Discussion Papers In Economics
And Business

Employment and Hours of Work
Noritaka Kudoh

Masaru Sasaki

Discussion Paper 07-35

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEmployment and Hours of Work
Noritaka Kudoh

Masaru Sasaki

Discussion Paper 07-35

October 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEmployment and Hours of Work∗
Noritaka Kudoh†

Masaru Sasaki‡

Hokkaido University

Osaka University

October 1, 2007

Abstract
This paper develops a dynamic model of the labor market in which the degree of substitution
between employment and hours of work is determined as part of a search equilibrium. Each firmchooses the demand for working hours and the number of vacancies, and the hourly wage rate isdetermined by Nash bargaining. A firm increases the demand for hours as recruitment becomes
more costly. Labor market tightness influences the composition of labor demand through its
impact on the wage rate. Restricting working hours can expand employment, but doing so is
not necessarily eﬃcient. When there are two industries that diﬀer in their equipment costs,
workers employed by firms with higher equipment costs work longer and earn more.
JEL classification: J21, J23, J31, J64.
Keywords: employment, hours of work, search frictions.

∗

We thank Hiromi Nosaka and participants in seminars at Kyoto University, Tohoku University, Osaka Prefecture

University, Nagoya University, the JEA Meeting, and the Australasian Meeting of the Econometric Society in Brisbane
for their helpful comments. Part of this research is financially supported by the Japan Society for the Promotion of
Science (Grant Number 18730164).
†
E-mail: kudoh@econ.hokudai.ac.jp‡
E-mail: sasaki@econ.osaka-u.ac.jp1

1

Introduction

There are extensive and intensive margins for adjusting labor input: the number of workers andhours of work per employee.1 Understanding how firms utilize these two margins is crucial forunderstanding the long-run trend in hours of work, the movements of employment and hours
over the business cycle, the likely eﬀect of regulation regarding hours of work, and cross-sectional
diﬀerences in hours of work, to name a few.
Despite the importance of these issues, there seems to be no consensus among economists
regarding how hours of work should be modeled. Prescott (2004), for example, argued that working
hours should be understood by the conventional neoclassical framework with labor supply derived
by households’ labor—leisure choice (Lucas and Rapping, 1969). Pissarides (2000, 2007) developed
an equilibrium matching model in which workers choose hours of work. With or without search
frictions, the view of these authors is that hours of work are determined from the supply side ofthe labor market.
On the other hand, the labor adjustment literature such as Sargent (1978) and Hamermesh(1993) considers that hours of work are determined by the demand side. In this literature, a firm
faces an exogenous cost of adjusting employment and optimally chooses employment and hours
of work. An intermediate case is considered by Marimon and Zilibotti (2000), who constructeda general equilibrium with search frictions in which a firm and a worker bargain over the hourly
wage rate and hours of work.
In this paper, we construct a dynamic equilibrium model of labor demand under search frictions,focusing on how firms utilize employment and hours of work. The idea is that hours of work can
be chosen instantly whereas employment adjustment is frictional. As in the labor adjustment
literature, we assume that firms, not workers, choose hours of work. The novel feature of thispaper is that the source of labor adjustment costs is search frictions, and that the adjustment costis influenced by labor market tightness. Thus, the degree of substitution between employment and1

Hamermesh (1993) asserted, “Assuming the labor is the only input into production is nonsensical. Similarly silly

is the assumption that the employment—hours split can be ignored in analyzing adjustment.” (p. 209.)2

hours of work is determined as part of the equilibrium. This sharply contrasts with the traditionalmodels of working hours in which the degree of substitution between working hours and employment
is structurally given by the production technology (Hoel, 1986; Booth and Schiantarelli, 1987;
Calmfors and Hoel, 1988; Hamermesh, 1993; Cahuc and Zylberberg, 2004).
The basic model is an extension of Smith (1999), which is particularly useful for our purpose
because the number of workers at each firm is determined endogenously, although modeling the firm
size in a search equilibrium is generally a formidable task. We incorporate the choice of working
hours into Smith’s (1999) framework. Each firm chooses the number of vacancies and hours ofwork per employee to maximize the value of the firm. The hourly wage rate is determined by Nashbargaining. We derive the bargaining outcome explicitly and show that the hourly wage rate is
positively related to the firm’s marginal product of labor. In addition, as is standard in the labormarket search literature, the wage rate increases with the tightness of the labor market.
In addition to the choice of hours, we modify Smith’s (1999) model such that the wage bargaining takes place before each firm chooses the number of vacancies and hours of work. This meansthat a firm takes the hourly wage rate as given when it chooses the composition of employment.
The order of events has a nontrivial consequence. As a firm does not have any incentive to employ
more workers to influence the bargaining outcome, Smith’s (1999) bargaining externality is notpresent.
Our main focus is on the interaction between search frictions and the determination of hours ofwork. The model suggests that a firm has an incentive to increase hours of work if it faces greater
search frictions. In particular, a firm increases hours of work if labor market tightness increases,
exogenous recruitment cost increases, or the job separation rate increases. These are all considered
as measures of frictions in adjusting the number of employees. A firm substitutes employment forhours when these frictions increase.
Free entry of firms and the absence of the bargaining externality ensure that the equilibrium
level of total labor inputs at each firm is eﬃcient. However, we show that firms overemploy. The
social planner does not take into account the hourly wage rate when the optimal hours of work3

