7.1 Expected utility function

z Simple gambles

Let {}n a a A ,...,1= be the set of outcomes. Then G , the set of simple gambles (on

A), is given by ()⎭

⎬⎫⎩⎨⎧=≥∑i i i n n p p a p a p 1,0|,..,11o o Compound gambles

z Axioms of choice under uncertainty

1. Completeness

2. Transitivity

1a ≿ 2a ≿ … ≿ n a

3. Continuity. For any gamble g in G, there is some probability, ]1,0[∈α, such

that )

)1(,(~1n a a g o o αα−. In words, continuity means that small changes in probabilities do not change the nature of the ordering between two gambles.

4. Monotonicity. For all probabilities ]1,0[,∈βα,))1(,(1n a a o o αα−≿

))1(,(1n a a o o ββ− iff βα≥.

Monotonicity implies n a a f 1.

5. Substitution. If ),...,(11k k g p g p g o o =, and ),...,(11k k h p h p h o o = are in G ,

and if i g h i i ∀~, then

g h ~. 6. Reduction to simple gambles. The decision maker cares about only the

effective probability. Hence, it can not model the behavior of vacationers in Las Vegas!

7. Independence axiom. For any three gambles 321,,g g g and )1,0(∈α, we

have 1g ≿ 2g iff 31)1(g g αα−+ ≿ 32)1(g g αα−+.

Question: does the preference under certainty satisfy this axiom? Why?

z V on Neumann-Morgenstern Utility

Utility functions possessing the expected utility property is VNM utility functions.

Expected utility property:

The utility function :u G Æ R has the expected utility property if, for every g ∈G , ∑==n

i i i a u p g u 1)()(, where ),...,(11n n a p a p o o is the simple gamble induced by g.

Theorem 7.1 Existence of a VNM function on G

A preference over gambles in G satisfying axioms above can be represented by at least one utility function which has the expected utility property.

Proof: 1. construct a utility function: )(g u is the number satisfying )))(1(,)((~1n a g u a g u g o o −. By axiom 3, there exists such a number, by axiom 4, this number is unique. Hence, we can define a real-valued function in this way.

2. u represents ≿. Because of axiom 2, we have g ≿ g’ iff )))(1(,)((1n a g u a g u o o −≿)))'(1(,)'((1n a g u a g u o o −. And axiom 4 tells us that )))(1(,)((1n a g u a g u o o −≿)))'(1(,)'((1n a g u a g u o o − iff )'()(g u g u ≥.

3. we must show ∑==n

i i i a u p g u 1)()(, where g is a simple gamble. By axiom

6, we can easily extend our result to any gambles.

By definition, i n i i i q a a u a a u a ≡−)))(1(,)((~1o o . Hence, we can know by

axiom 5, that g a p a p q p q p g n n n n =≡),...,(~),...,('1111o o o o .

By axiom 6, the reduced gamble 's g and g’ is indifferent:

'~)(1,)('111

g a a u p a a u p g n n i i i n i i i s ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛=∑∑==o o Hence, g g s ~'. However, )

))(1(,)((~1n a g u a g u g o o −. Therefore, we conclude: ∑==n

i i i a u p g u 1)()(.

Theorem 7.2 VNM utility functions are unique up to positive affine transformations

Suppose )(•u is VNM utility function, then the VNM utility function )(•v represents the same preferences iff 0)()(>+=ββαg u g v .

Proof:

1. Sufficiency. Obvious!

2. Necessity.

Because )(•u represents the preference over G , we have

)()...(...)(1n i a u a u a u ≥≥≥

Hence, there exists a unique ]1,0[∈i α such that

)()1()()(1n i i i a u a u a u αα−+=

Now, because )(•u has the expected utility property:

))1(,(~1n i i i a a a o o αα−

Because )(•v represents the same preference and has the expected utility property:

]1,0[)()1()()(1∈−+=i n i i i a v a v a v ααα

Together, we have:

)

()()()(1)()()()(11n i i i i n i i a v a v a v a v a u a u a u a u −−=−=−−αα )()()()(,)()()()()()(),()(11111n n n n n i i a u a u a v a v a u a u a u a v a v a u where a u a v −−≡−−≡

+=⇒βαβα

This theorem tells us that VNM utility functions are not completely unique, nor are they entirely ordinal.

