Discussion Papers In Economics
And BusinessLand development, search frictions, and city structure
Yasuhiro Sato and Wei Xiao

Discussion Paper 13-12

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLand development, search frictions, and city structure
Yasuhiro Sato and Wei Xiao

Discussion Paper 13-12

June 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLand development, search frictions, and city structure∗
Yasuhiro Sato†and Wei Xiao‡
June 25, 2013

Abstract
This paper analyzes the interactions between labor and housing (and land) markets in a city.
We develop a monocentric city model involving land development and frictional unemployment.
Unemployment, the spatial structure of a city, land development, housing demand, prices of
housing and land are all endogenously determined in the model. We then characterize two
diﬀerent spatial configurations, spatial mismatch equilibrium in which unemployed workers arelocated far from jobs and integrated equilibrium in which unemployed workers live in areas close
to jobs. To better understand how two equilibria are aﬀected by labor market parameters, suchas search intensity, wage, job finding rate, job destruction rate, and so on, we implement a
comparative steady state analysis. We further explored the eﬀects of policies such as a tax onland development to subsidize residents, a subsidy to reduce residents’ commuting costs, and a
subsidy to improve unemployed workers’ benefits.
JEL Classification: R14, R21, R28
Key words: Land development, City structure, Search frictions, Spatial mismatch

1

Introduction

It is commonly observed that the intensity of land development varies a lot between diﬀerent areasof modern cities: buildings close to a city center are usually taller than those in a city fringe. It isalso observed that the incidence of unemployment is unevenly distributed between diﬀerent areas
within cities. In several U.S. large cities, the unemployment rates are higher in city centers than∗

We thank Marcus Berliant, Ping Wang, and Yves Zenou for their comments and suggestions. Of course, we areresponsible for any remaining errors. We acknowledge the financial support from RIETI, the Japan Society for the
Promotion of Science, through a Grant-in-Aid for Young Scientists (B).
†
Graduate School of Economics, Osaka University, email: ysato@econ.osaka-u.ac.jp‡
Corresponding author, Department of Economics, Stockholm University, email: wei.xiao@ne.su.se1

in suburbs.1 In other countries, we also observe the variation of unemployment rate in diﬀerent
areas within a city, although it becomes more complex.2 For instance, in Japan, the three largest
metropolitan areas (MAs) exhibit diﬀerent patterns:3 in the second largest city, Osaka MA, theunemployment rate is higher in the city center than in suburbs (in 2005, it is 0.108 in the central cityand 0.068 in the suburb cities). In the third largest city, Nagoya MA, the opposite holds true (theunemployment rate in 2005 is 0.055 in the central city and 0.044 in suburb cities). In the largest
city, Tokyo MA, we observe no systematic spatial diﬀerence in the unemployment rate (in 2005, it
is 0.056 in the central city and 0.056 in the suburb cities). As first shown by Wasmer and Zenou[21] and further investigated by subsequent studies (see Zenou [26]), interactions between land andlabor markets can explain these spatial diﬀerences in the unemployment rate. They also provedthat such interactions result in market ineﬃciency, indicating possible positive roles of policies in
improving welfare of residents.
In this paper, we uncover eﬀects of various policies on land use patterns and welfare by constructing a monocentric city model that involves unemployment and land use. To be more precise,
we consider a city where all jobs are located in the unique central business district (CBD). Employed workers and unemployed workers coexist in the city due to search frictions in the labormarket. Developers rent land from absentee landlords and supply housing service. Each worker
obtains utility from the numèraire and housing consumption while she/he needs to commute to
the CBD in order to work when employed and in order to search for a job when unemployed. By
comparing the maximum housing price that each type of workers is willing to pay at each location,
we obtain two types of equilibrium: spatial mismatch equilibrium in which unemployed workers
locate far away from the CBD and integrated equilibrium where unemployed workers are close to
the CBD. By using this framework, we analyze policy eﬀects on land use/development and welfareof residents in the two types of equilibrium. Here, we focus on the following three policies: (a) aland development policy that taxes housing development in the city to subsidize residents, (b) atransportation policy that improves the transportation infrastructure in the city, and (c) an income
transfer from employed workers to unemployed workers.
Our paper is related with long tradition of the literature of monocentric city models. This1

The discussion about this phenomenon can be found in the literature on spatial mismatch initiated by the seminal

work of Kain [10].
2Zenou [25] made a careful comparative review of the spatial distributions of unemployment in U.S. cities andEuropean cities.
3
Here, we define the Japanese MAs as the Urban Employment Areas proposed by Kanemoto and Tokuoka [12].

2literature dates back at least to the classic monocentric city model by Alonso [1], Mills [14], andMuth [15], which has become the standard framework to explain the observed regularities in thereal world cities, such as the variation of land development intensity and housing (land) price.
Brueckner [3] provides a unified treatment of these models, which is built around the key observationthat diﬀerence of commuting costs within an urban area is balanced by the diﬀerence of houisng
prices. It is shown that the model in Brueckner [3] does an excellent job of predicting the internal
structure of cities and explaining intercity diﬀerences of spatial structure. To account for theobserved pattern of higher-income groups locating more peripherally, Hartwick [6] and Wheaton[23] extends the monocentric city model with homogeneous workers to incorporate multiple income
groups. When the income elasticity of land consumption exceeds the income elasticity of commuting
costs, households of greater income select locations near the city fringe as a compensation to landconsumption. Although these studies provided complete analysis of city structure under a perfectlycompetitive labor market, unemployment was not introduced in them, implying that there is no
scope for welfare improving policies.
Given the prevailing spatial variations in labor market conditions, recent studies examinedinteractions between the labor market and the housing (land) market. Wasmer and Zenou [21]developed an urban search model by introducing a land market into the search-matching model.4
In their model, workers’ search eﬃciency is negatively aﬀected by their distance to jobs. The
endogeneous location of workers in the city reflects the trade-oﬀ between commuting costs, land
rents, and the surplus associated with search. They show that there are two possible spatialconfigurations in the city. The unemployed workers may be located close to the CBD or the city
fringe far away from jobs. Sato [18][19] considered the heterogeneity of workers in the backgroundof urban labor market. In all these models, the land and housing markets are not fully modelled:they simply assumed that there is no land development in the city and each worker consumes fixedunits of land. Coulson et al. [4] explained the spatial mismatch by developing a search matchingmodel for a city with central and suburban labor markets. They assume that search is costly andthe costs of setting up a firm is higher in the central markets. These two key assumptions ensurethat unemployment rate is higher in the central area of a city than the suburban area. However,the city structue is still exogeneously given in their model. Our model is the most closely related toSmith and Zenou [20] and Xiao [24]. Smith and Zenou [20] extended the model described in Wasmer
and Zenou [21] by endogenizing the job search intensity and housing consumption whereas they4

A complete introduction of the search-matching framework can be found in Pissarides [16].

3treated land development exogenously, and Xiao [24] endogenized land development in the context
of a monocentric city with search frictions in the labor market whereas he assumed fixed housing
consumption.5 In contrast, we endogenized both demand and supply sides of the housing market
in order to examine the full relationship between housing and labor markets, which is indispensable
to policy analysis.
The paper proceeds as follows. Section 2 proposes the model. Section 3 characterizes two
spatial configurations, spatial mismatch equilibrium and integrated equilibrium. Section 4 explains
the results of comparative steady state analysis that are usefull in understanding the policy eﬀects.
Section 5 explores the performance of policies. Section 6 concludes.

2

Model

2.1

Spatial structure

We extend the basic framework of an urban search model developed by Smith and Zenou [20].
Consider a closed city where there is a continuum of workers of size one. Workers are either employedor unemployed. An employed worker works and obtains wage income whereas an unemployedworker seaches for a job. We follow Smith and Zenou [20] in assuming the city structure andcommuting behaviors of each type of workers: we consider a linear monocentric city, normalize theland endowment at each location to one, and assume that land is owned by absentee landlords.
An employed worker commutes to the CBD to work and her/his commuting cost is tx, where t
is a positive constant and x is the distance from the CBD. An unemployed worker commutes to
the CBD to search for a position and get interviewed by firms posting vacancies. She/he bears
the commuting cost stx, where s ∈ (0, 1) represents the search intensity (such as frequency of job
interviews). Because our primary purpose is to analyze the housing development in the monocentriccity with job search, we simplify the framework of Smith and Zenou [20] by assuming that s isexogenous whereas we endogenize the supply of housing service.

2.2

Developers

Housing service is supplied by developers, who rent land L from absentee landlords and combine
it with capital K to supply housing service. The production function of housing service is definedas G(x) = Lγ K 1−γ , where γ is a positive constant satisfying 0 < γ < 1. The housing service
5

For the earlier studies relating the labor market to the land (housing) market, see Zenou [26].

4produced at location x can be written as g(x) = S 1−γ in which S = K/L is the capital-land ratio.
A developer at location x, behaving as a price taker, determines the supply of housing service inorder to maximize its profit:max Π(x) = R(x)S 1−γ − rS − φ(x),S

where R(x), r, and φ(x) are the price of housing service at location x, the capital price, and theland rent at x, respectively. We assume that the capital market is global and the city is suﬃciently
small that the capital price is exogenously given.
The standard profit maximization yields the investment per acre of land and the supply of
housing service as6¸(1 − γ) R(x) 1/γ
S (x) =,r¸∙
(1 − γ) R(x) (1−γ)/γg(x) =
.
r
∙

(1)
(2)

The constant returns to scale in housing service production imply that the equilibrium profit ofeach developer is zero. From this zero profit condition, the bid rent, i.e., the maximum land rent
that a developer can pay, Φ(x), is determined by R(x)S (x)1−γ − rS (x) − Φ(x) = 0. Inserting (1)into it, we obtain
Φ(x) = γµ1−γ
r¶(1−γ)/γR (x)1/γ .

