In the Name of Godgsme.sharif.edu/profs/fatemi/wp-content/uploads/sites/24/...In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics

In the Name of God Sharif University of Technology

Graduate School of Management and Economics

Microeconomics 1 44715 (1391-92 1st term)

Dr. S. Farshad Fatemi

Chapter 6:

Choice under Uncertainty

Microeconomics 1 Dr. F. Fatemi Page 136 Graduate School of Management and Economics – Sharif University of Technology

Expected Utility Theory

In many cases, individuals face some degree of risk when making their

decisions. The general theory of choice established earlier in the course can

be applied to these situations.

Definition (MWG 6.B.1): A simple lottery is a list ( ) with

and ∑ ; where is interpreted as the probability of

outcome occuring.

Definition (MWG 6.B.2): Given simple lotteries (

),

and probabilities with ∑ , the compound

lottery ( ) is the risky alternative that yields the simple

lottery with probability for .


For any compound lottery, a simple lottery can be calculated which is called

the reduced lottery.

Definition (MWG 6.B.3): The preference relation ≿ on the space of

simple lotteries is continuous if for any , the sets

{ [ ] ( ) ≿ } [ ]

and

{ [ ] ≿ ( ) } [ ]

are closed.

The continuity axiom implies the existence of a utility function representing

the preference relation ≿.

≿ ( ) ≿ ( )


Definition (MWG 6.B.4): The preference relation ≿ on satisfies the

independence axiom if for all , and [ ] we have:

≿ ( ) ≿ ( )

Definition (MWG 6.B.5): The utility function has an

expected utility form if there is assignment of numbers ( ) to the

outcomes such that for every simple lottery ( ) we have:

( ) .

A utility function with the expected utility form is called v.N-M

(von Neumann-Morgenstern) expected utility function.


Proposition (MWG 6.B.1): A utility function has an

expected utility form if and only if it is linear, that is, if and only if satisfies

the property that

(∑

) ∑ ( )

for any lotteries , , and the probabilities

( ) , ∑ .

Note: Unlike what we have seen so far about the utility function properties,

the expected utility property is a cardinal property of utility functions.


Proposition (MWG 6.B.2): Suppose that is a v.N-M

expected utility function for the preference relation ≿ on . Then

is another v.N-M utility function for ≿ if and only if there are

scalars and such that for every .

Using this property the differences of utilities have meaning.

If a preference relation ≿ is representable by a utility function which has the

expected utility form, then since a linear function is continuous, ≿ is

continuous and satisfies the independence axiom.


Proposition (MWG 6.B.3) Expected Utility Theorem: Suppose that ≿ on

satisfies the continuity and independence axiom. Then ≿ admits a utility

representation of the expected utility form. That is we can assign a number

to each outcome in such a manner that for any two lotteries

( ) and (

), we have

≿ ∑

∑

Examples:

Expected utility as a guide to introspection

The Allais paradox

Machina’s paradox

Induced preferences


Money Lotteries and Risk Aversion

A monetary lottery can be described by a cumulative distribution ( ):

[ ]

where is the amounts of money.

In terms of density function ( ), the distribution can be written as:

( ) ∫ ( )

From here we assume that the lottery space to be the set of distribution

functions over nonnegative amounts of money.


Then if ≿ represents a rational preference of a decision maker and satisfies

the continuity and independence axiom, there is an assignment of utility

values ( ) to nonnegative amounts of money with the property that any

( ) can be evaluated by a utility function ( ) of the form:

( ) ∫ ( ) ( )

To distinguish between these two utility functions they are called:

( ) : The von Neumann-Morgenstern expected utility function

and

( ) : The Bernoulli utility function.

( ) is increasing and continuous.


Definition (MWG 6.C.1): A decision maker

i) is risk averse if for any lottery ( ):

∫ ( ) ( ) (∫ ( ))

ii) is risk neutral if for any lottery ( ):

∫ ( ) ( ) (∫ ( ))

iii) is strictly risk averse if the indifference holds only when the two lotteries

are the same.

The inequality which is called Jensen’s inequality is the definition of a concave

function.


Definition (MWG 6.C.2): Given a Bernoulli utility function ( ) we define

the following concepts:

i) The certainty equivalent of ( ) is ( ) where:

( ( )) ∫ ( ) ( )

ii) For any and positive number , the probability premium is ( )

where:

( ) (

( )) ( ) (

( )) ( )

Note: For a risk averse decision maker (recall ( ) is nondecreasing):

( ) ∫ ( ) ( ( )) (∫ ( )) ∫ ( ) ( ) (∫ ( ))


Proposition (MWG 6.C.1): Suppose a decision maker is an expected

utility maximize with a Bernoulli utility function ( ) on amounts of money.

Then the following properties are equivalent:

i) The decision maker is risk averse.

ii) ( ) is concave.

iii) ( ) ∫ ( ) for all ( ).

iv) ( ) for all .

Examples:

Insurance

Demand for a risky asset

General asset problem


Definition (MWG 6.C.3): Given a twice differentiable Bernoulli utility

function ( ) on amounts of money, the Arrow-Pratt coefficient of

absolute risk aversion at is defined as

( ) ( )

( )

Comparison of Payoff Distribution in Terms of

Return and Risk

Definition (MWG 6.D.1): The distribution ( ) first-order stochastically

dominates ( ) if for every nondecreasing function , we have

∫ ( ) ( ) ∫ ( ) ( )


Proposition (MWG 6.D.1): The distribution of monetary payoffs ( )

first-order stochastically dominates the distribution ( ) if and only if

( ) ( )

Definition (MWG 6.D.1): For any two distributions ( ) and ( ) with the

same mean, ( ) second-order stochastically dominates(or is less risky

than) ( ) if for every nondecreasing concave function , we

have

∫ ( ) ( ) ∫ ( ) ( )

Documents

In the Name of Godgsme.sharif.edu/profs/fatemi/wp-content/uploads/sites/24/...In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics