Marshallian factor market externalities and the dynamics of industrial localization

JOURNAL

OF URBAN

ECONOMICS

28, 349-370 (1990)

Marshallian Factor Market Externalities and the Dynamics of Industrial Localization PAUL A. DAVIDANDJOSHUA

L. ROSENBLOOM*

Department of Economics, Stanford University, Stanford, California 94305-6072; and Department of Economics, University of Kansas, Lawrence, Kansas 66045-2113

Received January 26, 1988; revised April 25, 1988. When an industry has thus chosen a locality for itself, it is likely to stay there long: so great are the advantages which people following the same skilled trade get from near neighborhood to one another.. . A localized industry gains a great advantage from the fact that it offers a constant market for skill. Employers are apt to resort to any place where they are likely to find a good choice of workers.. . ; while men seeking employment naturally go to places where there are many employers who need such skill as theirs.. . . The advantages of variety of employment are combined with those of localized industries in some of our manufacturing towns, and this is a chief cause of their continued growth. Alfred Marshall, Principles of Economics, 8th ed., pp. 271-272 (1920). o 1990 Academic PKSS, IX.

Why do people and firms tend to congregate spatially? Typically, the geographical localization of industry and the rise of cities are explained as consequences of the attraction created by site-specific resources and/or the existence of technical economies of scale in production [lo, 81. Sometimes these scale economies are supposed to arise from nonconvexities at the level of the individual directly productive establishments; at other times the lumpiness of urban “overhead” facilities is seen as causing unit total costs of production to decline (over some range) with increases in the density of population and the volume of production carried on at the place in question. This paper focuses attention on a logically distinct explanation for the growth of urban-industrial agglomerations, one that takes as its premise the existence of what may be called “Marshallian” factor market externalities. Unlike technical economies of increasing city size, the forces with which we are concerned take the form of pecuniary externalities that tend to reduce the prices at which primary inputs can be purchased as more and more of those inputs come to be assembled at the locale in question. *We are indebted to Moses Abramovitz, W. Brian Arthur, Louis Cain, Paul Krugman, and Gavin Wright for comments on earlier formulations of the material contained in this paper. Financial support for this research, under grants to the High Technology Impact Program of the Center for Economic Policy Research at Stanford University, is gratefully acknowledged. 349 0094-1190/90 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

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The essential dynamic features of the localization of economic activities based on Marshallian externalities can be illustrated in a simple model which postulates constant returns to scale in production. Although no site-specific inputs are used, and the industrial activities are of the sort sometimes described in the literature of location economics as “footloose,” the urban site as a fixed factor nevertheless plays a key role in the dynamic behavior of the model. Externalities arising from the density of the labor force congregating at the site, and from the size of the capital stock assembled thereupon, are assumed to influence the rates of adjustment of those input variables, and through these channels to affect the growth of production and population at the site. Despite widespread agreement that one of the distinctive features of the urban economy is the presence of externalities arising from the concentration of economic activities at a physically constrained site, few attempts have been made to model explicitly the effects of city size on the dynamics of urban growth [15, pp. 4-121. Von Rabenau [20] examined the effects of introducing nonconstant returns to scale in production, finding stable city size to exist under decreasing returns to scale, whereas under increasing returns urban size was unstable-population and production either increased without limit or fell to zero, depending upon the initial conditions. A model somewhat closer in spirit to that considered here has been investigated by Dendrinos [7], who hypothesizes that city size exerts a direct influence on the rate at which capital and labor accumulate at a particular location. ’ Dendrinos, however, focuses attention exclusively on congestion, a negative, growth-inhibiting externality; whereas, the model developed here gives greater emphasis to the operation of positive externalities of the sort that might account for features observed during earlier phases of urban development. The first section of this paper examines a simple deterministic model of local symmetric positive externalities characterizing the local markets for two distinct types of workers who cooperate in producing a good at constant marginal cost. In the absence of any other countervailing forces, these conditions are shown to give rise to highly unstable (and, e.x ante, rather unpredictable) patterns of urban growth. Several possible modifications of this simple model are considered. The second section introduces a model featuring asymmetric externalities, intended to capture the special characteristics of capital market externalities affecting industrial localiza‘Although Dendrinos [7] presents an analysis of the consequences of introducing a nonlinearity in the dynamic behavior of the system, the model he actually specifies does not in fact generate the behavior which he discusses. His model could be easily altered to achieve the mathematical properties which he supposes it to have, but it would seem appropriate then to offer a different rationale to support the new specification. See Rosenbloom 1181.

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tion. Because of the congestion effects arising eventually from the interdependence of capital investments at a fixed location, this modification removes some of the instabilities arising from the (hypothesized) existence of positive externalities in the market for labor. The dynamic behavior of the urban economy remains extremely sensitive to alterations ‘in the external environment, however, especially to changes in the attractiveness of other urban locations to mobile factors of production. In the third section the consequences for urban growth of several types of alterations in the external environment are considered. A brief concluding section emphasizes the sensitivity of dynamic behavior to initial conditions (“history”) in the models examined here, and relates this to other recent work modeling nondeterministic processes that are capable of generating spatial agglomerations in economic activity. 1. LABOR

MARKET URBAN

EXTERNALITIES GROWTH

AND

(a) Labor Markets and “Risk-Pooling”

