in a risky investment that will return Y t J t in one period where the Y t 0

# In a risky investment that will return y t j t in one

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, in a risky investment that will return Y t · J t in one period, where the Y t 0 are i.i.d. random variables. Felicity is derived from consumption in any given period according the increasing function c 7→ u ( c ), and the objective is to maximize the expected discounted value of t =0 δ t u ( C t ), 0 < δ < 1. We are going to discuss this problem using some results that we will prove later. We will 111
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 if u ( s ) is concave using iterative ap- proximations to the value function. (3) Assume that u ( c ) = c r for some r (0 , 1), and show that the optimal policy allocates a constant proportion of A t to each of the three alternatives and that the value function V ( s ) is a multiple of u ( s ). (4) Show that if u ( · ) is concave and E Y < r then J t 0. 8. Rubinstein-St˚ ahl bargaining Two people, 1 and 2, are bargaining about the division of a cake of size 1. They bargain by taking turns, one turn per period. If it is i ’s turn to make an offer, she does so at the beginning of the period. The offer is α where α is the share of the cake to 1 and (1 - α ) is the share to 2. After an offer α is made, it may be accepted or rejected in that period. If accepted, the cake is divided forthwith. If it rejected, the cake shrinks to δ times its size at the beginning of the period, and it becomes the next period. In the next period it is j ’s turn to make an offer. Things continue in this vein either until some final period T , or else indefinitely. 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 perfect equilibrium. Suppose that 1 is going to make the next to next last offer, find the subgame perfect equilibrium. Note the contraction mapping aspect and find the unique solution for the infinite length game in which 1 makes the first offer. Problem 10.15 . The Joker and the Penguin have stolen 3 diamond eggs from the Gotham museum. If an egg is divided, it loses all value. The Joker and the Penguin split the eggs by making alternating offers, if an offer is refused, the refuser gets to make the next offer. Each offer and refusal or acceptance uses up 2 minutes. During each such 2 minute period, there is an independent, probability r , r (0 , 1) , event. The event is Batman swooping in to rescue the 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 enjoy their ill-gotten gains. Question: What does the set of subgame perfect equilibria look like? [Hint: it is not in your interest to simply give the Rubinstein bargaining model answer. That model assumed that what was being divided was continuously divisible.] 9. Optimal simple penal codes Here we are going to examine the structure of the subgame perfect equilibria of infinitely repeated games.