Employment and Hours of Work∗
Noritaka Kudoh†

Masaru Sasaki‡

Hokkaido University

Osaka University

January 16, 2009

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. Eachfirm chooses the demand for working hours and the number of vacancies, and the wage profile
is determined by Nash bargaining. The wage profile is generally nonlinear in hours of work,
and it defines the trade-oﬀ between employment and hours of work. Hours of work are longer
than optimal because only part of the cost is reflected in the wage profile. The Hosios condition
is not enough for eﬃciency. When there are two industries, workers employed by firms withhigher recruitment costs work longer and earn more. That is, “good jobs” require longer hours
of work. Interestingly, technology diﬀerentials cannot account for working hours diﬀerentials.
JEL classification: J21, J23, J31, J64.
Keywords: employment, hours of work, search frictions.

∗

We thank the associate editor and two referees for their extremely helpful comments. We also thank Hiromi

Nosaka and participants in seminars at Hitotsubashi University, Kansai University, Ritsumeikan University, 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 isfinancially supported by KAKENHI (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. Understanding how firms utilize these two margins is crucial for understanding the long-run trend in hours of work (Prescott, 2004; Pissarides, 2007), the movementsof employment and hours over the business cycle (Burnside et al., 1993), the likely eﬀect of regulation regarding hours of work (Hoel, 1986; Booth and Schiantarelli, 1987; Hunt, 1998; Marimon
and Zilibotti, 2000), and cross-sectional diﬀerences in hours of work (Hamermesh, 1993), to namea few.
The last few decades have witnessed declines in hours of work in major developed economies(Rogerson, 2006; Pissarides, 2007). Despite a decline in average working hours, a disparity inworking hours across workers has gradually widened. This disparity is particularly distinguished
for workers in their 30s. The Japanese Labour Force Survey (2006) documented that the share ofmale employees aged between 35 and 39 years who work more than 60 hours per week rose from
18.9% in 1993 to 23.5% in 2003.1 Similarly, the share of those who work less than 35 hours 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?
In this paper, we construct a dynamic equilibrium model of labor demand under search frictions
and focus 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. A novel feature of this paper
is that the source of labor adjustment costs is search frictions, and that the cost is influenced bylabor market tightness. Thus, the degree of substitution between employment and hours of work
is determined as part of the search equilibrium. This sharply contrasts with the traditional models
of working hours (Hoel, 1986; Booth and Schiantarelli, 1987; Calmfors and Hoel, 1988; Cahuc andZylberberg, 2004) and the labor adjustment models (Sargent, 1978; Hamermesh, 1993), in which
the degree of substitution between working hours and employment is structurally given by the1

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).

2

production function or the adjustment cost function.
To understand the role of search frictions, consider the standard neoclassical model (Lucas andRapping, 1969; Prescott, 2004), in which workers face a labor—leisure choice and choose hours ofwork optimally, taking the market wage rate as given. In the neoclassical framework, firms canemploy any quantity of labor at the market wage rate. Thus, from the firms’ viewpoint, it does
not matter whether workers work longer or shorter hours because the competitive market ensuresthat the quantity of labor demanded equals the quantity of labor supplied. On the other hand,with search frictions and bilateral trading, if employees work shorter, then the firm must pay extra
search costs to maintain its labor input. Thus, firms do care about working hours.
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 thefirm size in a search equilibrium is generally a formidable task.2 We incorporate the choice ofworking hours into Smith’s (1999) framework. Each firm chooses the number of vacancies andhours of work per employee to maximize the value of the firm.
A novel feature of the model is that wage bargaining determines the wage profile, which relatesearnings and hours of work. In particular, the wage profile is shown to be a nonlinear function ofhours of work, and this reflects the concave production technology and the workers’ convex utility
cost of longer hours of work. Interestingly, the wage profile is consistent with the standard upwardsloping labor supply curve, and it defines the trade-oﬀ the firm faces when choosing employment
and hours of work.
Smith’s (1999) main result is that firms overemploy. The mechanism is intuitive. The social
planner does not take into account the wage rate when the optimal employment is chosen because
wages are simply transfers among the members of the society. However, firms do care about
the wage rate. When the production technology is concave, an increase in employment results
in a reduction in the wage rate. Firms have incentives to exploit this opportunity, resulting inoveremployment. This mechanism also works in our framework.
2

Bertola and Caballero (1994), Bertola and Garibaldi (2001), Cahuc and Wasmer (2001), and Cahuc et al. (2008)

also developed search models with large firms.

3

However, it is not a trivial matter to assess whether hours of work are too long or too short.
This is because there are two opposite eﬀects at work. One eﬀect comes directly from Smith’s(1999) overemployment result because higher employment tends to induce shorter hours of work.
The other eﬀect comes from the choice of hours. When the social planner chooses hours of work,
he or she is concerned with the disutility form longer hours of work because it constitutes the social
cost of longer hours of work. However, when firms choose hours of work, they are concerned onlywith the wage rates. This implies that firms care about only part of disutility that is reflected in thewage profile. As a result, firms choose longer hours of work than the social planner does. We showthat overall, hours of work are too long if firms choose working hours. Interestingly, hours of work
are too short when workers choose working hours. This is a result of overemployment; it reduces
the marginal product and hence the marginal hourly wage rate, reducing workers’ incentives towork longer.
It has become increasingly important to ask whether regulating hours of work increases employment (Hoel, 1986; Booth and Schiantarelli, 1987; Hunt, 1998; Marimon and Zilibotti, 2000).
According to our model, under regulation of working hours, a reduction of hours of work generally increases employment at each firm. Whether it expands the aggregate employment depends
crucially on the initial condition. Suppose that firms are to choose hours of work, in which casehours of work are too long under laissez-faire. Starting at laissez-faire, regulating hours of work
increases the aggregate employment and reduces unemployment.
We also study two other forms of regulation. One is about wage regulation. We consider ascenario in which the wage profile is perfectly regulated, and find that a wage profile that mimics
the worker’s disutility function implements the eﬃcient levels of employment and hours of work.
The key is to match the marginal hourly wage rate to the marginal disutility from longer hours ofwork. The other regulation we consider is entry regulation. This issue is becoming very importantin the recent literature (Bertrand and Kramarz, 2002; Blanchard and Giavazzi, 2003; Fang andRogerson, 2007). We consider a scenario in which the number of firms is regulated, and findthat a decrease in the number of firms as a result of tighter regulation leads to shorter hours of4

