Microeconomic Theory
Econ 101A
Fall 2008
GSI: Eva Vivalt
Section Notes 2: Consumer Theory
1
Utility Maximization Problem
Problem setup:
•
Two goods:
x
1
and
x
2
with prices
p
1
and
p
2
.
•
Consumers have a ﬁxed income,
I
, and thus are constrained to spend less or equal to the amount
I
on goods
x
1
and
x
2
. Thus the budget constraint can be expressed as:
p
1
x
1
+
p
2
x
2
≤
I
.
•
It is impossible to consume negative amounts of goods, so
x
1
≥
0
,x
2
≥
0.
•
Consumers maximize utility
U
(
x
1
,x
2
) which is increasing in both
x
1
and
x
2
and quasiconcave in
(
x
1
,x
2
).
•
Consumers are price takers, which means that they take prices as ﬁxed (i.e. no impact of their
individual demands on prices. This can be justiﬁed by postulating a large number of consumers).
Since utility is increasing in both
x
1
and
x
2
, we can safely assume that the budget constraint holds with
equality. Also, we will assume that the nonnegativity constraints are slack, so that
x
1
>
0 and
x
2
>
0.
Setting the inequalities to equality, combining the two constraints, and rearranging a bit we get the
standard utility maximization problem (or UMP):
max
x
1
,x
2
U
(
x
1
,x
2
) s.t.
p
1
x
1
+
p
2
x
2
=
I
We can solve this constrained maximization problem by writing the Lagrangian:
L
(
x
1
,x
2
,λ
) =
U
(
x
1
,x
2
)

λ
[
p
1
x
1
+
p
2
x
2

I
]
and ﬁnding the FOCs:
∂L
∂x
1
=
U
1
(
x
*
1
,x
*
2
)

λ
*
x
1
= 0
∂L
∂x
2
=
U
2
(
x
*
1
,x
*
2
)

λ
*
x
2
= 0
∂L
∂λ
=

[
p
1
x
*
1
+
p
2
x
*
2

I
] = 0
The solution yields the Lagrange multiplier
λ
*
=
λ
*
(
p
1
,p
2
,I
) and the ordinary demand functions
x
*
1
=
x
*
1
(
p
1
,p
2
,I
) and
x
*
2
=
x
*
2
(
p
1
,p
2
,I
).
1
1
These are actually uncompensated demand functions. The functions are “uncompensated” because price changes will
cause utility changes – a situation which does not occur with compensated demand curves.
1