The characterization of external effects as “separable” has played an important role in the development of the theory of externalities. The separable case is particularly well behaved when procedures for achieving an optimum allocation of resources in the presence of externalities are examined. Davis and Whinston (1962) find that separability assures the existence of a certain kind of equilibrium in bargaining between firms which create externalities, and that equilibrium does not exist without separability. Kneese and Bower (1968) argue that with separability the computation of Pigovian taxes to remedy externalities is particularly simple. Marchand and Russell (1974) demonstrate that certain liability rules regarding external effects lead to Pareto optimal outcomes if and only if externalities are separable. In each of these cases the problem is posed in terms of two firms related by technological externalities, and separability is defined in terms of a cost function. In this paper, we will characterize that class of production functions which give rise to separable cost functions, and show that the relation between production functions and separable cost functions is by no means as trivial as has been claimed

Montgomery, W. David

Caltech Authors - Main

CALI F O R N I A  I N S TITU T E  O F  T E CHN OLOGY 
Division of the Humanities and Social Sciences 
Pasadena, California 91109 
SEPARABLE EXTERNALITIES IN 
COST AND PRODUCTION FUNCTIONS 
W. David Montgomery 
Social Science Working Paper 
Number 42 
Jtme 1974 
SEPARABLE EXTERNALITIES IN 
COST AND PRODUCTION FUNCTIONS 
W. David Montgomery 
California Institute of Technology 
The characterization of external effects as 11separable11 
has played an important role in the development of the theory of 
externalities. The separable case is particularly well behaved when 
procedures for achieving an optimum allocation of resources in the 
presence of externalities are examined. Davis and Whinston ( 1962) 
find that separability assures the existence of a certain kind of 
equilibrium in bargaining between firms which create externalities, 
and that equilibrium does not exist without separability. Kneese and 
Bower ( 1968) argue that with separability the computation of Pigovian 
taxes to remedy externalities is particularly simple. Marchand and 
Russell ( 1974) demonstrate that certain liability rules regarding 
external effects lead to Pareto optimal outcomes if and only if 
externalities are separable. In each of these cases the problem is 
posed in terms of two firms related_ by technological externalities, 
and separability is defined in terms of a cost function. In this paper 
we will characterize that class of production functions which give rise 
to separable cost functions, and show that the relation between production 
functions and separable cost functions is by no means as trivial as has 
been claimed. 
Let C ( Y l' Y z) be the cost function of a firm which produces
Y1, and suffers an external diseconomy which is a function of Y2.
That cost function is defined in terms of a production func'tion 
F( X . • •  X , Y2), in the following manner:1 n 
C(Yl' Y2) min I: w.x.. ' ' ' 
subject to Y1 = F(X1 • .  · Xn
, Y2 ).
Some precise definitions and lemmas regarding separability 
are needed: 
Definition 1: A cost function C(Yl' Y2) is separable if and only if
if can be written as c1(Y1) + C2(Y2).
2 
Definition 2: A production function F(X1 • . .  Xn' Y2) is separable
if and only if it can be written as g( x1 . • .  Xn) + h( Y2 ).
Lemma l: 
everywhere. 
Proof: 
2 
A cost function is separable if and only if d� �y 
l 2 
twice and 
Necessity is proved by differentiating C = c1 + c22 
observing that �� 
l
�y 2 = 0. Sufficiency is proved
1 
by
0 
observing that the general solution of the second-order partial differential 
2 
equation O� �y = 0 is of the form C = C1(Y1) + C2(Y2).1 2 
1. Lester Ford, [ 1955], p. 251 derives this result. 
We assume throughout that C and F are continuously twice 
differentiable. The problem is to find what general class of fi.mctions F 
3 
give rise to cost functions with the property 
o2c
It has o. oY1oY2
been claimed by Marchand and Russell that a cost function is separable 
if and only if the production frunction from which it is derived is 
separable. This conjecture is false in both directions. We begin by 
giving a simple example of a separable production function which does 
not generate a separable cost function. It will be seen later that the 
class of separable functions does not exhaust the class_ of functions 
giving rise to separable cost functions. 
Consider a production function of the form 
F • Ii x1x2 -cY2 (I) 
where a. + f3 < 1. We find the cost function by solving the cost-
minimization problem and using the first-order conditions and the 
production function to eliminate the inputs from the cost equ,ation. 
From the first-order conditions we have 
wl 
wz
.xz
8X1
Solving for x1 and substituting in ( 1) gives 
•+8 (Wz• )" YI = Xz W18 CY2.
( 2) 
( 3)  
Solving ( 3) for x2, and substituting the resulting expression for x2
in (2) enables us to express x1 and x2 in terms of Y1 and Y2 alone. 
Substitution in C = w1x1 + w2x2 gives the cost function 
• 
c 
( ·)fW 2 •-)•+8 . Wz l+�\w\ 8 (YI+ 
Clearly ( 4) is not separable if a. + f3 � 1. 
_I_ 
CY )o+P 2 • (4) 
4 
To f:ind a production function which does generate a separable 
2 
cost fnnction we express a: d;. in terms of the derivatives of the 1 2 
production function, and then find a general solution of the partial 
2 
differential equation which results when d; d� is set equal to zero. 
I 2 
The general relation between cost and production functions is 
fo1.md by adopting the approach of Samuelson1s Foundations. Consider 
the constrained cost minimization problem 
Minimize L:w.x. subject to Y1 - F (X1, • • .  X ) . 1 i n ' 
We adopt the following abbreviations 
oF oF 
ax. = Fi cY = FY
' 
2 
o. 
czF " F ;ix.;ixi •i
....£.£__ = F oX.oY iY . ' 
Form the Lagrangian expression 
L. = L;w ix i + >. ( y I - F ( XI • • . X n' y 2) ) . ' 
First-order conditions are 
w i >.F. ' 0
Y1 - F o. 
We perturb the solution by varying Y1 and Y2. Totally differentiat:ing 
the first-order conditions gives the system of equations 
F II .
Fn l.
FI .
F ln F 1 
F F nn n 
F 0n 
Let A= 
dX1 
dX n 
d)JI. 
Fil.
Fnl.
FI .
=
dW1 
I. 
dW 
FIYdYZ 
__ 
n
_ F dY A nY 2 
dY1 - FydY2 
F ln Fl 
F F nn n 
F 0n 
Further let !::. • •  be the i, j th co-factor of /J.. lJ 
Since c = Lwkx , k k 
� 
oY2 
oXk = L:wkoY k 2
Solving for dXk using Cramer's rule gives 
( 6) 
dXk = 
n 
l
' dW. ) � \T - FiYdYZ Llik + (dY1 - F�dY2)An+l,k] 
A 
We assume that F is strictly quasi-concave in X1 . . •  Xn' so that 
A 1 0. 
Then 
oXk -- = oY2 
-�FiY6ik - FYAn+l,
1 
A 
5 
and oc 
CIYz
- ��k[� FiYAik + FY6 n+1,k]
A 
Since Wk = AFk, 
� 
oY2 
-i:JF;y>-
_
(�FkAik)]- AFYt FkAn+l,k
A 
6 
But tFkAn+l, k =A, a.nd LF !::. • = 0 since it is a
n expansion by 
k K ik 
alien co-factors. Therefore 
ac 
oY2 = -
AFy.
(7) 
Differentiating ( 7) with respect to Y 1 gives 
From (6), 
Therefore 
(IZC 
oY 2aY 1
ax; 
->.L;FiYoY11 
F 21_YoY1 
� 
oY1 
o2C 
oY2?Y1 
= A�1+l, n+lA 
oXi
oY1
An+l, i 
A 
---�(L:F A + A  F )fl . iY n+l. i n+l,n+l Y · 1 
7 
Characterizing the class of production functions which generate 
separable cost functions reduces to finding the general form of the 
solution of 
L;F 6 . + 6 F = O. . iY n+l, i n+l,n+l Y ( 8) 
' 
Note first that if 6. l n+l, n+ 0 and FiY 0 for all i, then ( 8) is
sat isfied. 
Theorem l; If F is of the form g( X1 . . •  Xn} + h( Y z) where g is 
homogeneous of degree one, then F generates a separable cost function. 
Proof: Obviously F iY = 0 for all i. If g is homogeneous of degree
one then \g . .  \ = 6 1 1 = 0 wherever evaluated (Quirk and Saposnik, lJ n+ ,n+ · 
1963]. Therefore F satisfies ( 8 ) , 
We begin by finding a general solution for ( 8) when there
is just one input, denoted X. Then ( 8) becomes
FXFYX -
FYFXX 0. 
