So by doing this the firm’s output cannot decrease, hence its revenue cannot de
crease. Since the original ¯
z
is profit maximizing by assumption, the revenue must
not increase with the same cost.
Therefor the new input combination must be
another profit maximizing combination.
4. Q4
(a)
The CES function exhibits constant returns to scale, i.e. is homogeneous of degree
1 since
f
(
tz
) =
"
K
X
k
=1
a
k
(
tz
k
)
α
#
1
α
=
"
t
α
K
X
k
=1
a
k
(
z
k
)
α
#
1
α
=
t
"
K
X
k
=1
a
k
(
z
k
)
α
#
1
α
=
tf
(
z
)
(b)
Let us first show that a CES function is concave
2
(thus quasiconcave in particular)
to justify the FOC analysis we do later.
3
Since it exhibits CRS, it suffices to
show that
f
is quasi concave, and so it is enough to show that a monotonic
transformation of
f
is a concave function. If
α >
0, then
t
7→
t
α
is increasing so it
is enough to show that [
f
(
z
)]
α
=
∑
K
k
=1
a
k
(
z
k
)
α
is concave. But this is immediate
since this is a sum of concave functions. If
α <
0,
t
7→ 
t
α
is increasing, thus it
suffices to show that

[
f
(
z
)]
α
=

∑
K
k
=1
a
k
(
z
k
)
α
is concave. This follows since
for each
k
,

a
k
(
z
k
)
α
is concave and so again this is a sum of concave functions.
4
2
When
α >
1, this function is convex, which is a consequence of the Minkowski inequality.
3
This property can be shown by a careful examination of the Hessian matrix but I hope that you appreciate
the following argument based on general observations on concave functions.
4
The role of assumption of
α
≤
1 is now also clear: it guarantees that the quasiconcavity of the production
function: if
α >
1, the function is quasiconvex and the fator demand will be determined by corner solutions.
6
(c)
Since this is a CRS technology, it suffices to analyze the minimum cost function
given factor prices
w
=
(
w
1
, ..., w
K
)
and the required output level
y
= 1, since the
cost function is homogeneous of degree one in output
y
. So the cost minimization
problem we are interested in is:
min
z
X
k
w
k
z
k
subject to
f
(
z
)
≥
1
,
which is equivalent to if
α >
0
,
min
z
X
k
w
k
z
k
subject to
K
X
k
=1
a
k
(
z
k
)
α
≥
1
,
and if
α <
0
min
z
X
k
w
k
z
k
subject to 0
>

K
X
k
=1
a
k
(
z
k
)
α
≥ 
1
,
It is easy to see (check!) that the constraint is binding at the minimum. Also the
objective function is convex, and the constraint is quasiconcave, so the following
first order condition is necessary and sufficient condition for cost minimization:
λw
k
=
αa
k
(
z
k
)
α

1
, k
= 1
, ..., K
K
X
k
=1
a
k
(
z
k
)
α
= 1
,
where
λ
is a positive constant. From the first set of equations, we get
w
k
w
l
=
a
k
a
l
z
k
z
l
α

1
,
for any
k
and
l
. So,
ˆ
w
k
ˆ
w
l
1
α

1
=
z
k
z
l
,
where ˆ
w
k
=
w
k
/a
k
for
k
= 1
, ..., K
. Then, from the last equation,
1 =
K
X
k
=1
a
k
(
z
k
)
α
=
a
k
(
z
k
)
α
+
X
l
6
=
k
a
l
z
k
ˆ
w
k
ˆ
w
l

1
α

1
!
α
=
(
z
k
)
α
"
K
X
l
=1
a
l
ˆ
w
k
ˆ
w
l
ρ
#
,
where
ρ
=
α
1

α
. So we get the minimizer as
z
k
(
w,
1) =
h
∑
K
l
=1
a
l
ˆ
w
k
ˆ
w
l
ρ
i

1
α
, thus
7
the factor demand for input
k
given output level
y
is, for
k
= 1
, ..., K
,
z
k
(
w, y
) =
y
h
∑
K
l
=1
a
l
w
k
/a
k
w
l
/a
l
ρ
i
1
α
=
"
w
k
a
k
ρ
K
X
l
=1
a
l
w
l
a
l

ρ
#

1
α
y,
=
w
k
a
k

1
1

α
"
K
X
l
=1
a
l
w
l
a
l

α
1

α
#

1
α
y.
(5)
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 Summer '19
 Economics, Derivative, Convex function, Lj