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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.
11.2 Uses the MR–MC condition to illustrate third degree price discrimination. Instructors
might point out the general result here (which is discussed more fully in Chapter 13) that,
assuming marginal costs are the same in the two markets, marginal revenues should also
be equal and that implies price will be higher in the market in which demand is less
elastic.
11.3 An algebraic example of a profit function with one input. The problem asks the student
to derive the supply and input demand functions from this profit function using
Shephard’s lemma.
11.4 A problem in the theory of supply under uncertainty. This example shows that setting
expected price equal to marginal cost does indeed maximize expected revenues, but that,
for risk-averse firms, this may not maximize expected utility. Part (d) asks students to
calculate the value of better information.
11.5 A simple use of the profit function with fixed proportions technology.
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.
11.8 This is a conceptual examination of the effect of changes in output price on input
demand.
CHAPTER 11:
Profit Maximization
Chapter 11: Profit Maximization
115
Analytical Problems
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.
11.10 Some envelope results. This problem describes some additional mathematical
relationships that can be derived from the profit function.
11.11 Le Châtelier’s principle. This problem demonstrates this central principle of economics
in various contexts. The principle compares long-run to short-run changes. The logic
behind the principle is that in the long-run, there are more margins to adjust, so a “better”
outcome can be produced than in the short run. Whether this “better” outcome involves a
bigger or smaller change in the variable of interest depends on the nature of the
optimization, whether maximization or minimization. In maximization problems (as in
parts (a) and (b)), the long-run change will generally be bigger. In minimization
problems [as in parts (c) and (d)], the change will generally be smaller in the long run.
11.12 More on derived demand with two inputs. This problem shows how an industry’s
demand for an input can be computed and why that demand will depend on the elasticity
of demand for the good being produced. This is a nice problem therefore for tying
together input and output markets.
11.13 Cross-price effects in input demand. This is a continuation of Problem 11.11 to
consider cross-price effects. The problem attempts to clarify how input cost shares enter
into input demand elasticities.
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
students work through a simple tweak of that numerical example. The tweak has
independent interest, showing that vertical integration between the car body and assembly
can be beneficial if the assembly’s investment is important enough.
Solutions
Chapter 11: Profit Maximization
116
c.
11.2 Total cost is
( )
2
2AL
0.25 0.25 ,C q q q= = +
Revenues are
Chapter 11: Profit Maximization
117
2
Solving these simultaneously gives
11.3 a. Since
2,ql=
24= l.
q
c. Profit maximization requires
d. From the production function,
24.lq=
Replacing
q
from the supply function,
e. Intuitive properties include the following:
Chapter 11: Profit Maximization
118
11.4 a. Expected profits are
b. In the two states of the world, profits are
c. Output levels between 13 and 19 all yield greater utility than does
20. q=
d. If the firm can predict
,P
P MC=
11.5 a. In order for the second-order condition for profit maximization to be satisfied,
Chapter 11: Profit Maximization
119
Profit maximization requires
Rearranging,
e.
11.6 Without any tax,
( ) ( ) ( ).q R q C q
=−
With a lump sum tax
,T
Chapter 11: Profit Maximization
120
11.7 a. If
,q a bP=+
c. The constant elasticity demand curve is
b,q aP=
b
is the price elasticity
d. If
,0
qP
e
(downward sloping demand curve), then marginal revenue will be less
Chapter 11: Profit Maximization
121
e.
11.8 a. With marginal cost increasing, an increase in
P
will be met by an increase in
.q
b. The Cobb–Douglas case is best illustrated in two of the examples in Chapter 11.
Chapter 11: Profit Maximization
122
is an inferior input.
Analytical Problems
11.9 A CES profit function
c. σ determines how easily firms can adapt to differing input prices. The higher is
d.
1 1 1 (1 )( 1)
1
(1 ) ( ) .q K P v w
− − − −−
−
= = − +
11.10 Some envelope results
a. We have
22 .
lk
v v w w v w
===
c. We have
Chapter 11: Profit Maximization
123
d. Because it seems likely that
0lP
(see Problem 11.8), we can conclude that
11.11 Le Châtelier’s principle
a., b. Totally differentiate both sides of the definitional relation with respect to
P
:
By analogy to part (c) of Problem 11.10,
**
.
qk
vP
=−
c. Totally differentiate the definitional relation with respect to
w
:
**
ss
l l l k
Chapter 11: Profit Maximization
124
implying
*
s
l l v
Substituting in succession into the initial derivative,
( )( )
**
*
s
l v k w
ll
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=
Chapter 11: Profit Maximization
125
11.12 More on derived demand with two inputs
b. Under the assumption of constant returns to scale,
c. Because costs are homogenous of degree 1, the derivatives of
C
are
homogeneous of degree 0. Hence,
Chapter 11: Profit Maximization
126
d. We have
.
wv
wv
CC
CC
=
f. The terms
l
s
−
and
k
s
−
are a mathematical representation of the substitution
effect. Because the sign of
is positive and
s
is positive, the overall sign of the
Chapter 11: Profit Maximization
127
11.13 Cross-price effects in input demand
a. Similarly to 11.11 part (b),
( ( , ,1)) ,
vv
k QC D C v w C
==
and
,
kw
kw
ewk
=
Chapter 11: Profit Maximization
128
b. In the formula for cross-price elasticity of input demand, both the elasticity of
substitution and the elasticity of demand for the output are weighted by the share
c. From Euler’s theorem,
11.14 Profit functions and technical change
The profit function is
11.15 Property rights theory of the firm
First, consider keeping the assets separate. Fisher Body maximizes
Chapter 11: Profit Maximization
129
4 4 16 16 16
Next, consider GM ownership. Fisher Body of course sets
** 0.
F
x=
GM maximizes
Comparing the joint surpluses under the two ownership structures, GM ownership is
