ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E17-2

Employment and Hours over the Business
Cycle in a Model with Search Frictions
byNoritaka Kudoh
Hiroaki Miyamoto
Masaru Sasaki

May 2017

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Employment and Hours over the Business Cycle in a Modelwith Search Frictions1
Noritaka Kudoh2

Hiroaki Miyamoto3

Masaru Sasaki4

Nagoya University

International Monetary Fund

Osaka University

May 9, 2017

1 We thank Yukiko Abe, Jim Albrecht, Junichi Fujimoto, Shigeru Fujita, Takatoshi Ito, Tokuo Iwaisako,

Ryo Kambayashi, Daiji Kawaguchi, John Kennan, W. Bentley MacLeod, Daisuke Oyama, Hugh Patrick, Etsuro Shioji, Nori Tawara, David E. Weinstein, and seminar participants at Columbia University, 2015 Econometric Society World Congress, Hitotsubashi University, Hokkaido University, JEA2014, Kyoto University,
Meiji University, Nagoya University, Otaru University of Commerce, SWET2013, Tokyo Labor Economics
Workshop, University of Tokyo, and Yokohama National University for comments and suggestions. We
also thank Makoto Kakinaka and Nao Sudo for providing data. Part of this research is financially supported
by KAKENHI (26380230 and 26380248).
2 E-mail: kudoh@soec.nagoya-u.ac.jp
3 E-mail: HMiyamoto@imf.org
4 E-mail: sasaki@econ.osaka-u.ac.jp

Abstract
This paper presents a labor market search-matching model with multi-worker firms to investigate
how firms utilize employment and hours of work over the business cycle. Each firm and its
employees bargain over an earnings schedule that maps hours into earnings, and this scheduledetermines the costs of utilizing the two margins of labor adjustment. We calibrate the model for
the Japanese labor market, in which fluctuations in hours of work account for 79 percent of thevariations in the aggregate labor input. We find that the aggregate labor input in the model is as
volatile as that in the data, and is 25 times as volatile as that in the model without hours of work.
Interestingly, with bargained hours of work, the model generates significantly small fluctuations
in employment, unemployment, and vacancies, and is outperformed by the model with constanthours of work.
JEL classification: E32, J20, J64.
Keywords: search, hours of work, employment, business cycles, multi-worker firms.

1Introduction

Firms adjust their labor inputs over the business cycle through the intensive margin (hours ofwork per employee) and the extensive margin (the number of employees). In their survey, Hall etal. (2000) found that 62 percent of firms consider overtime as the primary reaction to a demandboom. In a perfectly competitive labor market, firms do not need to utilize overtime because
they can employ extra workers instantly at the going wage rate. This suggests the importance offrictions in understanding how firms utilize the intensive and extensive margins over the businesscycle.
In this paper, we study the composition of labor demand over the business cycle by focusing
on the fact that firms need to engage in time-consuming search-matching process when hiring
new employees while changes in hours of work per employee are instantaneous. Our basic modelbuilds on a recent development in labor market search models with multi-worker firms.1 A novel
property of our model is that firms facing productivity shocks choose both vacancies and hours
of work per employees.
Introduction of endogenous working hours per employee changes the search-matching modelin a nontrivial manner because total compensation depends on hours of work as well as the hourly
wage rate. Further, the hourly wage rate itself may depend on hours of work. To capture thepossibility of nonlinear compensation scheme, we assume that a firm and its employees bargain
over a state-contingent contract in the form of an earnings schedule which maps hours of work
per employee into earnings (Cooper et al., 2007; Kudoh and Sasaki, 2011). The earnings scheduleturns out to be a convex function of hours of work, and this specifies the marginal hourly wage rate
for the firm of choosing hours of work.
In our framework, there is a clear distinction between earnings and the wage rate. The earnings equation is similar to the wage equation in the standard search-matching model, and thus
1 A partial list of related contributions is Bertola and Caballero (1994), Smith (1999), Cahuc and Wasmer (2001),

Koeniger and Prat (2007), Cahuc et al. (2008), Felbermayr and Prat (2011), Kudoh and Sasaki (2011), and Acemoglu andHawkins (2014). Leading examples of business cycle studies using the multi-worker paradigm are Cooper et al. (2007),
Elsby and Michaels (2013), Krause and Lubik (2013), and Fujita and Nakajima (2016).

1

depends on the current labor market tightness as well as productivity. However, the marginalhourly wage rate is independent of labor market tightness. As a result, the steady-state hours
of work per employee are determined primarily by the level of TFP without any reference to theparameters related to search frictions. This suggests that search frictions are irrelevant for understanding the long-run trend of hours per worker. In contrast, the steady-state level of employment
is influenced by both TFP and search frictions because the marginal gain from creating an additional vacancy depends directly on the vacancy-filling probability, which is determined by the
degree of search frictions.
We calibrate our model to match the Japanese labor market facts. An important characteristicof labor market fluctuations in Japan is that the intensive margin accounts for a particularly large
proportion of cyclical fluctuations in the aggregate labor input. In our empirical analysis, wefind that variations in hours of work per employee account for 79 percent of the variations in theaggregate labor input, while variations in the number of employees account for 21 percent of thevariations.2 Clearly, in understanding labor market fluctuations in Japan, it is a serious omission
if one uses a model without variable hours of work. In our numerical analysis, we show that
our model replicates much of the observed magnitudes of fluctuations in total labor input, hours
of work per employee, employment, unemployment, and vacancies. In particular, variations inhours of work per employee account for 84 percent of the variations in the aggregate labor input,
while variations in the number of employees account for 19 percent of the variations.3
To clarify the importance of hours of work, we study our model without hours of work, andfind that the aggregate labor input in the model with variable hours of work fluctuates 25 times asmuch as that in the model without hours of work. Introduction of hours of work also magnifies
labor adjustment along the extensive margin: unemployment and vacancies in the model with variable hours of work fluctuate 4 times as much as those in the model without hours of work. These
results are obtained without wage rigidity or additional shocks. This is in sharp contrast to the2 This is in sharp contrast with the labor market fluctuations in the U.S., in which 79 percent of the variations areaccounted for by the extensive margin.
3 Our decomposition of variations admits some discrepancy so that the sum of variations may deviate from unity.

2

existing literature, which suggests inability of the textbook search-matching model in generating
the observed magnitudes of fluctuations in unemployment and vacancies in the US, known as theunemployment volatility puzzle (Shimer, 2005).
Using a model nearly identical to ours, but without hours of work, Krause and Lubik (2013)
investigate whether multi-worker paradigm with Stole and Zwiebel’s (1996) intra-firm bargaining
helps resolve the unemployment volatility puzzle, and conclude that while there is an improvement, the effect is negligible. Why do the two similar models generate distinct results?
Our model’s ability to generate high volatilities in unemployment and vacancies comes partly
from the presence of disutility from hours of work. We find that disutility from hours of work
plays a role similar to having a high target value of unemployment benefit in calibration because
it raises the level of compensation and reduces the firm’s profit, making the job-creation incentive more exposed to external shocks. In addition, introduction of hours of work introduces the
Frisch elasticity parameter into the analysis, and this magnifies the link between hours of work anddisutility. Indeed, we verify that the magnitudes of fluctuations in unemployment and vacancies
depend significantly on the Frisch elasticity.
Interestingly, to generate fluctuations along the extensive margin, it matters who chooses hours
of work. We show that if hours of work are determined in bargaining, then the volatilities in
employment, unemployment, and vacancies become about 1/3 of those from the basic model.
The model in which hours of work are chosen by workers shares the same property. Further, thesemodels are outperformed by the model in which hours of work are held constant. We therefore
conclude that our maintained assumption that firms choose hours of work from the bargained
contract serves as a useful benchmark for understanding hours of work over the business cycle.
Since this paper deals with general issues, there is a long list of related studies. Therefore, thedescription of the literature below must necessarily be partial.
We intend our empirical analysis to be a contribution to the growing literature on re-examinationof hours of work using new models and new datasets. This includes Rogerson (2006) and Ohanianand Raffo (2012), to name a few. Particularly relevant is Ohanian and Raffo (2012), who find that3the intensive margin is increasingly more important for labor adjustment in 14 OECD countries
they studied. Using the dataset we built for the Japanese labor market, our empirical analysis
confirms the importance of the intensive margin.
In the labor hoarding literature and more generally the factor utilization literature, it is often argued that real business cycle models with costly labor adjustment display realistic cyclicalfluctuations. Burnside et al. (1993), for instance, assume that it takes one period to adjust labor input. Our model naturally possesses the same property because the level of employment increases
only after firms post vacancies and the matching process takes one period. Contrary to the factor
utilization literature, in which the cost function for factor utilization is exogenously given, in ourmodel, the cost that a firm faces when choosing longer hours of work is endogenously determinedby intra-firm bargaining.
In terms of the structure of the model, particularly related to our study are Cooper et al. (2007)
and Kudoh and Sasaki (2011), in which determination of hours of work is considered in the context of a search-matching model with multi-worker firms.4 Cooper et al. (2007) study both employment and hours of work over the business cycle using a model similar to ours. They emphasize the importance of nonlinear cost of posting vacancies, and as a result, wage determination
is simplified by assuming a take-it-or-leave-it offer protocol. While we restrict our analysis to alinear vacancy cost, our model is more general in that we adopt the Stole-Zwiebel bargaining, as
in a steady-state economy developed by Kudoh and Sasaki (2011). The resulting earnings schedule reflects changes in the current state variables. As a result, our model closely replicates themagnitudes of fluctuations in both earnings and the hourly wage rate in the data, both of which
fluctuate as much as TFP.
Our model is also closely related to the model developed by Fang and Rogerson (2009), whostudy the intensive and extensive margins using the framework of Merz (1995) and Andolfatto
4 In a recent, independent contribution, Trapeznikova (2015) takes up the issue similar to ours. The model developedin Trapeznikova (2015) introduces on-the-job search with an intra-firm bargaining that is different from the StoleZwiebel to ensure efficiency of the bargaining outcome, and the model is calibrated to match the Danish labor market
to replicate the firm-level facts.

