It is early Monday morning and Darryl Dawdle must write a term paper. Darryl’s instructor does not accept late papers and it is crucial for Darryl to meet the deadline. The paper is due on Thursday morning, so Darryl has three days to work on it. He knows that it will take him 12 hours to do the research and write the paper. Darryl hates working on papers and likes to postpone unpleasant tasks. But he also knows that it is less painful to spread the work over all three days rather than doing it all on the last day. For any day, t, let xt be the number of hours that he spends on the paper on day t, and xt+1, and xt+2 the number of hours he spends on the paper the next day and the day after that. At the beginning of day t, Darryl’s preferences about writing time over the next 3 days are described by the utility function
U(xt, xt+1, xt+2) = −x2t – 1/2 x2t+1 – 1/3 x2t+2.
(a) Suppose that on Monday morning, Darryl makes a plan by choosing xM, xT, and xW to maximize his utility function
U(xM, xT, xW) = −x2M – 1/2 x2T – 1/3 x2W
subject to the constraint that he puts in a total of 12 hours work on the paper. This constraint can be written as xM + xT + xW = 12. How many hours will he plan to work on Monday? xM = _____________ Tuesday? xT = __________ Wednesday? xW = __________
(b) On Monday, Darryl spent 2 hours working on his term paper. On Tuesday morning, when Darryl got up, he knew that he had 10 hours of work left to do. Before deciding how much work to do on Tuesday, Darryl consulted his utility function. Since it is now Tuesday, Darryl’s utility function is
U(xT, xW, xTh) = −x2T – 1/2 x2W – 1/3 x2Th,
where xT , xW, and xTh are hours spent working on Tuesday, Wednesday and Thursday. Of course work done on Thursday won’t be of any use. To meet the deadline, Darryl has to complete the remaining work on Tuesday and Wednesday. Therefore the least painful way to complete his assignment on time is to choose xT and xW to maximize
U(xT , xW, 0) = −x2T – 1/2 x2W
subject to xT + xW = 10. To do this, he sets his marginal disutility for working on Tuesday equal to that for working on Wednesday. This gives the equation __________ Use this equation and the budget equation xT + xW = 10 to determine the number of hours that Darryl will work on Tuesday _________ and on Wednesday __________ On Monday, when Darryl made his initial plan, how much did he plan to work on Tuesday? _________ On Wednesday? ___________ Does Darryl have time-consistent preferences? _________
(c) Suppose that on Monday morning Darryl realizes that when Tuesday comes, he will not follow the plan that maximizes his Monday preferences, but will choose to allocate the remainder of the task so as to maximize
U(xT, xW, 0) = −x2T – 1/2 x2W
subject to the constraint that xT + xW = 12 − xM. Taking this into account, Darryl makes a new calculation of how much work to do on Monday. He reasons as follows. On Tuesday, he will choose xT and xW so that his marginal disutility of working on Tuesday equals that on Monday. To do this he will choose xW/xT = ____________ Darryl uses this equation, along with the constraint equation to xT + xW = 12 – xM to solve for the amounts of work he will actually do on Tuesday and Wednesday if he does xM hours on Monday. When he does this, he finds that if he works xM hours on Monday, he will work xT (xM) = 1/3 (12−xM) and xW(xM) = 2 3 (12 − xM) hours on Wednesday. Now, for each possible choice of xM, Darryl knows how much work he will do on Tuesday and Wednesday. Therefore, on Monday, he can calculate his utility as the following function of xM
UM(xM) = −x2M – 1/2 xT(xM)2 – 1/3 xW(xM)2.
Set the derivative of this expression with respect to xM to find that Darryl maximizes his utility by working ____________ hours on Monday.
(d) Does Darryl’s three-period utility function have exponential discounting or hyperbolic discounting? ___________ If exponential, what is the discount rate δ; if hyperbolic, what is the parameter k? ___________