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Economics
Date
Feb 20, 2024
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Problem Set 2
Economics of Information and Uncertainty
ECON 447, Winter 2024
The solution is
due on February 13
. Please upload your solution as a single
pdf-file to the corresponding assignment on mycourses before the deadline.
Late submissions will not be accepted. You can work in groups, but everyone
has to hand in her/his own solution. Copied solutions and ”failed to submit”
will receive 0% for this particular problem set. I will post a sample solution
shortly after the submission due date. The TA will use the TA office hours
on February 15 for a conference to discuss the solution.
1. (taken from Edoardo Gallo, Cambridge, Winter 2018)
Suppose an individual has to pay
2 to enter a competition. The prize
is
19 and the probability that he wins is 1/3. He is an expected utility
maximizer with vNM utility function
u
where
u
(
x
) is the utility value
that the individual attaches to having wealth level
x
. His current wealth
is
10. Calculate (i) the certainty equivalent and (ii) the risk premium
for each of the following cases: (a)
u
(
x
) =
log
(
x
), (b)
u
(
x
) =
x
2
, and
(c)
u
(
x
) =
x
. Will he enter the competition if
u
(
x
) =
log
(
x
)?
2. A risk-lover.
Suppose that there are two states of the world and that each state
realizes with probability 1/2.
A decision maker receives $
x
1
≥
0 in
state 1 and $
x
2
≥
0 in state 2.
The decision makers is an expected
utility maximizer and he has vNM utility function
v
(
x
) =
x
2
.
(a) In the 2-states of the world diagram, draw the indifference curve
of the decision maker through the lottery
P
= (
x
1
, x
2
) = (8
,
4).
(b) Determine
CE
(
P
) and
R
(
P
) and show them in the diagram.
1
(c) If the decision maker can choose exactly one lottery of all lotteries
(
x
1
, x
2
) that have expected outcome
6 (i.e. 1
/
2
x
1
+ 1
/
2
x
2
= $6),
which one does he choose? You can explain your answer referring
to the diagram.
3. For each of the following vNM utility functions, determine the Arrow
Pratt measure of absolute risk aversion, and whether the risk aversion
is increasing, decreasing or constant in
x
.
(a) Quadratic vNM utility function:
v
(
x
) =
x
−
αx
2
,
α >
0
(b) Log utility function:
v
(
x
) = ln(
x
)
(c) Power utility function:
v
(
x
) =
x
1
−
σ
1
−
σ
for
σ >
0
(d) Exponential utility function:
v
(
x
) =
−
e
−
α
·
x
for
α >
0
4. Degrees of risk aversion.
(a) Consider two decision makers who both have expected utility pref-
erences over the lotteries over the non-negative real line. Decision
maker 1 has vNM utility function
v
1
(
x
) =
x
1
/
4
+ 5 and decision
maker 2 has
v
2
(
x
) =
x
1
/
2
. Is one of the decision makers more risk-
averse than the other? How does your response change if instead
v
1
(
x
) =
x
1
/
4
+ 5
x
?
(b) (van Zandt, Exercise 5.5) A risk-averse expected utility decision
maker has decreasing absolute risk aversion. Her certainty equiv-
alent for a lottery that pays
0 and
800 with probabilities 1
/
3
and 2
/
3, respectively, is
500. Which does she prefer, to get
400
and
1200 with probabilities 1
/
3 and 2
/
3, respectively, or to get
900 for sure? Explain.
5. First order stochastic dominance.
2
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