work and a greater number of employees at each firm. However, it results in higher unemployment
because there are fewer firms. In other words, expansions in employment at individual firms cannot
compensate for the decrease in the number of firms.
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 themining industry, 40.8 hours for manufacturing, 32.6 hours for services, and 28.8 hours for retailtrade. We extend the model to ask why some individuals work longer than others. This finding
is interesting because it suggests that there are large diﬀerences in working hours even amongindustries, each of which consists of a variety of jobs. Accordingly, we focus on job characteristics
rather than worker characteristics as determinants of dispersion in hours of work.
We first address within-industry diﬀerentials in hours of work, such as the full-time-part-time
diﬀerentials. To account for the diﬀerentials, we adopt a model of a single industry in which thelabor market is pooled for two types of jobs. The key to working hours diﬀerentials is the diﬀerencesin recruitment costs. Hamermesh (1993), for example, documented that firms spend much higher
costs to recruit workers with skills or education than to fill jobs that do not require much skill ortraining. Our result implies that jobs with high recruitment costs are characterized as “good jobs”(Acemoglu, 2001), and workers with such jobs work longer than those with “bad jobs”.
Finally, we address interindustry diﬀerentials in hours of work. There are two industries andthe labor markets are segmented. Workers apply to one of the industries, so arbitrage dictates
between the two markets. We show that the industry with a higher recruitment cost pays more
and requires longer hours of work. The jobs in this industry are considered as “good jobs”, andtightness of the labor market in this industry is lower because more workers apply to this industry.
The key finding is that recruitment costs account for interindustry dispersion in hours of work.
Interestingly, we find that diﬀerences in technology cannot account for working hours diﬀerentials.

5

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.

2.2

Firms

The production technology is given by f (L), where L ≡ hl is total labor input, h denotes hours
of work per employee and l is the number of employees at each firm. We assume that f 0 (L) >0 > f 00 (L) and f (0) = 0. In particular, we assume that production technology is specified byf (L) = ALα , where A > 0 and α ∈ (0, 1).
The instantaneous payoﬀ to a firm with l employees is given by f (hl) − W (h, l)l − kv − c, where
v is the number of vacancies, k > 0 is the cost of each vacancy, which may include a recruitment
6

cost and an equipment cost (Acemoglu, 2001), and c ≥ 0 is a fixed cost. Each worker earns W (h, l),where W (h, l) is the wage function or wage profile to be determined as part of equilibrium. The
wage profile W (h, l) is negotiated in each period, so the profile is treated as a function of the statevariable l. It is important to note that w(h, l) ≡ W (h, l)/h is the average hourly wage rate, whichis generally diﬀerent from the marginal hourly wage rate Wh (h, l) if the wage profile is a nonlinear
function of h. For the firm’s decision making, what matters is the marginal hourly wage rate, so
we focus on W (h, l) rather than w(h, l).
Let J(l) be the value of an operating firm with l employees. The Bellman equation is given by
J (l) = max {f (hl) − W (h, l)l − kv − c + δJ (l+1 )} ,
h,v

(3)

where l+1 = (1 − λ)l + q(θ)v is the next period’s employment level, δ ≡ 1/(1 + r) is the discountfactor, and λ is the exogenous rate of job destruction. The first-order conditions with respect to hand v are:f 0 (hl) − Wh (h, l) = 0,(4)

−k + q(θ)δJ 0 (l+1 ) = 0,(5)respectively. The envelope condition yields
J 0 (l) = f 0 (hl) h − W (h, l) − Wl (h, l)l + (1 − λ) δJ 0 (l+1 ) .

(6)

In any steady state, the flow into unemployment equals the flow into employment:
λl = q(θ)v.

(7)(r + λ)k= f 0 (hl)h − W (h, l) − Wl (h, l)l.
q(θ)

(8)

Then, (5) and (6) imply

2.3

Workers and Wages

The value of being employed by a firm of size l, denoted by J E (l), satisfies
J E (l) = W (h, l) − e (h) + λδJ U + (1 − λ)δJ E (l+1 ),7(9)

where δ ≡ 1/(1 + r) is the discount factor, λ is the exogenous separation rate, e(h) is disutility
from work, and J U is the values of being unemployed. We assume that e0 (·) > 0, e00 (·) > 0 andlimh→∞ e(h) = ∞.
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 (l) − 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 (l) − J U ).
In the limit as ∆ → 0,¤£
βJ 0 (l) = (1 − β) J E (l) − J U .

(10)This is the key equation for rent sharing.3 Note that this amounts to maximizing the generalized
Nash product [J 0 (l)]1−β [J E (l) − J U ]β with respect to W (h, l), taking J U as given.
We need to solve the bargaining outcome (10) for the wage rate. First, use (5) and (6) to obtain
J 0 (l) =¤1+r £ 0f (hl) h − W (h, l) − Wl (h, l)l .
r+λ

(11)

Similarly, from (9), we obtain
E

J (l) − J

U

·
¸
r
1+r
U
W (h, l) − e (h) −
J ,
=
r+λ
1+r

(12)

Substituting (11) and (12) into (10), we obtain£¤(1 − β) r U
J ,W (h, l) = β f 0 (hl) h − Wl (h, l)l + (1 − β) e (h) +
1+r

(13)

which is a diﬀerential equation about the unknown wage function for each h and θ.
Proposition 1 The wage function is given by
W (h, l) = l
3

−1/β

Z l

¤
£
1
(1 − β) r U
J .
i β −1 f 0 (hi) h di + (1 − β) e (h) +1+r0Smith (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

Proof. In Appendix.
It is easy to verify that ∂W (h, l)/∂J U > 0. Thus, as in Pissarides (2000), the wage rate increases
with the value of unemployment. Since the value of unemployment is increasing in labor market
tightness, this implies that the wage rate increases with the tightness.
For f (L) = ALα , the wage function reduces toW (h, l) =αAhα lα−1
α + 1−ββ+ (1 − β) e (h) +(1 − β) r U
J ,
1+r

(14)

from which it is easy to verify that
Wh (h, l) =Wl (h, l) =α2 A [hl]α−1
α + 1−ββ+ (1 − β) e0 (h) > 0,

(α − 1) αAhα lα−2
α + 1−β
β

< 0.

(15)
(16)

These two results are worth emphasizing. First, the wage profile is not a linear function of h. For
α suﬃciently close to unity, Whh > 0 holds. In other words, as long as the production technology
is suﬃciently less concave, the marginal hourly wage rate is increasing in hours of work, justifying
the observed positive association between hours of work and the hourly wage rage. Murphy andTopel (1997), for example, found a positive cross-sectional relationship between working hours andthe hourly wage rate, and interpreted this relationship as the standard labor supply curve basedon a worker’s labor—leisure choice. Our result suggests that the positive association between hours
of work and the wage rate represents the wage profile as a result of wage bargaining, and this isderived without a worker’s labor—leisure choice.
Finally, (16) states that the wage rate is decreasing in the number of employees. As pointed
by Smith (1999), this eﬀect is due to the concavity of the production technology. We will see that
because of this eﬀect, firms have incentives to overemploy to reduce the wage rates. It is easy to
see that this term is zero for the linear production technology (α = 1).4
4

We do not consider the case with α = 1 because the return function in the Bellman equation will be linear in land hence unbounded. See Stokey and Lucas (1989).

