Chapter 17 Choice Making Under Uncertainty

© 2005 Pearson Education Canada Inc.17.1

Chapter 17Chapter 17

Choice Making Under Choice Making Under UncertaintyUncertainty


Calculating Expected Monetary ValueCalculating Expected Monetary Value

The The expected monetary valueexpected monetary value is simply the is simply the weighted average of the payoffs (the weighted average of the payoffs (the possible outcomes), where the weights are possible outcomes), where the weights are the probabilities of occurrence assigned to the probabilities of occurrence assigned to each outcome.each outcome.


Expected ValueExpected Value

Given: Two possible outcomes having Given: Two possible outcomes having payoffs payoffs XX11 and and XX2 2 and and Probabilities of Probabilities of

each outcome given by each outcome given by PrPr1 1 & & PrPr22..

The expected value (EV) can be The expected value (EV) can be expressed as:expressed as:

EV(X) = PrEV(X) = Pr11XX11+ Pr+ Pr22XX22


Expected Utility HypothesisExpected Utility Hypothesis Expected utility is calculated in the same way Expected utility is calculated in the same way

as expected monetary value, except that the as expected monetary value, except that the utility associated with a payoff is substituted for utility associated with a payoff is substituted for its monetary value. its monetary value.

With two outcomes for wealth ($200 and $0) With two outcomes for wealth ($200 and $0) and with each outcome occurring ½ the time, and with each outcome occurring ½ the time, the expected utility can be written:the expected utility can be written:

E(E(uu) = (1/2)) = (1/2)UU($200) + (1/2)($200) + (1/2)UU($0)($0)


Expected Utility HypothesisExpected Utility Hypothesis

If a person prefers the gamble previously If a person prefers the gamble previously

described, over an amount of money $M with described, over an amount of money $M with

certainty then:certainty then:

(1/2)(1/2)UU($200) + (1/2)($200) + (1/2)UU($0) > U(M)($0) > U(M)


Defining A ProspectDefining A Prospect

The remainder of the chapter will be The remainder of the chapter will be talking about lotteries which will be talking about lotteries which will be referred to as referred to as prospectsprospects which offer three which offer three different outcomes.different outcomes.

The term prospect will refer to any set of The term prospect will refer to any set of probabilities (qprobabilities (q11, q, q22, q, q33: and their assigned : and their assigned

outcomes ($10 000, $6 000 and $1 000).outcomes ($10 000, $6 000 and $1 000). Note that the probabilities must sum to 1.Note that the probabilities must sum to 1.


Defining A ProspectDefining A Prospect

Such a Such a prospectprospect will be denoted as: will be denoted as:

(q(q11, q, q22, q, q33: 10 000, 6 000, 1 000) : 10 000, 6 000, 1 000)

or simply:or simply:

(q(q11, q, q22, q, q33))


Deriving Expected Utility FunctionsDeriving Expected Utility Functions

Continuity assumption:Continuity assumption:For any individual, there is a unique number e*, For any individual, there is a unique number e*, (0<e*<1), such that he/she is indifferent between the (0<e*<1), such that he/she is indifferent between the two prospects (0, 1, 0) and (e*, 0, 1-e*).two prospects (0, 1, 0) and (e*, 0, 1-e*).

This assumptions guarantees that persons are This assumptions guarantees that persons are willing to make tradeoffs between risk and willing to make tradeoffs between risk and assured prospects. Note that e* will vary across assured prospects. Note that e* will vary across individuals.individuals.


von Neuman-Morgenstern von Neuman-Morgenstern Utility FunctionUtility Function

Given any two numbers a and b with a>b, Given any two numbers a and b with a>b, we could let U(10 000)=a and U(1 000)=b. we could let U(10 000)=a and U(1 000)=b. We would then have to assign a utility We would then have to assign a utility number to $6 000 as follows: number to $6 000 as follows:

U(6 000) =ae*+b(1-e*)U(6 000) =ae*+b(1-e*)


von Neuman-Morgenstern von Neuman-Morgenstern Utility FunctionUtility Function

With the continuity assumption (and others) satisfied With the continuity assumption (and others) satisfied and the utility function constructed as shown, these and the utility function constructed as shown, these important results are applicable:important results are applicable:

1.1. If an individual prefers one prospect to another, then If an individual prefers one prospect to another, then the preferred prospect will have a larger utility.the preferred prospect will have a larger utility.

2.2. If an individual is indifferent between two prospects, If an individual is indifferent between two prospects, the two prospects must have the same expected the two prospects must have the same expected utility.utility.


