1. Consider the utility function given by U x = ln x and
the set of gambles with the possible outcomes $10, $20
and $30. For each exercise, it may be useful to use a
spreadsheet or other numerical tools.
(a) Graph the indifference curves implied by expected
utility in the Marschak–Machina triangle. What is
the slope of the indifference curves?
(b) Now suppose that the decision maker maximizes
probability weighted utility, with theweights given by
π p =
i + 1 −pi
Graph a few examples of indifference curves. What
does the probability weighting do to the shape of
these curves relative to those in part a?
(c) Repeat the exercise in b, now assuming the decision
maker maximizes rank-dependent expected
utility. Thus, now the probability weighting function
is applied to cumulative probabilities.
(d) Finally, consider the regret utility function given by
U x, y =
x −y 2 if x≥ y
− x −y 2 if x < y
Plot the implied indifference curves supposing the
alternative choice would yield $19 with certainty and
compare the shape of these curves to those of the
other models considered. How do these curves
change when the foregone gamble is altered?
2. Suppose that there were three possible states of nature
as represented in the table below:
States of Nature 1 2 3
Probability 0.35 0.4 0.25
Gamble 1 $1,000 $2,000 $3,000
Gamble 2 $1,800 $1,800 $1,800
Gamble 3 $2,500 $1,500 $1,500
(a) What conditions would be required for the regret
theory utility function to predict preference cycling
when choosing between pairs of the three possible
gambles? Graph an example of a function that
would satisfy these conditions.
(b) Is it possible for preference cycling to occur given the
same choices under expected utility maximization
with probability weights? Show why or why not.
(c) Would it be possible for preference cycling to occur
given the same choices under rank-dependent expected-
utility maximization? Show why or why not.
3. Confirm that the common ratio effect as found in
Example 9.4 could be explained either by probability
weighting or by regret theory. To do this, find utility
and weighting functions that satisfy the models and
that lead to the choices found in Example 9.4.
4. Consider two gambles, each with outcomes $10, $20, $30,
and $40. Gamble 1 has probabilities p10, p20, p30,
1 −p10 −p20 −p30 for these outcomes and gamble 2
has probabilities q10, q20, q30, and 1−q10 − q20 − q30.
Suppose that Gamble 2 stochastically dominates
Gamble 1. Thus, p10 >q10, p10 +p20 > q10 + q20, and
p10 +p20 +p30 >q10 +q20 +q30. Show that anyone who
maximizes rank-dependent expected utility must prefer
Gamble 2 to Gamble 1.
5. Suppose that you are a policymaker considering
instituting a tax to combat climate change. The current
climate research is conflicting as to the probability that
carbon emissions will lead to catastrophic climate
change. Suppose that the probability of a catastrophic
climate change is equal to 0.3 × cϕ, where c represents
carbon dioxide emissions and, depending on which
scientists you listen to, ϕ can be as low as 0.1 or as
high as 0.9. As a policymaker you wish to maximize
expected social welfare. If a catastrophic climate
change occurs, social welfare will be equal to 0, no
matter how much other production occurs. If a
catastrophic climate does not occur, then social welfare
is given by the profit of the emitting industry,
π = p− t y −k y = 3 −t y− 0.15y2, where y is
output, p is the output price, k . is the cost of production,
and t is the tax you impose. The firm chooses
y to maximize profits, according to y= 3− t 0.3.
Carbon dioxide emissions are given by c =y.
(a) Suppose you display α-maxmin expected-utility
preferences, with α=1 (fully ambiguity averse).
What tax will you choose? What level of social
welfare will be realized if a catastrophic climate
change is not realized? You may use a spreadsheet
to determine the answer if needed.
(b) Suppose α=0 (fully ambiguity loving). What tax
will you choose? What level of social welfare will
be realized if a catastrophic climate change is not
(c) Suppose that a definitive study shows that ϕ=0.2.
What is the expected social welfare–maximizing
tax? What is the resulting expected social welfare
and the social welfare resulting if no catastrophic
climate change is realized?