Question

Consider a homogeneous product industry with inverse demand function p = 36 - 4Q. There are two identical firms in the market, each of them facing the total cost function C = 12q.

(a) Firms compete in prices according to the Bertrand model, find the Bertrand-Nash equilibrium.

Answer 1

This is a case of simultaneous price setting in a homogeneous market (Bertrand competition).

P = 36 - 4Q

and C = 12Q

Thus MC = $\frac{\mathrm{d} C }{\mathrm{d} Q} = 12$

Each firm knows that products are homogeneous i.e. perfect substitutes for each other. Thus consumer would buy from the producer which charges the minimum price.

Both firms would want to undercut their prices to capture the maximum market share. But the minimum they can charge is P = MC = 12 since below it, marginal cost is more than marginal benefits and thus firm has negative profits.

Hence the Bertrand Nash equilibrium is given by (p1, p2) = (12, 12).