are chosen. However, for firms, the hourly wage rate is an additional marginal cost of increasing
hours of work. As hours of work are too short and the total labor input is eﬃcient, employment isnecessarily too high. Thus, even though free entry induces the eﬃcient level of labor inputs, thepresence of search frictions distorts the composition of the labor demand.
It has become increasingly important to ask whether regulating hours of work increases employment (Booth and Schiantarelli, 1987; Marimon and Zilibotti, 2000). To address this issue, we solve
the model by treating hours of work as a policy parameter to see if there is an employment eﬀect
from restricting working hours. The answer is generally yes, as long as the marginal product oflabor exceeds the marginal loss associated with longer hours. However, since hours of work are lessthan the eﬃcient level, reducing hours from the equilibrium level will deviate from the optimum
further.
Many authors have documented declines in hours of work in major developed economies(Fitzgerald, 1996; Hunt, 1998; Pissarides, 2007). Despite a decline in average working hours,
a disparity in working hours across workers has gradually widened. This disparity is particularly
distinguished for workers in their 30s. The Japanese Labour Force Survey (2006) documented thatthe share of male employees aged between 35 and 39 years who work more than 60 hours per weekrose from 18.9% in 1993 to 23.5% in 2003.2 Similarly, the share of those who work less than 35hours per week also rose from 6.4% to 7.1%. An important question is, what is the major cause of
the dispersion in hours of work?
To address this issue, we extend the basic model to generate cross-sectional diﬀerentials inhours of work. The modeling strategy is quite straightforward. We consider a two-industry version of the model to find parameters that are responsible for generating diﬀerentials in hours ofwork. Interestingly, the analysis reveals that the cost of recruitment is solely responsible for hours
diﬀerentials.

2

For the US economy, Kuhn and Lozano (2005) investigated this issue and documented that “highly educated,

high-wage, salaried men” had the greatest increase in long working hours. See also Hamermesh (1993).

4

2

The Model

2.1

Environment

This section constructs a dynamic model of the demand for hours and employment with search
frictions. The model developed in this section is an extension of Smith (1999). Consider an economy
consisting of workers and firms. The measure of workers is normalized to one. There is a large
number of firms, with the exact number to be determined by free entry. Time is discrete and allagents discount the future at the common discount rate r.
The number of matches M is determined by the standard constant returns to scale matching
technology: M = m(U, V ), where U is the total number of job seekers and V is the number ofaggregate job vacancies. A vacancy is matched with a worker during a period with probability q,
where
M
q≡
=m
V

µ

¶
U
,1 .
V

(1)

It is easy to verify that an increase in labor market tightness V /U ≡ θ decreases the matchingprobability q. Similarly, the probability that a worker is matched with a vacancy is given by
M/U = m(1, V /U ) = θq(θ). Let λ be the exogenous rate of job destruction. Then, in any steadystate, the flow into employment θq(θ)U must equal the flow into unemployment λ(1 − U ). Thus:m (U, V ) = λ(1 − U ),(2)which defines the Beveridge curve, as shown in Figure 1.
The production technology is given by f (hl), where h denotes hours of work per employee andl is the number of employees at each firm. We assume that f 0 (.) > 0 > f 00 (.) and f (0) = 0. The
instantaneous payoﬀ to a firm with l employees is given by f (hl) − whl − e(h)l − kv − π, where w
is the hourly wage rate, v is the number of vacancies, k > 0 is the cost of each vacancy, which may
be considered as an equipment cost (Acemoglu, 2001), π ≥ 0 is a fixed cost, and e(h) ≥ 0 is thecost associated with longer working hours. Function e(.) is meant to capture the time constraint
for each worker and the productivity loss due to longer working hours. We assume that e0 (h) > 0,e00 (h) > 0 and limh→∞ e(h) = ∞. It is important to note that without function e(.), employment
5and working hours will be perfect substitutes.3 In eﬀect, function e(.) precludes the scenario in
which a firm chooses hours of work per employee to be arbitrarily large.

2.2

Firms

Timing of events has a nontrivial consequence. In particular, as documented in Pissarides (2000,
chapter 7), whether the working hours are determined before or after wage determination has
nontrivial consequences. In this section, we assume that wage bargaining takes place before hours
of work and employment are determined. In other words, each firm takes the wage rate as given
when it chooses the composition of the labor demand. The firm chooses working hours and thenumber of vacancies to maximize the value of the firm, whereas wage bargaining determines how thetotal surplus is split between the firm and workers. We solve the model backwards. We first solve
the firm’s optimization in terms of hours and vacancies. Then, we analyze the wage determination.
Let J(l) be the value of an operating firm with l employees. The Bellman equation is given by¾½
1
J ((1 − λ) l + qv) ,(3)
J (l) = max f (hl) − whl − e(h)l − kv − π +h,v1+rwhere λ is the exogenous rate of job destruction. The first-order conditions with respect to h andv are:
f 0 (hl) − w − e0 (h) = 0,−k +q
J 0 ((1 − λ) l + qv) = 0,
1+r

(4)
(5)

respectively. The envelope condition yields
J 0 (l) = f 0 (hl) h − wh − e(h) +1−λ 0J ((1 − λ) l + qv) .
1+r

(6)

In any steady state, the flow into unemployment equals the flow into employment:
λl = qv.
3(7)

Booth and Schiantarelli (1987), Calmfors and Hoel (1988), Hoel (1986), and Holzer (1994), for example, consid-

ered the production function of the form f (g(h)hl), where g(h) is the average productivity per hour. According tothis modeling, employment and working hours are imperfect substitutes because the eﬃciency of labor depends on
working hours. Our modeling is meant to capture both changes in eﬃciency and other direct costs.

