Heterogeneous labor and agglomeration over generations

Regional Science and Urban Economics 77 (2019) 367–381

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Heterogeneous labor and agglomeration over generations☆ Ryusuke Ihara Faculty of Economics, Asia University, 5-8 Sakai, Musashino-shi, Tokyo, 180-8629, Japan

A R T I C L E

I N F O

JEL classification: R12 R23 J61 Keywords: Heterogeneous labor Generational transition Migration Agglomeration economies

A B S T R A C T

Productivity in cities is enhanced by diverse workers from various regions and countries. However, agglomeration can homogenize the workers over time. To investigate the transition of labor diversity in the agglomeration process, this paper presents a two-region non-overlapping generations model. Workers are assumed to be differentiated in terms of their birthplaces, and the distribution of the birthplaces depends on their previous generation’s residency choices. As a main result, this paper shows that the generational transition changes the birthplace distribution, which allows the workers to keep migrating to the core region. The agglomeration of workers results in a loss of labor diversity. On the other hand, the social welfare is maximized by an even distribution of birthplaces, which involves a persistent interregional circulation of workers. In addition, the following two extensions of the model explore the possibility to achieve the social optimum: A housing consumption results in a dispersion of the birthplaces that maximizes the social welfare; Additional agglomeration economies due to the amount of labor results in an over-concentration of the birthplaces, compared with the social optimum.

1. Introduction Can cities benefit from labor diversity forever? With the progress of urbanization coupled with a structural change in industries, recent decades have witnessed an increasing interest in labor diversity. As Jacobs (1961) pointed out, labor diversity is a source of productivity in urban areas. Cities attract heterogeneous workers who have different values, cultures, and ways of thinking, which enhances urban productivity. In fact, as is widely noted, the prosperity of Silicon Valley is reinforced by foreign workers from various countries. However, such a benefit might not last forever because workers in cities could become homogenized over time. As Berliant and Fujita (2008, 2012) pointed out, the heterogeneity of workers will decline if they communicate and collaborate together for a long time because their knowledge and way of thinking are gradually homogenized. In

other words, if the workers concentrate in a certain city for a long time, it will reduce labor diversity. Such a negative aspect of agglomeration gets more serious when we consider the transition of generations. Some of the significant parts of labor heterogeneity are formed under the influence of one’s regional environment of origin, e.g., culture, lifestyle, habits, language, etc. Therefore, even though heterogeneous workers from various regions and countries migrate to a city, their children (i.e., the next generation workers) will become homogenous because they grow up in the same circumstances. That is to say, the generational transition results in the loss of labor diversity via agglomeration. The negative aspect of agglomeration may be particularly apparent in countries with an over-concentration of population in certain cities, which is typically observed in Asian countries such as Japan and South Korea. For instance, Fig. 1 shows the transition of the population ratios

☆ This study is conducted as a part of the project “Innovation Enhancing Regional Economic Structure and Evolution of Cities” undertaken at the Research Institute of Economy, Trade and Industry (RIETI). This study is also supported JSPS KAKENHI Grants Numbers 16K17118 and 16K04626. I am grateful to the anonymous referees and the participants of the presentations at the Applied Regional Science Conference (the University of Tokyo), the European Regional Science Association (the University College of Cork), and the workshops in the RIETI. E-mail address: [email protected].

https://doi.org/10.1016/j.regsciurbeco.2019.06.003 Received 24 May 2018; Received in revised form 22 May 2019; Accepted 11 June 2019 Available online 23 June 2019 0166-0462/© 2019 Elsevier B.V. All rights reserved.

R. Ihara

Regional Science and Urban Economics 77 (2019) 367–381

Fig. 1. Transition of the population ratios in Japan (%).

in Japan.1 Over the past four decades, the scale of inter-prefectural migration has decreased by half, which suggests that more people have been staying in the same cities for a longer period. As a typical example, we observe that in Osaka M.A., the ratio of nonnative residents has been decreasing, while the population of the city has remained almost unchanged. Tokyo M.A. also has a similar tendency. Fig. 1 shows that the ratio of nonnatives significantly decreased during the year when the population size grew slowly. In those cases, stagnant interregional migration has caused a reduction in the diversity of workers based on their regional origin.

There have been numerous recent studies on the positive aspect of labor diversity.2 For instance, Ottaviano and Peri (2005, 2006) and Bellini et al. (2013) studied the effects of cultural heterogeneity on urban productivity in the United States and Europe, respectively. Sparber (2009) also showed that racial diversity has a positive effect on wages in the United States, especially in legal services, computer manufacturing, and computer software. Iranzo et al. (2008) and Navon (2010) explained that diversity in skills and knowledge is beneficial for firms and plants. Furthermore, Florida (2002) emphasized that regional creativity is explained by cultural heterogeneity and people’s tolerance to it. In a theoretical study, Berliant and Fujita (2008, 2012) presented the micro-foundation of knowledge creation using an idea of “tables at a Chinese restaurant” and pointed out the role of culture and communication costs in it. The role of labor diversity in agglomeration has been studied mainly in the context of labor pooling. For instance, Duranton and Puga (2004) grouped the sources of agglomeration into three categories: sharing, matching, and learning. In a seminal work on the matching in labor market, Kim (1987) developed a model with heterogeneous workers and firms. Since this work, Kim (1991) related labor heterogeneity with the size of city, and Helsley and Strange (1990) studied the inefficiency in the system of cities. Abdel-Rahman and Wang (1995) also explored the interregional disparity in a core-periphery structure. On the learning effect, Combes and Duranton (2006) compared the cost and benefit of poaching workers having heterogeneous knowledge. Additionally, the agglomeration process of economic activities is adequately investigated in the field of new economic geography (NEG). Originating with Krugman (1991) and Fujita et al. (1999), numerous

1 Tokyo M.A. consists of Tokyo, Kanagawa, Saitama, and Chiba prefectures. Osaka M.A. consists of Osaka, Kyoto, Hyogo, and Nara prefectures. The ratio of nonnative residents in each metropolitan area is derived from the differences in the population according to the census and the Koseki (which means family register). The population census refers to the number of the current residents in each prefecture. On the contrary, Koseki is a system unique to Japan in which the population is calculated from the location noted in the family register and is usually a person’s birthplace or the place registered as their homeland. Therefore, Koseki provides helpful information and is the only indicator that describes the distribution of people’s origin or native identity in Japan. In Fig. 1, the ratio of nonnative residents in Tokyo is decreasing, although the population of Tokyo is still increasing (despite the declining birthrate). One of the reasons is that some of the immigrants in Tokyo changed the location of their Koseki to Tokyo. For example, people have to create a new Koseki when they get married; some people change their Koseki for legal convenience or because of their attachment to their place of residence, and so on. People’s identity is affected not only by their place of origin but also by the places they reside in for a long time. Therefore, the decline in the ratio of nonnatives in Tokyo in Fig. 1 means an increase in the ratio of longtime residents. Such a trend suggests the homogenization of workers in Tokyo.

2

Labor heterogeneity could be of little consequence (or sometimes even harmful) to some traditional production sectors and industries, while it is still quite important and positively affects creative industries such as R&D and hightech industries. This paper largely focuses on the positive effects. 368

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Regional Science and Urban Economics 77 (2019) 367–381

where ritK and wjit are the reward to capital and the wage for labor, respectively, and p is the price of consumption goods. The profit maximization yields the following factor prices:

studies have attempted to explain the cumulative agglomeration of economic activities in ongoing globalization processes. Relating to labor heterogeneity, Ottaviano and Prarolo (2009) demonstrated that multicultural cities emerge when interregional communication is easy, and Amiti and Pissarides (2005) introduced the matching process in labor market into the framework of the NEG. While these works mainly consider the positive aspects of labor diversity in agglomeration economies, this paper adds a new viewpoint: agglomeration reduces labor diversity over time. The key element is the transition of generations. Supposing workers are differentiated in terms of their birthplaces, the residential choice of workers in each generation affects the next generation’s (i.e., their children’s) labor diversity. In such a case, the birthplace distribution continues to change over generations, which allows the workers to keep migrating to a certain region in the long run. This results in an over concentration: all the workers will stay in the core region, which leads to diminished labor diversity. To see the role of generational transition in determining labor diversity in the agglomeration process, this paper presents a two-region nonoverlapping generations model. The remainder of this paper is organized as follows. Section 2 presents the framework of the model, and Section 3 considers the essential case that will aid in providing the main results of the paper. Sections 4 and 5 expand the analysis by introducing housing consumption and agglomeration economies due to the amount of labor. Finally, Section 6 concludes this paper.

