Lecture 1. Preferences and Utility

7/30/2019 Lecture 1. Preferences and Utility

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Nicholson and Snyder, Copyright 2008 by Thomson South-Western. All rights reserved.

Preferences and Utility

Chapter 3


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Axioms of Rational Choice Completeness

if A and B are any two situations, an

individual can always specify exactly one ofthese possibilities:

A is preferred to B

B is preferred to A

A and B are equally attractive


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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 individuals choices are

internally consistent


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Axioms of Rational Choice Continuity

if A is preferred to B, then situations suitably

close to A must also be preferred to B The assumption ensures the continuity of

the consumers preferences and rules out

jumps.


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Utility Given these assumptions, it is possible to

show that people are able to rank 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)


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Utility Utility is affected by

the consumption of physical commodities

psychological attitudes peer group pressures

personal experiences

the general cultural environment


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Utility Economists generally devote attention to

quantifiable options while holding

constant the other things that affect utility ceteris paribusassumption


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Utility Assume that an individual must chooseamong consumption goods x1, x2,, xn

We can show his rankings using a utilityfunction of the form:

utility = U(x1, x2,, xn; other things)

Often other things are held constant

utility = U(x1, x2,, xn)


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Utility We can assume the individual isconsidering two goods, xand y

utility = U(x,y)


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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

1

UUdx

dyMRS


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Marginal Rate of Substitution MRS changes as xand ychange

reflects the individuals willingness to trade yforx

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

yto gain additional units ofx

At (x2, y2), the indifference curveis flatter. The person would be

willing to give up less yto gainadditional units ofx


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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


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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 A

because B contains more

xand ythan 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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Utility and the MRS Suppose an individuals preferences forhamburgers (y) and soft drinks (x) canbe represented by

yx 10utility

Solving fory, we get

y= 100/x

Solving for MRS = -dy/dx(along U1):MRS= -dy/dx= 100/x2


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Utility and the MRSMRS= -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(x,y)

The total differential ofUis

dyy

Udx

x

UdU

Along any indifference curve, utility is

constant (dU= 0)


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Deriving the MRS Therefore, we get:

yUxU

dxdyMRS

constantU

MRSis the ratio of the marginal utility ofxto the marginal utility ofy


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Convexity of Indifference

Curves Suppose that the utility function is

yxutility

We can simplify the algebra by taking the

logarithm of this function

U*(x,y) = ln[U(x,y)] = 0.5 ln x+ 0.5 ln y


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Convexity of IndifferenceCurves

x

y

y

x

yU

xUMRS

5.0

5.0

*

*

Thus,


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Convexity of IndifferenceCurves

If the utility function is

U(x,y) = x + xy + y

There is no advantage to transforming

this utility function, so

x

y

yU

xUMRS

1

1


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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)

when

1,

0

utility = U(x,y) = x/ + y/

Perfect substitutes = 1

Cobb-Douglas = 0

Perfect complements = -


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Important Points to Note: If individuals obey certain behavioralpostulates, 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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Important Points to Note: The negative of the slope of the

indifference curve measures the marginal

rate of substitution (MRS)

the rate at which an individual would trade

an amount of one good (y) for one more unitof another good (x)

MRSdecreases as xis substituted fory individuals prefer some balance in their

consumption choices


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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

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Lecture 1. Preferences and Utility