6

Then, (5) and (6) imply (r + λ)k/q = f 0 (hl)h − wh − e(h). With (4), this expression reduces to(r + λ) k= e0 (h)h − e(h).
q(8)

Since e00 (h) > 0, the right-hand side of (8) is increasing.
Equation (8) plays a central role in this paper. It connects the four measures of search frictions
and hours of work. One is obviously q. A decrease in q means that a firm finds it harder to filla vacancy. Similarly, an increase in k increases frictions by raising the cost of each vacancy. An
increase in the discount rate r also implies greater search frictions because a firm becomes lesspatient, so (8) implies that impatience raises working hours. Finally, an increase in the separation
rate also implies longer working hours because a firm needs to open more vacancies to maintaina steady-state level of employment. Interestingly, (r + λ)k/q measures the degree of frictions inlabor adjustment.

2.3

Workers and Wages

Let J U and J E denote the values of being unemployed and employed, respectively. The probability
that a worker finds a job is M/U = θq (θ). Then, the value of being unemployed is writtenrecursively as J U = θq(θ)δJ E +(1−θq(θ))δJ U , where δ ≡ 1/(1+r) is the discount factor. We assume
there are no unemployment benefits. The value of being employed is J E = wh+λδJ U +(1−λ)δJ E ,where λ is the exogenous separation rate, and wh is the instantaneous payoﬀ to the worker. We
ignore the utility cost of longer hours of work. From these two Bellman equations, we obtain
JE − JU =(1 + r) wh.
r + λ + θq (θ)(9)

We assume that workers and a firm share the total surplus. In addition, we assume that workers
are not unionized. Consider a bargaining process between a firm and workers of measure ∆. The
threat point for the firm is J (l − ∆) because failing to agree on a contract implies losing theworkers. The total match surplus is therefore J(l) − J(l − ∆) + ∆(J E − J U ). If the firm’s shareof the surplus is given by 1 − β ∈ [0, 1], then we have β[J(l) − J(l − ∆)] = (1 − β)∆(J E − J U ). In
the limit as ∆ → 0,¡¢
βJ 0 (l) = (1 − β) J E − J U .
7(10)

This is the key equation for rent sharing.4 Substituting the envelope condition (6) and (9) into
(10), we obtainf 0 (hl) h − wh − e(h) =1 − β (r + λ) wh,
β r + λ + θq (θ)(11)

from which it is easy to verify that the hourly wage rate w and the marginal product f 0 (hl) are
positively related. Using (4), we rewrite (11) asw=

β r + λ + θq (θ) e0 (h) h − e(h)
.
1−β
r+λ
h

(12)

This equation determines the wage rate w as a function of h and θ. It is easy to verify that∂w/∂θ > 0 and ∂w/∂h > 0. Using (8), we rewrite (12) further asW ≡ wh =βk r + λ + θq (θ)
,
1−β
q (θ)

(13)

where W is the earnings. Note that dW/dθ > 0. That is, as in Pissarides (2000), the earnings
increase with the labor market tightness. A new finding here is that as (8) implies that dh/dθ > 0,working hours and earnings are positively related. Murphy and Topel (1997) found a positive crosssectional relationship between working hours and the hourly wage rate. Murphy and Topel (1997)
interpreted this positive relationship as the standard labor supply curve based on a worker’s labor—
leisure choice. According to our result, the positive association between hours and the wage rate is
the result of market tightness. To be precise, an increase in labor market tightness raises the wage
rate as a result of Nash bargaining, and it also increases working hours because of (8). Because
an increase in the tightness raises both the wage rate and working hours, these two variables are
positively related. However, unlike the labor—leisure choice model, the two variables are related
only indirectly.
The key assumption of this paper is that firms, not workers, choose the hours of work. The
conventional view is that workers choose how long they work. For example, in standard neoclassicalmacroeconomics (Lucas and Rapping, 1969; Prescott, 2004), workers face a labor—leisure choice4

Smith (1999) also considered wage determination in a model similar to ours, and derived the wage rate using theintuition from the sequential bargaining theory (Osborne and Rubinstein, 1990). In contrast, our method exploits
the Nash sharing rule together with the envelope condition in the dynamic programming problem.