)

()()()(1)()()()(11n i i i i n i i a v a v a v a v a u a u a u a u −−=−=−−αα reflects the fact that the ratios of the differences between the preceding utility numbers are uniquely determined by i α, which again is determined by the preference of the decision maker. Hence, the ratio of utility differences has inherent meaning regarding the preference. That is why we say that VNM utility functions provide more than ordinal information. However, we still can not use VNM utility functions for interpersonal comparisons of well-being.

7.2 Discussion of the theory of expected utility

Example 1 : (Allais 1953)

There are three possible monetary prizes. (so the number of outcomes is 3)

First prize: 60,000 dollars

Second prize: 50,000 dollars

Third prize: 0 dollars

)0,1,0(1=g )01.0,.0,1.0(2=g Which one do you like?

)9.0,0,1.0(3=g ).0,11.0,0(4=g Which one?

Intuitive explanation: the value of certainty and uncertainty.

Example 2: Ellsberg Paradox

There are 90 balls. 30 are red, and others are either black or white.

You are asked to pick a ball, whether you will win depends on whether you have chosen the right color.

Which color(s) do you prefer to be the right color?

1. red or white?

2. “red or black” or “black or white”?

Intuitive explanation:

Subjective versus objective probability

What is “unknown”?

Example 3: Machina’s Paradox

“a trip to Venice” ≿ “ watching an excellent movie about Venice” ≿ “staying home”

)0,01.0,99.0(1=g )01.0,0,99.0(2=g

If you anticipate that in the event of not getting the trip to Venice, your tastes over the other two outcomes will change: you will be so disappointed and feel miserable watching a movie about Venice, hence, it also is rational to choose 2g although you like this movie in general.

Example 4: “Framing problem”

)00,41(1o o =g )067.0,1633.0(1o o =g

Which one do you like?

For which will you pay more?

7.3 Risk aversion

Example:

{}2,4,10A −= ()()109.0,403.0,207.0,

108.0,42.021o o o o o −==g g

⎪⎩⎪⎨⎧−====2046.0101)(x if x if x if x u

Then, the decision maker prefers 1g because

expected utility )()(21g u g u > However, the expected value )()(21g E g E <.

Risk aversion:

Let )(•u be an individual’s VNM utility function for gambles over nonnegative levels of wealth. Then for the simple gamble ),...,(11n n w p w p g o o =, the individual is said to be

1. risk averse at g if )())((g u g E u >

2. risk neutral at g if )())((g u g E u =

3. risk loving at g if )())((g u g E u <

Certainty Equivalent and Risk Premium

The certainty equivalent of any simple gamble g over wealth levels is an amount of wealth, CE, offered with certainty, such that )CE ()(u g u =. The risk premium is an amount of wealth, P, such that )P -E(g)()(u g u =. Clearly, CE )(−=g E P .

The Arrow-Pratt Measure of absolute risk aversion )

(')

(")(w u w u w R a −≡ 1. A Risk averter has a positive Arrow-Pratt Measure.

2. Any positive affine transformation of utility will leave the measure unchanged.

3. It is a local measure. Decreasing absolute risk aversion (DARA) is generally a

sensible restriction to impose. It says that the individual is less averse to taking small risks at higher levels of wealth.

4. It is an effective measure, i.e., a consumer with larger Arrow-Pratt measures have

a lower CE.

To see this, suppose two consumers with two utility functions )(w u and )(w v , respectively. We have

0)

(')

(")(')("≥∀−>−w w v w v w u w u , and we want to show

21CE CE <, where ∑∑==)()()()(21i i i i w v p CE v w u p CE u .

Define 0))(()(1≥∀=−x x v u x h we can show it is a concave function. Noting

1

1111))(())(())(())(()(−−−−−⎟⎟⎠⎞⎜⎜⎝⎛===dx x v d x v d dx

x v d x v dv dw w dv , thus, 0))

(('))

((')('11>=

−−x v v x v u x h and 0"()())

())(()()

())(()()(2211CE u CE v h w v p h x p h x h p x v u p w u p CE u i i i i i i i i i i ===<===∑∑∑∑∑−

Where the inequality, called Jensen’s inequality , follows because )(x h is a concave function.

Finally, we have 21CE CE < because utility functions are increasing functions.

