling of variables:

c t

C t z 1=g t ;i t I t z 1=g t :e27TThe planning problem is:

max c t ;H t ;i t f g E 0X

1t ?0b t log c t àH 1tf t 1tf "#;subject to c t ti t ?H g t i 1àg t :e28T

The problem,(28),is that of a planner who allocates resources efficiently across interme-diate goods and who does not permit monopoly power distortions.Because there is no

296Lawrence J.Christiano et al.

state variable in the problem,it is obvious that the choice variables that solve Eq.(28)are constant over time.This implies that the C t and I t that solve the planning problem are a fixed proportion of z1=g t over time.It turns out that the allocations that solve Eq.(28)also solve the Ramsey optimal policy problem of maximizing Eq.(1)with respect to the11 variables listed in Eq.(25)subject to the9equations listed before Eq.(25).4 Because inflation,p t,fluctuates in equilibrium,Eq.(23)suggests that p?t fluctuates too.It turns out,however,that p?t is constant to a first-order approximation.To see this,note that the absence of inflation in the steady state also guarantees there is no price dispersion in steady state in the sense that p?t is at its maximal value of unity (see Eq.23).With p?t at its maximum in steady state,small perturbations have a zero first-order impact on p?t.This can be seen by noting that p t is absent from the log-linear expansion of Eq.(23)about p?t?1:

^p?t?x p^p?tà1:e29THere,a hat over a variable indicates:

^%t?d%t

%

;

where%denotes the steady state of the variable,%t,and d%t?%tà%denotes a small perturbation in%t from steady state.We suppose that in the initial period,^p?tà1?0, so that,to a first-order approximation,^p?t?0for all t.

Log-linearizing Eqs.(15),(16),and(18)we obtain the usual representation of the Phillips curve:

^p t?e1àbx pTe1àx pT

x p

^s ttb E t^p tt1:e30T

Combining Eq.(3)with Eq.(10),taking into account Eq.(27)and the setting of n in Eq.(26),real marginal cost is:

s t?1

l f

1àctc R t

1àctc R

1

1àg

1àg

c t H f t

g

!g

:

Then,

^s t?gef^H tt^c tTtc^R t:e31TSubstituting out for the real wage in Eq.(19)using Eq.(3)and applying Eq.(27),

4The statement in the text is strictly true only in the case where the initial distortion in prices is zero,that is p?

tà1?1.

If this condition does not hold,then the statement still holds asymptotically and may even hold as an approximation after a small number of periods.297

DSGE Models for Monetary Policy Analysis

H ft1 t c t?

g

1àg

i t:e32T

Similarly,scaling Eq.(24):

c tti t?H g t i1àg t:

Using Eq.(32)to substitute out for i t in the above expression,we obtain:

c tt1àg

g

H ft1

t

c t?H g t

1àg

g

H ft1

t

c t

!1àg

:

Log-linearizing this expression around the steady state implies,after some algebra,

^c t?^H t:e33TSubstituting the latter into Eq.(31),we obtain:

^s t?ge1tfT^c ttc

e1àcTbtc ^R

t:e34T

In Eq.(34),c?t is the percent deviation of c t from its steady-state value.Since this steady-state value coincides with the constant c t that solves Eq.(28)for each t,c?t also corre-sponds to the output gap.The notation we use to denote the output gap is x https://www.wendangku.net/doc/6714201501.html,ing this notation for the output gap and substituting out s?t in the Phillips curve,we obtain:

^p t?k p ge1tfTx ttc

e1àcTbtc ^R

t

!

tb E t^p tt1;e35T

where

k p e1àbx pTe1àx pT

x p

:

When g?1and c?0,Eq.(35)reduces to the“Phillips curve”in the classic New Keynesian model.When materials are an important factor of production,so that g is small,then a given jump in the output gap,x t,has a smaller impact on inflation.The reason is that in this case the aggregate price index is part of the input cost for intermediate good producers.So,a small price response to a given output gap is an equilibrium because individual intermediate good firms have less of an incentive to raise their prices in this case.With c>0,Eq.(35)indicates that a jump in the interest rate drives up prices.This is because with an active working capital channel a rise in the interest rate drives up marginal cost.5

5Equation(35)resembles equation(13)in Ravenna and Walsh(2006),except that we also allow for materials inputs,

i.e.,g<1.

