Discussion Papers In Economics
And Business

Precautionary Demand for Labor in Search Equilibrium

Noritaka Kudoh

Masaru Sasaki

Discussion Paper 07-34

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPrecautionary Demand for Labor in Search Equilibrium

Noritaka Kudoh

Masaru Sasaki

Discussion Paper 07-34

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Precautionary Demand for Labor in Search Equilibrium∗
Noritaka Kudoh†

Masaru Sasaki‡

October 1, 2007

Abstract
This paper studies firms’ job creation decisions in a labor market with search frictions. A
simple labor market search model is developed in which a firm can search for a second employee
while producing with a first worker. A firm expands employment even if the instantaneous
payoﬀ to a large firm is less than that of staying small–a firm has a precautionary motive toexpand its size. In addition, this motive is enhanced by a greater market tightness. Because ofthis eﬀect, firms’ decisions become interdependent–a firm creates a vacancy if it expects otherfirms to do the same, creating strategic complementarity among firms and thereby self-fulfilling
multiple equilibria.
JEL classification: E24, J23
Keywords: labor demand, firm size distribution.

∗

We would like to thank Ryoichi Imai, Takashi Shimizu and the participants of the Japanese Economic Association

Meetings at Chuo University and the DC conference at Keio University for their helpful comments and discussions.
Part of this research is financially supported by the Japan Society for the Promotion of Science.
†
Department of Economics, Hokkaido University; E-mail: kudoh@econ.hokudai.ac.jp‡
Department of Economics, Osaka University; E-mail: sasaki@econ.osaka-u.ac.jp1

1

Introduction

Why do firms want to be larger at some points in time and smaller at diﬀerent points in time? This
paper extends the standard search-matching model of the labor market to investigate the firms’ job
creation decision regarding whether to expand employment or to stay small, and looks for factorsthat influence the decision. This paper demonstrates the existence of precautionary labor demandin search equilibrium.
The standard search-matching model, which is summarized by Pissarides (2000) and Rogerson
et al. (2005), has received much attention as the framework for studying various labor market
issues. In the standard model, a match, not a firm, is the production unit. Each job and worker
engages time-consuming search in the labor market, and each matched pair produces. This is
novel in itself because researchers can focus on the number of jobs created in the market instead ofstudying the employment policy of each firm. In the basic model, the number of jobs is determinedby free entry. However, such a job creation process ignores the decision of each incumbent firm
regarding whether to expand employment or not. We find this issue very important and worth
exploring.
Papers that are most related to this study are those of Bertola and Caballero (1994) and Smith(1999), who studied models with endogenous firm size. Bertola and Caballero (1994) considereda search-matching model in which firms can open many vacancies at once and job destruction is
endogenous. The business conditions shock causes labor demand to fluctuate, and firms may havean incentive to fire some of their employees. However, Bertola and Caballero (1994) found thatfirms hoard labor during recessions because the good time will come sooner or later and hiring takes
time. Smith (1999) employed a simple representative model of a firm and demonstrated that, withconcave production functions, firms overemploy because the wage rate declines as more workers
are employed.
In contrast, we stay as close to the standard Pissarides (2000) economy as possible, but modify
it by assuming that a firm can employ up to two workers. We focus on a firm’s discrete choiceproblem regarding whether to expand employment or to stay small. In a frictionless static economy,2the answer to the question would be evident. A firm expands employment if the instantaneous
payoﬀ of being large exceeds that of being small.
The answer becomes less evident when we address the question in a dynamic economy withsearch frictions. The main finding of this paper is that a firm of size one creates a vacancy andtries to expand employment even if the instantaneous payoﬀ to a large firm is less than that ofstaying small. We call this result precautionary demand for labor. The mechanism that drives theresult is similar to that found in Pissarides (1994), in which a worker can engage in on-the-job
search. If on-the-job search is possible, an unemployed worker’s reservation wage can be below theunemployment benefit because the value of employment includes the option of finding a better job
in the future. A similar mechanism works in this paper, but in the opposite direction. Even if theinstantaneous payoﬀ to be a size-one firm is high, the value of this state is not very good because
the firm is exposed to the risk of losing the only employee and the opportunity to produce.
The notion of precautionary labor demand slightly diﬀers from that of labor hoarding foundin Bertola and Caballero (1994). labor hoarding occurs when firms decide not to cut employment
during a recession, waiting for a better aggregate economic condition to come. Firms do so because
hiring takes time due to the search friction. In contrast, the precautionary demand for labor arises
when firms expand employment in anticipation that employees will leave sooner or later. The key
is that firms are exposed to the risk of losing productive workers. With this mechanism in mind, we
conjecture that precautionary motives for under-employment are also possible, if firms are exposedto the risk of (unwillingly) retaining unproductive workers. The literature on firing restriction deals
with this issue.
We also find that this precautionary motive is enhanced by an increase in market tightness.
In other words, a firm is more willing to expand employment when the market is tighter. Such amechanism is absent in the standard model. With this eﬀect, firms’ decisions become interdependent: a firm’s employment policy depends on the degree of market tightness and at the same timeit influences the tightness of the market. It is shown that the firms’ decisions are connected inthat they strategically complement each other. As a result, there is scope for self-fulfilling multiple3equilibria, and the precautionary labor demand plays a crucial role for the outcome. Suppose thata firm of size one expects that other firms will expand employment. This expectation leads to thebelief that there will be greater congestion and higher market tightness. Thus, it will take longer tofind a replacement once it has lost the employee. Since all matches are productive, the opportunity
lost by being vacant is significant. Thus, the firm wishes to reduce the risk of being vacant, andtherefore prefers to be large. The same reasoning will imply that if a firm believes other firms will
stay small, then it will prefer to remain small.
We demonstrate these results using a model that modifies the standard labor market searchmodel. Although we try to keep the model simple and stay as close to the standard model as
possible, some additional assumptions are necessary. Some of them are made to simplify the exposition while others play crucial roles in generating the results. We present a section for clarifying
the importance of the way we modify the standard framework, and we also present a section for
discussing the results in detail.