(3)Because we assumed that landlords live outside of the city, land rents received by them can be seen
as a leakage of welfare in the city.

2.3

Workers

There are e employed workers and u unemployed workers, implying that e+u = 1. When employed,each worker obtains the wage income w whereas she/he obtains b when unemployed. b represents
the income of self-employment or unemployment benefits. We assume that both w and b are
exogenous.
Workers obtain utility from consumption of housing service and the numéraire. We apply the
instantaneous utility function of the Cobb-Douglas form:u = α ln h + (1 − α) ln z,(4)where 0 < α < 1. h and z describe the levels of housing and numéraire consumption, respectively.
6

Notice that the land endowment at each location x is normalized to one.

5

2.4

Asset values and steady state conditions

Letting R(x) denote the price of housing service, the budget constraint of a worker is
I = z + R(x)h + τ x,
where I and τ are given by
½
w
I=
b

if employed,
if unemployed

τ=

½

t
st

if employed.
if unemployed

Although our framework is dynamic, consumption levels are determined by instant utility maximization. Each worker, taking prices and g(x) as given, maximizes u with respect to z and h subject
to the above budget constraint. Utility maximization leads to the standard demand functions:z = (1 − α)(I − τ x),h=α(I − τ x)
.
R(x)

(5)

The resulting indirect utility is given byv(x) = B + ln(I − τ x) − α ln R(x),

(6)

£
¤
where B ≡ ln αα (1 − α)1−α .

Time is continuous and we assume the oﬀ-the-job search. The opportunity of landing a job

for an unemployed worker arrive according to a Poisson process at an exogenous rate sλ (> 0).7
Each job is destroyed according to a Poisson process at an exogenous rate δ (> 0). Let W (x) andU (x) denote the asset value of an employed worker residing at x and that of an unemployed worker
residing at x, respectively. W (x) and U (x) are given byρW (x) = v(x) + δ(Umax − W (x)),(7)

ρU (x) = v(x) + sλ(Wmax − U (x)).
where ρ is the exogenous discount rate. Umax and Wmax represent the results of workers’ maximization regarding location choice (i.w., Umax = maxx U(x) and Wmax = maxx W (x)). Equation(7) implies that the flow capital cost of an employed or unemployed worker at location x is equal
to the instantaneous utility plus the utility derived from the change of her/his economic status.
We focus on the steady state. From the assumptions on the job matching and destruction
process, we have the steady state condition the job flows:δe = sλu.
7

λ can be endogenized in the same way as Smith and Zenou [20] without altering our main results.

6

Combined with e + u = 1, this yieldse=sλ,δ + sλu=δ.
δ + sλ(8)

It can be seen that an increase in job destruction rate δ reduces the number of employed workers
and raises the number of unemployed workers. For search eﬀort s and job finding rate λ, they havean opposite eﬀect on e and u. An increase of s and λ leads to a larger e and a smaller u.

2.5

Bid rent functions and spatial structure

Workers can relocate costlessly within the city, which is a standard assumption in urban economics.8
This assumption implies that there is no incentive for workers to relocate in equilibrium. Therefore,in equilibrium, all employed workers enjoy the same level of value (W (x) = W max = W ) and thisalso holds true for unemployed workers (U (x) = U max = U ).
In order to determine the equilibrium location of workers, we use the concept of bid rents,
defined as the maximum price of housing service at location x that each type of worker is willingto pay in order to reach her/his respective level of equilibrium utility (in this paper, asset value).
Plugging (6) into (7), the bid rents Ω of employed and unemployed worker can be expressed asfollows:
α ln Ωw (x) = B + ln(w − tx) + δ(U − W ) − ρW,(9)α ln Ωu (x) = B + ln(b − stx) + sλ(W − U ) − ρU.
In order to see the location pattern in equilibrium, it is suﬃcient to examine the slope of the two
bid rent functions at the intersection of Ωw (x) and Ωu (x). Diﬀerentiating Ωw (x) and Ωu (x) andevaluating them at Ωw (x) = Ωu (x) = Ω(x), we can see that
Ω0w (x) = −Ωw (x) t,
α w − tx

Ω0u (x) = −Ωu (x) st,
α b − stx

which imply that Ω0w (x) < 0 and Ω0u (x) < 0. Therefore, we have two bid rent functions which are
downward sloping with respect to the distance from the CBD. We can see which is steeper thanthe other at the intersection of the two bid rent functions:¤¯£
sgn Ω0w (x) − Ω0u (x) ¯Ωw (x)=Ωu (x)=Ω(x) = sgn [sw − b] .

From this, we have the following proposition:98
9For the eﬀects of relocation costs in urban job search models, see Zenou [26] and Kawata [13].
When sw = b, both employed and unemployed workers reside in mixed. Although we can characterize such

equilibrium, this case yields no interesting results.

7

Proposition 1 Employed workers reside closer to the CBD than unemployed workers if sw < b.
The opposite holds true if sw > b.
The intuition of Proposition 1 is straightforward. Given wage and unemployment benefits, thespatial structure of the city is determined by unemployed workers’ search eﬀort s. If the unemployedworkers search less frequently, the commuting costs induced by search are lower. Then, employedworkers have stronger incentive to reduce commuting costs and propose higher bid rents for theapartments close to the CBD than the unemployed workers. Therefore, the area near the CBD is
occupied by employed workers and the area near the edge of the city is allocated to unemployedworkers. However, if unemployed workers search more frequently, we then obtain a city with adiﬀerent structure in which unemployed workers reside in the area close to the jobs while employedworkers live far away from the CBD. Following the tradition in the urban job search models (seeZenou [26]), we call the former spatial structure the spatial mismatch equilibrium and the latterthe integrated equilibrium.10 In both cases, the residential area is divided into two zones: in one
zone, only employed workers reside, and unemployed workers live in the other zone. We call theformer the employment zone (EZ) and the latter the unemployment zone (U Z).
In the following analysis, we focus on the two possibilities in Proposition 1.11

3Equilibrium

As noted before, in equilibrium, all employed workers enjoy the same utility level W and all unemployed workers enjoy the same utility levels U. Plugging Wmax = maxx W (x) = W , Umax =maxx U (x) = U and (2) into (7), we obtain
ρW = B + ln(w − tx) − α ln R(x) + δ(U − W ),(10)

ρU = B + ln(b − stx) − α ln R(x) + sλ(W − U ),We have two diﬀerent spatial configurations (i) spatial mismatch equilibrium and (ii) integrated
city equilibrium. We will deal with these two cases sequentially.
10

The standard spatial mismatch hypothesis argues that the spatial disconnection between residential locations ininner cities and suburban job opportunities account for the adverse labor market outcomes of low skilled minorities(see the seminal work by Kain [10] and the empirical surveys by Jencks and Mayer [9], Holzer [7], Kain [11], Ihlanfeldt
and Sjoquist [8]).
11
If we endogenize the search intensity, much richer equilibrium configurations are possible. See Smith and Zenou[20] for details.

8

3.1

(i) Spatial mismatch equilibrium

We start from characterizing the spatial mismatch equilibrium (the case of sw < b). In thisequilibrium, the market housing price is determined by½
for x ∈ (0, xb]Ωw (x),
R(x) =
for x ∈ (bx, x]Ωu (x)(11)

where xb is the intersection of Ωw (x) and Ωu (x). Hence, the market housing price at the center ofthe city x = 0 is determined by the bid rent of an employed worker, and that at the edge of thecity x = x is determined by the bid rent of an unemployed worker. Here, we set the opportunity
cost of land use to one. Therefore, we haveR(0) = Ωw (0),R(x) = Ωu (x) = 1.

Substituting these equations and (11) into (10), we obtain
ρW = B + ln w − α ln Ωw (0) − δ(W − U )ρU = B + ln(b − stx) + sλ(W − U ).
We can rewrite the bid rent functions (9) by using these asset values as¶¶µµw − tx 1/αb − stx 1/αΩw (x) = Ωw (0),Ωu (x) =.
wb − stx(12)

This shows that the bid rents of the employed and the unemployed workers are decreasing with theb is determined as
distance to the CBD. Solving Ωw (x) = Ωu (x), we can see that xxb=w [(b − stx) Ωw (0)α − b].
t [(b − stx) Ωw (0)α − sw](13)

The remaining endogenous variables to be fixed are Ωw (0) and x, which are determined by
the population conditions. The population density d(x) at location x is described by the housing
supply over the housing demand g(x)/h(x). From (5) and (2), we obtain d(x) as³´(1−γ)/γ½ 1 1−γΩw (0)1/γ w−1/(αγ) (w − tx)1/(αγ)−1
for x ∈ (0, xb]αr.
d(x) =³´(1−γ)/γ−1/(αγ)1 1−γ1/(αγ)−1
(b
−
stx)
(b
−
stx)
for
x
∈
(b
x
,
x]
α
r

(14)

We know that all employed workers reside in the employment zone (0, xb] and all unemployed workers

live in the unemployment zone (bx, x]. Therefore, the numbers of employed and unemployed workers
are described as
e=

Z x

d(x)dx,

u=

0

9

Z x
x


d(x)dx(15)

Substituting (13) and (14) into (15) and solving the two equations, we obtain Ωw (0) and x as
Ωw (0) = [1 + φt (su + e)]γ ,b − stx =sw
b − sw,αγ +[1 + φt (su + e)][1 + φtsu]αγ(16)where φ = γ1 [r/ (1 − γ)](1−γ)/γ .

