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Market Shares: Some Power Law Results and Observations


  • Raaj Sah
  • Rajeev Kohli


The authors report an empirical regularity in the market shares of brands. They propose a theoretical explanation for this observed regularity and identify additional consequences of their explanation. Empirical testing of these consequences supports the proposed explanation. The empirical regularity is obtained using cross-sectional data on market shares of brands in 91 product categories of foods and sporting goods sold in the United States. In total, the data set has 1171 brands. The key feature of the empirical regularity is that, in each category, the difference between pairs of successively-ranked market shares forms a decreasing series. In other words, the decrease in the market share between two successively-ranked brands becomes smaller as one progresses from higher-ranked to lower-ranked brands. The observed patterns of market shares are represented by the power law to a remarkable degree of accuracy, far surpassing the fits typically observed in predictive models of market shares. The authors then propose an explanation for the regularity in terms of a model of consumer purchases. At the heart of this model is a special case of the well-known Dirichlet-multinomial model of brand purchases. This special case corresponds to Bose-Einstein statistics, and has an intuitive implication that the purchase probability of a brand is proportional to the number of units of that brand already purchased. A notable feature of the model is that it considers multiple product categories, and allows for the entry and exit of brands over time. The authors then describe, and test using the data set, two predictions of the proposed model regarding patterns of market shares across product categories. For both of these predictions, which appear counterintuitive, the authors report reasonable empirical support. Overall, there appears to be good evidence supporting the proposed model and the observed regularities. The authors discuss some potential implications of the regularities for marketing practice and research. They also offer an interpretation of the previously known square-root relationship between market share and the order of entry of firms into an industry.

Suggested Citation

  • Raaj Sah & Rajeev Kohli, 2004. "Market Shares: Some Power Law Results and Observations," Working Papers 0401, Harris School of Public Policy Studies, University of Chicago.
  • Handle: RePEc:har:wpaper:0401

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    References listed on IDEAS

    1. Gurumurthy Kalyanaram & William T. Robinson & Glen L. Urban, 1995. "Order of Market Entry: Established Empirical Generalizations, Emerging Empirical Generalizations, and Future Research," Marketing Science, INFORMS, vol. 14(3_supplem), pages 212-221.
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    4. P.J. Lamberson, 2016. "Winner-take-all or long tail? A behavioral model of markets with increasing returns," System Dynamics Review, System Dynamics Society, vol. 32(3-4), pages 233-260, July.

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