ch10

78

CHAPTER 10 COST FUNCTIONS

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 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.

10.2 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.

10.3 A simplified numerical Cobb-Douglas example in which one of the inputs is held

fixed.

10.4 A fixed proportion example. The very easy algebra in this problem may help to solidify basic concepts.

10.5 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.

10.6 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.

10.7 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.

10.8 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 79

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 Support the draftsman. It's geometrically obvious that SAC cannot be at minimum

because it is tangent to AC at a point with a negative slope. The only tangency occurs at minimum AC.

10.2 a. By definition total costs are lower when both q1 and q2 are produced by the

same firm than when the same output levels are produced by different firms [C(q1,0) simply means that a firm produces only q1].

b. Let q = q1+q2, where both q1 and q2 >0. Because 1 2 1 1( , ) / ( ,0) /C q q q C q q< by

assumption, 1 1 2 1( , ) / ( ,0)q C q q q C q< . Similarly 2 1 2 2( , ) / (0, )q C q q q C q< . Summing yields 1 2 1 2( , ) ( ,0) (0, )C q q C q C q< + , which proves economies of scope. 10.3 a. 150 = q 25 = J J 30 = J900 = q 5.05.0

J = 100 q = 300 J = 225 q = 450 b. Cost = 12 J= 12q2/900

dC 24q 2qMC = = = dq 900 75

q = 150 MC = 4 q = 300 MC = 8 q = 450 MC = 12

10.4 q = min(5k, 10l) v = 1 w = 3 C = vk + wl = k + 3l


a. In the long run, keep 5k = 10, k = 2l

0.5 0.5 0.55lC = 2l + 3l = 5l q AC = = MC = .10l

=

b. k = 10 q = min(50, 10l)

0.3l < 5, q = 10l C = 10 + 3l = 10 + q

0.310AC = + q

If l > 5, q = 50 C = 10 + 3l 10 + 3lAC = 50

MC is infinite for q > 50.

MC10 = MC50 = .3.

MC100 is infinite.

10.5 a. ,q = 2 kl k = 100, q = 2 100 l q = 20 l

2q ql = l = 20 400

2 2q qSC = vK + wL = 1(100) + 4 = 100 + 400 100

⎛ ⎞⎜ ⎟⎝ ⎠

SC 100 qSAC = = + q q 100

b. 225If q = 25, SC = 100 + = 106.25

100qSMC = .

50⎛ ⎞⎜ ⎟⎝ ⎠

.50 = 5025 = SMC4.25 =

10025 +

25100 = SAC

If q = 50, SC = 100 + 125 = 100502

⎟⎟⎠

⎞⎜⎜⎝

⎛

1 = 5050 = SMC2.50 =

10050 +

50100 = SAC

If q = 100, SC = 100 + 200 = 1001002

⎟⎟⎠

⎞⎜⎜⎝

⎛


. 2 = 50

100 = SMC2 = 100100 +

100100 = SAC

If q = 200, SC = 100 + 500 = 1002002

⎟⎟⎠

⎞⎜⎜⎝

⎛

. 4 = 50200 = SMC2.50 =

100200 +

200100 = SAC

c.

d. As long as the marginal cost of producing one more unit is below the average-cost curve, average costs will be falling. Similarly, if the marginal cost of producing one more unit is higher than the average cost, then average costs will be rising. Therefore, the SMC curve must intersect the SAC curve at its lowest point.

e. 2 2so = 4 = / 4q qq = 2 kl kl l k

2SC = vk + wl = vk + /4kwq

f. 2 0.5 0.5so = 0.52SC = v /4k = 0 k qw vwq

k−∂

−∂

g. 0.50.5 0.5 0.5 0.5 0.50.5 0.5C = vk + wl = q + q = qww v w v v (a special case of Example10.2)

h. If w = 4 v = 1, C = 2q

( ) 2SC = k = 100 = 100 + /100q , SC = 200 = C for q = 100

( ) 2SC = k = 200 = 200 + /200q , SC = 400 = C for q = 200

SC =800 = C for q = 400


10.6 a. 1 2total = q qq + . 1 1 21 2 = 25 = 5 = 10q ql l l

12 2

1 21 2 = 25 + = 25 + /25 S = 100 + /100q qSC l C

2 21 2

total 1 2 = + = 125 + + 25 100q qSC SCSC

To minimize cost, set up Lagrangian: 1 2£ ( )SC q q qλ= + − − .