3.2(ii) Integrated equilibrium

In the integrated equilibrium (the case of sw > b), the market housing price is determined by½
for x ∈ (0, xb]Ωu (x),(17)
R(x) =
for x ∈ (bx, x]Ωw (x)

where xb is again the intersection of Ωw (x) and Ωu (x). Hence, the market housing price at thecenter of the city x = 0 is determined by the bid rent of an unemployed worker, and that at theedge of the city x = x is determined by the bid rent of an employed worker:
R(0) = Ωu (0),R(x) = Ωw (x) = 1.

Similarly to the previous section, these equations and (10) lead toρW = B + ln(w − tx) + δ(U − W ),(18)

ρU = B + ln b − α ln Ωu (0) + sλ(W − U ).
By substituting (18) into (9), we obtainΩw (x) =µw − txw − tx¶1/α,Ωu (x) = Ωu (0)µb − stx
b

¶1/α

.

(19)

b is determined as
Solving Ωw (x) = Ωu (x), it can be seen that xxb=b [(w − tx)Ωu (0)α − w].
t [s(w − tx)Ωu (0)α − b](20)

From (5) and (2), the population density d(x) at location x is determined byd(x) =

½1

α
1
α

³

1−γ
r

³

´(1−γ)/γ1−γr´(1−γ)/γΩu (0)1/γ b−1/(αγ) (b − stx)1/(αγ)−1(w − tx)−1/(αγ) (w − tx)1/(αγ)−1

for x ∈ (0, x
b]

.

(21)

for x ∈ (bx, x]

The location of two types of workers in the integrated equilibrium is opposite to that in the spatial
mismatch equilibrium. Using (21), the numbers of employed and unemployed workers are described
as
e=

Z x
x


d(x)dx,

u=

Z x
0

10

d(x)dx.

From these equations, we can see that Ωu (0) and x are determined by½¾
b1
sw − bγ.
w − tx =+Ωu (0) = [1 + φt(su + e)] ,s [1 + φt(su + e)]αγ[1 + φte]αγ(22)

The following proposition summarizes the above arguments:
b, x, R(0), e, u):
Proposition 2 The equilibrium is characterized by a tuple (g(x), Φ (x) , R(x), Ωw (x), Ωu (x), x
the numbers of two types of workers (e, u) are determined by (8). In the spatial mismatch equi-

librium, R(0) = Ωw (0), and (16) determines (x, Ωw (0)). The bid rent functions (Ωw (x), Ωu (x))
and xb are given by (12) and (13), respectively, which determines the market housing priceR(x) as in (11). This , in turn, determines the development level g(x) and the market land

price Φ (x) as described in (2)and (3). In the integrated equilibrium, R(0) = Ωu (0), and (22)b, which,
determines (x, Ωw (0)). Equations (19) and (20) determines (Ωw (x), Ωu (x)) and x

combined with (17), (2), and (3), gives R(x), g(x), and Φ (x).
The two equilibrium city configurations are shown in Figure 1.

[Figure 1 around here]

4

Comparative Steady States

In this section, we explain the results of comparative steady states that help us understand thepolicy eﬀects. The goal of our analysis is to deduce the impacts of search intensity, wage, andb), housing prices (Ω0 (x), Ωw (x),unemployment benefits on the spatial structure of the city (x and x

and Ωu (x)), land prices (Φw (x) and Φu (x)), land development (gw (x) and gu (x)), employed andunemployed workers’ lifetime utilities (W and U ). These arguments are keys to understand thepolicy analysis that will be given in the next section. The results of comparative steady states forother parameters, t, δ, and λ, are given in Appendix.12 Our results are comparable to those shown
in Wheaton [22][23] and Brueckner [3].

4.1

An increase in the job search intensity

The results of comparative steady states regarding s are summarized in the following proposition.
12

The proof of Proposition 3, 4 and A1 are given in a technical appendix that is available upon request.

11

Proposition 3 The eﬀects of changes in s are summarized as follows:[Table 1 around here]

Henceforth, we use ‘+’ to represent a positive impact, ‘-’ a negative impact, ‘0’ no impact, and‘?’ an ambiguous impact. Now let’s explain the impacts on all variables.
In the spatial mismatch equilibrium, the eﬀects of an increase in the job search intensity s areshown in Fig. 2.

[Fig. 2 around here]

When s increases, employment in the city increases, which intensifies the competition for housingsb]. This can be confirmed in (16)in the employment zone, EZ, and raises Ωw (x) for x ∈ (0, xand (12). Facing a higher housing price, the profit-maximizing developers then provide more

housing. Therefore, land development in the EZ, gw (x), becomes larger. To produce more housing,developers increase the input of land. Hence, the land price in the EZ, Φw (x), also becomes higher.
With an increase in land input, the threshold between EZ and U Z (the unemployment zone) moves
from xb to a more distant location from the CBD, x
b0 , and the employment zone gets larger fromEZ to EZ 0 in Fig. 2.

The impact of s in the U Z on housing prices, Ωu (x), land prices, Φu (x), and housing supplies,
gu (x), is ambiguous. Using (12), it can be shown that the housing price curve in the U Z moves up
and rotates in a clockwise direction with a rising s. The eﬀect of s on Ωu (x) depends on its impact
on the size of the city, x. If x decreases with the increase of s, then housing prices, land prices,and housing supplies increase in the area close to the EZ. In the area close to the edge of the city,Ωu (x), Φu (x), and gu (x) decrease. However, if x becomes larger with an increase in s, then Ωu (x),Φu (x), and gu (x) increase at each location in the U Z. In Fig. 2, we only show the case in which x
becomes smaller.
Although the comparative static analysis is complicated in the U Z, the intuition is simple. An
increase in s means that an unemployed worker increases the frequency of commuting to the CBD.
To save commuting costs, she/he prefers a closer location to the EZ. As a result, housing prices insuch locations are bid up, which in turn encourages developers to increase housing supplies there.
Land prices in these locations also become higher. In the locations close to the city fringe, an
12unemployed worker would like to propose a higher bid rent if and only if the city becomes larger
and commuting costs become higher.
Now consider the impacts on lifetime utilities of employed workers and unemployed workers,
W and U. From (10), we can show that an increase in s reduces the instantaneous utility of
employed workers by increasing housing prices in the EZ. However, its eﬀect on the expected lossof becoming unemployed (U − W ) is ambiguous. The net eﬀect on W is then ambiguous. For
unemployed workers, the eﬀects of s on their instantaneous utility and the expect gain from being
employed in future are ambiguous. Hence, the net eﬀect on U is ambiguous.
In the integrated equilibrium, with an increase in s, the housing price curve moves up androtates in a clockwise direction, which then leads to an ambiguous impact on Ωu (x), Φu (x), gu (x),and xb. Intuitively, an increase in s raises the number of employed workers, which raises the demands

for housing in the EZ. Then Ωw (x), Φw (x), and gw (x) become larger. It can be shown that thecity becomes larger when s increases.

4.2

An increase in income

The next proposition summarizes the results of comparative steady states regarding w and b.
Proposition 4 The eﬀects of changes in w and b are summarized as follows:[Table 2 around here]

An increase in w
The eﬀects of an increase in the wage rate w in the spatial mismatch equilibrium are described inFig. 3.

[Fig. 3 around here]

Using (16), it can be shown that an increase in w has no impact on the housing price at x = 0 andleads to a larger x. Changes in the housing prices associated to a rise in w can be explained byusing (12): in the EZ, an increase in w makes the housing price curve rotate in a counterclockwise
direction around its intercept at the vertical axis. In the unemployment zone (U Z), it moves the13housing price curve up and then rotates it in a clockwise direction. In the appendix, we show thatthe threshold x
b becomes larger as w increases. Intuitively, a higher wage raises the housing demandsof employed workers, which then increases housing prices in the EZ, Ωw (x). Facing higher housing
prices, developers would like to supply more housing to the market. To do so, they need to increase
the input of land and propose higher bids for land. Hence, both gw (x) and Φw (x) increase. The

