On testing the convexity of hedonic price functions

Abstract

Recent work on hedonic price functions for the urban housing market has centered on the choice of an appropriate functional form. For example, the work of Goodman [2], Linneman [3], and Quigley [4] has suggested use of the Box-Cox transformation of the dependent variable to estimate hedonic functions. While these and other writings on the topic emphasize the choice of an optimal functional form and the resulting misspecification bias if the optimal form is not used, another reason for interest is to determine the curvature of the corresponding budget constraint faced by the housing consumer. The important work of Rosen [5] has demonstrated that the hedonic price function is a joint envelope of bid functions of consumers and offer functions of producers. Consumers’ bids are expected to be concave, reflecting convex preferences (the consumer being willing to bid a higher price for “mean” homes than “extreme” homes). On the other hand, we expect producers’ offers to be convex, perhaps indicating lower cost of producing mean homes than extreme homes. The joint envelope of the consumer’s bid function and the producer’s offer function is the hedonic price function, which may be concave or convex. The reason for interest in its curvature lies in the corresponding nonlinearity of the budget constraint for the consumer of housing. Whether the budget constraint is concave or convex in characteristics space is important since equilibrium is determined by the tangency of a convex indifference curve and this budget constraint. Quigley illustrates the budget constraint as possibly having both convex and concave segments but does not investigate the question of estimating the degree of convexity or concavity of the constraint since his focus is on determining the demand for housing components. It is the purpose of this note to present a method by which hedonic functions estimated using Box-Cox transformations can be tested for convexity. Specific examples are taken from Linneman [3] and Anderson [l]. A Box-Cox transformation of house price (or rent) p(‘) = ( px 1)/A can be used as the dependent variable to determine the optimal functional form of the hedonic function, giving the estimating equation