NBER WORKING PAPER SERIES

SHOOTING THE AUCTIONEER

Roger E. A. FarmerAndrew Hollenhorst

Working Paper 12584http://www.nber.org/papers/w12584

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138October 2006

This research was supported by NSF grant SBR 0418174. The authors wish to thank Stephanie SchmittGrohe and Martin Uribe for helpful and extensive comments on an earlier draft. We also wish to thankparticipants at the NBER summer institute in 2005 and seminar particiapnts at Duke University andthe University of British Columbia. The views expressed herein are those of the author(s) and do notnecessarily reflect the views of the National Bureau of Economic Research.

© 2006 by Roger E. A. Farmer and Andrew Hollenhorst. All rights reserved. Short sections of text,not to exceed two paragraphs, may be quoted without explicit permission provided that full credit,including © notice, is given to the source.

Shooting the AuctioneerRoger E. A. Farmer and Andrew HollenhorstNBER Working Paper No. 12584October 2006JEL No. E32,J6

ABSTRACT

Most dynamic stochastic general equilibrium models of the macroeconomy assume that labor is tradedin a spot market. Two exceptions by David Andolfatto and Monika Merz combine a two-sided searchmodel with a one-sector real business cycle model. These hybrid models are successful, in some dimensions,but they cannot account for observed volatility in unemployment and vacancies. Following suggestionsby Robert Hall and Robert Shimer, this paper shows that a relatively standard DSGE model with stickywages can account for these facts. Using a second-order approximation to the policy function we simulatemoments of an artificial economy with and without sticky wages and we document the dependenceof unemployment and vacancy volatility on two key parameters; the disutility of effort and the degreeof wage stickiness. We compute the welfare costs of the sticky wage equilibrium and find them tobe small.

Roger E. A. FarmerUC, Los AngelesDepartment of EconomicsBox 951477Los Angeles, CA 90095-1477and NBERrfarmer@econ.ucla.edu

Andrew HollenhorstDepartment of Economics UCLA8283 Bunche Hall: Box 951477Los Angeles CA 90095-1477ahollen@ucla.edu

Shooting the Auctioneer�

Roger E. A. Farmerrfarmer@econ.ucla.edu

Andrew Hollenhorstahollen@ucla.edu

February 2003 (This draft: August 2006)

Abstract

Most dynamic stochastic general equilibrium models of the macro-economy assume that labor is traded in a spot market. Two exceptionsby David Andolfatto and Monika Merz combine a two-sided searchmodel with a one-sector real business cycle model. These hybrid mod-els are successful, in some dimensions, but they cannot account forobserved volatility in unemployment and vacancies. Following sug-gestions by Robert Hall and Robert Shimer, this paper shows that arelatively standard DSGE model with sticky wages can account forthese facts. Using a second-order approximation to the policy func-tion we simulate moments of an arti�cial economy with and withoutsticky wages and we document the dependence of unemployment andvacancy volatility on two key parameters; the disutility of e¤ort andthe degree of wage stickiness. We compute the welfare costs of thesticky wage equilibrium and �nd them to be small.

1 Introduction

Most dynamic stochastic general equilibrium models (DSGE) of the macro-economy are built around a spot market for labor. Two exceptions, Andol-fatto [2] and Merz [17], combine the two-sided search model of Mortensen

�This research was supported by NSF grant SBR 0418174. The authors wish to thankStephanie Schmitt Grohé and Martín Uribe for helpful and extensive comments on anearlier draft. We also wish to thank participants at the NBER summer institute in 2005and seminar particiapnts at Duke University and the University of British Columbia.

1

and Pissarides [21], [20], [22] with a one-sector real business cycle model.These hybrid models are successful in some dimensions at explaining howunemployment and vacancies move over the business cycle but they cannotaccount for observed volatility in unemployment and vacancies. This papershows that a DSGE model with rigid wages can account for these facts.Shimer [25] suggests that the problem with search theoretic models is that

they are typically closed with a Nash bargaining solution. Nash bargaining,as a wage-setting mechanism, allows too much wage �exibility relative tothe data. Hall [10], [11] has explored Shimer�s suggestion that models withrigid or partially adjusting wages may be more successful than �exible wageeconomies at explaining the facts. This paper builds on the Hall-Shimerapproach by constructing a fully speci�ed dynamic general equilibrium modeland studying the properties of alternative wage determination mechanisms.We construct a version of a real business cycle model in which we add

a two-sided matching technology similar to those studied by Andolfatto andMerz. We use this arti�cial economy to study the properties of three alterna-tive equilibria. In the �rst, the wage is chosen to mimic the social planningsolution: We call this a �exible wage economy. In the second equilibriumthe real wage grows at the rate of underlying technological progress but isunresponsive to current productivity shocks. We call this the rigid wage so-lution. Finally, we study an economy in which the real wage adjusts 19% ofthe way towards the e¢ cient solution in every quarter. We call this a stickywage economy.In the context of the Mortensen-Pissarides model, Hagedorn andManovskii

[9] have shown that the Hall-Shimer volatility puzzle can be solved by choos-ing a high value for the outside option of the worker. Although our model ofthe labor market is embedded into a fully speci�ed DSGE environment we�nd that a disutility of e¤ort parameter, related to Hagedorn and Manovskii�soutside option, plays an important role in in�uencing unemployment and va-cancy volatility. We are not, however, able to resolve the Hall-Shimer puzzlewith this parameter alone. Since our model allows for variable search inten-sity we �nd that the social planning optimum has counterfactual implicationsfor the Beveridge curve. In order to generate the observed negative corre-lation between unemployment and vacancies and simultaneously to generatethe magnitude of unemployment and vacancy volatility observed in the datawe need to choose the disutility of e¤ort and the degree of wage stickinesstogether.Since we need sticky wages to explain the data it seems reasonable to ask

2

if the welfare cost of failing to adjust the real wage is large. Using a secondorder approximation to the utility function we �nd that when wages adjust19% of the way to their optimal value in each period that the welfare costrelative to the �rst best is roughly 40 cents per person per quarter in termsof foregone consumption. It seems likely that a cost of this magnitude couldbe explained by menu costs.To summarize; our paper makes three contributions to existing literature.