EZ is also extended to EZ 0 .
From (12), we see that w has no direct impact on housing prices in the U Z. However, it stillaﬀects Ωu (x) indirectly through its eﬀects on the size of the city. With a rising w, x becomes larger
and the city expands to x0 . It implies that each commuting trip to the CBD becomes more expensive
for unemployed workers. To reduce expenditures on commuting, unemployed workers located in
the U Z prefer apartments close to the EZ. Therefore, they propose higher bids for apartments inthe area near the EZ. In the technical appendix, we show that an increase in w leads to an increasein Ωu (x) at all locations in the U Z. On the supply side, as those in the EZ, developers increase
the housing supply gu (x) and raises their bids for land Φu (x).
The impacts of w on workers’ lifetime utilities, W and U, are ambiguous. An increase of w has
two opposite eﬀects on W (U ). On the one hand, a higher w leads to an increase (decrease) of
the employed (unemployed) workers’ instantaneous utility, which has a positive (negative) impact
on W (U ). One the other hand, it increases the diﬀerence W − U, which is a loss (gain) for theemployed (unemployed) workers in future. As a result, if the job destruction rate δ is low, anincrease in w raises the employed workers’ lifetime utility W. If the job acquisition rate sλ is high,an increase in w also raises the unemployed workers’ lifetime utility.
In the integrated equilibrium, a change in w doesn’t aﬀect unemployed workers’ housing demand,b is unaltered. In the EZ,so Ωu (x), Φu (x), and gu (x) in the U Z are not aﬀected. As a result, x
housing prices Ωw (x) increases with a rising w. According to (2), (3) and (22), gw (x), Φw (x), andx also increase. It can be shown that an increase in w also raises W and U.
An increase in b
In the spatial mismatch equilibrium, an increase in the unemployment benefit b has no impacts
on Ωw (x), gw (x), Φw (x), and x. This is because of our simple market, in which unemployment isfrom search frictions and unemployment benefits have no impact on employment. In the U Z, an
increase in b rotates unemployed workers’ bid rent curve in a counterclockwise direction around theintersection of two bid-rent curves. Hence, it raises Ωu (x), which in turn raises Φu (x), and gu (x).
14The city also becomes larger. As for U, an increase in b has a positive impact on an unemployedworker’s instantaneous utility and a negative impact on her expected gain from being employed.
However, the net eﬀect is always positive. Finally, although b has no impact on an employedworker’s instantaneous utility, an increase of b still increases W by decreasing the expected loss
from the change of her economic status.
In the integrated equilibrium, the impact of b in the integrated city is similar to the impact of
w in the spatial mismatch equilibrium. Applying the same argument, we show that an increaseb and x. The impacts on W and U arein b increases Ωu (x), Ωw (x), Φu (x), Φw (x), gu (x), gw (x), x
ambiguous.

5

Policy issues

With the results of comparative steady states in hand, we move to the analysis of policy eﬀects
on housing development and welfare in this economy. Here, we consider three policies: land development tax, transportation policy, and income transfer. In so doing, we relate the performanceof each policy to the search intensity of unemployed workers. Because the model in the previous
section is already complex and becomes intractable once we introduce these policies, we resort tonumerical analysis in this section.
The values of parameters we use are as follows. The wage rate of employed workers is w = 30
and the value of leisure is b = 10. Therefore, we have a spatial mismatch equilibrium when s ≤ 1/3
and an integrated equilibrium when s > 1/3. The job finding rate is λ = 2/5 while the job
destruction rate is δ = 1/50. The discount rate is ρ = 0.05. Since the amount of non-land input
can be measured in any unit, we normalize r to 1. Commuting cost per unit of distance is t = 1/10.
The parameter in workers’ utility function is α = 0.15. In this paper, we set the search intensity
as s = 0.25(< 1/3) or s = 0.8(> 0.8). We can obtain the qualitatively same results under diﬀerent
values of s.13 In Table 3, we summarize the two equilibira under these parameter values.

[Table 3 around here]

Note here that the unemployment rate is higher in the spatial mismatch equilibrium (16.7%) where
we set the search intensity as s = 0.25 than in the integrated equilibrium (5.9%) where we set the13

The results under diﬀerent values of s are available upon request.

15

search intensity as s = 0.8. This is consistent with the conventional image that cities with spatial
mismatch suﬀer from unemployment more than cities without it.

5.1

Land development tax

We first consider the eﬀects of a land development tax. Consider a city government that collects
taxes from land development and redistributes them to consumers in a lump-sum fashion. The
main question is whether such development tax can possibly improve the welfare.
Denote by ηL the tax rate on housing development and by ζ the lump-sum transfer to a worker.
Then, the profit function of a developer at x becomes as¡¢Π(x) = 1 − η L R(x)S 1−γ − rS − φ(x).

By solving the profit maximization problem and using the housing production function, we obtainthe supply of housing as follows¸(1−γ)/γ¢(1 − γ) ¡L.
g(x) =
1 − η R(x)
r
∙

(23)

By using the zero-profit condition, we obtain the following bid-rent function
¶
µ
¢
¤1/γ
1 − γ (1−γ)/γ £¡1 − η L R (x)Φ(x) = γ.
r(24)

We can see that the development tax reduces the supply of housing service at each location x given

housing price R(x). However, as shown in the following analysis, the development tax also indirectly
aﬀects housing supplies through its eﬀects on housing prices. Assume that tax revenues are evenly
redistributed among residents by the city government. The lump-sum subsidy ζ L is determined by
the city government’s budget constraint
Z x
L
R (x) g(x)dx = ζ L .
η
0With the lump sum transfer ζ L , the budget constraints of the employed workers and the unemployedworkers are respectively given byw + ζ L = z + R (x) h + tx,
b + ζ L = z + R (x) h + stx.
Now we can derive the equilibrium under land development policy. As before, we have two
diﬀerent equilibria. In the case of the spatial mismatch equilibrium, the housing prices at x = 0
16and x are given byhiγ¡¢−(1−γ)/γt (e + su) ,(25)Ωw (0) = 1 + φ 1 − η L¢¡
s w + ζLb − sw + (1 − s) ζ L
iαγ + hiαγ .
b + ζ L − stx = h
1 + φ (1 − ηL )−(1−γ)/γ t (su + e)
1 + φ (1 − η L )−(1−γ)/γ tsu

Substituting (25) into (12), (11), (13) and replacing w by w + ζ L and b by b + ζ L , we obtain theb when there is a land development policy. In
equilibrium solutions of Ωw (x) , Ωu (x) , R (x) , and x
the integrated equilibrium, the housing price at x = 0 and x are equal tohiγ¢−(1−γ)/γ¡Ωu (0) = 1 + φ 1 − η Lt (e + su) ,
(26)
⎫
⎧
⎬
1⎨
sw − b − (1 − s) ζ L
b + ζLhiαγ + hiαγ .
w + ζ L − tx =⎭s ⎩ 1 + φ (1 − η L )−(1−γ)/γ t (su + e)
1 + φ (1 − ηL )−(1−γ)/γ teb can be obtained by substituting
Similarly, the equilibrium solutions of Ωw (x) , Ωu (x) , R (x) , and x(26) into (19), (17), (20) and then replacing w by w + ζ L and b by b + ζ L .

Using the parameters given above, we examine the eﬀect of land development tax by evaluating
the derivatives of endogenous variables with respect to ζ L at ζ L = 0. If these derivatives are
positive, the land development tax aﬀects endogenous variables positively. In Table 3, we report
the eﬀects of land development tax on asset values (W , U , and SW ≡ eW + uU ), total land rent
(T LR), land development (g(x)), and threshold locations (bx and x) for two values of job searchintensity, s.

[Table 4 around here]

The first and second columns show the eﬀects of development tax on the spatial mismatch andintegrated equilibrium, respectively. The impacts of land development policy are complicated. On
the one hand, the subsidy ζ L increases w and b. According to the results in Proposition 4, anincrease in w has ambiguous impacts on (resp. increases) W and U while an increase in b raises(resp. has ambiguous impacts on) W and U in the spatial mismatch equilibrium (resp. in the
integrated equilibrium). On the other hand, the tax rate ηL has a direct influence on housing
prices, which aﬀects workers’ utilities through changing their housing demands. As shown in Table
4, the land development policy raises W, U, and SW for both types of equilibrium.
Table 4 also shows the impacts on T LR received by landlords, or the leakage from urban welfare.
We found that the tax on land development reduces the leakage for all levels of s. In the spatial
17

mismatch equilibrium, on the one hand, according to Proposition 4, an increase in w leads to ahigher Ωw (x) and a higher Ωu (x) while an increase in b has no impact on Ωw (x) and positive
impacts Ωu (x). In the integrated equilibrium, the eﬀects of w and b get reversed. By increasing
w or b, an increase in η L raises land rents at all locations across the city. On the other hand, as
shown in (24), an increase in ηL reduce land rents directly. Our numerical analysis shows that thenegative eﬀect of an increase in η L dominates its positive eﬀect for both types of equilibrium.
Let us now study housing supply g (x) expressed by (23). Our focus is on housing supplies in
the center of the EZ and the U Z. As in the analysis of the eﬀect on T LR, an increase of ηL has
two opposite eﬀects on housing supplies. The result of our numerical analysis indicates that in thespatial mismatch equilibrium, the positive eﬀect dominates the negative eﬀect in the center of the
U Z while the latter dominates the former in the center of the EZ. The development tax encourages
land development in the suburbs whereas it discourages land development in the center of the city.
Put diﬀerently, land development tax makes the city spatially dispersed in the spatial mismatch
equilibrium. Unlike in the spatial mismatch equilibrium where the EZ is close to the CBD, it is
the U Z that is close to the CBD in the integrated equilibrium. In such a case, the marginal impactof η L on housing supply in the center of the U Z is negative while it is positive in the center of theEZ.
The city dispersion is confirmed by the results regarding x
b and x: the land development tax

encourages developers to develop more housings at locations far from the CBD and raises x
b and xfor both types of equilibrium although the impacts on x
b are much smaller than those on x.