Alfred Marshall took up the economic advantages of spatial agglomeration in Book IV, Chapter X of his Principles of Economics [13], where he examined “those very important external economies which can often be secured by the concentration of many small businesses of a similar character in particular localities: or, as is commonly said, by the localization of industry.” For present interests the focal point in Marshall’s discussion is not the underlying technological and organizational sources of increasing returns to scale that are internal to establishments, business firms, or even industries, but rather his recognition-as in the passage reproduced at the head of this paper-of the disadvantageous terms on which small and spatially isolated firms are obliged to compete for workers, and the corresponding labor market benefits to be derived by employers from the agglomeration of business at any particular locale.* One important microeconomic source of positive pecuniary externalities associated with increasing local labor market size may be found in the insurance or “risk-pooling” effects of increasing numbers of both workers 2The same theme was elaborated upon in Industry rind Trade [14, Book II, vi, 6, pp. 284-871, where Marshall takes note of the fact that places such as Sheffield and Solingen “have acquired industrial ‘atmospheres’ of their own; which yield gratis to the manufacturers of cutlery great advantages that are not easily to be had elsewhere: and an atmosphere cannot be moved. . The leadership in a special industry, which a district derives from an industrial atmosphere, such as that of Sheffield or Solingen, has shown more vitality than might have seemed probable in view of the incessant changes of technique.. . [Hlistory shows that a strong center of specialized industry often attracts much new shrewd energy to supplement that of native origin, and is thus able to expand and maintain its lead.”

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and employers3 In an uncertain world, in which the fluctuating fortunes of individual firms do not display strong positive correlation even in the same industry, the presence of large numbers of employers at a given location obviously will tend to reduce the magnitude of temporal variations in aggregate labor demand expected at that locale. Further, if a number of diverse industries whose patterns of temporal demand variations are not positively correlated should gather at the same site, the extent of fluctuations in the aggregate demand for labor at the place would be further attenuated.4 Other things equal, the smaller variance in earnings that workers would experience will make these “larger” (more diversified) labor markets more attractive to the risk averse among potential migrants. On the other side of the market, spatial proximity can create positive externalities for employers because workers’ search costs (in the event of layoff) are smaller and the expected relocation costs will be much less substantial where there is a greater probability of re-employment in the same local labor market. Employers who cannot perfectly foresee changes in their own demands for labor will incur greater “wage costs” where workers are obliged to bear higher risks of being laid off or expect to have to spend longer periods (in search) between jobs. It has been observed that “implicit contracts” between firms and workers offer reduced layoff risks in exchange for lower wage rates. But, such arrangements are less likely to be sought where, by locating in proximity to other potential employers with uncorrelated labor demands, a firm can provide its workers with the benefits of risk-pooling. To the extent that this risk-pooling relieves employers of some of the costs that would otherwise be entailed-either in compensating individual workers for greater (unpooled) risk of layoffs or in having a work force that is “contractually” hxed although demand is variable-firms whose labor demands are not positively correlated would tend to be attracted to each others’ neighborhoods. Marshall’s characteristically compendious discussion notes a number of other conditions, including greater possibilities for specialization in the 3We are grateful to Paul Krugman for suggesting “risk-pooling” as the most straightforward formulation of the microeconomic conditions that might produce positive labor market externalities. A more complicated argument linking lower average wage rates with the greater diversity of employers found in large urban labor markets can be derived from the model of migration and local job-search in David [6]. See Lebergott [12, pp. 242-451 for evidence of the importance of externalities associated with labor market size in the development of the American economy. 4The effects of these two sources of diversity will clearly depend on the extent to which the firms require workers with industry-specific skills. Presumably the workers who benefit the most from the agglomeration of diverse industries are those with very general skills, while workers in possession of industry-specific skills are more likely to be attracted by the localization of a single industry.

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provision of intermediate inputs, a finer division of labor, and the more rapid diffusion of innovation among specialized producers and workers, which also may favor the localization of an industry at particular (urban) sites.5 But, these seem to be more in the nature of technical and organization economies of scale that are internal to an industry, although quite possibly external to its individual constituent firms. As such they are left outside the realm of the present discussion, and in the following analysis will be explicitly assumed away by taking each type of input and the production technology to be identical in all locations. (b) Production and Labor Force Dynamics

We begin by considering, in effect, a productive activity that employs only inputs which are symmetrical with regard to the factor market externalities that affect their supply prices to employing firms at the location in question. For the purposes of analysis it is assumed that such capital as may be required is supplied perfectly elastically at an exogenously determined rental rate. Firms produce real net output (V) using two kinds of labor (L,, L,)-e.g., engineers and mechanics-according to a well-behaved production function which is homogeneous of degree one,

I,’ = F(L,, L,), where SF/SL, = F, > 0, SF/SL, = F, > 0; Fll, Fz2 < 0; and F,, = F,, > 0. We assume the output is tradeable, and outside markets are sufficiently large relative to the capacity of the city’s industries for the output price to be treated as a parameter. The local markets for industrial labor are also taken to be competitive, so that firms must treat the real wage rate as given and maximize profits by adjusting their employment of both kinds of labor until their respective marginal productivities everywhere are equal to the respective wage rates: (2)

For simplicity we assume that it is never economical to transform one type of worker into the other type, and we ignore the possibility of new workers being created at the site by natural increase. Net immigration therefore is the sole source of changes in the labor force. Migrants of either type are presumed to be attracted by a positive differential between the local real wage rate, wi, and the real wage rate offered at other equally accessible locations, wi. On the assumption that the income and substitu‘These various possibilities are discussed at some length in Marshall Chap. Xl.