2

The Model

2.1

Environment

The model is a straightforward extension of the standard labor market search model presented in
Pissarides (2000). Thus, we follow Pissarides (2000) as much as possible. In order to investigate theincumbent firms’ decision regarding whether to expand employment or not, we make the following
modifications to the standard model:• There is a fixed number of firms.
• A firm can employ up to two workers.
• The instantaneous payoﬀs to the firms are exogenous.
There is a fixed number of firms and workers, K firms and L workers, in the economy. It is
assumed that L is suﬃciently greater than K. In particular, L cannot be below 2K. The firm size4

in this paper is determined by the number of employees in the firm, and a firm with n workers
is called a firm of size n. The firm size in this model can be either 0, 1, or 2. Note that we are
not investigating the optimal firm size here. Thus, the presence of an upper bound to firm size
does not influence the issue we are investigating. In order to focus on a size-one firm’s decision, we
assume that no firm can post two vacancies at once.
Time is continuous.1 In addition, we focus on a steady state throughout this paper. Let us
start with describing the matching technology. As is standard, the total number of matches in thelabor market, denoted by M , is determined by a constant-returns-to-scale matching function:
M = m (uL, vL) ,(1)

where u is the unemployment rate while v is the number of vacant jobs as a fraction of the labor
force. The rate at which a firm with a vacancy meets an unemployed worker during a unit of timeis M/vL = m (uL/vL, 1) ≡ q (θ), where θ ≡ v/u. It is easy to check that q (θ) is decreasing in θ.
In addition, we assume that limθ→∞ q(θ) = 0 and limθ→0 q(θ) = ∞. Similarly, the rate at which a
job seeker meets a vacancy during a unit of time is M/uL = θq (θ). In addition, limθ→∞ θq(θ) = ∞and limθ→0 θq(θ) = 0.
Define kn as the proportion of firms of size n. Then, k0 + k1 + k2 = 1 must hold. A firm of size
one is either searching for additional worker or not searching. Thus, k1 = k1s + k1ns , where k1s is themeasure of size-one firms that engage in search activity while k1ns is the measure of size-one firmsthat are not searching. Thus:
k0 + k1s + k1ns + k2 = 1.

(2)

The total number of employees in this economy, E, is given by
E = [k1s + k1ns + 2k2 ]K.

(3)

The unemployment rate is defined by
u=
1

L−E
.
L

(4)

In a previous version of this paper, we have used a discrete-time model. The main messages of the two versions

are the same, while the exposition in this paper is much simpler.

5

The total number of vacancies in the market is given by
vL = k0 K + k1s K.

