Managerial Economics and Strategy, 2/e

Appendix 11A Cournot Oligopoly with Many Firms

We derive the Nash-Cournot equilibrium for n identical firms with constant marginal cost, m, no fixed cost, and a general linear inverse demand function, $p=a-bQ=a-b\left[q_1+q_2+\mathrm{...}+q_n\right].$ [&p|=|a|-|bQ|=|a|-|b[q_{1}|+|q_{2}|+||cdots||plus| q_{n}].&] Consequently, Firm 1’s revenue function is

$$R_1=p{q}_1=\left(a-b\left[q_1+q_2+\mathrm{...}+q_n\right]\right)q_1=a{q}_1-b{q}_1{2}^{}-b\left[q_2+q_3+\mathrm{...}+q_n\right]q_1.$$ (11A.1)

By differentiating the revenue function with respect to $q_1$ [&q_{1}&] while holding the other firms’ quantities constant, we obtain Firm 1’s marginal revenue function:

$$M{R}_1=a-2b{q}_1-b\left[q_2+q_3+\mathrm{...}+q_n\right].$$ (11A.2)

To determine Firm 1’s profit-maximizing ...