Utility Functions
If – U is a utility function that
represents a preference relation f~
and – f is a strictly increasing function, then V = f(U) is also a utility function
A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).
Some Other Utility Functions and Their Indifference Curves
Continuity means that small changes to a consumption bundle cause only small changes to the preference level.
Utility Functions
A utility function U(x) represents a
北京大学国家发展研究院2014年秋季学期 本科双学位课程
中级微观经济学(2班)
第三讲： 效用、选择
任课教师：李力行
第四章-效用
Utility function (效用函数） – Definition – Monotonic transformation （单调转换） – Examples of utility functions and their indifference curves
V(x1,x2) = x1 x23 (a = 1, b = 3)
Cobb-Douglas Indifference
x2
Curves
x1
Some Other Utility Functions and Their Indifference Curves
- A Cobb-Douglas utility function could be transformed into the following forms while still representing the same preference
Indifference Curves
x2 45o
W(x1,x2) = min{x1,x2}
8
min{x1,x2} = 8
5
min{x1,x2} = 5
3
min{x1,x2} = 3
35 8
x1
Some Other Utility Functions and Their Indifference Curves
Consider
V(x1,x2) = x1 + x2.
What do the indifference curves for this “perfect substitution” utility function look like?
Perfect Substitution Indifference
This complete collection of indifference curves completely represents the consumer‟s preferences.
Utility Functions & Indiff. Curves
x2
x1
Utility Functions & Indiff. Curves
A utility function of the form
U(x1,x2) = f(x1) + x2
is called quasi-linear 拟线性. E.g. U(x1,x2) = 2x11/2 + x2.
Quasi-linear Indifference Curves
x2
Each curve is a vertically shifted copy of
Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer‟s preferences.
Utility Functions & Indiff. Curves
x2
U6 U4 U2
x1
Utility Functions & Indiff. Curves
Comparing all possible consumption bundles gives the complete collection of the consumer‟s indifference curves, each with its assigned utility level.
Marginal utility （边际效用） Marginal rate of substitution 边际替代率
– MRS after monotonic transformation
Utility Functions 效用函数
A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.
Comparing all possible consumption bundles gives the complete collection of the consumer‟s indifference curves, each with its assigned utility level.
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). U(x1,x2) = x1x2 → (2,3) (4,1) ~ (2,2).
bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y.
p
Utility Functions & Indiff. Curves
Consider the bundles (4,1), (2,3) and (2,2).
U(x1,x2) = x1a/(a+b) x2b/(a+b) U(x1,x2) = alnx1 + blnx2
But the bundle (2,3) is in the indifference curve with U 6.
Utility Functions & Indiff. Curves
x2
(2,3) (2,2) ~ (4,1)
p
U6 U4
x1
Utility Functions & Indiff. Curves
consider
W(x1,x2) = min{x1,x2}.
What do the indifference curves for this “perfect complementarity” utility function look like?
Perfect Complementarity
Curves
x2
x1 + x2 = 5 13
x1 + x2 = 9 9
x1 + x2 = 13 5
V(x1,x2) = x1 + x2.
5 9 13
x1
All are linear and parallel.
Some Other Utility Functions and Their Indifference Curves
preference relation f~ if and only if:
p
x‟ x”
U(x‟) > U(x”)
x‟ p x”
U(x‟) < U(x”)
x‟ ~ x”
U(x‟) = U(x”).
Utility Functions
Utility is an ordinal 序数 concept. E.g. if U(x) = 6 and U(y) = 2 then
Utility Functions & Indiff. Curves
Equal preference same utility level.
All bundles in an indifference curve have the same utility level.
So bundles (4,1) and (2,2) are in the indifference curve with U
p p
p p
Utility Functions
Define V = U2. Then V(x1,x2) = x12x22 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. Define W = 2U + 10. Then W(x1,x2) = 2x1x2+10 W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again, (2,3) (4,1) ~ (2,2). W represents the same preferences as U and V