Chapter Four
Utility
效用
What Do We Do in This
Chapter?
? We create a mathematical measure
of preference in order to advance
our analysis.
Utility Functions
?A preference relation that is
complete,reflexive,transitive can be
represented by a utility function.
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.
?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
?All bundles in an indifference curve
have the same utility level.
?U(x1,x2)=Constant is the equation of
an indifference curve.
Utility Functions & Indiff,Curves
U ? 6
U ? 4
(2,3) (2,2) ~ (4,1)
x1
x2 p
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.
Utility Functions
?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
?Consider
V(x1,x2) = x1 + x2.
What do the indifference curves for
this,perfect substitution” utility
function look like?
Perfect Substitution Indifference
Curves
5
5
9
9
13
13
x1
x2
x1 + x2 = 5
x1 + x2 = 9
x1 + x2 = 13
V(x1,x2) = x1 + x2.
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.
Some Other Utility Functions and
Their Indifference Curves
?Consider
W(x1,x2) = min{x1,x2}.
What do the indifference curves for
this,perfect complementarity” utility
function look like?
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
W(x1,x2) = min{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}
Some Other Utility Functions and
Their Indifference Curves
?A utility function of the form
U(x1,x2) = f(x1) + x2
is linear in just x2 and is called quasi-
linear.
?E.g,U(x1,x2) = 2x11/2 + x2.
Quasi-linear Indifference Curves
x2
x1
Each curve is a vertically shifted
copy of the others.
Some Other Utility Functions and
Their Indifference Curves
?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
All curves are hyperbolic,
asymptoting to,but never
touching any axis.
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 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? ?
?
?
?
?
U
x
dx U
x
dx
2
2
1
1? ?
rearranged is
Marginal Utilities and Marginal
Rates-of-Substitution
?
?
?
?
U
x
dx U
x
dx
2
2
1
1? ?
rearranged is
And
d x
d x
U x
U x
2
1
1
2
? ? ? ?
? ?
/
/
.
This is the MRS.
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 Functions
?MRS = - f (x1) does not depend upon
x2 so the slope of indifference curves
for a quasi-linear utility function is
constant along any line for which x1
is constant,What does that make
the indifference map for a quasi-
linear utility function look like?
?
Marg,Rates-of-Substitution for
Quasi-linear Utility Functionsx
2
x1
Each curve is a vertically
shifted copy of the others.
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
?More generally,if V = f(U) where f is a
strictly increasing function,then
M 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.
The Key to this Chapter
? The indifference curve of a
consumer preference can be
represented by a utility function
based equation:
U(x1,x2) = k,a constant.