Chapter Four

Utility

效用

Structure

4.1 Cardinal utility vs. Ordinal utility 4.2 Utility function (效用函数） 4.3 Positive monotonic transformation （正单调转换）

4.4 Examples of utility functions and their indifference curves

4.5 Marginal utility （边际效用） and Marginal rate of substitution (MRS) 边际替代率

4.1 Cardinal utility vs. ordinal utility Cardinal Utility Theory

utility is measurable Important concepts: total utility (TU) and marginal

utility (MU)

Recall: TU and MU

TU: the sum of utility you gain from consuming each unit of product.

MU: the gain in utility obtained from consuming an additional unit of good or service.

Diminishing marginal utility: MU decreases.

Relationship between TU and MU

TU is usually positive, MU can be positive or negative.

TU increases if MU>0 but decreases if MU<0.

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

Utilityfunction

Ordinal Utility Theory

Ordinal utility is the ranking of alternatives as first, second, third, and so on.

More realistic and less restrictive.

4.2 Utility Function

A preference relation that is complete, transitive and continuous can be represented by a continuous utility function.

Utility function is a way of representing a person‘s preferences

Continuity means….

Utility Functions

Definition: A utility function U(x):X->R represents a preference relation if and only if:

x’ x” U(x’) U(x”)≧

~

~

Utility Functions & Indiff. 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 Functions & Indiff. Curves All bundles in an indifference curve have

the same utility level.

Utility Functions & Indiff. Curves

U 6U 4

(2,3) (2,2) (4,1)

x1

x2

Utility Functions & Indifference map

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

An indifference map is equivalent to a utility function.

Utility Functions & Indiff. Curves

U 6U 4U 2

x1

x2

4.3 Ordinal property of utility functions Proposition: Suppose u is a utility

function that represents a preference relation , f(u) is a strictly increasing function (i.e. f(u) is a positive monotonic transformation of u), then f(u) is a utility function that represents the same preference relation as u. Proof:

Examples of positive monotonic transformation

4.4 Examples of Utility Functions and Their Indifference Curves Perfect substitute

u(x1,x2) = x1 + x2.

Perfect complement u(x1,x2) = min{x1,x2}

Quasi-linear utility function (拟线性效用函数 ) U(x1,x2) = f(x1) + x2

Cobb-Douglas Utility Function U(x1,x2) = x1

a x2b

Perfect Substitution Indifference Curves

5

5

9

9

13

13

x1

x2

x1 + x2 = 5

x1 + x2 = 9

x1 + x2 = 13

u(x1,x2) = x1 + x2.

Perfect Complementarity Indifference Curves

x2

x1

45o

min{x1,x2} = 8

3 5 8

35

8

min{x1,x2} = 5

min{x1,x2} = 3

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

Quasi-Linear Utility Functions A utility function of the form

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

is linear in just x2 and is called quasi-linear （拟线性） .

Quasi-linear Indifference Curvesx2

x1

Cobb-Douglas Utility Function Any utility function of the form

U(x1,x2) = x1a x2

b

with a > 0 and b > 0．

Cobb-Douglas Indifference Curvesx2

x1

4.5 Marginal utility (MU) and MRS The marginal utility of commodity i is the rate-

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

MUUxii

Marginal Utility

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

2

Derivation of MRS

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

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

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

So MRS=f’(x1).

Ux

f x1

1 ( ) Ux2

1

Marg. Rates-of-Substitution for Quasi-linear Utility Functionsx2

x1

MRS is a constantalong any line for which x1 isconstant.

MRS = f’(x1’)

MRS = f’(x1”)

x1’ x1”

Monotonic Transformations & MRS What happens to MRS when a positive

monotonic transformation is applied?

Monotonic Transformations & MRS

For U(x1,x2) = x1x2 the MRS = x2/x1.

Create V = 2U; i.e. V(x1,x2) =2x1x2. What is the MRS for V?

MRS does not change.

Monotonic Transformations & MRS More generally, if V = f(U) where f is a

strictly increasing function, then MRS is unchanged by a positive monotonic transformation.

Proof: