Question

Consider the Bertrand duopoly discussed in class. Assume each firm has constant marginal cost c = 10 and zero fixed cost. Each firm chooses a price Pi ≥ 0. The market demand is given by Q = 130 − P, where P = min{P1, P2} is the market price. Refer to the notes for more details on how payoffs are computed.

a) (2pts) If firm 1 is the only firm on the market, what price would this monopolist charge?

b) (2pts) In the Bertrand model with both firms on the market, if firm 2 chooses a price higher than the monopoly price, what is firm 1’s best response?

c) (3pts) Use payoffs to explain why (P1 = 10, P2 = 10) is Nash equilibrium.

d) (3pts) Suppose both firms choose the monopoly price. How much profit would each firm make? Explain why it is not Nash equilibrium.

Answer 1

In the Bertrand duopoly model with constant marginal cost of 10, a single monopolist firm would choose a price where marginal revenue equals marginal cost. When both firms are present, the Nash equilibrium is set at prices equivalent to the marginal cost, leading to zero economic profit. Choosing a monopoly price would not be a Nash equilibrium as firms would have an incentive to lower prices to capture the market.

Step-by-step explanation:

When firm 1 is the only firm on the market, it becomes a monopolist.

To maximize profit, the monopolist would set a price where marginal revenue equals marginal cost. Given the linear demand curve Q = 130 - P and the constant marginal cost (MC) (which is also the average cost (AC)) of 10, the profit-maximizing quantity (Qm) is where MR = MC.

If Firm 2 sets a price above the monopoly price, firm 1's best response is to set their price just below firm 2's price (P2), thus capturing the entire market.

The scenario where both firms set prices at the marginal cost (P1 = 10, P2 = 10), they end up with zero economic profit, which is a condition for Nash equilibrium in the Bertrand model.

No firm has an incentive to change their strategy unilaterally as lowering the price further would result in losses, and increasing the price would cause the firm to lose the market to the competitor.

If both firms choose the monopoly price, they would theoretically split the market demand and earn profits, but this is not a Nash equilibrium.

A single firm has an incentive to slightly undercut the monopoly price to capture the entire market, leading to a lower price in a competitive environment.