The �rst is the �nding that a sticky wage DSGE economy does a good job ofexplaining the time series properties of unemployment and vacancies in theU.S. Second, to resolve the Hall-Shimer volatility puzzle in a DSGE modelwith variable search intensity we need both sticky wages and the abilityto pick the reservation wage. Finally we �nd the welfare costs of the stickywage equilibrium are small. Hence we are able to show that the main featuresof the Hall-Shimer sticky wage equilibrium continue to hold in a standardproduction economy with risk averse consumers and capital accumulationwhilst preserving the ability of the standard RBC model to explain otherfeatures of the data.

2 Related Literature

In the Andolfatto [2] and Merz [17] models, unemployment and vacanciesenter di¤erently into the social objective function since vacancies use unitsof commodities but unemployment uses labor as an input. In the model de-veloped below we have made them symmetric (unemployment and vacanciesboth impose a time cost) to emphasize a stark implication of the RBC model.In the social planning solution both vacancies and unemployment should beprocyclical.1 Our reason for modifying the Adolfatto-Merz approach is thattheir models have di¢ culty in explaining the volatility and cyclical propertiesof unemployment and vacancies. This problem was documented in U.K. databy Millard, Scott and Sensier [18] and in U.S. data by Shimer [25] who pointsout that when search models are closed with a Nash bargaining solution, theydeliver counterfactual labor market predictions.

1In order to generate negatively correlated unemployment and vacancies, both Mertzand Andolfatto study versions of their respective models in which search intensity byworkers is �xed. Merz studies a version of her model with variable search intensity in whichshe �nds (Merz [17] Table 3 page 282) that unemployment and vacancies are positivelycorrelated.

3

We are not the only people currently working on the problem of com-bining search with an RBC model and closely related papers include thoseby Blanchard and Galí [4], Costain and Reiter [5], Gertler and Trigari [8],den Haan, Ramey Watson [6], Fujita and Ramey [7], Hall [12] and Veracierto[27]. Although our model is di¤erent in some respects from those studiedby each of these authors our �ndings are complementary. Like Blanchardand Galí [4], Gertler and Trigari [8] and Hall [12], we study the e¤ects ofclosing the model with a sticky wage although we do not attempt to providea microfoundation for this assumption as in Menzio [16]. Instead, we adoptMoen�s [19] approach of competitive search equilibrium in which competitivemarket makers announce the wage at which �rms and workers must agree ifsearch is to take place at their location.In Moen�s work competition between market makers enforces a �rst best

equilibrium. In our view this assumption is strong and the process by whicha competitive wage is established is likely to take time. If one identi�esthe competitive market maker of Moen with the Walrasian auctioneer, ourapproach is one where the auctioneer has been removed and replaced eitherby a �xed or slowly adjusting wage; hence the title of our paper �shootingthe auctioneer.�

3 The Social Planning Problem

In this section we describe an arti�cial economy that adapts the standardone-sector real-business-cycle model by adding a search technology for mov-ing labor between leisure and productive activities. We solve for the socialplanning optimum and show how the model with unemployment and vacan-cies is related to a standard environment with a spot market for labor.

3.1 Setting up the social planning problem

The social planner maximizes the discounted present value of the function

Jt = max1Xt=s

�t�sEs

�log (Ct)� �

L1+ t

1 + � b (Ut + Vt)

�:

The �rst term in the square bracket represents the utility of consumptionwhich we take to be logarithmic. The second term represents the disutility

4

of working in market activity and the third is the utility cost of searching fora job. The cost of search has two components; Ut is time spent searching bya worker for a job, and Vt is time spent by the representative family in itsrole as an employer searching for workers.The stock of employment evolves according to the expression

Lt = (1� �L)Lt�1 +Mt; (1)

where we assume that matches separate exogenously at rate �L. The term

Mt = B (Ut)� (Vt)

1�� (2)

is the matching function which we take to be Cobb-Douglas with weight �.The problem is constrained by a sequence of capital accumulation con-

straints,

Kt+1 = Kt (1� �K) + Yt � Ct; t = 1:::;

and by a production function,

Yt = At (Kt)� �(1 + g)t Lt�(1��) : (3)

Output, Yt is produced using labor Lt and capital Kt which depreciates atrate �K : The term (1 + g)

t measures exogenous technological progress and Atis an autocorrelated productivity shock which follows the stationary process

At = A�t�1 exp ("t) ; 0 < � < 1;

Et�1 ("t) = 0:

We assume that f"tg1t=1 is a Markov process with bounded support and welet "t be the history of shocks, de�ned recursively as;

"t = "t�1 � "t;

"1 = "1:

The assumption of bounded support is required in Section 5 in which wecompute a second order approximation to the policy function.

