What offer should A make to maximize its expected profit? "11. Suppose two firms, X and Y, are bargaining over how to split a total stake of $120,000. As in the earlier management-labor example, the firms alternate offers. Delay is very costly to both parties. During the time between offers, the value of the stake shrinks by 50 percent. When X makes the first offer, $120,000 is to be split; failing an immediate agreement, by the time Y makes the next offer, the stake is $60,000; at X's next turn, the stake is $30,000; and so on.

a. Suppose three offer rounds (X, then Y, then X) are allowed. What equilibrium offers will the parties make? Which offer will be accepted?

b. What if there are four offer rounds? Is making the first offer advantageous?

c. Suppose offers can be made indefinitely, with the stake continually shrinking. Show that X demands $80,000 for itself in the first round and Y agrees, that is, accepts $40,000. In other words, X claims two- thirds of the total stake. Hint: As in the earlier examples, X demands a fraction of the stake (call this z) such that Y is indifferent to the options of agreeing, that is, accepting $120,000(1-2), or waiting one turn and making its own offer. Note that, at its turn, Y will demand $60,000 (z) for itself, leaving the remainder for X. Use these facts to solve for z.