ritK = (1 − 𝛼)pAit Kit−𝛼

2 ∑

l𝛼jit ,

(3)

j =1 1 wjit = 𝛼 pAit Kit1−𝛼 l𝛼− . jit

(4)

Capital is assumed to be freely mobile between the regions. Letting K K = r2t yields the capital distribution: K = K1t + K2t , r1t A1t (l𝛼11t + l𝛼21t )1∕𝛼 K1t = 1∕𝛼 . 1∕𝛼 𝛼 K A1t (l11t + l𝛼21t )1∕𝛼 + A2t (l𝛼22t + l𝛼12t )1∕𝛼 1∕𝛼

(5)

This means more capital is located in a more populous region. Workers can also choose their residential regions. However, the workers migrating from their home region to the other (foreign) one must incur adjustment costs for cultural differences (such as language, customs, lifestyle, habits). Supposing the adjustment costs take an iceberg form, the amount of foreign effective labor that is available for production is rewritten as ljit = Ljit 𝜏,

for i ≠ j,

(6)

where Ljit is the actual number of foreign workers, and 𝜏 ∈ (0, 1) is the easiness to adjust. Consequently, the effective wage that each worker receives is written as

2. The model To investigate the relation between labor diversity and the agglomeration process, this section develops a two-region non-overlapping generations model, whose structure is summarized as follows. The model is divided into two stages, the short-run and the long-run. The short-run stage considers the workers’ choice of residence, while the long-run stage considers the generational transition. The workers are differentiated in terms of their birthplaces, and the residential distribution of each generation determines the birthplace distribution of the next generation. In this sense, the transition of generations means the transition of labor diversity. The sequence of the residential choices over periods leads the economy to a steady-state.

Wjit = wjit 𝜏,

for i ≠ j.

(7)

On the other hand, local workers remaining in their home region do not incur any adjustment costs, thus liit = Liit and Wiit = wiit . As a result, effective wages are expressed as 1 W11t = 𝛼 pA1t K 1−𝛼 L𝛼− 11 1∕𝛼

[

×

]1−𝛼

(L𝛼11 + (L21 𝜏)𝛼 )1∕𝛼

,

A1t (L𝛼11 + (L21 𝜏)𝛼 )1∕𝛼 + A2t (L𝛼22 + (L12 𝜏)𝛼 )1∕𝛼 1∕𝛼

1∕𝛼

(8)

W12t = 𝛼 pA2t K 1−𝛼(L12 𝜏)𝛼−1 𝜏 1∕𝛼

2.1. Production sector

[

×

Following Ottaviano and Peri (2005), homogenous consumption goods are produced using labor and capital. Assuming labor is differentiated in terms of workers’ origins, the production function is expressed as Qit = Ait Kit1−𝛼

2 ∑

l𝛼jit ,

1∕𝛼

(9)

1∕𝛼

[

×

(1)

wjit ljit ,

,

1∕𝛼

1 W22t = 𝛼 pA2t K 1−𝛼 L𝛼− 22

]1−𝛼

(L𝛼22 + (L12 𝜏)𝛼 )1∕𝛼

, (10)

A1t (L𝛼11 + (L21 𝜏)𝛼 )1∕𝛼 + A2t (L𝛼22 + (L12 𝜏)𝛼 )1∕𝛼 1∕𝛼

1∕𝛼

W21t =𝛼 pA1t K 1−𝛼 (L21 𝜏)𝛼−1 𝜏 1∕𝛼

where Ait means the level of technology in region i in period t, Kit is the amount of capital located in region i, ljit is the number of workers who were born in region j but reside and work in region i, and 𝛼(< 1) means the labor intensity. Note that this function exhibits a love of variety in terms of labor input. For instance, comparing the two cases, (i) l1it = l2it = l and (ii) l1it = 2l, l2it = 0, it is evident that the former obtains a larger production amount than the latter.3 The profit function is expressed as 2 ∑

]1−𝛼

A1t (L𝛼11 + (L21 𝜏)𝛼 )1∕𝛼 + A2t (L𝛼22 + (L12 𝜏)𝛼 )1∕𝛼

j =1

𝜋it = pQit − ritK Kit −

(L𝛼22 + (L12 𝜏)𝛼 )1∕𝛼

[

×

(L𝛼11 + (L21 𝜏)𝛼 )1∕𝛼

A1t (L𝛼11 + (L21 𝜏)𝛼 )1∕𝛼 + A2t (L𝛼22 + (L12 𝜏)𝛼 )1∕𝛼 1∕𝛼

1∕𝛼

]1−𝛼

.

(11)

The terms in squared parentheses represent the effect of capital distribution on productivity, and Lij on the right edge represent the direct effect of labor supply. In the final section of this paper, we will examine the effect of introducing agglomeration economies that depends on the amount of labor. For this purpose, we introduce the following equation:

(2)

j =1

Ait = (1 + Liit + Ljit 𝜏)𝛾 .

(12)

This equation means that the technology level of each region is increasing with the total amount of effective labor supply, where 𝛾 captures the degree of the agglomeration economies according to the amount of labor.

3

In this production function, labor input is in a CES form, where the elasticity of substitution of labor is assumed to equal to 1∕(1 − 𝛼) to make the model tractable. Such a restriction maintains the generality of the results in this paper. 369

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2.2. Consumption behavior

sity of the next generation is endogenously determined by the residential distribution of the current generation. Based on this equation, the interregional distribution of births will converge to a steady state.

Consumption behavior is described as follows. The utility function of each worker born in region j and resides in region i is 1−𝜇 𝜇

Ujit = cjit hjit ,

3. Essential analysis

(13)

where cjit and hjit represent the amounts of consumption goods and housing land, respectively, and 𝜇(< 1) is the consumption share for H h , where housing. The budget constraint is given by yjit = pcjit + rjit jit yjit represents personal income and ritH represents housing rent. Utility maximization yields the following optimum consumption functions:

We first consider an essential case which eliminates both the housing land consumption and agglomeration economies due to the amount of labor. This means 𝜇 = 0 and 𝛾 = 0 in this section. Although we will expand the analysis in the next sections, the main result of this paper is derived in this section.

cjit = (1 − 𝜇)yjit ∕p,

(14)

3.1. Short-run analysis of residential choice

hjit = 𝜇 yjit ∕ritH .

(15)

Workers born in each region compare the indirect utilities they can enjoy at their home region with those in the foreign region; thus, their residential distribution in equilibrium is determined by equating the indirect utilities they can enjoy in the two regions. Note that the migration behavior of the workers born in each region takes the other region’s residential distribution (as well as the birthplace distribution) as exogenous. The equilibrium distribution, 𝜆∗1t and 𝜆∗2t , is determined by the interaction between the residential choices in the two regions. More specifically, the residential choices in the two regions are derived from the equations V11t = V12t and V22t = V21t , which are, respectively, given by ( )𝛼 [ ( )𝛼 ] (1 − 𝜆2t )(1 − L1t )𝜏 𝜆2t (1 − L1t ) = 1+ 𝜏 1∕(1−𝛼) , (20) 1+ 𝜆1t L1t (1 − 𝜆1t )L1t 𝜏 ( )𝛼 [ ( )𝛼 ] (1 − 𝜆1t )L1t 𝜏 𝜆1t L1t 1+ = 1+ 𝜏 1∕(1−𝛼) . (21) 𝜆2t (1 − L1t ) (1 − 𝜆2t )(1 − L1t )𝜏

In a housing market, the rent is given by ritH = 𝜇

Liit yiit + Ljit yjit Hi

,

(16)

where Hi is the total amount of land for housing. As a result, the indirect utility is given by )1−𝜇 ( 𝜇 yjit Hi 1−𝜇 Vjit = . (17) p (Ljj yjj + Lij yij )𝜇 Assuming that both capital and land for housing are equally owned by all the workers, the income of each worker is expressed as yjit = Wjit + RKt + RHt , ∑2

RKt =

K j=1 rjt Kjt , ∑2 ∑2 i=1 j=1 Lji

∑2

H j=1 rjt Hjt . ∑2 i=1 j=1 Lji

RHt = ∑2

(18)