[14, Book IV,

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tion effects of higher real wage rates upon the supply of labor offset each other, labor inputs are taken to grow at the same rate as the industrial labor force. The size of the local market for each type of labor, measured by Li, relative to that of the largest equally accessible industrial employment center, Ai, also exerts a positive effect upon the net immigration rate. The effects of increasing labor market size are not, however, unbounded in their ability to attract immigrants. The possibilities for further diversification and consequent risk-pooling gains diminish as the number of workers and employers increases, so the attractiveness of the larger local labor market may reasonably be supposed to be bounded from above.6 Accordingly, the growth rate of each type of labor may be described by a function of the form

; =rw +N(L)

- C(w,A),

where, for clarity, we suppress the subscripts for each type of labor. To represent the boundedness of the labor market externality, it will be assumed that N(L) asymptotically approaches some finite limit, N*, as L increases. Furthermore, to prevent the possibility that the labor force continues to grow when wages reach zero, N* must be less than or equal to C(ty, h) for w > 0. Finally, when w and L are both zero it is assumed that L/L = 0. By combining (3) with the profit maximizing condition from (2) we may characterize the dynamic behavior of the two kinds of labor in terms of a pair of nonlinear, first-order differential equations. Substituting for w, (3) may be rewritten as

2 I = Mi(Lt,L,)

= n&(L,,L,)

+ N(Li)

- C(W~,A,),

i = 1,2.

(4) Given specific functional forms, it is possible in principle to solve the system of differential equations simultaneously to find the growth path of each kind of labor as a function of time, initial conditions, and the 6This is not a feature peculiar to the case in which labor market externalities arise from risk-pooling. In other models, too, the city size effects should eventually diminish in their ability to attract immigrants. For example, in a migration and local job search model, the effects of increasing wage offer dispersion upon the expected value of the accepted offer would be bounded, even if the variance of the wage offer distribution itself were to increase linearly with increasing city size (cf. David [6]). Workers are also constrained in the extent of the job searches they can conduct, and therefore a point will be reached beyond which new immigrants cannot benefit individually from any further increase in the volume of job openings created by normal turnover at a location where there are more jobs.

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0kc’ FIG.

AND INDUSTRIAL

LOCALIZATION

355

hlfiL,=o, 0/ +7 0 0 00 Ll

1. Phase diagram for L,.

parameters of the model. Even without such information, it is still possible to determine the general qualitative features of those solutions by examining the phase diagram for this simple system. (c) Phase Diagram for Industrial Labor Force Growth Setting M(L,, L,) for the first type of labor (hereafter denoted as M,) equal to zero implicitly defines the locus of points in the L,-L, plane along which the size of the L, labor market is constant. Suppose that we begin at a point on this locus, and increase the number of type 2 workers at the urban site: this raises the marginal product, and hence the wages of type 1 workers, increasing the site’s attractiveness and inducing an inflow of type 1 labor. Similarly, a reduction in L, will induce an outflow of type 1 labor. These dynamic characteristics are depicted by the horizontal arrows pointing toward the M, = 0 locus in Fig. 1. As drawn in Fig. 1, M, = 0 is generally rising, and eventually becomes asymptotic to a ray through the origin. To verify that this is the correct shape for the locus of points along which the number of type 1 laborers is constant, consider the restriction imposed on L, and L, by the condition that M, = 0 in (4). This may be expressed as

From the assumption of zero degree homogeneity, it follows that F&L,, L,) depends only on the ratio of L, to L,, and so may be replaced by FI(L2/L1). Since F1(LZ/LI) is a monotonically increasing function of its argument, application of the inverse function rule allows (5) to be rewrit-

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ten as 2L = F; 1 C(w174) - N(k) n Ll [

17

(6)

where F-’ denotes the inverse function of F. When the labor force is large, Nt< L,) approaches zero and N(L,) asymptotically approaches some constant N*. As a result, L,/L, must approach a constant. That is, for L sufficiently large, the locus M, = 0 becomes asymptotic to a ray through the origin. Similarly, as L, approaches zero, the argument of F-’ approaches a finite value, so the ratio of L, to L, must be bounded from above, implying that the number of laborers of each type declines to zero. As the labor force increases from zero, the behavior of the M = 0 locus is complicated by the behavior of the labor force externality term, N(L). The slope along the constant labor force locus may be found by totally differentiating (4) and rearranging terms as

dL, dL,

= M=O

%(L,, -

L,) - N,(L)

nF,,(L,,

L,)

’

The denominator of the expression on the right is always positive, but the terms in the numerator are of opposite sign, so dL,/dL, may be positive, negative, or zero along the A4 = 0 locus. As long as the positive effect on the labor market externality of adding an additional worker is outweighed by the negative effect on the marginal productivity of labor, M, = 0 will be positively sloped. If, however, in some range, the labor force externality outweighs the reduced marginal productivity of type 1 labor, it is possible for the site to continue to attract laborers of type 1 even though the type 2 labor force is constant or falling. Therefore, as is suggested in Fig. 1, the constant labor force locus may pass through one or more local maxima before becoming asymptotic to a ray through the origin.’ The position of the M, = 0 locus in the plane is affected by the strength of the “outside” competition for immigrants. Holding L, constant, an increase in C(w,, hi) increases the right-hand side of (41, requiring an increase in L, to maintain the equality of the two sides. Thus, M = 0 is shifted upward at every point by an increase in the attractiveness of the alternative locations. Such a shift in C(o,, A,) also raises the slope of the line to which the M, = 0 locus converges as L, approaches infinity, as can be seen from inspection of (6). Recall that C increases both with the level of prevailing outside wage rates, and also with the size of the industrial ‘To simplify the discussion, we may assume that the N(L)-function takes a form excludes the possibility of more than a single local maximum in the M = 0 loci.

which

MARSHALLIAN

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INDUSTRIAL

357

LOCALIZATION

:O

Ll

2 (W

2 (4 FIG. 2.