8

and choose hours of work optimally, taking the market wage rate as given. Such models requirethat working hours are divisible and that firms can employ any quantity of labor at the market
wage rate. From the firms’ viewpoint, it does not matter whether workers work longer or shorter
hours because the competitive market ensures that the quantity of labor demanded equals thequantity of labor supplied.
With search frictions and bilateral trading, if workers choose shorter working hours, then a firmmust pay extra search costs to maintain its labor input. Thus, firms do care about working hours,
and we believe this is realistic. We argue that it is plausible to assume that firms choose hours ofwork. Our framework is related to some extent to the indivisible labor literature (Hansen, 1985;
Rogerson, 1988), where workers must choose either to work full time or to be unemployed. In theindivisible labor literature, the standard working hours are treated as exogenous. In this paper,
the length of “full time” is determined by individual firms.

3

Equilibrium

3.1

Characterization

Following Smith (1999), we assume free entry of firms and that a firm entering the market opens
vacancies to achieve immediately the steady state level of employment l, so that there are notransitional dynamics or (transitory) size distribution of firms. Because the job finding rate is q,
in order to achieve l in the next period, the firm must create exactly l/q vacancies today. Thus,
the value of entry is given by
J (0) = −kl
1−π+J (l) .
q
1+r

(14)

Equation (14) is interpreted as follows. The new firm creates l/q vacancies in order to employ lworkers in the next period, and pays kl/q and π. Because the firm will employ l workers in thenext period, the continuation value is J(l). Free entry of firms therefore requires that J(0) = 0.
Thus, from (3) and (14), we obtainf (hl) − whl − e(h)l −9r+λkl = (1 + r) π.
q(15)

Equation (7) can be rewritten as λl = [m(U, V )/V ] × v, which implies that, assuming symmetricequilibrium, the number of firms isN≡V
m (U, V )
1−U
=
=
.
v
λl
l

(16)

Other steady-state conditions are given by (4)—(8) and (15), which can be reduced to a system of
equations in h and l:
f 0 (L) = w + e0 (h),(17)

f (L) = f 0 (L)L + (1 + r) π,(18)

where L ≡ hl. Equation (17) states that the marginal product of labor must equal the marginal
cost, and (18) asserts that the total output is distributed to labor and the fixed cost. Note that (18)determines the steady-state value of L. This suggests that with free entry of firms, the total demand
for labor inputs L is determined without any reference to the market tightness. This is because
the speed of adjusting the number of firms is infinite although the labor market is frictional. The
choice of h is influenced by the market tightness. However, since total labor input L is determinedby (18), the labor market tightness θ aﬀects the choice of h and l only through changes in thehourly wage rate w.
Let L∗ denote the solution to (18). Then, the general equilibrium of the economy is characterized by f 0 (L∗ ) = w + e0 (h), (8), and (12), which reduce to:(r + λ) kq (θ)= e0 (h)h − e(h),f 0 (L∗ ) =β r + λ + θq (θ) e0 (h) h − e(h)+ e0 (h),
1−β
r+λ
h

(19)
(20)

which determine the equilibrium values of θ and h. These equations have important interpretations.
Equation (19) relates search frictions and hours of work. Equation (20) comes from the wage
equation.
Proposition 1 There exists a unique steady-state equilibrium if −θq 0 (θ) /q (θ) ∈ (0, 1).

10

Proof. Equation (19) implicitly defines θ = F (h), and F is monotone-increasing. Totallydiﬀerentiating (20), we obtainihθq 0 (θ)
∙
¸
q
(θ)
1
+
q(θ)
β r + λ + θq (θ) e00 (h)e0 (h) h − e(h)
β
00
dθ +
+ e (h) dh.
0=
1−β
r+λ
h
1−β
r+λ
h
Thus, if −θq 0 (θ) /q (θ) ∈ (0, 1), then (20) implies a negative relationship between h and θ. Cobb—
Douglas matching technology, for example, satisfies this condition. Thus, (20) implicitly defines
θ = G(h), and G is monotone-decreasing. Therefore, there is a unique equilibrium.
Determination of equilibrium is depicted in Figure 2. The θ = G(h) locus is decreasing if andonly if 1 + θq0 (θ) /q (θ) > 0. This condition is violated if the matching technology exhibits a strong
complementarity, in which case the locus is upward sloping and the uniqueness of the equilibrium
is not guaranteed. In what follows we focus on the case in which 1 + θq 0 (θ) /q (θ) > 0.
Other equilibrium values can easily be computed. The hourly wage rate is given by (12) andemployment is given by l = L∗ /h, where L∗ is the solution to (18). Since θ = V /U is determined,
U and V are determined by the Beveridge curve (2), as in Figure 1, and the equilibrium numberof firms is given by (16).
Proposition 2 Specify the production technology as f (L) = ALα , with α ∈ (0, 1). (a) An increase
in A raises h, reduces l, and raises θ. This implies that earnings wh increase, V increases, U
decreases, and the number of firms N increases. (b) An increase in k raises h, and reduces land θ. This implies that earnings wh decrease, V decreases and U increases. The eﬀect on N isambiguous.
Proof. Under this specification, f 0 (L) = αALα−1 and (18) reduces to∙(1 + r)πL =(1 − α)A
∗

¸1

α

.