Subjective ProbabilitiesSubjective Probabilities

The expected utility theory is often applied The expected utility theory is often applied in risky situations in which the probability in risky situations in which the probability of any outcome is not of any outcome is not objectively knownobjectively known or or there exists there exists incomplete information.incomplete information.

The ability to apply expected-utility theory The ability to apply expected-utility theory is such scenarios is to use is such scenarios is to use subjective subjective probabilities.probabilities.


The Expected Utility FunctionThe Expected Utility Function

Assume there are 2 states of wealth (wAssume there are 2 states of wealth (w11

and wand w22) which could exist tomorrow and ) which could exist tomorrow and

they occur with probabilities (q and 1-q) they occur with probabilities (q and 1-q) respectively.respectively.

The expected utility function for tomorrow:The expected utility function for tomorrow:

U(q,1-q:wU(q,1-q:w11ww22) = qU(w) = qU(w11)+(1-q)U(w)+(1-q)U(w22))


The Expected Utility FunctionThe Expected Utility Function

Two key features of this utility functions:Two key features of this utility functions:

1.1. The U functions are cardinal, meaning The U functions are cardinal, meaning that the utility values have specific that the utility values have specific meaning in relation to one another.meaning in relation to one another.

2.2. This expected utility function is linear in This expected utility function is linear in its probabilities (which simplifies MRS).its probabilities (which simplifies MRS).


Figure 17.1 Indifference curves in state spaceFigure 17.1 Indifference curves in state space


From Figure 17.1From Figure 17.1

Figure 17.1 shows an indifference curve Figure 17.1 shows an indifference curve for utility level u. Wealth in state 1(today) for utility level u. Wealth in state 1(today) and state 2 (tomorrow) are on each axis.and state 2 (tomorrow) are on each axis.

q and (1-q) are fixed.q and (1-q) are fixed. The MRS (slope of uThe MRS (slope of u00) shows the rate at ) shows the rate at

which an individual trades wealth in state 1 which an individual trades wealth in state 1 for wealth in state 2, before either of these for wealth in state 2, before either of these states occur.states occur.



The slope of the indifference curve is The slope of the indifference curve is equal to the ratio of the probabilities times equal to the ratio of the probabilities times the ratio of the marginal utilities.the ratio of the marginal utilities.

Each marginal utility however is function of Each marginal utility however is function of wealth in only one state since the utility wealth in only one state since the utility functions are the same in each state.functions are the same in each state.

Therefore the MRS equals the ratio of the Therefore the MRS equals the ratio of the probabilities.probabilities.



Hence, along the 45 degree line, where Hence, along the 45 degree line, where wealth in the two states are equal, the wealth in the two states are equal, the slope of uslope of u00 is q/(1-q). is q/(1-q).

If q is large relative to (1-q) then uIf q is large relative to (1-q) then u00 is is relatively steep and vice versa.relatively steep and vice versa.

In other words, if you believe state 1 is In other words, if you believe state 1 is very likely (q is high) then you will prefer very likely (q is high) then you will prefer your wealth in state one rather than state your wealth in state one rather than state two.two.


Figure 17.2 Preferences towards riskFigure 17.2 Preferences towards risk


Optimal Risk BearingOptimal Risk Bearing

Now that different attitudes toward risk Now that different attitudes toward risk have been defined, it is necessary to have been defined, it is necessary to illustrate how attitudes toward risk affect illustrate how attitudes toward risk affect choices over risky prospects.choices over risky prospects.

An An expected value lineexpected value line shows prospects shows prospects with the same expected value. Note with the same expected value. Note however that along this line, the risk of however that along this line, the risk of each prospect varies. each prospect varies.


Figure 17.3 The expected monetary value lineFigure 17.3 The expected monetary value line


From Figure 17.3 From Figure 17.3

At point A there is no risk and that risk At point A there is no risk and that risk increases as the prospects move away from increases as the prospects move away from the 45 degree line.the 45 degree line.

The slope of the expected value line equals The slope of the expected value line equals the ratios of the probabilities (relative prices)the ratios of the probabilities (relative prices)

Utility will be maximized when the individual’s Utility will be maximized when the individual’s MRS equals the ratios of the probabilities. MRS equals the ratios of the probabilities.


Figure 17.4 Optimal risk bearingFigure 17.4 Optimal risk bearing


Optimal Risk BearingOptimal Risk Bearing The optimal amount of risk that a person bears The optimal amount of risk that a person bears

in life depends on his/her aversion to risk.in life depends on his/her aversion to risk. The choices of risk averse persons tend toward The choices of risk averse persons tend toward

the 45 degree line where wealth is the same no the 45 degree line where wealth is the same no matter what state arises.matter what state arises.

