Heteroscedastic Exponomial Choice

Abstract

We investigate analytical and empirical properties of the Heteroscedastic Exponomial Choice (HEC) model to lay the groundwork for its use in theoretical and empirical research that build demand models on a discrete choice foundation. The HEC model generalizes the Exponomial Choice (EC) model by including choice-specific variances for the random components of utility (the error terms). We show that the HEC model inherits some of the properties found in the EC model: closed-form choice probabilities, demand elasticities and consumer surplus; optimal monopoly prices that are increasing with ideal utilities in a hockey-stick pattern; and unique equilibrium oligopoly prices that are easily computed using a series of single-variable equations. However, the HEC model has several key differences with the EC model that show variances matter: the choice probabilities (market shares) as well as equilibrium oligopoly prices are not necessarily increasing with ideal utilities; and the new model can include choices with deterministic utility or choices with zero probability. However, because the HEC model uses more parameters, it is harder to estimate. To justify its use, we apply HEC to grocery purchase data for thirty product categories and find that it significantly improves model fit and generally improves out-of-sample prediction compared to EC. We go on to investigate the more nuanced impact of the variance parameters on oligopoly pricing. We find that the individual and collective incentives differ in equilibrium: Firms individually want lower error variability for their own product, but collectively prefer higher error variability for all products – including their own – because higher error variability softens the price competition.

Keywords: Discrete choice theory, random utility models, Exponomial Choice model, demand modeling, demand elasticity, consumer surplus, maximum likelihood estimation, pricing, price equilibrium