1. Chapter Four
Utility 效用

2. 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

3. 4.1 Cardinal utility vs. ordinal utility
Cardinal Utility Theory
utility is measurable
Important concepts: total utility (TU) and marginal utility (MU)

4. 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.

5. 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.

6. 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).

7. Goods, Bads and Neutrals
Utility
Utility function
Units of water are goods
Units of water are bads
Water x’

8. Ordinal Utility Theory
Ordinal utility is the ranking of alternatives as first, second, third, and so on.
More realistic and less restrictive.

9. 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….

10. Utility Functions f ~ f ~
Definition: A utility function U(x):X->R represents a preference relation if and only if: x’ x” U(x’) ≧ U(x”) ~ f ~ f

11. 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. p

12. Utility Functions & Indiff. Curves
All bundles in an indifference curve have the same utility level.

13. Utility Functions & Indiff. Curvesx2
(2,3) (2,2) ~ (4,1) p
U º 6
U º 4 x1

14. 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.

15. Utility Functions & Indiff. Curvesx2
U º 6
U º 4
U º 2 x1

16. 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:

17. Examples of positive monotonic transformation

18. 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) = x1a x2b

19. Perfect Substitution Indifference Curvesx2
x1 + x2 = 5 13
x1 + x2 = 9 9
x1 + x2 = 13 5
u(x1,x2) = x1 + x2. 5 9 13 x1

20. Perfect Complementarity Indifference Curvesx2
45o
u(x1,x2) = min{x1,x2} 8
min{x1,x2} = 8 5
min{x1,x2} = 5 3
min{x1,x2} = 3 3 5 8 x1

21. 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）.

22. Quasi-linear Indifference Curvesx2 x1

23. Cobb-Douglas Utility Function
Any utility function of the form U(x1,x2) = x1a x2b with a > 0 and b > 0．

24. Cobb-Douglas Indifference Curvesx2 x1

25. 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.

26. Marginal Utility
E.g. U(x1,x2) = x11/2 x22

27. Derivation of MRS
The general equation for an indifference curve is U(x1,x2) º k, a constant.

28. 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).

29. Marg. Rates-of-Substitution for Quasi-linear Utility Functionsx2
MRS = f’(x1’)
MRS = f’(x1”)
MRS is a constant along any line for which x1 is constant.
x1’
x1” x1

30. Monotonic Transformations & MRS
What happens to MRS when a positive monotonic transformation is applied?

31. 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.

32. 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: