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1 Question 1
(15 points) Suppose there are I firms in an industry. Each firm can try to convince Congress to implement a subsidy in this industry. Let hi ≥ 0 denote the hours of efort put in by firm i. When the efort levels of the firms are h = (h1 , h2,..., hI ), the value of the subsidy that gets approved is:
where a, b ≥ 0 are constants. The cost of efort to firm i is c(hi ) = whi(2)
where w > 0 is a constant. Firm i’s payof is si − c(hi ).
(a) (10 points) For what set of parameters a,b,w do the firms have a dominant strategy?
(b) (5 points) Make a prediction about the size of the subsidy when your conditions in (i) are satisfied.
2 Question 2
(15 points) Consider a two-player game in which player 1 chooses a number (n1 ) on the interval [0, ∞ ) and player 2 chooses a number (n2 ) on the interval ( −∞ , 0] and the payofs to each player are ⇧ = n1 n2
(a) (3 points) (i) Find strictly dominated strategies for each player, if any. (ii) Does IESDS yield a unique solution to this game?
(b) (12 points) Now, consider weakly dominated strategies. (i) Find weakly dominated strategies for each player, if any. (ii) Does IEWDS yield a unique solution to this game? If so, does the order of deletion matter?
3 Question 3
(10 points) Assume x ≥ 0. Find all pure strategy NE of the following game. Can you make a prediction about the outcome of the game?
4 Question 4
(20 points) The price in the market is given by P(q1 , q2 ) = 1 − q1 − q2 , where q1 and q2 are the quantities produced by the two firms. The cost function of each firm is given by:
(a) (10 points) Suppose α = β = 1. Find the Nash equilibrium output levels when the firms choose outputs simultaneously.
(b) (10 points) Suppose β = 1 is held fixed. What do you expect to happen to equilibrium quantities when (i) α increases from 1 (ii) α decreases from 1? Draw best responses to support your claim. (No need to solve for actual quantities)