(5)That is, the total number of vacancies equals the number of firms of size zero plus the number offirms of size one seeking a second worker. From (3), (4) and (5), we obtainθ≡v
k0 K + k1s K
k0 + k1s
=
=
,
L
s
ns
u
L − [k1s + k1ns + 2k2 ]K
K − k1 − k1 − 2k2(6)where L/K is constant. This suggests that the degree of labor market tightness θ is determined
jointly with the distribution of firms.
Let λ be the Poisson arrival rate at which a match exogenously dissolves. Then, in any steadystate, the flows into and out of the unemployment pool must be the same. Thus, λ(1 − u) =θq(θ)u must hold, from which it is easy to show that u = λ/[λ + θq(θ)]. Thus, the equilibrium
unemployment rate is determined if market tightness θ is determined. In addition, the conventional
negative relationship between u and θ holds.

2.2

Stationary Distribution of Firms

This paper focuses on the steady-state equilibrium in which the distribution of firms is constantover time. In any steady state, the number of firms that flow into and out of a particular statemust be balanced. Let σ̄ denote the fraction of size-one firms that engage in search activity while
σ is a firm’s choice variable; σ = 1 if a particular firm of size one opens a vacancy and σ = 0 if it
does not. Then:
λk1s + λk1ns = q(θ)k0 ,(7)

σ̄q(θ)k0 + σ̄λk2 = λk1s + q(θ)k1s ,(8)(1 − σ̄)q(θ)k0 + (1 − σ̄)λk2 = λk1ns ,(9)q(θ)k1s = λk2 .

(10)According to (7), the measures of firms that flow into and out of the state of size zero must
be balanced. The firms that flow into this state are those of size-one firms that have just lost the6employee, which amount to λk1s + λk1ns . The outflow equals q(θ)k0 because a firm in this statefinds an employee with the rate q(θ). Two comments are in order. First, a size-two firm does
not lose two employees at once because in a continuous-time environment, the probability of two
independent events occurring is negligible. Second, we assume that no firm can post two vacancies
at once. This precludes the flow of size-zero firms jump into the state of size two.
The second equation, (8), balances the inflow and the outflow regarding the state in which afirm is of size one and opens a vacancy. The inflow equals σ̄q(θ)k0 + σ̄λk2 , which equals the measureof size-zero firms that have found an employee and opens another vacancy plus the measure of sizetwo firms that have lost an employee and opens another vacancy. The outflow equals λk1s + q(θ)k1s ,which is the sum of measures of size-one firms that have lost an employee and found a second
employee.
The third equation, (9), balances the inflow and the outflow regarding the state in which afirm is of size one and does not open a vacancy. The inflow equals (1 − σ̄)q(θ)k0 + (1 − σ̄)λk2 ,which equals the measure of size-zero firms that have found an employee but do not open anothervacancy plus the measure of size-two firms that have lost an employee but do not open anothervacancy. Since they do not search, the outflow is simplyλk1ns . Finally, (10) states that the inflow
into the state of size two is q(θ)k1s and the outflow is λk2 .

2.3

Firms

In what follows, we describe each firm’s behavior. There are four states to consider, each of whichis assigned a value function. Let V denote the value to an empty firm of searching for an employee.
J1s is the value to a size-one firm of engaging in search activity, while J1ns is the value of a size-one
firm that does not engage in search activity. Finally, J2 is the value of being a size-two firm.
Let r be the discount rate. The value of a firm without an employee, V , is given by
rV = −c + q(θ) max [σ (J1s − V ) + (1 − σ) (J1ns − V )] ,σ∈[0,1](11)

where c ≥ 0 is the cost of maintaining a vacancy. Equation (11) is interpreted as follows. As longas a firm is empty, it has to engage in search activity. With the Poisson arrival rate q(θ), the firm7

finds an employee and becomes a size-one firm. The firm must choose whether or not it will engage
in search activity to employ a second worker.
A firm of size one is in either two states, searching or not searching. Consider the value to asize-one firm of engaging search activity while producing. The value is given by
rJ1s = R1 − c + λ[V − J1s ] + q(θ) [J2 − J1s ] ,(12)where R1 is the net instantaneous payoﬀ to the firm of producing with one employee. Thus, R1
equals the total revenue minus the cost of employing one worker, including wage bills and otherbenefits. We do not follow the routine of determining the share of revenue with Nash bargaining.
We merely postulate that the total revenue is somehow split between the firm and the worker, and
R1 is the payoﬀ the firm gets.
Equation (12) may be interpreted as follows. A firm of size one receives R1 and pays c as thecost of posting a vacancy. With Poisson arrival rate λ, the employee leaves the firm. With arrival
rate q(θ), the firm finds a second worker.
The value to a size-one firm of not engaging in search activity is given by
rJ1ns = R1 + λ[V − J1ns ].