4

(1996). In this model, the production unit is a matched worker-job pair, and this rules out the issue
of intra-firm bargaining. In addition, the Merz-Andolfatto paradigm has a utility-maximizing
household, who optimally chooses the level of consumption. Fang and Rogerson (2009) show
that, with a concave utility function, there is a rich interaction between employment and hours
through consumption, and as a result, an increase in the vacancy cost decreases employment andincreases hours of work. This rich interaction comes from the labor-supply side. In contrast,
we focus on the labor-demand side, and show that an increase in the vacancy cost decreases thesteady-state level of employment, but it has no effect on hours of work in the steady state.
Using frictionless, competitive business cycle models, Kydland and Prescott (1991) and Cho
and Cooley (1994) study hours and employment over the business cycle, and calibrate their models to match the US labor market facts. Braun et al. (2006) adopt the Cho-Cooley framework to
study the labor market in Japan. In the Appendix, we compare a version of the Cho-Cooley modelwith our model, and show that, in the steady state, hours of work per employee are longer in ourfrictional economy than in the frictionless economy. Thus, while determination of the steady-statehours of work is independent of the parameters capturing the degree of search frictions, the very
presence of frictions affects hours of work in the long-run.
The remainder of the paper is organized as follows. In Section 2, we empirically examine
how labor inputs are adjusted over the business cycle in Japan, and present some other cyclicalcharacteristics of the labor market in Japan. Section 3 describes our basic model, followed bycharacterization of equilibrium of the model in Section 4. In Section 5, we calibrate the model
parameters and present the business cycle properties of our model. Section 6 presents the results
under alternative parameter values and assumptions, and Section 7 concludes. Proofs and some
additional results are found in the Appendix.

5

2

Labor Market Facts

2.1

Data

This section presents some empirical results highlighting the cyclical properties of the Japaneselabor market. We obtain the series of the number of employed workers and the number of labor
force from the Labour Force Survey (LFS), conducted by the Statistics Bureau and the DirectorGeneral for Policy Planning. The series of the average monthly hours worked per worker are
obtained from the Monthly Labour Survey (MLS) conducted by the Ministry of Health, Labourand Welfare (MHLW). We construct our measure of the aggregate labor input as the product of theaverage monthly hours worked per worker and the number of employed workers, normalized by
the labor force. These measures are consistent with those used in Ohanian and Raffo (2012).
The TFP series are from Braun et al. (2006).5 We obtain the unemployment rate series fromthe LFS. The vacancy rate series are obtained from the Monthly Report on Employment Service(Shokugyo Antei Gyomu Tokei) conducted by the MHLW. Following Miyamoto (2011) and Lin
and Miyamoto (2012), we construct the job finding and separation rates from the LFS. The series ofreal earnings and real wages are constructed from the nominal earnings series from the Monthly
Labour Survey (MLS) conducted by the MHLW. Following the convention, we use the consumer
price index (CPI) to obtain the real series.
Our data are quarterly, which, when necessary, are obtained by averaging or aggregating thecorresponding monthly series. The sample covers 1980Q2–2010Q4. All series are seasonally adjusted using the Census Bureau’s X12 ARIMA procedure and transformed by taking natural logarithms. Since our focus is on cyclical fluctuations in the series, the low-frequency movements inthe data are filtered out by using the Hodrick-Prescott (HP) filter with smoothing parameter 1600.

2.2

Labor Market Fluctuations in Japan

Figure 1 plots the cyclical components of the aggregate labor input, hours of work per worker,
and the number of employed workers. The figure shows that the aggregate labor input and its
5 The TFP series are extended to 2010 by Nao Sudo, and we use the extended series.

6

0.025
Total hours worked
Hours worked per worker
Number of employed workers

0.02

Deviation from trend

0.015
0.01
0.005
0
-0.005-0.01-0.015-0.02-0.025
1980

1985

1990

1995
Year

2000

2005

2010

Figure 1: Total hours worked and its components over business cycles in Japan.
Note: The solid line indicates the cyclical component of total hours of worked. The dash-dottedline indicates the cyclical component of hours of worked per worker. The dashed line indicatesthe cyclical components of employed workers. The cyclical components are obtained by using theHP filter with smoothing parameter 1600. Sample covers 1980Q2-2010Q4.

7

components fluctuate significantly over the business cycle, and both hours and employment comove with the aggregate labor input. It also indicates that the aggregate labor input comoves
more closely with hours of work per worker than with employment, and that employment is lessvolatile than hours per worker.
Table 1: Summary statistics, quarterly Japanese data, 1980-2010
Variables

û

v̂

fˆ

ŝ

Ŵ

ŵ

t̂

ĥ

lˆ

Â

ŷ

Standard deviation

0.059

0.096

0.091

0.092

0.012

0.010

0.009

0.008

0.004

0.011

0.016

û

1

-0.777-0.4200.449-0.317-0.084-0.526-0.325-0.558-0.281-0.734

v̂

-

1

0.320

-0.517

0.656

0.113

0.728

0.644

0.378

0.470

0.781

fˆ

-

-

1

-0.344

0.192

0.123

0.201

0.139

0.185

0.005

0.215

ŝ

-

-

-

1

-0.3650.035-0.416-0.422-0.108-0.371-0.501

Ŵ

-

-

-

-

1

0.751

0.554

0.507

0.253

0.287

0.436

ŵ

-

-

-

-

-

1

-0.023-0.121

0.193

0.002

0.003

t̂

-

-

-

-

-

-

1

0.902

0.485

0.359

0.741

ĥ

-

-

-

-

-

-

-

1

0.060

0.392

0.602

lˆ

-

-

-

-

-

-

-

-

1

0.038

0.493

Â

-

-

-

-

-

-

-

-

-

1

0.680

ŷ

-

-

-

-

-

-

-

-

-

-

1

Correlation matrix

Table 1 quantifies what we see in Figure 1 and provides more information by summarizing
cyclical characteristics of the key labor market variables such as the unemployment rate (u), thevacancy rate (v), the job-finding rate ( f ), the separation rate (s), total earnings (W), the real wage
rate (w), total hours of work or the aggregate labor input (t), hours of work per employee (h),the number of employed workers (l), total factor productivity (A), and output (y). All variables
are with hat, meaning that they are in percentage deviations from their trend levels. Since theseries are in natural logarithms, the standard deviations can be interpreted as mean percentage

8deviations from their trend levels.

Hours and Employment

Consistent with Figure 1, Table 1 shows a strong positive relationshipbetween total hours worked and hours worked per worker. The correlation between them is 0.90.
We also find a positive relationship between total hours and employment, with a correlation of
0.49. This is significantly smaller than the correlation between total hours and hours per worker.
As Braun et al. (2006) pointed out, there is no strong correlation between hours worked per worker
and employment. The correlation between them is 0.06. Table 1 also shows that the aggregate
labor input and its components comove positively with TFP and output, and comove negatively
with the unemployment rate, indicating that they are all pro-cyclical. Interestingly, the correlation
between employment and TFP is 0.04 while the correlation between hours per worker and TFP
is 0.39. The standard deviation of total hours worked is 0.9 percent, that of hours worked per
worker is 0.8 percent, and that of employment is 0.4 percent. Thus, hours of work per worker are
twice as volatile as employment. Since the standard deviations of TFP and output are 1.1 percent
and 1.6 percent, respectively, the aggregate labor input and its components are less volatile than
TFP or output.

Unemployment and Vacancy Table 1 shows that the unemployment rate is counter-cyclical andthe vacancy rate is pro-cyclical. The correlation between the unemployment rate and TFP is -0.28
and the correlation between the unemployment rate and output is -0.73. The correlation between
the vacancy rate and TFP is 0.47 and the correlation between the vacancy rate and output is 0.78.
Since the unemployment rate is counter-cyclical and the vacancy rate is procyclical, these twoseries comove negatively. The correlation between them is -0.78, which implies the Beveridge
curve. Both the unemployment rate and the vacancy rate are significantly more volatile than
TFP or output. While the standard deviations of TFP and output is 1.1 percent and 1.6 percent,respectively, the standard deviations of the unemployment and vacancy rates are 5.9 percent and9.6 percent, respectively.

9

Job Finding Rate and Separation Rate

The job finding rate is acyclical or procyclical and theseparation rate is counter-cyclical. The correlation between the job finding rate and TFP is 0.005,
which is very weak. However, the correlation between the job finding rate and output is 0.22.
The correlation between the separation rate and TFP is -0.37 and the correlation between theseparation rate and output is -0.50. The standard deviations of the job finding rate and separation
rate are 9.1 percent and 9.2 percent, respectively. Both the job finding rate and separation rate are
more volatile than TFP or output, but the magnitudes are similar.

Real Earnings and Real Wage Rate In this paper, we clearly distinguish between earnings andthe hourly wage rate. While real earnings are procyclical, the real hourly wage rate is acyclical.
The correlation between real earnings and TFP is 0.29 and the correlation between real earnings
and output is 0.44, indicating that real earnings are procyclical. However, the correlation between
the real hourly wage rate and TFP is 0.002 and the correlation between the real hourly wage rate
and output is 0.003. Real earnings and the real hourly wage rate are both positively correlatedwith employment. Interestingly, while real earnings are positively correlated with hours of work,
the real hourly wage rate is negatively correlated with hours. This negative correlation between
hours and hourly wage rate might look inconsistent with the presence of overtime payment. The
key to understanding the result is that we can only compute the series of the average hourly wage
rate, and the series of the marginal hourly wage rate are not available. The two measures do notcoincide unless earnings per worker take the following form: W = wh.

2.3

Decomposition of Fluctuations

We now study the relative contributions of the intensive and extensive margins to fluctuations inˆ variance of total hours worked can be decomposed as
the total hours worked. With t̂ = ĥ + l,
Var (t̂) = Var (ĥ) + Var (lˆ) + 2Cov(ĥ, lˆ) = Cov(t̂, ĥ) + Cov(t̂, lˆ).

(1)

The term Cov(t̂, ĥ) gives the amount of variations in t̂ that derived from variations in ĥ andˆ Similarly, the term Cov(t̂, lˆ) is the amount of variations in t̂ thatthrough its comovement with l.
10derived from variations in lˆ and through its comovement with ĥ. By dividing the both sides of (1)by Var (t̂), we obtain
1=

Cov(t̂, ĥ) Cov(t̂, lˆ)+= βh + βl ,Var (t̂)Var (t̂)(2)where βh and βl are the relative contributions of variations in ĥ and lˆ to variations in t̂. These
measures are an application of the “beta value” in finance.6
From the data, we find that βh = 0.79 and βl = 0.21. In other words, the intensive margin
explains 79 percent of variations in total hours worked and the extensive margin accounts for 21
percent of the variations. This implies that over the business cycle, Japanese firms adjust labor inputs both by the intensive and extensive margins, but they use the intensive margin more heavily
than the extensive margin.7

3

The Model

3.1

Environment

In this section, we present our basic model for explaining cyclical behaviors of employment andhours of work. The key feature of the model is that it captures the fact that while hours of work
per worker can change within a period, firms need to open (costly) vacancies in a frictional labormarket to hire new employees. To focus on the composition of the labor demand at each firm, webuild a labor market search-matching model with multi-worker firms.
Consider an economy consisting of a large number of workers and firms. The measure ofworkers is normalized to unity. Both workers and firms are homogeneous. Time is discrete andall agents discount the future at the common discount rate r.
The production technology for each firm is given by At Lαt k1t α , where 0 < α < 1, At denotes
6 Petrongolo and Pissarides (2008) and Fujita and Ramey (2009) apply this measure to decompose unemployment

fluctuations into inflow fluctuations and outflow fluctuations.
7 This result is in sharp contrast with what we find from the U.S. data. We utilize the dataset constructed by Ohanianand Raffo (2012) and find that βh = 0.21 and βl = 0.79 for the U.S. labor market. Thus, U.S. firms adjust labor inputs
mainly through the extensive margin.

