Darcy is going to camp for two full days; for the sake of argument, let’s suppose the camp starts early Saturday and ends late Sunday. Darcy has been given an allowance of A = 1000; 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 = 60 units of immediate culinary joy. If he instead eats decadently, his immediate culinary joy is D = 120 but he will be ill the following day, which causes him I = 80 units of additional disutility on that following day.1 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 = 0.85 and I-don’t-really-care-about- tomorrow discount factor β = 0.75. 1. 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. PDUF((H,H,H,H,H,H,...),(A/2,A/2,0,0,0,0,0,...)) 

b. PDUF((D,H,H,H,H,H,...),(A/2,A/2,0,0,0,0,0,...)) 

c. PDUF((H,D,H,H,H,H,...),(A/2,A/2,0,0,0,0,0,...)) 

d. PDUF((D,D,H,H,H,H,...),(A/2,A/2,0,0,0,0,0,...)) 

We’ve started the first one for you. To save on pen- or keystrokes, write Darcy’s PDU from the eating and spend- ing profiles (e1,e2,e3,e4,e5,...) and (s1,s2,s3,s4,s5,...) as PDUF((e1,e2,e3,e4,e5,...),(s1,s2,s3,s4,s5,...)) = pdueF(e1,e2,e3,e4,e5,...) + pdu$F(s1,s2,s3,s4,s5,...) where pdueF measures present discounted utility from just the eating portion of Darcy’s behaviour and pdu$F his present discounted utility from just the spending portion, and the subscript Fs indicate that this calculation is being made on Friday. Thus we have the following equalities: a. PDU((H,H,H,H,H,H,...),(A/2,A/2,0,0,0,0,0,...)) = β[δH + δ2H + δ3H + δ4H + ...]︸ ︷︷ ︸=pdue F (H,H,H,H,...) + β[δ√A/2 + δ2√A/2 + δ3√0 + δ4√0 + ...]︸ ︷︷ ︸ =pdu$ F (A/2,A/2,0,0,...) = β[δH + δ2H + δ3H + δ3 ∑∞ t=1 δtH]+ β[δ√A/2 + δ2√A/2 ] = 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.)