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Econ 101 Summer 2024

Problem Set 2

Due: 09/10/2024 at 01:00 pm PST

1. Repeated Game

Consider the following normal form game.

(a) Suppose this game is played once. Find all pure strategy Nash equilibria.

(b) Now suppose the game is played two times without discounting. Can (D, A) be achieved in the first stage in a pure-strategy subgame perfect Nash equilibrium, by utilizing the following set of strategies?

(c) Now suppose the game is played T times without discounting, where T is a finite number. Find the smallest T such that (D, A) can be achieved in the first period.

2. Stackelberg

Firms 1, 2 and 3 compete in quantities. The inverse market demand is given by

p = 400 − (q1 + q2 + q3),

where q1, q2 and q3 are the quantities produced by firms 1, 2 and 3, respectively. The marginal cost for firm 1 is c1 = 20, and the marginal cost for both firm 2 and firm 3 is 40. The order of play is as follows: Firm 1 first sets a quantity q1. Then, firms 2 and 3 observe q1 and simultaneously set q2 and q3. Each firm sets its quantity to maximize its own profits.

(a) Let us first consider the optimal action of firm 2. Given a value q1 initially set by firm 1 and a value of q3 set by firm 3, what is the value of q2 that maximizes the profits of firm 2? Hint: Your answer should provide q2 in terms of q1 and q3.

(b) Notice that, after firm 1 sets q1, a subgame starts, in which firms 2 and 3 simultaneously set q2 and q3. Given q1, find the values of q2 and q3 that firms 2 and 3 must set in a Subgame Perfect Nash Equilibrium. Hint: Your answer should provide q2 and q3 in terms of q1.

(c) Find the value of q1 that firm 1 sets in the Subgame Perfect Nash Equilibrium of this game. What is the equilibrium price? And what are the equilibrium profits of each firm?

3. Repeated Game

Consider the game in which the stage game depicted below is infinitely repeated and in which both players discount future payoffs with discount factor δ ∈ [0, 1].

(a) Suppose that both players play the Grim Trigger strategy. This strategy is characterized by exerting effort (playing E) in the first round. In every future period, play E as long as (E, E) has always been played in the past, and play S if anything other than (E, E) has been played at any point in the past. Then, for what values of δ can Grim Trigger be supported on the equilibrium path of play?

(b) Suppose that both players play the tit-for-tat strategy. This strategy is characterized by exerting effort (playing E) in the first round. Then in future rounds, a player copies the action chosen by his opponent in the previous period (i.e if Player 2 shirks (plays S) in period 2, then Player 1 plays S in period 3). Then, for what values of δ can tit-for-tat be supported on the equilibrium path of play?

(c) Suppose that both players play the perfect tit-for-tat strategy. This strategy is charac-terized by exerting effort (playing E) in the first round. Then in future rounds, a player plays E unless the actions are disagreed in the previous period. Then, for what values of δ can perfect tit-for-tat be supported on the equilibrium path of play?

4. Insurance

You are pondering whether to buy insurance for your new car. The car is worth $50,000. Apart from that, your wealth sums up to $40,000. Your utility takes the form u(w) = √w. There is 0.5% chance that your car will be stolen. With the insurance in place, you will get fully reimbursed if this tragedy happens.

(a) If the insurance premium is $400, would you buy the insurance?

(b) What is the largest premium you are willing to pay for the car insurance?

(c) If your utility is instead given by u(w) = log(w), how does your answer to part (2) change?