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Unformatted text preview: Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Subadditity of Cost Functions: Lecture XX
Charles B. Moss1
1 University of Florida November 3, 2011 Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale 1 Concepts of Subadditivity
Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region 2 Composite Cost Functions and Subadditivity 3 Economies of Scope 4 Economies of Scale Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Evans, D. S. and J. J. Heckman. 1984. A Test for
Subadditivity of the Cost Function with an Application to the
Bell System. American Economic Review 74, 61523.
The issue addressed in this article involves the emergence of
natural monopolies.
Speciﬁcally, is it possible that a single ﬁrm is the most
costeﬃcient way to generate the product.
In the speciﬁc application, the researchers are interested in the
Bell System (the phone company before it was split up). Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Mathematics of te Cost Function
Until this lecture, we have considered cost functions of single
outputs.
The the cost frontier is easily extended to consider multiple
outputs
C ( w 1 , w 2 , y1 , y2 ) = α 0 + α 1 w 1 + α 2 w 2 +
1
α11 w1 w1 + α12 w1 w2 + α22 w2 w2 + β1 y1 + β2 y2 +
2
1
1
β11 y1 y1 + β12 y1 y2 + β22 y2 y2
2
2
γ11 w1 y1 + γ12 w1 y2 + γ21 w2 y1 + γ22 w2 y2 Charles B. Moss Subadditity of Cost Functions: Lecture XX (1) Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region The cost function C (y ) is subadditive at some output level if
and only if
C (y ) <
n
n
C ( yi ) i =1 (2) qi = q i =1 which states that the cost function is subadditive if a single
ﬁrm could produce the same output for less cost.
As a mathematical nicety, the point must have at least two
nonzero ﬁrms. Otherwise the cost function is by deﬁnition the
same. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Developing a formal test, Evans and Heckman assume a cost
function based on two input
C (ai y1 , bi y2 ) > C (y1 , y2 ) i = 1, · · · n
i
(3)
ai ,
bi = 1, ai ≥ 0, bi ≥ 0
i i Thus, each of i ﬁrms produce ai percent of output y1 and bi
percent of the output y2 . Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region A primary focus of the article is the region over which
subadditivity is tested.
The cost function is subadditive, and the technology implies a
natural monopoly.
C ( a i y1 , b i y 2 ) > C ( y 1 , y 2 ) .
(4)
i The cost function is superadditive, and the ﬁrm could save
money by breaking itself up into two or more divisions.
C ( a i y 1 , b i y 2 ) < C ( y1 , y 2 )
(5)
i The cost function is additive if
C ( a i y 1 , b i y 2 ) = C ( y1 , y 2 )
i Charles B. Moss Subadditity of Cost Functions: Lecture XX (6) Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region The notion of additivity combines two concepts from the cost
function: Economies of Scope and Economies of Scale.
Under Economies of Scope, it is cheaper to produce two
goods together.
The example I typically give for this is the grazing cattle on
winter wheat. However, we also recognize following the concepts of Coase,
Williamson, and Grossman and Hart that there may
diseconomies of scope.
The second concept is the economies of scale argument that
we have discussed before. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Subadditivity and the Translog
As stated previously, a primary focus of this article is the
region of subadditivity.
In our discussion of cost functions, I have mentioned the
concepts of Global versus local.
To make the discussion more concrete, let us return to our
discussion of concavity. From the properties of the cost function, we know that the
cost function is concave in input price space. Thus, using the
Translog form
1
ln (C ) = α0 + α ln (w ) + ln (w ) ln (w )
2
1
β ln (y ) + ln (y ) B ln (y ) + ln (w ) Γ ln (y )
2 Charles B. Moss Subadditity of Cost Functions: Lecture XX (7) Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Gradient
Continued
The gradient vector for the Translog cost function is then s1
.
∇w C (w , y ) = exp [ln (C )] . .
sn α1 + A·q ln (w ) + Γ · y
1 .
.
= exp [ln (C )] .
αn + A·n ln (w ) + Γ · y
n Denoting the vector of share equations as φ (w , y ) α1 + A·q ln (w ) + Γ1· ln (y ) .
.
φ (w , y ) = .
αn + A·n ln (w ) + Γn· ln (y )
Charles B. Moss Subadditity of Cost Functions: Lecture XX (8) (9) Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Hessian α1 + A·q ln (w ) + Γ1· ln (y ) .
.
∇2 C (w , y ) = exp [ln (C )] .
xx
αn + A·n ln (w ) + Γn· ln (y ) α1 + A·q ln (w ) + Γ1· ln (y )
A11 · · · . .
..
.
× + exp [ln (C )] .
.
.
.
αn + A·n ln (w ) + Γ · ln (y )
A 1n · · ·
n
C × A + φ (w , y ) φ (w , y )
Charles B. Moss Subadditity of Cost Functions: Lecture XX A 1n
.
.
.
Ann (10) Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Given that the cost is always positive, the positive versus
negative nature of the matrix is determined by
A + φ (w , y ) φ (w , y ) (11) Comparing this results with the result for the quadratic
function, we see that
∇2 C ( w , y ) = A
ww (12) Thus, the Hessian of the Translog varies over input prices and
output levels while the Hessian matrix for the Quadratic does
not. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region In this sense, the restrictions on concavity for the Quadratic
cost function are globalthey do not change with respect to
output and input prices.