Phase diagrams

for L, and ~5,.

labor market at the largest equally accessible alternative employment locale. Hence, even if the outside wage rate is unchanged, the growth of a “rival” city in the same catchment region for mobile workers causes the M, = 0 locus to slide upward and toward the left. The existence of local labor market “externality effects” means that the growth of employment at a rival location poses a dire threat to the prospects for continued growth, and may even replace growth with contraction, if the M, = 0 locus is shifted past the city’s original position. Negative “between-market externalities” in this model, therefore, are dual to positive “within-market externalities” of location decisions by firms and workers. Since the growth of L, is described by an equation identical in form to that for L, we need simply to reflect the locus M(L,, L,) about the 45” line to obtain the phase diagram for L,. The complete system may now be analyzed by combining the phase diagrams for the two kinds of labor. Two fundamental configurations are possible for the case in which there is a single local maximum in the h4 = 0 locii, as depicted in the two panels of Fig. 2. Figure 2a depicts a situation in which the strength of outside competition for both kinds of labor has shifted the constant labor force locii so far that there is no possibility for growth. No matter what the initial endowment of labor, the urban concentration faced with this situation is doomed to eventual disappearance. In Fig. 2b, on the other hand, the area below M2 and above M, is a region of unbounded expansion. This zone may be entered by cities above or below the two constant labor force curves, provided that their initial endowments of one or the other type of labor are sufficiently large.’ If a city’s endowment of labor of type 1 is too small, given the labor endowment of type 2, however, the direction of motion will lead into the regiqn *Note

that the M,

= 0 locus must be crossed

vertically

and the M2 = 0 locus horizontally.

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below M, and above M,-whence there is no escape. Once in this zone of the phase plane the city’s labor force must eventually dwindle to zero. The dashed boundary SS’ in Fig. 2b indicates the “separatrix” of this dynamic system: it is the locus of points CL,, L2) to the left of which movement proceeds inexorably downward, and to the right of which movement follows an uninterrupted course upward. The economic destiny of any particular locale may thus be said to be ruled completely by “initial conditions” in this model, for these determine where it is positioned in reference to the separatrix of the system. As Fig. 2b indicates, the model implies that once the city enters the region of uninterrupted growth it continues to grow unchecked. This is implausible, and reflects the absence of forces counteracting the postulated positive within-labor market externalities. Either of two possible modifications of the model suffices to close off the “growth corridor.“’ First, it could be supposed that the outside competition initially encountered arose from neighboring urban places, but that having vanquished its nearby initial challengers, the growing city would have to contend with stronger and more distant competition. Second, if decreasing returns to scale in production set in after some point, the M, and M, locii will curve back toward each other eventually, resulting in a stable equilibrium at the upper end of the growth corridor. A third line of extension which also may resolve the problem of instability involves abandoning the input symmetry restriction and modifying the model to include externalities-both positive and negative-which may be hypothesized to exist in the provision of capital at a particular urban site. It is to this possibility that we turn in the next section of the paper. II. URBAN

GROWTH MARKET

WITH ASYMMETRIC EXTERNALITIES

FACTOR

(a) Production and Capital Stock Dynamics

With the introduction of capital, the various labor inputs are subsumed into a single homogeneous labor force, denoted by L. Real net output now is produced by L and capital, K, according to a well behaved, first-order homogeneous production function, I/ = F(L, K), where FL, FK > 0; FLL, FKK < 0; and FKL > 0. The stock of urban capital includes buildings and 91f the specification of the functions N(L) and F(L,, L,) are such as to allow there to be more than a single maximum in the respective M = 0 loci, it is possible for a “zone of stable growth,” leading to a stable equilibrium city size to exist. For example, if the M = 0 loci had two local maxima, rather than the one shown in Fig. 1, the analogue of Fig. 2b could exhibit as many as three nonzero intersection points between M, = 0 and Mz = 0. The second of these nonzero intersections will be a “stable attractor” in this system.

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structures used both directly and indirectly by the industrial sector, such as directly productive plant and equipment, social overhead capital (e.g., streets, sewers, water supply, bridges, and harbors), as well as private overhead capital in the form of housing for the industrial work force. The local market for capital is assumed to be competitive, and producers adjust their use of capital so that FK = I-, the real (net) rental rate. The growth rate of the urban capital stock is taken to be a positive function of the real net rate of return (realized by industry) on the stock already installed. A higher rate of return induces a higher rate of flow of net investment from outside sources of finance; higher net profit rates may permit local authorities to use higher tax rates to issue municipal debt bearing higher coupon yields; alternatively, if the capital stock is thought to be privately owned by resident capitalists, and if wage earners saved nothing, a higher rate of return would in effect provide the resident owners of K with a bigger fund from which they could make additional investments. The effect of a given real rate of return upon the rate of capital formation expenditures at the location is taken to vary positively with the extent of experience which the investing community (whether local or “extramural”) has acquired with the business of that particular place. Two dimensions of new investment experience must be distinguished. The first is quantitative, and can be measured by the integral of the past net additions that have been made to the stock, which is to say by the present size of K itself. These effects of location-specific “learning” by investors will be supposed to follow a typical concave learning-function: increasing with K, but at a diminishing rate. The second aspect of investor familiarity is qualitative, and concerns the diversity of the range of investment projects that have been undertaken. This is likely to increase slowly until the scope of activities conducted at the location reaches some threshold size, K’, after which it becomes more diversified. The combined effect of these two forces is to make the response of gross investment, I, to net earnings on the existing capital stock a positive S-shaped function of K itself, Z = (S( K))rK,

where S,,(K)

S(K) > 0; S,(K) > 0; and S,,(K) > 0 for < 0 for K > K’, for some 0 < K’ < 00.

(8) K < K’

and

As with the labor force externality, it is assumed that the positive effects of an expanding capital stock are bounded, so that S(K) asymptotically approaches some finite limit S* as K increases. While increasing K increases the proportionate gross capital formation response to a given real rate of return, I-, given new capital formation expenditure undertaken

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in the locale, the actual net additions that can be made to the stock of urban capital are adversely affected by the degree to which the site has become “congested” as the result of past capital formation activity. A city can be regarded as a system of interrelated functions serving its directly productive activities, such as the housing and provisioning of the labor force, its transportation to and from places of employment and recreation, the generation and transmission of energy to carry out these activities, the disposal of wastes generated in the process, and so forth. The technical interrelatedness of the capital goods used in these sub-processes is heightened when they must be located at a spatially constrained site. Hence, as Veblen [19] suggested, the (city) economy that has been built up already may suffer serious “penalties of taking the lead,” in the form of capital congestion effects which either raise the relative price of new units of capacity, or dissuade investors from undertaking at previously developed sites capital expansion projects that would otherwise be financially attractive. The foregoing suggests also that the maintenance and replacement of units of the existing stock of capital would become more costly as urban precincts become more densely packed with structures and capital facilities above and below ground. Both aspects can be represented in the replacement term, R = gK*, g > 0, which imposes a rapidly rising “drag” on the growth of net additions to urban capital (k = I - R) as the stock of K itself rises. Combining the rate of return effect given by (8) with this “capital congestion” effect leads to the following description of the growth rate of the (net) stock of urban capital, where we have substituted the marginal productivity condition specified earlier:

; =B(L,K)

= (S(K)}F,(L,K)

-gK.

(9)

(b) Phase Diagram for the Growth of the Capital Stock

The capital stock is maintained at a constant level if B(L, K) = 0. Starting on the B = 0 locus, an increase in the labor force increases the marginal productivity of capital, causing a positive local investment demand effect which overcomes the drag on capital accumulation imposed by the existing capital stock. So, to the right of the B = 0 locus new capital formation takes place. A movement in the opposite direction, on the other hand, leads to a running down of the local aggregate stock of capital. Setting I&!,, K) equal to zero and rearranging terms, we derive the following restriction on K and L along the constant capital stock locus: F,(L

sK K) = S(K)

.

(10)

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By zero degree homogeneity, F,(L, K) depends only on the capital labor ratio. Furthermore, since F&r,, K) is a monotonically decreasing function of K/L it may be inverted to express the capital labor ratio along B = 0 as K -=F$ L

[S(K) 1’ E

(11)

For K sufficiently large, S,(K) approaches zero and S(K) approaches a constant S*, so the term inside the brackets increases linearly with K. As a result, K/L falls as K increases along B = 0. To compensate for the ever increasing drag imposed by the accumulation of capital the real interest rate must rise, which requires that the capital-labor ratio decline. On the other hand, as K approaches zero along B = 0, gK/S(K) also approaches zero. Since this implies that K/L must be increasing, L must also be approaching zero, so the B = 0 locus passes through the origin. For the range of values of K in which S(K) is increasing, the operation of capital market externalities means that the slope of B = 0 need not always be positive. Totally differentiating (9) with respect to K and L, and rearranging terms, the slope along the constant capital stock locus is dK WW’,,(L K) z B=o = - S,(K)F,(L,K) + S(K)F,,(L,K)

-g’

(12)

By assumption, the numerator is always positive, but the terms in the denominator are of opposite signs, with the first term positive and the second two negative. The slope of the B = 0 locus thus depends on the relative magnitude of the positive effect on capital formation due to the capital market externality, and the negative effects of the decline in the marginal productivity of capital and the added drag of additions to the capital stock in place. Where these two effects are precisely in balance, the denominator will be equal to zero and the B = 0 locus will become vertical. Thus, the B = 0 locus may be fully S-shaped (with a region of negative slope) as drawn in Fig. 3.” From the specification of

“For the B = 0 locus to become negative it is necessary, of course, that it pass through that point of vertical slope at which the denominator of (12) equals zero, and, K and L also satisfy the restrictions imposed by (9). Multiplying the denominator in (12) by K and setting the resulting expression equal to the right-hand side of (9) implies the following condition that must be met for B = 0 to become vertical:

S,(K)K=1S(K)

4db KW FK(-L K)

That is, the elasticity of the capital market externality at this point must exceed 1 minus the elasticity of the marginal productivity of capital with respect to K.

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3. Phase diagram for K.

S(K) as an increasing sigmoid-shaped function of K, there will be some range of values in which the capital market externality is growing quite rapidly, making it likely that dK/dL will become negative within some region along B = 0. The position of the B = 0 locus in the plane is affected by the magnitude of the “capital congestion” parameter, g. As g is increased, the drag of existing capital increases, and the higher must be the marginal productivity of capital (and hence the larger must be L) to maintain the growth of the capital stock at any steady rate. Thus, an increase in g results in a rightward displacement of the B = 0 locus. (b) The Complete System in the Phase Plane

To describe the motion of the complete system in the K - L plane it is necessary to combine the phase diagram for the capital stock derived above with that for the industrial labor force. The dynamics of labor force growth are taken to be those postulated for each of the (symmetrical) labor inputs in Section I. Therefore, we can immediately obtain the expression describing the growth rate of the labor force from (4) by replacing L, with K. In orienting the M = 0 and B = 0 locii vis-a-vis one another, it is helpful to note that since as K increases K/L is falling along B = 0, the constant capital stock locus will eventually be below the constant labor force locus. Thus, if B = 0 is ever above M = 0 it eventually must cut it from above. Taking account of this fact, there are three possible configurations of the pair of curves, which are illustrated in Fig. 4. In Fig. 4a the B = 0 locus lies everywhere below M = 0. The economic prospects for any locale confronted with the combination of conditions that give rise to this situation are especially depressing. No matter what

MARSHALLIAN

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363

r ko 3 ho J L!z

K

K ko

5 Lo J 0E

L 4 (4 FIG.

