Problems in this chapter consist mainly of applications of the
P MC
rule for profit
maximization by a price-taking firm and some examination of the firm’s derived demand for
inputs. A few of the problems (13.2–13.5) ask students to work through derivations related to
marginal revenue, but this concept is not really used in the monopoly context until Chapter 14.
The last problem provides practice with the new material on the “theory of the firm.”
Comments on Problems
11.1 A very simple application of the P = MC rule. Results in a linear supply curve.
11.3 An algebraic example of a profit function with one input. The problem asks the student
expected price equal to marginal cost does indeed maximize expected revenues, but that,
11.6 Easy problem that shows that a tax on profits will not affect the profit–maximization
output choice unless it affects the relationship between marginal revenue and marginal
cost.
11.7 Practice with calculating the marginal revenue curve for a variety of demand curves.
Analytical Problems
CHAPTER 11:
Profit Maximization
11.9 A CES profit function. A very brief introduction to the CES profit function. Deriving
the function involves a lot of algebra, but seeing how the parameters of the underlying
production function enter this profit function is quite instructive.
optimization, whether maximization or minimization. In maximization problems (as in
parts (a) and (b)), the long-run change will generally be bigger. In minimization
together input and output markets.
11.14 Profit functions and technical change. Applies the envelope theorem to derive a result
useful for empirical work on the measuring the impact of technical progress.
11.15 Property rights theory of the firm. The material from the Extensions on “theories of
the firm” is somewhat more philosophical than most of the rest of the book, so the
numerical example in that part of the text can be quite instructive. This problem has
11.1 a.
0.2 10.MC C q q
Setting
20,MC P
yields
*50.q
b.
1000 800 200.Pq C
c.
0.5 .
L A L
MC q q
Setting
AA
MR MC
and
LL
MR MC
yields
50 0.5 0.5 ,
25 0.5 0.5 .
2
A A L
LAL
q q q
qqq
30 ,
A
q
35,
A
P
10,
L
q
22.5.
L
P
Also,
1,050 225 400 875.
24= l.
Profits increase with output price and decrease with wages. The function
is homogeneous of degree 1 in the prices.
Supply increases with output price and decreases with the wage. The
function is homogeneous of degree 0 in input and output prices.
Labor demand increases with output price and decreases with the wage.
Input demand is homogeneous of degree 0 with respect in output and input
prices.
0.5 30 ( ) +0.5 20
E q C q q C q
500 400 100.E E P q C q
b. In the two states of the world, profits are
and
When
20,P
15 q
and
12.5.
112.5.E
( ) 0.5 212.5 0.5 12.5 9.06.E U = =
2,
100
q
kl
2()
.
100
q v w
C vk wl
Profit maximization requires
()
q v w
2
2
( , , )
50 50
v w P Pq C
P P v w
12 .
dP q a q a
MR = P + q = + q =
dq b b b
1
() .
b
q a P
P MR bb
This is positive because
0.b
d. If
,0
qP
e
(downward sloping demand curve), then marginal revenue will be less
becomes
qb
as in part (b).
e.
11.8 a. With marginal cost increasing, an increase in
P
will be met by an increase in
.q
To produce this extra output, more of each input will be hired (unless an input is
inferior).
b. The Cobb–Douglas case is best illustrated in two of the examples in Chapter 11.
In Example 11.4, the short-run profit function exhibits a positive effect of
P
on
labor demand. A similar result occurs in Example 11.5, where holding a third
Commented [CE1]: COMP: In this graph, set Parts a, b, d
as Parts (a), (b), (d).