Labor Markets and Business Cycles1
Robert Shimer
University of Chicago
robert.shimer@gmail.com

January 31, 2009

1 This manuscript was originally prepared for the CREI Lectures in Macroeconomics, 2008. I haverevised it in light of the comments I received at those lectures and from students at the Gerzensee
Advance Doctoral Course in Economics and at the Massachusetts Institute of Technology. The
manuscript will be published by Princeton University Press. I am grateful for excellent research
assistance from Katarina Kotovova. This research is supported by a grant from the National Science
Foundation.

Overview
The goal of this book is to consolidate and extend recent results on the role of search frictions
and wage rigidities in explaining the cyclical dynamics of labor markets. Since 2002, the yearthat the working paper versions of Shimer (2005), Hall (2005), and Costain and Reiter (2008)
first circulated, there has been a profusion of research in this area, but the underlying questionis as old as macroeconomics: “why do employment and unemployment fluctuate so much atbusiness cycle frequencies?”
Lucas and Rapping’s theory of intertemporal substitution in labor supply is the starting
point for any modern analysis of employment fluctuations, including the Real Business Cycle
model and the New Keynesian model. The key assumption is that workers decide how much
to work at each point in time, taking as given the prevailing wage. To the extent that labor
supply is elastic, hours of work fluctuate with movements in the wage.
While models based on intertemporal substitution in labor supply are qualitatively consistent with the movement of hours of work over the business cycle, they run into at least two
problems. First, a number of authors have argued that a labor-market clearing model cannot
explain the magnitude of the observed fluctuations in hours worked. In a frictionless environment, the marginal rate of substitution between consumption and leisure should be equal tothe marginal product of labor, after adjusting for labor and consumption taxes. When theylooked at data, Parkin (1988), Rotemberg and Woodford (1991) and (1999), Hall (1997),
Mulligan (2002), and Chari, Kehoe, and McGrattan (2007) found that this relationship does
not hold.
In Chapter 1, I reaffirm this finding, verifying that there is a wedge between the marginal
rate of substitution and the marginal product of labor, the labor wedge, and that wedge
varies cyclically. During almost every recession, the labor wedge increases sharply. From theperspective of a frictionless model, there are two ways to interpret this finding: recessions may
be times when labor income taxes and consumption taxes rise, discouraging workers fromsupplying labor; or they may be times when the disutility of work increases. In a reducedform model, both would dissuade workers from working, causing countercyclical increases in
the measured labor wedge. But unfortunately neither possibility is empirically tenable.
1

2
The second problem with the frictionless model is that, in an environment where workers
can decide how much to work at each point in time, it is possible to generate movements
in hours worked but impossible to generate unemployment, i.e. non-employed workers who
would like to work at the prevailing wage. This omission potentially has important implications for welfare, since a worker who cannot find a job at the prevailing wage but would
like to have one is, by revealed preference, worse off than if she simply chose not to workat that wage. It potentially also has important consequences for the positive analysis ofbusiness cycles, since most cyclical movements in the aggregate amount of hours worked areaccounted for by movements between employment and unemployment, not by movements inhours worked by employed workers.
Equilibrium search and matching models provide an ideal laboratory for understanding
unemployment and have been used extensively for this purpose. The models build on the ideathat it takes workers time to find a job. Thus a worker entering the labor market or a worker
who loses her job necessarily experiences a spell of unemployment. Moreover, unemployedworkers are worse off than employed workers because they are unable to work until they finda job. In this sense, search and matching provides a theory of unemployment, not just of
non-employment.
Search and matching models also often assume that firms must expend resources in order
to find a suitable worker. A matching function determines the number of workers and firmsthat meet as a function of the unemployment rate and firms’ recruiting effort. Fluctuationsin the profitability of hiring a worker, possibly due to fluctuations in aggregate productivity,
induce fluctuations in recruiting. When firms recruit harder, unemployed workers find jobs
faster, pulling down the unemployment rate. Thus search and matching models naturally
generate movements in unemployment duration, an important component of the observed
fluctuations in unemployment at business cycle frequencies.
But the question remains whether search and matching models are quantitatively consistent with the observed behavior of labor market outcomes. There is a good reason to expectthat they are not. Recall that a competitive labor market model cannot explain all of theobserved fluctuations in the labor wedge. Viewed through the lens of a frictionless model,
recessions look like periods when the labor wedge rises, reducing labor supply.
Now consider introducing a labor adjustment cost into a competitive model, making it
costly for firms to increase their employment level. This will directly lower the volatilityof employment. Firms will increase employment by less during expansions because hiring iscostly. They will also be less willing to reduce employment during recessions in order to avoid
future hiring costs, when desired employment returns to normal. Thus hours worked will tend
to be more stable over the business cycle when adjustment costs are larger. If real-world data3
were generated by an economy with labor adjustment costs but an economist ignored theexistence of those costs, he would be surprised at how stable observed hours worked are overthe business cycle. Measuring the labor wedge with data generated by the economy, he would
conclude that the wedge rises during expansions and falls during recessions. This is exactly
the opposite of what we observe in the data.
But search frictions act, at least in part, like a labor adjustment cost, since they imply
that it takes unemployed workers time to find a job and it takes firms time to hire workers. If
this reduces the volatility of employment, the labor wedge will tend to be positively correlatedwith employment. Such a model of search frictions will not be useful in explaining the cyclicalbehavior of labor markets.
The bulk of this book confirms the thrust of this argument. Search frictions per se do
not help to explain fluctuations in the labor wedge, but rather they exacerbate the problemsof the frictionless model. However, I also find that subsidiary assumptions, especially alternative assumptions on wage setting, may help to explain why the measured labor wedge iscountercyclical and why employment is so volatile.
To understand this last statement, note that in matching models based on Pissarides(1985) and Mortensen and Pissarides (1994), search frictions create a bilateral monopoly
situation between workers and firms. Workers and firms engage in time consuming searchfor partners before negotiating a wage. Once they have sunk this cost, there is a range ofwages at which both prefer to match rather than breakup. Loosely speaking, each will agree
to any wage larger than the marginal rate of substitution between consumption and leisurebut smaller than the marginal product of labor, if the alternative is breaking up.
A critical question is how wages are determined. A common assumption in the search andmatching literature is that the worker and firm bargain over the gains from trade, splitting
the surplus according to the Nash bargaining solution (Nash, 1953). In Chapter 2, I provethat under this wage-setting assumption, the wage, the marginal rate of substitution, andthe marginal product of labor are all proportional to current productivity under particularassumptions on preferences (balanced growth and additive separability between consumption
and leisure) and under the assumption that output is produced using only labor. Productivity
shocks affect neither the labor wedge nor the (un)employment rate. This neutrality result
is inspired by Blanchard and Galı́ (2006), who reach a similar conclusion in a model wherefirms face a labor adjustment cost.
In Chapter 3, I break this neutrality result in several ways. First, I allow for moregeneral preferences with substitutability between consumption and leisure. The resulting
fluctuations are minuscule. Second, I introduce capital into the model. While the resulting
framework generates cyclical movement in employment and the labor wedge, it is inconsistent4with the data. In particular, I verify that employment is positively correlated with themeasured labor wedge in the model, for the reason described above: search frictions dampen
fluctuations in employment, which viewed through the lens of the frictionless model suggeststhat expansions are periods when labor tax rates are higher. Third, I consider other shocks,
for example reallocation shocks that change the probability of an employed worker becoming
unemployed. This has little effect on the results. I conclude that the (counterfactual) positive
comovement of the labor wedge and employment is a robust feature of search models whenwages are set via Nash bargaining.
Chapter 4 considers an alternative wage setting procedure that is no less plausible thanthe Nash bargaining solution and has vastly different implications for the behavior of themodel. I assume wages are backward-looking. I find that this form of wage rigidity canpotentially explain why employment is so volatile even if the elasticity of labor supply isrelatively small. If wages do not fall following a negative productivity shock, firms will be
reluctant to hire workers, pushing up unemployment duration and the unemployment rate.
This type of wage rigidity is based on ideas developed by Hall (2005).1 In a framework
similar to Shimer (2005), Hall shows that if wages are rigid, unemployment is extremely
sensitive to underlying shocks. He stresses that this type of wage rigidity is not susceptible
to the Barro (1977) critique. That is, no matched worker and firm would mutually prefer torenegotiate their wage. Similarly, Blanchard and Galı́ (2006) consider a real wage rigiditythat makes the wage move less than one-for-one with the shock. Firms respond to relatively
low wages during booms by creating many new jobs, driving down the unemployment rate.
However, this also implies that part of the productivity increase is spent on additional job
creation. Consumption then increases by less than productivity, generating a countercyclicallabor wedge. Gertler and Trigari (forthcoming) reach a similar conclusion in a model withoverlapping wage contracts that are not contingent on the path of productivity shocks.
I intend for this book to provide a stand-alone treatment of search and matching models.
It should be suitable for advanced graduate students and other researchers familiar withmodern recursive methods, for example at the level of Ljungqvist and Sargent (2004). At thesame time, the book is far from exhaustive. I develop one particular model of search frictions,
integrating the search model with a standard real business cycle model. I focus exclusively
on business cycle issues, neglecting fascinating topics such as European unemployment thatmany others have addressed using search models. I abstract from important (but difficultand controversial) issues like the role of incomplete markets in search models with aggregate
fluctuations (Bils, Yongsung, and Kim, 2007; Krusell, Mukoyama, and Şahin, 2007; Nakajima,
1

One may also think of this as a modern attempt to integrate search theory with disequilibrium macroeconomics (Barro and Grossman, 1971; Malinvaud, 1977; Benassy, 1982).

52008). I do not discuss the possibility of using microeconomic data to uncover some of themore important structural parameters of the model, although some of the most exciting
work in search theory today exploits the extraordinary richness of microeconomic data to dothat.2 Perhaps most importantly, I do not attempt to review the burgeoning literature on
the business cycle properties of search models, mentioning only a few papers from which I
knowingly borrow ideas.3 My excuse is that the scope of this project, originally conceived
to accompany three lectures at the Centre de Recerca en Economia Internacional (CREI),
prevents me from doing so.

2

For recent negative assessments of the evidence for rigid wages, see Haefke, Sonntag, and van Rens (2008)
and Pissarides (forthcoming).
3
A non-exhaustive reading list would certainly include Yashiv (2006), Krause and Lubik (2007), Mortensen
and Nagypál (2007), Rudanko (forthcoming), Farmer and Hollenhorst (2006), Kennan (2006), Rotemberg(2006), Rudanko (2008), and the papers collected in a special issue of the Scandinavian Journal of Economics
entitled “Macroeconomic Fluctuations and the Labor Market” (2007, volume 107, issue 4).

6Chapter 1
The Labor Wedge
Throughout this book, I study the interaction of optimizing households and firms in a closed
economy. I begin in this chapter by developing a competitive, representative agent versionof the model. The chapter has two objectives. First, I introduce much of the notation that
I rely on throughout the book. Because of this, I include details in this chapter that are notreally necessary for the second, more substantive objective: I use the model to measure andanalyze the behavior of the labor wedge, the wedge between the marginal rate of substitution
between consumption and leisure and the marginal product of labor. I confirm the wellknown result that the labor wedge tends to rise during recessions, so the economy behaves
as if there is a countercyclical tax on labor. The remainder of the book explores whether
extending the model to incorporate labor market search frictions can explain the behavior of
the labor wedge.
I start the chapter by laying out the essential features of the model: optimizing households,
optimizing firms, a government that sets taxes and spending, and equilibrium conditions thatlink the various agents. In Section 1.2, I use pieces of the model to derive a static equationthat relates hours worked, the consumption-output ratio, and the labor wedge. Section 1.3
discusses how I measure the first two concepts and uses these measures to calculate theimplied behavior of the labor wedge in the United States. I establish the main substantive
result, that the labor wedge rose strongly during every recession since 1970. I establish therobustness of my results to alternative specifications of preferences in Section 1.4 and discuss
the possibility that the results are driven by preference shocks in Section 1.5. I finish thechapter with a brief discussion in Section 1.6 on the empirical relationship between the hours
fluctuations that I analyze here and fluctuations in employment and unemployment, whichis the main topic of subsequent chapters.

7

8

1.1

CHAPTER 1. THE LABOR WEDGE

Representative Agent ModelI denote time by t = 0, 1, 2, . . . and the state of the economy at time t by st . Let st ={s0 , s1 , . . . , st } denote the history of the economy and Π(st ) denote the time-0 belief about
the probability of observing an arbitrary history st through time t. Exogenous variables likeaggregate productivity, government spending, and distortionary tax rates may depend on
the history st . At date 0, there is an initial capital stock k0 ≡ k(s0 ) and an initial stock ofgovernment debt b0 = b(s0 ). The capital stock is owned by firms, while households hold thedebt.

HouseholdA representative household is infinitely-lived and has preferences over history-st consumptionc(st ) and history-st hours of work h(st ). To start, I assume that preferences are ordered by
the utility function


∞ X
X
γε
t
t
t
t 1+ε
h(s ) ε ,(1.1)
β Π(s ) log c(s ) −
1
+
ε
t
t=0
s

where β ∈ (0, 1) is the discount factor, γ > 0 measures the disutility of working, and, asI show below, ε > 0 is the Frisch (constant marginal utility of wealth) elasticity of labor
supply.
This formulation imposes that preferences are additively separable over time and acrossstates of the world. It also imposes that preferences are consistent with balanced growth—doubling a household’s initial assets and its income in every state of the world doubles
its consumption but does not affect its labor supply. This is consistent with the absenceof a secular trend in hours worked per household, at least in the United States (Aguiarand Hurst, 2007; Ramey and Francis, forthcoming). I maintain both of these assumptions
throughout this book. The formulation also imposes that the marginal utility of consumption
is independent of the worker’s leisure. This restriction is more questionable and so I relax it
in Section 1.4 below.
The household choose a sequence for consumption and hours of work to maximize utility
subject to a single lifetime budget constraint,
a0 =

∞ X
X
t=0

st


q0 (st ) c(st ) − (1 − τ (st ))w(st)h(st ) − T (st ) .

(1.2)

The household has initial assets a0 = a(s0 ). In addition, τ (st ) is the labor income tax rate,w(st) is the hourly wage rate, and T (st ) is a lump-sum transfer in history st , all denominated9

1.1. REPRESENTATIVE AGENT MODELin contemporaneous units of consumption.1 Thus c−(1−τ )wh−T represents consumption inexcess of after-tax labor income and transfers, which is discounted back to time 0 according to
the price q0 (st ). This then represents the cost at time 0 of purchasing one unit of consumption
in history st , denominated in units of history-s0 consumption. Put differently, q0 (st ) is thehistory-s0 price of an Arrow-Debreu security that pays one unit of consumption in history st
and nothing otherwise.
Also define the assets of the household following history st asta(s ) =∞ X
X

t′ =t st′ |st

′
′
′
′
′
′ 
qt (st ) c(st ) − (1 − τ (st ))w(st )h(st ) − T (st ) ,′′where the notation st |st indicates that the summation is taken over histories st that are con′tinuation histories of st , i.e. st ≡ {st , st+1 , st+2 , . . . , st′ } for some states {st+1 , st+2 , . . . , st′ }.
′′Then qt (st ) is the price of a unit of consumption in history st = {st , st+1 , st+2 , . . . , st′ } paidin units of history-st consumption. No arbitrage requires that q0 (st )qt (st+1 ) = q0 (st+1 ) for all

st and st+1 ≡ {st , st+1 }. Equivalently, the lifetime budget constraint implies a sequence ofintertemporal budget constraints,a(st ) + (1 − τ (st ))w(st)h(st ) + T (st ) = c(st ) +X

qt (st+1 )a(st+1 ),(1.3)

st+1 |stso assets plus labor income plus transfers in history st is equal to consumption plus purchasesof assets in continuation histories st+1 .

Firm
The representative firm owns the capital stock k0 = k(s0 ) and has access to a Cobb-Douglas
production function, producing gross output z(st )k(st )α hd (st )1−α in history st , where z(st ) is
history-contingent total factor productivity,2 hd (st ) is the labor it demands, and α ∈ [0, 1) isthe capital share of income. A fraction δ of the capital depreciates in production each period,
while at the end of period t, the firm purchases any capital that it plans to employ in periodt + 1. That is, history-st+1 ≡ {st , st+1 } capital k(st+1 ) is purchased in history st and so must1

One can easily extend the model to include a consumption tax. Then τ (st ) measures the total tax wedge,
the cost to an employer of providing its worker with one unit of the consumption good.
2
Although I do not place explicit restrictions on the productivity process, I do assume that worker’s
expected utility is finite so her optimization problem is well-behaved. This is ensured if productivity isbounded but is true under substantially weaker conditions, if productivity does not grow too fast.

10

CHAPTER 1. THE LABOR WEDGE

be measurable with respect to st . The present value of the firm’s profits is then given by0J(s , k0 ) =

∞ X
X
t=0

t



t

t α d

t 1−α

q0 (s ) z(s )k(s ) h (s )stt

+ (1 − δ)k(s ) − k(st+1) − w(s )h (s ) . (1.4)
tdtNote that this expression presumes that the firm does not pay any taxes. I do this fornotational simplicity alone. In particular, any payroll taxes are rolled into the labor income
tax rate τ . The firm chooses the sequences hd (st ) and k(st+1 ) to maximize J.
I can also write the value of the firm’s profits from history st on as
J(st , k(st )) =∞ X
Xt′ =t st′



′
′
′
′
′
′
′
′
qt (st ) z(st )k(st )α hd (st )1−α + (1 − δ)k(st ) − k(st +1 ) − w(st )hd (st ) .

This implies the recursive equation
J(st , k(st )) = z(st )k(st )α hd (st )1−α + (1 − δ)k(st ) − k(st+1 ) − w(st )hd (st )X+
qt (st+1 )J(st+1 , k(st+1 )). (1.5)
st+1 |stThe value of a firm that starts history st with capital k(st ) comes from current productionz(st )k(st )α hd (st )1−α , minus the cost of investment k(st+1 ) − (1 − δ)k(st ), minus labor costs
w(st)hd (st ), plus the value of starting the following period in history st+1 ≡ {st , st+1 } withk({st+1 }) units of capital.

Government
A government sets the path of taxes, transfers, and government debt to fund some spendingg(st). I assume government spending is wasteful or at least is separable from consumption
and leisure in preferences. The government faces a budget constraint in any history st ,tb(s ) =∞ X
X

t′ =t st′ |st

′
′
′
′
′
′ 
qt (st ) τ (st )w(st )h(st ) − g(st ) − T (st ) ,(1.6)

so debt b(st ) is equal to the present value of future tax receipts in excess of spending andlump-sum transfers. Again, this is equivalent to a sequence of budget constraints of the formb(st ) + g(st ) + T (st ) = τ (st )w(st )h(st ) +X

st+1 |stqt (st+1 )b(st+1 ),(1.7)

1.2. DERIVING THE LABOR WEDGE

11

so initial debt plus current spending and transfers is equal to current tax revenue plus new
debt issues.

Equilibrium
An equilibrium consists of paths for consumption c(st ), labor supply h(st ), labor demandhd (st ), capital k(st ), assets a(st ), debt b(st ), taxes τ (st ), transfers T (st ), government spendingg(st), intertemporal prices q0 (st ) and the wage rate w(st ) such that: {c(st )}, {h(st )}, and{a(st )} solve households’ utility maximization problem, maximizing equation (1.1) subject tothe budget constraint (equation 1.2) given {q(st )}, {w(st )}, {τ (st )}, and {T (st )}; {hd (st )}and {k(st )} solve firms’ profit maximization problem in equation (1.4) given {q0 (st )} and{w(st)}; the government budget is balanced, so equation (1.6) holds; the labor market clears
in all histories, h(st ) = hd (st ); the capital market clears in all histories, a(st ) = J(st , k(st )) +b(st ); and the goods market clears,z(st )k(st )α hd (st )1−α + (1 − δ)k(st ) = c(st ) + g(st) + k(st+1 ),i.e. output plus undepreciated capital is equal to consumption plus government spending
plus next period’s capital stock. One can confirm that goods market clearing is implied by
the household budget constraint (equation 1.3), the firm’s value function (equation 1.5), thegovernment budget constraint (equation 1.7), and capital and labor market clearing.

1.2

Deriving the Labor Wedge

To see the implications of this model for the labor wedge, I focus on a subset of the equilibrium
conditions. First, consider the household’s choice of history-st consumption and labor supply.
The first order conditions are
β t Π(st )11= λq0 (st ) andc(st )

β t Π(st )γh(st ) ε = λq0 (st )(1 − τ (st ))w(st )

(1.8)
(1.9)

where λ is the Lagrange multiplier on the budget constraint, equation (1.2). Note from thesecond equation that a one percent increase in the after-tax wage (1 −τ )w raises labor supply
h by ε percent, holding fixed the Lagrange multiplier λ and the intertemporal price q0 (st ).
Thus ε is the Frisch elasticity of labor supply, a key parameter in this chapter. In any history
with positive probability, Π(st ) > 0, eliminate λq0 (st )/β t Π(st ) between these equations and12

CHAPTER 1. THE LABOR WEDGE

solve for the wage:
1

γc(st )h(st ) εw(s ) =.
1 − τ (st )t(1.10)

This states that the wage is equal to the tax-adjusted marginal rate of substitution between
consumption and leisure (MRS).
Next turn to the firm’s choice of history-st labor demand. From equation (1.4), the firstorder condition isy(st )w(st ) = (1 − α) d t ,(1.11)h (s )
where y(st ) = z(st )k(st )α hd (st )1−α is the firm’s gross output. Equation (1.11) states that thewage is equal to the marginal product of labor (MPL).
Eliminate the state-contingent wage between equations (1.10) and (1.11) and impose labormarket clearing, hd (st ) = h(st ). Solving for τ (st ) givesτ (st ) = 1 −1+εγc(st )/y(st) h(st ) ε .
1−α(1.12)

This static equation explains how the tax rate τ affects the consumption-output ratio c/yand hours worked h. It is worth stressing that this relationship holds even though productivity, government spending, and distortionary taxes may be time-varying or stochastic.
Expectations of these changes are all captured by the current consumption-output ratio.
For example, if productivity is currently below trend, the consumption-output ratio will
be high and, to the extent that labor supply is elastic, labor supply will be low. An increasein government spending without a corresponding change in contemporaneous taxes will tend
to reduce the consumption-output ratio and raise hours worked in an offsetting manner.
Prescott (2004) uses a version of equation (1.12) to examine the effect of tax variation
over time and across countries on labor supply. More precisely, he uses a slightly different
functional form for preferences, with period utility function log ct + γ log(100 − ht ), where
100 represents the available amount of time per week. He then calibrates γ to match theaverage number of hours worked across a broad set of countries, h̄ ≈ 20. With this functional
form, the Frisch elasticity of labor supply is 100/h − 1, or about 4 on average. My choiceof functional forms brings the issue of the elasticity of labor supply to the forefront of the
discussion.
In addition, I focus on a different implication of this equation. Under the hypothesis thatbusiness cycle fluctuations are not primarily due to changes in taxes and transfers, I interpret
cyclical variation in τ (st ) as the labor wedge, the wedge between the MRS and MPL. Moreprecisely, I measure hours and the consumption-output ratio at quarterly frequencies. By
making appropriate assumptions about the disutility of working γ, the capital share α, and1.3. MEASUREMENT

13

the Frisch elasticity of labor supply ε, I back out the labor wedge from equation (1.12). Thisapproach builds upon a substantial body of research, including Parkin (1988), Rotemberg
and Woodford (1991) and (1999), Hall (1997), Mulligan (2002), and Chari, Kehoe, andMcGrattan (2007).

1.3

Measurement

To measure the labor wedge for the U.S. using equation (1.12), I need time series of theconsumption-output ratio and hours worked, as well as values for the parameters ε, γ, andα. Nominal consumption and output data are available at quarterly frequencies from the
National Income and Product Accounts, Table 1.1.5. Output is Gross Domestic Product,
while consumption is personal consumption expenditures on nondurable goods and services.
I focus on the most comprehensive available series on hours, Prescott, Ueberfeldt, andCociuba’s (2008) measure of total hours worked relative to the noninstitutional population
ages 16 to 64, available quarterly since 1959. This series is based primarily on data originallycollected as part of the Current Population Survey (CPS), a monthly survey of households
that is used to construct the unemployment rate. Total hours is equal to the product of
the number of civilians at work and the average hours worked by a person at work,3 plus
the number of military personnel, who are assumed to work for 40 hours per week. The
population is equal to the civilian noninstitutional population ages 16 to 64 plus the numberof military personnel.
I compare the results with those based on a measure of hours paid per adult in the civilian noninstitutional population from the Current Employment Statistics (CES), a monthly
survey of business establishments.4 The main drawback to this survey is that hours data areunavailable for the government (and military) sector as well as for farm workers, proprietors,
unpaid family workers, and supervisors. In addition, the CES measures hours paid rather
than hours worked, and thus includes vacation time, sick days, and so on. This series isavailable since 1964.
I also use an unpublished series for hours worked constructed by the Bureau of Labor
Statistics as part of the Major Sector Productivity and Costs program.5 I again deflate thisseries by the adult noninstitutional population. Although this measure relies primarily on
3

See http://www.bls.gov/cps/, series LNU02005053 and LNU02005054 for the monthly data since June
1976. Prescott, Ueberfeldt, and Cociuba (2008) obtained earlier data from Table A-24 of the BLS publication
Employment and Earnings.
4
See http://www.bls.gov/ces/, series CES0500000034.
5
See http://www.bls.gov/lpc/, series PRS84006033, for the business sector. The series I use augments
this with estimates of hours worked in the government sector. I am grateful to Simona Cociuba for providing
me with this data and clarifying the relationship between the different series for hours.

14

CHAPTER 1. THE LABOR WEDGE
60

Labor Wedge

50

40

30

ε=4
ε=1
ε = 12

20

10
1955

1960

1965

1970 19751980 1985 1990 1995 2000 2005
Year
Figure 1.1: The U.S. labor wedge from equation (1.12) using the CPS measure of hours. The
solid blue line shows ε = 4, the dashed red line shows ε = 1, and the dotted green line shows
ε = 12 . In each case, I fix the remaining parameters to ensure that the average labor wedge
is 0.40. The gray bands show NBER recession dates.
data from the CES, it also uses data from the CPS to estimate the hours worked by workers
who are not covered by the establishment survey. It also adjusts the CES data to convert
hours paid into hours worked. In principle the coverage of this series should be similar to
Prescott, Ueberfeldt, and Cociuba (2008).
Rather than take a stand on the Frisch labor supply elasticity ε, I consider a range ofpossible values and report three of them, ε = 0.5, 1, and 4. The lowest value is towards theupper range of elasticities for prime age mens’ hours that many microeconomists consider
plausible; see Blundell and MaCurdy (1999). The highest value is in line with the elasticities that macroeconomists frequently use in representative agent business cycle and growthmodels. For each value of the elasticity, I set the ratio γ/(1 − α) so that the average laborwedge is 0.4 from 1959 to 2007, consistent with the tax wedge that Prescott (2004) reports.
The results are similar if the average labor wedge is 0.3 or 0.5.
Figure 1.1 shows the implied behavior of the labor wedge using the CPS measure ofhours.. Two patterns stand out. First, there has been a trend decline in the labor wedge
since around 1980. Arguably this reflects underlying movements in labor and consumption
taxes. Second, the labor wedge is countercyclical. I indicate NBER recession dates with15

1.3. MEASUREMENT

Deviation from Trend

20

10

0

−10

−20
1955

1960

1965

1970 19751980 1985 1990 1995 2000 2005
Year
Figure 1.2: Deviation of the labor wedge from log trend, HP filter with parameter 1600,
using the CPS measure of hours. The solid blue line shows ε = 4, the dashed red line shows
ε = 1, and the dotted green line shows ε = 21 . In each case, I fix the remaining parameters
to ensure that the average labor wedge is 0.40. The gray bands show NBER recession dates.
gray bands. Regardless of the elasticity of labor supply, the labor wedge rose during every
recession except the first, with more pronounced fluctuations when labor supply is less elastic.
This effect does not disappear even when labor supply is infinitely elastic.6
To emphasize this pattern, Figure 1.2 shows the difference between log τ and its trend,
where I measure the trend using a Hodrick-Prescott filter with a standard smoothing parameter, 1600 for quarterly data. It is easy to see a sharp increase in the labor wedge during
every recession except in 1960. The magnitude of the implied cycles depends on the elasticityof labor supply. For example, with ε = 1, the period around the 1990 recession is associatedwith a ten percent increase in the labor wedge relative to trend, while with ε = 4, the increase
was almost six percent.7 Higher values of the labor supply elasticity only slightly dampen
the implied fluctuations—even if the Frisch elasticity is infinite, the labor wedge rose by four
percent during this time period. Conversely, smaller values of the Frisch elasticity amplify
6

Observe that the elasticity of labor supply enters equation (1.12) as (1 + ε)/ε. This means that anelasticity of 4 and an infinite elasticity have nearly the same effect on the labor wedge.
7
With ε = 1, the labor wedge was five percent below trend in the third quarter of 1990 and rose to five
percent above trend by the second quarter of 1992. With ε = 4, it rose from three percent below trend to
three percent above trend during the same period.

16

CHAPTER 1. THE LABOR WEDGE
30

Annual Growth Rate

20
10
0
−10
−20
−30
1955

1960

1965

1970 19751980 1985 1990 1995 2000 2005
Year
Figure 1.3: Annual growth rate of the labor wedge using the CPS measure of hours. The
solid blue line shows ε = 4, the dashed red line shows ε = 1, and the dotted green line shows
ε = 12 . In each case, I fix the remaining parameters to ensure that the average labor wedge
is 0.40. The gray bands show NBER recession dates.
fluctuations in the labor wedge.
Figure 1.3 displays the same findings slightly differently, depicting the annual growthrate of the labor wedge. Again, the labor wedge grew during every recession except the firstin 1960. The magnitude of fluctuations in the growth of the labor wedge depends on theelasticity of labor supply.
Table 1.1 summarizes these results. The first row in the top panel shows the standard
deviation of the detrended labor wedge for four different labor supply elasticities while thefirst column shows the standard deviation of the detrended consumption-output ratio anddetrended hours. When the elasticity is small, the labor wedge is four times more volatile
than hours and five times more volatile than the consumption-output ratio, while the relative volatilities of the labor wedge and hours are similar when the elasticity is large. The
remaining entries show the correlation between the labor wedge (for different elasticities) andthe consumption-output ratio and hours. The correlation with the consumption-output ratio
disappears when the elasticity is high enough, but the labor wedge is strongly negatively
correlated with hours, regardless of the elasticity of labor supply. The bottom panel shows
the analogous results for the annual growth rate of the labor wedge, the consumption-output

1.3. MEASUREMENT

17

labor wedge
detrended
ε = 0.5ε=1
ε=4 ε=∞
s.d.
0.055
0.031
0.018
0.014
c/y
0.010
0.338
0.278
0.049 −0.131h0.013 −0.795 −0.835 −0.745 −0.628
annual growth ε = 0.5ε=1ε=4 ε=∞
s.d.
0.079
0.045
0.027
0.022
c/y
0.015
0.256
0.163 −0.088 −0.260h0.018 −0.803 −0.835 −0.733 −0.617
Table 1.1: Comovement of the labor wedge, hours, consumption-output ratio for four different
values of the labor supply elasticity using the CPS measure of hours. In the top panel, allseries are detrended with an HP filter with parameter 1600. The bottom panel shows annual
growth rates.
ratio, and hours. They are quantitatively very similar. It looks as if hours growth is negative
when the labor income tax rate is rising, regardless of the elasticity of labor supply.
There are several different ways to understand these results. Viewed through the lensof a model with a competitive labor market, fluctuations in the labor income tax rate drive
fluctuations in hours. If in reality the tax rate is constant, the model underpredicts fluctuations in hours worked at business-cycle frequencies, given the observed time path of theconsumption-output ratio. This is an old critique of competitive models of the labor market:
such models can only explain part of the cyclical fluctuations in hours worked, particularly
when labor is supplied relatively inelastically.
Table 1.2 shows that the main conclusions hold with the alternative measures of hours.
I assume that the elasticity of labor supply is ε = 1 and examine how alternative measuresof hours affect the behavior of the labor wedge. These other measures raise the volatilityof the labor wedge, and by more so than they raise the volatility of hours. In addition,
the correlation between the two series is, if anything, increased. I find similar results withother values of the labor supply elasticity and so conclude that this result is robust to theexact measure of hours. The competitive model cannot explain all of the movement in hours
relative to the consumption-output ratio if the labor income tax rate is constant.
One possible explanation for this pattern is that labor tax rates are in fact countercyclical.
This hypothesis has some supporters. For example, in a recent paper, Mertens and Ravn
(2008) measure tax shocks using the Romer and Romer (2007) narrative analysis of tax
policy. They conclude that tax shocks account for 18 percent of the variance of output at

18

CHAPTER 1. THE LABOR WEDGE
detrended
CPS
CES
BLS
std. dev. h 0.013 0.018 0.014
std. dev. τ 0.031 0.049 0.035
correl. (h,τ ) -0.835 -0.868 -0.883

annual growth
CPS
CES
BLS
std. dev. h 0.018 0.026 0.020
std. dev. τ 0.045 0.069 0.051
correl. (h,τ ) -0.835 -0.872 -0.879
Table 1.2: Comovement of the labor wedge and hours with labor supply elasticity ε = 1. In
the top panel, all series are detrended with an HP filter with parameter 1600. The bottom
panel shows annual growth rates. The first column uses the hour series from Prescott,Ueberfeldt, and Cociuba (2008) from 1959 to 2007, constructed primarily from the CPS. The
second uses hours data from the CES from 1964 to 2007. The third uses unpublished hours
data constructed by the BLS to measure labor productivity from 1959 to the first quarter of2006.
business cycle frequencies. Perhaps most provocatively, they find that the 1982 recession wascaused by workers’ anticipation of future tax cuts. Of course, I have shown that expectationsof future tax cuts may affect both the consumption-output ratio and the hours worked, but
not the labor wedge, so such behavior cannot easily explain the patterns in the data. In
any case, most economists seem to be skeptical that tax movements alone can explain theobserved variation in the labor wedge.

1.4

Alternative Specification of Preferences

A second possible explanation for the behavior of the labor wedge is that in this model
either the MRS or MPL is misspecified. The specification of the MPL depends only on
the assumption of a Cobb-Douglas aggregate production function. Macroeconomists are
justifiably reluctant to abandon that assumption because it ensures that the capital andlabor shares of national income as well as the interest rate are constant, consistent with the
Kaldor (1957) growth facts.
The specification of household preferences is also tightly constrained by long-run restrictions. Maintain the assumption that preferences are separable across time and states of theworld, but relax the assumption of additive separability between consumption and leisure.
To be consistent with balanced growth—the absence of a long-run trend in hours—and a19

1.4. ALTERNATIVE SPECIFICATION OF PREFERENCES
constant Frisch elasticity, preferences over consumption and leisure must be ordered by
1+ε

∞ X
X
t=0

s

γε
h(st ) εc(st )1−σ 1 + (σ − 1) 1+εttβ Π(s )
1−σ
t

σ

−1

.

(1.13)As before, γ > 0 parameterizes the disutility of work and ε > 0 is the Frisch labor supply
elasticity. The new parameter σ > 0 determines the substitutability between consumption
and leisure. The limit as σ → 1 nests the separable case in equation (1.1). The case whereσ > 1 is of particular interest, since this implies the marginal utility of consumption is higher
when households work more, consistent with standard models of time allocation (Becker,1965). In any case, utility is increasing and concave in consumption and decreasing andconcave in hours of work.
With these preferences, the first order conditions for consumption and hours are
1+ε σγεh(st ) ε= λq0 (st ),1+ε
1+ε σ−11γεh(st ) εqβ t Π(st )c(st )1−σ 1 + (σ − 1)σγh(st ) ε = λq0 (st )(1 − τ (st ))w(st).
1+εβ t Π(st )c(st )−σ 1 + (σ − 1)

The Frisch demand system expresses consumption c(st ) and hours h(st ) as functions of the
Lagrange multiplier λ, the intertemporal price q0 (st ), and the wage rate w(st ). Eliminatingc(st ) between the first order conditions gives1εσγh(st ) =

λq0 (st )β t Π(st ) σ1(1 − τ (st ))w(st ),so the Frisch elasticity of labor supply, the elasticity of hours with respect to the wage holding
fixed the intertemporal price and the Lagrange multiplier, is in fact ε. Moreover, eliminate
λq0 (st )/β t Π(st ) between the first order conditions to get1σγc(st )h(st ) εtw(s ) =1+ε

γε
(1 − τ (st )) 1 + (σ − 1) 1+εh(st ) ε

whenever Π(st ) > 0. Eliminate the wage using equation (1.11) and impose market clearing,h(st ) = hd (st ). This gives
1+ετ (st ) = 1 −γσ(c(st )/y(st ))h(st ) ε1+εγε(1 − α) 1 + (σ − 1) 1+εh(st ) ε.

(1.14)

This is a modest generalization of equation (1.12). Once again, one needs only to know

20

CHAPTER 1. THE LABOR WEDGE
10
8

Deviation from Trend

6
4
2
0
−2
−4
−6
−8
−10
1955

1960

1965

1970 19751980 1985 1990 1995 2000 2005
Year
Figure 1.4: The deviation of the labor wedge from log trend, HP filter with parameter 1600,
using equation (1.14) and the CPS measure of hours. The dashed red line shows σ = 1, thedotted green line shows σ = 2, and the solid blue line shows σ = 4. In each case I set ε = 1and α = 1/3 and fix the remaining parameter γ to ensure that the average labor wedge is0.40. The gray bands show NBER recession dates.
the consumption-output ratio, hours worked, and the value of four parameters in order tocompute the labor wedge.
To understand the quantitative implications of this expression, I use the hours series
from Prescott, Ueberfeldt, and Cociuba (2008). I fix the labor share at the conventional
value of 1 − α = 2/3 and for different values of the substitutability parameter σ and theelasticity of labor supply ε, choose the disutility of work parameter γ to ensure an averagelabor wedge of 0.40. Figure 1.4 shows the time series behavior of the labor wedge with the
Frisch elasticity fixed at 1. The dashed red line corresponds to the limit as σ converges to
1, the additively separable case that I analyzed before, while the remaining two lines show
σ = 2 and σ = 4. Raising the substitutability between consumption and leisure modestly
reduces the magnitude of fluctuations in the labor wedge but does not qualitatively change
the results. I do not show the results with a higher elasticity of labor supply, but they too
are similar.
Additionally, the microeconomic behavior of the model is unreasonable when σ is muchlarger than 1. Consider the following thought experiment: two workers normally work and21

1.5. PREFERENCE SHOCKS

consume the same amount. One year, however, one is unable to work, for example becauseof an idiosyncratic shock to the disutility of work γ, while the other continues to work theaverage number of hours that I observe in the data. With complete markets, the two workers
want to keep their marginal utility of consumption λ equal through this episode. How much
lower is the consumption of the unemployed worker, cu , compared to the consumption of theemployed, ce ? The first order condition for consumption impliesce
γε 1+ε= 1 + (σ − 1)
h ε ,
cu
1+ε
where h is the number of hours worked by the employed worker. With σ = 1, consumption
is equal for the two workers. With ε = 1 and σ = 1.5, consumption falls by 13 log points forthe unemployed. With σ = 2, consumption should fall by 20 log points, and with σ = 4 itshould fall by 31 log points.
Are these numbers reasonable? Aguiar and Hurst (2005) provide some guidance. Theyfind that food consumption expenditures drop by about 17 percent at retirement, accompanied by a 53 percent increase in the time spent on food production. This is consistent
with values of σ in the range of 2. Much stronger substitutability is inconsistent with thenumbers in Aguiar and Hurst (2005) and do not resonate introspectively. In any case, evenwith infinite substitutability between consumption and leisure, σ → ∞, the results do notchange appreciably. With the labor supply elasticity fixed at ε = 1, the standard deviationof the annual growth rate of the labor wedge is more than twice the corresponding number
for hours and the correlation between the two series is −0.76.

