Chapter Four
Utility
效用
Structure
? Utility function (效用函数）
– Definition
– Monotonic transformation （ 单调转换）
– Examples of utility functions and their
indifference curves
? Marginal utility （ 边际效用）
? Marginal rate of substitution 边际替代率
– MRS after monotonic transformation
Utility Functions
?A utility function U(x) represents a
preference relation if and only if:
x’ x” U(x’) > U(x”)
x’ x” U(x’) < U(x”)
x’ ~ x” U(x’) = U(x”).
~f
p
p
Utility Functions
?Utility is an ordinal (i.e,ordering)
concept,[序数效用 ]
?E.g,if U(x) = 6 and U(y) = 2 then
bundle x is strictly preferred to
bundle y,But x is not preferred three
times as much as is y.
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
Utility Functions & Indiff,Curves
?An indifference curve contains
equally preferred bundles.
?Equal preference ? same utility level.
?Therefore,all bundles in an
indifference curve have the same
utility level.
Utility Functions & Indiff,Curves
?So the bundles (4,1) and (2,2) are in
the indiff,curve with utility level U ???
?But the bundle (2,3) is in the indiff,
curve with utility level U ? 6.
?On an indifference curve diagram,
this preference information looks as
follows:
Utility Functions & Indiff,Curves
U ? 6
U ? 4
(2,3) (2,2) ~ (4,1)
x1
x2 p
Utility Functions & Indiff,Curves
?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
U ? 6
U ? 4
U ? 2
x1
x2
Utility Functions & Indiff,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
?There is no unique utility function
representation of a preference
relation.
?Suppose U(x1,x2) = x1x2 represents a
preference relation.
?Again consider the bundles (4,1),
(2,3) and (2,2).
Utility Functions
?U(x1,x2) = x1x2,so
U(2,3) = 6 > U(4,1) = U(2,2) = 4;
that is,(2,3) (4,1) ~ (2,2).p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?Define V = U2.
p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?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.
p
p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?Define W = 2U + 10.
p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?Define W = 2U + 10.
?Then W(x1,x2) = 2x1x2+10 so
W(2,3) = 22 > W(4,1) = W(2,2) = 18,
Again,
(2,3) (4,1) ~ (2,2).
?W preserves the same order as U and V
and so represents the same preferences.
p
p
Utility Functions,Monotonic
Transformation
?If
–U is a utility function that
represents a preference relation
and
– f is a strictly increasing function,
? then V = f(U) is also a utility function
representing,
~f
~f
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 of
water are
goods
Units of
water are
bads
Around x’ units,a little extra water is a neutral.
Utility
function
Some Other Utility Functions and
Their Indifference Curves
? Perfect substitute
– V(x1,x2) = x1 + x2.
? Perfect complement
– W(x1,x2) = min{x1,x2}
? Quasi-linear
– U(x1,x2) = f(x1) + x2
? Cobb-Douglas Utility Function
– U(x1,x2) = x1a x2b
? What do the indifference curves for these
utility functions look like?
Perfect Substitution Indifference
Curves
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 Complementarity
Indifference Curvesx
2
x1
45o
min{x1,x2} = 8
3 5 8
3
5
8
min{x1,x2} = 5
min{x1,x2} = 3
All are right-angled with vertices on a ray
from the origin.
W(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 （ 准线性）,
?E.g,U(x1,x2) = 2x11/2 + x2.
Quasi-linear Indifference Curves
x2
x1
Each curve is a vertically shifted
copy of the others.
Cobb-Douglas Utility Function
?Any utility function of the form
U(x1,x2) = x1a x2b
with a > 0 and b > 0 is called a Cobb-
Douglas utility function.
?E.g,U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
V(x1,x2) = x1 x23 (a = 1,b =
3)
Cobb-Douglas Indifference
Curvesx2
x1
Marginal Utilities
?Marginal means,incremental”.
?The marginal utility of commodity i is
the rate-of-change of total utility as
the quantity of commodity i
consumed changes; i.e.MU U
xi i
? ?
?
Marginal Utilities
?If U(x1,x2) = x11/2 x22 then
MU
U
x
x x
MU
U
x
x x
1
1
1
1 2
2
2
2
2
1
1 2
2
1
2
2
? ?
? ?
??
?
?
?
/
/
Marginal Utilities and Marginal
Rates-of-Substitution
?The general equation for an
indifference curve is
U(x1,x2) ? k,a constant.
Totally differentiating this identity gives?
?
?
?
U
x
dx U
x
dx
1
1
2
2 0? ?
Marginal Utilities and Marginal
Rates-of-Substitution?
?
?
?
U
x
dx U
x
dx
1
1
2
2 0? ?
rearranged isd x
d x
U x
U x
2
1
1
2
? ? ? ?
