In one of the most influential papers written on the subject of externalities, Otto Davis and Andrew Whinston argue that corrective taxes and private bargaining are

likely to achieve an optimum in the presence of mutual externalities between two firms only when externalities are separable, in the sense that marginal cost is independent of the level of externality. Further analysis of the concept of separability reveals that even this conclusion is too optimistic. I shall argue below that the assumptions needed to make taxes and negotiations work properly rule out the possibility of having externalities in any observable situation

Montgomery, W. David

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Reprinted from 
Technology Social Science Working 
Paper# l ct 
THE AMERICAN ECONOMIC REVIEW 
© The American Economic Association 
Separability and Vanishing Externalities 
By w. DAVID MONTGOMERY* 
In one of the most influential papers 
written on the subject of externalities, Otto 
Davis and Andrew Whinston argue that 
corrective taxes and private bargaining are 
likely to achieve an optimum in the presence 
of mutual externalities between two firms 
only when externalities are separable, in the 
sense that marginal cost is independent of 
the level of externality. Further analysis of 
the concept of separability reveals that even 
this conclusion is too optimistic. I shall a:i;gue 
below that the assumptions needed to 1nake 
taxes and negotiations work properly rule 
out the possibility of having externalities in 
any observable situation. 
The crucial step in Davis and Whinston's 
argument is the proposition that separability 
" ... implies the game theoretic concept of 
dominance" (p. 247). They contend that
when the cost function is separable, the 
pro£t-maxi1nizing output of a firm affected 
by an externality is determined indepen­
dently of the output choice of the firm caus­
ing the externality. Thus, when the process 
of mutual accommodation to externality is 
formulated as a noncooperative game, choos­
ing that output level is a dominant strategy 
for the affected firm. Moreover, when the 
cost function is nonseparable, each firm must 
base its strategy on expectations about how 
the other firm will respond to its choices, and 
the game may fail to have an equilibrium 
point. 
As stated by Davis and Whinston, the ar­
gument that separability of the cost function 
implies the existence of a dominant strategy 
is incorrect. A strategy should specify both 
inputs and outputs. In general, the functional 
relation between inputs and outputs depends 
on the level of externality. If the input com-
*California Institute of Technology. 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 Tech­
nology. 
174 
bination which minimizes the cost to one 
firm of producing a fixed level of output 
changes as the output of the other firm 
changes, then it will not be true that the 
complete optimal strategy for one firm is 
independent of the actions of the other. The 
simplest example is the one input, one output 
case. Suppose an output Y* is optimal for 
firm 1 under all actions of firm 2, but that the­
input required to obtain this output varies 
as the output of firm 2 varies. Then firm 1 
cannot decide on an input-output pair with­
out knowledge of what firm 2 will do.
Separability of the cost function implies 
only that the output choice is independent 
of externality. A further condition is needed 
to guarantee that input choice is also inde­
pendent. For input choice to be independent 
of externality, the marginal productivity of 
each input must be independent of the level 
of externality. If this is to be the case, the 
production function must also be separable. 
Writers following Davis and \i\Thinston, 
including James Marchand and Keith Rus­
sell, have assumed that separability of the 
cost function is equivalent to separability of 
the production function. It will be estab­
lished that this assumption is false, and that 
it is only possible to have separability of both 
functions in a special case. I will give two 
counter examples. 
Let C( Y1, Y2) be the cost function of a 
firm which produces Y1 and suffers an ex­
ternal diseconomy which is a function of Y2. 
The cost function is dual to a production 
function F(X1, ... , X1., Y2).
Definition 1: A cost function C( Y1, Y2) is 
separable if and only if it can be written as 
C1(Y1)+C2(Y,). 
Definition 2: A production function 
F(X1, ... , X., Y,) is separable if and only
if it can be written as g(X11 • • •  1 Xn) 
+h(Y2). 
VOL. 66 NO. 1 MONTGOMERY: VANISHING EXTERNALITIES 175 
LEMMA 1: A cost function is separable if
and only if a2C/aY,aY,=O everywhere. A 
production function is separable if and only if 
a2F/iJXiaY2=0, i= 1 . . .  n, everywhere.1
Consider the separable production function 
. ' F = X,X, - h(Y,) 
where a+�< 1. We find the cost function by
solving the cost-minimization problem and 
using the first-order conditions and the pro­
duction function to eliminate the inputs from 
the cost equation. From the first-order con­
ditons we have 
(1) 
w, 
w, 
aX2 
�x, 
Solving for X1 and substituting in the pro­
duction function gives 
(2) 
_ •+• / w,a
)
"
Y, = X2 (-- - h(Y2) 
\ W ,� 
Solving (2) for X,, and substituting the re­
sulting expression for X2 in (1) enables us to
express X1 and X2 in terms of Y1 <t.Ild Y2 
alone. Substitution in C= W1X1+ W2X2 gives 
the cost function 
(3) C = W2 ( 1 + �) (�:;r·f(•H> 
· ( Y, + h( Y2))1t<•+'1 
Clearly (3) is not separable if a+�r' 1. 
Now consider the nonseparable production 
function 
F = (X1 - h,(Y2))•(X2 - h2(Y2))'
A mathematical development similar to that 
of the first example yields the separable cost 
function 
( ") (°' W,)-•/(•Hl !/(•+Pl C = W2 1 +- - - Y1 � � w, 