(21)

(a) Equation (20) implies that an increase in A shifts the θ = G(h) locus upward. It is thenevident that h and θ increase. Now, (21) implies that an increase in A decreases L∗ . Thus,l = L∗ /h decreases. From (16), it is now evident that N increases. (b) It is evident that anincrease in k shifts the θ = F (h) locus downward. Thus, h increases and θ decreases. Since L∗ is11

not aﬀected, l = L∗ /h decreases. Since U increases and l decreases, the eﬀect on the number offirms N is ambiguous.
The results are intuitive. An increase in productivity A induces potential firms to enter themarket, which tightens the labor market and raises the expected cost of a vacancy. Firms will
respond to this by substituting l for h. Greater tightness leads to higher earnings wh. Similarly,
an increase in the cost of vacancy k directly raises the expected cost of a vacancy. Thus, firms will
respond by substituting l for h, reducing job creation. Labor market tightness decreases and, as aresult, the earnings decrease.

3.2

WelfareIn this section, we consider the eﬃciency of the economy. Following Smith (1999), we consider thesocial planner’s problem for r = 0 and focus on the steady-state welfare: N [f (hl) − e(h)l − π] − kV .
Thus,
1−U
[f (hl) − e(h)l − π] − kV subject to m(U, V ) = λ(1 − U ).
h,l,U,Vlmax

The first-order conditions are:f 0 (L) = e0 (h),(22)

f (L) = f 0 (L)L + π,

(23)f (L) − π− e(h) =lk(mU + λ)
,
mV

(24)

m(U, V ) = λ (1 − U ) ,(25)

where L ≡ hl. The eﬃciency conditions (22)—(25) are comparable with the decentralized conditions.
First, we compare (23) with (18). With r = 0, these two conditions are identical, suggesting thatthe total labor input is eﬃcient. Consider (22) and (17). These conditions coincide if and only
if w = 0. This implies that the equilibrium hours of work are too short because each firm faces
an additional marginal cost from increasing hours, w, whereas the social planner faces only e0 (h).
Thus, the total labor input is eﬃcient, the equilibrium hours of work are shorter than optimal, andthe equilibrium employment is less than optimal.
12

3.3

The Employment Eﬀect of Restricting Working Hours

So far, we have seen that the hours of work implied by this model are less than optimal. However,
we cannot take this result at face value. Note that we have assumed that a worker’s instantaneous
payoﬀ is simply wh. Thus, in the analyses in the preceding sections, we have underestimated thewelfare loss associated with long working hours, and it is quite possible that hours of work must
be reduced. In addition, there is a policy debate about whether regulating hours of work can
increase the aggregate employment (Booth and Schiantarelli, 1987, Marimon and Zilibotti, 2000).
Suppose that hours of work can be perfectly controlled by regulation. Does restricting working
hours stimulate employment? The purpose of this section is to address this issue.
The Bellman equation is given by½
J (l) = max f (hl) − whl − e(h)l − kv − π +v¾1
J ((1 − λ) l + qv) .
1+r

Here, hours of work h is a policy parameter (Marimon and Zilibotti, 2000). The first-order conditions with respect to v is −k + q(1 + r)−1 J 0 (l) = 0, and the envelope condition is given by (6).
From these conditions, we have(r + λ) k= f 0 (hl) h − wh − e(h).
q(26)

From the free-entry condition, (r + λ)kl/q = f (hl) − whl − e(h)l − (1 + r)π. We combine these
two to obtain f (hl) = f 0 (hl)hl + (1 + r)π, which is identical to (18). This expression determines
the total labor input of each firm L∗ . This immediately implies that a reduction in h increases l.
In other words, there is a positive employment eﬀect of restricting hours of work at the individual
firm level.
Consider the general equilibrium eﬀects of restricting hours of work. From (11),
∙
¸−1
£ 0 ∗
¤
1−β
(r + λ)f (L ) h − e(h) .
wh = 1 +β r + λ + θq (θ)
Substituting this into (26) and arranging terms, we obtain1−β(r + λ) kβ (r + λ) q (θ)=≡ Ω (θ) .
f 0 (L∗ ) h − e(h)r + λ + θq (θ) + 1−β (r + λ)
β

13

(27)

It is easy to verify that Ω (θ) is decreasing. Equation (27) determines the equilibrium labor market
tightness, as depicted in Figure 3. Consider a reduction in h. This reduces the left-hand sideof (27) if and only if f 0 (L∗ ) − e0 (h) > 0. In this case, the equilibrium tightness θ increases as
h decreases. From the Beveridge curve (Figure 1), it is evident that the aggregate number ofvacancies V increases and the unemployment rate U decreases. To summarize:
Proposition 3 Suppose working hours h are perfectly regulated. A reduction in hours of work
increases the number of employees at each firm. It increases the aggregate vacancies and decreases
unemployment if and only if f 0 (L∗ ) − e0 (h) > 0.

