1. Do it now or do it later. My friend Paco Vivalavida has to do an activity once between now (t = 1) and T = 4. That is, he has to choose a vector of decisions (x1, x2, x3, x4), xt {Y, N}, where xt = Y means do the activity in period t if he has not already done it at t 0 < t, and xt = N means not do the activity at t. For instance, (N, Y, N, Y ) means Paco does not do the activity at t = 1, does the activity af t = 2 if it was not done before, does not do it at t = 3 if it has not been done before and does it at t = 4 if it has not been done before. Such a strategy leads to him actually doing the activity at t = 2. Rewarding Activity. Paco loves movies and he just got a free movie ticket that he can use at the local movie theater in one of the next 4 Saturdays. Each Saturday a different movie is shown, with better movies scheduled at later dates. In particular, Pacos (undiscounted) utilities associated to each of the movies are (u1, u2, u3, u4) = (3, 5, 8, 13), while he gets 0 if he does not watch a movie at t. He has time preferences with = 1. For instance, his utility at t = 1 from (N, Y, N, Y ) is u2, while her period-2 self derives utility u2. Similarly, his period-1 self gets u4 when he chooses (N, N, N, Y ) while her period-3 self gets u4. (a) If Pacos time preferences are given by = 1, what is his optimal choice of (x1, x2, x3, x4)? (b) For the remainder of the problem assume Paco is present-biased with = 0.5. If he could commit to (x1, x2, x3, x4) (e.g. by giving the coupon to a friend and instructing him to deliver it at a specific t), what would his choice be? (c) If Paco is sophisticated, what would he choose? Does he preproperate (do an activity too soon)? (d) Now assume that Paco is naive, that is, he thinks that her future selves have = 1 while in reality they have the same = 0.5 as Paco does in period 1. For instance, this implies that, while his period-3 self gets u4 when he chooses (N, N, N, Y ), Paco wrongly believes in periods t = 1 and 2 that his period-3 self gets u4. What would his choice be under noncommitment? Given your answers, is it always good to be sophisticated?