Behavioural Economics ECON5324 Problem Set 2
Due 8pm (Sydney time) 17 March, 2023
1. Rosario has to finish her dissertation within 10 days, that is, at time t = 1, t = 2, ...,
or t = 10. It takes one day to finish the dissertation, and on the day Rosario
does so, she incurs an instantaneous disutility cost equivalent to $10. Rosario is a
hyperbolic discounter with β = 0.85 and δ = 1. Her (instantaneous) utility function
is u(x) = x.
(a) [7pts] Suppose the university has a system in which it charges Rosario $1 in
fees for every day she does not finish her dissertation (paid each day that it is
incurred). E.g., finishing on day 2 incurs a cost of $1 paid on day 1. When does
Rosario finish if she is naive? How much does she pay in penalties? (Hint, past
penalties are sunk, e.g., from the perspective of t = 2 self, any penalties paid
in t = 1 are sunk, and do not factor into decisions or utilities going forward.)
(b) [5pts] Still in the $1/day system, when does Rosario finish if she is sophisti-
cated?
(c) [5pts] Now suppose that the university has a deadline system: Rosario incurs
a penalty of $10 (paid on the day the dissertation is completed) if she does not
finish her dissertation before day 10 (so finishing on day 9 does not trigger the
penalty, but finishing on day 10 does). There are no daily penalties. When
does Rosario finish in this system if she is naive? How much does she pay in
penalties?
(d) [5pts] When does Rosario finish in the deadline system if she is sophisticated?
(e) [4pts] Does it make a big difference to a naive hyperbolic discounter whether
she is in a day-by-day-penalty or deadline system? Explain intuitively.
(f) [4pts] Does it make a big difference to a sophisticated hyperbolic discounter
whether she is in a day-by-day-penalty or deadline system? Explain intuitively.
2. Mike lives for seven periods, t = 1, 2, 3, 4, 5, 6, 7. In period 7, he enjoys the fruits
of the human capital, e, that he has amassed up to that point. In period 7 his
instantaneous utility for human capital is u7(e) = e. Mike does not derive utility
from his human capital in periods 1 through 6, nor does his stock of human capital
depreciate in any way. At the beginning of period 1, he starts off with a stock of
250 units of human capital. During periods 1 through 6 Mike has three options:
Option A Write no documents
Option B Write documents in Microsoft Word
Option C Write documents in LATEXusing Overleaf
Mike starts period 1 in the default option A, which does not builds human capi-
tal. Option B builds 5 units of human capital per period (up to period 6, i.e., no
1
additional human capital is built in period 7), and Mike can switch costlessly from
option A to option B at the beginning of any period. Option C builds 40 units of
human capital per period (up to period 6), but it takes a one-time immediate effort
cost of 90 units to switch from option A to option C (because it is painful and tricky
to set up LATEX). Suppose that there is no way to switch between options B and C,
and that once Mike makes any switch he cannot switch again.
Mike is a hyperbolic discounter with β = 1
2
and δ = 1. He cannot commit his future
behavior. He decides at the beginning of each period, including period 1, whether
to switch from option A to some other option, and if so, to which other option. If
he switches at the beginning of period t, he will build human capital at the rate
given by the new option during period t, e.g., if he switches to option C in period
3, he will build 40 units of human capital in periods 3, 4, 5, and 6.
(a) [14pts] Suppose Mike only has access to options A and B. If he is naive, when
does he switch to option B, if at all? What about if he is sophisticated?
(b) [14pts] Suppose Mike only has access to options A and C. If he is naive, when
does he switch to option C, if at all? What about if he is sophisticated?
(c) [12pts] Suppose Mike has access to all three options. Show that if he is naive
he waits until period 4 and then switches to option B at the beginning of
that period. Explain intuitively why he waits so long to switch to a superior
option when he could have costlessly switched all along. Show that if Mike
were sophisticated he would switch to option C in period 1.
3. A patient lives for two periods, 1 and 2. Her well-being in period 2 depends on her
state of health as well as some action t ≥ 0 taken in period 1. Suppose the patient’s
state of health can be represented by a real number s ≥ 0. For example, higher
numbers could represent better health and higher potential lifetime utility. The
patient’s initial belief is that s = s1 = 25 with probability one-half and s = s2 = 36
with probability one-half.
The patient derives utility from two sources. First, she would like to take the
appropriate action. Formally, if her state of health is s and she takes action t,
her “instrumental utility” is −|s − t|. This means that in terms of instrumental
utility, it is optimal to align the action perfectly with the state—to set t = s.
Because lower values of s represent worse health, corresponding low values of t
could represent taking health problems more seriously, for instance by having a
better diet or exercising; in other words, a more serious health condition calls for a
more serious response.
Second, the person derives anticipatory utility from her beliefs. Her anticipatory
utility depends on the average state of health given her beliefs. Specifically, if
she thinks the probability of the state s1 is p, then her anticipatory utility is
20
√
ps1 + (1− p)s2. The patient’s overall utility, which combines expected instru-
mental utility and anticipatory utility, is then
20
√
ps1 + (1− p)s2 − p|s1 − t| − (1− p)|s2 − t|
The patient has the option of visiting a doctor in period 1 to get diagnosed. The
visit is free. If she visits the doctor, she will know the true s with certainty. If she
does not visit the doctor, she does not learn any information about s, and will keep
believing that the probability of state s1 is one-half. If she visits the doctor, she can
choose t after the visit.
2
(a) [4pts] Write the patient’s expected utility as a function of t if she does not visit
the doctor. What t or range of t does she choose? What is her expected utility
given the optimal t?
(b) [4pts] Write the patient’s utility as a function of t if she visits the doctor and
learns that her state of health is s1 = 25. What t does she choose? What is
her utility given the optimal t?
(c) [4pts] Repeat the exercise in part (b) for the case when the patient visits the
doctor and learns that her state of health is s2 = 36.
(d) [4pts] Write the patient’s total expected utility from visiting the doctor. This
is the weighted sum of the utilities in parts (b) and (c), with the weights equal
to the probabilities of the two possible states of health. Does the patient visit
the doctor?
(e) [10pts] Now suppose that s1 = 0, so that the patient’s possible problem is more
serious. The other possibility is still s2 = 36, with the two health states still
being equally likely. Using the same steps as in parts (a) through (d), solve for
whether the patient goes to the doctor.
(f) [4pts] Conventional economic wisdom says that when information is more im-
portant for making decisions—such as above, when a patient’s health problem
is potentially more serious—a person is more likely to seek out that informa-
tion. How does the consideration of anticipatory utility qualify this insight?