( 9) 
( 9) is equal to the numerator 
3 (FY) of the expression 
OX F X 
. Thus the 
solution of ( 9) v.·ill be a function such that the ratio of F y to F X is
independent of X. 
F 
Let -.:£.FX 
Then for any fixed value of F, 
dX 
dY = 
-•< Y >· 
0( Y). 
8 
In Figure 1, these level s.urfaces are illustrated. Each is simply a
horizontal displacement cif some other level surface (isoquant). Denote 
each isoquant C(A). Thein on any isoquant X = A +  f(Y) where f is 
some arbitrary function. The function F which solves { 9) is an 
arbitrary function of A, say F = A(>.). Thus F = A (x + h(Y)) , where 
A and h are arbitrary functions, 
y 
x 
which may be restricted to preserve the convexity and strict quasi­
concavity of :x:2• We check that this solution works by differentiating: 
z. 
' 
FX = A
• 
FY = hA 
FY = h0(Y).FX 
I am indebted to Joel Franklin for this demonstration. 
This demonstration suggests that the solution to { 8) may have 
the form 
A(g(Xl ... Xn ) + h(Yzl)
9 
where A and h are arbitrary functions and jg1) = O. We demonstrate 
that this is the case by establishing a relation between co-factors of 
and co-factors of 
A . . 
'J 
A. 
J 
gij 
gj 
A. ' 
0 
gi 
0 
IAI 
!cl. 
It is well kno"Wn that [A! (A')n+llc/.
Write 
\ I A .. A. 
>J ' 
A. o I \ ' 
" A 
" A 
2 r n 1 gl +A gll ... A glgn + A gin 
' 
gngl +A gnl · 
1
1 2 
, .A gn +Agnn
' Agl . . . A gn 
' A gl
' A gn 
0 
Then ( 8) bee om es 
" ' I I D. hgiAn+l ,i i +A, h 1IAln+l,n+l
= 0. 
Since LA1 gi\Aln+l, ii 
!Al, we have, if h f; 0,
Lemma 3: If 
then 
A 
A 
" 
IAJ + A
'
IAJn+l,n+I = 0.
�c .. : c
)
C= .. 1�·�····1 ' 
c. : 0 J • 
le I= -I;L:c.c.!c .. J ..i j l J lJ l, J 
( 10) 
where Jc .. j .  is the co-factor of C . . in the co-factor I C I 1 1 •lJ lJ lJ n+ , n+ . 
�: Expand IC j by its last column, obtaining 
Jcj =l:ci/c li,n+I
' 
c21 · · · cznl en • . .  ctn 
=c1I · n+Z 1(-l)Zn+l • (-!) + . .. +c en! c n c c nr1 n-1 n-l1n c
1 c cl . c n n 
10 
II 
Now expand each determinant in this sum by its last row, obtaining 
C22 d • • CZn 
Jc!= c1c1 i :
lc21 · • • c2,n-l 
<-1)n+2<-1t+l+ • . •  + c1cn : I r-1)n+2(-1)2n
.• cnZ D • •  cnn 
c12 c1n 
1c I " . C I n n,n-
en. c 1, n-1
+ c c n I (-l)Z
n+l(-l)n+l+ . . • + C C n n
c . .n-1, 2 
en . 
= L:L:c .  c. j ci-1. . l J 
i J c1+1 
c n-1,n 
c1 ·-1'c1 ·+1·• J , J c1n 
c ............ c n nn
Now ell. 
J c ii Iii (-l)i+j
C. I I 1- ' 
c i+l, 1 
c ..n-1, l 
Zn+l+i+j ( -1) 
cl ·-1'c1 ·+1 •• J • J 
cn-1.n-l 
Gin 
C. I 1- ,n
C. I 1+ ,n 
c n, I .c .. c n, J-l n+l c nn 
Therefore 
I c I= -��cici jciil i,j · 1 J 
(-l)2n+\-!)2n
Quirk and Ruppert [ 1968] have shown that for a matrix of 
the form of A, using thEO: rule for evaluating the determinant of the 
sum of two matrices gives 
(Aln+l,n+l (A1 )
n
lGln+l,n+l
n " ' n-1,.. lei + A (A ) "-' gj n+l,j. j=l 
Using our lemma, 
IAln+l,n+l = (Ar )
n
!G]n+l,n+l +
" ' n-1'<" lei A (A ) "-'gj n+l, jJ 
(A1 )njcJn+l,n+l A
"(A' )n-ljej
' nj I " ' -21 [ = (A ) e'n+l,n+l - A (A ) A .