(13)

The value of a firm that employs two employees, J2 , is given by
rJ2 = R2 + λ max [σ (J1s − J2 ) + (1 − σ) (J1ns − J2 )] ,σ∈[0,1](14)

where R2 is the net instantaneous payoﬀ to the firm. Again, R2 is exogenous. With Poisson arrival
rate λ, one of the employees leaves the firm, in which case the firm needs to choose whether it willseek a second employee or not.
It is assumed that the firm size cannot exceed two. Of course, one could follow Bertola andCaballero (1994) and Smith (1999) to allow firms to choose any size. As emphasized earlier, we
are interested in the very choice that a firm faces regarding whether it should expand its size or
it should stay small. We believe that focusing on the choice faced by a size-one firm is enough togive us important insights regarding the employment policy of a firm.
8From (12) and (13), (r + λ)(J1s − J1ns ) = −c + q(θ)[J2 − J1s ] holds. Thus, J1s > J1ns holds if
and only if J2 − J1s > c/q(θ). This condition compares the net gain from becoming a size-two firmwith the expected total fixed cost of vacancy. Solve (11)-(14) for J2 − J1s as a function of θ andsubstitute it into the above inequality. After some algebra, it is easy to show that J1s > J1ns holds
if and only if
R2 − R1 +(r + λ) c(R1 + c) λ>.
r + λ + q(θ)
q(θ)

(15)

An interesting special case is obtained by letting c = 0. In the conventional Pissarides (2000)
model, c > 0 is necessary for determinacy of equilibrium because of the free entry of jobs. Here,the presence of the cost of maintaining a vacancy is not an essential part of the model. With c = 0,
it is evident that J1s > J1ns ⇐⇒ J2 > J1s , implying that a firm tries to expand employment if andonly if the value of being a size-two firm is greater than that of being a size one that engages insearch activity. After some algebra, this condition is expressed by
R2 > φ(θ)R1 ,φ(θ) ≡(16)

r + q(θ)∈ (0, 1).
r + λ + q(θ)

Proposition 1 In the limiting case with c = 0, a size-one firm creates a vacancy if and only if
R2 > φ (θ) R1 where φ (θ) ∈ (0, 1).
Corollary 2 A firm expands its size even if R2 < R1 .
Proposition 1 asserts that firms have precautionary motives to hire extra workers because firms
are willing to create vacancies even if the instantaneous payoﬀ to a size-two firm is below thecurrent instantaneous payoﬀ. The mechanism is better understood by recalling the labor market
search model with on-the-job-search, in which an unemployed worker’s reservation wage is below
the unemployment benefit.2 This is because the worker foresees that there is a chance of getting
a better job in the future; even though the wage is low, the value of the job can be very high.
Quite similarly, a firm of size one is willing to create a vacancy to be of size two because even if
2

A simple exposition of this result appears in Rogerson et al. (2005).

9

R2 < R1 , the value of being a size-one firm can be very small. Here, the value of being a size-two
firm comes from the fact that it faces no risk of being empty in the next instant. Remember thatin this economy, separations are exogenous and firms do not want to lose their employees because
all matches are productive. Thus, we can conclude that in the real world, we should expect toobserve precautionary labor demand in an economy in which workers quit while firms do not wish
to lose them.
Proposition 3 φ0 (θ) < 0.
Figure 1 shows the determination of σ under c = 0. Let θ̂ solve R2 /R1 = φ (θ). Propositions 3
and 1 suggest that a firm expands employment if and only if θ ≥ θ̂. When θ is large, a firm finds
it hard to fill its vacancy. In other words, the expected duration of vacancies is relatively long.
In this case, the firm wishes to be large so that it is not exposed to the risk of being empty fromthe stochastic separation. The importance of Proposition 3 is that it works in the direction thatenhances the precautionary labor demand.
When θ is small, on the other hand, a firm finds it easy to fill its vacancy. Thus, there is noneed to avoid being empty. It is important to point out that firms are worse oﬀ with a higher θ.
In fact, an increase in θ reduces the value of being a size-two firm. However, the value of being asize-one firm is even more significantly reduced because size-one firms are more exposed to the riskof devoid of employees.3
That a firm’s decision depends on the tightness of the market has an important implication.
Since market tightness is influenced by the decision itself, the decision depends on what otherfirms do. As described later, this causes strategic complementarity among firms and opens up the
possibility of self-fulfilling multiple equilibria.
Consider the general case in which c ≥ 0. In this case, a size-one firm creates a vacancy if andonly if (15) holds. Rewrite the condition as
R2 > φ(θ)R1 +3[r + λ] λ + [r + λ + q(θ)] rc.
[r + λ + q(θ)] q(θ)