5.2

Transportation policy

The second policy to be considered is to reduce the commuting costs in the city. Suppose that thecity government collects taxes ηT from each worker and use them as investment in transportation
infrastructure in order to improve the commuting traﬃc conditions. With the transportation policy,
the commuting cost per unit t is assumed to be reduced to t/(1 + ηT ). Then, an employed worker’s
budget constraint can be written as
w − η T = z + R (x) h +tx,
1 + ηT

while an unemployed worker has the following budget constraintb − η T = z + R (x) h +
18

stx
.
1 + ηTReplacing w by w − ηT , b by b − η T , and t by t/(1 + η T ) in the equilibrium conditions that
we have solved in Section 3, we then obtain the spatial mismatch equilibrium and the integrated
equilibrium under the transportation policy. Using the parameters given above, let’s show how theinvestment to transportation infrastructure influences the equilibria.
The third and fourth columns of Table 4 report the results. Again, they can be explainedusing the results of comparative steady states. Indeed, an increase in η T reduces w, b, and t.
According to Proposition 4, a decrease in w has ambiguous impacts on (resp. decreases) W and U
while a decrease in b reduces (resp. has ambiguous impacts on) W and U in the spatial mismatch
equilibrium (resp. in the integrated equilibrium). As described in Appendix, a decrease in thecommuting cost, t, leads to a rise in W and U. The net eﬀects on W , U , and SW are thenambiguous. However, our numerical analysis shows that the positive eﬀects dominate the negativeeﬀects, and the transportation policy raises W , U , and SW for both types of equilibrium.
The transportation policy aﬀects T LR through reducing workers’ income and commuting costs.
We have shown that a decrease in w (resp. b) reduces housing prices across the city in the spatial
mismatch equilibrium (resp. in the integrated equilibrium) while a decrease in b (resp. w) only
reduces housing prices in the U Z (resp. EZ) in the spatial mismatch equilibrium (resp. in the
integrated equilibrium). According to (3), land rents are aﬀected in the same way. We have also
shown in Appendix that a decrease in t ambiguously aﬀects housing prices and hence, T LR. As aresult, the net eﬀects of η T are ambiguous. Still, in our numerical analysis, ηT decreases T LR.
Similarly, η T has ambiguous eﬀects on housing supply g (x) . However, our numerical exercise
indicates that an investment in transportation infrastructure induces developers to supply lesshousing.
b and x are also ambiguous. On the one hand, a decrease in w
Theoretically, the impacts of η T on x

(resp. b) decreases x
b and x in the spatial mismatch equilibrium (resp. in the integrated equilibrium)

while a decrease in b (resp. w) only decreases x in the spatial mismatch equilibrium (resp. in the
integrated equilibrium). On the other hand, a decrease of commuting cost has ambiguous impacts

on xb and x. Hence, the net impact is not clear. However, the transportation policy increases bothbxb and x in a similar way in the spatial mismatch equilibrium whereas it increases x more than x
in the integrated equilibrium. This implies that the transportation policy enlarges EZ.

19

5.3

Transfer to unemployed workers

We finally consider a policy that aims at subsidizing unemployed workers. Suppose that the subsidy
to unemployed workers is financed by taxes on employed workers. Denote by ηw the tax rate andby ζ w the subsidy rate. Then an employed worker’s budget constraint can be written as
w − η w = z + R (x) h + tx,
whereas an unemployed worker faces the following budget constraintb + ζ w = z + R (x) h + stx.
In equilibrium, the revenue of the city government is equal to its expenditure
ζw =

eη w
.
u

The model is exactly as before but we now replace w by w − η w and b by b + eηw /u. If η w = 0, wethen goes back to the model introduced in Section 2. Now let’s analyze the impact of an increasein s on the equilibrium outcomes when there is a transfer policy.
The eﬀects on the spatial equilibrium outcomes are shown in the fifth and sixth columns. The
transfer from employed workers to unemployed workers reduces w and raises b. The comparative
steady states show that the net impacts on W and U are ambiguous. Here, we find that an increasein ηw raises W, U, and SW . Similarly, the comparative steady states show that the net eﬀectof transfer on T LR is analytically ambiguous. However, our numerical analysis finds negligible
impacts of η w on T LR in the spatial mismatch equilibrium and negative eﬀects in the integrated
equilibrium.
Finally, let us analyze the impacts on the city structure. The transfer from employed workers
to unemployed workers reduces housing demands of employed workers and increases that of unemployed workers. However, the housing supply is mostly unaﬀected. Hence, the EZ shrinks andthe U Z expands. This is represented by decreases in x
b in the spatial mismatch equilibrium and byincreases in it in the integrated equilibrium.

6

Concluding remarks

This paper developed a monocentric city model with search frictions in a labor market and development in a land market. Our model captures some important features in the real world cities:
20

in equilibrium, we obtain two diﬀerent spatial configurations, a segregated city and an integrated
city, which are roughly correspond to “old cities” and “new or edge cities” in the US (Wasmer andZenou [21]). We then explore the interactions between the labor market and the land (housing)market. We also investigated how policies that are likely to improve welfare are associated toland development and city structure. Analysis showed that the eﬀects of policies on development
patterns and on land rents (welfare of landlords) can be quite diﬀerent between the two spatialconfigurations.
To better understand cities in the real world, some extensions of our model could be made. For
example, a detailed labor market structure with endogenous search frictions could be introducedin the monocentric model. Another possible extension is to relax the assumption of monocentriccity and consider a city with a subcenter.

References[1] Alonso, W., 1964, Location and Land Use, Cambridge: Harvard University Press.
[2] Anderson, J.E., 1993, Land Development, Externalities, and Pigouvian Taxes, Journal of
Urban Economics 33(1),1-9.
[3] Brueckner, J.K., 1987, The Structure of Urban Equilibria: A Unified Treatment of the MuthMills Model, ch. 20, 821-845 in: E.S. Mills, eds., Handbook of Regional and Urban Economics,
vol. 2, Elsevier.
[4] Coulson, E., D. Laing, and P. Wang, 2001, Spatial Mismatch in Search Equilibrium, Journalof Labor Economics 19, 949-972.
[5] Fujita, M., 1989, Urban Economic Theory, Cambridge: Cambridge University Press.
[6] Hartwick, J., U. Schweizer, and P. Varaiya, 1976, Comparative Static Analysis of Residential
Economy with Several Classes, Journal of Economic Theory 13, 396-413.
[7] Holzer, H., 1991, The Spatial Mismatch Hypothesis: What Has the Evidence Shown?, Urban
Studies 28, 105-122.
[8] Ihlanfeldt, K. and D. Sjoquist, 1998, The Spatial Mismatch Hypothesis: A Review of Recent
Studies and Their Implications for Welfare Reform, Housing Policy Debate 9, 849-892.
21

[9] Jencks, C. and S. Mayer, 1990, Residential Segregation, Job Proximity, and Black Job Opportunities, in: L.E. Lynn and M.G.H. Mcgeary, eds., Inner-city Poverty in the United States,
187—222. Washington, DC: National Academy Press.
[10] Kain, J.F., 1968, Housing Segregation, Negro Employment, and Metropolitan Decentralization,
Quarterly Journal of Economics 82, 175-797.
[11] Kain, J.F., 1992, The Spatial Mismatch Hypothesis: Three Decades Later, Housing Policy
Debate 3, 371-460.
[12] Kanemoto, Y., and K. Tokuoka, 2002, The Proposal for the Standard Definition of theMetropolitan Area in Japan [In Japanese.], Journal of Applied Regional Science 7, 1—15.
[13] Kawata, K., 2012, Explicit Spatial Friction and Worker Heterogeneity in Urban Searchmatching Model, mimeo.
[14] Mills, E.S., 1967, An Aggregate Model of Resource Allocation in a Metropolitan Area, American Economic Review, Papers and Proceedings 57, 197-210.
[15] Muth, R.F., 1969, Cities and Housing, Chicago: University of Chicago Press.
[16] Pissarides, C.A., 2000, Equilibrium Unemployment Theory, Second Edition, M.I.T Press, Cambridge.
[17] Rossi-Hansberg, E., P.-D. Sarte, and R. Owens, 2010, Housing Externalities, Journal of Political Economy 118, 485-535.
[18] Sato, Y., 2001, Labor Heterogeneity in an Urban Labor Market, Journal of Urban Economics
50, 313-337[19] Sato, Y., 2004, City Structure, Search, and Workers’ Job Acceptance Behavior, Journal of
Urban Economics 55, 350-370.
[20] Smith, T.E. and Y. Zenou, 2003, Spatial Mismatch, Search Eﬀort, and Urban Spatial Structure,
Journal of Urban Economics 54, 129-156.
[21] Wasmer, E. and Y. Zenou, 2002, Does City Structure Aﬀect Job Search and Welfare?, Journalof Urban Economics 51,515-541.

22

[22] Wheaton, W.C., 1974, A Comparative Static Analysis of Urban Spatial Structure, Journal of
Economic Theory 9, 223-237.
[23] Wheaton, W.C., 1976, On the Optimal Distribution of Income among Cities, Journal of Urban
Economics 3, 31-44.
[24] Xiao, W., 2012, Search Frictions, Unemployment, and Housing in Cities: Theory and Policies,”
mimeo.
[25] Zenou, Y., 2000, Unemployment in Cities. in: J.-M. Huriot and J.-F. Thisse, eds., Economics
of Cities. Theoretical Perspectives, Cambridge: Cambridge University Press, ch. 10, 343-389.
[26] Zenou, Y., 2009, Urban Labor Economics, Cambridge: Cambridge University Press.

Appendix: Full Results of Comparative Steady States
In this appendix, we explain the results of basic comparative steady states analysis regarding t,
δ, and λ. The goal of our analysis is to deduce the impacts of these parameters on the spatialb), housing prices (Ω0 (x), Ωw (x), and Ωu (x)), land prices (Φw (x) andstructure of the city (x and x

Φu (x)), land development (gw (x) and gu (x)), employed and unemployed workers’ lifetime utilities(W and U ).
The next proposition summarizes the results of comparative steady states in the spatial mismatch equilibrium and integrated equilibrium.
Proposition A1 The eﬀects of changes in t, δ, and λ are summarized as follows:WUΩw (x)Ωu (x)R(0)Φw (x)

Φu (x)xxb

gw (x)

gu (x)

spatial mismatch equilibrium
t

-

-

?