1.5

Preference Shocks

A third theoretical possibility is that the representative agent’s disutility of work, γ, isstochastic. This modifies equation (1.12) to readτ (st ) = 1 −
1+εγ(st )c(st )/y(st) h(st ) ε ,1−αwhere γ(st ) is the history-contingent disutility of work. Then an economist who ignored
variation in the disutility of work would falsely conclude that there are fluctuations in thelabor wedge, even if labor taxes are constant in the data and the model is otherwise correct.
Many recent quantitative macroeconomic models allow for such a preference shock. An
unobserved demand shock plays an important role in explaining aggregate fluctuations in
Rotemberg and Woodford (1997). They interpret the shock as a combination of a preference

22

CHAPTER 1. THE LABOR WEDGE

shock and a shock to government spending.8 Erceg, Henderson, and Levin (2000) and Smets
and Wouters (2003) also have a quantitatively important preference shocks in their models
of monetary policy. More recently, Galı́ and Rabanal (2004) find that a preference shock
explains 57 percent of the variance of output and 70 percent of the variance of hours in theirestimated dynamic stochastic general equilibrium model.9
A closely related theoretical possibility is that workers have time-varying market power.
This is often formalized by assuming that each household is the monopoly supplier of aheterogeneous type of labor and sets the wage to maximize its utility. Recessions are periods
when different types of labor are poor substitutes, so households are better able to exploit
their market power, reducing hours to drive up wages. A number of recent papers haveemphasized time-varying wage markups as an important source of business cycle shocks,
including Smets and Wouters (2003) and (2007). In Smets and Wouters (2007), the wage
markup shock accounts for twenty percent of the variance in output and over half the variancein inflation at a ten-quarter horizon. Galı́, Gertler, and López-Salido (2007) also find an
important role for markup fluctuations, but reach a different conclusion. They argue thatsome other, unspecified, primitive shock causes countercyclical fluctuations in markups, which
in turn generates a countercyclical labor wedge.
Like many economists, I have a strong prior belief that the changes in the disutility oflabor and changes in wage markups do not drive business cycle fluctuations.10 Although
households may differ in their disutility of work and the disutility may change over time
for some households, one would expect those movements to average out in a large economy.
Still, the empirical success of preference and markup shocks is revealing. They work because,viewed through the lens of a market-clearing model, recessions look like times when workers
choose to supply less labor, and expansions like times when they choose to supply more labor,
than is predicted by the model.
8

Unlike preference shocks, government spending can be measured and is not strongly correlated with thelabor wedge. In any case, the model I have developed here allows for government spending shocks, but these
do not affect the labor wedge equation.
9
Galı́ and Rabanal (2004, p. 271) write that the preference shock can be “interpret[ed] more broadly as a(real) demand shock.”10
For example, Modigliani (1977, p. 6) writes “Sargent (1976) has attempted to remedy this fatal flawby hypothesizing that the persistent and large fluctuations in unemployment reflect merely corresponding
swings in the natural rate itself. In other words, what happened to the United States in the 1930’s was asevere attack of contagious laziness! I can only say that, despite Sargent’s ingenuity, neither I nor, I expect,most others at least of the nonmonetarists’ persuasion are quite ready yet to turn over the field of economic
fluctuations to the social psychologist.” Mankiw (1989, p. 82) writes “Alternatively, one could explain theobserved pattern without a procyclical real wage by positing that tastes for consumption relative to leisurevary over time. Recessions are then periods of ‘chronic laziness.’ As far as I know, no one has seriously
proposed this explanation of the business cycle.”1.6. FROM HOURS TO UNEMPLOYMENT

23

4
3

Deviation from Trend

2
1
0
−1
−2
−3
−4
−5
1955

1980 1985 1990 1995 2000 2005
Year
Figure 1.5: Deviation of per capita hours and the e-pop ratio from log trend, HP filter withparameter 1600. The solid blue line shows the deviation of hours from log trend and thedashed red line shows the deviation of employment. The gray bands show NBER recession
dates.

1.6

1960

1965

1970 1975

From Hours to Unemployment

In the remainder of the book, I explore whether search and matching models, based on
Pissarides (1985) and Mortensen and Pissarides (1994), can help to explain the behavior of
the labor wedge. In doing so, I switch from a focus on the behavior of hours worked to a focus
on the behavior of employment and unemployment. This is because search costs introduce
a nonconvexity into households’ decision problem which emphasizes the binary decision ofwhether to work, rather than the continuous decision of how many hours to work eachweek. The data indicate that this focus is, for the most part, appropriate. Figure 1.5 showsthat the correlation between detrended employment-population (e-pop) ratio and detrended
hours per adult is 0.97 and detrended hours are only slightly more volatile than detrended
employment, with a relative standard deviation of 1.3.11 In words, most business cycle
frequency fluctuations in hours are accounted for by fluctuations in employment, rather thanfluctuations in the number of hours per employee.
11

I compute hours per member of the non-institutional population aged 16 to 64 using the measure ofhours in Prescott, Ueberfeldt, and Cociuba (2008). I use their measure of employment as well, and divide by
the same population measure.

24

CHAPTER 1. THE LABOR WEDGE

Absolute Deviation from Trend

2

1

0

−1

−2
1955

1960

1965

1970 19751980 1985 1990 1995 2000 2005
Year
Figure 1.6: Deviation of the e-pop and u-pop ratios from trend, HP filter with parameter
1600. The dashed red line shows the deviation of the e-pop ratio from trend and the solid
blue line shows the deviation of the u-pop ratio. The gray bands show NBER recession dates.
Like many search models, the book also focuses on the margin between employment andunemployment, abstracting away from entry and exit from the labor force. Again, this isempirically reasonable at business cycle frequencies. Figure 1.6 shows that the correlation
between the absolute deviation of the e-pop ratio from trend and the absolute deviation of
the unemployment-population (u-pop) ratio from trend is −0.90, with a relative standard
deviation of 1.4.12 When employment falls below trend, most of the workers show up as
unemployed, rather than dropping out of the labor force.

12

I use the standard Bureau of Labor Statistics measure of unemployment, based on the CPS.

Chapter 2
Benchmark Search ModelIn this chapter I develop a benchmark search model. A representative firm allocates workers
between two activities, production and recruiting. Recruiters enable the firm to attract moreworkers, while producers generate revenue by creating a consumption good. There is no
capital in the model, so production uses only labor. Employed workers are periodically hit
by shocks that leave them unemployed, while unemployed workers find jobs when they contact
a recruiter. Workers’ preferences are additively separable over time and between consumption
and leisure, and consistent with balanced growth. A government levies a labor income tax
and rebates the proceeds lump-sum to households. Finally, wages are set according to the
Nash bargaining solution.
I find that aggregate productivity shocks have no effect on employment or the labor wedge.
To understand why, suppose productivity is temporarily above trend and firms take advantageof this by shifting workers into production. Since there is no capital, this necessarily raises
consumption. But workers would like to maintain smooth consumption, which puts downward
pressure on interest rates. This makes recruiting, a form of investment that is costly today
but generates revenue in the future, more profitable. With the particular assumption on
preferences, firms in fact find it optimal to maintain a constant division of labor between
recruiting and production. With a constant number of recruiters, employment is constant.
With a constant number of producers, the consumption-output ratio is constant. Thus themodel implies that the labor wedge, as measured in equation (1.12), should be constant as
well. The benchmark search model therefore is unable to explain the data that I discussed
in Chapter 1.
I start in Section 2.1 with a steady state version of the model in order to introduce theenvironment and notation. Section 2.2 allows for productivity shocks, Section 2.3 establishes
conditions under which the equilibrium solves an associated planner’s problem, and Section 2.4 shows that the neutrality result is robust to a number of generalizations, including25

26

CHAPTER 2. BENCHMARK SEARCH MODELa labor force participation margin and an endogenous choice of hours for employed workers.
Loosely speaking, if a version of the model without search frictions does not generate fluctuations in employment or hours worked, search frictions will not explain why labor market
outcomes fluctuate cyclically. Thus this neutrality result is a natural extension of known
results in competitive business cycle models.

2.1

Steady State

There are three types of economic actors: households composed of many individuals who
consume, search for jobs, and supply labor in order to maximize expected utility subject
to a budget constraint; firms, which allocate workers between production and recruiting inorder to maximize the expected present value of profits; and a government, which taxes labor
earnings and rebates the proceeds to households lump-sum. There are two technologies, one
for matching unemployed workers seeking job openings to recruiters seeking workers andanother for producing the consumption good using labor.
Denote time by t = 0, 1, 2, . . . and assume for now that there are no shocks, so in thenotation of the previous chapter, Π(st ) is degenerate.

2.1.1

Firms

A representative firm employs a measure n0 workers at time 0. The firm has access to twotechnologies. The first is a constant returns to scale production technology: each worker
who uses the technology produces z units of the consumption good per period. Let ℓt denotethe measure of producers, i.e. workers who use the production technology in period t. The
second is a constant returns to scale recruiting technology: a worker who uses the recruiting
technology in period t attracts an average of µ(θt ) unemployed workers to the firm at thestart of period t + 1, where θt is the ratio of the measure of recruiters to the measure ofunemployed workers in the aggregate economy, hereafter the recruiter-unemployment ratio.
Let vt denote the measure of recruiters, i.e. workers who use the recruiting technology inperiod t.
vt µ(θt ) represents the measure of new employment relationships, or matches, in the economy in period t. This is essentially a matching function (Pissarides, 1985), but the inputs into
matches are unemployed workers and recruiters, rather than unemployed workers and vacantjobs. In particular, the matching function exhibits constant returns to scale and should be
increasing in both the measure of unemployed workers and the measure of vacancies. Thus
I assume µ : R+ → R+ is continuous and nonincreasing with limθ→0 µ(θ) = ∞. It will be27

2.1. STEADY STATE

useful to define f (θ) ≡ θµ(θ), which I argue below is the probability that an unemployedworker finds a job. This is nondecreasing and concave, with f (0) = 0 and f (θ) ≤ 1 for all θ,
so in particular limθ→∞ µ(θ) = 0.
The present value of the firm’s profits, discounted back to time-0 and indexed by the
initial firm size n0 , isJ(0, n0 ) =

∞
X
t=0


q0t zℓt − wt (ℓt + vt ) ,(2.1)

where q0t is the price of one unit of consumption in date t, denominated in units of date 0
consumption and wt is the wage in units of the contemporaneous consumption good. The
firm can freely switch workers between production and recruiting, but total employmentnt ≡ ℓt + vt is constrained by past recruiting. That is, n0 is given, while in any period t ≥ 0,
nt+1 = vt µ(θt ) + (1 − x)nt ,(2.2)

where x ∈ (0, 1) is the “employment exit probability,” i.e. the exogenous rate of job separations. Note that since limθ→0 µ(θ) = ∞ and µ is continuous, µ(θ) > x for small θ. Thisimplies that a firm can always grow by putting enough of its workers into recruiting as longas the recruiter-unemployment ratio is not too large. The firm chooses the sequences {ℓt }and {vt } in order to maximize the present value of profits in equation (2.1), taking as given
the constraint on firm growth in equation (2.2), the initial level of employment n0 , the pathof prices wt and q0t , and the path of the recruiter-unemployment ratio θt .

2.1.2

Households

There is a representative household consisting of a continuum of expected-utility-maximizing,
infinitely-lived individuals with measure 1. Each household member i ∈ [0, 1] has timeseparable preferences over her consumption cit and her labor supply nit . She may be either
employed by a firm (nit = 1) or unemployed (nit = 0). Her period felicity is log cit if she isunemployed and log cit − γ if she is employed. She discounts future felicity with factor β < 1.
The household allocates total consumption in period t, ct , in order to maximize the sumof household utility, and so equalizes the marginal utility of consumption across individuals. With additive separability between consumption and leisure, this implies the household
equalizes consumption across individuals, acting as if it has a utility function
∞
X
t=0


β t log ct − γnt ,

(2.3)

28

CHAPTER 2. BENCHMARK SEARCH MODEL

where nt is the fraction of household members who are employed in period t. Effectively
the large household is able to insure its members against all idiosyncratic shocks, mimicking
the complete markets allocation. This approach to finding the complete markets allocationin search models is due originally to Merz (1995); see Andolfatto (1996) for an alternative
approach to modeling complete markets in a search economy.
The household faces two types of constraints. First, there is a single budget constraint,
which states that initial assets must equal the difference between the present value of consumption and the present value of after-tax labor income plus transfers,
a0 =

∞
X
t=0

q0t (ct − (1 − τ )wt nt − Tt ).

(2.4)

The household earns pre-tax income wt nt , pays a proportional labor income tax τ , receives
a lump-sum transfer Tt , and discounts period-t income at the same rate q0t as firms.
Second, employment is determined by household members’ flow in and out of jobs. Employed workers lose their job with probability x each period, while unemployed workers finda job with probability f (θt ) = θt µ(θt ), the product of the recruiter-unemployment ratio andthe ratio of matches to recruiters, i.e. the ratio of matches to unemployed workers. Put differently, the number of workers finding jobs, f (θt )(1 − nt ), is equal to the number of workers
recruited by firms, µ(θt )vt (n0 ), since f (θ) = θµ(θ) and θt = vt (n0 )/(1 − nt ). The household
takes n0 as given and recognizes thatnt+1 = (1 − x)nt + f (θt )(1 − nt ).

(2.5)

Let V (0, a0 , n0 ) denote the time-0 value of maximizing utility in equation (2.3) subject to thebudget constraint in equation (2.4) and the law of motion for employment in equation (2.5).

2.1.3

Government

The government runs a balanced budget in each period, rebating tax revenue lump-sum tohouseholds:
Tt = τ wt nt .

(2.6)

I assume for now that there is no government spending.

2.1.4

Wages

At the start of each period, each employed worker bargains with her employer over her wage.
If bargaining fails, the match breaks up; the worker is unemployed in the current period but

29

2.1. STEADY STATE

has a chance to search for a job, while the firm loses the job. If bargaining succeeds, theworker is paid the pre-tax bargained wage w this period and is deployed by the firm either
producing output or recruiting other workers. Moreover, both the worker and firm anticipate
agreeing on the equilibrium wage wt in any future period t when they are matched. In
equilibrium, the bargained wage equals the equilibrium wage in every period.
Rather than write down a particular model of bargaining, I follow Pissarides (1985) andposit that the outcome of bargaining satisfies several reasonable properties and look at theirimplications. More precisely, I use the axiomatic Nash bargaining solution (Nash, 1953), but
relax the symmetry assumption. Let Ṽn (t, w) denote the marginal utility for a householdwith the equilibrium level of assets and employment of having one worker employed at awage w in period t rather than unemployed. Also let J˜n (t, w) denote the marginal profit toa firm with the equilibrium number of employees of employing one worker at w. In bothcases, the subscript ‘n’ refers to partial derivatives, to stress that these are marginal values.
I compute Ṽn and J˜n in Section 2.1.6 below.
Assuming there are gains from trade, that is wages w satisfying both Ṽn (t, w) ≥ 0 andJ˜n (t, w) ≥ 0, the bargained wage maximizes the asymmetric Nash product,Ṽn (t, w)φ J˜n (t, w)1−φ ,(2.7)

where φ ∈ (0, 1) represents workers’ bargaining power. If there are no gains from trade, theworker becomes unemployed.

2.1.5

Equilibrium

For an arbitrary tax rate τ , an equilibrium consists of paths for consumption ct , employment
nt , producers ℓt , recruiters vt , transfers Tt , intertemporal prices q0t , the wage rate wt , and theratio of recruiters to unemployed workers θt such that: {ℓt } and {vt } solve firms’ profit maximization problem, maximizing equation (2.1) subject to the law of motion for employment
in equation (2.2) given {q0t }, {wt }, and {θt }; {ct } and {nt } solve households’ utility maximization problem, maximizing equation (2.3) subject to the budget constraint (equation 2.4)
and the law of motion for employment in equation (2.5) given {q0t }, {wt }, {θt }, and {Tt }; thegovernment budget is balanced, so equation (2.6) holds; wt maximizes the Nash product inequation (2.7) for all t; the labor market clears, nt = ℓt + vt for all t; and the goods market

clears, zt ℓt = ct for all t.
I look for a steady state equilibrium where consumption, employment, producers, recruiters, transfers, and wages are all constant, while q0t = q̄ t for some q̄ < 1, so intertemporal
prices decline geometrically. Note that for such a steady state to exist, I require that n0 = n̄,30

CHAPTER 2. BENCHMARK SEARCH MODEL

where
n̄ =

f (θ̄)f (θ̄) + x(2.8)

and θ̄ is the steady state ratio of recruiters to unemployed workers. From equation (2.5), this
implies nt = n̄ for all t. I drop irrelevant time subscripts in what follows.

2.1.6

Characterization

Households
I begin by expressing the household’s problem recursively. A household starts a period withassets a and employment n. Suppressing the time argument in the value function, its lifetime
utility satisfies
V (a, n) = max
log a + (1 − τ )w̄n + T̄ − q̄a − γn + βV a , (1 − x)n + f (θ̄)(1 − n) , (2.9)
′
a



′



′

where w̄ is the equilibrium wage and T̄ is the equilibrium transfer. The household decides
this period how much to save a′ , determining its state at the start of the next period (a′ , n′ ),where n′ = (1 − x)n + f (θ̄)(1 − n) solves equation (2.5). Savings determines the consumption
available this period via the intertemporal budget constraint, c = a + (1 − τ )w̄n + T̄ − q̄a′ .
Together with a natural borrowing limit, which ensures that a household can always service
the interest on its debt,a≥−(1 − τ )w̄ + T̄,
1 − q̄

this is equivalent to the original household budget constraint in equation (2.4).
The household’s behavior is standard. Differentiate the value function with respect toinitial assets to get the envelope condition1
Va (a, n) = ,cwhere the subscript denotes a partial derivative and c = a + (1 − τ )w̄n + T̄ − q̄a′ . Moreover,the first order condition with respect to next period’s assets impliesq̄= βVa (a′ , n′ ).
c

In steady state a = a′ = ā and n = n′ = n̄, and in particular the marginal value of assets isconstant,
Va (a, n) = Va (a′ , n′ ).

31

2.1. STEADY STATE

Eliminating Va between the envelope and first order conditions then pins down the intertemporal price, q̄ = β. Then the budget constraint determines consumption, equal to the annuity
value of assets, (1 − β)ā, plus the after-tax wage times the employment rate, plus the lumpsum transfer. For a household that starts at steady state employment, n = n̄,c̄ = (1 − β)ā + (1 − τ )w̄n̄ + T̄ .

(2.10)To characterize the wage using the Nash bargaining solution in equation (2.7), I need tocompute the marginal value to the household of having a worker employed at w̄ rather thanunemployed. Differentiating equation (2.9) gives
Vn (a, n) =(1 − τ )w̄
− γ + β(1 − x − f (θ̄))Vn (a, n).
c̄(2.11)

Now consider the value of a household that has assets ā, has n̄ workers paid w̄ and has asmall measure, say ǫ > 0, workers paid an arbitrary wage w this period. All wages revert tow̄ next period. Modifying equation (2.9), the value of the household isV̂ (w, ǫ) = max
log ā + (1 − τ )(ǫw + n̄w̄) + T̄ − q̄a′ − γ(n̄ + ǫ)′a
+ βV a , (1 − x)(n̄ + ǫ) + f (θ̄)(1 − n̄ − ǫ) .
′Differentiate this with respect to ǫ and evaluate at ǫ = 0 to get the marginal value of an
employed worker at w:
V̂ǫ (w, 0) =(1 − τ )w− γ + β(1 − x − f (θ̄))Vn (ā, n̄) ≡ Ṽn (w).
c̄

Combining this with equation (2.11) gives a convenient expression for the marginal value of
having a worker employed at w rather than unemployed:Ṽn (w) =(1 − τ )(w − w̄)+ Vn (ā, n̄).
c̄

(2.12)Vn (ā, n̄) is the marginal value of having a worker employed at the equilibrium wage rather
than unemployed. To this one adds the worker’s incremental after-tax income from receiving
an arbitrary wage rather than the equilibrium wage, (1−τ )(w−w̄), multiplied by the marginalutility of income 1/c̄.

32

CHAPTER 2. BENCHMARK SEARCH MODEL

Firms
Next I turn to the firm. It is again useful to express its problem recursively, suppressing timearguments:J(n) = max z(1 − ν) − w̄ n + q̄J (1 − x + νµ(θ̄))n ,(2.13)
ν∈[0,1]

where ν = v/(v + l) is the share of employees who are recruiters and the size of the firm next
¯ for all n,period is n′ = (1 − x + νµ(θ̄))n. The firm’s value function is linear in n, J(n) = Jn
where

 
J¯ = max z(1 − ν) − w̄ + q̄ 1 − x + νµ(θ̄) J¯ .
(2.14)
ν∈[0,1]The value function is linear because a firm that is twice as large as another but devotes thesame share of employees to recruiting earns twice as much current profits and is twice aslarge next period.1
Since the objective function in equation (2.14) is linear in ν, the solution is typicallyextreme, either ν = 0 or ν = 1. The former implies that firms shrink over time withattrition, while the latter implies that there is no production, and so neither is consistent
with a steady state equilibrium. There is an interior solution only if¯
z = βµ(θ̄)J,(2.15)

where I use q̄ = β. A producer yields output z today, while a recruiter generates µ(θ̄) addi¯ tomorrow. For workers to be engaged in both activities,
tional employees, each valued at J,
these must be equal.
Eliminate J¯ between equations (2.14) and equation (2.15):z = βµ(θ̄)

z − w̄
.
1 − β(1 − x)(2.16)The left hand side is the current output of a worker using the production technology. The
right hand side is the expected number of additional workers attracted by a recruiter times thepresent value of profits produced by those workers if they are all employed in the production
technology. If equation (2.16) holds, all firms find any value of ν ∈ [0, 1] optimal.
From equation (2.2), employment is constant ifν̄ = x/µ(θ̄).

(2.17)

More precisely, any individual firm earns the same present value of profits for any choice of1

I give a formal proof that the firm’s value is linear in the stochastic model in Section 2.2.

33

2.1. STEADY STATE{νt }; however, in the entire economy, a fraction ν̄ of all employed workers must be recruitersin any particular period, so as to keep employment constant. Note that since ν̄ ∈ [0, 1] andµ(θ) is nondecreasing, this equation places an upper bound on θ̄, µ(θ̄) ≥ x.
Now consider a firm that employs n̄ workers at w̄ and ǫ workers at w this period. All
wages revert to w̄ next period. Modifying equation (2.13) slightly, the value of such a firm isˆ ǫ) = max z(1 − ν)(n̄ + ǫ) − w̄n̄ − wǫ + q̄ J¯ 1 − x + νµ(θ̄) (n̄ + ǫ) .
J(w,νDifferentiate this with respect to ǫ and evaluate at ǫ = 0 to get the marginal value of
employing a worker at w:
Jˆǫ (w, 0) = z − w + β(1 − x)J¯ ≡ J˜n (w).
Since J¯ = z − w̄ + β(1 − x)J¯, I can also express the marginal value of an employee paid w as
J˜n (w) = w̄ − w + J¯,(2.18)

analogous to equation (2.12). The marginal value of employing a worker at an arbitrary wage
¯ plusw this period is the marginal value of employing a worker at the equilibrium wage, J,
the wage savings w̄ − w.
Wage
I next characterize the wage using equation (2.7). The first order condition for the equilibrium
wage isJ˜′ (w̄)Ṽ ′ (w̄)+ (1 − φ) n= 0.
φ n
Ṽn (w̄)J˜n (w̄)From equation (2.12), Ṽn′ (w̄) = (1 − τ )/c̄ and Ṽn (w̄) = Vn (ā, n̄), while equation (2.18) implies¯ Therefore maximizing the Nash product is equivalent to making
J˜′ (w̄) = −1 and J˜n (w̄) = J.
nthe gains from trade proportionate:(1 − φ)Vn (ā, n̄)c̄ = φ(1 − τ )J¯.

(2.19)

To find the equilibrium wage w̄, eliminate Vn and J¯ using equations (2.11) and (2.15):(1 − φ)(1 − τ )w̄ − γc̄z= φ(1 − τ ).
1 − β(1 − x − f (θ̄))βµ(θ̄)

34

CHAPTER 2. BENCHMARK SEARCH MODEL

Eliminate 1 − β(1 − x) from the left hand side using equation (2.16) and simplify:(1 − φ)(1 − τ )w̄ − γc̄z= φ(1 − τ ).
z−w̄
µ(θ̄) z + f (θ̄)µ(θ̄)

Recalling that f (θ) = θµ(θ), solve for the after-tax wage:(1 − τ )w̄ = φ(1 − τ )z(1 + θ̄) + (1 − φ)γc̄.

(2.20)

The after-tax wage is a weighted average of (1−τ )z(1+ θ̄) and γc̄, with the weight on the firstterm equal to workers’ bargaining power. If a worker is employed, she can directly produce(1 − τ )z units of after-tax output. In addition, by marginally reducing the unemployment
rate, she frees up θ̄ workers to produce output rather than recruit, since θ̄ is the aggregate
recruiter-unemployment ratio. That is, these additional workers can produce while keeping
the recruiter-unemployment ratio constant. Thus (1 − τ )z(1 + θ̄) measures the additional
after-tax output generated by moving a worker from unemployment to employment, i.e. themarginal product of labor. The worker captures a fraction φ of the MPL. On the other hand,
if a worker is employed, she suffers disutility γ. To value this in consumption units, divide by
the marginal utility of consumption 1/c̄. Thus γc is the marginal rate of substitution between
consumption and leisure. The worker keeps a fraction 1 − φ of the MRS. In a frictionlessmodel, the MRS and MPL are equal and both are equal to the wage. With search frictions,
there are typically gains from trade, so the MPL is larger than the MRS and the wage is aweighted average of the two, with weights determined by bargaining power.
Equation (2.20) states that a worker with higher consumption, or equivalently a lower
marginal utility of wealth, is paid a higher wage. This is due to the increase in her threat
point when evaluated in consumption-equivalent units. Yet when I solved the household’s
consumption problem, I implicitly assumed that her wealth does not affect her wage. One
possible interpretation is that a firm cannot observe a household’s wealth when bargaining.
While a worker would like to claim to be rich so as to boost her wage, firms bargain as if
they are facing a worker with the representative household’s marginal utility of wealth 1/c̄.
Another interpretation is that each firm bargains simultaneously with all its workers and theyagree on a common wage. Assuming households spread their workers across many firms, asingle household’s decision to save more will not affect the wage that its members receive,
eliminating any incentive to distort its savings.

35

2.1. STEADY STATE

2.1.7

Equilibrium

Five objects completely characterize an equilibrium: consumption c̄, employment n̄, thewage w̄, the recruiter-unemployment ratio θ̄, and the fraction of employed workers who arerecruiters ν̄. I have obtained three relationships between these variables: equation (2.8),
which gives the steady state employment rate; equation (2.16), which gives a condition for
an interior solution and equation (2.20), which determines the wage. To close the model, I
use two market clearing conditions. First, consumption is equal to output:c̄ = zn̄(1 − ν̄).

(2.21)For the representative household, a fraction n̄ workers are employed, and of those, a fraction
1 − ν̄ each produce z units of the consumption good. Second, there are ν̄ n̄ recruiters and1 − n̄ unemployed workers, so the ratio of recruiters to unemployed workers is
θ̄ =

ν̄ n̄
.
1 − n̄

(2.22)

Given a solution to these five equations, it is easy to solve for any other variable of interest.
To find the equilibrium, first eliminate ν̄ from equation (2.21) using equation (2.22) andthen eliminate n̄ using equation (2.8). This implies that consumption is proportional toproductivity,z(f (θ̄) − xθ̄)c̄ =.
f (θ̄) + x
Next eliminate w̄ from equation (2.16) using equation (2.20) and then eliminate c̄ using thepreceding equation to obtain an equation in θ̄ alone, T (θ̄) = 0 whereT (θ) ≡

θ(1 − β(1 − x))(1 − φ)γ(f (θ) − xθ)− 1 + φ(1 + θ) +.
βf (θ)(1 − τ )(x + f (θ))(2.23)

To prove an equilibrium exists, observe that T is continuous. Moreover, T (0) = −(1 −φ) < 0

since by assumption limθ→0 f (θ)/θ = ∞. On the other hand, T (θ̃) = (1−β)/(βx)+φ(1+ θ̃) >0, where f (θ̃)/θ̃ = x. By the intermediate value theorem, there must exist a θ̄ ∈ (0, θ̃) suchthat T (θ̄) = 0. For some functional forms there may be multiple solutions to this equation,
but in particular cases it is easy to verify uniqueness. Indeed, I have not found an example
with multiple solutions to this equation.
An important observation is that productivity z does not appear in equation (2.23) andso the equilibrium ratio of recruiters to unemployed workers is independent of productivity.
The key to this result is that income and substitution effects cancel. In an economy with highproductivity, workers earn higher wages. The substitution effect makes leisure less attractive,

36

CHAPTER 2. BENCHMARK SEARCH MODEL

while the income effect makes it more attractive. With these preferences, or balanced growth
preferences more generally, the two effects balance and so there is no change in equilibrium
recruiting effort.
For a given value of the ratio of recruiters to unemployed workers θ̄, recover employment n̄ from equation (2.8) and then the fraction of employees who are recruiters ν̄ fromequation (2.22), proving both are independent of productivity. Also use equation (2.21) tofind consumption c̄ and equation (2.20) to get the wage w̄, both proportional to productivity.
Last, one can use equation (2.6) to compute the level of transfers T̄ .
Finally, one can verify that the household’s assets are equal to the value of the jobs
that it owns, ā = J¯n̄. Combining the household budget constraint, equation (2.4), and thegovernment budget constraint, equation (2.6), and using q̄ = β givesā =c̄ − w̄n̄.
1−βEliminate c̄ using equation (2.21), w̄ using equation (2.16), and ν̄ using equation (2.22):
zn̄
ā =
1−β



θ̄(1 − n̄) 1 − β(1 − x)−+n̄
βµ(θ̄)

.

Finally, note that (1 − n̄)/n̄ = x/f (θ̄) and µ(θ̄) = f (θ̄)/θ̄. This gives ā = zn̄/βµ(θ̄), and so
the result follows from equation (2.15). This is just a version of Walras’s law: when all othermarkets clear, the asset market must clear as well.

2.1.8

The Measured Labor Wedge

Consider an economist who believes that households have preferences given by equation (1.1),
with disutility of leisure γ̂ and Frisch labor supply elasticity ε̂, that the production function
is Cobb-Douglas with capital share α̂, and that the labor market clears without any search
frictions. Then from equation (1.12), he would measure the labor wedge asτ̂ = 1 −1+ε̂γ̂(c/y)n ε̂ .
1 − α̂

But if the model in this section were the true data generating process, consumption would
equal output and employment would be given by equation (2.8). Thus he would measureγ̂τ̂ = 1 −1 − α̂f (θ̄)f (θ̄) + x 1+ε̂
ε̂.

372.2. PRODUCTIVITY SHOCKS

In particular, if the economist correctly understood that production used only labor (α̂ = 0)
and that the Frisch labor supply elasticity was infinite (ε̂ = ∞), he would measureτ̂ = 1 − γ̂f (θ̄).
f (θ̄) + x(2.24)

It is not surprising that the labor wedge is constant in steady state. The important point isthat it bears little resemblance to the true tax rate τ . The economist would attribute lessthan-full employment to taxes or the disutility of work, when in fact it is also a consequenceof the search frictions.
A particularly acute example occurs if γ = 0, so there is no disutility of work. In thiscase, equation (2.23) implies that the ratio of recruiters to unemployed is independent of the
tax rate τ , because the pre-tax wage and hence the incentive to create jobs is independentof the tax rate (equation 2.20). It follows that any difference in the labor income tax rate
in different countries does not cause differences in the employment rate. But an economist
viewing the world through the lens of the frictionless model would conclude that there must
be some disutility of labor, γ̂ > 0; otherwise he would expect everyone to work. Using
equation (2.24), he would then have difficulty understanding why there is no difference in
employment rates across countries despite large differences in taxes.2

2.2

Productivity Shocks

I now extend the model to allow for shocks to the production technology. As before, denote
time by t = 0, 1, 2, . . . and the state of the economy at time t by st . Let st = {s0 , s1 , . . . , st }denote the history of the economy and Π(st ) denote the time-0 belief about the probability ofobserving an arbitrary history st through time t. I allow productivity and lump-sum transfers
to be history-dependent. Each producer in history st produces output z(st ) and each worker,
employed or unemployed, receives a lump-sum transfer T (st ), set at a level to ensure thatthe government runs a balanced budget in each period. Assume z(st ) is strictly positive forall st and, to ensure utility is finite, assume that
−∞ <
2

∞ X
X
t=0

st

β t Π(st ) log z(st ) < ∞.

(2.25)

Of course, this problem is not present in the data. There are substantial differences in tax rates acrosscountries, but these can only partially explain the differences in employment rates, even if the elasticity oflabor supply is large; see, for example, Prescott (2004).

38

CHAPTER 2. BENCHMARK SEARCH MODEL

In this section, I prove that consumption c(st ) and the wage w(st ) are each proportional
to contemporaneous productivity z(st ), while the recruiter-unemployment ratio θ(st ), theemployment rate n(st ), and the fraction of employed workers in recruiting ν(st ) are constantand independent of the stochastic process for productivity. The notion of equilibrium is thesame as in the model without productivity shocks and so I proceed somewhat less formally,presenting the model and the key equilibrium relationships together.

2.2.1

Firms

A representative firm employs n0 = n(s0 ) workers at time 0 and subsequently assigns employees to the stochastic production technology and the deterministic recruiting technology.
Its value solves
0J(s , n0 ) =

∞ X
X
t=0

st


q0 (s ) z(s )(n(s , n0 ) − v(s , n0 )) − w(s )n(s , n0 ) ,
t



t

t

t

t

t

(2.26)

where q0 (st ) is the price of an Arrow-Debreu security that pays one unit of consumption
following history st , denominated in units of history-s0 consumption. Output in history st isz(st ) times the number of employees not devoted to recruiting, n − v, while the wage is w(st )per employee. In addition, the firm’s size evolves according ton(st+1 , n0 ) = v(st , n0 )µ(θ(st )) + (1 − x)n(st , n0 )(2.27)

for all st+1 = {st , st+1 }. Each recruiter attracts µ(θ(st )) new workers, while a fraction x ofthe existing labor force leaves. The firm chooses history-contingent recruiting v(st , n0 ) to
maximize the value of profits J(s0 , n0 ).
A simple argument proves that the firm’s value is linear in n0 . Take any two initial sizes,
n0 > n′0 > 0. Suppose a firm with initial size n′0 sets recruiting to (n′0 /n0 )v(st , n0 ) in eachhistory st . Then using induction on equation (2.27), one can verify that the firm’s size is(n′0 /n0 )n(st , n0 ) in each history. From equation (2.26), the firm’s profits are (n′0 /n0 )J(s0 , n0 ).
Since behaving optimally generally yields more profits, J(s0 , n′0 ) ≥ (n′0 /n0 )J(s0 , n0 ). A
symmetric argument, that a firm with n0 workers can mimic a scaled version of the optimal
policy for a firm with n′0 workers, establishes that J(s0 , n0 ) ≥ (n0 /n′0 )J(s0 , n′0 ). Combining¯ 0 ), so the value of the firm perthese inequalities implies J(s0 , n0 )/n0 = J(s0 , n′ )/n′ = J(s00employee is independent of firm size. The same logic implies that ν(st ) = v(st , n0 )/n(st , n0 ),the fraction of workers devoted to recruiting, is independent of n0 .
This allows me to simplify the analysis by focusing on the firm’s value per employee.
¯ t ) denote the present value of the firm’s profits per employee following history st ,Let J(s39

2.2. PRODUCTIVITY SHOCKS
evaluated in time t, history st units of consumption. This solves the recursive equationJ¯(st ) = maxν∈[0,1] X
t+1 ¯ t+1
z(s )(1 − ν) − w(s ) + νµ(θ(s )) + 1 − x
qt (s )J(s ) ,ttt(2.28)

st+1 |stwhere qt (st+1 ) is the price of a unit of consumption in history st+1 ≡ {st , st+1 }, paid in unitsof history-st consumption. The firm chooses the fraction of its workers who are recruiters
ν. It earns revenue z(st )(1 − ν) per worker this period, pays a wage w(st ), and then, fromequation (2.27), grows by a factor νµ(θ(st )) + 1 − x. The summation gives the expecteddiscounted value of continuation profits from each employee.
Since both the production and recruiting technologies are linear, an individual firm’s
choice of whether to produce or recruit is typically indeterminate; however, aggregate production and recruiting is determined in equilibrium. To see this, note from equation (2.28)
that an interior solution, 0 < ν(st ) < 1, requiresz(st ) = µ(θ(st ))X¯ t+1 ),
qt (st+1 )J(s(2.29)

st+1 |stso the foregone profits from devoting a worker to recruiting rather than production areexactly offset by the increase in the continuation value of the firm. If this condition holds,
the Bellman equation (2.28) simplifies to read


1
−
x
¯ ) = z(s ) 1 +J(s− w(st ).
µ(θ(st ))tt(2.30)Part of the revenue from a job comes from the output that the worker can produce, z(st ). In
addition, suppose the firm has a target size next period. An additional employee this period
remains at the firm next period with probability 1 − x. Since each recruiter can attractµ(θ(st )) new workers, the presence of an additional employee frees (1 − x)/µ(θ(st )) other
employees from recruiting, enabling them to produce z(st ) units of output each. The valueof a job is the sum of these two components, net of the wage w(st ) that the worker receives.
Finally, compute the value of a job that pays an arbitrary wage w in history st andthen pays w(sτ ) if it survives until a continuation history sτ , τ > t. Analogous withequation (2.18),¯ t ).
J˜n (st , w) = w(st ) − w + J(s(2.31)
This simply increments the value of a job by the reduction in the wage below the equilibrium,w(st) − w.

40

2.2.2

CHAPTER 2. BENCHMARK SEARCH MODEL

Households

The representative household allocates total consumption following history st , c(st ), in orderto maximize the sum of household utility, acting as if it has a utility function
∞ X
X
t=0

st


β t Π(st ) log c(st ) − γn(st ) ,(2.32)

where n(st ) is the fraction of household members who are employed following history st . The
household faces a single lifetime budget constraint,
a0 =

∞ X
X
t=0

st


q0 (st ) c(st ) − (1 − τ )w(st )n(st ) − T (st ) .

(2.33)

Here a0 is the household’s initial level of assets in units of consumption at time 0.
The household also faces a constraint on unemployment. The probability an unemployedworker finds a job depends on the contemporaneous ratio of recruiters to unemployed workers,f (θ(st )), while the probability an employed worker loses her job is a constant x. Thus thehousehold takes n0 as given and recognizes thatn(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st ))(2.34)

for any st+1 = {st , st+1 }. Note that the household cannot control its employment rate, which
depends on parameters and on the endogenous ratio of recruiters to unemployed workers.
The household chooses a path for consumption c(st ) to maximize equation (2.32) subject tothe budget constraint in equation (2.33), taking as given the path of its employment rate inequation (2.34).
Define the representative household’s assets following any history st as the present valueof future consumption in excess of labor income,ta(s ) =

∞ X
X
t′ =t

t′

s |st

′
′
′
′
′ 
qt (st ) c(st ) − (1 − τ )w(st )n(st ) − T (st ) ,(2.35)′

denominated in units of consumption in history st . Here qt (st ) is the price of one unit of′consumption in continuation history st , paid in units of consumption in history st . Thisimplies a standard sequence of intertemporal budget constraints:a(st ) + (1 − τ )w(st )n(st ) + T (st ) = c(st ) +X

st+1 |stqt (st+1 )a(st+1 ),(2.36)

41

2.2. PRODUCTIVITY SHOCKS

Following any history, initial assets plus current period after-tax labor income and transfers
must equal consumption plus the purchase of one period Arrow securities that provide next
period’s assets.
As in the case of firms, I express the household’s problem recursively. Let V (st , a, n)denote the household’s utility in history st , given assets a and employment n. This solves
the Bellman equationt

V (s , a, n) = max
t+1
{a(s

)}



log c − γn + βX Π(st+1 )st+1 |stΠ(st )

V s

t+1

, a(s

t+1


), n ,′

(2.37)

where a(st+1 ) is the household’s purchase of Arrow securities that pay off in history st+1 ={st , st+1 }; consumption c solves the intertemporal budget constraint, equation (2.36):
c = a + (1 − τ )w(st )n + T (st ) −X

qt (st+1 )a(st+1 );st+1 |stand next period’s employment solves equation (2.34):
n′ = (1 − x)n + f (θ(st ))(1 − n).
The envelope condition for current assets is
Va (st , a, n) =1c̃(st , a, n),

where c̃(st , a, n) is the consumption of a household that starts history st with assets a andemployment n. The first order condition for next period’s assets is
qt (st+1 )Π(st+1 )=βVa (st+1 , a(st+1 ), n′ ).
c̃(st , a, n)Π(st )
Combining these and looking at a household with initial assets a(st ) and employment n(st )gives the Euler equation that prices an Arrow security:
qt (s

t+1

Π(st+1 )c(st ))=βΠ(st )c(st+1 )(2.38)

for any st+1 = {st , st+1 }, where c(st ) ≡ c̃(st , a(st ), n(st )) is the household’s equilibrium levelof consumption. This is, of course, a completely standard asset pricing equation in any modelwith complete markets.
The envelope condition for current employment for a household with assets a(st ) and42

CHAPTER 2. BENCHMARK SEARCH MODEL

employment n(st ) in history st isVn (st , a(st ), n(st )) =(1 − τ )w(st )−γc(st )+ β(1 − x − f (θ(st )))X Π(st+1 )st+1 |stΠ(st )Vn (st+1 , a(st+1 ), n(st+1 )). (2.39)

Finally, compute the marginal value to a household with the equilibrium level of employment
and assets of having one employed worker in a job that pays an arbitrary wage w in history
st and then pays w(sτ ) if it survives until a continuation history sτ , τ > t. Analogous withequation (2.12), this solvesṼn (st , w) =(1 − τ )(w − w(st ))+ Vn (st , a(st ), n(st )).
c(st )(2.40)

This increments the value of an employed worker by the increase in the after-tax wage,
measured in units of marginal utility.

