Chapter 3: Preferences and Utility
°
There are
n
commodities (also include time and location). A
consumption bundle
is denoted as
X
= (
x
1
; x
2
; : : : ; x
n
)
2
R
n
+
:
°
A consumer °should be able±to make simple choice between two consumption bundles,
known as
preference relation
.
°
If the consumer likes bundle
A
better than bundle
B
, then °
A
is (strictly) preferred
to
B
±, denote as
A
±
B
.
°
If the consumer thinks that
A
and
B
are equally attractive, then °
A
is indi/erent
from
B
±, denote as
A
²
B
.
°
If either
A
±
B
or
A
²
B
, then °
A
is weakly preferred to
B
±, or
A
is at least as
good as
B
, denote as
A
%
B:
°
Axioms of Rational Choice
° Complete:
For all
A
and
B
, either
A
±
B
, or
B
±
A
, or
A
²
B
.
° Transitive:
If
A
±
B
and
B
±
C
, then
A
±
C
.
° Continuous:
If
A
±
B
, and
C
is °very close to±
A
, then
C
±
B:
Lexicographic preference is not continuous!
°
U
(
³
) :
R
n
!
R
is a
utility function
if
U
(
A
)
> U
(
B
)
if and only
A
±
B
, and
U
(
A
) =
U
(
B
)
if and only
A
²
B
.
°
If there is a utility function, then preference must be complete and transitive.
°
If preference is complete, transitive, and
continuous
, then there is
a
utility function.
°
Any monotonic transformation of a utility function is also a utility function.
°
With
n
commodities, denote a bundle as
A
= (
x
1
; x
2
; : : : ; x
n
)
, and utility as
U
(
A
) =
U
(
x
1
; x
2
; : : : ; x
n
)
:
With two commodities (
n
= 2
), simply write a utility function as
U
(
x; y
)
.
°
An
indi/erence curve
is a set of all bundles that are indi/erent from each other.
°
All bundles on the same indi/erence curve have the same utility.
°
Given utility function
U
(
x; y
)
, an indi/erence curve is
f
(
x; y
) :
U
(
x; y
) =
°
g
,
where
°
is the °common utility±of the bundles on the indi/erence curve.
°
How to ²nd an indi/erence curve?
Set
U
(
x; y
) =
°
, then solve
y
as a function of
x
(and
°
).
°
Marginal Rate of Substitution
(
MRS
)
°
The slope of an indi/erence curve (at a point/bundle)
MRS
=
´
dy
dx
°
°
U
=
°
:
1