?

+

?

?

?

?

?

?

δ

?

?

-

?

-

-

?

?

-

-

?

λ

?

?

+

?

+

+

?

?

+

+

?

integrated equilibrium
t

-

-

?

?

+

?

?

?

?

?

?

δ

?

?

-

-

-

-

-

-

+

-

-

λ

?

?
+
+
+
+
+
+ +
Table A1. Comparative steady states regarding t, δ, and λ+23

An increase in t
We start from the spatial mismatch equilibrium. In the EZ, an increase in the commuting cost t
aﬀects housing prices Ωw (x) in two diﬀerent ways. On the one hand, it intensifies the competition
for housing at x = 0, which then increases Ωw (0), the housing price in the CBD. According to (12),an higher Ωw (0) moves the housing price curve in the EZ up. On the other hand, the housing
b isprice curve rotates in a clockwise direction with a rising t. Therefore, the impact on Ωw (x) and x

ambiguous. However, we still can conclude that an increase in t at least raises the prices of housing
at locations near the CBD. In turn, housing supplies and land prices in this area also increase.
Intuitively, when t increases, the commuting trip becomes more expensive for employed workers.
To reduce their commuting costs, employed workers prefer to live close to the CBD. Therefore,
they propose higher bids for apartments in these locations. Housing supplies and land prices inthese locations also increase with a rising t. In the U Z, the bid rent curve of unemployed workers
also moves up and rotates in a clockwise direction with a rising t. Thus, the impacts on Ωu (x),gu (x), and x are ambiguous. Concerning the utility level of workers, we show that an increase ofcommuting costs always reduces W and U .
An increase in t in the integrated equilibrium has the same impacts as in the spatial mismatch
equilibrium.
An increase in δ and λFinally, take another two labor market parameters δ and λ. In the spatial mismatch equilibrium, anincrease in job destruction rate δ leads to a decrease of employment, which weakens the competition
for housing in the EZ and reduces housing prices Ωw (x). In turn, housing supplies, gw (x), landb, also become smaller with a rising δ. From (16), we see thatprices, Φw (x), and the size of EZ, x

the eﬀect of δ on x is ambiguous. According to (19), (2), and (3), the eﬀects of δ on housing prices,

housing supplies, and land prices in the U Z are also ambiguous due to the indeterminate change ofx. The impact of λ is just the opposite of the impact of δ. Both δ and λ have ambiguous impacts
on W and U .
In the integrated equilibrium, an increase in δ leads to a decrease in the total number of employedworkers and searchers, e + su, which then reduces the housing price at x = 0. Using (19), (2) and(3), we show that Ωu (x), Φu (x), and gu (x) decrease. An increase in the number of unemployedworkers increases the size of U Z, x
b. According to (22), however, its impact on x is negative, which24

in turn decreases Ωw (x), Φw (x), and gw (x). As in the spatial mismatch equilibrium, the changes
following an increase of λ are just the reverse of the impacts of an increase in δ.

25

W

U

Ωw (x)Ωu (x)R(0)Φw (x)

Φu (x)xxb

gw (x)

gu (x)

?

+

+

?

+

?

+

?

spatial mismatch equilibrium
s

?

?

+

?

+

+

?

integrated equilibrium
s

?

?

+

?

+

+

?

Table 1. Comparative steady states regarding sWUΩw (x)Ωu (x)R(0)Φw (x)

Φu (x)xxb

gw (x)

gu (x)

spatial mismatch equilibrium
w

?

?

+

+

0

+

+

+

+

+

+

b

+

+

0

+

0

0

+

+

0

0

+

integrated equilibrium
w

+

+

+

0

0

+

0

+

0

+

0

b

?

?

+

+

0

+

+

+

+

+

+

Table 2. Comparative steady states regarding w and b

spatial mismatch (s = 0.25)

integrated (s = 0.8)

Asset value of employed (W )

56.55

57.95

Asset value of unemployed (U )

50.12

55.14

Social welfare (SW )

55.48

57.79

Unemployment rate (u)

0.167

0.059

Housing supply of EZ (gw (x))

0.54

0.54

Housing supply of UZ (gu (x))

0.50

0.59

Table 3. Comparison between the spatial mismatch equilibrium and integrated equilibrium
Notes: Housing supplies, gw (x) and gu (x) are evaluated at the centers of EZ and UZ, respectively.

26

27

28

29

Technical Appendix (NOT FOR PUBLICATION)
Yasuhiro Sato∗and Wei Xiao†
June 25, 2013This appendix presents the proof of the results summarized in Proposition 3, 4 and A1. We
start from the spatial mismatch equilibrium.

1

The spatial mismatch equilibrium

1.1

Search intensity

Eﬀect on xb and xUsing (16), we getα(b − stx) Ωw (0) = sw + (b − sw)µ

1 + φt (su + e)
1 + φtsu¶αγ.

Plugging the above equation into (13) yields
½
∙
¸αγ ¾
1 + φtsu
w
1−
.
x
b=
t
1 + φt (su + e)

(A1)

Let

Γ≡

1 + φtsu.
1 + φt (su + e)

We then havexb=w
(1 − Γαγ ) .
tsλδPlugging e = δ+sλand u = δ+sλ
into the expression of Γ and diﬀerentiating it with respect to s

yieldsδλ∗†φt (δ+sλ)2∂Γ= −hi < 0,∂ss(δ+λ) 2
1 + φt δ+sλ

Graduate School of Economics, Osaka University, email: ysato@econ.osaka-u.ac.jp
Corresponding author, Department of Economics, Stockholm University, email: wei.xiao@ne.su.sexwhich implies ∂
∂s > 0.

LetΛ≡b − sw
sw+.
[1 + φt (su + e)]αγ[1 + φtsu]αγ(A2)Diﬀerentiating (16) with respect to s yields
∙
¸
1 ∂Λ
∂x
=−
+ tx ,
∂s
st ∂s
where
∂Λ
∂s

∙
¸
¸−αγ
∙
s (δ + λ) −αγsδ= w 1 + φt
− w 1 + φt
δ + sλδ + sλ⎧⎫⎪⎪⎬αγφtδ ⎨(b − sw) δ
sw (δ + λ)
−
+
h
i
h
i
αγ+1
αγ+1 ⎪(δ + sλ)2 ⎪sδ⎩ 1 + φt s(δ+λ)⎭
1 + φt δ+sλ
δ+sλ
⎫
⎧
⎬
⎨
1
1
h
i
h
i
< w
−
αγ
⎭
⎩ 1 + φt s(δ+λ) αγ
1 + φt sδ
δ+sλ

δ+sλ

< 0.

Therefore, the sign of ∂x∂s is indeterminate.
Eﬀect on Ωw (x), Φw (x) and gw (x)w (0)> 0. Then diﬀerentiating (12) with respect to s
Using (16), it is easy to show that ∂Ω∂s
w (0)
w (x)
> 0 in the result, we have ∂Ω∂s
> 0, which together with (2) and (3) implies
and using ∂Ω∂s∂gw (x)w (x)> 0 and ∂Φ∂s
> 0 for x ∈ (0, xb].
∂s

Eﬀect on Ωu (x), Φu (x) and gu (x)Diﬀerentiating (12) yields

1111∂Ωu (x)tx∂ (b − stx)
1
= − (b − stx) α −1 (b − stx)− α − (b − stx) α (b − stx)− α −1
,
∂s
α
α
∂s

∂Ωu (x)where ∂(b−stx)= ∂Λ
is indeterminate. In turn, the impact of s∂s∂s < 0. Therefore, the sign of∂son Φu (x) and gu (x) is also ambiguous.
Eﬀect on W and U
Plugging (12) into (10) yields
ρWρU

α (ρ + sλ)δρ+δln w −
ln Ωw (0) +ln(b − stx),
ρ + δ + sλ
ρ + δ + sλ
ρ + δ + sλ
αsλ
ρ+δ
sλ
ln w −
ln Ωw (0) +ln(b − stx).
= B+ρ + δ + sλ
ρ + δ + sλ
ρ + δ + sλ

= B+

∂U
The sign of ∂W∂s and ∂s is then indeterminate.

(A3)
(A4)

1.2

Wage

Eﬀect on xb and xDiﬀerentiating (A1) with respect to w, we obtain1 − Γαγ
∂b
x
=
> 0.
∂w
t

Diﬀerentiating (16) with respect to w yields
½
¾
1
1
1
∂x
=−
> 0.
−
∂w
t [1 + φt (su + e)]αγ[1 + φtsu]αγ
Eﬀect on Ωw (x), Φw (x) and gw (x)Diﬀerentiating (12) with respect to w yieldsΩw (0) tx∂Ωw (x)
=
∂w
α w2

µ

w − txw¶1/α−1

> 0.

w (x)
w (x)
We then have ∂Φ∂w
> 0 and ∂g∂w
> 0.

