1. Consider a model of Cournot competition, with 2 firms and a linear inverse demand function P(Q) = a – Q (where Q = q_{1} + q_{2} is the total quantity produced by the two firms and a is a parameter). The firms have different marginal costs: c_{1} for Firm 1 and c_{2} for Firm 2.
(a) Find the Nash equilibrium.
(b) Assume Firm 1’s marginal cost is larger (c_{1} > c_{2}). Which firm produces more in equilibrium? How do the quantities produced in equilibrium change if Firm 1 improves its technology, leading to a lower c_{1} (while c_{2} 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(x_{1}, x_{2}). 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(x_{1}, x_{2})/2 - c(x_{i}).
Draw the players Best Responses, and find the Nash equilibria when
(i) f(x_{1}, x_{2}) = 3x_{1}x_{2} and c(x_{i}) = x_{i}^{2}, for i = 1, 2.
(ii) f(x_{1}, x_{2}) = 4x_{1}x_{2} 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.