Problem 1. Envy: a capital sin
Consider the following ultimatum bargaining game. There is 1 unit of a good and player 1 offers a split (x, 1 - x), where x ∈ [0, 1] is chosen by player 1. Player 2 accepts the offer
(Y ) or refuses it (N ). If player 2 accepts the offer, then player 1 gets x and player 2 gets 1 - x. If player 2 refuses the offer, then both players get 0. We assume that when player 2
is indifferent between accepting and refusing, then he accepts.
(a) Suppose that each player maximizes his payoff. Find the Backward Induction Equi- librium [easy; we have done it in class already].
Suppose now a more interesting case: players are envious. More precisely, the utility of player 1 is equal to his payoff minus β times the payoff of player 2 and the utility of player 2 is equal to his payoff minus β times the payoff of player 1, with β > 0. The parameter β can therefore be interpreted as a measure of "envy".
(b) Find the Backward Induction Equilibrium as a function of β.
(c) What happens as β increases? Interpret the result.

Problem 2. Tough negotiations
A firms revenue function when it uses L units of labor is given by

A union that represents workers presents a wage offer w to the firm. The firm observes the offer and either accepts or rejects it. If the firm accepts the offer, it then chooses the number L of workers to employ (which, for simplicity, you should take to be a continuous variable). If it rejects the offer, no one is hired. The revenue of firm and workers is then 0.
(a) Write down the firm's profit function.
Suppose that the union cares both about the total number of employees hired L and the wage they each receive w. More precisely the union maximizes the following payoff function:

The payoff of the union is also 0 if no one is hired.
(b) Find the subgame perfect equilibrium (equilibria?) of this game.
(c) Can you find one outcome of the game that both parties prefer to any subgame perfect equilibrium outcome? Interpret.