9

3Equilibrium

3.1

Characterization

Following Smith (1999), we assume free entry of firms and that a firm entering the market opens alarge number of vacancies to achieve the steady state level of employment l in the next period, sothat there are no transitional dynamics or (transitory) size distribution of firms. Because the rateof filling a vacancy is q(θ), in order to achieve l in the next period, the firm must create exactlyl/q(θ) vacancies today. Thus, the value of entry is given by
J (0) = −kl
− c + δJ (l) .
q(θ)

(17)

Equation (17) is interpreted as follows. The new firm creates l/q(θ) vacancies in order to employ lworkers in the next period, and pays kl/q(θ) and c. Because the firm will employ l workers in thenext period, the continuation value is exactly J(l). The number of firms is determined by the freeentry condition J(0) = 0. Thus, from (3) and (17), we obtainf (hl) − W (h, l)l =r+λ
kl + (1 + r) c.
q(θ)

(18)

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

(19)

Consider the unemployed. The probability that an individual finds a job is M/U = θq(θ).
Then, the value of being unemployed is written recursively as
J U = b + θq(θ)δJ E (l+1 ) + (1 − θq(θ))δJ U ,(20)

where b is the (exogenous) unemployment benefit. Substitute (10) and (5) into (20) to obtain theequilibrium value of J U as
£
¤
r
β
β
J U = b + θq(θ)δ J E (l) − J U = b + θq(θ)δ
J 0 (l) = b +
θk.
1+r
1−β
1−β
10

(21)

Definition 2 A steady-state equilibrium under free entry is a set of variables h, l, θ, U , and V
that satisfy θ = V /U , (2), (4), (8), and (18), equipped with the wage profile (14).
Under free entry, U and V are determined by the Beveridge curve after θ is determined, so we
focus on the determination of h, l, and θ. The equilibrium is then summarized by the following:f 0 (hl) = Wh (h, l),(22)f (hl) = f 0 (hl)hl − Wl (h, l)l2 + (1 + r) c,(23)= Wh (h, l)h − W (h, l) − Wl (h, l)l,(24)

(r + λ)kq(θ)

where the wage function is given by (14) and J U is given by (21). Equation (22) is from the firm’s
choice of hours of work (4), (23) is derived by substituting (8) into the free entry condition (18),and (24) is derived by substituting (4) into the choice of employment (8). With f (L) = ALα , theseequations reduce to:αALα−1 = e0 (h) ,αβ + 1 − β(1 − β) (1 − α)ALα = (1 + r) c,
αβ + 1 − β£¤(r + λ) k+ βθk = (1 − β) e0 (h) h − e (h) − b ,
q(θ)

(25)
(26)
(27)

where L ≡ hl is the total labor input. It is easy to see that the equilibrium is uniquely determined.
First, the equilibrium labor input L is determined by the free entry condition (26). Given L, (25)determines h. Finally, (27) determines θ. The determination of equilibrium is depicted in Figure
2. The upward-sloping curve is from (27), which represents the optimal choice of employment:
as labor market tightness increases, the firm substitutes away from employment. However, thechoice of hours is independent of tightness because the the firm’s demand for hours of work is
determined by comparing the marginal product and the marginal wage rate Wh (h, l), both ofwhich are independent of tightness. This is reflected by the horizontal line in this figure.
Proposition 3 Suppose e(h) = εh2 . (a) An increase in A increases h and θ. (b) An increase in
k has no eﬀect on h and decreases θ. (c) An increase in ε decreases h and θ.
11

Proof. (a) Equation (26) implies that an increase in A reduces L. Since (25) and (26) imply
e0 (h)L = (1 + r)cα/(1 − β)(1 − α), it is evident that L and h are negatively related. Thus, anincrease in A increases h. (b) Note that k appears only in (27). An increase in k will shift theemployment curve upward in Figure 2. (c) From (26), changes in ε have no eﬀect on L. Given this,(25) and (27) are rewritten as H = 2εh and (r + λ)k/q(θ) + βθk = (1 − β)[εh2 − b], where H is aconstant. Eliminate h from these two expressions to obtain (r+λ)k/q(θ)+βθk = (1−β)[H 2 /4ε−b],from which it is easy to verify that dθ/dε < 0.
The results are intuitive. An increase in productivity 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 employment for hours of work. An increase in the cost of vacancy
directly raises the expected cost of a vacancy. This has two eﬀects, direct and indirect. The
direct eﬀect is that firms respond this increase in the vacancy cost by substituting employment forhours of work. The other, indirect, eﬀect is to discourage entry, which reduces tightness and firms
substitute hours for employment. The overall eﬀect of an increase in recruitment cost on hours
of work is neutral. Finally, an increase in the marginal disutility from work will reduce hours ofwork even though hours of work is the firm’s choice variable. The key is that the wage profile isdetermined by bargaining. An increase in the marginal disutility of work is reflected in the wage
profile, so firms face a greater marginal wage rate for choosing longer hours of work.
Here we present a numerical example. The parameters are A = 0.9, α = 0.8, β = 0.4, r = 0.01,
λ = 0.2, c = 0.1, k = 0.1, b = 0.1, e(h) = h2 , and q = θ−0.5 . The equilibrium is given by L = 0.83,
h = 0.41, l = 2.04, and θ = 0.58. Figure 3 shows the equilibrium wage profile. Since we have chosen
the parameters so that the disutility function is suﬃciently convex and the production function is
not too concave, the wage profile is convex as shown in the figure.

3.1.1

Neutrality Result

Does it matter who faces the cost of longer hours of work? Here, we briefly discuss an alternativeenvironment in which firms face the cost of longer hours of work. Let J(l) be the value of an
operating firm with l employees. The Bellman equation is given by J(l) = maxh,v {f (hl) − e(h)l −12W (h, l)l − kv − c + δJ(l+1 )}, where l+1 = (1 − λ)l + qv. The first-order conditions with respect
to h and v are f 0 (hl) − e0 (h) − Wh (h, l)h = 0 and −k + qδJ 0 (l+1 ) = 0, respectively. The envelope
condition yields J 0 (l) = f 0 (hl) h − e(h) − W (h, l) − Wl (h, l)l + (1 − λ)δJ 0 (l+1 ). The wage equation
becomes W (h, l) = β[f 0 (hl)h − e(h) − Wl (h, l)l] + (1 − β)rJ U /(1 + r). The corresponding wage
function is
W (h, l) =r
αβAhα lα−1
− βe(h) + (1 − β)JU .
αβ + 1 − β
1+rThe equilibrium conditions are: f 0 (hl) = e0 (h)+Wh (h, l), f (hl) = f 0 (hl)hl−Wl (h, l)l2 +(1+r)c,and (r + λ)k/q = e0 (h)h − e(h) + Wh (h, l)h − W (h, l) − Wl (h, l)l. It is easy to verify that these
conditions reduce to (25)—(27).
Proposition 4 It does not matter who faces the cost of longer hours of work.

