Economics 202:
Intermediate Microeconomic Theory
1. HW #5 on website. Due Tuesday.
Expected Value
• Suppose that Smith and Jones decide to flip a coin
– heads (x1)  Jones will pay Smith $1
– tails (x2)  Smith will pay Jones $1
• From Smith’s point of view,
E( X )  1x1  2 x2
1
1
E ( X )  ($1)  ( $1)  0
2
2
• Games which have an expected value of zero (or cost their
expected values) are called actuarially fair games
– a common observation is that people often refuse to participate in
actuarially fair games
– except if the stakes are small or they gain utility from playing the
game (like state lottery or slot machines). We focus on risk aspect
not consumption aspect of gambling.
St. Petersburg Paradox
• A coin is flipped until a
head appears
• If a head appears on the
nth flip, the player is paid
$2n
x1 = $2, x2 = $4, x3 =


 1
E ( X )   i x i   2  
2
i 1
i 1
i
E( X )  ?
$8,…,xn = $2n
• The probability of getting
of getting a head on the ith
trial is (½)i
1=½, 2= ¼,…, n= 1/2n
• Would you play?
i
Expected Utility
• Individuals do not care directly about the dollar values of the
prizes
– they care about the utility that the dollars provide
• If we assume diminishing marginal utility of wealth, the St.
Petersburg game may converge to a finite expected utility value
– this would measure how much the game is worth to the individual
Expected Utility
• Expected utility can be calculated in the same manner as
expected value
n
E ( X )   i U ( xi )
i 1
• Because utility may rise less rapidly than the dollar value of
the prizes, it is possible that expected utility will be less than
the monetary expected value
Expected Utility Maximization
• A rational individual will choose among gambles based on
their expected utilities (the expected values of the von
Neumann-Morgenstern utility index)
• Consider two gambles:
– first gamble offers x2 with probability q and x3 with probability (1q)
expected utility (1) = q · U(x2) + (1-q) · U(x3)
– second gamble offers x5 with probability t and x6 with probability
(1-t)
expected utility (2) = t · U(x5) + (1-t) · U(x6)
• The individual will prefer gamble 1 to gamble 2 if and only
if
q · 2 + (1-q) · 3 > t · 5 + (1-t) · 6
Risk Aversion
• Two lotteries may have the same expected value but differ in
their riskiness
– flip a coin for $1 versus $1,000
• Risk refers to the variability of the outcomes of some
uncertain activity
• When faced with two gambles with the same expected value,
individuals will usually choose the one with lower risk
Risk Aversion
Utility (U)
U(W) is a von Neumann-Morgenstern
utility index that reflects how the individual
feels about each value of wealth
U(W)
The curve is concave to reflect the
assumption that marginal utility
diminishes as wealth increases
Wealth (W)
Risk Aversion
Utility (U)
Suppose that W* is the individual’s current
level of income
U(W)
U(W*)
U(W*) is the individual’s
current level of utility
W*
Wealth (W)
Risk Aversion
• The person will prefer current wealth to that wealth via a
fair gamble
• The person will also prefer a small gamble over a large one
Risk Aversion
U(W*) > Uh(W*) > U2h(W*)
Utility (U)
U(W)
U(W*)
Uh(W*)
U2h(W*)
W* - 2h
W* - h
W*
W* + h
W* + 2h
Wealth (W)
Utility Functions & Attitudes toward Risk
• Total Utility curve plots the utility of
different return levels
• The return is unknown beforehand
and  uncertain
• Prob[Return = 2%] = 0.5
Prob[Return = 8%] = 0.5
Expected return = 2(.5) + 8(.5) = 5%
300
Total Utility
Z
U[E(r)]=225
U
E[U(r)]=200
• Expected Utility is the average utility
you can expect to get from different
possible returns
E[U(r)] = 100(.5) + 300(.5) = 200
• Utility from a certain return = U[E(r)]
• Risk-aversion is when a person prefers
a certain return to an uncertain return
giving the same expected return:
U[E(r)] > E[U(r)]
100
A
Return
2% E(r)=5%
8%
Utility Functions & Attitudes toward Risk
• Risk-aversion  TU has positive,
diminishing slope  Diminishing
Marginal Utility
• Risk-aversion is common
– Fire-Insurance
300
Total Utility
Z
U[E(r)]=225
U
E[U(r)]=200
– U[E(r)] = utility of $1,000 premium
– E[U(r)] = average utility of
(-$100K*.01) + ($0*.99) = -$1,000
• Depends on the probabilities and
amount of loss
100
A
Return
2% E(r)=5%
8%
Utility Functions & Attitudes toward Risk
• Risk-Neutrality  TU has positive,
constant slope  constant marginal
utility
• Investor is indifferent between the
certain return and the risk of
2% or 8% return
U[E(r)] = E[U(r)]
• Risk-loving  TU has positive,
increasing slope  increasing
marginal utility
• Investor prefers the risk of 2% or
8% return
• U[E(r)] < E[U(r)]
• Example could be 10 lottery tickets
• Utility from $0 for sure < -50%
return
• Decreases as stakes increase: Vegas!
