GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.173

On-the-job search in urban areas

Keisuke Kawata and Yasuhiro Sato

February 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanOn-the-job search in urban areas
Keisuke Kawata∗

Yasuhiro Sato†

January 21, 2010

Abstract
This study develops an on-the-job search model involving spatial structure. In this model,
workers are either employed and commuting frequently to a central business district (CBD)or unemployed and commuting less frequently to the CBD to search for a job. When an
unemployed worker succeeds in oﬀ-the-job search, the quality of the job match is determined stochastically: a good match yields high-productivity whereas a bad match yields
low-productivity. Although a high-productivity worker does not search for a new job, a lowproductivity worker decides whether to conduct an on-the-job search, which would require
additional commuting to the CBD. Analysis of this model demonstrates that in equilibrium, the relocation path of workers corresponds to their career path, while welfare analysis
demonstrates that such a spatial structure distorts firms’ decision regarding the posting of
vacancies.

JEL classification: J64; R14; R23
Keywords: City structure; On-the-job search; Unemployment; Eﬃciency; Relocation and career
paths;

1

Introduction

There is no doubt that urban areas currently play a dominant role as areas of employment
and residence throughout the world.1 In many cities, people live in suburbs and commute
from there to business districts. How does the structure of a city and commuting relate to job
creation and unemployment within it? Recently, several studies have addressed this question
by combining urban and labor economics. Some of these studies have succeeded in providing
answers to a certain part of this question by developing search and matching models within
urban structures.2
Wasmer and Zenou [15] introduced a monocentric city structure a la Alonso [1] into a job
search model based on the key assumption that workers’ search intensity is negatively aﬀectedby access to jobs that are concentrated within a Central Business District (CBD). Such anassumption leads to multiple urban configurations in equilibrium, including that of unemployedworker residing close to jobs or that of unemployed workers residing distant from jobs, with the∗Corresponding author, Graduate School of Economics, Osaka University, e-mail: hge005kk@mail2.econ.osakau.ac.jp†
Graduate School of Economics, Osaka University, e-mail: ysato@econ.osaka-u.ac.jp
1
United Nations [14] reports that in 2000, 76 percent of the population in developed countries and 39 percentof the population in developing countries resided and worked in urbanized areas.
2
There are also studies that have addressed this question by using other types of models and have yieldedinteresting findings. For instance, Zenou [17] adopted an eﬃciency wage model involving an urban structure. He
found that firms do not recruit workers who reside at an excessive distance from the place of work because theirproductivity is lower than that of workers residing more proximate to the place of work.

later configuration being consistent with the well-known spatial mismatch hypothesis.3 Smithand Zenou [13] demonstrated that when search intensity is determined endogenously, anothertype of configuration is obtained in equilibrium: there are two areas where unemployed workers
reside, one of which is close to the CBD and the other of which is distant from the CBD.
Employed workers live in between these two areas.
Crampton [3] analyzed worker’s simultaneous decision regarding job search and residential
areas. Rouwendal [9] demonstrated the possibility of excessive commuting due to the existenceof information asymmetry in the job search process. Sato [12] analyzed how city structure aﬀects
workers’ job-acceptance behavior and the labor market by introducing city structure into a searchmodel with workers’ decision to accept a job oﬀer. He found that reductions in the urban costsof living such as commuting costs, increase the likelihood that job seekers will accept job oﬀers.
Zenou [19] proposed a spatial search model in which both job creation and job destruction are
endogenous. He demonstrated that in equilibrium, workers with high productivity and wages
live close to jobs, have low per distance commuting costs, and pay high land rents. He also
showed that higher per distance commuting costs and higher unemployment benefits lead tomore job destruction. Zenou [20] developed a spatial search model in which firms post wages.
He found by simulation that when workers have diﬀerent value imputed to leisure and diﬀerent
equilibrium wages, a reduction in commuting costs for all workers reduces the unemployment
rate of high-wage workers and the profit of all firms while increasing the wages of all workers
and the proportion of firms paying the high wage.45
This study focuses on a diﬀerent significant aspect of job search, namely, on-the-job search,i.e., we introduce the possibility that employed workers search for a new job. It is often observedthat employed workers search for a new job when they are unsatisfied with their current jobs.
We also observe that people change their locations when they land a new job and obtain higher
wage income. This study explores the possible interactions between such a career path in thelabor market and a relocation path in the urban land market. To do so, the monocentric citymodel a la Alonso [1] is introduced in the on-the-job search model developed by Pissarides [7],in which workers are either employed or unemployed. An employed worker frequently commutes
to the CBD to work while an unemployed worker commutes less frequently to the CBD in order
to search for a job. When an unemployed worker succeeds in obtaining a job (i.e. succeeds
in oﬀ-the-job search), the quality of the job match is determined stochastically: a good match
yields high productivity whereas a bad match leads to low productivity.
Although high productivity workers do not search for a new job, low productivity workers decide whether or not she/he looks for a new job. On-the-job search requires additional commuting
to the CBD, which leads to the following urban configuration: currently employed job seekers(on-the-job searchers) live closest to the CBD, unemployed workers (oﬀ-the-job searchers) live
most distant from the CBD, and employed workers not searching for a new position (non-jobsearchers) live in between the former two. In this model, spatial relocation corresponds to acareer path, with the career path from unemployment to employment or from on-the-job search
3First stated by Kain [5], the spatial hypothesis posits that job decentralization to the suburbs without theresidential movement of African Americans has led to a high unemployment rate and low wages paid in inner-cityneighborhoods, where African Americans are concentrated. Since then, a large number of empirical studies testing
this hypothesis have oﬀered much evidence for its support (see Preston and McLaﬀerty [8] for a survey of recent
empirical studies).
4
For a comprehensive survey of this literature, see Zenou [18], among others.
5
Sato [10] [11] also developed a job search model involving a monocentric city structure. However, he assumed
that because all types of workers have the same commuting costs and the same level of housing consumption, theresidential location of a particular type of worker cannot be determined. Therefore, he focused on the eﬀects of
the overall cost of living, which depends on the population size of the city, on the local labor market. Sato [10]investigated the relationship between wage and unemployment diﬀerences among diﬀerent cities in inter-urban
equilibrium. Sato [11] found a link between agglomeration economies and the worker-firm matching process, as
did Wheeler [16] in a separate study using a diﬀerent model. Sato [11] provided the conditions of the matchingprocess necessary for the existence of agglomeration economies as well.

2to a good-match entailing relocation from the outskirts of a city to an inner part of the city orfrom the innermost area of the city to the suburbs.
In this spatial equilibrium, the wages of low productivity workers are lower when they live
closer to the CBD. In the model proposed here, residing proximate to the CBD signals that aworker is an on-the-job seeker, which decreases the bargaining position of the worker and reduces
her/his wages. Spatial structure also impacts the eﬃciency of the properties of the model. Thismodel proposes that whereas decision regarding an on-the-job search is eﬃcient, firms’ decision
regarding job vacancies is distorted even under the Hosios condition (Hosios [4]), which contrasts
with the findings of related studies of spatial oﬀ-the-job search models, such as those of Wasmer
and Zenou [15] and Zenou [18].
This paper is structured as follows. Section 2 introduces the basic structure of the model.
Section 3 demonstrates the existence of and the properties of a unique equilibrium. Section 4
addresses issues regarding welfare and Section 5 presents the study conclusions.

