Discussion Papers In Economics
And Business

Analyzing the impact of labor market integration
Keisuke Kawata, Kentaro Nakajima, and Yasuhiro Sato

Discussion Paper 13-29

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAnalyzing the impact of labor market integration
Keisuke Kawata, Kentaro Nakajima, and Yasuhiro Sato

Discussion Paper 13-29

November 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAnalyzing the impact of labor market integration∗
Keisuke Kawata†, Kentaro Nakajima‡, and Yasuhiro Sato§November 19, 2013

Abstract
We develop a competitive search model involving multiple regions, geographically mobile workers, and moving costs. Equilibrium mobility patterns are analyzed and characterized, indicating thatshocks to a particular region, such as a productivity shock, can propagate to other regions throughworkers’ mobility. Moreover, equilibrium mobility patterns are not efficient due to the existence ofmoving costs, implying that they affect social welfare not only because they are costs but also because they distort equilibrium allocation. By calibrating our framework to Japanese regional data,
we demonstrate that the impacts of eliminating migration costs are comparable to those of a 30%
productivity increase.

Keywords: geographical mobility of workers, competitive job search, moving costs, labor market integration
JEL Classification Numbers: J61, J64, R23∗

We thank In-Koo Cho, Gilles Duranton, Michal Fabinger, Masahisa Fujita, Esteban Rossi-Hansberg, Yoshitsugu

Kanemoto, Akihiko Matsui, Daisuke Oyama, Pierre Picard, Artyom Shneyerov, Takatoshi Tabuchi, Ryuichi Tanaka, Asher
Wolinsky, Yuichiro Yoshida, Dao-Zhi Zeng, Takashi Kunimoto, and the participants of various seminars and conferences forvery useful comments and discussions. We acknowledges the financial support from RIETI, and the Japan Society for the
Promotion of Science through a Grant-in-Aid for Young Scientists (B).
†
Hiroshima University, Japan, e-mail: keisuke@hiroshima-u.ac.jp‡
Tohoku University, Japan, e-mail: nakajima.kentaro@gmail.com§
Osaka University, Japan, e-mail: ysato@econ.osaka-u.ac.jp

1

1 Introduction
This study analyzes the possible impacts of regional labor market integration on local and national labormarkets and social welfare. As observed in many countries, there exists considerable labor mobility
within a country, and such migration has been shown to be sensitive to local labor market conditions.1
We then naturally expect that migration should eventually eliminate regional differences in labor market
conditions, such as those in wages and unemployment rates. However, contrary to this expectation, we
observe persistent and significant differences in such labor market outcomes. For instance, Lkhagvasuren[8] showed that the magnitude of cross-state unemployment differences is approximately identical withthe cyclical variation in the national unemployment rate.2
Migration sensitivity to labor market conditions and the persistent regional differences in labor market outcomes imply that regional labor markets are only imperfectly integrated, which is primarily attributed to the existence of moving costs in general. Such moving costs include those related to job
turnover, which depends on institution and regulations affecting labor markets, such as mutual recognition of professional degrees among different regions and occupational licenser requirements, those ofmoving, selling, and finding houses, which may depend on transportation and communication technology, and those of adjusting to a new environment and re-constructing social networks. The aim of thispaper is to qualify and quantify the effects of moving costs on local and national labor markets.3
We develop a competitive search model involving multiple regions and moving costs. As modeledin Acemoglu and Shimer [1] [2] and Moen [14], firms post wages when opening their vacancies andjob search is directed.4 Search is off-the-job and only unemployed workers can move between regions.
Although job seekers can search jobs (i.e., can access information on vacancies) both within and outside
their places of residence, a new job in a region different from their initial places of residence incurs
moving costs.
Our analysis first examines the qualitative effects of moving costs on migration patterns. The intriguing result is that a change in moving costs results in spillover effects through migration responses,
resulting in a counter-intuitive result: better access from one region to another that is characterized as
1

For earlier contributions on this issue, see Blanchard and Katz [4], Borjas et al [5], and Topel [24] among others. Recent

contributions include Hatton and Tani [7], Kennan and Walker [9], and Rabe and Taylor [21].
2
The same holds true for Japanese prefectures. Population census of Japan reports prefectural unemployment rates every
five years. The coefficients of variation of cross-prefecture unemployment in 1985, 1995, and 2005 are approximately 0.35,
0.31, and 0.23, respectively. That of time series unemployment from 1985 to 2005 is 0.27.
3
In the international context, the degree of labor market integration also depends on the formation of political and economic
unions such as the European Union. Although we base our arguments on migration within a nation in this study, our framework
is applicable to such unions as well.
4
See, among others, Rogerson et al [22] for recent developments in the literature on job search models that include acompetitive search model.

2having better economic conditions, such as higher productivity, may negatively affect the source region.
It increases job settlements from the source region to the better region whereas it decreases job settlements to other regions, which may result in a higher unemployment rate in the source region. Hence, anaccess improvement between two regions may widen the difference between the two regions.
Second, equilibrium of the model is shown to be inefficient: Migration flow is insufficient when
the destination (resp. source) region poses a relatively high (resp. low) asset value of an unemployedworker. A high asset value of an unemployed worker in the destination region implies that in-migrationof job seekers to the region is socially beneficial. However, firms in the destination region ignore such
migration benefits when opening their vacancies, resulting in insufficient job settlements and migration.
When the asset value of an unemployed worker in the source region is low, out-migration of job seekers
from the region is socially beneficial. Again, firms in the destination region ignore such benefits when
opening vacancies, resulting in insufficient migration. Thus, migration costs reduces social welfare not
only because they decrease social surplus when migration takes place but also because they distorts theequilibrium allocation.
Furthermore, we demonstrate how to quantify losses from moving costs. For this, we calibrate ourframework to Japanese prefectural data and then consider a counterfactual experiment in which local
labor markets are fully integrated and moving costs do not exist. The counterfactual analysis indicatesthat such integration has a significant impact on unemployment and welfare, which are comparable tothose caused by a 30% productivity increase.
Several previous studies investigated the role of migration and possible effects of labor market integration. Lkhagvasuren [8] extended the island model of Lucas and Prescott [10] by introducing job search
frictions in each island as modeled in the Mortensen-Pissarides model.5 In Lkhagvasuren’s model, aworker’s productivity is subject to a shock specific to the worker-location match. Therefore, a job seeker
hit by a negative productivity shock may have the incentive to move to other islands even if her/his
current location offers high probability of finding jobs, leading to a possibility of simultaneous in- andout-migration. Using this framework, he showed that regional differences in the unemployment rate may
persist, regardless of high labor mobility between regions, and that labor mobility is procyclical. Although our model is similar to that developed in Lkhagvasuren [8] in the sense that both exhibit labormobility and regional unemployment differences simultaneously, they are different in focus: We uncover
the possible role of moving costs in determining migration patterns, whereas he examined the role ofproductivity shocks.
In immigration literature, Ortega [18] developed a two-country job search model in which workers
could decide where to search for jobs. The workers need to incur moving costs if they search for jobs
5

For details on the Mortensen-Pissarides model, see, among others, Mortensen and Pissarides [16] and Pissarides [20].

