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Ch 2. Budget Constraint
The economic theory of consumers is to answer the question: How does a consumer
choose
the best
bundle of goods she
can afford
.
This chapter will examine how to describe what a consumer can afford
; and the next
chapter will explore how the consumer determines what is the best
.
2.1 The Budget Constraint
In reality, there are many goods to consume, but it is convenient for the time being to
consider only the twogood case since we can make use of graphical illustration.

We use
X
or (
x
1
, x
2
) to indicate a
consumption bundle
. For
i = 1, 2
,
x
i
denotes the
quantity of good
i
chosen by the consumer.

We use
P
or (
p
1
, p
2
) to indicate the marketset
prices
of both goods. For
i = 1, 2
,
p
i
denotes the price of good
i
.

We use
m
to denote the consumer°s
income
used for consumption.

The
budget constraint
of a consumer requires that the amount of money spent on
both goods be no more than her income. That is,
p
1
x
1
+ p
2
x
2
≦
m
 (2.1)

A bundle of both goods satisfying this constraint is referred to as an
affordable
bundle. The set of affordable bundles at
P
&
m
is called the consumer°s
budget
set
, which includes all points that satisfy inequality (2.1)
2.2 The Budget Line
To know better about the properties of the budget constraint, let°s study the
budget line
(associated with (2.1)) first, which is the set of bundles that cost exactly
m
:
p
1
x
1
+ p
2
x
2
= m
 (2.3)
budget set
figure 2.1
budget line with
slope =
2
1
p
p
−
2
p
m
x
2
1
p
m
x
1
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Given
P
and
m
, use eqn (2.3) to derive the horizontal and
vertical intercept
s (
m / p
1
&
m / p
2
), connecting both intercepts gives you the budget line, as shown in figure
2.1.
To achieve this, set
x
i
= 0
in (2.3) to have
x
j
= m / p
j
(where
i
≠
j
), meaning that the
consumer could buy
m / p
j
units of good
j
if she spends all of her money
m
on that
good.
Change equation (2.3) to a function with
x
1
as the independent variable:
1
2
1
2
2
x
p
p
p
m
x
−
=
 (2.4)
We then know that the
slope
of the budget line is
° p
1
/ p
2
, which measure the rate at
which the market substitutes good 1 for good 2 because
p
is set only by the market,
not by individual consumers.
To see this more clearly, consider a case where the consumer changes her choice of
both goods from (
x
1
, x
2
) to (
x
1
±, x
2
±
) that just exhaust her income
m
(where
x
i
± = x
i
+
△
x
i
).
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 Spring '09
 Gu

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