2

Model

Consider a closed, linear, and monocentric city whose land is owned by absentee landlords. It
has one CBD, the location of which is approximated by one point and within which all firms are
assumed to be exogenously located.6 A continuum of risk-neutral workers of size N live within
the city, each of which exits the labor market according to a random Poisson process with theexogenous aggregate rate δ > 0. Such an exit is replaced by an entry of a new worker, who firstsearches for a job as an unemployed worker. This assumption keeps the total number of workers
constant. Although identical ex ante, the workers become heterogeneous after entering thelabor market due to the occurrence of stochastic events, with u of N being unemployed workers
and N − u being employed workers. While unemployed workers search for a job, employedworkers earn wage income that depends on their productivity y, which is either high or low anddetermined randomly upon the job match, with a good match yielding productivity yh and abad match yielding productivity yl . A subscript h (l) represents that the variable is related toa good (bad) match. We assume that yh > yl and Pr(y = yl ) = Pr(y = yh ) = 1/2. Moreover,
it is assumed that yl is suﬃciently large such that even if a match yields low productivity, theworker prefers to be employed rather than unemployed, and her/his employer does not dismiss
her/him. We assume that on-the-job search is possible, such that a low-productivity worker cansearch for a new job while working.
In the city, all workers occupy the same amount of land (normalized to 1) outside the CBD.
We assume that the density of land is 1, implying that x units of housing are located within adistance x of the CBD.
In this paper, it is assumed that the sole commuting cost is that of the time expended indoing so. Each worker is endowed with one unit of time, that she/he uses for working, searching,commuting and leisure. Assume that an employed worker must expend a fixed amount of timehw (> 0) on work activities and that the job search requires a fixed amount of time hs (> 0).
Moreover, we assume that the commuting time to the CBD is σw x for an employed worker
and σu x for an unemployed worker. Here, commuting is necessary for an unemployed worker
to be interviewed. We assume that hw > hs and σw ≥ σu : working at a job requires moretime and frequent commuting than does searching for a job. A worker is assumed to obtain
the instantaneous utility ul (l) from leisure, where u0l > 0 and u00l < 0. The time for leisure isgiven by lu = 1 − hs − σu x for an unemployed worker, le1 = 1 − hw − hs − σw x for an employedworker who is searching for a new position (i.e., an on-the-job searcher), le0 = 1 − hw − σw x foran employed worker who is not searching.
6

This model of a centralized city can easily be modified to describe a decentralized city by locating all firms
within a suburban business district that is located at one end of a linear city. Such a modification would not alter
the results of this study.

3

Based on these assumptions, the decrease in utility resulting from living more distantly fromthe CBD can be expressed as follows:∂ul (lu )= −σu u0l (1 − hs − σu x),∂x∂ul (le1 )= −σw u0l (1 − hw − hs − σw x),∂x∂ul (le0 )= −σw u0l (1 − hw − σw x).
∂x
Assuming that hw > hs and σw ≥ σu implies that lu > le0 > le1 for a given x. Because
σu u0l (lu ) < σw u0l (le0 ) < σw u0l (le1 ) is readily known from u00 < 0, the marginal cost of commuting
is the highest for on-the-job searchers and the lowest for unemployed workers, as expressed by
the followings:∂ul (le1 )∂ul (le0 )∂ul (lu )
>
>
.
0>
∂x
∂x
∂x
For the sake of analytical tractability, we adopt the simplest way of describing commuting cost.
Specifically, commuting cost is described by linear functions of the distance x from the CBDsuch that it is τ x for an employed worker not searching for a new job, (1+s∗ )τ x for an on-the-job
searcher, and sτ x for an unemployed worker, where τ , s∗ and s are exogenous and satisfy thatτ > 0 and 0 < s∗ ≤ s < 1.78
Because of these assumptions, as is demonstrated in Section 3, the equilibrium urban configuration is such that on-the-job searchers reside most proximate to the CBD, unemployedworkers reside most distant from the CBD, and employed workers not searching for a job reside
between the first two groups. The cost of living in the city is the sum of the residential land
rent R(x) and the commuting cost: it is R(x) + τ x for an employed worker not searching for ajob, R(x) + (1 + s∗ )τ x for an on-the-job searcher, and R(x) + stx for an unemployed worker.
Although workers are identical ex ante, they are heterogeneous in terms of place of residencex, employment status (employed or unemployed), productivity (high or low), and job search
activity (no search, on-the-job search, or oﬀ-the-job search). Following Burdett and Mortensen[2] and Pissarides [7], on-the-job search by firms is assumed to be impossible, i.e., firms cannot
search for a new worker for an already filled job.

2.1

Matching framework

There are v firms that have vacancies and search for workers. Each of them posts one vacancy,
which can be filled by only one worker. Let es denote the number of on-the-job searchers andφ denote the eﬃciency of their on-the-job search. We normalize the eﬃciency of oﬀ-the-job
search to one. Job matches are generated by a Poisson process with the aggregate rate of
M = m(u + φes , v), where u + φes represents the number of job searchers in terms of eﬃciency
units. m(·, ·), which is defined on R+ × R+ , is assumed to be strictly increasing in botharguments, twice diﬀerentiable, strictly concave, and homogeneous of degree one. We also
assume that m(·, ·) satisfies 0 ≤ M ≤ min[u + φe, v], m(u + φes , 0) = m(0, v) = 0 and the Inada
condition for both arguments.
7

More precisely, it is assumed that the value of leisure measured is given byul (lu ) = b − sτ x,ul (le1 ) = bb − (τ + s∗ )x,ul (le0 ) = bb − τ x,

where b and bb are constants that represent the value of leisure at location x = 0. We normalize bb to zero.
8s and s∗ can be interpreted as indicators of search intensity. For a discussion on the endogenous determinantof search intensity, see Smith and Zenou [13] and Wasmer and Zenou [15].

4Job matching occurs at the rate of p(θ) = M/(u + φes ) = m(1, θ) for each unemployedworkers, φp(θ) for each on-the-job searchers, and q(θ) = M/v = m(1/θ, 1) for a firm seeking tofill a vacancy, where θ is the measure of labor market tightness defined as θ = v/(u + φes ).From
the assumptions regarding m(·, ·), we obtain that p(θ)(u+φe) = q(θ)v, dp/dθ > 0 and dq/dθ < 0
for any θ ∈ (0, +∞). We can also see that limθ→0 p = 0, limθ→∞ p = ∞, limθ→0 q = ∞ andlimθ→∞ q = 0. If φ = 0, this model becomes a model without the presence of on-the-job search.
For the sake of descriptive simplicity, an on-the-job searcher is assumed to move to a new job
if the new job provides her/him with the same or a larger asset value as does the current job.
As is standard in on-the-job search models, it is assumed that firms, upon meeting a worker, do
not know whether she/he is an on-the-job searcher or an oﬀ-the-job searcher.