+ W1h1(Y2) + W2h2(Y2) 
To find conditions under which the pro-
1 Lester Ford, p. 251, proves sufficiency. Necessity is 
trivial. 
duction function and its dual cost function 
are both separable, we express d2C/dY1dY2 in
terms of the derivatives of the production 
function. We adopt the following abbrevia­
tions 
Let 
aF 
-=Fi 
ax, 
a2F 
---- = Fij 
ax,ax, 
Li = 
aF 
-=Fy 
av, 
a2F
--- =FiY 
ax,a v, 
F1 . • .  F. 0 
Further let Li,,. be the i, jth cofactor of Li. 
THEOREM 1. A cost function is separable if
and only if it is derived from a prodi.tction 
function which satisfies 
(4) L Firll.n+l, i  + Lln+1,n+1FY = 0
at every point which is a proper cost minimum.2 
It follows that the proper statement of 
the relation between separability and domi­
nance is that separability of the cost func­
tion and of the production function implies 
the game-theoretic concept of dominance. 
Moreover, the two separability hypotheses are 
satisfied simultaneously only when the pri­
vate part g of the production function has a 
specific form. 
COROLLARY 1: If the cost function C(Y,, 
Y2) and the production function F(X,, ... , 
X., Y2) are both separable and Fyr'O every­
where, then ) gijl =0, where I giil is the Hessian
determinant of g. 
PROOF: 
By separability of the cost function (4) 
holds. Separability of the production func­
tion implies F iY = 0 for all i. Therefore 
2 For proof, see Appendix. 
176 THE AMERICAN ECONOMIC REVIEW MARCH 1976 
COROLLARY 2: If the cost function and
production function are separable and Fy�O, 
then marginal cost ac/aY1 is constant in Y1 
and Y2. 
PROOF: 
From the Appendix and Paul Samuelson, 
p. 67,
Constancy of marginal cost has some 
startling consequences. If no externality 
were present, then the supply function would 
be such that for any price of output greater 
than ac;av, the firm will produce un­
bounded output. At any price less than 
ac/aY,, it will produce zero output. Only 
when price of output equals ac;av, will the 
firm produce finite output. But when C2(Y2) 
;;:eo there is a fixed cost, imposed on the firm 
by externality. Therefore even if price is 
exactly equal to aC/oY1 (which is a constant
function of Y1 and independent of Y2), the 
firm will be losing money and will produce 
zero output But if price is at all higher than 
ac/aYi, the firm can earn unbounded profits 
by producing infinite output. Thus no equi­
librium involving finite, nonzero output by 
a firm which suffers from an externality and 
has separable cost and production functions 
can exist unless the firm which causes the 
externality is producing zero output. 
That is, when the conditions for the exis­
tence of dominant strategies in a two-firm 
externality game are satisfied, the only 
equilibrium possible is one in which one of 
the firms is out of business. It follows that it 
is impossible ever to observe a firm with 
separable cost and production functions suf­
fering an externality, since either it or the 
firm causing the externality will always be 
driven out of business in equilibrium. If each 
firm creates an externality affecting the 
other, then only one can survive in equilib­
rium, and it will not suffer or cause any ex­
ternality in equilibrium. Moreover, if the 
two firms produce the same output from the 
same input, this equilibrium is a Pareto opti­
mum. Since the marginal costs of all firms 
are constant by hypothesis, one firm can pro­
duce any output as efficiently as many. The 
fixed cost imposed by externality implies 
that when more than one firm is in operation, 
more input is needed to produce a given out­
put than is needed when only one firm op­
erates. 
The consequence of this analysis is the in­
tensification of the pessimism expressed by 
Davis and Whinston regarding the possibil­
ity of using either corrective taxes or private 
bargaining to correct externalities. The only 
circumstances in which they thought such 
policies workable define a vacuous case, one 
in which externalities will never be observed 
or need to be remedied. 
APPENDIX 
Proof of Theorem 1 
By Lemma 1 the cost function is separable 
if and only if a2C/aY1aY,=O. We express
a2C/aY1aY2 in terms of the production func­
tion as follows. Form the Lagrangian expres­
sion 
L = L: wixi + '(Y, - F(x, ... x", Y,)) 
First-order conditions are 
Wi - AFi = 0 
Y1 - F = 0 
We perturb the solution by varying Y1 and 
Y2• Totally differentiating the first-order 
conditions gives the system of equations 
(Ai)
F11 ... Fin F1 dX1 
iF1 . . .  F" 0 
dX" 
dA/A J 
(dW, )-- - FlYdY, 
A 
l dw., · 1-,- - F"ydY2
I dY, - FydY, 
VOL. 66 NO. 1 MONTGOMERY: VANISHING EXTERNALITIES 177 
Solving for dXk using Cramer's rule gives 
We assume that F is strictly quasi concave 
in X1 ... X., so thatl:.,,<O. Then 
Since 
ax, 
av, 
�'.'.__ = _1_ {- t [F;yA ( L F,l:.ik
'
)] 
a Y 2 A i=1 k 
- J..Fy � F,l:.n+1,k} 
But .Lk FkAn+i,k=A, and 2:1.: Fk!::.i1.:=0 since
it is an expansion by alien cofactors. There­
fore 
(A2) 
ac 
av, 
Differentiating (A2) with respect to V1 gives 
a'c ax, a1-
___ = - A L F;y-- - Fy--
av,av, , av, av, 
From (Al), 
at.. An+1,n+1 
-- = ).----, av, t. 
Therefore 
a v,a v, 
ax, 
a V1 
REFERENCES 
.6.n+l,i 
0. Davis and A. Whinston, "Externalities,
Welfare, and the Theory of Games," !. 
Polit. Econ., June 1962, 70, 241-62. 
L. Ford, Differential Equations, New York 
1955. 
J. R. Marchand and K. P. Russell, "Externali­
ties, Liability, Separability1 and Resource 
Allocation," Amer. Econ. Rev.: Sept. 1973, 
63, 611-20. 
P. A. Samuelson, Foundations of Economic 
Analysis, Cambridge, Mass. 1966. 


Separability and vanishing externalities

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Separability and vanishing externalities

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