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 (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, concep ts, 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 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, shows, in 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 the production function can be recovered in such cases.
78
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. Letq = q1+q2, where both q1 and q2 >0. Because C ( q1 , q2 ) / q < C ( q1, 0) / q1 by assumption, q1C ( q1, q2 ) / q < C (q1, 0) . Similarly q2C ( q1, q2 ) / q < C (0, q2 ) . Summing yields C ( q1 , q2 ) < C ( q1 ,0) + C (0, q2 ) , which proves economies of scope.
10.3
a.
b.
q = 9000.5 J 0.5 = 30 J J = 25 q = 150 J = 100 q = 300 J = 225 q = 450 J = 12q2/900 Cost = 12 dC 24q 2q MC = = = 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
Chapter 10: Cost Functions
80
a.
In the long run, keep 5k = 10,
C = 2l + 3l = 5l = 0.5q b.
k = 2l 5l AC = = 0.5 10l
MC = 0.5 .
k = 10 q = min(50, 10l) l < 5, q = 10l C = 10 + 3l = 10 + 0.3q 10 AC = +0.3 q 10 + 3l If l > 5, q = 50 C = 10 + 3l AC = 50 MC is infinite for q > 50. MC 10 = MC 50 = .3. MC 100 is infinite.
10.5
a. q = 2 kl , k = 100, q = 2 100 l q = 20 l
q l= 20
2
q l= 400
2 ⎛ q2 ⎞ q SC = vK + wL = 1(100) + 4⎜ ⎟ = 100 + 100 400 ⎝ ⎠
SAC = b.
SC 100 q = + q q 100
q ⎛ 252 ⎞ SMC = . If q = 25, SC = 100 + ⎜ ⎟ = 106.25 50 ⎝ 100 ⎠ SAC =
100 25 + = 4.25 25 100
SMC =
25 = .50 50
⎛ 502 ⎞ If q = 50, SC = 100 + ⎜⎜ ⎟⎟ = 125 ⎝ 100 ⎠ SAC =
100 50 + = 2.50 50 100
SMC =
50 =1 50
⎛ 100 2 ⎞ If q = 100, SC = 100 + ⎜⎜ ⎟⎟ = 200 ⎝ 100 ⎠
Chapter 10: Cost Functions
SAC =
100 100 + =2 100 100
SMC =
81
100 =2. 50
⎛ 2002 ⎞ If q = 200, SC = 100 + ⎜⎜ ⎟⎟ = 500 100 ⎝ ⎠ SAC =
100 200 + = 2.50 200 100
SMC =
200 =4. 50
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.
q = 2 kl so q = 4kl
2
2
l = q / 4k 2
SC = vk + wl = vk + wq /4k f.
2 ∂SC 2 = v − wq /4k = 0 so k = 0.5qw0.5v −0.5 ∂k
0.5
g.
C = vk + wl = 0.5q w0.5v 0.5 + 0.5q w0.5 v0.5 = qw v0.5 (a special case of Example10.2)
h.
If w = 4
v = 1,
C = 2q 2
SC = ( k = 100) = 100 + q /100 , SC = 200 = C for q = 100 2
SC = ( k = 200) = 200 + q /200 , SC = 400 = C for q = 200 SC =800 = C for q = 400
Chapter 10: Cost Functions
82
10.6
a.
q total = q1 + q2 . q1 = 25l 1 = 5 l 1 2
q 2 = 10 l 2 2
SC 1 = 25 + l 1 = 25 + q1 /25
S C 2 = 100 + q 2 /100 2
2
q1 q + 2 SC total = SC1 + SC2 = 125 + 25 100 To minimize cost, set up Lagrangian: £ = SC + λ (q − q1 − q2 ) .
∂£ 2q1 = − λ = 0 ∂ q1 25 ∂ £ 2q 2 − λ = 0 = ∂ q 2 100 Therefore q1 b.
4 q1 = q 2
q1 = 1/5 q
= 0.25q2 . q 2 = 4/5 q
2
q SC = 125 + 125 SMC (100) =
SMC =
SAC =
125 q + q 125
200 = $1.60 125
SMC (125) = $2.00 c.
2q 125
SMC (200) = $3.20
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.
10.7
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.
From Shephard's Lemma 1/ 3
a.
∂C 2 ⎛ v ⎞ l= = q ∂w 3 ⎜⎝ w ⎟⎠
∂C 1 ⎛ w ⎞ k= = q ∂v 3 ⎜⎝ v ⎟⎠
2/3
Chapter 10: Cost Functions
83
Eliminating the w/v from these equations:
b.
⎛3⎞ q=⎜ ⎟ ⎝2⎠
2/3 1/3
( 3) l 2 / 3 k 1/ 3 = Bl 2 / 3k1/ 3
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,
=
k b.
∂C ∂C = Bv −0.5q l = = Bw −0.5q. ∂v ∂w
From part a,
q k
=
v 0.5 , B
q l
=
w0.5 B
so
q k
q l
+ = 1 or k −1 + l −1 = q −1
The production function then is q = (k −1 + l −1 ) −1 . 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.
C
= q1 γ [( v a )1−σ + ( w b)1−σ ]1 1−σ .
b.
C
= qa − ab − bv a wb .
c.
wl vk
d.
=b a.
(v / a) σ ] so wl vk = (v w)σ −1 (b a ) σ . Labor’s ( w b) 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.
k l
= RTS σ or l k = [
Chapter 10: Cost Functions
84
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, el c , w = ek c ,v = 0 (because
q is held constant). For the Cobb-Douglas, el c , w = − α α + β , ek c ,v = − 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 lwc w + lvc v = 0 . Dividing by l c gives the result.
c.
