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Unformatted text preview: Economics 1200 Spring 2012 Homework # 2
Write your answers to the following questions on separate sheets of paper. Your answers are due
in class on Tuesday, February 7. Late homeworks are not accepted.
1. Consider the two-round home bargaining game discussed in class. The minimum the
seller will sell his home for is $188,000 and the maximum the buyer is willing to pay is
$200,000. Both players know these two amounts and are bargaining over the difference,
M=$12,000. Assume the disagreement values are 0 for both players. Suppose the buyer
moves first by making a proposal and the seller can accept or reject it. If the seller rejects
the buyer’s proposal, the seller gets to make a counterproposal, which the buyer can then
accept or reject. The game is then over. Suppose that both players discount future income
using a period discount factor of =.2.
a. Use rollback to find the equilibrium for this 2-round game. What is the sale price
of the home? Which player buyer or seller gets the larger share of M?
b. Suppose the buyer’s discount factor was b=.8 while the seller’s discount factor
remained at s=.2? How does your answer to part a change in this case?
c. Return to the case where both have the same discount factor of =.2. Suppose
now that there is no limit to the number of alternating bargaining rounds and the
buyer continues to move first. Use rollback reasoning to find the equilibrium
price in this case. How does an unlimited number of bargaining rounds affect the
share of the first mover—the buyer—relative to the 2- or 3-round case? What
intuition can you offer for this difference?
2. In each of the following three games, each player can choose between two actions,
“cooperate” or “defect”. Suppose that in all three games, higher payoff numbers are
preferred to lower payoff numbers. For each game, find all of the pure strategy Nash
equilibria. Show/explain how you found these equilibria.
a. Prisoner’s Dilemma Player Cooperate 1 Player 2
10,80 Defect 80,10 40,40 b. Stag Hunt Player Cooperate 1 Player 2
5,40 Defect 40,5 40,40 c. Chicken Player Cooperate 1 Player 2
50,80 Defect 80,50 1 40,40 3. Suppose that Pat and Sam intended to communicate with each other about what to do
tonight but the message never got through. Now each has to simultaneously and
independently decide where to show up (communication is no longer possible). There are
just two possibilities – a ball game or a concert. Other things equal, Pat likes ball games
better, and Sam likes concerts better. Both Pat and Sam agree that either event would be
more fun if the other person were also there. However, Pat and Sam differ in their
attitudes about how important it is that they be there together. Since Pat is choosing
between the game and the concert and Sam is facing the same two choices, there are four
possible outcomes. The table below shows how Sam and Pat rank these four outcomes.
Worst Pat’s Ranking
Pat at game, Sam at game
Pat at game, Sam at concert
Pat at concert, Sam at concert
Pat at concert, Sam at game Sam’s Ranking
Sam at concert, Pat at concert
Sam at game, Pat at game
Sam at concert, Pat at game
Sam at game, Pat at concert a. Write down the normal form of this game. Choose payoffs that are consistent with
the rankings given in the table above. Assume there are no ties (e.g. “best” is strictly
better than “second best”, which is strictly better than “third best” which is strictly
better than “worst”).
b. Find all pure strategy Nash equilibria for this game. Does either player have a
dominant strategy? Explain.
4. Two players, Jack and Jill are put in separate rooms. Each is then told the rules of the
game. Each is to pick one of six letters, G, K, L, Q, R or W. If they happen to choose the
same letter, both get payoffs as indicated in the table below (higher numbers=higher
payoffs). Otherwise, if they choose different letters, both earn a payoff of 0.
Jill’s Payoff G
1 a. Illustrate this game in normal form using a game table. What are the Nash
equilibria in pure strategies?
b. Can one of the equilibria be a focal point? If so, which one and why?
5. Consider the continuous version of the Cournot duopoly game discussed in class. There
are two firms, 1 and 2, which manufacture a homogeneous good. Each firm chooses
quantities to produce, q1 and q2, respectively so as to maximize profits. The price they
receive per unit of the good is given by p = max [a-b(q1+q2), 0]. Each firm’s constant
marginal (per unit) cost is c>0.
a. Show that in the Nash equilibrium of the Cournot game, the quantities that each
firm brings to market are: q1 q 2 ac
3b the equilibrium price p=(1/3)a + (2/3)c, and the profits earned by each firm are: 2 1 2 (a c) 2
9b Hint: use the best response functions derived in class to get the quantities, then
use these to determine price and finally, firm profits.
b. Now consider a “cartel” version of the same game, where firms 1 and 2 collude
(and act as though they are a joint monopolist). In this case, the firms solve the
following profit maximization problem: Maxq1q ,q2 [a b(q1 q2 ) c](q1 q2 )
Notice that the only difference between the cartel and the duopoly profit
maximization problem is that in the cartel, the two firms acknowledge that their
profits depend on total production (q1+q2), and not on individual production
alone, so the maximization problem is a little different. The quantity that each
firm produces in the cartel can be thought of as a production quota (as in OPEC).
i. Write down the first order conditions from the above joint profit
maximization problem (i.e. find the expressions: d / dq1 0, d / dq 2 0
ii. Solve these two equations for the cartel quantities (quotas) q1, q2. Then find
the cartel price p, and the profit earned by each firm.
iii. Compare the cartel quantity, price and profits with the Cournot Nash
equilibrium quantity, price and profits you found in part a. Explain in words
why these amounts differ between the two cases. 3 ...
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This note was uploaded on 03/04/2012 for the course ECON 1200 taught by Professor Staff during the Fall '08 term at Pittsburgh.
- Fall '08