3abroad. Differences in the job separation rate may incentivize workers in the high job separation country
to migrate to the low job separation country. Because wages are determined by Nash bargaining, firms
expect to make low wage payments to immigrants having high search costs, thereby incentivizing themto increase vacancies. Thus, incentives to migrate and increase vacancies reinforce each other, resulting
in Pareto-ranked multiple equilibria. In contrast, we employ a competitive search model in which wages
are posted and search is directed. This modeling strategy results in a unique equilibrium, enabling us tofocus on the analysis of geographical mobility patterns.
The following studies highlight the positive effects of labor market integration on human capital accumulation and specialization. Miyagiwa [12], in the context of immigration between countries, showedthat if economies of scale exist in education, skilled worker migration benefits the host region by increasing the skilled labor ratio, whereas it negatively influences the source region by discouraging skill
formation. In such an environment, regional integration represented by reductions in moving costs induces people in the host region to invest more in human capital whereas it discourages people in thesource region from investing in it. Wildasin [25] presented a multi-region model in which human capital
investment increases specialization but exposes skilled workers to region specific earnings risk. He thenshowed that the skilled workers’ mobility across regions mitigates such risk and improves efficiency, andexamined how the ways of financing investments, such as local taxes, affect efficiency. However, thesimple treatment of migration decisions in these studies fail to provide a solid and detailed analysis of
migration patterns and their efficiency properties, which we focus on in this study.
Our quantitative analysis is also related to recent studies such as Bayer and Juessen [3], Coen-Pirani[6], and Kennan and Walker [9]. Bayer and Juessen [3] and Kennan and Walker [9] estimated partial equilibrium models in which worker’s moving decisions are motivated by idiosyncratic and location-specificfactors. Bayer and Juessen [3], in particular, share common aspects with our quantitative analysis: Theyobtained a moving cost estimate, which is approximately two-thirds the average annual household income, and considered a counterfactual experiment in which moving costs are set to zero. They focus on
the effects on moving flows: Moving cost elimination increases the U.S. interstate migration rate from
3.7% to 12.6% in the baseline case. In contrast, we focus on the general equilibrium effects of moving
costs, which is in common with Coen-Pirani [6]. Coen-Pirani [6] developed a general equilibrium modelof migration based on the island model of Lucas and Prescott [10] to show that the model can replicate
several stylized facts regarding moving patterns in the United States. In contrast, we investigate thequantitative impacts of labor market integration on unemployment and welfare.
The remainder of this study is organized as follows. Section 2 presents the basic setups. Section 3
analyzes the equilibrium geographical mobility patterns. Section 4 presents the efficiency property ofequilibrium. Section 5 quantifies the effects of moving costs. Section 6 concludes.

4

2 General settings
Consider H regions (region 1, 2, ..., H) in which there is a continuum of risk-neutral workers of size
N . Workers are either employed or unemployed. While employed, workers can not move between
regions, whereas unemployed workers can by bearing moving costs tij . They can seek employment
opportunities both beyond and in their region of residence, however, they incur moving costs tij in casethey get employed beyond their region of residence.67 We employ the following standard assumptions
regarding moving costs: (i) finding a job in the current region of residence incurs no moving cost tii = 0,(ii) moving costs are symmetric, tij = tji , and (iii) moving costs satisfy the triangle inequality, tij ≤tih +thj . Such moving costs include the costs of selling and buying/renting a house and any psychologicalcosts incurred on renewing social networks. This study primarily analyzes the impacts of existence andchanges in such moving costs on labor market outcomes and welfare.8
We assume that only unemployed workers seek employment opportunities. Once a worker is employed by a firm, the firm-worker pair in region i produces output yi , where without loss of generality,
we assume that a region with a larger number is associated with higher productivity, yi+1 ≥ yi . A worker
exits the economy according to a Poisson process with rate δ (> 0), who is replaced by a new worker
thereby keeping the total population size, N , constant. A new worker enters the economy as an unemployed worker in the same region as her/his parent. The following figure summarizes the structure of themodel:[Figure 1 around here]

2.1 Matching framework
Because arguments are based on a competitive search model, the overall job search market is divided
into sub-markets, each of which is characterized by a wage rate, and hence, by a geographical mobility
pattern, known as the "block recursivity" (Menzio and Shi [13]; Shi [23]). Job matches accompanied by6

We later show that an unemployed worker may move only when she/he gets employed. While being unemployed, she/he

has no incentive to move.
7
Alternatively, we can assume that workers can search for employment opportunities only locally, referred to as the "movethen search" regime. In our framework, workers can move between regions while searching for jobs, implying the applicabilityof this regime. In addition, workers can search for jobs outside of their region of residence, implying that the "search thenmove" regime is also possible. However, as shown later, only the "search then move" regime emerges in equilibrium. See
Molho [15] for a comparison of equilibrium unemployment rates between the "move then search" regime and the "search thenmove" regime.
8
One may suspect that migration costs are different across people. Under a competitive search framework, such heterogeneity does not alter our results qualitatively because of the block recursivity that we will refer to in the next section.

5

migration from region i to region j are generated by a Poisson process with rate Mij = μj m(uij , vij ),where uij and vij are the number of unemployed workers who seek employment in region j whileliving in region i, and the number of vacancies directed at such job searchers, respectively. We callthis sub-market as sub-market ij. μj represents location-specific matching efficiency. μj m(·, ·) is thematching function defined on R+ × R+ , and assumed to be strictly increasing in both arguments, twice
differentiable, strictly concave, and homogeneous of degree one. Moreover, we assume that μj m(·, ·)satisfies 0 ≤ Mij ≤ min[uij , vij ], μj m(uij , 0) = μj m(0, vij ) = 0 and the Inada condition for botharguments.
In each sub-market, worker-job matching occurs at the rate of pij = p(θij ) = Mij /uij = μj m(1, θij )for a job seeker, and qij = q(θij ) = Mij /vij = μj m(1/θij , 1) for a firm seeking to fill a vacancy. θij
is the measure of labor market tightness in sub-market ij defined as θij = vij /uij . From the assumptions regarding μj m(·, ·), we obtain that pij uij = qij vij , dpij /dθij > 0 and dqij /dθij < 0 for any
θij ∈ (0, +∞). We can also see that limθij →0 pij = 0, limθij →∞ pij = ∞, limθij →0 qij = ∞, andlimθij →∞ qij = 0. Moreover, we assume that the elasticity of the firm’s contact rate with respect tomarket tightness, ηij ≡ −(θij /qij )dqij /dθij = 1 − (θij /pij )dpij /dθij , is constant and common acrossall submarkets (ηij = η, ∀i, j).9

2.2 Asset value functions
Let ρ (> 0) denote the discount rate and define r as r = δ + ρ. When locating region i, the asset
value functions for an employed worker, Wi (w), an unemployed worker, Ui a firm with a filled position,
Ji (w), a firm with a vacancy, Vji , are given by (1)-(4), respectively.
rWi (w) = w,

(1)

rJi (w) = yi − w,

(2)

rUi = b +HX
h=1

pih (Wh (wih ) − Ui − tih ) ,rVji = −k + qji (Ji (wji ) − Vji ) ,

(3)
(4)

where b and k represent the flow utility of an unemployed worker, including the value of leisure andunemployment benefits, and the cost of posting a vacancy, respectively. We assume that yi > b, ∀i.
Moreover, region i represents the region where agents (i.e., workers and firms) are located, and regions
h and j represent the region in which unemployed workers seek employment and firms post vacancies,
respectively. Note that the block recursivity divides the labor market into sub-markets, and each submarket ij is characterized by the combination of the place of residence, i, and the place of job search,
9This assumption leads to a set of functions that include the Cobb-Douglass function, which is standard in the literature on

theoretical and empirical search models (See Petrongolo and Pissarides [19]).

6

j. The wage rate may differ between sub-markets within a region and hence the asset values Wi (w) and¡¢¡¢Ji (w) may also differ: We may observe that wji 6= wj 0 i , Wi (wji ) 6= Wi wj 0 i , and Ji (wji ) 6= Ji wj 0 i(j 6= j 0 ). In (3), the second term represents the sum of expected gains in the asset values from finding

jobs net of moving costs. Thus, the moving costs are described as reductions in asset values.10 In (4),Vji depends on the firm’s location, i, and the location of posting a vacancy, j.

2.3 Equilibrium
Because this is a competitive search model, that is, firms post wages and search is directed, the job searchmarket in each region is divided into sub-markets according to the migration pattern. An unemployedworker in region i chooses sub-markets to search for jobs in order to maximize her/his asset value. In sodoing, she/he can search for jobs in multiple sub-markets.11 In equilibrium, the asset value in region itakes the same value Ui regardless of the submarkets that she/he choose.
A firm providing a vacancy determines its wage to post while anticipating the market response:
it regards Ui as given and takes the relationship between wij and θij that is determined by (3) into
consideration. The firm’s decision is described as follows:
max Vij

wij ,θij

s.t. (3), where Ui is treated as given.