2.2.3

Government

The government runs a balanced budget in each period, rebating tax revenue lump-sum tohouseholds:
T (st ) = τ w(st )n(st ).

2.2.4(2.41)Wages

Wages are determined at the start of each period by bargaining between households andfirms. If they agree on a wage w following history st , the household’s utility increases byṼn (st , w) and the value of the job is J˜n (st , w). The wage w(st ) maximizes the weighted
geometric average of the gains from trade,w(st ) = arg max Ṽn (st , w)φ J˜n (st , w)1−φ ,w(2.42)

where φ ∈ [0, 1] represents workers’ bargaining power.
¯ t ). Equation (2.40)
Equation (2.31) implies ∂ J˜n (st , w)/∂w = −1 and J˜n (st , w(st )) = J(simplies ∂ Ṽn (st , w)/∂w = (1 − τ )/c(st ) and Ṽn (st , w(st )) = Vn (st , a(st ), n(st )). Maximizing

equation (2.42) is therefore equivalent to setting(1 − φ)Vn st , a(st ), n(st ) c(st ) = φ(1 − τ )J¯(st ).

43

2.2. PRODUCTIVITY SHOCKS

Use this to eliminate Vn (st , a(st ), n(st )) and Vn (st+1 , a(st+1 ), n(st+1 )) from equation (2.39):¯ t ) = (1 − φ) (1 − τ )w(st ) − γc(st )φ(1 − τ )J(s

+ φ(1 − τ )β(1 − x − f (θ(st )))sX Π(st+1 )c(st )Π(st )c(st+1 )t+1 t¯ t+1 ).
J(s|s¯ t ) using equation (2.30) and J(s¯ t+1 ) using equation (2.29), with qt (st+1 )Next eliminate J(sgiven from equation (2.38). After simplifying I obtain the wage equation(1 − τ )w(st ) = φ(1 − τ )z(st )(1 + θ(st )) + (1 − φ)γc(st ).

(2.43)

This is only a slight generalization of equation (2.20), with the wage depending on contemporaneous values of productivity, the ratio of recruiters to unemployed workers, andconsumption, rather than steady state values. The interpretation of the wage is similarly
unchanged.

2.2.5

Market Clearing

Close the model using two market clearing conditions. First, consumption is equal to output:c(st ) = z(st )n(st )(1 − ν(st )).

(2.44)For the representative household, a fraction n(st ) workers are employed, and of those, afraction 1 − ν(st ) each produces z(st ) units of the consumption good. Second, the ratio of
recruiters to unemployed workers isθ(st ) =2.2.6

ν(st )n(st ).
1 − n(st )(2.45)

Equilibrium

Let θ̄ solve equation (2.23) and suppose the initial employment level solves equation (2.8),i.e.
n0 = n̄ =f (θ̄).
f (θ̄) + x

I claim that there is an equilibrium with the desired properties: consumption, wages, andtransfers are proportional to contemporaneous productivity, c(st ) = c̄z(st ), w(st ) = w̄z(st ),and T (st ) = T̄ z(st ); while the recruiter-unemployment ratio, the share of recruiters in em-

44

CHAPTER 2. BENCHMARK SEARCH MODEL

ployment, and the employment rate are constant, θ(st ) = θ̄, ν(st ) = ν̄, and n(st ) = n̄.3
To prove this, note that if θ(st ) = θ̄ for all st , the initial condition n0 = n̄ and thelaw of motion for employment, equation (2.34), ensure that n(st ) = n̄ for all st . Thenequation (2.45) implies
ν(st ) = x/µ(θ̄) ≡ ν̄for all st ; equation (2.44) implies c(st ) = c̄z(st ), wherec̄ = n̄(1 − ν̄) =f (θ̄) − xθ̄;f (θ̄) + x(2.46)and equation (2.43) implies w(st ) = w̄z(st ), wherew̄ = φ(1 + θ̄) + (1 − φ)γc̄/(1 − τ ).

(2.47)

To confirm that this is an equilibrium, substitute these results into the condition for an¯ t+1 ) given by equation (2.30):
interior equilibrium, equation (2.29), with J(sz(st ) = 1 − x + µ(θ̄)(1 − w̄) X

qt (st+1 )z(st+1 ).

st+1 |stt+1t)z(s ). Substitute this into theEquation (2.38) and c(st ) = c̄z(st ) imply qt (st+1 ) = β Π(sΠ(st )z(st+1 )tprevious equation and eliminate z(s ) to get
1 = β 1 − x + µ(θ̄)(1 − w̄) .

(2.48)

Eliminating w̄ and then c̄ using equations (2.47) and (2.46) delivers equation (2.23). Finally,
condition (2.25) ensures that the household’s utility is finite.
How can productivity shocks have no effect on employment? Consider a temporary increase in productivity. Since this does not alter the efficiency of hiring workers, all else equal,
firms would shift workers from recruiting to production, raising consumption and reducing
hiring, i.e. future employment and output. But there is an offsetting general equilibrium
effect. Since productivity is temporarily high, consumption is temporarily high. For thisto be an equilibrium, the interest rate must be low, i.e. the intertemporal price q mustbe high, which makes future output more valuable, encouraging hiring. With the prefer3

If n0 6= n̄, the model has some transitional dynamics. I show how to deal with this in Chapter 3, inmodels where productivity shocks necessarily affect the unemployment rate. In practice, I find that theeconomy converges rapidly to steady state. Also, there may be other equilibria of the model. For example, ina deterministic discrete time version of Pissarides (1985), a deterministic cycle can arise. In that environment,
one can prove uniqueness if the period length is sufficiently short.

2.2. PRODUCTIVITY SHOCKS

45

ences in equation (2.32), these effects exactly offset, regardless of the stochastic process
for productivity. Indeed, the stochastic process for productivity does not itself affect therecruiter-unemployment ratio or the unemployment rate.
This finding builds on Blanchard and Galı́’s (2006) neutrality result in a model with laboradjustment costs. It also recalls some results from frictionless business cycle models. Take astandard framework, as in Chapter 1, but assume there are two production technologies, one
for producing the consumption good and one for producing the investment good. If there isno government spending and if households have preferences as in equation (1.1), productivity
shocks in the consumption-good-producing sector do not affect employment or the allocationof labor across the two sectors. Again, this is because income and substitution effects exactly
offset. On the other hand, shocks to the investment good sector, which here would beequivalent to shocks to the matching function µ, cause movements in both employment andinvestment.
The assumption that the recruiting technology uses labor rather than goods is critical forthis neutrality result. If the recruiting technology used goods, a positive productivity shock
would effectively reduce the cost of recruiting, inducing an increase in hiring during booms.
Many authors assume recruiting is in fact goods-intensive; see Merz (1995) and Andolfatto(1996) for examples. On the other hand, the textbook matching model (Pissarides, 1985,
2000) assumes that, while recruiting costs are denominated in goods, a productivity shockraises the recruiting shock proportionately. This is technically equivalent to assuming that therecruiting technology uses labor and is unaffected by the productivity shock. This assumption
seems empirically more plausible. Recruiting is, in fact, a time-intensive activity. Moreover,it seems a priori implausible that shocks to the cost of recruiting drive cyclical fluctuations
in employment.
The neutrality result is stronger than the finding in Shimer (2005), where I argued that
productivity shocks have a quantitatively small effect on employment in a calibrated version of
the Pissarides (1985) model. There are several important differences between the frameworks.
First, in the earlier paper, I assumed households have linear preferences over consumption
and leisure and so there are no general equilibrium effects on interest rates. When future
productivity is anticipated to be low relative to current productivity, firms reduce hiring.
Second, in the earlier paper I assumed that recruiting uses goods rather than labor. Moving
away from linear preferences and recognizing that recruiting is a labor-intensive activity bothseem like desirable modifications to the model.4
4
It is worth stressing that the finding in Shimer (2005) is quantitative while the neutrality result here
is exact. Some authors have argued with the calibration in that earlier paper; see especially Hagedorn andManovskii (2008), who argue for a different calibration of the value of leisure and the bargaining parameter
φ. The neutrality result holds here regardless of bargaining powers and regardless of how much workers like

46

2.2.7

CHAPTER 2. BENCHMARK SEARCH MODEL

The Measured Labor Wedge

Since productivity shocks do not affect employment and the consumption-output ratio isfixed at 1, such shocks do not affect the measured labor wedge. Again consider an economist
who believes households have preferences given by equation (1.1), with disutility of leisureγ̂ and Frisch labor supply elasticity ε̂, that the production function is Cobb-Douglas withcapital share α̂, and that the labor market clears without any search frictions. If this model
were the data generating process, he would measureγ̂τ̂ = 1 −1 − α̂f (θ̄)f (θ̄) + x 1+ε̂
ε̂.

(2.49)The measured labor wedge would be constant. Thus this model fails both to generate fluctuations in employment and fluctuations in the labor wedge.

2.3

Planner’s Problem

Consider a hypothetical social planner who wants to maximize the utility of the representative
household. He takes as given an initial level of employment n0 = n(s0 ) and decides how many
workers to allocate to production and how many to recruiting in each state. The measure of
recruiters determines future employment, while the measure of producers determines current
consumption.
Formally, it is easiest to assume that the planner chooses the history-contingent share of
recruiters in employment {ν(st )} and the history-contingent recruiter-unemployment ratio{θ(st )} to maximize the household’s utility
∞ X
X
t=0

st



β t Π(st ) log z(st )n(st )(1 − ν(st )) − γn(st ) ,where n(st ) is employment in history st and z(st )n(st )(1 − ν(st )) is consumption in that
history, subject to the usual constraint on the evolution of employment,n(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st )),where st+1 = {st , st+1 } and θ(st ) = ν(st )n(st )/(1 − n(st )) is the recruiter-unemployment
ratio.

leisure, i.e. regardless of the value of γ.

472.3. PLANNER’S PROBLEM

Taking advantage of the properties of the log function, express the objective function as
∞ X
X
t=0

st

t

t



0

Z(s ) ≡

t

∞ X
X
t=0



t



β Π(s ) log n(s )(1 − ν(s )) − γn(s ) + Z(s0 ),

where

t

β t Π(st ) log z(st )stis a finite constant. Note that the objective function is additively separable between the exogenous productivity process, summarized through Z(s0 ), and the planner’s choice variables.
Moreover, productivity does not appear in the constraint on the evolution of employment.
It is immediate that the solution to the planner’s problem is independent of Z(s0 ), i.e. ν(st ),θ(st ), and n(st ) do not depend on the path of productivity shocks. This parallels the finding
in the decentralized economy. The household’s utility in any history st is then equal to thesum of two terms, W (n), which depends on its current employment rate but not st , and′Z(st ), the expected present value of the sum of future log z(st ).
I proceed by expressing W (n) recursively. This solves
W (n) = max log(n − θ(1 − n)) − γn + βW (1 − x)n + f (θ)(1 − n) ,θwhere n − θ(1 − n) = n(1 − ν) since ν = θ(1 − n)/n. One can think of W (n) as the maximized
utility of a household if z(st ) = 1 for all st . The first term is utility from consumption, thesecond is disutility from work, and the third is the continuation value, which depends onnext period’s employment.
The first order and envelope conditions are
1= βf ′(θ)W ′ (1 − x)n + f (θ)(1 − n) andn − θ(1 − n)
1+θW ′(n) =− γ + β(1 − x − f (θ))W ′ (1 − x)n + f (θ)(1 − n) .
n − θ(1 − n)In steady state, n = (1 − x)n + f (θ)(1 − n) and so the previous equations reduce to T ∗ (θ̄) = 0
where
1 − β(1 − x − f (θ))
f (θ) − θxT ∗ (θ) =− (1 + θ) + γ.
′
βf (θ)f (θ) + xT ∗ is continuous in θ with T ∗ (0) = −1 and T ∗ (θ̃) = (1 − β)/(βf ′ (θ̃)) + f (θ̃)/θ̃f ′ (θ̃) − 1 (1 +θ̃) > 0, where f (θ̃)/θ̃ = x.5 By the intermediate value theorem, there must exist a θ̄ ∈ (0, θ̃)such that T ∗ (θ̄) = 0.
5

Concavity of f ensures that f (θ) ≥ θf ′ (θ) for all θ > 0.

48

CHAPTER 2. BENCHMARK SEARCH MODEL
In general, the social planner’s recruiter-unemployment ratio is not an equilibrium allo-

cation, but it is under one set of conditions. Assume τ = 0, so there are no distortionary
taxes, and φ = 1 − θf ′ (θ)/f (θ), so workers’ bargaining power is equal to the elasticity of thenumber of meetings with respect to the unemployment rate, the Mortensen (1982)–Hosios(1990) condition. Then one can verify that T (θ) = (θf ′ (θ)/f (θ))T ∗ (θ) in equation (2.23), so
the any solution to T ∗ (θ̄) also solves T (θ̄) = 0. If the Mortensen-Hosios condition is violated(and τ = 0), the equilibrium level of unemployment will be inefficient.
The relationship between the equilibrium and the solution to the social planner’s problem
is not purely a normative issue. Moen (1997) and Shimer (1996) consider an environment
where firms can post wage contracts in an effort to attract workers. Workers observe all
the contracts and decide where to apply, trading off the probability of getting a contractf (θ) against the terms of the contract. Thus when one firm offers a relatively high wage, its
recruiters are more productive. Those papers find that the allocation in a competitive searchequilibrium without taxes solves a social planner’s problem, maximizing the utility of therepresentative household. Establishing this result in the current environment would requirea significant investment in new notation and so I omit the proof. But the point is that there
are reasons to believe that a decentralized economy may achieve the planner’s allocation.

2.4

Extensions

2.4.1

Labor Force Participation

In the benchmark model, a household has no control over its employment rate, which is determined by the balance between the exogenous separation rate x and the endogenous—but
exogenous to the household—job finding rate f (θ(st )), in equation (2.34). This section generalizes the model by giving households control over their future employment rate through theallocation of nonemployed workers between two tasks: unemployment, which allows workers
to find jobs; and inactivity, which provides more leisure. This extension is interesting for two
reasons. First, it shows that the neutrality result does not rely on households being passive
actors in the labor market. Second, Figure 1.6 shows that the employment-population ratio
is somewhat more volatile than the unemployment-population ratio, so some fluctuations in
employment induce workers to drop out of the labor force, rather than become unemployed.
Thinking about labor force participation potentially allows the model to address this fact.
The representative household allocates total consumption following history st , c(st ), in49

2.4. EXTENSIONS
order to maximize the sum of its individuals’ utility, acting as if it has a utility function
∞ X
X
t=0

st


β t Π(st ) log c(st ) − γn n(st ) − γu u(st ) ,(2.50)

where n(st ) is the fraction of household members who are employed following history st
and u(st ) is the fraction of household members who are unemployed. The remaining 1 −n(st ) − u(st ) household members are inactive. Implicit in this formulation, the utility ofan employed household member who consumes c is log c − γn ; the utility of an unemployedhousehold member who consumes c is log c − γu ; and the utility of an inactive household
member who consumes c is log c. I assume that γu > 0, so workers suffer some disutility fromunemployment compared to the omitted third category, inactivity. In addition, it may benatural to assume that γn ≥ γu , so unemployed workers enjoy more leisure than do employedworkers; however, that restriction is not necessary for what follows.
The household faces a single lifetime budget constraint, identical to equation (2.33) andrepeated here for convenience.
a0 =

∞ X
X
t=0

st


q0 (st ) c(st ) − (1 − τ )w(st )n(st ) − T (st ) .

(2.51)The household can freely move workers between unemployment and inactivity. However,workers can only find jobs by going through unemployment, which constrains employment.
The probability that an unemployed worker finds a job depends only on the contemporaneous
ratio of recruiters to unemployed workers, θ(st ), while the probability an employed worker
loses her job is a constant x. Inactive workers cannot find jobs. Thus the household takes n0
as given and recognizes thatn(st+1 ) = (1 − x)n(st ) + f (θ(st ))u(st )(2.52)

for any st+1 = {st , st+1 }. The household can raise its future employment rate by increasing its current unemployment rate, at the cost of higher current disutility. The householdchooses a path for consumption c(st ) and unemployment u(st ) to maximize equation (2.50)
subject to the budget constraint in equation (2.51) and the law of motion for employment inequation (2.52).
To express this problem recursively, let V (st , a, n) denote the value of a household in

50

CHAPTER 2. BENCHMARK SEARCH MODEL

history st with assets a and n employed members. This solvest

V (s , a, n) =max{a(st+1 )},u∈[0,1−n]

log c − γn n − γu u+βX Π(st+1 )Π(st )

st+1 |st

V s

t+1

, a(s

t+1


), n , (2.53)′

with c solving the intertemporal budget constraint,c = a + (1 − τ )w(st )n + T (st ) −X

qt (st+1 )a(st+1 ),st+1 |stand next period’s employment determined from current employment and unemployment as
n′ = (1 − x)n + f (θ(st ))u.
As before, the first order condition for a(st+1 ),qt (st+1 )Π(st+1 )
t+1
t+1
t+1
=
β
V
s
,
a(s
),
n(s
)
,
a
c(st )
Π(st )and the envelope condition for a(st ),Va (st , a(st ), n(st )) =1,c(st )

yield the standard Euler equation for the price of an Arrow security, equation (2.38) andrepeated here,Π(st+1 )c(st )qt (st+1 ) = β.
(2.54)
Π(st )c(st+1 )Assuming an interior solution for next period’s employment, so some workers are unemployed
and some are inactive, the first order condition for u(st ),γu = βf (θ(st ))X Π(st+1 )st+1 |stΠ(st )

Vn st+1 , a(st+1 ), n(st+1 ) ,and the envelope condition for n(st ),Vn (st , a(st ), n(st )) =X Π(st+1 )(1 − τ )w(st )
t+1
t+1
t+1
−
γ
+
β(1
−
x)
V
s
,
a(s
),
n(s
)
,
n
n
c(st )
Π(st )
t+1 t
s

|s

51

2.4. EXTENSIONS
reduce totγu = βf (θ(s ))X Π(st+1 )  (1 − τ )w(st+1 )st+1 |stΠ(st )c(st+1 )γu (1 − x)
− γn +f (θ(st+1 )).

(2.55)The left hand side is the current cost of switching a marginal worker from inactivity tounemployment, foregone leisure. The right hand side is the benefit: next period, if theworker is hired, with probability f (θ(st )), the household will earn more labor income, valued
at the marginal utility of wealth 1/c(st+1 ). From this, subtract the disutility of working
γn . Moreover, the household can offset the increase in employment at t + 1 by reducing
unemployment and increasing inactivity by enough to return the employment rate to its
normal value in period t + 2. This generates leisure from additional inactivity in period t + 1.
The rest of the household’s problem is unchanged; in particular, the marginal value tothe household of an employed worker in a job that pays an arbitrary wage w in history st andthen pays w(sτ ) if it survives until a continuation history sτ is still given by equation (2.40),
rewritten here asṼn (st , w) =(1 − τ )(w − w(st ))+ Vn (st , a(st ), n(st )).
c(st )(2.56)

The firm’s problem is unchanged and so the wage solves(1 − τ )w(st ) = φ(1 − τ )z(st )(1 + θ(st )) + (1 − φ)γn c(st ),(2.57)

identical to equation (2.43) except in the notation for the marginal rate of substitution
between consumption and leisure. For firms to be indifferent between current production
and recruiting, it must be the case thatz(st ) = µ(θ(st ))X¯ t+1 ).
qt (st+1 )J(s(2.58)

st+1 |stThis ensures that the value of employing a worker is equal to the amount she produces plus
the number of recruiters that she frees up times the amount they produce minus the wage:


1−x
t
t
¯
− w(st ).
J(s ) = z(s ) 1 +µ(θ(st ))(2.59)

Finally, market clearing imposes that consumption is equal to output,c(st ) = z(st )n(st )(1 − ν(st )),(2.60)

52

CHAPTER 2. BENCHMARK SEARCH MODEL

and the ratio of recruiters to unemployed workers isθ(st ) =ν(st )n(st ).
u(st )(2.61)As in the benchmark model, I claim that there is an equilibrium with a constant ratioof recruiters to unemployed workers, θ(st ) = θ̄, a constant share of workers in recruiting,ν(st ) = ν̄, constant employment and unemployment, n(st ) = n̄ and u(st ) = ū, with the wage
and consumption proportional to productivity, w(st ) = w̄z(st ) and c(st ) = c̄z(st ). To provethis, observe that in such an equilibrium equation (2.55) reduces toγu =βf (θ̄) ((1 − τ )w̄ − γn c̄),(1 − β(1 − x))c̄(2.62)

while the wage equation (2.57) becomes(1 − τ )w̄ = φ(1 − τ )(1 + θ̄) + (1 − φ)γn c̄.
Eliminating c̄ between these equations gives(1 − β(1 − x))γu + βf (θ̄)γn φ(1 + θ̄)w̄ =.
(1 − β(1 − x))γu + βφf (θ̄)γn(2.63)

This defines the ratio of the wage to productivity as a function of model parameters and theratio of recruiters to unemployed workers. To close the model, proceed as in the case withouta labor force participation margin: equations (2.54), (2.58), and (2.59) imply
1 = β 1 − x + µ(θ̄)(1 − w̄) ,(2.64)

identical to equation (2.48), a second relationship between the wage and the ratio of recruiters
to unemployed workers. Eliminate w̄ between these two equations to get an expression for θ̄
alone, T (θ̄) = 0 where(1 − β(1 − x))γu + βf (θ)γn φ(1 + θ)
θ(1 − β(1 − x))
T (θ) ≡−1+,
βf (θ)(1 − β(1 − x))γu + βφf (θ)γn(2.65)a modest variant of equation (2.23). Observe that since limθ→0 f (θ)/θ = ∞, T (0) = −1+φ <0. At the other end, limθ→∞ f (θ)/θ = 0 implies limθ→∞ T (θ) = ∞. Since T is continuous on(0, ∞), there is a solution to T (θ) = 0 on this interval. Moreover, each of the terms in T is
increasing in θ, which ensures that the equilibrium is unique.
Once I have found θ̄, it is straightforward to recover the remaining variables of interest.

53

2.4. EXTENSIONS

Obtain w̄ from equation (2.63) and c̄ from equation (2.62). In the proposed equilibrium,
equation (2.52) reduces to xn̄ = f (θ̄)ū, while equation (2.61) becomes θ̄ = ν̄ n̄/ū. Eliminatingn̄/ū from the latter equation using the former gives down ν̄ = xθ̄/f (θ̄); this ensures that firms’
hiring rate ν̄µ(θ̄) just offsets their attrition rate x. Finally, pin down the employment levelfrom equation (2.60), n̄ = c̄/(1 − ν̄), and the unemployment level ū from ū = ν̄ n̄/θ̄.
A curious property of the equilibrium is that θ̄, w̄, and ν̄ are independent of the tax
rate τ , while c̄, n̄, and ū are proportional to 1 − τ . Raising taxes to fund lump-sum benefits
discourages participation in the labor force but does not distort activities for workers who arein the labor force. More relevant to the theme of this book, this model has both equilibrium
unemployment and inactivity, but neither is affected by productivity shocks.

2.4.2

Variable Hours

I now extend the model to consider an intensive margin for hours. More precisely, I assume
that employed workers and firms bargain both over an hourly wage and over the numberof hours worked. This ensures that the choice of hours is bilaterally efficient, so workers
and firms exploit all the gains from trade. Again, the objective is two-fold: to verify therobustness of the neutrality result; and to allow the model to address the data along additional dimensions. Figure 1.5 shows that the correlation between detrended employment anddetrended hours is 0.97, but the standard deviation of detrended hours is 1.3 times as large
as the standard deviation of detrended employment. In other words, during periods whenemployment is high relative to trend, hours per employee is typically also above trend. For
simplicity I abstract from the decision to participate in the labor market, although one could
allow for both margins.
The representative household chooses a sequence for {c(st )} to maximize
∞ X
X
t=0

s



γε
t
t
t 1+ε
β Π(s ) log c(s ) −n(s )h(s ) ε ,
1
+
ε
t
t

t

(2.66)

where n(st ) is the fraction of household members who are employed and h(st ) is the hours
worked by each employed household members following history st . I continue to think ofthis as representing the sum of individual utilities, where the utility of an individual who
1+ε

γεconsumes c and works h hours (possibly h = 0) is log c − 1+εh ε . The household faces asingle lifetime budget constraint,

a0 =

∞ X
X
t=0

st


q0 (st ) c(st ) − (1 − τ )w(st )n(st )h(st ) − T (st ) ,(2.67)

54

CHAPTER 2. BENCHMARK SEARCH MODEL

where w(st ) is the hourly wage, while its employment rate evolves exogenously according ton(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st )).

(2.68)

Below I discuss how employed workers and firms bargain over wages and hours.
Let V (st , a, n) denote the value of a household with assets a and n employed workers inhistory st . This solvest

V (s , a, n) = max
t+1
{a(s

)}




X Π(st+1 )

γε
t 1+ε
t+1
t+1
′
log c −
nh(s ) ε + β
V s , a(s ), n , (2.69)
t)
1+ε
Π(s
t+1 t
s

|s

where c satisfies the intertemporal budget constraint,c = a + (1 − τ )w(st )nh(st ) + T (st ) −X

qt (st+1 )a(st+1 ),st+1 |stand next period’s employment is determined from current employment and unemployment
asn′ = (1 − x)n + f (θ(st ))(1 − n).
As usual, the first order condition for next period’s assets and the envelope condition for
current assets yield the Euler equation,
qt (s

t+1

Π(st+1 )c(st ).
)=βΠ(st )c(st+1 )(2.70)The envelope condition for employment is
Vn (st , a(st ), n(st )) =1+ε
(1 − τ )w(st )h(st )γε−h(st ) ε
t
c(s )
1+ε
X Π(st+1 )+ β(1 − x − f (θ(st )))Vn (st+1 , a(st+1 ), n(st+1 )), (2.71)
t)
Π(s
t+1 t

s

|s

a generalization of equation (2.39). Also let Ṽn (st , w, h) denote the marginal value to ahousehold with the equilibrium level of assets and employment of having a worker employed
at a wage w and working h hours rather than unemployed in history st . If she remains
employed in continuation history st+1 , she earns the equilibrium wage w(st+1 ) and works the55

2.4. EXTENSIONS
equilibrium number of hours h(st+1 ). This satisfies(1 − τ ) wh − w(st )h(st )Ṽn (s , w, h) =c(st )tγε  1+εt 1+εεε− h(s )+ Vn (st , a(st ), n(st )). (2.72)−h
1+εThe first term gives the after-tax income generated by a worker employed at (w, h) rather
than at the equilibrium wage-hour pair (w(st ), h(st )), evaluated in utils by multiplying by
marginal utility 1/c(st ). The second term gives the utility cost of the extra hours of work.
The third term is the marginal value of having the worker employed at the equilibrium wage
and hours rather than unemployed.
The representative firm earns profit z(st ) from each hour of labor devoted to production
and recruits µ(θ(st )) workers with each hour of labor devoted to recruiting, where I now
interpret θ(st ) as the ratio of the number of recruiting hours to the number of unemployedworkers. The firm’s value in history s0 with n0 workers is
J(s0 , n0 ) =

∞ X
X
t=0

st



q0 (st ) z(st )(n(st ) − v(st )) − w(st )n(st ) h(st ),(2.73)

where v(st ) is the number of employees devoted to recruiting in history st . The firm employsn(st ) − v(st ) producers, each of whom works h(st ) hours and produces z(st ) per hour inhistory st . In addition, it pays a wage w(st) for each hour of work. Firm growth satisfiesn(st+1 ) = v(st )h(st )µ(θ(st )) + (1 − x)n(st ),(2.74)

where st+1 = {st , st+1 }. Each of the v(st ) recruiters works h(st ) hours and recruits µ(θ(st ))per hour of work. In addition, a fraction 1 − x of the old labor force remains for anotherperiod.
As usual, the firm’s value is linear in its employment rate and so its value per employee
solves the recursion¯ t ) = max
J(s
ν




z(st )(1 − ν) − w(st ) h(st ) X
t+1 ¯ t+1
+ νh(s )µ(θ(s )) + 1 − x
qt (s )J(s ) , (2.75)ttst+1 |stwhere ν is the fraction of the workforce devoted to recruiting. The condition for an interior

56

CHAPTER 2. BENCHMARK SEARCH MODEL

equilibrium for recruiting isz(st ) = µ(θ(st ))X¯ t+1 ),
qt (st+1 )J(s(2.76)

st+1 |stunchanged from equation (2.29). If this holds, the Bellman equation (2.75) reduces to


1−x
t
t
t
¯
J(s ) = z(s ) h(s ) +− w(st )h(st ).
µ(θ(st ))(2.77)

An additional employee can produce z(st )h(st ) units of output. In addition, with probability
1 − x she remains at the firm next period, so the firm can reduce the number of recruiters by(1 − x)/h(st )µ(θ(st )) while keeping its size unchanged. The workers released from recruiting
instead produce z(st )h(st ) units of output. Finally, the worker is paid an hourly wage w(st ).
Again let J˜n (st , w, h) denote the marginal value to the firm of having a worker employedat an arbitrary wage w and working an arbitrary number of hours h in history st , and thenthe equilibrium wage and hours thereafter. This satisfies¯ t ).
J˜n (st , w, h) = (z(st ) − w)h − (z(st ) − w(st ))h(st ) + J(s(2.78)

This is the marginal value of an employed worker at the equilibrium wage and hours plus the
additional profits from paying the worker w for h hours of work, rather than the equilibrium
values (w(st ), h(st )).
I assume that the household and firm bargain jointly over both wages and hours, so theequilibrium wage and hours satisfies the Nash bargaining solution(w(st), h(st )) = arg max Ṽn (st , w, h)φ J˜n (st , w, h)1−φ .
w,h(2.79)

This ensures that the household and firm agree on a Pareto optimal division of the gains
from trade.6 Under equation (2.79), the first order condition for wages is(1 − φ)Vn (st , a(st ), n(st ))c(st ) = φ(1 − τ )J¯(st ),(2.80)

unchanged from the model without an hours margin. The first order condition for the choice6
One can conceive of other assumptions here. For example, the household and firm may bargain over
the wage and then the firm unilaterally sets hours. Typically this will lead to a Pareto inefficient outcome.
This alternative does not affect the main result, neutrality of productivity shocks. I focus on this assumption
because it may not be plausible that workers and firms who repeatedly interact would get stuck at an
inefficient allocation.

57

2.4. EXTENSIONS
of hours is
1
φ γc(st )h(st ) ε − (1 − τ )w(st )(1 − φ)(z(st ) − w(st ))=.
¯ t)Vn (st , a(st ), n(st ))c(st )J(sSolve these two equations for the choice of hours:th(s ) =

(1 − τ )z(st )γc(st )

ε.

(2.81)

The choice of hours does not directly depend on workers’ bargaining power but rather is set
at a bilaterally efficient level to maximize the monetary gains from trade between a worker
and firm, Ṽn (st , w, h)c(st ) + (1 − τ )J˜n (st , w, h). Instead, the worker and firm use the wage todivide the gains from trade.

As in the model without hours, it is possible to solve equations (2.70), (2.71), (2.76),(2.77), and (2.80) for the equilibrium wage:1
θ(st )
γε(1 − τ )w(s ) = φ(1 − τ )z(s ) 1 ++ (1 − φ)c(st )h(st ) ε ,
t
h(s )
1+ε
t

t

a generalization of equation (2.43). The after-tax hourly wage is a weighted average of themarginal product of labor—the increase in output from having an additional worker, bothbecause of the output she produces and because of the recruiters who can be reallocated to
production—and the marginal rate of substitution between consumption and leisure. Eliminating h(st ) from the last term using equation (2.81) simplifies this further: θ(st )
εw(s ) = φ 1 ++ (1 − φ)z(st ).
h(st )
1+εt(2.82)To close the model, I use the two market clearing conditions: consumption is equal to
output,c(st ) = z(st )n(st )h(st )(1 − ν(st )),(2.83)and the ratio of hours of recruiters’ time to unemployed workers isθ(st ) =ν(st )n(st )h(st ).
1 − n(st )(2.84)

I look for an equilibrium with a constant ratio of recruiting hours to unemployed workers,θ(st ) = θ̄, a constant share of workers in recruiting, ν(st ) = ν̄, constant employment, n(st ) =n̄, constant hours, h(st ) = h̄, with the wage and consumption proportional to productivity,
w(st) = w̄z(st ) and c(st ) = c̄z(st ). To find the six constants, use equations (2.70), (2.76),

58

CHAPTER 2. BENCHMARK SEARCH MODEL

and (2.77) to prove that if such an equilibrium exists,
1 = β 1 − x + h̄µ(θ̄)(1 − w̄) .

The algebra is similar to that behind equation (2.48). Then eliminate w̄ using this expression
and equation (2.82). Solving for hours gives:
1+ε
h̄ =
1−φ




1 − β(1 − x)+ φθ̄ .
βµ(θ̄)(2.85)

Similarly, in an equilibrium with this characterization, eliminate c̄, n̄, and ν̄ between theemployment equation (2.68), the hours equation (2.81), the market clearing condition inequation (2.83), and the definition of θ in equation (2.84). This gives(1 − τ )(x + f (θ̄))
γ µ(θ̄)h̄ − x θ̄

h̄ =

!ε

.

(2.86)

Eliminating h̄ between equations (2.85) and (2.86), I find that an equilibrium is determined
as a θ̄ solving T (θ̄) = 0 where1+ε
T (θ) =1−φ


1 − β(1 − x)
+ φθ
βµ(θ)

−

ε

β(1 − φ)(1 − τ )(x + f (θ))
 .
γθ (1 + ε) 1 − β(1 − x) + βφf (θ) − β(1 − φ)x

The first term is increasing in θ, rising from 0 at θ = 0 (since µ(0) = ∞) to ∞ at θ = ∞.
The second term is decreasing in θ, since (x + f (θ))/θ is decreasing while the rest of thedenominator is nondecreasing in θ. It evaluates to ∞ at θ = 0 and then drops to 0 at θ = ∞.
Since both terms are continuous in θ, T is continuous, increasing, and maps the positive realline into the real line. The equation T (θ̄) = 0 therefore has a unique solution.

Using this solution it is straightforward to recover h̄, c̄, n̄, ν̄, and w̄ from the expressions
used to construct equations (2.85)–(2.86). Since productivity shocks affect neither employment nor hours worked, I again conclude that they do not affect the measured labor wedge.

2.4.3

Other Extensions

Many other extensions do not alter the neutrality result. I mention three here but omit theproofs, since they simply repeat the structure of the previous arguments.

59

2.4. EXTENSIONSFirst, I have assumed that unemployed workers get leisure but no monetary transfer. In
reality, however, many also receive unemployment benefits. Suppose that each unemployedworker gets a benefit that is proportional to the average wage, say b̄w(st ) in history st forsome b̄ ∈ (0, 1). Assuming benefits are taxed, the household budget constraint reads
a0 =

∞ X
X
t=0

st



q0 (st ) c(st ) − (1 − τ )w(st ) n(st ) + b̄(1 − n(st )) − T (st ) .

The government uses tax revenue to pay both benefits and transfers, with constraint
T (st ) + (1 − τ )b̄w(st )(1 − n(st )) = τ w(st )n(st ).
I find that unemployment benefits put upward pressure on wages and so reduce recruiting
and hence employment; however, the neutrality result stands.7
Second, I have assumed that the government rebates all of its tax revenue to households.
In reality, government spending also represents a significant part of the budget. Suppose therepresentative household has preferences over consumption, labor supply, and government
spending g(st ),∞ XXβ t Π(st ) log c(st ) − γn(st ) + ψ(g(st )) ,t=0

stand faces the usual budget constraint, equation (2.33). The government uses tax revenue tofund both spending and lump-sum rebates:
T (st ) + g(st) = τ w(st )n(st ).
The resource constraint, equation (2.44), must be modified to account for the additional use
of output:c(st ) + g(st ) = z(st )n(st )(1 − ν(st )).
If government spending is proportional to productivity, g(st ) = ḡz(st ) for all histories st , theneutrality result carries over to this environment. Consumption, spending, and wages are all
proportional to productivity, while employment is constant.
Is the assumed proportionality of government spending and productivity reasonable? It
is optimal if ψ(g) ∝ log g, in which case the government should keep public and private
spending proportional. Moreover, g(st) = ḡz(st ) is a plausible description of how governments
in fact behave. The assumption implies that government spending is also proportional to7

In reality, unemployment benefits are usually indexed to past, not current wages. This would create somenonneutrality in response to productivity shocks.

60

CHAPTER 2. BENCHMARK SEARCH MODEL

tax revenue, g(st ) = g̃τ w(st )n(st ), where ḡ ≡ g̃τ w̄n̄. While this ignores the possibility of
government spending shocks, it is consistent with procyclical spending induced by a balanced
budget requirement. Again, the neutral case seems like a reasonable benchmark.
Third, I have assumed production uses only labor. Suppose there are two types of firms

or, equivalently, two production technologies that are available to all firms.8 A capital-good
producer converts ℓk (st ) units of labor and kk (st ) units of capital into zk kk (st )αk ℓk (st )1−αkunits of capital, which it sells at a price p(st ), denominated in units of the consumption
good. A consumption-good producer converts ℓc (st ) units of labor and kc (st ) units of capital
into zc (st )kc (st )αc ℓc (st )1−αc units of the consumption good, with relative price 1. Both typesof firms also recruit new workers, with each recruiter attracting µ(θ(st )) workers in thefollowing period. Capital partially depreciates in production and capital used in history
st+1 = {st , st+1 } is purchased in history st .
In this environment, I find that shocks to the technology for producing the consumption good, zc (st ), affect neither the amount of capital used nor the number of producers
or recruiters in either sector. Instead, consumption, the wage and the price of the capital
good are all proportional to the productivity shock. Effectively capital and labor are now
both acquired through investment technologies—the capital production technology and therecruiting technology—that are not subject to shocks. Given the symmetry between the twotechnologies, it is not surprising that the neutrality result in the model with only a recruiting
technology carries over to this more general model.

2.5Discussion

Search and matching models help explain why there is unemployment. In the absence of
unemployment, recruiting would be unproductive and wages would be high. Both forces
would deter firms from putting resources into recruiting new workers, which would push up
the unemployment rate. The models that I have developed in this chapter show exactly how
the equilibrium unemployment rate is determined.
But it does not logically follow that search and matching models can explain why there
are fluctuations in unemployment. In the models in this chapter, the stochastic process for
productivity and the realization of productivity shocks have no impact on the unemployment
rate. This finding is useful for two reasons. First, it makes solving the models relatively
straightforward, and in particular ensures that a closed-form solution is available. This is8

In Chapter 3 I consider a one-sector model with capital, where a single technology is used to produce
both the consumption and the capital good, and find that this breaks the neutrality result. The key differenceis that in the one-sector model, a temporary increase in productivity makes investment more productive andso encourages capital accumulation. That effect is absent from the two-sector model.

2.5. DISCUSSION

61

convenient for understanding how the model works. Second, the neutrality result provides aconvenient benchmark. In the remainder of the book, I look at extensions to this model thatbreak the result, but it will always be useful to refer back to this benchmark to understandthe behavior of those more complicated models.

