Advanced Microeconomics (ES30025)
Seminar Three: Bargaining
Advanced Microeconomics (ES30025)
Seminar Three: Bargaining
1. Consider a Nash bargain between you and a firm for whom you might work. If you are
employed, the firm will earn £300. If the firm does not employ you: (a) you will earn £100
elsewhere; and (b) the firm would use someone from an employment agency who would cost
the firm £150 and whose work would earn £225 for the firm. How much will you be paid if?
(i)
(ii)
(iii)
(iv)
You and the firm have equal bargaining power?
The firm has twice as much bargaining power as you?
The firm has all the bargaining power?
You have all the bargaining power?
2. Consider the following alternating offers game of bargaining without impatience: There
are 2 players: a buyer, Bill (B), and a seller, Sally (S). They are negotiating a sales contract.
The maximum price Bill is willing to pay is $300. The minimum price Sally is willing to
accept is $200. The price P must therefore lay in the range $200 < P < $300. To simplify the
discussion, assume that price will be at least $200, and so bargaining can be phrased in terms
of the division of the surplus $100. Thus, if P = $245, this is an allocation of $45 of the
surplus to the seller Sally and $55 of the surplus to the buyer Bill - i.e., the surplus $100 is
the ‘cake’ being divided. We write lower case p to represent price in excess of $200: that is, p
= P - $200.
Assume that Sally moves first, proposing price p1S . Bill accepts or rejects. If he
accepts, sale takes place at price p1S . If he rejects, he proposes an alternative price p2B . Sally
accepts or rejects his proposal. If she accepts, sale takes place at price p2B . If she rejects,
bargaining ends, there is no sale and they have both lost the opportunity to gain.
(i)
(ii)
Find the sub-game perfect equilibrium of this game.
Now suppose that the last round (in which Sally either accepts Bill's offer or the sale falls
through) is deleted. So in the new final round, if Bill rejects Sally's offer, the sale falls
through. Solve the game.
3. Continue with the game described in Question 2 but now assume now that there are 100
rounds and that Bill and Sally are impatient, each applying a discount rate of 10 percent. That
is, a player is indifferent between having $1 in one round, or having $1.10 one round later.
Assume that Bill moves first.
(i)
(ii)
(iii)
Work out what would happen in the 100th, 99th and 98th rounds.
What is the discount factor if the discount rate is 10%?
If this game were amended so that there were no limit on the number of rounds of bargaining,
what would the solution be?
4. Mr. One and Mr. Two are negotiating over the division of a £1 coin. If the sum of their
agreed shares is greater than 1, or if they fail to reach an agreement, then the £1 coin will
evaporate. Mr. One and Mr. Two have initial wealth mi > 0 . i = 1, 2 and utility functions,
( )
ui xi = ln xi , where xi is Mr. i’s wealth including his share of the £1 coin, and i = 1, 2:
(i)
Construct and solve the Nash bargaining solution to this game.
1
Advanced Microeconomics (ES30025)
(ii)
Seminar Three: Bargaining
How does initial wealth impact on the division of the £1 coin?
2