Last, the claim takes the form
0
)
,
,...,
,
,
(
)
,
,...,
,
,
(
3
2
1
2
3
2
1
1
>
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
+
+
dc
u
p
p
c
p
c
p
h
u
p
p
c
p
c
p
h
d
L
L
, where
h
j
is the representative
consumer’s Hicksian demand for good
j
,
j
= 1,2.
Examine the validity of the claim for the following two cases. Comment, comparing 1.3 with 1.2 above.
Case 1.
Quasilinear utility function
L
L
j
L
j
j
j
L
x
x
b
x
b
a
x
x
x
u
+
−
=
∑∑
−
=
−
=
1
1
1
1
2
2
1
2
1
)
,...,
,
(
~
,
where
a
> 0, and
b
> 0 (defined on the domain where all marginal utilities are positive).
Case 2
. CES utility function
1
1
1
2
1
)
,...,
,
(
~
~
−
σ
σ
=
σ
−
σ
⎟
⎠
⎞
⎜
⎝
⎛
=
∑
L
j
j
L
x
x
x
x
u
, where
σ
> 0,
σ
≠
1. Hint
. The
corresponding expenditure function is
( )
σ
−
=
σ
−
∑
=
1
1
1
1
)
,
(
J
j
j
p
u
u
p
e
.
Question 2. Human capital externalities in production
There are
J
identical firms, each producing the same good (called output) by using
M
inputs, which are
interpreted as labor of various skills due to varying amounts of human capital. For
m
= 1, …,
M
, skill type
m
is
defined by a positive real number
h
m