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The problems in this chapter focus mainly on the relationship between production and
cost functions. Most of the examples developed are based on the Cobb–Douglas function
(or its CES generalization), although a few of the easier ones employ a fixed proportions
assumption. Two of the problems (10.7 and 10.8) make use of Shephard’s lemma since it
is in describing the relationship between cost functions and (contingent) input demand
that this envelope-type result is most often encountered. The analytical problems in this
chapter focus on various elasticity concepts, including the introduction of the Allen
elasticity measures.
Comments on Problems
10.1 An introduction to the concept of “economies of scope.” This problem illustrates
the connection between that concept and the notion of increasing returns to scale.
10.2 A simplified numerical Cobb–Douglas example in which one of the inputs is held
fixed.
10.3 A fixed proportion example. The very easy algebra in this problem may help to
solidify basic concepts.
10.4 This problem derives cost concepts for the Cobb–Douglas production function
with one fixed input. Most of the calculations are very simple. Later parts of the
problem illustrate the envelope notion with cost curves.
10.5 Another example based on the Cobb–Douglas with fixed capital. Shows that in
order to minimize costs, marginal costs must be equal at each production facility.
Might discuss how this principle is applied in practice by, say, electric companies
with multiple generating facilities.
10.6 This problem focuses on the Cobb–Douglas cost function and shows, in a simple
way, how underlying production functions can be recovered from cost functions.
10.7 This problem shows how contingent input demand functions can be calculated in
the CES case. It also shows how the production function can be recovered in such
cases.
CHAPTER 10:
Cost Functions
10.8 Famous example of Viner’s draftsman. This may be used for historical interest or
as a way of stressing the tangencies inherent in envelope relationships.
Analytical Problems
10.9 Generalizing the CES cost function. Shows that the simple CES functions used
in the chapter can easily be generalized using distributional weights.
10.10 Input demand elasticities. Develops some simple input demand elasticity
concepts in connection with the firm’s contingent input demand functions (this is
demand with no output effects).
10.11 The elasticity of substitution and input demand elasticities. Ties together the
concepts of input demand elasticities and the (Morishima) partial elasticity of
substitution concept developed in the chapter. A principle result is that the
definition is not symmetric.
10.12 The Allen elasticity of substitution. Introduces the Allen method of measuring
substitution among inputs (sometimes these are called Allen/Uzawa elasticities).
Shows that these do have some interesting properties for measurement, if not for
theory.
Solutions
10.1 a. By definition, total costs are lower when both
1
q
and
2
q
are produced by
b. Let
12
,q q q=+
12
, 0. qq
By assumption,
Chapter 10: Cost Functions
102
10.2 a. Substituting into the production function,
b. Because Smith’s effort is sunk, to compute marginal cost we only need to
consider Jones’ effort in the cost function. To produce
q
pages requires
10.3 Given
( )
min 5 ,10 .q k l=
a. In the long run, no input should be wasted. Hence,
5 10 ,k l q==
implying
2 5.k l q==
Thus,
Chapter 10: Cost Functions
103
b.
( )
min 50,10 ,ql=
when
10.k=
There are two cases to consider. First, if
5,l
then
10 , q = l
implying
50.q
Hence,
Finally, right at
50,q=
we have the same formula for total cost as
above:
c. Substituting
1v=
and
3w=
into the formulae from the previous parts, in
the long run,
1 2.AC = MC =
In the short run, for
50,q
10.4 Given
2,q = kl
100.k =
Chapter 10: Cost Functions
104
b. We have
If
50,q=
If
100,q=
If
200,q=
Chapter 10: Cost Functions
105
c.
d. As long as the marginal cost of producing one more unit is below the
e. Since
1
2,q = k l
2
1
= 4 ,kl
q
implying
f. Deriving the first-order condition from the previous expression,
g. Substituting first for
l
and then for
1
k
into the cost function,
and
Chapter 10: Cost Functions
106
10.5 a. Total output is
= ,q q q+
with
To minimize cost, set up Lagrangian:
b. Since
12
15qq=
24 5.qq=
Chapter 10: Cost Functions
107
c. In the long run, given constant returns to scale, location doesn’t really
d. If there are decreasing returns to scale with identical production functions,
10.6 a. From Shephard’s lemma,
b. Eliminate the
10.7 a. As for many proofs involving duality, this one can be algebraically messy
Chapter 10: Cost Functions
108
b. From part (a),
c. This is a CES production function with
1. r=−
Hence,
Analytical Problems
10.9 Generalizing the CES cost function
Chapter 10: Cost Functions
109
Thus, labor’s relative share is
Labor’s relative share depends on
.
If
1,
labor’s share moves in the
10.10 Input demand elasticities
a. The elasticities can be read directly from the contingent demand functions
in Example 10.2. For the fixed proportions case,
b. Because cost functions are homogeneous of degree 1 in input prices,
c. Use Young’s theorem:
Chapter 10: Cost Functions
110
d. Multiplying by shares in part (b) yields
10.11 The elasticity of substitution and input demand elasticities
a. If
i
w
b. If
j
w
does not change,
c. The cost function will be (similar to Equation 10.29)
Chapter 10: Cost Functions
111
By Shephard’s lemma,
10.12 The Allen elasticity of substitution
a. By Shephard’s lemma:
Chapter 10: Cost Functions
112
,
ij
j
i
sp
ji
p
s
eps
=
c. In the Cobb–Douglas case,
1
,C q Bv w
+ + +
=
where
Chapter 10: Cost Functions
113
