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5 An Economic Mode} of Urban Growth MARTIN J. BECKMANN Infrastructures are basic to all economic life. The urban infrastructure is one of the most diverse and complex. To name only the most important components, it includes streets and public transportation; water supply and sewage removal; police and fire protection; judicial, educational, and health facilities; and parks and other recreational facilities. The study of infrastructure opens up an approach to a whole class of economic prob lems. Characteristically, both private and public agents demand infrastructure, but most infrastructure is supplied by the public sector. Yet there are a great variety of institutional arrangements for the creation and management of infrastructure, offering a fruitful field for economic analysis and one that has been insufficiently explored. This chapter will focus on the de- mand side of urban infrastructure. No exact relationship will be established between the size or population of cities and the demand for infrastructure. Rather, we will consider generically the principal forces behind this demand those forces that drive the growth or decline of individual cities and the urban system as a whole. Explaining the rise and fall of cities is one of the fundamental problems of urban economics. Cities may exist and grow for noneconomic reasons, particularly political ones. Madrid and Washington are examples of cities that were located for purely political reasons; the rise of centralized gov- ernment is the principal driving force behind their growth. In a market economy the economic fortunes of a city clearly depend on two factors. The first is the value of goods and services that can be 98

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AN ECONOMIC MODEL OF URBAN GROWTH 99 sold to residents with sources of income that do not derive from the city's current productive activities. This first factor includes the economic base associated with retirement income, certain welfare payments, remittances transferred from other geographic regions, and some other forms of un- earned income and the use of accumulated wealth. The second factor is the value of goods and services that can be exported to the rest of the economy. More exactly, the gross income of a city is determined by the value added in production and trade. It is this added value that generates the demand for labor and thus determines the population that can be so supported. Urban population is, in fact, an increasing function of such added value strict proportionality would result if per capita income were constant. Exporting goods and services to locations outside the city is of greater overall importance than exogenous sources of income in the urban economy. These exports may go to the immediate neighborhood (rura centers) or to the region of which the city is the economic capital. They may also be sold nationwide or worldwide, in which case the city's advantage derives either from localized natural resources or from spe- cialized labor resources that, often for historical reasons, choose to reside in this location. When a city exports goods and services to a surrounding region, it is known as a central place and serves as a trade and production center of a particular region. Subregions aggregate into regions, which leads to a nesting of regions and to a hierarchy of central places (see chapter 31. The size of a central place depends on its rank in the hierarchy of central places because this determines the size of the region to which it exports- and on the income of the region of which the city is the capital. The more prosperous the region, the greater the demand for services from its regional capital. Each central place supplies a range of goods and services, some to its immediate neighborhood, in its role as a lowest order central place. Some goods and services are produced in every city, but others are pro- duced only in central places above a certain rank. The rank of a good is the minimum rank of a central place that supplies this good. The growth of central places is a result of the growth of a region's income and of shifts in the specialization for goods and services among centers of dif- ferent ranks. Cities that serve as supply points to distant markets produce spe- cialized products from specialized resources. When such production is concentrated in only a few cities, it is usually because of economies of scale; the joint location of several related specialized activities arises from agglomeration economies that is, economies of joint location- wherein producers use each other's products or related or joint facilities

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100 MARTIN J. BECKAlANI!: (see chapter 41. These cities change in size when the demand for these specialized products shifts or when natural resources are discovered or exhausted. "Resources" include capital stock as well as natural assets. Thus, when General Motors decides to set up an assembly plant in some location deemed favorable as a supply point for a regional market (identified by the company), this specialized capital (the plant) gives the location an advantage, at least temporarily, for producing a certain type of automobile. When a city specializes in one product, its growth or decline is usually a result of the market demand for this product. The larger the potential (world) market, the stronger the potential for growth or decline. There are any number of possible products or resources to which a city may look for economic success. Spectacular growth rates for settlements have occurred based on areas as diverse as mining and defense work. In mining, growth results from the more or less accidental discovery of extractable resources, and decline occurs when the resources are depleted or sudden shifts of demand take place. In defense work, growth is mainly due to political decisions about where to locate defense contracts. Predictions of long-term demand and demand shifts for particular prod- ucts, however, have been notoriously incorrect. Prosperous cities may fall on hard times when their product is no longer in demand and when no alternative uses have been found for their specialized resources. Decline may occur either because a competitor city is producing a similar good more efficiently or because in general the market for the good has faded. Economic theory is able to predict the future of cities with specialized industries only in quite general terms. Oscillations, or cycles of boom and bust, are probably more characteristic of specialized urban economies than of diversified ones. We can provide somewhat more illumination in the task of predicting expected changes in central places over periods that extend beyond the life span of a particular industry. These changes can be summarized as a decline of smaller central places and growth of the metropolis. Metro- politan growth is often accompanied by a decentralization or spread of the metropolitan area itself. But it appears that most new economic ac- tivities are launched in or near metropolitan centers and then are filtered down or diffused through a competition among higher order central places. Ease of transportation and other factors affect the pattern that emerges. A deeper economic understanding of the forces behind urbanization can be gained by looking at individual cities and particular cases. Yet at the same time, we must try to understand what drives the expansion of the urban system as a whole. To keep the analysis as simple as possible, we consider as an example an economy with two goods that are produced