Using (1), (2), and (4), this optimization is written asµ¶yj − wij− Vij
max −k + qij
wij ,θij
r
H
³w
´
X
ih
− Ui − tih , where Ui is treated as given.
pihs.t. rUi = b +
r

(5)

h=1

The related first-order conditions are given by
0 = −qij − λpij ,µ¶´dpij ³ wij
dqij yj − wij− Vij − λ− Ui − tij .
0=
dθij
r
dθij
r
We assume free entry and exit of firms, which drives the asset value of posting a vacancy to zero: Vij = 0.
The first-order conditions then yield the wage rate posted by a firm in sub-market ij:
wij = ηyj + (1 − η) r (Ui + tij ) .

(6)

Thus, for a given market tightness, the wage rate becomes higher as the productivity, yj , asset value ofan unemployed worker, Ui , and moving cost, tij , increase. A higher yj enables a firm to offer a higher
10

Alternatively, we can assume that a mover need to pay flow costs of moving until she/he exits the economy. This alternation

does not change any of our results.
11
From the assumption of the Poisson process, the probability that an unemployed worker obtains multiple offers at a time is
zero.

7

wage rate whereas a higher Ui or tij requires a firm to pay higher compensation in order to attract job
applicants. Plugging (6) into the zero-profit condition, Vij = 0, we obtainrk = qij (1 − η) (yj − rUi − rtij ) .

(7)

Of course, there may be some region j where yj − rtij − rUi ≤ 0. In such a case, no vacancy is posted
in sub-market ij and pij = 0.
We focus on the steady state. Although the total population is constant, the population in each region
may change over time. Here, the steady state requires that the unemployment rate in each region, uni ,P
is constant. The dynamics of the unemployment rate are given by duni /dτ = δ − uniτ (δ + h pihτ ),where τ represents time. This yields the steady state level of unemployment rate as
uni =

δ+

δ
PH

h=1 pih.

(8)

Once the asset value of an unemployed worker, Ui , is given, other endogenous variables are welldetermined: (7) uniquely determines the market tightness, θij . Then, (6) and (8) give the wage and unemployment rates, wij and uni , respectively. The asset values other than Ui are determined accordingly.
The wage equation (6) is rewritten as(1 − η)(yj − rtij − rUi ) = yj − wij .

(9)Using this, we can rearrange the zero-profit condition (7) as
rk = qij (yj − wij ).

(10)

Substituting (1), (10), and qij = pij /θij into (3), we can rewrite the asset value of an unemployed worker,(3), as
H h
³y
´
i
X
h
pih
− Ui − tih − kθih .
rUi = b +
r

(11)

h=1

Equations (9) and (10) imply that θij is a function of Ui for all j. Thus, (11) implicitly determines Ui .
The following proposition establishes the existence and uniqueness of the solution:
Proposition 1 The steady state equilibrium exists and is unique.
Proof. See Appendix A.

3 Equilibrium properties
Here, we summarize several equilibrium properties that are worth listing, thereby focusing on interior
solutions though we can obtain qualitatively the same results as those obtained below even if we allow
corner solutions.
83.1 Migration patterns
In equilibrium, we can confirm that unemployed workers, while searching for a job, do not have the
incentive to migrate:
Proposition 2 The difference between the asset value of an unemployed worker in region i and that inregion j is smaller than the moving costs between the two regions:tij ≥ |Ui − Uj | ,∀i, j ∈ H,

where the equality holds true if and only if tij = 0.
Proof. See Appendix B.
Thus, we know that migration takes place only when unemployed workers find jobs. Moreover, this
proposition implies that if there is no moving cost (tij = 0, ∀i, j), the asset value of an unemployedworker is the same across regions. From (9) and (10), θij and pij are also the same across regions,which, combined with (8), results in equalization of regional unemployment rates.
The probability of such migration depends on the difference between the social gains from making amatch, yj − rtij − rUi , which is the output of a match minus the value of an unemployed worker and therelated moving costs, as well as the matching efficiency of the destination region, μj :Proposition 3 The job finding rate associated with migration from region i to region j increases as thesocial gains from a match and the location specific matching efficiency increase:pij > pi0 j 0

if yj − rtij − rUi > yj 0 − rti0 j 0 − rUi0 and μj ≥ μj 0 .

Proof. See Appendix C.
This proposition has several implications. First, a particular destination attracts people more from a
region with low moving costs and a low asset value of an unemployed worker (i.e., in destination j, thejob finding rate from region i, pij , is higher than that from region i0 , pi0 j , if ti0 j + Ui0 > tij + Ui ).
Second, a destination with low moving costs, high productivity and a high matching efficiency attracts
more employed workers from a particular region (i.e., for a job seeker in region i, the job finding rate inregion j, pij , is higher than that in region j 0 , pij 0 , if yj − rtij > yj 0 − rtij 0 and μj > μj 0 ). Finally, thenet migration from region i to j is positive when the productivity, asset value of the unemployed worker,
and matching efficiency are higher in region j than in region i (i.e., pij > pji if yj + rUj > yi + rUi andμj > μi ).

93.2 Effects of labor market integration and spillover effects of productivity shocks throughmigration
Nest, we examine the effects of regional labor market integration. In our framework, the labor market
integration is described by a reduction in moving costs tij .
Proposition 4 A reduction in moving costs from region i to region j, tij , (i) increases the asset value ofan unemployed worker in region i, Ui , (ii) increases the job finding rate from region i to region j, pij ,but decreases that from region i to region j 0 6= j, pij 0 (j 0 6= j), (iii) decreases the wage rate when findinga job in region j from region i, wij , but increases the wage rate when finding a job in other regions, wij 0 ,and (iv) has ambiguous effects on the unemployment rate in region i, uni .
Proof. See Appendix D.
A reduction in moving costs tij increases job searchers’ gains in region i from a job match in region
j, increasing the asset value, Ui . From (7), we can see that a reduction in tij directly increases θij (adirect effect) and influences θij through changes in Ui (an indirect effect). Although the direct effectpositively influences θij and increases pij and the indirect effect has a opposite impact, the direct effectdominates the indirect effect in region j. In other regions, we observe no direct effect, implying that pij 0(j 0 6= j) unambiguously declines. The wage rate wij is lower for a lower tij because firms only have to
pay lower compensation in order to attract job seekers from region i to region j, which in turn, implies
that firms in other regions need to pay higher wages in order to attract workers from region i. Althougha lower moving costs, tij , implies a higher job finding rate from region i to region j, pij , it leads tolower job finding rates to other regions, pij 0 (j 0 6= j), through changes in Ui . The former effect lowers
the unemployment rate in region i, uni , whereas the latter effect raises it. When yj − rtij is sufficiently
large, a change in tij significantly affects Ui and hence it becomes possible that the latter effect dominates
the former. Put differently, a better access from region i to a region with good job opportunities, i.e., aregion with high yj , may reduce job placement flows to other regions and increase the unemployment
rate in region i. This is counter-intuitive since we normally expect that such a better access would lower
the unemployment rate in the source region. The spillover effects on the job finding rate in other regions
give rise to this intriguing result.
Moreover, due to responses of migration flows, a productivity shock in a particular region spills overto other regions.
Proposition 5 An increase in productivity in region j , yj , (i) increases the asset value of an unemployedworker in region i (i 6= j), Ui , (ii) increases the job finding rate from region i to region j, pij , but
decreases that from region i to region j 0 6= j, pij 0 (j 0 6= j), (iii) increases not only the wage rate when10

finding a job in region j from region i, wij , but also the wage rate when finding a job in other regions,wij 0 , and (iv) has ambiguous effects on the unemployment rate in region i, uni .
Proof. See Appendix D.
Productivity improvement in region j increases the employment flows from all regions into region j,pij , ∀i, which increases the asset values of an unemployed worker in these sending regions, Ui . However,it decreases the employment flows to other regions, i.e., region j 0 , (i 6= j 0 , j 0 6= j), pij 0 . In contrast, it
increases the wage rate in all regions while such an effect is most prominent in the region where theproductivity shock arises. With higher productivity in region j, firms can afford to post higher wages,
forcing
firms in other regions to pay higher wages in order to attract workers. The effect on the unemployment rate, uni , is again ambiguous because of the opposing effects of changes in pij and changes in pij 0
on uni . This finding is in contrast to the results of standard job search models with no moving cost,where a positive productivity shock always lowers the unemployment rate (see Rogerson et al [22], forinstance).