3.2

WelfareIn this section, we consider the eﬃciency of the equilibrium. Following Smith (1999), we consider
the social planner’s problem for r = 0 and focus on the steady-state welfare: N [f (hl) − e(h)l − c] +bU − kV , where N is the number of firms in steady state. Thus,maxh,l,U,V1−U[f (hl) − e(h)l − c] + bU − kV subject to m(U, V ) = λ(1 − U ).
l

It is easy to establish that the eﬃcient allocation must satisfy:
f 0 (L) = e0 (h),(28)

f (L) = f 0 (L)L + c,

(29)f (L) − c− e(h) − b =lmU + λk,mV(30)m(U, V ) = λ (1 − U ) .

(31)

The eﬃciency conditions (28)—(31) are comparable with the decentralized conditions. First, we
compare (29) with (23). Even with r = 0, these two conditions are not identical because of theterm Wl (h, l) < 0.

13

By Wl (h, l) < 0, we can conclude that the equilibrium L is larger than the eﬃcient one,
replicating Smith’s (1999) result.5 The term Wl (h, l) < 0 is the source of 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 source of this externality is easily understood by the objective function for the social
planner. For the social planner, the wages are transfers among the members of the society. Thus,
for eﬃciency, wages should not influence agents’ decisions. However, for individual agents, wages
matter in a crucial manner. The eﬀect of overemployment on h is negative. However, the opposite
eﬀect is at work.
The same mechanism works for the eﬃciency of hours of work. The social planner is only
concerned with the marginal disutility from longer hours of work. However, because of the concave
production technology, an increase in hours of work has a negative impact on the wage rate. This
will encourage firms to choose longer hours of work. In addition, as the wage function (14) clarifies,
only a fraction of e0 (h) is reflected in the wage profile, which also encourages firms to expand hours
of work.
Proposition 5 h is above the optimal level h∗ .
Proof. Let h∗ and L∗ denote their eﬃcient levels. (26) and (29) imply L/L∗ = [1−α/(1−β)]1/α .
Since e0 (h∗ )/e0 (h) = (1 − β)L/L∗ , we have e0 (h∗ )/e0 (h) = (1 − β)[1 − α/(1 − β)]1/α < 1, which
implies h∗ > h.
Proposition 6 The equilibrium allocation is eﬃcient if and only if η(θ) = β and β = 0 hold,where η(θ) ≡ −q0 (θ)θ/q(θ).
Proof. When β = 0, (25) and (26) reduce to αALα−1 = e0 (h) and (1 − α)ALα = (1 + r)c.
These conditions coincide with (28) and (29) at r = 0. Note that [1 − η(θ)]q(θ) = mV and[1 − ξ(θ)]θq(θ) = mU hold, where η(θ) ≡ −q 0 (θ)θ/q(θ) is the elasticity of q(θ) with respect to θ,
5

Remember that in Smith (1999), there is no employment—hours split, so too large L implies overemployment.

14and ξ(θ) = 1 − η(θ) is the elasticity of θq(θ) with respect to θ. It is then easy to rewrite (30) as[1 − ξ(θ)]θq(θ) + λ
f (L) − c− e(h) − b =k.
l[1 − η(θ)]q(θ)(32)

Now consider (18). At r = 0, this is rewritten as
αβAhα lα−1βθq(θ) + λf (hl) − c−− e (h) − b =k.
(1 − β) l(1 − β) [αβ + 1 − β]
(1 − β) q(θ)
It is evident that this expression is identical to (32) if and only if η(θ) = β holds and β = 0.
Interestingly, the Hosios (1990) condition (η(θ) = β) is not enough for eﬃciency. The reason
is the presence of Smith’s (1999) externality, which exists as long as the production function isconcave. To eliminate it, one needs to impose β = 0, under which the firm takes all the bargaining
surplus. In such a case, firms no longer have incentives to overemploy to cut the wage rates because
externality is internalized.

3.2.1

Does it matter who chooses hours of work?

Does it matter who chooses hours of work? To address this question, suppose that each employedchooses hours of work to maximize the value of employment. Then, J E (l) = maxh {W (h, l)−e(h)+λδJ U + (1 − λ)δJ E (l+1 )}. The first-order condition is Wh (h, l) = e0 (h), or
α2 A [hl]α−1= e0 (h) .
αβ + 1 − β(33)Other equilibrium conditions aref (hl) = f 0 (hl) hl −(α − 1) αA [hl]αα + 1−ββ+ (1 + r) c,£¤
r+λk + βθk = (1 − β) e0 (h) h − e (h) − b + f 0 (hl) h − e0 (h) h.
q(θ)
It is evident that for L, it does not matter who chooses hours of work. By comparing (33) with(25) and (28), we can conclude that hours of work are shorter than optimal.
Now turn to the scenario in which hours of work are determined in the bargaining stage. In particular, consider the case in which the wage rate and hours of work are determined so as to maximize
the generalized Nash product. The Nash bargaining problem is given by maxW,h [J 0 (l)]1−β [J E (l) −15

J U ]β subject to (6) and (9), taking J 0 (l+1 ), J U , and J E (l+1 ) as given. The first-order conditions
with respect to W and h are (10) and¤£ EU
0 (l)]
¤
£ E
(l)
−
J
∂
J
∂
[J
+ βJ 0 (l)= 0.
(1 − β) J − J U
∂h
∂h

(34)

£
¤
∂ [J 0 (l)] ∂ J E (l) − J U
+
= 0,
∂h
∂h

(35)

Reduce these expressions towhere ∂[J 0 (l)]/∂h = f 0 (hl)+f 00 (hl)hl−Wh (h, l)−Wlh (h, l)l and ∂[J E (l)−J U ]/∂h = Wh (h, l)−e0 (h).
The wage function is the same as before and is given by (14). Thus, (35) becomes f 0 (hl)+f 00 (hl)hl−
Wlh (h, l)l − e0 (h) = 0, or
f 0 (hl) + f 00 (hl) hl =(α − 1) α2 A [hl]α−1
α + 1−ββ+ e0 (h) .

With f (L) = ALα , we obtain [αβ + 1 − β]−1 α2 ALα−1 = e0 (h), which is identical to (33). From
this and other equilibrium conditions, it is easy to establish that employment and hours of workin this economy coincide with those when hours of work are chosen by each worker.
Let h∗ denote the eﬃcient level of hours of work, hw denote hours of work chosen by workers,
hb denote hours of work chosen in bargaining, and hf denote hours of work chosen by firms. Then
we can summarize the results as follows.
Proposition 7 hw = hb < h∗ < hf .
Thus, it does matter who is to make the choice of hours of work. Its policy implication isimportant. If hours of work are determined by workers, then, for eﬃciency, policies must be
designed to induce longer hours of work. On other hand, if firms choose hours of work, then policesthat reduce hours of work may improve eﬃciency.