5

3.2 Solving the social planning problem

The social planner can alter the stock of workers in productive activitiesby varying the time spent searching for jobs by workers or the time spentsearching by �rms for workers. Since the stock of labor can only be in-creased by hiring, the inclusion of employment as a state variable adds anadditional propagation mechanism for shocks. Although this mechanism ispotentially important, in practice the separation rate from �rms is so highthat the contribution of this additional component is not large and in our cal-ibrated model most movements in employment at business cycle frequenciesare caused by variations in time spent searching by �rms or by workers2.To model the movements in unemployment and vacancies that would be

observed in an e¢ cient allocation we solve the social planning problem. Tomove labor into and out of productive activity the planner chooses contingentsequences fUt ("t) ; Vt ("t)g. The �rst-order conditions for the choice of thesevariables are given by Equations (4) and (5);

�Mt

Ut�t = b; (4)

(1� �)Mt

Vt�t = b; (5)

where �t is the Lagrangian multiplier on the labor accumulation constraint(1). Dividing (4) by (5) implies that the ratio Vt=Ut is constant

VtUt=1� �

�:

We de�ne 't = Vt=Ut to be labor market tightness since when 't is highthere are many �rms looking for workers but few workers searching for jobs.From a planning perspective, there is an optimal level of tightness and themost e¢ cient way to increase the labor stock Lt is to increase unemploymentand vacancies together. If the data were generated by a social planningsolution one would expect to observe that movements in Ut are perfectlycorrelated with movements in Vt.

2Shimer [25] cites data from Abowd and Zellner [1] and from the Job Openings andLabor Turnover Survey, to argue that separations occur at a rate of approximately 10%per quarter in the U.S. data. This is a big number - it implies that 40% of the labor forceseparates from employment in a year.

6

A consequence of constant labor market tightness is that �t, the shadowprice of increasing the stock of labor, is also constant. By the de�nition ofthe matching function, Equation (4), and constancy of tightness we have

�t =b

�

Ut

B (Ut)� (Vt)

1�� =1

B

b

�� (1� �)1��: (6)

By moving Ut and Vt together the social planner maintains a constant mar-ginal utility cost of creating new matches.The �rst order condition for the choice of capital Kt+1 is given by

1

Ct= �Et

�1

Ct+1

�1� �K + �

Yt+1Kt+1

��; (7)

and the �rst order conditions for choosing fLtg is

(1� �)YtCtLt

� �L t = �t � � (1� �L)Et [�t+1] :

Combining Equation (8) with (6) leads to the expression

1

Ct

�(1� �)

YtLt� Ct�L

t

�= �; � = (1� � (1� �L))

1

B

"b

�� (1� �)1��

#;

(8)which is closely related to the static optimizing condition for labor that arisesfrom a standard RBC model. The parameter � drives a wedge between themarginal product of labor and the disutility of employment and as � ! 0,Equation (8) converges to the familiar �rst order condition for labor in theone-sector RBC model. For small values of �, the time paths of capital,gdp, consumption and hours are close to the solutions obtained from anRBC economy with a spot market for labor.The parameter � is key to our discussion below of the ability of this model

to replicate unemployment dynamics. Although our model is richer than thestandard search model studied by Hagedorn and Manovskii [9], � plays thesame role as the reservation wage in their environment. In a search modelwith Nash bargaining, � would place a lower bound on the wage that the �rmcould o¤er. We will return to the role of � later in the paper when we discussits interaction with wage stickiness in helping to generate the Beveridge curveand simultaneously to generate realistic unemployment volatility.

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4 A Decentralized Model

In this section we study a decentralized version of the model. We assumethat a representative worker/�rm takes the real wage as given and choosescapital, unemployment and vacancies to maximize expected utility. To closethe model we introduce three alternative solution concepts to determine thereal wage.In the �rst concept we adopt the idea that competition between market

makers forces the wage to maximize the expected utility of potential workers,that is, the wage is chosen to implement the social planning optimum. Wecompare this solution with an alternative, suggested by Hall [10], in whichthe real wage is unresponsive to current market conditions. We implementthis solution by assuming that the real wage is that which would prevail alongthe non-stochastic balanced growth path. Since the �xed wage solution leadsto �uctuations in unemployment and vacancies that are too volatile relativeto the data, we also consider a third equilibrium concept in which the realwage adjusts partially each period towards its optimal value.

4.1 Setting up the agent�s problem

The decentralized economy is populated by a unit measure of householdswho optimally choose labor e¤ort LS, search e¤ort U , and capital holdingsK to maximize utility and operate the production technology for which theychoose employment of labor LD, search e¤ort V , and the amount of capitalto rent from households. We assume markets exist that allow households toperfectly share consumption risk. The solution to this decentralized problemis equivalent to the solution of the problem of a representative agent whoacts both as a household and as a �rm.In his role as a household, the agent supplies labor LSt . His utility function

is

Jt = max1Xt=s

�t�sEs

�log (Ct)� �

L1+ t

1 + � b (Ut + Vt)

�and labor supply in period t is related to search e¤ort Ut and lagged laborsupply by the expression

LSt = (1� �L)LSt�1 + Ut

�Mt

�Ut: (9)

8

�Mt= �Ut is the increase in employment when the household increases its searchintensity, Ut, by one unit. This probability is parametric to the household,but is determined in equilibrium as the ratio of aggregate matches �Mt toaggregate search intensity �Ut.The representative worker/�rm faces the following sequence of budget

constraints;3

Kt+1 = Kt (1� �K)+AtK�t

�(1 + g)t LDt

�1��+WtL

St �WtL

Dt �Ct: t = 1; :::

(10)The household can increase its stock of workers, LDt by incurring a utilitycost �bVt of search. Every additional unit increase in Vt leads to an increasein the stock of employed workers of �Mt= �Vt where �Vt is aggregate searchintensity by all other �rms and �Mt is the aggregate number of matches. Thisleads to the following expression

LDt = (1� �L)LDt�1 + Vt

�Mt

�Vt; (11)

for the accumulation equation faced by the household in its role as a labordemander.

4.2 Solving the agent�s problem

The representative agent chooses state contingent sequences�Kt+1

�"t�; Ut�"t�; Vt�"t�1

t=1;

taking as given the production function (3) and the accumulation constraints(9), (10) and (11). The �rst order condition for the choice of capital leads tothe Euler equation;

1

Ct= �Et

�1

Ct+1

�1� �K + �

Yt+1Kt+1

��: (12)

3Although we have modeled an economy with a single asset, storable capital, nothing ofsubstance would be added by including a complete set of contingent claims markets. Sincethis is a representative agent economy, additional markets would serve only to determinethe prices for additional assets at which the representative agent would choose not totrade.