3.4

Discussion

Smith (1999) considered the case of overemployment as a result of a bargaining externality. The
mechanism is simple. As we demonstrated in (11), the wage rate and the marginal product oflabor are positively related. With concave production technology, this implies that a larger labor
input reduces the marginal product and hence the wage rate. The key is that in Smith (1999), thewage rate is determined after the firm’s decision, leaving room for the firm to influence the wage
bargaining outcome by employing more workers. This is the source of over-employment in Smith(1999). In the preceding sections, we have assumed that wage bargaining takes place before thefirm chooses hours of work and vacancies. As a result, the total labor input in equilibrium turned
out to be eﬃcient.
This section demonstrates how the model may be modified to consider the bargaining externality
in Smith (1999). Since the key is that the firm’s choice will influence the bargaining outcome, theorder of events must be modified as follows. At the beginning of each period, each firm chooses
hours of work and the number of vacancies. Then, the bargaining takes place.
Consider the bargaining outcome. From (9) and (10), we havewh =r + λ + θq (θ) β
J 0 (l) ≡ ω(l).
1+r
1−β

(28)

In the first stage, the firm chooses hours and vacancies, taking into account how employment l

14

influences the bargaining outcome (28).5 Thus, the Bellman equation is:½
J (l) = max f (hl) − ω(l)l − e(h)l − kv − π +h,v¾
1
J ((1 − λ) l + qv) .
1+r

The first-order conditions with respect to h and v are:f 0 (hl) − e0 (h) = 0,−k +q
J 0 ((1 − λ) l + qv) = 0,
1+r

(29)
(30)

Similarly, the envelope condition is given by
J 0 (l) = f 0 (hl) h − ω(l) − ω0 (l)l − e(h) +1−λ 0J ((1 − λ) l + qv) .
1+r

(31)

Conditions (30) and (31) imply thatJ 0 (l) = f 0 (hl) h − wh − e(h) +(1 − λ) k
.
q

(32)

Free entry of firms implies that any entering firm earns zero profit:
J (0) = −kl
1−π+J (l) = 0.
q
1+r

(33)

Arranging these expressions, we obtain the equilibrium conditions as
f 0 (L) = e0 (h),(34)

f (L) = f 0 (L) L + (1 + r)π − ω 0 (l)l2 ,

(35)

r+λ
k = e0 (h)h − e(h) − ω(l) − ω0 (l)l,q(36)

where we have imposed the steady-state condition λl = qv. From (32) it is easily verified that
J 00 (l) = f 00 (hl) h2 < 0, implying that ω 0 (l)l < 0 above.
First, (34) is identical to (22). The key equation is (35), which must be comparable to (23).
By ω0 (l)l < 0, we can conclude that the equilibrium L is larger than the eﬃcient one, replicating
Smith’s (1999) result.6 Since L is too large, (34) implies that h is less than its optimum. Therefore,
Since J(l) is maximized with respect to h, we can ignore the eﬀect of h on J 0 (l).
6Remember that in Smith (1999), there is no employment—hours split, so too large L implies overemployment.
515

l is too large. The term ω0 (l)l < 0 is the source of the bargaining externality in Smith (1999),
in the sense that a firm has an incentive to overemploy to reduce the wage rates of all employees
by exploiting the relationship between the marginal product of labor and the wage rate. The
model presented in Section 2 does not possess this property because wage bargaining takes place
before the choice of hours and vacancies and therefore the firm cannot influence the wage rate byhiring more employees, even though the wage rate and the marginal product of labor are positively
related.
Interestingly, the analysis in Section 3.2 suggests that a firm may have another incentive tooveremploy. Without the bargaining externality, the total labor input is eﬃcient. However, since
the firm faces the wage rate as an extra marginal cost of increasing hours, the demand for hours
of work is less than the optimal. This causes the level of employment to be too large.

4

Interindustry Diﬀerentials in Working Hours

4.1

Characterization

Hamermesh (1993) documented that there are sizable diﬀerences in working hours across industries.
For example, in 1990 in the US, the average weekly hours of work were 44 hours for the mining
industry, 40.8 hours for manufacturing, 32.6 hours for services, and 28.8 for retail trade. Thisfinding is interesting because it suggests that there are large diﬀerences in working hours evenamong industries, each of which consists of a variety of jobs. Hamermesh’s (1993) insight is that
there are interindustry diﬀerentials in the relative costs of workers and hours.
Armed with the model developed in the preceding sections, we now investigate possible determinants of dispersion in working hours. There could be a variety of reasons why people work
diﬀerently, and we do not intend to give a comprehensive list of those reasons. Instead, we present
a simple two-industry model and explore the possibility of interindustry diﬀerentials in working
hours. The key question in this section is: which structural parameters are responsible for generating working hours diﬀerentials, and which parameters are not?
Consider an environment where there are two types of firms, type 1 and type 2. As before, the16matching technology is m(U, V ), and we define the market tightness as θ ≡ V /U . The probability
that a vacancy is filled is q(θ) ≡ m(U/V, 1), and the probability that a worker finds a job is θq(θ).
The Bellman equation for a firm of type i = 1, 2 is given by½
Ji (li ) = max Ai (hi li )α − wi hi li − e(hi )li − ki vi − π i +hi ,vi¾
1
Ji ((1 − λ) li + q (θ) vi ) .
1+r

Notice that both types of firms face the same market tightness. This reflects the assumption thatthe two industries are not segmented and that workers will accept all types of jobs.7 In addition,
we assume that all firms face the same separation rate.8 The first-order conditions with respect tohi and vi are:αAi (hi li )α−1 = wi + e0 (hi ),(37)(1 + r) ki
.
q (θ)

(38)

(r + λ) ki= e0 (hi )hi − e(hi ).
q (θ)

(39)

Ji0 (li ) =From the envelope condition, and (38), we obtain

Free entry requires that J(0) = 0, from which it is easy to establish that (1−α)Ai (hi li )α = (1+r)π i .
1Thus, total labor input in sector i is determined as hi li = [(1 + r)π i /(1 − α)Ai ] α . Substituting this
into (37), we obtain
1−α

1

α−1

α (1 − α) α Aiα [(1 + r) π i ] α = wi + e0 (hi ).