300
U
TU
Z
U[E(r)]=
E[U(r)]
A
2% E(r)=5%
8%
Return
U
TU
Z
E[U(r)] >
U[E(r)]
A
2% E(r)=5% 8%
Return
Methods to Minimize Risk
(1) Insurance (fire, health, car, etc.)
•
•
•
•
•
•
•
What is the max price a risk-averse
person will pay?
For an expected loss of $1,000, they will
pay > $1,000 Why?
They value $98,300 for sure the same as
$99,000 expected income  $1,700 max
Green line is expected utility line
Will the insurance co. provide a policy?
Pr(fire) = .01 so 1 out of 100 will burn
Minimum ins. co. will accept = $1,000
per homeowner since 1 will burn.
Room for mutually beneficial exchange
(2) Diversification (variety of risks)
• Stocks in different industries, start-ups
U
Total Utility
Z
U*
A
0
Income
98,300
100K
99K
Example: Risk Preference & Insurance
In Japan, golfers who get a “hole-in-one” are expected to give gifts to
relatives, fellow workers, and friends. These gifts can cost them the
equivalent of thousands of dollars. The cost is so great that there
actually exists a market for “hole-in-one” insurance. A Japanese golfer,
who is an expected utility maximizer, has the following utility function,
U(I) = I½ where I is monthly income. Our golfer, who has a monthly
income of $10,000, is quite accomplished and has a probability of
making a hole-in-one equal to 1 in 100. If our golfer does indeed ace a
hole, the typical gift expenditure is $3600.
(a) Is the golfer risk-averse, risk-neutral, or risk-loving? Support your
answer mathematically and explain.
(b) Calculate the expected income and expected utility (indicate your units).
Example: Risk Preference & Insurance
(c) Compute the maximum premium that would be paid to fully
insure against the expected costs of getting a hole-in-one.
(d) If the golfer’s monthly income were only $8,000, what would be
the maximum he or she is willing to pay? Explain the
relationship between income and the risk premium.
Segue
• In uncertain situations, economists typically assume
individuals are concerned with the expected utility
associated with various outcomes
– if they obey the von Neumann-Morgenstern axioms, they will
make choices in a way that maximizes expected utility
Kahneman-Tversky Experiments
•
•
Is expected utility a reasonable assumption?
Choice
Dollars
Probability
A
$1,000,000
1
B
5,000,000
.10
0
.01
1,000,000
.89
Kahneman-Tversky Experiments
•
Choice
C
D
Dollars
$1,000,000
0
5,000,000
0
Probability
.11
.89
.10
.90
Kahneman-Tversky Experiments
•
Choice
E
F
Dollars
$30
45
0
Probability
1
.80
.20
Kahneman-Tversky Experiments
•
•
Flip a coin and if 2 heads in a row you can choose below:
Choice
Dollars
Probability
G
$30
1
H
45
.80
0
.20
•
•
No coin flip
Choice
I
J
Dollars
$30
45
Probability
.25
.20
Kahneman-Tversky Experiments
•
•
•
People don’t simply care about the final rewards and their associated
probabilities.
They seem to care about how these are achieved which is inconsistent
with expected utility hypothesis.
Choose the correct answer:
Assume Johnny is a risk-averse expected utility-maximizer who
always turns down 50-50 win $11/lose $10 bet.
What is the largest value of Y for which Johnny will turn down a 50-50
lottery in which the outcomes are lose $100/win $Y?
(a) $110
(b) $221
(c) $2,000
(d) $20,242
(e) $1.1 million
(f) $2.5 billion
(g) He will turn it down no matter what Y is
(h) Need more information
Kahneman-Tversky Experiments
•
•
•
Answer =
Logic is that, under the expected utility
framework, turning down a moderatestakes gamble means that the MU of
money must diminish very rapidly.
“Expected utility is an ex-hypothesis!”
•
•
Rabin and Thaler (2001), “Anomalies: RiskAversion,” Journal of Economic Perspectives,
v. 15, no. 1, pp. 219-232.
Another viewpoint: Loss Aversion &
Mental Accounting