© John Riley
28 July 2011
ANSWERS TO EVEN NUMBERED EXERCISES IN APPENDIX C
SECTION C.1: TWO VARIABLES
Exercise C.12: Firm with interdependent demands
A firm sells two products.
Demand prices for these products are as follows.
11
2
2
1
2
22
180
,
180
p
qq
p
q
q
=−
−
=−−
The cost of production is
2
2
()
Cq q
qq q
α
=+
+
.
(a) If
1
=
, show that the profit function is concave and solve for the outputs that satisfy the
first order conditions.
(b) Show that the first order conditions for profit maximization are satisfied at
(20,20) if
4
q
==
.
(c) Are the second order (necessary) conditions satisfied at
(20,20)
q
=
?
(d) Show that there are two corner solutions to the FOC.
Explain why profit must be maximized
at these outputs.
(e) Show that, fixing
2
q
, profit is a concave function of
1
q
and hence solve for
*
12
, the
profit maximizing output of commodity 1.
(f) Totally differentiate
*
12 2
(()
,)
Π
and explain why
**
d
dq
q
Π∂
Π
=
∂
Hence show that
*2
1
2
4
(
,
) 180 45(1
) (4
(1
) )
)
d
q
dq
q
αα
Π
−
+
−
−
+
∂
(g)
Hence characterize the function
*
Π
for different values of
.
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28 July 2011
ANSWER
(a)
22
12
1
1
( ) 180(
)
R
qq
q
q
q
q
q
=+
−
−
−
,
11
2
2
()
Cq q
qq q
α
+
Hence
1
1
2
2
( ) 180(
) 2
(1
)
2
q
q
q
q
q
Π=
+−
−
+
−
.
The matrix of second partial derivatives is
4(
1
)
)
4
−−
+
⎡
⎤
⎢
⎥
−+
−
⎣
⎦
.
If
1
=
the determinant of the
nth principal minor has sign ( 1)
n
−
so the function is concave.
The FOC for an interior maximum
are as follows.
1
180 4
)
0
q
∂Π
=−−
+
=
∂
2
180 (1
)
4
0
q
∂Π
=−
+
−=
∂
.
If
1
=
the solution is
*
(30,30)
q
=
.
Since the function is concave the FOC are sufficient.
(b)
If
4
=
,
0
(20,20)
q
=
satisfies the FOC.
However the maximand is not concave so we
cannot be sure that this is the solution.
(c)
The second order necessary conditions are that the determinants of the principal minors
should alternate in sign.
This is NOT the case with
4.
=
Thus
0
q
is not a local maximum.
(d) Since there is no interior maximum there must be a corner solution.
Suppose that only
1
0
q
>
.
The FOC are
1
1
180 4
0
q
q
∂Π
=−=
∂
1
2
180 (1
)
0
q
q
∂Π
+
≤
∂
.
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 Fall '10
 riley
 Derivative, Mathematical analysis, Convex function, John Riley

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