3.3

Implications for Regulation

3.3.1

The Employment Eﬀect of Restricting Working Hours

There is a policy debate about whether regulating hours of work can increase the aggregate employment (Booth and Schiantarelli, 1987, Marimon and Zilibotti, 2000). According to Proposition 7,
16

regulating hours of work may be welfare-enhancing if firms choose hours of work under laissez-faire.
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 by J(l) = maxv {f (hl) − W (l)l − kv − c + δJ(l+1 )}, wherel+1 = (1 − λ)l + q(θ)v. Here, hours of work h is a policy parameter (Marimon and Zilibotti, 2000).
The first-order conditions with respect to v is −k + q(θ)δJ 0 (l+1 ) = 0, and the envelope conditionis given by (6). From these equations, (r + λ)k/q = f 0 (hl)h − W (l) − W 0 (l)l. With free entry, we
have f (hl) = W (l)l + (r + λ)kl/q + (1 + r)c = f 0 (hl)hl − W 0 (l)l2 + (1 + r)c. The wage function is
given by (14) and J U is given by (21). After some algebra, it is easy to show that the equilibrium
is a pair of L and θ that satisfy(1 − β) (1 − α) ALα= (1 + r) c,
αβ + 1 − βr+λ(1 − β) αALα−1
k + βθk =h − (1 − β) [e (h) + b] ,q(θ)αβ + 1 − βwhere L = hl.
Proposition 8 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 αALα−1 > [αβ + 1 − β]e0 (h).
This implies that, starting at the laissez-faire, regulating hours of work will have no impact
on the aggregate number of vacancies and unemployment because αALα−1 = [αβ + 1 − β]e0 (h)holds at laissez-faire. As h reduces further, the left-hand side of the condition decreases and thecondition is satisfied.

3.3.2

Regulated Wages

Suppose that the wage rates are regulated. In particular, we assume that the wage profile W (h)is written by the regulator and is exogenous. To simplify the analysis, we assume that the wage
function is continuous and diﬀerentiable with respect to h. In addition, W 0 (h) > 0.

17

The Bellman equation is given by J(l) = maxh,v {f (hl) − W (h)l − kv − c + δJ(l+1 )}, wherel+1 = (1 − λ)l + q(θ)v. First-order conditions are f 0 (hl) − W 0 (h) = 0 and −k + q(θ)δJ 0 (l+1 ) = 0,
respectively. The envelope condition is J 0 (l) = f 0 (hl) h − W (h) + (1 − λ)δJ 0 (l+1 ). The key is thatabsence of wage bargaining, we have Wl (h) = 0, which rules out the bargaining externality because
firms have no incentive to overemploy to reduce the wage rates. With free entry, the equilibrium
is given by f 0 (hl) = W 0 (h), f (hl) = f 0 (hl) hl + (1 + r)c, and (r + λ)k/q(θ) = W 0 (h)h − W (h).
Proposition 9 The following wage function implements the eﬃcient levels of employment andhours of work: W (h) = e(h) + W̄ , where W̄ is constant.
Proof. It is evident that total employment is eﬃcient at r = 0. The choice of hours of work
will be eﬃcient if h satisfies f 0 (hl) = e0 (h). Thus, the regulator must make sure that W 0 (h) = e0 (h).
Take integral on both sides to obtain W (h) = e(h) + W̄ , where W̄ is an arbitrary constant.
This proposition suggests that, for eﬃciency, the regulator must set the marginal increase in
the wage rate to match the marginal disutility from longer hours of work. Regulators are usuallyconcerned with the minimum level of earnings, W̄ . However, it does not matter for eﬃciency of
employment and hours of work. From (r + λ)k/q(θ) = W 0 (h)h − W (h), we can conclude thatan increase in W̄ decreases θ, leading to a higher unemployment rate. In other words, the policy
maker faces a trade-oﬀ between W̄ and unemployment.

3.3.3

Entry Regulation

Some critics argue that entry regulation can serve as a useful labor market policy because lower
product market competition may increase employment. Using a model of monopolistic competition
and rent sharing, Blanchard and Giavazzi (2003) showed that tougher entry regulation leads tolower employment. Bertrand and Kramarz (2002) argued that stronger entry regulation slowed
down employment growth in France. Does entry regulation increase employment or reduce it? To
address this important issue, we modify the model so that the number of firms is regulated.
Suppose that the number of firms N is constant, and is suﬃciently large. We treat N as a18

continuous variable for the ease of exposition. Equation (19) in this environment must beNX
i=1

li = 1 − U (θ),(36)

where U 0 (θ) < 0 from the Beveridge curve. Thus,αAlα−1 = h1−αe0 (hi ) ,iαβ + 1 − β i£¤(r + λ) k+ βθk = (1 − β) e0 (hi ) hi − e (hi ) − b
q(θ)

(37)
(38)

hold for each firm i = 1, ..., N , and the free entry condition is replaced with (36). We look for a
Nash equilibrium in which individual li is determined as the solution to (36)—(38), taking as given
all other firms’ employment levels. It is easy to verify that (37) and (38) define li = Φ(θ) for all iin the symmetric equilibrium, where Φ0 (θ) < 0. Thus, (36) reduces to N Φ(θ) = 1 − U (θ), which
determines the equilibrium tightness.
Proposition 10 A decrease in N decreases h and θ, and increases l and U .
Proof. The symmetric equilibrium must satisfy N Φ(θ) = 1 − U (θ), where the left-hand sideis decreasing and the right-hand side is increasing. Thus, a decrease in N decreases θ. Expression(38) implies that a decrease in θ is associated with a decrease in h for all firms. Since l = Φ(θ)
is decreasing, a decrease in θ must be associated with an increase in l for all firms. U increases
because it is decreasing in θ.
Since hours of work are longer than optimal when the condition in Proposition 5 is satisfied,entry regulation is useful in reducing hours of work. However, although it expands employment at
each individual firm, it increases aggregate unemployment because the number of firms is limited.
An alternative way to model entry regulation is to introduce a fixed cost of entry while maintaining the assumption free entry. In this case, the free entry condition is replaced with J(0) = F ,where F > 0 is the cost of entry, which is the policy parameter. The equilibrium is characterizedby (25)—(27) and (26) is replaced with
(1 − β) (1 − α)ALα = (1 + r) c + rF.
αβ + 1 − β19It is then easy to establish that an increase in F increases L and reduces h and θ. In other words,
if entry regulation is implemented as a fixed entry fee, either pecuniary or non-pecuniary, then astronger regulation reduces hours of work and unemployment. Thus suggests that the exact formof entry regulation matters: a direct control of the number of firms reduces employment while
entry cost expands employment.

