Chapter Four
Utility
效用
Structure
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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
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Cardinal Utility Theory
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utility is measurable
Important concepts: total utility (TU) and marginal
utility (MU)
Recall: TU and MU
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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
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TU is usually positive, MU can be positive or
negative.
TU increases if MU>0 but decreases if MU<0.
Goods, Bads and Neutrals
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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
Units of
water are
goods
x’
Utility
function
Units of
water are
bads
Water
Ordinal Utility Theory
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Ordinal utility is the ranking of alternatives as
first, second, third, and so on.
More realistic and less restrictive.
4.2 Utility Function
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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
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Definition: A utility function U(x):X->R
represents a preference relation f
if
and
~
only if:
f
x’ ~ x”
U(x’) ≧ U(x”)
Utility Functions & Indiff. Curves
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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
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Utility Functions & Indiff. Curves
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All bundles in an indifference curve have
the same utility level.
Utility Functions & Indiff. Curves
(2,3)
p
x2
(2,2) ~ (4,1)
U6
U4
x1
Utility Functions & Indifference map
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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
x2
U6
U4
U2
x1
4.3 Ordinal property of utility functions
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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
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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
a b
 U(x1,x2) = x1 x2
Perfect Substitution Indifference
Curves
x2
x1 + x2 = 5
13
x1 + x2 = 9
9
x1 + x2 = 13
5
u(x1,x2) = x1 + x2.
5
9
13
x1
Perfect Complementarity Indifference
Curves
x2
45o
u(x1,x2) = min{x1,x2}
min{x1,x2} = 8
8
min{x1,x2} = 5
min{x1,x2} = 3
5
3
3 5
8
x1
Quasi-Linear Utility Functions
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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 Curves
x2
x1
Cobb-Douglas Utility Function
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Any utility function of the form
U(x1,x2) = x1a x2b
with a > 0 and b > 0．
Cobb-Douglas
Indifference Curves
x
2
x1
4.5 Marginal utility (MU) and MRS
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The marginal utility of commodity i is the rateof-change of total utility as the quantity of
commodity i consumed changes; i.e.
U
MU i 
 xi
Marginal Utility
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E.g. U(x1,x2) = x11/2 x22
Derivation of MRS
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The general equation for an indifference
curve is
U(x1,x2)  k, a constant.
MRS for Quasi-linear Utility Functions
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A quasi-linear utility function is of the form
U(x1,x2) = f(x1) + x2.
U
 f ( x1 )
 x1
So MRS=f’(x1).
U
1
 x2
Marg. Rates-of-Substitution for Quasilinear Utility Functions
x2
MRS =
f’(x1’)
MRS = f’(x1”)
x1’
x1”
MRS is a
constant
along any line
for which x1 is
constant.
x1
Monotonic Transformations & MRS
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What happens to MRS when a positive
monotonic transformation is applied?
Monotonic Transformations & MRS
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For U(x1,x2) = x1x2 the MRS = x2/x1.
Create V = 2U; i.e. V(x1,x2) =2x1x2. What
is the MRS for V?
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MRS does not change.
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Monotonic Transformations & MRS
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More generally, if V = f(U) where f is a
strictly increasing function, then MRS is
unchanged by a positive monotonic
transformation.
Proof: