Zeitschrift fiir National6konomie 35 (1975), 125--148 9 by Springer-Verlag 1975 Urban Structure and Alternative Transportation Systems* By Joseph E. Haring, Vienna, Jeffrey Chapman, Los Angeles, Calif., USA, and Thomas Slobko, Los Angeles, Calif., USA, with the assistance of Sabina Doval (Received January 10, 1975) Section I: The Mills -- von Thiinen Model The Mills model has two major sectors: (1) a housing market; (2) a transportation market. The two sectors are linked by a differential equation which is in equilibrium when the marginal costs of housing and transportation are equal (and of opposite sign) for each household. This condition determines the locational equi- librium of households. Before we go into specifics about the equa- tions, however, we should list the major assumption of the model. The Assumptions of the Model 1. The Urban area has a well-defined central business district (CBD) with location and size given. 2. The CBD is circular with a radius of one. 3. The model contains no analysis of what happens inside CBD. 4. The labor force of N workers is given and entirely employed in the CBD. 5. Land is used only for worker's residences and commuting. 6. Land is available for urban uses in all directions from the CBD, except for a pie slice of 2 H-~ radians. * An earlier version of this paper was read at the Econometric Society Meetings in New York, December 1973. Valuable suggestions by Edwin Mills, William Vickrey, David Pines, A. H. Studenmund, and Hermann Schuster are gratefully acknowledged. Alan Levit provided computer programming. 126 J.E. Haring et al.: 7. Land is available for urban uses to the extent necessary to satisfy equilibrium conditions below. 8. Land rent outside urban area has rent Ra (exogenous). 9. The urban area stops at UA distance from center of CBD (model determines UA). 10. Housing services are produced with capital and land. 11. The rental rate on capital is exogenous. 12. The rental rate on land is endogenous and varies with dis- tance from city center. 13. The function that determines land rent is to be determined by the model. 14. Competition is assumed throughout. In the following system, Eqs. 1 through 6 represent the housing market, with aggregate supply represented by a Cobb-Douglas pro- duction function. Individual demand equations are assumed to exist for each worker's family, and are then summed to generate aggregate demand. Supply of and demand for transportation to and from work are embraced in Eqs. 8 through 10. Eqs. 11 through 1.3 define some of the institutional circumstances and limitations of the model. Eq. 7 is a differential equation embracing implicitly all elements of the model and determining the locational equilibrium for each worker's household; here the marginal cost of housing is just balanced by the (negative) marginal cost of transportation. Equations 1. Supply of housing services X2s (u) =A2L2 (u) ~ K~ (u) 1-~ 2. Rental rate of land R (u) = 3. Rental rate of capital t-- (1-ec~)P2(u) X~.(u) ~2 P~ (u) X~8 (u) L.,. (u) K~ (u) 4. Demand for housing services per family living u miles from center x2D (u) = B2P2 (u)O2W O~ Urban Structure and Alternative Transportation Systems 5. Total housing service demand at u 127 X2D (u)=x2~ (u) N (u) 6. Housing market X2D (u) = X~ s (u) 7. Locational equilibrium x.,D (u) P2' (u) + P3' (u) = 0 8. Transportation system design capacity (Supply) X3 s (u) -- A3 L3 (u) 9. Transportation demand X3o (u) =N- ~N (u') du 10. Relationship between travel cost, number of users, and design capacity of road t~3 11. All available land (less CBD) in urban area is used for housing or transportation 1 L2 (u) +L3 (u) =#u 12. Land is bid away from agriculture R (ua) = Ra 13. All workers housed in area (outside CBD) u A 8 .f N (u) du =N The following is a list of variables used, with definitions. ~2 = elasticity of housing service production with respect to land input. = daily rate of return per dollar's worth of capital used in the production of housing services. t \"Within a small interval du miles wide centered on a distance u miles from the center, L2 (u) du square miles are used for housing and L3 (u) du square miles for transportation. Since a pie slice of 2 x-~ radians is un- available for use, the total available land in the interval u is ~u du. (Eqs. 1 to 11 must hold for each value of u in the interval e ~u
