Economics 500, Fall 2021
Problem Set 6, Due Tuesday, October 14
Question 1 We start with some issues in risk attitudes.
1.1 Consider a Qsimple form of the Cobb-Douglas utility function, given
by U (x) = L α
1 xℓ . The simplification here is that the exponent α is
common across good. Derive the indirect utility function. Identify
for what values of α the function is concave in w, and for which it is
convex in w. Us this to characterize when this person is risk averse,
neutral or seeking in income.
1.2 Give the definition of risk aversion and risk seeking in income (or
wealth) in terms of the comparison of the expected utility of a lottery
and the utility of its expected value. Offer a similar definition in terms
of risk aversion over prices. Use your indirect utility function from [1.1]
to identify when this person is risk averse, neutral or seeking in prices.
1.3 Consider the expenditure minimization problem for a fixed utility level
u. Give the definition of risk aversion and risk seeking in prices in
terms of the comparison of the expected expenditure of a lottery and
the expenditure of its expected value. Establish whether or when the
expenditure minimizer will be risk averse neutral or seeking in prices.
1.4 Consider an expected utility maximizer. As is customary, assume there
is a finite set of alternatives X and a Bernoulli utility function u : X →
R. Now introduce a finite state space Θ, where the probability of state
θ is p(θ), and associate with each state a lottery over X. Now think
about how the expected utility maximizer evaluates these lotteries.
Give the definition of risk aversion and risk seeking in probabilities
in terms of the comparison of the expected utility of a lottery over
probabilities and the expected utility of the expected probabilities.
Establish whether or when the expected utility maximizer will be risk
averse neutral or seeking in probabilities. If it helps, it is fine to keep
things simple by assuming there are only two states.
1.5 Let Θ be a finite set of states and X a set of alternatives. The function
u : Θ × X → R is Bernoulli a utility function with u(θ, x) giving the
utility of alternative x in state θ. The probability of state θ is p(θ).
Explain how the Bernoulli utility functions here and in [1.4] differ.
Give an example of a case in which each is applicable.
1.6 Continuing with the setting in [1.5], suppose that an alternative must
be chosen before the state is known, and write the attendant utility
maximization problem. Then suppose that an alternative x can be
chosen after the state is known, and write the utility maximization
problem. Show that the expected utility in the latter case is always
larger than in the former. Construct an example in which the differ-
ence in expected utilities is strict. Explain how these results can be
interpreted as showing that “more information is always better.”
1.7 Repeat your answer to [1.4] for the utility function introduced in [1.5]
and the two cases considered in [1.6]. Explain when the person is risk
averse, neutral and seeking.
1.8 Identify the principle that explains your answer to the preceding ques-
tions.
Question 2. Now we focus on risk aversion.
2.1 We defined the Arrow-Pratt measure of risk aversion. This is often
referred to as a measure of absolute risk aversion. Write the cor-
responding measure of relative risk aversion. Explain how these two
measures are related—if one is constant in income or wealth, how does
the other behave?
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2.2 Consider the Bernoulli utility function u(x) = 1−θ x1−θ . What are rea-
sonable values of θ, and for what values of θ does this utility function
exhibit risk averse, risk neutral, and risk-seeking preferences? Calcu-
late the coefficient of relative risk aversion and explain how it depends
on θ, and indicate why this is called a constant-relative-risk-aversion
utility function. Explain how you can work with the case of θ = 1.
−θx
2.3 Consider the Bernoulli utility function u(x) = 1−eθ . What are rea-
sonable values of θ, and for what values of θ does this utility function
exhibit risk averse, risk neutral, and risk-seeking preferences? Calcu-
late the coefficient of absolute risk aversion and explain how it depends
on θ, and indicate why this is called a constant-absolute-risk-aversion
utility function. Explain how you can work with the case of θ = 0.
2.4 It seems intuitive that for a risk aversion person, it is good for a lottery
over income or wealth to have a higher mean and a low variance.
Let’s investigate a setting in which we can make this precise. The
Bernoulli utility function is given by u(x) = −e−θx . Relate this utility
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function to those considered in [2.2] and [2.3] and interpret θ. Now
consider a lottery over x given by a normal distribution with mean µ
and variance σ 2 . Then show that the expected utility of this lottery
can be written as a linear function of the mean µ and variance σ 2 , and
explain how this mean-variance representation captures differences in
risk aversion. (To do this, first write the expression for expected utility.
This will give you an integral over an expression whose form is e raised
to some rather unwieldy looking expression. Separate this expression
into those terms that do not contain x, and hence can be taken our of
the integral, and those that contain x and hence must remain. Then
argue the remaining integral can be shown to equal one, and then
simplify what’s left.)
2.5 Let Θ be a finite set of states and p a prior distribution over θ. Consider
a person whose Bernoulli utility function u : R → R exhibits risk
aversion. This person’s current allocation ix given by x(θ). Suppose
this person can purchase, at zero cost, any lottery z(θ) that has zero
expected value. Characterize the utility-maximizing lottery. In light
of your result, assess the following statement: “Risk aversion people
in competitive markets will fully insure.”
Question 3 Here, we look at some of the challenges to expected utility
theory.
3.1 In response to the Allais paradox, it is often suggested that axioms A1-
A3 should be weakened. The typical target is the independence axiom.
One suggested axiom is the betweenness axiom: If p and q are lotteries
with p ∼ q and α ∈ [0, 1], then αp + (1 − α)q ∼ p. Give your best
argument for why you expect betweenness to hold. Present an example
in which you think independence is problematic but betweenness is
reasonable. Show that independence implies betweenness, but that
the converse can fail. Show that the choices associated with the Allais
paradox are consistent with betweenness.
3.2 Suppose an urn contains 100 balls, each of which is either red or green,
but you have no information as to the proportion of red or green balls.
One ball is to be drawn from the urn. You must choose one of the
following three lotteries: (a) receive a payoff of 1 (add some zeros to
make it interesting if you would like) with probability .49, no matter
what color ball is drawn; (b) you receive a payoff of 1 if a red ball is
drawn; (c) you receive a payoff of 1 if a green ball is drawn. Many
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people are indifference between lotteries b and c, but prefer lottery a
to both b and c. Now suppose you encounter the spirit of Howard
Raiffa, who argues, “It is irrational to prefer a to b or c. If you could
announce you choice of b or c until after a ball was drawn but before its
color was revealed, you could flip a coin between b and c and ensure
a payoff of 1 with probability half, no matter what the color of the
ball, giving an outcome that first-order stochastically dominates a and
hence which you would surely prefer to a. But then it cannot matter
whether you flip the coin before or after the ball is drawn, and so
flipping a coin and then choosing b or c as appropriate must be better
than a.” Explain why, since there are comparing lotteries with only
two outcomes, risk attitudes are irrelevant here. Then explain why
you do or do not find this reasoning convincing.