3.4

Endogenous Entry

In the basic model, we have assumed that J(0) = 0. This implies that there is entry of firms as longas J(0) > 0. However, from a potential entrant’s point of view, the opportunity cost of starting
up a company must be the value of being a worker (Fonseca et al, 2001). Thus, we assume thatan entrepreneur starts up a company if and only if J(0) ≥ J U . We maintain the assumption of aunit measure of workers. Thus, if J(0) < J U , then firms exit without increasing total measure ofworkers.6
Under endogenous entry, we replace J(0) = 0 with J(0) = J U . Thus, J U = −kl/q − c + δJ(l),so (23) is replaced with f (hl) = f 0 (hl)hl − Wl (h, l)l2 + (1 + r)c + rJ U , where J U is given by (21).
The equilibrium under endogenous entry is characterized by (25), (27), and¸·(1 − β) (1 − α)βαAL = (1 + r) c + b +θk ,αβ + 1 − β
1−β

(39)

from which we define L = L (θ) with L0 > 0. Thus, the equilibrium is characterized by a pair of hand θ that satisfyαA[L(θ)]α−1 = e0 (h) ,αβ + 1 − β£¤(r + λ) k+ βθk = (1 − β) e0 (h) h − e (h) − b .
q(θ)

(40)
(41)

Equilibrium determination is depicted in Figure 4. Equation (41) gives the upward-sloping curve as
in Figure 2. Equation (40) gives the downward-sloping curve. The intuition is roughly as follows.
6

Note that there is a continuum of workers while the number of entrepreneurs equals the number of firms, which

is finite. Thus, it requires a mass of entrepreneurs to change the total measure of workers, so we can safely assume
that the measure of workers is invariant to entrepreneurs’ decisions.

20

An increase in labor market tightness induces firms to substitute employment for longer hours
of work, which defines the upward-sloping curve. On other hand, an increase in tightness makes
entrepreneurship less attractive, which reduces the number of firms. Incumbent firms’ reactions
are to increase the total labor input, which reduces the marginal product of labor. Since firms
want to match the marginal product with the marginal disutility of longer hours, the demand forhours decreases.
Comparative statics results are summarized below.
Proposition 11 (a) An increase in A increases h and θ. (b) An increase in k has an ambiguous
eﬀect on h and decreases θ. (c) An increase in λ increases h and decreases θ.

4Dispersion in Hours of Work

4.1

Preliminaries

According to the Groningen Growth & Development Centre (GGDC) 60-Industry Database,7 in2003 in the US, the annual hours of work per employee were 2,306 hours for the mining industry,
1,615 hours for retail trade, and 1,350 hours for hotel and catering. In 2003 in the European Union,
the annual hours work were 1,738 hours for the mining industry, 1,515 hours for retail trade, and1,503 hours for hotel and catering. In 2002 in Japan, the annual hours of work per employee were2,055 hours in the mining industry, 1,661 hours for retail trade, and 1,661 hours for hotel andcatering.
Armed with the model developed in the preceding sections, we investigate possible determinantsof 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 focus on job characteristics (rather than worker characteristics such as diﬀerences in preferences) as determinantsof dispersion in hours of work. In particular, we present a simple model with two types of firms
and explore the possibility of diﬀerentials in working hours. The key question in this section is:
760-Industry Database is available at http://www.ggdc.net/index-dseries.html. For a comprehensive account of

this database, see Rogerson (2006), whose focus is on (the evolution of) cross-country diﬀerentials in hours of work.

21

which structural parameters are responsible for generating working hours diﬀerentials, and whichparameters are not?
In this section and the next, we consider two models of hours dispersion. First, we develop amodel of within-industry hours dispersion. This model assumes that the labor market is pooled
for all jobs, so a job seeker can receive a job oﬀer from a firm of either industry. Next, we consider
a model of interindustry hours dispersion. This model assumes segmented labor markets, so a job
seeker in an industry cannot receive a job oﬀer from a firm of the other industry.

4.2

Within-Industry Diﬀerentials

Suppose that there is a single industry, but there are two types of firms, type 1 and type 2. Asbefore, the 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). We assume a single labor market. Thus, all firms and workers face the same labormarket tightness θ.
The Bellman equation for a firm of type j = 1, 2 is given by Jj (lj ) = maxhj ,vj {Aj [hj lj ]α −jj
W j (hj , lj )lj − kj vj − cj + δJj (l+1)}, where l+1= (1 − λ)lj + q(θ)vj . Notice that both types offirms face the same market tightness. This reflects the assumption that the two firms are in thesame industry so the markets are not segmented, and that workers will accept all types of jobs.8
We assume that the bargaining weight is the same for all firms.9 In addition, we assume that
all firms face the same separation rate.10 The first-order conditions with respect to hj and vj ,and the envelope condition are αAj [hj lj ]α−1 = Whj (hj , lj ), q(θ)δJj0 (lj ) = kj , and (r + λ)δJj0 (lj ) =αAj [hj lj ]α−1 hj − W j (hj , lj ) − Wlj (hj , lj )lj , respectively.
8

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 ci should not be too large.
9Here, we choose to avoid focusing on the bargaining weight because the foundation of this parameter is therelative frequency of making an oﬀer in the context of strategic bargaining (Osborne and Rubinstein, 1990).
10
It is evident that firms facing greater separation rates would choose longer working hours. A model with on-thejob search (Burdett and Mortensen, 1998) is required to generate a search equilibrium with endogenously dispersed
separation rates.

22

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 bargaining requires βJj0 (lj ) = (1 − β)(JjE − J U ) for
j = 1, 2, where the value of being employed by a firm of type j is JjE = Pj + λδJ U + (1 − λ)δJjE ,where Pj ≡ W j (hj , lj ) − e(hj ). It is easy to establish that the wage function is given byj

W (hj , lj ) =αAj hαj ljα−1
α + 1−ββ+ (1 − β) e (hj ) +(1 − β) r U
J .
1+r

(42)

Let φ denote the equilibrium fraction of type 1 vacancies. The probability that a job seeker
finds a position is θq(θ). Given that a job seeker finds a position, he or she is employed by a firm oftype 1 with probability φ. The value of being unemployed is J U = b + θq(θ)[φδJ1E + (1 − φ)δJ2E ] +(1 − θq(θ))δJ U , or
¢
¡
¢¤
£ ¡
r
J U = b + θq (θ) φδ J1E − J U + (1 − φ) δ J2E − J U .
1+r

(43)

Noting JjE − J U = (1 − β)−1 β(1 + r)kj /q(θ), we haverβθ [φk1 + (1 − φ) k2 ]JU = b +
.
1+r
1−β

(44)

Lemma 12 P1 > P2 holds if and only if k1 > k2 .
Proof. From the employee’s value functions, it is easy to verify that J1E − J2E = (r + λ)−1 (1 +r)[P1 − P2 ]. Similarly, the bargaining outcome and the vacancy choice imply JjE − J U = (1 −β)−1 β(1 + r)kj /q(θ) for i = 1, 2. From these equations, we obtain
P1 − P2 =β r+λ(k1 − k2 ).
1 − β q(θ)

It is then easy to establish that P1 > P2 holds if and only if k1 > k2 .
In other words, jobs with higher recruitment costs are “good jobs” (Acemoglu, 2001).