Use Young’s Theorem: ∂l c ∂ 2C ∂ 2C ∂k c vwl c vwk c = = = Now multiply left by c right by . ∂v ∂v∂w ∂w∂v ∂w lC k cC Multiplying by shares in part b yields sl el c , w + sl el c ,v = 0 . Substituting from
d.
part c yields sl el c ,w + sk ek c , w e.
=0.
All of these results give important checks to be used in empirical work.
10.11 The elasticity of substitution and input demand elasticities If wi does not change, s i , j = ∂ ln( xic / x jc ) / ∂ ln( w j / wi ) = ∂ ln( xic / x jc ) / ∂ ln( w j )
a.
e x c , w
= ∂ ln xic / ∂ ln w j
e x c , w
= ∂ ln x jc / ∂ ln w j
e x c , w
− e x ,w = ∂ ln xic / ∂ ln w j − ∂ ln x jc / ∂ ln w j = ∂ ln( xic / x jc ) / ∂ ln w j = si , j
i
j
i
b.
j
j
c j
j
does not change, w j If s j ,i = ∂ ln( x jc / xic ) / ∂ ln(wi / w j ) = ∂ ln( x jc / xic ) / ∂ ln( wi )
e x c , w
= ∂ ln x jc / ∂ ln wi
e x c , w
= ∂ ln xic / ∂ ln wi
e x c , w
− e x ,w = ∂ ln x jc / ∂ ln wi − ∂ ln xic / ∂ ln wi = ∂ ln( x jc / xic ) / ∂ ln wi = s j ,i
j
i
j
c.
j
i
i
i
c i
i
The cost function will be (similarly to equation 10.26):
Chapter 10: Cost Functions
n
C ( w1 , w 2 ,..., w n , q )
= q (∑ w
85
( ρ −1) / ρ ρ /( ρ −1) k
)
k =1
n
Let B
= ( ∑ w ρ k /( ρ −1) ) k =1
By Shephard ' s lemma :
= ∂C ( w1 , w 2 ,..., w n , q ) / ∂w i = qB −1 / ρ w i1 /( ρ −1) x jc ( w1 , w 2 ,..., w n , q ) = ∂ C ( w1 , w 2 ,..., w n , q ) / ∂ w j = qB −1 / ρ w j1 /( ρ −1) x ic ( w1 , w 2 ,..., w n , q )
e x c , w
= ( ∂ x ic / ∂w j )( w j / x ic ) = [ − 1 /( ρ − 1)] B −1 w ρ j /( ρ −1)
e x c , w
= ( ∂ x jc / ∂w j )( w j / x jc ) = 1 /( ρ − 1) − [1 /( ρ − 1)] B −1 w ρ j /( ρ −1) = −σ + e x
e x c , w
= ( ∂ x jc / ∂w i )( w i / x jc ) = [ − 1 /( ρ − 1)] B −1 w ρ i /( ρ −1)
i
j
j
j
j
e x c , w i
s i , j
i
i
= ( ∂ x ic / ∂w i )( w i / x ic ) = 1 /( ρ − 1) − [1 /( ρ − 1)] B −1 w ρ i /( ρ −1) = −σ + e x
= s j ,i = e x
c i
, w j
− e x
c j
, w j
= e x
c j
, wi
− e x
c i
, wi
= σ
10.12 The allen elasticity of substitution a.
= ∂C / ∂wi = C i = (∂ xic / ∂w j )( w j / xic ) = (∂C / ∂wi ∂w j )( w j / xic ) = C ij ( w j / C i )
By Shephard ' s lemma : xic e x c , w i
j
= w j x jc / C = w j C j / C e x , w / s j = C ij ( w j / C i )C /( w j C j ) = C ij C / C i C j = Ai , j s j
c i
j
b.
esi , p j
= (∂si / ∂ p j )( p j / s i ) = [∂ ( pi C i / C ) / ∂ p j ][ p j /( pi C i / C )] =
= [ pi ∂ (C i / C ) / ∂ p j ]( p j C / pi C i ) = pi [(C ji C − C i C j ) / C 2 ]( p j C / pi C i ) = = (C ji C − C i C j ) p j / C i C = (C ji C / C j C i − 1) p j C j / C = s j ( Ai , j − 1) c. The Cobb-Douglas case:
c j
c i
, w j
, wi
Chapter 10: Cost Functions
86
C = q 1 /( α + β ) Bv α /(α + β ) w β /( α + β ) , where B
= (α + β )α −α / α + β β − β / α + β
= ∂C / ∂w = q 1 /(α + β ) B[ β /(α + β )]v α /(α + β ) w −α /(α + β ) C k = ∂C / ∂v = q 1 /(α + β ) B[α /(α + β )]v − β /(α + β ) w β /(α + β ) C l , k = ∂C k / ∂w = q 1 /(α + β ) B[αβ /(α + β ) 2 ]v − β /(α + β ) w −α /(α + β ) Ak ,l = C k ,l C / C k C l = 1 C l
The CES case:
C = q1 / γ (v1−σ
+ w1−σ )1 /(1−σ ) C l = ∂C / ∂w = q 1 / γ (v1−σ + w1−σ ) σ /(1−σ ) w −σ C k = ∂C / ∂v = q1 / γ (v1−σ + w1−σ ) σ /(1−σ ) v −σ C k ,l = ∂C k / ∂w = q1 / γ σ (v1−σ + w1−σ ) ( 2σ −1) /(1−σ ) v −σ w −σ Ak ,l = C k ,l C / C k C l = σ