After some algebra, one can show that J2 − J1s is increasing in θ while J2 is decreasing with θ.

10

It is evident that as the vacancy cost increases, a size-one firm becomes more reluctant to engage
in search activity. Therefore, the precautionary demand for labor exists only for suﬃciently smallvalues of c.

2.4

Discussion of the Key Assumptions

One vacancy at a time: Related models such as Bertola and Caballero (1994) and Smith(1999) allow firms to create an arbitrary amount of vacancies at a time. Of course, one could
consider an alternative version of the model in which a firm can post two vacancies at once. Sucha model would be interesting in itself. Using such a model, one could ask, for example, whether afirm opens one vacancy at a time or tries to fill two vacancies at once. In this paper, we are simply
asking whether firms want to be large or not, rather than how fast they want to be large.

Upper bound of firm size: That a firm cannot employ more than two workers is too
restrictive an assumption. However, note that we are not investigating the optimal firm size. We
are focusing on the binary choice that a firm faces regarding whether to expand employment or
not. In the standard search-matching model, the size of a firm is, by construction, one. By allowing
the firm size to be two, the model of this paper generates the simplest possible endogenous size
distribution of firms.

Exogenous separations: An important implication of exogenous separation is that a match
dissolves while it is productive. In other words, firms lose employees even though they do not want
to. We suspect that the precautionary labor demand would not arise if job destruction were a choicevariable to the firms. Thus, an appropriate interpretation of separation in this model is that workers
engage in on-the-job search. In this case, it is possible that a firm loses its (productive) employee.
Alternatively, one could simply assume that workers retire. How plausible is the assumption ofexogenous separation? Hall (2005) finds that fluctuations in unemployment are not a result of
massive layoﬀs. Hall’s (2005) finding at least partially supports the use of a framework that does
not incorporate endogenous job destruction.

11

No entry/exit of firms: Most of the standard labor market search models assume free entryof jobs.4 Models with endogenous firm size commonly employ a framework in which the numberof firms is predetermined. On behalf of these models, we emphasize that there is an importantdistinction between free entry of jobs and free entry of firms. Although there is no entry or exitof firms, the number of jobs can change as σ changes. In addition, job creation is actually freewhen c = 0. In the standard search-matching model, the free entry assumption seems the mostappropriate choice, as the economy is viewed as a collection of jobs. Free entry of firms, on theother hand, is less obvious because firms face start-up costs (Fonseca et al. (2001)), capital market
imperfection (Acemoglu (2001)), and regulation (Blanchard and Giavazzi (2003)), to name a few.

Zero vacancy cost: It is important to emphasize that letting c = 0 does not pose any
problem in our analysis because the number of firms is fixed. In the standard labor market searchmodel, there is free entry of jobs, and a positive vacancy cost plays a crucial role in generating asearch equilibrium. In fact, letting c = 0 does cause a problem, for example, in Pissarides (2000).
In particular, an infinite number of vacancies opens, and the value of an occupied job will be zero,
implying that the worker takes all the available profit. This is a Walrasian outcome.

3

Equilibrium

An important property of the model is that there is a non-trivial distribution of firms that is drivenby each firm’s optimal employment policy. As demonstrated earlier, a firm’s decision depends on
what other firms would do. Such feedback is possible partly because there is a fixed number offirms, so it is impossible for a firm’s decision to depend on the behavior of firms that have not yet
entered the market. In order to characterize the firms’ behavior in full, we need to compute the4

One can argue that a model with free entry should be interpreted as a model describing the long-run behavior of

the labor market while a model with a fixed number of firms should be interpreted as describing short-run behavior.
Although such an interpretation is possible and may be plausible, we do not necessarily regard the model as one
describing some transitory state of the economy.

