indifference curves.s05

7/31/2019 Indifference Curves.s05

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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


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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

1UU

dX

dYMRS


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Marginal Rate of Substitution MRS changes as Xand Ychange

reflects the individuals willingness to trade Y

forX

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


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Indifference Curve Map Each point must have an indifference

curve through it

Quantity of X

Quantity of Y

U1

U2U3 U1 < U2 < U3

Increasing utility


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Transitivity Can two of an individuals 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 individual

should be indifferent between A and B

But B is preferred to Abecause B contains more

Xand Y than A


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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 is

equivalent to the assumption that all

combinations ofXand Ywhich are

preferred to X* and Y* form a convex set

X*

Y*


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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 preferred

to bundles that are heavily weighted toward one

commodity

(X1 + X2)/2

(Y1 + Y2)/2


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Utility and the MRS

Suppose an individuals preferences for

hamburgers (Y) and soft drinks (X) can

be represented by

YX 10utility

Solving forY, we get

Y= 100/X

Solving for MRS = -dY/dX:

MRS= -dY/dX= 100/X2


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Utility and the MRS

MRS= -dY/dX= 100/X2

Note that as Xrises, MRSfalls

When X= 5, MRS= 4 When X= 20, MRS= 0.25


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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 ofX1 = MUX1 = U/X1

The marginal utility is the extra utility

obtained from slightly more X1 (all else

constant)


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Marginal Utility

The total differential ofUis

n

n

dXX

UdX

X

UdX

X

UdU

...

2

2

1

1

nXXXdXMUdXMUdXMUdU

n

...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


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Deriving the MRS

Suppose we change Xand Ybut keep

utility constant (dU= 0)

dU= 0 = MUXdX+ MU

YdY

Rearranging, we get:

YU

XU

MU

MU

dX

dY

Y

X

/

/

constantU

MRSis the ratio of the marginal utility of

Xto the marginal utility ofY


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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 MRSrequires that the utilityfunction be quasi-concave

This is independent of how utility is measured

Diminishing marginal utility depends on howutility is measured

Thus, these two concepts are different


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

5050 ..utility YXYX

The marginal utility of a soft drink ismarginal utility = MUX= U/X= 0.5X

-

0.5Y0.5

The marginal utility of a hamburger is

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

0.5X

Y

YX

YX

MU

MU

dX

dYMRS

Y

X

5050

5050

5

5

..

..

constantU .

.


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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


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Perfect Substitutes

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

Quantity of X

Quantity of Y

U1U2

U3

The indifference curves will be linear.

The MRS will be constant along the

indifference curve.


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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


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Examples of Utility Functions CES Utility (Constant elasticity of

substitution)

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

when 0 and

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

when = 0 Perfect substitutes = 1

Cobb-Douglas = 0

Perfect complements = -


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Examples of Utility Functions CES Utility (Constant elasticity of

substitution)

The elasticity of substitution () is equal to

1/(1 - )

Perfect substitutes =

Fixed proportions = 0


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Homothetic Preferences

If the MRSdepends 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 MRSis the same at

every point

Perfect complements MRS= ifY/X>/, undefined ifY/X= /, and MRS= 0 if

Y/X< /


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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 MRSdepends on the

amount ofYconsumed, the utility function

is not homothetic


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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


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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)

MRSdecreases as Xis substituted forY This is consistent with the notion that

individuals prefer some balance in their

consumption choices


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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

indifference curves.s05

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