62

CHAPTER 2. BENCHMARK SEARCH MODEL

Chapter 3
Real Effects of Productivity Shocks
I now extend the benchmark search model to study environments in which productivity shocks
affect employment. The assumptions that drive the exact neutrality result in Chapter 2 are
special, in the sense that almost any conceivable modification would break the finding. I startin Section 3.1 with what is perhaps the easiest modification of the benchmark model, an alternative formulation of preferences. More precisely, I maintain the assumption that preferences
are consistent with balanced growth, but I relax the assumption that the intertemporal elasticity of substitution in consumption is equal to one, or equivalently that preferences are
additively separable in consumption and leisure. If households are less willing to substituteintertemporally, then interest rates need to rise less in response to a temporary increase inproductivity. This raises the profitability of recruiting new workers, so firms shift employees
from production to recruiting. This behavioral response both moderates the initial consumption response and induces a prolonged rise in employment. But in a calibrated version of themodel, I find that this change in preferences causes only tiny movements in employment andthe labor wedge.
In Section 3.2, I reintroduce physical capital into the model. This is not only a realistic
assumption, but it is qualitatively important because it gives households a savings technology.
Even with the preferences in Chapter 2, a temporary increase in productivity causes a smaller
reduction in interest rates when households can save some of their additional income. Thisagain raises the profitability of recruiting new workers, pulling down the unemployment rate.
But in a calibrated version of the model, the response of employment and the labor wedge
to productivity shocks remains very small. Moreover, I confirm the intuition that search
frictions create a positive correlation between employment and the labor wedge, in contrast
to the data in Table 1.1. Effectively search frictions moderate the response of employment
to the shock. An economist who looked at data generated by the model, but who did notrecognize the existence of the frictions, would incorrectly conclude that the labor wedge rises
63

64

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

in booms, dampening what would otherwise be a larger increase in employment.
In Section 3.3, I introduce a shock to the employment exit probability x, imperfectly
correlated with the productivity shock. I do this mainly for realism. Since the seminal
book by Davis, Haltiwanger, and Schuh (1996), a number of authors have stressed that thejob destruction rate rises sharply in downturns. Building on this, Mortensen and Pissarides(1994) extend the Pissarides (1985) search and matching model to allow for an endogenous
increase in the employment exit probability during downturns. Although I maintain thesimpler assumption that the employment exit probability is exogenous, I can still examine
whether fluctuations in x significantly affect the behavior of the model. My conclusion isnegative. Fluctuations in employment and the labor wedge are modest and the correlation
between the two outcomes is positive, not negative as in the data. Intuitively, an increase in
the employment exit probability has little effect on the desired employment rate, but instead
leads to an increase in recruiting effort as households and firms work to restore this level of
employment. This change in the balance between recruiting and production has only a small
impact on the outcomes that I study in this book.

3.1

General Preferences

I start by relaxing the assumption that the marginal utility of consumption is independentof work effort, allowing for more general balanced growth preferences. I find that, unlessthe stochastic process for productivity satisfies a particular condition, productivity shocks
have real effects on employment because income and substitution effects no longer cancel.
To evaluate the size of these effects, I calibrate the model and linearize it around a stochastic
steady state. Since I will build on this calibration and on the numerical approach throughout
the remainder of the book, I describe them carefully here. I conclude that the impact of
productivity shocks is quantitatively minuscule.

3.1.1

Households
1−σ

σ

−1
if she consumes
Assume the period utility function of household member i is c (1+(σ−1)γ)1−σ1−σc and is employed, and c 1−σ−1 if she consumes c and is not employed, where σ > 0 describes

the substitutability of consumption and leisure and γ > 0 describes the disutility of work.
The utility function for an employed household member recalls equation (1.13), specialized
to the case with indivisible labor.
A representative household maximizes the sum of utilities of its members, acting as if it

65

3.1. GENERAL PREFERENCES
has utility function
∞ X
X
t=0

β t Π(st )stce (st )1−σ 1 + (σ − 1)γ
1−σ

σ

−1

!
t 1−σ
c
(s
)
−
1
u
n(st ) +(1 − n(st )) ,
1−σ

(3.1)

where ce (st ) is the consumption of each employed household member, cu (st ) is the consumption of each unemployed household member, and n(st ) is the household employment rate inhistory st .
The household faces a single lifetime budget constraint, that initial assets must be equal
to the difference between the present value of consumption and the present value of after-tax
labor income plus transfers,
a0 =

∞ X
X
t=0

st


q0 (st ) ce (st )n(st ) + cu (st )(1 − n(st )) − (1 − τ )w(st )n(st ) − T (st ) .

(3.2)

The budget constraint is unchanged from the previous chapter, e.g. equation (2.33), except
for the distinction between the consumption of the employed and the consumption of the unemployed. As before, the evolution of employment is exogenous to the household, determinedby the law of motionn(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st ))(3.3)

for any st+1 = {st , st+1 }. This too is unchanged from equation (2.34). The household
maximizes equation (3.1) subject to equation (3.2) and equation (3.3).
The new piece of this problem is the choice of the relative consumption of employed andunemployed household members. Place a multiplier λ on the lifetime budget constraint andtake the first order condition for the choice of ce and cu to get
β t Π(st )ce (st )−σ 1 + (σ − 1)γσ= β t Π(st )cu (st )−σ = λq0 (st ),proving that the consumption of employed workers is always proportional to the consumptionof unemployed workers, ce (st )/cu (st ) = 1 + (σ −1)γ. Let c(st ) ≡ ce (st )n(st ) + cu (st )(1 −n(st ))denote average consumption. Thenc(st )
cu (s ) =1 + (σ − 1)γn(st )tc(st ) 1 + (σ − 1)γand ce (s ) =.
1 + (σ − 1)γn(st )tSubstituting these results into equations (3.1) and (3.2) implies that the household acts as if

66

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

it has a utility function defined over average consumption c(st ) and labor supply n(st ),
∞ X
X
t=0

s

c(st )1−σ 1 + (σ − 1)γn(st )β t Π(st )
1−σ
t

σ

−1

,

(3.4)

faces the same budget constraint as in the previous chapter,
a0 =

∞ X
X
t=0

st


q0 (st ) c(st ) − (1 − τ )w(st )n(st ) − T (st ) ,(3.5)

and faces the same law of motion for employment, equation (3.3). Notably household preferences over consumption and the fraction of the household that works are identical to thebalanced growth, constant Frisch elasticity case I studied in equation (1.13), with an infinite
Frisch elasticity of labor supply.
To proceed further, write the household’s problem recursively:
σ

 1−σ
X Π(st+1 )c
1 + (σ − 1)γn − 1
t+1
t+1
′
+β
V s , a(s ), n ,V (s , a, n) = max{a(st+1 )}
1−σΠ(st )
t+1 t
t

s

|s

(3.6)

where consumption solves the intertemporal budget constraintc = a + (1 − τ )w(st )n + T (st ) −X

qt (st+1 )a(st+1 ),st+1 |stand next period’s employment solves
n′ = (1 − x)n + f (θ(st ))(1 − n).
Using the envelope condition for assets and the first order condition for a(st+1 ), I obtain an
expression for the price of an Arrow security:Π(st+1 )qt (st+1 ) = βΠ(st ) !σc(st ) 1 + (σ − 1)γn(st+1 ).
c(st+1 ) 1 + (σ − 1)γn(st )(3.7)

This modification of equation (2.38) recognizes that the marginal utility of consumption
depends on the household’s employment rate when σ 6= 1.
Next write the envelope condition for employment for a household with the equilibrium

67

3.1. GENERAL PREFERENCES
level of assets and employment:ttt

Vn (s , a(s ), n(s )) =c(st )
1 + (σ − 1)γn(st )−σt(1 − τ )w(s ) −c(st )
1 + (σ − 1)γn(st )

1−σ

σγ

 X Π(st+1 )

t+1
t+1
t+1
+ β 1 − x − f (θ(s ))
V
s
,
a(s
),
n(s
)
. (3.8)nΠ(st )
t+1 t
t

s

|s

The first term gives the increase in utility from the additional consumption afforded by having
one more working household member. The second term gives the decrease in utility from thelower leisure. The third term is the continuation utility. Finally, the marginal value of having
a worker employed at an arbitrary wage w this period and the equilibrium wage thereaftersatisfiestṼn (s , w) =c(st )
1 + (σ − 1)γn(st )−σ(1 − τ )(w − w(st)) + Vn (st , a(st ), n(st )),since the marginal utility of consumption is c(st )/(1 + (σ − 1)γn(st ))result follows the same logic as is behind equation (2.12).

3.1.2

−σ

(3.9)

. The proof of thisFirms

The change in household preferences does not affect the firm’s problem. I repeat the key
equations here for convenience. First, there is an interior solution for recruiting effort if
productivity is equal to the expected value of the jobs created by a recruiter:z(st ) = µ(θ(st ))X¯ t+1 ).
qt (st+1 )J(s(3.10)

st+1 |stSecond, the value of a job is equal to the workers’ productivity plus the productivity of theother workers released from recruiting minus the wage:


1−x
t
t
¯
J(s ) = z(s ) 1 +− w(st ).
µ(θ(st ))(3.11)

Finally, the marginal value of a job paying an arbitrary wage w this period and the equilibrium
wage thereafter is equal to the marginal value of a job paying the equilibrium wage plus the
reduction in the wage w(st ) − w:
¯ t ).
J˜n (st , w) = w(st ) − w + J(s(3.12)

68

3.1.3

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

Wages

As in the benchmark model, I assume wages are determined at the start of each period bybargaining between households and firms. If they agree on a wage w following history st ,household utility increases by Ṽn (st , w) and the value of the job is J˜n (st , w). The wage w(st )maximizes the weighted geometric average of the gains from trade,w(st ) = arg max Ṽn (st , w)φ J˜n (st , w)1−φ ,(3.13)

w

where φ ∈ [0, 1] represents workers’ bargaining power.

Take the first order condition for w. Using equations (3.9) and (3.12) to compute Ṽn (st , w),
J˜n (st , w), and their derivatives with respect to w, I obtainttt(1 − φ)Vn s , a(s ), n(s )c(st )
1 + (σ − 1)γn(st )σ¯ t ).
= φ(1 − τ )J(sUse this to eliminate Vn from equation (3.8). Simplifying with the Euler equation (3.7) gives¯ t ) = (1 − φ)(1 − τ )w(st ) −φ(1 − τ )J(s(1 − φ)σγc(st )1 + (σ − 1)γn(st )+ φ(1 − τ ) 1 − x − f (θ(st )) X¯ t+1 ).
qt (st+1 )J(sst+1 |st¯ t+1 ) using equation (3.10) and J(s¯ t ) using equation (3.11):
Eliminate J(s(1 − τ )w(st ) = φ(1 − τ )z(st )(1 + θ(st )) +(1 − φ)γσc(st ).
1 + (σ − 1)γn(st )(3.14)

This is a modest generalization of equation (2.43).

3.1.4

Equilibrium

The remainder of the model is unchanged. Consumption is equal to the output created by
the n(st )(1 − ν(st )) producers,c(st ) = z(st )n(st )(1 − ν(st )),(3.15)and θ is the recruiter-unemployment ratio:
θ(st ) =ν(st )n(st ).
1 − n(st )(3.16)

69

3.1. GENERAL PREFERENCES
In addition, the government budget constraint pins down the transfer,
T (st ) = τ w(st )n(st ).

(3.17)The notion of equilibrium is also unchanged.

3.1.5

A Special Case

Under some conditions, there is an equilibrium in which productivity shocks do not affect
employment. To prove this, I again look for an equilibrium with a constant ratio of recruiters
to unemployed workers, θ(st ) = θ̄, a constant share of workers in recruiting, ν(st ) = ν̄, andconstant employment, n(st ) = n̄, with the wage and aggregate consumption proportional toproductivity, w(st ) = w̄z(st ) and c(st ) = c̄z(st ).
If there is such an equilibrium, the wage equation (3.14) reduces to(1 − τ )w̄ = φ(1 − τ )(1 + θ̄) +(1 − φ)γσc̄
.
1 + (σ − 1)γ n̄

Eliminate c̄, ν̄, and n̄ using steady state versions of equations (3.3), (3.15), and (3.16):
(1 − φ)γσ f (θ̄) − xθ̄ .
w̄ = φ(1 + θ̄) +(1 − τ ) x + f (θ̄) 1 + (σ − 1)γ(3.18)

Then substitute equation (3.11) into equation (3.10), eliminating qt (st+1 ) using equation (3.7):
t 1−σ

z(s )

= βµ(θ̄)X Π(st+1 )st+1 |stΠ(st )z(st+1 1−σ)
1 − x + µ(θ̄)− w̄ .
µ(θ̄)(3.19)

There is an equilibrium of this form if and only if this equation can be satisfied in any history
st . That in turn requires that there exists a number s̄ satisfying
es̄ = X Π(st+1 )z(st+1 )1−σ

st+1 |stΠ(st )z(st )1−σ

1
 1−σ



(3.20)

for all histories st . That is, the expected growth rate of z 1−σ must be history-independent.
This restriction is trivially satisfied when σ = 1, the case I studied in Chapter 2. It also
holds whenever productivity growth is independently and identically distributed over time,so z(st+1 ) = z(st )est+1 , where st+1 is independently and identically distributed. In that case,equation (3.20) defines s̄ so that the expected value of e(1−σ)s̄ is equal to the expected value

70

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

of e(1−σ)st+1 . Otherwise the restriction is violated.
If equation (3.20) holds, eliminate w̄ from equation (3.19) using equation (3.18) to getthat an equilibrium is characterized by T (θ̄) = 0 where(1 − φ)γσ f (θ) − xθ
e−s̄(1−σ) − β(1 − x) .
T (θ) ≡
− 1 + φ(1 + θ) +βµ(θ)(1 − τ ) x + f (θ) 1 + (σ − 1)γ(3.21)

This generalizes equation (2.23) to the case where preferences are not separable between
consumption and leisure. However, the generalization only works under a particular joint
restriction on preferences and on the stochastic process for productivity.

3.1.6

Productivity and Unemployment

Even when equation (3.20) holds, the productivity growth rate affects the employment rate.
For example, suppose σ = 0, so workers are risk-neutral. Also assume productivity follows ageometric random walk, so the growth rate of productivity is independently and identically
distributed over time, with expectation s̄. Then an equilibrium is characterized by a zero ofT , which reduces toe−s̄ − β(1 − x)
T (θ) =
− 1 + φ(1 + θ).
βµ(θ)
In this case T is increasing in θ, ensuring that there is a unique equilibrium. Moreover, it isdecreasing in s̄. It follows that higher productivity growth raises the ratio of recruiting effort
to unemployment and hence lowers unemployment. To understand this result, think of therelated social planner’s problem, where the planner’s objective is maximizing workers’ utility.
Since the intertemporal elasticity of substitution in consumption is infinite when σ = 0, theplanner does not much mind delaying consumption to take advantage of future productivity
growth. He does this by putting more workers into recruiting. In the decentralized economy, the intertemporal price qt (st+1 ) does not depend on consumption growth when σ = 0(equation 3.7). Then faster productivity growth raises the value of a job next period andencourages firms to recruit more workers today (equation 3.10).
To proceed further, I calibrate the parameters of the model. I spend some time here
discussing the choice of parameters because I use them throughout the remainder of thebook. First it is necessary to set the length of a time period. Unemployment duration in
the United States is typically quite short, so most unemployed workers manage to find ajob within a quarter. To avoid a corner solution for workers’ matching probability f (θ),
I think of a time period as a month. I set the discount factor at β = 0.996, just underfive percent annually. The Bureau of Labor Statistics measures multifactor productivity
growth annually. I fix the productivity growth rate at s̄ = 0.0012, about 1.4 percent per

71

3.1. GENERAL PREFERENCES

year, consistent with the annual measures of multifactor productivity growth in the private
business sector constructed by the Bureau of Labor Statistics.1 I fix the tax rate at τ = 0.4,
the average marginal rate constructed by Prescott (2004).
I turn next to the parameters that determine flows between employment and unemployment. In Shimer (2005), I measure the average exit probability from employment tounemployment in the United States at x = 0.034 per month and I stick with that number
here for comparability. Although there are many estimates of the matching function f in theliterature (see the survey by Petrongolo and Pissarides, 2001), most papers assume that firms
create job vacancies in order to attract workers and so estimate matching functions using
data on unemployment and vacancies. The technology in this paper is slightly different, withfirms using workers to recruit workers. Unfortunately there is no good time series showing
the number of workers (or hours of work) devoted to recruiting, and so the choice of f issomewhat arbitrary. Still, following much of the search and matching literature, I focus on
an isoelastic function, f (θ) = µ̄θη , and look at the symmetric case, η = 0.5.2
To pin down the efficiency parameter in the matching function µ̄, I build on evidence
in Hagedorn and Manovskii (2008) and Silva and Toledo (2009). Those papers argue that
recruiting a worker uses approximately 4 percent of one worker’s quarterly wage, i.e. a recruiter can attract approximately 25 new workers in a quarter. Using this and the fact thatthe unemployment rate in the United States was five percent on average during the post war
period, I determine µ̄. I proceed in several steps. First, from equation (3.3), the steady state
employment rate satisfies
f (θ)
n=
.
x + f (θ)
Setting n = 0.95 and x = 0.034, this implies f (θ) = 0.646 in steady state. Second, thefunctional form f (θ) = µ̄θη implies
µ̄ =

f (θ)
= f (θ)1−η µ(θ)η ,ηθwhere the second equation follows because µ(θ) ≡ f (θ)/θ. From this equation, I set µ̄ = 2.32,
consistent with f (θ) = 0.646, µ(θ) ≈ 8.33, and η = 1/2. Note that this implies that therecruiter-unemployment ratio is θ = f (θ)/µ(θ) ≈ 0.078. Using equation (3.16), the share of

recruiters in employment is ν = θ(1 − n)/n ≈ 0.004, with 99.6 percent of employees devoted
to production. Thus in this calibration, the implicit hiring costs are small, at least on average.
1

See ftp://ftp.bls.gov/pub/special.requests/opt/mp/prod3.mfptablehis.zip, Table 4. Between
1948 and 2007, productivity grew by 0.818 log points, or approximately 0.014 log points per year.
2
Formally, I assume f (θ) = min{µ̄θη , 1} to ensure that it is a proper probability. In steady state f (θ) < 1
and so when I linearize around the steady state, the constraint f (θ) ≤ 1 is not binding.

72

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS
parameter
discount factor βmean productivity growth s̄
tax rate τ
employment exit probability xmatching function f (θ)
workers’ bargaining power φconsumption-labor complementarity σ
disutility of work γ
autocorrelation of productivity growth ρstandard deviation of productivity growth ς

value
0.996
0.00120.4
0.034
2.32θ1/2
0.5
2
0.399
0.4
0.00325

Table 3.1: Calibrated parameter values with consumption-labor complementarities.
Finally, I turn to bargaining and preference parameters. I impose symmetry in the bargaining problem, φ = 0.5. This assumption is important, and so I discuss the implicationsof other choices for φ in various places throughout the book. I set the preference parameter
governing consumption-labor complementarities at σ = 2, consistent with a moderate difference between the consumption of the employed and non-employed (see Section 1.4). Finally,
I set the parameter governing the taste for leisure to match a five percent unemployment
rate along the balanced growth path; this imposes γ ≈ 0.399. Obviously this last parameteris sensitive to the extent of consumption-labor complementarity. Table 3.1 summarizes theseparameter choices. Note that the last two rows are irrelevant with deterministic productivity
growth.
I use the calibrated model to examine how responsive the unemployment rate is to steadystate growth. Consider an economy with the same parameters but s̄ = 0.01, i.e. twelve
percent annual growth, a rate normally associated with a growth miracle. Because theincome effect outweighs the substitution effect when σ > 1, expectations of higher future
growth raises interest rates by enough to discourage firms from recruiting, raising current
production and the unemployment rate. Quantitatively, however, this effect is trivial. In
steady state, the unemployment rate rises from 5 to 5.05 percent from this enormous change
in productivity growth. Changes in other parameters do not much alter this conclusion. For
example, set the complementarity parameter σ = 4 and re-calibrate the disutility of work
to have a five percent unemployment rate. One percent monthly productivity growth raises
the unemployment rate to 5.16 percent. At the other extreme, with σ → 0 and γ again
re-calibrated, the unemployment rate remains at 5.00 percent when growth increases to one
percent per month.

73

3.1. GENERAL PREFERENCES

3.1.7

Productivity Shocks

In general, comparative statics need not be informative about how the economy responds
to shocks. I therefore next examine the behavior of the model when productivity growth isstochastic. To be concrete, I assume productivity has a stochastic trend,log z(st+1 ) = log z(st ) + st+1 ,where st+1 = {st , st+1 } and st follows a first order Markov process: Π(st+1 )/Π(st ) = π(st+1 |st )is the probability of state st+1 next period conditional on state st this period, where st ≡{st−1 , st } and st+1 ≡ {st , st+1 }. If productivity growth is serially correlated, so the distribution of st+1 in fact depends on st , the restriction in equation (3.20) fails. In this case, therecruiter-unemployment ratio and employment vary with the state of the economy. I estab-lish these results by solving the model numerically in a neighborhood of a balanced growthpath.
One might alternatively assume that productivity has a deterministic trend, log z(st ) =s̄t + st , where st follows a stationary first order Markov process with mean 0 and s̄ is trend
productivity growth. This structure is common to much of the real business cycle literaturesince Kydland and Prescott (1982). The results are similar and so I omit them for the sakeof brevity; however, in the remainder of the book I compare the behavior of economies withdeterministic and stochastic trends.
To describe the equilibrium, I reduce the system of equations describing the equilibrium
to pair of stationary equations that implicitly define how employment and the recruiterunemployment ratio depend productivity growth. Substitute equation (3.15) for c(st ) inequation (3.14) and write the wage relative to productivity asw(st )(1 − φ)γσn(st )(1 − ν(st ))
t
.
=
φ(1
+
θ(s
))
+
z(st )(1 − τ ) 1 + (σ − 1)γn(st )(3.22)

Then substitute equation (3.11) into equation (3.10), eliminating qt (st+1 ) using equation (3.7)
and c(st ) and c(st+1 ) using equation (3.15). Since by assumption z(st+1 )/z(st ) = est+1 ,1 = βµ(θ(st))

X

st+1 |stπ(st+1 |st )est+1 (1−σ)

 !σn(st )(1 − ν(st )) 1 + (σ − 1)γn(st+1 )n(st+1 )(1 − ν(st+1 )) 1 + (σ − 1)γn(st )1−xw(st+1 )× 1+−. (3.23)µ(θ(st+1 ))z(st+1 )Eliminate w(st+1 )/z(st+1 ) using equation (3.22). Finally, ν(st ) and ν(st+1 ) are determined

74

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

from equation (3.16) asν(st ) =θ(st )(1 − n(st ))θ(st+1 )(1 − n(st+1 ))
t+1
and
ν(s
)
=
,
n(st )n(st+1 )(3.24)

while n(st+1 ) satisfies equation (3.3). After making the various substitutions, I obtain an
equation relating current and future values of θ and n to current and future values of productivity growth:
1 = βµ(θ(st))

X

st+1 |st×π(st+1 |st )est+1 (1−σ)
 !σn(st ) − θ(st )(1 − n(st )) 1 + (σ − 1)γn(st+1 )
n(st+1 ) − θ(st+1 )(1 − n(st+1 )) 1 + (σ − 1)γn(st )
×

1+

1−x
− φ(1 + θ(st+1 ))µ(θ(st+1 ))(1 − φ)γσ n(st+1 ) − θ(st+1 )(1 − n(st+1 ))−(1 − τ ) 1 + (σ − 1)γn(st+1 )!

. (3.25)

I close the model using the law of motion for employment in equation (3.3), which also relates
current θ and n to future n.
Equations (3.3) and (3.25) define the recruiter-unemployment ratio and employment asfunctions of the history of shocks. However, past shocks directly enter the equations only
through the current level of productivity growth st , which in turn affects the distribution ofnext period’s productivity growth st+1 . This motivates me to look for a solution to theseequations in which the recruiter-unemployment ratio depends only on current productivity
growth rate and current employment, say θ(st ) = Θ(st , n(st )). Substitute this conjecture andequation (3.3) into equation (3.25) to obtain an equation of the form
T (θ(st ), st , n(st )) = 0.

(3.26)

The functional form of T is messy and so I do not write it out explicitly. The point is that tofind an equilibrium, we must solve this functional equation. Note that in such an equilibrium,
employment depends only on lagged productivity growth and lagged employment, n(st+1 ) =N(st , n(st )).
Although I cannot solve these equations exactly, I can easily approximate them in a

3.1. GENERAL PREFERENCES

75

neighborhood of the stochastic steady state. I assume
st+1 = s̄ + ρ(st − s̄) + ςυt+1 ,(3.27)

where υ is a white noise innovation with mean 0 and standard deviation 1 and so ς is thestandard deviation of the productivity shock, while ρ ∈ (0, 1) is its autocorrelation. I thenlook for a log-linear approximation to the policy function Θ(s, n):log θ = log θ̄ + θs (s − s̄) + θn (log n − log n̄).

(3.28)The parameter θ̄ is the recruiter-unemployment ratio in the stochastic steady state, whileθs represents the elasticity of the recruiter-unemployment ratio with respect to productivity
growth and θn represents the corresponding elasticity with respect to current employment.
To find these elasticities, eliminate θ(st ) from equation (3.26) using equation (3.28):
T (θ̄eθs (s−s̄)+θn (log n−log n̄) , s, n) = 0.

(3.29)

I impose that T (θ̄, s̄, n̄) = 0 and that the derivatives of T with respect to s and n are zero at

the steady state as well. Together with the equation for the steady state employment rate,n̄ = f (θ̄)/(x + f (θ̄)), this pins down the four unknowns n̄, θ̄, θs , and θn .3

Compared to the deterministic model, there are two new parameters, the autocorrelationand standard deviation of shocks to the productivity growth rate. I again pin these downusing annual measures of multifactor productivity growth in the private business sector.
The unconditional standard deviation of annual productivity growth is 0.0180, while theannual autocorrelation is 0.043, barely positive. Because of time aggregation, determining
these as functions of the monthly productivity growth process takes a bit of algebra.4 Since
monthly productivity growth is a first-order autoregressive process, its unconditional varianceis E(s − s̄)2 = ς 2 /(1 − ρ2 ). I use this to compute the unconditional variance of annual
productivity growth.5 Using s+t to denote the t-period ahead value of productivity growth,3More precisely, I fix n̄ = 0.95 and treat the disutility of work γ as an unknown parameter.
A naı̈ve calculation might suggest a correlation of ρ12 between productivity growth in consecutive years,
since ρ is the monthly autocorrelation. This is incorrect because it ignores the comparatively high correlation
between growth rates in December of one year and January of the next year. The formulas that follow
explicitly account for this.
5
For a thorough treatment of stochastic linear difference equations, see Ljungqvist and Sargent (2004,
Section 2.4).
4

76

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

annual productivity growth is

E

11
X
t=0

P11

t=0 s+t and so its unconditional variance is(s+t − s̄)

!2

=

12 +

11
X
t=1

2(12 − t)ρt!

ς2.
1 − ρ2

Similarly, the unconditional covariance of productivity growth in consecutive years is

E

11
X
t=0

(s+t − s̄)

!

23
X

t=12

(s+t − s̄)

!

=ρ

11
X

ρt

t=0

!2

ς2
.
1 − ρ2

To obtain these expressions, repeatedly substitute for s+t using equation (3.27) to getts+t = s̄ + ρ (s − s̄) + ςtX′ρt−t υt′ .

t′ =1

Use this to expand the products in the expressions for the variance and covariance. Since2Eυ+t υ+t′ = Eυ+t s = 0 for all t′ 6= t > 0, the only remaining terms take the form Eυ+t
= 1 andEs2 = ς 2 /(1 − ρ2 ). A bit more algebra then delivers the final expressions. Finally, the ratioof the covariance to the variance gives the unconditional correlation between productivity

growth in consecutive years,

P11 t 2
ρ
t=0 ρ
.
P11
12 + t=1 2(12 − t)ρt

Using this expression and the target annual autocorrelation of 0.043, I pin down ρ = 0.4.
I then turn to the expression for the unconditional variance and use the annual target forthe standard deviation, 0.0180, to pin down ς = 0.00325. I leave the remaining parameters
unchanged at their values in Table 3.1.
With these parameter values, I solve the log-linearized system and find thatlog θ = −2.557 − 0.779(s − 0.0012) + 0.228(log n − log 0.95),
which describes how the current recruiter-unemployment ratio depends on current productivity and employment. Since consumption and leisure are substitutes, higher productivity
encourages firms to produce rather than recruit, reducing future employment. Note thatfor these parameters, higher employment also raises the recruiter-unemployment ratio; thishappens only with sufficiently high values of the substitutability parameter σ. When moreworkers are employed, firms put more workers into recruiting, which mitigates the rise incurrent production and hence consumption but raises future employment and consumption.
This takes advantage of the complementarity between consumption and employment in pref-

77

3.1. GENERAL PREFERENCES
erences.

Using the expression for θ, I then log-linearize equation (3.3) to characterize employment
next period as a function of current productivity and employment,
log n+1 = log 0.95 − 0.013(s − 0.0012) + 0.324(log n − log 0.95).
Finally, I use equation (3.27) to write next period’s productivity as
s+1 = 0.0012 + 0.4(s − 0.0012) + 0.00325υ+1.
These last two equations are a linear dynamic system. A key question is their stability. Thatis, if productivity and employment start away from their steady state values, do they tend toreturn to them? To answer this, let m ≡ {s− s̄, log n−log n̄} denote the state of the economy
and represent the transitional dynamics as m+1 = Am + Dυ+1 , where A is a 2 × 2 transition

matrix and so Am is the expected value of m+1 conditional on m and D determines how theshock υ+1 affects the state of the system. For this choice of parameters,
A=

0.4

0

−0.013 0.324

and
D=

ς
0

!

!

,

where ς is the standard deviation of the innovation to the productivity process and the 0 in thesecond entry for D indicates that productivity shocks have no direct effect on employment.
The dynamic system is locally stable, converging towards the steady state from any nearby
value of m = {s − s̄, log n−log n̄}, if and only if the eigenvalues of A all lie between −1 and 1.
Given the triangular structure of the transition matrix, the eigenvalues are just the diagonal
elements, 0.4 and 0.324 and so the system is in fact locally stable. Moreover, the size of theeigenvalues tells us about the speed of convergence. The larger eigenvalue, 0.4, is associatedwith the adjustment of productivity growth back to its normal level. The smaller eigenvalue,
0.324, is associated with the slightly faster adjustment of employment. This indicates that
if productivity is at its normal value but employment is not, 68 percent of the deviation of
employment from steady state is wiped out within a month. This finding of fast employment
adjustment dynamics is endemic to the search and matching literature. A number of papers
have modified the model in an effort to explain why the unemployment rate is in fact quitepersistent (Hall, 1995; Cole and Rogerson, 1999; Pries, 2004), but I do not pursue that route

78

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

in this book.
Next I compute the unconditional variance and covariance of the state variables, Σ ≡E(mm′ ). Since this is an unconditional variance-covariance matrix, the expected value ofm+1 m′+1 is also Σ. Then using m+1 = Am + Dυ+1 , I obtain′Σ = E m+1 m′+1 = E (Am + Dυ+1 )(m′ A′ + υ+1
D ′ ) = AΣA′ + DD ′,where the last equality uses the fact that υ+1 is a shock and hence orthogonal to m,′′E(mυ+1) = E(υm′+1 ) = 0, and has unit variance, E(υ+1 υ+1) = 1. This equation is easily solved numerically for Σ:Σ=

1.1905 −0.0072−0.0072

0.0003!

ς 2.

In particular, the unconditional standard deviation of employment istiny number.

√

0.0003ς = 0.00006, aWith a larger value of the complementarity parameter, σ = 4, the response of employment
is bigger but still quantitatively unimpressive. The policy function becomeslog θ = −2.557 − 2.297(s − 0.0012) + 1.884(log n − log 0.95),
and the state equation islog n+1 = log 0.95 − 0.039(s − 0.0012) + 0.352(log n − log 0.95).
Now the unconditional standard deviation of employment is 4.8 percent as large as theunconditional standard deviation of monthly productivity growth. While this is about threetimes larger than before, it is still a tiny number and it relies on an implausibly strong
complementarity between consumption and leisure.
Finally, I can compute the labor wedge. Imagine an economist who ignores the existence oflabor market frictions and so measures the wedge between the marginal rate of substitutionof consumption for leisure and the marginal product of labor using equation (1.14). He
estimates the disutility of leisure γ̂ so as to imply an average labor wedge of τ̂ = 0.4, sets thecapital share of income at α = 0, and recognizes that the Frisch elasticity of labor supply is
ε = ∞ and the parameter governing the substitutability of consumption and leisure to σ = 2.
Then using employment data, with the employment rate averaging n̄ = 0.95, he measures79

3.2. CAPITAL
the labor wedge in period t asτ̂ (st ) = 1 −γ̂σn(st ),1 + (σ − 1)γ̂n(st )(3.30)

so the average labor wedge satisfiesτ̂ = 1 −γ̂σn̄.
1 + (σ − 1)γ̂ n̄

He would solve this equation for the disutility of work and conclude γ̂ = 0.451, higher than
the true value of γ = 0.399. This is because he fails to realize that workers are unemployed
not only because they have a taste for leisure but also because of the search frictions.
The economist would also uncover movements in the labor wedge that are negatively
correlated with the employment rate, as in the data. However, the size of these movements would be small. To see this, log-linearize τ̂ around the stochastic steady state. Fromequation (3.30),
log τ̂ = log 0.4 − 1.050(log n − log 0.95).
The economist would conclude that whenever employment is above its normal level, the laborwedge is roughly the same proportion below its normal level. This is broadly consistent withthe evidence in Table 1.1 when the elasticity of labor supply is infinite. Still, the fact that themodel yields only very small movements in the employment rate implies that it yields equallysmall movements in the labor wedge, both inconsistent with the data. I conclude that the
search model with substitutability between consumption and leisure is a quantitative failure
at explaining the behavior of the employment rate and the labor wedge.

3.2

CapitalI now reintroduce productive capital—and shocks to the production technology of the capital
good—into the model. In the benchmark search model in Chapter 2, the economy can smoothshocks to the production technology only by changing the allocation of workers between
recruiting and production. Since the productivity of the recruiting technology is constant,a temporary improvement in the efficiency of the production technology induces offsetting
effects: on the one hand, it is a good time to produce rather than recruit; on the other hand,
the marginal utility of consumption is low, encouraging firms to defer production. With thepreferences in equation (2.32), these effects cancel and so recruiting is acyclic. With the moregeneral preferences in equation (3.1), they do not quite cancel, but I found recruiting wasstill nearly constant.

80

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS
Capital accumulation, like recruiting, is an investment. The key difference between the

two types of investment is that it is reasonable to assume that shocks to the technology for
producing the consumption good do not affect the efficiency of the recruiting technology;
however, many shocks that improve the efficiency of producing the consumption good arelikely to also improve the efficiency of producing the investment good. This implies that when
the productivity is high, it is a good time to produce investment goods. This endogenous
propagation mechanism has been well-understood since the advent of the real business cycle
model (Kydland and Prescott, 1982), but is neglected in any business cycle model without
capital.

3.2.1

Households

The household’s problem is unchanged from Section 2.2, i.e. the case with logarithmic utility
over consumption and separability between consumption and leisure. From this problem, I
again obtain three key results: the Euler equation,
qt (st+1 ) = β

Π(st+1 )c(st );Π(st )c(st+1 )(3.31)

the marginal value of an employed household member,(1 − τ )w(st )Vn (s , a(s ), n(s )) =−γc(st )ttt+ β(1 − x − f (θ(st )))X Π(st+1 )st+1 |stΠ(st )Vn (st+1 , a(st+1 ), n(st+1 )); (3.32)

and the marginal value of having a worker employed in a job that pays an arbitrary wage w
in history st and the equilibrium wage thereafter,Ṽn (st , w) =(1 − τ )(w − w(st ))+ Vn (st , a(st ), n(st )).
tc(s )(3.33)

I omit the derivation of these results because they do not depend on whether there is capitalin the economy; see Section 2.2.2 for details.

3.2.2

Firms

I start by describing the firm’s problem verbally. A representative firm employs n0 = n(s0 )workers and owns capital k0 = k(s0 ) at time 0. In history st , it assigns a fraction ν(st ) of itsn(st ) workers to recruiting and the remaining n(st )(1 − ν(st )) workers to production. The

81

3.2. CAPITAL

1−αproducers use the capital k(st ) to generate z(st )k(st )α n(st )(1 − ν(st ))of a single goodthat is used both for consumption and for investment. A fraction δ of the capital depreciates

during production. The recruiters each attract µ(θ(st )) workers to the firm, while a fraction
x of the workers leave the firm, thus determining n(st+1 ). Finally, the firm can freely buy
or sell capital in history st , determining k(st+1 ). Because there is a single technology for
producing the capital and consumption good, the relative price of the capital good is always
equal to 1.
More formally, the present value of the firm’s profits are
0J(s , n0 , k0 ) =

∞ X
X
t=0

st


1−α
q0 (s ) z(st )k(st )α n(st )(1 − ν(st ))t+ (1 − δ)k(st ) − k(st+1 ) − w(st )n(st ) , (3.34)

where firm growth satisfiesn(st+1 ) = n(st ) ν(st )µ(θ(st )) + 1 − x .

(3.35)

This extends equation (1.4) to recognize that in the search and matching model, the levelof employment is a state variable in the firm’s problem. Note that k(st+1 ) and n(st+1 ), thecapital stock and total workforce in history st+1 , are determined in the prior history st . Thisimplies that k({st , st+1 }) = k({st , s′t+1 }) and n({st , st+1 }) = n({st , s′t+1 }) for all st , st+1 , ands′t+1 .
To characterize the firm’s behavior, let J(st , n, k) denote the value of a firm that starts
history st with n workers and k units of capital. This is homogeneous of degree 1 jointly inn and k because a firm with twice as many workers and twice as much capital can produce
twice as much, invest twice as much, and hire twice as much at all future dates, yielding
twice as much profits. Moreover, J solves the recursive equationtJ(s , n, k) = maxz(st )k α n(1 − ν)
′
ν,k

+

1−α

X

st+1 |st+ (1 − δ)k − k ′ − nw(st )
qt (s

t+1

)J s

t+1


 ′
, n νµ(θ(s )) + 1 − x , k . (3.36)t

In each period, the firm chooses the fraction of its workforce that are recruiters and its capitalstock next period in order to maximize current revenue net of investment costs and laborcosts plus the expected continuation value of the firm.

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CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS
Assuming an interior solution for the share of recruiters ν, I obtain the first order conditiont(1 − α)z(s )k(st )n(st )(1 − ν(st ))α= µ(θ(st ))X

st+1 |stqt (st+1 )Jn st+1 , n(st+1 ), k(st+1 ) . (3.37)

The left hand side is the marginal product of a producer. The right hand side is the expectedvalue of the additional workers attracted by a recruiter. Also write the envelope condition
for employment asαk(st )Jn (s , n(s ), k(s )) = (1 − α)z(s )(1 − ν(st )) − w(st)n(st )(1 − ν(st )) X+ ν(st )µ(θ(st )) + 1 − x
qt (st+1 )Jn st+1 , n(st+1 ), k(st+1 ) .
t

t

t

t



st+1 |stEliminate the continuation value using equation (3.37) to obtain
t

t

t

t

Jn (s , n(s ), k(s )) = (1 − α)z(s )k(st )n(st )(1 − ν(st ))

α 

1−x
1+
− w(st).
µ(θ(st ))(3.38)

This extends equation (2.30) to an environment where the marginal product of labor depends
on the capital-producer ratio. As before, the term 1 + (1−x)/µ(θ(st )) recognizes that when afirm employs an additional worker, this raises current output directly and also raises output
because the firm can afford to move workers from recruiting to production this period, while
keeping the same size next period.
Next turn to the first order condition for next period’s capital stock. From equation (3.36),
1=X

st+1 |stqt (st+1 )Jk st+1 , n(st+1 ), k(st+1 ) .

Purchasing a unit of capital reduces current profit by 1. This must equal the increase in thecontinuation value of the firm. The envelope condition for capital is
t

t

t

t

Jk (s , n(s ), k(s )) = αz(s )k(st )n(st )(1 − ν(st ))α−1+ 1 − δ.

Evaluate this in history st+1 and substitute into the first order condition for next period’s
capital to getk(st+1 )1=
qt (st+1 ) αz(st+1 )n(st+1 )(1 − ν(st+1 ))
t+1 t
s

X

|s



α−1

+1−δ

!

.

(3.39)

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3.2. CAPITAL

Firms are willing to invest in capital if the cost of capital this period is equal to the expected
net marginal product of capital next period, including the value of reselling the undepreciatedportion of the capital.
Finally, compute the marginal profit of employing a worker at an arbitrary wage w inhistory st and the equilibrium wage thereafter, rather than losing the worker. For a firm withthe equilibrium level of employment n(st ) and capital k(st ), this isJ˜n (st , w) = w(st ) − w + Jn (st , n(st ), k(st )).

(3.40)

This is essentially unchanged from equation (2.31).

3.2.3

Wages

The Nash bargaining solution states that the wage maximizes the Nash productṼn (st , w)φ J˜n (st , w)1−φ .

