The Interaction Between Local Government and Urban Residential Location: Reply and Further Analysis
回应了Helpman、Pines和Borukhov对作者早期模型的推广，并进一步分析在家庭效用水平存在强制差异时，收入分配如何影响最优城市形态，发现效用分布变化会改变城市密度梯度。
Elhanan Helpman, David Pines, and Eli Borukhov (H-P-B) provide an interesting generalization of my earlier paper (1974b). Their less specific characterization of local public services permits them to consider both completely congestible club goods,.I which are the types of services implicitly considered in my earlier model,2 and pure local public services. H-P-B also draw a useful distinction between public goods that act as substitutes for increased land occupancy (complements for increased population density) and can be viewed as people-enhancing services, and those public goods that are complementary to large land holdings a-nd can be considered as property-enhancing services.3 A case can be made that the specific assumptions in my paper are realistic approximations for most urban areas in the United States, and that none of H-P-B's results contradict my conclusions within that framework of restricted optimality; nevertheless, the power of what I would call the planninganalytic approach4 in dealing with very generalized functional relationships is obvious. What is equally obvious is that H-P-B have not considered the parallel case to my analysis where income levels vary according to some known distribution. It may be appropriate to argue that ultimate urban policy prescriptions should not be tied to specific assumptions about functional forms, but similarly, a useful analysis should also incorporate the reality of tolerated differences in the income (or utility) levels and at the very least demonstrate their effect on solutions.5 H-P-B have substantially overcome the first difficulty, and the present paper adopts their technique to account for the effects of mandated dispersions in household utility levels. The interesting conclusions show how changes in the mandated distribution of utility level alter the optimal urban form. I begin by establishing the importance of a city's income distribution in determining its shape. Previously developed spatial equilibrium market solutions are used to numerically simulate the sizable differences in a city's density gradient6 under different' assumed distributions of income. Next, assumptions of specific functional forms are relaxed and the planning-analytic approach is Ipursued under several assumed distributions of household utility. In particular, sufficient conditions for a rising density gradient in the city are established, and the possibility ofhaving an optimal allocation of public *Assistant professor of economics and of environmental engineering, Cornell University. I wish to thank Yoshitsugu Kanemoto for his helpful comments. 'See James Buchanan. 2Both Yoshitsugu Kanemoto and John Wile have also pointed out the restrictions that are inherent in my previous local public goods formulation. 'A Cobb-Douglas or log-linear utility function such as I used in my paper specifies the public good as the borderline case, neither substitute nor complement. 4The distinction is between a straightforward optimal solution to the urban spatial problem, as analyzed by H-P-B, versus a two-step procedure in which equilibrium market solutions are developed and then a social optimum is computed based upon this restricted market framework. Robert Solow used the latter procedure, and I used it in my paper, and it may or may not lead to a global optimum. One advantage of theplanning-analytic approach is the ease with which externalities may be identified and therefore second best solutions in a market analysis can be recognized. 5The prior works of Martin Beckmann and Aldo Montesano have also pursued this problem within a market framework. John Riley, James Mirrlees, and Avinash Dixit have explored slightly more limited cases using the planning-analytic approach. As an. example Dixit allows unequal utilities to exist, weighted by a negative exponential welfare function. Dixit's model also assumes specific Cobb-Douglas utility and production functions., 'The density gradient is the relationship between population density,and distance from the city's central business district (CBD).