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Solving Share Equations in Logit Models Using the LambertW Function

Author

Listed:
  • Aravindakshan Ashwin

    (University of California, Davis, aaravind@ucdavis.edu)

  • Ratchford Brian

    (University of Texas at Dallas, btr051000@utdallas.edu)

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.

Suggested Citation

  • Aravindakshan Ashwin & Ratchford Brian, 2011. "Solving Share Equations in Logit Models Using the LambertW Function," Review of Marketing Science, De Gruyter, vol. 9(1), pages 1-17, April.
    • Handle: RePEc:bpj:revmkt:v:9:y:2011:i:1:p:1-19:n:1
    DOI: 10.2202/1546-5616.1120
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    References listed on IDEAS

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    1. Peter M. Guadagni & John D. C. Little, 1983. "A Logit Model of Brand Choice Calibrated on Scanner Data," Marketing Science, INFORMS, vol. 2(3), pages 203-238.
    2. de la Croix, David & Licandro, Omar, 1999. "Life expectancy and endogenous growth," Economics Letters, Elsevier, vol. 65(2), pages 255-263, November.
    3. Aneel Karnani, 1985. "Strategic Implications of Market Share Attraction Models," Management Science, INFORMS, vol. 31(5), pages 536-547, May.
    4. David Besanko & Sachin Gupta & Dipak Jain, 1998. "Logit Demand Estimation Under Competitive Pricing Behavior: An Equilibrium Framework," Management Science, INFORMS, vol. 44(11-Part-1), pages 1533-1547, November.
    5. Suman Basuroy & Dung Nguyen, 1998. "Multinomial Logit Market Share Models: Equilibrium Characteristics and Strategic Implications," Management Science, INFORMS, vol. 44(10), pages 1396-1408, October.
    6. Kauffman, Stuart & Lobo, Jose & Macready, William G., 2000. "Optimal search on a technology landscape," Journal of Economic Behavior & Organization, Elsevier, vol. 43(2), pages 141-166, October.
    7. Thomas S. Gruca & K. Ravi Kumar & D. Sudharshan, 1992. "An Equilibrium Analysis of Defensive Response to Entry Using a Coupled Response Function Model," Marketing Science, INFORMS, vol. 11(4), pages 348-358.
    Full references (including those not matched with items on IDEAS)

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