11

the level of total factor productivity (TFP), k t denotes the stock of capital at each firm, and Lt
denotes the labor input (i.e., total hours worked). The level of TFP is stochastic. Assuming homogeneous labor, we postulate that Lt

ht lt , where ht is hours of work per employee and lt is thenumber of employees at each firm. Thus, output yt is given by yt = At hαt ltα k1t α .
It will be shown that, with this constant-returns-to-scale technology, the (steady-state) valueof operating a firm exactly cancels out the cost of firm entry, and as a result, firms are indifferent
between entry and exit, as in the zero-profit result in the standard neoclassical model (Proposition
2). Thus, we adopt the conventional remedy for indeterminacy of the number of firms by normalizing the measure of firms to be unity. In effect, we can ignore entry and exit of firms over thebusiness cycle and focus on our main theme, employment versus hours over the business cycle.
Note, however, that our aggregate data contains variations in employment due to changes in thenumber of firms.
Each firm possesses a technology that converts one unit of the final consumption good into aunit of investment good. Let xt be the level of investment made in period t. Then the stock of
capital evolves according to k t+1 = (1δk )k t + xt , where δk is the rate of capital depreciation.

All workers are risk neutral, and maximize the expected lifetime utility, given by∞E0 ∑ δt [ Ite (ht )] ,t =0

where δ1/(1 + r ) is the discount factor, It denotes income, and e(ht ) represents disutility ofwork. We assume that e0 ( ) > 0, e00 ( ) > 0 and limh!∞ e(h) = ∞. Our specification of disutility
function ise ( h ) = e0h 1+ µ,1+µ(3)

where e0 > 0 and 1/µ is the Frisch elasticity.
The labor market is frictional. The number of matches in period t is determined by the matching technology m0 U ξ V 1 ξ , where m0 > 0 and 0 < ξ < 1 are parameters, Ut is the total numberof job seekers, and Vt is the number of aggregate job vacancies. Let Vt /Utθt denote the labormarket tightness. A vacancy is matched to a worker during a period with probability qt , whereξ

1 ξ

qt = m0 Ut Vt/Vt = m0 θt
12ξq (θt ) .

(4)

This is referred to as the vacancy filling rate. It is easy to verify that an increase in labor market
tightness θt decreases this probability. Similarly, the probability that a worker is matched with aξ1 ξvacancy, or the job finding rate, is given by m0 Ut Vt

1 ξ/Ut = m0 θt= θt q(θt ). This probability is

increasing in θt .
We assume exogenous separations. At the end of each period, a fraction λ of the current
employees are assumed to leave the firm. Since the firm creates vt units of vacancies, the numberof new employees for the next period is q(θt )vt . These new employees are not hit by the separation
shock. Thus, the number of employees at each firm evolves according to lt+1 = (13.2

λ ) lt + q ( θ t ) v t .

TimingLet St = ( At , lt , k t , Ut ) be the set of state variables. Among the state variables, the level of At isrevealed at the beginning of each period. Given the state variables, each firm and its employees
bargain over a state-contingent contract that specifies an earnings schedule, which maps h into an
amount of compensation W (Kudoh and Sasaski, 2011). At this point, the level of earnings W
is not realized because hours of work are determined only after the earnings schedule is agreed
upon. The bargaining outcome is summarized by W (ht ; St ). With this schedule, the firm chooses
hours of work per employee (ht ), vacancies to create (vt ), and the level of capital investment (xt ).
Then, production takes place and output yt is realized. Finally, λlt of the current employees leave
the firm, and q(θt )vt workers are newly employed. It is important to note that a firm cannot
choose hours of work before the bargaining stage because the cost structure is realized only after
the earnings schedule is agreed upon.8

3.3

Firms

We solve the firm’s optimization problem by stationary dynamic programming, taking this period’s bargaining outcome W (h; S) as given, which reflects the fact that the set of state variables
8 In addition, a firm has no incentive to choose h in advance because doing so cannot manipulate the bargaining

outcome. In contrast, a firm does have an incentive to manipulate the bargaining outcome by choosing a large l,
generating the well-known overhiring result (Smith, 1999).

13

S and the state-contingent contract W (h; S) are known at the time of the firm’s optimization. The
W (h; S)linstantaneous payoff to a firm is given by yx, where γ > 0 is the (constant)γv

cost of posting a vacancy. Let l+1 and k +1 denote the levels of employment and capital in the next
period, respectively. The value of a firm J (S) satisfies the following Bellman equation:n
J (S) = max Ahα l α k1 αh,v,xW (h; S)lox + δEJ (S+1 ) ,

γv

(5)

where
l +1 = (1λ)l + q(θ )v,(6)

k +1 = ( 1δk ) k + x.

(7)

We assume that, as in Cooper et al. (2007), the firm chooses hours of work per employee.9 Models with alternative assumptions regarding how hours of work are determined are presented in
Section 5.3.
The first-order conditions with respect to h, v, and x imply
αAhα 1 l α 1 k1 α = Wh (h; S),
δEJl (S+1 ) =

(8)

γ
,
q(θ )

(9)

δEJk (S+1 ) = 1.

(10)

The envelope conditions yieldJl (S) = αAhα l α 1 k1 α
Jk (S) = (1W (h; S)α) Ahα l α k αWl (h; S)l + (1Wk (h; S)l + (1λ) δEJl (S+1 ) ,δk ) δEJk (S+1 ) .

(11)
(12)

9 Although direct evidence is hard to come by, we note here that there is a large discrepancy between the estimates ofthe labor supply elasticity. Namely, the micro elasticity, which primariry reflects the intensive margin of labor supply,is much smaller than the macro elasticity, which reflacts both the intensive and extensive margins. We intrepret thisfact as indirect evidence that while labor market paricipation (extensive margin of labor supply) is chosen by eachindividual, labor supply conditional on employment (i.e., hours of work per employee) is driven by the demand side.

14

Substitute (9) and (10) into (11) and (12) to obtain
Jl (S) = αAhα l α 1 k1 α
Jk (S) = (1W (h; S)α) Ahα l α k αWl (h; S)l +Wk (h; S)l + 1

(1

λ) γ
,
q(θ )

(13)

δk ,

(14)

λ)δEJ E (S+1 ) ,(15)

respectively.

3.4

Workers

The value of being employed, J E (S), satisfies
J E (S) = W (h; S)e (h) + λδEJ U (S+1 ) + (1where J U (S) is the value of being unemployed. Note that hours of work are determined by thefirm, and therefore the worker takes h and the level of disutility e(h) as given. The value of being
unemployed can be written as
J U (S) = b + θq(θ )δEJ E (S+1 ) + (1

θq(θ ))δEJ U (S+1 ) ,(16)

where b is the unemployment benefit. Since disutility from long working hours is captured by
the disutility function, our b does not include the value of leisure. Thus, it primarily reflects theunemployment insurance provided by the government.

3.5

Earnings Schedule

We assume that at the beginning of each period, workers and a firm bargain over a state-contingent
contract, which takes in the form of an earnings schedule W (h; S) that maps hours of work into earnings. It is important to emphasize that the exact amount of earnings W and hours of work h are
not determined in the bargaining stage. We assume that workers are not unionized, and eachworker is treated as a marginal worker (Stole and Zwiebel, 1996).
Consider a bargaining between a firm and a group of workers of measure ∆. The threat point
for the firm is J ( A, l∆, k, U ) because failing to agree on a contract implies losing the workers.

The total match surplus is therefore J ( A, l, k, U )15

J ( A, l∆, k, U ) + ∆( J E (S)J U (S)). If theβ 2 [0, 1], then the bargaining outcome must satisfy

firm’s bargaining power is given by 1β[ J ( A, l, k, U )∆, k, U )] = (1J ( A, lβ)∆[ J E (S)βJl (S) = (1J U (S)]. In the limit as ∆ ! 0,hβ) J E (S)i
J U (S) .

(17)

This is the key equation for rent sharing. Note that this amounts to maximizing the asymmetric
Nash product [ Jl (S)]1 β [ J E (S)J U (S)] β with respect to W (h; S).

Proposition 1 The earnings schedule is given by
W (h; S) =αβAhα l α 1 k1 α+ (1αβ + 1 ββ) [e (h) + b] + βγθ.

(18)

Proof. See Appendix A.
The earnings schedule (18) is a natural extension of the one derived in Kudoh and Sasaki(2011). If the worker’s bargaining power β is zero, then (18) takes a very simple form: W (h; S) =e (h) + b. This makes the worker indifferent between the states of employment and unemployment. Similarly, if the firm’s bargaining power is zero (β = 1), then we obtain W (h; S)l = y + γθl,forcing the firm to pay more than what it produces. This suggests an upper bound on β for existence of equilibrium.
From (18), we obtain
Wh (h; S) =α2 βAhα 1 l α 1 k1 α+ (1αβ + 1 ββ) e0 (h) > 0,

Whh (h; S) =(1α)α2 βAhα 2 l α 1 k1 α+ (1αβ + 1 β

Wl (h; S) =(1α)αβAhα l α 2 k1 α< 0,
αβ + 1 β

Wk (h; S) = (1α)αβAhα l α 1 k α> 0.
αβ + 1 ββ) e00 (h) ,

(19)
(20)
(21)
(22)

The key result here is that the marginal hourly wage rate given by (19) is nonlinear in hours ofwork, and is influenced by the marginal product of hours per worker (αAhα 1 l α k1 α /l) and themarginal disutility from longer hours of work. The influence of the former (latter) increases (decreases) with β. Expression (20) suggests that, when the disutility function is sufficiently convex
16and the worker’s bargaining power β is sufficiently small, the earnings schedule itself becomes
convex in hours of work (Whh > 0).
Another key result is that the level of earnings is decreasing in the number of employees(Wl < 0). This property induces the firm to employ too many workers in order to cut the wage
rate, known as the overhiring effect (Smith, 1999). It is interesting to observe from (18) and (19)that labor market tightness θ has no effect on the marginal hourly wage rate, while it influences
the level of earnings. This will imply that the labor market conditions summarized by θ has nodirect impact on the choice of hours of work.
Using (19)–(22), we rewrite (13) and (14) asJl (S) =(1 β ) αAhα l α 1 k1 α (1 β) [e (h) + b]
αβ + 1 β(1 β ) (1 α ) α α αJk (S) =Ah l k + 1
αβ + 1 β

βγθ +

(1

λ) γ
,
q(θ )

δk .

