Chapter Tow
2. Choice Under Uncertainty
(Uncertainty and Risk)
1
Introduction
the behavior of a consumer’s choice is either under
conditions of certainty or uncertainty.
In this chapter we consider choice under uncertainty
– how situations where the utility from a chosen
action is to some extent unpredictable.
– why individuals usually dislike risks that
uncertainty poses and are willing to pay
something to reduce them.
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2.1. Probability and Lottery
• Probability: the chance of occurrence of sth.
• The first task is describing the set of choices
• The probability of a repetitive event happening
is the relative frequency with which it will occur
– E.g. probability of obtaining a head on the fair-flip of
a coin is 0.5
• If a lottery offers n distinct prizes and the
probabilities of winning the prizes are i (i=1,n)
n
then

i 1
i
1
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 We imagine that the choices facing the consumer take
the form of lotteries….
 used to represent risky alternatives.
 A lottery is denoted by px+(1-p)y.
 This notation means: "the consumer receives prize x with
probability p and prize y with probability (1 - p).“
The prizes may be money, bundles of goods, or even
further lotteries.
 Most situations involving behavior under risk can be
put into this lottery framework.
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Assumptions
Assumptions about the consumer's perception of the
lotteries open to him are:
 Certainty:- Getting a prize with probability one is the same
as getting the prize for certain.
 Independence of Order:- The consumer doesn't care
about the order in which the lottery is described.
 Compounding:- A consumer's perception of a lottery
depends only on the net probabilities of receiving the various
prizes.
 preferences are complete, reflexive, and transitive.
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Expected Value
• For a lottery (X) with prizes x1,x2,…,xn and the
probabilities of winning 1,2,…n, the
expected value of the lottery is
E ( X )  1x1  2 x2  ...  n xn
E( X ) 
n
 x
i 1
i
i
• The expected value is a weighted sum of
the outcomes
– the weights are the respective probabilities
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• Suppose that Abebe and Bekele decide to flip
a coin
– heads (x1)  Abebe will pay Smith 1 birr
– tails (x2)  Bekele will pay Jones 1 birr
• From Abebe’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
are called actuarially fair games
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Common observation is that people often refuse to
participate in actuarially fair games
There may be a few exceptions
– when very small amounts of money are at stake
– when there is utility derived from the actual
play of the game
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2.3. Expected Utility
• Individuals do not care directly about the dollar
values of the prizes
– they care about the utility that the dollars provide
• The expected utility property says that the utility of a
lottery is the expectation of the utility from its prizes
and such an expected utility function is called von
Neumann Morgenstern utility function.
• Assuming diminishing marginal utility of wealth
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• Expected utility can be calculated in the same
manner as expected value
n
E ( X )  1u1   2u2  ...   nun    iU ( 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
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Expected Utility Maximization
• A rational individual will choose among
gambles based on their expected utilities
• Consider two gambles:
– first gamble offers x2 with probability q and x3
with probability (1-q)
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)
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• Substituting the utility index numbers gives
expected utility (1) = q · 2 + (1-q) · 3
expected utility (2) = t · 5 + (1-t) · 6
• The individual will prefer gamble 1 to
gamble 2 if and only if
q · 2 + (1-q) · 3 > t · 5 + (1-t) · 6
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3.3. Different preference for risk
– Risk averse:- prefer outcomes with low uncertainty to
outcomes with high uncertainty. Has diminishing marginal
Utility of Wealth, not make the bet
– Risk neutral:- constant MU of wealth, Individual is
indifferent between making the bet and not
– Risk lover:- …increasing MU of wealth, choose to make
the fair bet
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2.4. Risk Aversion
• Risk refers to the variability of the outcomes
of some uncertain activity
• Two lotteries may have the same expected
value but differ in their riskiness
• When faced with two gambles with the same
expected value, individuals will usually
choose the one with lower risk
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In general, we assume that the marginal
utility of wealth falls as wealth gets larger
E.g. flip a coin for 1 birr Vs 1,000 birr
– a flip of a coin for 1,000 birr promises a small
gain in utility if you win, but a large loss in utility
if you lose
– a flip of a coin for 1 birr is inconsequential as the
gain in utility from a win is not much different as
the drop in utility from a loss
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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)
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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)
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• Suppose that the person is offered two fair
gambles:
– a 50-50 chance of winning or losing $h
Uh(W*) = ½ U(W* + h) + ½ U(W* - h)
– a 50-50 chance of winning or losing $2h
U2h(W*) = ½ U(W* + 2h) + ½ U(W* - 2h)
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Utility (U)
The expected value of gamble 1 is Uh(W*)
