Lecture 9
Consumer Theory I
AAE 635
Fall 2010
9.1 Introduction
So far we have explored the production behavior of cost minimizing firms and profit maximizing
firms. We have also investigated the short run production equilibrium and the long run production
equilibrium. The production behavior (and its comparative statics) under optimization gives the
supply side story of the market, and for the demand side story, we need to examine the
consumption behavior by rational consumers.
Consider a household choosing
n
consumption goods
x
= (
x
1
, …,
x
n
)
T
. The household has
preferences represented by a utility function
u
(
x
). Also, the household faces the budget constraint:
p
T
x
≤
y
or
Σ
i
p
i
x
i
≤
y
where
y
> 0 is household income, and
p
= (
p
1
, …,
p
n
)
T
is a
n
price vector
for
x
,
p
i
> 0 denoting the market price of
x
i
,
i
= 1, …,
n
. Assume that economic rationality for
household decisions is represented by the following utility maximization problem
V
(
p
,
y
) =
.
(1)
{()
:
,
0
,
}
Tn
Max
u
y
≤≥
∈
x
xpx
x
xR
This simply states that the households make consumption decisions so as to maximize their utility
subject to a budget constraint. Denote the solution of the optimization problem (1) by
x
*(
y
,
p
).
Here,
x*
(
y
,
p
) are utility maximizing decision rules for household consumption. They are also
called
Marshallian
demand functions. The function
V
(
y
,
p
) in (1) is the indirect utility function
satisfying
V
(
y
,
p
) =
u
(
x
*(
y
,
p
)).
9.2 Consumption decisions
Throughout our discussion, we will make the following (intuitive) assumption.
Assumption A1
: (nonsatiation)
For any consumption bundle
x
≥
0 and for every
ε
> 0, there is a bundle
x
' satisfying
u
(
x
') >
u
(
x
),
x
'
≥
x
,
x
'
≠
x
, and 
x

x
' <
.
Nonsatiation means that the household can always increase its utility by increasing the
consumption of some commodity. This generates the following key result.
Under (A1), the budget constraint is necessary binding at the optimum
x
*:
p
T
x
*(
y
,
p
) =
y
.
Proof
: Assume that the budget constraint is not binding at
x
*:
p
T
x
*(
y
,
p
) <
y
. Under assumption
A1 and given
p
> 0, there exists a consumption bundle
x
' such that
p
T
x
'
≤
y
and
u
(
x
') >
u
(
x
*(
y
,
p
)). But this contradicts
x
*(
y
,
p
) being utility maximizing. Thus, the budget constraint must be
binding at
x
*.
This means that, under nonsatiation, the utility maximization problem (1) can be written as
.
(2)
(,)
:
,
0
,
}
T
VyM
a
x
u
y
==
≥
x
px
p
x
x
x
n
∈
R
Expression (2) is a standard constrained maximization problem. It can be analyzed using the
Lagrangean approach. Define the associated Lagrangean
L
(
x
,
λ
,
y
,
p
) =
u
(
x
) +
[
y

p
T
x
],
where
is a Lagrange multiplier corresponding to the budget constraint
y

p
T
x
= 0.
We have the following result:
Given
p
> 0, the constraint qualification (CQ: rank(
p
) = 1) is always satisfied.
Assume that the utility function
u
(
x
) is twice continuously differentiable. Then, given an interior
solution
x
*, the maximization problem (2) implies the FONC
1