Example 1 : Investment

An risk-averse investor has wealth w , and the risky investment is ()i i r p o . How much (β) will he invest in this risky project?

The final wealth is: βββi i r w r w +=++−)1( So, the investor maximizes his utility:

w

t s r w u p i i ≤≤+∑βββ

0..)

(max

The case of corner solution: 00)('0≤⇔≤+=∑∑∗

∗i i i i i

r p r r w u p iff β

β

The interior solution requires that the expected return rate is nonnegative. FOC is: 0)('=+∑∗

i i i r r w u p β

SOC is:

0)("2

<+∑∗

i i i

r r w u p β

To show that risky assets are “good” rather than “bad”…

2)(")("i

i i i

i i r r w u p r r w u p dw d ∑∑∗∗

∗++−=βββ i i i a i i i i

r r w u r w R p r r w u p ∑∑∗∗∗

++−=+)(')()("βββ

Under DARA, 0

)()(0)()(><+<>+∗∗i a i a i a i a r if

w R r w R r if w R r w R ββi a i i a r w R r r w R )()(<+⇒∗β

Hence,

0)(')()(')()("=+<++=+−∑∑∑∗∗∗∗i i a i i i i a i i i i r r w u w R p r r w u r w R p r r w u p ββββ

0>∗

dw d β I.e., for DARA, more wealth will be invested into risky projects as wealth increases.

Example 2: Insurance

An investor has initial wealth w , the VNM utility function is )(w u . The probability of an accident is ()1,0∈α and the loss is L when it occurs. If the insurance is totally fair, i.e., the price of insurance is α, how much insurance will she buy?

Let us say, she will buy insurance x . Thus, the expected utility is:

)()1()(x w u x L x w u αααα−−++−−

Suppose she is an expected utility maximizer, then FOC:

0)(')1()(')1(0

0)(')1()(')1(=<−−−+−−−>=−−−+−−−x if

x w u x L x w u x if x w u x L x w u αααααααααααα For a risk averter, 0"−−w u L w u , hence, 0>x and L x =. For a risk neutral investor, 0"=u , ?=x For a risk loving investor?

7.4 Comparison of payoff distributions in term of return and risk

z Pay off distribution )(•F

z First-order stochastic dominance: the distribution )(•F first-order stochastically

dominates the distribution )(•G if, for every nondecreasing function R R u →:,

we have ∫∫≥)()()()(x dG x u x dF x u .

z Theorem 7.3 The distribution )(•F first-order stochastically dominates the

distribution )(•G if and only if x x G x F ∀≤)()(.

Proof: “only if” part. Denote )()()(x G x F x H −=. Suppose that 0)(>x H for some x . Then we can define a nondecreasing function ⎩⎨

⎧>=otherwise x

x if x u 01)(. This function has the property that ∫<−=0()()(x H x dH x u . It implies that

∫∫<)()()()(x dG x u x dF x u , which is contradiction because )(•F first-order

stochastically dominates the distribution )(•G .

“if” part. Only for differentiable utility functions )(x u . Integrating by parts, we have: dx x H x u dx x H x u x H x u x dH x u )()(')()(')]()([)()(0∫∫∫−=−=∞ Hence, if 0)(≤x H and )(x u is increasing, then 0)()(≥∫x dH x u .

z Second-order stochastic dominance: for any two distributions with the same mean,

)(•F second-order stochastically dominates )(•G if, for every nondecreasing concave function R R u →+:, we have ∫∫≥)()()()(x dG x u x dF x u .

z Mean-Preserving Spreads: consider the following compound lottery. In the first

stage, we have a lottery over x distributed according to )(•F . In the second stage, we randomize each possible outcome x further so that the final payoff is x+z, where z has a distribution function H with a mean of 0 [i.e.,

0)(=∫z dH z ]. Thus,

the mean of x+z is x. Let the resulting reduced lottery be denoted by G , we say that G is a mean-preserving spread of F.

z Theorem 7.4 consider two distributions with the same mean, we say that G is

a mean-preserving spread of F is equivalent to say that F second-order stochastically dominates G.

Proof: suppose u is a concave function.

()()∫∫∫∫∫∫=+≤+=)

()()()()()()()()()(x dF x u x dF z dH z x u x dF z dH z x u x dG x u下载本文