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

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

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