An accurate description of each indirect utility is shown in Appendix A. To understand the characteristic of the short-run stage, Fig. 2 numerically illustrates these equations as implicit functions in the (𝜆1t , 𝜆2t ) plane. The three panels, respectively, show the cases for different distributions of the birthplaces: L1t = 0.5, 0.55, 0.7. Other parameters are 𝛼 = 0.4 and 𝜏 = 0.5. The horizontal and vertical arrows are the migration choices of workers born in regions 1 and 2, and the resulting equilibrium is indicated by E. Considering the directions of the arrows, we can see that point E is stable. In addition, as L1t increases, the residential choices gradually concentrate in region 1, and finally, it becomes a full concentration in region 1, as shown in the right panel. To understand the basic characteristics of Fig. 2, let us focus on the residential distribution of the workers born in region 1 (see also equation (20)). First, note that labor productivity is affected by the relation between the distributions of capital and labor. Moreover, capital distribution depends on the residential distributions of the workers born in region 1 and in region 2. Therefore, if we consider a case where more workers reside in their home region, the rise in 𝜆1t decreases their productivity in the home region, since the capital movement is restricted to some degree by the exogenous 𝜆2t . This effect is shown by the left-hand side of equation (20), which is decreasing in 𝜆1t . Second, what is the effect of the increase in the number of births in region 1? The rise in L1t reduces the influence of the workers born in region 2 on capital distribution, which reduces the effect of 𝜆1t on their productivity. This effect is shown in equation (20), where the second term of the left-hand side and that of the square parentheses on the right-hand side are decreasing in L1t . On the other hand, the adjustment cost in the foreign region remains unchanged, which

2.3. Short-run and the long-run stages Finally, we have to distinguish the short-run and the long-run stages of this model. The short-run stage analyzes the residential choice of workers in each period. Since workers migrate to regions where they can enjoy higher indirect utility, the short-run equilibrium is given as the residential distribution that equalizes indirect utilities found in the two regions. Now, let Lit be the number of births (i.e., the number of workers who were born) in region i, and 𝜆it be the share of workers who choose to remain in their home region (and work as local workers). Then we can rewrite the number of workers as Liit = 𝜆it Lit and Lijt = (1 − 𝜆it )Lit . Consequently, the share of workers in the short-run equilibrium, 𝜆∗it , can be derived from Viit = Vijt , taking Lit as given. Next, let us organize the notation for analytical simplicity. The total amounts of labor and capital are set as follows, respectively: L1t + L2t = 1 and K1t + K2t = 1. Similarly, the total amount of housing land in each region is set as follows: H1 = H2 = 1. In addition, consumption goods are considered as the numéraire: p = 1. The long-run stage considers the generational transition. The transition of the number of births is given by the following difference equation: L1t +1 = f (L1t ) = 𝜆∗1t (L1t ) · L1t + (1 − 𝜆∗2t (L1t )) · (1 − L1t ).

(19)

That is, the number of workers residing in region 1 in period t, expressed as f(L1t ), equals to the number of workers who will be born in region 1 in the next period, L1t +1 .4 This means that the labor diver4 The difference equations of the two regions are expressed below. Substituting L2t = 1 − L1t in these equations, we can confirm that the equations are the same.

L2t +1 = 𝜆∗2t (L1t , L2t ) · L2t + (1 − 𝜆∗1t (L1t , L2t )) · L1t .

L1t +1 = 𝜆∗1t (L1t , L2t ) · L1t + (1 − 𝜆∗2t (L1t , L2t )) · L2t , 370

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Fig. 2. Residential choice.

makes the home region relatively attractive. Therefore, 𝜆1t is increasing in L1t . Third, we consider the interregional interaction between the residential choices of the workers born in two regions. In Fig. 2, the residential distribution in equilibrium is given by the intersection of the lines Viit = Vijt , which indicates that 𝜆∗1t > 𝜆∗2t for the case of L1t > 1∕2. Note that the interaction between the two regions stimulates the concentration of residents. In other words, an increase in L1t increases 𝜆1t and reduces 𝜆2t . Then the increase in 𝜆1t reduces 𝜆2t more, and vice versa. As a result of the sequence of the interaction, the degree of the concentration of residents becomes larger than that of their birthplaces. Focusing on the symmetric structure allows us to examine the behavior of this model more precisely. First, solving Viit = Vijt under L1t = 1∕2 yields the residential distribution in equilibrium as

𝜆∗it =

1 1 > , 2 1 + 𝜏 𝛼∕(1−𝛼)

Fig. 3. Transitional dynamics of birthplace distribution.

(22) population in a region attracts more workers to migrate to there. In other words, the residential choice becomes more sensitive to the differences in the birthplace distribution. These features are confirmed by equations (22) and (24).5 In addition, note that the adjustment cost is assumed to be exogenously fixed in this study. As Carrington et al. (1996) argued, however, it can be decreasing in the number of immigrants. In other words, as interregional migration increases, interregional communication becomes smoother. We consider this case in Appendix D, which explains that the endogenous adjustment cost enlarges the tendency of concentration. That is, labor heterogeneity causes agglomeration and agglomeration is promoted by the decrease in the communication costs.

and the total derivative of Viit = Vijt yields ( )2 1 − 𝜏 𝛼∕1−𝛼 d𝜆2t || = − 1 − < −1. d𝜆1t ||L1t =1∕2,𝜆it =𝜆∗ 2𝜏 𝛼∕1−𝛼

(23)

it

This equation means that the equilibrium point is stable because the slope of V11t = V12t is steeper than that of V22t = V21t . Appendix A shows the derivation of this equation. On the other hand, the effect of changing the birthplace distribution on the residential distribution is given as follows: d𝜆∗1t | d𝜆 ∗ | 4𝜏 𝛼∕1−𝛼 | = − 2t || = > 0. | dL1t |L1t =1∕2,𝜆it =𝜆∗ dL1t |L1t =1∕2,𝜆it =𝜆∗ 1 − 𝜏 2𝛼∕1−𝛼 it it

(24)

3.2. Long-run analysis of generational transition

Appendix B also explores the corner point (𝜆1t , 𝜆2t ) = (1, 0), where the slope of V22t = V21t is always steeper than that of V11t = V12t . In addition, the core-periphery structure in which all workers are born in one of the two regions is sustainable: V11t ∕V12t ∣L1t =1,𝜆1t =1 = 1∕𝜏 > 1. Finally, let us confirm how a decrease in the adjustment costs affects residential allocation. The important point here is that the lines Viit = Vijt are concave to the origin, and become linear as the adjustment costs fall. To understand this, consider the cases of 𝜏 = 0 and 𝜏 = 1. In the former case, the adjustment cost is infinity, which leads the residential distribution to a perfect division: 𝜆1t = 𝜆2t = 1. In the latter case, the adjustment cost is zero, which makes the equilibrium distribution arbitrary, keeping the relation of 𝜆1t = 1 − 𝜆2t . Intuitively speaking, the decrease in the adjustment costs gives the workers more incentives to migrate to the foreign regions. In addition, the fact that the lines get close to a linear function means that the relation between 𝜆1t and 𝜆2t becomes elastic (in a region with more residents). Therefore, as the adjustment costs fall, a little increase in the births

In the long-run stage, we examine the transitional dynamics of birthplace distribution over periods. As mentioned in Section 2, equation (19) relates the residential distribution of the current generation, f(L1t ), and the birthplace distribution of the next generation, L1t +1 . Fig. 3 numerically shows the residential distribution in relation to the birthplace distribution in each period, for the case of 𝛼 = 0.7 and 𝜏 = 0.6. Considering the generational transition in this figure, we can see that

5 Labor heterogeneity plays an important role in the concentration of residents. To understand this, let us consider a counterexample of this model where all the workers are homogenous while there is an adjustment cost in the foreign region. In such a case, the residential distribution in the equilibrium will be 𝜆∗1t = 𝜆∗2t = 1, which means there are no immigrants in the foreign regions. If we consider the generational transition of workers in this case, the initial distribution of birthplaces remains unchanged over periods.

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the symmetric structure is not stable in the long run, and the birthplace distribution converges to the core-periphery structure across generations. Let us investigate the stability of this symmetric structure in detail. First, we can easily confirm that the symmetric structure is in a steady state, by substituting L∗1t = 1∕2 and equation (22) into equation (19). Then, using equation (24), the total derivative of equation (19) yields d𝜆 ∗ d𝜆 ∗ df || = L1t 1t − (1 − L1t ) 2t + 𝜆∗1t + 𝜆∗2t − 1 | ∗ dL1t |L1t =1∕2,𝜆it =𝜆 dL1t dL1t it

=

1 + 𝜏 𝛼∕(1−𝛼) > 1. 1 − 𝜏 𝛼∕(1−𝛼)

(25)

This equation means that the slope of the line f(L1t ) is larger than 1 at L1t = 1∕2 in Fig. 3. Hence, we obtain the following proposition. Proposition 1. Equal divisions of birthplaces are always unstable in the long run. As explained above, if the birthplace distribution incidentally deviates from the symmetry in a period, the residential distribution of workers becomes more concentrated than that in the beginning. Since residency distribution is turned over to the next generation’s birthplace distribution, accumulation of generational changes leads workers to concentrate in one of the two regions step-by-step. Consequently, the distribution of workers converges to the steady-state equilibrium of L∗1 = 0 or L∗1 = 1. The important point to note is that the cumulative transition of generations keeps the workers migrating to a core region, and extinguishes the labor diversity in the long run.

Fig. 4. Social welfare.

of residence is given by 𝜆∗it = 1∕2, i.e., equal employment in home and foreign regions. The ratio of employment in the home region rises as the adjustment cost increases. Equation (30) shows, on the contrary, the case where the residents concentrate in their home region. Because the concentration of residents results in the loss of labor diversity of the next generation, welfare apparently worsens. Equation (31) implies that social welfare is maximized by choosing 𝜆it = 𝜆∗it for each region. Fig. 4 illustrates the numerical example for the case of 𝛼 = 0.6, 𝜏 = 0.7, and 𝛿 = 0.5. Note that the case of 𝜆1 = 𝜆2 = 1 has a particular characteristic that there is no migration between the regions, as explained in Appendix C. Here a comparison with Proposition 1 gives us the following proposition:

3.3. Social optimum Next, we consider a social welfare function to explore what kind of distributions can maximize the social welfare.6 In this paper, the social welfare function, S W, is given by the sum of the discounted indirect utilities of all the workers in future: SW =

∞ ∑

𝛿 t (S t ) ,

Proposition 2. The social optimum cannot be achieved as a stable steadystate equilibrium.

(26)

In other words, while the social welfare is maximized at the symmetric distribution of the birthplace, the birthplace gradually concentrates in one of the two regions in the long run. In conclusion, generational transition results in an over-concentration of population, which implies the loss of labor diversity through agglomeration. It is helpful to compare the private marginal benefit (PMB) from migration with the social marginal benefit (SMB) in order to understand the reason for the market failure. The PMB of migrating from region j to region i is shown by Viit − Vijt . As examined in the shortrun analysis earlier, workers migrate to region i as long as the PMB is positive, and it becomes zero in equilibrium. On the contrary, considering the SMB, we should notice that the externality exists in the generational transition. As mentioned earlier, parents’ choice of residence determines their children’s birthplace distribution through equation (19). However, parents consider their own indirect utility alone, but do not consider their children’s utility. To the see this, let V Gijt = Vijt + 𝛿(𝜆jt +1 V Gjjt +1 + (1 − 𝜆jt +1 )V Gjit +1 ) be the intergenerational joint utility that evaluates the utility of their children. In this case, the SMB from migration is expressed as V Giit − V Gijt , and the difference between PMB and SMB is the factor in the market failure. More precisely, the effect of the residential distribution on their children’s utility can be examined by the following total derivatives.

t =0

St = V11t 𝜆1t L1t + V12t (1 − 𝜆1t )L1t + V22t 𝜆2t L2t + V21t (1 − 𝜆2t )L2t ,

(27)

where 𝛿(< 1) is a discount factor. Consider a government that chooses 𝜆1 and 𝜆2 to maximize S W in a steady state. Omitting t from equation (19) and solving it for L1 , the birthplace distribution in the steady state is given by

̃ L1 =

1 − 𝜆2 , 2 − 𝜆1 − 𝜆2

(28)

and the social welfare is expressed as S W = S∕(1 − 𝛿). Though the social welfare function S W is somewhat complicated, some of the features can be identified by the following equations. SW ∣L =̃L ,𝜆 =𝜆 = (𝜆1 )𝛼 + ((1 − 𝜆1 )𝜏)𝛼 ∕(1 − 𝛿), 1 1 2 1

(29)

SW ∣L =̃L ,𝜆 =1 = 1∕(1 − 𝛿), 1 1 i

(30)

dSW ∕d𝜆i ∣L =̃L ,𝜆 =𝜆∗ = 0. 1 1 i it

(31)

Equation (29) shows the value of S W in the symmetric structure

̃ L1 = 1∕2. If we consider the case in which there is no adjustment cost (i.e., if 𝜏 = 1), then this equation implies that the optimal distribution

𝜕 Viit+1 d𝜆it+1 𝜕 Viit+1 d𝜆jt+1 𝜕 Viit+1 dL1t+1 dViit +1 = + + , d𝜆it 𝜕𝜆it+1 d𝜆it 𝜕𝜆jt+1 d𝜆it 𝜕 L1t+1 d𝜆it

(32)

𝜕 Vijt+1 d𝜆it+1 𝜕 Vijt+1 d𝜆jt+1 𝜕 Vijt+1 dL1t+1 + + . 𝜕𝜆it+1 d𝜆it 𝜕𝜆jt+1 d𝜆it 𝜕 L1t+1 d𝜆it

(33)

dVijt +1

6

d𝜆it

The social welfare function approach has been broadly adopted in the framework of two-region core-periphery models. For instance, see Charlot et al. (2006), Pflüger and Südekum (2008), and Baldwin et al. (2003). The analysis in this paper follows the line of these articles.

=

These equations show the effect of a slight deviation of residence from the symmetric distribution on the indirect utilities of the next gener372

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Regional Science and Urban Economics 77 (2019) 367–381

Fig. 5. Residential choices.

ation. Note that the sign of the last terms of equations (32) and (33) is negative (see Appendix A for details). This means that an increase in the number of residents has a negative effect on their children through the increase in the number of births. However, the workers do not consider the negative externality in their choice of residence, which leads to the over-concentration of population as a market failure. Considering the fact that workers’ residential choice affects their children’s utility, it might be reasonable to assume that the parents care about their children. For instance, Razin and Ben-Zion (1975) presented a model where the parents’ utility depends on that of their children. In the same manner as their work, if we replace the utility function with the intergenerational joint function, then the externality would be internalized. In such a case, workers hesitate to concentrate in one region because it would diminish their children’s welfare. Consequently, a social optimum would be achieved as a stable steady-state equilibrium.

( )2 𝛼(1 − 𝛼)(1 − 𝜇) 1 − 𝜏 𝛼∕(1−𝛼) + 4𝜇𝜏 𝛼∕(1−𝛼) d𝜆2t || = −1 − . | d𝜆1t |L1t =1∕2,𝜆it =𝜆∗ 2[𝛼(1 − 𝛼) − (1 + 𝛼(1 − 𝛼))𝜇]𝜏 𝛼∕(1−𝛼) it

(34) This implies the slope of the line V11t = V12t at point E becomes positive when 𝜇 > 𝜇 ∗ ≡ 𝛼(1 − 𝛼)∕(1 + 𝛼(1 − 𝛼)). In other words, rise in housing rent caused by the inflow of foreigners drives native-born workers out to the other region. Observing that the slope is larger than unity in this case, we can conclude that the equilibrium is stable for all cases of 𝜇. In addition, the effect on the equilibrium share of a slight change in the symmetrical birthplace distribution is expressed as d𝜆 ∗ d𝜆∗1t | | = − 2t |L =1∕2,𝜆 =𝜆∗ | it dL1t |L1t =1∕2,𝜆it =𝜆∗ dL1t 1t it it

(

)

4(𝛼(1 − 𝛼) − (1 + 𝛼(1 − 𝛼))𝜇)𝜏 𝛼∕(1−𝛼) 1 − 𝜏 𝛼∕(1−𝛼) )( = ( ( )2 ), 𝛼(1 − 𝛼)(1 − 𝜇) 1 − 𝜏 𝛼∕(1−𝛼) + 4𝜇𝜏 𝛼∕(1−𝛼) 1 + 𝜏 𝛼∕(1−𝛼)

4. Housing consumption

(35) The following Sections 4 and 5 extend the above analysis to explore the relation between the steady-state equilibrium and the social optimum. First, we consider a case of 𝜇 > 0 and 𝛾 = 0, where housing consumption is positive. In this case, the location choices of workers is influenced by the change that workers have to pay a housing rent while they earn the rent income.

which is negative when 𝜇 > That is, a region with a larger births population attracts workers from the other region as discussed in Section 3. However, if housing consumption is sufficiently large, the negative effect of the rise in housing rent pushes out more residents than the rise in the number of births.

4.1. Short-run analysis of residential choice

4.2. Long-run analysis of generational transition

The effect of housing consumption on the short-run equilibrium is as follows. A greater dispersion tendency will be observed compared with the previous section, because of the additional dispersion force. To observe this, first confirm the residential choice under the symmetric birthplace distribution. Numerical examples are illustrated in Fig. 5, for the case of 𝛼 = 0.6, 𝜏 = 0.3, and L1t = 0.5. Although the slopes of the lines change from negative to positive, depending on the value of 𝜇, the direction of the arrows reveals that the equilibrium points are stable in both the cases. More precisely, the choice of residence in equilibrium is given by equation (22), the same as that in Section 3, and the stability of the equilibrium point E is examined by the following equation:

The generational transition is examined here. As shown in the shortrun equilibrium, the introduction of housing consumption reinforces the dispersion force in this model. Consequently, two different patterns of transitional dynamics emerge, as shown by Fig. 6. The left panel is for a lower level of housing consumption, 𝜇 = 0.05, and the right panel is for a higher level of it, 𝜇 = 0.2. Other parameters are set as 𝛼 = 0.6 and 𝜏 = 0.5. In the left panel, the symmetric structure is unstable, and the birthplace distribution approaches to the steady-state equilibrium, numerically given by 𝜆∗1 ≃ 0.998, 𝜆∗2 ≃ 0.001 and L∗1 ≃ 0.998, or their reversed equations. however, the right panel shows that, if housing consumption is sufficiently large, the dispersion structure becomes stable. In this case, the steady-state equilibrium is given by 𝜆∗1 = 𝜆∗2 ≃ 0.739 thus L∗1 = 0.5.

𝜇∗ .

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Fig. 6. Transitional dynamics of birthplace distribution.

Fig. 8. Social welfare. Fig. 7. The value of df(Lit )∕dLit .

Similarly to Section 3, the total derivative of equation (19) and equation (35) derive the slope of the line f(L1t ) at the symmetry as ( ) 𝛼(1 − 𝛼)(1 − 𝜇) 1 − 𝜏 2𝛼∕(1−𝛼) df || = . (36) ( ) | 2 dLit |L1t =1∕2,𝜆it =𝜆∗ 𝛼(1 − 𝛼)(1 − 𝜇) 1 − 𝜏 𝛼∕(1−𝛼) + 4𝜇𝜏 𝛼∕(1−𝛼) it

𝛼(1 − 𝛼)(1 − 𝜏 𝛼∕(1−𝛼) ) . 2 + 𝛼(1 − 𝛼)(1 − 𝜏 𝛼∕(1−𝛼) )

(38)

SW ∣L =̃L ,𝜆 =1 = 1∕(1 − 𝛿), 1 1 i

(39)

and Fig. 8 depicted for the case of 𝛼 = 0.6, 𝜇 = 0.1, 𝜏 = 0.6, and 𝛿 = 0.5. Comparing the social optimum with the steady-state equilib-

This function is apparently decreasing in 𝜇 , and is less than unity when

𝜇 > 𝜇 ∗∗ ≡

SW ∣L =̃L ,𝜆 =𝜆 = 2𝜇 ((𝜆1 )𝛼 + ((1 − 𝜆1 )𝜏)𝛼 )1−𝜇 ∕(1 − 𝛿), 1 1 2 1

rium, the following proposition is obtained.

(37)

Proposition 3. When the housing consumption is sufficiently large, social welfare is maximized in a steady-state equilibrium.

That is, the symmetric birthplace distribution becomes stable when the share of housing consumption is sufficiently large. Fig. 7 shows equation (36) as a function of 𝜇 for the case of 𝛼 = 0.6 and 𝜏 = 0.5. Note also that 𝜇∗∗ is decreasing in 𝜏 . This implies that a fall in adjustment costs increases the tendency to dispersion. Intuitively, as adjustment costs reduce, workers’ choice of residence becomes indifferent between the two regions in comparison to the dispersion force from housing consumption. Consequently, it can be confirm that the housing consumption introduces an additional dispersion force to this model.

In this case, the number of births is equal between the regions. The housing consumption as a dispersion force can make steady-state equilibrium socially optimal. To understand what this result means, noticed that the intergenerational externality does not always exist in this section. Recall the discussion about the difference between the PMB and the SMB from migration. As discussed in Section 3, a deviation from the symmetric residential distribution 𝜆1t = 𝜆2t = 𝜆∗it changes the number of births from the symmetry ̃ L1 = 1∕2. For instance, an increase in the number of births in region 1 affects their indirect utilities in the following two ways: The first is the same as that in the previous section, which worsens their indirect utilities regardless of their residential regions. The second is through housing consumption, which worsens (raises) the indirect utilities of the workers residing in region 1 (region 2). (See the total

4.3. Social optimum Adopting the same analysis as in Section 3, it can be observed that the welfare is maximized in the symmetric structure with 𝜆∗it = 1∕(1 + L1 = 1∕2, as indicated by 𝜏 𝛼∕(1−𝛼) ) and ̃

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Fig. 9. Residential choice.

differentials in Appendix A for details.) Thus, as examined earlier, a sufficiently large housing consumption pushes out the residents to a less populous region, which could make the dispersion of residence stable. In this case, the second effect restricts the first effect. In other words, a generation’s residential choice does not affect the future distribution of births.

that there is only one equilibrium where residents disperse among regions, while the middle and right panels have multiple equilibria due to the agglomeration economies introduced to this short-run stage. More precisely, in the middle panel with 𝛾 = 0.5, there are five equilibria: one stable dispersion equilibrium, two unstable non-full concentration equilibria, and two sustainable concentration equilibria. In the right panel with 𝛾 = 1, the dispersion equilibrium becomes unstable even in the short run. As a result, the residential distribution tends to concentrate in one of the two regions, as the degree of agglomeration economies increases.7 The stability of the symmetric birthplace distribution is examined as follows. The choice of residence in the symmetric equilibrium, 𝜆∗it , is given by equation (22), as in the previous sections, but the equilibrium is no longer always stable. The slope of the line V11t = V12t at the symmetric structure is expressed as

5. Agglomeration economies due to the amount of labor The above Sections 3 and 4 have explained that a symmetric birthplace distribution maximizes social welfare. However, can the social optimum not be achieved by a concentration structure in any cases? To explore the possibility that the core-periphery structure of birthplace distribution may become a social optimum, Section 5 introduces agglomeration economies due to the amount of labor into

d𝜆2t || = d𝜆1t ||L1t =1∕2,𝜆it =𝜆∗ it

( )( ) 3 + 𝜏 𝛼∕(1−𝛼) + (1 + 𝜏)𝜏 𝛼∕(1−𝛼) 𝜏 𝛼∕(1−𝛼) + 𝜏 −𝛼∕(1−𝛼) ( ) . 𝛾(1 + 𝜏)(1 + 𝜏 𝛼∕(1−𝛼) ) + 𝛼(1 − 𝛼) 3 + 𝜏 𝛼∕(1−𝛼) + (1 + 𝜏)𝜏 𝛼∕(1−𝛼)

𝛾(1 + 𝜏)(1 + 𝜏 𝛼∕(1−𝛼) ) − 𝛼(1 − 𝛼)

the short-run stage, by considering equation (12) with 𝛾 > 0. On the other hand, we eliminate the housing consumption by setting 𝜇 = 0 for the tractability of the model.

(40)

See Appendix A for the derivation. The value of equation (40) becomes larger than −1 when 𝛾 gets sufficiently large, which indicates that the symmetric equilibrium becomes unstable as the degree of agglomeration economies increases. (Note that the case of 𝛾 = 0 corresponds to that in Section 3.) In addition, if 𝛾 becomes even more larger, the sign of equation (40) becomes positive. This means the slope of the line L11t = L12t in Fig. 9 becomes positive at the intersection with the other line, and the line becomes a broken line. Next, the marginal effect of a deviation of the births distribution from the symmetry is given by the following equation:

5.1. Short-run analysis of residential choice Because of the introduction of the new agglomeration economies due to the amount of labor, there can exist multiple equilibria in the stage of the residential choice. For a better understanding, Fig. 9 numerically shows the residential allocation for three different levels of the agglomeration economies: 𝛾 = 0.2, 0.5, and 1. The other parameters are set as 𝛼 = 0.6 and 𝜏 = 0.4. The solid line indicates that the residential allocation is stable, while the broken line indicates that it is unstable in the short run. The unstable allocation implies that a small group of workers moving from the distribution on the line would, in turn, stimulate more migration. Thus, the unstable allocation may be broken by any incidental event. Comparing these panels, we can see that the equilibrium structure has the following characteristics. The left panel with 𝛾 = 0.2 shows

7 Considering the change in the level of adjustment costs, it can be seen that the residential distribution changes from dispersion to concentration as the adjustment costs reduce. As discussed in Section 3, the fall in the adjustment costs stimulates the concentration of residents, and this is reinforced by the agglomeration economies due to the amount of labor.