Eﬀect Ωu (x), Φu (x) and gu (x)Diﬀerentiating (12) with respect to w yields
1∂x∂Ωu (x)st= (b − stx)1/α (b − stx)− α −1
,
∂w
α
∂w
u (x)
u (x)
u (x)
∂x
> 0. We then have ∂Ω∂w
> 0. In turn, we have ∂Φ∂w
> 0 and ∂g∂w
> 0.
where ∂w

Eﬀect on W and U
Diﬀerentiating (A3) and (A4) yields
ρ

ρ+δ
1
δ
∂ (b − stx)
∂W
=
+
,
∂w
ρ + δ + sλ w ρ + δ + sλ
∂w

ρ

∂U
sλ
1
ρ + δ ∂ (b − stx)=+,∂w
ρ + δ + sλ w ρ + δ + sλ∂w

∂U
where ∂(b−stx)< 0. Therefore, the sign of ∂W∂w∂w and ∂w is then indeterminate.

1.3

Unemployment benefits

Eﬀect on xb and x

From (A1), it can be seen that b has no impact on xb.
Diﬀerentiating (16) yields

1
∂x
=
∂b
st

µ
¶
∂Λ
1−
,
∂b

where
∂Λ
1
=
< 1.
∂b
(1 + φtsu)αγ
Therefore, we have ∂x∂b > 0.
Eﬀect on Ωw (x), Φw (x) and gw (x)From (12), we see that b has no impact on Ωw (x). Then it has no impact on Φw (x) and gw (x)either.
Eﬀect on Ωu (x), Φu (x) and gu (x)Diﬀerentiating (12) yeilds∂Ωu (x)
∂b

=
>

∙
¸
1
1
b − stx
1−b − stx(1 + φtsu)αγ b − stxµ¶1/α−1∙¸
1b − stbx
1 b − stx
1
1−
α b − stxb − stx(1 + φtsu)αγ b − stx1αµb − stxb − stx¶1/α−1u (x)for x > xb. Plugging (A1) into the above inequality, we obtain ∂Ω∂b
> 0. In turn, we have∂Φw (x)u (x)> 0 and ∂g∂b
> 0.
∂b

Eﬀect on W and U
Diﬀerentiating (A3) and (A4) yieldsρδ
1 ∂ (b − stx)
∂W
=
,
∂b
ρ + δ + sλ b − stx
∂b

ρ

∂U
ρ+δ
1 ∂ (b − stx)=.
∂b
ρ + δ + sλ b − stx∂b∂USince ∂(b−stx)> 0, we have ∂W∂b∂b > 0 and ∂b > 0.

1.4

Commuting cost

Eﬀect on xb and xDiﬀerentiating (A1) with respect to t, we obtain

where

∂b
x
w (1 − Γαγ ) wαγΓαγ−1 ∂Γ
=−
,
−
∂t
t2
t
∂t
sλ
φ δ+sλ
∂Γ
=−
< 0.
∂t
[1 + φt (su + e)]2x
Therefore, the sign of ∂∂t is indeterminate.

Diﬀerentiating (16) with respect to t yields
∙
¸
1 ∂Λ
∂x
=−
+ sx ,∂tst ∂twhere∂Λ∂tno= −αγ sw [1 + φt (su + e)]−αγ−1 φ (su + e) + (b − sw) [1 + φtsu]−αγ−1 φsu< 0.

Therefore, the sign of ∂x∂t is indeterminate.
Eﬀect on Ωw (x), Φw (x) and gw (x)Diﬀerentiating (12) with respect to t, we obtainµ¶¶µw − tx 1/α−1∂Ωw (0) w − tx 1/αt∂Ωw (x)=Ωw (0)
−
,
∂t
∂t
w
αw
w
w (0)
w (x)
> 0. Therefore, the sign of ∂Ω∂t
is indeterminate. The impact on Φw (x) and gw (x)where ∂Ω∂tis then ambiguous.
Eﬀect on Ωu (x), Φu (x) and gu (x)Diﬀerentiating (12) with respect to t yields
"
#
1
α
1
1
∂Ωu (x)
1
∂
(b
−
stx)
(b
−
stx)
−α
−1
=−
(b − stx) (b − stx) α sx +
,
1
∂t
α
∂t
(b − stx) α +1u (x)< 0. Therefore, the sign of ∂Ω∂t
is indeterminate. The sign of Φu (x) and gu (x) is
where ∂(b−stx)∂tindeterminate either.
Eﬀect on W and U
Diﬀerentiating (A3) and (A4) with respect to t yieldsρα (ρ + sλ) 1 ∂Ωw (0)δ
1 ∂(b − stx)
∂W
=−
+
∂t
ρ + δ + sλ Ωw (0) ∂t
ρ + δ + sλ b − stx
∂t

ρ

αsλ
1 ∂Ωw (0)ρ+δ
1 ∂(b − stx)
∂U
=−
+
∂t
ρ + δ + sλ Ωw (0) ∂t
ρ + δ + sλ b − stx
∂t

w (0)
∂U
Note that ∂Ω∂t
> 0 and ∂(b−stx)< 0. Therefore, we have ∂W∂t∂t < 0 and ∂t < 0.

1.5

Job destruction

Eﬀect on xb and xDiﬀerentiating (A1) yields

wherewαγ αγ−1 ∂Γ
∂b
x
=−
Γ
,
∂δ
t
∂δ
sλ
sδ
1 + sφt δ+sλ
+ φt δ+sλ
∂Γ
sλ
= φt
> 0.
³
´
2
∂δ
(δ + sλ)2
1 + φt s(δ+λ)δ+sλxTherefore, we have ∂∂δ < 0.

Diﬀerentiating (16) yields
1 ∂Λ
∂x
=−
,
∂δ
st ∂δ
∂x
where the sign of ∂Λ∂δ is indeterminate. Therefore, the sign of ∂δ is indeteterminate either.

Eﬀect on Ωw (x), Φw (x) and gw (x)Diﬀerentiating (12) yields∂Ωw (x)∂δµ¶w − tx 1/α ∂Ω0 (x)
=
w
∂δ
µ
¶1/α
w − txs(1 − s)λ= −γ[1 + φt (su + e)]γ−1 φtw(δ + sλ)2
< 0

w (x)
w (x)
< 0 and ∂g∂δ
< 0 for s ∈ (0, 1).
for s ∈ (0, 1) . We then have ∂Φ∂δEﬀect on Ωu (x), Φu (x) and gu (x)Diﬀerentiating (12) yields∂Ωu (x)
1
=
∂δ
α

µ

b − stxb − stx¶1/α−1 ∙¸∂ (b − stx)b − stx.
−2∂δ(b − stx)

u (x)
u (x)
is ambiguous, the sign of ∂Ω∂δis indeterminate. The sign of ∂Φ∂δandSince the sign of ∂(b−stx)∂δ∂gu (x)is indeterminate either.
∂δEﬀect on W and U
The impact of δ on W and U is ambiguous.

1.6

Job acquisition

Eﬀect on xb

Diﬀerentiating (A1), we get
∂Γ
∂b
x
w
= − αγΓαγ−1 ,
∂λ
t
∂λ

where

∂
x
> 0.
We then have ∂λφtsδ
1 + sφt∂Γ=−³´ < 0.
2∂λ(δ + sλ) 1 + φt s(δ+λ) 2
δ+sλEﬀect on Ωw (x), Φw (x) and gw (x)Diﬀerentiating Ωw (x) with respect to λ yields∂Ωw (x)∂λµ¶w − tx 1/α ∂Ω0 (x)
=
w
∂λ
¶1/α
µ
w − txs (1 − s) δ
= γ
[1 + φt (su + e)]γ−1 φt.
w(δ + sλ)2

w (x)
w (x)
w (x)
> 0 for s ∈ (0, 1) . Then, ∂Φ∂λ> 0 and ∂g∂λ
> 0.
Therefore, we have ∂Ω∂λEﬀect on x, Ωu (x), Φu (x), gu (x), W and U.
We find ambiguous impact of λ on x, Ωu (x), gu (x), W and U.

2

The integrated equilibrium

2.1

Search intensity

Eﬀect on xb and xUsing (22), we get(w − tx) Ωu (0)α =

1
s

½
µ
¶ ¾
1 + φt(su + e) αγb + (sw − b).
1 + φteInserting this equation into (19), we have
∙
µ
¶αγ ¸
1 + φte
b
1−
.
x
b=
st
1 + φt(su + e)

(A5)

sλ
δ
Plugging e = δ+sλand u = δ+sλ
into (A5) and diﬀerentiating it with respect to s leads to!αγ # Ã!αγ−1"
Ãαγφbδ 2sλsλ
1 + φt δ+sλ
1 + φt δ+sλ∂bxbs(δ+sλ)2=− 2 1−
+
h
i2 ,
∂s
s t
1 + φt s(δ+λ)
1 + φt s(δ+λ)
1 + φt s(δ+λ)
δ+sλ
δ+sλ
δ+sλ

x
which implies that the sign of ∂∂s is indeterminate.

