Chapter 3

PREFERENCES AND UTILITY

Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON

Axioms of Rational Choice

• Completeness– If A and B are any two situations, an

individual can always specify exactly one of these possibilities:

• A is preferred to B• B is preferred to A• A and B are equally attractive

Axioms of Rational Choice

• Transitivity– If A is preferred to B, and B is preferred to

C, then A is preferred to C– Assumes that the individual’s choices are

internally consistent

Axioms of Rational Choice

• Continuity– If A is preferred to B, then situations “close

to” A must also be preferred to B– Used to analyze individuals’ responses to

relatively small changes in income and prices

Utility• Given these assumptions, it is possible to

show that people are able to rank in order all possible situations from least desirable to most

• Economists call this ranking utility– If A is preferred to B, then the utility assigned

to A exceeds the utility assigned to B

U(A) > U(B)

Utility• Utility rankings are ordinal in nature

– They record the relative desirability of commodity bundles

• Because utility measures are nonunique, it makes no sense to consider how much more utility is gained from A than from B

• It is also impossible to compare utilities between people

Utility• Utility is affected by the consumption of

physical commodities, psychological attitudes, peer group pressures, personal experiences, and the general cultural environment

• Economists generally devote attention to quantifiable options while holding constant the other things that affect utility– ceteris paribus assumption

Utility• Assume that an individual must choose

among consumption goods X1, X2,…, Xn

• The individual’s rankings can be shown by a utility function of the form:

utility = U(X1, X2,…, Xn)

• Keep in mind that everything is being held constant except X1, X2,…, Xn

Economic Goods• In the utility function, the X’s are assumed

to be “goods”– more is preferred to less

Quantity of X

Quantity of Y

X*

Y*

Preferred to X*, Y*

Worsethan

X*, Y*

?

?

Indifference Curves• An indifference curve shows a set of

consumption bundles among which the individual is indifferent

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

Combinations (X1, Y1) and (X2, Y2)provide the same level of utility

Marginal Rate of Substitution• The negative of the slope of the

indifference curve at any point is called the marginal rate of substitution (MRS)

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

1UUdX

dYMRS

Marginal Rate of Substitution• MRS changes as X and Y change

– reflects the individual’s willingness to trade Y for X

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

At (X1, Y1), the indifference curve is steeper.The person would be willing to give up more

Y to gain additional units of X

At (X2, Y2), the indifference curveis flatter. The person would bewilling to give up less Y to gainadditional units of X

Indifference Curve Map• Each point must have an indifference curve through it

Quantity of X

Quantity of Y

U1

U2

U3 U1 < U2 < U3

Increasing utility

Transitivity• Can two of an individual’s indifference curves intersect?

Quantity of X

Quantity of Y

U1

U2

A

BC

The individual is indifferent between A and C.The individual is indifferent between B and C.

Transitivity suggests that the individualshould be indifferent between A and B

But B is preferred to Abecause B contains more

X and Y than A

Convexity• A set of points is convex if any two points can be joined by a

straight line that is contained completely within the set

Quantity of X

Quantity of Y

U1

The assumption of a diminishing MRS isequivalent to the assumption that allcombinations of X and Y which are preferred to X* and Y* form a convex set

X*

Y*

Convexity• If the indifference curve is convex, then the combination (X1 + X2)/2, (Y1 + Y2)/2

will be preferred to either (X1,Y1) or (X2,Y2)

Quantity of X

Quantity of Y

U1

X2

Y1

Y2

X1

This implies that “well-balanced” bundles are preferredto bundles that are heavily weighted toward onecommodity

(X1 + X2)/2

(Y1 + Y2)/2

Utility and the MRS

• Suppose an individual’s preferences for hamburgers (Y) and soft drinks (X) can be represented by

YX 10 utility • Solving for Y, we get

Y = 100/X

• Solving for MRS = -dY/dX:MRS = -dY/dX = 100/X2

Utility and the MRS

MRS = -dY/dX = 100/X2

• Note that as X rises, MRS falls– When X = 5, MRS = 4– When X = 20, MRS = 0.25

Marginal Utility• Suppose that an individual has a utility

function of the form

utility = U(X1, X2,…, Xn)

• We can define the marginal utility of good X1 by

marginal utility of X1 = MUX1 = U/X1

• The marginal utility is the extra utility obtained from slightly more X1 (all else constant)

Marginal Utility• The total differential of U is

n

n

dXX

UdX

X

UdX

X

UdU

...2

2

1

1

nXXX dXMUdXMUdXMUdUn

...21 21

• The extra utility obtainable from slightly more X1, X2,…, Xn is the sum of the additional utility provided by each of these increments

Deriving the MRS• Suppose we change X and Y but keep

utility constant (dU = 0)

dU = 0 = MUXdX + MUYdY

• Rearranging, we get:

