3
1. Homework Bob has a quasi-hyperbolic discounted utility
U (c0 , c1 , c2 ) = u(c0 ) + u(c1 ) + 2 u(c2 ).
Let u(c) = c.
(a) Suppose that Bob is indierent between the three proles
c0 = c1 = c2 = 10
c0 = 10, c1 = 9, c2 = 11
c0 = 20, c1 = c2 = 0.
Find and .
(b) Suppose that at t = 1, he has a quasi-hyperbolic discounted utility
U1 (c1 , c2 ) = u(c1 ) + u(c2 ).
Argue that Bobs preferences are time inconsistent. (Give an example
when at t = 1 he reverts his previous choices that he made at t = 0.)
2. Ann has three days left before an exam. Each day she has a hyperbolic utility
with = 1 and = 0.5. Each day she can either work hard and get a delayed
reward of 10, or have fun with an immediate reward of 7. (Note that unlike the
examples in the lecture she can work 1, 2, or 3 days and get 10, 20, or 30 in
delayed rewards respectively.)
(a) What will choose she to do in each of these days?
(b) Suppose that on the rst day, she can promise to pay a ne if she does
not work hard for the other two days. Suppose that paying the ne has
an immediate cost of 10. Will she want to make this obligation?
(c) What if the ne has an immediate cost 1?
3. Chris needs to decide whether to go home or go to the library. At home, his
menu is H = {a, b, c} where a is study hard, b is play a video game, and c is surf
the Internet. In the library, his menu is L = {a, c}. He has a costly self-control,
and his utility is
U (A) = max[u(x) max[v (y ) v (x)]] = max(u(x) + v (x)) max v (y ).
xA
y A
xA
y A
Suppose that u(a) = 10, u(b) = u(c) = 0 and v (a) = 0, v (b) = 15 and v (c) = 5.
Which of the two menus H and L will Chris choose? What choices does he plan
to make in the menus H and L?
4. Case 6.3 on p. 263.