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Chapter 17
Choice Making Under
Uncertainty
17.1
© 2005 Pearson Education Canada Inc.
Calculating Expected Monetary Value
 The
expected monetary value is simply the
weighted average of the payoffs (the
possible outcomes), where the weights are
the probabilities of occurrence assigned to
each outcome.
17.2
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Expected Value
 Given:
Two possible outcomes having
payoffs X1 and X2 and Probabilities of
each outcome given by Pr1 & Pr2.
 The
expected value (EV) can be
expressed as:
EV(X) = Pr1X1+ Pr2X2
17.3
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Expected Utility Hypothesis
 Expected
utility is calculated in the same way
as expected monetary value, except that the
utility associated with a payoff is substituted for
its monetary value.
 With
two outcomes for wealth ($200 and $0)
and with each outcome occurring ½ the time,
the expected utility can be written:
E(u) = (1/2)U($200) + (1/2)U($0)
17.4
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Expected Utility Hypothesis
If a person prefers the gamble previously
described, over an amount of money $M with
certainty then:
(1/2)U($200) + (1/2)U($0) > U(M)
17.5
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Defining A Prospect
 The
remainder of the chapter will be
talking about lotteries which will be
referred to as prospects which offer three
different outcomes.
 The term prospect will refer to any set of
probabilities (q1, q2, q3: and their assigned
outcomes ($10 000, $6 000 and $1 000).
 Note that the probabilities must sum to 1.
17.6
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Defining A Prospect
 Such
a prospect will be denoted as:
(q1, q2, q3: 10 000, 6 000, 1 000)
or simply:
(q1, q2, q3)
17.7
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Deriving Expected Utility Functions
Continuity assumption:
For any individual, there is a unique number e*,
(0<e*<1), such that he/she is indifferent between the
two prospects (0, 1, 0) and (e*, 0, 1-e*).
This assumptions guarantees that persons are
willing to make tradeoffs between risk and
assured prospects. Note that e* will vary across
individuals.
17.8
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von Neuman-Morgenstern
Utility Function
 Given
any two numbers a and b with a>b,
we could let U(10 000)=a and U(1 000)=b.
We would then have to assign a utility
number to $6 000 as follows:
U(6 000) =ae*+b(1-e*)
17.9
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von Neuman-Morgenstern
Utility Function

With the continuity assumption (and others) satisfied
and the utility function constructed as shown, these
important results are applicable:
1.
If an individual prefers one prospect to another, then
the preferred prospect will have a larger utility.
If an individual is indifferent between two prospects,
the two prospects must have the same expected
utility.
2.
17.10
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Subjective Probabilities
 The
expected utility theory is often applied
in risky situations in which the probability
of any outcome is not objectively known or
there exists incomplete information.
 The ability to apply expected-utility theory
is such scenarios is to use subjective
probabilities.
17.11
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The Expected Utility Function
 Assume
there are 2 states of wealth (w1
and w2) which could exist tomorrow and
they occur with probabilities (q and 1-q)
respectively.
 The expected utility function for tomorrow:
U(q,1-q:w1w2) = qU(w1)+(1-q)U(w2)
17.12
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The Expected Utility Function

