The latest craze sweeping Pcoria is a new kind of exotic chocolate imported from the faraway alpine nation of Chplandia. Chplandian chocolate is delicious and cheap: each bar produces $2 worth of utility and only costs $1. But unfortunately, it is not very healthy: if a Pcorian eats x bars on any given day, she suffers negative health effects the next day that amount to −$¹/4x². But she does not suffer any ill effects on the day she eats the chocolate.

a. Assume Pcorians are beta-delta discounters with β = 1 and δ = 1, and that each day is a period. Interpret these assumptions about β and δ. Are Pcorians patient? Are they time-consistent?

b. What is the optimal number of Chplandian chocolate bars x∗ for each Pcorian to eat each day under these assumptions?

c. Now assume δ = 2/3 instead. How does this affect optimal chocolate intake x∗? Explain intuitively why x∗ changes the way it does when δ falls.

d. Bob, a chocolate-loving Pcorian, is doing his grocery shopping for the week and deciding how much Chplandian chocolate to buy. How many bars of chocolate does Bob think he wants to eat tomorrow?

e. When tomorrow actually arrives, will Bob change his mind about how many bars he wants to eat? Explain whether Bob is time-consistent or time-inconsistent in his preferences.

f. Now assume δ = 2/3 and β = 0.5. Explain what β represents and what sort of effect this new β will have on the Pcorians.

g. What is the number of Chplandian chocolate bars ˆx that each Pcorian eats each day under these new assumptions, assuming each Pcorian works to maximize the present self’s utility function?

h. Assume Bob is a naive hyperbolic discounter. How many chocolate bars does Bob think he will want tomorrow? How many will he actually want?