University of Wollongong
Research Online
Faculty of Business - Economics Working Papers
Faculty of Business
1990
Ray scale economies and multiproduct cost
functions
Eduardo Pol
University of Wollongong, [email protected]
Recommended Citation
Pol, Eduardo, Ray scale economies and multiproduct cost functions, Department of Economics, University of Wollongong, Working
Paper 90-8, 1990, 8.
http://ro.uow.edu.au/commwkpapers/322
Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library:
[email protected]
THE UNIVERSITY OF WOLLONGONG
DEPARTMENT OF ECONOMICS
RAY SCALE ECONOMIES AND MULTIPRODUCT COST
FUNCTIONS
E.J. Pol
Department of Economics
The University of Wollongong
Wollongong NSW 2500 Australia
Working Paper 90S,
Co-ordinated by Dr Charles Harvie & Di Kelly
PO Box 1144 [Northfieids Avenue], Wollongong NSW 2500 Australia
Phone: [042] 270 725 or 270 555. Telex 29022. Cable: UNIOFWOL. Fax [042] 270 477
ISSN 1035-4581
ISBN 0-86418-122-1
ABSTRACT
The purpose of this paper is to show that if the ray scale economies depend on the
output vector alone, it is possible to derive the full behaviour of costs as output
bundles change. The formal proofs provide operational criteria for generating
multiproduct cost functions for which ray scale economies depend either on cost
level alone, or on output proportions alone, in a presassigned manner.
1.
Introduction
In the m u l t i p r o d u c t
and t h e re fo re
(1977),
cost cannot
the s t a n d a r d d e f i n i t i o n
cannot be applied.
e co nom ies
case a v e r a g e
An a n a l y t i c a l l y
in a m u l t i p r o d u c t
tr ac tab le
ec on om ie s
co nc ep t
(1977)- is ter me d
e c o n o m i e s ” . It is a s t r a i g h t f o r w a r d
of costs
scale
of scale
s e t t i n g - i n t r o d u c e d by Baumol
and Pan za r and W i l l i g
of s i n g l e - p r o d u c t
of
be defined,
scale ec on om ie s
"ray scale
ex t e n s i o n of
and ind ic at es
the conce pt
the be h a v i o r
as the p r o d u c t i o n levels of a g i v e n b u n d l e of
ou tpu ts cha ng e e q u i p r o p o r t i o n a t e l y . Since ray e c o n o m i e s
scale may be d i f f e r e n t at d i f f e r e n t
m ay be us ef u l
to c h a r a c t e r i z e
for w h i c h ec on om ie s
o u tp ut v e c t o r alone.
scales of operati on,
the c la ss es
of
it
f u nc ti ons
in terms of the
this p a p e r is
that a k n o w l e d g e of a ray scale e c o n o m i e s
this p r o p e r t y allows
of cost
of scale can be d e s c r i b e d
The pu r p o s e
of
to sho w
function having
the d e r i v a t i o n of the m u l t i p r o d u c t
cost
function.
2. Oirectional elasticity and ray average cost
Let C: Y — *R be a m u l t i p r o d u c t
cost
fun ct io n d e f i n e d by the
fo rmu la C = C ( y ) = C(yi ,...,y „ ), wh er e yt
for the ph y s i c a l q u a n t i t y of o u t p u t
(i = l ,...,n)
i, Y for
the n o n n e g a t i v e
o r t h a n t of R“ , and R for the set of n o n n e g a t i v e
By d i r e c t i o n a l
e l a s t i c i t y of
e v a l u a t e d at an d in
the
total
the d i r e c t i o n
of
stands
real numbers.
cost f u n c t i o n
the o u t p u t
C = C(y)
v e c t o r y it
is m e a n t
y
a (y)
=* --C(y)
C(9y)
lim
Q-+1
- C(y)
0>0 ,
9-1
(1)
if the limit exists. When C(y)
limit
is differentiable at y,
the
(1) exists and it turns out to be the sum of all
partial cost elasticities,
i.e.
1
a (y ) =
R a y aver a g e cost
*
^C(y)
--- 2 _
C(y) t = l
yi -----^ yi
(RAC)
is
(2)
the cost of an out pu t v e ct or
of fixed pr o p o r ti o n s
d iv i d e d by an h o mo gen eou s m e a s u r e
of
size of the outputs.
Ana ly ti ca ll y,
the
measure
9 ev a l u a t e d
the RAC in terms of
the
at the o u tp u t vec to r y is
C(0y)
RAC = ------
,
0>O
(3)
0
It is not d i f f i c u l t
determines
the sign of
d i f f e r e n t i a t i o n of
0 = 1 ,
to show that
(3)
the n u m er ic al v al ue
the slope of the RAC.
with r e s p e c t
to 0,
of a(y)
In fact,
the
f o l l o w e d by s e t t i n g
yields
dRAC(y)
--------
= a ( y )-1
(4)
de
In particular,
RAC
a s uf fi c i e n t
is that the d i r e c t i o n a l
[Cf.
Baumol et al.
(1982,
c o n d i t i o n of s tr ic tl y d e c r e a s i n g
e l a s t i c i t y be less
p.51)].
From now on,
than un ity
a(y)
wi l l be
r e f e r r e d to as a r a y scale e c o n o m i e s function.
3. Homotheticity and ray scale economies functions
Be f o r e g o i n g into the proof
of an o p e r a t i o na l m e t h o d
g e n e r a t i n g m u l t i p r o d u c t cost f u n ct io ns with v a r i a b l e
economies
denote
some ad di t i on al n o m e n c l a t u r e
is in order.
the class of all p o s i t i v e l y h o m o ge ne ou s
d e g r e e m.
Let f £ H
be a m u l t i p r o d u c t
scale
Let H
fu n c t i o n s
cost function.
for
A
of
continuously differentiable
and g' (f) >0
for 0^f<°°,
( H C F ) . Finally,
to mak e
in a m u l t i p r o d u c t
term ed
explicit
setting
st r o n g e r d e f i n i t i o n of
consider
is
fu nc t i o n C = git),
a h o m o t h e t i c cost f u n c t i o n
contac t with scale ec on om ie s
it is n e c e s s a r y to in di ca te
a(y).
Let C = C(y)
a continuously differentiable
composite
f u nc ti on
function,
if
a = h«g«f:
y<7C(y)
Given a homogeneous
ec on om ie s
where g(0)=0
cost
a
be an HCF and
function h:
R — >R. The
Y— *R is a r a y s c a l e e c o n o m i e s
= a (y)C (y)
(5)
f u n c t i o n and a ray scale
fu n c t i o n it is s h o w n b e l o w h o w these fu n c t io ns
be t r a n s f o r m e d
to y i e l d
an HCF
can
e x h i b i t i n g the p r e a s s i g n e d
sc ale economies.
Theorem 1
Let f £ H .
T h e re exi st s
scale e c o no mie s
sa t i s f i e s
an HCF,
function
C = C(y),
a = a(y),
if and only if,
the f o l l o w i n g d i f f e r e n t i a l
dlogC
h(C)
d l o gf
m
with p r e a s s i g n e d
C
equation:
Proof
Necess ity.
If there
exi sts
an HCF,
C = C(y),
w i t h h(C)
preassigned,
yVC(y)
But since f £ H ,
it fo l l o w s
=
dC
y — Vf
df
= h(C)C
fr o m Euler' s
dC
—
mf=h(C)C,
df
i.e.
the d e s i r e d result.
(7)
theor em that
(8)
Sufficiency. The general solution of equation
(6) can be
written as:
exp [<f> (y) ] = kf
(9)
where
0(y)
and k is a c o n s t a n t
A p p e n d i x that
differential
(9)
of
= mJdC/h(C)C
i n te gr at io n.
is the g e n e r a l
equation
(5).
Thus
(10)
It is shown
in the
s o l u t i o n of the partial
sufficiency
is proved.
4- Inhomothetic cost functions.
HCFs have
contours,
scale
economies
so that w h e n
the p r e c e d i n g
a(y)
i n v ar ia nt
is ho mo g e n e o u s
theorem breaks
generating constant
a along every ray,
down.
c a n n o t be
alone will be t er m e d
economies
of deg re e
In fact,
a a l o n g each ray,
for w h i ch ray scale
to y along isocost
an HCF.
a cost
zero,
function
but di f f e r e n t values
Those cost
functions
d e p e n d on output pr op ort ion s
i n h o m o t h e t i c cost f u n c t i o n s
( ICF).
