Question

Suppose the utility function of a decision maker for the amount of money x is given by U(x) = x2.

(a) This decision maker is considering the following two lotteries: A: With probability 1, he gains 3000. B: With probability 0.4, he gains TL 1000, and with probability 0.6, he gains TL 4000. Which of the two lotteries will the decision maker prefer? What is the certainty equivalent (CE) for lottery B? Based on the CE for B, is the decision maker risk-averse, risk-neutral or risk- seeking?

(b) This decision maker is now considering the following three lotteries: C: With probability 0.3, he gains TL 5000, with probability 0.4, he gains TL 4000, and with probability 0.3 he gains nothing. D: With probability 0.5, he gains TL 4000, and with probability 0.5, he gains TL 500. E: With probability 0.2, he gains TL 6000, with probability 0.5, he gains TL 3000, and with probability 0.3 he gains nothing. Which of the three lotteries will the decision maker prefer? What is the CE for lottery D? What is the CE for lottery E? Show that the lottery that has higher utility has higher CE.

Answer 1

ROBABILITY MODELS FOR ECONOMIC DECISIONS by Roger Myerson
excerpts from Chapter 3: Utility Theory with Constant Risk Tolerance
3.1. Taking account of risk aversion: utility analysis with probabilities
In the decision analysis literature, a decision-maker is called risk-neutral if he (or she) is
willing to base his decisions purely on the criterion of maximizing the expected value of his
monetary income. The criterion of maximizing expected monetary value is so simple to work
with that it is often used as a convenient guide to decision-making even by people who are not
perfectly risk neutral. But in many situations, people feel that comparing gambles only on the
basis of expected monetary values would take insufficient account of their aversion to risks.
For example, imagine that you had a lottery ticket that would pay you either $20,000
or $0, each with probability 1/2. If you are risk neutral, then you should be unwilling to sell this
ticket for any amount of money less than its expected value, which is $10,000. But many risk
averse people might be very glad to exchange this risky lottery for a certain payment of $9000.
Given any such lottery or gamble that promises to pay you an amount of money that will
be drawn randomly from some probability distribution, a decision-maker's certainty equivalent
(abbreviated CE) of this gamble is the lowest amount of money-for-certain that the decision-
maker would be willing to accept instead of a gamble. That is, saying that $7000 is your
certainty equivalent of the lottery that would pay you either $20,000 or $0, each with probability
1/2, means that you would be just indifferent between having a ticket to this lottery or having
$7000 cash in hand.
In these terms, a risk-neutral person is one whose certainty equivalent of any gamble is
just equal to its expected monetary value (abbreviated EMV). A person is risk averse if his or
her certainty equivalent of a gamble is less than the gamble's expected monetary value. The
difference between the expected monetary value of a gamble and a risk-averse decision-maker's
certainty equivalent of the gamble is called the decision-maker's risk premium (abbreviated RP)
for the gamble. Thus,
RP = EMV!CE.
So if a lottery paying $20,000 or $0, each with probability 1/2, is worth $7000 to you, then your
risk premium for this lottery is $10,000!$7000 = $3000.
When you have a choice among various gambles, you should choose the one for which
you have the highest certainty equivalent, because it is the one that is worth the most to you. Butwhen a gamble is complicated, you may find it difficult to assess your certainty equivalent for it.
The great appeal of the risk-neutrality assumption is that, by identifying your certainty equivalent
with the expected monetary value, it makes your certainty equivalent something that is
straightforward to compute or estimate by simulation. So what we need now is to find more
general formulas that risk-averse decision-makers can use to compute their certainty equivalents
for complex gambles and monetary risks.
A realistic way of calculating certainty equivalents must include some way of taking
account of a decision-maker's personal willingness to take risks. The full diversity of formulas
that a rational decision-maker might use to calculate certainty equivalents is described by a
branch of economics called utility theory. Utility theory generalizes the principle of expected
value maximization in a simple but very versatile way. Instead of assuming that people want to
maximize their expected monetary values, utility theory instead assumes that each individual has
a personal utility function that assigns a utility value to every possible monetary income level that
the individual might receive, such that the individual always wants to maximize the expected
value of his or her utility. For example, suppose that you have to choose among two gambles,
where the random variable X denotes the amount of money that you would get from the first
gamble and the random variable Y denotes the amount of money that you would get from the
second gamble. A risk-neutral decision-maker would prefer the first gamble if E(X) > E(Y). But
according to utility theory, when U(x) denotes your "utility" for getting any amount of money x,
you should prefer the first gamble if E(U(X)) > E(U(Y)). (Recall from Chapter 2 that, when X
has a discrete distribution, the expected value operator E(C) can be written
E(U(X)) = 3 P(X=x)*U(x), x
where the summation range includes every number x that is a possible value of the random
variable X.) Furthermore, your certainty equivalent CE of the gamble that will pay the random
monetary amount X should be the amount of money that gives you the same utility as the
expected utility of the gamble. Thus, we have the basic equation
U(CE) = E(U(X)).
Utility theory can account for risk aversion, but it also is consistent with risk neutrality or
even risk-seeking behavior, depending on the shape of the utility function. In 1947, von
Neumann and Morgenstern gave an ingenious argument to show that any consistent rational
decision maker should choose among risky gambles according to utility theory. Since thedecision analysts have developed techniques to assess individuals' utility functions. Such
assessment can be difficult, because people have difficulty thinking about decisions under
uncertainty, and because there are so many possible utility functions. But we may simplify the
process of assessing a decision-maker's personal utility function if we can assume that the utility
