A Cournot oligopoly has 19 firms, and inverse market demand P = 60 - Q. All firms have marginal cost, . The equilibrium output for each firm will be

Question

BlackTom AI · Accepted Answer

We model the situation as a Cournot oligopoly with n firms, linear inverse demand P = a - bQ, and constant marginal cost c. Here a = 60, b = 1, c = 20, and n = 19. In Cournot competition, each firm chooses its output qi to maximize profit given the others’ outputs. The standard best-response for firm i is qi = (a - c - b Q_-i) / (2b), where Q_-i is the total output of the other firms. In a symmetric equilibrium, all firms produce the same amount q, so Q = n q and Q_-i = (n - 1) q. Substituting into the best-response gives q = (a - c - b (n - 1) q) / (2b). Solve for q: q = (a - c) / [b (n + 1)].

Plugging in the numbers: a - c = 60 - 20 = 40, b = 1, n = 19, so q = 40 / (19 + 1) = 40 / 20 = 2. Therefore, each firm produces 2 units.

Now evaluate the answer options:
- 1 unit: This would yield a smaller total quantity than the Cournot equilibrium. With Q = 19, price would be P = 60 - 19 = 41, but each firm would earn less profit than at q = 2, given the concave profit function and the reaction of other firms. Thus 1 unit is inconsistent with the Cournot solution.
- 2 units: This matches the calculated symmetric Cournot equilibrium quantity per firm, given the demand, costs, and number of firms. It follows directly from the formula q = (a - c) / [b (n + 1)].
- 3 units: If each firm produced 3, total Q = 57, price P = 60 - 57 = 3. The disproportionate rise in quantity would reduce price drastically and profits would not be maximized given the reaction of other firms; this is not the Cournot equilibrium.
- 4 units: Similarly, producing 4 each would yield Q = 76, which would drive price negative under this demand, contradicting the equilibrium conditions and indicating that such a level cannot sustain in equilibrium with the stated costs and beliefs about rivals’ behavior.

In summary, the option that aligns with the Cournot equilibrium derivation for these parameters is the one where each firm produces 2 units, as it satisfies the symmetric equilibrium condition q = (a - c) / [b (n + 1)].

A Cournot oligopoly has 19 firms, and inverse market demand P = 60 - Q. All firms have marginal cost, . The equilibrium output for each firm will be单项选择题

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