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Review of Marketing Science

Ed. by Rao, Ram C.

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Solving Share Equations in Logit Models Using the LambertW Function

Ashwin Aravindakshan
  • 1University of California, Davis,
/ Brian Ratchford
  • 2University of Texas at Dallas,
Published Online: 2011-04-18 | DOI: https://doi.org/10.2202/1546-5616.1120

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.

Keywords: logit models; discrete choice models; LambertW; equilibrium prices

Published Online: 2011-04-18

Citation Information: Review of Marketing Science. Volume 9, Issue 1, ISSN (Online) 1546-5616, DOI: https://doi.org/10.2202/1546-5616.1120, April 2011

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