Chapter 6
NAME
Demand
Introduction.
In the previous chapter, you found the commodity bundle
that a consumer with a given utility function would choose in a speciFc
price-income situation. In this chapter, we take this idea a step further.
We Fnd demand
functions
, which tell us for
any
prices and income you
might want to name, how much of each good a consumer would want. In
general, the amount of each good demanded may depend not only on its
own price, but also on the price of other goods and on income. Where
there are two goods, we write demand functions for Goods 1 and 2 as
x
1
(
p
1
,p
2
,m
)and
x
2
(
p
1
2
).
∗
When the consumer is choosing positive amounts of all commodities
and indi±erence curves have no kinks, the consumer chooses a point of
tangency between her budget line and the highest indi±erence curve that
it touches.
Example:
Consider a consumer with utility function
U
(
x
1
,x
2
)=(
x
1
+
2)(
x
2
+ 10). To Fnd
x
1
(
p
1
2
x
2
(
p
1
2
), we need to Fnd a
commodity bundle (
x
1
2
) on her budget line at which her indi±erence
curve is tangent to her budget line. The budget line will be tangent to
the indi±erence curve at (
x
1
2
) if the price ratio equals the marginal
rate of substitution. ²or this utility function,
MU
1
(
x
1
2
)=
x
2
+10 and
2
(
x
1
2
x
1
+ 2. Therefore the “tangency equation” is
p
1
/p
2
=
(
x
2
+ 10)
/
(
x
1
+ 2). Cross-multiplying the tangency equation, one Fnds
p
1
x
1
+2
p
1
=
p
2
x
2
+10
p
2
.
The bundle chosen must also satisfy the budget equation,
p
1
x
1
+
p
2
x
2
=
m
. This gives us two linear equations in the two unknowns,
x
1
and
x
2
. You can solve these equations yourself, using high school algebra.
You will Fnd that the solution for the two “demand functions” is
x
1
=
m
−
2
p
1
p
2
2
p
1
x
2
=
m
p
1
−
10
p
2
2
p
2
.
There is one thing left to worry about with the “demand functions” we
just found. Notice that these expressions will be positive only if
m
−
2
p
1
+
10
p
2
>
0and
m
p
1
−
10
p
2
>
0. If either of these expressions is negative,
then it doesn’t make sense as a demand function. What happens in this
∗
²or some utility functions, demand for a good may not be a±ected by
all of these variables. ²or example, with Cobb-Douglas utility, demand
for a good depends on the good’s own price and on income but not on the
other good’s price. Still, there is no harm in writing demand for Good
1 as a function of
p
1
,
p
2
,and
m
. It just happens that the derivative of
x
1
(
p
1
2
) with respect to
p
2
is zero.