use the standard bargaining solution to find the outcomes of the following bargaining problems.

ECON 402. Fall 2014

HOMEWORK 6

Due in class on Thursday, November 20th

1. Use the standard bargaining solution to find the outcomes of the following

bargaining problems. Player 1’s payoff is u1 = v1(z) + t and player 2’s payoff is

u2 = v2(z) − t. In each case report the chosen z and t as well as players’ individual

payoffs.

(a) z ∈{5, 10, 15}, v1(z) = 12 ·z, v2(z) = z, d1 = 1, d2 = 0, π1 = 1 3 , and π2 =

2 3 .

(b) z ∈{0, 2, 10, 20}, v1(z) = − √ z, v2(z) = 2 ·z, d1 = 0, d2 = 0, π1 = 12 , and π2 =

1 2 .

(c) z ∈{1, 3, 5, 7, 9}, v1(z) = z2, v2(z) = 3 ·z−3, d1 = 2, d2 = 4, π1 = 47 , and π2 = 3 7 .

(d) z ∈ [1, 15], v1(z) = 12 ·z, v2(z) = z 1/3, d1 = 1, d2 = 0, π1 =

3 4 , and π2 =

1 4 .

2. A contract is being negotiated between a firm (player F) and a worker (player W).

The contract specifies two things: Job description and salary t. Regarding the job

description, W can be either a production supervisor or a maintenance supervisor. If

W works as a production supervisor the payoff to W is t− 5, 000 and the payoff to F

is 55, 000−t. If W works as a maintenance supervisor, his payoff is t−15, 000 and the

payoff to F is 70, 000 − t. If W and F fail to reach an agreement, W has an outside

job opportunity worth 15, 000 and F has an alternative job candidate whose services

are worth 5, 000 to F .

(a) Let πF and πW denote the bargaining weights of W and F respectively. Solve this

bargaining problem using the standard bargaining solution. Note: The solution

needs to specify a job description and the salary t.

(b) Suppose πF = 2 3

and πW = 1 3 . What is the salary t predicted by the standard

bargaining solution?

(c) Suppose πF = πW = 1 2 . What is the salary t predicted by the standard bargaining

solution?

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3. Consider a three-player bargaining game, where players are negotiating over how to

split a surplus of $1. The game begins with player 1 proposing a three-way split of the

surplus. Then player 2 must decide whether to accept the proposal or to substitute

player 1’s proposal with his own alternative proposal. Finally, player 3 must decide

whether to accept or reject the current proposal (which would be player 1’s original

proposal if it was accepted by player 2, OR player 2’s alternative proposal if player 2

decided to replace the original proposal made by player 1). If player 3 accepts, then

the players obtain the specified shares of the surplus. If player 3 rejects, then each

player gets 0.

(a) Draw the extensive form of this game.

(b) Determine the subgame perfect equilibria.

4. Suppose that you are attempting to buy a property, and that you are bargaining with

the current owner of the property over the sale price. The property is worth 200, 000

to you and 100, 000 to the current owner. Assume that bargaining takes place with

alternating offers and that each stage of bargaining takes a full day to complete.

Suppose that if an agreement is not reached after ten days of bargaining, then the

sale with definitely not take place and no further bargaining can occur. Suppose that

you and the current owner have the same discount factor of δ per day. The real estate

agent has allowed you to decide whether you will make the first offer.

(a) Suppose δ < 1 4

(both you and the owner are relatively “impatient”). Should you

make the first offer or should you let the current owner make the first offer? Why?

(b) Suppose instead that δ is closer to one, so both you and the owner are relatively

“patient”. Specifically suppose δ > ( 1 2

)1 9 (note, this implies δ > 0.925874712).

Should you make the first offer or should you let the current owner make the first

offer? Why?

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