DUALITY IN CONSUMPTION I
We have considered the utility maximization problem:
max U=U(x
1
, x
2
)
s
.
t
.
p
1
x
1
+p
2
x
2
=m
The first order condition are solved to derive the demand functions:
x
1
*=x
1
(p
1
, p
2
, m)
and
x
2
*=x
2
(p
1
, p
2
, m).
For any given pair of prices (p
1
, p
2
) and income m, (x
1
*
,x
2
*) will be the bundle that
maximize the consumer’s utility subject to the budget constraint. The corresponding
utility level would be
V=U[x
1
*(p
1
, p
2
, m), x
2
(p
1
, p
2
, m)].
Note that any change in the parameters (p
1
, p
2
, m) will lead to a change in (x
1
*
,x
2
*) and
therefore the maximally attained level of utility V will also change.
We may express U[x
1
*(p
1
, p
2
, m), x
2
(p
1
, p
2
, m)] directly as a function of (p
1
, p
2
,
m):
V=U[x
1
*(p
1
, p
2
, m), x
2
(p
1
, p
2
, m)]= V(p
1
, p
2
, m).
This is because as the
Indirect Utility Function
, the utility function ranks consumption
bundles according to the consumer’s preferences. The indirect utility function ranks price
income combinations (p
1
, p
2
, m). For every combination of (p
1
, p
2
, m), there is a specific
budget line and the preferred bundle on that line. If the preferred bundle on a different
budget line lies on a higher indifference curve, the consumer in better off with the latter
budget line.
The budget line A
0
B
0
(Figure 1) corresponds to (p
1
0
, p
2
0
, m
0
). The best point on
A
0
B
0
is K. The line A
1
B
1
corresponds to (p
1
1
, p
2
1
, m
1
) and the best point on A
1
B
1
is L.
Because L is on a higher indifference curve, the consumer can reach a higher level of
utility given (p
1
1
, p
2
1
, m
1
) than from (p
1
0
, p
2
0
, m
0
).
Figure 1
1