Risk inclined persons move away from the 45 Risk inclined persons move away from the 45 degree line and are willing to take the chance degree line and are willing to take the chance that they will be better off in one state that they will be better off in one state compared to the other.compared to the other.


Pooling RiskPooling Risk

Risk Pooling is a form of insurance aimed Risk Pooling is a form of insurance aimed at reducing an individual’s exposure to risk at reducing an individual’s exposure to risk by spreading that risk over a larger by spreading that risk over a larger number of persons.number of persons.

Suppose the probability of either Abe or Suppose the probability of either Abe or Martha having a fire is 1-q, the loss from Martha having a fire is 1-q, the loss from such a fire is L dollars and wealth in period such a fire is L dollars and wealth in period t denoted as wt denoted as wtt. .



Abe’s expected utility is:Abe’s expected utility is:

u(q, L,wu(q, L,w00) = qU(w) = qU(w00)+(1-q)U(w)+(1-q)U(w00-L).-L).

If Abe’s house burns his wealth is wIf Abe’s house burns his wealth is w00-L, -L,

and his utility U(wand his utility U(w00-L). If it does not burn, -L). If it does not burn,

his wealth is whis wealth is w0 0 and utility is U(w and utility is U(w00). ).



If Abe and Martha pool their risk (share any If Abe and Martha pool their risk (share any loss from a fire), There are now three relevant loss from a fire), There are now three relevant events:events:

1.1. One house burns.One house burns.Probability = 2q(1-q), Abe’s Loss=L/2Probability = 2q(1-q), Abe’s Loss=L/2

2.2. Both houses burn. Both houses burn. Probability = (1-q)Probability = (1-q)22 , Abe’s Loss=L , Abe’s Loss=L

3.3. Neither house burns. Neither house burns. Probability = qProbability = q22 , Abe’s loss = 0 , Abe’s loss = 0


Risk PoolingRisk Pooling Abe’s expected utility with risk pooling:Abe’s expected utility with risk pooling:

(1-q)(1-q)22U(wU(woo-L)+2q(1-q)U(w-L)+2q(1-q)U(w00-L/2)+q-L/2)+q22U(wU(w00)) Rearranging and factoring Abe’s individual and Rearranging and factoring Abe’s individual and

risk pooling utility function shows he is better off risk pooling utility function shows he is better off if he is risk averse as:if he is risk averse as:

U(wU(w00-L/2)>(1/2)U(w-L/2)>(1/2)U(w00-L)+(1/2)U(w-L)+(1/2)U(w00)) When individuals are risk averse, they have When individuals are risk averse, they have

clear incentives to create institutions that allow clear incentives to create institutions that allow them to share (pool) their risks.them to share (pool) their risks.


Figure 17.5 Optimal risk poolingFigure 17.5 Optimal risk pooling


The Market for InsuranceThe Market for Insurance

What is Abe’s What is Abe’s reservation demand price reservation demand price for insurance for insurance (the maximum he is willing to (the maximum he is willing to pay rather than go without)?pay rather than go without)?

Set his expected utility without insurance Set his expected utility without insurance equal to the equal to the certainty equivalent certainty equivalent (assured (assured prospect wprospect wcece) in Figure 17.6.) in Figure 17.6.


Figure 17.6 The demand for insuranceFigure 17.6 The demand for insurance


The Market for InsuranceThe Market for Insurance

On the assumption that insurance On the assumption that insurance companies are risk neutral, what is the companies are risk neutral, what is the lowest price they will offer full coverage?lowest price they will offer full coverage?

This is the This is the reservation supply pricereservation supply price, , denoted by Idenoted by Iss in Figure 17.6 in Figure 17.6

Ignoring any administrative costs, the Ignoring any administrative costs, the expected costs are (1-q)L and the firm will expected costs are (1-q)L and the firm will write a policy if revenues (I) exceed costs. write a policy if revenues (I) exceed costs.


The Market for InsuranceThe Market for Insurance As shown in Figure 17.6, there is a viable insurance As shown in Figure 17.6, there is a viable insurance

market because the reservation supply price Imarket because the reservation supply price Is s =(1-q)L =(1-q)L

is less than the reservation demand price (distance is less than the reservation demand price (distance ww00-w-wcece).).

Abe trades his risky prospect for the assured prospect Abe trades his risky prospect for the assured prospect and reaches indifference curve u*.and reaches indifference curve u*.

If no resources are required to write and administer If no resources are required to write and administer insurance policies and if individuals are risk-averse, insurance policies and if individuals are risk-averse, there is a viable market for insurance.there is a viable market for insurance.

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Chapter 17 Choice Making Under Uncertainty

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