Heterogeneity of labor markets and city size in an open spatial economy

Regional

Science

and Urban

Economics 21 (1991) 109-126. North-Holland

Heterogeneity of labor markets and city size in an open spatial economy Sunwoong Kim* University of Wisconsin - Milwaukee, Milwaukee, WI 53201, USA Received

May 1988, final version received June 1990

This paper presents a model of city size determination in an open spatial economy. It argues that the city size is determined by two interacting forces. First, holding other things constant, the wage rate increases with the size of the city, because the larger the city, the better the match between the diverse job requirements and the diverse labor pool. The higher wage of a city will in turn attract more workers from its hinterland. However, the growth of the city will be limited by the physical consequence of the concentration of economic activities. Since the city has to accommodate more people, the rent and the density of the city will rise. Holding wage level constant, the higher price of land will decrease the welfare of the city resident. The size of the city will be determined where the higher wage the worker receives in the city is exactly compensated by the higher rent he has to pay so that he gets the same level of utility as a worker elsewhere.

1. Introduction

Since Alonso’s seminal work (1964), the monocentric land use model has been used extensively in urban economics. Due to its simplicity and elegance,

it has been a workhorse model in urban economics. For the last two decades, the model has been extended in many directions in order to incorporate more realistic features such as externalities, local public goods, or durability of housing stock [see Fujita (1986) or Wheaton (1979) for an extensive survey on the topic]. One critical assumption of the monocentric land use model is that all employment is concentrated in the center of the city, sometimes referred to as the central business district (CBD). Urban residents commute to this CBD to work and earn an exogenously-given wage. This author finds it rather puzzling that the model has paid very little attention, if any, to the local labor market and the determination of the wage. If the working of the model *The author would like to acknowledge helpful comments and suggestions by Masahisa Fujita, Francisco Rivera-Batiz, Tony Smith, Konrad Stahl, an anonymous referee, and members of the Regional Science Theory Workshop at the University of Pennsylvania. Any remaining errors are mine. 0166-0462/91/%03.50

0

1991-Elsevier

Science Publishers

B.V. (North-Holland)

110

S. Kim, Heterogeneity of labor markets and city size

critically depends on the fact that workers have to commute between their residence and CBD, then we should have a better understanding of the urban labor market. By explicitly considering the urban labor market in a monocentric spatial economy, this paper attempts to analyze the relationship between the wage level and the size of the city in an open spatial economy. There is ample empirical evidence suggesting that the wage level is positively correlated with city size. The reduced form estimation of wage functions (nominal wage as dependent variable and the characteristics of workers and the characteristics of local labor markets as independent variables) clearly indicates that the workers in a larger city are paid more than those in a smaller city ceteris paribus [see Fuchs (1967) and Hoch (1972) for details]. Several theories have been suggested in order to explain the apparent wage differentials among different city sizes. Some argued that the higher cost of living in larger cities will drive up the nominal wage to equate the real wages among regions. Others argued that the high wages in large cities are nothing more than a monetary compensation for the poor environment such as pollution and congestion of such areas [see Goldfarb and Yezer (1976) for an empirical evaluation of alternative theories]. However, as Rosen (1979) has pointed out, such explanations will not be complete without a discussion of the labor demand, because the nominal wage differential would disappear in the long run unless the large cities have higher labor productivity. For example, if the nominal wages are higher in larger cities, the firms in these cities tend to use higher capital-labor ratios implying that nominal returns to capital are lower in larger cities. But given the footlooseness of capital any differentials in returns to capital will quickly disappear. Thus, capital will outmigrate from larger cites, reducing wages in these cities and raising it in smaller ones. Thus, in the long run, nominal wages will be equalized among cities of different city sizes. Indeed empirical evidence on scale economies of city size strongly supports the hypothesis that large cities are more productive than small cities.’ The basic theoretical underpinning of such studies is that the production function shifts out as the city size grows as was suggested by Chipman (1970). In order to make the competitive economic system compatible with aggregate increasing returns to scale he suggested that there exists an external scale economy which each firm does not fully take into consideration. An individual firm acts as if it had constant or decreasing returns to scale with ‘Using CES (constant elasticity of substitution) production function, Sveikauskas (1975) estimated that the labor productivity increases by six percent as the city size doubles. Segal (1976) tried to distinguish the agglomeration effect from the increasing returns to scale effect by estimating a generalized Cobb-Douglas production function with a shift parameter of the population size. And he concluded that agglomeration economies are more evident than increasing returns to scale. A critical review and re-estimation of the production function by

S. Kim, Heterogeneity

of labor markets and city size

111

an exogenous parameter which increases with the aggregate size of the economy. The individual firms do not internalize such external scale economy since they cannot control the size of the market. Thus, they will act as if they were not aware of such external scale economies. Although there have been numerous suggestions that why there exist such agglomeration economies, only a few rigorous attempts have been made in order to understand the natue of such external scale economies [see the collection of papers in Regional Science and Urban Economics 18, no. 1 (1988)]. In a recent paper, Kim (1990) has argued that average productivity increases with the size of the labor market because average match between the skill characteristics of workers and the job requirements of firms improves as the size of the market increases. Moreover, in a related paper, Kim (1989) argued that the larger the size of the market, the more specialization of labor occurs resulting in further increase of productivity. However, there are two major limitations in these papers. First, the models describe closed labor markets in which no interurban movement of workers is permitted. In reality, there is substantial movement of labor as well as goods making the closed city assumption inadequate. Second, there is no concept of space in the model. One may feel awkward that we try to explain agglomeration economies without any specific reference to space. After all, agglomeration is a spatial phenomenon. This paper is an attempt to alleviate such shortcomings by considering labor specialization in an open spatial economy. 2. The model We shall describe a model of an open spatial urban economy. There two final consumption goods: a composite good and land. Only composite good is produced. More specifically, it can be produced either ‘rural (i.e., agricultural)’ technology which uses land and labor as inputs by ‘urban (i.e., industrial)’ technology which uses only labor. The markets

are the by or for

Moomaw (1981) suggested that Sveikauskas’ result was exaggerated because of the overestimation of capital stock in older (and, thus larger) cities. His revised estimate is that two and a half percent of increased productivity is achieved by doubling the city size. Henderson (1986) tried to distinguish the effects of localization economies and those of agglomeration economies by using trans-log production functions. Following the usual convention, he defined localization economy as a scale economy external to firms, but internal to industries, and agglomeration economy as external to industries as well. His results are basically that the effect of localization economies is usually stronger than that of agglomeration economies. He also finds that the localization economies level of quite soon. But he confined his study in the manufacturing industries only, in which industrial linkages and availability of the labor pool tend to be more important. Presumably, service industries require more face-to-face contact and fast digestion of continuous information flow. Thus his conclusions about the significance of localization economies over agglomeration economies would be exaggerated if we included all types of economic activities in cities. Nakamura (1985) has done a similar study in Japan and came up with a similar set of conclusions.

112

S. Kim, Heterogeneity of labor markets and city size

the composite good and land are competitive. The structure of the labor market will be spelled out later. The interurban movement of labor and the composite good is costless. The composite good is available everywhere with a uniform price. Since we take the composite good as a numeraire, the price of the good can be normalized to one without any loss of generality. There are two groups of economic agents in the model: workers-cum-consumers (we will call them just workers for convenience) and firms. We will describe their behavior separately.

2. I. Workers Workers have identical tastes. Their utility depends on land consumption 4 (whose rent is r) and the composite commodity x (whose price is normalized to unity). Their income (y) is divided into expenditure on the composite good, land and commuting cost. Commuting is applicable to urban residents only. The income is exclusively generated by the wage.2 The travel cost is assumed linear and equal to k times distance t. The unit cost of travel k is taken to be exogenous.3 Urban workers choose the location of the residence, the amount of land consumption, and the amount of the composite good consumption. In short, the urban worker’s consumption behavior is summarized by maximizing the utility function4

u = 4x3 41,

(1)

subject to the budget constraint x + r(t)q + kt =y.

