Problems for the …rst seminar
ECON4260 Behavioral Economics –Fall semester 2011.
Solutions to the problems will be presented at the seminars in week 37.
Please direct any question to Kjell Arne Brekke (Room ES1032, Tel 228 41169,
E-mail: k.a.brekke@econ.uio.no)
Problem 1
Consider the pair of lotteries (all values are in kroner).
A: ( 50; 50%; +70; 50%) and B:(0)
C: ( 50; 50%; +90; 50%) and D:(0)
Suppose that Ola is ranking lotteries according to prospect theory with
(p) = p (outcomes are weighted by their probabilities) and
v(x) =
x if x
x if x
0
0
That is the reference point is r = 0 as mostly assumed in Kahneman and
Tversky’s original paper.
a) For what values of will Ola prefer A to B? And for what values of will
he prefer C to D?
Now consider the choice where you are …rst given 100 kroner and then choose
C’: ( 50; 50%; +90; 50%) and D’:(0)
and the lotteries with no initial payout
E: (+50; 50%; +190; 50%) and F:(100)
Kari is an expected utility maximizer and mentally adds gains and losses in
the lotteries to her existing wealth.
b) Show that if Kari prefer E to F, she must also prefer C’to D’.
c) If we know Ola’s ranking of the lotteries C’and D’, can we infer his ranking
for E and F?
Now, Ola and Kari are o¤ered to choose between lottery C played twice and D
twice.
d) Show that this yields the lotteries
G: ( 100; 25%; + 40; 50%; + 180; 25%) and H:(0)
Now assume that = 2 for Ola, and that Kari’s ranking of lotteries is
independent of wealth for small changes in wealth (within 1000 kroner).
e) Show that Ola will change the ranking of C and D when they are played
twice, while Kari will maintain her ranking. (Hint: Show that if Kari prefer D
to C, she will prefer the lottery sequence (C,D) to (C,C).)
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Problem 2
This problem was given in the …rst lecture. Each one of you should ask 4
students the question below. I want you to ask two sophisticated students
(with some formal training in mathematical statistics) and two
non-sophisticated students (with no courses in mathematical statistics). You
may collaborate and attend lectures for …rst year students and students at
intermediate/advanced courses in statistics at the math department.
Send me the results prior to the …rst seminar.
The question:
"A dice has four Green (G) faces and two Red (R) faces. The dice will be
rolled 20 times, and the result (R or G) will be written down. This will
produce a sequence of 20 letters.
You can choose one of the three short sequences below:
RGRRR
GRGRRR
GRRRRR,
Suppose that if your chosen sequence appears in the sequence of 20 letters, you
would win 500 kroner. Which one of the sequences 1.-3. would you prefer? "
Problem 3
Suppose a person chooses lottery (B) over lottery (A) where
Lottery A :
Lottery B :
(4000; 0:3; 3500; 0:65; 0; 0:05)
(3500; 1)
Suppose further that the same person chooses lottery (C) over lottery (D),
where
Lottery C :
Lottery D :
(4000; 0:30; 0; 0:70)
(3500; 0:35; 0; 0:65)
a) Does this person violate expected utility?
b) Does this person violate expected utility with risk aversion?
Explain your answers
Problem 4
In this problem we use Közegi and Rabin’s variant of prospect theory, to
derive predictions for a recent experiment (Abeler, Falk, Goette and Hu¤man
(2011): Reference Points and E¤ort Provision, American Economic Review,
April 2011, Vol 101, pp. 470-492)
A person count zeros in tables, paid at a rate w per correct table. The number
of correct tables n(e) depend on the e¤ort e provided (we assume n0 > 0 and
n00 < 0). The e¤ort cost c(e) is unobservable (and we assume c0 > 0 and
2
c00 > 0). The person gets paid either a …xed amount, F = 10, or the amount
earned by counting tables. Thus the payment is
Payment,
=
2
1 =F
= wn(e)
with probability 50% (p1 )
with probability 50% (p2 )
The persons utility is
U ( jr) = m( ) + n( jr) where we assume
m( ) =
(
r)
if
r
n( jr) =
with > 1
(
r) if
r
r is the reference point which equals expected payo¤. The person has rational
expectations, thus
Reference point, r =
r1 = F
r2 = wn(e)
with probability 50% (q1 )
with probability 50% (q2 )
a) Consider …rst as a reference a standard model with n( jr) = 0:Show that
expected utility is
1
(F + wn(e))
2
and that the person will provide e¤ort until the point
1 0
wn (e) = c0 (e)
2
Now we return to the model above and consider the expected utility with
reference point
2
2 X
X
U ( i jrj )pi qj
EU ( jr) =
j=1 i=1
b) Show that if wn(e) < F , the expected utility is
1
1
(F + wn(e)) +
(wn(e) F )
2
4
c) What is the marginal bene…t of e¤ort?
c) Show that if wn(e) > F the marginal bene…t of e¤ort is
=
1
2
1
4
wn0 (e)
d) Since cost is not observable we cannot predict the e¤ort or the number of
sorted tables n(e). When the experiment is conducted we observe n(e) for all
the subjects in the experiment. The parameters F and w are determined by
the researcher running the experiment, and can be varied in di¤erent
treatment. Can the results from the experiment be used to test the two
models? (the traditional one in a) and the one considered in b)-c)? ) What
kind of observation is inconsistent with the traditional model and what kind of
observation is inconsistent with the last one?
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