2.2

Value functions

Let Wj (x), U (x), Jj (x) and V denote the asset value of an employed worker, an unemployedworker, a firm with a filled job, and a firm with a vacancy, respectively. j represents the
productivity of a match (i.e., high (h) or low (l), j ∈ {h, l}). x describes the worker’s residential
location (x ∈ [0, ∞)), which is the distance to the CBD. In Jj (x), x refers to the residential
location of a worker employed by the firm.
The asset value of an unemployed worker who lives at x is, therefore, given by(r + δ)U (x) = b +¢p(θ) X ¡ max− U (x) − R(x) − sτ x,
Wj
2

(1)

j∈{h,l}where Wjmax = maxx∈[0,∞] Wj (x). r, b, and R(x) represent the discount rate, the value of leisure,and the market land rent, respectively, with r and b being positive constants and R(x) being
determined endogenously. It is assumed that even a bad match yields a value suﬃciently lager
than the value of leisure. The first term of (1) is the instantaneous utility of being unemployed
while the second term is the expected capital gain from oﬀ-the-job search. An unemployedworker is matched to a job at the rate of p(θ), with the match yielding high (low) productivity
at a probability of 1/2 (1/2), leading to an asset value of Whmax (Wlmax ). The third and fourthterms are the land rent and the location-dependent cost of oﬀ-the-job search, respectively. As
previous explained, it is assumed that an unemployed worker will accept any type of job when
she/he has a chance of making a match.
The asset value of an employed worker with productivity j is described as(2)(r + δ)Wj (x) = wj (x) + λ(U max − Wj (x))⎧ ⎛⎞⎫⎬⎨ φp(θ) X£¤+ max i ⎝max Wjmax− Wj (x), 0 − s∗ τ x⎠ − R(x) − τ x,0⎭2i∈{0,1} ⎩0j ∈{h,l}where U max = maxx∈[0,∞] U , wj (x) is the wage and λ is the exogenous job separation rate. The
first and second terms are the instantaneous utility obtained from the wage income and thecapital loss from job separation, respectively. The third term represents the expected
net returni
hP
max
from on-the-job search, which is the expected capital gain (φp(θ)/2) j 0 max Wj 0 − Wj (x), 0
from on-the-job search minus the cost s∗ τ x of it. If the net return is non-negative, an employedworker will search for a new job (i = 1), but will not do so if it is negative (i = 0).
While it is assumed that firms do not know whether employees are searching for a new job,
it is assumed that firms know where employees live. Based on the knowledge of its employees’
residential location and productivity, a firm can makes the belief regarding whether its employee5

is doing an on-the-job search. The asset values of firms are given byrJj (x) = yj − wj (x) + (λ + δ + i∗ φp(θ))(V − Jj (x)),q(θ) XrV = −c +(Jj (x) − V ),
2

(3)
(4)

j∈{h,l}where yj is the output of a match yielding productivity j, c is the cost of posting a vacancy, andi∗ is the firm’s belief regarding whether an employee is looking for a new job. As later discussed
in more detail, such a belief must be consistent with the relationship among the productivity ofthe match, the residential location of the employee, and her/his job search status. In (3), thefirst and second terms describe instantaneous profits while the third term represents capital loss
from job destruction resulting from job separation (λ) of the employee, her/his exit (δ) fromthe labor market, and her/his job turnover (i∗ φp(θ)). In (4), the first and second terms are the(instantaneous) cost of recruitment and the capital gain from filling a vacancy.
Equilibrium is determined by (i) the outcome of wage bargaining (Section 2.3), (ii) labor market conditions (the free entry of firms and on-the-job search decision of employees, as describedin Section 2.4), (iii) land market conditions (locational arbitrage and bid rent, as described in
Section 2.5) and (iv) steady state conditions.

2.3

Wage determination

Following Pissarides [7], wages are assumed to be determined by the first-order condition of the
Nash bargaining conducted between a firm and an employee:(1 − β)(Wj (x) − U max ) = β(Jj (x) − V ).

(5)

This relationship can also be expressed as Wj (x) − U max = β(Wj (x) + Jj (x) − U max − V ). Asβ represents the labor share of the total surplus, it indicates the bargaining power of a worker.
Note that as employee’s job searching status is private information, it does not aﬀect her/his
wages directly, but does so indirectly because it is determined by her/his residential location,
which is public information and a factor considered in wage bargaining.

2.4

Labor market conditions

As will be formally described in Section 2.5, it is assumed that workers can move freely within
the city and face no costs of relocation, implying that the asset value of a worker is the sameacross locations by locational arbitrage, as is expressed in (6):Wj (x) = Wj

and U (x) = U,

∀x.

(6)

Moreover, from the wage determination rule (5), we obtain
Jj (x) = Jj ,

∀x.

(7)

Assuming the free entry and exit of firms, when the value V of posting a vacancy is positive,
more firms will post vacancies, and when V is negative, some firms will stop posting vacancies.
In equilibrium, V is driven to zero. Combined with V = 0 and (7), (4) leads to the following
free entry condition:
X
2c
Jj .
(8)
=
q(θ)
j∈{h,l}From (5), (6) and (7), the value (1) of an unemployed worker becomes(r + δ)U = b +βp(θ) X
(Jj − V ) − R(x) − sτ x,2(1 − β)j∈{h,l}6

which, combined with V = 0 and (8), leads to(r + δ)U = b +βcθ− R(x) − sτ x.
1−β

(9)

Similarly, (6) and (7) implies that the value (2) of an employed worker becomes(r + δ)Wj = wj (x) + λ(U − Wj )
⎞⎤
⎡ ⎛
X
£
¤
φp(θ)
+ max ⎣i ⎝max Wj 0 − Wj , 0 − s∗ τ x⎠⎦ − R(x) − τ x.
2i∈{0,1}0(10)

j ∈{h,l}Whether an employed worker searchesPfor a new job£depends on ¤whether her/his return fromon-the-job search, expressed as φp(θ)/2 j 0 ∈{h,l} max Wj 0 − Wj , 0 , is larger than the cost of£¤P
doing so, expressed as s∗ τ x. Because j 0 max Wj 0 − Wh , 0 is always zero, we readily knowthat a high-productivity worker will not search for a new job. In contrast, a low-productivity
worker will search for a job if and only ifφp(θ)(Wh − Wl ) ≥ s∗ τ x.
2

(11)This equation, the left hand side of which expresses the expected capital gain from on-the-job
search and the right hand side is the cost of doing so, is very similar to the equation describing
on-the-job search conditions presented in Pissarides [7]. The diﬀerence between the two is thatin (11), the search cost includes the spatial factor whereas the equation in [7] does not do so.
As is often assumed in search models, workers (firms) are assumed to regard θ as given, andhence, p(θ) (θ and q(θ)) as given, which implies that they take the left hand side of (11) asgiven. This implies that a threshold x∗ exists such that a low-productivity worker who lives atx ≤ x∗ searches for a new job, where x∗ is defined byx∗ =φp(θ)(Wh − Wl )
.
2s∗ τ

(12)

Firms know the productivity and location of their workers, and use this information in
forming their belief i∗ . In equilibrium, a firm forms a belief that only unemployed workers
and low-productivity workers with x ≤ x∗ are searching for a job. Letting wl1 (x) and wl0 (x)denote the wages of a low productivity worker who is searching for a new job and that of a low
productivity worker who is not, respectively, leads to the following lemma.
Lemma 1 The equilibrium wages satisfy that wh (x) = wh , wl1 (x) = wl1 , and wl0 (x) = wl0 , ∀x.
Furthermore, the on-the-job search reduces wages by φp(θ)Jl , i.e., wl0 = wl1 + φp(θ)Jl .
Proof See Appendix A.
Intuitively, the asset value of a firm matched with a worker who is searching a new job islower than that of a firm matched with a worker who is not by the expected loss φp(θ)Jl causedby job turnover. The wage diﬀerentials compensate for such diﬀerentials in the asset value.

2.5

Land market conditions

Based on the assumption that workers can change their locations of residence without incurringcosts, there is no incentive for workers to relocate in equilibrium. Therefore, as shown in (6), in
equilibrium, all employed workers enjoy the same level of value (Wj (x) = Wjmax = Wj ) and all
unemployed workers enjoy the same level of value (U (x) = U max = U ). These two conditions
and the wage determination rule (5) imply that the value of a firm is also independent of thelocation of its employee (Jj (x) = Jj ) as is described in (7).
7In order to determine the equilibrium location of workers, we use the concept of bid rents,
defined as the maximum land rent at location x that each type of worker is willing to pay inorder to reach her/his respective level of equilibrium utility (in this paper, asset value). Notethat in such a framework, the equilibrium asset value of a firm and that of its employee do
not depend on whether the employee is doing on-the-job search. This is due to the adjustment
through land rent and wages. From the definition of bid rents and consideration of Lemma 1
and equations (12), (9) and (10), the bid rents Ω of unemployed and employed worker can be
expressed as follows:
Unemployed workers:Ωu (x) = b − (r + δ)U − sτ x,
Employed low-productivity workers:½ 1Ωl (x) ≡ wl1 + λU − (r + δ + λ + φp(θ)/2)Wl + φp(θ)Wh /2 − (1 + s∗ )τ x
Ωl (x) =Ω0l (x) ≡ wl0 + λU − (r + δ + λ)Wl − τ x
if x ≥ x∗(13)

if x < x∗(14)

Employed high-productivity workers:Ωh (x) = wh + λU − (r + δ + λ)Wh − τ x.

(15)

Diﬀerentiating (13) and (15) with respect to x, we obtain∂Ωu= −sτ,
∂x

∂Ωh
= −τ.
∂x

Ωl (x) is a relatively complex function of x whose slope is described by∂Ω0l
if x ≤ x∗ ,=−(1 + s∗ )τ,
∂x
∂Ω1l
=−τ,
if x > x∗ .
∂x
Ωl (x) is discontinuous at x∗ and shifts upwards by φp(θ)Jl , reflecting the decreases in wages
resulting from the firm’s belief that a low-productivity worker who lives at x ≤ x∗ is searchingfor a new job and the diﬀerence in the bid rent balancing the diﬀerence in wages. Put differently, from the viewpoint of workers, low productivity employees can commit themselves tonot searching for a job by living farther away from the CBD than x∗ . The decline in bid rents
reflects the value of their commitment.
Figure 1 summarizes the above arguments regarding bid rents.
[Figure 1 around here]
In this figure, the horizontal and vertical axis represent the distance from the CBD and the bid
rent, respectively, and the market land rent is the upper envelope of the bid rent curves. From
this figure, the equilibrium locational pattern in the model can be derived as follows:
Proposition 1 In equilibrium, the residential area is separated into four areas. The area closest
to the CBD is occupied by low-productivity workers searching for a new job, the area second
closest is occupied by high-productivity workers, the third closest area by both high-productivity
workers and low-productivity workers not searching for a new job, and the fourth closest (themost distant) area by unemployed workers.
8,

The underlying mechanism of this proposition is similar to that of Wasmer and Zenou [15]:workers locate in descending order of the losses that they incur from living distant from the CBD.
Because Ωl (x) is discontinuous at x∗ , there is an area where only high-productivity workers live
between two areas where low productivity workers live.
Let el denote the number of low-productivity workers. Because high-productivity workers
don’t look for a new job, there are only two possibilities: either (i) only a proportion of lowproductivity workers search for a new job (es < el ) or (ii) all low-productivity workers search
for a new job (es = el ). Based on the possibility of these situations, the bid rents in Figure 1
satisfy the followings:Ω1l (es ) = Ωh (es )Ω1l (es ) = Ωh (es )and Ωh (N − u) = Ω0l (N − u) = Ωu (N − u)and Ωh (N − u) = Ωu (N − u)if es < el ,if es = el .

Equations from (13) to (15) together with Ω1l (es ) = Ωh (es ) and Ωh (N − u) = Ω0l (N − u) yieldφp(θ)(Wl − Wh ) + s∗ τ es = 0,2
wh − wl0 − (r + δ + λ)(Wh − Wl ) = 0.

wh − wl1 − (r + δ + λ)(Wh − Wl ) +

(16)
(17)

Substituting (17) into (16), we obtainwl0 − wl1 +φp(θ)(Wl − Wh ) + s∗ τ es = 0.
2

From (12) and Lemma 1, this equation determines the number es of on-the-job searchers as∙¸∙¸φp(θ)(Wh − Wl − 2Jl )φp(θ)Jl∗es = min(18),el = min x −
, el
2s∗ τ
s∗ τ
In equilibrium with the interior solution es < el , a certain proportion of the low-productivity
workers are crowded out by high-productivity workers from the interval [x∗ − φp(θ)Jl /(s∗ τ ), x∗ ],where they would have searched for a new job if they had resided there. Put diﬀerently, x∗ −es represents the number of low-productivity workers who are crowded out from on-the-job
search. Such a crowding out eﬀect does not exist in the non-spatial on-the-job search modela la Pissarides [7]. Hereafter, we assume the interior solution (i.e., es < el ), whose suﬃcientcondition will be provided in Lemma 5.