(3.41)

Replace J˜n (st , w) using equation (3.40) and Ṽn (st , w) using equation (3.33). The first ordernecessary and sufficient condition for a maximum of the Nash product is(1 − φ)Vn st , a(st ), n(st ) c(st ) = φ(1 − τ )Jn (st , n(st ), k(st )).

(3.42)

Use this to eliminate Vn (st , a(st ), n(st )) and Vn (st+1 , a(st+1 ), n(st+1 )) from equation (3.32):
φ(1 − τ )Jn (st , n(st ), k(st )) = (1 − φ)(1 − τ )w(st ) − (1 − φ)γc(st )X+ φ(1 − τ )(1 − x − f (θ(st )))qt (st+1 )Jn (st+1 , n(st ), k(st+1 )).
st+1 |stReplace Jn (st , n(st ), k(st )) using equation (3.38) andusing equation (3.37), giving the wage equationPk(st )(1 − τ )w(s ) = φ(1 − τ )(1 − α)z(s )n(st )(1 − ν(st ))tt

st+1 |st qt (s

α

t+1

)Jn (st+1 , n(st+1 ), k(st+1 ))(1 + θ(st )) + (1 − φ)γc(st ), (3.43)

a natural generalization of equation (2.43). The after-tax wage is a weighted average of
two terms: the after-tax marginal product of labor both of this worker and of the workers
released from recruiting by her employment; and the marginal rate of substitution between
consumption and leisure.

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3.2.4

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

Equilibrium

I assume the government levies labor taxes and rebates the proceeds lump-sum to households
in every period. For notational simplicity, I impose a balanced budget,T (st ) = τ w(st )n(st ).

(3.44)

Ricardian equivalence holds in this environment, so alternative assumptions on government
debt would not affect the equilibrium allocations.
I then close the model with a set of market clearing conditions. The economy’s resource
constraint implies that next period’s capital stock isk(st+1 ) = z(st )k(st )α n(st )(1 − ν(st ))1−α+ (1 − δ)k(st ) − c(st ).

(3.45)

Next period’s employment is determined from current employment and current recruiting,n(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st )).

(3.46)

And the recruiter-unemployment ratio θ satisfies
ν(st )n(st )θ(s ) =.
1 − n(st )t(3.47)The definition of equilibrium is now standard and so I omit it.

3.2.5

Balanced Growth

Suppose there is deterministic Hicks-neutral productivity growth at rate s̄, log z(st+1 ) =log z(st ) + s̄. I claim that there is an equilibrium where consumption, capital, and wages
grow at rate s̄/(1 − α), while employment, the ratio of recruiters to unemployed, and the11share of recruiters in employment are constant. That is, c(st ) = c̄z(st ) 1−α , k(st ) = k̄z(st ) 1−α ,1w(st) = w̄z(st ) 1−α , n(st ) = n̄, θ(st ) = θ̄, and ν(st ) = ν̄. This result is useful both forunderstanding how the economy works and also later for characterizing stochastic deviations
from the balanced growth path due to technology shocks.
To prove this claim, I assume that there is such an equilibrium and show how to solve forthe unknown constants. Eliminate the intertemporal price qt (st+1 ) from equation (3.39) using
equation (3.31) and eliminate the share of recruiters ν(st ) and ν(st+1 ) using equation (3.47).

85

3.2. CAPITALThis gives
1 = βe

−s̄
1−α

α



k̄
n̄ − θ̄(1 − n̄)

α−1

!

+1−δ .

(3.48)

Then eliminate Jn (st+1 , n(st+1 ), k(st+1 )) from equation (3.37) using equation (3.38) evaluatedin history st+1 . Again eliminate q and ν using equations (3.31) and (3.47):k̄w̄ = (1 − α)n̄ − θ̄(1 − n̄)α 

1 − β(1 − x)1−
βµ(θ̄)

.

(3.49)

The wage equation (3.43) and equation (3.47) imply(1 − τ )w̄ = φ(1 − τ )(1 − α)k̄n̄ − θ̄(1 − n̄)α(1 + θ̄) + (1 − φ)γc̄.

(3.50)

Finally, the resource constraint equation (3.45) and the law of motion for employment inequation (3.46) each simplify slightly:s̄

e 1−α k̄ = k̄ α n̄ − θ̄(1 − n̄)
and
n̄ =

1−α

+ (1 − δ)k̄ − c̄

(3.51)f (θ̄).
x + f (θ̄)

(3.52)The preceding five equations, equations (3.48)–(3.52), have five unknowns, the wage constant w̄, the capital constant k̄, the consumption constant c̄, employment n̄, and the recruiterunemployment ratio θ̄. It is possible to reduce them to an implicit equation for θ̄. Eliminatew̄ between equations (3.49) and (3.50), then eliminate k̄ using equation (3.48), next c̄ using
equation (3.51), and finally eliminate n̄ using equation (3.52). This algebra yields T (θ̄) = 0,
where


1 − β(1 − x)
T (θ) ≡ (1 − τ )(1 − α) 1 −− φ(1 + θ)
βµ(θ)
s̄

− (1 − φ)γ(1 − αβ)e 1−α − (1 − α)β(1 − δ)s̄e 1−α − β(1 − δ)!

f (θ) − θxf (θ) + x.

Observe that T is continuous. When θ = 0, T (0) = (1 − τ )(1 − α)(1 − φ) > 0. Also let θ̃ > 0
solve f (θ̃) = θ̃x, so µ(θ̃) = x. Evaluating gives T (θ̃) = −(1−τ )(1−α) 1−β+φ(1+ θ̃) < 0. Byβx
the intermediate value theorem, there is a solution to T (θ) = 0, an equilibrium, in between
these two points.
Once I obtain θ̄, it is straightforward to compute the value of the other variables of86

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

interest along a balanced growth path. Note that since both capital and employment are statevariables, the economy is on a balanced growth path only with particular initial conditions
for the capital stock and employment. I defer an analysis of the stability of the system, i.e.
whether it converges to the balanced growth path from nearby initial conditions, until afterI discuss the full model with a productivity shock.

3.2.6

Productivity Shocks: Deterministic Trend

Suppose log z(st ) = s̄t + st where st follows a stationary first order Markov process withmean 0. Thus s̄ is the deterministic growth rate while st represents a transitory deviationfrom trend. Let π(st+1 |st ) ≡ Π(st+1 )/Π(st ) denote the probability of state st+1 next period
conditional on state st this period, where st ≡ {st−1 , st } and st+1 ≡ {st , st+1 }. This nests thebalanced growth path as a special case with st = 0 for all t.
Stationary Equilibrium
Inspired by the balanced growth path, I define consumption, capital, and wages relative
s̄t

s̄t

s̄t

to trend productivity as c̃(st ) ≡ c(st )e− 1−α , k̃(st ) ≡ k(st )e− 1−α , and w̃(st ) ≡ w(st )e− 1−α .
Along a balanced growth path, I found that c̃(st ), k̃(st ), and w̃(st ) are constant. I look
for an equilibrium of the stochastic growth model where they are stationary. Similarly, since
employment, the share of recruiters in employment, and the ratio of recruiters to unemployed
are all constant along a balanced growth path, I look for an equilibrium of the stochastic
growth model where these variables are stationary.
To proceed, I look for a stationary system of equations in these new variables. Useequation (3.31) to eliminate the intertemporal price q from the condition for an interior
solution for capital, equation (3.39), and then use equation (3.47) to eliminate the share of
recruiters in employment ν:s̄

e 1−α = βXst+1 |stt

π(st+1 |st )

c̃(s )  st+1αec̃(st+1 )

t+1

t+1

t+1

k̃(s

t+1

)

n(st+1 ) − θ(st+1 )(1 − n(st+1 ))

!α−1



+ 1 − δ .

(3.53)

Similarly eliminate Jn (s , n(s ), k(s )) from the condition for an interior solution forrecruiting, equation (3.37), using the envelope condition for employment, equation (3.38),

87

3.2. CAPITAL
evaluated in history st+1 . Then eliminate q and ν using equations (3.31) and (3.47):(1 − α)est

k̃(st )n(st ) − θ(st )(1 − n(st ))st+1(1 − α)e!α= βµ(θ(st ))k̃(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))X

st+1 |stπ(st+1 |st )!α c̃(st )c̃(st+1 )

1−x
1+µ(θ(st+1 ))− w̃(st+1!

) . (3.54)

Next use equation (3.47) to rewrite the wage equation (3.43), evaluated in history st+1 , as
(1 − τ )w̃(st+1 ) = (1 − φ)γc̃(st+1 )k̃(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))+ φ(1 − τ )(1 − α)est+1!α(1 + θ(st+1 )). (3.55)

Again using equation (3.47), the resource constraint equation (3.45) becomess̄k̃(st+1 )e 1−α = est k̃(st )α n(st ) − θ(st )(1 − n(st ))1−α+ (1 − δ)k̃(st ) − c̃(st ).

(3.56)

Equation (3.46) is unaffected by this notation.
Equations (3.46) and (3.53)–(3.56) describe a stationary set of relationships between thevariables of interest and so may in fact admit a stationary solution. I look for a solution withthe property that the policy variables θ(st ) and c̃(st ) depend on the exogenous state st , onemployment n, and on the capital stock relative to trend k̃. Using the Markov property ofthe state variable and equations (3.46) and (3.56), this implies that st+1 , n(st+1 ), and k̃(st+1 )depend on the same objects. To characterize such a solution, eliminate w̃(st+1 ) between
equations (3.54) and (3.55):(1 − α)est

k̃(st )n(st ) − θ(st )(1 − n(st ))c̃(st )
(1 − φ)γc̃(st+1 )−c̃(st+1 )1−τ
st+1 |st
!α 
!
t+1
k̃(s )1−x+ 1 − φ − φθ(st+1 )
.
t+1
t+1
t+1
t+1
n(s ) − θ(s )(1 − n(s ))µ(θ(s ))= βµ(θ(st ))+ (1 − α)est+1!αXπ(st+1 |st )(3.57)

Now let the functions Θ and C denote the equilibrium recruiter-unemployment ratio and equilibrium consumption as functions of the current state (s, n, k̃), so θ(st ) = Θ(st , n(st ), k̃(st ))88

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS
parameter
discount factor βmean productivity growth s̄
tax rate τ
employment exit probability xmatching function f (θ)
workers’ bargaining power φdisutility of work γ
capital share α
depreciation rate δ
autocorrelation of detrended productivity ρ
standard deviation of detrended productivity ς

value
0.996
0.00120.4
0.034
2.32θ1/2
0.5
0.471
0.33
0.00280.98
0.005

Table 3.2: Calibrated parameter values in model with capital, deterministic trend.
and c̃(st ) = C(st , n(st ), k̃(st )). After substituting these functions into equations (3.53)
and (3.57), eliminate k̃(st+1 ) using equation (3.56) and n(st+1 ) using equation (3.46) toobtain a pair of nonlinear equations that implicitly define Θ and C.
Calibration
As in Section 3.1, I solve the model by log-linearizing around a stochastic steady state. I
assume that the deviation of productivity from trend follows a linear process,st+1 = ρst + ςυt+1 ,(3.58)

where υ is a white noise innovation with mean 0 and standard deviation 1 and so ς is thestandard deviation of the productivity shock, while ρ ∈ (0, 1) is its autocorrelation. I thenlook for a log-linear approximation to the policy functions,log θ = log θ̄ + θs s + θn (log n − log n̄) + θk (log k̃ − log k̄),(3.59)

log c̃ = log c̄ + cs s + cn (log n − log n̄) + ck (log k̃ − log k̄).

(3.60)In these equations, θ̄ is the recruiter-unemployment ratio, c̄ is consumption relative to productivity, n̄ is employment, and k̄ is capital relative to productivity in the stochastic steadystate. The remaining six parameters represent the elasticity of the two policy functions withrespect to the three state variables.
The introduction of capital changes neither the strategy for calibrating most of the parameters nor the results, and so I keep their values fixed at those I used in the model with
consumption-labor complementarity. Of course, moving from a stochastic to a deterministic

89

3.2. CAPITAL

trend changes the productivity process. I continue to assume that mean productivity growthis s̄ = 0.0012 and set the autocorrelation of productivity growth to ρ = 0.98 and the standard deviation to ς = 0.005. These values are similar to standard calibrations of total factor
productivity (Cooley and Prescott, 1995), with an adjustment to account for the fact thattime periods are one month long.
Since the model now assumes consumption and leisure are additively separable, I need tochange the disutility of work so as to keep the average employment rate fixed at 0.95. Thisrequires setting γ = 0.471.
The model with capital has two new parameters, the capital share α and the monthly
depreciation rate δ. I fix α = 0.33 to match the capital share of income in the National Income
and Product Accounts. I set δ = 0.0028 per month, which pins down the capital-output ratio
in the stochastic steady state. To see how, rewrite equation (3.48) along a balanced growth
path as ȳ−s̄
1 = βe 1−α α + 1 − δ ,k̄1where y(st ) ≡ z(st )k(st )α (n(st ) − θ(st )(1 − n(st )))1−α is gross output and ȳ ≡ y(st )z(st )− 1−αis output relative to trend. Then this choice of parameter values implies that the ratio of
capital to monthly output, k̄/ȳ, is 38.3. Equivalently, the ratio of capital to annual output
is 3.2, the average capital-output ratio in the United States since 1948.6 The remaining
parameters are unchanged; I show their values in Table 3.2 for convenience.
This leaves nine unknowns in equations (3.59) and (3.60): the recruiter-unemployment
ratio θ̄, consumption relative to productivity c̄, and capital relative to productivity k̄ in thestochastic steady state, and the six elasticities of the policy functions with respect to thestate variables. In addition, I effectively treat the value of leisure γ as a residual, a tenthunknown variable. I pin these down using ten equations. As described above, I substitutethe log-linear approximations θ(st ) = Θ(st , n(st ), k̃(st )) and c(st ) = C(st , n(st ), k̃(st )) into
equations (3.53) and (3.57), and then eliminate k̃(st+1 ) using equation (3.56) and n(st+1 )using equation (3.46). This gives two equations, which I log-linearize around the steadystate {s, n, k} = {0, n̄, k̄}. The resulting system yields eight equations. The two remaining
equations come from steady state versions of the state equations (3.46) and (3.56):n̄ = (1 − x)n̄ + f (θ̄)(1 − n̄)1−αk̄e= k̄ α n̄ − θ̄(1 − n̄)+ (1 − δ)k̄ − c̄.
s̄
1−α

6

More precisely, I use the Bureau of Economic Analysis’s Fixed Asset Table 1.1, line 1 to measure thecurrent cost net stock of fixed assets and consumer durable goods. I use National Income and ProductAccounts Table 1.1.5, line 1 to measure nominal Gross Domestic Product.

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CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

I solve these ten equations numerically.
Solution
I find that the policy functions arelog θ = log 0.078 + 7.387s − 0.480(log n − log 0.95) − 2.779(log k̃ − log 218.2),(3.61)

log c̃ = log 4.696 + 0.250s + 0.014(log n − log 0.95) + 0.603(log k̃ − log 218.2),
where 0.078 is the recruiter-unemployment ratio, 0.95 is the employment rate, 218.2 is capital
relative to trend, and 4.696 is consumption relative to trend, all in the stochastic steady state.
When productivity is above trend, the recruiter-unemployment ratio and consumption
both increase. The increase in the recruiter-unemployment ratio raises next period’s employment rate. Since the elasticity of consumption with respect to current productivity isless than 1, physical capital investment also rises with productivity. Thus both endogenous
state variables increase with current productivity. I confirm this by looking at the implied
log-linear approximation to the state equations (3.46) and (3.56):
log n+1 = log 0.95 + 0.126s + 0.312(log n − log 0.95) − 0.047(log k̃ − log 218.2),(3.62)

log k̃+1 = log 218.2 + 0.020s + 0.019(log n − log 0.95) + 0.991(log k̃ − log 218.2).
To understand these dynamics, focus on households’ desire to maintain smooth consumption.
Following a positive productivity shock, households would like to save more. This pushes
down the interest rate (raises qt (st+1 )), which encourages firms to invest both in capital and in
recruiting workers. The increase in employment raises the marginal product of capital, which
encourages more investment, and also enables firms to devote more resources to recruiting.
Eventually, however, the wealth effect from the increase in the capital stock raises wages,
which puts downward pressure on recruiting. This reduces employment and returns the
economy to steady state.
To examine these dynamics more systematically, express the log-linear approximation to
the model’s dynamics as m+1 = Am + Dυ+1 where m = {s, log n − log n̄, log k̃ − log k̄} is thestate of the system, A is the transition matrix



0.980
0
0


A =  0.126 0.312 −0.047  ,0.020 0.019

0.991

91

3.2. CAPITAL
and D translates the univariate white noise shock υ into next period’s state



ς
 
D =  0 .
0The eigenvalues of A are 0.99, 0.98, and 0.31, and so the system is locally stable. The largest
eigenvalue, 0.99, reflects the slow adjustment of the capital stock to steady state due to thelow depreciation rate. The second eigenvalue is equal to the persistence of the productivity
shock. The smallest is associated with the rapid adjustment of employment to steady state.
Next I compute the unconditional variance and covariance of the state variables. As I
discussed in Section 3.1, this is determined by Σ solving Σ = AΣA′ + DD ′ . I obtain
25.253 3.175 19.505

Σ =  3.175 0.560 0.469  ς 2 .
19.505 0.469 46.550
This confirms that deviations of productivity from trend are positively correlated with deviations of both employment and capital from trend. However, the volatility of employment√
is still small, with an unconditional standard deviation equal to 0.560ς = 0.004. Put
differently, suppose that productivity suddenly increases to st = 0.01, approximately fourstandard deviations above trend, and gets stuck there. The unemployment rate will quickly
but modestly decline from its steady state value of 5.00 percent to 4.83 percent. Moreover,the capital stock will start to increase and this wealth effect will partially reverse the change
in unemployment. Asymptotically, the unemployment rate will rise back to 4.98 percent,scarcely different from where it started.
Labor Wedge
A particularly useful way to understand the stability of employment is to look at the laborwedge. Once again, suppose an economist ignores the existence of labor market frictions andso measures the wedge between the marginal rate of substitution and the marginal productof labor using equation (1.12). He recognizes that the capital share of income is α = 0.33 andthe Frisch labor supply elasticity is infinite since labor is indivisible. Then he would measureτ̂ (st ) = 1 −γ̂c(st )/y(st) n(st ).
1−α

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CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

Using evidence from the stochastic steady state, he would conclude that if the true tax factor
is 0.4, the disutility of work is γ̂ = 0.513. Again, this is slightly higher than the true value
γ = 0.471, since our economist does not understand that unemployment is governed by search
frictions.
He could also measure the cyclical dynamics of the labor wedge τ̂ . To see how this would
be done using model-generated data, note that both n(st ) and c(st )/y(st ) are stationary, thelatter because c(st )/y(st) = c̃(st )/ỹ(st ), wheres̄tỹ(st ) ≡ y(st)e− 1−α = est k̃(st )α n(st )(1 − ν(st ))1−αis detrended output. I can therefore log-linearize the labor wedge around its steady statevalue to obtainlog τ̂ = log 0.4 + 1.095s − 0.431(log n − log 0.95) − 0.398(log k̃ − log 218.2).
The measured labor wedge is low when productivity is below trend, employment is above
trend, or the capital stock is above trend. When productivity is below trend, the desire
to maintain smooth consumption keeps the consumption-output ratio high, which lowers
the measured labor wedge. High capital also raises the consumption-output ratio due to
its wealth effect, and so has the same implication. High employment directly reduces themeasured labor wedge.
Finally, the economist might look at the comovement between the labor wedge and thestate variables in the economy. In general, consider any set of variables m̃ that are linear
functions of the state variables m, say m̃ = Ãm for some matrix Ã. Then the variancecovariance matrix of these variables satisfies E(m̃m̃′ ) = E(Ãmm′ Ã′ ) = ÃΣÃ′ , where I simplify
this using the definition of the variance-covariance matrix Σ = E(mm′ ). Critically, he wouldfind that the correlation between the detrended measured labor wedge τ̂ and detrended
employment is 0.96. In the data, the correlation is sharply negative; see for example Table 1.1,
which shows that the correlation is always larger in magnitude than −0.6. He would alsofind a strong negative correlation between the labor wedge and the consumption-output ratio,
while in the data the correlation is much weaker and sometimes even switches signs.
It is easy to understand why the model generates a positive correlation between themeasured labor wedge and employment. Search frictions act like an adjustment cost onlabor. Our economist ignores the existence of that adjustment cost and so expects to see
relatively large movements in employment in response to the observed shocks. When he does
not, he interprets this, through the lens of the competitive model from Chapter 1, as evidence
that the labor wedge rises during expansions, discouraging households from supplying much

93

3.2. CAPITAL
ỹ
c̃
θ
k̃
n wn/yc/y1 0.717 4.391 0.990 0.109 0.012 0.515
1 0.871 0.714 0.748 0.713 -0.886 -0.729

τ̂
0.614
0.728

s
0.729
0.971

c̃

—

1 0.279 0.978 0.293 -0.563 -0.299

0.298

0.729

θ

—

—

1

1.000

0.860

k̃

—

—

—

1

0.092 -0.379 -0.091

0.090

0.569

Correlations n

—

—

—

—

1.000 -0.877 -0.976

0.962

0.844

wn/y

—

—

—

—

—

1

0.938 -0.947 -0.967

c/y

—

—

—

—

—

—

1 -0.998 -0.870

τ̂

—

—

—

—

—

—

—

1

0.870

s

—

—

—

—

—

—

—

—

1

Relative s.d.
ỹ

0.070 0.966 -0.938 -0.999

Table 3.3: Model with capital, deterministic trend. Co-movements of variables in an infinite
sample.
more labor, and falls during recessions, encouraging households not to contract their labor
supply too much. But the failure of the frictionless model occurred because in the data
employment fluctuates too much, not too little. An adjustment cost is unlikely to explain
this observation.
Comovements: Detrended Variables
Although I focus on the comovement of employment and the labor wedge, the model has
predictions along many other dimensions. Table 3.3 shows the co-movement of some key
economic outcomes. Table A.1 in Appendix A shows comparable numbers constructed from
historical United States data. The first row shows the ratio of the standard deviation ofa detrended variable to the standard deviation of detrended output, ỹ(st ). I look at consumption, the recruiter-unemployment ratio, capital, employment, the labor share (wn/y),
the consumption-output ratio, and the measured labor wedge. For example, the standard
deviation of log consumption relative to trend is 0.72 times the standard deviation of log
output relative to trend, so consumption is less volatile than output. This reflects the desire
to smooth consumption. The lower part of the table shows the contemporaneous correlation
between different variables; the correlation between consumption and output is strongly positive, 0.87. Together this implies that when output is above trend, the consumption-output
ratio is low, which I verify in the seventh column.

94

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

Comovements: Growth Rates
One possible explanation for the gap between the model and data is that I do not measure
the same objects in the two environments. In the model, I measure the level of stationary
variables including employment, the recruiter-unemployment ratio, the consumption-output
ratio, and the labor wedge; and I measure the the ratio-to-trend of trending variables including output, consumption, capital, and wages. In contrast, I cannot observe the trend in realworld data. Instead, I estimate the trend using a Hodrick-Prescott filter.
To understand whether this difference in measurement is an issue, I compute annual
growth rates for the various variables of interest in the model and compare this with the datain the lower panel of Table 1.1. This has the added advantage that the annual growth rate oftrending variables—consumption, capital, output, and wages—is equal to the annual growthrate of their detrended counterparts plus a constant, and so are easy to compute.
I start with some algebraic preliminaries. Again let m̃ = Ãm and now let m̃+i − m̃ denotethe i-period growth rate of m̃. Then this solves

m̃+i − m̃ = Ã(m+i − m).
Moveover, since m+1 = Am + Dυ+1 , a simple induction argument impliesim+i = A m +i−1
X

Aj Dυ+(i−j).

j=0

Combining these, I get
E(m̃+i − m̃)(m̃+i − m̃)′
= EÃ (Ai − I)m +

i−1
X
j=0

Aj Dυ+(i−j)!

(Ai − I)m +i−1
X

Aj Dυ+(i−j)

j=0

!′

Ã′ ,

where I is the identity matrix. Since υ is serially uncorrelated and υ+(i−j) is uncorrelatedwith m for all i > j, this reduces toE(m̃+i − m̃)(m̃+i − m̃)′ = Ã (Ai − I)Σ(Ai − I)′ +

i−1
X
j=0

Aj DD ′(Aj )′

!

Ã′ ,

where as usual Σ = E(mm′ ). This covariance matrix is straightforward to calculate.
Using this approach, I compute the annual growth rates of the variables of interest andshow comovements in Table 3.4; Table A.2 in Appendix A shows comparable numbers con-95

3.2. CAPITALỹc̃θ
1 0.303 6.552
1 0.858 0.973k̃
n wn/yc/y0.300 0.158 0.019 0.756
0.192 0.912 -0.931 -0.979

τ̂
0.921
0.971

s
0.900
0.994

c̃

—

1

0.722

0.668 0.712 -0.706 -0.734

0.720

0.804

θ

—

—

1

-0.033 0.889 -0.968 -0.997

1.000

0.992

k̃

—

—

—

1 0.067

0.019

0.014 -0.034

0.094

Correlations n

—

—

—

—

1

-0.751 -0.920

wn/y

—

—

—

—

—

1

0.948 -0.975 -0.961

c/y

—

—

—

—

—

—

1 -0.995 -0.992

τ̂

—

—

—

—

—

—

—

1

0.991

s

—

—

—

—

—

—

—

—

1

Relative s.d.
ỹ

0.876

0.897

Table 3.4: Model with capital, deterministic trend. Co-movements of annual growth rates in
an infinite sample.
structed from historical United States data. The results that I emphasized in Table 3.3 areunchanged. The correlation between the annual growth rate of the labor wedge and employment is 0.88, while the correlation with the consumption output ratio is −0.99. This isinconsistent with the data, as shown in the bottom panel of Table 1.1. Moreover, none of theother numbers change to an economically significant extent when moving from deviations
from trend to annual growth rates. For example, among the correlations, only the correlation of the capital stock with four other variables changes signs. In three of those cases,
the correlation is economically insignificant, smaller than 0.1 in absolute value under either
measure.
Impulse Response Functions
Another way to summarize the results is through impulse responses. Suppose that productivity, employment, and capital are all at their stochastic steady state values in period −1.
p
In period 0, productivity jumps up by one standard deviation, or ς/ 1 − ρ2 = 2.5 percent,before reverting slowly back to steady state, with an autocorrelation of 0.98. The solid blue
dots in Figure 3.1 show how the economy responds. Consumption jumps up upon the impactof the shock but soon starts to fall. Since workers would otherwise like to maintain smoothconsumption, the decline in consumption must be induced by a decline in the interest rate(a rise in qt (st+1 )). This encourages firms both to recruit workers and to invest in capital,
leading to the increase in employment n and capital k̃. But with higher consumption anda higher marginal product of labor, the wage increases (equation 3.43). This moderates the

96

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

incentive for firms to recruit more workers and eventually returns the economy to steadystate.
The figure also confirms that employment and hence unemployment is less volatile than
output. Following the shock, the unemployment rate falls rapidly from 5.0 to 4.6 percent
before reversing course, eventually rising slightly above its initial value. While this is an
improvement on the earlier search models in this book, the volatility of employment is still
modest.
One might expect that breaking the link between wages and the marginal product of labor
would help to generate fluctuations in employment, but in practice the labor share wn/y isnearly constant. In the frictionless economy, the labor share is constant at 1 − α = 0.6700.
Search frictions reduce it very slightly, to 0.6697 in the stochastic steady state. Figure 3.1
shows that, upon the impact of a positive productivity shock, the labor share falls by by 0.08
percent, to 0.6692, and then recovers rapidly. In this sense, wages are almost exactly equal
to the marginal product of labor even in the presence of search frictions.
Sensitivity
I consider the sensitivity of these results along two dimensions. First, one might be concernedthat I have taken too small a departure from the frictionless model. In steady state, only 0.4
percent of employed workers are engaged in recruiting, while the remainder are producers.
In fact, intuition suggests that more severe search frictions will only dampen fluctuations in
employment, and simulations bear that out. I recalibrate the model with more severe search
frictions, µ̄ = 1, so recruiters can only contact 1.5 new workers per period in the stochastic
steady state. I also change the value of leisure so as to leave the steady state employment rateunchanged. This modification raises the steady state share of employed workers in recruiting
to 2.2 percent, increasing the role of search frictions in labor market outcomes.
The solid red line in Figure 3.1 shows the resulting impulse response. More severe search
frictions moderate fluctuations in employment and the recruiter-unemployment ratio, butotherwise have little effect on outcomes. Notably, the critical pattern, that high productivity
raises employment and the measured labor wedge, is unchanged. Conversely, I find that whenrecruiting is more effective, so the model approaches the frictionless benchmark, employment
is more volatile.7 Search frictions alone do not help to explain the behavior of the measured
labor wedge.
Second, my choice of workers’ bargaining power, φ = 0.5, was arbitrary. Notably, in a
version of this model with linear utility and no capital, Hagedorn and Manovskii (2008) argue
7

The search model does not quite converge to the frictionless benchmark when µ̄ → ∞ because employment
is set a period in advance.

97

3.2. CAPITAL
1.2
bbbb

output ỹ

2.5

bb

b

bb

bb

bb

bb

bb

bb

bb

2.0

bb

bbb

bbb

bbb

bbb

bbb

bbb

1.5

bbb

bbb

bbb

bbb

bbbb

bbbb

bbbb

bbbb

1.0

bbbb

bbbbb

bbbbb

bbbbb

bbbbbbbbbbbbbbbbbbb

0.5

bbbbbbb

bbbbbbb

consumption c̃

3.0

0.9

20

40

60

80

100 120

0

bb

0.3

bb

40

bb

1.2
bb

bb

bb

bb

0.2

bb

bb

bb

bb

bb

bb

bb

bb

bbb

0.1
0

bbb

bbb

bbb

capital k̃

b

b

bbb

bbb

bbbb

bbbb

b

−0.1

bbbb

bbbbb

0.8
0.4

bbbbbbbbbbbbb

bbbbbbbbb

bbbbbbbbbbbb

0
0

20

40

60

80

100 120

b
b

b

b
b

b
b

b
b

b
b

b
b

bbbbbbbbbbbbbb

60

b
b

b
b

0

b
b

b
b

b
b

b

bb
b

bb

b
bb

b
bb

20

b
bb

bbbb

bbb

bbb

bb
bb

bbbbbb

bbbbb

bbbbb

bbbbb

40

b
b

b

15

b
b

b
b

bb

bb

bb

10

bb

bb

bb

bb

bb

bb

bb

bb

5

bb

bb

bb

bbb

bbb

bbb

bbb

bbb

bbb

bbbb

0
0

20

40

bbbb

bbbbb

bbbbb

bbbbbb60

bbbbbbbb

bbbbbbbbbb

80

bbbbbbbbbbbbbbbbb

bbbbbbb

−0.02
−0.04

bbbbb

bbbbb

bbbb

bbbb

bbbb

bbbb

60

labor wedge τ̂

b
b

b
b

b

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bb

bbb

bbb

bbb

bbb

bbb

bbb

bbbb

bbbb

bbbbb

bbbbb

bbbbbbbbbbbbb

bbbbbbbbbb

bbbbbbbbbbbbbbb

bbb

bbb

bbbb

bbbb

bbbb

bbbbbb80
bbbbbbb

bbbbbbbb

bbbbb

bbbbb

bbbbb

bbbb

bbbb

bbb

100 120

bbbbbbbbbb

bbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbb

bbb

b

−0.08

2.5
b

bb

bbbb

bbbb
bbbbb

bbbbbbb−0.10

100 120

bb

bb

bbb

100 120

−0.06

b
b

b

bb

b
bb

bb
bbb

80

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

bbbbb

0
b

labor share wn/yrecruiting/unemployment θbproductivity s

employment n

bbbb

0.4

3.0
2.5
2.0
1.5
1.0
0.5
0
−0.5

20

1.6

0.5

−5

bbbbbbb

0
0bb

bbbbbbbbbbbbbbbbbbbbbbbb

bbbbbbbb

0.3

0

20

b

b

0.6

b
bb

bb

b
bb

bb

b

b
bb

bbbbb

bbbb

bbb

bbb

b
bb

b
b

b

2.0
1.5

b
b

b
b

0
bb

bb

20

bb

bb

bb

bb

bb

bb

bb

1.0

bb

bb

bb

bb

40

bb

bb

bb

bb

bb

bbb

0.5

bbb

bbb

60

bbb

bbb

bbb

bbbb

bbbb

bbbb

80

bbbb

bbbbb

bbbbb

bbbbbbbbbbb

bbbbbb100 120

bbbbbbbbbbbbb

bbbbbbbb

bbbbbbb

0
0

20

40

60 80
month

100 120

0

20

40

60 80
month

100 120

Figure 3.1: Model with capital, deterministic trend. Response to a 2.5 percent increase inproductivity at t = 0. All variables are expressed as deviation from trend in log points. The
blue dots show µ̄ = 2.32 and the red lines show more frictions, µ̄ = 1.

98

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

for a lower bargaining power and a higher value of leisure. They find that this significantly
amplifies employment fluctuations. I confirm that their results carry over to this setting. At
one extreme, suppose we insist that the value of leisure is zero, γ = 0, so unemployment istruly involuntary, caused only be the search frictions. Fixing all the other parameter, theeconomy has five percent unemployment if workers’ bargaining power is very high, φ = 0.924.
With these parameters, the unconditional standard deviation of employment falls from 0.004
to 0.00002, so the model generates virtually no unemployment fluctuations.
At the other extreme, reduce workers’ bargaining power to φ = 0.05 and raise the valueof leisure to γ = 0.511, so as to maintain a five percent unemployment rate in the stochastic
steady state. For comparison, recall that in a frictionless economy, a value of leisure ofγ̂ = 0.513 results in the same nonemployment rate, while still higher values of leisure areinconsistent with five percent unemployment. Thus this is nearly the highest possible valuefor γ and the lowest possible value for φ. As expected, these changes raise the unconditional
standard deviation of employment by a factor of five, to 0.020. Moreover, they raise therelative volatility of employment from 11 percent of output to 45 percent. While theseoutcomes are promising, the behavior of the labor wedge is counterfactual. The correlation
between the labor wedge and employment is only slightly reduced, to 0.87 in levels and 0.68
in growth rates, far from the negative correlations that we observe in the data. Similarly, thetheoretical correlation between the labor wedge and the consumption-output ratio remains
strongly negative, −0.92 in levels and −0.81 in growth rates. It thus seems that the choiceof workers’ bargaining power does not drive my main finding, the counterfactual behavior of
the comovement of the labor wedge with employment and the consumption-output ratio.

3.2.7

Productivity Shocks: Stochastic Trend

I turn next to the possibility that there are shocks to the trend of productivity, rather
than transitory fluctuations around the trend. Assume log z(st+1 ) = log z(st ) + st+1 wherest+1 = {st , st+1 } and st follows a stationary first order Markov process. Let π(st+1 |st ) again
denote the probability of state st+1 next period conditional on state st this period. While
this change in the nature of the productivity shock alters many properties of the model, I
show in this section that it does not substantially affect the comovement of the labor wedge
with employment and the consumption-output ratio.
Again inspired by the balanced growth path, I look for an equilibrium where appropriately scaled versions of consumption, capital, and wages are stationary. Define rela−1−1tive consumption, capital, and wages as c̃(st ) ≡ c(st )z(st ) 1−α , k̃(st ) ≡ k(st )z(st ) 1−α , and−1w̃(st ) ≡ w(st )z(st ) 1−α . Similarly, since employment, the share of recruiters in employment,
and the ratio of recruiters to unemployed are all constant along a balanced growth path, I99

3.2. CAPITAL
look for an equilibrium in which these variables are stationary.

To find a set of equilibrium conditions relating stationary variables, I eliminate the intertemporal price q using equation (3.31) and the share of recruiters ν using equation (3.47)
from the remaining equilibrium conditions. Then the condition for an interior solution for
capital, equation (3.39), reduces to1=βX

st+1 |stt

π(st+1 |st )c̃(s )ec̃(st+1 )

st+1
− 1−α



α

k̃(s

t+1

)

n(st+1 ) − θ(st+1 )(1 − n(st+1 ))

!α−1



+ 1 − δ .

(3.63)

Eliminating Jn (st+1 , n(st+1 ), k(st+1 )) from the condition for an interior solution for recruiting,equation (3.37), using the envelope equation (3.38) for employment, evaluated in history st+1 ,yields(1 − α)k̃(st )n(st ) − θ(st )(1 − n(st ))×(1 − α)!α= βµ(θ(st ))X

st+1 |stk̃(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))π(st+1 |st )c̃(st )c̃(st+1 )

!α 

1−x
1+
µ(θ(st+1 ))!

− w̃(st+1 ) .

The wage equation (3.43) evaluated in history st+1 reduces to(1 − τ )w̃(st+1 ) = (1 − φ)γc̃(st+1 )+ φ(1 − τ )(1 − α)

k̃(s

t+1

)

n(st+1 ) − θ(st+1 )(1 − n(st+1 ))!α(1 + θ(st+1 )).

Eliminating w̃(st+1 ) between the last two equations gives(1 − α)k̃(st )n(st ) − θ(st )(1 − n(st ))βµ(θ(st ))X

st+1 |st+ (1 − α)!α=π(st+1 |st )k̃(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))c̃(st )c̃(st+1 )!α −(1 − φ)γc̃(st+1 )1−τ

1−x+ 1 − φ − φθ(st+1 )µ(θ(st+1 ))!

. (3.64)

Finally, the resource constraint equation (3.45) becomesst+1k̃(st+1 )e 1−α = k̃(st )α n(st ) − θ(st )(1 − n(st ))1−α+ (1 − δ)k̃(st ) − c̃(st ).

(3.65)

100

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

Once again, equation (3.46) is unchanged.
As in the model with a deterministic trend, I look for a solution to equations (3.46)
and (3.63)–(3.65) with the property that the policy variables θ(st ) and c̃(st ) depend on theexogenous state st , on employment n, and on the capital stock relative to trend k̃. Using theMarkov property of the state variable and the state equations (3.46) and (3.65), this implies
st+1 , n(st+1 ), and k̃(st+1 ) depend on the same objects.
Again let the functions Θ and C denote the equilibrium recruiter-unemployment ratio
and equilibrium consumption relative to trend as functions of the current state (s, n, k̃), soθ(st ) = Θ(st , n(st ), k̃(st )) and c̃(st ) = C(st , n(st ), k̃(st )). After substituting these functions
into equations (3.63) and (3.64), eliminate k̃(st+1 ) using equation (3.65) and n(st+1 ) using
equation (3.46) to obtain a pair of nonlinear equations that implicitly define Θ and C.
I solve the model by calibrating it and log-linearizing around a stochastic steady state. I
leave most of the calibration unchanged from the model with a deterministic trend (Table 3.2),
except the stochastic process for productivity, which I borrow from my earlier analysis of astochastic trend in the economy where consumption and leisure are substitutes. Thus I
assume that productivity growth follows a linear process, st+1 = s̄ + ρ(st − s̄) + ςυt+1 . I
set average monthly productivity growth rate to s̄ = 0.0012, the autocorrelation of monthly
productivity growth to ρ = 0.4, and the standard deviation of monthly productivity growthto ς = 0.00325. These values are consistent with the observed annual behavior of multifactor
productivity in the private business sector in the United States between 1948 and 2007.
I then solve for a log-linear approximation to the equilibrium policy functions, findinglog θ = log 0.078 + 1.548(s − 0.0012) − 0.480(log n − log 0.95) − 2.779(log k̃ − log 218.2),
log c̃ = log 4.696 + 0.381(s − 0.0012) + 0.014(log n − log 0.95) + 0.603(log k̃ − log 218.2).
An increase in the productivity growth raises the recruiter-unemployment ratio θ and increases consumption c̃. Future productivity is expected to be much higher than current
productivity and households take advantage of that by increasing consumption immediately.
This puts upward pressure on interest rates (downward pressure on q), discouraging investment both in capital and recruiting. Still, because the shock is sufficiently transitory, theinterest rate response is modest and recruiting actually rises.8
Using the equilibrium policy functions, I then find the log-linear approximation to the8Note that the responses of the recruiter-unemployment ratio and consumption to employment and to
detrended capital are identical in the model with stochastic and deterministic trends, at least to the sixth
decimal point.