(2)

If the worker belongs to the rural sector, his/her utility is given by (3) *In other words, rent is not a part of real income. It can be sustained either by assuming that land is owned by absentee landlords or that any rental income is taxed away by the government. 3The unit travel cost k can be determined within the model if we consider the local government explicitly. Then, we have to specify the production function of the transportation service, and the decision rule of the local government regarding to the level of transportation service. This will be a much richer model enabling us to analyze the role of local government and public investment in transportation facilities. However, the inclusion of the local government sector will grossly complicate the model. Thus, we will defer such an extension to the future. 41t is assumed that the utility function is differentiable and increasing at all x >O and q >O; and indifference curves are strictly convex and smooth, and do not cut axes.

S. Kim, Heterogeneity of labor markets and city size

113

where qr is per capita rural farmland and r. is the rural rent. The rural worker chooses q1 to maximize his/her utility (3), whereas rO adjusts so that it reaches the exogenous level of U. Every worker supplies one unit of labor inelastically. The worker can work either in the rural sector or in the urban sector, If the worker works in the rural sector, the per capita output, thereby his/her wage (under the perfect competition assumption) is wO. Production technology in the urban sector will be described below. In order to determine city size endogenously, we close the model by assuming that w. is also exogenous. If there are N workers in the city, they are indexed on the circumference of a circle of unit length with uniform density. The index represents the skill characteristics of workers.5 There is no a priori superiority or inferiority among workers’ skills. They are just different.6 For example, one worker is good at cooking, and another is good at singing. If they worked in a restaurant then the former would be more productive than the latter. However, if they worked for an opera company, the latter would be more productive. The size of the difference between the indices of two workers (s) represents how different they are. Clearly, the difference ranges from zero to one half.

2.2. Firms and urban technology Profit maximizing firms produce the composite good by hiring workers. It is assumed that firms use no land and locate where the accessibility is the best: the center of the city. There are a number of production technologies available. Each technology can be distinguished by its job requirement, which represents the skill characteristic of the worker who has the highest productivity (m) with the firm. Thus, firms also can be indexed on the same unit circle or workers’ skill characteristics without any ambiguity. If a potential worker does not have the job requirement of the firm, then the worker has to be trained to meet the requirement. More specifically, the marginal productivity of such a worker is assumed to be x(s) = m - cs,

(4)

where s is the difference between the index of the worker’s skill characteristic and that of the firm’s job requirement, and parameter c represents the degree 5Notice the asymmetrical treatment of labor between the two sectors. In the rural sector labor is treated as an homogeneous input, whereas it is treated heterogeneous in the urban sector. 6The difference on workers’ skill characteristics is analogous to the notion of horizontal product differentiation in industrial organization literature.

S. Kim, Heterogeneity of labor markets and city size

114

of substitutability of workers of different skill characteristics. High that substitution of workers is more dificult.7 All technologies are assumed to require an identical minimum scale (M). Specifically, a representative firm’s production function is

Y=

X=

0

if

X
X-M

if

XZM,

1 x(s) ds. SE.7

Y is output, X is the total labor workers who work for the firm.

2.3.

c means efficient

(6)

input

in efficiency

units,

and S is the set of

Working of the model

There are three markets in the economy: the composite good, labor, and land. With the assumption of perfect mobility (without any cost), the discussion of the composite good is immaterial. Thus, we shall consider the labor market and the land market only. We shall adopt an open-city assumption so that the utility level of a worker is taken to be exogenous (U). If the utility level of the worker in the city (in question) is greater than U for some reason, workers from its hinterland will migrate into the city. The increase of the population reduces land consumption per worker and increases commuting cost. Thus, the utility level decreases. The migration will stop when the utility level of the city reaches U. In particular, we shall assume that the migration process is non-self-selecting so that the distribution of the skill characteristic remain uniform.’ Given the size of the labor market (number of workers in the city) and the parameters of the technology (minimum efficient scale, marginal productivity of the worker who has the firm’s job requirement, and degree of labor substitutability) we can determine the equilibrium wage and equilibrium number of firms by examining the labor market equilibrium conditions. ‘Kim (1989) analyzed the model that takes m and c as the worker’s decision variables on human capital investment. More specifically, m can be regarded as the depth of human capital indicating the ability to do a specific task. On the other hand, c can be viewed as the breadth of human capital indicating the ability to learn similar tasks. A high value of m implies high productivity when the job requirement is the same as the worker’s skill characteristic. A low value of c means that the worker’s productivity does not suffer very much when the worker’s skill characteristic is different from the job requirement. *Notice that this assumption implies that all cities have the same distribution of labor characteristics, and hence precludes the possibility of the formation of a city system of specialized labor pools.

S. Kim, Heterogeneity

of labor markets and city size

115

Given the wage, exogenous utility level, and rural land rent, we can determine the size of the city (both its physical size and the number of workers that the city will accommodate) by examining the land market equilibrium conditions. The key variable that connects the two market is the number of workers in the city, and it will be determined by the equilibrium conditions of the two markets. 3. Labor market equilibrium’ Given the heterogeneity of the local labor market and the number of workers in the city, we can determine the equilibrium number of firms and the wage by examining the labor market. In the short run, firms cannot choose technologies (job requirements). Given the job requirement of firms every worker chooses the firm that offers the highest wage. Wage determination rule is discussed later. In the long run, firms enter or exit depending on whether existing firms make positive or negative profits. We assume that the adjustment is costless. Thus, the number of firms will be determined by the zero profit condition. We are mainly concerned about the zero profit symmetric equilibrium in which the difference between the job requirements of any two firms that have the most similar requirements are the same. 3.1. Wage determination The wage will be determined by the bargaining between workers and firms. The key assumption that we adopt is that the wage will be determined by an axiomatic bargaining solution. The bargaining solution is a rule that determines the bargaining outcome with any given bargaining environment.” More specifically, each party knows exactly how much it will gain by having the employment contract, and the bargaining outcome (i.e., wage) will be determined at the mid-point where the worker’s surplus of having the employment contract over his threat point, the payoff of the second best alternative when the bargaining is not materialized, is the same as the firm’s marginal profit of having the worker. Negotiation is costless, and collective bargaining by workers or coalition formation by firms is not allowed. Since we shall maintain the static framework, we also assume that no pair of agents will miss a potentially beneficial bargaining opportunity. We shall only consider the case in which every worker in the economy has at least two viable job alternatives (i.e., the marginal productivity of the worker is greater than the reservation wages for at least two firms in the ‘This section is heavily drawn from Kim (1990). Interested readers may want to consult the paper for more detail. “See Binmore, Rubinstein, and Wolinsky (1986) and Kim (1987) for using axiomatic bargaining solution for economic modeling.

S. Kim, Heterogeneity

116

of labor markets and city size

m

m - cs

m - cH

m - c(2H-s)

m - 2H

1 0 Fig. 1. Productivity

I I

I

s

H

of a representative

worker

I 2H

(s) for a pair of representative

firms.

market) and that the worker’s threat point is the highest possible wage in the negotiation with other firms. Although this specification seems to overstate the bargaining power of the worker, it is compatible with our equal division surplus assumption.’ 1 Consider a pair of firms whose job requirements are so slightly different that the indices of any other firms’ job requirement do not lie in between the pair. Take the job requirement of the firm on the left as an origin and consider a representative worker whose skill characteristic is different from the job requirement by s [refer to fig. 11. If we denote the difference of the two job requirements as 2H, the worker’s marginal productivity for the left firm is m-cs, whereas his productivity for the right firm is m -c(2H -s). If s
S. Kim, Heterogeneity

of labor markets and city size

117

w(s) = [m - cs]/2 + [m - c(2H - s)]/2 = m - cH. We shall call H the firm’s acceptable skill characteristic range. It is interesting to note that the wage is independent of s (the difference being the job requirement and worker’s skill characteristic). The worker who has a good match with one firm can demand a higher wage because he has high productivity. However, since the worker has a poor match with the other firm, his bargaining position will be weakened. With the assumption of the linear training costs, these two effects will exactly offset each other, resulting in a flat wage schedule.

3.2. Symmetric zero-profit condition The short-run wage function includes the firm’s acceptable skill characteristic range H, which is determined in the long-run zero-profit condition.” As the situation is symmetric in both directions, the zero-profit condition of a representative firm becomes 2mNH-&2Nj{cs+w(s)}ds=O.

(8)

0

Substituting the wage function into the zero-profit equilibrium H:

H=,/m.

condition,

we get the

(9)

The acceptable range H becomes smaller, if the size of the market is larger, if minimum efficient scale (fixed cost) is smaller, and if technology requires more specific labor. i3 Substituting the equilibrium H into the wage function, the equilibrium wage can be written as

w=m-JG+. The equilibrium wage will be higher if the size of the labor market is larger, and marginal productivity of the best worker is higher. The wage will be lower if the minimum efficient scale is larger and if the degree of specific labor requirement is higher. ‘*Notice that we have restricted our analysis within symmetric equilibrium in which acceptable skill characteristic of all firms are identical in size. ?he size of the market in this paper is equivalent to the density of labor skill characteristic distribution, not the variety of labor skills. The latter is constant for different labor market sizes.

118

S. Kim, Heterogeneity

of labor markets and city size

Since there will be full employment in equilibrium, the equilibrium of firms (n) will be determined by the equilibrium H (refer to fig. 1):

number

(11) The equilibrium number of firms will be larger, if the size of the market is _ larger, if minimum efficient scale is smaller, and if technology requires more specific labor. The assumption that all workers have at least two viable employment opportunities implies that14 m-2cHzw,.

(12)

H into the above By substituting equilibrium parameter region for such a case as follows: (m - wO)’ 2 &M/N.

equation,

we can express

the

(13)

Given the values of m, c, M, and wO, this condition will be satisfied if N is large enough. It gives an interesting interpretation of the minimum city size for a given industry. Industries which have a large minimum efficient scale (M) should locate in a large city. Similarly, industries which require very specific job requirement (c) will locate in a large city. It can be shown that the number of employees per firm is 2,/m, and output per firm is 2mJm-2M. The size of the firm, measured either by employment or output, will be larger if the size of the labor market is larger. The firm size will be smaller if the technology requires more specific kinds of labor.

4. Land market equilibrium The consumer The well-known are:

chooses the level of x and 4 in order to maximize utility. first-order conditions for the utility maximization problem

k= --r’(t)q(t),

(14)

au/a4-,(+(Y-x-kt)

au/ax “‘Otherwise opportunity.

some workers’

’

4

.

(15)

threat point is w0 and they have only one viable employment

S. Kim, Heterogeneity

of labor markets and city size

119

If the income is identical among workers, the competitive market assumptions will bring about identical utility among workers. Moreover, the open city assumption assures that the workers will enjoy the exogenous utility level U. Thus, the bid rent function I and the equilibrium quantity of x and q can be solved in terms of U, t, y, and k by the two first-order conditions along with the utility function and the budget constraint. In an open city, the supply of land will reach from the city center to the boundary b, where the marginal urban worker’s bid rent equals the rural rent:

rt%b,Y, k) = ro,

(16)

which is equivalent to y-kb=w,,

(17)

as the rural rent adjusts to a level that rural workers’ utility reaches U. Assuming that the city does not have any physical barriers for urban development, we can determine the number of workers that the city can accommodate:

2r;t/q@, t, y, k) dt = N.

(18)

0

Applying Leibniz’s Theorem shown that

to the above equation,

Wheaton

dN/dii= -27~; t/q2(aq/X)dt+2nb/q(b)dbldii
(1974) has

(194

0

dN/dy = - 27~;t/q*(aq/ay) dt + 2nb/q(b) db/dy > 0,

(19b)

0

dN/dk = - 2nj t/q’(aq/ak) dt + 2xb/q(b) db/dk < 0,

(19c)

0

dN/dw, = 2xb/q(b) dbldw,
(19d)

Holding income and transportation rate constant, higher background utility level reduces the equilibrium population size. The reduction of population will reduce the density and increase the utility of the urban

