© 2009 Pearson Education Canada 17/1 Chapter 17 Choice Making Under Uncertainty

© 2009 Pearson Education Canada17/1

Chapter 17Chapter 17

Choice Making Under Choice Making Under UncertaintyUncertainty


Calculating Expected Monetary ValueCalculating Expected Monetary Value

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


Expected ValueExpected 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


Expected Utility HypothesisExpected Utility Hypothesis Expected utilityExpected utility is calculated the same is calculated the same

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

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

E(u)E(u) = (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 described, over an amount of money $M

with certainty then:with certainty then:

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


Defining a ProspectDefining a Prospect The remainder of the chapter will cover The remainder of the chapter will cover

lotteries, which will be referred to as lotteries, which will be referred to as prospectsprospects which offer three different which offer three different outcomes.outcomes.

The term prospect will refer to any set of The term prospect will refer to any set of probabilities (probabilities (qq11, q, q22, q, q33: and their assigned : and their assigned outcomes ($10 000, $6000 and $1000).outcomes ($10 000, $6000 and $1000).

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:

((qq11, q, q22, q, q33: 10 000, 6000, 1000) : 10 000, 6000, 1000)

or simply:or simply:

((qq11, q, q22, q, q33))


Deriving Expected Utility FunctionsDeriving Expected Utility Functions Continuity assumption:Continuity assumption:

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

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


von Neuman-Morgenstern Utility Function

Given any two numbers a and b with Given any two numbers a and b with a>ba>b, we could let , we could let UU(10 000)=a and (10 000)=a and UU(1 000)=(1 000)=bb. We would then have to . We would then have to assign a utility number to $6000 as assign a utility number to $6000 as follows: follows:

UU(6000) =(6000) =ae*+b(1-e*)ae*+b(1-e*)


von Neuman-Morgenstern Utility 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 in such scenarios is to use in such scenarios is to use subjective subjective probabilitiesprobabilities..


The Expected Utility FunctionThe Expected Utility Function

Assume there are 2 states of wealth (Assume there are 2 states of wealth (ww11 and and ww22) which could exist tomorrow and ) which could exist tomorrow and they occur with probabilities (they occur with probabilities (qq and 1 and 1-q-q) ) respectively.respectively.

The expected utility function for The expected utility function for tomorrow: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 Two key features of this utility

functions:functions:1.1. The The UU functions are cardinal, meaning 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 for utility level u. Wealth in state 1(today) . Wealth in state 1(today) and state 2 (tomorrow) are on each axis.and state 2 (tomorrow) are on each axis.

q q and (1-and (1-qq) are fixed.) are fixed. The MRS (slope of The MRS (slope of uu00) shows the rate at ) shows the rate at

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


From Figure 17.1From Figure 17.1 The slope of the indifference curve is equal The slope of the indifference curve is equal

to the ratio of the probabilities times the to the ratio of the probabilities times the ratio of the marginal utilities.ratio of the marginal utilities.

Each marginal utility, however, is function Each marginal utility, however, is function of wealth in only one state, since the utility of 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.


From Figure 17.1From Figure 17.1

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 slope of uu00 is is q/(1-qq/(1-q).).

If If qq is large relative to is large relative to (1-q(1-q) then ) then uu00 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 (very likely (qq is high) then you will prefer 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 Martha having a fire is 1-q1-q, the loss from , the loss from such a fire is such a fire is L L dollars and wealth in period dollars and wealth in period t t denoted as denoted as wwtt..



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 If Abe’s house burns, his wealth is ww00--LL, and his utility , and his utility U(wU(w00-L-L). If it does ). If it does not burn, his wealth is wnot burn, his wealth is w0 0 and utility and utility is is U(wU(w00).).



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

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

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

3.3. Neither house burns. Neither house burns. Probability = Probability = qq22 , 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(w00L/2)+qL/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 risk pooling utility function shows he is better off if he is risk averse as:off 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))


Risk PoolingRisk Pooling

When individuals are risk averse, they When individuals are risk averse, they have clear incentives to create have clear incentives to create institutions that allow them to share institutions that allow them to share (pool) their risks.(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 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 (assured prospect (assured prospect wwcece 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 companies are risk neutral, what is the lowest price they will offer full the lowest price they will offer full coverage?coverage?

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


The Market for InsuranceThe Market for Insurance Ignoring any administrative costs, Ignoring any administrative costs,

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


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

insurance market because the reservation insurance market because the reservation supply price supply price IIs s =(1-q)L=(1-q)L is less than the is less than the reservation demand price (distance reservation demand price (distance ww00-w-wcece).).

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

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

Documents

© 2009 Pearson Education Canada 17/1 Chapter 17 Choice Making Under Uncertainty