101

3.2. CAPITAL

Relative s.d.
ỹ

ỹ
c̃
θ
1 2.092 9.636
1 0.995 -0.997

k̃n3.478 0.240
1.000 -0.999

wn/y
0.128
0.013

c/y
τ̂
s
1.102 1.298 0.534
0.981 -0.971 -0.067
0.996 -0.990c̃ —1 -0.983

0.995 -0.996

0.115

θ

—

—

1

-0.996

0.068 -0.962

k̃

—

—

—

1 -0.999

Correlations n

—

—

—

—

1 -0.034 -0.984

wn/y

—

—

—

—

—

c/y

—

—

—

—

τ̂

—

—

—

s —

—

—

0.994

0.018

0.948

0.036
0.147

0.982 -0.972 -0.062
0.975

0.046

1

0.207 -0.253

0.997

—

—

1 -0.999

0.128

—

—

—

—

1 -0.175

—

—

—

—

—

1

Table 3.5: Model with capital, stochastic trend. Co-movements of variables in an infinite
sample.
state equations:
log n+1 = log 0.95 + 0.026(s − 0.0012) + 0.312(log n − log 0.95) − 0.047(log k̃ − log 218.2),
log k̃+1 = log 218.2 − 0.605(s − 0.0012) + 0.019(log n − log 0.95) + 0.991(log k̃ − log 218.2).
This confirms that higher productivity growth raises employment but reduces the detrended
capital stock by encouraging current consumption.
I next stack the state variables as m ≡ {s − s̄, log n − log n̄, log k̃ − log k̄}. I write

the transition equation as m+1 = Am + Dυ+1 and compute the eigenvalues of the transition
matrix A. The smallest and largest are unchanged at 0.99 and 0.31, corresponding to the slow
adjustment of capital and the fast adjustment of employment. The intermediate eigenvalue,
0.4, reflects the modest autocorrelation of shocks.
Finally, I construct other detrended variables as linear combinations of state variables,m̃ = Ãm, and compute the variance-covariance matrix E(m̃m̃′ ) = ÃΣÃ′ , where Σ = E(mm′ )solves Σ = AΣA′ + DD ′. Table 3.5 shows the relative standard deviations and the correlation
matrix for the same set of variables as was in Table 3.3.
Some results appear quite different and are superficially encouraging. For example,−1
the volatility of employment, n(st ), relative to detrended output, ỹ(st ) = y(st)z(st ) 1−α =1−αk̃(st )α n(st ) − θ(st )(1 − n(st )), more than doubles from 0.11 to 0.24. On the other hand,the stochastic trend reverses the correlation between these variables, so employment is counterfactually high when output is low relative to productivity. This is because an increase inproductivity growth leads to a jump up in consumption, which is partially accommodated

102

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

Relative s.d.
ỹ

ỹ
c̃
θ
1 2.243 9.577
1 0.956 -0.969

k̃n
3.465 0.235
1.000 -0.988

wn/y
0.394
0.120

c/y
τ̂
1.319 1.668
0.867 -0.819

s
1.638
0.093

0.975 -0.951

0.380

c̃ —

1 -0.854

0.961 -0.968

0.405

θ

—

—

1

-0.9650.130 -0.718

k̃

—

—

—

1 -0.992

Correlations n

—

—

—

—

1 -0.189 -0.897

wn/y

—

—

—

—

—

1

0.597 -0.668

1.000

c/y

—

—

—

—

—

—

1 -0.996

0.575

τ̂

—

—

—

—

—

—

—

1 -0.647

s —

—

—

—

—

—

—

—

0.942

0.136

0.651

0.157

0.876 -0.828

0.109

0.851 -0.163

1

Table 3.6: Model with capital, stochastic trend. Co-movements of annual growth rates in an
infinite sample.
by moving workers from recruiting to production. In addition, recall that I measure output
relative to the level of productivity, i.e. ỹ, not y. I find that when productivity growth increases, output rises but by less than productivity growth. Similar forces drive the negative
correlation between employment and capital relative to productivity.
Other results are more robust across models, especially the comovement of the variables Iam most interested in, employment, the consumption-output ratio, and the measured laborwedge. The correlation between the labor wedge and employment rises from 0.96 to 0.97,
while the correlation between the labor wedge and the consumption output ratio remains
at −1.00. As I have noted before, both predictions are counterfactual. Still, from theperspective of understanding the predictions of the model, it is reassuring that this finding
does not appear to depend on the specification of the stochastic process for productivity.

Table 3.6 computes the same statistics for the annual growth rates of various variables.
Computing growth rates rather than levels has a quantitatively significant impact on manyof the results. Still, the correlation of the labor wedge with employment remains strongly
positive, 0.88, and the correlation of the labor wedge with the consumption-output ratio
strongly negative, −1.00. Again these are inconsistent with the data in the bottom panel ofTable 1.1.
Figure 3.2 shows the impulse response to a one standard-deviation increase in the productivity growth rate s from s̄ = 0.0012 to s̄ + √ ς 2 = 0.0047 at t = 0. The cumulative1−ρ

impact of this shock is to raise the level of productivity by about 0.6 percent. Since thecapital share is α = 0.33, this eventually raises output, consumption, and the capital stock3.2. CAPITAL

103

by about 1/(1 − α) times as much, 0.9 percent.

One advantage to looking at the impulse responses is that I can easily plot the behaviorof nonstationary variables. Indeed, Figure 3.2 shows the behavior of output y, rather thandetrended output ỹ, and similarly for the other nonstationary variables, consumption andcapital. I find that consumption jumps upon the impact of the shock, but by less than theincrease in output. During the subsequent adjustment, the consumption-output ratio falls
unusually low before returning to trend. During this intervening period, firms invest heavily
in physical capital, propagating the initial shock.
Following the shock, employment rises modestly for five months, peaking about 0.06 percent above the initial level. In then gradually reverts to trend. This mirrors the patternof the consumption-output ratio, driving the strong negative correlation between these twovariables. Although the labor share responds slightly more in this model than with a deterministic trend, the changes are still tiny. Upon the impact of the shock, it jumps up from0.6697 to 0.6702 before returning gradually to its original value.
The bottom left panel in Figure 3.2 shows the behavior of the labor wedge and confirmsthat it follows virtually the same pattern as employment. While the magnitude of fluctuations in the labor wedge are significantly greater than the magnitude of the fluctuations in
employment, the comovement of the two variables is the opposite of what is in the data.
Once again, the intuition that search frictions act like a labor adjustment cost but otherwise have little effect on the model is borne out. The solid red lines in Figure 3.2 show what
happens when search frictions are more severe, µ̄ = 1. This mainly muffles the response of
employment and the recruiter-unemployment ratio and slightly amplifies the impact on thelabor wedge.
Similarly, other calibrations of the model, e.g. with workers’ bargaining power at φ = 0.05
and the value of leisure at γ = 0.511, also have little impact on the key result. While thissubstantially increases the volatility of employment, until it is 3.6 times as volatile as output(3.5 times in growth rates), it does not reverse the positive correlation between the laborwedge and employment (0.68 in levels, 0.45 in growth rates), nor the negative correlation
between the labor wedge and the consumption-output ratio (−0.80 in levels, −0.78 in growth
rates). Extreme calibrations of the model do not seem to overturn this relationship.

3.2.8

Summary

In a search model with capital, productivity shocks affect employment, much as they doin the frictionless real business cycle model. But while search frictions provide a notion ofunemployment that is absent from the frictionless model, they moderate rather than amplify
fluctuations in employment by acting as a labor adjustment cost. The positive comovement

104

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS
1.0

output y

0.8
0.6

bbbbbbbbbbb
bb

bbbbbbbbbbb

bbbbbbbbbb

bbbbbbbb
bbbbbbbb

bbbbbbbbbbbb

bbbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbb

bbbbb

bbbbbbbbbbbbbbbbbbbb

b

0.4
b

0.2

consumption c

1.0

0
20

40

60

80

bbbb

bbbb

bbbb

bbbb

bbbb

bbb

bbb
bbbbb

bbbbb

bbbbbb

bbbbb

bbbbb

bbb

bbbbbbbbbbbbbb

bbbbbbbb

bbbbbbbbb

bbb

bbbbbbbbbbb

bbbbbbbbb

0.2
0

20

40

60

80

100 120

1.0
bbbbbbb

0.8
bbb

bbb

b

0.04

bbb

bbb

bbb

bbb

bbbb

bbbb

bbbb

bbbb

bbbb

b

bbbb

bbbb

bbbbb

bbbbb

bbbbb

bbbbb

bbbbbbbbbbbb0.02
0

bbbbbbbbbbbbb

bbbbbbb

capital k

0.06

bbbbbbb

bbbbbb

0.6
0.4
0.2

0

20

40

60

80

100 120

bbb

bbbb

0
b

bb
bbb

bb

0

bbb

bbb

bbb

bbb

bb

bb
bbb

20

bbb

bbb

bbb

bbb

bbbb

b

bbb
bbbb

40

bbbb

bbbb

bbbb

bbbb

60

bbbbb

bb

bbbb
bbbbb

bbbbb

80

bbbbbbbbbbbbbbbbbbbbbbbbbbb

100 120

0.12
bbbbbb
b

2

bbb

b

bbb

bbb

bbb

bbb

bbb

bbb

bbb

bbb

bbb

bbbb

bbbb

bbbb

bbbb

bbbb

bbbb

bbbbb

bbbbb

1

bbbbb

bbbbb

bbbbb

bbbbbbbbbbbbbbbbbbbbbbbbb

bbbbbbb

labor share wn/y

3

bb

0

20

40

60

80

100 120

0.4
0.3

b
b

0.2

bbbbb

bbb

bbb

bbb

bbb

bbb

bbb

b

bbb

bbb

bbb

bbb

bbb

bbbb

bbbb

bbbb

bbbb

bbbb

bbbb

bbbb

bbbbb

bbbbb

bbbbb

0.1
0

b

0

20

40

60 80
month

bbbbb

bbbbb

bbbbbbbbbbbbbbbbbbbbbbbbbb

0.08

100 120

b

0.04
b

b
b

0
−0.04

0

productivity growth s

employment n

0.4

100 120

0.08

recruiting/unemployment θ

0.6

bbbbb
bbbbbbb00labor wedge τ̂

0.8

0bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

20

40

60

80

100 120

0.4
b

0.3
0.2
b

0.1
b
b

0
0

bb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

20

40

60 80
month

100 120

Figure 3.2: Model with capital, stochastic trend. Response to an increase in the productivity
growth rate at t = 0. All variables are expressed as deviation from initial trend in log points,
except the productivity growth rate, which is measure in percentage points. The blue dotsshow µ̄ = 2.32 and the red lines show more frictions, µ̄ = 1.

3.3. SHOCKS TO THE EMPLOYMENT EXIT PROBABILITY

105

between employment and the labor wedge appears to be a robust measure of the model’s
failure along this dimension. Since the frictionless model already has trouble explaining why
hours worked are so volatile and predicts no movement in the labor wedge, introducing search
frictions is at this point a step in the wrong direction.

3.3

Shocks to the Employment Exit Probability

3.3.1

Other Shocks

The search model with capital can easily be extended to allow for other shocks, many ofwhich are important in practice. For example, in reality, there are shocks both to the growthrate and to the level of productivity. In reality, the government consumes real resources andthe amount of that spending fluctuates over time.9 In reality, the technology for producing
capital goods has improved more rapidly than the technology for producing consumption
goods during the postwar period, resulting in a systematic decline in the price of capital
equipment (Krusell, Ohanian, Rı́os-Rull, and Violante, 2000), but there are fluctuations in
the relative improvements in these two technologies, i.e. investment-specific technology shocks(Fisher, 2006).
Since the methodology for introducing these additional shocks should by now be straightforward, I do not formally develop the models in this book. Still, the effects of these typesof shocks seem predictable. Without search frictions, each would generate some fluctuations
in hours worked and the consumption-output ratio, but none would break the equivalence
between the marginal product of labor in the consumption goods sector and the tax-adjusted
marginal rate of substitution between consumption and leisure. In other words, the laborwedge would be constant. Search frictions then dampen fluctuations in employment or hours
worked and so tend counterfactually to cause a positive correlation between the labor wedge
and employment and a negative correlation between the labor wedge and the consumptionoutput ratio.

3.3.2

Model

Shocks to the employment exit probability x deserve more serious consideration. Such shocks
have a direct impact on the employment and unemployment rates by affecting the flow of
workers from employment to unemployment. Although these shocks do not play any role in amodel without labor market frictions, since they can be costlessly reversed, many economists
believe that they are an important contributor to aggregate employment fluctuations. For
9

For a recent attempt to identify government spending shocks, see Blanchard and Perotti (2002).

106

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

example, the research by Davis, Haltiwanger, and Schuh (1996) showed that the job destruction rate, measured as the net decline in employment at plants where employment is
contracting, is more volatile than the job creation rate, the net increase in employment at
plants where employment is expanding, at least in the United States manufacturing sector.
Given the close theoretical link between the job destruction rate and the employment exit
probability—the fraction of workers who are employed one month but not the next—many
economists have presumed that shocks to x, or endogenous fluctuations in x, must be criticalin a search model of the business cycle. More recent research using worker flow data has
confirmed that variation in the employment exit probability accounts for one-quarter to onehalf of the overall fluctuations in unemployment, while fluctuations in the job finding ratef (θ(st )) account for the remaining portion.10
There are two basic approaches to introducing fluctuations in the employment exit probability into this model. The first is to generate the fluctuations endogenously. In Mortensen
and Pissarides (1994), employed workers are subject to idiosyncratic productivity shocks andquit their job when the idiosyncratic shock is too bad. Shocks to aggregate productivity
change the endogenous threshold for exiting employment, a potentially important amplification mechanism. The second, simpler, approach treats the employment exit probability as
an exogenous shock, possibly correlated with productivity, say x = x(st ).
I follow the second approach here, allowing for correlated shocks to the deviation of
productivity from trend and the deviation of the employment exit probability from its normal
value.11 Formally, it is useful to think of the state st as a vector, st = {sz,t , sx,t }, where sz,tdetermines the current deviation of productivity from trend, log z(st ) = s̄t + sz,t, and sx,tdetermines the deviation of the employment exit probability from its normal value, log x(st ) =log x̄ + sx,t . The innovations to the two state variables and their subsequent evolution may
be correlated.
This modification has little effect on the notion of equilibrium or the equations describings̄tit. Again define consumption relative to trend, c̃(st ) ≡ c(st )e− 1−α , capital relative to trend,
s̄t

s̄t

k̃(st ) ≡ k(st )e− 1−α , and wages relative to trend, w̃(st ) ≡ w(st )e− 1−α . I look for an equilibrium
where c̃(st ), k̃(st ), and w̃(st ) are stationary, as are employment n(st ), the share of recruitersin employment ν(st ), and the recruiter-unemployment ratio θ(st ).
Following the same logic as in Section 3.2, an equilibrium is described by equations (3.53)
and (3.56) and a suitably modified version of equations (3.46) and (3.57). First, the employ10

The lower bound on the role of the employment exit probability comes from Shimer (2007). The upper bound is from Fujita and Ramey (forthcoming), while Elsby, Michaels, and Solon (2007) provide anintermediate estimate.
11
One can also solve the model with a stochastic trend in productivity.

3.3. SHOCKS TO THE EMPLOYMENT EXIT PROBABILITY

107

ment rate evolves asn(st+1 ) = (1 − x(st ))n(st ) + f (θ(st ))(1 − n(st )).

(3.66)

Second, the combination of the firm’s first order and envelope conditions and the wage
equation reduce tosz,t(1 − α)ek̃(st )n(st ) − θ(st )(1 − n(st ))c̃(st )
(1 − φ)γc̃(st+1 )−c̃(st+1 )1−τ
st+1 |st!α !
1 − x(st+1 )k̃(st+1 )+ 1 − φ − φθ(st+1 ).
n(st+1 ) − θ(st+1 )(1 − n(st+1 ))µ(θ(st+1 ))= βµ(θ(st ))+ (1 − α)esz,t+1!αXπ(st+1 |st )(3.67)

These last two equations replace the previously-constant employment exit probability with
its history-dependent counterpart.

3.3.3

No Productivity Shocks

As usual, I solve the model by log-linearizing around a stochastic steady state. To illustrate
the effect of shocks to the employment exit probability x(st ), I first calibrate the model with
only that shock. That is, I assume productivity grows deterministically, log z(st ) = s̄t andso sz,t = 0 for all t. On the other hand, the deviation of the logarithm of the employment
exit probability from its average value, sx,t = log x(st ) − log x̄, follows a linear process,sx,t+1 = ρx sx,t + ςx υx,t+1 ,(3.68)

where υx,t+1 is a white noise innovation with mean 0 and standard deviation 1. I look for alog-linear approximation to the policy functions,log θ = log θ̄ + θx sx + θn (log n − log n̄) + θk (log k̃ − log k̄),log c̃ = log c̄ + cx sx + cn (log n − log n̄) + ck (log k̃ − log k̄).
Here θx is the elasticity of the recruiter-unemployment ratio with respect to the deviation of
the employment exit probability from trend and cx is the analogous elasticity of consumption.
Most of the calibration of the model is unchanged, but I need values for two new pa-

108

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

rameters, the autocorrelation and standard deviation of the shock to the employment exit
probability, ρx and ςx . I calibrate these using data on the monthly employment exit probability; see Shimer (2007) for details on the construction of this time series and the associatedseries for the monthly job finding probability.12 After detrending using a Hodrick-Prescott filter with smooth parameter 1600, I find that the quarterly autocorrelation of the employment
exit probability is 0.56, and so I calibrate the monthly autocorrelation to ρx = 0.83 ≈ 0.561/3 .

I set the standard deviation to ςx = 0.034 so as to match the unconditional standard deviationof the detrended employment exit probability, √ ςx 2 = 0.061. The remaining parameters1−ρx

are fixed at their levels in Table 3.2, except that I shut down the productivity shocks.
When I simulate the model, I find thatlog θ = log 0.078 + 0.098sx − 0.480(log n − log 0.95) − 2.779(log k̃ − log 218.2),
log c̃ = log 4.696 − 0.001sx + 0.014(log n − log 0.95) + 0.603(log k̃ − log 218.2).
Notably, an increase in the employment exit probability raises the recruiter-unemployment
ratio. One might intuitively have expected the opposite response. When the employment
exit probability increases, firms do not expect jobs to last as long. This reduces the value ofjobs and encourages firms to use their current workers for production rather than recruiting.
On the other hand, if firms do not increase their recruiting effort, employment and hence
production will be unusually low in subsequent periods. This reduces the interest rate andgives firms an incentive to recruit more workers. Moreover, wages fall in response to workers’
worsening situation, which raises the value of a job and further encourages recruiting. In
equilibrium, these last two forces dominate the first.
This has two important implications. First, part of an increase in the employment exit
probability is offset by an endogenous increase in the job finding probability f (θ(st )), reducing
the impact on employment. In practice, this effect is small, however. I find that the standard
deviation of the log job finding probability is only 0.07 times the standard deviation of
the log employment exit probability. In the data, the job finding probability is 1.4 times
as volatile. Second, the employment exit probability is positively correlated with the job
finding probability. In the model, the correlation of the logarithm of two variables is 0.91
and the correlation of the annual growth rate of the two variables is 0.98. In the data, thecorrelation between the two series is −0.43 when detrended and −0.46 when measured as
annual growth rates. The model with only a shock to the employment exit probability fails
along both dimensions.
12

The data are available from http://robert.shimer.googlepages.com/flows. I use quarterly averagesof the underlying monthly data because the monthly data are contaminated by measurement error.

3.3. SHOCKS TO THE EMPLOYMENT EXIT PROBABILITY

3.3.4

109

Two Shock ModelI next extend the model in two dimensions: I allow the shock to the employment exit probability also to affect productivity; and I introduce a second shock that only affects productivity.
I leave the stochastic process for deviation of the employment exit probability from its average value, equation (3.68), unchanged. In addition, I assume that the deviation of log
productivity from trend, sz,t = log z(st ) − s̄t, follows a linear process,sz,t+1 = ρz sz,t + ςz υz,t+1 − ςzx υx,t+1 ,(3.69)

where υz,t+1 is a second white-noise innovation with mean 0 and standard deviation 1, independent from υx,t+1 as well as from all leads and lags of that shock.
It may be useful to think of υx as a reallocation shock that affects both the employment
exit probability and productivity, in opposite directions, while υz is an aggregate shockthat only has an impact on productivity. This approach seems broadly consistent with
the idea that some shocks reduce the productivity in all firms, while others reduce average
productivity but hurt some firms more than others, leading to an increase in the exit ratefrom employment (Blanchard and Diamond, 1989). Still, labeling one shock “reallocation”
and the other “aggregate” is arbitrary, as is the introducing correlation between sx and sz
through the direct impact of the reallocation shock on productivity.
I leave the stochastic process for the employment exit probability unchanged, ρx = 0.83
and ςx = 0.034. I also keep the autocorrelation of productivity fixed at its value in Section 3.2.6, ρz = 0.98. I then use two targets to set the remaining parameters, the weights
on the innovations ςz and ςzx . First, the monthly standard deviation of the deviation ofp
2 ; I keep that fixed at approximately 0.005.
productivity from trend is now equal to ςz2 + ςzx
Second, I find that these two parameters are critical for the correlation between the job
finding and exit probabilities, −0.43 in the detrended data.13 If ςz is large relative to ςzx ,the job finding and employment exit probabilities may be positively correlated. When ςzx isrelatively large, the two probabilities are too strongly negatively correlated. I set ςz = 0.0037
and ςzx = 0.0034 to hit these two targets.
With this calibration, the policy functions becomelog θ = log 0.078 + 7.387sz + 0.098sx − 0.480(log n − log 0.95) − 2.779(log k̃ − log 218.2),
log c̃ = log 4.696 + 0.250sz − 0.001sx + 0.014(log n − log 0.95) + 0.603(log k̃ − log 218.2).
13

In the model, I continue to assume f (θ) = µ̄θη and so log f and log θ are perfectly correlated. Thus thecorrelation between the employment exit probability and the recruiter-unemployment ratio is also −0.43.

110

CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

An increase in reallocation (υx,t+1 > 0) raises sx and reduces sz . In this calibration, the latter
effect dominates and so the recruiter-unemployment ratio and consumption both fall. An
adverse aggregate shock reduces sz and so has the same effect on the two policy variables.
Turning to the state equations, I findlog n+1 = log 0.95 + 0.126sz − 0.032sx + 0.312(log n − log 0.95) − 0.047(log k̃ − log 218.2),
log k̃+1 = log 218.2 + 0.020sz + 0.000sx + 0.019(log n − log 0.95) + 0.991(log k̃ − log 218.2).
An increase in reallocation reduces next period’s employment and slightly increases next
period’s capital stock.14 An adverse aggregate shock has the same effect.
As usual, I can express the state of the economy as m ≡ {sz , sx , log n−log n̄, log k̃ −log k̄}and represent the transitional dynamics as m+1 = Am + Dυ+1 , where A is a 4 × 4 transition
matrix and so Am is the expected value of m+1 conditional on m and D determines how theshock υ+1 = {υz,+1, υx,+1} affects the state of the system. For this choice of parameters,


0.980

0

0

0






0
0.830
0
0 


A=

 0.126 −0.032 0.312 −0.047 
0.020
0.000 0.019
0.991
and



ςz −ςzx


 0
D=
 0

0




ςx 
.
0 

0

The eigenvalues of A are effectively unchanged and remain between 0 and 1. Two of them
represent the persistence of the shocks, 0.98 for productivity and 0.83 for the employment
exit probability. The other two eigenvalues represent the slow adjustment of the capital stock
to steady state, 0.99, and the comparatively fast adjustment of employment to steady state,
0.33.
The next step is to compute the unconditional variance-covariance matrix of the statevariables, Σ ≡ AΣA′ + DD ′ , and of any associated variables m̃ ≡ Ãm. I confirm thatthe unconditional standard deviation of the employment exit probability is 0.061 and thecorrelation between the employment exit probability and the job finding probability is −0.43.
Both numbers are consistent with the detrended data. Moreover, the model predicts that thejob finding probability is slightly more volatile than the employment exit probability, with a
14

The coefficient on sx is a negligible 0.00002.

111

3.3. SHOCKS TO THE EMPLOYMENT EXIT PROBABILITY

Relative s.d.
ỹ

ỹ
1
1

c̃
θ
k̃
n wn/yc/y0.711 4.152 0.984 0.156 0.013 0.525
0.865 0.714 0.742 0.674 -0.826 -0.733

τ̂
0.578
0.726

sz
sx
0.71 1.713
0.974 -0.365

c̃

—

1

0.272 0.978 0.266 -0.519 -0.293

0.292

0.734 -0.136

θ

—

—

1 0.066 0.869 -0.877 -0.991

0.999

0.853 -0.432

k̃

—

—

—

1 0.084 -0.347 -0.089

0.087

0.576 -0.036

Correlations n

—

—

—

—

1 -0.718 -0.925

0.855

0.767 -0.644

wn/y

—

—

—

—

—

1

c/y

—

—

—

—

—

—

1

τ̂

—

—

—

—

—

—

—

1

0.863 -0.437

sz

—

—

—

—

—

—

—

—

1 -0.402

sx

—

—

—

—

—

—

—

—

—

0.871 -0.895 -0.905-0.988 -0.861

0.591
0.512

Table 3.7: Model with shocks to productivity and to the employment exit probability, deterministic trend. Co-movements of variables in an infinite sample.
standard deviation of 0.074. In the detrended data, the standard deviation of the job finding
probability is 0.083. Since I did not target this parameter, this may be viewed as a successof the model.
I turn next to the usual matrix of relative standard deviations and correlations in Table 3.7. The results are similar to those in Table 3.3, the results from the same model without
the shock to the employment exit probability. Shocks to the employment exit probability
raise the relative standard deviation of employment from 11 percent of the standard deviationof output to 16 percent. They also slightly reduce the correlation between employment andall other variables. None of the other numbers in the table change appreciably. In particular,
the correlation between the labor wedge and employment remains strongly positive (0.86
instead of 0.96) and the correlation between the labor wedge and the consumption output
falls slightly from −1.00 to −0.99. Although I do not show the behavior of annual growth
rates, the changes from Table 3.4 are similarly modest.
The bottom line is that, although fluctuations in the employment exit probability may

explain some of the movements in employment, they do not explain the cyclical behavior of
the labor wedge. To get some intuition for this finding, think back again to a frictionlessmodel. Suppose, for some reason, there was a shock that forced some household membersout of their job. The household would immediately respond by sending other members
to work, leaving employment and the labor wedge unchanged. With search frictions, thehousehold cannot do this instantaneously, but shocks to the employment exit probability

1

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CHAPTER 3. REAL EFFECTS OF PRODUCTIVITY SHOCKS

still act mainly to increase labor turnover and have little effect on the behavior of the laborwedge. The conclusion that search frictions act like a procyclical tax on labor is unchanged.
I need to look beyond this model for an explanation of the behavior of the labor wedge.

Chapter 4
Rigid Wages
The search models in Chapters 2 and 3 make two significant departures from the frictionlessmodel in Chapter 1. The first is of course the introduction of search frictions. In the
search model, firms use some of their employees to recruit new workers, and the difficulty ofrecruiting depends on the ratio of recruiters to unemployed workers. The second departureis the determination of wages. Although perhaps more subtle, this assumption is no lessimportant. In the presence of search frictions, matched workers and firms are in a bilateralmonopoly situation. Standard arguments that competitive forces equate the wage both to themarginal rate of substitution between consumption and leisure and to the marginal productof labor are inapplicable. I have instead assumed that the wage is set by Nash bargaining andwas able to express it as a weighted average of the MRS and MPL, e.g. in equation (3.43).
Starting with Shimer (2005) and Hall (2005), a recent body of research has assessed theimportance the Nash bargaining assumption by considering the role of wage rigidities in searchmodels. In some sense, one can think of this research as a modern effort to capture certain
aspects of disequilibrium macroeconomic models (Barro and Grossman, 1971; Malinvaud,
1977; Benassy, 1982). In particular, a search model with rigid wages provides a way ofanalyzing “classical unemployment,” i.e. unemployment caused by high wages.
It is not easy to model wages that are too high in a competitive framework. Loosely
speaking, the equilibrium of a competitive business cycle model like that in Chapter 1 is
described by κ equations in κ unknown variables, one of which is the wage. Since the wage
is an equilibrium object, it does not make sense within the context of the model to ask what
happens if it is too high. That is, we know that if the wage is not equal to its equilibrium
level, one (or more) of the equations describing the equilibrium must be dropped to ensure
a solution to the remaining system of κ − 1 equations in κ − 1 variables. But a competitive
business cycle model provides no guidance about which equation should be dropped, andhence does not inform us about the disequilibrium values of the κ − 1 other variables.
113

114

CHAPTER 4. RIGID WAGESThe search model provides the necessary guidance. In the benchmark model, the wage isa weighted average of the MRS and the MPL, but that restriction is no more plausible thana myriad of other possibilities. Replacing the Nash bargaining solution with an alternative
assumption on wage setting substitutes one equation for a different one. I show in this chapterthat the model is still well-behaved, and so we can use it to ask what happens if wages are
too high, i.e. higher than the wage predicted by the Nash bargaining solution. I find that
this may help to explain the behavior of the labor wedge.

4.1

Wage Indeterminacy

My starting point is the observation that there is typically a set of wages that a worker iswilling to accept, Ṽn (st , w) ≥ 0, and that a firm is willing to pay, J˜n (st , w) ≥ 0. Take, forexample, the model in Section 3.2 with capital. Using equations (3.38), (3.40), and (3.47),
J˜n (st , w) ≥ 0 if and only ift(1 − α)z(s )k(st )n(st ) − θ(st )(1 − n(st ))

α 

1−x
1+
µ(θ(st ))

≥ w.

The left hand side is the marginal product of a producer plus the marginal product of theworkers freed from recruiting by the possibility of the worker is still around next period,
the MPL. The firm will employ a worker as long as this exceeds the wage. Similarly, using
equations (3.33), (3.38), (3.42), and (3.43), Ṽn (st , w) ≥ 0 if and only if
γc(st )
φw≥−
(1 − α)z(st )
1−τ
1−φ



k(st )n(st ) − θ(st )(1 − n(st ))α 

1 − x − f (θ(st ))µ(θ(st ))

.

The first term is the marginal rate of substitution between consumption and leisure divided
by the share of labor income that the household keeps. The second term incorporates theincrease in the probability of employment next period from having a job this period; this drops
out if 1 − x = f (θ(st )), so next period’s employment status is independent of this period’s. In
the calibrated model, 1 − x > f (θ(st )), meaning that a worker is more likely to be employednext month if she is employed this month. This implies that a worker will accept a job
paying an after tax wage less than γc(st ), the current marginal rate of substitution between
consumption and leisure, in order to increase the likelihood that she has a job next period.
To get a sense of the size of band of wage indeterminacy, I evaluate these inequalities along
a balanced growth path, as in Section 3.2.5, with the usual calibration of model parameters.
I find that a worker is willing to work in history st if w > 0.88w(st), while a firm is willingto employ a worker if w < 1.12w(st). These leaves a lot of room for wages to deviate from

4.2. NO CAPITAL

115

the equilibrium wage w(st ) without eliminating the bilateral gains from trade.
Within an employment relationship, the level of the wage does not have any allocational
effects and so there is no possibility of a bilateral Pareto improvement for a matched worker
and firm. Indeed, since households own the firms, the wage simply affects the distribution ofcapital and labor income, without altering consumption. But future wages determine whether
firms allocate workers to production or recruiting. If they expect wages to be low relative to
productivity, they will allocate more workers to recruiting, pulling down the unemployment
rate. This is the key reason why the rigid wage model that I develop in this chapter cansignificantly affect the unemployment rate.
I start my formal analysis in the next section by extending the benchmark model without
capital to allow for rigid wages. I distinguish between a target wage w ∗(st ), which is determined by axiomatic Nash bargaining, and the actual wage w(st ), which is a weighted averageof the past actual wage adjusted for average productivity growth and the current target wage,
w(st) = rw(st−1)es̄ + (1 − r)w ∗ (st ). When r = 0, the actual wage and target wage are equal,

as in the benchmark model. When r = 1, the actual wage grows with productivity. I thenreintroduce capital. I find that rigid wages significantly amplify employment fluctuations andmay be able to explain the behavior of the labor wedge. A transitory increase in productivity
growth leads to a persistent increase in employment because wages take some time to catch
up with productivity. During much of the adjustment process, the measured labor wedge isnegative, consistent with the empirical correlation between the labor wedge and employment.
I close the chapter by noting that the approach I take to rigid wages is somewhat arbitrary.
That is, there are many alternative models of rigid wages, each of which would potentiallygive a different prediction for the behavior of employment. Following Hall (2009), I showthat the rest of the model’s structure still puts some restrictions on the data it can generate.
Effectively, rigid wages can make it look as if the elasticity of labor supply is infinite along
the extensive margin. That is, the search model with rigid wages can potentially explain why
it looks as if the elasticity of labor supply is very high, even though households would prefera smoother path of employment at the prevailing wage.

4.2

No Capital

I return to the model without capital from Section 2.2. The households’ and firms’ problems
are the same, with only the wage-setting procedure changed. I repeat the main results fromthose problems here before turning to wages.

116

4.2.1

CHAPTER 4. RIGID WAGESHouseholds

I assume the period utility function is log c for an unemployed individual who consumes c andlog c−γ for an employed individual who consumes c. The household problem then gives three
key results. First is the consumption Euler equation with log preferences over consumption:
qt (s

t+1

Π(st+1 )c(st ))=β.
Π(st )c(st+1 )(4.1)

Second, the value of having an additional worker employed is equal to her after-tax wage,
measured in units of marginal utility, minus the disutility of working, plus the discounted
value of having an additional worker employed next period, multiplied by the increase in theprobability that the worker is employed next period if she has a job this period:Vn (st , a(st ), n(st )) =(1 − τ )w(st )−γc(st )t+ β(1 − x − f (θ(s )))X Π(st+1 )st+1 |stΠ(st )Vn (st+1 , a(st+1 ), n(st+1 )). (4.2)

Finally, the marginal value of having a worker employed at an arbitrary wage w rather thanthe unemployed isṼn (st , w) =(1 − τ )(w − w(st ))+ Vn (st , a(st ), n(st )),
t
c(s )

(4.3)

the increment in the after-tax wage, measured in utils, plus the marginal value of an employedworker.

4.2.2

Firms

The firm’s problem also delivers three key results. First, for firms to be willing to engage bothin recruiting and production, the returns from the two activities must be equal. Producing
yields output z(st ) today, while recruiting yields µ(θ(st )) new employees tomorrow, each
¯ t+1 ), sovalued at J(sX¯ t+1 ).
z(st ) = µ(θ(st ))qt (st+1 )J(s(4.4)
st+1 |stSecond, by employing one more worker today, the firm gets an additional z(st ) units of
output. It may also reallocate (1 − x)/µ(θ(st )) workers from recruiting to production andstill have the same expected employment next period, further raising current production. On
the other hand, it must pay the wage. Putting this together, when the firm is indifferent

117

4.2. NO CAPITAL

between recruiting and producing, the marginal value of employing an additional worker is


1−x
t
t
¯
J(s ) = z(s ) 1 +− w(st ).
µ(θ(st ))(4.5)

Finally, the marginal value of employing a worker at an arbitrary wage w rather than losing
the worker is¯ t ),J˜n (st , w) = w(st ) − w + J(s(4.6)
the decrement in the wage plus the marginal value of a job.

4.2.3

Market Clearing

Next there are two market-clearing conditions. The employment level next period is equal to
the number of employed workers who keep their job plus the number of unemployed workers
who find a job,n(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st )),(4.7)

and consumption is equal to productivity times the number of employed workers who are notrecruiters,c(st ) = z(st ) n(st ) − θ(st )(1 − n(st )) ,(4.8)

where the number of recruiters is the ratio of recruiters to unemployment θ times unemployment 1 − n. Note that this last condition combines equations (2.44) and (2.45).

4.2.4

Wages

I turn now to the new part of the model, backward-looking wage setting. I assume that thetarget wage solves the Nash bargaining problem,w ∗ (st ) = arg max Ṽn (st , w)φ J˜n (st , w)1−φ ,
w

(4.9)

while the actual wage is a weighted average of last period’s wage, adjusted for productivity
growth. and the current target wage,
w(st ) = rw(st−1 )es̄ + (1 − r)w ∗(st ),(4.10)

where r ∈ [0, 1) is a parameter and s̄ is long run average productivity growth,s̄ ≡ ′limt →∞X Π(st′ ) log z(st′ ) − log z(st )st′ |stΠ(st )

t′ − t.

118

CHAPTER 4. RIGID WAGESThis notation and the discussion that follows presumes that s̄ is well-defined and independentof the initial history st , consistent with both a deterministic and a stochastic trend in productivity. Note that the benchmark model imposes r = 0, while r = 1 corresponds to wages
that do not respond to shocks, in which case the wage path is determined by an exogenous
initial condition.
I use these equations to solve for the target wage. First, equations (4.3) and (4.6) imply1−τ
(1 − τ )(w ∗ (st ) − w(st ))∂ Ṽn (st , w)
t
∗ t
=
,
Ṽ
(s
,
w
(s
))
=
+ Vn (st , a(st ), n(st ))n∂wc(st )c(st )∂ J˜n (st , w)
¯ t ).
= −1, and J˜n (st , w ∗ (st )) = w(st ) − w ∗ (st ) + J(s∂w
Use these expressions to write the necessary and sufficient first order condition from maximizing equation (4.9) as¯ t ) + w(st ) − w ∗ (st ) .
(1 − φ)Vn st , a(st ), n(st ) c(st ) = (1 − τ ) φJ(s(4.11)

Combining this with equation (4.2) and using the Euler equation (4.1) implies¯ t ) + w(st ) − w ∗ (st ) = (1 − φ)(1 − τ )w(st ) − (1 − φ)c(st )γ(1 − τ ) φJ(sX¯ t+1 ) + w(st+1 ) − w ∗ (st+1 ) .
+ (1 − τ )(1 − x − f (θ(st )))qt (st+1 ) φJ(sst+1 |st¯ t ) using equation (4.5) and J(s¯ t+1 ) using equation (4.4). After simplifying
Next eliminate J(sI obtain an equation for the wage target,γc(st )1−τX+ (1 − x − f (θ(st )))qt (st+1 ) w ∗ (st+1 ) − w(st+1 ) . (4.12)

w ∗(st ) = φz(st )(1 + θ(st )) + (1 − φ)st+1 |stThe first two terms are the usual weighted average of the current marginal product of laborand the current marginal rate of substitution, while the third term is new. The wage rigidity introduces the possibility that the actual and target wages differ, w(st+1 ) 6= w ∗ (st+1 ).
Assuming 1 − x > f (θ(st )), so a worker is more likely to be employed next period if she isemployed this period, the expectation that the actual wage will be below the target wage next
period puts upward pressure on the target wage this period. Nash bargaining automaticallycompensates for the loss in future wages.
To understand wage dynamics when r ∈ (0, 1), suppose for a moment that productivity,119

4.2. NO CAPITAL

consumption, and the ratio of recruiters to unemployed workers are constant. Also assumes̄ = 0. Then equation (4.12) reduces towt∗ = φz̄(1 + θ̄) + (1 − φ)γc̄∗
+ β(1 − x − f (θ̄)) wt+1
− wt+1 .
1−τTogether with the wage adjustment equation (4.10), this is a pair of first order linear difference
equations for the actual wage wt and the target wage wt∗ . The eigenvalues of the resulting
system are both positive, with one bigger than unity and one smaller than unity, consistent
with saddle path dynamics. Since both eigenvalues are positive, the actual and target wages
γc̄
converge monotonically towards the steady state w̄ = φz̄(1 + θ̄) + (1 −φ) 1−τ
. Moreover, along
the saddle path, there is a decreasing relationship between wt and wt∗ . That is, w0 ≷ w̄ implies
wt ≷ w̄ ≷ wt∗ for all t. Along the adjustment path, the wage solves wt+1 = r̃wt + (1 − r̃)w̄
where

r̃ ≡

1 + rβ(1 − x − f (θ̄)) −q

1 + rβ(1 − x − f (θ̄))

2rβ(1 − x − f (θ))

2

− 4r 2 β(1 − x − f (θ̄)).

One can confirm that 0 < r̃ < r, so the adjustment towards steady state is faster than underthe “naı̈ve” rule wt+1 = rwt + (1 − r)w̄. This is because the target wage is forward-looking,rising above steady state when next period’s wage is expected to be below next period’s
target.

