In most applications that follow we suppress the
h
superscript
of
υ
h
and
υ
h
,
and use instead
υ
and
υ
i
.
Given factor rental
rates
w
,
the household’s income is given by
w
·
υ
,
which is used
to purchase
q
j
units of consumption good
j
at market price
p
j
,
j
= 1
, ..., M.
Then, the household’s budget constraint is given
by
w
·
υ
≥
p
·
q
where
q
= (
q
1
, ..., q
M
)
∈
R
M
++
.
In other words, each household
consumes a strictly positive level of each consumption good.
Consumer preferences over goods are represented by the util-
ity function
u
:
R
M
++
→
R
+
,
defined as
u
(
q
)
.
Assumption 1
u
(
q
)
satisfies the following properties:

2.
The Preliminaries
11
1.
u
(
q
) is increasing and strictly concave in
q
,
2.
u
(
q
) is everywhere continuous, and everywhere twice dif-
ferentiable,
3.
u
(
q
) is homothetic.
Assumption 1.1 yields indifference curves that are convex, As-
sumption 1.2 ensures Marshallian demands are continuous func-
tions, while Assumption 1.3 yields Marshallian demands that are
separable in prices and income.
Two indirect functions emerge from the consumer’s problem:
the indirect utility function and the expenditure function. The
indirect utility function
gives the household’s maximum attain-
able utility given income
w
·
υ
, defined as
V
(
p
,
w
·
υ
)
≡
max
q
{
u
(
q
) :
w
·
υ
≥
p
·
q
}
The indirect utility function inherits the following properties
from the direct utility function (see Cornes, pp. 67–70):
V1.
Homogeneous of degree zero in
p
and
w
·
υ
;
V
(
θ
p
, θ
w
·
υ
)
=
V
(
p
,
w
·
υ
)
, θ >
0,
V2.
V
(
p
,
w
·
υ
) is convex in
p
,
V3.
V
(
p
,
w
·
υ
) is continuous and differentiable in
p
and
w
·
υ
,
V4.
V
(
p
,
w
·
υ
) =
v
(
p
)
w
·
υ
: separable in
p
and
w
·
υ
,
By V4, the marginal utility of an additional unit of income is
v
(
p
)
.
V5.
Given differentiability, Marshallian demands follow from
Roy’s identity,
q
j
(
p
) (
w
·
υ
) =
−
v
p
j
(
p
)
v
(
p
)
w
·
υ
(2.1)
where, throughout the text, the subscript on a function
indicates a partial derivative, e.g.,
v
p
j
=
∂v
(
p
)
/∂p
j
and
v
p
1
p
2
=
∂
2
v
(
p
)
/∂p
1
∂p
2
.

12
2.
The Preliminaries
Since consumers face the same prices and have identical pref-
erences, the “community” indirect utility function is given by
V
=
v
(
p
) (
w
·
V
)
while the total domestic Marshallian demand for good
j
is
Q
j
=
q
j
(
p
) (
w
·
V
)
,
∀
j
∈ {
1
,
· · ·
, M
}
(2.2)
These functions are the simple aggregation of individual con-
sumer welfare and demands. It also follows from V1 that (2.2)
is homogeneous of degree minus one in prices
p
and of degree
one in income.
The
expenditure function
gives the minimum cost of achieving
utility level
q
∈
R
at given prices
p
,
and is defined as
E
(
p
, q
)
≡
min
q
{
p
·
q
:
q
≤
u
(
q
)
}
The expenditure function inherits from
u
(
·
)
,
the following prop-
erties:
E1.
E
(
p
, q
)
>
0 for any
p
and
q >
0
,
E2.
E
(
p
, q
) is non-decreasing in
p
and
q,
E3.
E
(
p
, q
) is concave and continuous in
p
,
E4.
E
(
λ
p
, q
) =
λE
(
p
, q
)
, λ >
0
:
homogeneous of degree 1 in
p
,
E5.
E
(
p
, q
) =
E
(
p
)
q
:
separable in
p
and
q,
E6.
Shephard’s lemma: If
E
(
p
, q
) is differentiable in
p
, then
q
j
=
E
p
j
(
p
, q
) =
E
p
j
(
p
)
q, j
= 1
, ..., M
E
1 says purchasing a strictly positive consumption bundle is
costly.
E
2 says, all else equal, (i) if the price of a consump-

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- Spring '14
- Roe,TerryLee
- Economics, Supply And Demand, rental rate, yj