0

L

4 lb)

4 (c)

4. Three possible phase diagrams for the complete dynamic model.

initial conditions obtain, the place is bound to disappear as a productive site. The strength of outside competition in the labor market, combined with the capital congestion effects faced by the city make it impossible for it to retain any of its initial endowment of mobile factors of production. In Fig. 4b, on the other hand, where the B = 0 locus is initially above the M = 0 locus, there is a single nonzero point of intersection of the two curves. That this equilibrium is stable is clear from inspection of the dynamics of motion around point E; the city will tend to converge to this point from any initial position. In the third possible case, where the constant capital stock locus rises temporarily above the constant labor force locus there are at least two equilibria, denoted by (L*, K*) and CL**, K**), respectively.” Only the latter of these intersections CL**, K**) is dynamically stable. Were a “shock,” say a major fire, to destroy a potion of the capital stock when the urban economy was poised at CL*, K*), the blow would prove catastrophic -setting in motion a gradual loss of population and the gradual erosion of the remaining stock of urban capital. On the other hand, an exogenous shift in the supply of finance sufficient to push the local capital stock above K*, say the flight of funds from other places, would set in motion a cumulative expansion process, leading eventually to the higher equilibrium at CL**, K*). As Fig. 4c shows, the region between the two intersections of the M = 0 and the B = 0 loci is a “corridor of stable growth” which may be entered from above or below by city economies that, given the size of their labor market, are initially either too capital intensive or not capital intensive enough. The former will attract workers while using up some of their “If additional turning points exist in the constant labor force and capital stock locii, then additional intersections-each corresponding to an equilibrium-are possible. Dynamically stable equilibria arise wherever the B = 0 locus cuts the M = 0 locus from above.

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capital endowment, while the latter will shed some of their labor force while building up their capital facilities to the point that both labor and capital inputs can begin to grow concurrently. The stable growth corridor cannot be reached from every point in the plane, however. In fact there will exist a separatrix-such as the locus indicated by the heavy dashed line in Fig. 4c-separating the region from which the growth corridor can be reached from the zone from which it cannot. Thus, while town T can “use up” or “run down” an initial capital endowment that was greater than K * to pass into the growth corridor, hamlet H, beginning with a smaller labor force is doomed to pass eventually into the region in which B = 0 is below M = 0, thereby being consigned to economic oblivion. Of course, such will be the prospect facing all those other places which find themselves initially caught in the latter region. Thus, unlike the situation depicted in Fig. 4b, the economic life of small settlements under these conditions remains precarious, and their ultimate fate will be difficult to predict oc ante even though their initial endowments of capital per worker may be relatively large. This aspect of the model accords well with the widely varied experience of “boom towns” based upon local natural resource endowments. Each of the situations depicted in Fig. 4 may be relevant for some places or time periods, but, since our concern is primarily with the successful establishment and growth of cities, the latter two cases appear to be the more interesting ones. As a descriptive tool, the dynamic characteristics depicted in Fig. 4c seem to offer especially rich possibilities. Because of the presence of both stable and unstable regions in the phase diagram, the dynamic behavior of the local economy is sensitive to alterations in the external environment, especially changes in the relative attractiveness of other urban centers which are competing for the same scarce resources. III.

DYNAMIC RESPONSE TO CHANGE ECONOMIC ENVIRONMENT

(a) Reduced Transport

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Since the production function F(L, K) is a net output, or value-added function, its level would be shifted upward by an increase in the volume of purchased intermediate inputs that were combined with labor and capital at the urban site. Assuming that there are substitution possibilities-either within activities or through altering the mix of urban production activities -the degree of purchased input-intensity of the local industrial sector will be increased by a reduction of the c.i.f. price of raw materials relative to the f.o.b. price of the city’s exportable products. An improvement in the locale’s transportation connections with the outside world, by reducing the wedge between the c.i.f. price of raw materials delivered to local produc-

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ers and the f.o.b. price they received after allowing for freight carriage of their output to extramural consumers, would be tantamount to an upward shift of P(L, K). If the transport improvement is one that especially favors the place in question, rather than occurring symmetrically throughout the system of cities, its effects will be both more profound and more readily predictable. Just how F(L, K) will be shifted by comparative transport improvement depends in general upon the relative substitutability of raw materials for L and K. But, in the simple case specified here, the effect would be equivalent to a Hicks-neutral improvement in the efficiency of L and K: the K/L-ratio would not be altered. Neutral factor-augmenting change of this kind can be represented by leaving the B = 0 and M = 0 loci undisturbed and simply resealing the L- and K-axes of the phase diagram equiproportionally downward. In other words, we can leave Fig. 4 alone, but associate smaller absolute values with the natural unit measures of L and K; if, instead, L and K were measured in “efficiency units,” nothing would need to be altered. The main effect of an improvement in the location’s comparative position in the transportation network thus will be to reduce the levels of L and K at which it can enter the corridor of stable growth. But it is rather unrealistic to suppose that an improvement in the transportation network would be so locationally specific that other places would not benefit. To assess the full impact, therefore, it is important to consider the consequences of the growth of economic activity at rival locations. (b) Labor Force Growth at Rival Locations

