Question

In the Bertrand model with product differentiation, suppose that the two Bertrand firms face the following symmetric demand curves:

q₁ = 96 - 2p₁+1/2 p₂

q₂ = 96 - 2p₂ + 1/2p₁

where q1, q2 ≥ 0 and p1, p2 ≤ 48. MC for both firms is 12.

1. Is product differentiation more or less significant in this example than in the example given in the text in Equations 10.3A and 10.3B? Why?
2. Find the Bertrand equilibrium.

Answer 1

In class, we considered Cournot competition where two firms choose quantities and let the price be fixed by the market. We consider here a different model of competition on prices called Bertrand competition. Consider 2 firms producing identical products (i.e., that are perfect substitutes) with a constant marginal cost c. Each firm i choose the price pi ∈ [0, 1]. We assume that the demand curve is linear so that the total demand is Q(p) = 1 − p where p = min(p1, p2) and is entirely directed to the firm with the smallest price. If both prices are equal, the demand is equally shared between the two firms. We denote by qi the demand to firm i. For instance, the demand to firm 1 is q1(p1, p2) =    1 − p1 if p1 < p2, 0 if p1 > p2, (1 − p1)/2 if p1 = p2.