Estimation of Pure Characteristics Demand Models with Pricing∗

A pure characteristics model is a class of discrete-choice random-coefficients demand models in which there is no idiosyncratic logit error term in a consumer’s utility. The absence of the logit error term and the use of numerical integration to approximate the integral in aggregate market shares lead to a nonsmooth formulation of approximated market share equations. As a result, solving the approximated market share equations for the unobserved product characteristics and estimating the model by using the nested fixed-point approach as proposed in the existing econometrics literature become computationally intractable. To overcome this difficulty, we first characterize consumers’ purchase decisions by a system of complementarity constraints. This new characterization leads to smooth approximated market share equations and allows us to cast the generalized method of moments (GMM) estimation of a pure characteristics model essentially as a quadratic program with linear complementarity constraints, parameterized by an exponential, thus nonlinear, function of the structural coefficient of price. We also extend this estimation problem to incorporate an endogenous pricing mechanism that captures the competitive profit maximization behavior of the producing firms. We provide existence results of a solution for the GMM estimator and present numerical results to demonstrate the computational effectiveness of our approach.