(23)
(24)

Expression (23) determines the marginal value of the firm.

4

Equilibrium

4.1

Definition

We look for a rational expectations equilibrium in which TFP follows an exogenous stochastic
process. Below, we define equilibrium of the model as a system of stochastic difference equations.
From (8) and (19), we obtainααβ + 1βAt htα 1 ltα 1 k1t α = e0 (ht ) ,(25)

which governs ht . Substitute (23) into (9) to obtain the job creation equation:Et(1 β ) αAt+1 hαt+1 ltα+11 k1t+1ααβ + 1 β(1β ) [ e ( h t +1 ) + b ]βγθt+1 +(1 λ ) γq ( θ t +1 )=(1 + r ) γ,q(θt )(26)

which determines the demand for lt . Equations (25) and (26) summarize the firm’s optimal choice
regarding hours of work and employment. The evolution of employment follows lt+1 = (1λ)lt + q(θt )vt , which determines the the firm’s vacancy vt .
17

Similarly, substitute (24) into (10) to obtainEt(1 β ) (1 α )At+1 hαt+1 ltα+1 k t+α1 + 1
αβ + 1 βδk= 1 + r.

This determines the demand for capital. The evolution of capital stock is given by k t+1 = (1(27)δk ) k t +xt , which determines investment xt .
The aggregate variables are determined as follows. In this economy, the number of the unemployed Ut equals the the rate of unemployment because the labor force is normalized to unity.
In each period, θt q (θt ) Ut job seekers find jobs. Similarly, the aggregate number of employees is
Ut , from which the aggregate number of separations is λ(11Ut ). Thus, the number of theunemployed evolves according to
Ut+1

Ut = λ (1Ut )θt q (θt ) Ut .

(28)

In any steady state, the flow into employment θq(θ )U must equal the flow into unemploymentλ (1U ), or m(U, V ) = λ(1U ), which defines the Beveridge curve. Labor market tightness isgiven by θt = Vt /Ut .
Since the number of firms is normalized to unity, we obtain
Ut

1
lt

= 1,

(29)

where the numerator is the aggregate number of employees and the denominator is the numberof employees at each firm, so the ratio defines the number of firms in the economy. For the same
reason, the aggregate number of vacancies equals the number of vacancies created by each firm,or Vt = vt . Similarly, the aggregate output, or GDP of the economy Yt , is given by Yt = yt .

18

4.2

Steady State Equilibrium

From (25)-(29), we obtain the equations that determine a non-stochastic steady state:(1 β ) (1 α )AK α = r + δk ,αβ + 1 βα
AK1 α = e0 (h) ,αβ + 1 β(r + λ ) γ+ βγθ = (1 β) e0 (h) h
q(θ )
λ
l=1
,
λ + θq (θ )

(30)
(31)
e (h)

b ,

(32)
(33)

where K = k/hl is the capital-labor ratio.10 For existence of a steady-state equilibrium, β must
be strictly less than one, otherwise total compensation exceeds output. Since the right-hand sideof (32) must be positive, parameters must be chosen to satisfy e0 (h)he(h)

b > 0 (or µ(1 +µ) 1 e0 h1+µ > b) and β < 1.
Uniqueness of the steady state is verified as follows. First, the steady-state capital-labor ratio
K is determined by (30), which comes from (27). Given K, the steady-state hours of work h isdetermined by (31), which is from (25). Given h, (32), which is from (26), determines θ. Given θ,
the steady state level of l is determined by (33), which comes from (28) and (29). Finally, given thevalues of K, h, and l, we can derive the value of k by k = Khl.
Proposition 2 For an appropriately chosen cost of entry, in any steady-state equilibrium, firms are indifferent between entry and exit.
This result verifies that our assumption of a unit measure of firms causes no loss of generality.
The cost of entry is chosen to make the size distribution of firms degenerate, as in Smith (1999)
and Kudoh and Sasaki (2011). The detail is found in Appendix B.
Proposition 3 (a) An increase in A increases K, h, θ, l, and k. (b) An increase in β decreases K, h, θ, l,and k. (c) An increase in γ has no effect on K and h, and decreases θ, l, and k. (d) An increase in λ has no
10 Appendix C discusses the relashionship between our model and Cho and Cooley (1994), which we think serves as

the frictionless counterpart of our model.

19effect on K and h, and decreases θ, l, and k. (e) An increase in m0 has no effect on K and h, and increases θ,l, and k.
Proof of Proposition 3 is straightforward, and is omitted. Among the results, the most notable
is that changes in γ, λ, and m0 , which are the key parameters determining the labor market frictions, have no effect on the steady state level of hours of work, while these parameters influence
the steady-state level of employment and labor market tightness. An important implication of theresult is that search frictions are irrelevant for understanding the long-run trend of hours of work.
To make this point clearer, eliminate h from (30) and (31) to obtain0e (h) = α ( aβ + 1

β)

1
α

(1

r + δkβ ) (1

1

α)

1
α

1

Aα,

from which it is clear that the steady-state hours of work are determined by TFP, since all othervariables are structural parameters.

5

Quantitative Analysis

5.1

Calibration

In this section, we study a quantitative version of the basic model. Specifically, we calibrate themodel to match the selected long-run Japanese labor market facts summarized in Section 2. We
then solve the quantitative model by approximating the equilibrium conditions around the nonstochastic steady state, and simulate it to obtain the model’s cyclical properties.
We choose the model period to be a quarter and set the discount rate to be r = 0.01, which
implies the discount factor to be δ = 1/(1 + r ) = 0.99. This choice of the parameter is somewhat a
priori, but is consistent with other studies such as Braun et al. (2006). In the production function,
we set α = 2/3 to target the labor share.11 Following Esteban-Pretel et al. (2010), we set the
deprecation rate to be δk = 0.028.
11 We aware that in our frictional economy, α is not necessarily the labor’s share of national income. Nonetheless, we

assume α to take the same value as in the perfectly competitive economy.

20

The matching function is Cobb-Douglas, given by m(U, V ) = m0 U ξ V 1 ξ , where m0 is thematching constant and ξ is the matching elasticity with respect to the number of job-seekers. Lin
and Miyamoto’s (2014) estimate of the elasticity ξ for the Japanese labor market is 0.6. We adopt
the Lin-Miyamoto estimate to set ξ = 0.6. This value lies in the plausible range of 0.5–0.7, which isreported by Petrongolo and Pissarides (2001). Following the convention, we use the Hosios (1990)
condition to pin down the worker’s bargaining power, so β = ξ = 0.6.
Using the panel property of the monthly LFS, Miyamoto (2011) and Lin and Miyamoto (2012)
construct the job-finding rate and the separation rate in Japan. Miyamoto (2011) also reports themean value of the vacancy-unemployment ratio to be 0.78. Given this, we target the vacancyunemployment ratio to be θ = 0.78. We use the monthly job-finding rate 0.142 and the vacancyunemployment ratio to pin down the scale parameter m0 . In particular, m0 is the solution tom0 (0.78)1 0.6 = 3

0.142. We also set the exogenous separation rate λ = 0.014 = 3

0.0048 from

Miyamoto (2011) and Lin and Miyamoto (2012).
We choose µ = 1.8 or the Frisch elasticity is 1/µ = 0.56, which is consistent with the micro
evidence that the Frisch elasticity is less than one.12 Our parameter value is also consistent withthe evidence that the Frisch elasticity for males in Japan is in the range of 0.2–0.7 (Kuroda and Yamamoto, 2008). To generate a realistic magnitude of fluctuations in hours of work per employee,
we need a small µ (or, a large Frisch elasticity). At the same time, we need to constrain our choiceof µ to be within the range of the set of parameters that supports a steady state equilibrium toexists (namely, µ(1 + µ) 1 e0 h1+µ > b), which requires a large µ. We will discuss the sensitivity ofthe model to the choice of µ.
We target the steady-state value of hours worked to be 1/3.13 With h = 1/3, (30) and (31)jointly determine the implied value of e0 , which is 12.576.
In our model, the value of b reflects mostly the unemployment benefit provided by the government because the value of leisure is captured by the disutility from work e(h). According to
12 In a similar environment to ours, Cooper et al. (2007) calibrated the value of µ to be 1.9 for the US economy.
13 In other words, we target h to be 8 hours per day or 40 hours per week (or 5 business days).

quantitative results are independent of the choice of the target level for h.

21

However, ourNickell, (1997), the benefit replacement rate in Japan is about 60 percent.14 We thus adopt this estimate to target the unemployment benefit b to satisfy b = 0.6W. Given this, we determine b andγ by solving (18) and (32) with targets θ = 0.78 and h = 1/3. The implied values are b = 0.348
and γ = 0.020.
Finally, we assume that TFP follows a first order autoregressive process. Specifically, log At
satisfies log At

log A = ρ (log At 1

log A) + ε t ,where 0 < ρ < 0 and ε t

N (0, σ2 ). We set

ρ = 0.612 and σ = 0.0085 to match the first-order autocorrelation and standard deviation of (thebusiness cycle component of) total factor productivity (TFP) in the data. Thus, for all simulations,the standard deviation of the percentage deviation of TFP from its steady-state level is 0.011, as in
the data.
The parameter values for the benchmark analysis are summarized in Table 2. Note that thevalues of parameters m0 , e0 , b, and γ are endogenous in the sense that the values of these parameters are re-calibrated to match the target moments for each set of purely exogenous parameters.

5.2

Main Results

We now compare the selected business cycle statistics from the simulated series with those fromthe corresponding time series data. To this end, we shall primarily focus on the magnitude offluctuations in each variable measured by the standard deviation. Table 3 reports the standard
deviations of the unemployment rate, the vacancy rate, earnings, the (average) hourly wage rate,
the aggregate labor input, hours of work per employee, and employment, scaled by the standard
deviation of TFP. We also report the relative contributions of the intensive and extensive margins
calculated by formula (2).
For all variables listed in Table 3, the standard deviations obtained from our basic model areclose to those obtained from the data. The relative standard deviations generated by the modelof the aggregate labor input, hours per employee, and employment are 0.75, 0.63, and 0.15, respectively. The corresponding relative standard deviations from the data are 0.80, 0.70, and 0.35.
14 See also Martin (2000).

22

Table 2: Parameter values
Parameter

Description

Value

Source/TargetrInterest rate

0.01

Esteban-Pretel et al. (2010)

δDiscount rate 1/(1 + r )

0.99

-

α

Parameter in production function

2/3

Labor sharem0

Matching efficiency

0.471

Monthly job-finding rateξMatching elasticity

0.6

Lin and Miyamoto (2014)

λExogenous separation rate

0.014

Monthly separation ratee0

Parameter in disutility function

12.576h = 1/3

δkDepreciation rate

0.028

Esteban-Pretel et al. (2010)µInverse of Frisch elasticity

1.8

Kuroda and Yamamoto (2008)b

Unemployment benefits

0.348

Replacement rate = 60 percent

βWorker’s bargaining power

0.6

β = ξ (Hosios condition)γ

Vacancy cost

0.020

v

A

Productivity

1.0

NormalizationρAR-coefficient of shock

0.612

Data

σ

Standard deviation of the shock

0.0085Datau ratio = 0.78

While the magnitude of fluctuations in employment for the model is less than a half of that forthe data, the model replicates much of fluctuations in the labor demand and its compositions. In
particular, the model captures the observation that much of fluctuations in the aggregate labor
input is accounted for by fluctuations in hours of work per employee. Indeed, by applying (2) todecompose the variations in the aggregate labor input, we find that βh = 0.84 and βl = 0.19.
The inability of our model in generating enough labor market fluctuations along the extensive
margin can be partly explained by the absence of the other extensive margin, namely, changes in
employment that are driven by entry and exit of firms over the business cycle. While this margin
is beyond the scope of this paper, we speculate that a model with this third margin can generatea higher volatility in employment. With this limitation in mind, we conclude that our model
23

Table 3: Hours of Work and Labor Market Fluctuations
Relative standard deviations
Û

V̂

Ŵ

ŵ

t̂

ĥ

lˆ

βh

βl

Data

5.45

8.92

1.16

0.91

0.80

0.70

0.35

0.79

0.21

Basic model

4.33

12.81

1.55

0.91

0.75

0.63

0.15

0.84

0.19

Model w/o hours

0.96

2.85

1.22

1.22

0.03

-

0.03

-

-

Model with h = 1/3

4.33

12.81

0.96

0.96

0.15

-

0.15

-

-

Bargained hours

1.36

4.02

1.51

0.88

0.67

0.63

0.05

0.95

0.07

captures the labor market dynamics in Japan fairly well.15
It is often reported that the textbook search-matching model of the labor market fails to account for the observed high volatility of the unemployment and vacancy rates, often referred toas the unemployment volatility puzzle (Shimer, 2005).16 Interestingly, Table 3 shows that ourmodel does replicate fluctuations in the unemployment rate and the vacancy rate with realistic
or even greater magnitudes. The relative standard deviation of the unemployment rate from themodel is 4.33 while that from the data is 5.45, and the relative standard deviation of the vacancy
rate from the model is 12.81 while that from the data is 8.92. The relative standard deviation of thelabor market tightness, the ratio of the vacancy rate and the unemployment rate, for the model is
14.66 while that for the data is 12.92. We emphasize that these results are obtained without wage
rigidity or additional shocks.
To clarify the importance of hours of work, we consider the polar case in which the intensive
margin is shut down. To be more concrete, we study a model in which h is removed. Thus, thelevel of labor input is now Lt = lt so that the production function is replaced with At ltα k1t α . In
15 We also note that there is a sizable proportion of employees under short-term contracts in Japan, referred to as non-regular employment (Miyamoto, 2016). Labor adjustment along the extensive margin for non-regular employment is
considered to be less frictional. This partly explains why fluctuations along the extensive margin in the data are greaterthan those implied by the model.
16 Esteban-Pretel et al.

(2011) and Miyamoto (2011) show that the unemployment volatility puzzle holds for theJapanese economy.

24

addition, we remove disutility term e(ht ) from the model. This corresponds to the model studied
by Krause and Lubik (2013), in which the cyclical properties of a model with multi-worker firms
are compared to those of a textbook search-matching model to clarify whether the multi-worker
paradigm helps resolve the unemployment volatility puzzle and conclude that while the multiworker paradigm helps increase the magnitudes of fluctuations in unemployment and vacancies,
there is no sizable quantitative improvement.
The results are reported in Table 3. Without hours of work, the relative standard deviation of
the aggregate labor input is 0.03. The corresponding value from the basic model is 0.75, which
is about 25 times as large as that from the model without hours of work. Similarly, in the model
without hours of work, the relative standard deviations of unemployment and vacancies are 0.96
and 2.85, respectively. The corresponding values from the basic model are respectively 4.33 and12.81, which are about 4 times as large as those from the model without hours of work. Thus, theextensive margin alone cannot account for fluctuations in the aggregate labor input. In addition,
introduction of the intensive margin magnifies labor adjustment along the extensive margin.
To understand the impact of the intensive margin on the magnitude of fluctuations along theextensive margin, we report an additional result in Table 3. Instead of removing hours of workfrom the model, we study the model in which hours of work are fixed at our calibration target
h = 1/3. Thus, in the steady state, this model is identical to the basic model, implying thatworkers incur disutility from long hours of work as in the basic model. However, the intensive
margin is shut down in the sense that firms cannot change hours per employee in response to
shocks. Interestingly, in this model, unemployment and vacancies fluctuate as much as those of
the basic model.
From the comparison of the two models, we conclude that introduction of disutility from longhours of work per se helps generate high magnitudes of fluctuations in unemployment and vacancies, irrespective of whether the intensive margin is active or not. This is somewhat consistent
with studies that emphasize the importance of including the value of leisure in calibrating theflow value of unemployment (Hagedorn and Manovskii, 2008; Hall and Milgrom, 2008). As is25

evident from (18), the level of disutility e(h) works in the same direction as b does: it increases theequilibrium level of total compensation, increasing the firm’s exposure to external shocks.

5.3

Bargained Hours of WorkIn the basic model, we have assumed that the firm and its employees bargain over an earnings
schedule and that the firm chooses hours of work per employee from the state-contingent contract
W (h; S). A possible criticism on this setup is that, from a theoretical perspective, it is more natural
to assume that both hours of work and earnings are determined in bargaining, as in Fang andRogerson (2009). Although we have chosen our basic model to understand the labor market
fluctuations through the lens of labor demand, it is nonetheless useful to present an alternativemodel in which hours of work are bargained.
We consider an alternative model in which hours of work per employee are determined in
bargaining. Specifically, we assume that the level of compensation W (not the earnings schedule) and hours per employee h are determined so as to maximize the asymmetric Nash product:[ Jl (S)]1 β [ J E ((S)(1β)[ J E (S)J U (S)] β . The first-order conditions with respect to W and h imply βJl (S) =J U (S)] and ∂[ Jl (S)]/∂h + ∂[ J E (S)]/∂h = 0, where∂ [ Jl (S)](1 β ) α2=
Ahα 1 l α 1 k1 α∂hαβ + 1 β∂ J E (S)α2 Ahα 1 l α 1 k1 α
=
1 β
∂h
α+

(1

β) e0 ( h) ,βe0 (h) .

βThus, (25) is replaced withα2
αβ + 1βAt htα 1 ltα 1 k1t α = e0 (ht ) .

(34)

The difference comes from the evaluation of the marginal benefit of an additional hour of work:
the marginal benefit for the worker is less than that for the firm by factor of α < 1. Equation (34)is nearly identical to (25), making the qualitative properties of the model nearly identical to thebasic model.
By comparing (34) with (25), one might argue that our basic model generates a small employment volatility because it overestimates the marginal benefit of an additional hour of work (be26cause hours of work are chosen only to maximize the firm’s value), making the extensive margin
relatively less attractive.
Table 3 shows that the standard deviations of unemployment, vacancies, and employment
obtained from the model with bargained hours of work are about 1/3 of those from the basicmodel. The marginal product of an additional hour is reduced by factor of α, which reduces
the firm’s incentive to utilize the intensive margin. However, for the model to match the target
steady-state hours of work per employee, h = 1/3, the endogenous parameters need to be recalibrated. The implied values are now e0 = 8.384 and γ = 0.075. The smaller e0 offsets theimpact of the reduction in the marginal product, while the increase in γ increases the marginal
cost of posting a vacancy. As a result, the magnitudes of fluctuations along the extensive margin,
namely, fluctuations in employment, vacancies, and unemployment, are all reduced, keeping thestandard deviation of hours of work per employee virtually unchanged.

Hours of Work Chosen by Workers For completeness, we consider yet another scenario in
which hours of work per employee are chosen by workers. As in the basic model, each firm
and its employees bargain over an earnings schedule at the beginning of each period. Given theearnings schedule W (h; S), each employee chooses hours of work in each period. For this model,
we replace (15) withn
J E (S) = max W (h; S)he (h) + λδEJ U (S+1 ) + (1oλ)δEJ E (S+1 ) .

The first-order condition requires Wh (h; S) = e0 (h), from which we obtain the expression identicalto (34). Thus, the model in which hours of work are chosen by workers and the model in whichhours of work are bargained are observationally equivalent, as in a related steady-state economy
studied by Kudoh and Sasaki (2011).