12

firm distribution as well as the market tightness.5
We focus on symmetric equilibria in which all firms in the same state choose the same employment policy. There are potentially two types of equilibria, one in which all size-one firms choose to
create a vacancy and another in which all firms choose to be small. First, consider the equilibrium
in which all firms of size one wish to expand employment. Such an equilibrium is characterizedby σ̄ = 1. From (7)-(10), the firm distribution of this equilibrium is given by λk1s = q(θ)k0 ,q(θ)k0 + λk2 = λk1s + q(θ)k1s , q(θ)k1s = λk2 , and k0 + k1s + k2 = 1. Solve these equations as functionsof θ to obtain
k0 =

λ2
λq(θ)
[q(θ)]2
s
,
k
=
,
k
=
.
2
1
λ2 + λq(θ) + [q(θ)]2λ2 + λq(θ) + [q(θ)]2λ2 + λq(θ) + [q(θ)]2After some algebra, (6) reduces to
λ2 + λq(θ)¤θ=£ 2≡ ΩL (θ) .
λ + λq(θ) + [q(θ)]2 L/K − λq(θ) − 2[q(θ)]2(17)

The properties of the function ΩL (θ) are summarized as follows.

Lemma 4 limθ→0 ΩL (θ) = 0; limθ→∞ ΩL (θ) = K/L; Ω0L (θ) ≥ 0 holds if and only if θ ≤ θ̄, whereθ̄ is defined below.
Proof. It is convenient to rewrite (17) as ΩL (θ) = [λ2 /q(θ) + λ]{[λ2 /q(θ) + λ + q(θ)]L/K −λ − 2q(θ)}−1 . It is then easy to verify that limθ→0 ΩL (θ) = 0. It is easy, though tedious, tocompute the derivative of ΩL (θ) as Ω0L (θ) = [λ2 − (L/K − 2)(2λ + q(θ))q(θ)]λq 0 (θ)D−2 , whereD ≡ λ2 L/K + λq(θ)(L/K − 1) + [q(θ)]2 (L/K − 2). Then, it is evident that Ω0L (θ) ≥ 0 if and onlyif
λ2 ≤ (L/K − 2) (2λ + q(θ)) q(θ).

(18)

Such a requirement is satisfied as long as L/K is suﬃciently large and λ is not too large. L/K > 2
is necessary. Consider the right-hand side of (18). For any q > 0, it is strictly increasing. This5

In a previous version of this paper, we have also studied the equilibrium with free entry of firms. With free entry,

each firm’s decision is independent of those of other firms because the degree of market tightness is determined byfree entry, not by the employment policy of the firms.

13

implies that there exists a unique q that satisfies (18) at equality. Let such value denote q̄. Then,the condition (18) is satisfied for q ≥ q̄. Additionally, define θ̄ as the tightness that corresponds toq̄. Then, it is evident that (18) holds if and only if θ ≤ θ̄.
Figure 2 depicts the typical configuration of the function ΩL (θ). A candidate equilibrium isgiven by a fixed point of the mapping.
Next, consider the equilibrium in which all firms wish to stay small. Thus, σ̄ = 0. In thisequilibrium, the distribution of firms is given by λk1ns = q(θ)k0 , q(θ)k0 = λk1ns , and k0 + k1ns = 1.
Solve these to obtain
k0 =

λ
q(θ)
, kns =.
λ + q(θ) 1
λ + q(θ)

After some algebra, (6) reduces toθ=λ≡ ΩS (θ) .
[λ + q(θ)] L/K − q(θ)(19)

