Duopoly Competition with Differentiated Products

This Demonstration illustrates the best response functions in the Singh and Vives model of product differentiation under Bertrand competition [1] as opposed to the homogeneous goods Cournot and Stackelberg duopoly model [2, Chapter 4], which is treated in the Cournot and Stackelberg Equilibria and Best Responses Demonstration. There are two firms operating in a market characterized by the inverse demand functions and , where is the quantity of firm and is the quantity of firm . Firms face marginal costs of and , respectively. The parameters of the demand functions are (demand sensitivity to own price), (demand sensitivity to rival's price) and (determines complements/substitute in consumption). The intersection (marked by a red dot) displays the Bertrand equilibrium. For comparison, the prices implied by the equilibrium under Cournot competition are also illustrated in the graph. Note that the equilibrium prices for the Bertrand competition are always less than the corresponding equilibrium prices for the Cournot competition under the same conditions. Depending on the parameter , prices are either strategic complements or strategic substitutes (or neither) in this setting.

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Reference
[1] N. Singh and X. Vives, "Price and Quantity Competition in a Differentiated Duopoly," The RAND Journal of Economics, 15(4), 1984 pp. 546–554.
[2] P. Belleflamme and M. Peitz, Industrial Organization: Markets and Strategies, 2nd ed., Cambridge, UK: Cambridge University Press, 2015.
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