Problems for the …rst seminar
ECON4260 Behavioral Economics –Spring semester 2013.
Solutions to the problems will be presented at the seminars in week 5. 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
Each one of you should ask 4 students the questions below. You should have a
sample of two sophisticated students (with some formal training in
mathematical statistics) and two non-sophisticated students (with no courses
in mathematical statistics). It is a good idea to collaborate, and you could
collect all the data by attend lectures for …rst year students and students at
intermediate/advanced courses in statistics at the math department.
Send the results to Kristo¤er Midttømme kristo¤er.m@gmail.com, at the
latest on Friday, January 25th.
The questions to ask:
a) "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? "
b) Divide here the sample randomly in two, (one sophisticated and one
non-sophisticated in each group)
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Ask one group:
"Suppose you attend an experiment where you get 50 kroner in show-up fee.
Then for each line in the list below, the choice is between a lottery or a safe
outcome. The lottery is always:
L=
100 kroner probability 75%
0
probability 25%
The safe outcome get increasingly better:
Choice Lottery Safe outcome I chose safe from (starting with):
1
L
0 kroner
2
L
10 kroner
3
L
20 kroner
4
L
30 kroner
5
L
40 kroner
6
L
50 kroner
7
L
60 kroner
8
L
70 kroner
9
L
80 kroner
10
L
90 kroner
11
L
100 kroner
From what choice do you prefer the safe outcome.
Ask the other group:
"Suppose you attend an experiment where you get 100 kroner in show-up fee.
Then for each line in the list below, the choice is between a lottery or a safe
outcome. Note that if the outcome is negative the amount is taken from your
show-up fee. The lottery is always:
L=
50 kroner probability 75%
50 kroner probability 25%
Choice Lottery Safe outcome I chose safe from (starting with):
1
L
50 kroner
2
L
40 kroner
3
L
30 kroner
4
L
20 kroner
5
L
10 kroner
6
L
0 kroner
7
L
10 kroner
8
L
20 kroner
9
L
30 kroner
10
L
40 kroner
11
L
50 kroner
From what choice do you prefer the safe outcome.
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What results do you expect based on the work by Kahneman and Tversky
presented in this class?
Problem 3
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
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 X
2
X
EU ( jr) =
U ( i jrj )pi qj
j=1 i=1
b) Show that if wn(e) < F , the expected utility is
=
1
(F + wn(e)) +
2
4
1
4
(wn(e)
F)
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?
Problem 4 (Discussed at the seminar if there is time left.)
Suppose a person chooses lottery (B) over lottery (A) where
Lottery A :
Lottery B
:
4000 probability 30%
3500 probability 65%
0
probability 5%
3500 for sure
Suppose further that the same person chooses lottery (C) over lottery (D),
where
Lottery C
:
Lottery D
:
4000 probability 30%
0
probability 70%
3500 probability 35%
0
probability 65%
Consider a two stage lottery. In the …rst stage
First stage
Go to second stage
Y
with probability 35%
with probability 65%
and in the second stage the person either play:
Lottery 1: :
4000 probability 6/7
0
probability 1/7
or
Lottery 2: 3500 for sure
a) Show that if Y = 3500, then the two stage lottery yields Lottery A if
Lottery 1 is played in the second stage, and Lottery B if Lottery 2 is played in
the second stage.
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b) Show that we can derive Lottery C and D similarly by choosing a di¤erent
Y.
c) Does the stated behavior violate the independence axiom of expected
utility?
d) Does the person violate expected utility?
Explain your answer
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