Chapter 4Utility

2

Introduction

Last chapter we talk about preference, describing the ordering of what a consumer prefers.

For a more convenient mathematical treatment, we turn this ordering into a mathematical function.

3

Utility Functions

A utility function U: R+nR maps each

consumption bundle of n goods into a real number that satisfies the following conditions: x’ x” U(x’) > U(x”)

x’ x” U(x’) < U(x”)

x’ x” U(x’) = U(x”).

4

Utility Functions

Not all theoretically possible preferences have a utility function representation.

Technically, a preference relation that is complete, transitive and continuous has a corresponding continuous utility function.

5

Utility Functions

Utility is an ordinal (i.e. ordering) concept. The number assigned only matters about ranking, but the sizes of numerical differences are not meaningful.

For example, if U(x) = 6 and U(y) = 2, then bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y.

6

An Example

Consider only three bundles A, B, C. The following three are all valid

utility functions of the preference.

7

Utility Functions There is no unique utility function

representation of a preference relation. Suppose U(x1,x2) = x1x2 represents a

preference relation. Consider the bundles (4,1), (2,3) and

(2,2). U(2,3) = 6 > U(4,1) = U(2,2) = 4.

That is, (2,3) (4,1) (2,2).

8

Utility Functions Define V = U2. Then V(x1,x2) = x1

2x22 and

V(2,3) = 36 > V(4,1) = V(2,2) = 16.So again, (2,3) (4,1) (2,2).

V preserves the same order as U and so represents the same preferences.

9

Utility Functions Define W = 2U + 10. Then W(x1,x2) = 2x1x2+10. So,

W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again,

(2,3) (4,1) (2,2). W preserves the same order as U and V

and so represents the same preferences.

10

Utility Functions If

U is a utility function that represents a preference relation and

f is a strictly increasing function,

then V = f(U) is also a utility functionrepresenting .

Clearly, V(x)>V(y) if and only if f(V(x)) > f(V(y)) by definition of increasing function.

~

~

11

12

Ordinal vs. Cardinal

As you will see, for our analysis of consumer choices, an ordinal utility is enough.

If the numerical differences are also meaningful, we call it cardinal.

For example, money, weight, height are all cardinal.

13

Utility Functions & Indiff. Curves An indifference curve contains equally

preferred bundles.

Equal preference same utility level.

Therefore, all bundles on the same indifference curve must have the same utility level.

14

Utility Functions & Indiff. Curves

15

Utility Functions & Indiff. Curves

U=6

U=5

U=4U=3U=2U=1

16

Goods, Bads and Neutrals A good is a commodity which

increases your utility (gives a more preferred bundle) when you have more of it.

A bad is a commodity which decreases your utility (gives a less preferred bundle) when you have more of it.

A neutral is a commodity which does not change your utility (gives an equally preferred bundle) when you have more of it.

17

Goods, Bads and NeutralsUtility

Waterx’

Units ofwater aregoods

Units ofwater arebads

Around x’ units, a little extra water is a neutral.

Utilityfunction

18

Some Other Utility Functions and Their Indifference Curves Consider

V(x1,x2) = x1 + x2.

What do the indifference curves look like?

What relation does this function represent for these two goods?

19

Perfect Substitutes

5

5

9

9

13

13

x1

x2

x1 + x2 = 5

x1 + x2 = 9

x1 + x2 = 13

V(x1,x2) = x1 + x2.

These two goods are perfect substitutes for this consumer.

20

Some Other Utility Functions and Their Indifference Curves Consider

W(x1,x2) = min{x1,x2}. What do the indifference curves look

like? What relation does this function

represent for these two goods?

21

Perfect Complementsx2

x1

45o

min{x1,x2} = 8

3 5 8

35

8

min{x1,x2} = 5min{x1,x2} = 3

W(x1,x2) = min{x1,x2}

These two goods are perfect complements for this consumer.

22

Perfect Substitutes and Perfect Complements In general, a utility function for

perfect substitutes can be expressed as u (x, y) = ax + by;

And a utility function for perfect complements can be expressed as: u (x, y) = min{ ax , by }for constants a and b.

23

Some Other Utility Functions and Their Indifference Curves A utility function of the form

U(x1,x2) = f(x1) + x2

is linear in just x2 and is called quasi-linear.

For example, U(x1,x2) = 2x11/2 + x2.

24

Quasi-linear Indifference Curvesx2

x1

Each curve is a vertically shifted copy of the others.