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AN ECONOMIC MODEL OF URBAN GROWTH 101 from two resources, land and labor. Labor is assumed to be mobile, and both land and labor are assumed to be of uniform quality. The first or "agricultural" good is land intensive and characterized by constant returns to scale. The second or "industrial" good requires little land and has increasing returns to scale. (In economic parlance, "returns to scale" can be explained as follows. If all inputs are doubled and output is also doubled, returns are constant; if all inputs are doubled and output is more than doubled, returns to scale are increasing.) In addition, this economy includes transportation activity that uses only labor. (Note that capital has been omitted. Because under mild assumptions it can be shown that capital and labor are used in constant proportions in each activity, the two factors may be conveniently aggregated into a single factor called labor.) To simulate the operation of market forces in a competitive economy, we postulate an optimum location of available labor on an unlimited amount of land. This results in maximum welfare as measured by the aggregate of consumer utilities due to the consumption of the two goods. Labor supply by each household is considered fixed, and income con- straints are disregarded. This model has been analyzed at length (Beck- mann and Puu, 19851. This combination of variables produces a predictable result. If the total population is not too large, the production of the good with increasing returns is entirely concentrated in one small circular area, which we may call a city. This city is surrounded by a larger circle in which the agri- cultural good is produced. The density of land use declines with distance from the city and ends at a well-defined distance at which the product price in the center equals wage costs plus transportation costs to the market in the center. This is a modified version of a so-called von Thunen system (von Thunen, 1826) composed of a single metropolis and its rural hin- terland. In a central-place system, centers of each rank constitute the focal point of such von Thunen systems, and the spatial economy founded on an environment with uniform resource endowments may be understood as the combination of central places with their urban satellites and rural hinterlands. A study of economic growth in such a system would have to explain the differential growth rates of centers of various orders. The goal in this chapter is to try something simpler, namely, to examine the growth of city and rural hinterland for a simple von Thunen system. As it turns out, this is sufficiently challenging and can serve as a first step in the direction of a full analysis of economic growth in a spatial economy. Our main concern is in determining the relative growth rates of the urban and rural population. To analyze the process of economic growth in a von Thunen economy,

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102 AJAR TIN J. BECKMANN we explore an aggregate production function for the agricultural sector, describing agricultural output delivered to a city as a function of aggregate inputs of labor and transportation into the agricultural sector. The appendix to this chapter presents the mathematical argument for the conclusions that follow. Under reasonable assumptions, decreasing returns to scale in agriculture result from the necessity of using more and more transportation as the agricultural zone is expanded. The question of the development of the spatial economy can thus be shown to be equivalent to studying a growth process in a two-sector economy in which an urban good is pro- duced with increasing returns to scale and an agricultural good is produced with decreasing returns to scale. If we assume that the market achieves an efficient allocation of total available labor between urban and agricultural production, we find that urban population increases relative to rural population as total population grows. The driving force behind urban growth is the increasing returns in the production of urban goods, combined with the substitution of urban for agricultural goods, which in the aggregate are produced under dimin- ishing returns to scale. Figure 5-1 shows the degree of urbanization as a function of total population when ax = 3/4, 5~ = 11/10, and ~ = 1/4. This result should be compared with the actual proportion of urban population in the United States in the last 200 years (Figure 5-21. Thus, the prediction of economic theory is that, under reasonable as- sumptions, the urban sector must always grow relative to the agricultural 1.0 .8 .6 ~1~ .4 .2 .0 - 1o-2 10-1 101o1 1o2 103 Lo + L1 FIGURE 5-1 Urban population (L1) as a fraction of total population (Lo + L1), calculated. Lo rural population.

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AN ECONOMIC MODEL OF URBAN GROWTH 1.0 _ .8 - Q o - o .6 c o a Q lo. '4 C o 103 - / / - / 40 80 120 160 200 240 Total Population (mililons) FIGURE 5-2 Urban population as a fraction of total population in the United States, 1790-1980. Data source: U.S. Bureau of the Census, Census of Popu- lation. Washington, D.C., various years. sector and that this process will continue in the future. The substitution of services for industrial production does not change this prediction be- cause many urban services, like industry, enjoy increasing returns to scale. From this perspective, any past reversals of the urbanization process must be viewed as resulting from economic setbacks that occurred because of wars and other catastrophes. If we accept such a viewpoint, the challenge of providing urban infrastructure can only become greater. REFERENCES Beckmann, M. J., andT. Puu. 1985. SpatialEconomics: Flow, Density and Potential. Amsterdam: North-Holland. Thunen, J. H. von. 1826. Der isolierte Staat in Beziehung auf Landwirtschaft und Na- tionalokonomie. Stuttgart: Gustav Fischer, 1966 (reprint). APPENDIX To analyze the process of economic growth in a von Thunen economy, we must first construct an aggregate production function for the agricultural