4 Inefficiency arising from the moving costs
Now we characterize the efficiency of equilibrium. We use the social surplus, S, as the efficiency criterion, which is standard in job search models (See Pissarides [20]). S is the sum of total output and flow
utility of unemployed workers minus the costs of posting vacancies and migration:
#
"
Z ∞X
H
H
X
(kθihτ + pihτ tih ) e−ρτ dτ
yi (Niτ − uiτ ) + buiτ − uiτ
S≡
0

i=1

(12)

h=1

We start by describing the planner’s problem. The social planner maximizes the social surplus subject
to the laws of motion of regional population and unemployment:maxθijτ ,Niτ ,uiτ

S

(13)

H

H

X
X
dNiτ
=
uhτ phiτ − uiτ
pihτ
dτ
h=1
h=1
!
ÃH
X
duiτ
= δNiτ − uiτ
andpihτ + δ
dτ

s.t.

h=1

where τ represents time. Changes in regional population arise from social changes (differences betweenPP
in-migration uhτ phiτ and out-migration uiτpihτ ). Inflows to the unemployment pool are newcom-

ers to the economy and outflows from it are those who get employed. We relegate the derivation of
the optimal conditions to Appendix E. After evaluating the first-order conditions for the social planner’s
maximization at the steady state, we obtain the following proposition:
11

Proposition 6 Define Dij as#"HX(1 − η) qij
Dij ≡pih (Uh − Uj ) .
r (Ui − Uj ) + ηP
r+η H
h=1 pihh=1(14)

Equilibrium market tightness θij is socially optimal if and only if Dij = 0 in equilibrium. If and only if
Dij > 0, θij is greater than the optimal tightness. The opposite holds true if and only if Dij < 0.
Proof. See Appendix E.
Therefore, the equilibrium market tightness θij and the job finding rate pij are insufficient when thedestination region has a relatively high asset value of an unemployed worker, Uj , or when the source
region has a relatively low Ui . A high asset value of an unemployed worker in the destination region
implies that inflows of job seekers to the region are socially beneficial. However, firms ignore such
benefits of migration in opening their vacancies, leading to insufficient market tightness. In contrast,
when the asset value of an unemployed worker in the source region is low, outflows of job seekers fromthe region are socially beneficial. Again, firms ignore such benefits in opening vacancies, resulting ininsufficient market tightness.
In case of identical moving costs for all migration patterns (tij = t, i 6= j, ∀i, j), migration from any
region i to region H, where productivity is the highest, is always too small and that to region 1, where
productivity is the lowest, is always too large, and there exists a threshold region b
j(i) for which flows toregion j > b
j(i) are too small and flows to region j ≤ bj(i) are too large.12

Moreover, Proposition 2 implies that the absence of migration costs (tij = 0, ∀i, j) implies thatUi = Uj ∀i, j and hence Dij = 0:Corollary 7 If there are no migration costs, i.e., tij = 0, ∀i, j, equilibrium is socially optimal.
In the absence of moving costs, our framework becomes a standard competitive search model, of which
equilibrium is socially optimal (see Moen [14] and Rogerson et al [22], among others). Thus, moving
costs reduce the social surplus not only because they reduce the movers’ asset values but also because
they distort the equilibrium.
12

We can prove the result as follows. We readily know that Ui = Uj if yi = yj . Moreover, (16) proves that
dUi
pjj
pij
dUj


−
=
−
.
dyj
dyj
r + h pjh
r + h pih

Proposition 3 implies that pii = pjj > pij = pji and pih = pjh if yi = yj , which lead todUi 
dUj
−
> 0.
dyj
dyj yj =yi

Hence, the continuity of Ui with respect to yj , ∀i, j, proves that Ui > Uj if yi > yj . From the assumption that yH > · · · >yi+1 > yi > · · · > y1 , we know that UH > · · · > Ui+1 > Ui > · · · > U1 . From (14), we readily know that DiH < 0 andj(i) for which Dij < 0 for j > 
j(i) and Dij < 0 for j ≤ j(i).
Di1 > 0 for all i, and there exists a threshold region 

12

5 Quantitative analysis
In this section, we demonstrate how our framework can be used to quantify the overall losses frommoving costs. This exercise also serves to reveal the impacts of regional integration on the economy.
Here, we calibrate our model to Japanese prefectural data, and provide counterfactual analysis regarding
changes in moving costs.
We use data on Japanese prefectures for 2000-2009.13 In calibrating our model, we focus on thelong-run characteristics of regional labor markets in Japan to ensure that the calibration is consistent
with the steady state analysis given in the previous section. More concretely, we focus on the level andregional variation of the unemployment rate averaged over these periods, which are represented in thefollowing figure.
[Figure 2 around here]
The overall unemployment rate of these 46 prefectures averaged over 2000-2009, unN , is 0.0455, and theunemployment rate of each prefecture ranges from 0.0305 (Fukui prefecture) to 0.064 (Osaka prefecture)(Population Census, Ministry of Internal Affairs and Communications). The degree of dispersion can beqP2measured by the coefficient of variation: CV = (1/un) (1/46) 46
i=1 (uni − un) where un is theaverage of regional unemployment rates. CV for the regional unemployment rate averaged over 20002009 is 0.182, which is somewhat lower than that in the United States.14 We will examine the extent
to which moving costs affect the overall unemployment rate, the dispersion of regional unemployment
rates, and welfare.

5.1 Calibration
In the following analysis, we normalize the total population, N , to one. The values of the job separation rate, δ, the regional output per capita, yi , and the distance between regions, zij , are taken from theJapanese data: δ is set to 0.16, which is the annual job separation rate in Japan averaged over the years
2000-2009 (Survey on Employment Trends, Ministry of Health, Labour and Welfare). We employ the percapita gross prefectural domestic product (in million yen, Prefectural Accounts, Department of National
13

We excluded Okinawa prefecture and used the data on the remaining 46 prefectures. This is because Okinawa prefecture

comprises islands and is located extremely far from other prefectures, making it an outlier. In fact, the distance between it andthe neighboring prefecture is around 650km whereas in most cases, the distance between two neighboring prefectures is lessthan 100km. Note here that the distance between prefectures is measured by the distance between the locations of prefectural
governments. This elimination reduces the coefficient of variation regarding regional unemploment. For instance, the figure forthe year 2000 decreases from 0.232 to 0.172.
14
Lkhagvasuren [8] reported that between January 1976 and May 2011, the coefficient of variation of cross-state unemployment rates in the United States ranges from 0.175 to 0.346 with an average of 0.237.

13

Accounts, Cabinet Office) as yi . zij is measured as the distance (in 100km) between prefectural governments (which is taken on February 20, 2013 from http://www.gsi.go.jp/KOKUJYOHO/kenchokan.html,
Geographical Information Authority of Japan). We normalize the flow utility of an unemployed worker,
b, to one.
We set the value of the discount rate, ρ, to 0.0151, which comes from the average annual interest
rate of 10-year national bond of Japan during 2000-2009 (which is taken on February 20, 2013 fromhttp://www.mof.go.jp/jgbs/reference/interest_rate/data/jgbcm_2000-2009.csv, Ministry of Finance). In
existing studies such as Coen-Pirani [6], Lkhagvasuren [8], and Kennan and Walker [9], this value is set
to 0.04 − 0.05. We will verify the robustness of our results against a higher value of ρ (ρ = 0.05).
We specify moving costs, tij , as a linear function of the distance between prefectures i and j, thatis, tij = tzij , where zij is the distance between regions and t is a positive constant. We will verify therobustness of our results against a different functional form for moving cost.
In the following quantitative analysis, we employ a Cobb-Douglas form of the matching function,1−η, where μj and η are constants satisfying that μj > 0 and 0 < η < 1.
given by μj m(uij , vij ) = μj uηij vij

