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 MarschakMachina 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 xy

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 1q10 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 welfaremaximizing

tax? What is the resulting expected social welfare

and the social welfare resulting if no catastrophic

climate change is realized?

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