Question

1. Consider the following oligopoly model. The market demand is p(Q) = 100−Q. There are three identical firms 1, 2 and 3 producing the homogeneous product. Each firm has a constant marginal cost of 0. The three firms choose their outputs simultaneously , without observing the quantity decisions by others. Find the Cournot-Nash equilibrium in this model. Obtain the profits in equilibrium for each firm.

Answer 1

For a demand curve of the firm P = a - b(q1 + q2 + .... + qn) where Q = q1 + q2 + .... + qn, and Marginal cost = c,

in Nash equilibrium,

Firm output (q) = (a - c) / [(n + 1) x b]

Market output (Q) = n x q

Here, P = 100 - Q, so a = 100 and b = 1

n = Number of firms = 3

MC = c = 0

So,

(I) q = (100 - 0) / [(3 + 1) x 1] = 100 / (4 x 1) = 25

(II) Q = 3 x 25 = 75

(III) P = 100 - 75 = 25

(IV) For each firm, Profit = q x (P - c) = 25 x (25 - 0) = 25 x 25 = 625