120

S. Kim, Heterogeneity of labor markets and city size

residents to the background level in order to restore the equilibrium. On the other hand, rising income increases demand for land. Consequently, the bid rent function will rise everywhere. The increase of land price not only expands the border, but reduces individual land consumption. Thus, the size of the labor force (population) must increase through in-migration. Greater travel cost has the opposite effect on density and the city boundary resulting in out-migration. Higher rural productivity or larger per capita farmland reduces the physical size of the city, whereas the density and rent functions remain unaffected. Consequently, the size of population the city can accommodate decreases. 5. Determination

of city size and comparative

statics

Given the wage level (along with transportation cost k, the rural rent rO, and the background utility level ti), the land market equilibrium condition (18) determines the size of the labor force and physical city size. On the other hand, given the size of the labor force along with the marginal productivity m, the labor substitutability c, and the minimum efficient scale M, the labor market equilibrium condition (10) determines the wage level, the number of firms and the output level. Since we assumed that the wage provides the only income of the workers, we can determine the equilibrium city size and wage by solving the labor market equilibrium condition and the land market equilibrium condition simultaneously. Since the model is presented in a static long-run framework, the discussion of the dynamic process is necessarily crude. However, in order to examine the existence and the stability of the model, it would be useful to specify the adjustment process more precisely. Let us consider the following adjustment process. Any changes in the land market parameters (k, U,wo) are absorbed by the land market first, that is to say, the city size will adjust while the wage stays constant. Then the new city size will determine the wage level in the labor market, and the new wage will determine a new city size. This process will go on until the new equilibrium is achieved. Similarly, any changes in the labor market parameters (m,c, M) will be first absorbed by wage adjustments. The new wage then determines city size, and the process goes on until the new equilibrium is reached. Let us refer to fig. 2 for illustration. There are two curves in each graph, where the horizontal axis is the size of the city (IV) and the vertical axis is the wage (w) or income (y). The curves labeled as ‘labor’ represent the labor market equilibrium conditions (10). They show the wage paid to urban workers given the productivity of urban technology and the size of the labor market. The curves labeled as ‘land’ are the land market equilibrium condition (18). They represent the number of urban workers the city can accomodate given the transportation cost and rural utility level. Notice that

S. Kim, Heterogeneity

of labor markets

121

and city size

w- Y

land

m

market

/ A ____---

labor

market

-

(a)

w. Y

4cM (m-w,,)

N 2

land

market

m arket

(b)

N

w.

Y

land

market

m labor

market

Cc)

N

Fig. 2. Existence

and stability

of equilibria

the wage function in the labor market approaches m (the marginal productivity of the worker who has the skill characteristic exactly equal to the firm’s job requirement) asymptotically as N goes to infinity. It reflects the fact that agglomeration economies peter out eventually, and that maximum wage is limited by the efficiency of urban technology. If the two curves meet, then an equilibrium with finite city size is achieved (e.g., points A, B, and C). The

S. Kim, Heterogeneity of labor markets

122

and city size

land

labor

w

market

market

o

N

Fig. 3. The effect of more productive

technology

in stable equilibrium.

equilibrium is locally stable if the slope of the labor market equilibrium condition is smaller in absolute value than that of the land market at the equilibrium point: WdN

1labor <

dyldN

1land.

(20)

Case (a) exhibits an unique stable equilibrium (point A); case (b) does not have any equilibria; and case (c) has multiple equilibria with one stable equilibrium point (C) and one unstable equilibrium point (B). Generally speaking, the equilibrium does not exist if the urban wage is not high enough to compensate the higher rent and commuting cost for urban life. Although the exact condition for existence will depend on the functional form of the utility function, an example will shed some light. For instance, a sufficient condition for the existence of a stable equilibrium when the utility function takes the log-linear form of u (x, q) = f In x + 3 In q is e_2i(m-wo)4(m+5w,),~ cMk2 = x ’

(21)

The above condition is more likely to be satisfied if c, M, k, and U are small and if (m - wo) is large. In other words, a stable equilibrium with finite city size will come to an existence: (1) if urban technology is sufficiently more efficient than the rural technology; (2) if rural utility is sufficiently low; and (3) if transportation cost is sufficiently low.

