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, in a risky investment that will returnYt·Jtin one period, where theYt≥0 are i.i.d. random variables.Felicity is derived fromconsumption in any given period according the increasing functionc7→u(c), and the objectiveis to maximize the expected discounted value of∑∞t=0δtu(Ct), 0< δ <1. We are going todiscuss this problem using some results that we will prove later. We will111
Chapter 10.10(1) Set this up as a stochastic dynamic programming problem and give the Bellmanequation.(2) Show that the value function,V(s) is concave ifu(s) is concave using iterative ap-proximations to the value function.(3) Assume thatu(c) =crfor somer∈(0,1), and show that the optimal policy allocates aconstant proportion ofAtto each of the three alternatives and that the value functionV(s) is a multiple ofu(s).(4) Show that ifu(·) is concave andE Y < rthenJt≡0.8. Rubinstein-St˚ahl bargainingTwo people, 1 and 2, are bargaining about the division of a cake of size 1. They bargainby taking turns, one turn per period.If it isi’s turn to make an offer, she does so at thebeginning of the period.The offer isαwhereαis the share of the cake to 1 and (1-α)is the share to 2.After an offerαis made, it may be accepted or rejected in that period.If accepted, the cake is divided forthwith. If it rejected, the cake shrinks toδtimes its sizeat the beginning of the period, and it becomes the next period. In the next period it isj’sturn to make an offer. Things continue in this vein either until some final periodT, or elseindefinitely.Suppose that person 1 gets to make the final offer. Find the unique subgame perfect equi-librium. Suppose that 2 is going to make the next to last offer, find the unique subgame perfectequilibrium. Suppose that 1 is going to make the next to next last offer, find the subgameperfect equilibrium. Note the contraction mapping aspect and find the unique solution for theinfinite length game in which 1 makes the first offer.Problem10.15.The Joker and the Penguin have stolen3diamond eggs from the Gothammuseum. If an egg is divided, it loses all value. The Joker and the Penguin split the eggs bymaking alternating offers, if an offer is refused, the refuser gets to make the next offer. Eachoffer and refusal or acceptance uses up2minutes. During each such2minute period, thereis an independent, probabilityr,r∈(0,1), event. The event is Batman swooping in to rescuethe eggs, leaving the two arch-villains with no eggs (eggsept the egg on their faces, what a yolk).However, if the villains agree on a division before Batman finds them, they escape and enjoytheir ill-gotten gains.Question: What does the set of subgame perfect equilibria look like? [Hint: it is not in yourinterest to simply give the Rubinstein bargaining model answer. That model assumed that whatwas being divided was continuously divisible.]9. Optimal simple penal codesHere we are going to examine the structure of the subgame perfect equilibria of infinitelyrepeated games.