PROPERTIES OF UTILITY

One way to begin an analysis of individuals choices that we say are characterized in a utility function is to state a basic set of postulates, or axioms, that characterize what we call “rational behavioro Completeness: If A and B are any two situations, the individual can always specify exactly one of the following possibilities:

A is preferred to B

B is preferred to A

A and B are equally attractive

here, people are not paralyzed by indecision --- this rules out that A is preferred to B, and, B is preferred to A

Transitivity: If an individual reports A is preferred to B and that B is preferred to C, then A is preferred to C --- choices are internally consistent

Continuity: If an individual reports A is preferred to B, the situations suitably close to A must also be preferred to B --- this helps us analyze relatively small changes in income and prices

UTILITY

Given the assumptions of completeness, transitivity, and continuity, it is possible to show formally that people are able to rank in order all possible situations from least desirable to most desirable

This ranking we call “utility” after the inventor, Jeremy Bentham, a 19th century

The utility of situation A and situation B would be denoted U(A,B)

Bentham suggested the utilitarian approach as “more is better”

Therefore, if a person prefers situation A to situation B, then U(A) is greater than U(B)

We could attach numbers to these utility rankings, but these numbers will not be unique

Any set of numbers we arbitrarily assign that accurately reflects the original preference ordering will imply the same set of choices

It is the same to say that U(A) = 5 and U(B) = 4, as U(A) = 1million, U(B) = 0.5

The nonuniqueness of utility measures suggest that it is not possible to compare utilities between people

So utility = U(q1, q2, . . ., qn; other things) In finance we are usually dealing with

intertemporal utility as U(Co, C1s), where Co = current consumption and C1s = next period consumption, but that consumption can come in different states, s.

Trades and Substitution

Most economic activity involves voluntary trading between individuals

When someone buys bread, they voluntarily give up something, usually money (in this age)

They give up the money for something of greater value

INDIFFERENCE CURVES

Hence we come to “INDIFFERENCE CURVES”

An indifference curve shows a set of consumption bundles among which the individual is indifferent – that is, the bundles all provide the same level of utility

INDIFFERENCE CURVES

Co

C1

U

C1 C1΄

Co

Co΄

The bundle (Co,C1) has the same utility level as the bundle (Co΄,C1΄)

Increasing Utility in the NE direction

Co

C1

U

C1 C1΄

Co

Co΄

Increasing utility

Inconsistent preferences

Co

C1

U1

C1 C1΄

Co

Co΄

U2

Intersecting indifference curves imply inconsistent preferences

MRS – marginal rate of substitution

Co

C1

U1

C1 C1΄

Co

Co΄

The negative slope of an indifference curve at some point is termed the marginal rate of substitution (MRS)

MRS = -dCo/dC1, given U = U1

MRS tells us something about the trades a person voluntarily makes

The MRS diminishes between bundles (Co,C1) and (Co΄,C1΄)

The indifference curve becomes flatter or less steep --- one can trade less Co and get more C1 – either in a static sense or intertemporally

Intertemporal substitution

Co

C1

U1

C1 C1΄

Co

Co΄

With Co = current consumption and C1 = future consumption, MRS is intertemporal substitution --- a tradeoff in time for goods

MRS = -dCo/dC1, given U = U1

Indifference curves are convex --- utility is a convex set

Co

C1

U1

C1 C1΄

Co

Co΄

Utility set

Utility frontier

We can connect points on the frontier with a line – any point on the line is also in the set

Any point on a linear combination of point is within the set

Nonconvexity – insconsistent preference

Co

C1

U1

C1 C1΄

Co

Co΄

Utility set

Utility frontier

We can connect points on the frontier with a line – any point on the line is also in the set

Examples

U = Co1/2C11/2 = CoC1 We could square this Utility function

and preserve the ordering of preference as CoC1

Suppose CoC1 = U = 100, then CoC1 = 10

Find the MRS of CoC1 = 100, by transforming to C1 = 100/Co

Then MRS = -dC1/dCo = 100/Co2

Clearly MRS declines as Co increases CoC1 is actually a rectangular

hyperbola as you recall from geometry, and therefore, the slope of the indifference curve is the same at all points

MRS and Marginal Utility

Let utility be in general, U = (Co,C1) We can totally differentiate this

function as, dU = (∂U/∂C1)dC1 + (∂U/∂Co)dCo Along any particular indifference curve

dU = 0 (same utility level)

So manipulate dU = (∂U/∂C1)dC1 + (∂U/∂Co)dCo = 0

To get MRS =-dC1/dCo= (∂U/∂C1) / (∂U/∂Co) MRS = the ratio of the marginal utility

of C1 to the marginal utility of Co

Utility increasing at a decreasing rate

If utility increases at a decreasing rate, then if utility is U(Co, C1)

∂U/∂Co > 0, but ∂2U/∂Co2 < 0 That is the first derivative of utility

with respect to an argument, such as Co, is positive, but the second derivative of utility with respect to that same argument is negative

This suggests that utility is increasing as the argument, say Co, increases, but the rate of increase is diminishing as Co increases

Examples

Suppose we develop the utility of wealth, say W as

U(W) = Ln(W) --- a natural logarithmic utility function

U΄= ∂U/∂W = 1/ W, and is > 0 for any value of wealth, W

U΄΄= ∂2U/∂W2 = -1/ W2 < 0

Look at quadratic utility U(W) = a + bW + cW2

for a, b, c being parameters U΄ = b + 2cW > 0 for all W > 0 U΄΄ = 2c, a constant If c < 0 and 2c < b, then we have U΄>0

and U΄΄< 0

And now exponential utility as U(W) = -e(-aW) = -EXP[-aW] For a being a positive constant U΄ = ae-aW > 0 and is a constant U΄΄ = -a2e(-aW) and is also a constant Because the dedikind, e, is a constant

We use all these properties when we discuss Expected Utility, Risk Aversion, and applications to finance and insurance