Therefore ( 10) bee omeu 
" 
.!'!_ 
A' - A=- IAIA 
' n+11 I (A ) e'n+l,n+l IA! + 0. ( ll)
But jGL+l,n+l = O by hypothesis, so that the left hand The expression A(g(X) + h(Y)) , side of ( 11 )  vanishes identically. 
containing two arbitrary functions, is the general solution of the 
differential eql,lation ( 8 }.
12 
If F = A (g(Xl + h(Y)} where I gijl = 0, and if A and h are
suitably restricted to preserve the quasi-concavity and concavity of F n 
in X, then the resulting cost function will be separable. Separability 
of the production function is neither necessary nor sufficient for 
separability of the cost function. 
Acknowledgments 
I am indebted to Joel Franklin, David Grether, and 
James Quirk for substantial mathematical assistance and moral 
support. Research support was provided by the Environmental 
Quality Laboratory of the California Institute of Technology. 
13 
References 
Davis, 0. A., and 'Vhinston, A. [ 1?62]. nExternalities, Welfare, 
and the Theory of Games. 11 Journal of Political Economy, 
vol. 70 (June 1962). 
Ford, Lester [ 1955 J. Differential Equations. New York, 1955.
Kneese, S., and Bower, :B. [ 1968], Managing Water Quality: 
Economies. Technology, Institution. Baltimore, 1968. 
Marchand, J. R., and Russell, K. P. [ 1973]. 11Externalities, 
Liability, Sepairability, and Resource Allocation. 11 
American Economic Review, vol. 63 (September 1973), 
Quirk, J. P., and Ruppert, R. [ 1968]. 11Maximization and the 
Qualitative Calculus. 11 Papers in Quantitative Economics. 
Edited by Quirk and Zarley. Lawrence, Kansas, 1968. 
14 
Quirk, J. P., and Saposnik, R. ( 1963]. "Homogeneous Production 
Functions and 1v!arginal Cost C urves. 11 Institute Paper No. 41, 
Institute for Quantitative Research in Economics and 
Management, F'urdue University. Lafayette, Indiana, 1963. 


Separable Externalities in Cost and Production Functions

CALI F O R N I A  I N S TITU T E  O F  T E CHN OLOGY 
Division of the Humanities and Social Sciences 
Pasadena, California 91109 
SEPARABLE EXTERNALITIES IN 
COST AND PRODUCTION FUNCTIONS 
W. David Montgomery 
Social Science Working Paper 
Number 42 
Jtme 1974 
SEPARABLE EXTERNALITIES IN 
COST AND PRODUCTION FUNCTIONS 
W. David Montgomery 
California Institute of Technology 
The characterization of external effects as 11separable11 
has played an important role in the development of the theory of 
externalities. The separable case is particularly well behaved when 
procedures for achieving an optimum allocation of resources in the 
presence of externalities are examined. Davis and Whinston ( 1962) 
find that separability assures the existence of a certain kind of 
equilibrium in bargaining between firms which create externalities, 
and that equilibrium does not exist without separability. Kneese and 
Bower ( 1968) argue that with separability the computation of Pigovian 
taxes to remedy externalities is particularly simple. Marchand and 
Russell ( 1974) demonstrate that certain liability rules regarding 
external effects lead to Pareto optimal outcomes if and only if 
externalities are separable. In each of these cases the problem is 
posed in terms of two firms related_ by technological externalities, 
and separability is defined in terms of a cost function. In this paper 
we will characterize that class of production functions which give rise 
to separable cost functions, and show that the relation between production 
functions and separable cost functions is by no means as trivial as has 
been claimed. 
Let C ( Y l' Y z) be the cost function of a firm which produces
Y1, and suffers an external diseconomy which is a function of Y2.
