A Constructive Approach to Estimating Pure Characteristics Demand Models with Pricing

Discrete-choice demand models are important and fundamental tools for understanding consumers’ choice behavior, and for analyzing firms’ operations and pricing strategies. In these models, products are often described as a vector of observed characteristics. A consumer chooses the product that maximizes her utility, assumed to be a function of the observed product characteristics and the consumer’s preference over these product characteristics. One central task in the demand estimation literature is to, based on observed data, infer consumers’ preferences on product characteristics. We consider such an estimation problem for pure characteristics models, a class of random-coefficients demand models without the idiosyncratic logit error term in a consumer’s utility function. 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 and estimating the model by using existing methods proposed in the econometrics literature remain 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 corresponding generalized method of moments (GMM) estimation problem essentially as a quadratic program with linear complementarity constraints, parameterized by an exponential, thus nonlinear, function of the structural parameter on price. We also extend this estimation framework 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.