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place distribution for the case of 𝛼 = 0.6, 𝜏 = 0.3 and 𝛾 = 1. The solid and broken lines of f(L1t ) represent the stable and unstable distributions of residents in the short-run stage, respectively. As a result, we can conclude that the residential choices in the short-run stage reinforce the concentration of birthplaces in the long-run stage. 5.3. Social optimum To examine the social optimum, it should be noticed that there are two types of externalities in this section. The first is the positive externality arising from the agglomeration economies due to the amount of labor, which implies that the optimal residential distribution should be more concentrated than the equilibrium. The second is the intergenerational negative externality, which implies that the optimal births distribution should be more even than the equilibrium. If the degree of agglomeration economies due to the amount of labor is not sufficiently large, then the latter effect dominates the former one. In this case, the

Fig. 10. Transitional dynamics of birthplace distribution.

d𝜆∗1t | d𝜆 ∗ | 4𝜏 𝛼∕(1−𝛼) | ) = − 2t || = ( | dL1t |L1t =1∕2,𝜆it =𝜆∗ dL1t |L1t =1∕2,𝜆it =𝜆∗ 1 + 𝜏 𝛼∕(1−𝛼) it it ( )( ) ( )( ) 𝛼(1 − 𝛼) 3 + 2𝜏 𝛼∕(1−𝛼) + 𝜏 1∕(1−𝛼) 1 − 𝜏 𝛼∕(1−𝛼) + 𝛾 1 − 𝜏 1∕(1−𝛼) 1 + 𝜏 𝛼∕(1−𝛼) × ( )( )2 ( ), 𝛼(1 − 𝛼) 3 + 2𝜏 𝛼∕(1−𝛼) + 𝜏 1∕(1−𝛼) 1 − 𝜏 𝛼∕(1−𝛼) − 2𝛾(1 + 𝜏)𝜏 𝛼∕(1−𝛼) 1 + 𝜏 𝛼∕(1−𝛼) which is increasing in 𝛾 . That is, the tendency that a region with a larger number of births attracts foreign workers is reinforced by the agglomeration economies due to the amount of labor. Finally, if all the workers are born in one region, it is evident that the concentration of residents in their home region is always sustainable because V11t ∕V12t ∣L1t =1,𝜆1t =1 = 2𝛾∕𝛼 ∕𝜏 > 1.

result is the same as that in Section 3, which implies that the symmetric structure is socially optimal while the concentration structure is obtained as the steady-state equilibrium. However, if the degree of agglomeration economies becomes sufficiently large, then the former exceeds the latter. In such a case, the symmetric structure no longer maximize the social welfare. This brings us to the final question: can the full agglomeration structure be socially optimal? Although the social welfare function is still complex, the following examination can provide some of its characteristics. )𝛾 (𝜆 )𝛼 + ((1 − 𝜆1 )𝜏)𝛼 ( 1 1 1 + 𝜆1 + (1 − 𝜆1 )𝜏 , (43) SW ∣L =̃L ,𝜆 =𝜆 = 1 1 1 2 1 1−𝛿 2 2

5.2. Long-run analysis of generational transition The short-run analysis above reveals that the population can be concentrated in one of the two regions when the degree of the agglomeration economies due to the amount of labor is sufficiently large. In such a case, the birthplaces are also concentrated in the region in the steady state. However, even if the residential distribution disperses in the short-run equilibrium, the accumulation of the residential choices across periods can lead the birthplace distribution to a concen-

SW ∣L =̃L ,𝜆 =1 = 2𝛾 ∕(1 − 𝛿), 1 1 i

( ) (𝜆𝛼2 + ((1 − 𝜆2 )(1 − 𝜏))𝛼 )1∕𝛼 2𝛾 𝜏 𝛼 𝛼 lim SW ∣L =̃L = lim − − 1 1 𝜆1 →1 𝜆1 →1 (1 − 𝛿)(1 − 𝜆1 )1−𝛼 2𝛾(1−𝛼)∕𝛼 (1 − 𝛿)(1 − 𝜆2 ) ( )) ( 2𝛾 𝛾((1 − 𝜆2 )(1 − 𝜏) + 1) + 2 𝛼 + (1 − 𝛼)2−𝛾∕𝛼 (𝜆𝛼2 + ((1 − 𝜆2 )𝜏)𝛼 )1∕𝛼

+

(41)

2(1 − 𝛿)(1 − 𝜆2 )

(44)

(45)

= −∞.

It is important to note that the social welfare certainly deteriorates as 𝜆i → 1. This indicates the negative effect of loosing labor diversity

trated structure in the long run, as shown in Section 3. Using equation (41), the total derivative of equation (19) at the symmetric structure is expressed as df || 1 + 𝜏 𝛼∕(1−𝛼) = dLit ||Lit =1∕2,𝜆it =𝜆∗ 1 − 𝜏 𝛼∕(1−𝛼) (

× 1+

it

4𝛾𝜏 𝛼∕(1−𝛼) (1 + 𝜏 1∕(1−𝛼) ) ( )( )2 ( ) 𝛼(1 − 𝛼) 3 + 2𝜏 𝛼∕(1−𝛼) + 𝜏 1∕(1−𝛼) 1 − 𝜏 𝛼∕(1−𝛼) − 2𝛾(1 + 𝜏)𝜏 𝛼∕(1−𝛼) 1 + 𝜏 𝛼∕(1−𝛼)

The value of this equation is larger than 1 even when 𝛾 is sufficiently small, which means that the symmetric structure is always unstable. Fig. 10 shows a numerical example of the transitional dynamics of birth-

)

(42)

.

exceeds the agglomeration economies due to the amount of labor, as the population gets concentrated. In other words, a moderate concentration is needed for the social optimal. Here, we obtain the final proposition:

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Fig. 11. Social welfare.

Proposition 4. Even if there exist agglomeration economies in the shortrun stage, the birthplaces distribution in the steady-state equilibrium will still be concentrated compared with the social optimum.

More precisely, the short-run analysis on residential distribution and the long-run analysis on the generational transition derived the following results. (i) Even if the residential choice of workers in each period kept them dispersed between the two regions, the cumulative residential choices over periods gradually led the birth population to concentrate in one of the two regions. Consequently, all workers continued to reside in one region in a steady-state equilibrium, which would remove labor diversity from this economy. (ii) Social optimum was given by an equal division of births between the regions, which was not achieved by a steady-state equilibrium. In other words, a dynamic inefficiency due to the generational transition was observed. (iii) When we introduced housing land consumption, social welfare could be maximized in a steady-state equilibrium wherein the number of births was evenly divided between the regions. (iv) By contrast, even when we introduced agglomeration economies in the short-run stage, the birthplace distribution in the steady-state equilibrium was still over-concentrated compared with that in the social optimum. Result (iv) explains well the current problem of Japan. The population share of the Tokyo Metropolitan Area (composed of Tokyo, Kanagawa, Chiba, and Saitama prefectures) gradually increased from 13.7% in 1920 to 28.4% in 2015. The expansion of the Tokyo metropolitan area may be interpreted as the path to a steady-state equilibrium of concentration. Considering that the social optimum requires a more moderate concentration, we have a concern about the decline in labor diversity in Japan. Then how can we achieve the socially optimal distribution? Our results can suggest a couple of measures for this problem. The first measure is to introduce a taxation system. This model has two sources of market failure. The first is the agglomeration economies due to the amount of labor, which stimulates the residential concentration in the short-run stage. However, there exists a more important source of market failure in the long-run stage: the overgenerational externality. In particular, each generation’s residential choice affects their children’s birthplace distribution, that is, the next generation’s labor heterogeneity. However, if each generation does not consider the effect on the next generation, a “dynamic inefficiency” emerges. In such a case, a Pigovian tax (or subsidy) can be effective in handling market failure. If a government levies sufficient tax on the residents of a populous region, it will reduce the incentive of residential concentration. Managing the birthplace distribution of the next generation by this policy, the socially optimal dispersion could be obtained as a steady-state equilibrium.

