Please answer all the questions as completely as possible and show your work. Please write neatly and circle or underline your answers so that there is no confusion as to what your answer to the question is. I can be reached at my office 372-7238, my home 943-5523 or via e-mail at achaudh@tricity.wsu.edu. Good luck and all the best. Regards and all the best

There are multiple parts to this question - read the question carefully.

Consider an industry with two firms. The two firms behave as Cournot Duopolists and compete in quantities - i.e. they play a simultaneous move game in which each firm decides on what output to produce without knowing the output produced by the other firm.

The firms face an inverse demand function of the form P = 12 - Q where Q is the sum of the outputs produced by the two firms i.e. Q = Q₁+ Q₂. The two firms have zero marginal cost of production.

1.) If the two firms operate as Cournot duopolists, calculate the output produced by each firm and the profit that each firm makes.

3.) You should find when you answer the previous two parts that the profit for an individual firm is higher when they form a cartel and operate as a monopoly. (If not then you need to rethink your answers!) Now one of the problems with sustaining a viable cartel is that there is always an incentive to cheat on the part of the individual firms. We discussed this in class that if one of the firm produces the output that it should as a part of the cartel then the other firm can cheat and produce a little more output and increase its profits.

Using the proof that we discussed in class - prove this proposition. Assume that firm 2 continues to produce the output that it should produce as a member of the cartel (which is the answer you found in Part 2). Show that in that case firm 1 can increase its output by a small amount and make a higher profit.

Consider the following two person game: (In each cell the number on the left is the pay-off to Player 1 and the number on the right is the pay-off to Player 2 - the usual stuff.)

				
					Player 2
			Left 		Middle		Right

		Top	3,3		0,3		0,0

Player 1	Center	3,0		2,2		0,2

		Bottom	0,0		2,0		1,1

Are there strategies for Player 1 that are clearly dominated? Are there strategies for Player 2 that are clearly dominated? (Hint: Can you eliminate all dominated strategies at once or do you have to do an iterated elimination of dominated strategies?) Keep on eliminating dominated strategies for each player.

Consider a monopolist who produces in two different markets. The inverse demand functions are given by P_i = a_j - b_jQ_j where j = 1,2. The monopolist has a constant marginal cost of production "c"

i.e. the cost function is C= c.Q. Show that the monopolist will charge a higher price in the market with lower price elasticity of demand and vice versa i.e. P₁ > P ₂ if |e₁| < |e₂| and vice versa.

Now let us take a numerical example of the same problem.

P₁ = 10 - Q₁ and P₂ = 20 - Q₂

Marginal Cost (constant) = $2

How much output should the monopolist produce in each market?

Consider the following two player game:

				Player 2

			Left		Right

		Top	2, 18		2, 18

Player 1

		Bottom	0,0		4,2

(1) Suppose the two players move simultaneously. Identify all the Nash Equilibria of this game.

(2) Suppose player 1 moves first and player 2 moves second after observing player 1's move. Draw the extensive form of the game and solve for the subgame perfect equilibrium.

(3) When you look at the normal form of the game (the way I have depicted it) you can identify a dominant strategy Nash Equilibrium - identify that.

Consider the following two person game:

				Player 2

				Cooperate	Defect

		Cooperate	6,6		0,8

Player 1
		Defect		8,0		2,2

From our discussion in class, we know that this game is an example of the Prisoner’s Dilemma and that the unique Nash Equilibrium of this game is (Defect,Defect).

A. Now, suppose player 1 moves first and chooses Cooperate or Defect. Player 2 moves next and after having observed Player 1's choice, Player 2 decides on whether to cooperate or defect. Draw the extensive form of the game. Is (Defect,Defect) still the equilibrium outcome of the extensive form of the game? Explain your answer very clearly.

B. Based on our discussion in class, we know that if this game is repeated infinitely, then (Cooperate, Cooperate) may emerge as the equilibrium outcome if the players value the future enough. Show that there exists a discount factor (d) such that (Cooperate,Cooperate) may emerge as the equilibrium and calculate the value of that discount factor.

					Calvin Column


				Wrestling		Bowling

		Wrestling	3,1			0,0

Rhonda Row

		Bowling		0,0			1,3