Solution
Selling Imperfect Substitutes
Show the problem again
Two firms each sell one variety of a good, with marginal cost 2. Firm i faces demand qᵢ = 20 − 2pᵢ + pⱼ (for pᵢ, pⱼ in the relevant range, so quantities are positive). The firms choose prices simultaneously. What is the price in the symmetric Nash equilibrium?
Worked solution
Firm 1 maximises (p₁ − 2)(20 − 2p₁ + p₂). The first-order condition is 20 − 4p₁ + p₂ + 4 = 0, so p₁ = (24 + p₂)/4. By symmetry p = (24 + p)/4, hence 3p = 24 and p = 8. Each firm sells 20 − 16 + 8 = 12 units for a profit of 6 × 12 = 72. Unlike homogeneous Bertrand, prices stay above marginal cost 2 because undercutting only captures part of the rival's demand, and prices are strategic complements.
Source: Original problem written for AxiomIQ; differentiated-product Bertrand model in the style of Singh and Vives (1984).