As surveyed by Petrongolo and Pissarides [19], the Cobb-Douglas matching function is very standard
in the literature of theoretical and empirical search models. We rearrange the matching function asln[μj m(uij , vij )/uij ] = ln[μj ] + (1 − η) ln[vij /uij ] = ln[μj ] + (1 − η) ln[θij ], and estimate it by using
the data on job applicants, job openings, and job placements (Monthly Report of Public Employment Security Statistics, Ministry of Health, Labour and Welfare). Note here that the job seekers’ job finding rate
μj m(uij , vij )/uij is given by the number of job placements per job applicant, and the market tightness
θij is given by the number of job openings per job applicant. Our spatial units are Japanese prefectures.15
The Monthly Report of Public Employment Security Statistics reports the number of active job applicants, active job openings, and job placements in every month. To eliminate seasonal volatility, we aggregate monthly data into annual data by taking averages. Because figures for job placements within prefectures are available, we can estimate the matching function ln[μj m(uij , vij )/ujj ] = ln[μj ]+(1−η) ln[θjj ]to obtain η = 0.512 and μj . Details of the estimation are provided in Appendix F. In the benchmark case,
we estimate the matching function using the fixed effects (FE) model. We will verify for the possible
bias arising from the endogeneity of θjj .
The remaining two parameters, the moving cost parameter, t, and the cost of providing a vacancy, k,
are chosen by targeting the coefficient of variation of the unemployment rate and the overall unemployment rate, which results in t = 5.348 and k = 0.0196 in the benchmark case. Tables 1 and 2 summarize
the parameter values and calibration results, respectively.
[Tables 1 and 2 around here]
15

Here, again, we eliminated Okinawa prefecture from our sample.

14

In Table 2, we also report the value of social surplus given by (12). Using this calibrated model, we will
execute a counterfactual analysis regarding moving costs.

5.2 Counterfactual analysis
In order to see the quantitative impacts of moving costs, we consider the following counterfactual analysis. Assume now that regional labor markets are fully integrated and that there is no moving cost. Thus,
we set t = 0, keeping other parameters fixed as described in Table 1, and run a counterfactual simulation.
We compare the resulting unemployment rate and welfare to the calibrated values shown in the previous
sub-section. Eliminating moving costs has the following two effects. First, it directly implies the disappearance of losses from moving and increases the social surplus. Second, as shown in Proposition 6 andCorollary 7, it restores the efficiency of equilibrium by making people more mobile between regions,
thereby improving the distribution of the labor force to enhance the job creation efficiency and increase
the social surplus.
The results of counterfactual analysis are reported in Table 3.
[Table 3 around here]
As shown in Table 3, in the benchmark case, the overall unemployment rate, unN , drops by 0.0199
points from 0.0455 to 0.0256, which corresponds to 43.7% decreases. As shown before, when t = 0,
the unemployment rate is the same across regions, and hence, the coefficient of variation for the regional
unemployment rate, CV , becomes zero. The social surplus, S, increases by 104.5 points from 362.0
to 466.5, which corresponds to 28.8% increases. Such large welfare gains arise from the two effects
explained above.
In order to appreciate the magnitude of such impacts, we consider an additional counterfactual inwhich productivity increases in all regions, and compare the changes in the two counterfactuals. In fact,
we can see that the effects of labor market integration are comparable to those of a 30% productivity
increase. In Table 3, we provide the results of the counterfactual analysis, where output per capita ineach region increases by 30%. Such productivity changes results in a 27.9% decrease in unN , 30.7
percent decrease in CV , and 33.6% increase in S. This result shows that losses from moving costs canbe highly significant in a quantitative sense.

5.3 Robustness check
In this section, we discuss the robustness of our results against possible alternative settings.

15

Endogeneity bias in estimating the matching function
First, as is well known, market tightness, θjj , i.e., the independent variable in estimating the matching
function, ln[m(ujj , vjj )/ujj ] = ln[μj ]+(1−η) ln[θjj ], is also an endogenous variable in search models.
Such endogeneity may bias the estimated coefficient obtained by the standard fixed effects (FE) model.
In order to verify the robustness against endogeneity, we conducted a fixed effects instrumental variable(FEIV) estimation. We follow several recent studies that estimated the matching function in using lagsof market tightness as instruments (see e.g., Yashiv [26]). As we explain in Appendix F, we used thetwo-period and three-period lags of market tightness as instruments, and obtained 0.575 as the estimatedvalue of η. Table 4 reports the parameter values in the robustness check.
[Table 4 around here]
In Table 4, the column of Robustness check (1) presents parameter values in the case where thematching function is estimated by FEIV method. Because we obtained 0.512 in the benchmark case(i.e., under FE estimation), FEIV estimation yields a slightly higher value. Still, the main results arehighly similar to those of the benchmark case. The results of calibration and counterfactual analysis
are provided in the column of Robustness check (1) in Tables 2 and 3. Here, in the absence of moving
costs, the unemployment rate, unN , would be lower by 42.8%, and social surplus, S, would be higher
by 29.3%. These figures are again comparable to the effects of a 30% productivity increase, which has
effects of lowering unN by 26.8% and the coefficient of variation of the regional unemployment rates,
CV , by 31.3%, and of raising S by 33.6%.
Concave moving costs
Second, we need to examine the degree to which our results depends on the specification of moving
costs. In the benchmark case, we specified the moving costs as a linear function of the distance between
regions, i.e., tij = tzij . However, the marginal moving costs may decline with distance because the costdifference between not moving and moving 10km would be significant whereas that between moving
100km and moving 110km may not be so. In order to represent this possibility, we assume a concave
function of the distance between regions as the moving costs. More specifically, we use a logarithmic
function, i.e., tij = t ln[zij ]. Parameter values in this case are shown in the column of Robustness
check (2) in Table 4. The calibration results and counterfactual analysis are presented in the column ofRobustness check (2) of Tables 2 and 3. In this case, if there were no moving cost, unN would be lower
by 72.0%, and S would be higher by 41.2%. In contrast, a 30% productivity increase lowers unN by
53.6% and CV by 75.2%, and increases S by 25.1%. Thus, we observe that the effects of moving costs
are even more significant in this case than in the benchmark case.
16Higher discount rate
Third, the value of discount rate, ρ, that we use (ρ = 0.0151) is lower than that used in existing studies
such as Coen-Pirani [6], Lkhagvasuren [8], and Kennan and Walker [9] (ρ = 0.04 or 0.05). This isbecause the interest rate in Japan was at a unprecedentedly low level in the 2000s. In order to confirm that
our results are not attributable to this low discount rate, we run a counterfactual simulation in which thediscount rate is higher (ρ = 0.05). Parameter values in this case are shown in the column of Robustness
check (3) in Table 4. The results of calibration and counterfactual analysis are given in the column ofRobustness check (3) of Tables 2 and 3. In this case, if there were no moving cost, unN would be lower
by 44.1%, and S would be higher by 23.3%. In contrast, a 30% productivity increases lowers unN by
27.9% and CV by 30.7%, and increases S by 33.2%. Again, we confirm the robustness of our results.
Difference in periods
Finally, we check whether our results change depending on the analysis periods. Here, we divide thesample into two periods (2000-2004 and 2005-2009). As we explain in Appendix F, we obtained η =0.456 for 2000-2004 and η = 0.608 for 2005-2009. Parameter values for 2000-2004 and those for 20052009 are shown in the columns of Robustness check (4) and (5) in Table 4, respectively. The calibration
results and counterfactual analysis for 2000-2004 and those for 2005-2009 are presented in the columnsof Robustness check (4) and (5) of Tables 2 and 3, respectively. For 2000-2004, elimination of moving
costs lowers unN by 49.5% and increases S by 31.7% whereas a 30% productivity increase lowers
unN by 28.0% and CV by 4.27%, and increases S by 33.3%. For 2005-2009, elimination of moving
costs lowers unN by 46.4% and increases S by 31.8% whereas a 30% productivity increase lowers unN
by 25.5% and CV by 25.0%, and increases S by 33.6%. Thus, the effects of moving costs are arevery similar over these periods and comparable to the effects of a 30% productivity increase. The only
difference between these periods lies in the effect of productivity improvements on the unemployment
differential, which is smaller for the early 2000s than for the late 2000s.