S. Kim, Heterogeneity

of labor markets and city size

123

If we assume that the stability condition is satisfied, the effects of the parameter changes are all straightforward. Notice that the slopes of the two market equilibrium conditions in fig. 3 are such that the stability condition is satisfied. Any changes in parameters can be represented by shifting either of the two curves. For example, a more productive technology (higher m, lower c, or lower M) will shift the labor equilibrium condition upwards resulting in an increase of the equilibrium city size as well as the equilibrium wage. A drop in transportation cost will cause the land market equilibrium condition to shift to the right, resulting in a higher wage, higher density, higher rent, and larger city size. A reduction in background utility level or a lower agricultural productivity will have a similar impact. Hence comparative static results can be summarized as follows: dw/dm > 0,

dwJdc < 0,

dwfdii < 0,

dwfdw, < 0,

dN/dm > 0,

dNfdc < 0,

dN/dii < 0,

dN/dw, < 0,

dw/dM c 0,

dwfdk c 0, Wa)

dNfdM < 0,

dNfdk < 0, (22b)

These results seem to be consistent with many stylized facts in urbanization and the growth of cities. It is well known that industrialization is closely related to rapid urbanization and growth of cities. Our model predicts that the increase in industrial productivity raises urban wages and city sizes. Modern industrial technologies are characterized by large minimum efficient scale, increasing returns to scale and specialized labor requirement as was envisioned in the model. As long as urban technologies are more productive than rural technology, the formation and growth of cities are promoted by the agglomeration economies in the local labor market. The concentration of workers in the city provides a large local labor market in which diverse job requirements of firms and skill characteristics of workers are matched. Given the availability of specialized jobs in the urban labor market, the match between jobs and workers will improve as the size of the market increases. In other words, firms will be able to get workers that fit to their descriptions better in a larger labor market. The better matching will improve the productivity, and raise wages. Higher wages will attract more workers from the hinterland. Moreover, as was analyzed in Kim (1989), workers in large labor markets tend to invest in more specialized human capital. Such specialization further increases the labor productivity. However, the cost of intra-urban transportation and the desirability for land consumption restrict the city size. It is well documented that urban rent increases with the size and density of the urban area. Holding wage level constant, the higher price of land will decrease the welfare of city residents.

124

S. Kim, Heterogeneity

of labor markets

and city size

The size of the city will be determined when the higher wage of the worker in the city is exactly compensated by higher rents so that the worker achieves the same level of utility as workers elsewhere. Reduction in transportation cost also increases the city size. The innovations in transportation technology enable the city to spread out, and thus accommodate more people in a daily commuting range. For example, the massive suburbanization in American cities since World War I is largely due to the emergence of automobile technology and highway network. The large local labor market, in turn, increases productivity and wages through better job matching and specialization. A drop in the per capita rural output also causes a city to grow. There is a large volume of literature in the twentieth-century urbanization in developing countries that the rapid growth of large cities in those countries is due to rural push (e.g., high population growth rate, poverty, etc). As living conditions worsen in rural areas, workers are willing to go to the city resulting in a larger city size. Although the equilibrium wage in our model also rises, the increase will not be very large when N is already very large. Moreover, under the open city assumption, the utility level of the urban residents will drop to the background level. 6. Conclusions and further extensions What is the raison d’etre of the city? The city is a place where the diverse needs of many people are met. The more the economy is specialized, the greater the need for matching and coordination among agents. In particular, the modern technology available since the industrial revolution calls for not only large scale production but very specialized labor. The city provides the diverse labor force to support such specialized production activities in the economy. Such concentration of economic activities increases the competition for land and raises urban rent. Thus, city size is determined by the two interacting forces of agglomeration economies in labor market and agglomeration diseconomies in land market. The model presented in this paper emphasizes the importance of the local labor market as a centralizing force in the formation and the growth of cities. As Alonso’s urban land use model was adopted from von Thiinen’s classical work on agricultural land use which takes the location of the town center given, the monocentric urban land use model is not very useful in understanding the suburbanization of firms and the formation of megalopolis. In the framework suggested in this paper, multi-centered spatial structures will be generated if we relax the assumption that firms do not use land. If we allow more than one produced good, then it is logically possible to have cities of different sizes. For example, suppose there are two produced goods. Cities may produce only one good or two goods depending on the