That cost function is defined in terms of a production func'tion 
F( X . • •  X , Y2), in the following manner:1 n 
C(Yl' Y2) min I: w.x.. ' ' 
' 
subject to Y1 = F(X1 • .  · Xn, Y2 ).
Some precise definitions and lemmas regarding separability 
are needed: 
Definition 1: A cost function C(Yl' Y2) is separable if and only if
if can be written as c1(Y1) + C2(Y2).
2 
Definition 2: A production function F(X1 • . .  Xn' Y2) is separable
if and only if it can be written as g( x1 . • .  Xn) + h( Y2 ).
Lemma l: 
everywhere. 
Proof: 
2 
A cost function is separable if and only if d� �y 
l 2 
twice and 
Necessity is proved by differentiating C = c1 + c22 
observing that �� 
l
�y 2 = 0. Sufficiency is proved
1 
by
0 
observing that the general solution of the second-order partial differential 
2 
equation O� �y = 0 is of the form C = C1(Y1) + C2(Y2).1 2 
1. Lester Ford, [ 1955], p. 251 derives this result. 
We assume throughout that C and F are continuously twice 
differentiable. The problem is to find what general class of fi.mctions F 
3 
give rise to cost functions with the property 
o2c
It has o. oY1oY2
been claimed by Marchand and Russell that a cost function is separable 
if and only if the production frunction from which it is derived is 
separable. This conjecture is false in both directions. We begin by 
giving a simple example of a separable production function which does 
not generate a separable cost function. It will be seen later that the 
class of separable functions does not exhaust the class_ of functions 
giving rise to separable cost functions. 
Consider a production function of the form 
F • Ii x1x2 -cY2 (I) 
where a. + f3 < 1. We find the cost function by solving the cost-
minimization problem and using the first-order conditions and the 
production function to eliminate the inputs from the cost equ,ation. 
From the first-order conditions we have 
w
l 
wz
.xz
8X1
Solving for x1 and substituting in ( 1) gives 
•+8 (Wz• )" YI = Xz W18 CY2.
( 2) 
( 3)  
Solving ( 3) for x2, and substituting the resulting expression for x2
in (2) enables us to express x1 and x2 in terms of Y1 and Y2 alone. 
Substitution in C = w1x1 + w2x2 gives the cost function 
• 
c 
( ·)fW 2 •-)•+8 . Wz l+�\w\ 8 (YI+ 
Clearly ( 4) is not separable if a. + f3 � 1. 
_I_ 
CY )o+P 2 • (4) 
4 
To f:ind a production function which does generate a separable 
2 
cost fnnction we express 
a: d;. in terms of the derivatives of the 1 2 
production function, and then find a general solution of the partial 
2 
differential equation which results when d; d� is set equal to zero. 
I 2 
The general relation between cost and production functions is 
fo1.md by adopting the approach of Samuelson1s Foundations. Consider 
the constrained cost minimization problem 
Minimize L:w.x. subject to Y1 - F (X1, • • .  X ) . 1 i n ' 
We adopt the following abbreviations 
oF oF 
ax. 
= Fi cY = FY' 
2 
o. 
c
z
F " F 
;ix.;ixi •i
....£.£__ = F 
oX.oY iY . ' 
Form the Lagrangian expression 
L. = L;w ix i + >. ( y I - F ( XI • • . X n' y 2) ) . ' 
First-order conditions are 
w
i 
>.F. ' 0
Y1 - F o. 
We perturb the solution by varying Y1 and Y2. Totally differentiat:ing 
the first-order conditions gives the system of equations 
F II .
Fn l.
FI .
F ln F 1 
F F 
nn n 
F 0n 
Let A= 
dX1 
dX n 
d)JI. 
Fil.
Fnl.
FI .
=
dW1 
I. 
dW 
FIYdYZ 
__ n_ F dY A nY 2 
dY1 - FydY2 
F ln Fl 
F F 
nn n 
F 0n 
Further let !::. • •  be the i, j th co-factor of /J.. lJ 
Since c = Lwkx , k k 
� 
oY2 
oXk = L:wkoY k 2
Solving for dXk using Cramer's rule gives 
( 6) 
dXk = 
n 
l
' dW. ) � \T - FiYdYZ Llik + (dY1 - F�dY2)An+l,k] 
A 
We assume that F is strictly quasi-concave in X1 . . •  Xn' so that 
A 1 0. 
