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 specific
priceincome situation. In this chapter, we take this idea a step further.
We find 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
, p
2
, m
).
∗
When the consumer is choosing positive amounts of all commodities
and indifference curves have no kinks, the consumer chooses a point of
tangency between her budget line and the highest indifference curve that
it touches.
Example:
Consider a consumer with utility function
U
(
x
1
, x
2
) = (
x
1
+
2)(
x
2
+ 10).
To find
x
1
(
p
1
, p
2
, m
) and
x
2
(
p
1
, p
2
, m
), we need to find a
commodity bundle (
x
1
, x
2
) on her budget line at which her indifference
curve is tangent to her budget line. The budget line will be tangent to
the indifference curve at (
x
1
, x
2
) if the price ratio equals the marginal
rate of substitution. For this utility function,
MU
1
(
x
1
, x
2
) =
x
2
+ 10 and
MU
2
(
x
1
, x
2
) =
x
1
+ 2.
Therefore the “tangency equation” is
p
1
/p
2
=
(
x
2
+ 10)
/
(
x
1
+ 2).
Crossmultiplying the tangency equation, one finds
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 find that the solution for the two “demand functions” is
x
1
=
m
−
2
p
1
+ 10
p
2
2
p
1
x
2
=
m
+ 2
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
>
0 and
m
+2
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
∗
For some utility functions, demand for a good may not be affected by
all of these variables. For example, with CobbDouglas 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
, p
2
, m
) with respect to
p
2
is zero.
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68
DEMAND
(Ch.
6)
case is that the consumer will choose a “boundary solution” where she
consumes only one good. At this point, her indifference curve will not be
tangent to her budget line.
When a consumer has kinks in her indifference curves, she may choose
a bundle that is located at a kink.
In the problems with kinks, you
will be able to solve for the demand functions quite easily by looking
at diagrams and doing a little algebra.
Typically, instead of finding a
tangency equation, you will find an equation that tells you “where the
kinks are.”
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 Summer '09
 JOHNG.SESSIONS
 Microeconomics, Utility, Twinkies

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