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Now consider the intertemporal Euler equation.Expressing (4)in terms of scaled variables,

1?E t b c t

c t t1m 1g z ;t t1R t p t t1;m z ;t t1 z t t1z t

:Log-linearly expanding about steady state and recalling that c ?t corresponds to the output gap:0?E t x t àx t t1à1g ^m z ;t t1

t^R t à^p t t1 !;or,x t ?E t x t t1à^R t à^p

t t1à^R ?t àá??;e36TWhere ^R ?t 1g E t ^m z ;t t1:e37T

We suppose that monetary policy,when linearized about steady state,is characterized by the following Taylor rule:

^R t ?r p E t ^p t t1tr x x t :e38T

The equilibrium of the log-linearly expanded economy is given by Eq.(35)to (38).

2.3Frisch labor supply elasticity

The magnitude of the parameter,f ,in the household utility function plays an impor-tant role in the analysis in later sections.This parameter has been the focus of much debate in macroeconomics.Note from Eq.(3)that the elasticity of H t with respect to the real wage,holding C t constant,is 1/f .The condition,“holding C t constant,”could mean that the elasticity refers to the response of H t to a change in the real wage that is of very short duration,so short that the household’s wealth —and,hence,con-sumption —is left unaffected.Alternatively,the elasticity could refer to the response of H t to a change in the real wage that is associated with an offsetting lump-sum transfer payment that keeps wealth unchanged.The debate about f centers on the interpreta-tion of H t .Under one interpretation,H t represents the amount of hours worked by a typical person in the labor force.With this interpretation,1/f is the Frisch labor supply elasticity.6This is perhaps the most straightforward interpretation of 1/f given our 6The Frisch labor supply elasticity refers to the substitution effect associated with a change in the wage rate.It is the percent change in a person’s labor supply in response to a change in the real wage,holding the marginal utility of consumption fixed.Throughout this chapter,we assume that utility is additively separable in consumption and leisure,so that constancy of the marginal utility of consumption translates into constancy of consumption.

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assumption that the economy is populated by identical households,in which H t is the labor effort of the typical household.An alternative interpretation of H t is that it repre-sents the number of people working,and that1/f measures the elasticity with which marginal people substitute in and out of employment in response to a change in the wage.Under this interpretation,1/f need not correspond to the labor supply elasticity of any particular person.The two different interpretations of H t give rise to very differ-ent views about how data ought to be used to restrict the value of f.

There is an influential labor market literature that estimates the Frisch labor supply elasticity using household level data.The general finding is that,although the Frisch elasticity varies somewhat across different types of people,on the whole the elasticities are very small.Some have interpreted this to mean that only large values of f(say, larger than unity)are consistent with the data.Initially,this interpretation was widely accepted by macroeconomists.However,the interpretation gave rise to a puzzle for equilibrium models of the business cycle.Over the business cycle,employment fluctu-ates a great deal more than real wages.When viewed through the prism of equilibrium models the aggregate data appeared to suggest that people respond elastically to changes in the wage.But,this seemed inconsistent with the microeconomic evidence that indi-vidual labor supply elasticities are in fact small.At the present time,a consensus is emerging that what initially appeared to be a conflict between micro and macro data is really no conflict at all.The idea is that the Frisch elasticity in the micro data and the labor supply elasticity in the macro data represent at best distantly related objects.

It is well known that much of the business cycle variation in employment reflects changes in the quantity of people working,not in the number of hours worked by a typical household.Beginning at least with the work of Rogerson(1988)and Hansen (1985),it has been argued that even if the individual’s labor supply elasticity is zero over most values of the wage,aggregate employment could nevertheless respond highly elas-tically to small changes in the real wage.This can occur if there are many people who are near the margin between working in the market and devoting their time to other activities.An example is a spouse who is doing productive work in the home,and yet who might be tempted by a small rise in the market wage to substitute into the market.

Another example is a teenager who is close to the margin between pursuing additional education and working,who could be induced to switch to working by a small rise in the wage.Finally,there is the elderly person who might be induced by a small rise in the market wage to delay retirement.These examples suggest that aggregate employ-ment might fluctuate substantially in response to small changes in the real wage,even if the individual household’s Frisch elasticity of labor supply is zero over all values of the wage,except the one value that induces a shift in or out of the labor market.7

7See Rogerson and Wallenius(2009)for additional discussion and analysis.

