Symmetry Constraints and Variable Returns to Scale in Logit Models

Abstract

This article addresses two problems with earlier applications of the linear logit model of cost shares. The first problem is that symmetry holds only for one set of cost shares. In this article, an iterative, nonlinear estimation procedure is used to impose symmetry for all predicted shares. The second problem is that analytical expressions for the integrals of log-linear cost shares cannot be derived analytically, although they exist, given the continuity of the logit share equations. This article proposes a procedure for approximating the underlying cost function by using the share predictions as stochastic, numerical approximations of these integrals. The model has constant but unequal elasticities of substitution. This feature does not contradict the Uzawa impossibility theorem, which holds for integrable functions, because the logit model is essentially a numerical approximation of the constant elasticity of substitution model. Another interesting property is that the concavity conditions are relativ...