1. Consider a model of Cournot competition, with 2 firms and a linear inverse demand function  P(Q) =  a –  Q (where  Q  =  q₁ +  q₂ is the total quantity produced by the two firms and  a is a parameter). The firms have different marginal costs: c₁ for Firm 1 and c₂ for Firm 2.

(a) Find the Nash equilibrium.

(b) Assume Firm 1’s marginal cost is larger (c₁ >  c₂). Which firm produces more in equilibrium? How do the quantities produced in equilibrium change if Firm 1 improves its technology, leading to a lower c₁ (while c₂ is unchanged)?

(c) Find the total quantity produced and each firm’s profit in equilibrium. Describe what happens to these when Firm 1 changes its technology as above.

2. Two people are engaged in a joint project. If each person i puts in the effort xi, the outcome of the project is worth  f(x₁, x₂). Each person’s effort level xi is a number between [0,1], and effort costs c(x_i). The worth of the project is split equally between the two people, regardless of their effort levels, so the net payoff of each player is  f(x₁, x₂)/2 - c(x_i).

Draw the players Best Responses, and find the Nash equilibria when

(i) f(x₁, x₂) = 3x₁x₂ and c(x_i) = x_i², for i = 1, 2.
(ii) f(x₁, x₂) = 4x₁x₂ and c(x_i) = x_i, for i = 1, 2.

Provide a brief interpretation in each case.
 
3. Find all the Nash equilibria (both pure and mixed) of the 2-player game below.
 
       L       M       R
A  -1,1    3,1     1,2
B    3,3   2,-1    0,0
C    0,0   0,2     3,3
 
4. Consider the following 2-player game:
 
      L     R
U  2,2   1,2
D  5,1   0,0
 
(a) Allowing for mixed strategies, graph the Best Responses.
(b) Give all the Nash equilibria (pure and mixed) of this game.