4.2.5

Balanced Growth

Suppose productivity grows deterministically at rate s̄ > 0, log z(st+1 ) = log z(st ) + s̄. Thereis an equilibrium where consumption and actual and target wages grow at rate s̄, while
employment and the ratio of recruiters to unemployed are constant. That is, c(st ) = c̄z(st ),w(st) = w̄z(st ), w ∗ (st ) = w̄ ∗ z(st ), n(st ) = n̄ and θ(st ) = θ̄. The proof of this claim is simple.
Under the conjectured functional forms, equation (4.10) implies actual and target wages areequal, w̄ = w̄ ∗ , and hence a comparison of equations (2.43) and (4.12) indicates that bothare equal to their value in the flexible wage economy. The remaining equations are therefore
also unchanged.
The assumption that the wage is a weighted average of last period’s wage adjusted for
productivity growth and the current target wage is important for this result. Without theadjustment, faster productivity growth would reduce the current wage relative to the target
wage. This would act like a reduction in workers’ bargaining power, giving firms an incentive
to shift workers into recruiting and reducing unemployment along the balanced growth path.

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4.2.6

CHAPTER 4. RIGID WAGES

Productivity Shocks: Deterministic Trend

Now suppose log z(st ) = s̄t + st where st follows a stationary first order Markov process withmean 0. Thus s̄ is the deterministic growth rate while st represents a transitory deviationfrom trend. Let π(st+1 |st ) ≡ Π(st+1 )/Π(st ) denote the probability of state st+1 next period
conditional on state st this period, where st ≡ {st−1 , st } and st+1 ≡ {st , st+1 }. As usual
define consumption, actual wages, and target wages relative to the underlying growth rateas c̃(st ) ≡ c(st )e−s̄t , w̃(st ) ≡ w(st )e−s̄t , and w̃ ∗ (st ) ≡ w ∗(st )e−s̄t . I look for an equilibrium
where detrended consumption, detrended actual and target wages, and employment and therecruiter-unemployment ratio are stationary.
In the stochastic model, equations (4.1), (4.4), and (4.5) implyc̃(st )
e = βµ(θ(s ))π(st+1 |st ) t+1c̃(s )
t+1 t
st

t

s

X

|s



t+1
))
st+1 1 − x + µ(θ(st+1e− w̃(s ) .
µ(θ(st+1 ))Eliminate c̃(st ) using equation (4.8), c̃(st ) = est n(st ) − θ(st )(1 − n(st )) :1 = βµ(θ(st))

X

st+1 |stπ(st+1 |st )n(st ) − θ(st )(1 − n(st ))×n(st+1 ) − θ(st+1 )(1 − n(st+1 ))
1−xw̃(st+1 )1+− st+1. (4.13)µ(θ(st+1 ))e

Similarly, eliminate c̃(st ) from equation (4.12) for the target wage:γn(st ) − θ(st )(1 − n(st ))1−τ ∗ t+1Xn(st ) − θ(st )(1 − n(st ))w̃ (s ) − w̃(st+1 )t+ β(1 − x − f (θ(s )))π(st+1 |st ).
n(st+1 ) − θ(st+1 )(1 − n(st+1 ))
est+1
t+1 t
w̃ ∗(st )e−st = φ(1 + θ(st )) + (1 − φ)s|s(4.14)The state variables n and w̃ are in turn governed by equation (4.7), which is unchanged, andby equation (4.10), which reduces tow̃(st ) = r w̃(st−1 ) + (1 − r)w̃ ∗(st ).

(4.15)

Equations (4.7) and (4.13)–(4.15) relate stationary variables and so admit a solution in which
the policy functions and the evolution of employment and wages relative to trend depend only
on the current value of the exogenous state variable st , the current level of employment n(st ),and the previous period’s wage relative to trend w̃(st−1 ). That is, the policy functions take the4.2. NO CAPITAL

121

form θ(st ) = Θ(st , n(st ), w̃(st−1 )) and w̃ ∗ (st ) = W ∗ (st , n(st ), w̃(st−1 )) and the state variables
evolve according to n(st+1 ) = N(st , n(st ), w̃(st−1 )) and w̃(st ) = W (st , n(st ), w̃(st−1 )).
I log-linearize equations (4.7) and (4.13)–(4.15) around a stochastic steady state for(s, n, w̃−1) and look for policy functions and laws of motion that are linear in the deviation of the state variables from their stochastic steady state values. The model has one newparameter, the persistence of wages r. I initially set r = 0.9, so wages adjust about half way
to the stochastic steady state in six months, but I consider the sensitivity of my results tothat parameter. The remaining parameters are unchanged from their values in Table 3.2, except the capital share, which is implicitly α = 0, and the value of leisure, which I recalibrate
to γ = 0.579 to ensure a five percent unemployment rate in steady state.
With this calibration, the equilibrium policy functions arelog θ = log 0.078 + 24.351s − 0.743(log n − log 0.95) − 26.034(log w̃−1 − log 0.995)

log w̃ ∗ = log 0.995 + 2.471s + 0.476(log n − log 0.95) − 1.574(log w̃−1 − log 0.995),

where w̃−1 is last period’s wage, 0.078 is the steady state recruiter-unemployment ratio,n̄ = 0.95 is the steady state employment rate, and w̄ = 0.995 is the steady state ratio of wages
to productivity. In contrast to the flexible wage version of this model, where productivity
has no real effects, positive productivity shocks now raise the recruiter-unemployment ratio,
while high wages have the opposite effect. On the other hand, if last period’s wage was above
trend, the current target wage will be below trend, helping to pull the actual wage back totrend. This is the saddle path dynamics of the actual and target wage that I discussed before.
The state variables evolve as
n+1 = log 0.95 + 0.414s + 0.307(log n − log 0.95) − 0.443(log w̃−1 − log 0.995),w̃ = log 0.995 + 0.247s + 0.048(log n − log 0.95) + 0.743(log w̃−1 − log 0.995).
An increase in productivity immediately raises employment and wages. Higher employment
continues to put upward pressure on wages, while higher wages are self-reinforcing because of
the slow adjustment process. Eventually higher wages start to reduce employment, returning
the economy to steady state.
As usual, I can stack the state variables, m = {s, log n − log n̄, log w̃−1 − log log w̄}, andexpress the transitional dynamics as m+1 = Am+ Dυ+1 . The eigenvalues of the matrix A are
all positive and less than one, confirming that the equilibrium is locally stable. The largest,
0.980, is equal to the autocorrelation in productivity. The lowest, 0.363, is determined by the
rapid adjustment of employment to steady state. The intermediate one, 0.687, reflects the

122

CHAPTER 4. RIGID WAGES

speed of adjustment in wages. Notably this is significantly faster than the naı̈ve persistencemeasure r = 0.9 and suggests that the autocorrelation of wages may be low despite theirstrong backward-looking nature.
Finally, I can compute the unconditional variance-covariance matrix Σ = AΣA′ + DD ′.
Using this, I find that the unconditional standard deviation of log employment is about 15
percent of the unconditional standard deviation of the productivity shock, or 0.004. Conversely, the unconditional standard deviation of log wages falls slightly, from 0.025 in theflexible wage model (r = 0) to 0.023 in the model with rigid wages.
When wage rigidities are more severe, say r = 0.99, all the results are amplified. In thiscase, the linearized state variables satisfy
n+1 = log 0.95 + 1.570s + 0.338(log n − log 0.95) − 1.822(log w̃−1 − log 0.995),w̃ = log 0.995 + 0.066s + 0.006(log n − log 0.95) + 0.925(log w̃−1 − log 0.995).
Compared to the case of r = 0.9, employment is more responsive and wages are less responsive to shocks, while both variables respond more to past wages since that state is increasingly
relevant. The local dynamics remain stable, with eigenvalues 0.980, 0.906, and 0.356. Although the middle eigenvalue is higher than before, the persistence of the root associated withwage dynamics remains far below unity. Nevertheless, this significantly alters the variancecovariance matrix. The unconditional standard deviation of log employment is slightly larger
than the unconditional standard deviation of the productivity shock, or 0.025. This still does
little to mute the behavior of wages, whose standard deviation falls only slightly more, to0.020.
These results suggest that rigid wages may significantly amplify productivity shocks without much affecting the behavior of wages. Although it is possible to further explore the model
without capital, for example by studying a stochastic trend in productivity, the implicationsof rigid wages for labor market outcomes are best understood by reintroducing savings andinvestment. I return to the more general model now.

4.3

CapitalI extend the model with capital in Section 3.2 to allow for rigid wages. Once again, thehouseholds’ and firms’ problems are unchanged. I repeat the main results from those problems
before describing wage determination.

123

4.3. CAPITAL

4.3.1

Households

The implications of household optimization are summarized by equations (3.31)–(3.33). I
repeat them for convenience. Consumption satisfies the Euler equation, determining theintertemporal price,
qt (st+1 ) = β

Π(st+1 )c(st ).
Π(st )c(st+1 )(4.16)The marginal value of an additional employed household member is the after tax wage expressed in units of marginal utility, minus the disutility of work, plus the increase in theprobability that the worker is employed the following period, multiplied by the expectedmarginal value of an employed household member in that period:Vn (st , a(st ), n(st )) =(1 − τ )w(st )−γc(st )+ β(1 − x − f (θ(st )))X Π(st+1 )st+1 |stΠ(st )Vn (st+1 , a(st+1 ), n(st+1 )). (4.17)

Finally, the marginal value of having a worker employed at an arbitrary wage w rather
than the unemployed is the value of receiving w rather than the equilibrium wage w(st ),discounting taxes and expressed in units of marginal utility, plus the value of an additional
employed household member:Ṽn (st , w) =4.3.2

(1 − τ )(w − w(st ))+ Vn (st , a(st ), n(st )).
c(st )(4.18)Firms

The implications of firm optimization are given by the four conditions in equations (3.37)–(3.40). Again, I repeat them for convenience, eliminating the share of recruiters in employment using the analog of equation (3.47). First, if firms are at an interior solution both forrecruiting,k(st )(1 − α)z(s )n(st ) − θ(st )(1 − n(st ))
t



α

= µ(θ(st ))X

qt (st+1 )Jn (st+1 , n(st+1 ), k(st+1 )).

st+1 |st(4.19)The left hand side is the output generated by a producer in the current period, where n −θ(1 − n) is the number of producers, i.e. workers minus recruiters. The right hand side is the
future value generated by a recruiter.

124

CHAPTER 4. RIGID WAGES

Second, the marginal value of an employed worker in history st is
t

t

t

t

Jn (s , n(s ), k(s )) = (1 − α)z(s )k(st )n(st ) − θ(st )(1 − n(st ))

α 

1−x
1+
µ(θ(st ))− w(st ).

(4.20)
The worker can produce this period. In addition, she frees (1 − x)/µ(θ(st )) other workers
from having to recruit, while allowing the firm to maintain its size. Finally, she is paid a
wage.
Third, the firm must be indifferent about investing,1=X

qt (st+1 ) αz(st+1 )st+1 |st

k(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))

α−1

+1−δ

!

.

(4.21)The left hand side is the cost of buying a unit of capital. The right hand side is the expected
net marginal product of capital next period.
Finally, the marginal profit of employing a worker at an arbitrary wage w in history st
and the equilibrium wage thereafter, rather than losing the worker, is equal to the difference
between the equilibrium wage and her actual wage plus the marginal value of an employedworker,J˜n (st , w) = w(st ) − w + Jn (st , n(st ), k(st )).
(4.22)

4.3.3

Market Clearing

There are two market clearing conditions. Employment next period isn(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st )).

(4.23)

And the capital stock next period isk(st+1 ) = z(st )k(st )α n(st ) − θ(st )(1 − n(st ))1−α+ (1 − δ)k(st ) − c(st ).

(4.24)

Again these are unchanged from the flexible wage model.

4.3.4

Wages

As in the first part of this chapter, I again assume that the target wage solves the Nashbargaining problem,w ∗ (st ) = arg max Ṽn (st , w)φ J˜n (st , w)1−φ ,(4.25)w

125

4.3. CAPITAL

while the actual wage is a weighted average of last period’s wage adjusted for productivity
growth and the current target wage,s̄

w(st ) = rw(st−1 )e 1−α + (1 − r)w ∗(st ),(4.26)

where r ∈ [0, 1) and s̄ is the average productivity growth rate. Note that if total factor
productivity grows at a steady rate s̄, consumption, output, and wages all grow at rates̄/(1 − α), so the adjustment keeps wages in line with average wage growth.
I characterize the target wage using the same approach as in the first part of the chapter.
In particular, use equations (4.18) and (4.22) to determine Ṽn (st , w ∗ (st )), J˜n (st , w ∗(st )), andtheir partial derivatives. Then use these to write the necessary and sufficient first ordercondition from maximizing equation (4.25) as(1 − φ)Vn st , a(st ), n(st ) c(st ) = (1 − τ ) φJn (st , n(st ), k(st )) + w(st ) − w ∗ (st ) .

(4.27)

Combining this with equation (4.17) and using the Euler equation (4.16) implies(1 − τ ) φJn (st , n(st ), k(st )) + w(st ) − w ∗(st )= (1 − φ)(1 − τ )w(st ) − (1 − φ)γc(st )X+ (1 − τ )(1 − x − f (θ(st )))qt (st+1 ) φJn (st+1 , n(st+1 ), k(st+1 )) + w(st+1 ) − w ∗ (st+1 ) .
st+1 |stEliminate the current marginal value of a worker Jn (st , n(st ), k(st )) using equation (4.20)
P
and the expected future marginal value of a worker st+1 |st qt (st+1 )Jn (st+1 , n(st+1 ), k(st+1 ))using equation (4.19). This gives an equation for the target wage,∗tt

w (s ) = φ(1 − α)z(s )k(st )n(st ) − θ(st )(1 − n(st ))αγc(st )
(1 + θ(s )) + (1 − φ)
1−τ
X

t
t+1
∗ t+1
+ (1 − x − f (θ(s )))qt (s ) w (s ) − w(st+1 ) . (4.28)
tst+1 |stThis is a generalization of equation (3.43) to the case where w(st ) and w ∗(st ) are not necessarily equal, or equivalently a generalization of equation (4.12) to the case where the marginal
product of labor is not simply z(st ).

126

CHAPTER 4. RIGID WAGES

4.3.5

Balanced Growth

As in the model without capital, rigid wages do not affect the balanced growth path. Thatis, if productivity grows deterministically at rate s̄ > 0, log z(st+1 ) = log z(st ) + s̄, then there
is an equilibrium in which consumption, capital, and actual and target wages grow at rates̄/(1−α), while employment and the ratio of recruiters to unemployed are constant. To provethis, assume that the actual and target wages both grow at rate s̄/(1 − α), w(st ) = w̄es̄t/(1−α)

and w ∗ (st ) = w̄ ∗ es̄t/(1−α) . Then equation (4.26) implies w̄ = r w̄ + (1 − r)w̄ ∗, or w̄ = w̄ ∗.
But if the actual wage is equal to the target wage, which in turn solves the Nash bargaining
problem, the actual wage is unchanged from the flexible wage model. Since nothing else haschanged, the entire equilibrium is unchanged.

4.3.6

Productivity Shocks: Deterministic Trend

To introduce shocks into the model, assume as usual that productivity fluctuates arounda deterministic trend, log z(st ) = s̄t + st , where st follows a stationary first order Markov
process with mean 0. Let π(st+1 |st ) ≡ Π(st+1 )/Π(st ) denote the probability of state st+1

next period conditional on state st this period, where st ≡ {st−1 , st } and st+1 ≡ {st , st+1 }.
Define consumption, capital, actual wages, and target wages relative to the underlying growthrate as c̃(st ) ≡ c(st )e−s̄t/(1−α) , k̃(st ) ≡ k(st )e−s̄t/(1−α) , w̃(st ) ≡ w(st )e−s̄t/(1−α) , and w̃ ∗ (st ) ≡w ∗(st )e−s̄t/(1−α) .
I rewrite the equations in terms of stationary variables. The first two are unchanged fromthe flexible wage model. Replace qt (st+1 ) in equation (4.21) using equation (4.16) to obtains̄

e 1−α = βXst+1 |stt

π(st+1 |st )

c̃(s )  st+1αec̃(st+1 )

k̃(s

t+1

)

n(st+1 ) − θ(st+1 )(1 − n(st+1 ))

!α−1



+ 1 − δ ,(4.29)

identical to equation (3.53). Similarly eliminate Jn between equations (4.19) and (4.20),
giving(1 − α)est

k̃(st )n(st ) − θ(st )(1 − n(st ))(1 − α)est+1!α= βµ(θ(st ))k̃(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))X

st+1 |st!α π(st+1 |st )c̃(st )c̃(st+1 )

1−x
1+µ(θ(st+1 ))!

− w̃(st+1 ) , (4.30)

identical to equation (3.54).
The expression for the target wage, equation (4.28), is new to this model. Expressed in127

4.3. CAPITAL
terms of stationary variables, it becomes!αtγc̃(st )k̃(s)
tw̃ ∗(st ) = φ(1 − α)est
(1
+
θ(s
))
+
(1
−
φ)
n(st ) − θ(st )(1 − n(st ))1−τXc̃(st )+ β(1 − x − f (θ(st )))π(st+1 |st ) t+1 w̃ ∗ (st+1 ) − w̃(st+1 ) . (4.31)
c̃(s )
t+1 ts|s

The state equations (4.24) and (4.26) reduce tos̄k̃(st+1 )e 1−α = est k̃(st )α n(st ) − θ(st )(1 − n(st ))1−α+ (1 − δ)k̃(st ) − c̃(st )(4.32)

for the capital stock, identical to equation (3.56), andw̃(st ) = r w̃(st−1 ) + (1 − r)w̃ ∗ (st )(4.33)

for the actual wage. Equation (4.23) for the employment rate is unchanged.
Equations (4.23) and (4.29)–(4.33) implicitly define the equilibrium policies θ(st ), c̃(st ),and w̃ ∗ (st ) as functions of the state variables st , n(st ), k̃(st ), and w̃(st−1 ). As usual, I
solve the model by looking for a linear approximation to the optimal policy functions in aneighborhood of the steady state. Most of the calibration is by now standard and uses thevalues from the flexible wage model (Table 3.2), but there is one new parameter, the extent
of wage rigidity r. Since there is no clear empirical counterpart to this, I use the model fortwo purposes. First, I examine whether wage rigidities help to align the model’s predictions
with the data. Second, I examine how changes in this parameter affect the behavior of themodel. I start by setting r = 0.95 and later consider alternative values.
The linear approximation to the equilibrium policy functions arelog θ = log 0.078 + 40.825s − 0.630(log n − log 0.95)+ 10.441(log k̃ − log 218.2) − 38.184(log w̃−1 − log 4.017),
log c̃ = log 4.696 + 0.259s + 0.014(log n − log 0.95)+ 0.607(log k̃ − log 218.2) − 0.023(log w̃−1 − log 4.017),

log w̃ ∗ = log 4.017 + 2.974s − 0.215(log n − log 0.95)+ 1.146(log k̃ − log 218.2) − 2.321(log w̃−1 − log 4.017).
Comparing with equation (3.61), the response of detrended consumption to the shock, to employment, and to detrended capital is virtually unchanged. On the other hand, the recruiterunemployment ratio is five times as responsive to the shock in the rigid wage model. It is

128

CHAPTER 4. RIGID WAGES

somewhat more responsive to employment. And the sign of the response of the recruiterunemployment ratio to the detrended capital stock is reversed. The explanation for this lastresult is instructive. In the flexible wage economy, the wealth effect from a high capital stock
put upward pressure on wages, which discouraged firms from recruiting. In the rigid wage
model, that effect is weakened, so the complementarity between capital and labor dominates.
When the capital stock is high, the marginal product of labor is high, which encourages more
recruiting at a given wage.
The policy functions also show that the recruiter-unemployment ratio falls sharply withthe lagged wage, while consumption falls modestly. The response of the recruiter-unemployment ratio reflects the persistence of wages, since high future wages discourage hiring
today. Consumption falls with the lagged wage because low wages make it a profitable time
to recruit, which comes at the expense of current consumption.
Finally, the third policy function shows that when actual wages are above trend, thetarget wage is far below trend, helping to realign the two variables. High capital and highproductivity each increase the target wage by raising the marginal product of labor, whilehigh employment has the opposite effect.
I turn next to the linear approximation to the state equations:
log n+1 = log 0.95 + 0.694s + 0.309(log n − log 0.95)+ 0.178(log k̃ − log 218.2) − 0.649(log w̃−1 − log 4.017),
log k̃+1 = log 218.2 + 0.018s + 0.019(log n − log 0.95)+ 0.990(log k̃ − log 218.2) + 0.003(log w̃−1 − log 4.017),
log w̃ = log 4.107 + 0.149s − 0.011(log n − log 0.95)+ 0.057(log k̃ − log 218.2) + 0.834(log w̃−1 − log 4.017).
Higher productivity raises employment, capital, and wages. Employment responds five times
more to the shock than it does in the flexible wage model (equation 3.62), as one might expectfrom the recruiter-unemployment ratio policy function. Employment is persistent, positively
influenced by past capital, and negatively influenced by past wages. The response of thecapital stock is largely unchanged from the flexible price model. Finally, wages fall with
employment and rise with capital due to pressure from the marginal product of labor. They
are also strongly related to past wages, although the coefficient on w̃−1 , 0.83, is significantly
below the coefficient in equation (4.33), r = 0.95. The forward-looking target wage greatly
moderates the exogenous wage rigidity.
I then stack the four state variables to compute the eigenvectors, 0.990, 0.980, 0.854,
0.290. As usual, the largest eigenvector corresponds to the persistence of capital, the second

129

4.3. CAPITALỹc̃θ
1 0.688 11.207
1 0.844 0.514

k̃
0.962
0.720

n wn/y0.267 0.144
0.559 -0.530

c/y
0.559
-0.75

τ̂
0.509
0.794

s
0.691
0.982

c̃

—

1

0.068

0.980

0.083 -0.210 -0.278

0.393

0.751

θ

—

—

1

-0.1040.856 -0.943 -0.837

0.703

0.596

k̃

—

—

—

1 -0.088 -0.069 -0.082

0.205

0.603

Correlations n

—

—

—

—

1 -0.663 -0.898

0.690

0.606

wn/y

—

—

—

—

—

1

0.690 -0.612 -0.590

c/y

—

—

—

—

—

—

1 -0.938 -0.833

τ̂

—

—

—

—

—

—

—

1

0.893

s —

—

—

—

—

—

—

—

1

Relative s.d.
ỹTable 4.1: Model with capital and rigid wages (r = 0.95), deterministic trend. Co-movementsof variables in an infinite sample.
to the persistence of productivity shocks, and the smallest to the persistence of employment.
The third largest eigenvector represents the new state variable, the lagged wage. Since all
the eigenvectors lie in the unit circle, the model exhibits locally stable dynamics.
Table 4.1 shows the comovement of the key detrended economic variables. It is perhaps
best read by comparing with Table 3.3, which shows the behavior of the same variables in
the flexible wage model. Looking at the first row, rigid wages raise the standard deviation of
employment and the recruiter-unemployment ratio relative to output by almost a factor ofthree. This increases the volatility of the labor share wn/y but, perhaps surprisingly, reduces
the volatility of the labor wedge τ̂ .
Turning to correlations, the rigid wage model substantially weakens the correlation between employment and the labor wedge, from 0.96 to 0.69. In the data, the correlation isstrongly negative. It similarly weakens the correlation between the consumption-output ratio
and the labor wedge, from −1.00 to −0.94, while the data indicates that the correlation isclose to zero. Thus this wage rigidity reduces but does not eliminate the gap between modeland data.

Finally, the model implies that the labor share wn/y is negatively correlated with employment, although less so than in the flexible wage model. When productivity rises above
trend, the rigidity prevents the wage from rising as much as it would in the flexible wage
model, driving the negative correlation. Acting against this, employment responds to the
low wage, reducing output per worker and keeping the labor wedge relatively constant.
Table 4.2 shows the annual growth rates of the same variables. The basic picture remains

130

CHAPTER 4. RIGID WAGESỹc̃θ
1 0.259 21.884
1 0.804 0.805

k̃
0.265
0.162

n wn/yc/y0.513 0.300 0.807
0.888 -0.653 -0.982

τ̂
0.551
0.915

s
0.717
0.982

c̃

—

1

0.452

0.704

0.497 -0.366 -0.676

0.790

0.816

θ

—

—

1

-0.2520.803-0.96 -0.853

0.750

0.820

k̃

—

—

—

1 -0.157

0.261

0.025

0.164

0.164

Correlations n

—

—

—

—

1 -0.607 -0.941

0.670

0.806

wn/y

—

—

—

—

—

1

0.692 -0.671 -0.708

c/y

—

—

—

—

—

—

1 -0.881 -0.955

τ̂

—

—

—

—

—

—

—

1

0.972

s —

—

—

—

—

—

—

—

1

Relative s.d.
ỹTable 4.2: Model with capital and rigid wages (r = 0.95), deterministic trend. Co-movementsof annual growth rates in an infinite sample.
unchanged. For example, rigid wages reduce the correlation between the annual growth ratesof the labor wedge and employment and the annual growth rates of the labor wedge and theconsumption-output ratio, but do not break or reverse either relationship.
These comovements stand out clearly in Figure 4.1, where I again contrast the baselinemodel, with µ̄ = 2.32, to the model with more search costs, µ̄ = 1. Comparing this with
Figure 3.1, the response of employment to the productivity shock is both larger and faster.
Indeed, employment reaches its maximum value in the period immediately after the shock.
But while this dampens the response of the labor wedge, it does not reverse it. Indeed, thelabor wedge remains positive during most of the subsequent adjustment. This drives the
positive correlation between it and employment.
As I mentioned before, the choice of the wage rigidity parameter r was arbitrary. Indeed,
the results I have reported are sensitive to this choice. To see that, Figure 4.2 shows thecorrelation between the labor wedge and both employment and the consumption-output ratio
as a function of r, both in levels and in annual growth rates. When the rigidity parameter
is approximately 0.993, the correlations are negligible, while for higher rigidities, the laborwedge tends to be low when employment is above trend or the consumption-output ratio is
above trend.
To explore the behavior of extreme values of the wage rigidity parameter more seriously,
I set r = 0.993. Table 4.3 shows the comovements of the variables in this case; the behavior
in annual growth rates is similar. Not surprisingly, more wage rigidity increases the volatilityof the recruiter-unemployment ratio and makes employment two-thirds as volatile as output.

131

4.3. CAPITAL

output ỹ

b
b

b
b

3

b
b

b
b

bb

bb

bb

bb

b

2

bb

bb

bbb

bbb

bbb

bbb

bbb

bbb

bbbb

bbbb

bbbb

bbbb

bbbbb

1

bbbbb

bbbbb

bbbbbbbbbbbbbbbbbbb

bbbbbbb

bbbbbbbb

bbbbbbbbb

bbbbbbbb

consumption c̃

1.2

4

0
20

40

60

80

b

0

b
b
b
b

1.0

b
b

b
b

0.5
0

b
b

b

bb

bb

bb

bbb

bbbb

bbbbb

bbbbbbbb−0.5

bbbbbbbbbb

bbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

20

40

1.5
1.0
0.5

bbbbbbbbbbbbbbbbbbbbbbb

20

40

60

80

100 120

b

b

b
b

b
b

b
b

b
b

b

b
b

0
0

b
b

0

b
b

0
b

−0.5

b
b
b
b

50

b

25

b
b

b
b

bbbbb

bbbbb

bbbbb

bbbb

bbbb

bbbb

bbbb

bbbb

bbbb

bb

60

80

100 120

bb

bb

bb

bbb

0

bbbb

bbbbbbb

bbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

bb

b
bb

b
bb

b

b

b

b

b

bb

b
bb

bbb

bbbb

bbb

bbb

bbbb

bbb

bbbbbbbbbb

bbbbb

40
bbbbbbbb

bbbbbbbbbbbbbbbbbbbbb

60

bbbbbbbbbbbbbbbbb

bbbbbbbb

bbbbbb80

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

bbbbbb

bbbbb

bbbbb

bbbbb

bbbb

100 120
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b
b
b

−1.0

b

bb

b
bb

b
bb

20

labor share wn/y

125
75

bbbbb

b

1.5

100

bbbbbb0.3

100 120

capital k̃

employment nbbbbbbbb

2.0

2.0

b

−1.5
b

−25

0

2.5

20

60

100 120

2.5
b

b

bb
bbb bbbb

bb

bb

bb

bb

bb

bb

bb

1.0
0.5

bb

bb

bb

bb

bb

bb

bb

bb

bbb

0
−0.5

80

b

2.0
1.5

40

−2.0

bbb

bbb

bbb

bbb

bbb

bbbb

bbbb

bbbb

bbbbb

bbbbbbbbbbbbb

productivity s

recruiting/unemployment θ

b

b

0.6

b
b

b
bb

b
bb
b

b
bb

bbbbbbbbbbbbbbbbb

bbbbbbbbbbb

bbbbb

bbbb

bbb

bbb

0
0

labor wedge τ̂

0.9

b
bb

bbbbbbbbb

bbbbbbbbbbbbb

bbbbbbbbb

b

0
b
b

2.0
1.5

b
b

b

20
b

bb

bb

bb

bb

bb

bb

bb

bb

bb

1.0

bb

40

bb

bb

bb

bb

bb

bb

bb

bb

bbb

0.5

bbb

60

bbb

bbb

bbb

bbb

bbbb

bbbb

80

bbbb

bbbb

bbbbb

bbbbb

100 120

bbbbbbbbbbbbbbbbbbb

bbbbbbbb

bbbbbbb

0
0

20

40

60 80
month

100 120

0

20

40

60 80
month

100 120

Figure 4.1: Model with capital and rigid wages (r = 0.95), deterministic trend. Response
to a 2.5 percent increase in productivity at t = 0. All variables are expressed as deviationfrom trend in log points. The blue dots show µ̄ = 2.32 and the red lines show more frictions,µ̄ = 1.

132

CHAPTER 4. RIGID WAGES

1.0

correlation in levels

corr(n,τ̂ )
0.5

0

−0.5
corr(c/y,τ̂)

correlation in annual growth rates

−1
0.90

0.92

0.94
0.96
wage rigidity r

0.98

1.00

0.98

1.00

1.0
corr(n,τ̂ )
0.5

0

−0.5
corr(c/y,τ̂)
−1
0.90

0.92

0.94
0.96
wage rigidity r

Figure 4.2: Model with capital and rigid wages, deterministic trend. Correlation between thelabor wedge, employment, and the consumption-output ratio for various values of the wage
rigidity parameter r. The top graph shows correlations between levels and the bottom shows
correlations between annual growth rates.

133

4.3. CAPITALỹc̃θ
1 0.600 26.902
1 0.788 0.735

k̃
0.912
0.636

n wn/yc/y0.663 0.114 0.643
0.763 -0.749 -0.819

τ̂
0.114
0.280

s
0.501
0.990

c̃

—

1

0.188

0.975

0.206 -0.453 -0.293

0.681

0.817

θ

—

—

1

-0.0330.954 -0.838 -0.967 -0.131

0.719

k̃

—

—

—

1 -0.006 -0.276 -0.079

Correlations n

—

—

—

wn/y

—

—

c/y

—

τ̂

Relative s.d.
ỹ

0.727

0.67

—

1 -0.673 -0.994 -0.309

0.708

—

—

—

1

—

—

—

—

—

1

—

—

—

—

—

—

—

1

0.398

s —

—

—

—

—

—

—

—

1

0.742 -0.417 -0.807
0.2 -0.777

Table 4.3: Model with capital and rigid wages (r = 0.993), deterministic trend.
movements of variables in an infinite sample.

Co-

On the other hand, the increase in wage rigidity substantially reduces the volatility of thelabor wedge. This is because employment and the consumption-output ratio are nearly
equally volatile and very strongly negatively correlated. In contrast, the data in Table A.1
indicate that while employment and the consumption-output ratio are equally volatile, theircorrelation is only −0.6 and so the labor wedge is in fact more volatile than either of thesecomponents. It is also worth noting that the increase in wage rigidity dampens fluctuationsin the labor share wn/y, compared to Table 4.1. While this calibration is not an unqualified

success, it certainly shows that the model can generate extreme volatility in recruiting effort
and a negative correlation between employment and the labor wedge without substantiallyaltering the comovement of other variables, e.g. the joint behavior of output, consumption,and capital, as summarized in Table 3.3.

4.3.7

Productivity Shocks: Stochastic Trend

I obtain similar results with a stochastic trend in productivity, so log z({st , st+1 }) = log z(st )+st+1 where st follows a stationary first order Markov process. As usual, I look for an
equilibrium where an appropriately scaled version of consumption and wages are stationary when growth is stochastic. Define relative consumption, capital, actual wages, and
−1

−1

−1

target wages as c̃(st ) ≡ c(st )z(st ) 1−α , k̃(st ) ≡ k(st )z(st ) 1−α , w̃(st ) ≡ w(st )z(st ) 1−α , and−1w̃ ∗(st ) ≡ w ∗ (st )z(st ) 1−α . Similarly, since employment and the share of recruiters in employ-ment are constant along a balanced growth path, I look for an equilibrium of the stochastic
growth model where these variables are stationary.

134

CHAPTER 4. RIGID WAGES

Following the same argument as in the flexible wage model, replace qt (st+1 ) in equation (4.21)
using equation (4.16) to obtainc̃(st ) − st+1 
1=βπ(st+1 |st ) t+1 e 1−α αc̃(s )
t+1 t
s

X

|s

k̃(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))

!α−1



+ 1 − δ .

(4.34)

Similarly eliminate Jn between equations (4.19) and (4.20):(1 − α)×k̃(st )n(st ) − θ(st )(1 − n(st ))(1 − α)!α= βµ(θ(st ))X

st+1 |stk̃(st+1 )n(st+1 ) − θ(st+1 )(1 − n(st+1 ))π(st+1 |st )!α c̃(st )c̃(st+1 )

1−x
1+µ(θ(st+1 ))− w̃(st+1!

) . (4.35)

The target wage equation (4.28) becomes
!α
t
k̃(s
)
γc̃(st )tw̃ ∗(st ) = φ(1 − α)
(1
+
θ(s
))
+
(1
−
φ)
n(st ) − θ(st )(1 − n(st ))1−τXc̃(st )+ β(1 − x − f (θ(st )))π(st+1 |st ) t+1 w̃ ∗ (st+1 ) − w(s̃t+1 ) . (4.36)c̃(s )
t+1 ts|s

The state equations (4.24) and (4.26) reduce tost+1k̃(st+1 )e 1−α = k̃(st )α n(st ) − θ(st )(1 − n(st ))
and
s̄−st

1−α

+ (1 − δ)k̃(st ) − c̃(st )w̃(st ) = r w̃(st−1 )e 1−α + (1 − r)w̃ ∗ (st ).

(4.37)(4.38)

Equation (4.23) for the employment rate is unchanged.
My calibration follows the flexible price version of the model in Section 3.2.7. In particular,
I still set s̄ = 0.0012, ρ = 0.4, and ς = 0.00325, where st+1 = s̄ + ρ(st − s̄) + ςυt+1 . I again
initially set the wage rigidity parameter at r = 0.95. The other parameters are unchanged.

1354.3. CAPITAL
I start by reporting the equilibrium policy functions:log θ = log 0.078 + 81.417(s − 0.0012) − 0.630(log n − log 0.95)+ 10.441(log k̃ − log 218.2) − 38.184(log w̃−1 − log 4.017),
log c̃ = log 4.696 + 0.436(s − 0.0012) + 0.014(log n − log 0.95)+ 0.607(log k̃ − log 218.2) − 0.023(log w̃−1 − log 4.017),

log w̃ ∗ = log 4.017 + 5.096(s − 0.0012) − 0.215(log n − log 0.95)+ 1.146(log k̃ − log 218.2) − 2.321(log w̃−1 − log 4.017).
An increase in the productivity growth rate raises recruiting, consumption, and the target
wage on impact. These findings are qualitatively unchanged from the flexible wage model,
although the response of the recruiter-unemployment ratio is 68 times larger. The responsesof the equilibrium policies to the three state variables are unchanged from the case with adeterministic trend and the interpretation is likewise unaltered.
I turn next to the state equations:
log n+1 = log 0.95 + 1.384(s − 0.0012) + 0.309(log n − log 0.95)+ 0.178(log k̃ − log 218.2) − 0.649(log w̃−1 − log 4.017),
log k̃+1 = log 218.2 − 0.612(s − 0.0012) + 0.019(log n − log 0.95)+ 0.990(log k̃ − log 218.2) + 0.003(log w̃−1 − log 4.017),
log w̃ = log 4.017 − 1.163(s − 0.0012) − 0.011(log n − log 0.95)+ 0.057(log k̃ − log 218.2) + 0.834(log w̃−1 − log 4.017).
An increase in productivity growth raises employment but lowers capital and actual wages relative to trend. The increase in employment reflects the surge in the recruiter-unemployment
ratio, while the decline in capital is a consequence of rising interest rates that encourage
current consumption at the expense of investment. The relative wage falls for mechanicalreasons: when productivity growth increases, the actual wage cannot keep up because of thewage rigidity. The response of employment, capital, and wages to each other are unchanged
from the model with a deterministic trend. It follows that three of the eigenvalues are unchanged as well, while the fourth, the one associated with the exogenous shock, declines to
0.4.
Table 4.4 shows the comovement of variables in an infinite sample, comparable to Table 3.5. The standard deviation of employment and the recruiter-unemployment ratio increase by a factor of 9. Moreover, the correlation between employment and the labor wedge

136

CHAPTER 4. RIGID WAGESỹc̃θ
1 1.727 83.803
1 0.228 0.497

k̃
2.944
0.186

c̃

—

1

-0.4030.988 -0.569

θ

—

—

1

k̃

—

—

—

1 -0.625

Correlations n

—

—

—

—

1 -0.810 -0.911 -0.449

wn/y

—

—

—

—

—

1

0.584

c/y

—

—

—

—

—

—

1

τ̂

—

—

—

—

—

—

—

1 -0.591

s —

—

—

—

—

—

—

—

Relative s.d.
ỹ

-0.523n wn/yc/yτ̂
1.998 0.867 1.787 1.238
0.645 -0.535 -0.339 -0.827
0.295

s
0.565
0.097

0.839 -0.437

0.073

0.853 -0.993 -0.668 -0.619

0.702

0.423

0.851 -0.329 -0.076
0.229

0.697 -0.750
0.040

0.016
1

Table 4.4: Model with capital and rigid wages (r = 0.95), stochastic trend. Co-movementsof variables in an infinite sample.
is reversed, falling from 0.97 in the flexible wage model to -0.45 here. Similarly, the correlation of the consumption-output ratio with the labor wedge rises from −1.00 to 0.04. Table 4.5

shows the corresponding annual growth rates. The conclusions are similar. The model does
have some counterfactual predictions, however, including the negative correlation between
detrended employment and detrended consumption or capital.

Figure 4.3 shows the impulse response to a one time increase in productivity growth.
Recall that this figure shows the behavior of variables in levels, not relative to current productivity, making it easier to interpret the behavior of nonstationary variables like output,
consumption, and capital. I find that an increase in the productivity growth rate leads
to an immediate increase in consumption and an even larger increase in output. Thus
the consumption-output ratio falls upon the impact of the shock before slowly reverting totrend. Since wages are slow to respond to the shock, employment and recruiting also increasestrongly before reverting slowly back to their normal level. The amplification compared to
the flexible wage model in Figure 3.2 is clear.
One might expect wage rigidities to create sharp movements in the labor share, but in fact
the changes are fairly modest. The labor share falls by −0.2 percent on the impact of this large
increase in productivity growth. This reflects the boom in output with fixed employment
and rigid wages. As employment starts to grow, the labor share quickly recovers, following anonmonotonic pattern back to steady state. This observation is important for microeconomic
studies that attempt to assess the extent of wage rigidity by directly comparing the behaviorof wages and labor productivity (e.g. Pissarides, forthcoming; Haefke, Sonntag, and van Rens,

137

4.3. CAPITAL
1.0

1.0

bb
b
b

output y

0.8

b

bb

bbb

bbbbbbbbbbbbbbbbbbbbbbbbbb

bbbbbbbbbbbbb

bbbbbbbbbbbbbb

bbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbb

0.6
0.4
b

0.2

consumption c

b

0
20

40

60

80

0.4

0.8

0

capital k

b

b

0.4
b

b
b
b
b

0.2

b
b

bb

bb

bbb

20

40

0.4

bbbbb

bbbbbbbbbbb

20

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

40

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

60

80

100 120

b

bb

bb

bbb

0

bbb

bbb

bb

b
bb

0

labor share wn/y

b

b

−0.1

b
b
b
b
b
b
b
b
b
b

bb

bb

bbb

bbbb

bbbbbbb

bbbbbbbbbbbbbbbbb

0
0

20

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

40

60

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

80

bbb

60

80

100 120

b

b

b

bb

bbb

bbbbbbbbbbbbbbbbbb

bbbbbbb

bbbbbbbb

bbbbbbbb

bbbbbbbb

bbbbbbbbb

bbbbbbbbbb

bbbbbbbbbbb

b

0
b
b

b

0

20

40

60 80
month

bbbbbbbbbbbbb

bbbbbbbbbbbbbb

bbb

100 120

bbb

b
bb

b

b

b

b

b

b

bb

bb

bbb

bbb

bbb

bbb

bbbb

bbbb

b

bbb
bbbb

40
bbbb

bbbbbbbbbbbbbbb

bbbb

bbbb

bbbbb

bbbbb

60

bbbbb

b

bbbb
bbbbb

bbbbbb80

bbbbbbbbbbbbbbbbbbb

bbbbbbbb

100 120

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b

b

−0.3

100 120

0.4
0.2

b

−0.2

productivity growth s

b

10

bbb

bb
bbb

20

0

30

−0.4

bb

0.6

0.2

b

0

−0.2

bbbbbbbbbbbbbbbbb

bbbbbbbbbbb

bbbbbbbbbb

0.8
b

0.6

20

bbbbb

bbbbbbbbb

1.0

0

labor wedge τ̂

bbb

bbbb

bbb

bbb

bb

bbbbb

bbbb
bbbb

bbbbbbbbb

bbbbbbbb

0.2

100 120

1.0
employment n

0.6

bbbbbbb

bbbbbbbbbbbb00recruiting/unemployment θ

0.8

0

20

40

60

80

100 120

0.4
b

0.3
0.2
b

0.1
b
b

0
0

bb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

20

40

60 80
month

100 120

Figure 4.3: Model with capital and rigid wages (r = 0.95), stochastic trend. Response to anincrease in the productivity growth rate at t = 0. All variables are expressed as deviationfrom initial trend in log points. The blue dots show µ̄ = 2.32 and the red lines show more
frictions, µ̄ = 1.

