Solving Share Equations in Logit Models Using the LambertW Function

Abstract

Though individual demand and supply equations can readily be expressed in logit models, closed-form solutions for equilibrium shares and prices are intractable due to the presence of products of polynomial and exponential terms. This hinders the employment of logit models in theoretical studies, and also makes it difficult to develop reduced-form expressions for share and price as a function of exogenous variables for use in empirical studies. In this paper we propose that a mathematical function called the ‘LambertW’ be employed in solving logit models for equilibrium shares and prices. We derive closed form solutions for price and share in both the monopoly case as well as in the presence of competition. In the competitive case, the prices of the focal firm and the competitor are dependent on each other; hence the equilibrium prices are endogenous and need to be determined simultaneously. To solve this issue, we provide a simple technique that researchers can employ to derive the optimal prices for both the focal firm and the competitor simultaneously.