It was noted in Section I that the growth of the labor force at a rival location would, by increasing C, stiffen the competition for mobile workers and cause the M = 0 locus to shift upward and to the left. Fig. 5a shows this happening to a young city which, in an environment of weak outside competition indicated by the dashed M = 0 locus, had managed to expand L and K until it reached Y on the phase plane. At that point, we imagine the labor force of a rival center suddenly expands, increasing C(w, A,) to C(w, A,) and shifting the stationary-L locus outward to a new position indicated by the solid line. The overtaken city would then begin to lose its work force, but, as it would continue to make additions to its capital stock, there would be a rapid rise of capital per unit of labor input in the industrial sector. Two outcomes are possible, depending on the magnitude of the shift in the M = 0 locus, and the initial position of the overtaken city. If the labor force growth in the rival center is sufficiently large, it may shift the M = 0 locus to a point at which the overtaken city can no longer re-enter the stable growth corridor; indeed the stable growth corridor may even shrink to nonexistence. If the increment in the labor force at the

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other location is more moderate, however, the effect of the shift in the M = 0 locus only retards the growth of the overtaken city, as it moves to a higher capital-labor ratio before re-entering the stable growth corridor. In an open region in which one place may gain workers without imposing direct population losses upon other urban economies, a one-time shock of this sort is not inconceivable. It would, however, be more plausible to think of a rival growing more steadily by tapping some large pool of migrants, in which case we would have not a single shock, but a succession of graduated leftward shifts of the M = 0 locus. Since C,, < 0, by previous assumption, these would become smaller and smaller until the position of the M = 0 locus for our city had stabilized, even though the labor force size of its largest rival was continuing to expand. Assuming that the effect of this growth at the rival location was not sufficient to doom our city to unending contraction, the presence of a continuously growing rival center would in this case retard the rate of labor force growth and rapidly force up the capital-labor ratio along the city’s expansion path, as is depicted in Fig. 5b.12 Of course, in a closed region in which, say, two urban places were competing with each other for the same mobile work force, the contraction of the labor force at one locale would imply its further expansion at the other center. Movements in the M = 0 locus cannot, in this instance, be treated as exogenous, but must be directly incorporated into the model. This requires the construction of a more elaborate dynamic model, in which the rate of capital and labor accumulation in both locations is simultaneously determined. I3 It may be seen that the transfer of workers between the two cities will have two contradictory effects on the position of the M = 0 locus facing each city. On the other hand, the expansion of ‘*The implications of this latter point are rather far-reaching and need to be explored in greater depth elsewhere. Briefly stated, however, the present model suggests that a growing metropolitan center would enjoy labor market externalities that permitted it to draw in industrial workers while expanding along a comparatively low K/L path, where real wages also remained comparatively low. Its competitive pull upon the pool of industrial workers who might otherwise be attracted to a smaller, provincial manufacturing center, would imply that the only path along which the latter could expand was one that for some time entailed higher real wage rates and a higher and more rapidly rising (or more slowly declining) level of K/L. The possible relevance of this as a basis for reinterpreting the comparative development of manufacturing at different locations in the Atlantic economy in the nineteenth century undoubtedly will occur to those who have wrestled with the so-called Rothbard-Habakkuk problem: why should (provincial) US manufacturing industries have been characterized by a higher capital-labor ratio, and by a faster rate of labor productivity growth than the industries of (metropolitan) Britain? Cf., e.g., David [6, chap. I]. 13Such a model will not in general be susceptible to analysis with the simple sort of phase diagrams employed here because it requires the simultaneous determination of more than two variables.

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FIG. 5. Effects of the growth of a rival city: Open region case.

the growing city’s labor force increases the size of the labor market externality at that location, shifting the M = 0 locus faced by its rival up and to the right. On the other hand, however, the transfer of workers reduces the capital-labor ratio-and hence wages-in the growing city, while raising the capital-labor ratio in the other city, thereby causing an offsetting shift in the M = 0 locus. As a result, the effects of labor force growth at a rival location are no longer unambiguous. In the situation in which the larger of the two urban centers has entered the region of stable growth, however, and is thus drawing in both capital and labor, the adverse effect on real wages of labor force growth is likely to be attenuated. In this case the effect of increasing labor market size in the larger city unambiguously results in a leftward shift of the M = 0 locus facing its smaller rival. Meanwhile, as the labor force at the smaller city shrinks, the M = 0 locus facing the larger city will be shifting down and to the right (although this movement will be moderated by the rising capital intensity and hence wage rates in the smaller place) thereby displacing the stable equilibrium point outward to the right. This set of developments is depicted in Fig. 6. This competition may end in the eventual extinction of the smaller city (case c’), if the own-market labor force externality effect continues to be important for large values of L. If, on the other hand, the effects of the own-market labor force externality are rapidly exhausted, both cities (case c” and c) may be able to reach their respective stable equilibria at E’ and E. Because of the movement of their respective constant labor force loci, however, these will be at different sizes. The existence of an articulated regional hierarchy of urban places suggests that a growing city may encounter early competitive challenges from neighboring rivals whom it may vanquish, because it attracts its growing labor force largely at their expense. At the outset the existence of

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FIG. 6. Effects of the growth of a rival city: Closed region case.

a spatially more remote and larger rival, although growing, will not necessarily impede the young city’s ability to sustain a high immigration rate if the common pool from which they both draw is a large one. But at some point they will be big enough to effect each other, and the renewal of an effective outside challenge could bring about another phase of retarded labor force growth. IV. CONCLUSION:

MARSHALLIAN EXTERNALITIES AND HISTORY This paper has explored the possibilities of an explanation of urban growth based solely on the existence of externalities operating through the size of urban markets for labor and capital. Although the deterministic models developed in this paper are stylized, they are capable of reproducing a number of the phenomena observed in actual urban systems. Among these are the great diversity of growth paths followed by the smaller urban centers, and the emergence of a hierarchy of urban places. Closer examination of the extent to which models incorporating Marshallian factor market externalities are able to reproduce empirically noted regularities clearly is in order. The existence of multiple equilibria in models of this kind, and the sensitivity of the ultimate dynamic outcomes to small differences in initial conditions, or to relatively small shocks, allows realistic scope for historical events to play a role in the dynamics of spatial systems. Thus, the same forces that promote the agglomeration of footloose industries allow the details of seemingly transient and adventitious circumstance to exert an enduring influence upon the spatial distribution of economic activity and population. Although this is a point that still seems to need underscoring, it is scarcely a novel insight; it was appreciated fully a half-century ago by a number of

