The theory of the firm and industry equilibrium

Martin J. Osborne
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7.3 Exercises on cournot's duopoly model

  1. An industry contains two firms, one whose cost function is TC(y) = 30y and another whose cost function is TC(y) = y2. The inverse demand function for the firms' output is p = 120 − Q, where Q is the total output. What are the firms' outputs in a Nash equilibrium of Cournot's model?

    Solution

    The best response function of firm 1 is b1(y2) = (90 − y2)/2 (see example) and the best response function of firm 2 is b2(y1) = (120 − y1)/4 (see example).

    Thus the Nash equilibria are the solutions of the equations

    y1 = (90 − y2)/2
    y2 = (120 − y1)/4.
    These equations have a single solution, (y1, y2) = (240/7, 150/7). Thus the game has a unique Nash equilibrium, (y1, y2) = (240/7, 150/7).