Model of cournot competition

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(xi). 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(xi).

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

(i) f(x1, x2) = 3x1x2 and c(xi) = xi2, for i = 1, 2.
(ii) f(x1, x2) = 4x1x2 and c(xi) = xi, 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.