? ?
/
/
.
This is the MRS.
Marg,Utilities & Marg,Rates-of-
Substitution; An example
?Suppose U(x1,x2) = x1x2,Then
?
?
?
?
U
x
x x
U
x
x x
1
2 2
2
1 1
1
1
? ?
? ?
( )( )
( )( )
M RS d x
d x
U x
U x
x
x
? ? ? ? ?2
1
1
2
2
1
? ?
? ?
/
/
.
so
Marg,Utilities & Marg,Rates-of-
Substitution; An exampleMRS x
x
? ? 2
1
MRS(1,8) = - 8/1 = -8
MRS(6,6) = - 6/6 = -1.
x1
x2
8
6
1 6
U = 8
U = 36
U(x1,x2) = x1x2;
Marg,Rates-of-Substitution for
Quasi-linear Utility Functions
?A quasi-linear utility function is of
the form U(x1,x2) = f(x1) + x2.
so
?
?
U
x
f x
1
1? ? ( )
?
?
U
x 2
1?
M RS d x
d x
U x
U x
f x? ? ? ? ? ?2
1
1
2
1
? ?
? ?
/
/
( ).
Marg,Rates-of-Substitution for
Quasi-linear Utility Functionsx
2
x1
MRS = - f (x1) does not
depend upon x2.
MRS is a
constant
along any line
for which x1 is
constant.
MRS =
- f(x1’)
MRS = -f(x1”)
x1’ x1”
Monotonic Transformations &
Marginal Rates-of-Substitution
?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?
Monotonic Transformations &
Marginal Rates-of-Substitution
?For U(x1,x2) = x1x2 the MRS = - x2/x1.
?Create V = U2; i.e,V(x1,x2) = x12x22,
What is the MRS for V?
which is the same as the MRS for U.
M RS
V x
V x
x x
x x
x
x
? ? ? ? ? ?
? ?
? ?
/
/
1
2
1 2
2
1
2
2
2
1
2
2
Monotonic Transformations &
Marginal Rates-of-Substitution
?More generally,if V = f(U) where f is a
strictly increasing function,thenM RS V x
V x
f U U x
f U U x
? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
/
/
( ) /
' ( ) /
1
2
1
2? ? ? ?
? ?
U x
U x
/
/
.1
2
So MRS is unchanged by a positive
monotonic transformation.
Utility
效用
Structure
? Utility function (效用函数）
– Definition
– Monotonic transformation （ 单调转换）
– Examples of utility functions and their
indifference curves
? Marginal utility （ 边际效用）
? Marginal rate of substitution 边际替代率
– MRS after monotonic transformation
Utility Functions
?A utility function U(x) represents a
preference relation if and only if:
x’ x” U(x’) > U(x”)
x’ x” U(x’) < U(x”)
x’ ~ x” U(x’) = U(x”).
~f
p
p
Utility Functions
?Utility is an ordinal (i.e,ordering)
concept,[序数效用 ]
?E.g,if U(x) = 6 and U(y) = 2 then
bundle x is strictly preferred to
bundle y,But x is not preferred three
times as much as is y.
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
Utility Functions & Indiff,Curves
?An indifference curve contains
equally preferred bundles.
?Equal preference ? same utility level.
?Therefore,all bundles in an
indifference curve have the same
utility level.
Utility Functions & Indiff,Curves
?So the bundles (4,1) and (2,2) are in
the indiff,curve with utility level U ???
?But the bundle (2,3) is in the indiff,
curve with utility level U ? 6.
?On an indifference curve diagram,
this preference information looks as
follows:
Utility Functions & Indiff,Curves
U ? 6
U ? 4
(2,3) (2,2) ~ (4,1)
x1
x2 p
Utility Functions & Indiff,Curves
?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
U ? 6
U ? 4
U ? 2
x1
x2
Utility Functions & Indiff,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
?There is no unique utility function
representation of a preference
relation.
?Suppose U(x1,x2) = x1x2 represents a
preference relation.
?Again consider the bundles (4,1),
(2,3) and (2,2).
Utility Functions
?U(x1,x2) = x1x2,so
U(2,3) = 6 > U(4,1) = U(2,2) = 4;
that is,(2,3) (4,1) ~ (2,2).p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?Define V = U2.
p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?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.
p
p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?Define W = 2U + 10.
p
Utility Functions
?U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
?Define W = 2U + 10.
?Then W(x1,x2) = 2x1x2+10 so
W(2,3) = 22 > W(4,1) = W(2,2) = 18,
Again,
(2,3) (4,1) ~ (2,2).
?W preserves the same order as U and V
and so represents the same preferences.
p
p
Utility Functions,Monotonic
Transformation
?If
–U is a utility function that
represents a preference relation
and
– f is a strictly increasing function,
? then V = f(U) is also a utility function
representing,
~f
~f
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 of
water are
goods
Units of
water are
bads
Around x’ units,a little extra water is a neutral.
