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Analytical Problems
11.9 A CES profit function
b. Diminishing returns is required if MC is to be increasing—the required second-
order condition for profit maximization.
11.10 Some envelope results
a. We have
22 .
lk
v v w w v w
c. We have
22 .
ql
w w P P w P
11.11 Le Châtelier’s principle
a., b. Totally differentiate both sides of the definitional relation with respect to
P
:
**
.
ss
q q q k
P P k P
c. Totally differentiate the definitional relation with respect to
w
:
**
.
ss
l l l k
w w k w
**
*
*
2
*
*
.
s
s
s
l v k w
ll
w w k v
lv
l
w k v
l
w
d. It is difficult to use the methods from parts (a)–(c) here. Let’s see what happens
when we try. Start from the definitional relation
( , , ) ( , , , ( , , )).
c
C v w q SC v w q k v w q
Totally differentiate with respect to
w
:
.
c
C SC SC k
w w k w
( , , ) ( , , ) ( , , , ) ( , , , )
C v w q C v w q SC v w q k SC v w q k
11.12 More on derived demand with two inputs
a. By Shephard’s lemma, each partial derivative gives the quantity of input
demanded to produce one unit of output. Multiplication by
Q
gives total industry
demand.
b. Under the assumption of constant returns to scale,
, ,1 .P MC C v w
So in equilibrium,
( ) ( ( , ,1)).Q D MC D C v w
Furthermore,
( ( , ,1)) ,
( ( , ,1)) ,
vv
ww
k QC D C v w C
l QC D C v w C
implying
2
2
,
.
v vv
w ww
kD C C Q
v
lD C C Q
w
c. Because costs are homogenous of degree 1, the derivatives of
C
are
homogeneous of degree 0. Hence,
0,
vv wv
vC wC
implying
.
vv wv
w
CC
v
Similarly,
.
ww wv
v
CC
w
d. We have
.
wv
wv
CC
CC
Rearranging,
.
wv
wv
CC
CC
e.
,,
2,
l w k l Q P
l w vk D wlP
e s s e
w l QC Q P
,,
2.
k v l k Q P
k v wl D vkP
e s s e
v k QC Q P
f. The terms
l
s
and
k
s
are a mathematical representation of the substitution
effect. Because the sign of
is positive and
l
s
is positive, the overall sign of the
substitution effect will be negative. The size of the effect increases when the
11.13 Cross-price effects in input demand
a. Similarly to 11.11 part (b),
( ( , ,1)) ,
( ( , ,1)) ,
vv
ww
k QC D C v w C
l QC D C v w C
implying
,
.
v w vw
w v wv
kD C C C Q
w
lD C C C Q
v
From 11.11 part (c),
substitution and the elasticity of demand for the output are weighted by the share
of the other input. This happens because the effect of a change in the price of the
other input will depend primarily on the importance of this other input. In the
case of
,,
kw
e
.
ww wv
ll
w w w w
C C vC C
AC C wC C
Multiplying this by
vv
CC
and using Shephard’s lemma yields
.
wv k
ll kl
w v l
vkC C s
AA
wlC C s
11.14 Profit functions and technical change
The profit function is
,
( , , , ) max ( , , ) .
kl
P v w t Pf k l t vk wl
By the envelope theorem,
ln .
t
ttt
tPf f f
Pq t Pq q f
11.15 Property rights theory of the firm
First, consider keeping the assets separate. Fisher Body maximizes
1/2 1/2
1.
2F G F
x ax x
The first-order condition is
1/2
11 0,
4F
x
*116.
4 4 16 16 16
** 0.
2
3
3.
a
a
