Income distribution and the residential density gradient

W. C. Wheaton ( Journal of Economic Theory 9, 223–237, 1974 ) asserted that an increase in incomes lowers central densities raising suburban densities in closed monocentric cities with identical resident incomes and tastes. We show that with identical Cobb-Douglas tastes and identical incomes, a sufficient condition for densities to fall everywhere when incomes rise is that the land rent bill of the innermost resident be larger than the commuting bill of the outermost resident. Estimating this model using data in E. S. Mills (“Studies in the Structure of the Urban Economy,” Johns Hopkins Press, Baltimore/London, 1972), we show that densities are predicted to fall everywhere in Baltimore, Denver, Milwaukee, Philadelphia, Rochester, and Toledo when income is increased by 10% circa 1960. But, when we disaggregate the model using the 1960 census income distribution (keeping tastes identical), a 10% increase in incomes rotates densities around distances which vary from 3.3 miles in Toledo to 9.8 miles in Philadelphia. In addition to empirically vindicating Wheaton's assertion, the disaggregated model gives approximately negative exponential densities consistent with spatial equilibrium, and without resorting to R. F. Muth's (“Cities and Housing”, University of Chicago Press, 1969) assumption of a unitary compensated price elasticity of demand.