Utility
function
Some Other Utility Functions and
Their Indifference Curves
? Perfect substitute
– V(x1,x2) = x1 + x2.
? Perfect complement
– W(x1,x2) = min{x1,x2}
? Quasi-linear
– U(x1,x2) = f(x1) + x2
? Cobb-Douglas Utility Function
– U(x1,x2) = x1a x2b
? What do the indifference curves for these
utility functions look like?
Perfect Substitution Indifference
Curves
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 Complementarity
Indifference Curvesx
2
x1
45o
min{x1,x2} = 8
3 5 8
3
5
8
min{x1,x2} = 5
min{x1,x2} = 3
All are right-angled with vertices on a ray
from the origin.
W(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 （ 准线性）,
?E.g,U(x1,x2) = 2x11/2 + x2.
Quasi-linear Indifference Curves
x2
x1
Each curve is a vertically shifted
copy of the others.
Cobb-Douglas Utility Function
?Any utility function of the form
U(x1,x2) = x1a x2b
with a > 0 and b > 0 is called a Cobb-
Douglas utility function.
?E.g,U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
V(x1,x2) = x1 x23 (a = 1,b =
3)
Cobb-Douglas Indifference
Curvesx2
x1
Marginal Utilities
?Marginal means,incremental”.
?The marginal utility of commodity i is
the rate-of-change of total utility as
the quantity of commodity i
consumed changes; i.e.MU U
xi i
? ?
?
Marginal Utilities
?If U(x1,x2) = x11/2 x22 then
MU
U
x
x x
MU
U
x
x x
1
1
1
1 2
2
2
2
2
1
1 2
2
1
2
2
? ?
? ?
??
?
?
?
/
/
Marginal Utilities and Marginal
Rates-of-Substitution
?The general equation for an
indifference curve is
U(x1,x2) ? k,a constant.
Totally differentiating this identity gives?
?
?
?
U
x
dx U
x
dx
1
1
2
2 0? ?
Marginal Utilities and Marginal
Rates-of-Substitution?
?
?
?
U
x
dx U
x
dx
1
1
2
2 0? ?
rearranged isd x
d x
U x
U x
2
1
1
2
? ? ? ?
? ?
/
/
.
This is the MRS.
Marg,Utilities & Marg,Rates-of-
Substitution; An example
?Suppose U(x1,x2) = x1x2,Then
?
?
?
?
U
x
x x
U
x
x x
1
2 2
2
1 1
1
1
? ?
? ?
( )( )
( )( )
M RS d x
d x
U x
U x
x
x
? ? ? ? ?2
1
1
2
2
1
? ?
? ?
/
/
.
so
Marg,Utilities & Marg,Rates-of-
Substitution; An exampleMRS x
x
? ? 2
1
MRS(1,8) = - 8/1 = -8
MRS(6,6) = - 6/6 = -1.
x1
x2
8
6
1 6
U = 8
U = 36
U(x1,x2) = x1x2;
Marg,Rates-of-Substitution for
Quasi-linear Utility Functions
?A quasi-linear utility function is of
the form U(x1,x2) = f(x1) + x2.
so
?
?
U
x
f x
1
1? ? ( )
?
?
U
x 2
1?
M RS d x
d x
U x
U x
f x? ? ? ? ? ?2
1
1
2
1
? ?
? ?
/
/
( ).
Marg,Rates-of-Substitution for
Quasi-linear Utility Functionsx
2
x1
MRS = - f (x1) does not
depend upon x2.
MRS is a
constant
along any line
for which x1 is
constant.
MRS =
- f(x1’)
MRS = -f(x1”)
x1’ x1”
Monotonic Transformations &
Marginal Rates-of-Substitution
?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?
Monotonic Transformations &
Marginal Rates-of-Substitution
?For U(x1,x2) = x1x2 the MRS = - x2/x1.
?Create V = U2; i.e,V(x1,x2) = x12x22,
What is the MRS for V?
which is the same as the MRS for U.
M RS
V x
V x
x x
x x
x
x
? ? ? ? ? ?
? ?
? ?
/
/
1
2
1 2
2
1
2
2
2
1
2
2
Monotonic Transformations &
Marginal Rates-of-Substitution
?More generally,if V = f(U) where f is a
strictly increasing function,thenM RS V x
V x
f U U x
f U U x
? ? ? ? ? ?
?
? ?
? ?
? ?
? ?
/
/
( ) /
' ( ) /
1
2
1
2? ? ? ?
? ?
U x
U x
/
/
.1
2
So MRS is unchanged by a positive
monotonic transformation.