25

Some Other Utility Functions and Their Indifference Curves Any utility function of the form

U(x1,x2) = x1a x2

b

with a > 0 and b > 0 is called a Cobb-Douglas utility function.

For example, U(x1,x2) = x11/2 x2

1/2, (a = b = 1/2), and V(x1,x2) = x1 x2

3 , (a = 1, b = 3).

26

Cobb-Douglas Indifference Curves

x2

x1

All curves are hyperbolic,asymptoting to, but nevertouching any axis.

27

Cobb-Douglas Utility Functions By a monotonic transformation V=ln(U):

U( x, y) = xa y b implies V( x, y) = a ln (x) + b ln(y).

Consider another transformation W=U1/(a+b)

W( x, y)= xa/(a+b) y b/(a+b) = xc y 1-c so that the sum of the indices becomes 1.

28

Marginal Utilities

Marginal means “incremental”. The marginal utility of commodity i is

the rate-of-change of total utility as the quantity of commodity i consumed changes; i.e.

MU

Uxii

29

Marginal Utilities

For example, if U(x1,x2) = x11/2 x2

2, then

22/1

12

2

22

2/11

11

2

2

1

xxx

UMU

xxx

UMU

30

Marginal Utilities and Marginal Rates of Substitution The general equation for an

indifference curve is U(x1,x2) k, a constant.

Totally differentiating this identity gives

Uxdx

Uxdx

11

22 0

31

Marginal Utilities and Marginal Rates of Substitution

Uxdx

Uxdx

11

22 0

Uxdx

Uxdx

22

11

It can be rearranged to

32

Marginal Utilities and Marginal Rates of Substitution

Uxdx

Uxdx

22

11

can be furthermore rearranged to

And

d xd x

U xU x

2

1

1

2

//

.

This is the MRS (slope of the indifferencecurve).

33

A Note In some texts, economists refer to

the MRS by its absolute value – that is, as a positive number.

However, we will still follow our convention.

Therefore, the MRS is2 1 1

1 2 2

/

/U

dx U x MUMRS

dx U x MU

34

MRS and MU Recall that MRS measures how many

units of good 2 you’re willing to sacrifice for one more unit of good 1 to remain the original utility level.

One unit of good 1 is worth MU1. One unit of good 2 is worth MU2.

Number of units of good 2 you are willing to sacrifice for one unit of good 1 is thus MU1 / MU2.

35

An Example Suppose U(x1,x2) = x1x2. Then

Ux

x x

Ux

x x

12 2

21 1

1

1

( )( )

( )( )

so 2 1 2

1 2 1

/.

/

dx U x xMRS

dx U x x

36

An Example

MRS(1,8) = -8/1 = -8 MRS(6,6) = - 6/6 = -1.

x1

x2

8

6

1 6U = 8

U = 36

U(x1,x2) = x1x2;2

1

xMRS

x

37

MRS for Quasi-linear Utility Functions A quasi-linear utility function is of the

form U(x1,x2) = f(x1) + x2.

Therefore,

Ux

f x1

1 ( )Ux2

1

2 11

1 2

/( ).

/

dx U xMRS f x

dx U x

38

MRS for Quasi-linear Utility Functions MRS = - f′(x1) depends only on x1 but

not on x2. So the slopes of the indifference curves for a quasi-linear utility function are constant along any line for which x1 is constant.

What does that make the indifference map for a quasi-linear utility function look like?

39

MRS for Quasi-linear Utility Functions

x2

x1

Each curve is a vertically shifted copy of the others.

MRS is a constantalong any line for which x1 isconstant.

MRS = -f(x1’)

MRS = -f(x1”)

x1’ x1”

40

Monotonic Transformations & Marginal Rates of Substitution Applying a monotonic (increasing)

transformation to a utility function representing a preference relation simply creates another utility function representing the same preference relation.

What happens to marginal rates of substitution when a monotonic transformation is applied?

41

Monotonic Transformations & Marginal Rates of Substitution For U(x1,x2) = x1x2 , MRS = -x2/x1.

Create V = U2; i.e. V(x1,x2) = x12x2

2. What is the MRS for V?

which is the same as the MRS for U.

21 1 2 2

22 1 2 1

/ 2

/ 2

V x x x xMRS

V x x x x

42

Monotonic Transformations & Marginal Rates of Substitution More generally, if V = f(U) where f is

a strictly increasing function, then

The MRS does not change with a monotonic transformation. Thus, the same preference with different utility functions still show the same MRS.

1 1

2 2

/ ( ) /

/ ' ( ) /

V x f U U xMRS

V x f U U x

1

2

/.

/

U x

U x