9

The �rst order conditions for the choice of time spent searching in his capacityas a worker and a �rm leads to the following two �rst-order conditions;

1

CtWt � �

�LSt� = b

� �Ut�Mt

� � (1� �L)Et

� �Ut+1�Mt+1

��; (13)

1

Ct

�(1� �)

YtLDt

�Wt

�= b

� �Vt�Mt

� � (1� �L)Et

� �Vt+1�Mt+1

��: (14)

In a competitive equilibrium, the model is closed by the market equilib-rium conditions

Lt � LSt = LDt ; �Ut = Ut; �Vt = Vt;

and by the de�nition of the aggregate matching function

�Mt =Mt = BU �t V1��t :

Using these market clearing conditions and the de�nition of tightness, inequilibrium Equations (13) and (14) become

1

Ct(Wt � �L tCt) =

b

B

�'��1t � � (1� �L)Et

�'��1t+1

��; (15)

1

Ct

�(1� �)

YtLt�Wt

�=

b

B

�'�t � � (1� �L)Et

�'�t+1

��: (16)

Since there are two ways of moving labor between leisure and employ-ment, but only one price, the model as it stands is missing an equilibriumcondition. Typically, a model of this kind would be closed by adding a Nashbargaining equation to �x the real wage, Wt. The Nash bargaining solution,for appropriate choice of bargaining weights, can be shown to implement thesocial planning solution.4 Alternatively one might appeal to Moen�s idea of

4The generalized Nash bargaining solution divides the surplus of a match in proportionto an exogenous bargaining weight. For the case of a Cobb-Douglas matching function,this solution implements the social planning optimum when the bargaining weight is equalto the elasticity parameters � of the matching function. This result is a generalization ofthe Hosios condition [14] to a model with more general utility functions.

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competitive market makers to argue that the wage will be chosen to maxi-mize the expected utility of the representative worker. In either case, theimposition of the e¢ cient solution leads to a wage equation of the form

Wt = � (1� �)YtLt+ (1� �)Ct�L

t : (17)

Combining Equation (17) with the equilibrium conditions and the �rst-orderconditions of the competitive model, Equations (13) and (14), one arrives atequations for unemployment and vacancies that mimic the �rst-order condi-tions of the social planner, Equations (4) and (5).In the following analysis we will study three di¤erent equilibrium con-

cepts. In the �rst we choose the wage according to Equation (17) to mimicthe social planning optimum. We also consider a �xed wage equilibrium inwhich we allow the real wage to grow at the underlying rate of growth ofthe economy but we do not allow it to respond to productivity shocks. Inthe third equilibrium concept we allow the real wage to adjust each periodby a fraction � of the way back towards the e¢ cient solution in each period.The equations that de�ne the wage are explained more fully in Section 5.2after we de�ne a set of stationary variables that allow us to �nd approximatesolutions to our model.

5 Computational Issues

This section describes the procedure that we used to compute the propertiesof equilibria in the arti�cial economy. We begin by describing the solutionalgorithm that we used to compute the properties of arti�cial time seriesgenerated by the model. We then describe the alternative wage determinationmechanisms that we used to close the model.

5.1 The Solution Algorithm

To compute solutions to the model we used a second order approximationto the policy function due to Schmitt-Grohé and Uribe [24]. Their proce-dure requires that the variables be separated into a set of non predeterminedvariables pt and a set of predetermined variables qt. To implement this pro-cedure, one must �rst �nd a representation of the model in which all of thevariable are stationary.

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To compute a stationary transformation of the model, we de�ned thefollowing variables

kt = log

�Kt

(1 + g)t

�; at = log (At) ;

yt = log

�Yt

(1 + g)t

�ct = log

�Ct

(1 + g)t

�; lt = log (Lt) ;

ut = log (Ut) ; vt = log (Vt) ;

zt = log

�YtLt

�; wt = log

�Wt

(1 + g)t

�; jt =

Jt

(1 + g)t

1��+�

(1��)2

!:

The vector qt consists of the predetermined variables

qt = fat; kt; lt�1; wt�1g ;

and the vector of nonpredetermined variables, pt is given by

pt = fyt; ct; ut; vt; zt; jtg :

We assume that all uncertainty arises from stochastic productivity shocksthat take the form

at+1 = �at + �"t+1

where "t+1 s N (0; 1) and is independent across time and � is the standarddeviation of the innovation to the productivity shock. The model is a set ofequations

Etf (pt+1; pt; qt+1; qt) = 0; (18)

where the function f consists of identities, model de�nitions and �rst-orderconditions. These equations are de�ned in Appendix A.

5.2 Alternative wage determination mechanisms

To compute the decentralized solution under alternative wage determinationmechanisms we solved for the time path of wt in the social planning optimum.This is given by the expression;

wSPt = log�(1� �) �eyt�lt + � (1� �) ect�lt

�:

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Next we computed the steady state value �w;

�w = log�(1� �) �ey�l + � (1� �) ec�l

�:

To compute alternative equilibria we simulated sequences for a set of equa-tions in which the wage is given by the expression

wt =(1� �)

(1 + g)wt�1 + �wSPt :

By setting � = 1 this solution implements the social planning optimum.Alternatively, setting � = 0 and choosing an appropriate initial condition�xes the wage equal to its unconditional mean along the balanced growthpath. Choosing any other value of � in the interval (0; 1) implements apartial adjustment mechanism in which the logarithm of the real wage adjustsa fraction � of the way towards the social planning optimum in any givenperiod.The solution to the model, when it exists and is unique, is of the form

pt6�1

= g (qt; �) ;

qt+14�1

= h (qt; �) + ��"t+1;

where � is the standard deviation of the shock "t and � is the column vector

� = [1; 0; 0; 0]0 :

Schmitt-Grohé and Uribe provide code that generates analytic �rst and sec-ond derivatives of the function f in Equation (18).5 Evaluating these deriv-atives at the point

�p = g (�q; 0) ; �q = h (�q; 0)

leads to the second order approximation

~pt6�1

= �p6�1+ gq6�6

~qt6�1+1

2Gqq6�16

~mt16�1

; (19)

~qt+14�1

= �q4�1+ hq4�4

~qt4�1+1

2Hqq4�16

~mt16�1

+ �4�1�"t+1: (20)

5The exact relationship between these expressions and the Schmitt-Grohé-Uribe codeis explained in Appendix B.

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The terms~qt = (qt � �q) ; ~pt = (pt � �p)

are deviations of pt and qt from their non-stochastic steady states and

�p =1

2g���

2; �q =1

2h���

2

are bias terms that cause ~pt and ~qt to di¤er from zero when the model isnonlinear. The variable

~mt = ~qt ~qt � vec (~qt~q0t)

is a vector of cross product terms.The Schmitt-Grohé-Uribe solution has a number of advantages over al-

ternative algorithms. First, it uses the symbolic math feature of Matlab tocompute analytic derivatives of a user speci�ed set of functions. This fea-ture mechanizes the process of solving for derivatives by hand and removesa potential source of error. Second, the program computes a second orderapproximation to the policy function which is essential if one is interested ina welfare comparison of alternative wage determination mechanisms.

6 Taking the model to the data

This section describes the procedures we used to pin down key parametersof the model. We begin by describing parameters that are in common withstandard RBC models and move on to describe some novel features that arisefrom our version of a search model of the labor market.

6.1 Standard features of the calibration

Table 1a lists the values of six key moments that we used to calibrate para-meters.

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Table 1a: Product Market: Moments from the U.S. data

Moment DescriptionValue in baselinecalibration

gAverage quarterly growthrate of per capita gdp 0:0045

rAverage quarterly realinterest rate 0:0160

cyAverage ratio of consumptionto gdp 0:75

ls Labor�s share of gdp 0:66� Autocorrelation of TFP 0:99�" Standard deviation of TFP 0:007

Since the arti�cial economy is based on a Solow growth model, the para-meter g which represents the quarterly growth rate of technological progressis equal to the quarterly per capita growth rate of gdp. This was set at0:0045 which implies an annual per capita growth rate of 1:8%, equal to theU.S. average for the past century. To calibrate the elasticity of capital inproduction; �, we used the assumptions of competitive labor markets andconstant returns-to-scale which imply that � is equal to 1 � ls, where ls islabor�s share of gdp.To compute the time series properties of the productivity shock we com-

puted a time series for total factor productivity in the data using the expres-sion

TFP =Yt

K�t L

1��t

:

We regressed the log of TFP on its own lagged value and computed the �rstorder autocorrelation coe¢ cient and the standard deviation of the residual.This led to values of � = 0:99 and �" of 0:007:The procedure for solving for the steady state levels of output, consump-

tion, capital and the parameters � and �k relies only on the steady stateresource constraint and capital Euler equation and is identical to the pro-cedure followed in a standard RBC model. This is a consequence of thefact that the friction introduced in the labor market distorts decisions onlyas the economy transitions from one state to another and is inconsequen-tial in a nonstochastic economy in steady state. To compute the quarterly

15

depreciation rate we solved the steady state equations,

(1 + r) = 1� �K +�

ky; (21)

(g + �K) ky = 1� cy; (22)

for ky, the steady state capital to gdp ratio and �K , the depreciation rate asfunctions of r, �; g and cy. Equation (21) is a no-arbitrage relationship inthe asset market and (22) is the steady state gdp accounting identity.The unknowns r and cy were set equal to their historical averages in the

data; we set r = 0:0162 which represents an annual rate of 6:6% (computed asthe average annual yield on the S&P 500) and cy = 0:75, which is the averageratio of consumption to gdp when government consumption is included aspart of consumption. The value of � was computed from the steady statecapital Euler equation

�(1 + r)

(1 + g)= 1;

which gives a value of � = 0:99. Table 1b lists the values for the parameters�, �K and �, implied by this exercise.

Table 1b: Parameter values implied by Moments from Table 1a

Parameter DescriptionValue in baselinecalibration

�Quarterly discount factor

0:99

�KQuarterly depreciation ratefor physical capital

0:0314

�Elasticity of capital in production

0:34

6.2 Labor market parameters calibrated from the steadystate

We now turn to features of our model that di¤er from standard RBC economies.Table 2a lists some of the moments from data and parameter values that wereused to calibrate the labor market portion of the model. We set the sepa-ration rate, �L, at 10% per quarter based on Shimer�s [25] interpretation of

16

the JOLT data, the unemployment rate at 5:8%; and the participation rateto equal 70%.

