Instructor: Martin J. Osborne
Time: 1 hour 50 minutes. Answer all questions. The numbers in brackets at the start of each question are the numbers of points the questions are worth. To obtain credit, you must give arguments to support your answers.
X Y Z X
2,2 0,2 3,1 3,0 0,0 4,0 4,2 0,1 5,1 Y Z
X Y X
$5,$0 $50,$40 $10,$9 $80,$100 Y
Each player cares not about the amount of money she gets, but about the difference between the amount of money she gets and the amount of money the other player gets. (That is, she cares about the amount of money she gets minus the amount of money the other player gets.)
u1(a1, a2) = a1(2a2 a1)and person 2's preferences are represented by the payoff function
u2(a1, a2) = a2(2 2a1 a2)
i(q1, q2) = qiP(q1 + q2) Ci(qi),where P is the inverse demand function and Ci is firm i's cost function.
Find the Nash equilibrium (equilibria?) of the strategic game that models this situation when the inverse demand function is linear, given by
the cost function of firm 1 is C1(q1) = cq1, the cost function of firm 2 is C2(q2) = q22, and c < (3/4).
P(Q) =
Q if Q 0 if Q > ,
(Carefully present the steps in your argument. Throughout your analysis, ignore the case in which the total output is so high that the price is zero--just assume that the price is always given by P(Q) = Q.)
Specifically, the total demand facing firm i is the sum of Ri(pi) (the demand from i's regular customers) and the demand from customers who compare prices, which is the same as it is in Bertrand's model (D(pi) if pi < pj, (1/2)D(pi) if pi = pj, and 0 if pi > pj).
The functions R1, R2, and D are continuous and decreasing (a higher price reduces demand in each case), and R1(c) > 0, R2(c) > 0, and D(c) > 0. The cost function of firm i is defined by Ci(qi) = cqi for i = 1, 2.