YU

XU

MU

MU

dX

dY

Y

X

/

/

constantU

• MRS is the ratio of the marginal utility of X to the marginal utility of Y

Diminishing Marginal Utility and the MRS

• Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS– Diminishing MRS requires that the utility

function be quasi-concave• This is independent of how utility is measured

– Diminishing marginal utility depends on how utility is measured

• Thus, these two concepts are different

Marginal Utility and the MRS• Again, we will use the utility function

5050 .. utility YXYX • The marginal utility of a soft drink is

marginal utility = MUX = U/X = 0.5X-0.5Y0.5

• The marginal utility of a hamburger is

marginal utility = MUY = U/Y = 0.5X0.5Y-0.5

X

Y

YX

YX

MU

MU

dX

dYMRS

Y

X

5050

5050

5

5..

..

constantU .

.

Examples of Utility Functions• Cobb-Douglas Utility

utility = U(X,Y) = XY

where and are positive constants

– The relative sizes of and indicate the relative

importance of the goods

Examples of Utility Functions• Perfect Substitutes

utility = U(X,Y) = X + Y

Quantity of X

Quantity of Y

U1

U2

U3

The indifference curves will be linear.The MRS will be constant along the indifference curve.

Examples of Utility Functions• Perfect Complements

utility = U(X,Y) = min (X, Y)

Quantity of X

Quantity of YThe indifference curves will be L-shaped. Only by choosing more of the two goods together can utility be increased.

U1

U2

U3

Examples of Utility Functions• CES Utility (Constant elasticity of

substitution)utility = U(X,Y) = X/ + Y/

when 0 andutility = U(X,Y) = ln X + ln Y

when = 0– Perfect substitutes = 1– Cobb-Douglas = 0– Perfect complements = -

Examples of Utility Functions• CES Utility (Constant elasticity of

substitution)– The elasticity of substitution () is equal to

1/(1 - )

• Perfect substitutes = • Fixed proportions = 0

CES Utility Function

One of the assumptions of the Ordinal Theory is the assumption of continuity or substitutability. How easy it is to substitute one good for another has an impact on the shape of the utility function, thus, the indifference curve.

CES Utility Function

Along an indifference curve the MRS decreases as the Y/X ratio decreases. we wish to define some parameter that measures this degree of responsiveness. If the MRS does not change at all as the ratio of Y/X changes, we might say that substitution is easy—perfect substitutes where MRS = α / β and σ = ∞.

CES Utility Function

Alternatively, if the MRS changes rapidly for small changes in Y/X ratio, we would say the substitution is difficult, because minor variations in the Y/X ratio will have a substantial effect on the good’s relative utility, thus, the shape of the indifference curve.

CES Utility Function

A scale-free measure of this responsiveness is provided by the elasticity of substitution. Elasticity of substitution (σ) is:

€

=%Δ(

y

x)

%Δ(MRS)=d(y

x)

d(MRS)*MRSy

x

€

€

=1

1−δ

€

CES Utility Function

For Cobb-Douglass Utility Function:

δ = 0 σ = 1

For Perfect Substitutes:

δ = 1 σ = ∞

For Perfect Complements:

δ = -∞ σ = 0

σ is always positive because Y/X ratio and MRS change in the same direction.

Homothetic Preferences• If the MRS depends only on the ratio of

the amounts of the two goods, not on the quantities of the goods, the utility function is homothetic– Perfect substitutes MRS is the same at

every point– Perfect complements MRS = if Y/X > /, undefined if Y/X = /, and MRS = 0 if Y/X < /

Nonhomothetic Preferences• Some utility functions do not exhibit

homothetic preferences

utility = U(X,Y) = X + ln Y

MUY = U/Y = 1/Y

MUX = U/X = 1

MRS = MUX / MUY = Y

• Because the MRS depends on the amount of Y consumed, the utility function is not homothetic

Monotonic Transformation

A monotonic transformation is a way of transforming a set of numbers to another set of numbers in a way that preserves the order of the numbers.

A monotonic transformation of a utility function is a utility function that represents the same preferences as the original utility function.

Important Points to Note:• If individuals obey certain behavioral

postulates, they will be able to rank all commodity bundles– The ranking can be represented by a utility

function– In making choices, individuals will act as if they

were maximizing this function

• Utility functions for two goods can be illustrated by an indifference curve map

Important Points to Note:• The negative of the slope of the

indifference curve measures the marginal rate of substitution (MRS)– This shows the rate at which an individual

would trade an amount of one good (Y) for one more unit of another good (X)

• MRS decreases as X is substituted for Y– This is consistent with the notion that

individuals prefer some balance in their consumption choices

Important Points to Note:• A few simple functional forms can capture

important differences in individuals’ preferences for two (or more) goods– Cobb-Douglas function– linear function (perfect substitutes)– fixed proportions function (perfect

complements)– CES function

• includes the other three as special cases