function is in some mathematically natural family of utility functions.
For practical decision analysis, the most convenient utility functions to use are those that
have a special property called constant risk tolerance. Constant risk tolerance means that, if we
change a gamble by adding a fixed additional amount of money to the decision-maker's payoff in
all possible outcomes of the gamble, then the certainty equivalent of the gamble should increase
by this same amount. This assumption of constant risk tolerance is very convenient in practical
decision analysis.
One nice consequence of constant risk tolerance is that it allows us to evaluate
independent gambles separately. If you have constant risk tolerance then, when you are going to
earn money from two independent gambles, your certainty equivalent for the sum of the two
independent gambles is just the sum of your certainty equivalent for each gamble by itself. That
is, the lowest price at which you would be willing to sell a gamble is not affected by having other
independent gambles. This independence property only holds if you have one of these constant-
risk-tolerance utility functions.
If a risk-averse decision-maker's preferences over gambles satisfy this assumption of
constant risk tolerance, then the decision-maker must have a utility function U in a simple one-
parameter family of functions that are defined by the mathematical formula:
U(x) = !EXP(!x'J),
where the parameter J is called the risk-tolerance constant. (Here EXP is a standard function in
Excel.)
Thus, if we can assume that a decision-maker has constant risk tolerance, then we only
need to measure this one risk-tolerance parameter J for the decision-maker. By asking the
decision-maker to subjectively assess his or her personal certainty equivalent for one simple
gamble, we can get enough information to compute the risk tolerance that accounts for this
personal assessment. Then, once we have found an appropriate risk tolerance for this decision-
maker, we will be able to use it to estimate the decision-maker's expected utility and certainty
equivalent for any gamble that we can simulate, no matter how complex.assessment can be difficult, because people have difficulty thinking about decisions under
uncertainty, and because there are so many possible utility functions. But we may simplify the
process of assessing a decision-maker's personal utility function if we can assume that the utility
function is in some mathematically natural family of utility functions.
For practical decision analysis, the most convenient utility functions to use are those that
have a special property called constant risk tolerance. Constant risk tolerance means that, if we
change a gamble by adding a fixed additional amount of money to the decision-maker's payoff in
all possible outcomes of the gamble, then the certainty equivalent of the gamble should increase
by this same amount. This assumption of constant risk tolerance is very convenient in practical
decision analysis.
One nice consequence of constant risk tolerance is that it allows us to evaluate
independent gambles separately. If you have constant risk tolerance then, when you are going to
earn money from two independent gambles, your certainty equivalent for the sum of the two
independent gambles is just the sum of your certainty equivalent for each gamble by itself. That
is, the lowest price at which you would be willing to sell a gamble is not affected by having other
independent gambles. This independence property only holds if you have one of these constant-
risk-tolerance utility functions.
If a risk-averse decision-maker's preferences over gambles satisfy this assumption of
constant risk tolerance, then the decision-maker must have a utility function U in a simple one-
parameter family of functions that are defined by the mathematical formula:
U(x) = !EXP(!x'J),
where the parameter J is called the risk-tolerance constant. (Here EXP is a standard function in
Excel.)
Thus, if we can assume that a decision-maker has constant risk tolerance, then we only
need to measure this one risk-tolerance parameter J for the decision-maker. By asking the
decision-maker to subjectively assess his or her personal certainty equivalent for one simple
gamble, we can get enough information to compute the risk tolerance that accounts for this
personal assessment. Then, once we have found an appropriate risk tolerance for this decision-
maker, we will be able to use it to estimate the decision-maker's expected utility and certainty
equivalent for any gamble that we can simulate, no matter how complex.1
2
3
4
5
6
7
8
9
10
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13
14
15
16
17
18
19
20
21
22
23
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25
26
27
28
29
30
AB C D E F G H
RiskTolerance
36.49 *
EU U(CE)
-0.9466642 -0.9466642
$X UTIL(X,RT) UINV(U,RT)
-20 -1.7299782 -20
-15 -1.5084485 -15
-10 -1.3152864 -10
-5 -1.1468593 -5
0 -1 0
5 -0.8719465 5
10 -0.7602907 10
15 -0.6629328 15
20 -0.578042 20
25 -0.5040217 25
30 -0.4394799 30
35 -0.383203 35
40 -0.3341325 40
45 -0.2913457 45 FORMULAS
50 -0.2540378 50 A2. =RISKTOL(20,-10,2)
55 -0.2215074 55 B4. =-EXP(-(20)/A2)*0.5+-EXP(-(-10)/A2)*0.5
60 -0.1931426 60 C4. =-EXP(-(2)/A2)
65 -0.16841 65 B7. =-EXP(-A7/$A$2)
70 -0.1468445 70 C7. =-$A$2*LN(-B7)
B7:C7 copied to B7:C25.
Chart plots (A7:A25,B7:B25).
*Uses add-in Simtools.xla, available at
http://home.uchicago.edu/~rmyerson/addins.htm
-2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-.8
-.6
-.4
-.2
.0
-20 -10 0 10 20 30 40 50 60 70
Utility, with RiskTol = 36.49
Monetary Income
4
Figure 3.1. A utility function with constant risk tolerance.
It is natural that this numerical measure of "risk tolerance" may seem mysterious at first.
The meaning of these risk-tolerance numbers will become clearer as you get practical experience
using them.
...To avoid the confusion of trying to interpret expected utility numbers, we should
always convert them back into monetary units by asking what sure amount of money would also
yield this same expected utility. This amount of money is then the certainty equivalent of the
lottery. Recall basic certainty-equivalent formula U(CE) = EU. With constant risk tolerance J,
the utility of the certainty equivalent becomes U(CE) = !EXP(!CE'J). So the certainty
equivalent satisfies !EXP(!CE'J) = EU. But the inverse of the EXP function is the natural
logarithm function LN(). So with constant risk tolerance J, the certainty equivalent of a gamble
can be computed from its expected utility by the formula
CE = !J*LN(!E(U(X))).