# Time inconsistency and chocolate. The latest craze sweeping Pcoria is a new kind of exotic chocolate

Time disproportion and chocolate. The concluding craze sweeping Pcoria is a new skin of strange chocolate adventitious from the faraway alpine nation of Chplandia. Chplandian chocolate is exquisite and cheap: each bar produces \$2 merit of service and simply consumes \$1. But unfortunately, it is not very healthy: if a Pcorian eats x bars on any given day, she suffers negative health goods the instant day that total to −\$1 /4x2 . But she does not suffer any ill goods on the day she eats the chocolate.

a Usurp Pcorians are beta-delta discounters after a while β = 1 and δ = 1, and that each day is a determination. Interpret these assumptions about β and δ. Are Pcorians unrepining? Are they time-consistent?

b What is the optimal compute of Chplandian chocolate bars x∗ for each Pcorian to eat each day subordinate these assumptions? [Hint: Balance the marginal avail and marginal consume.]

c Now usurp 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 abundantly Chplandian chocolate to buy. How abundant bars of chocolate does Bob reckon he scantinesss to eat tomorrow?

e When tomorrow substantially arrives, accomplish Bob modify his mind about how abundant bars he scantinesss to eat? Explain whether Bob is time-consistent or time-inconsistent in his preferences.

f Now usurp δ = 2 /3 and β = 0.5. Explain what β represents and what description of issue this new β accomplish entertain on the Pcorians.

g What is the compute of Chplandian chocolate bars xˆ that each Pcorian eats each day subordinate these new assumptions, stiff each Pcorian works to maximize the exhibit self’s service character?

h Usurp Bob is a easy hyperbolic discounter. How abundant chocolate bars does Bob reckon he accomplish scantiness tomorrow? How abundant accomplish he substantially want?