138

CHAPTER 4. RIGID WAGES

Relative s.d.
ỹ

ỹ
c̃
θ
k̃
1 0.790 88.160 1.273
1 -0.510 0.743 -0.654-0.191n wn/yc/yτ̂
2.071 0.942 1.559 1.210
0.969 -0.707 -0.900 -0.748

0.922 -0.628

0.133

0.834

s
0.667
0.108

c̃

—

1

0.002

0.445

θ

—

—

1 -0.541

0.807 -0.998 -0.573 -0.965

0.717

k̃

—

—

—

1 -0.803

0.493

0.347

0.066

Correlations n

—

—

—

—

1

-0.770 -0.940 -0.751

0.169

wn/y

—

—

—

—

—

1

0.521

c/y

—

—

—

—

—

—

1

0.481

0.156

τ̂

—

—

—

—

—

—

—

1

-0.736

s

—

—

—

—

—

—

—

—

1

0.887

0.970 -0.756

Table 4.5: Model with capital and rigid wages (r = 0.95), stochastic trend. Co-movementsof annual growth rates in an infinite sample.
2008). The labor share is the ratio of the wage w to the measured average product of labory/n. The finding that the labor share declines only modestly during an expansion implies thatlabor productivity rises only slightly more than wages following an increase in productivity
growth. This small gap may be difficult to discern using the best available microeconomic
data, and so a failure to observe greater volatility of labor productivity than wages may say
little about the relevance of wage rigidities for labor market fluctuations.
I again explore the role of the wage rigidity parameter r by graphing the correlations
between the labor wedge, employment, and the consumption-output ratio as a function ofr in Figure 4.4. Compared to the version of the model with a deterministic trend, I findless need for an extreme value of this parameter. Indeed, to match the slightly negative
correlation between the labor wedge and both employment and the consumption-output
ratio, I can allow for even smaller values of r.

4.3.8

Evaluation

These experiments help to illustrate the ability of wage rigidities to moderate the gap between model and data, but they are not definitive. As I have emphasized, the wage rigidity
parameter, r, is a free parameter. One might try to address this by using economic theory
to pin down the value of r. Efforts in this direction include Hall and Milgrom (2008) and
Gertler and Trigari (forthcoming).
A related issue is that the required amount of wage rigidity may depend on the natureof the shocks hitting the economy. I argued in the previous chapter that the flexible wage

139

4.3. CAPITAL

correlation in levels

1.0

0.5

corr(c/y,τ̂)

0−0.5
corr(n,τ̂ )correlation in annual growth rates

−1
0.90

0.92

0.94
0.96
wage rigidity r

0.98

1.00

0.98

1.00

1.0
corr(c/y,τ̂)
0.5

0

−0.5
−1
0.90

corr(n,τ̂ )

0.92

0.94
0.96
wage rigidity r

Figure 4.4: Model with capital and rigid wages, stochastic trend. Correlation between thelabor wedge, employment, and the consumption-output ratio for various values of the wage
rigidity parameter r. The top graph shows correlations between levels and the bottom shows
correlations between annual growth rates.

140

CHAPTER 4. RIGID WAGES

economy could not match the data regardless of the nature of shocks, e.g. productivity or
government spending, aggregate or reallocational. Although I have not considered othershocks in the rigid wage model, it seems likely that both the extent of wage rigidity and thenature of shocks may be important for understanding the comovement of aggregate variables.
But the bottom line is that, while wage rigidity may be important for understanding thecyclical behavior of the labor market, it is hard to measure. I close the book with an effort
to sidestep this question, asking whether the model has additional restrictions that holdregardless of the true value of r and regardless of the nature of shocks hitting the economy.
Following Hall (2009), I tentatively conclude in the next section that it does.

4.4

Using Hours Data to Test the Model

The tenet of the rigid wage model is that, within the interval of wages that firms are willingto pay and that workers are willing to accept, there is no good reason to predict which wage
will be set in equilibrium. The actual wage that will be paid depends on details of wage
negotiations between workers and firms. But while this choice does not affect the survivalof existing employment relationships, it does affect firms’ willingness to recruit, thereby
changing the unemployment rate.
Despite this indeterminacy, Hall (2009) argues that a search model with rigid wages makessome predictions about the comovement of the consumption-output ratio and hours worked,
at least in an environment where the number of hours worked per employed worker, h(st ), is
endogenous. Matched workers and firms should agree on a joint surplus maximizing level ofhours and use other elements of the wage contract to divide the gains from trade. This idea
puts some discipline on the model’s predictions.
To be concrete, suppose that a worker’s income is linear in her hours worked, w0 (st ) +w1 (st )h(st ). Workers and firms bargain efficiently over the marginal wage w1 (st ) and hours
worked h(st ), but I will be deliberately vague about how the fixed component of wages w0 (st )is set. Efficient bargaining over the marginal wage and hours ensures that workers’ and firms’
choice of hours maximizes the joint gains from trade, a prediction that I show has empiricalcontent.
More precisely, I flesh out the details of this model and prove that it is mathematicallyequivalent to a frictionless one in which workers have a finite elasticity of hours supply but
an infinite elasticity of labor supply. In other words, a search model with rigid wages canpotentially explain why it looks as if workers are willing to accept substantial variation in
employment over the business cycle, even though in reality the elasticity of labor supply is
not large.

141

4.4. USING HOURS DATA TO TEST THE MODEL

4.4.1

Households

I assume throughout this section that the marginal utility of consumption may differ for
employed and unemployed workers. The period utility function of a typical household member
is

1+ε

γε
c1−σ 1 + (σ − 1) 1+εh ε1−σif she consumes c and is employed for h hours. It isσ−1c1−σ − 11−σif she is unemployed. A representative household decides how much each of its members may
consume in order to maximize the sum of utilities of its members, acting as if it has utility
function
1+ε

σ

!
t 1−σ
c
(s
)
−
1
u
β t Π(st )n(st ) +(1 − n(st )) ,1−
σt=0 st(4.39)
where ce (st ) is the consumption of each employed household member and cu (st ) is the con∞ XXγεce (st )1−σ 1 + (σ − 1) 1+εh(st ) ε1−σ−1

sumption of each unemployed household member in history st .
The household faces a single lifetime budget constraint, that initial assets must be equal
to the difference between the present value of consumption and the present value of after-tax
labor income plus transfers,
a0 =

∞ X
X
t=0

st




q0 (st ) c(st ) − (1 − τ ) w0 (st ) + w1 (st )h(st ) n(st ) − T (st ) ,(4.40)

where c(st ) ≡ ce (st )n(st ) + cu (st )(1 − n(st )) is total consumption. As discussed before, I
assume that a worker who works for h(st ) hours in history st earns pre-tax income w0 (st ) +w1 (st )h(st ). In addition, employment is determined by the law of motionn(st+1 ) = (1 − x)n(st ) + f (θ(st ))(1 − n(st ))(4.41)

for all st+1 = {st , st+1 }. The household chooses {ce (st ), cu (st )} to maximize equation (4.39)
subject to equation (4.40) and equation (4.41), taking as given initial assets a0 , bargained
hours {h(st )}, prices {w0 (st ), w1 (st ), q0 (st )}, taxes τ , and transfers {T (st )}.
Focus temporarily on the choice of consumption for each employed and unemployed mem-

142

CHAPTER 4. RIGID WAGES

ber. Combining the first order conditions with the definition of total consumption, I obtain1+εce (st ) =cu (st ) =γεc(st ) 1 + (σ − 1) 1+εh(st ) ε1+εγε
1 + (σ − 1) 1+εh(st ) ε n(st )c(st )1+εγεh(st ) ε n(st )1 + (σ − 1) 1+ε

and.

Substituting this into the utility function equation (4.39) gives
1+ε

∞ X
X
t=0

s

γε
h(st ) ε n(st )c(st )1−σ 1 + (σ − 1) 1+ε
β t Π(st )
1−σ
t

σ

−1

,

(4.42)

so the household chooses total consumption {c(st )} to maximize this expression subject toequation (4.40) and equation (4.41), taking as given initial assets a0 , bargained hours {h(st )},prices {w0 (st ), w1 (st ), q0 (st )}, taxes τ , and transfers {T (st )}.
Now let V (st , a, n) denote the value of a household with assets a and n employed workers
in history st . This solves


 1−σ
γε
t 1+ε
t σ
ε n(s )
c
1
+
(σ
−
1)
h(s
)
−1
1+ε
V (st , a, n) = max
t+1
{a(s
)}
1−σ

X Π(st+1 )

t+1
t+1
′
V s , a(s ), n , (4.43)+βΠ(st )
t+1 ts|s

where c satisfies the intertemporal budget constraint,X
c = a + (1 − τ ) w0 (st ) + w1 (st )h(st ) n + T (st ) −qt (st+1 )a(st+1 ),st+1 |stand next period’s employment is determined from current employment and unemployment
asn′ = (1 − x)n + f (θ(st ))(1 − n).
As usual, the first order condition for next period’s assets and the envelope condition for
current assets yield the Euler equation,
Π(st+1 )qt (st+1 ) = βΠ(st ) !σ1+εγεc(st ) 1 + (σ − 1) 1+εh(st+1 ) ε n(st+1 )
.

1+ε
γε
c(st+1 ) 1 + (σ − 1) 1+εh(st ) ε n(st )(4.44)

143

4.4. USING HOURS DATA TO TEST THE MODEL
The envelope condition for employment is
Vn (st , a(st ), n(st )) =−!−σ

(1 − τ ) w0 (st ) + w1 (st )h(st )

!1−σ

σ

c(st )1+εγε
1 + (σ − 1) 1+εh(st ) ε n(st )c(st )1+εγεh(st ) ε n(st )1 + (σ − 1) 1+ε+ β 1 − x − f (θ(st ))1+εγεh(st ) ε1+ε X Π(st+1 )

t+1
t+1
t+1
V
s
,
a(s
),
n(s
)
. (4.45)nΠ(st )
t+1 ts|s

Also let Ṽn (st , w1 , h) denote the marginal value to the household with the equilibrium levelof assets and employment of having a worker employed at an arbitrary hourly wage w1 andworking an arbitrary number of hours h, rather than be unemployed in history st . This
notation suppresses the fixed component of wages. This solves1+ε
γσε  t 1+εtεεṼn (s , w1 , h) =
h(s )
−h
1+ε

c(st )1+εγε
1 + (σ − 1) 1+εh(st ) ε n(st )+ (1 − τ ) w1 h − w1 (st )h(st )!1−σc(st )1+εγε
1 + (σ − 1) 1+εh(st ) ε n(st )!−σ+ Vn (st , a(st ), n(st )). (4.46)

The first term is the utility benefit of any reduction in hours of work. The second term is theutility value of the after-tax income generated by a worker employed at (w1 , h) rather than at
the equilibrium marginal wage-hour pair (w1 (st ), h(st )). The third term is the marginal valueof having the worker employed at the equilibrium wage and hours rather than unemployed.

4.4.2

Firms

A representative firm employs n0 = n(s0 ) workers and owns capital k0 = k(s0 ) at time 0.
In history st , it assigns a fraction ν(st ) of its n(st ) workers to recruiting and the remainingn(st )(1 − ν(st )) workers to production. Each worker works h(st ) hours. The producers
1−α
use the capital k(st ) to generate output z(st )k(st )α h(st )n(st )(1 − ν(st )). The single
produced good is used both for consumption and for investment. A fraction δ of the capitaldepreciates during production. The recruiters each attract h(st )µ(θ(st )) workers to the firm,
while a fraction x of the workers leave the firm, thus determining n(st+1 ). Finally, the firm
can freely buy or sell capital in history st , determining k(st+1 ).

144

CHAPTER 4. RIGID WAGES

Writing this formally, the firm’s value in history s0 with n0 workers and capital k0 is
J(s0 , n0 , k0 ) =

∞ X
X
t=0

st


1−α
q0 (st ) z(st )k(st )α h(st )n(st )(1 − ν(st ))t+ (1 − δ)k(s ) − k(st+1) − w0 (s ) + w1 (s )h(s ) n(s ) , (4.47)
t

t

t



t

so flow profits is the sum of output and undepreciated capital minus the cost of new capitaland the wage bill. Firm growth satisfiesn(st+1 ) = n(st ) h(st )ν(st )µ(θ(st )) + 1 − x ,(4.48)

where st+1 = {st , st+1 }. The firm chooses its capital stock k(st ) and the allocation of workers to recruiting ν(st ) to maximize profits in equation (4.47) subject to the law of motion for employment in equation (4.48), taking as given bargained hours {h(st )} and prices{w0 (st ), w1 (st ), q0 (st )}.
Let J(st , n, k) denote the value of a firm that starts history st with n workers and k unitsof capital. The value function satisfies the recursive equationtJ(s , n, k) = maxz(st )k α h(st )n(1 − ν)
′
ν,k

1−α


+ (1 − δ)k − k ′ − n w0 (st ) + w1 (st )h(st )

X
 ′
t+1
t+1
t
t
+
qt (s )J s , n νh(s )µ(θ(s )) + 1 − x , k . (4.49)
st+1 |stAssuming an interior solution for the share of recruiters ν, I obtain the first order conditionk(st )(1 − α)z(s )h(st )n(st )(1 − ν(st ))
t



α

= µ(θ(st ))X

st+1 |stqt (st+1 )Jn st+1 , n(st+1 ), k(st+1 ) .

(4.50)The left hand side is the marginal product of an hour of production labor. The right handside is the expected value of the additional workers attracted by an hour of recruiting labor.
Also write the envelope condition for employment asαk(st )Jn (s , n(s ), k(s )) = (1 − α)z(s )h(st )(1 − ν(st ))ttth(s )n(s )(1 − ν(s ))t− w0 (s ) + w1 (st )h(st ) X+ ν(st )h(st )µ(θ(st )) + 1 − x
qt (st+1 )Jn st+1 , n(st+1 ), k(st+1 ) .
t

t

t

t



st+1 |st145

4.4. USING HOURS DATA TO TEST THE MODEL
Eliminate the continuation value using equation (4.50) to obtain
t

t

t

t

Jn (s , n(s ), k(s )) = (1 − α)z(s )k(st )h(st )n(st )(1 − ν(st ))

α 

1−x
t
h(s ) +µ(θ(st ))− w0 (st ) + w1 (st )h(st ) . (4.51)

The value of employing an additional worker is the sum of the worker’s marginal product as
a producer and the marginal product of the workers who are released from recruiting becauseof the worker’s presence, minus the wage.
Next turn to the first order condition for next period’s capital stock,
1=

X

st+1 |stqt (st+1 )Jk st+1 , n(st+1 ), k(st+1 ) .

Purchasing a unit of capital reduces current profit by 1. This must equal the increase in thecontinuation value of the firm. The envelope condition for capital is
t

t

t

t

Jk (s , n(s ), k(s )) = αz(s )k(st )h(st )n(st )(1 − ν(st ))α−1+ 1 − δ.

Evaluate this condition in history st+1 and substitute into the first order condition for next
period’s capital to get
1=

X

qt (s

t+1

) αz(st+1

st+1 |st)

k(st+1 )h(st+1 )n(st+1 )(1 − ν(st+1 ))

α−1

!

+1−δ .

(4.52)

This ensures that firms are willing to invest in capital, so the cost of capital this period isequal to net marginal product of capital next period.
Finally, compute the marginal profit of employing a worker at an arbitrary marginal wage
w1 for an arbitrary number of hours h in history st and the equilibrium wage thereafter, rather
than losing the worker. For a firm with the equilibrium level of employment n(st ) and capitalk(st ), this isJ˜n (s , w1 , h) = (1 − α)z(s )ttk(st )h(st )n(st )(1 − ν(st ))α(h − h(st ))+ (w1 (st )h(st ) − w1 h) + Jn (st , n(st ), k(st )). (4.53)

The first term is the change in output, the marginal product of labor times the change inhours. The second term is the change in the wage bill. The final term is the value of a worker

146

CHAPTER 4. RIGID WAGES

at the equilibrium wage and hours.

4.4.3

Wage Setting and the Labor Wedge

The key assumption is that workers and firms bargain over the marginal wage and hours,
satisfying the Nash bargaining procedure:(w1 (st ), h(st )) = arg max Ṽn (st , w1 , h)φ J˜n (st , w1, h)1−φ .

(4.54)

w1 ,hI do not, however, put any restrictions on the base wage w0 (st ).
Using equations (4.46) and (4.53), the first order condition of equation (4.54) for themarginal wage evaluated at (w1 (st ), h(st )) isφ(1 − τ )tVn (s , a(st ), n(st ))c(st )1+εγε
1 + (σ − 1) 1+εh(st ) ε n(st )

!−σ

=

Jn

1−φ

(st , n(st ), k(st )).

(4.55)The same equations imply that the first order condition for the choice of hours evaluated at(w1 (st ), h(st )) isφ1γσh(st ) εtttVn (s , a(s ), n(s ))c(st )1+εγε
1 + (σ − 1) 1+εh(st ) ε n(st )c(st )− (1 − τ )w1 (st )

1+ε

!1−σ
!−σ !

γε
1 + (σ − 1) 1+εh(st ) ε n(st )


α

1−φ
k(st )tt=(1 − α)z(s )− w1 (s ) .
Jn (st , n(st ), k(st ))h(st )n(st )(1 − ν(st ))Eliminate Vn from this expression using the previous one:γσh(st )1εc(st )1+εγε
1 + (σ − 1) 1+εh(st ) ε n(st )!
t= (1 − τ )(1 − α)z(s )k(st )h(st )n(st )(1 − ν(st ))α. (4.56)

The marginal wage w1 (st ) then determines how those gains are split.
To reinterpret equation (4.56), I write down the monetary gains from trade at an arbitrary
wage (w0 , w1 ) and hours h. This is the sum of the after-tax value of a job to a firm and themarginal value of an additional employed worker in the household, evaluated (for expositional

147

4.4. USING HOURS DATA TO TEST THE MODEL
convenience only) in units of goods rather than utility:(1 − τ )J˜n (st , w1 , h) + Ṽn (st , w1 , h)c(st )1+εγεh(st ) ε n(st )1 + (σ − 1) 1+ε!σ.

Now suppose hours worked maximizes this joint gains from trade. Using equations (4.46)
and (4.53) to eliminate Ṽn and J˜n , I find that hours worked must maximizet(1 − τ )(1 − α)z(s )αk(st )(h − h(st ))ttth(s )n(s )(1 − ν(s ))!
1+εγσε  t 1+εc(st )h(s ) ε − h ε
+
.
1+ε
1+ε
1 + (σ − 1) γε h(st ) ε n(st )1+εBut taking the necessary and sufficient first order condition with respect to h and evaluating
at h = h(st ), I obtain equation (4.56). Thus hours worked maximizes the joint gains fromtrade, generalizing one of the conclusions from Section 2.4.2 to this environment.
I use the insight that hours are chosen to maximize the gains from trade to obtain an
expression for the labor wedge. Lety(st ) ≡ z(st )k(st )α h(st )n(st )(1 − ν(st ))denote aggregate output. Then rewrite equation (4.56) as

1−α


1+ε
γσ (1 − ν(st ))c(st )/y(st) h(st ) ε n(st )
τ =1−
 .
1+ε
γε
(1 − α) 1 + (σ − 1) 1+εh(st ) ε n(st )(4.57)

This is analogous to equation (1.14) in the frictionless model with nonseparable preferences.
There are only two differences. First, there is no analog of the share of producers 1 − ν(st )in the frictionless model. With search frictions, measured production only accounts for afraction 1 − ν(st ) of output; the remaining output is the unmeasured investment in future
hiring through recruiting. Thus total (measured and unmeasured) output is y(st )/(1 − ν(st ))and so (1 − ν(st ))c(st )/y(st) measures the ratio of consumption to total output.
Second, in the frictionless model, the labor supply elasticity ε governed the entire hours

choice; there was no distinction between hours per worker and the employment-population
ratio. With search frictions, the labor supply elasticity governs the choice of hours per worker,
but the model behaves as if the elasticity for the employment-population ratio is infinite.
Quantitatively, the first difference between the two equations is likely to be small. In mybaseline calibration, a fraction ν = 0.004 of workers are recruiters while the remainder are

148

CHAPTER 4. RIGID WAGES

producers. Even if ν doubles during a boom, this reduces 1 − ν from 0.996 to 0.992. Fromequation (4.57), this scarcely affects the labor wedge.
The second difference is much more significant and suggests a testable implication. Suppose that by varying w0 (st ) appropriately, I can perfectly match the empirical behavior of
the employment rate n(st ). Equation (4.57) shows that the model still makes predictions
about the comovement of employment, the consumption-output ratio, and hours per employed worker, each of which can be measured in the data. For example, fix the labor supply
elasticity at ε = 1 and the capital share at α = 1/3. For different values of the complementarity parameter σ, I set γ(1 − ν) so that the labor wedge is equal to 0.4 on average.1 I thenuse data on the consumption-output ratio c(st )/y(st ), on hours per employed worker h(st ),and on employment n(st ) to compute the right hand side of equation (4.57).2

Figure 4.5 shows the results, expressed as deviations from log trend, for three different
values of σ. The results are relatively insensitive to the exact choice of σ, although fluctuations
are somewhat smaller when σ is larger. The figure shows a significant increase in the laborwedge during all but one of the post-war recessions; however, comparing the magnitude of
the increases with the analogous numbers from the frictionless model, it is clear that the
search model reduces the extent of the anomalies.
For example, the standard deviation of the detrended labor wedge in the rigid wage model,
measured by equation (4.57), with ε = 1 and σ = 2 is 1.5 percent. The standard deviationof the annual growth rate is 2.3 percent. If I measure the labor wedge using the frictionlessmodel, equation (1.12), with ε = ∞, the standard deviation of the detrended wedge is 1.4
percent and the standard deviation of the annual growth rate is 2.2 percent. Effectively, thesearch model with rigid wages and a relatively small labor supply elasticity behaves similarly
to a frictionless model with an infinite labor supply elasticity. Figure 4.6 displays this pointgraphically.
This figure does not prove that wage rigidity is important for understanding the behaviorof the labor market at business cycle frequencies. It does, however, suggest that the models
of wage rigidity can be tested using the same approach that is effective in evaluating both thefrictionless model and the flexible wage model. Obviously hours of work is only one marginchosen by matched workers and firms, so the general approach suggested by Hall (2009) andoutlined here may yet yield additional predictions.

1

This approach ignores fluctuations in the share of recruiters ν(st ). In principle, one could deduce thesefrom the behavior of employment, for example using equation (4.48). As I discussed above, these movements
are unlikely to be quantitatively significant and so I ignore them here.
2
I use data on hours and employment from Prescott, Ueberfeldt, and Cociuba (2008).

4.4. USING HOURS DATA TO TEST THE MODEL

149

4

Deviation from Trend

2

0

−2
−4
−6
1955

1960

1965

1970 19751980 1985 1990 1995 2000 2005
Year
Figure 4.5: The deviation of the labor wedge from log trend, HP filter with parameter 1600,
using equation (4.57). The dotted green line shows σ = 1, the dashed red line shows σ = 2,
and the solid blue line shows σ = 4. In each case I set ε = 1 and α = 1/3 and fix theremaining parameter γ(1 − ν) to ensure that the average labor wedge is 0.40. The graybands show NBER recession dates.

150

CHAPTER 4. RIGID WAGES

4

Deviation from Trend

2

0

−2
−4
−6
1955

1960

1965

1970 19751980 1985 1990 1995 2000 2005
Year
Figure 4.6: The deviation of the labor wedge from log trend, HP filter with parameter 1600.
Both lines set α = 1/3. The dashed red line shows the rigid wage model with ε = 1 andσ = 2, measuring the labor wedge using equation (4.57). The solid blue line shows thefrictionless model with ε = ∞ and σ = 1, measuring the labor wedge using equation (1.12).
The gray bands show NBER recession dates.

Appendix A
Data
Although this book focuses on the behavior of the labor wedge, I also discuss the comovementof a number of other macroeconomic time series. Each of these has an empirical counterpart.
• Output y: I use a quantity-weighted measure of real Gross Domestic Product, National
Income and Product Accounts Table 1.1.3, line 1. I express this in per capita terms,
dividing by the population series from Prescott, Ueberfeldt, and Cociuba (2008).
• Consumption: c I use a quantity-weighted measure of real consumption of nondurables
and services, National Income and Product Accounts Table 1.1.3, Rows 4 and 5. I
express this in per capita terms, dividing by the population series from Prescott, Ueberfeldt, and Cociuba (2008).
• Recruiter-unemployment ratio θ: I proxy the number of recruiters with the Conference
Board help-wanted advertising index, available directly from the Conference Board. I
divide this by the number of unemployed workers, BLS series LNS13000000.
• Capital stock k: I measure the capital stock using the Bureau of Economic Analysis’s
Fixed Asset Table 1.1, line 1, the current cost net stock of fixed assets and consumer
durable goods. This is an annual series, which I interpolate rowarly to obtain a quarterly
series. I divide by the population series from Prescott, Ueberfeldt, and Cociuba (2008).
• Employment n: I use the measure of employment from Prescott, Ueberfeldt, and Cociuba (2008), divided by population from the same paper.

• Labor share wn/y: I measure the labor share using National Income and ProductAccounts Table 1.10. Labor income is “Compensation of employees, paid” (line 2).
Capital income is consumption of fixed capital (line 23) plus net operating surplus ofprivate enterprises (line 11) minus proprietors’ income (line 15). Labor share is labor
151

152

APPENDIX A. DATA

Relative s.d.
y

y
c
θ
k
n
1 0.565 15.300 0.205 0.655
1 0.825 0.895 0.396 0.781

wn/y
c/y
τ̂
0.520 0.648 0.946
0.000 -0.867 -0.219

c

—

1

0.764 0.331 0.670 -0.006 -0.537 -0.391

θ

—

—

1 0.470 0.868 -0.075 -0.729 -0.440

k

—

—

—

1 0.509 -0.086 -0.202 -0.427

Correlations n

—

—

—

—

1 -0.138 -0.616 -0.633

wn/y

—

—

—

—

—

1 -0.025

0.242

c/y

—

—

—

—

—

—

1

-0.131

τ̂

—

—

—

—

—

—

—

1

Table A.1: Detrended U.S. data, 1959–2007.
y
c
θ
k
n
1 0.581 14.704 0.247 0.639
1 0.813 0.882 0.448 0.690

wn/y
c/y
τ̂
0.550 0.666 1.028
0.007 -0.839 -0.088

c

—

1

0.052 -0.479 -0.219

θ

—

—

1 0.419 0.819 -0.057 -0.720 -0.311

k

—

—

—

1 0.329 -0.043 -0.233 -0.141

Correlations n

—

—

—

—

1 -0.068 -0.538 -0.597

wn/y

—

—

—

—

—

1 -0.033

0.193

c/y

—

—

—

—

—

—

1

-0.260

τ̂

—

—

—

—

—

—

—

1

Relative s.d.
y0.697 0.480 0.512

Table A.2: Annual Growth rates, U.S. data, 1959–2007.
income divided labor plus capital income. This implicitly assigns the same labor shareto proprietors’ income and to production and important taxes (line 9).
• Consumption-output ratio wn/y: I measure the nominal consumption-output ratio.
Consumption is nondurables and services, National Income and Product Accounts Table 1.1.5, Rows 4 and 5. Output is Gross Domestic Product, line 1.

• Labor Wedge τ̂ : I measure the labor wedge using equation (1.12), assuming ε = 1 andsetting the disutility of work to ensure that the labor wedge is 0.4 on average.

I construct quarterly data from 1959 to 2007. I detrend the data using an HP filter withparameter 1600 (Table A.1). I also express the data as annual growth rates (Table A.2).

Bibliography
Aguiar, Mark and Erik Hurst. 2005. “Consumption versus Expenditure.” Journal of Political
Economy 113 (5):919–948.
———. 2007. “Measuring Trends in Leisure: The Allocation of Time over Five Decades.”Quarterly Journal of Economics 122 (3):969–1006.
Andolfatto, David. 1996. “Business Cycles and Labor-Market Search.” American Economic
Review 86 (1):112–132.
Barro, Robert J. 1977. “Long-term contracting, sticky prices, and monetary policy.” Journalof Monetary Economics 3 (3):305–316.
Barro, Robert J. and Herschel I. Grossman. 1971. “A general disequilibrium mode of income
and employment.” American Economic Review 61 (1):82–93.
Becker, Gary S. 1965. “A Theory of the Allocation of Time.” The Economic Journal
75 (299):493–517.
Benassy, Jean-Pascal. 1982. The Economics of Market Disequilibrium. Academic Press New
York.
Bils, Mark, Chang Yongsung, and Sun-Bin Kim. 2007. “Comparative Advantage in Cyclical
Unemployment.” NBER Working Paper 13231.
Blanchard, Olivier and Roberto Perotti. 2002. “An Empirical Characterization of the Dynamic Effects of Changes in Government Spending and Taxes on Output.” Quarterly
Journal of Economics 117 (4):1329–1368.
Blanchard, Olivier J. and Jordi Galı́. 2006. “A New Keynesian Model with Unemployment.”
MIT Mimeo.
Blanchard, Olivier Jean and Peter Diamond. 1989. “The Beveridge Curve.” Brookings Papers
on Economic Activity 1989 (1):1–76.
153

154

BIBLIOGRAPHY

Blundell, Richard and Thomas MaCurdy. 1999. “Labor Supply: A Review of Alternative
Approaches.” Handbook of Labor Economics 3 (1):1559–1695.
Chari, V.V., Patrick J. Kehoe, and Ellen R. McGrattan. 2007. “Business Cycle Accounting.”
Econometrica 75 (3):781–836.
Cole, Harold L. and Richard Rogerson. 1999. “Can the Mortensen-Pissarides Matching Model
Match the Business-Cycle Facts?” International Economic Review 40 (4):933–959.
Cooley, Thomas F. and Edward C. Prescott. 1995. “Economic Growth and Business Cycles.”In Frontiers of Business Cycle Research, edited by Thomas F. Cooley. Princeton University
Press, 1–38.
Costain, James S. and Michael Reiter. 2008. “Business Cycles, Unemployment Insurance,
and the Calibration of Matching Models.” Journal of Economic Dynamics and Control
32 (4):1120–1155.
Davis, Steven J., John C. Haltiwanger, and Scott Schuh. 1996. Job Creation and Destruction.
MIT Press.
Elsby, Michael W., Ryan Michaels, and Gary Solon. 2007. “The Ins and Outs of Cyclical
Unemployment.” NBER Working Paper No. 12853.
Erceg, Christopher J., Dale W. Henderson, and Andrew T. Levin. 2000. “Optimal Monetary Policy with Staggered Wage and Price Contracts.” Journal of Monetary Economics
46 (2):281–313.
Farmer, Roger E. A. and Andrew Hollenhorst. 2006. “Shooting the Auctioneer.” NBER
Working Paper 12584.
Fisher, Jonas D. M. 2006. “The Dynamic Effects of Neutral and Investment-Specific Technology Shocks.” Journal of Political Economy 114 (3):413–451.
Fujita, Shigeru and Garey Ramey. forthcoming. “The Cyclicality of Separation and Job
Finding Rates.” International Economic Review .
Galı́, Jordi, Mark Gertler, and J. David López-Salido. 2007. “Markups, Gaps, and the Welfare
Costs of Business Fluctuations.” The Review of Economics and Statistics 89 (1):44–59.
Galı́, Jordi and Pau Rabanal. 2004. “Technology Shocks and Aggregate Fluctuations: How
Well Does the RBC Model Fit Postwar US Data?” NBER Macroeconomics Annual 19:225–
318.

155

References

Gertler, Mark and Antonella Trigari. forthcoming. “Unemployment Fluctuations with Staggered Nash Wage Bargaining.” Journal of Political Economy .
Haefke, Christian, Marcus Sonntag, and Thijs van Rens. 2008. “Wage Rigidity and Job
Creation.” IZA Discussion Paper 3714.
Hagedorn, Marcus and Iourii Manovskii. 2008. “The Cyclical Behavior of Equilibrium Unemployment and Vacancies Revisited.” American Economic Review 98 (4):1692–1706.
Hall, Robert E. 1995. “Lost Jobs.” Brookings Papers On Economic Activity 1:221–256.
———. 1997. “Macroeconomic Fluctuations and the Allocation of Time.” Journal of Labor
Economics 15 (1):223–250.
———. 2005. “Employment Fluctuations with Equilibrium Wage Stickiness.” American
Economic Review 95 (1):50–65.
———. 2009. “Reconciling Cyclical Movements in the Marginal Value of Time and the
Marginal Product of Labor.” Stanford University Mimeo.
Hall, Robert E. and Paul R. Milgrom. 2008. “The Limited Influence of Unemployment on
the Wage Bargain.” American Economic Review 98 (4):1653–1674.
Hosios, Arthur J. 1990. “On the Efficiency of Matching and Related Models of Search and
Unemployment.” Review of Economic Studies 57 (2):279–98.
Kaldor, Nicholas. 1957.
67 (268):591–624.

“A Model of Economic Growth.”

The Economic Journal

Kennan, John. 2006. “Private Information, Wage Bargaining and Employment Fluctuations.”
NBER Working Paper 11967.
Krause, Michael U. and Thomas A. Lubik. 2007. “The (Ir) relevance of Real Wage Rigidity
in the New Keynesian Model with Search Frictions.” Journal of Monetary Economics
54 (3):706–727.
Krusell, Per, Toshihiko Mukoyama, and Ayşegül Şahin. 2007. “Labor-Market Matching with
Precautionary Savings and Aggregate Fluctuations.” Princeton University Mimeo.
Krusell, Per, Lee E. Ohanian, José Vı́ctor Rı́os-Rull, and Giovanni L. Violante. 2000.
“Capital-skill Complementarity and Inequality: A Macroeconomic Analysis.” Econometrica 68 (5):1029–1053.

156

BIBLIOGRAPHY

Kydland, Finn E. and Edward C. Prescott. 1982. “Time to Build and Aggregate Fluctuations.” Econometrica 50 (6):1345–1370.
Ljungqvist, Lars and Thomas J. Sargent. 2004. Recursive Macroeconomic Theory. MIT
Press, second ed.
Malinvaud, Edmond. 1977. The Theory of Unemployment Reconsidered. Halsted Press.
Mankiw, N. Gregory. 1989. “Real Business Cycles: A New Keynesian Perspective.” Journalof Economic Perspectives 3 (3):79–90.
Mertens, Karel and Morten O. Ravn. 2008. “The Aggregate Effects of Anticipated andUnanticipated US Tax Policy Shocks: Theory and Empirical Evidence.” Cornell University
Mimeo.
Merz, Monika. 1995. “Search in the Labor Market and the Real Business Cycle.” Journal of
Monetary Economics 36 (2):269–300.
Modigliani, Franco. 1977. “The Monetarist Controversy or, Should We Forsake Stabilization
Policies?” American Economic Review 67 (2):1–19.
Moen, Espen R. 1997. “Competitive Search Equilibrium.” Journal of Political Economy
105 (2):385–411.
Mortensen, Dale T. 1982. “Property Rights and Efficiency in Mating, Racing, and Related
Games.” American Economic Review 72 (5):968–979.
Mortensen, Dale T. and Éva Nagypál. 2007. “More on Unemployment and Vacancy Fluctuations.” Review of Economic Dynamics 10 (3):327–347.
Mortensen, Dale T. and Christopher A. Pissarides. 1994. “Job Creation and Job Destructionin the Theory of Unemployment.” The Review of Economic Studies 61 (3):397–415.
Mulligan, Casey B. 2002. “A Century of Labor-Leisure Distortions.” NBER Working Paper
8774 .
Nakajima, Makoto. 2008. “Business Cycles in the Equilibrium Model of Labor Market Searchand Self-Insurance.” University of Illinois at Urbana-Champaign Mimeo.
Nash, John. 1953. “Two-Person Cooperative Games.” Econometrica 21 (1):128–140.

References

157

Parkin, Michael. 1988. “A Method for Determining Whether Parameters in Aggregative
Models Are Structural.” Carnegie-Rochester Conference Series on Public Policy 29:215–
252.
Petrongolo, Barbara and Christopher A. Pissarides. 2001. “Looking into the Black Box: A
Survey of the Matching Function.” Journal of Economic Literature 39 (2):390–431.
Pissarides, Christopher A. 1985. “Short-Run Equilibrium Dynamics of Unemployment, Vacancies, and Real Wages.” The American Economic Review 75 (4):676–690.
———. 2000. Equilibrium Unemployment Theory. Cambridge, MA: MIT Press, second ed.
———. forthcoming. “The Unemployment Volatility Puzzle: Is Wage Stickiness the Answer?” Econometrica .
Prescott, Edward C. 2004. “Why Do Americans Work So Much More Than Europeans?”
Federal Reserve Bank of Minneapolis Quarterly Review 28 (1):2–13.
Prescott, Edward C., Alexander Ueberfeldt, and Simona Cociuba. 2008. “U.S. Hours andProductivity Behavior: Using CPS Hours Worked Data: 1959-I to 2007-IV.” Mimeo,
February.
Pries, Michael J. 2004. “Persistence of Employment Fluctuations: A Model of Recurring Job
Loss.” Review of Economic Studies 71 (1):193–215.
Ramey, Valerie A. and Neville Francis. forthcoming. “A Century of Work and Leisure.”
American Economic Journal: Macroeconomics .
Romer, Christina D. and David H. Romer. 2007. “A Narrative Analysis of Postwar Tax
Changes.” University of California, Berkeley Mimeo.
Rotemberg, Julio J. 2006. “Cyclical Wages in a Search-and-Bargaining Model with Large
Firms.” NBER Working Paper 12415.
Rotemberg, Julio J. and Michael Woodford. 1991. “Markups and the Business Cycle.” NBER
Macroeconomics Annual :63–129.
———. 1997. “An Optimization-Based Econometric Framework for the Evaluation of Monetary Policy.” NBER Macroeconomics Annual 12:297–346.
———. 1999. “The Cyclical Behavior of Prices and Costs.” In Handbook of Macroeconomics,
Handbook of Macroeconomics, vol. 1, edited by John B. Taylor and Michael Woodford,
chap. 16. Elsevier, 1051–1135.

158

BIBLIOGRAPHY

Rudanko, Leena. 2008. “Aggregate and Idiosyncratic Risk in a Frictional Labor Market.”
Boston University Mimeo.
———. forthcoming. “Labor Market Dynamics under Long-Term Wage Contracting.” Journal of Monetary Economics .
Shimer, Robert. 1996. Essays in Search Theory. Ph.D. thesis, Massachusetts Institute ofTechnology.
———. 2005. “The Cyclical Behavior of Equilibrium Unemployment and Vacancies.” American Economic Review 95 (1):25–49.
———. 2007. “Reassessing the Ins and Outs of Unemployment.” University of Chicago
Mimeo.
Silva, José and Manuel Toledo. 2009. “Labor Turnover Costs and the Cyclical Behavior of
Vacancies and Unemployment.” Macroeconomic Dynamics Forthcoming.
Smets, Frank and Rafael Wouters. 2003. “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area.” Journal of the European Economic Association 1 (5):1123–
1175.
———. 2007. “Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach.”American Economic Review 97 (3):586–606.
Yashiv, Eran. 2006. “Evaluating the Performance of the Search and Matching Model.”European Economic Review 50 (4):909–936.