The Bergson Social Welfare
Function with Externality*
In Bergson’s original formulation [111 of the social welfare
function, the problem of externality is largely ignored, or implicitly
assumed away. The purpose of this exercise is to revise Bergson’s
formulation to take account of externality. In view of the importance
of the Bergson social welfare function and because the problem of
externality is usually treated as a separate problem, not directly
formulated into a coherent welfare maximization scheme, the present
exercise may prove to be useful.
It will be shown that both the Samuelson condition for public
goods [8] and Evans’ equation for external goods [2] can be derived as
special cases of the general formulation. Moreover, with this general
formulation, it is shown that the existence of some pure private good
is, in general, necessary for formulating the Pareto optiraality conditions in terms of marginal rates of substitution. Hence the paradox
of universal externality which we discuss in the last section of this
paper. It is believed that this paradox is important and may open up
interesting fields of investigation.
The social welfare function may be expressed in the form
W = W ( q ,y17 a;,b’j, a:, by,, . . . , x,, yn, a:, b:, a;,K,C”, Ly’, C)’,
P’r, 8 , t , . . .)
(1)
“Here C” and D” are the amounts of the non labor factors of production
C and D employed in the production unit producing the consumers9)good
X ; Q and D y are the amounts of these factors employed in the production
unit producing the consumers’ good P ; x iand y i are the amounts of S and
Y consumed by the ith individual; and a;, b;, a;, and 6: are the amounts
of each kind of work performed by him for each production unit during
the given period of time. The symbols, T , s, t , . . . , denote . . . other . . .
elements affecting the welfare of the community”. [l,p. 81.
Assuming that, for relatively small changes in the ‘economic’
variables, other elements in welfare are not significantlp affected,
so that these elements may be taken as given, Bergson gets
* I am grateful to Professor Abram Bergson for his comments. on the first
draft of this paper. For the sake of easy comparison, I have followed his original
paper closely.
1 Numbers in square brackets refer to references listed at the end ‘of the article.
517
518
THE ECONOMIC RECORD
DEC.
Then Bergson writes the production constraints as
X = X(A", B", C", D"); Y = Y(AY,BY, CY, DY)
(3)
where A", B" and AY,By are the amounts of two kinds of labour used in
the X and P production units respectively. It can clearly be seen
that equation (3) assumes away externality between production units.
In order to take externality into consideration, we replace (3) by
X = X ( A X B",
, C", D", AY, BY, C y ,D y )
(3')
Y = Y(AY,BY, CY, DY, A", B", C", D")
Assuming that E varies continuously with xl, yl,. . ., the general
condition for a position of maximum economic welfare, subject to the
production constraint (3') and the given amounts of resources, is
dE = 0
(4)
By equations (2), (3'), (4) and the given amounts of resources, we
can deduce the necessary conditions for a maximum welfare.
The first group of conditions do not differ from Bergson's. They
relate to the consumption and supply of services by each individual
in the community. They require that the marginal welfare of each
commodity and the marginal dis-welfare of each type of work be the
same with respect to each individual in the community.
where It's are Lagrangean multipliers.
When we come to the conditions relating to production, we get a
different set of equations from that of Bergson's.
ax
1' d-x+A
ay
It - + 1
'aAy
ax
11
It
ay
-x=
'aA
Its
ax
-=It4
'aAY
ay
W + 1 2 aB"
=
2.5
2 In the case of two commodities, (3) could be interpreted as capable of taking
account of externality between production units, as A' uniquely determines A*, etc.
But as the model is meant to be applicable to multiple commodities, externality is
in fact assumed away. Moreover, if externality is not assumed away by (3) as
such, it is assumed away in Bergson derivation of the maximum condition by treating such terms as BX/BA' as zero.
1972
THE BERGSOX SOCIAL WELFARE FUNCTION
519
Thus, in Bergson’s equations (11)-(14), the second terms in our
equations (11’)-(14’) ) are all missing. Take (ll’),if there is no
externality, aY/aAw = 0, and our equation (11’) is reduced to
Bergson’s equation (11).Equation (11’)-(14’) require that the aggregate welfare of the consumers ’ goods produced directly 01’ indirectly
(through external effects) must equal the negative of the dis-welfare of
that increment of work.