S. Kim, Heterogeneity

of labor markets and city size

125

nature of local labor pool and the interurban transportation costs of the goods. If it is costless to transport the goods and if workers in different industries are not substitutable, then it is easy to see that cities will specialize in only one good. To see this, let us assume that a city has two unrelated industries. Then the rent of the city should be higher than that of the city specializing in one industry, because the former should have fewer workers than the latter. Thus, the utility of the worker in the city with two industries will be lower than that of the city specializing in one. Workers will outmigrate from the city with two industries. In equilibrium, every city will specialize in one industry. If the production technologies of two goods differ in terms of exogenous parameters (c, m, M), it is possible that cities may differ in terms of their sizes.i5 However, if the transportation is not costless, incomplete specialization will be possible. Another very restrictive assumption of the model is the non-selective migration process. If we allow that workers are aware of different job opportunities among different cities and migrate in order to take advantage of them, a system of specialized cities can emerge as an equilibrium configuration. The development of such a model will be deferred for the future. ‘%ee Henderson

(1974) for a similar argument.

References Alonso, W., 1964, Locations and land use (Harvard University Press, Cambridge, MA). Binmore, K., A. Rubinstein and A. Wolinsky, 1986, The Nash bargaining solution in economic modelling, The Rand Journal of Economics 17, 176188. Chipman, J.S., 1970, External economies of scale and competitive equilibrium, Quarterly Journal of Economics 84, 347-385. Fuchs, V.R., 1967, Differentials in hourly earning by region and city size, 1959, Occasional paper 101 (National Bureau of Economic Research, New York). Fujita, M., 1986, Urban land use theory, in: J.J. Gabszewicz et al., Location theory (Harwood Academic Publishers, London) 73-149. Goldfarb, R.S. and A.M.J. Yezer, 1976, Evaluating alternative theories of intercity and interregional wage differentials, Journal of Regional Science 16, 345362. Henderson, J.V., 1974, The sizes and types of cities, American Economic Review 64, 64&656. Henderson, J.V., 1986, Efficiency of resource usage and city size, Journal of Urban Economics 19, 47-70. Hoch, I., 1972, Income and city size, Urban Studies 9. Kim, S., 1987, A product differentiation model with outside good and price bargaining, Economics Letters 24, 305-309. Kim, S., 1989, Labor specialization and the extent of the market, Journal of Political Economy 97, 692-705. Kim, S., 1990, Labor heterogeneity, wage bargaining, and agglomeration economies, mimeographed, Bryn Mawr College, Journal of Urban Economics 28, 160-177. Mills, E.S., 1972, Studies in the structure of the urban economy (Johns Hopkins University Press, Baltimore, MD). Moomaw, R.L., 1976, Productivity and city size: A critique of the evidence, Quarterly Journal of Economics 96, 675-688.

R.s.u.E.-

E

126

S. Kim, Heterogeneity

of labor markets and city size

Nakamura, R., 1985, Agglomeration economies in urban manufacturing industries: A case of Japanese cities, Journal of Urban Economics 17, 108-124. Rosen, S., 1979, Wage-based indexes of urban quality of life, in: P. Mieszkowski and M. Straszheim, eds., Current issues in urban economics (Johns Hopkins Press, Baltimore, MD). Segal, D., 1976, Are there returns to scale in city size?, The Review of Economics and Statistics 58, 339-350. Shefer, D., 1973, Localization economies in SMSA’s: A production function approach, Journal of Regional Science 13, 55-61. Sveikauskas, L., 1975, The productivity of cities, Quarterly Journal of Economics 89, 393-413. Wheaton, W., 1974, A comparative static analysis of urban spatial structure, Journal of Economic Theory 9,223-237. Wheaton, W., 1979, Monocentric models of urban land use: Contributions and criticisms, in: P. Mieszkowski and M. Straszheim, eds., Current issues in urban economics (Johns Hopkins Press, Baltimore, MD).