Computing General Equilibrium Prices for Spatial Economies

Abstract

THE monocentric urban area is without doubt the best known theoretical construct in the literature of urban economics. Many variations exist, but all share certain basic features: a city is located about a node on an otherwise featureless plain. This node, the Central Business District (CBD), is the only concentrated place-or the only place-of employment and shopping. Workers live in housing located outside the CBD and travel to and from their workplaces a fixed number of times each day over a uniformly available, radial transportation system. Transportation costs increase with distance of the residence from the CBD and may include either or both money and time. Households purchase their housing and other commodities to maximize a utility function, not including transportation as an input, subject to a budget constraint net of transportation expenses. As is well known, a number of propositions can be derived from these principal assumptions and some subsidiaries; for example, the existence of downward sloping land rent and housing price surfaces and the spatial segregation of households by income class and preferences (Muth, 1969). What is possibly as well known as this basic model is the difficulty of making the assumptions more realistic. As Beckmann (1974, p. 99) has noted, even minor variations tend to run into a thicket of mathematical difficulties. The effects of multiple workplaces, restrictive zoning, local amenities, and racial discrimination are only a few of the features of real urban areas that can be incorporated into the basic model only with great difficulty or not at all. Since analytical solutions to more realistic models have been difficult or impossible, there has been some recent effort to use computer simulation to examine their properties. Mills, for example, has examined the characteristics of a complex model both with congestion costs and with an efficiently priced transportation system, finding the city's land area in the latter case to be greater than in the former. This result, he suggested, would surprise some persons (1972, ch. 8). Hartwick (1974) has used the same model to study the conditions for spatial segregation or integration of various productive activities, asking, in effect, in what circumstances will persons commute to distinct work areas American-style rather than live above local shops European-style. Zeller (1971) has devised a computer simulation model to portray the bid-rent curves of firms making intrametropolitan location decisions. A particularly interesting possibility has become available recently from the work of various persons, notably Scarf (1973), on the computation of equilibrium prices for competitive economies. The algorithms developed are capable of calculating equilibrium prices, demands and supplies for an economy given only initial resource endowments and demand and supply functions. No more than quite modest restrictions need be imposed on the functions, and the characterization of the economy by number of commodities, types of consumers, and interrelationships in production can be quite rich. Since the characteristics of the various urban models are simply those of competitive economic systems in equilibrium, there would seem to be an obvious opportunity to explore the properties of quite complex models with a very general technique. As might be expected, however, efforts to apply the algorithms to models of spatial economies encounter their own problems, both technical and conceptual. To my knowledge, the first successful application to spatial Received for publication October 3, 1975. Revision accepted for publication September 14, 1976. * An earlier version of this article was presented at the European Conference on Housing Markets, Mons, Belgium, June 1976. I am indebted to Curtis Harris for careful reading and perceptive comments on earlier drafts. Herbert Scarf, Barbara Bergmann, Charles Clotfelter, Charles Lieberman, Michaei Murray, and others have also made helpful comments. Computer funds were provided in a faculty research grant from the University of Maryland Computer Science Center. Copies of the algorithm described in this paper are available upon request.