The properties of the function ΩS (θ) are summarized as follows.
Lemma 5 limθ→0 ΩS (θ) = 0; limθ→∞ ΩS (θ) = K/L; Ω0S (θ) > 0.
Proof. From (19), it is easy to compute the derivative of ΩS (θ) as Ω0S (θ) = −λ(L/K −1)q0 (θ)[λL/K + (L/K − 1)q(θ)]−2 > 0.
Figure 3 depicts the typical configuration of the function ΩS (θ). A candidate equilibrium isgiven by a fixed point of the mapping.
Proposition 6 Let θL and θS solve θ = ΩL (θ) and θ = ΩS (θ), respectively. Then, θS < θL holds.
Proof. Notice that (17) can be rewritten as#−1"£¤
λ2 + λq(θ) + [q(θ)]2 L
λ + 2q(θ)ΩL (θ) = λ ×−q(θ).
λ + q(θ)
K
λ + q(θ)
Then, it is easy to see that ΩL (θ) > ΩS (θ) holds if and only if£ 2¤
λ + λq(θ) + [q(θ)]2 LL
λ + 2q(θ)[λ + q(θ)]− q(θ) >−q(θ),
K
λ + q(θ)
K
λ + q(θ)

14which reduces to λL/K > −q(θ) and this is true for any positive q. This proves ΩL (θ) > ΩS (θ)
for all θ.
Since u is decreasing in θ, the unemployment rate associated with θS is higher than the otherone. Thus, the result suggests that the unemployment rate is higher in the equilibrium where firms
stay small.
Since the general condition for σ̄ = 1, given in (15), is complicated, we focus on the special
case in which c = 0. Since all size-one firms choose to expand employment if and only if θ > θ̂, astationary equilibrium is given by a fixed point of θ = Ω(θ), where⎧⎪⎨ ΩL (θ) for θ > θ̂Ω(θ) =⎪⎩ ΩS (θ) for θ ≤ θ̂.

Figures 4—6 depict the three possible configurations of Ω (θ). Consider Figure 4. This case arises
if θ̂ < θS , or R is suﬃciently large. As shown in the figure, the unique steady-state equilibrium isfound from the map θ = ΩL (θ) and is θL . Consider the case with θ̂ > θL (R is suﬃcicently small).
In this case, the steady state is found by a map θ = ΩS (θ), as depicted in Figure 5. There is aunique equilibrium and is θS . In either case, the stationary equilibrium is uniquely determined.
An interesting case arises when θS < θ̂ < θL . This case is depicted in Figure 6. Since themaps θ = ΩL (θ) and θ = ΩS (θ) are both valid, so are θL and θS . Thus, we encounter thecase of multiple equilibria. The multiplicity obtained here is worthy of comment. We have seenthat the precautionary labor demand has the important property that a firm’s employment policy
depends on other firms’ employment policies. That is, strategic complementarity resulting from theprecautionary labor demand generates the multiplicity of equilibria. Suppose that a firm expects
all other firms to expand employment (i.e., σ̄ = 1). It is then reasonable for the firm to expectthat the market tightness θ will be high and that it will be harder to fill its vacancy. Because
the precautionary demand is enhanced by high θ (Proposition 3), the firm’s choice reinforces thisreasoning. Therefore the firm creates a vacancy.
Similarly, suppose that the firm expects that all other firms will stay small (i.e., θ will be low).
Then the value of being vacant is expected to be high because it will be easier for the firm to find a15

match. Thus, the firm does not need to reduce the chance of being devoid of employees. Therefore,
the precautionary motive for larger size will be weak and the firm is likely to stay small.

4

Discussion of the Results

In the standard search-matching model, there is no precautionary labor demand at work. Sincemarket tightness is determined by free entry, each firm ignores its impact on market tightness whencompeting for unemployed workers. To put it diﬀerently, in the standard model, firms do not take
into account the congestion eﬀect they cause. Dropping the free entry assumption changes that.
Since market tightness is determined solely by the firms’ employment policy, each firm faces thecongestion eﬀect. Thus, for precautionary labor demand to arise, the free entry assumption must
be dropped.6
The key to understanding the precautionary labor demand is that firms might lose their employees when they do not wish to. Remember that job destruction in this model occurs exogenously,
so a separation occurs even though a match is productive. The opportunity lost by a separation isgreater when labor market tightness is high because it takes more time to fill a vacancy.
Is it possible to get the opposite result? In other words, is it equally plausible that firms have aprecautionary motive to hire fewer workers? Our conjecture is yes. Imagine an alternative versionof the model in which firms cannot fire workers when they need to. As the recent research on the
search model with firing costs suggests, firms are less likely to hire when there is a firing restriction(Blanchard and Summers (1988); Saint-Paul (1995)). However, as Bentolila and Bertola (1990)
claim, the firing costs do not always generate under-employment.
Bertola and Caballero (1994) construct a labor market search model with endogenous firm sizeand endogenous job destruction to find labor hoarding behavior. Firms tend to fire too little duringa recession because aggregate business conditions might improve in the future and hiring takes time.
The precautionary demand for labor found in this paper can be understood as a counterpart of labor
6

Indeed, in an earlier version of the paper we have verified that adding the free entry condition (V = 0) leads tono precautionary labor demand.

16

hoarding. To be more precise, labor hoarding arises in a model with endogenous job destruction,and it works at the firing margin. Precautionary labor demand arises at the other end: hiring
margin. They both arise from a single fact: separation is immediate and hiring takes time.
In this paper we have assumed that firms’ instantaneous payoﬀs are constant. How importantis this assumption? To take a quick look at the consequence of introducing endogenous wages,
suppose that the firms’ payoﬀs can be written as R1 = p1 − w1 (θ) and R2 = p2 − 2w2 (θ), where
pn is the profit to be shared with n employees. The wage rate at a firm of size n is denoted bywn (θ), which is assumed to be increasing in θ. That is, the wage rate increases as market tightness
increases, which is consistent with the standard model with wage bargaining. Now, if we denote
R(θ) ≡ [p2 − 2w2 (θ)]/[p1 − w1 (θ)], then, with c = 0, (16) suggests that a firm of size one tries toexpand employment if and only if R(θ) > φ(θ). The shape of R(θ) could be anything; there is no
guarantee that it is even monotonic. However, quite a lot can be said from this short-cut model.
Consider the slope of R(θ). With exogenous payoﬀs, R(θ) is a straight line, as in Figure 1. SupposeR(θ) is increasing in θ. Such a case occurs if an increase in the market tightness raises w1 much
more than w2 . In this case, the result found in Proposition 1 survives.
Alternatively, suppose that R(θ) is decreasing. If the slope is relatively flat, then the results
would not change. A completely new possibility arises when the slope of R(θ) is negative and is
steeper than that of φ(θ). In this case, a firm would expand employment if and only if the market
tightness is low instead of high. There is still precautionary labor demand in the sense that a firm
expands employment even if R2 < R1 . However, an increase in market tightness does not enhance
the precautionary motive. Without this channel, the strategic complementarity among firms ceases
to exist and therefore no multiplicity of equilibria can arise. This suggests that no-free-entry is
not enough for multiple equilibria; precautionary labor demand and especially its enhancement(Proposition 3) are crucial.

17

5

Conclusion

This paper has studied the choice of job creation using a stylized model of a labor market withsearch frictions. Two findings are worthwhile. One is that firms have precautionary motives toexpand employment. One of the factors that generate the precautionary labor demand is thatfirms face the risk of losing (productive) workers. The precautionary demand is enhanced bygreater market tightness, implying that firms’ employment policies influence each other throughimpacts on market tightness. This leads to the other main finding of this paper. Self-fulfilling
multiple equilibria are possible due to the strategic complementarity among firms. The analysis
suggests that precautionary labor demand should be observed in an economy in which (i) entry offirms is to a certain degree limited and (ii) firms unwillingly lose their employees.
We conclude by suggesting some directions for future research. In this paper, we have employed
the assumption of exogenous separations. It is worthwhile to explore a model in which workers’
choice of separation is introduced along the line of Burdett and Mortensen (1998) to see if there isanother source of strategic complementarity. We have found that precautionary labor demand isenhanced by market tightness. It is worth exploring this eﬀect in the context of the business cycle.
This framework might help explain the cyclical behavior of market tightness. Finally, it is very
important to consider other margins within which labor input may be adjusted. In particular, it
is worth investigating a model in which a firm can choose between the extensive margin (numberof employment) and intensive margin (hours of work). The model presented in this paper provides
a reasonable framework for studying these issues.

18

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Figure 1: Threshold level of θ

R2
R1

φ (θ )
Firms create a vacancy

Firms stay smallθθˆFigure 2: Function ΩLΩ L (θ )θLθ

Figure 3: Function ΩSΩ S (θ )θSθFigure 4: Stationary Equilibrium (Case 1)

Ω L (θ )Ω S (θ )

θ

θL

θˆ

Figure 5: Stationary Equilibrium (Case 2)

Ω L (θ )Ω S (θ )

θS

θ

θˆ

Figure 6: Multiple Equilibria

Ω L (θ )Ω S (θ )θSθˆ θ Lθ