27

6

Robustness

6.1

Parameter Values

To clarify the extent to which the model’s ability in generating the magnitude of fluctuations
needed to match the data depends on the choice of parameters, we provide some sensitivity
analyses in terms of the labor supply elasticity and the unemployment benefit. We take thesetwo parameters for our sensitivity analyses because these parameters are known to have a rangeof estimates and calibrated values.
Table 4: Alternative Parameter Values
Relative standard deviations
Û

V̂

Ŵ

ŵ

t̂

ĥ

lˆ

βh

βl

Data

5.45

8.92

1.16

0.91

0.80

0.70

0.35

0.79

0.21

Basic model

4.33

12.81

1.55

0.91

0.75

0.63

0.15

0.84

0.19

µ = 1.9

3.36

9.95

1.52

0.92

0.69

0.60

0.11

0.87

0.16

µ = 1.7

6.26

18.53

1.58

0.91

0.83

0.67

0.21

0.79

0.25

b/W̄ = 0.4

0.77

2.28

1.56

0.92

0.65

0.63

0.03

0.97

0.04

b/W̄ = 0.8, µ = 5.0

5.53

16.35

1.18

0.94

0.39

0.24

0.19

0.57

0.48

β = 0.5

3.89

11.50

1.46

0.83

0.73

0.63

0.13

0.86

0.18

For many countries, the Frisch labor supply elasticity, 1/µ, has a range of estimates. The
empirical literature suggests that the elasticity from the micro data is much smaller than thatfrom the macro data because the macro elasticity includes variations in labor market participation(or the extensive margin of labor supply). Further, the micro evidence suggests that the Frischelasticity is less than one. Our benchmark model employs µ = 1.8, which is close to the value
used in Cooper et al. (2007), µ = 1.9. Table 4 presents the results under two values of µ, 1.9 and1.7.
For µ = 1.9 (or, 1/µ = 0.53) the values of e0 is re-calibrated to match the target h = 1/3, toobtain e0 = 14.036. Similarly, the values of γ is re-calibrated to be 0.026. These values are greater
28

than those under µ = 1.8. While a higher µ implies a greater marginal hourly wage rate, a higher
γ implies a greater marginal cost of posting a vacancy. Thus, fluctuations in the aggregate labor
input, hours per employee, and employment are all less than those from the basic model because
the marginal costs for the both margins increased. Overall, the effect on the extensive margin
dominates the other, and the relative importance of the intensive margin increases to 0.87. Forµ = 1.7 (or, 1/µ = 0.59), the mechanism is reversed, and the magnitudes of fluctuations in allmeasures are greater than those for the benchmark case.
In our benchmark calibration, we choose the value of b so that b = 0.6W, where 60 percent
is the actual replacement rate in Japan reported in Nickell (1997). The choice of the parametervalue for b has been the subject of discussion in the literature. For the US labor market, Shimer(2005) sets b so that the replacement rate is 40 percent to target the actual replacement rate in theUS, while Hagedorn and Manovskii (2008) argue that Shimer’s b is too low and that with a muchhigher b, search-matching models can replicate unemployment and vacancy fluctuations withrealistic magnitudes. With this debate in mind, we consider alternative levels of b that correspondto the replacement rates of 40 percent and 80 percent.
Under the replacement rate of 40 percent, the corresponding value of b is b = 0.230, and theimplied value for the vacancy cost is γ = 0.112. Since the vacancy cost is 5.6 times as large as thatfor the basic model, the incentive to utilize the extensive margin reduces significantly. As a result,the magnitude of fluctuations in employment decreases significantly, keeping the magnitude offluctuations in hours per employee unchanged, as shown in Table 4. Consequently, the relative
importance of the intensive margin increased from 0.84 to 0.97. Further, Table 4 suggests that asmaller b cause unemployment and vacancies to be much less volatile. These results are in linewith Hagedorn and Manovskii (2008).17
Hagedorn and Manovskii (2008) argued that, given that individuals typically have access toalternative sources of income such as home production, the flow value of unemployment should
be close to the flow value of employment. With this in mind, we consider yet another set of pa17 The relationship between the magnitude of fluctuations and the match surplus is clarified by Hornstein et al. (2005)

and Elsby and Michaels (2013).

29

rameters which include a very high replacement rate within a set of parameters that support a
steady state (i.e., µ(1 + µ) 1 e0 h1+µ > b, which implies that we need a higher µ to have a large
replacement rate). Thus, we consider the replacement rate being 80 percent with µ = 5 (or the
Frisch elasticity to be 0.2). The implied parameter values are e0 = 422.978 and γ = 0.016. Table
4 reports the results. With a smaller marginal cost of posting a vacancy, the magnitude of fluctuations in employment increases and gets closer to the data. However, the increased marginalhourly wage rate reduces the incentive to utilize the intensive margin significantly and reduced
the magnitude of fluctuations in hours per employee. Thus, we conclude that while a model witha high replacement rate helps generate high volatilities in unemployment and vacancies, it cannot
generate fluctuations in the aggregate labor input and hours per employee.
In the basic model, we follow the convention to choose β = ξ to meet the Hosios condition.
Another useful benchmark is to assume the symmetric Nash product, which implies β = 0.5 inour framework. With this value, the implied parameter values are e0 = 13.225 and γ = 0.031.
This case is somewhat similar to the case under a lower replacement rate. The implied valuesof e0 and γ are both greater than those under the basic model, making both the intensive andextensive margins more costly. However, the value of e0 is only slightly greater than that for thebasic model while the value of γ is about 1.6 times as large as that for the basic model. This makes
the extensive margin more costly for firms.

6.2

Extensions

In this section, we present the results from some extensions.

Decreasing Efficiency

In the benchmark model, we have assumed that while longer hours ofwork cause greater disutility, there is no loss in productivity. Using the UK micro data, Pencavel(2014) found that the relationship between output and hours of work per worker is concave.
This raises a question that our basic model might have overestimated the marginal benefit of anadditional hour of work, making the extensive margin relatively less attractive. To capture thepossibility that the productivity of hours per employee is concave, we modify our basic model
30

Table 5: Alternative Assumptions
Relative standard deviations
Û

V̂

Ŵ

ŵ

t̂

ĥ

lˆ

βh

βl

Data

5.45

8.92

1.16

0.91

0.80

0.70

0.35

0.79

0.21

Basic model

4.33

12.81

1.55

0.91

0.75

0.63

0.15

0.84

0.19

Decreasing efficiency

1.62

4.80

1.39

0.81

0.62

0.58

0.05

0.93

0.09

Capital adjustment

3.01

8.22

1.18

0.69

0.56

0.49

0.10

0.87

0.17

Stochastic separations

7.39

13.04

1.55

0.91

0.81

0.63

0.25

0.76

0.26

Contract rigidity

5.87

17.37

1.48

0.56

1.07

0.92

0.20

0.86

0.18

such that the effective labor input satisfies L = hη l, where η

1 captures the efficiency of hours

per employee. The model with η = 1 corresponds to the benchmark economy. The value of a firm
is replaced withn
J (S) = max Ahαη l α k1 αh,v,xW (h; S)lγvox + δEJ (S+1 ) ,and the optimization conditions need to be modified accordingly. The earnings function for thiseconomy isW (h; S) =αβAhαη l α 1 k1 α+ (1αβ + 1 ββ) [e (h) + b] + βγθ.

For our numerical exercise, we adopt Pencavel’s (2014) estimate, η = 0.8. As shown in Table 5,
the cyclical properties of this model are similar to those of the model with bargained hours ofwork. The standard deviations of unemployment, vacancies, and employment obtained from thismodel are about 1/3 of those from the basic model. The marginal product of an additional houris reduced by factor of η, which reduces the firm’s incentive to utilize the intensive margin. This
results in a smaller standard deviation of hours per employee (relative to the standard deviationof TFP): 0.58 from this model and 0.63 from the basic model. To replicate the same target moments,
the endogenous parameters need to be re-calibrated, and the implied value for the vacancy cost
is now γ = 0.066, which is three times as large as that for the basic model. The increased vacancy
cost significantly reduces the volatility in employment, vacancies, and unemployment.
31

Costly Capital Adjustment

Our benchmark model includes capital as a factor of production.

With capital, production technology exhibits constant returns to scale, which safely rules out
variations in vacancies along the margin of entry and exit. A possible side effect is that, since adjustment in capital is perfectly flexible, the very presence of a more flexible margin of adjustment
might make the cost of labor adjustment relatively more costly, generating too little fluctuationsin the aggregate labor input. It is therefore useful to investigate how introduction of costly capitaladjustment influence the volatilities of labor input and its components.18 In modeling the cost ofcapital adjustment, we follow the convention as much as possible. Namely, with costly capitaladjustment, the value of the firm is given byn
J (S) = max Ahα l α k1 αh,v,xW (h; S)lγvxo
C (k, k +1 ) + δEJ (S+1 ) ,where the cost of capital adjustment takes the following standard form:
1
C (k, k +1 ) =
2

x
k

2

1
k=
2

k +1 k
+ δk
k

2

k.

In this economy, adjustments of capital and employment are both costly. Table 5 shows that, withcostly capital adjustment, the magnitudes of fluctuations in all variables are smaller than thosefrom the basic model as well as those from the data. This result is due to the property that the costof capital adjustment reduces the steady-state level of capital. With a smaller stock of capital, themarginal product of hours of work per employee is reduced, and this dominates the reductionin the marginal hourly wage rate. As a result, the firm’s incentive to utilize the intensive margin
declines. Similarly, a reduction in capital reduces the marginal product of employment, which
reduces the firm’s incentive to utilize the extensive margin. Overall, introduction of capital adjustment cost reduces the magnitudes of fluctuations in both the intensive and extensive margins.
In other words, the presence of perfectly flexible capital does not make labor adjustment morecostly.
18 It is important to note that, in the basic model, there is little room for improving the magnitude of fluctuations incapital. In the data, the standard deviation of capital is 0.012 while that of the basic model is 0.017.

32

Stochastic Separations Recent empirical studies demonstrate that both unemployment inflow
and outflow significantly contribute to the unemployment dynamics in Japan (Miyamoto, 2011;
Lin and Miyamoto, 2012). In particular, in the data, TFP and the separation rate are negatively
correlated. This tends to amplify fluctuations in the unemployment rate. Shimer (2005) finds thatwhile introduction of separations shocks helps increase the magnitudes of fluctuations in unemployment and vacancies, it destroys the negative correlation between the unemployment rate andthe vacancy rate, or the Beveridge curve. To assess the importance of the unemployment inflow
channel in generating fluctuations in employment and unemployment within our framework andto see if the Beveridge curve is preserved, we study a model that incorporates the separation ratethat follows a more realistic stochastic process. Specifically, we assume that the separation rate
follows a first-order autoregressive process of log λt
0 < ρλ < 1 and ε λ,tlog λ = ρλ (log λt 1

log λ) + ε λ,t , where

N (0, σλ2 ). From the data, we set ρλ = 0.1575 and σλ = 0.0907. Table 5shows that introduction of stochastic separations significantly increases the standard deviationsof unemployment and vacancies. However, the magnitudes of fluctuations are too large to matchthe observed standard deviations of unemployment and vacancies. Another finding is that themodel with separation shocks accounts for much of the observed fluctuations in employment.
The standard deviation of employment (relative to that of TFP) is 0.25 while that from the data
is 0.35. However, as in the literature, stochastic separations weaken the negative correlation between unemployment and vacancies. An important implication of the results is that, while ourmodel precludes the firing margin of labor adjustment, introduction of the firing margin per se
may not improve the performance of the model.

