Ann and Bob are playing an infinite horizon alternating offers bargaining game over a pie of size 1. Let si denote player i's share of the pie. Shares must be non-negative (si 0) and must add to one (sa + sb = 1). Ann and Bob are both altruistic. That is, Ann, for example, gets utility both from her own consumption and Bob's. If there is an agreement in period t assigning shares sa and sb then Ann gets uA = t1(sa + sb) and Bob gets uB = t1(sb + sa). We assume that 0 < < 1 and 0 < < 1. Hence, Bob, for example, cares about Ann, but cares more about himself. In odd periods t = 1,3,..., Ann makes an offer sa = xt and sb = 1xt and Bob accepts or rejects. In even periods t = 2,4,... Bob makes an offer sa = yt and sb = 1 yt and Ann accepts or rejects. If an agent accepts, then all agents are assigned their shares, and the game ends. Otherwise, we move to the next period. If no offer is ever accepted, then all players get a payoff of zero.

We look for a stationary SGPNE for which the following hold:

(A) for each odd period t, there is an x t such that Bob accepts if and only if xt x t,

(B) for each even period t, there is an y t such that Ann accepts if and only if yt y t, and

(C) there are values x,y,x , and y such that xt = x and x t = x in odd periods, and yt = y and y t = y in even periods.

(1) Find a relationship between x and y that must hold in a stationary SGPNE. (Hint: say that xt = x . What relationship must hold between the utilities for accepting and rejecting xt? What are the expressions for those utilities?) Use this relationship to find a simplified expression for x as a function of y. Find a relationship between y and x that must hold in a stationary SGPNE. Use this relationship to find a simplified expression for y as a function of x.

(2) Explain why the relationship you found between x and y must hold in a SGPNE. That is, why should Bob reject in period t if xt > x ? Why should he accept if xt x ?

(3) What relationship should hold between x and x in a SGPNE? What relationship should hold between y and y . Explain

(4) Use the relationships you found in parts (1) and (3) to solve for the SGPNE values of x and y. These equilibrium values should be expressions which depend on only , , and . (Hint: In order to write simplified expressions, you should solve for (1 + ) x = .... Similarly, you should solve for an expression (1 + ) y = ....)

(5) Verify that if = = 0, then we get the exact same expression as we found in lecture when agents were not altruistic.