However, the concavity restrictions on the Translog are
localﬁxed at a speciﬁc point, because they depend on prices
and output levels.
Note that this is important for the Translog.
Speciﬁcally, if we want the cost function to be concave in
input prices
x A + φ (w , y ) φ (w , y ) x ≤ 0∀x
⇒ x Ax + x φ (w , y ) φ (w , y ) ≤ 0
(13)
⇒ x Ax + φ (w , y ) x φ (w , y ) x ≤ 0
But φ (w , y ) x φ (w , y ) x ≥ 0.
Thus, any discussion of subadditivity, especially if a Translog
cost function is used (or any cost function other than a
quadratic), needs to consider the region over which the cost
function is to be tested.
Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region C q1 Admissible Region q2 Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Thus, much of the discussion in Evans and Heckman involve
the choice of the region for the test.
Speciﬁcally, the test region is restricted to a region of observed
point.
∗
Deﬁning y1m as the minimum amount of y1 produced by any
∗
ﬁrm and y2m as the minimum amount of y2 produced, we can
deﬁne alternative production bundles ytB ∗
∗
ytA = (φy1t + y1m , ω y2t + y2m )
∗
∗
= ((1 − φ) y1t + y1m , (1 − ω ) y2t + y2m )
0 ≤ φ ≤ 1, 0 ≤ ω ≤ 1 Charles B. Moss Subadditity of Cost Functions: Lecture XX (14) Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Mathematics of the Cost Function
Subadditivity and the Translog
Admissible Region Thus, the production for any ﬁrm can be divided into two
components within the observed range of output.
Thus, subadditivity can be deﬁned as
CtA (φ, ω ) = C ytA = C ym + ytA
CtB (φ, ω ) =C ytB = C ym + ytB
Ct =C ytA + ytB = C (yt )
Ct − CtA (φ, ω ) − CtB (φ, ω )
Subt (φ, ω )
Ct (15) If Subt (φ, ω ) is less than zero, the cost function is subadditive,
if it is equal to zero the cost function is additive, and if it is
greater than zero, the cost function is superadditive
Consistent with their concept of the region of the test, Evans
and Heckman calculate the maximum and minimum
Subt (φ, ω )for the region. Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Composite Cost Functions and Subadditivity
Pulley, L. B. and Y. M. Braunstein. 1992. A Composite Cost
Function for Multiproduct Firms with an Application to
Economies of Scope in Banking. Review of Economics and
Statistics 74, 22130.
Building on the concept of subadditivity and the global nature
of the ﬂexible function form, it is apparent that the estimation
of subadditivity is dependent on functional form
Pulley and Braunstein allow for a more general form of the
cost function by allowing the BoxCox transformation to be
diﬀerent for the inputs and outputs.
φ
y −1
y (φ ) =
: φ = 0
(16)
φ
= ln (y ) : φ = 0
Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale The composite cost function form C (φ ) 1
= exp α0 + α q + q (π) Aq (π) + q (π) Ψr
2
(φ)
1
(π )
exp β0 + β r + r Br + q Υr
2
f (φ) (q , ln (r ))
(π ) (τ ) (17) Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale If φ = 0, π = 0, and τ = 1 the form yields a standard
Translog with normal share equations.
If φ = 0 and τ = 1 the form yields a generalized Translog
1
ln (C ) = α0 + α q (π) + q (φ) Aq (π) + q (π) Ψ ln (r ) +
2
1
β ln (r ) + ln (r ) B ln (r )
2
s = Ψq (π) + β + B ln (r ) (18) If π = 1, τ = 0, and Ψ = 0, the speciﬁcation becomes a
separable quadratic speciﬁcation
1
(φ )
C
=
α0 + α q + q Aq
2
(φ)
(19)
1
× exp β0 + β ln (r ) + ln (r ) B ln (r )
2
s = β + B ln (r )
Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale The demand equations for the composite function is
1
s = α0 + α q + q Aq + q Ψ ln (r )
2
B ln (r ) + Υ q
Charles B. Moss −1 Ψq + Subadditity of Cost Functions: Lecture XX (20) Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale Economies of Scope
Given the estimates, we can then measure Economies of
Scope in two ways. The ﬁrst measures is a traditional measure Scope = C (q1 , 0, · · · 0, r ) + · · · C (0, 0, · · · qn , r ) − C (q1 , q2 , · · · qn , r )
C ( q1 , q2 , · · · qn , r )
(21) Another measure suggested by the article is “quasi”
economies of scope Qscope = C ({ 1 − (m − 1) } q1 , q2 , · · · r ) · · · − C (q1 , q2 , · · · qn , r )
C ( q1 , q2 , · · · r )
(22)
Charles B. Moss Subadditity of Cost Functions: Lecture XX Concepts of Subadditivity
Composite Cost Functions and Subadditivity
Economies of Scope
Economies of Scale The Economies of Scale are then deﬁned as
Scale =
i Charles B. Moss C (q , r )
∂ C (q , r )
qi
∂ qi Subadditity of Cost Functions: Lecture XX (23) ...
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