The ideas in the previous paragraphs can be illustrated in our model.We adopt the technically convenient assumption that the household has a large number of members, one for each of the points on the line bounded by0and1.8In addition,we assume that a household member only has the option to work full time or not at all.A household member’s Frisch labor supply elasticity is zero for almost all values of the wage.Let l2 [0,1]index a particular member in the family.Suppose this member enjoys the follow-ing utility if employed:

log C tàl f;f>0;

and the following utility if not employed:

log C t:

Household members are ordered according to their degree of aversion to work.Those with high values of l have a high aversion(e.g.,small children,and elderly or chroni-cally ill people)to work,and those with l near zero have very little aversion.We sup-pose that household decisions are made on a utilitarian basis,in a way that maximizes the equally weighted integral of utility across all household members.Under these cir-cumstances,efficiency dictates that all members receive the same level of consumption, whether employed or not.In addition,if H t members are to be employed,then those with0l H t should work and those with l>H t should not.For a household with consumption,C t,and employment,H t,utility is,after integrating over all l2[0,1]:

log C tàH1tf

t

1tf

;e39T

which coincides with the period utility function in Eq.(1).Under this interpretation of the utility function,Eq.(3)remains the relevant first-order condition for labor.In this case,given the wage,W t/P t,the household sends out a number of members,H t,to work until the utility cost of work for the marginal worker,H f t,is equated to the corresponding utility benefit to the household,(W t/P t)/C t.

Note that under this interpretation of the utility function,H t denotes a quantity of workers and f dictates the elasticity with which different members of the households enter or leave employment in response to shocks.The case in which f is large corre-sponds to the case where household members differ relatively sharply in terms of their aversion to work.In this case there are not many members with disutility of work close to that of the marginal worker.As a result,a given change in the wage induces only a small change in employment.If f is very small,then there is a large number of

8Our approach is most similar to the approach of Gali(2010a),although it also resembles the approach taken in the recent work of Mulligan(2001)and Krusell,Mukoyama,Rogerson,and Sahin(2008).301

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household members close to indifferent between working and not working,and so a small change in the real wage elicits a large labor supply response.

Given that most of the business cycle variation in the labor input is in the form of numbers of people employed,we think the most sensible interpretation of H t is that it measures numbers of people working.Accordingly,1/f is not to be interpreted as a Frisch elasticity,which we instead assume to be zero.

3.SIMPLE MODEL:SOME IMPLICATIONS FOR MONETARY POLICY

Monetary DSGE models have been used to gain insight into a variety of issues that are important for monetary policy.We discuss some of these issues using variants of the simple model developed in the previous section.A key feature of that model is that it is flexible,and can be adjusted to suit different questions and points of view.The classic New Keynesian model,the one with no working capital channel and no mate-rials inputs(i.e.,g?1,c?0)can be used to articulate the rationale for the Taylor principle.But variants of the New Keynesian framework can also be used to articulate challenges to that principle.Sections3.1and3.2below describe two such challenges.

The fact that the New Keynesian framework can accommodate a variety of perspec-tives on important policy questions is an important strength.This is because the frame-work helps to clarify debates and to motivate econometric analyses so that data can be used to resolve those debates.9

Sections3.3and3.4below address the problem of estimating the output gap.The output gap is an important variable for policy analysis because it is a measure of the effi-ciency with which economic resources are allocated.In addition,New Keynesian models imply that the output gap is an important determinant of inflation,a variable of particular concern to monetary policymakers.We define the output gap as the per-cent deviation between actual output and potential output,where potential output is output in the Ramsey-efficient equilibrium.10

We use the classic New Keynesian model to study three ways of estimating the out-put gap.The first uses the structure of the simple New Keynesian model to estimate the output gap as a latent variable.The second approach modifies the New Keynesian model to include unemployment along the lines indicated by CTW.This modification of the model allows us to investigate the information content of the unemployment rate for the output gap.In addition,by showing one way that unemployment can be integrated into the model,the discussion represents another illustration of the versatility

9For example,the Chowdhury,Hoffmann,and Schabert(2006)and Ravenna and Walsh(2006)papers cited in the

previous section,show how the assumptions of the New Keynesian model can be used to develop an empirical

characterization of the importance of the working capital channel.

10In our model,the Ramsey-equilibrium turns out to be what is often called the“first-best equilibrium,”the one that is not distorted by monopoly power or flexible prices.

of the New Keynesian framework.11The third approach which is studied in section 3.4explores the HP filter as a device for estimating the output gap.In the course of the analysis,we illustrate the Bayesian limited information moment matching procedure discussed in the introduction.

