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Microeconomics: MT12Lectures on Risk and Expected utility:Lecture 1

Sujoy Mukerji

Oxford University

November 13, 2012

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What guides choice under risk?

We are aiming for a theory that will help us understand how agents decidebetween risky alternatives

More precisely, agent is choosing between lotteries.

A lottery is a random variable, X; it pays o an amount xi withprobability pi, i=1, 2, ..., I.

Notation: (x1 , ..., xI; p1, ..., pI)

Consider the following lotteries,

`1 =(100,0.5; 0.5, 0.5)`2 =(200,

100; 0.5, 0.5)

`3 =(20000,10000; 0.5, 0.5)Would you want to take on any of these gambles? How would yourank them?

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What guides choice under risk?

Expected value of the rst lottery isE(`1)=100 0.5+ ()0.5 0.5=49.75

E(`2)= 50; E(`3)=5000.

While clearly having a greater expected value is an attractive thing, it

is just as clear form the way most people respond to these lotteries,that it is by no means the only feature that guides choice. `1 has theleast expected value but the one most likely to chosen.

The feature that makes `3 so unattractive to most people is thatthere is an even chance of the extremely unpleasant outcome of losing10000

and, seemingly, not compensated by the even chance of the pleasant20000.

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What guides choice under risk?

People seem to be bothered by spread of outcome and bothered more

by the downside risk than the upside.

A way to model this eect is to think of individuals as assessingoutcomes in terms of some utility function and then evaluating theexpected utility of the gamble oered:

Example

Suppose Smith has an utility function given by u(M) =p

M. If he has aninitial wealth, w, of 10000, how would he rank `1, `2, `3?Eu(`1)=0.5

u(w+100)+0.5

u(w

0.5)=

0.5 p10100+0.5 p9999.5=100.248Eu(`2)=0.5

p10200+0.5

p9900=100.247

Eu(`3)=0.5 p

30000+0.5 p

0= 86.603.And so, `1 is the most attractive of the three for Smith.

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What guides choice under risk?The general point: a concave utility function

u(.)

Final wealthw+x2w+x1 E(w+x)CE(w+x)

u(E(w+x))

Eu(w+x)

utils

Concavity of the utility function implies:

u(w+x2) u(E(w+X)) E(w+X).

And, for a linear u,

CE(w+X) =E(w+X).

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(Absolute) Risk attitudes

Notation Given a lottery `, `E denotes the lottery (E(`); 1).

The lottery `

Esimply yields the expected value of `

as the sure outcome.Denition

We say an agent is risk averse if, at any wealth level w, and for anylottery `, she prefers `E to `.The agent is risk seeking if she prefers ` to`E; she is risk neutral if she is indierent between `E and `.

Theorem

Consider an agent A with utility function uA(.).The following areequivalent:

1 Agent A is risk averse

2 uA(.) is concave (or u00 < 0, if u is dierentiable)3 CE(w+X; uA) < E(w+X).

Proof.

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Risk premiumThe cost of risk

u(.)

Final wealthw+x2w+x1 E(w+x)CE(w+x)

u(E(w+x))

Eu(w+x)

utils

RP

Risk Premium=RP(w+X) =E(w+X) CE(w+X).

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(Comparative) Risk attitudes

We say an agent A is more risk averse than agent B if, holding initialwealth constant across agents, for any lottery `, A is will ask for agreater risk premium than B.

It can be shown, ifA and Bhave dierentiable utility functions, uA

and uB,

AuA(w)=u00A(w)

u0A(w) u

00B(w)

u0B(w) =AuB(w).

AuA is called the Arrow-Pratt index. It is a measure of the degree ofrisk aversion. Risk costs more the more risk averse you are, thegreater your degree of risk aversion.

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Caveat: utility is cardinal!

The utility function u(.) is cardinal, not simply ordinal. Dierences inutility levels matter; recall, when trying to understand the nature ofrisk aversion we emphasized the inequality,

u(w+x2) u(E(w+X)) < u(E(w+X)) u((w+x1),

Since utility functions embody risk attitudes, transformations ofutilities which do not preserve risk attitudes are not permitted.Increasing linear transforms dopreserve risk attitudes represented. Tosee this, let v(x) =u(x) +, > 0. Notice, v0(x) =u0(x) andv00 (x)=u00(x). Hence,

Av(w)=v00 (w)

v0 (w) =

u

00 (w)u0 (w)

=Au(w).

However, arbitrary (non-linear) increasing transforms do not preserverisk attitude in general, e.g., v(x) =[u(x)]2 .

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