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Harvard University - ECONOMICS 1051PS8_08spr-solutions.

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Problem Set 8 Solutions Economics 1051 Spring 2009 1. Bargaining. It is November 1st, and Angus and Bronwen are arguing over how to divide the remainder of their left-over trick or treat candy. They decide on the following rules. On Nov. 1st, Angus will make Bronwen an offer [sA(1); sB(1)] where sA(1) + sB(1) = 1. If Brownen accepts then Angus gets a share sA(1) of the candy and Brownen gets a share sB(1). If Brownen refuses, the noone gets any candy on Nov 1st. In this case, on Nov. 2nd, Brownen gets to make Angus an offer [sA(2); sB(2)] where sA(2)+sB(2) = 1. If Angus accepts then Angus gets a share sA(2) of the candy and Brownen gets a share sB(2). If Angus refuses, then noone gets a candy on Nov 2nd. In this case, on Nov. 3rd, Angus will make Bronwen an offer [sA(3); sB(3)] where sA(3) + sB(3) = 1. If Brownen accepts then Angus gets a share sA(3) of the candy and Brownen gets a share sB(3). If Brownen refuses, the noone gets any candy on Nov 3rd. In this case, on Nov. 4th, Brownen gets to make Angus an offer [sA(4); sB(4)] where sA(4) + sB(4) = 1. If Angus accepts then Angus gets a share sA(4) of the candy and Brownen gets a share sB(4). If Angus refuses, however, then all the candy is thrown away by their parents. There is a catch. Both Angus and Bronwen are hungry and impatient for candy so they discount candy tomorrow. For Angus, each day, one candy tomorrow is worth only δA candy today where δA < 1. For Bronwen, each day, one candy tomorrow is worth only δB candy today where δB < 1. It is commonly known that Angus and Bronwen are rational and that they are not interested in distributive justice. You may also assume that each of Angus and Bonwen accepts an offer whenever he or she is indifferent. a) What do you expect to happen, and what allocation of candy would arise if δA = δB? Explain your reasoning carefully. Solution. We use the backward induction. On day 4, Brownen will offer [sA(4); sB(4)] = [0; 1]: indeed, Angus must accept any offer ["; 1 − "] for " > 0, and is indifferent between accepting and rejecting offer [0; 1]; hence, equilibrium requires that Brownen offers [0; 1] and Angus accepts. On day 3, Angus anticipates this, and knows that he must offer Brownen at least δB = δ: if he offers less, then Brownen would prefer 1 on day 4 to this smaller amount on day 3. Hence, [sA(3); sB(3)] = [1 − δ; δ].

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