Contract Rigidity Shimer (2005) suggests that introduction of some wage rigidity helps increase
the magnitudes of fluctuations in unemployment and vacancies. Although our basic model perfectly replicates the observed volatility in hourly wage rate, earnings in the model fluctuate morethan those in the data. Thus, there is a (small) room for improvement. A caveat here is that,with variable hours of work and a nonlinear earnings schedule, the concept of wage rigidity is
not well-defined, as it assumes to have a model in which the labor contract takes in the form of33

hourly wage rate. Thus, we instead study the model with contract rigidity. Our modeling strategy
is to modify the model as little as possible, rather than to write down a full-fledged micro-foundedmodel of rigidity. To be more specific, we introduce an ad-hoc earnings schedule with rigidity thatpossesses the following two properties. One is that the rigid earnings schedule is identical to thebenchmark earnings schedule (18) in the steady state. In other words, while ad-hoc, the rigid
earnings schedule does not alter the steady state of the basic model. The other is that the current
contract does not fully reflect the current TFP (Pissarides, 2009). Let StP = ( AtP , lt , k t , Ut ) be the perceived state of the economy in period t, where AtP is the perceived level of TFP which is assumed
to satisfy
AtP = φAt + (1

φ) A.

(35)

Thus, the perceived TFP is given by the weighted average of the true current TFP and its steadystate level.19 We assume that in the bargaining stage, the true state is only partially verifiable, andas a result, the contract is conditional only on the perceived state. The resulting earnings schedulein period t is given by

W (ht ; StP ) =αβAtP hαt ltα 1 k1t α+ (1αβ + 1 ββ) [e (ht ) + b] + βγθt .

(36)

It is important to note that, while the contract is rigid, it does not guarantee that the wage rate isalso rigid because changes in hours of work do change the level of earnings. For our numericalanalysis, we choose φ to be 0.359 from a structural estimation by Lin and Miyamoto (2014). Table
5 shows that the model with contract rigidity increases the magnitudes of fluctuations in unemployment and vacancies. However, the magnitudes are too large. While the standard deviation of
the unemployment rate is about the same as that of the data, the standard deviation of the vacancy
rate is about twice as large as that of the data. Introduction of contract rigidity significantly decreases the standard deviation of the average hourly wage rate (relative to that of TFP) from 0.91
to 0.56. On the other hand, the standard deviation of earnings (relative to that of TFP) decreased
19 Our formulation of wage rigidity is inspired by Hall (2005) and in particular Krause and Lubik (2007).

These

authors assume an ad-hoc wage equation in which the actual current wage rate is given by the weighted average of
the Nash bargained wage rate and a reference wage rate, such as the past wage rate and the steady-state level.

34

only slightly, from 1.55 to 1.48. This is because the level of earnings fluctuates as hours of work
per employee changes, and the magnitude of fluctuations in hours of work per employee is about
1.5 times as large as that from the basic model and about 1.3 times as large as that from the data.
Since contract rigidity increases the magnitude of fluctuations in hours of work per employee toomuch, the relative importance of the extensive margin is reduced in the model with rigidity even
if the standard deviation of employment is higher than that of the basic model.

7

Conclusion

In this paper, we showed empirically that while both employment and hours per employee areimportant for explaining cyclical fluctuations in the Japanese labor market, fluctuations in hours
of work explain 79 percent of variations in the aggregate labor input. We then introduced variable
hours of work into a search-matching model with multi-worker firms to investigate how firms
utilize hours of work and vacancy creation over the business cycle, and calibrated the model’s
parameters to mimic the Japanese labor market facts. We showed that our model accounts formuch of fluctuations in the aggregate labor input, hours of work, employment, unemployment,
and vacancies, albeit it is less successful in generating the magnitude of fluctuations in employment. Overall, we conclude that introduction of hours of work is an important direction in the
study of labor market dynamics over the business cycle.
The limitation of our model is that we did not consider the other extensive margin, namely,changes in employment that are driven by entry and exit of firms over the business cycle. For thisreason, our framework is best used for economies in which fluctuations in employment associatedwith firm entry and exit play a relatively minor role. A possible future research is to introduce thismargin into our framework to study how these three margins fluctuate over the business cycle.
An important line of future research is to identify the institutional characteristics across countries and provide a unified framework that captures the observed cross-country differences in thecomposition of labor demand over the business cycle. For this investigation, one needs a modelwith endogenous firing. With a micro-founded model of firing, one can study the impact of em35

ployment protection such as firing restriction on the importance of the intensive and extensive
margins over the business cycle.20 An important recent contribution along this line is Llosa etal. (2014), in which a frictionless model with firing costs is developed. Our framework, whenmodified accordingly, will provide a frictional counterpart of their model. A caveat is that, as wedemonstrated, the model with separation fluctuations does not perform well; in particular, it does
not produce a strong negative correlation between unemployment and vacancies.
Another important line of future research is to consider differences in types of employment.
While we assumed a single employment contract for all workers, the labor market in Japan is best
understood as being polarized into two groups of workers, those with well-protected long-term
contracts and those with less-paid, less-protected “non-regular” employment contracts, underwhich firms may terminate contracts at will and hours of work are more flexible. Workers undersuch non-regular employment contracts have been increasing, and they amount to 40 percent of
total employees in Japan (Miyamoto, 2016). Our framework has an advantage in investigating
this important issue because it is designed to study the composition of labor demand.

20 It is certainly easy to introduce a separation cost into our model. However, we believe that such a model cannot

approximate the reality of an economy with firing costs. A useful model of firing costs must possess both involuntary
separations (i.e., firing) and involuntary labor hoarding (i.e. restricted firing).

36

Appendix
A

Proof of Proposition 1

Subtract (16) from (15) to obtain
J E (S)J U (S) = W (S, h)b + [1e (h)λhθq(θ )] δEJ E (S+1 )i
δEJ U (S+1 ) .

(37)

Observe that, (9) and (17) imply
1
β

βh

i
δEJ U (S+1 ) =δEJ E (S+1 )

γ
.
q(θ )

Use this to rewrite (37) as follows:
J E (S)J U (S) = W (S, h)

e (h)

b+

1

θq(θ ) βγ
.
β
q(θ )

λ
1

(38)

Substitute (13) and (38) into (17) to obtainβ αAhα l α 1 k1 α

= (1β) [W (S, h)W (S, h)b ] + [1e (h)which reduces to W (S, h) = β[αAhα l α 1 k1 α
Wl (S, h)l +Wl (S, h)l +λWl (S, h)l ] + (1

(1

λ) γ
q(θ )
βγ
θq(θ )],q(θ )β)[e (h) + b] + βγθ, or

1
1 β
W (S, h) = αAhα l α 1 k1 α +[e (h) + b] + γθ.
β
β

(39)

This is a differential equation about the unknown earnings function. This equation satisfies for all
l0, along with the condition that
W (S, h)l

Ahα l α k1 α ,(40)which requires that the total wage payment does not exceed the firm’s revenue. It is useful toobserve that
i
1
1
1
∂ h
1
W (S, h)l β = Wl (S, h)l + W (S, h) l β
∂l
β
1
1 β
1
= αAhα l α 1 k1 α +[e (h) + b] + γθ l β
β
1
1
1 β
= αAhα l α+ β 2 k1 α +[e (h) + b] + γθ l β
β
37

1

.

1

Since (40) implies W (S, h)l βAhα lα+ β1 1 1 α
k , we have lim

l !0 W ( S, h ) l1
β= 0. Thus, it followsthat1

W (S, h)l β =

Z l
0

=

αAhα iαAhα k1 α
α + β11lα+ β1 2 1 αkα+ β1 1

+

+ [(1

1

β
β

1

1
di
[e (h) + b] + γθ i β1β) [e (h) + b] + βγθ ] l β .

Thus, we finally obtain
W (S, h) =αAhα k1 α α 1l+ (1α + β1 1β) [e (h) + b] + βγθ

as shown in the proposition.

B

Entry and Exit

We follow Smith (1999) and Kudoh and Sasaki (2011) to assume that each entrant must create
vacancies so that it operates with the steady-state level of employment l in the next period. This
assumption rules out the size distribution of firms. Because the rate of filling a vacancy is q(θ ),in order to achieve l+1 in the next period, the firm must create exactly l+1 /q(θ ) vacancies in thecurrent period. Thus, the value of entry is given by
J (0) =

γl+1
q(θ )

k +1 + δEJ (S+1 ) ,(41)

Therefore, the number of firms, Nt , is determined by J (0) = 0, or
γl+1+ k +1 = δEJ (S+1 ) .
q(θ )

(42)

In any steady state, the firm’s value of operation (without imposing (42)) is(1δ) J (S) = Ahα l α k1 α

W (S, h)l

γv

x

1 β
Ahα l α k1 α (1 β) [e (h) + b] l
αβ + 1 βrγ(1 β ) (1 α ) α α 1 α=
Ah l k+l δk kαβ + 1 β
q(θ )
γl
=r
+ rk,
q(θ )

=

38

βγθl

γ

λl
q(θ )

δk k

from which we obtain
δJ (S) =

γl
+ k.
q(θ )

Thus, in any steady state, the value of entry is
J (0) =γlq(θ )k + δJ (S) = 0.

Thus, firms are indifferent between entry and exit. As a result, the number of firms will be indeterminate in this economy in the sense that a free entry condition cannot pin down the number offirms. The same result holds in the standard neoclassical economy in which firms profits are zero.
The result did not arise in Smith (1999) or Kudoh and Sasaki (2011) because in these models, theproduction technology exhibits decreasing returns to scale.

C

Relation to the Frictionless Model

To understand the role of search frictions and wage bargaining, it is helpful to make a comparison
between our model and a model with a perfectly competitive labor market. The benchmark forour comparison is a version of Cho and Cooley (1994), in which there is a representative household who maximizes∞∑ δt [u (ct )e ( h t ) l t + b (1l t ) + g (1

lt )]

t =0

subject to k t+1 = Ahαt ltα k1t α + (11δk ) k t

ct , where g(.) is an increasing function of non-employment

lt . In this model, lt measures the days of work or the number of family members who par-ticipate in the labor market. As lt increases, more workers incur the utility cost e (ht ) and lose
the opportunity to receive b. All other costs of labor market participation is summarized by g(.).
To make this frictionless economy comparable with our model economy, we assume that g is
log-linear: g( x ) = e1 (ln xx ), where e1 is a positive parameter.