6 Concluding remarks
In this study, we developed a multi-region job search model and analyzed the impacts of moving costs
both qualitatively and quantitatively. By qualitative analysis, we showed that shocks to a particularregion, such as a productivity shock or improvement in access to another region, cause spillover effects
to other regions through migration responses. We proved that equilibrium is inefficient in the presenceof moving costs. Thus, moving costs reduce the social welfare not only because they decrease the social
surplus when migration takes place but also because they cause distortions. We also examined the overall

17

losses from moving costs quantitatively. We calibrated our framework to Japanese prefectural data anddemonstrated by a counterfactual simulation that the impacts of disappearance of moving cost on theeconomy would be comparable to those of a 30% productivity improvement in all regions.
We briefly mention the limitations and possible extensions of our model. First, in order to concentrate our attention on the analysis of migration patterns, we ignored one important dimensions related tomigration and labor market integration. As shown in Miyagiwa [12] and Wildasin [25], labor market integration enhances human capital accumulation and specialization. Moreover, it may affect investmentsof firms. Although incorporating these investment decisions into our framework would not change theefficiency results because investment decisions are know to be efficient in a competitive search model(e.g., Acemoglu and Shimer [2]; Masters [11]), it would amplify the effects of migration: a region receiving large migration or having better access from other regions enjoys the benefits of larger investments
whereas such benefits are absent in a region experiencing out-migration or having poor access from otherregions.
Second, we represented moving costs as a function of distance between regions in the quantitative
analysis. However, this is evidently a coarse approximation: A region having better transportation infrastructure such as a hub airport may be easier to move to and from than a region without it, for example.
Indeed, Nakajima and Tabuchi [17] discussed that there exist a case in which one should exclude distances when estimating moving costs (a case in which there is no employment uncertainty
and migration takes place based on utility differentials). Fortunately, our framework does not correspond to such a case. Still, it would be worth exploring a better description of moving costs than ours,
which may include the existence of distance-irrelevant costs.
Third, related to the second point, we may be able to endogenize moving costs. One possible way
is to introduce housing loans. Suppose that people buy houses by using mortgage loans. If negative
productivity shocks hit a region, the income level and housing price would decline. Then, people may
want to move to another region. This would require people to repay the mortgage loans. However, if
decreases in the income level and housing price were sufficiently large, people can not do so because
selling their houses at a sufficiently high price becomes difficult. Thus, mortgage loans may act as
moving costs in the face of economic fluctuations.
Finally ,our framework can be extended to represent the relationships between countries. For instance, we can consider an expansion of the European Union (EU). We would then be able to examine
the possible impacts of accession of a new member country on each member country’s labor market andoverall EU labor market. All these are important topics for future research.

Appendices
Appendix A: Proof of Proposition 1.
18

Define Γi as

H h
³y
´
i
X
h
− Ui − tih − kθih .
pih
Γi (Ui ) ≡ rUi − b −r
h=1

If the equation Γi (Ui ) = 0 has a unique solution for all i, we know that there exists a unique steady state
equilibrium. Equation (7) is rearranged ask=´dpij ³ yj− Ui − tij ,dθij r(15)

which, combined with the Inada condition of the matching function, implies that θij and pij are positive
when Ui is equal to zero and that θij and pij converge to zero as Ui goes to yj /r − tij . Hence, letting U idenote max[yi /r, maxj [yj /r − tij ]], we readily know thatΓi (0) < 0,
Γi (U i ) = rU i − b ≥ yi − b > 0.
Note that even though Γi (Ui ) may be kinked at Ui = yj /r − tij , it is continuous at Ui ∈ [0, U i ]. Thus,
Γi (Ui ) = 0 has at least one solution in [0, U i ], which shows the existence.
Γi (Ui ) may not be differentiable at Ui = yj /r − tij . However, except for these points, it is differentiable, and by differentiating Γi (Ui ) with respect to Ui , we obtainX
X ∂ [pih (yh /r − Ui − tih ) − kθih ] ∂θihdΓi (Ui )
=r+
pih −
dUi
∂θih
∂Ui
h
h
X
=r+
pih > 0,hwhere the second equality comes from (15). Combined with the continuity of Γi (Ui ), this proves that thesolution of Γi (Ui ) = 0 is unique.

Appendix B: Proof of Proposition 2.
From (1) and (3), we haverUi = b +

H h
³y
´
i
X
h
pih
− Ui − tih − kθih ,rh=1

which yields
Ui =

b+

P

/r − tih ) − kθih ]h [pih (yhP
r+

h pih

.

From (7), we know that θij = arg max Ui , ∀i, j ∈ H. Hence, we readily know thatPP
b + h [pjh (yh /r − tjh ) − kθjh ]b + h [pih (yh /r − tjh ) − kθih ]
P
P
≥
.
Uj =
r + h pjh
r + h pih
19

This implies thatUi − Uj ≤b+P/r − tih ) − kθih ]h [pih (yhPr + h pihp(t
ih jh − tih )P= hr + h pihP
pih tijhP≤r + h pih
P

−

b+

P

/r − tjh ) − kθih ]h [pih (yhP
r+

h pih

≤ tij ,

where the second inequality comes from the triangle inequality tjh ≤ tji + tih = tij + tih . Similararguments show that Uj − Ui ≤ tij .

Appendix C: Proof of Proposition 3.
Suppose temporarily that Ui is fixed. Differentiation of (7) with respect to yj − rUi − rtij yields
0=

∂θij
dqij
+ qij .
(yj − rUi − rtij )dθij∂ (yj − rUi − rtij )Plugging qij = pij /θij and (7) into this, we obtain∂θij
rkθij dqij /dθij+ pijqij (1 − η) ∂ (yj − rUi − rtij )∂θijrkη+ pij ,=−
1 − η ∂ (yj − rUi − rtij )0=which implies that∂θij
1 − η pij=
> 0.
∂ (yj − rUi − rtij )η rkAlso, differentiation of (7) with respect to μj gives³´−1
m 1, θij

∂θij
=−
> 0.
∂μj
dqij /dθij
Because dpij /dθij > 0, these inequalities imply that pij > pi0 j 0 if yj − rUi − rtij > yj 0 − rUi0 − rti0 j 0and μj > μj 0 .

Appendix D: Proof of Propositions 4 and 5.
We start by deriving the effect on the asset value of an unemployed worker, Ui . yj and tij affect Ui
only through changes in yj − rtij . Differentiating (11) with respect to yj − rtij and using (15), we obtain
p
∂Ui
P ij
=
> 0.
∂(yj − rtij )r + h∈Hi pih(16)

We readily see that ∂Ui /∂yj = ∂Ui /∂(yj − rtij ) > 0 and ∂Ui /∂tij = −r∂Ui /∂(yj − rtij ) < 0. The
effects on the job finding rate, pij , also appears through changes in yj − rtij . Differentiation of (7) with20

respect to yj − rtij , combined with (16), yields(1 − η)2 pij qij∂pij=∂ (yj − rtij )
ηk

Ã

1−

r+

pij
PH

h=1 pih

!

> 0,

(17)

(1 − η)2 pij 0 qij 0∂pij 0
pij
=−
< 0,
PH
∂ (yj − tij )ηkr + h=1 pih

which lead to ∂pij /∂yj > 0, ∂pij /∂tij < 0, ∂pij 0 /∂yj < 0, and ∂pij 0 /∂tij > 0. From (6), and by using(16), we obtain the effects on the wage rate:pij∂wij= η + (1 − η)
> 0,
PH
∂yj
r + h=1 pih∂wij 0pij= (1 − η)
> 0,
PH
∂yj
r + h=1 pih
Ã
!
∂wij
pij
= (1 − η) r 1 −
> 0,
P
∂tij
r+ H
h=1 pih∂wij 0pij= − (1 − η) r
< 0.
PH
∂tij
r + h=1 pih

Finally, from (17), we can see thatHX

H

(1 − η)2 pij qij X (1 − η)2 pih qih
pij
∂pih
=
−
PH
∂ (yj − rtij )ηkηkr + h=1 pih
h=1
h=1
P
P
∙
¸
(1 − η)2 pij (r + h pih ) qij − h pih qihP=ηkr + h pihP
(1 − η)2 pij rqij + h pih (qij − qih )P.
=ηkr + h pih

When yj − rtij is sufficiently large, market tightness θij is also large and qij is small, under whichPh ∂pih /∂ (yj − rtij ) is likely to be negative. Because the unemployment rate, uni , is given by (8),this raises uni .