3

Equilibrium

We focus on a steady state equilibrium in which the inflow into unemployment is equal to theoutflow from it:δN + λ(N − u) = (p(θ) + δ)u,which yields
δ+λ
u
=
.
N
δ + λ + p(θ)(19)

Lemma 2 For a given market tightness θ, the unemployment rate does not depend on the city
size N .

9The free-entry condition (8) can be rewritten as
c
q(θ)
|{z}

=

1
(Jh + Jl )2(20)

expected total search costs==

1 yl − wl0 + yh − wh
2
r +⎛
δ+λ
⎜
1
⎜
r+δ+λ⎝

yh + yl
2 }
| {z

−

expected output⎞

wh + wl0 ⎟
⎟
2 } ⎠
| {z

expected wage

which connects market tightness θ to wages wh and wl0 .
Wages wh and wl0 can be derived from asset value functions. Equations (3) and (10) can be
rewritten as the followings by using V = 0:
(r + δ + λ)Jh = yh − wh ,(21)

(r + δ + λ)Jl = yl − wl0∗

if x > x ,(r + δ + λ)(Wh − U ) = wh − (r + δ)U − τ x − R(x),(r + δ + λ)(Wl − U ) = wl0 − (r + δ)rU − τ x − R(x)if x > x∗ .

Substituting these equations and (9) into (5), we obtain9
wh = β(yh + cθ) + (1 − β) [b + (1 − s)τ (N − u)] ,(22)

wl0 = β(yl + cθ) + (1 − β) [b + (1 − s)τ (N − u)] .

In these equations, wages are increasing functions of market tightness θ. For a given θ, thewages increase with an increase in productivity and the value of leisure, in accordance with thefindings of Pissarides [7], as well as with an increase in the commuting cost τ and N − u, anddecrease with an increase in the search frequency of unemployed workers s, all of which arecommon characteristics of spatial search models (see Zenou [18]). Lemma 1 and (22) yield thewages of an on-the-job searcher wl1 :
wl1 = β(yl + cθ) + (1 − β) [b + (1 − s)τ (N − u)] − φp(θ)Jl(1 − β) [yl − b − (1 − s)τ (N − u)] − βcθ
= β(yl + cθ) + (1 − β) [b + (1 − s)τ (N − u)] − φp(θ),
r+δ+λwhich is a decreasing function of on-the-job search eﬃciency φ for a given θ.
The expected wage is obtained from (22) as follows:
wh + wl0β(yh + yl )=+ βcθ + (1 − β) [b + (1 − s)τ (N − u)] .
2
2

(23)

Equations (19), (20) and (23) are used to determine u, θ and (wh + wl0 )/2, and other variables(i.e. wl1 , es ) are well determined from them.
9

From (5) and V = 0, it is readily shown that workers prefer to be employed rather than unemployed if andonly if Jj > 0 (j ∈ {h, l}). From (21) and (22), the suﬃcient condition for this inequality to hold true is given byyl − wl0 > 0,
which can be rewritten as(1 − β) [yl − b − (1 − s)τ (N − u)] > βcθ.

Under this condition, Jh > Jl > 0 holds true, that is, even a bad match gives positive profits to a firm, implyingthat any type of match is maintained under this condition.

10

Substituting (19) into (23) yields∙µ¶¸
wh + wl0β(yh + yl )δ+λ=+ βcθ + (1 − β) b + (1 − s)τ N 1 −,22
δ + λ + p(θ)(24)

from which it can be observed that the market tightness θ aﬀects the expected wage via twochannels. In one channel, a higher θ implies the existence of better outside opportunities,
thus giving workers a better bargaining position. In the other channel, a higher θ leads to ahigher employment rate, which increases the total commuting costs of employed workers and,by increasing the relative attractiveness of outside opportunities, increases wages.
Figure 2 depicts (20) and (24) with the horizontal axis representing the market tightness
θ, vertical axis representing the expected wage (wh + wl0 )/2. The the downward-sloping curve
describes (20) and the upward-sloping curve describes (24).

[Figure 2 around here]

From the Inada condition regarding matching technology, the two curves always has theunique intersection, implying that the model has the unique steady state equilibrium.
Proposition 2 There exists the unique steady state equilibrium of the model.
Using Figure 2, the eﬀects of a change in exogenous variables can be determined,
Lemma 3 The labor market tightness θ decreases and the expected wage (wh + wl0 )/2 increases
as the city size N or the commuting cost τ increases, and as the commuting frequency s ofoﬀ-the-job search increases.
As described in (24), an increase in τ or N (a decrease in s) increases the expected wages
for a given θ, which reduces both the profit of posting a vacancy and θ.
This change in θ, in turn, aﬀects the threshold location x∗ of on-the-job searchers and thenumber es of on-the-job searchers. To examine this, we need to know the expression of Wh − Wl .
From (21), we know thatwh − wl0
r+δ+λβ(yh − yl )
=
.
r+δ+λ

Wh − Wl =Substituting this equation into (12), we obtainx∗ =φp(θ)β(yh − yl )> 0.
2s∗ τ (r + δ + λ)(25)

Lemma 4 There is always a non-empty interval [0, x∗ ] such that low-productivity workers wholive within it look for a new job. The threshold of the interval x∗ decreases with an increases in
the city size N , the commuting cost τ and the commuting frequency s∗ of on-the-job search, andincreases with an increase in the commuting frequency s of oﬀ-the-job search.
As is observed directly from (25), an increase in s∗ or τ leads to a higher cost of on-the-job
search and a decrease in the threshold x∗ . An increase in N , τ or s indirectly aﬀects x∗ bychanging θ.

11

¡
¢
From (21), it can be observed that Jl = yl − wl0 /(r + δ + λ). Substituting this equation
as well as (22) and (25) into (18), we obtain
¸
∙
φp(θ)
β (yh − yl )0(26)
− yl + wl
es = ∗s τ (r + δ + λ)
2
¾
½
φp(θ)
β (yh + yl )= ∗− yl + βcθ + (1 − β) [b + (1 − s)τ (N − u)] ,s τ (r + δ + λ)2which demonstrates the existence of three possible cases described in Lemma 5:
Lemma 5 In equilibrium, (i) a certain proportion of low-productivity workers search for a new
job:
¸
∙
φp(θ)
β (yh − yl )β (yh − yl )0
if wl0 > yl −− yl + wl ,,es = ∗s τ (r + δ + λ)22(ii) no employed worker looks for a new job:
es = 0,if

