Expected UtilityExpected Utility

Lecture I

Basic Utility

• A typical economic axiom is that economic agents (consumers, producers, etc.) behave in a way that maximizes their expected utility. The typical formulation is:

where x1 and x2 are consumption goods and Y is monetary income

Yxpxpst

xxUxx

2211

21,

,max21

• In decision making under risk, we are typically interested in the utility of income U(Y). How do these concepts relate?

• The linkage between these two concepts is the indirect utility function which posits optimizing behavior by the economic agent

– Specifically, assuming an Cobb-Douglas utility function the general utility maximization problem can be rewritten as:

– Due to the concavity of the utility function, the inequality can be replaced with an equality

Yxpxpst

xxxx

2211

21, 21

max

– The maximization problem can then be reformulated as a Lagrangian:

221121 xpxpYxxL

– The first order conditions are then

0

0

0

2211

22

21

2

11

21

1

xpxpYL

px

xx

x

L

px

xx

x

L

– Taking the ratio of the first two first order conditions yields

– Substituting this result into the third first order condition yields the demand for x1 as a function of prices and income

2

112 p

pxx

1211 ,,

p

YYppx

– Using this expression for x1 in the result derived from the ratio of the first order conditions yields the Marshallian demand for x2

2212 ,,

p

YYppx

Indirect Utility FunctionIndirect Utility Function

• The Marshallian demand curves can be used to derive the indirect utility function:

This indirect utility function relates utility directly to income and prices assuming optimizing behavior.

2121 ,,

pp

YYppV

• The expenditure function, like the indirect utility function, examines the implications of optimizing behavior.– However, the expenditure function determines

the minimum income required to provide a given level of utility. Mathematically, the expenditure function is given by the solution of

*21

2211, 21

min

Uxxst

xpxpxx

– Using the same Cobb-Douglas utility function described earlier, this minimization problem becomes

– The Lagrangian for this problem becomes

*21

2211, 21

min

Uxxst

xpxpxx

21*

2211 xxUxpxpL

– The first order conditions for this problem are then

0

0

0

21*

2

212

2

1

211

1

xxUL

x

xxp

x

L

x

xxp

x

L

– Again, taking the ratio of the first two first order conditions yields

2

12 p

px

– Substituting this result into the third first-order condition yields

1

2*1

2

11

*

12

11

*

0

0

p

pUx

p

pxU

xp

pxU

– The optimum level of x1 given a fixed level of utility and prices (the Hicksian demand curve) is then

– Similarly, the Hicksian demand for x2 is

1

21

211 ,,p

pUUppxh

2

11

212 ,,p

pUUppxh

– Substituting these expressions into the budget constraint yields the expenditure function

21

1

1

1

2

2

1

1

21

1

2

21

1

121 ,,

ppU

pUp

pUp

ppUp

ppUpUppe

• Note the duality between the functions. Starting with the expenditure function

1

1

2121

11

21

11

21

21

1

21

,,

,,

Y

ppYppV

YppU

YpUp

YppUUppe

Expected Utility

• The importance of these relationships for risk analysis involves the relationship between expected utility and the certainty equivalence function.– Next time we will discuss the concepts

involved in expected utility analysis. For now, assume that economic agents behave to maximize their expected utilities.

– Implicitly, this says that an economic agent is indifferent between two risky alternatives that yield the same expected utility.

– If we let one of the alternatives be a risk-free alternative then the economic agent is indifferent between two alternatives a0 and a1 if the utility from the certain action equals the expected utility from the uncertain action.

Expected Utility of a1

Utility of a0

U*

CertaintyEquivalent

U(Y)

Y

• Simplifying the indirect utility function by ignoring the price term yields power utility function:

r

YYU

r

1)(

1

– Assume that a risky investment has a Bernoulli distribution paying $150,000 with probability .6 and $50,000 with probability .4. What is this investment worth? First, assume r=.5 then the expected utility of the gamble is:

64.643

5.1

000,504.

5.1

000,1506.

5.15.1

YUE

– To compute the certainty equivalent of the gamble

20.569,10364.6435.

1

1

1

5.1

11

1

1

Y

YUr

YUr

r

YU

r

r

r

– Note that the expected value of the gamble is $110,000. This implies a risk premium of 110,000-103,569.20=$6,430.78.