Then 
oXk -- = oY2 
-�FiY6ik - FYAn+l,
1 
A 
5 
and oc 
CIYz
- ��k[� FiYAik + FY6 n+1,k]
A 
Since Wk = AFk, 
� 
oY2 
-i:JF;y>-
_
(�FkAik)]- AFYt FkAn+l,k
A 
6 
But tFkAn+l, k =A, a.nd LF !::. • = 0 since it is an expan
sion by 
k K ik 
alien co-factors. Therefore 
ac 
oY2 = -AF
y.
(7) 
Differentiating ( 7) with respect to Y 1 gives 
From (6), 
Therefore 
(IZC 
oY 2aY 1
ax; 
->.L;FiYoY11 
F 21_YoY1 
� 
oY1 
o2C 
oY2?Y1 
= A�1+l, n+lA 
oXi
oY1
An+l, i 
A 
---�(L:F A + A  F )fl . iY n+l. i n+l,n+l Y · 
1 
7 
Characterizing the class of production functions which generate 
separable cost functions reduces to finding the general form of the 
solution of 
L;F 6 . + 6 F = O. . iY n+l, i n+l,n+l Y ( 8) 
' 
Note first that if 6. l n+l, n+ 0 and FiY 0 for all i, then ( 8) is
sat isfied. 
Theorem l; If F is of the form g( X1 . . •  Xn} + h( Y z) where g is 
homogeneous of degree one, then F generates a separable cost function. 
Proof: Obviously F iY = 0 for all i. If g is homogeneous of degree
one then \g . .  \ = 6 1 1 = 0 wherever evaluated (Quirk and Saposnik, lJ n+ ,n+ · 
1963]. Therefore F satisfies ( 8 ) , 
We begin by finding a general solution for ( 8) when there
is just one input, denoted X. Then ( 8) becomes
FXFYX -
FYFXX 0. ( 9) 
( 9) is equal to the numerator 
3 (FY) of the expression 
OX F X 
. Thus the 
solution of ( 9) v.·ill be a function such that the ratio of F y to F X is
independent of X. 
F 
Let -.:£.FX 
Then for any fixed value of F, 
dX 
dY = 
-•< Y >· 
0( Y). 
8 
In Figure 1, these level s.urfaces are illustrated. Each is simply a
horizontal displacement cif some other level surface (isoquant). Denote 
each isoquant C(A). Thein on any isoquant X = A +  f(Y) where f is 
some arbitrary function. The function F which solves { 9) is an 
arbitrary function of A, say F = A(>.). Thus F = A (x + h(Y)) , where 
A and h are arbitrary functions, 
y 
x 
which may be restricted to preserve the convexity and strict quasi­
concavity of :x:2• We check that this solution works by differentiating: 
z. 
' 
F
X 
= A
• 
F
Y
= hA 
F
Y
= h0(Y).F
X 
I am indebted to Joel Franklin for this demonstration. 
This demonstration suggests that the solution to { 8) may have 
the form 
A(g(Xl ... Xn ) + h(Yzl)
9 
where A and h are arbitrary functions and jg1) = O. We demonstrate that this is the case by establishing a relation between co-factors of 
and co-factors of 
A . . 'J 
A. J 
gij 
gj 
A. ' 
0 
gi 
0 
IAI 
!cl. 
It is well kno"Wn that [A! (A')n+llc/.
Write 
\ I A .. A. >J ' 
A. o I \ ' 
" A 
" A 
2 r n 1 gl +A gll ... A glgn + A gin 
' gngl +A gnl · 
11 2 , .A gn +Agnn
' Agl . . . A gn 
' A gl
' A gn 
0 
Then ( 8) bee om es 
" ' I I D. hgiAn+l ,i i +A, h 1IAln+l,n+l
= 0. 
Since LA1 gi\Aln+l, ii 
!Al, we have, if h f; 0,
Lemma 3: If 
then 
A 
A 
" 
IAJ + A'IAJn+l,n+I = 0.
�c .. : c)C= .. 1�·�····1 ' c. : 0 J • 
le I= -I;L:c.c.!c .. J ..i j l J lJ l, J 
( 10) 
where Jc .. j .  is the co-factor of C . . in the co-factor I C I 1 1 •lJ lJ lJ n+ , n+ . 
�: Expand IC j by its last column, obtaining 
Jcj =l:ci/c li,n+I
' 
c21 · · · cznl en • . .  ctn 
=c1I · n+Z 1(-l)Zn+l • (-!) + . .. +c en! c n c c nr1 n-1 n-l1n c1 c cl . c n n 
10 
II 
Now expand each determinant in this sum by its last row, obtaining 
C22 d • • CZn 
Jc!= c1c1 i :
lc21 · • • c2,n-l 
<-1)n+2<-1t+l+ • . •  + c1cn : I r-1)n+2(-1)2n
.• cnZ D • •  cnn 
c12 c1n 
1c I " . C I n n,n-
en. c 1, n-1
+ c c n I (-l)Z
n+l(-l)n+l+ . . • + C C n n
c . .n-1, 2 
en . 
= L:L:c .  c. j ci-1. . l J 
i J c1+1 
c n-1,n 
c1 ·-1'c1 ·+1·• J , J c1n 
c ............ c n nn
Now ell. 
J c ii Iii (-l)i+j
C. I I 1- ' 
c i+l, 1 
c ..n-1, l 
Zn+l+i+j ( -1) 
cl ·-1'c1 ·+1 •• J • J 
cn-1.n-l 
Gin 
C. I 1- ,n
C. I 1+ ,n 
c n, I .c .. c n, J-l n+l c nn 
Therefore 
I c I= -��cici jciil i,j · 
1 J 
(-l)2n+\-!)2n
Quirk and Ruppert [ 1968] have shown that for a matrix of 
the form of A, using thEO: rule for evaluating the determinant of the 
sum of two matrices gives 
(Aln+l,n+l (A1 )
nlGln+l,n+l
n " ' n-1,.. lei + A (A ) "-' gj n+l,j. j=l 
Using our lemma, 
IAln+l,n+l = (Ar )
n!G]n+l,n+l +
" ' n-1'<" lei A (A ) "-'gj n+l, jJ 
(A1 )njcJn+l,n+l A
"(A' )n-ljej
' nj I " ' -21 [ = (A ) e'n+l,n+l - A (A ) A .
Therefore ( 10) bee omeu 
" 
.!'!_ 
A' - A=- IAIA 
' n+11 I (A ) e'n+l,n+l IA! + 0. ( ll)
But jGL+l,n+l = O by hypothesis, so that the left hand The expression A(g(X) + h(Y)) , side of ( 11 )  vanishes identically. 
containing two arbitrary functions, is the general solution of the 
differential eql,lation ( 8 }.
12 
If F = A (g(Xl + h(Y)} where I gijl = 0, and if A and h are
suitably restricted to preserve the quasi-concavity and concavity of F n 
in X, then the resulting cost function will be separable. Separability 
of the production function is neither necessary nor sufficient for 
separability of the cost function. 
Acknowledgments 
I am indebted to Joel Franklin, David Grether, and 
James Quirk for substantial mathematical assistance and moral 
support. Research support was provided by the Environmental 
Quality Laboratory of the California Institute of Technology. 
13 
References 
Davis, 0. A., and 'Vhinston, A. [ 1?62]. nExternalities, Welfare, 
and the Theory of Games. 11 Journal of Political Economy, 
vol. 70 (June 1962). 
Ford, Lester [ 1955 J. Differential Equations. New York, 1955.
Kneese, S., and Bower, :B. [ 1968], Managing Water Quality: 
Economies. Technology, Institution. Baltimore, 1968. 
Marchand, J. R., and Russell, K. P. [ 1973]. 11Externalities, 
Liability, Sepairability, and Resource Allocation. 11 
American Economic Review, vol. 63 (September 1973), 
Quirk, J. P., and Ruppert, R. [ 1968]. 11Maximization and the 
Qualitative Calculus. 11 Papers in Quantitative Economics. 
Edited by Quirk and Zarley. Lawrence, Kansas, 1968. 
14 
Quirk, J. P., and Saposnik, R. ( 1963]. "Homogeneous Production 
Functions and 1v!arginal Cost C urves. 11 Institute Paper No. 41, 
Institute for Quantitative Research in Economics and 
Management, F'urdue University. Lafayette, Indiana, 1963. 


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Separable Externalities in Cost and Production Functions

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