Assignment Task

1. Two firms produce an identical product and compete by simultaneously choosing output. Market demand is given by P(Q) = 40-Q, where P is the market price and Q is market output. Both firms have constant marginal costs of 10 and no fixed costs. Suppose the firms compete every period over an infinite horizon. Both firms have the same discount factor 8. Consider the following strategy:

• Each firm produces an output of 8 units as long as both firms have always produced 8 units in the past;
• If either firm ever produces something other than 8 units, both firms produce 10 units every period thereafter.
  00000

i. There is a subgame perfect Nash equilibrium based on the above strategies only if 8 ≥ 4/7.

ii. There is a subgame perfect Nash equilibrium based on the above strategies only if 8 > 0.5.

iii. There is a subgame perfect Nash equilibrium based on the above strategies only if 89/17..

iv. There is a subgame perfect Nash equilibrium based on the above strategies only if 8 > 3/7.

v. There is no subgame perfect Nash equilibrium based on the above strategies for any 8.

2. Two firms produce an identical product and compete by simultaneously choosing output. Market demand is given by P(Q) = 40-Q, where P is the market price and Q is market output. Both firms have constant marginal costs of 10 and no fixed costs. Suppose the firms compete in a single period only.

i. There is no Nash equilibrium to this game.

ii. In the Nash equilibrium to this game, each firm produces 12 units. In the Nash equilibrium to this game, each firm produces 10 units. In the Nash equilibrium to this game, each firm earns profits of 64. In the Nash equilibrium to this game, each firm produces 8 units.

3. Consider the Chioveanu and Zhou (2013) model of frame competition from class. Two firms produce an identical product and compete in a single period by simultaneously choosing a price between 0 and 1 and a message. Messages can be either simple (a) or complex (b). If both firms choose the simple message, all consumers can compare prices. A fraction 0.4 of consumers are unable to compare prices if both firms choose a complex message. A fraction 0.6 of consumers are unable to compare prices if exactly one of the firms chooses a simple message. Suppose the firms have marginal costs of 0.2 and no fixed costs. Suppose Firm 1 chooses message a.

i. If Firm 1 sets a price of 0.5, Firm 2 would like to set a price just below 0.5 and use message a.

ii. If Firm 1 sets a price of 0.5, Firm 2 would like to set a price of 1 and use message a.

iii. Firm 2 would like to use message b, no matter what price Firm 1 sets.

iv. If Firm 1 sets a price of 0.4, Firm 2 would like to set a price of 1 and use message a. If Firm 1 sets a price of 0.5, Firm 2 would like to set a price of 1 and use message b.

4. Firm 1 and Firm 2 have three strategies available, A, B, and C. The first entry in each cell contains the payoffs for Firm 1 and the second entry contains the payoffs for Firm 2. Suppose the game below is played once. In your answers, focus on pure strategies.

i. Playing A is a dominant strategy for both players.

ii. There are two Nash equilibria to the game.

iii. In the unique subgame perfect Nash equilibrium, both players play B.

iv. There are no Nash equilibria to the game.In the unique Nash equilibrium to the game, both players play A.

5. Consider the Chioveanu and Zhou (2013) model of frame competition from class. Two firms produce an identical product and compete in a single period by simultaneously choosing a price between 0 and 1 and a message. Messages can be either simple (a) or complex (b). If both firms choose the simple message, all consumers can compare prices. A fraction 0.4 of consumers are unable to compare prices if both firms choose a complex message. A fraction 0.6 of consumers are unable to compare prices if exactly one of the firms chooses a simple message. Suppose the firms have marginal costs of 0.2 and no fixed costs.

i. Because some consumers are able to compare prices, this gives firms an incentive to earn a high profit margin by setting high prices.

ii. Because some consumers are confused about prices, there is no Nash equilibrium in which both firms set price equal to marginal cost.

iii. Because some consumers are confused about prices, this gives firms an incentive to undercut their rival.

iv. Because of consumer confusion, the firm with the highest price will attract the highest market share.

v. In equilibrium, no consumers are able to compare prices.

6. Firm 1 and Firm 2 have three strategies available, A, B, and C. The first entry in each cell contains the payoffs for Firm 1 and the second entry contains the payoffs for Firm 2.

Suppose the game above is played twice. Both firms have the common discount factor, 8, where 0 < 8>

• in period 1: play B;
• in period 2: play C if both played B in period 1; otherwise play A.

i. There is a subgame perfect Nash equilibrium in which both players play the above strategy for any 8 > 1/3.

ii. Firm 1 has a subgame perfect Nash equilibrium in which they play A and Firm 2 has a subgame perfect Nash equilibrium in which they play B.

iii. In the unique subgame perfect Nash equilibrium, both players choose A in both periods.

iv. For any 8, there is no subgame perfect Nash equilibrium in which both players play the above strategy.

There is a subgame perfect Nash equilibrium in which both players play the above strategy for any 8.