Table 2a:Labor Market: Parameters chosen to match moments

from the U.S. data

Moment DescriptionValue in baselinecalibration

uAverage unemployment

rate0:058

p Average participation rate 0:7 Inverse labor supply elasticity 0

�Elasticity of the matching

function0:4

�LAverage quarterly separation

rate0:1

To pick the parameter we used the fact that real business cycle mod-els require high labor elasticity to generate su¢ cient volatility of hours. Inthe base-line calibration we picked = 0 which has become standard fol-lowing Hansen�s work [13] on indivisibilities. Finally, we set � according toBlanchard and Diamond�s [3] estimated value of � = 0:4.The steady state values of U and L are related to the unemployment rate

u, the participation rate p, and population size N by the de�nitions

p =L+ U

N; (23)

u =U

U + L: (24)

Since the model contains a single representative agent, we normalized thepopulation size to 1 and computed L and U from Equations (23) and (24).This led to a value of L = 0:66; and U = :041 which implies that therepresentative agent spends 66% of his time in paid employment and 4:1%searching for a job.To pin down steady state vacancies, new matches and the parameters B

and b we used the labor market �rst order conditions and the de�nition ofthe matching function. In the steady state, the number of new matches mustequal the number of jobs destroyed exogenously each period which implies,

m = �LL: (25)

17

From the steady state versions of Equations, (15) and (16), and notingthat the steady state wage w is given in terms of known quantities by thesteady state version of (17) we can solve for b and V as functions of �

b =w=c� �L

1� � + ��L

m

U; (26)

V =m

b (1� � + ��L)

�(1� �)

y=c

L� w=c

�: (27)

Finally, from the de�nition of the matching function we have

B =m

U �V 1�� : (28)

In the following table, b and V are calculated for � equal to 1:204, a valuethat we explain in the following section. The steady state values associatedwith the labor market are collected in table 2b.

Table 2b: Labor market parameter values implied by Tables 1a and 2a

Moment DescriptionValue in baselinecalibration

UFraction of timeunemployed

:041

VFraction of time searchingfor workers

:051

L Fraction of time working 0:66b Disutility of search e¤ort :83m Match parameter 0:066

BConstant of thematching function

1:43

6.3 Labor market parameters associated with the volatil-ity of unemployment and vacancies

There are two important parameters of the model that are not pinned downby the steady state calibration; these are �, a parameter that governs thedisutility of work and �, the degree of wage rigidity. The labor marketdynamics of the model are highly sensitive to the joint choice of these pa-rameters and they were calibrated to match several stylized facts regardingthe cyclical behavior of unemployment and vacancies.

18

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

-5

0

5

10

15

20

Lam bda

Chi = 1 .204

-.5

1

.5

2

1.5

0

corrσ2

-1

σ2U

σ2V

corr(U,Y)corr(V,Y)

Figure 1: Model statistics for di¤erent values of lambda

Figure 1 plots the variances of unemployment and vacancies and the cor-relations of each with GDP for di¤erent values of � holding �xed � at 1:204.Figure 2 plots these same variables against � holding � �xed at 0:19. Toselect the values 1:204 and 0:19 we searched over a grid of these parametersand chose values that came close to matching these four moments. This �g-ure illustrates the point that when � = 1, (the social planning solution) Uand V are both positively correlated with gdp and hence a model like thisone with highly elastic search intensity cannot explain the Beveridge curve.Figure 2 shows how the correlations of U and V with gdp depend on �.

Notably, these correlations switch sign as � increases above 1:325 and forvalues above 1:26 � becomes negative. This singularity when � approaches1:26 is present for all values of � 2 [0; 1]. These �gures make transparent theway in which the imposition of some degree of wage rigidity breaks the one toone correlation between U and V that obtains in the �exible wage economy.We conclude that both the volatilities and correlations with GDP of the key

19

1.1 1.15 1.2 1.25 1.3 1.35 1.4-10

-5

0

5

10

15

20

Chi

Lam bda = .19

σ2U

σ2V

corr(U,Y)corr(V,Y)

-.5

1

.5

2

1.5

σ2 corr

0

-1

Figure 2: Model statistics for di¤erent values of chi

labor market variables, U and V , are highly sensitive to the disutility of labor� and to the speed of wage adjustment �.

7 Matching the Data

We now turn to the performance of our model in certain key dimensionsby comparing two di¤erent calibrated economies with the U.S. data. Table3 reports the volatilities and correlations with gdp of gdp, consumption,investment, hours worked, labor productivity, the real wage, unemploymentand vacancies. The �rst column reports the moments of the quarterly datafrom 1955 �rst quarter, to 2002, fourth quarter. Consumption and investmentare both de�ned as the sum of private plus government components and allvariables are in 1996 U.S. dollars and de�ated by U.S. resident population.Hours is de�ned as employment per person multiplied by average hours where

20

employment is total non farm employment from the establishment survey.Productivity is gdp de�ated by hours and the real wage is computed ascompensation to employees divided by hours and de�ated by the 1996 gdpprice index. Unemployment is the U.S. unemployment rate for persons over16 years old and vacancies is an index of help wanted from the St. LouisFederal Reserve data base. All variables have been passed through the HP�lter with a smoothing parameter of 1600.The second two columns of Table 3 report the same moments for two

arti�cial economies. The middle panel is an economy in which the parameter� is set equal to 1 which allows the wage to adjust each period to a value thatcauses the decentralized solution to mimic the social planning optimum. Thethird panel reports data for an economy in which the log of the real wageadjusts by a fraction 0:19 towards the social planning optimum wage level inany given period.

Table 3Standard deviations in percent (a) and correlationswith Gdp (b) for U.S. and arti�cial economies

Quarterly U.S.time series

(1955.1-2002.4)

Flexible wageeconomy(� = 1)

Sticky wageeconomy(� = 0:19)

Series (a) (b) (a) (b) (a) (b)

Gdp1:59NA

1:00NA

1:43(0:18)

1:00(0:00)

1:58(0:22)

1:00(0:00)

Consumption1:04NA

0:85NA

0:71(0:10)

0:95(0:007)

0:75(0:11)

0:94(0:011)

Investment5:45NA

0:93NA

3:75(0:46)

0:98(0:003)

4:27(0:61)

0:98(0:004)

Hours1:80NA

0:92NA

0:79(0:10)

0:73(0:062)

1:05(0:18)

0:71(0:064)

Productivity0:74NA

�0:07NA

0:71(0:10)

0:95(0:007)

0:68(0:10)

0:87(0:039)

Real wage0:74NA

�0:24NA

0:71(0:10)

0:62(0:056)

0:42(0:08)