Let
1
Υ≡
s

½

b
sw − bαγ +[1 + φt(su + e)](1 + φte)αγ

Diﬀerentiating (22) with respect to s leads to
1 ∂Υ
∂x
=−
,
∂s
t ∂s

¾

.

where
b
= − 2
s

∂Υ
∂s

+

½

b
sw − b
+[1 + φt(su + e)]αγ[1 + φte]αγ

δλ−αγwφt (δ+sλ)
2

¾

+

⎧
b⎨

δ(δ+λ)−αγφt (δ+sλ)2s ⎩ [1 + φt(su + e)]αγ+1+⎫δλ
αγφt (δ+sλ)2 ⎬[1 + φte]αγ+1 ⎭[1 + φte]αγ+1δ(δ+λ)
δλ½¾
αγφt (δ+sλ)
αγφt (δ+sλ)
2
2 sw − bbbbsw − b
−
= − 2
−
αγ +
αγ
s
[1 + φt(su + e)][1 + φte]s [1 + φt(su + e)]αγ+1 [1 + φte]αγ+1 s< 0.
Therefore, ∂x∂s is positive.
Eﬀect on Ωw (x), Φw (x), and gw (x)Diﬀerentiating (19) with respect to s, yieldsµ¶¸∙
1 w − tx 1/α−1(w − tx) ∂ (w − tx)∂Ωw (x)=,−∂s
α w − tx∂s(w − tx)2∂Ωw (x)w (x)w (x)where ∂(w−tx)= ∂Υ> 0. In turn, ∂Φ∂sand ∂g∂s
are positive.
∂s∂s < 0. Therefore, we have∂sEﬀect on Ωu (x), Φu (x), and gu (x)Diﬀerentiating (19) with respect to s, we getµ¶µ¶∂Ω0 (x) b − stx 1/α Ω0 (x) b − stx 1/α−1 tx∂Ωu (x)
=
,
−
∂s
∂s
b
α
b
b
where the first term on the right-hand side is positive while the second term is negative. Therefore,u (x)is indeterminate. The impact on Φu (x) and gu (x) is also ambiguous.
the sign of ∂Ω∂s

Eﬀect on W and U
Using (18), we getρW = B +αδρ + sλδln b −ln Ωu (0) +ln(w − tx),
ρ + δ + sλ
ρ + δ + sλ
ρ + δ + sλ

(A6)

ρU = B +α (ρ + δ)sλρ+δln b −ln Ωu (0) +ln(w − tx).
ρ + δ + sλ
ρ + δ + sλ
ρ + δ + sλ

(A7)

∂U
The sign of ∂W∂s and ∂s is indeterminate.

2.2

Wage

Eﬀect on xb and x

From (A5), it can be seen that w has no impact on x
b. Diﬀerentiating (22) yields
µ
¶
1
1
∂x
=
1−
∂w
t
(1 + φte)αγ∂xTherefore, we have ∂w> 0.

Eﬀect on Ωw (x), Φw (x) and gw (x)Diﬀerentiating (19) yields∂Ωw (x)
∂w

=
=
>

∙
¸
1
∂ (w − tx) w − tx
1−
w − tx∂w
w − txµ¶1/α−1∙¸
1
w − tx
1 w − tx
1
1−
α w − tx
w − tx(1 + φte)αγ w − txµ¶∙¸
1 w − tx 1/α−1
1w − tbx
11−
α w − tx
w − tx(1 + φte)αγ w − tx
1αµw − txw − tx¶1/α−1w (x)w (x)for x > xb. Plugging (A5) into the above inequality, we obtain ∂Ω∂w
> 0. Then, the sign of ∂Φ∂ww (x)and ∂g∂w
is positive.

Eﬀect on Ωu (x), Φu (x) and gu (x)w has no impact on Ωu (x), Φu (x), and gu (x).
Eﬀect on W and U
Diﬀerentiating (A6) and (A7) yields
ρρ + sλ1 ∂(w − tx)
∂W
=
,
∂w
ρ + δ + sλ w − tx
∂w

ρ

∂U
sλ
1 ∂(w − tx)=,∂w
ρ + δ + sλ w − tx∂w

∂U
where ∂(w−tx)> 0. Therefore, we have ∂W∂w∂w > 0 and ∂w > 0.

2.3

Unemployment benefit

Eﬀect on xb and x

From (A5), we see that b has a positive impact on xb.

Diﬀerentiating (22) yields

1
∂x
=
∂b
st

½

1
1
αγ −
(1 + φte)[1 + φt(su + e)]αγ¾,

which implies ∂x∂b > 0.
Eﬀect on Ωw (x), Φw (x) and gw (x)Diﬀerentiating (19) yields
1∂Ωw (x)=−∂bαµw − txw − tx¶1/α−1w − tx ∂ (w − tx),∂b(w − tx)2

where∂ (w − tx)
1
=
∂b
s

½

1
1
−
[1 + φt(su + e)]αγ(1 + φte)αγ

¾

< 0.

w (x)
w (x)
w (x)
Therefore, ∂Ω∂b
> 0. In turn, we have ∂Φ∂b
> 0, and ∂g∂b
> 0.

Eﬀect on Ωu (x), Φu (x) and gu (x)Diﬀerentiating (19) with respect to b yields∂Ωu (x)Ωu (0)=∂bαµb − stxb¶1/α−1

stx
.
b2

u (x)
u (x)
u (x)
> 0. In turn, we have ∂Φ∂b
> 0 and ∂g∂b
> 0.
Therefore, we have ∂Ω∂b

Eﬀect on W and U
Diﬀerentiating (A6) and (A7) with respect to b, we getρδ1
ρ + sλ1 ∂ (w − tx)
∂W
=
+
,
∂b
ρ + δ + sλ b ρ + δ + sλ w − tx
∂b

ρ

∂U
ρ+δ 1
sλ
1 ∂ (w − tx)=+.
∂bρ + δ + sλ b ρ + δ + sλ w − tx∂b∂U
Since ∂(w−tx)< 0, we conclude that the sign of ∂W∂b∂b and ∂b is indeterminate.

2.4

Commuting cost

Eﬀect on xb and xDiﬀerentiating (A5) with respect to t leads to
¶αγ ¸
¶αγ−1
∙
µ
µ
1 + φte1 + φte
b
∂b
x
αγbφu
=− 2 1−.
+∂t
st
1 + φt(su + e)t [1 + φt(su + e)]2 1 + φt(su + e)x
Therefore, the sign of ∂∂t is indeterminate.

Diﬀerentiating (22) with respect to t yields∂x
x 1 ∂Υ
=− −
,
∂t
t
t ∂t
∂x
where ∂Υ∂t < 0. Therefore, the sign of ∂t is indeterminate.

Eﬀect on Ωw (x), Φw (x), and gw (x)Diﬀerentiating (19) with respect to t yieldsµ¶∙¸∂Ωw (x)
1 w − tx 1/α−1w − tx ∂ (w − tx)
1
=−
x+
.
∂t
α w − tx
w − txw − tx∂t
Since ∂(w−tx)< 0, the sign of Ωw (x) is indeterminate. In turn, the impact on Φw (x) and gw (x) is∂t
indeterminate.

Eﬀect on Ωu (x), Φu (x), and gu (x)Diﬀerentiating (19) with respect to t yields∂Ω0 (x)∂Ωu (x)=∂t∂tµ

b − stxb¶1/αΩu (0)−αµb − stxb¶1/α−1

sx
.
b

u (x)
u (x)
u (x)
Therefore, the sign of ∂Ω∂t
is indeterminate. In turn, the sign of ∂Φ∂tand ∂g∂t
is indeterminate.

Eﬀect on W and U
Diﬀerentiating (A6) and (A7) with respect to t yields
ραδ
1 ∂Ωu (0)
ρ + sλ1 ∂(w − tx)
∂W
=−
+
,
∂t
ρ + δ + sλ Ωu (0) ∂t
ρ + δ + sλ w − tx
∂t

ρ

∂U
α (ρ + δ)
1 ∂Ωu (0)sλ1 ∂(w − tx)
=−
+
,
∂t
ρ + δ + sλ Ωu (0) ∂t
ρ + δ + sλ w − tx
∂t

u (0)
∂U
> 0 and ∂(w−tx)< 0. Therefore, we have ∂W
where ∂Ω∂t∂t∂t < 0 and ∂t < 0.

2.5

Job destruction

Diﬀerentiating (A5) with respect to δ yields
Ã
!αγ−1
sλ
1 + φt δ+sλ
αγbφsλ1 + φt∂bx=³´
2s(δ+λ)∂δ(δ + sλ) 1 + φt s(δ+λ) 2
1 + φt δ+sλ
δ+sλ
> 0.

Diﬀerentiating (22) with respect to δ yields∂x∂Ωu (0)
< 0,
< 0.
∂δ
∂δ
w (x)
Diﬀerentiating (19) with respect to δ and using the above results yields, we have ∂Ω∂δ
< 0 and∂Ωu (x)
w (x)
u (x)
w (x)
u (x)
< 0, which in turn imply ∂Φ∂δ
< 0, ∂Φ∂δ< 0, ∂g∂δ< 0, and ∂g∂δ< 0.
∂δEﬀect on W and U
The eﬀects of δ on W and U are ambiguous.

2.6

Job asquisitionsλδ
Plugging e = δ+sλand u = δ+sλ
into (A5) and diﬀerentiating it with respect to λ yields

∂b
x
∂λ

= −
< 0.

Ã

sλ
1 + φt δ+sλ

1 + φt s(δ+λ)
δ+sλ

!−1

αγbφsδ
1 + φti2h(δ + sλ) 1 + φt s(δ+λ) 2
δ+sλ

Diﬀerentiating (22) yields∂Ωu (0)
∂x
> 0,
> 0.
∂λ
∂λ
w (x)
u (x)
> 0, ∂Ω∂λ> 0,
Diﬀerentiating (19) with respect to δ and using the above results, we obtain ∂Ω∂λ
w (x)
u (x)
w (x)
u (x)
> 0, ∂Φ∂δ> 0, ∂g∂λ> 0, and ∂g∂λ> 0.
which in turn imply ∂Φ∂δEﬀect on W and U
The eﬀects of λ on W and U are ambiguous.