Proposition 4 Suppose that utility is linear in consumption. (a) If β > 0, then the capital-labor ratio of
the economy with search and wage bargaining is less than that for the frictionless economy, and hours ofwork per employee in the economy with search and wage bargaining is longer than that for the frictionless39

economy. (b) The steady-state levels of the capital-labor ratio, hours of work, and employment for the two
economies coincide if β = 0, ξ = 1/2, and
e1 =

r+λ λ
γ .
m0 m0

(43)

Proof. From the first-order conditions, we obtain u0 (ct ) = δu0 (ct+1 )[(11δk ], e0 (ht )lt = u0 (ct )αAt hαt 1 ltα k1t α , and g0 (1lt ) = e 0 ( h t ) h te( ht )α) At+1 hαt+1 ltα+1 k t+α1 +b. Thus, a steady state isgiven by a set of f h, l, K g that satisfy(1α) AK α = r + δk ,(44)

αAK1 α = e0 (h),(45)

g 0 (1l ) = e0 ( h) h

e (h)

b,

(46)

where we have imposed u0 (c) = 1 to be consistent with our model, in which there is no consumption smoothing motive. It is interesting to observe that, (30) and (31) and coincide with (44)and (45) when β = 0. With β > 0, the capital-labor ratio (K) of the economy with search frictions
and wage bargaining is less than the one without, and hours of work per employee (h) for the
economy with search frictions and wage bargaining is longer than the one without. To facilitatecomparison of the levels of employment for the two economies, we assume g to be log-linear in
non-employment: g ( x ) = e1 (ln xx ), where e1 is a positive constant. Since g0 ( x ) = e1 (1x )/x,(46) reduces to
e1

l
1

l

= e0 ( h) h

e (h)

b.

(47)

This determines the relationship between h and l for the frictionless economy. In what follows welet β = 0. Eliminate θ from (32) and (33) to obtain
r+λ
γ
m0

λ l
m0 1 lξ1 ξ= e0 ( h) h

e (h)

b,

which reduces tor+λ λ lγ
= e0 ( h) h
m0 m0 1 l

40

e (h)

b

(48)

when ξ = 1/2. It is now clear that the steady-state levels of employment l for the two economies
coincide if
e1 =

r+λ λ
γ .
m0 m0

(47) and (48) imply that if search frictions are sufficiently severe, then the level of employment in
the economy with search frictions is below the one without frictions.
Condition (43) equates the marginal benefit of non-employment and an index of search costs.
A higher discount rate, a higher separations rate, a higher vacancy cost, or a smaller matchingcoefficient, each of which implies greater search frictions, increases the term of the right-handside of (43). If the right-hand side of (43) is greater than e1 , then the level of employment for the
economy with search frictions is below the one for the frictionless economy even when β = 0 andξ = 1/2. Proposition 4 indicates that our model contains a competitive, frictionless economy as a
special case.

41

References[1] Acemoglu, Daron, and William B. Hawkins. “Search with Multi-Worker Firms.” Theoretical
Economics 9 (2014) 538–628.
[2] Andolfatto, David. “Business Cycles and Labor-Market Search.” American Economic Review
86 (1996) 112–132.
[3] Bertola, Giuseppe, and Ricardo J. Caballero. “Cross-Sectional Efficiency and Labour Hoarding in a Matching Model of Unemployment,” Review of Economic Studies 61 (1994) 435–456.
[4] Braun, R. Anton, Julen Esteban-Pretel, Toshihiro Okada, and Nao Sudou. “A Comparison of
the Japanese and US Business Cycles.” Japan and the World Economy 18 (2006) 441–463.
[5] Burnside, Craig, Martin Eichenbaum, and Sergio Rebelo. “Labor Hoarding and the Business
Cycle,” Journal of Political Economy 101 (1993) 245–273.
[6] Cahuc, Pierre, Francois Marque, and Etienne Wasmer. “A Theory of Wages and Labor Demand with Intra-Firm Bargaining and Matching Frictions,” International Economic Review 49(2008) 943–972.
[7] Cahuc, Pierre, and Etienne Wasmer. “Does Intrafirm Bargaining Matter in the Large Firm’s
Matching Model?” Macroeconomic Dynamics 5 (2001) 742–747.
[8] Cho, Jan-Ok, and Thomas F. Cooley. “Employment and Hours over the Business Cycle.”
Journal of Economic Dynamics and Control 18 (1994) 411–432.
[9] Cooper, Russell, John Haltiwanger, and Jonathan L. Willis. “Search Frictions: Matching Aggregate and Establishment Observations.” Journal of Monetary Economics 54 (2007) 56–78.
[10] Elsby, Michael WL, and Ryan Michaels. “Marginal Jobs, Heterogeneous Firms, and Unemployment Flows.” American Economic Journal: Macroeconomics 5 (2013) 1–48.

42

[11] Esteban-Pretel, Julen, Ryo Nakajima, and Ryuichi Tanaka. “TFP Growth Slowdown and theJapanese Labor Market in the 1990s.” Journal of the Japanese and International Economies 24(2010) 50–68.
[12] Esteban-Pretel, Julen, Ryo Nakajima, and Ryuichi Tanaka. “Japan’s Labor Market Cyclicality
and the Volatility Puzzle.” RIETI Discussion Paper Series 11-E-040, 2011.
[13] Fang, Lei, and Richard Rogerson. “Policy Analysis in a Matching Model with Intensive andExtensive Margins.” International Economic Review 50 (2009) 1153–1168.
[14] Felbermayr, Gabriel, and Julien Prat. “Product Market Regulation, Firm Selection, and Unemployment.” Journal of the European Economic Association 9 (2011) 278–317.
[15] Fujita, Shigeru, and Makoto Nakajima. “Worker Flows and Job Flows: A Quantitative Investigation.” Review of Economic Dynamics 22 (2016) 1–20.
[16] Fujita, Shigeru, and Garey Ramey. “The Cyclicality of Separation and Job Finding Rates.”
International Economic Review 50 (2009) 415–430.
[17] Hagedorn, Marcus, and Iourii Manovskii. “The Cyclical Behavior of Equilibrium Unemployment and Vacancies Revisited.” American Economic Review 98 (2008) 1692–1706.
[18] Hall, Robert E. “Employment Fluctuations with Equilibrium Wage Stickiness.” American Economic Review (2005) 50–65.
[19] Hall, Robert E, and Paul R. Milgrom. “The Limited Influence of Unemployment on the Wage
Bargain.” American Economic Review 98 (2008) 1653–1674.
[20] Hall, Simon, Mark Walsh, and Anthony Yates. “Are UK Companies’ Prices Sticky?” Oxford
Economic Papers 52 (2000) 425–446.
[21] Hornstein, Andreas, Per Krusell, and Giovanni L. Violante. “Unemployment and Vacancy
Fluctuations in the Matching Model: Inspecting the Mechanism.” Economic Quarterly–Federal
Reserve Bank of Richmond 91 (2005) 19–51.
43

[22] Koeniger, Winfried, and Julien Prat. “Employment Protection, Product Market Regulation
and Firm Selection.” Economic Journal 117 (2007) F302–F332.
[23] Kudoh, Noritaka, and Masaru Sasaki. “Employment and Hours of Work.” European Economic
Review 55 (2011) 176–192.
[24] Kuroda, Sachiko, and Isamu Yamamoto. “Estimating Frisch Labor Supply Elasticity inJapan.” Journal of the Japanese and International Economies 22 (2008) 566–585.
[25] Kydland, Finn E., and Edward C. Prescott. “Hours and Employment Variation in Business
Cycle Theory.” Economic Theory 1 (1991) 63–81.
[26] Krause, Michael U., and Thomas A. Lubik. “The (Ir) relevance of Real Wage Rigidity in the
New Keynesian Model with Search Frictions.” Journal of Monetary Economics 54 (2007) 706–
727.
[27] Krause, Michael U., and Thomas A. Lubik. “Does Intra-Firm Bargaining Matter for Business
Cycle Dynamics?” Economic Quarterly–Federal Reserve Bank of Richmond 99 (2013) 229–250.
[28] Lin, Ching-Yang, and Hiroaki Miyamoto. “Gross Worker Flows and Unemployment Dynamics in Japan.” Journal of the Japanese and International Economies 26 (2012) 44–61.
[29] Lin, Ching-Yang, and Hiroaki Miyamoto. “An Estimated Search and Matching Model of theJapanese Labor Market.” Journal of the Japanese and International Economies 32 (2014) 86–104.
[30] Llosa, Gonzalo, Lee Ohanian, Andrea Raffo, and Richard Rogerson. “Firing Costs and Labor
Market Fluctuations: A Cross-Country Analysis.” mimeo, 2014.
[31] Martin, John P. “What Works among Active Labour Market Policies: Evidence from OECDCountries’ Experiences.” OECD Economic Studies 30 (2000) 79–113.
[32] Merz, Monika. “Search in the Labor Market and the Real Business Cycle.” Journal of Monetary
Economics 36 (1995) 269–300.

44

[33] Miyamoto, Hiroaki. “Cyclical Behavior of Unemployment and Job Vacancies in Japan.” Japanand the World Economy 23 (2011) 214–225.
[34] Miyamoto, Hiroaki. “Growth and Non-Regular Employment,” B.E. Journal of Macroeconomics
16 (2016) 523–554.
[35] Nickell, Stephen. “Unemployment and Labor Market Rigidities: Europe versus North America.” Journal of Economic Perspectives 11 (1997) 55–74.
[36] Ohanian, Lee E., and Andrea Raffo. “Aggregate Hours Worked in OECD Countries: New
Measurement and Implications for Business Cycles.” Journal of Monetary Economics 59 (2012)
40–56.
[37] Pencavel, John. “The Productivity of Working Hours.” Economic Journal 125 (2014) 2052–2076.
[38] Petrongolo, Barbara, and Christopher A. Pissarides. “Looking into the Black Box: A Surveyof the Matching Function.” Journal of Economic Literature 39 (2001) 390–431.
[39] Pissarides, Christopher A. “The Unemployment Volatility Puzzle: Is Wage Stickiness theAnswer?” Econometrica 77 (2009) 1339–1369.
[40] Rogerson, Richard. “Understanding Differences in Hours Worked.” Review of Economic Dynamics 9 (2006) 365–409.
[41] Shimer, Robert. “The Cyclical Behavior of Equilibrium Unemployment and Vacancies.”
American Economic Review 95 (2005) 25–49.
[42] Smith, Eric. “Search, Concave Production, and Optimal Firm Size.” Review of Economic Dynamics 2 (1999) 456–471.
[43] Stole, Lars A., and Jeffrey Zwiebel. “Intra-Firm Bargaining under Non-Binding Contracts.”
Review of Economic Studies 63 (1996) 375–410.
[44] Trapeznikova, Ija. “Employment Adjustment and Labor Utilization.” mimeo, 2015.

45