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104 MARTIN J. BECKAlANN sector, describing agricultural output ZO delivered to the city as a function of aggregate inputs of labor XO and of transportation T into the agricultural sector. Let r be the distance from the city and R the radius of the agricultural zone. The area occupied by the city will be neglected. The following notation will be used: r distance from the center R distance at which production intensity falls to zero Fir) labor employed in agricultural production per unit of land bx~r)~ output of agricultural product per unit of land (a Cobb- Douglas function) ZO aggregate agricultural output delivered to the city XO aggregate labor input into agriculture T aggregate transportation input Z1 = b1 Lid aggregate output of the urban good as a function of urban labor input LO These aggregates depend on the density of labor Output: rR ZO = Jo Herr bx~(r~dr Labor input: Transportation input: input Fir) as follows. (1) rR XO = J Barr x~r)dr. rR T = J 2=r2 b Xerox dr (2) (3) We maximize equation 1 subject to equations 2 and 3. The Lagrangian of this problem is fR fR fR J L(r~dr = J 2~rx~dr + AtL - ~ 2Trrxdr) OR + ANT- J 2=r2 x~dr). Jo Maximizing with respect to x under the integral yields C~^xcx- 1 or A- ~rc~x~-1 = 0

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AN ECONOMIC MODEL OF URBAN GROWTH x(r) = A (1 - or) 105 (4) This shows that labor input per unit area x(r) decreases with distance and is zero for r'R=- (5) Substituting for x(r) from equation 4 and R from equation 5 in equations 1, 2, and 3 for ZO, XO, and T yields Zo ~ A 1-~ ~-2 1 XO ~ A 1~ ~-2 T~ A 1-~ ~-3 (6) (7) (8) Eliminating A and ~ between equations 7 and 8 and substituting in equation 6 yields the aggregate production function for agriculture: ~2-20t Zo = B Xo3 - 2~ T3 - 2ot (9) Note that this is a Cobb-Douglas production function with decreasing returns to scale because the exponents add to less than 1: ot 2 - 20` 2-~ 3-20` 3-20` 3-20t for O < Ox < 1. The decreasing returns result from the necessity of using more and more transportation as the agricultural zone is expanded. Finally, consider the allocation of a fixed amount Lo of labor to agri- cultural production and transportation: or 2 - 206 Max B . X30-2~ T3-2~ Xo,T such that Xo + TCLo The efficiency conditions for this simple constrained maximum problem are

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106 and MARTIN J. BECKMANN CX X - 0 - 2 - (X 2 - 2Ot 2-~ Lo Lo (10) (1 1) and the aggregate production function in terms of aggregate labor Lo is then 2- Zo = be L3 . (12) The question thus has been shown to be equivalent to studying a growth process in a two-sector economy in which an urban good is produced with increasing returns to scale and an agricultural good is produced with decreasing returns to scale. (Transportation need no longer be considered explicitly.) In our model the urban good is either consumed in the city or transported to the city as back freight at no cost. We assume that the market achieves an efficient allocation of total available labor L between urban and agricultural production. Economic theory shows that this is true under perfect competition. The object of the market is then to maximize welfare as measured by an aggregate utility function. Let this be specified as U= aZO + (1-a)Z~ (13) in terms of aggregate outputs Zi of the two sectors. Furthermore, we have the production functions Zi = bi Licit where O< Fo< 1 < Pl. A given total labor force L is then allocated to achieve Max a be Lobe + (1-a)b~ Lath, Lo, such that The Lagrangian (14) (15) Lo + Li'L. a boLo + (1-a)b~ Last + A(N-Lo-Li) (16)

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AN ECONOMIC MODEL OF URBAN GROWTH is maximized when and Thus, From this or pro a be Lotte = A apt (1-a)bl L631 l = A. = L1 8~1 bl Al 1-a = Lo 83 be Q0 a 1 -~30 L1 = c Lo-~Q Ll = c Lo . Now equation 14 and 8Ql < 1 imply Lye 1, 107 (17) (18) (19) such that equation 18 shows a faster growth of urban population Ll com- pared to rural population Lo. For an illustrative calculation, let ax = 3/4 (a conventional value for labor's output elasticity), implying 2-cx Fo = 3 -2cx Let returns to scale in urban production be 10 percent (another conventional number). Then Al = 11/10. In the utility function, let ~ = 1/4, implying a sharply diminished marginal utility of consumption for either good when the other good is held constant. Then by = 95/87 = 1.097954. Urban population as a fraction of total population may be calculated from equation 18 for different levels of total population. It depends on both ~ and c. No econometric analysis has been attempted to obtain the optimal fit. Figures 5-1 and 5-2 compare actual calculated fractions.