Problem Set 1
Due 12:30pm January 31, 2019
微积分Midterm代考 Suppose the camp charges a fee of $ 10 in order to withold some of Darcy’s allowance; this fee would be taken out of the amount
Darcy is going to camp for two full days; for the sake of argument, let’s suppose the camp starts early Saturday and ends lat Sunday.
Darcy has been given an allowance of A = 200; he can allocate his spending of the allowance however he wants over the two days of camp, but he can’t take any money home with him, nor can he supplement it
with outside funds. Darcy’s instantaneous utility from spending st in period t is given by √st.
Camp has a full selection of foods, including healthy and decadent choices. If Darcy eats healthily on day t, on that day he gets H = 20 units of immediate culinary joy. If he insteads eats decadently, his immediate
culinary joy is D = 40 but he will be ill the following day, which causes him I = 30 units of additional disutility on that following day. Assume that, from Monday onward, Darcy has no choice but to eat healthily
and has no money to spend.
Darcy is a quasi-hyperbolic discounter with inter-day discount factor δ = 0.9 and I-don’t-really-care-abouttomorrow discount factor β = 0.6. 微积分Midterm代考
- Calculate Darcy’s present discount utility as viewed on Friday—the day before he goes to camp— from the following action profiles, where profile ((e1, e2, e3, e4, e5, ...),(s1, s2, s3, s4, s5, ...)) indicates how
Darcy eats (et) and spends (st) on each of the days. Be sure to show your work.
a. P DU((H, H, H, H, H, H, ...),(100, 100, 0, 0, 0, 0, 0, ...)) =
b. P DU((D, H, H, H, H, H, ...),(100, 100, 0, 0, 0, 0, 0, ...)) =
c. P DU((H, D, H, H, H, H, ...),(100, 100, 0, 0, 0, 0, 0, ...)) =
d. P DU((D, D, H, H, H, H, ...),(100, 100, 0, 0, 0, 0, 0, ...)) = Assuming Darcy ate healthily on Fridy night, what is Darcy’s optimal food consumption pattern as viewed at 11:59pm on Friday night? (I.e. you can ignore payoffs incurred on Friday itself).
- Explain whether/why Darcy wants to stay on this path when he actually arrives at camp.
4. Come Saturday, what food consumption pattern does Darcy want to pursue? Use calculations and formulae to show that Darcy’s PDU—as calculated on Saturday—of this alternate consumption pattern is higher than that from the plan he’d preferred as of Friday. (For the purposes of this comparison, you are welcome to use variables s1, s2 to represent spending—rather than specifying actual amounts, and you are welcome to assume that these are the same across the two plans.)
Use either calculus or trial-and-error to show that, as viewed on Friday, spending $100 on each of Friday and Saturday is suboptimal. In particular, show that the spending pattern ($ 200 1+δ 2 , $200 − $200
1+δ 2 ) maximizes Darcy’s present discounted utility as calculated on Friday. Show your work.Suppose the camp offers its patrons the following option. In advance of the start of camp, each camper can surrender some/part of his/her allowance and specify a set of rules dictating when and if she gets
the funds back. On Friday, Darcy considers the following plan: surrender $90 of his allowance to the camp with the condition that it be returned to him on Sunday morning if and only if he ate healthily on
Saturday. If Darcy executes this plan, come Saturday, will Darcy eat healthily? Use/show calculations to support your answer.Suppose the camp charges a fee of $ 10 in order to withold some of Darcy’s allowance; this fee would be taken out of the amount Darcy gets to spend on Sunday. Should Darcy pursue a surrender/withold
agreement with the camp? Explain why or why not.
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