wl0 ≤ yl −β (yh − yl ),2(iii) all low-productivity workers seek for a new job:
¸
∙
φp(θ)
β (yh − yl )
0
if
+
w
−
y
es = el ,ll ≥ el .
s∗ τ (r + δ + λ)
2
The remainder of this paper focuses on the most interesting case (i). From (26), the following
proposition can be derived:
Proposition 3 The number es of on-the-job searchers decreases as the commuting frequencys∗ of on-the-job search or the commuting cost τ increases, whereas the city size N and thecommuting frequency s of oﬀ-the-job search have an ambiguous eﬀect on es .
An increase in N or τ , or a decrease in s aﬀects es through two channels: (i) by leading to
a decrease in θ, which as shown in Lemma 3, or (ii) by leading to an increase in wage wl0 , asconfirmed by Lemma 3 and (22). Such changes in θ and wl0 have the following impacts on x∗
and the crowding out eﬀect x∗ − es , which is given byφp(θ)Jls∗ τ ¡¢
φp(θ) yl − wl0= ∗.
s τ (r + δ + λ)x∗ − es =From this and (25), we know that a decrease in θ lowers both x∗ and x∗ − es whereas an increasein wl0 has no impacts on x∗ but reduces x∗ − es . Hence, an increase in wl0 leads to an increasein es , while a decrease in θ has an ambiguous eﬀect on es . An increase in N or a decrease in s
aﬀects es only through these two channels, implying that the eﬀects of N and s are ambiguous.
An increase in τ has another negative direct eﬀect on es for any θ and this eﬀect dominates theother eﬀects. An increase in s∗ has only the direct negative eﬀect on es .
We next provide the comparative steady-state analysis regarding the importance of on-thejob search in the economy. To see this, we focus on the proportion of on-the-job searchersamong all workers and among low-productivity workers. From (19) and (26), the proportion of
on-the-job searchers among all workers becomes as
¸
∙
½ ∙
¸¾
b
φp(θ)
1 β (yh + yl )(1 − s)τ p(θ)
es= ∗− yl + βcθ + (1 − β)+,Ns τ (r + δ + λ) N
2N
δ + λ + p(θ)
from which the following proposition can be derived:
12

Proposition 4 An increase in the city size N , the commuting frequency s∗ of on-the-job search,or the commuting cost τ leads to a decrease in the proportion of on-the-job searchers among all
workers, whereas an increase in the commuting frequency s of oﬀ-the-job search has an ambiguous
eﬀect on it.
According to Proposition 3, an increase in s∗ or τ leads to a decrease in es , implying a lower
es /N . Although the eﬀect of N or s on es is ambiguous, the eﬀect of N on the denominator ofes /N is prominent and thus es /N decreases as N increases.
The steady-state condition for the number el of low-productivity workers becomesφp(θ)esup(θ)= (λ + δ)el +,22
where the left hand represents the inflow to the pool of low-productivity workers from the poolof unemployed workers and the right hand side represents the outflow of workers due to theexogenous shocks and job turnover. Combining this equation with (19) yields
µ
¶
p(θ)
N
φes
−
.
el =
2
δ + λ + p(θ) δ + λ
The steady-state condition for the number eh of high productivity workers is given byup(θ) φp(θ)x∗+= (λ + δ)eh ,22where the left hand side represents the inflow of low-productivity and unemployed workers into
the pool of high-productivity workers and the right hand side represents the outflow of highproductivity workers solely due to exogenous shocks.10 Combining this equation with (19) yields
µ
¶
p(θ)
N
φes
eh =
+
.
2
δ + λ + p(θ) δ + λHence, we now have the proportion of each type of employed workers among all workers:
µ
¶
el
p(θ)
u
φes
=
−
,
(27)
N
2(δ + λ) N
N
µ
¶
eh
p(θ)
u
φes
=
+
.
N
2(δ + λ) N
N
When the proportion of on-the-job searchers es /N increases, job turnover increases, raising the
proportion of high-productivity workers and lowering the proportion of low-productivity workers.
Finally, the proportion of on-the-job searchers among the low-productivity workers can bederived from (27) as follows:2(δ + λ)eses=elp(θ)(u − φes )2(δ + λ)(δ + λ + p(θ))es.
=p(θ) [(δ + λ)N − (δ + λ + p(θ))φes ]which readily leads to the following proposition:
Proposition 5 An increase in the commuting frequency s∗ of on-the-job search decreases the
proportion of on-the-job seekers among the low-productivity workers. So does an increase in thecity size N or in the commuting cost τ if (δ + λ + p(θ)) φp(θ) > (δ + λ)2 .
10Note that no high-productivity workers look for a new position.

13

Although an increase in s∗ leads to a decrease in es , it does not aﬀect θ, implying that anincrease in s∗ leads to a decrease in the proportion es /el . Having observed that an increase inτ or N leads to a decrease in both p(θ) and es /N , it can now be observed that a decrease in
es /N always leads to a decrease in es /el . From the partial diﬀerentiation of es /el with respect
to p(θ), the following can be derived to demonstrate that an increase in N or τ decreases the
proportion of on-the-job searchers among low-productivity workers, as stated in Proposition 5:∂es /el> 0 ⇔ (δ + λ + p(θ)) φp(θ) > (δ + λ)2 .
∂p(θ)
When the number of each type of worker can be determined, the market land rent function
can be derived as follows:⎧
if x ∈ (0, es )⎨ (1 + s)τ (es − x) + τ (N − u − es ) + sτ u.
R=
τ (N − u − x) + sτ u
if x ∈ [es , N − u]⎩sτ (N − u)if x ∈ (N − u, N ]4Eﬃciency

This section derives the first and second best optimal allocations and compares them to theequilibrium allocation described in the previous section, using the social surplus SS as thecriterion of welfare.

Second-best optimal allocation
We start from the analysis of the second-best optimal allocation, in which social planners can determine workers’ locations and whether they look for a new job although the market tightness isdetermined by the free-entry condition. As shown in Appendix B, the second-best optimal allocation requires that workers live in the same areas in which they would do so in the equilibrium.
Hence, the welfare function (the social surplus) is given by"Z)
exp[−rt] yh eh + yl el + bu−cθ(u + φe∗∗|{z}|{z s }

SS =output

−
|

Z e∗∗
s
0

∗

(1 + s )τ xdx −Z N −ue∗∗s{zsearch costs

τ xdx −

commuting costs

Z N

N −u

#

sτ xdx dt,}

where e∗∗s is the number of on-the-job searchers as determined by the social planner. SS consistsof the output from matches, search costs, and commuting costs. The social planner’s problem
in the second-best optimal allocation is then defined as
SS
max
∗∗

(28)

es

s.t.

deh
up(θ) φp(θ) ∗∗=+es − (δ + λ)eh ,
dt
2
2
du
= δN + λ(N − u) − (δ + p(θ))u,
dt
½
∙
¸
¾
1
yh + ylc=(1 − β)− b − (1 − s)τ (N − u) − βcθ ,
q(θ)
r+δ+λ
2

where the first constraint describes the law of motion of the number of high productivity workers,
the second constraint describes the law of motion of the number of unemployed workers, and thethird constraint is the free-entry condition. Here, it is assumed that unemployment is socially14

undesirable, or more precisely, that the shadow price of unemployment is negative.11 Without
this assumption, the social planner may prefer leaving workers unemployed rather than inducing
them to form bad matches and shut bad matches down.
As shown in Appendix C, the optimal number of on-the-job searchers is given by∙¸
φ p(θ)(yh − yl )∗∗− cθ ,(29)
es = ∗s τ 2 (r + δ + λ)which satisfies the following property:
Proposition 6 For any θ, the equilibrium number of on-the-job searchers is equal to the optimal
number (i.e., es = e∗∗s must hold true).
Proof See Appendix D.
In a standard on-the-job search model a la Pissarides [7], the number of on-the-job searchersin equilibrium is larger than that in the optimum. Here, spatial crowding out by high-productivity
workers reduces the number of on-the-job seekers, leading to the attainment of the second best
allocation, i.e., there is no distortion in decision regarding on-the-job search.
Put diﬀerently, the equilibrium decision regarding on-the-job search that is described in (18)maximizes the total surplus of forming a match with low-productivity workers, with the total
surplus given by(r + δ)(Wl (x) + Jl (x) − U )⎧yl − (r + δ)U + λ(U − Wl − Jl )⎪⎪⎨+ φp(θ) (Wh − Wl − 2Jl ) /2 − s∗ τ x=⎪⎪⎩yl − (r + δ)U + λ(U − Wl − Jl )if the employed worker
looks for a new job.

otherwise

Therefore, as long as φp(θ)(Wh −Wl −2Jl )/2 is larger than the search cost s∗ τ x, on-the-job searchincreases the total surplus, implying that the equilibrium decision regarding on-the-job searchmaximizes the surplus. In the absence of spatial structure, low-productivity (bad matched)workers initiate on-the-job search if the search cost is smaller than φp(θ)(Wh − Wl )/2, whichyields a larger number of on-the-job searchers than is optimal.

4.1

First-best optimal allocation

Having demonstrated that decision regarding on-the-job search is eﬃcient, the next question iswhether the free-entry condition leads to the optimal number of vacancies. This consideration isaddressed by comparing the equilibrium to the first-best optimal allocation, in which the social
planner can choose the market tightness θ as well as the number of on-the-job searchers e∗∗
s .
The corresponding problem can be expressed as:
SS
max
∗∗
es ,θ

up(θ) φp(θ) ∗∗deh=+es − (δ + λ)eh ,
s.t.
dt
2
2
du
= δN + λ(N − u) − (δ + p(θ))u.
dt
As shown in Appendix E, the optimality condition is given byθp0 (θ)(30)
r + δ + λ + p(θ) − uθp0 (θ)/(u + φe∗∗
s )
∙µ
¶
¸
φe∗∗
u
yh − yls p(θ)× r+δ+λ++ [yl − b − (1 − s)τ (N − u)].
u + φe∗∗2(r + δ + λ)u + φe∗∗
s
s

cθ =

11

See Appendix C.

15

In the absence of on-the-job search, as shown in Wasmer and Zenou [15] and Zenou [18], even withconsideration of the spatial structure, the Hosios condition (Hosios [4]) ensures the equilibrium
to be optimal, i.e., the free-entry condition (20) coincides to (30) when β = −θq 0 (θ)/q(θ) (=1 − θp0 (θ)/p(θ)) holds true: by substituting φ = 0 and β = 1 − θp0 (θ)/p(θ) into (20) and (30),
we obtain12∙¸
θp0 (θ)yh + ylcθ =− b − (1 − s)τ (N − u) .
(31)
r + δ + λ + p − θp0 (θ)
2However, in the presence of on-the-job search, the optimal θ under the Hosios condition is givenby (30) although (20) under the Hosios condition becomes (31), which determines the equilibrium
tightness. As shown in Appendix F, the right hand side of (30) is smaller than the right handside of (20) for a given θ. In contrast, the left hand side is the same for both the equilibrium
and the optimum. These loci are described in Figure 3.
[Figure 3 around here]

From Figure 3, we can readily see that the equilibrium value of θ is larger than the optimalvalue of θ.
Proposition 7 In the presence of on-the-job search, vacancies are over-provided in the equilibrium under the Hosios condition.
As social benefits derived from making a new job match is smaller for on-the-job search thanfor oﬀ-the-job search, the optimal value of the market tightness θ decreases when the on-the-job
search is possible. In contrast, the equilibrium value of θ is determined independently of whether
φ > 0, which is the result of spatial arbitrage among employed workers.

5

Concluding remarks

This study investigated the nature and impacts of the interaction between the structure of acity and the on-the-job search. In equilibrium, workers relocate within a city correspondingly
to their career turnover. Unemployed workers relocate from the outskirts of city to the inner
regions of the city when they land a job. Employed workers must determine whether to search
for a new job. If an employed worker searches, she/he lives most proximate to the CBD. If not,she/he resides inbetween the unemployed workers and on-the-job searchers. Thus, the model
proposed in this paper provides a tool to analyze the simultaneous determination of career andrelocation paths. Eﬃciency analysis demonstrated that although the decision to post a job
vacancy is distorted by the introduction of the spatial structure, decision regarding on-the-job
search is eﬃcient.
A few comments are in order. First, in the model presented here, the spatial dimension of onthe-job search appears only on the cost side of a job search. However, the spatial concentrationof job searchers and vacancies may lead to agglomeration economies (Sato [10] and Wheeler[16]), the importance of which has already been noted by Marshall [6]. As is always truefor economics analysis, balanced consideration of the costs and benefits must be given, andtherefore examination of the eﬀects of introducing the benefits of on-the-job search within a cityis warranted. Second, the model is a closed city model in which the population size of the city is
exogenous. However, if the benefits of concentration of job-seekers and vacancies outweighs thecosts of it, a large city will attract more workers and firms. It is therefore particularly important
to investigate the resulting distribution of economic activities in a multi-city world. Finally, forthe sake of analytical tractability, this model considered only two levels of productivity, high12Note that es = 0 when φ = 0.

16and low, which led to the development of an over-simplified wage distribution based on only
three possible values. Future research eﬀorts should introduce a continuous wage distribution ala Burdett and Mortensen [2]. All these are important directions in the future research.

Appendix A: Proof of Lemma 1
Start with the wages paid to a good-match worker. Because i∗ = 0 for Jh , (3), V = 0 and(7) imply thatwh (x) = wh ≡ yh − (λ + δ + r)Jh , ∀x.
Similar arguments for the wages paid to a bad-match worker imply that
wl1 (x) = wl1 ≡ yl − (λ + δ + r + φp(θ))Jl ,wl0 (x) = wl0 ≡ yl − (λ + δ + r)Jl ,

∀x.

∀x.

Hence, we obtain the results:
wl0 − wl1 = φp(θ)Jl .