U(W)
U(W*)
Uh(W*)
W* - h
W*
W* + h
Wealth (W)
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Utility (U)
The expected value of gamble 2 is U2h(W*)
U(W)
U(W*)
U2h(W*)
W* - 2h
W*
W* + 2h
Wealth (W)
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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)
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• The person will prefer current wealth to
wealth combined with a fair gamble
• The person will also prefer a small gamble
over a large one
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Risk Aversion and Insurance
• The person might be willing to pay some
amount to avoid participating in a gamble
• This helps to explain why some individuals
purchase insurance
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Utility (U)
W ” provides the same utility as
participating in gamble 1
U(W)
U(W*)
Uh(W*)
The individual will be
willing to pay up to
W* - W ” to avoid
participating in the
gamble
W* - h W ” W*
W* + h
Wealth (W)
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• An individual who always refuses fair bets is
said to be risk averse
– will exhibit diminishing marginal utility of
income
– will be willing to pay to avoid taking fair bets
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Willingness to Pay for Insurance
• Consider a person with a current wealth of
$100,000 who faces a 25% chance of losing
his automobile worth $20,000
• Suppose also that the person’s von
Neumann-Morgenstern utility index is
U(W) = ln (W)
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• The person’s expected utility will be
E(U) = 0.75U(100,000) + 0.25U(80,000)
E(U) = 0.75 ln(100,000) + 0.25 ln(80,000)
E(U) = 11.45714
• In this situation, a fair insurance premium
would be $5,000 (25% of $20,000)
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• The individual will likely be willing to pay more
than $5,000 to avoid the gamble. How much
will he pay?
E(U) = U(100,000 - x) = ln(100,000 - x) = 11.45714
100,000 - x = e11.45714
x = 5,426
• The maximum premium is $5,426
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Example:- A farmer owns a barn that has value of 50,000.
The probability of no fire is 0.99 and the probability of fire
where he will loss all the value of the barn is 0.01.
• The utility function of the farmers is given by:
u  w
1
2
a) What is the expected utility of the barn?
Eu  barn   p1  a1   1  p1  u  a2 
Eu  barn   0.99 50,000  0.01 0  221.37
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b) What is the certainty equivalent (CE) of The barn. CE is the
value of the sure thing (no risk) that has utility of 221.37.
1
2
221.37  w  w  49, 005
• The farmer should be indifferent between the barn (no
insurance) and 49,005.
c) What is the expected monetary value (EMV) of the barn?
E(MV )  0.99 50,000   0.1 0   49,500
d) Risk premium = Expected monetary value of the barn – CE
=
49, 500 - 49,005 = 495
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• This means to hold the barn uninsured rather than the sure
thing of equal utility, the Expected Value of the barn has to be
495 more than the sure thing.
e) The farmers has 0.01 chance of losing his barn, his expected
loss will be:
E (loss)=50,000(0.01)=500
• In the event of fire, the insurance company will pay the
farmer 50,000.
• Thus, ignoring administrative costs and profit, the insurance
company will charge a premium of at least 500.
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f) Will the farmer be willing to pay 500?
• If he does, he will have the barn insured less 500.
• So, we need to get the utility of 50,000-500=49,500.
1
2
u   49,500   222.5
• This utility is greater than the 221.3 of the uninsured barn, so
he will insure.
• The farmer would be willing to pay up to 995 (expected loss
plus risk premium).
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• Because;
1
2
u   50, 000  995  221.37
• The insurance company needs at least 500, but can charge up
to 995. However, competition would bring down the charge.
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2.5. Preference and Wealth
• Will the investor change her/his behavior as
wealth changes?
• Will more wealth imply greater investment in
risky assets? Will the individual invest less in
risky assets?
• To deal with these, we introduce the concepts
of absolute and relative risk aversion.
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Absolute Risk Aversion
• A measure of investor reaction to uncertainty
relating to dollar changes in their wealth. The
second derivative is negative so we take the
negative to make A(W) positive.
U '' ( w)
A(W )   '
U ( w)
• Now that we have A(W), we can take the first
derivative with respect to wealth. A’(W) tells
us how absolute risk aversion behaves with
respect to changes in wealth.
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Relative Risk Aversion
A measure of investor reaction to uncertainty
relating to percentage changes in their wealth.
''
WU ( w)
R(W )  
'
W ( w)
• Similar to absolute risk aversion, we can take
the first derivative of R(W) with respect to
wealth. R’(W) tells us how relative risk
aversion behaves with respect to changes in
wealth.
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Relative Risk Aversion:
RRA
Increasing
Constant
Decreasing
Def’n
% invested in risky
assets declines as
wealth increases
Property
R’(W) > 0
% invested remains
unchanged
R’(W) = 0
% invested
increases
R’(W) < 0
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Example 1:
• What are the characteristics of the Quadratic utility
function given below with respect to absolute and
relative risk aversion?
2
U (W )  w  bw
• Ans.:
u '' ( w)
2b
A(W )   '

u ( w) (1  2bw)
wu '' ( w)
w2b
R(W )   '

u ( w)
(1  2bw)
4b 2
A(W ) 
0
2
(1  2bw)
2b
R(W ) 
0
2
(1  2bw)
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