1. Suppose you have two nations, A and B, in a dispute over territo...

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1. Suppose you have two nations, A and B, in a dispute over territory (or something) of value Y in each time period. At the beginning of each time period they must spend eA and eB to maintain their bargaining position - - we'll assume that those influence their strength so that if they end up in a conflict the probabilities of winning are qa and QB. In each time period the nations must choose whether to come to a settlement (a division of Y) or to fight it out (once and for all). Conflict is destructive and the share 1 - 0 of Y is destroyed in the period conflict takes place (so in the conflict period 6Y remains but in future periods the land is still worth Y to the winner). Suppose in each bargaining round, A makes an offer (S, for subsidy) and B can either accept or reject. If B rejects, conflict ensues. As usually, future payoffs are discounted by s € [0, 1]. (a) Write the single period payoff to the players if they reach an agree- ment in which A pays B a subsidy S in each period. (b) Write the discounted sum of each players' payoff if they reach an agreement in which A pays B a subsidy S in each period. (c) Write the single period payoff to actor A if conflict occurs. (d) Write the discounted sum of i's payoff if conflict occurs. (Note that when i makes the decision to fight, it has already spent ei - SO it doesn't affect its payoff in the first period - and if conflict occurs the winner doesn't have to worry about maintaining its bargaining position in the future). (e) Using your answers to b) and d), derive the optimal subsidy S*. (f) For S* to be an equilibrium offer, A must be willing to make that offer. Show the conditions under which that is the case.

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