Utility

UTILITY FUNCTIONS

A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function (as an alternative, or as a complement, to the indifference “map” of the previous lecture).

Continuity means that small changes to a consumption bundle cause only small changes to the preference (utility) level.

UTILITY FUNCTIONS

A utility function U(x) represents a preference relation if and only if:

x0 x1 U(x0) > U(x1)

x0 x1 U(x0) < U(x1)

x0 x1 U(x0) = U(x1)

~

UTILITY FUNCTIONS

Utility is an ordinal (i.e. ordering or ranking) concept.

For example, if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. However, x is not necessarily “three times better” than y.

UTILITY FUNCTIONSand INDIFFERENCE CURVES

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

Suppose (2,3) > (4,1) (2,2). Assign to these bundles any

numbers that preserve the preference ordering;e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.

Call these numbers utility levels.

UTILITY FUNCTIONSand INDIFFERENCE CURVES

An indifference curve contains equally preferred bundles.

Equal preference same utility level. Therefore, all bundles on an

indifference curve have the same utility level.

UTILITY FUNCTIONSand INDIFFERENCE CURVES

So the bundles (4,1) and (2,2) are on the indifference curve with utility level U

But the bundle (2,3) is on the indifference curve with utility level U 6

UTILITY FUNCTIONSand INDIFFERENCE CURVES

U 6U 4

(2,3) > (2,2) = (4,1)

xx11

x2

x1

x2

UTILITY FUNCTIONSand INDIFFERENCE CURVES

Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’s preferences.

UTILITY FUNCTIONSand INDIFFERENCE CURVES

U 6U 4U 2xx1

1

x2x2

UTILITY FUNCTIONSand INDIFFERENCE CURVES

Comparing all possible consumption bundles gives the complete collection of the consumer’s indifference curves, each with its assigned utility level.

This complete collection of indifference curves completely represents the consumer’s preferences.

UTILITY FUNCTIONSand INDIFFERENCE CURVES

The collection of all indifference curves for a given preference relation is an indifference map.

An indifference map is equivalent to a utility function; each is the other.

UTILITY FUNCTIONS

If (i) U is a utility function that represents a preference relation; and (ii) f is a strictly increasing function,then V = f(U) is also a utility functionrepresenting the original preference function.Example? V = 2.U

GOODS, BADS and NEUTRALS

A good is a commodity unit which increases utility (gives a more preferred bundle).

A bad is a commodity unit which decreases utility (gives a less preferred bundle).

A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

GOODS, BADS and NEUTRALS

Utility

Waterx’

Units ofwater aregoods

Units ofwater arebads

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

Utilityfunction

UTILITY FUNCTIONS

0)b 0,(a , 2121 baxxxxU

bx, 2121 axxxU

min, 2,121 xxxxU

Cobb-Douglas Utility Function

Perfect Substitutes Utility Function

Perfect Complements Utility Function

Note: MRS = (-)a/b

Note: MRS = ?

UTILITYPreferences can be represented by a utility function if the functional form has certain “nice” properties

Example: Consider U(x1,x2)= x1.x2

1. u/x1>0 and u/x2>0

2. Along a particular indifference curve

x1.x2 = constant x2=c/x1

As x1 x2i.e. downward sloping indifference curve

UTILITY

Example U(x1,x2)= x1.x2 =16

X1 X2 MRS

1 16

2 8 (-) 8

3 5.3 (-) 2.7

4 4 (-) 1.3

5 3.2 (-) 0.8

3. As X1 MRS (in absolute terms), i.e convex preferences

COBB DOUGLAS UTILITY FUNCTION

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.

Examples

U(x1,x2) = x11/2 x2

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

3 (a = 1, b = 3)

COBB DOUBLAS INDIFFERENCE CURVES

x2

x1

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

PERFECT SUBSITITUTES

Instead of U(x1,x2) = x1x2 consider

V(x1,x2) = x1 + x2.

PERFECT SUBSITITUTES

5

5

9

9

13

13

x1

x2

x1 + x2 = 5

x1 + x2 = 9

x1 + x2 = 13

All are linear and parallel.

V(x1,x2) = x1 + x2

PERFECT COMPLEMENTS

Instead of U(x1,x2) = x1x2 or V(x1,x2) = x1 + x2, consider

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

PERFECT COMPLEMENTSx2

x1

45o

min{x1,x2} = 8

3 5 8

35

8

min{x1,x2} = 5

min{x1,x2} = 3

All are right-angled with vertices/cornerson a ray from the origin.

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

MARGINAL UTILITY

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

the rate-of-change of total utility as the quantity of product i consumed changes by one unit; i.e.

MUUxii

MARGINAL UTILITY

U=(x1,x2)

MU1=U/x1 U=MU1.x1

MU2=U/x2 U=MU2.x2

Along a particular indifference curve

U = 0 = MU1(x1) + MU2(x2)

x2/x1 {= MRS} = (-)MU1/MU2

MARGINAL UTILITY

E.g. if U(x1,x2) = x11/2 x2

2 then

22

2/11

11 2

1xx

x

UMU

MARGINAL UTILITY

E.g. if U(x1,x2) = x11/2 x2

2 then

MUUx

x x11

11 2

221

2

/

MARGINAL UTILITY

E.g. if U(x1,x2) = x11/2 x2

2 then

MUUx

x x22

11 2

22

/

MARGINAL UTILITY

E.g. if U(x1,x2) = x11/2 x2

2 then

MUUx

x x22

11 2

22

/

MARGINAL UTILITY

So, if U(x1,x2) = x11/2 x2

2 then

22/1

12

2

22

2/11

11

2

2

1

xxx

UMU

xxx

UMU

MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION

The general equation for an indifference curve is U(x1,x2) k, a constant

Totally differentiating this identity gives

Uxdx

Uxdx

11

22 0

MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION

Uxdx

Uxdx

11

22 0

Uxdx

Uxdx

22

11

rearranging

MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION

Uxdx

Uxdx

22

11

rearranging

2

1

1

2

/

/

xU

xU

xd

xd

This is the MRS.

MU’s and MRS: An example

Suppose U(x1,x2) = x1x2. Then

Ux

x x

Ux

x x

12 2

21 1

1

1

( )( )

( )( )

1

2

2

1

1

2

/

/

x

x

xU

xU

xd

xdMRS

so

MU’s and MRS: An example

MRSxx

2

1

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;

MONOTONIC TRANSFORMATIONS AND MRS

Applying a monotonic 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? (Hopefully, nothing)

MONOTONIC TRANSFORMATIONS AND MRS

For U(x1,x2) = x1x2 the 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.

MRSV xV x

x x

x x

xx

//

1

2

1 22

12

2

2

1

2

2

MONOTONIC TRANSFORMATIONS AND MRS

More generally, if V = f(U) where f is a strictly increasing function, then

MRSV xV x

f U U xf U U x

//

( ) /'( ) /

1

2

1

2

U xU x

//

.1

2So MRS is unchanged by a positivemonotonic transformation.