£ 1

1

2q = = 025q

λ∂−

∂

£ 2

2

2q = = 0100q

λ∂−

∂

Therefore 1 20.25q q= .

b. q 4/5 = q q 1/5 = q q = q4 2121

2 2q 125 qqSC = 125 + SMC = SAC = +

125 125 q 125

(100) 200SMC = = $1.60125

SMC(125) = $2.00 SMC(200) = $3.20

c. In the long run, can change k so, given constant returns to scale, location

doesn't really matter. Could split evenly or produce all output in one location, etc. C = k + l = 2q

AC = 2 = MC

d. If there are decreasing returns to scale with identical production functions,

then should let each firm have equal share of production. AC and MC not constant anymore, becoming increasing functions of q.

10.7 From Shephard's Lemma

a. 1/ 3 2 / 32 1

3 3C v C wl q k qw w v v∂ ∂⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠


b. Eliminating the w/v from these equations:

( ) 2 / 3 1/ 3 2 / 3 1/ 32/3

1/33q = l k = Bl k32

⎛ ⎞⎜ ⎟⎝ ⎠

which is a Cobb-Douglas production function.

10.8 As for many proofs involving duality, this one can be algebraically messy unless

one sees the trick. Here the trick is to let B = (v.5 + w.5). With this notation, C = B2q.

a. Using Shephard’s lemma,

0.5 0.5 .C Ck Bv q l Bw qv w

− −∂ ∂= = = =∂ ∂

b. From part a,

0.5 0.5

1 1 1, 1q v q w q qso or k l qk B l B k l

− − −= = + = + =

The production function then is 1 1 1( ) .q k l− − −= +

b. This is a CES production function with ρ = -1. Hence, σ = 1/(1-ρ) = 0.5. Comparison to Example 8.2 shows the relationship between the parameters of

the CES production function and its related cost function. Analytical Problems 10.9 Generalizing the CES cost function

a. 1 1 1 1 1[( ) ( ) ]C q v a w bγ σ σ σ− − −= + .

b. a b a bC qa b v w− −= .

c. wl vk b a= .

d. 1( / ) or [ ] so ( ) ( )( )v ak l RTS l k wl vk v w b aw b

σ σ σ σ−= = = . Labor’s

relative share is an increasing function of b/a. If σ > 1 labor’s share moves in the same direction as v/w. If σ < 1, labor’s relative share moves in the opposite direction to v/w. This accords with intuition on how substitutability should affect shares.


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, , ,

0c cl w k ve e= = (because

q is held constant). For the Cobb-Douglas, , ,

,c cl w k ve eα α β β α β= − + = − + .

Apparently the CES in this form has non-constant elasticities.

b. Because cost functions are homogeneous of degree one in input prices, contingent demand functions are homogeneous of degree zero in those prices as intuition suggests. Using Euler’s theorem gives 0c c

w vl w l v+ = . Dividing by cl gives the result.

c. Use Young’s Theorem:

2 2c cl C C k

v v w w v w∂ ∂ ∂ ∂

= = =∂ ∂ ∂ ∂ ∂ ∂

Now multiply left by right by c c

c c

vwl vwkl C k C

.

d. Multiplying by shares in part b yields , ,

0c cl ll w l vs e s e+ = . Substituting from

part c yields , ,

0c cl kl w k ws e s e+ = .

e. All of these results give important checks to be used in empirical work.