Appendix B: The location patterns in the first and second best optimal allocation
This appendix shows that in the optimal allocation, on-the-job searchers live closet to the
CBD, employed non-job searchers live next closest to the CBD, and unemployed workers live
most distant from the CBD.
Consider two workers A and B, whose commuting costs are τA and τB , respectively. Assume
that τA > τB and denote their locations by xA and xB , respectively. We show by contradictionthat xA < xB must hold true in the optimal allocation.
Suppose now that xA > xB holds true in the optimal allocation and let ∆ denote xA −xB > 0.
Then, by interchanging their location, the total commuting costs in the economy changes by−τA ∆ + τB ∆ = (τB − τA )∆ < 0. This contradicts the definition of the optimal allocation.
Hence, xA < xB must hold true, i.e., workers with higher commuting costs must locate closer
to the CBD in the optimal allocation, which is the same location pattern as that described in
Proposition 1.

Appendix C: Derivation of the second-best optimal allocation
As the free-entry condition in (28) determines θ as a function of u, it can be denoted as θ(u).
Using this expression, the welfare function can be rewritten as"Z
exp[−rt] yh eh + yl el + bu − cθ(u)(u + φe∗∗
s )

SS =
−

Z e∗∗
s
0

(1 + s∗ )τ xdx −Z N −ue∗∗sτ xdx −

Z N

N −u

#

sτ xdx dt,

which implies that (28) becomes

max
SS
∗∗
es

s.t.

deh
up(θ(u)) φp(θ(u)) ∗∗=+es − (δ + λ)eh ,
dt
2
2
du
= δN + λ(N − u) − (δ + p(θ(u)))u,dt

17

Noting that el = N − u − eh , the present-value Hamiltonian is defined as
"
Z e∗∗
s
∗∗
H = exp[−rt] yh eh + yl (N − u − eh ) + bu − cθ(u)(u + φes ) −(1 + s∗ )τ xdx−

Z N −ue∗∗sτ xdx −

Z N

N −u

#

sτ xdx + μg

0∙p(θ(u))(u + φe∗∗s ) − (δ + λ)eh2¸

+ μu [δN + λ(N − u) − (δ + p(θ(u)))u] .

Note here that the control variable is e∗∗s and the state variables are eh and u. The optimality
conditions are∂H
0 = ∗∗ ,
(32)
e∗∗
s :
∂es
dμg
∂H
eh :
,
=−
dt
∂eh
dμu
∂H
u:
=−
.
dt
∂u
By solving the three equations in (32), we obtain
0 = exp[−rt](−cθφ − s∗ τ e∗∗s ) + μg

φp(θ)
,
2

(33)

yh − yl,r+δ+λ(r + δ + λ) [yl − b + cθ − (1 − s)τ (N − u)] − p(θ)(yh − yl )/2
μu = − exp[−rt],(r + δ + λ)(λ + r + δ + p(θ))μg = exp[−rt]which determine the optimal number of on-the-job searchers:∙¸
φ p(θ)(yh − yl )∗∗es = ∗− cθ .
s τ 2 (r + δ + λ)The regularity condition that the social planner prefers employment to unemployment is thengiven byp(θ)(yh − yl )⇔(34)
< yl − b + cθ − (1 − s)τ (N − u),μu < 02(r + δ + λ)and we assume that this inequality holds true.

Appendix D: Proof of Proposition 6
Substituting the free entry condition in (28) into (29), yields
½
¾
φp(θ)
yh − ylyh + yl∗∗− (1 − β)+ βcθ + (1 − β) [b + (1 − s)τ (N − u)]es = ∗s τ (r + δ + λ)
2
2
¾
½
φp(θ)
β (yh + yl )= ∗− yl + βcθ + (1 − β) [b + (1 − s)τ (N − u)] ,s τ (r + δ + λ)2which is equal to (26).

Appendix E: Derivation of the first-best optimal allocation
In the first-best optimal allocation, the social planner can choose the market tightness θ aswell as the number of on-the-job searchers e∗∗
s . The corresponding problem can be expressed as:
max
SS
∗∗
es ,θ

s.t.

up(θ) φp(θ) ∗∗deh=+es − (δ + λ)eh ,
dt
2
2
du
= δN + λ(N − u) − (δ + p(θ))u,dt
18

The present-value Hamiltonian is defined as"
H = exp[−rt]−Z N −ue∗∗syh eh + yl (N − u − eh ) + bu − cθ(u + φe∗∗s )−

τ xdx −Z N#sτ xdx + μg

N −u

∙

Z e∗∗
s

(1 + s∗ )τ xdx

0p(θ)(u + φe∗∗s ) − (δ + λ)eh2¸

+ μu [δN + λ(N − u) − (δ + p(θ))u] .
The optimality conditions are
e∗∗
s :
θ:
eh :
u:

∂H
,
∂e∗∗
s
∂H
0=
,
∂θ
dμg
∂H
,
=−
dt
∂eh
dμu
∂H
=−
.
dt
∂u

0=

A new condition now existsθ : e−rt c =μg p0 (θ) μu p0 (θ)u.
−2u + φe∗∗s

Solving the optimality condition yields∙¸
cp(θ)(yh − yl )yh − yl1u,=+yl − b + cθ − (1 − s)τ (N − u) −0
p (θ)
2(r + δ + λ) r + δ + λ + p(θ)
2(r + δ + λ) u + φe∗∗s

Appendix F: The loci of the right hand sides of (30) and (31) in Figure 3
Note that (30) becomes (31) when φ = 0. When φ > 0, (30) can be rewritten asθp0 (θ)r + δ + λ + p(θ) − θp0 (θ)(1 − A)¾½
yh − yl+ (1 − A) [yl − b − (1 − s)τ (N − u)] ,× (r + δ + λ + p(θ)A)
2(r + δ + λ)

cθ =

(35)

∗∗
where A = φe∗∗s /(u + φes ). Note that ∂A/∂φ > 0 for a given θ. Let Φ denote the right handside of (35). For a given θ, a change in A aﬀects Φ in the following manner:¾½∂Φθp0 (θ)p(θ) (yh − yl )=− [yl − b − (1 − s)τ (N − u)] − Φ .
∂Ar + δ + λ + p(θ) − θp0 (θ)(1 − A) 2(r + δ + λ)Because (34) is assumed to hold, ∂Φ/∂A < 0 is obtained, and hence ∂Φ/∂φ = (∂Φ/∂A)(∂A/∂φ) <0 for a given θ. This fact, combined with the fact that (30) becomes (31) when φ = 0, demonstrates that the right hand side of (30) is smaller than that of (31) for any θ when φ > 0, as
described in Figure 3.

Acknowledgements
We thank Yoshihisa Asada, Daisuke Oyama, Komei Sasaki, Takaaki Takahashi, and the participants of 24th Annual Meeting of the Applied Regional Science Conference for their usefulcomments and discussions. Of course, we are responsible for any remaining errors. We acknowledge the financial support by the JSPS Grants-in-Aid for Scientific Research (S, A, B, and C)
and the MEXT Grant-in-Aid for Young Scientists (B).

19

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Figure 1: Bid rent curves

Figure 2: Equilibrium of the model
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Figure 3: The value of market tightness: Equilibrium v.s. the first best22