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contributors to the German literature on location theory. Critical of the essentially ahistorical approach deriving from Von Thiinen [21] and the early writings of Weber [22] which depicted industry location as more-orless foreordained by invariant geographical and technological considerations, later writers such as EnglHnder [9], Ritschl [17], and Palander 1161 viewed the determination of industrial location as being more closely akin to a geological process in which each successive layer that was deposited necessarily would build upon and adapt to the terrain that had been created by previous locational formations. But this historical approach, and the analytical insight that related the persisting influence of historical events to the operation of factor market externalities, can claim an even more illustrious provenance. It was clearly anticipated in Alfred Marshall’s remarks on the surprising permanence, even in the geographically fluid conditions of nineteenth century America, of numerous localized manufacturing industries at sites that had been fixed by “accidents of history”: It may be added that accidents, such as the arrival in times far back of energetic artisan immigrants, have founded industries, which have maintained their preeminence till now.. . . Thus, according to the [US] Census for 1900 Gloversville and its neighborhood which produce half of the leather gloves of the country, owe their skill to some Scotch [sic] glovers who settled there in 1760: Troy had a practical monopoly of detached collars; as the result of the enterprise of a working woman about 1830. Similarly, though less extreme cases are those of hosiery at Cohoes; of cheap jewelry at Attleboro; of fur hats at Danbury and Bethel. Industry and Trade, [4, Book II, vi, 6, pp. 287-881

The deterministic framework of analysis adopted here unfortunately remains illsuited to handle such phenomena. Explicit stochastic modeling along the lines explored in the recent work of Arthur, for example,14 seems to offer a natural, more attractive way to study the influence of accidents of history in this connection, and to deal constructively with the problem of indeterminacy and the dominance of initial conditions which have been highlighted by the analysis presented here. REFERENCES 1. W. B. Arthur, “Urban Systems and Historical Path-Dependence,” Food Research Institute, Stanford University, June (1987). 2. W. B. Arthur, “Industry Location Patterns and the Importance of History,” C.E.P.R. Paper 84, Stanford University (1986). 3. W. B. Arthur, “Competing Technologies and Lock-In by Historical Small Events: The Dynamics of Choice under Increasing Returns,” C.E.P.R. Paper 43, Stanford University (1985). 14See Arthur, Ermoliev, and Kaniovski [5], and Arthur [3] for a stochastic formulation of an analogous process with increasing returns to scale; and Arthur and David and Arthur [Z, l] for suggested applications to spatial economics.

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4. W. B. Arthur and P. A. David, “Agglomeration Economies and the Growth of High Technology Centers: A Proposed Program of Research,” Center for Economic Policy Research, Stanford University (1984). 5. W. B. Arthur, Y. M. Ermoliev, and Y. M. Kaniovski, A generalized urn problem and its applications, Cybernetics 19, 61-71 (1983). 6. P. A. David, “Technical Choice, Innovation and Economic Growth,” Cambridge Univ. Press, New York (19751. 7. D. Dendrinos, On the dynamic stability of interurban/regional labor and capital movements, J. Regional Sci. 22, 529-40 (1982). 8. D. Dendrinos and H. Mullaly, “Urban Evolution: Studies in the Mathematical Ecology of Cities,” Oxford Univ. Press, Oxford (1985). 9. 0. Engllnder, Kritisches and Positives zu einer allgemeinen reinen Lehre vom Standort, Z. Vo’olkwirtschati Sozialpolitick, Neue Folge, 5 (1926). 10. J. V. Henderson, “Economic Theory and the Cities,” Studies in Urban Economics, Academic Press, New York (1977). 11. J. V. Henderson, Efficiency of resource usage and city size, J. Urban Econom., 19, 47-70 (1986). 12. S. Lebergott, “Manpower in Economic Growth: the American Record Since 1800,” Economic Handbook Series, McGraw-Hill, New York (1964). 13. A. Marshall, “Principles of Economics,” 8th ed., MacMillan, London (1920). 14. A. Marshall, “Industry and Trade,” 3rd ed., MacMillan, London (1920). 15. T. Miyao, Urban growth and dynamics, in “Urban Dynamics and Urban Externalities,” (Richard Arnott, Ed.), Fundamentals of Pure and Applied Economics, Vol. 11, Regional and Urban Economics Section, Harwood Academic Publishers, New York/Chur, Switzerland (1987). 16. T. Palander, “Beitrlge zur Standortstheorie,” Almqvist 8c Wicksells, Stockholm (1935). 17. H. Ritschl, Reine und historische Dynamik des Standortes der Erzeugungszweige, Schmollers Jahrbuch, 51, 813-70 (19271. 18. J. L. Rosenbloom, Straightening out a nonlinearity in the dynamics of labor and capital movements: A comment on Dendrinos’ model, J. Regional Sci., to appear. 19. T. Veblen, “Imperial Germany and the Industrial Revolution,” New York (1915). 20. B. Von Rabenau, Urban growth with agglomeration economies and diseconomies, Geographia Polonica, 42, 77-90 (1979). 21. J. H. Von Thiinen, “Der Isolierte Staat in Beziehung auf Landwirtschaft und Nationalokonomie,” Hamburg (18261. 22. A. Weber, “Theory of the Location of Industries,” University of Chicago Press, Chicago (1929).