0:11(0:053)

Unemployment11:57NA

�0:86NA

5:91(0:36)

0:40(0:038)

11:43(1:50)

�0:72(0:067)

Vacancies13:00NA

0:90NA

5:91(0:36)

0:40(0:038)

13:67(1:17)

0:81(0:010)

21

Columns 2 and 3 of Table 3 were generated by simulating 100 runs of themodel for the baseline parameters setting � = 1 for column 2 and � = 0:19for column 3. We refer to the former as a �exible wage economy and tothe latter as a sticky wage economy. The disutility of labor � = 1:204 waschosen together with the wage �exibility parameter � = 0:19 to match thefour model moments described in the previous section to the correspondingsample moments in the data. For comparison purposes, we set � = 1:204in both calibrations. The numbers in parentheses are standard deviations ofthe reported variable over 100 simulations. In each case, the column labeled(a) reports the standard deviation of a variable and the column labeled (b)is its correlation with gdp. All arti�cial data has been passed through theHP �lter with a smoothing parameter of 1600 in the same way as the realworld data.There are two features of Table 3 that are important. Notice �rst, that

gdp, consumption investment and hours have the same statistical propertiesin the �xed wage and the �exible wage economies. In each case the corre-lations with gdp and the standard deviations of these series are within onestandard deviation of each other. The reasons for the di¤erences of these sta-tistics from the U.S. data are, by now, well understood and the model doesnot add much that is new in this dimension.6 The �xed and �exible wageeconomies di¤er substantially, however, in their predictions for the behaviorof unemployment and vacancies.In the data the standard deviations of unemployment and vacancies are

equal to 11:57 and 13:00. Vacancies are procyclical with a correlation co-e¢ cient with gdp of 0:9 whereas unemployment is countercyclical with acorrelation coe¢ cient of �0:86. In the social planning optimum, in con-trast, unemployment and vacancies each have a standard deviation of 5:91,they are perfectly correlated with each other and correlated with gdp witha coe¢ cient of 0:40: Contrast this with the sticky-wage arti�cial economy.Here, unemployment has a standard deviation of 11:43 and vacancies has astandard deviation of 13:67. As in the U.S. data, these variables are nega-tively correlated with each other. Vacancies is procyclical with a correlationcoe¢ cient with gdp of 0:81 and unemployment is countercyclical with a corre-lation coe¢ cient with gdp of �0:72. We conclude that the sticky wage modeldoes a reasonably good job of matching the labor market facts without com-

6For example, investment is too smooth in our simulated environment. This is typicallyaddressed by adding a cost of adjustment to the model.

22

promising the ability of the model to explain gdp, hours, consumption andinvestment.Although our model performs well in some dimensions its behavior with

respect to the real wage and productivity is disappointing. By adding anadditional shock to the model we might hope to reduce the model implicationsthat productivity and gdp are strongly correlated whereas the correlation inthe data is much weaker. The model also has counterfactual implications forthe correlation between the real wage and GDP although this may partlydepend on the timing of our labor adjustment equation. We assume thatnew workers can immediately contribute to gdp whereas a more standardassumption is to impose a one-period delay. We plan to explore this issue infuture research.

8 Welfare Costs of Sticky Wages

If the sticky wage economy does a good job of replicating the real world data,one might ask the question: What is the di¤erence in welfare between the�exible wage equilibrium and the equilibrium with sticky prices? To answerthis question, we computed a second order approximation to expected utilityfor values of � ranging from 0 to 1. When � = 0, the wage grows each periodat the rate g, but is unresponsive to innovations in the technology shock. Forour calibrated value � = 0:19, the wage adjusts each period by 19% of thedi¤erence between its previous value and the optimal wage for the period.We follow Lucas [15] and calculate the welfare cost of a sticky wage regime

parameterized by � as the fraction of consumption an agent would give upeach period in return for moving to the wage regime � = 1. To formalizethis, de�ne the expected utility of an agent in an economy parameterized by� and in the social planning economy (� = 1) respectively as

V � = E0

1Xt=0

�t

"log�C�t�� �

�L�t�1+

1 + � b�U�t + V �

t

�#; (29)

V SP = E0

1Xt=0

�t

"log�CSPt

�� �

�LSPt

�1+ 1 +

� b�USPt + V SP

t

�#; (30)

where the superscript � refers to the state contingent allocations in the stickywage economy and superscript SP refers to the social planning allocation.

23

The welfare cost of living in an economy with wage stickiness, which wedenote (�), is implicitly de�ned by

V � = E0

1Xt=0

�t

"log�(1� (�))CSPt

�� �

�LSPt

�1+ 1 +

� b�USPt + V SP

t

�#:

Since the period utility function is additively separable and logarithmic inutility we obtain the following expression for (�) in terms of the di¤erencein expected lifetime utility between the two regimes

(�) = 1� exp�(1� �)

�V � � V SP

��:

We used the Schmitt-Grohé-Uribe algorithm discussed above to obtain anapproximate di¤erence equation describing the evolution of lifetime utilityjt over time. That the approximation is second order accurate is especiallyimportant in the welfare calculation as in a �rst order approximation thecertainty equivalence property would obtain. Figure 3 plots the welfare costfor di¤erent values of � with all other model parameters �xed at the valuestabulated in the calibration section.We are most interested in the welfare cost in the economy where the

degree of wage stickiness, �, delivers plausible movements of unemploymentand vacancies over the business cycle, which we argued is the economy where� = 0:19. While there is still a welfare loss in this economy, we found it tobe much less than in the fully rigid case. The representative agent would bewilling to give up 0:0071% of consumption each period in order to live in the�exible wage economy. This is roughly $0:40 per person per quarter which isa relatively small number. This suggests that relatively low unmodelled costsof rapid wage adjustment, menu costs for instance, could serve as a plausibleexplanation for equilibrium wage stickiness. On the other hand, when wagesare made completely sticky the welfare cost is 0:37% of consumption or $20:86per person per quarter, a number that would require much higher costs tojustify.