3.1Taylor principle

A key objective of monetary policy is the maintenance of low and stable inflation.The classic New Keynesian model defined by g ?1and c ?0can be used to articulate the risk that inflation expectations might become self-fulfilling unless the monetary autho-rities adopt the appropriate monetary policy.The classic model can also be used to explain the widespread consensus that “appropriate monetary”policy means a mone-tary policy that embeds the Taylor Principle:a 1%rise in inflation should be met by a greater than 1%rise in the nominal interest rate.This subsection explains how the classic New Keynesian model rationalizes the wisdom of implementing the Taylor principle.However,when we incorporate the assumption of a working capital channel —particularly when the share of materials in gross output is as high as it is in the data —the Taylor principle becomes a source of instability.This is perhaps not surprising.When the working capital channel is strong,if the monetary authority raises the interest rate in response to rising inflation expectations,the resulting rise in costs produces the higher inflation that people expect.12

It is convenient to summarize the linearized equations of our model here:

^R ?t ?E t 1g ^m z ;t t1

e40T^p t ?k p g e1tf Tx t ta c ^R t ??tb E t ^p t t1e41T

11

For an alternative recent approach to the introduction of unemployment into a DSGE model,see Gali (2010a).Gali demonstrated that with a modest reinterpretation of variables,the standard DSGE model with sticky wages summarized in the next section contains a theory of unemployment.In the model of the labor market used there (it was proposed by Erceg et al.2000)wages are set by a monopoly union.As a result,the wage rate is higher than the marginal cost of working.Under these circumstances,one can define the unemployed as the difference between the number of people actually working and the number of people that would be working if the cost of work for the marginal person were equated to the wage rate.Gali (2010b)showed how unemployment data can be used to help estimate the output gap,as we do here.The CTW and Gali models of unemployment are quite different.For example,in the text we analyze a version of the CTW model in which labor markets are perfectly competitive,so Gali’s “monopoly power”concept of unemployment is zero in this model.In addition,the efficient level of unemployment in the sense that we use the term here,is zero in Gali’s definition,but positive in our definition.This is because in our model,unemployment is an inevitable by-product of an activity that must be undertaken to find a job.For an extensive discussion of the differences between our model and Gali’s,see Section F in the technical appendix to CTW,which can be found at https://www.wendangku.net/doc/6714201501.html,/$lchrist/research/Riksbank/technicalappendix.pdf .12Bruckner and Schabert (2003)made an argument similar to ours,although they do not consider the impact of materials inputs,which we find to be important.

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x t ?E t x t t1à^R t à^p t t1à^R ?t

àá??e42T^R

t ?r p E t ^p t t1tr x x t ;e43T

where

a c ?c e1àc T

b t

c :The specification of the model is complete when we take a stan

d on th

e law o

f motion for the exogenous shock.We do this in the followin

g subsections as needed.

We begin by reviewing the case for the Taylor principle using the classic New Keynesian model,with g ?1,c ?0.We get to the heart of the argument using the deterministic version of the model,in which ^R ?t 0.In addition,it is convenient to suppose that monetary policy is characterized by r x ?0.Throughout,we adopt the presumption that the only valid equilibria are paths for ^p

t ,^R t and x t that converge to steady state;that is,0.13Under these circumstances,Eqs.(41)and (42)can be solved forward as follows:

^p

t ?k p g e1tf Tx t tbk p g e1tf Tx t t1tb 2k p g e1tf Tx t t2t...e44Tand x t ?à^R t à^p t t1àáà^R t t1à^p t t2àáà^R t t2à^p t t3àáà...e45TIn Eq.(45)we have used the fact that in our setting a path converges to zero if,and only if,it converges fast enough so that a sum like the one in Eq.(45)is well defined.14Equation (44)shows that inflation is a function of the present and future output gap.Equation (45)shows that the current output gap is a function of the long term real interest rate (i.e.,the sum on the right of Eq.45)in the model.

Under the Taylor principle,the classic New Keynesian model implies that a rise in inflation expectations launches a sequence of events that ultimately leads to a 13

Although our presumption is standard,justifying it is harder than one might have thought.For example,Benhabib,Schmitt-Grohe,and Uribe (2002)presented examples in which some explosive paths for the linearized equilibrium conditions are symptomatic of perfectly sensible equilibria for the actual economy underlying the linear approximations.In these cases,focusing on the nonexplosive paths of the linearized economy may be valid after all if we imagine that monetary policy is a Taylor rule with a particular escape clause.The escape clause specifies that in the event the economy threatens to follow an explosive path,the monetary authority commits to switch to a monetary policy of targeting the money growth rate.There are examples of monetary models in which the escape clause monetary policy justifies the type of equilibrium selection we adopt in the text (see Benhabib et al.2002and Christiano &Rostagno,2001for further discussion).For a more recent debate about the validity of the equilibrium selection adopted in the text,see McCallum (2009)and Cochrane (2009)and the references they cite.14The reason for this can be seen below,where we show that the solution to this equation is a linear combination of terms like a l t .Such an expression converges to zero if,and only if,it is also summable.304Lawrence J.Christiano et al.