10.11 The elasticity of substitution and input demand elasticities

a. If wi does not change, )ln(/)/ln()/ln(/)/ln(, j

cj

ciij

cj

ciji wxxwwxxs ∂∂=∂∂=

jijcj

cij

cjj

ciwxwx

jcjwx

jciwx

swxxwxwxee

wxe

wxe

jcjj

ci

jcj

jci

,,,

,

,

ln/)/ln(ln/lnln/ln

ln/ln

ln/ln

=∂∂=∂∂−∂∂=−

∂∂=

∂∂=

b. If wj does not change,

)ln(/)/ln()/ln(/)/ln(, ici

cjji

ci

cjij wxxwwxxs ∂∂=∂∂=

ijici

cji

cii

cjwxwx

iciwx

icjwx

swxxwxwxee

wxe

wxe

icii

cj

ici

icj

,,,

,

,

ln/)/ln(ln/lnln/ln

ln/ln

ln/ln

=∂∂=∂∂−∂∂=−

∂∂=

∂∂=

c. The cost function will be (similarly to equation 10.26):


σ

σρρ

ρ

σρρ

ρ

ρρ

ρρ

ρρ

ρρ

ρρ

ρρ

ρρ

ρρρρ

=−=−==

+−=−−−=∂∂=

−−=∂∂=

+−=−−−=∂∂=

−−=∂∂=

=∂∂=

=∂∂=

=

=

−−

−−

−−

−−

−−

−−

=

−

−

=

−

∑

∑

icii

cjj

cjj

ci

icji

ci

icj

jcij

cj

jci

wxwxwxwxijji

wxiciii

ciwx

icjii

cjwx

wxjcjjj

cjwx

jcijj

ciwx

jjnncj

iinnci

n

kk

n

kkn

eeeess

ewBxwwxe

wBxwwxe

ewBxwwxe

wBxwwxe

wqBwqwwwCqwwwx

wqBwqwwwCqwwwxlemmasShephardBy

wBLet

wqqwwwC

,,,,,,

,)1/(1

,

)1/(1,

,)1/(1

,

)1/(1,

)1/(1/12121

)1/(1/12121

1

)1/(

/)1(

1

)1/(21

)]1/(1[)1/(1)/)(/(

)]1/(1[)/)(/(

)]1/(1[)1/(1)/)(/(

)]1/(1[)/)(/(

/),,...,,(),,...,,(

/),,...,,(),,...,,(:'

)(

)(),,...,,(

10.12 The allen elasticity of substitution a.

jijiijjjijijjwx

jjcjjj

ijijcijji

cijj

ciwx

iici

ACCCCCwCCwCse

CCwCxws

CwCxwwwCxwwxe

CwCxlemmasShephardBy

jci

jci

,,

,

/)/()/(/

//

)/()/)(/()/)(/(

/:'

===

==

=∂∂∂=∂∂=

=∂∂=

b.

)1(/)1/(/)(

)/](/)[()/](/)/([

)]//(][/)/([)/)(/(

,

2

,

−=−=−=

=−=∂∂=

=∂∂=∂∂=

jijjjijjiijjiji

iijjijiiiijjii

iijjiiijjips

AsCCpCCCCCCpCCCC

CpCpCCCCCpCpCppCCp

CCpppCCpsppseji

c.

The Cobb-Douglas case:


1/])/([/

)]/([/

)]/([/)(,

,,

)/()/(2)/(1,

)/()/()/(1

)/()/()/(1

//)/()/()/(1

==

+=∂∂=

+=∂∂=

+=∂∂=

+==

+−+−+

++−+

+−++

+−+−+++

lklklk

kkl

k

l

CCCCAwvBqwCC

wvBqvCC

wvBqwCCBwherewBvqC

βααβαββα

βαββαββα

βααβααβα

βαββααβαββααβα

βααβ

βαα

βαβ

βαβα

The CES case:

σσ σσσσσσγ

σσσσσγ

σσσσσγ

σσσγ

==

+=∂∂=

+=∂∂=

+=∂∂=

+=

−−−−−−

−−−−

−−−−

−−−

lklklk

klk

k

l

CCCCAwvwvqwCC

vwvqvCC

wwvqwCCwvqC

/)(/

)(/

)(/)(

,,

)1/()12(11/1,

)1/(11/1

)1/(11/1

)1/(111/1

ch10

Documents