Question

Question Consider an n firm homogeneous-good oligopoly with constant marginal cost, the same for all firms. Let-δ be the minimum value of the discount factor such that it is possible to sustain monopoly prices in a collusive agreement. Show that-δ is increasing in n. Interpret the result.

Consider an n firm homogeneous-good oligopoly with constant marginal cost, the same for all firms. Let ¯δ be the minimum value of the discount factor such that it is possible to sustain monopoly prices in a collusive agreement. Show that ¯δ is increasing in n. Interpret the result.

Answer 1

When there are n firms in the market all having a marginal cost 'm' and a demand function P = a - bQ where Q = q1 + q2 + q3 + ... + qn, each firm produces identical good given by q = (a - m)/(b(1 + n)) and charges a price P = (a + mn)/(1 + n). The profit earned by each firm is (a - m)^2/(b(n + 1)^2)

Now as a collusive monopolist, each firm earns a profit of (a – m)^2/4bn. If any firm cheats, it will be earning a profit of (a – m)^2(n + 1)^2/16bn^2.

Given that δ is the minimum value of the discount factor to sustain monopoly prices. When this happens, each firm continues to earn (a – m)^2/4bn for infinite period. In case any of the firms deviates in first period it will be able to secure (a – m)^2(n + 1)^2/16bn^2 in that period but will receive only (a - m)^2/(b(n + 1)^2) for each period forever. Hence the payoff is (1−δ)(a – m)^2(n + 1)^2/16bn^2 + δ(a - m)^2/(b(n + 1)^2)

No firm has an incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

(a – m)^2/4bn ≥ (1−δ)(a – m)^2(n + 1)^2/16bn^2 + δ(a - m)^2/(b(n + 1)^2)

simplified to δ ≥ ((n + 1)^2 - 4n) / ((n + 1)^4 - 16n)n

Here δ is increasing in n because the derivative is positive.