1.
2.
17.13
Two key features of this utility functions:
The U functions are cardinal, meaning
that the utility values have specific
meaning in relation to one another.
This expected utility function is linear in
its probabilities (which simplifies MRS).
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Figure 17.1 Indifference curves in state space
17.14
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From Figure 17.1
 Figure
17.1 shows an indifference curve
for utility level u. Wealth in state 1(today)
and state 2 (tomorrow) are on each axis.
 q and (1-q) are fixed.
 The MRS (slope of u0) shows the rate at
which an individual trades wealth in state 1
for wealth in state 2, before either of these
states occur.
17.15
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From Figure 17.1
 The
slope of the indifference curve is
equal to the ratio of the probabilities times
the ratio of the marginal utilities.
 Each marginal utility however is function of
wealth in only one state since the utility
functions are the same in each state.
 Therefore the MRS equals the ratio of the
probabilities.
17.16
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From Figure 17.1
 Hence,
along the 45 degree line, where
wealth in the two states are equal, the
slope of u0 is q/(1-q).
 If q is large relative to (1-q) then u0 is
relatively steep and vice versa.
 In other words, if you believe state 1 is
very likely (q is high) then you will prefer
your wealth in state one rather than state
two.
17.17
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Figure 17.2 Preferences towards risk
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Optimal Risk Bearing
 Now
that different attitudes toward risk
have been defined, it is necessary to
illustrate how attitudes toward risk affect
choices over risky prospects.
 An expected value line shows prospects
with the same expected value. Note
however that along this line, the risk of
each prospect varies.
17.19
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Figure 17.3 The expected monetary value line
17.20
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From Figure 17.3
 At
point A there is no risk and that risk
increases as the prospects move away from
the 45 degree line.
 The slope of the expected value line equals
the ratios of the probabilities (relative prices)
 Utility will be maximized when the individual’s
MRS equals the ratios of the probabilities.
17.21
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Figure 17.4 Optimal risk bearing
17.22
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Optimal Risk Bearing
 The
optimal amount of risk that a person bears
in life depends on his/her aversion to risk.
 The choices of risk averse persons tend toward
the 45 degree line where wealth is the same no
matter what state arises.
 Risk inclined persons move away from the 45
degree line and are willing to take the chance
that they will be better off in one state
compared to the other.
17.23
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Pooling Risk
 Risk
Pooling is a form of insurance aimed
at reducing an individual’s exposure to risk
by spreading that risk over a larger
number of persons.
 Suppose the probability of either Abe or
Martha having a fire is 1-q, the loss from
such a fire is L dollars and wealth in period
t denoted as wt.
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Pooling Risk
 Abe’s
expected utility is:
u(q, L,w0) = qU(w0)+(1-q)U(w0-L).
 If Abe’s house burns his wealth is w0-L,
and his utility U(w0-L). If it does not burn,
his wealth is w0 and utility is U(w0).
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Pooling Risk
If Abe and Martha pool their risk (share any
loss from a fire), There are now three relevant
events:
1. One house burns.
Probability = 2q(1-q), Abe’s Loss=L/2
2. Both houses burn.
Probability = (1-q)2 , Abe’s Loss=L
3. Neither house burns.
Probability = q2 , Abe’s loss = 0

17.26
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Risk Pooling
 Abe’s
expected utility with risk pooling:
(1-q)2U(wo-L)+2q(1-q)U(w0-L/2)+q2U(w0)
 Rearranging and factoring Abe’s individual and
risk pooling utility function shows he is better off
if he is risk averse as:
U(w0-L/2)>(1/2)U(w0-L)+(1/2)U(w0)
 When individuals are risk averse, they have
clear incentives to create institutions that allow
them to share (pool) their risks.
17.27
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Figure 17.5 Optimal risk pooling
17.28
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The Market for Insurance
 What
is Abe’s reservation demand price
for insurance (the maximum he is willing to
pay rather than go without)?
 Set his expected utility without insurance
equal to the certainty equivalent (assured
prospect wce) in Figure 17.6.
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Figure 17.6 The demand for insurance
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The Market for Insurance
 On
the assumption that insurance
companies are risk neutral, what is the
lowest price they will offer full coverage?
 This is the reservation supply price,
denoted by Is in Figure 17.6
 Ignoring any administrative costs, the
expected costs are (1-q)L and the firm will
write a policy if revenues (I) exceed costs.
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The Market for Insurance
As shown in Figure 17.6, there is a viable insurance
market because the reservation supply price Is =(1q)L is less than the reservation demand price
(distance w0-wce).
 Abe trades his risky prospect for the assured prospect
and reaches indifference curve u*.
 If no resources are required to write and administer
insurance policies and if individuals are risk-averse,
there is a viable market for insurance.

17.32
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