23

Assuming endogenous entry (J(0) = J U ),11 the equilibrium conditions areα
Aj Lα−1= e0 (hj ) ,jαβ + 1 − β¸·βθ [φk1 + (1 − φ) k2 ](1 − β) (1 − α)α
Aj Lj = (1 + r) cj + b +,αβ + 1 − β
1−β£¤(r + λ) kj+ βθkj = (1 − β) e0 (hj ) hj − e (hj ) − b ,
q(θ)

(45)
(46)
(47)

for j = 1, 2. The equilibrium is characterized by 6 equations in 6 unknowns, L1 , L2 , h1 , h2 , θ,and φ. Equation (47) suggests that, given the equilibrium labor market tightness θ, there is a
one-to-one relationship between kj and hj . It suggests that within-industry diﬀerentials in hours
of work can arise only when k1 6= k2 . Remember that kj captures the cost of creating a vacancy,
and it works similarly to the cost of creating a job in Acemoglu (2001), in which the diﬀerence inthe cost of equipment for each job creates the diﬀerence in job characteristics such as wages andproductivity. To emphasize:
Proposition 13 Within-industry diﬀerentials in working hours arise if and only if k1 6= k2 . In
particular, k1 > k2 implies that h1 > h2 . Technology diﬀerentials cannot account for diﬀerentials
in hours of work.
Jobs with a greater recruitment cost are “good jobs” as shown in Lemma 12. Further, Proposition 13 implies that good jobs are associated with longer hours work. Since there are two types ofjobs in a single industry, a possible interpretation of the result is that it describes the diﬀerentials
in hours of work between full-time and part-time jobs, or diﬀerentials between jobs that require
training and those that do not require training. To see the diﬀerence in recruitment costs, we quote
Hamermesh (1993, p. 208):12
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
11

Unfortunately, the basic model cannot be used to study heterogeneous firms because it cannot support anequilibrium with heterogeneous firms. Thus, the model we employ in what follows is the one with endogenous entry
presented in Section 3.4. The consequence of adoping the model with free entry is presented in Appendix B.
12
All values are in 1990 US dollars.

24

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.

4.3

Interindustry Diﬀerentials

To account for interindustry hours dispersion, this section studies a directed search version of themodel in the spirit of Acemoglu (2001, Section III.B) and Moen (1997). The key feature is that
there are two industries and their labor markets are segmented: once entered, a worker will neverreceive a job oﬀer from a firm of the other industry. However, each unemployed can freely choose
one of the industries at the beginning of each period. Arbitrage dictates and the values of seekinga job in the two industries must be balanced in equilibrium.
Suppose that the markets for the two types of jobs are segmented, and each worker can apply
to either industry 1 or industry 2. We assume that the two industries face the same matching
technology. Let Uj denote the number of unemployed workers applying to a firm of industry
j. Similarly, Vj is the number of vacancies of industry j. Thus, the probability that a worker
applying to industry j finds a job in that industry is θj q(θj ), where θj ≡ Vj /Uj . Let µ be
the fraction of workers in industry 1. Then, (2) is replaced with m (U1 , V1 ) = λ(1 − U1 )µ andm (U2 , V2 ) = λ(1−U2 )(1−µ), which define the Beveridge curves for industries 1 and 2, respectively.
Following Acemoglu (2001), we assume that each industry produces a distinct commodity. Let
p1 (µ) and p2 (µ) be the prices of the product in industries 1 and 2, respectively. We assume thatp1 (µ) is decreasing and p2 (µ) is increasing.13
As before, we assume that the bargaining weight is the same for all industries. In addition,
we assume that all firms face the same separation rate. The Bellman equation for a firm in
13

In general, the price function is more complex. However, we simply postulate the price function in order to avoid

further complication of the analysis. For a derivation of the price function, see Acemoglu (2001), for example.

25

jindustry j = 1, 2 is given by Jj (lj ) = maxhj ,vj {pj (µ)Aj [hj lj ]α − W j (hj , lj )lj − kj vj − cj + δJj (l+1)},
jwhere l+1
= (1 − λ)lj + q(θj )vj . The first-order conditions with respect to hj and vj , and theenvelope condition are αpj (µ)Aj [hj lj ]α−1 = Whj (hj , lj ), q(θj )δJj0 (lj ) = kj , and (r + λ)δJj0 (lj ) =αpj (µ)Aj [hj lj ]α−1 hj − W j (hj , lj ) − Wlj (hj , lj )lj , respectively.
The Nash bargaining requires βJj0 (lj ) = (1 − β)(JjE − JjU ) for j = 1, 2. The value of applying toindustry j = 1, 2 is JjU = b + θj q(θj )δJjE + (1 − θj q(θj ))δJjU . Similarly, the value of being employedis JjE = Pj + λδJjU + (1 −λ)δJjE for j = 1, 2, where Pj = Wj (hj , lj ) −e(hj ). From these expressions,
we obtainrβJU = b +
θj kj
1+r j
1−β
(see also (21)). For the two sectors to coexist, J1U = J2U must hold, which requires
k1 θ1 = k2 θ2(48)

in any equilibrium.
Lemma 14 P1 > P2 holds if and only if k1 > k2 .
Proof. From the employee’s value functions, it is easy to verify that J1E − J2E = (r + λ)−1 (1 +r)[P1 − P2 ]. Similarly, the bargaining outcome and the vacancy choice imply JjE − J U = (1 −β)−1 β(1 + r)kj θj /θj q(θj ) for i = 1, 2. From these equations, we obtain
¸
·
1
β
1
(r + λ)k1 θ1−.
P1 − P2 =1−β
θ1 q(θ1 ) θ2 q(θ2 )It is then easy to establish that P1 > P2 holds if and only if θ2 > θ1 . Finally, (48) implies k1 > k2 .

Jobs with higher recruitment costs are “good jobs” as in Lemma 12. The industry with a greater
recruitment cost faces a less tight labor market because this industry pays more and attract moreworkers.