(40)

Equation (39) has a very important implication. It suggests that interindustry diﬀerentials inhours of work can arise only when k1 6= k2 . Remember that ki captures the cost of creating avacancy, and it works similarly to the cost of creating a job in Acemoglu (2001), in which thediﬀerence in the cost of equipment for each job creates the diﬀerence in job characteristics such as
wages and productivity. To emphasize:
7

To ensure this, we need to focus on equilibria where there is a nonnegative surplus for a relationship to share.

This requires that the fixed cost πi should not be too large.
8
Imagine the case where the separation rate λ diﬀers across firms. Firms facing greater separation rates would
choose longer working hours. A model with on-the-job search (Burdett and Mortensen, 1998) is required to generatea search equilibrium with endogenously dispersed separation rates.

17

Proposition 4 Interindustry diﬀerentials in working hours arise if and only if k1 6= k2 . In particular, k1 > k2 implies that h1 > h2 .
We quote Hamermesh (1993, p. 208):9
A survey of employers in the Rochester, New York, area in 1965—66 found an averagehiring cost for all occupations of $910, but an average for professional and managerial
workers of $4,600. A survey in Los Angeles in 1980 found hiring and training costs of$13,790 for salaried workers, and $5,110 for production workers. [...] In a nationwide
survey of large employers in 1979 the cost of hiring a secretary was $680, but for acollege graduate was $2,200.
Thus, there is a sizable diﬀerence in recruitment costs, and jobs that require higher skills are
more costly to fill. The model suggests that such jobs pay more, and require longer hours of work.
In what follows, we assume that k1 > k2 .
Let φ denote the equilibrium fraction of type 1 vacancies. The probability that a job seeker
finds a position is θq(θ). Ignoring the unemployment benefit, the value of being unemployed is£¤
J U = θq (θ) φδJ1E + (1 − φ) δJ2E + (1 − θq (θ)) δJ U ,(41)

where δ ≡ 1/(1 + r). Similarly, the value of being employed by a firm of type i is
JiE = Wi + λδJ U + (1 − λ) δJiE ,(42)

where Wi is the instantaneous payoﬀ to the worker. From (42), it is easy to verify that
J1E − J2E =1+r
W1 − W2=(W1 − W2 ) .
1 − (1 − λ) δ
r+λ

(43)

This implies that W1 > W2 if and only if J1E > J2E . We solve (41) and (42) as

9

J1E − J U=

J2E − J U=

All values are in 1990 US dollars.

∙¸
1+r
W2 − W1
W1 − θq (θ) (1 − φ),r + λ + θq (θ)
r+λ
∙
¸
1+r
W1 − W2
W2 − θq (θ) φ.
r + λ + θq (θ)
r+λ
18

(44)
(45)

As before, we assume that workers and the firm share the rent. The firm’s share of rent isdenoted by 1 − β, where β ∈ [0, 1]. The Nash sharing rule requires that β[Ji (l) − Ji (l − ∆)] = (1 −β)∆(JiE −J U ), where ∆ is the measure of workers. In the limit as ∆ → 0, βJi0 (l) = (1−β)(JiE −J U )for i = 1, 2. With (38),¡¢β (1 + r) ki= (1 − β) JiE − J U .
q (θ)

(46)

Computing J1E − J2E from (46) and substituting into (43), we obtain
W1 − W2 =(r + λ) β k1 − k2
.
1−β
q (θ)

(47)

It is then easy to establish that W1 > W2 holds if and only if k1 > k2 . Substituting (46) and (47)
into (44) and (45), we obtain(r + λ) ki
1−β=
Wi − θ [φk1 + (1 − φ) k2 ]
q (θ)
β

(48)

for i = 1, 2. Remember that the worker’s instantaneous payoﬀ as Wi = wi hi . We use (39) torewrite (48) aswi =β e0 (hi )hi − e(hi ) + θ [φk1 + (1 − φ) k2 ]
.
1−β
hi

(49)

for i = 1, 2. This is the wage equation for this economy.
The steady-state equilibrium is characterized by(r + λ) ki
q (θ)
1−α

1

α−1

α (1 − α) α Aiα [(1 + r) π i ] α= e0 (hi )hi − e(hi ),=β e0 (hi )hi − e(hi ) + θ [φk1 + (1 − φ) k2 ]+ e0 (hi ).
1−βhiThese two expressions are the counterpart of (19)—(20). To simply the exposition, we specify thefunctional forms as m(U, V ) = aV b U 1−b and e(h) = h2 . Then, we obtain
α (1 − α)

1−α
α

1
α

Ai [(1 + r) π i ]

