809 Econ Spring 2006 Assignment 1. Time is discrete and the horizon is infinite; t = 0, 1, ..., . Each period is divided into two subperiods; stage 1 and stage 2 (day and night). There is a unit mass of infinitely-lived ex ante identical agents i [0, 1], with preferences defined over stochastic sequences: {et (i), ct (i), nt (i) : t 0} , where et (i) denotes effort during stage 1; ct (i) denotes consumption during stage 2; and nt (i) denotes effort during stage 2. These preferences are represented by an expected utility function: X E0 t [-et (i) + u(ct (i)) - t (i)g(nt (i))] ,
t=0
with 0 < < 1, > 0, and where t (i) { L , H }, 0 < L < H < . Let Pr[t (i) = L ] = 1/2 for all i and for all t. Assume that u00 < 0 < u0 and 0 < g 0 , g00 with g(0) = 0 and u(0) = u0 -. The available production technologies are as follows. Assume that et (i) R, so that effort first-stage is unbounded from above and below. Second stage output is produced according to yt (i) = nt (i). Assume that nt (i) 0 and that output is nonstorable. (a) Characterize the autarkic allocation and welfare level. (b) Characterize the solution to a planner's problem that maximizes ex ante welfare. Is this in any sense a `credit' economy? Describe the pattern of exchanges recommended by the planner. Could such an allocation be decentralized in any way? Explain. (c) Assume now that t (i) is private information. Ch...