Appendix E: Proof of Proposition 6.
The present-value Hamiltonian for the welfare maximization (13) is defined as
#
"
H
H
X
X
(kθihτ + pihτ tih ) e−ρτyi (Niτ − uiτ ) + buiτ − uiτ
Hτ =
+

i=1
H
X
i=1

λN
iτ

ÃH
X
h=1

h=1

phiτ uhτ − uiτ

H
X
h=1

pihτ

!

+

X
i

λuiτ

Ã

δNiτ − uiτ

H
X
h=1

pihτ − δuiτ!

.

uNote here that the control variables are θijτ , and the state variables are Niτ and uiτ . λNiτ and λiτ are the21co-state variables. The first-order conditions are given by¢¡¢dpijτ ¡ N
u
−ρτ
N
u
−ρτ
λjτ − λN= (1 − η) qijτ λN(18)iτ − λiτ − tij ejτ − λiτ − λiτ − tij edθijτ
yi e−ρτ + δλuiτ
(19)
λN
iτ =
r−δ
"
#
!
ÃH
H
H
H
X
X
X
X
−ρτ
N
N
u
0 = − yi − b +(kθihτ + pihτ tih ) e+
λhτ pihτ − λiτpihτ − λiτpihτ + r

ke−ρτ =

h=1

h=1

h=1

h=1

(20)

u
where (18) determines the optimal θijτ , and (19) and (20) can be solved to yield λNhτ and λiτ . We evaluate

these values at the steady state. Hence, we don’t need τ in the followings and Ni and ui are determinedby dNiτ /dτ = 0 and duiτ /dτ = 0.
Equations (18) and (20) yield(yi − b + η
λui = −

P−ρτ + ηh pih tih ) er+η

Moreover, (19) is rearranged asNλNi − λj =

P

h pih

P

h pih

¡ N
¢
λi − λNh´³(yi − yj ) e−ρτ + δ λui − λuj
r−δ

.

(21)

(22)

Plugging (19), (20) and (22) into (18), we obtainPP½(yi − b + η h pih tih ) + η h pih [(yi − yh ) + δ (λui − λuh ) eρτ ] /(r − δ)P
k = (1 − η) qij
r + η h pih´³⎫⎬(yj − yi ) + δ λuj − λui eρτ
− tij
+
⎭
r−δ
δ
= πij − Dij ,rwhere πij and Dij are defined asP∙¸b + η h pih (yh /r − tih )yjP− tij −
πij ≡ (1 − η) qij,(23)rr + η h pih
´
³
´
³
P
P
[yi − b − (r + η h pih ) tij ] − r λuj − λui eρτ − η h pih λuj − λuh − tih e−ρτ eρτP− k.
Dij ≡ (1 − η) qij
r + η h pih
In equilibrium, because pij = θij qij , (7) is rewrittenrkθij = (1 − η) pij (yj − rUi − rtij ) .
Summing up the both sides of it for j = 1...H, we obtain
rk

H
X
j=1

θij = (1 − η)

H
X
j=1

22

pij (yj − rUi − rtij ) ,which is rearranged as

µ

H
X

pij

rUi = b +

H
X

η

j=1

yj − rUi − rtij
r

¶

H

X
η
=
k
θij .
1−η
j=1

Plugging (1), (6) and the above equation into (3), the asset value of an unemployed worker in equilibrium
can be rewritten as∙

ηyj + (1 − η) r (tij + Ui )− Ui − tijrpijyj − rUi − rtij
r

pij

j=1

=b+η

H
X
j=1

¸

(24)

H

X
η
k
=b+
θij .
1−η
j=1

The second equality implies that
Ui =

b+η

P

(yh /r − tih )h pihP

r+η

h pih

.

Using this, we can rewrite the zero-profit condition (7) asPµ¶yjb + η h pih (yh /r − tih )P− tij −.
k = (1 − η) qijrr + η h pih

(25)

(26)

Plugging (25) into πij of (23), we can see that in equilibrium,³y´j− Ui − tij ,πij = (1 − η) qijrwhich, combined with (7), implies that πij = k holds true in equilibrium. From this, we know that theequilibrium market tightness is optimal if and only if Dij evaluated at the equilibrium is zero. Moreover,from the second-order condition of firm’s optimization (5), the equilibrium market tightness is larger
than the social optimum if and only if Dij evaluated at the equilibrium is positive, and the opposite holds
true if and only if it is negative.
From (18), we obtainX¡¢N
pih λN=h − λihX¡¢k X
θih e−ρτ +pih μui + tih e−ρτ .
1−η
h

h

Substituting this and (24) into (20), we know that in equilibrium,PPP(yi − b + η h pih tih ) e−ρτ − [η/(1 − η)] k h θih e−ρτ − η h pih (μui + tih e−ρτ )uPλi = −
r + η h pih
yi − rUi −ρτe=−r
Using this and (24), we can write Dij of (23) evaluated at the equilibrium as
Ã
(
Ã
!
!
´
³y
X
X
X
(1 − η) qij
ηjP− tij +
k
pih
θih −
θjh
Dij =
−b + r + η
r + η h pih
r
1−η
h
h
h
"
!#)
Ã
X
X
X
η k
−η
pih zih +θjh0 −
θhh0
− k.
1−η r
h

h

h

23

From (26), this can be further rewritten as
Ã
!#
"
X
X
X
X
(1 − η) qijηηXθih − kθjh −pih k
θjh0 − kθhh0
Dij = δ
.
k
P
1−η
r
r+η H
h=1 pihhhhhhFinally, from (24), we obtain Dij evaluated at the equilibrium as#"X(1 − η) qijPpih (Uh − Uj ) .
r (Ui − Uj ) + η
Dij =r + η h pihhAppendix F: Estimation of the matching function.
Data
Our spatial units are Japanese prefectures. For job status, we use the Monthly Report of Public Employment Security Statistics (Ministry of Health, Labour and Welfare). It contains numbers of active job
applicants, active job openings, and job placements in every month. Here, the number of job placements
is available for within prefectures and without prefectures. We use the former in estimating the matching
function. To eliminate seasonal volatility, we aggregate monthly data into annual data by taking average. In the analysis, Okinawa prefecture is excluded and hence we have 46 prefectures. We use data for2000-2009, implying that our sample size is 460. Here, we don’t take the average over years because therelationship represented in the matching function is not limited to the steady state. The following tableshows the descriptive statistics.
[Table 5 around here]
Empirical strategy
As we explained in Section 5.1, we employ a Cobb-Douglas form of the matching function:1−η,μj m(uij , vij ) = μjt uηijt vijt

where t represents time. From the assumption of the constant returns to scale, the matching function canbe redefined in terms of a job seeker’s job finding rate:1−η,fijt = μjt θijt

where fijt = μj m(uij , vij )/uijt is the job seeker’s job finding rate, and θijt = vijt /uijt is labor market
tightness. In the estimation, fijt is given by the ratio of the number of job placements to the numberof job applicants whereas θijt is given by the ratio of the number of job openings to the number of job
applicants. By taking the natural logarithm, we can rewrite the matching function asln[fijt ] = ln[μjt ] + (1 − η) ln[θijt ].
24

From this, we obtain an estimable equation as follows:
ln[fijt ] = ξj + (1 − η) ln[θijt ] + εjt .
We assume that the matching efficiency ln[μjt ] can be decomposed into a time-invariant term ξj and atime-variant term εjt . We assume that εjt satisfies the assumption of the standard error term. Because
our data are on the job placements within prefectures, the equation to be estimated becomesln[fjjt ] = ξj + (1 − η) ln[θjjt ] + εjt .

(27)

In the benchmark case, we estimate (27) by the fixed effect (FE) model. This allows us to deal withthe concern that the matching efficiency may be correlated with labor market tightness. For example,
existence of efficient matching intermediaries induces more job posting by local firms. If so, timeinvariant match efficiency, ξj , may be correlated to labor market tightness, ln[θjjt ]. FE model can beused even in the presence of such correlation between ξj and ln[θjjt ].
Further, one may be concerned that the time-variant matching efficiency, εjt , might also be correlatedwith labor market tightness, ln[θjjt ]. For example, firms may post their vacancies in response to changes
in the labor market’s matching efficiency in the current period. If so, ln[θjjt ] correlates with εjt and thestandard FE model does not work.To respond this concern, we use instrumental variables in estimating
the fixed effect model, which we refer to as FEIV model. We follow several recent studies that estimatedthe matching function in using lags of market tightness as instruments (see e.g., Yashiv, 2000):we use
two periods and three periods lagged labor market tightness, ln[θjjt−2 ] and ln[θjjt−3 ], as instruments forlabor market tightness, ln[θjjt ].16
Moreover, because we examine the difference between the early and late 2000s, in addition to thebaseline analysis that uses the full periods from 2000 to 2009, we separately estimate the matching
function (by FE model) for 2000-2004 and for 2005-2009.
Estimation Results
The estimation results are shown in Table 6.
[Table 6 around here]16

One may concern that the time-variant matching efficiency may serially correlated across periods. In that case, system gen-eralized method of moments (GMM) will work well. Our theoretical model, however, does not allow for the serial correlationof matching efficiency across periods. Because our purpose is counterfactual simulation by using a rigorously built theoreticalmodel, we do not allow serial correlation in matching efficiency, and we do not use system GMM for parameter estimation.

25

Column (i) shows the result by FE model. The point estimate of η is 0.512 and is significantly different
from zero. Column (ii) shows the result by FEIV model. The point estimate of η becomes slightly higher
under FEIV model than under FE model. Columns (iii) and (iv) show the results for 2000-2004 and for2005-2009, respectively. The estimated η is larger for the late 2000s than for the early 2000s.
In the quantitative analysis, we also need matching efficiency, which is captured by the estimatedprefectural fixed effects. Table 7 shows the descriptive statistics of the estimated fixed effects for eachcase.
[Table 7 around here]
On average, estimated matching efficiency is stable across estimation methods and periods.

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27

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1297-1322.

28

Parameters
δ
ρ
η
t
k
yi
zij
μi
b
N

Values0.16
0.01510.512
5.348
0.0196
Region specific
Specific between regions
Region specific1(Normalization)
1 (Normalization)

Description
Job separation rate
Discount rate
Parameter of the matching function
Moving cost per distance
Cost of posting a vacancy
Regional output per capita
Distance between regions i and j
Regional fixed components of the matching function
Flow utility of unemployment
Total number of workers

Table 1. Parameter values of the benchmark model.
Notes: The value of ρ comes from Japanese long-term interest rates. The values of δ, yi , and zij are taken from Japanese data. We estimated the Japanesematching function to obtain η and μi . We normalize the total population, N , and the flow utility of an unemployed worker, b, to one. The remaining two
parameters, t and k are chosen by targeting the data listed in Table 2.

Calibration targets
Overall unemp. rate, unN
Unemp. rate differences, CV
Social surplus, S

(a)

Data
(b)

(c)

Benchmark0.04550.182

0.04920.187

0.04180.188(1)0.04550.182
362.0

0.04550.182
362.2

Robustness check
(2)
(3)
(4)
0.04550.182
339.0

0.04550.182
114.2

0.0492
0.187
334.7

(5)
0.04180.188
404.0

Table 2. Calibration results.
Notes: Data columns represent different time periods: (a) Years 2000-2009, (b) Years 2004-2009, (c) Years 2005-2009. Benchmark and Robustnes check(1)-(3) calibrate Data (a). Robustness check (4) and (5) calibrate Data (b) and (c), respectively.

Benchmark
Counterfactual (no moving cost)
Overall unemp. rate, unN
Unemp. rate differences, CV
Social surplus, S
Counterfactual (30% productivity up)
Overall unemp. rate, unN
Unemp. rate differences, CV
Social surplus, S

Robustness check
(3)
(4)

(1)

(2)

(5)

0.0256(−43.7)0(−100)
466.5(28.8)

0.0260(−42.8)0(−100)
468.6(29.3)

0.0127(−72.0)0(−100)
478.7(41.2)

0.0254(−44.1)0(−100)
140.9(23.3)

0.0248(−49.5)0(−100)
440.9(31.7)

0.0224(−46.4)0(−100)
532.6(31.8)

0.0328(−27.9)0.126(−30.7)483.9(33.6)

0.0333(−26.8)0.125(−31.3)484.1(33.6)

0.0211(−53.6)0.0451(−75.2)424.2(25.1)

0.0328(−27.9)0.126(−30.7)152.2(33.2)

0.0354(−28.0)0.179(−4.27)446.4(33.3)

0.0311(−25.5)0.141(−25.0)540.1(33.6)

Table 3. Counterfactual results.
Notes: Robustness check columns represent different cases: (1) FEIV estimation of the matching function, (2) Concave moving costs, (3) Higher discountrate, (4) Years 2000-2004, (5) Years 2005-2009. Percentage changes are in parentheses.

Parameters
Benchmark
δ
ρ
η
t
k

0.16
0.01510.512
5.348
0.0196(1)
0.16
0.0151
0.574
4.997
0.0127

ValuesRobustness check
(2)
(3)
(4)
0.16
0.16
0.16
0.01510.05
0.0159
0.512
0.512
0.456
5.399
4.445
6.685
0.00587 0.0196 0.0292

(5)
0.16
0.01340.608
4.852
0.00851

Table 4. Alternative parameter values.
Notes: Columns represent different cases: (1) FEIV estimation of the matching function, (2) Concave moving costs, (3) Higher discount rate, (4) Years
2000-2004, (5) Years 2005-2009

Variables
Number of active job openings
Number of active job applicants
Number of job placements

Observations
460
460
460

Table 5. Descriptive statistics of data used in estimating the matching function

Mean
311451506855.727502.22

SD
353851.6475205.2
25779.06

Min
53409
99061
7273

Max
27155212630961
194951(i)FE
0.512∗∗∗(0.0140)(ii)
FEIV
0.574∗∗∗(0.0168)(iii)
FE
0.456∗∗∗(0.0195)(iv)
FE
0.608∗∗∗(0.0158)

Constant−2.574∗∗∗(0.00780)−2.579∗∗∗(0.00852)−2.568∗∗∗(0.0136)−2.582∗∗∗(0.00663)

Sample periods
Observations
Adjusted R2

2000 − 2009
460
0.794

2000 − 2009
322

2000 − 2004
230
0.841

2005 − 2009
230
0.851

Estimation peocedures
Estimated η

Table 6. Estimation results of the matching function.
Notes: Standard errors are in parentheses. "*", "**", and "***" represent p < 0.10, p < 0.05, and p < 0.01, respectively.

Estimation procedures
FE
FEIV
FE
FE

years
2000 − 20092000 − 20092000 − 2004
2005 − 2009

Observations
46
46
46
46

Table 7. Descriptive statistics of estimated regional matching efficiency.

Mean
1.035833
1.033825
1.04095
1.031262

SD
0.2626529
0.2502585
0.2944334
0.2372374

Min
0.5176032
0.514943
0.5219792
0.5140291

Max
1.691664
1.514431
2.035077
1.46129Structure of the model

Migration cost t12
Region H

Region 1

Region 2

Employed

Employed….

Employed

Unemployed

Unemployed….

Unemployed

Job searchFirms:
Wage posting
Free entry

Firms:
Wage posting
Free entry

Figure 1. Description of the model

Firms:
Wage posting
Free entry

0.07
0.065

Osaka

0.06

unemployment rate

0.055
0.05

Overall unemployment
rate (0.0455)

0.045

Tokyo

0.04
0.035
0.03

Fukui
0.025
0.02
3

5

7

9

11

13

15

output per employed worker (in million yen)

Figure 2: Prefectural output per employed worker and unemployment rates in Japan averaged
over the years 2000‐2009
Note: Dots represent the prefectural unemployment rates and the thick line represents theoverall unemployment rate