Eaton Micro 6e Ch17

7/27/2019 Eaton Micro 6e Ch17

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2005 Pearson Education Canada Inc.17.1

Chapter 17

Choice Making UnderUncertainty


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Calculating Expected Monetary Value

The expected monetary value is simply theweighted average of the payoffs (thepossible outcomes), where the weights are

the probabilities of occurrence assigned toeach outcome.


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Expected Value

Given: Two possible outcomes havingpayoffsX1andX2 andProbabilities ofeach outcome given by Pr1 & Pr2.

The expected value (EV) can be

expressed as:

EV(X) = Pr1X1+ Pr2X2


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Expected Utility Hypothesis

Expected utility is calculated in the same wayas expected monetary value, except that theutility associated with a payoff is substituted for

its monetary value.

With two outcomes for wealth ($200 and $0)and with each outcome occurring the time,

the expected utility can be written:E(u) = (1/2)U($200) + (1/2)U($0)


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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)


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Defining A Prospect

The remainder of the chapter will betalking about lotteries which will bereferred to asprospects which offer three

different outcomes.

The term prospect will refer to any set ofprobabilities (q1, q2, q3: and their assigned

outcomes ($10 000, $6 000 and $1 000).

Note that the probabilities must sum to 1.


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Defining A Prospect

Such aprospectwill be denoted as:

(q1, q2, q3: 10 000, 6 000, 1 000)

or simply:(q1, q2, q3)


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Deriving Expected Utility Functions

Continuity assumption:For any individual, there is a unique number e*,

(0


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von Neuman-MorgensternUtility 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 utilitynumber to $6 000 as follows:

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


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von Neuman-MorgensternUtility Function

With the continuity assumption (and others) satisfiedand the utility function constructed as shown, theseimportant results are applicable:

1. If an individual prefers one prospect to another, thenthe preferred prospect will have a larger utility.

2. If an individual is indifferent between two prospects,

the two prospects must have the same expectedutility.


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Subjective Probabilities

The expected utility theory is often appliedin risky situations in which the probabilityof any outcome is not objectively known or

there exists incomplete information.

The ability to apply expected-utility theoryis such scenarios is to use subjective

probabilities.


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The Expected Utility Function

Assume there are 2 states of wealth (w1and w2) which could exist tomorrow andthey 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)


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The Expected Utility Function

Two key features of this utility functions:

1. The U functions are cardinal, meaningthat the utility values have specificmeaning in relation to one another.

2. This expected utility function is linear inits probabilities (which simplifies MRS).


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Figure 17.1 Indifference curves in state space


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From Figure 17.1

Figure 17.1 shows an indifference curvefor 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 atwhich an individual trades wealth in state 1for wealth in state 2, before either of thesestates occur.


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From Figure 17.1

The slope of the indifference curve isequal to the ratio of the probabilities timesthe ratio of the marginal utilities.

Each marginal utility however is function ofwealth in only one state since the utilityfunctions are the same in each state.

Therefore the MRS equals the ratio of theprobabilities.


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From Figure 17.1

Hence, along the 45 degree line, wherewealth in the two states are equal, theslope of u0 is q/(1-q).

If q is large relative to (1-q) then u0 isrelatively steep and vice versa.

In other words, if you believe state 1 is

very likely (q is high) then you will preferyour wealth in state one rather than statetwo.


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Figure 17.2 Preferences towards risk


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Optimal Risk Bearing

Now that different attitudes toward riskhave been defined, it is necessary toillustrate how attitudes toward risk affect

choices over risky prospects.

An expected value line shows prospectswith the same expected value. Note

however that along this line, the risk ofeach prospect varies.


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Figure 17.3 The expected monetary value line


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From Figure 17.3

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

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

Utility will be maximized when the individuals

MRS equals the ratios of the probabilities.


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Figure 17.4 Optimal risk bearing


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Optimal Risk Bearing

The optimal amount of risk that a person bearsin 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 nomatter what state arises.

Risk inclined persons move away from the 45

degree line and are willing to take the chancethat they will be better off in one statecompared to the other.


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Pooling Risk

Risk Pooling is a form of insurance aimedat reducing an individuals exposure to risk

by spreading that risk over a larger

number of persons.

Suppose the probability of either Abe orMartha having a fire is 1-q, the loss from

such a fire is L dollars and wealth in periodt denoted as wt.


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Pooling Risk

Abes expected utility is:

u(q, L,w0) = qU(w0)+(1-q)U(w0-L).

If Abes 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 anyloss from a fire), There are now three relevantevents:

1. One house burns.Probability = 2q(1-q), Abes Loss=L/2

2. Both houses burn.

Probability = (1-q)2, Abes Loss=L3. Neither house burns.

Probability = q2, Abes loss = 0


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Risk Pooling

Abes expected utility with risk pooling:(1-q)2U(wo-L)+2q(1-q)U(w0-L/2)+q

2U(w0)

Rearranging and factoring Abes individual and

risk pooling utility function shows he is better offif he is risk averse as:

U(w0-L/2)>(1/2)U(w0-L)+(1/2)U(w0)

When individuals are risk averse, they haveclear incentives to create institutions that allowthem to share (pool) their risks.


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Figure 17.5 Optimal risk pooling


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The Market for Insurance

What is Abes reservation demand price

for insurance (the maximum he is willing topay rather than go without)?

Set his expected utility without insuranceequal to the certainty equivalent (assuredprospect wce) in Figure 17.6.


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Figure 17.6 The demand for insurance


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On the assumption that insurancecompanies are risk neutral, what is thelowest 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 willwrite a policy if revenues (I) exceed costs.


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2005 Pearson Education Canada Inc17 32


As shown in Figure 17.6, there is a viable insurancemarket because the reservation supply price Is =(1-q)L is less than the reservation demand price(distance w

0

-wce

).

Abe trades his risky prospect for the assured prospectand 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.

Eaton Micro 6e Ch17

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