Chapter 17
Choice Making Under
Uncertainty
17/1
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
Expected Utility Hypothesis
 Expected
utility is calculated 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
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
Defining a Prospect
The remainder of the chapter will cover
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, $6000 and $1000).
 Note that the probabilities must sum to 1.

17/6
© 2009 Pearson Education Canada
Defining a Prospect
 Such
a prospect will be denoted as:
(q1, q2, q3: 10 000, 6000, 1000)
or simply:
(q1, q2, q3)
17/7
© 2009 Pearson Education Canada
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 assumption guarantees that persons are
willing to make tradeoffs between risk and
assured prospects.
Note - e* will vary across individuals.
17/8
© 2009 Pearson Education Canada
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 $6000 as
follows:
U(6000) =ae*+b(1-e*)
17/9
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
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
in such scenarios is to use subjective
probabilities.
17/11
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
The Expected Utility Function

1.
2.
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).
17/13
© 2009 Pearson Education Canada
Figure 17.1 Indifference curves in state space
17/14
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
Figure 17.2 Preferences towards risk
17/18
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
Figure 17.3 The expected monetary value line
17/20
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
Figure 17.4 Optimal risk bearing
17/22
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
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.

17/24
© 2009 Pearson Education Canada
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).
17/25
© 2009 Pearson Education Canada
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
© 2009 Pearson Education Canada
Risk Pooling

Abe’s expected utility with risk pooling:
(1-q)2U(wo-L)+2q(1-q)U(w0L/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)
17/27
© 2009 Pearson Education Canada
Risk Pooling
 When
individuals are risk averse, they
have clear incentives to create
institutions that allow them to share
(pool) their risks.
17/28
© 2009 Pearson Education Canada
Figure 17.5 Optimal risk pooling
17/29
© 2009 Pearson Education Canada
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).
17/30
© 2009 Pearson Education Canada
Figure 17.6 The demand for insurance
17/31
© 2009 Pearson Education Canada
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
17/32
© 2009 Pearson Education Canada
The Market for Insurance
 Ignoring
any administrative costs,
the expected costs are (1-q)L and
the firm will write a policy if
revenues (I) exceed costs.
17/33
© 2009 Pearson Education Canada
The Market for Insurance
As shown in Figure 17.6, there is a viable
insurance market because the reservation
supply price Is =(1-q)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.

17/34
© 2009 Pearson Education Canada