Fig. 11 numerically illustrates SW with 𝛼 = 0.3, 𝜏 = 0.5, and 𝛿 = 0.5. The left panel shows the case of 𝛾 = 0.2, whose feature is similar to that in Section 3. In this case, the social welfare is maximized under the symmetric distribution, 𝜆1 = 𝜆2 ≃ 0.614 (implying ̃ L1 = 0.5). As the degree of agglomeration economies due to the amount of labor increases, the verge of SW on 𝜆i = 1 rises. When the agglomeration economies get sufficiently large, the peak of SW changes from the symmetric distribution to the concentration in one of the two regions. However, as equation (45) shows, the concentration in this situation is imperfect. The right panel of Fig. 11 shows the case of 𝛾 = 0.7 in which SW is maximized when 𝜆1 ≃ 0.909 and 𝜆2 ≃ 0.018 (implying ̃ L ≃ 0.915) or vice versa. Recalling that workers completely concentrate in one of the two regions in the steady-state equilibrium, we can conclude that the steady-state equilibrium does not maximize social welfare even if we introduce agglomeration economies in the short-run stage. To maintain labor heterogeneity, it is necessary to keep the interregional circulation of workers in every period. 6. Concluding remarks It has been discussed in the fields of urban and regional economics that economic activities benefit from labor diversity as agglomeration economies. However, agglomeration itself can reduce labor diversity in the long run. That is, if all the workers concentrate in a particular region, their descendants will be the homogenous residents grown up in the same circumstance. To investigate the transition of labor diversity in the agglomeration process, this paper presented a non-overlapping generations model. In this model, workers were assumed to be differentiated in terms of their origins, and foreign workers incurred an adjustment cost due to having to cope with cultural differences. They were born in each region at the beginning of each period, then chose their residency regions to work in, and finally left the job market at the end of the period. The distribution of birthplaces in each period was determined by the previous generation’s residential choices. The result obtained in this paper can be summarized as follows: generational transition allows the workers to keep migrating to the core region, which diminishes labor diversity endogenously.

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The second measure is to manipulate dispersion factors. As discussed in Section 4, it might be effective to reinforce incentives for housing consumption. Besides, fostering a preference in heterogeneity might also be another option to lead workers to dispersion. As stressed by Tabuchi and Thisse (2002) and Murata (2003), a taste heterogeneity for one’s residency location is a strong dispersion force. Conversely, if people have a similar preference for location, it might be one of the reasons for the excessive flow of population into Tokyo. In such a case,

it is desirable to better promote attachment to home regions. The last measure is to regulate interregional migration directly. The government of Japan has recently started attempting to reduce interprefectural migration by limiting the number of student enrollment at universities in inner Tokyo, so as to encourage high school students to choose their local university. However, when we consider such a migration policy, it is quite important to note that the social optimum needs a circulation of population between regions and countries; thus, migration itself should not be obstructed.

Appendix A. Fundamental equations and total differentials The fundamental equations of this model are expressed as follows. { } (1 − 𝜇)1−𝜇 𝛼ΓA11t∕𝛼 [(𝜆1t L1t )𝛼 + ((1 − 𝜆2t )(1 − L1t )𝜏)𝛼 ](1−𝛼)∕𝛼 (𝜆1t L1t )𝛼−1 + Rt V11t = { }𝜇 , 1∕𝛼 𝛼ΓA1t [(𝜆1t L1t )𝛼 + ((1 − 𝜆2t )(1 − L1t )𝜏)𝛼 ]1∕𝛼 + [𝜆1t L1t + (1 − 𝜆2t )(1 − L1t )]Rt { } 1∕𝛼 (1 − 𝜇)1−𝜇 𝛼ΓA2t [(𝜆2t (1 − L1t ))𝛼 + ((1 − 𝜆1t )L1t 𝜏)𝛼 ](1−𝛼)∕𝛼 ((1 − 𝜆1t )L1t 𝜏)𝛼−1 𝜏 + Rt V12t = { }𝜇 , 1∕𝛼 𝛼ΓA2t [(𝜆2t (1 − L1t ))𝛼 + ((1 − 𝜆1t )L1t 𝜏)𝛼 ]1∕𝛼 + [𝜆2t (1 − L1t ) + (1 − 𝜆1t )L1t ]Rt { } 1∕𝛼 (1 − 𝜇)1−𝜇 𝛼ΓA2t [(𝜆2t (1 − L1t ))𝛼 + ((1 − 𝜆1t )L1t 𝜏)𝛼 ](1−𝛼)∕𝛼 (𝜆2t (1 − L1t ))𝛼−1 + Rt V22t = { }𝜇 , 𝛼ΓA12t∕𝛼 [(𝜆2t (1 − L1t ))𝛼 + ((1 − 𝜆1t )L1t 𝜏)𝛼 ]1∕𝛼 + [𝜆2t (1 − L1t ) + (1 − 𝜆1t )L1t ]Rt { } 1∕𝛼 (1 − 𝜇)1−𝜇 𝛼ΓA1t [(𝜆1t L1t )𝛼 + ((1 − 𝜆2t )(1 − L1t )𝜏)𝛼 ](1−𝛼)∕𝛼 ((1 − 𝜆2t )(1 − L1t )𝜏)𝛼−1 𝜏 + Rt , V21t = { }𝜇 𝛼ΓA11t∕𝛼 [(𝜆1t L1t )𝛼 + ((1 − 𝜆2t )(1 − L1t )𝜏)𝛼 ]1∕𝛼 + [𝜆1t L1t + (1 − 𝜆2t )(1 − L1t )]Rt {

Γ = A1t [(𝜆1t L1t )𝛼 + ((1 − 𝜆2t )(1 − L1t )𝜏)𝛼 ]1∕𝛼 + A2t [(𝜆2t (1 − L1t ))𝛼 + ((1 − 𝜆1t )L1t 𝜏)𝛼 ]1∕𝛼 1∕𝛼

Rt ≡ RKt + RHt =

1∕𝛼

1 − 𝛼(1 − 𝜇) 𝛼∕(𝛼−1) Γ 1−𝜇

Ait = [1 + 𝜆it Lit + (1 − 𝜆jt )Ljt 𝜏]𝛾 ,

i, j = 1, 2,

}𝛼−1

,

(A.1)

(A.2)

(A.3)

(A.4)

(A.5) (A.6)

i ≠ j.

(A.7)

To avoid complex calculation, this paper divides the analysis into three sections. That is, we set 𝜇 = 0 and 𝛾 = 0 in Section 3; 𝜇 > 0 and 𝛾 = 0 in Section 4; and 𝜇 = 0 and 𝛾 > 0 in Section 5. Under the symmetric equilibrium of Lit = 1∕2 and 𝜆it = 𝜆∗it , the total differentials of the indirect utilities V11t and V12t are given by [( ) ( ) 𝜇 𝜇 𝛾(1 − 𝜇)(1 + 𝜏 𝛼∕1−𝛼 ) 𝛾(1 − 𝜇)(1 + 𝜏 𝛼∕1−𝛼 )𝜏 dV11t = Ψ −𝛼(1 − 𝛼)𝜏 𝛼∕1−𝛼 − d𝜆1t + −𝛼(1 − 𝛼) + d𝜆2t + − 𝛼∕ 1 −𝛼 1 ∕ 1 −𝛼 𝛼∕ 1 −𝛼 1 ∕ 1 −𝛼 1−𝜇 1−𝜇 3 + 2𝜏 +𝜏 3 + 2𝜏 +𝜏 (A.8) ) ] ( 𝜏 𝛼∕1−𝛼 𝜇 1 − 𝜏 𝛼∕1−𝛼 𝛾(1 − 𝜇)(1 − 𝜏 𝛼∕1−𝛼 ) dL + 2 −2𝛼(1 − 𝛼) − + , 1t 1 − 𝜇 1 + 𝜏 𝛼∕1−𝛼 1 + 𝜏 𝛼∕1−𝛼 3 + 2𝜏 𝛼∕1−𝛼 + 𝜏 1∕1−𝛼 [( ) ( ) 𝜇 𝜇 𝛾(1 − 𝜇)(1 + 𝜏 𝛼∕1−𝛼 )𝜏 𝛾(1 − 𝜇)(1 + 𝜏 𝛼∕1−𝛼 ) 𝛼(1 − 𝛼) + 𝜆 + 𝛼( 1 − 𝛼) − d d𝜆2t dV12t = Ψ − + 1t 1−𝜇 1−𝜇 𝜏 𝛼∕1−𝛼 3 + 2𝜏 𝛼∕1−𝛼 + 𝜏 1∕1−𝛼 3 + 2𝜏 𝛼∕1−𝛼 + 𝜏 1∕1−𝛼 (A.9) ( ) ] 𝜇 1 − 𝜏 𝛼∕1−𝛼 𝛾(1 − 𝜇)(1 − 𝜏 𝛼∕1−𝛼 ) 2𝛼(1 − 𝛼) +2 − + − , dL 1t 1 − 𝜇 1 + 𝜏 𝛼∕1−𝛼 1 + 𝜏 𝛼∕(1−𝛼) 3 + 2𝜏 𝛼∕1−𝛼 + 𝜏 1∕1−𝛼 ( )(1−𝜇)(1−𝛼−𝛾) ( )𝛾(1−𝜇) 3 + 2𝜏 𝛼∕1−𝛼 + 𝜏 1∕1−𝛼 Ψ = 2𝜇−𝛾(1−𝜇) 𝛼(1 − 𝜇) 1 + 𝜏 𝛼∕1−𝛼 . (A.10) In the equilibrium, substituting dL1t = 0 into dV11t = dV12t derives d𝜆2t ∕d𝜆1t , the slope of the line V11t = V12t at point E in Figs. 1, 4 and 8. On the other hand, substituting d𝜆2t = − d𝜆1t into dV11t = dV12t derives d𝜆1t ∕dL1t , which explains the effect on the residential choice of a small deviation of the birthplace distribution from the symmetry. In this case, the effect on the indirect utility level is given by substituting d𝜆1t ∕dL1t into the total derivative, dV11t ∕dL1t . Appendix BThe slope of line V iit = V ijt at the corner point We examine the relation between the slopes of V11t = V12t and V22t = V21t at the corner point (𝜆1t = 1, 𝜆2t = 0) in Figs. 1 and 8. Since Fig. 1 is derived as a specific case of 𝛾 = 0 for Fig. 8, we mainly consider Fig. 8. The relation between the two lines is calculated by the following steps. First, substituting L1t = 1∕2 into V11t = V12t in Section 5, we have