α−1
α

¸µ
¶
(r + λ) ki 1/2 1−b
β
+2
=
θ 2
1−β
a
µ
¶
β [φk1 + (1 − φ) k2 ] (r + λ) ki −1/2 1+b
+
θ 2
1−β
a
∙

(50)

for i = 1, 2. This expression implies that dθ/dφ < 0 holds for both i = 1, 2. The equilibrium of themodel is given as the values of φ and θ that solve (50).
19

4.2

Interindustry Diﬀerentials: Alternative Formulation

In Section 4.1, interindustry diﬀerentials in working hours occur as a result of the diﬀerent costs
for creating a job, ki . Equation (39) suggests another potential explanation for the dispersion ofworking hours, namely, the diﬀerent employment probabilities faced by firms. This section explores
this possibility by developing a directed search version of the model in the spirit of Acemoglu (2001,
Section III.B) and Moen (1997).
Suppose that the markets for the two types of jobs are segmented. A worker can apply toeither the type 1 sector or the type 2 sector. We assume that the two sectors face the same
matching technology. Let Ui denote the number of unemployed workers applying to a firm of typei. Similarly, Vi is the number of vacancies of type i. Thus, the probability that a worker applying
to sector i finds a job in that sector is θi q(θi ), where θi ≡ Vi /Ui . Then the value of applying tosector i = 1, 2 is JiU = θi q(θi )δJiE + (1 − θi q(θi ))δJiU , where δ ≡ 1/(1 + r). Similarly, the value ofbeing employed is JiE = Wi + λδJiU + (1 − λ)δJiE for i = 1, 2. From these equations,rJiU =

1+r(1 + r) θi q (θi ) Wi
= r+λ
Wi .
r + λ + θi q (θi )θ q(θ ) + 1
i

(51)

i

The two types of jobs coexist if and only if J1U = J2U ≡ J U . This requires that1+rr+λ
θ1 q(θ1 ) + 1

W1 =

1+r
r+λ
θ2 q(θ2 ) + 1

W2 .

Thus, it is evident that W1 > W2 ⇔ θ2 > θ1 . That is, workers face a tradeoﬀ between higher
payoﬀs and better job finding rates. The wage equation iswi =β r + λ + θi q (θi ) e0 (hi ) hi − e(hi )≡ w (θi , hi ) .
1−β
r+λ
hi

We substitute this into (51) to obtainrJiU = θi q (θi )¤ β (1 + r) ki
β 1+r £ 0
θi .
e (hi ) hi − e(hi ) =1−βr+λ1−βThus, J1U = J2U implies thatk1 θ1 = k2 θ2 .

20

(52)

Therefore, k1 > k2 ⇔ θ2 > θ1 ⇔ W1 > W2 . In other words, the industry with a greater equipment
cost faces a less tight labor market because this industry pays more and attract more workers.
From firms’ optimization and free entry, we obtain the equilibrium conditions: αAi (hi li )α−1 =w(θi , hi ) + e0 (hi ), (1 − α)Ai (hi li )α = (1 + r)π i , (r + λ)ki /q(θi ) = e0 (hi )hi − e(hi ) for i = 1, 2. Fromthe last expression, θi and hi are positively related. Thus:
Proposition 5 Interindustry dispersion in working hours arises if and only if k1 6= k2 . In particular, k1 > k2 implies that h2 > h1 .
This result is striking. If the labor markets for the two industries are pooled (as in Section4.1), then greater equipment costs imply longer working hours because they directly increase search
frictions for firms. On the other hand, if the two labor markets are segmented and workers choose
either of the two, then firms face diﬀerent equipment costs and tightness. As pointed out by
Acemoglu (2001), firms facing a greater equipment cost pay more. The same mechanism works in
our framework, and workers apply to the high-paying industry. As a result, the market tightness
in the high-k industry is lower, which means firms face smaller frictions. Interestingly, this eﬀect
dominates the other.

5

Conclusion

This paper has studied the demand for employment and hours of work in a dynamic model withsearch frictions. The key finding is that a firm’s demand for hours of work increases as it faces
greater search frictions, and the frictions are captured by a simple expression. Since there is freeentry of firms, the total labor input is determined without any reference to search frictions, and isat the eﬃcient level. However, search frictions distort the composition of the labor demand.
We have used this model to find the source of diﬀerences in working hours. We found that
diﬀerences in recruitment costs are solely responsible for dispersion in hours of work across industries. This finding is related to some extent to Acemoglu’s (2001), who argued that jobs with higher
equipment costs pay more. We have treated the recruitment (or equipment) costs as parameters.
21

The nature of these costs, and especially their determinants, must be investigated.
In this paper, we did not consider heterogeneity of workers. Kuhn and Lozano (2005) documented that the observed increase in hours of work in the US was concentrated among skilled
men. A deeper understanding of dispersion in hours of work requires a model with heterogeneous
skills. It is interesting to introduce skilled and unskilled workers into the model of this paper toingestigate whether the model can account for Kuhn and Lozano’s (2005) finding.

22

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24

Figure 1:V

V = θU

Beveridge curve

U

Figure 2:θθ = F (h)θ = G (h)hFigure 3:

Ω(θ )f ' ( L ) − e' ( h ) > 0

High hLow hΩ(θ )θ