Theorem 2
The fu nc tio na l
form of
an ICF is
a (y)
C {y ) =
w here B(y)
one,
is an a r b i t r a r y
and a(y)
[B(y)]
(11)
f u n c t i o n h o m o ge ne ou s
is h o m o g e n e o u s
of de gr e e
zero.
of degree
of
Proof
Using
the same
s o l u t i o n of
technique
(5)
as
is r e a d i l y
in the Appendix,
seen
the general
to be given by
F (K t .....K« ) = 0 ,
where
F(.)
is an a r b i t r a r y
(12)
function,
yl
Kl-t
----
( i - 2 , ----n)
(13)
yt
and
Kd
Equation
(12)
- C
[ y t ® > ]-t
mu s t be so l u b l e
(14)
as
a (y)
K„ = [J(Ki.... Kn-i )]
w h e r e J(.)
is an a r b i t r a r y
for Kt - i >0,
i=2,...,n.
(15)
f u n c t i o n tak in g pos iti ve values
Therefo re,
a (y)
C =
and the p ro of
[J(Ki ----- Ka-i )yi ]
(16)
is complete.
5. Conclusions
A d u a l i t y for the m u l t i p r o d u c t
s c a l e ec o n o m ie s
fi r m exists b e tw ee n
the ray
and
the cost
fu nctions.
provide operational
criteria
for g e n e r a t i n g m u l t i p r o d u c t
fu n c t i o n s w i t h v a r i a b l e
scale
e c on om ie s.
m u l t i p r o d u c t HCF a s s o c i a t e d w i t h
r a y scale e c o n o m i e s
differential
Moreover,
cost
a
the
f u n c t i o n can be o b t a i n e d by solving
the
e q u a t i o n of v a r i a b l e s
alone,
In fact,
2
a p a r t i c u l a r choice of
separated
if the ra y scale e c o n o m i e s
proportions
T h e o r e m 1 and
the class
of
form
(6).
d e p e n d on the output
ICF is d e s c r i b e d by
(11).
Several m et ho ds
de v e l o p e d
since
for m e a s u r i n g
the
1930s.
scale
The m e s s a g e
this paper
is that an e c o n o m e t r i c
p ara met er s
of
d e r i v a t i o n of
ec onomies
the ray scale
have been
that comes
across
with
es ti ma t i o n of the
economies
fu nc ti on permi ts
the u n d e r l y i n g m u l t i p r o d u c t
cost
the
function.
Appendix
The ge ner al s o l u t i o n of
pa rti al d i f f e r e n t i a l
the q u a s i - l i n e a r
equation
wh er e k is a constant,
0(y)
(5)
first order
is gi ve n by exp[0(y)]
= raJdC/h(C)C,
= Jcf,
and f £ H is an
a r b i t r a r y function.
Proof
The
technique of s o l u t i o n e m p l o y e d here
Sm irn ov
(1964,
esp.
p p . 310-316).
dyi/yi
is eq ui v a le nt
y 2 /yi
=
...
to e q u a t i o n
= ki , ...,
independent.
yn /yi
Then
is d i s c u s s e d in
T he d i f f e re nt ia l
system
= d y n /y a = d C / h ( C ) C
(5).
Clearly,
= kn -i,
system
(18)
(18)
the first inte gr al s
exp [ 0 ( y ) ] /y? = kn
has
a gener al
(19)
so lu t i o n of
the form
<3( yz /yi , ...,
wh er e G{
yn /yi , ex p[ 0(y)]/yi
1 is an a r b i t r a r y f u nc ti on.
for the last argument,
1=0,
Solvi ng
eq ua tio n
(20)
(20)
yields:
YV\
exp [0 (y) ] = yi
Thus,
J6H.
the proof
J ( y 2 /yt,
Let J = k f , w h e r e
is complete.
...,
yn /yi )
k is a c o n s t a n t and f £ H ,
(21)
and
REFERENCES
Baumol, William J., 1977, On the Proper Cost Tests for Natural Monopoly in a Multiproduct
Industry, American Economic Review 5, 809-22.
...... — ........ ; John C. Panzar and Robert D. Willig, 1982, Contestable Markets and the
Theory o f Industry Structure (Harcourt Brace Jovanovich, San Diego).
Panzar, John C. and Robert D. Willig, 1977, Economies of Scale in Multi-Output Production,
Quarterly Journal o f Economics 3, 481-493.
Smirnov, Vladimir I., 1964, A Course o f Higher Mathematics, Vol. IV (Pergamon Press,
London).
I wish to thank Professors William J. Baumol and Robert D. Willig for helpful comments and
suggestions on an earlier draft of this paper. In particular, I have found Professor W illig’s
insights most useful. The usual disclaimer applies.
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