Solution
The Firm That Stays Out
Show the problem again
Two firms choose quantities q₁, q₂ ≥ 0 simultaneously. The market price is P = max(0, 120 − q₁ − q₂). Firm 1 has marginal cost 0 and firm 2 has marginal cost 70; there are no fixed costs. Firms maximise profit and play the Nash equilibrium. What is firm 1's equilibrium profit?
Worked solution
The textbook formulas q₁ = (a − 2c₁ + c₂)/3 and q₂ = (a − 2c₂ + c₁)/3 give q₂ = (120 − 140)/3 < 0, which is impossible, so the interior solution is invalid. Firm 2's best response is q₂ = max(0, (50 − q₁)/2), which is zero whenever q₁ ≥ 50. Firm 1 therefore behaves as a monopolist if its monopoly output of 60 exceeds 50: it does, so q₁ = 60, P = 60 < 70 and firm 2 cannot profitably produce even a single unit. Firm 1's profit is 60 × 60 = 3600. The point is that a Cournot equilibrium must respect the non-negativity constraint; a firm with sufficiently high cost is simply shut out.
Source: Original problem written for AxiomIQ.