26

With endogenous entry, the equilibrium is characterized byαpj (µ)Aj Lα−1
= e0 (hj ) ,jαβ + 1 − β·¸(1 − β) (1 − α)βαpj (µ)Aj Lj = (1 + r) cj + b +θj kj
αβ + 1 − β
1−β£¤(r + λ) kj+ βθj kj = (1 − β) e0 (hj ) hj − e (hj ) − b ,q(θj )

(49)
(50)
(51)

k1 θ1 = k2 θ2 , m (U1 , V1 ) = λ(1 − U1 )µ, m (U2 , V2 ) = λ(1 − U2 )(1 − µ), θ1 = V1 /U1 , θ2 = V2 /U2 .
Thus, the equilibrium is the solution to the system of 11 equations in 11 unknowns, L1 , L2 , h1 , h2 ,
θ1 , θ2 , U1 , U2 , V1 , V2 , and µ.
Proposition 15 Suppose that all industries share the same β, λ, and e(h). Interindustry dispersion in working hours arises if and only if k1 6= k2 . In particular, k1 > k2 implies h1 > h2 .
Technology diﬀerentials cannot account for diﬀerentials in hours of work.
Proof. Equation (51) implies that hj and the left-hand side of (51) is positively related. Since(48) holds, the second term of the left-hand side of (51) is the same for industries 1 and 2. So we
can focus on comparing (r + λ)k1 /q(θ1 ) and (r + λ)k2 /q(θ2 ). Thus,¸·(r + λ) k2 θ2 (r + λ) k2
r+λ(r + λ) k1 (r + λ) k2
r+λ−== k2 θ2−>0−
q(θ1 )q(θ2 )q(θ1 ) θ1
q(θ2 )
θ1 q(θ1 ) θ2 q(θ2 )because k1 > k2 ⇔ θ2 > θ1 ⇔ θ2 q(θ2 ) > θ1 q(θ1 ). This establishes that the left-hand-side of (51) ishigher for industry 1 than industry 2.
If the labor markets are segmented for diﬀerent industries and workers choose either of the two,then firms face diﬀerent labor market tightness. As pointed out by Acemoglu (2001), firms facing
a greater recruitment cost (or equipment cost) pay more. Thus, more workers apply to the highpaying industry, reducing the tightness in that industry. Proposition 15 establishes that such jobs
require longer hours of work. Interestingly, diﬀerences in technology Aj cannot account for working
hours diﬀerentials; they account for diﬀerences in the total labor input Lj across industries. Since
employment at each firm satisfies lj = Lj /hj , both technology and recruitment costs account for
diﬀerences in firm size across industries.
27

5

Conclusion

This paper has studied the choice of employment and hours of work in a dynamic model withsearch frictions and wage bargaining. It revealed that the bargaining outcome defines the wage
profile that the firm faces when choosing the composition of employment and hours of work. We
found that diﬀerences in recruitment costs are responsible for dispersion in hours of work bothacross industries and within an industry. In particular, “good jobs” (Acemoglu, 2001) pay more
and require longer hours of work. Interestingly, technology diﬀerentials cannot account for working
hours diﬀerentials.
An advantage of our approach is that the wage profile is derived as part of an equilibrium.
Since the wage profile reflects both the production technology and the worker’s disutility fromlabor, hours of work may be determined either by the firm’s value maximization or by a worker’s
labor—leisure choice. In addition, the wage profile is consistent with the empirical labor supply
curve. We found that hours of work determined by a worker’s labor—leisure choice are too short.
In contrast, hours of work determined by the firm’s value maximization are too long, suggestingthat the regulator must induce shorter hours of work. Our model highlights the empirical question
as to who is to make the choice of hours of work. We also found that the Hosios condition is notenough for eﬃciency.
A limitation of our approach is that the basic model with free entry cannot support an equilibrium with heterogeneous firms. We have modified the basic model by replacing the free entry
assumption with endogenous entry. Generally, one needs a richer model of multiple industries toaccount for heterogeneity of job characteristics in equilibrium, and much of this line of research isleft for future research. A possible direction is suggested by Cahuc et al. (2008).
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 toinvestigate whether the model can account for Kuhn and Lozano’s (2005) findings.

28

In this paper we focused solely on the steady state. The choice of employment and hours
of work is important for understanding business cycles. It is often documented that during arecession, firms cut employment rather than reducing hours of work enough to maintain the levelof employment. Since employment adjustment is costly, it seems rational to hoard labor duringa recession if the aggregate business condition comes back shortly (Burnside et al., 1993; Bertola
and Caballero, 1994). This suggests that firms face a large and persistent negative shock during arecession. An important future work is to extend the model of this paper to explain the substitution
between employment and hours of work over the business cycle.

29Appendix
A

Proof of Proposition 1

Rewrite (13) as·¸
f 0 (hl) h 1 − βr1U+e (h) +J .
Wl (h, l) + W (h, l) =
βl
l
βl
1+r
Let y(l) ≡ W (h, l), a(l) ≡ B/l ≡ β −1 /l, andg(l) ≡·¸r
f 0 (hl) h 1 − βΠ+e (h) +J U ≡ π(l) + .
l
βl
1+r
l

Then the equation can be written more compactly as
y 0 + a(l)y = g(l),(52)

which is a linear ordinary diﬀerential equation with variable coeﬃcients, and its solution is known
to be easily derived. A useful reference is Bellman and Cooke (1995, pp. 40—41.), from which thegeneral solution to (52) is given by½ Z l¾ ·Z l½Z i¾¸
y(l) = exp −a(s)dsexp
a(s)ds g(i)di + C ,000where C is a constant. Since y(l) is the wage function, it is (economically) reasonable to assumey(0) = 0, which leads to C = 0. Thus,
½ Z l
¾Z l
½Z i
¾
½ Z l
¾
Z l
y(l) = exp −a(s)dsexp
a(s)ds g(i)di =exp −a(s)ds g(i)di.
0

0

0

0

i

Notice that
¾
¾
½
½ Z l
Z l
n
o
1
ds = exp −B [ln s]li = exp {−B [ln l − ln i]} = l−B iB .
a(s)ds = exp −B
exp −
i
i s
Thus,
y(l) =

Z l

l

0

= l

−B

¸
·
Z l
Z l
Π
−B
B
−B
di = li π(i) +i [π(i)] di + ΠliB−1 di
i
0
0

−B B

Z l
0

iB [π(i)] di +

Π
.
B

30

Thus, the wage function we are looking for is
W (h, l) = l

−1/β

Z l
0

i

1
−1
β

·
¤
£ 0
f (hi) h di + (1 − β) e (h) +

¸
r
U
J ,
1+r

which is a generalization of the wage functions derived by Bertola and Caballero (1994), Bertolaand Garibaldi (2001) and Cahuc and Wasmer (2001).

B

Firm Heterogeneity under Free Entry

This section briefly discusses why the free entry assumption must be replaced with endogenous
entry. Under free entry (J(0) = 0), the equilibrium conditions areα
Aj Lα−1= e0 (hj ) ,jαβ + 1 − β(1 − β) (1 − α)
Aj Lαj = (1 + r) cj ,αβ + 1 − β£¤(r + λ) kj+ βθkj = (1 − β) e0 (hj ) hj − e (hj ) − b .
q(θ)

(53)
(54)
(55)

Note that φ does not appear in (54), while (46) contains φ. It is easy to verify that (54) determines
Lj , and given this, (53) determines hj for j = 1, 2. However, (55) contradicts the fact that all firms
face the same tightness θ.

31

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34

V
Increase in θ

V = θU

Beveridge CurveU
Figure 1hOptimal employment choiceFree entry andoptimal choice ofhours

θFigure 2

Figure 3: Wage Profileh

Optimal employment choice

Endogenous entry
and optimal choiceof hoursθ
Figure 4