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CHAPTER 5: CHOICE
The optimization principle:
A rational consumer chooses a consumption bundle that is
preferred the most in the budget constraint.
The optimal choice
is a consumption bundle, also called
a demand bundle
,inth
e
budget constraint that is weakly preferred to any other bundle in the budget constraint.
In other words, there is no consumption bundle in the budget constraint that is strictly
prefered to a demand bundle.
In our notation, consumption bundle (
x
∗
1
,x
∗
2
) is a demand bundle if
p
1
x
∗
1
+
p
∗
2
x
2
≤
m
,and
for any consumption bundle (
x
1
,x
2
) such that
p
1
x
1
+
p
2
x
2
≤
m
,
(
x
∗
1
,x
∗
2
)
º
(
x
1
,x
2
)
⇐⇒
u
(
x
∗
1
,x
∗
2
)
≥
u
(
x
1
,x
2
)
.
This means that
u
(
x
∗
1
,x
∗
2
) is the maximum of the utility function over the budget constraint.
That is,
u
(
x
∗
1
,x
∗
2
)=m
a
x
x
1
,x
2
u
(
x
1
,x
2
)
,
subject to
p
1
x
1
+
p
2
x
2
≤
m.
This constrained optimization problem is referred as the
consumer problem
.
Under certain conditions, the budget line is tangent to an indi
f
erence curve at a demand
bundle:

6
x
1
x
2
0
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
s
x
∗
2
x
∗
1
An interior solution to the consumer problem.
The budget line and such an di
f
erence curve have the same slope at an interior demand
bundle. A necessary condition is
slope of indi
f
erence curve =
MRS
=
−
p
1
p
2
= slope of budget line.
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View Full Document On the graph, it is easy to argue that if these two rates of substitution are di
f
erent, then
the consumption bundle cannot be a demand bundle.
Suppose that MRS is
−
1but
−
p
1
p
2
=
−
2. The consumer is willing to exchange one unit of
good 1 with one unit of good 2. Good 1 costs twice as much as good 2.
Consider the following procedure which will make the consumer better o
f
:t
h
ec
o
n

sumer buys one unit less of good 1, with the money saved, the consumer will be able
to buy two more units of good 2.
Note that just one more unit of good 2 will make the consumer indi
f
erent for giving up
one unit of good 1. The additional one unit of good 2 will make the consumer better
o
f
. So the consumption bundle where
MRS
6
=
−
p
1
p
2
cannot be an optimal choice.
A few exceptions to this necessary condition.
1. MRS does not exist, such as perfect complements;
2. Noninterior solution, corner solution, such as perfect substitutes;

6
x
1
x
2
0
@
@
@
@
@
@
@
@
C
C
C
C
C
C
C
C
X
X
X
X
X
X
X
X
s
Kinky tastes.

6
x
1
x
2
0
H
H
H
H
H
H
H
H
s
Corner solution.
3. Nonconvex preference,

6
x
1
x
2
0
@
@
@
@
@
@
c
s
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This note was uploaded on 04/19/2008 for the course ECON 305 taught by Professor Souza during the Spring '08 term at Vanderbilt.
 Spring '08
 souza

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