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( 1+ (

= 1+

𝜆1t + (1 − 𝜆2t )𝜏

)𝛾∕1−𝛼 (

2

𝜆2t + (1 − 𝜆1t )𝜏 2

)𝛾∕1−𝛼 ( ( 1+

( 1+

(1 − 𝜆2t )𝜏 𝜆1t

𝜆2t (1 − 𝜆1t )𝜏

)𝛼 )

)𝛼 ) (B.1)

𝜏 𝛼∕1−𝛼 .

Secondly, manipulating the total derivative of equation (B.1) and the limit of equation (B.1) as 𝜆1t → 1 and 𝜆2t → 0, we obtain the slope of V11t = V12t at the end point, expressed as (( )1∕𝛼 ) 𝜆2t d𝜆2t 3 + 𝜏 𝛾∕1−𝛼 =− = lim − 𝜏 −𝛼∕1−𝛼 ≡ 𝜉1 (𝜏). (1 + 𝜏 𝛼 ) − 𝜏 𝛼∕1−𝛼 𝜆1t →1,𝜆2t →0 d𝜆1t 𝜆1t →1,𝜆2t →0 (1 − 𝜆1t ) 2 lim

Thirdly, rewriting V22t = V21t as )𝛾∕1−𝛼 ( ( )𝛼 ) ( (1 − 𝜆1t )𝜏 𝜆 + (1 − 𝜆1t )𝜏 1+ 1 + 2t 2 𝜆2t (

𝜆1t + (1 − 𝜆2t )𝜏

= 1+

2

)𝛾∕1−𝛼 ( ( 1+

𝜆1t (1 − 𝜆2t )𝜏

(B.2)

(B.3) )𝛼 )

𝜏 𝛼∕1−𝛼 ,

and the same manipulation of equation (B.3) yields lim

d𝜆2t

𝜆1t →1,𝜆2t →0 d𝜆1t

= −(

[

1 ≡ 𝜉2 (𝜏). )1∕𝛼 ]𝛾∕1−𝛼 (3 + 𝜏)∕2 𝜏 𝛼∕1−𝛼 (1 + 𝜏 −𝛼 ) − 𝜏 −𝛼∕1−𝛼

(B.4)

Note that the RHS of equation (B.4) should be larger than 1, whose condition is given by (

) ( ) 1 1 3 + 𝜏 𝛾∕1−𝛼 1 + 𝛼 − 𝛼∕1−𝛼 > 0. 2 𝜏 𝜏

(B.5)

Otherwise, the line of V22t = V21t does not pass the corner point. Finally, comparing the values of equations (B.2) and (B.4), we have ) (( )}1∕𝛼 ) ) ( ) 1 1 3 + 𝜏 𝛾∕1−𝛼 3 + 𝜏 𝛾∕1−𝛼 1 + 𝛼 − 𝛼∕1−𝛼 (1 + 𝜏 𝛼 ) − 𝜏 𝛼∕1−𝛼 2 𝜏 2 𝜏 ] {( ] [( ( )2𝛾∕1−𝛼 [ )𝛾∕1−𝛼 ) 1 3+𝜏 3 + 𝜏 𝛾∕1−𝛼 3+𝜏 = − 1− 𝜏 𝛼∕1−𝛼 − 1 2 2 2 𝜏 𝛼2 ∕1−𝛼

𝜉1 (𝜏) = 𝜉2 (𝜏)

{((

(

−

(B.6)

[ ( ) (( )]}1∕𝛼 ) ) ) ( 2 3 + 𝜏 𝛾∕1−𝛼 3 + 𝜏 𝛾∕1−𝛼 𝛼2 ∕1−𝛼 1 3 + 𝜏 𝛾∕1−𝛼 𝜏 − − 𝜏 𝛼 ∕1−𝛼 . 1− 𝛼∕ 1 −𝛼 2 2 2 𝜏

If 𝛾 = 0, the value of equation (B.6) is less than 1. That is, as shown in Fig. 1, the absolute value of the slope of V11t = V12t is smaller than V22t = V21t . On the other hand, as shown in Fig. 8, the rise in 𝛾 decreases the value of equation (B.6), thus the slope of V11t = V12t becomes larger than that of V22t = V21t . Moreover, we can easily see that the corner point itself is sustainable since V11t || ((3 + 𝜏)∕2)𝛾∕𝛼 (1 + 𝜏 𝛼 )(1−𝛼)∕𝛼 > 1. = V12t ||L1t =1∕2,𝜆1t =1,𝜆2t =0 𝜏

(B.7)

Appendix C. Social welfare at the corner point Considering the government’s choice of 𝜆i , we should draw attention to the peculiarity of the case where 𝜆1 = 1 and 𝜆2 = 1. In this case, there L1 .) In Fig. 4 (in is no labor circulation between the regions, thus the birthplace distribution is fixed at the initial distribution, L1 = L1o . (Thus L1 ≠ ̃ Section 3), the value of SW at the corner point (𝜆1 = 1, 𝜆2 = 1) equals 1∕(1 − 𝛿). In Fig. 8 (in Section 4), it becomes SW = (L11o−𝜇 + (1 − L1o )1−𝜇 )∕(1 − 𝛿),

(C.1)

and in Fig. 11 (in Section 5), it becomes ( )𝛼 SW = (1 + L1o )𝛾∕𝛼 L1o + (2 − L1o )𝛾∕𝛼 (1 − L1o ) ∕(1 − 𝛿).

(C.2)

Appendix D. Endogenous adjustment costs In this study, the interregional adjustment cost is assumed to be fixed exogenously. However, Carrington et al. (1996) claim that it can be decreasing in the number of immigrants. Although an accurate analysis is difficult because of the complexity of the model, we can explore the effect of endogenous adjustment costs through numerical examples. Based on the setting of Section 3, we replace the adjustment cost by the following:

𝜏̂t = 𝜏 + 𝛽(1 − 𝜏)(1 − 𝜆1t )(1 − 𝜆2t ),

(D.1)

where 𝛽 ∈ [0, 1] is the degree of the effect of immigrants on the adjustment costs. This describes that the interregional communication gets smooth as the share of workers working as a foreigner increases. The case of 𝛽 = 0 corresponds with that in Section 3. 379

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Fig. D1 shows the residential choice in the short-run stage. The broken line and solid line, respectively, indicate the residential distribution for the cases of 𝛽 = 0 and 𝛽 = 1. Points E indicates the equilibrium in the cases of 𝛽 = 0, and E′ indicates that in the case of 𝛽 = 1. Fig. D2 is the resulting dynamics of birthplace distribution in the long run. These figures show that the endogenous adjustment cost increases the tendency of concentration.

Fig. D1 Residential choice.

Fig. D2 Transitional dynamics of birthplace distribution.

An intuitive explanation follows. Recalling the discussion in the short-run analysis in Section 3, decrease in the adjustment costs gives an incentive for migration to foreign regions. This reduces the adjustment cost through equation (D.1). In addition, the reduction in the adjustment cost makes residential choice more sensitive to the birthplace distribution. This reinforces the workers’ tendency of concentration. In conclusion, labor heterogeneity causes agglomeration, and agglomeration is promoted by itself through the decrease in the adjustment cost. Finally, this modification does not alter the main result of the welfare analysis in Section 3, because the concentration of residents increases the adjustment cost. Appendix E. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.regsciurbeco.2019.06.003. References

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