9 Conclusion

We have shown that a relatively simple modi�cation to a standard real busi-ness cycle model, of the kind initially studied by Andolfatto and Merz, cannoteasily explain the properties of unemployment and vacancies in the U.S. data.

24

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2x 10-4 Welf are Cost

Lambda

Psi

Figure 3: The welfare cost of sticky prices

The problem with this model is the one identi�ed by Shimer: unemploymentand vacancies are not volatile enough and they have the wrong correlationwith gdp. We modi�ed the model using Hall�s suggestion that a model withrigid wages may provide a better representation of the data. As pointedout by Hall, the rigid wage model does not leave �rms or workers with anincentive to change their behavior, in e¤ect, because the search model has amissing market.Although the rigid wage model does better in some dimensions than the

�exible wage economy, it overshoots on unemployment volatility and leadsto gdp �uctuations that are too small. An intermediate model in whichthe real wage adjusts by 19% of the way each quarter towards the �exiblewage solution, does a much better job. This model performs as well as thestandard RBCmodel at capturing the volatility of hours, gdp, investment andconsumption. In addition it captures the observed volatility of unemploymentand has close to the correct volatility for vacancies. More important, we �nd

25

that unemployment is countercyclical and vacancies are procyclical, just asthey are in the U.S. data.We compared the welfare properties of alternative equilibria and found

that the rigid wage solution is associated with a welfare cost of roughly $20:86per person, a relatively large number. The partial adjustment equilibrium,on the other hand, is associated with a welfare cost of only $0:40 per quarter.This suggest that there may some small unmodelled cost of wage adjustmentthat is missing from the model, but which causes the sticky wage equilibriumto dominate, at least for �uctuations of the magnitude that we have observedin the post-war period.

26

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[3] Blanchard, Olivier Jean and Peter Diamond, (1989). "The BeveridgeCurve", Brookings Papers on Economic Activity 1, pp. 1�60.

[4] Blanchard, Olivier and Jordi Galí (2005). �Real Wage Rigidities and theNew Keynesian Model,�MIT Economics Department Working Paper05-14.

[5] Costain, James S. and Michael Reiter (2005). �Business Cycles, Unem-ployment Insurance and the Calibration of Matching Models,�mimeo,Universitat Pompeu Fabra.

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27

[12] __________ (2006). �The Labor Market and Macro Volatility: ANonstationary General-Equilibrium Analysis ,�mimeo, Stanford, Sep-tember 2006.

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28

[24] Schmitt-Grohé, Stephanie and Martín Uribe, (2004). �Solving dynamicgeneral equilibrium models using a second-order approximation to thepolicy function,� Journal of Economic Dynamics and Control 28, pp755�775.

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29

Appendix A: The function fProduction Function

f1 = yt � at � �kt � (1� �)lt: (A1)

Euler equation

f2 = e�ct � �

1 + ge�ct+1(1� �K + �eyt+1�kt+1): (A2)

Gdp accounting identity

f3 = (1 + g)ekt+1 � (1� �K)e

kt � eyt + ect : (A3)

Technology shock process

f4 = at � �at�1: (A4)

Utility function de�nition

f5 = ct � b (eut + evt)� �

1 + (elt)1+ + �ejt+1 � ejt : (A5)

Wage equation

f6 = ewt � (1� �)

1 + gewt�1 � �

��(1� �)eyt�lt + (1� �)�(elt) ect

�: (A6)

Vacancy �rst order condition

f7 = ewt�ct � (1� �) eyt�ct�lt +b

Be�(vt�ut) � b

B� (1� �L) e

�(vt+1�ut+1): (A7)

Unemployment �rst order condition

f8 = ewt�ct � ��elt� � b

Be(1��)(ut�vt) � b

B� (1� �L) e

(1��)(ut+1�vt+1): (A8)

Labor accumulation equation

f9 = elt � e�ut+(1��)vt � (1� �L)elt�1 : (A9)

30

De�nition of productivity

f10 = zt � yt + lt: (A10)

31

Appendix BThe Schmitt-Grohé-Uribe code generates arrays hx; hxx; h��; gx; gxx and

g��. The arrays gxx and hxx are three dimensional and may be unpacked intoten 4�4 and four 4�4matrices respectively. The second order approximationin matrix form can then be written as follows

~pt = g���2 + gq~qt +

264 ~q0tg1qq~qt...

~q0tg9qq~qt

375 ; (B1)

~qt+1 = h���2 + hq~qt +

264 ~q0th1qq~qt...

~q0th4qq~qt

375 ; (B2)

where ~qt is 4� 1 and ~pt is 9� 1. Using Kronecker product notation and thefact (see Hamilton [?] page 265) that

vec (ABC) = (C 0 A) vec(B); (B3)

it follows that

vec(~q0thiqq~qt) = ~q0t ~q0tvec(hiqq) = vec

�hiqq�0~mt; i = 1; ::4 (B4)

vec(~q0tgiqq~qt) = ~q0t ~q0tvec(giqq) = vec

�giqq�0~mt; i = 1; ::9 (B5)

where~mt16�1

= ~qt4�1 ~qt

4�1; (B6)

is a 16� 1 column vector. De�ning

Gqq =

264 vec�g1qq�0

...vec�g9qq�0375 ; (B7)

Hqq =

264 vec�h1qq�0

...vec�h4qq�0375 ; (B8)

and �q = h���2; �p = g���

2 leads to the expressions

~pt = �p + gq~qt +Gqq ~mt; (B9)

~qt+1 = �q + hq~qt +Hqq ~mt; (B10)

which corrspond to equations (19) and (20) in the text.

32