ECON 485

Problem Set 4

1. Consider a Stackelberg model of duopoly where p = 1 – q1 – q2, and firms have no cost of production.  Find the backwards induction outcome of the game.

2. Consider the following quantity-setting game with three firms i = 1,2 and 3.  The demand

function is P(Q) = 100 − Q, with Q = q1 + q2 + q3.  The production costs are: C(qi) = 20qi.

a.          Find the Nash equilibrium of this game when the firms choose their quantities simultaneously. Calculate each firm’sprofit.

b.          Find the Subgame Perfect Outcome if firm 1 is the leader and firms 2 and 3 move simultaneously after observing firm 1’s choice of q1 .  Calculate each firm’sprofit.

3. Reg is a key player for a basketball team owned by Geri.  Reg and Geri are embroiled in

negotiations over Reg’s salary for next year’s season. There are still three weeks remaining in the current year’s season, but Reg refuses to play until Reg and Jerry have reached an agreement.

The following information is commonly known. Reg’s value to the team is $8 million, i.e., the team would lose that amount if Reg were to leave the team.  Each week Reg refuses to play, the team loses $200,000, and Reg separately loses $100,000 in penalties.  This is the last year of Reg’s contract, so if no agreement is reached, Reg will (starting next season) play for a different team, which has already offered Reg $3 million more than Reg’s current salary provides.  Should Reg indeed leave, Geri will acquire a new star player, who will add a value of $4 million to the team.  Reg and Geri are infinitely patient (so δ = 1).

a.          Suppose Geri and Reg take turns making a take-it-or-leave-it offer at the start of each of the three remaining weeks. What is the outcome of these negotiations if Geri gets to make the last offer?  Specifically, identify the increase in Reg’s next-season salary.  How does this amount change if Reg makes the last offer?

b.          How do the amounts identified in part a change if Geri is unable to replace Reg?   What

does this say about the effect of “outside opportunities” on a player’s bargaining power?