1. (50 points) Rosario has to finish her dissertation within 100 days, that is, at time t = 1, t = 2, t = 3, .... or t = 100. It takes one day to finish the dissertation, and on the day Rosario does so, she incurs an instantaneous disutility cost equivalent to $10 (We will measure utility in dollar terms, as in willingness-to-pay, for this problem). Rosario is a hyperbolic discounter with 3 = 0.85 and 6 = 1. Thus, on day t = 1, Rosario seeks to maximize her discounted utility: t=100 U1 = ul + 38t-1 > ut t= 2 = uj + 0.85 * (u2 + u3 + us + ... + ugs + ugg + 4100) Suppose the university has a system in which it charges Rosario $1 in fees for every day she does not finish her dissertation (so if she finishes on day 2, she pays $1 in fees). The accumulated fees are assessed on day 100, so that is when they enter her instantaneous utility. Lastly, suppose Rosario is naive about her time inconsistency. (a) (3 points) On day 1, what is Rosario's discounted utility of finishing her dissertation immediately? (b) (3 points) On day 1, what is her discounted utility of planning to finish on day 2? (c) (3 points) On day 1, what is her discounted utility of planning to finish on day 3? (d) (3 points) Use your answers above to show that Rosario will not finish on day 1, and briefly say why. (e) (6 points) Now consider Rosario's decision on day 2. Use the same method above to show that she will not finish her dissertation on day 2. (f) (6 points) When will Rosario actually do the dissertation? How much will she pay in fees? (g) (8 points) Now suppose that the university has a deadline system: Rosario incurs a penalty of $10 if she does not finish her dissertation by day 10 (so finishing on day 9 does not trigger the penalty, but finishing on day 10 does). There are no daily penalties. When does Rosario finish in this system if she is naive? How much does she pay in penalties? (h) (8 points) Does it make a big difference to a naive hyperbolic discounter whether she is in a day-by-day-penalty or deadline system? Explain intuitively. (i) (8 points) Can you think of another example of a situation in which the costs of putting off an action grow slowly, day by day, and many people end up delaying the action much longer than their period 0 self would want? In your example, would it be possible for a hard deadline with a large cost to improve outcomes?(j) (2 points) When does Rosario finish in the first system (with $1 daily fees for delay) if she is sophisticated? Hint: Solve this problem starting from the end: Will she do it on day 99 if she hasn't done it yet? Will she do it on day 98 if she hasn't done it yet? For a sophisticate, a "plan" for doing the problem set later is only considered viable if it is realistic, i.e. her future self in that period would choose to do it. 2. (30 points) A student can go to a movie exactly once in four periods t = 0,1,2,3. The instantaneous utility, in monetary terms, from going to the movie in period t = 0 is 29, in period t = 1 it is 37, in period t = 2 it is 49, and in period t = 3 it is 67. A movie ticket costs $13, which has to be paid in the period in which the student goes to the movie (so the ticket price must be subtracted from instantaneous utility in the period in which the price is paid). The student is a hyperbolic discounter with 8 = ; and 6 = 1. (a) (10 points) When does a sophisticated student go to the movie? Show your work. (b) (10 points) When does a naive student go to the movie? Show your work. (c) (10 points) Explain intuitively why the naive student ends up better off than the sophis- ticated student in this case