Similarly, our equations (15’) and (16’) also differ from Bergson ’s.
ax
ax
aw a i l ~
These two conditions require that the sum of the direct and indirect
increments of welfare due to the shift of a marginal unit of factors
C and D from one production unit to another should equal the negative
of the dis-welfare caused by this adjustment. For the derivation of the
above see Appendix A.
It should be noted that Bergson’s formulation can take account
of externality between production and consumption3 by meitns of such
terms aE/aCx, etc. ‘Such an effect would arise, for example, through
a positive or negative evaluation of the relative amounts and kinds of
“factory smoke” emitted in the two production units for varying
amounts of one or the other factors employed in each unit.’ [ 1, p. 111.
It is convenient to designate Al, h2, X3, A4, h5, and he as, respectively,
the prices of X , Y,and the wages of Aw, AY, B”, and BY. Equations
( 5 ) and (6) thus require that the marginal welfare per ‘dollar’s
worth’ of each commodity be the Same f o r each commodity itnd for all
individuals in the ~ o m m u n i t y Similarly,
.~
equations ( 7 ) - ( l o ) , require
that the marginal welfare per ‘dollar’s worth’ of each kind of work
be the same with respect to each kind of work and each individual in
the community. Equations (11’)- (14’) require that the wages of each
type of labor equal the sum of the direct and indirect marginal value
productivity of that type of labour.5 Equations (15’) and (113’) require
that the sum of the direct and indirect marginal value productivity
equal the cost due to a shift in C o r D from one use to another.
3i.e. external effects of production activities on the consumers. T i i s does not
include the possible external effects of the consumption by each individual on prcduction. This aspect of externality (presumably less important than others) is not
even covered by our revision for the sake of simplicity. H a d we covered that,
equations (5)-(10) would have had t o be revised.
4Since I am using a monetary unit, the marginal welfare per dcillar’s worth
of each commodity is unity rather than the proportionality factor 0 in Bergson’s
formulation.
5 These requirements are strictly true only if the prices and wages are interpreted as shadow prices. For example, if wages are meant to be the actual
payment to workers, wages need not necessarily be made equal to marginal value
productivity unless we assume that wages are used as a means to allocate labour
and that each worker maximizes his o m utility in determining his supply of labour.
Bergson did not seem to take account of this point.
520
THE ECONOMIC RECORD
DEC .
M a x i m u m Conditions with Restrictions o n the W e l f a r e Function
The maximum conditions presented above are the general conditions for a position of maximum welfare, i.e. they are applicable t o
any social welfare function. The maximum conditions presented in
some welfare studies relate to a particular family of welfare functions.
Their derivation thus requires the introduction of restrictions on the
shape of the welfare function. Some examples are given below.
Indifference of Alternative Uses of Xon-Labour Resources:
A shift in a unit of any factor of production, other than labour,
from one production unit to another would leave welfare unchanged.
provided the amounts of all the other elements in welfare were constant.
It may be noted that this proposition renders the social welfare
function incapable of taking account of the possible different degrees
of external effects on non-labour factors of production in different uses
on the well-being of the consumers. The proposition, in effect, assumes
that the right-hand side of (15’) and (16‘) must equal zero. Hence, we
get
(17‘)
(18’)
which means that the sum of the direct and indirect marginal value
productivity on non-labour factors must be the same in every use.
Unlike the original Bergson’s (17) and (18), if we combine (17’)
and (IS’), we cannot eliminate the A’s. Thus, the ratio of the ‘social
marginal productivity’, as shown in (19’), can only be evaluated if
xse know the relative price of the products
(19‘)
Equation (19’) requires that the ratio of the social marginal productivity of a factor in one use to its social marginal productivitSof a factor in any other use be the same for all factors of production.
Bergson’s (19) can also be rewritten to show the equality of the
marginal rate of substitution of factors. The same can be done on our
(19’) provided we are prepared to call each side of (19’) the ‘social
marginal rate of substitution’.
ax A,ay d~ A l a s
F+;iTac.
- m+TJ@
ax
i2a1- - a y
i,ax
E+%aF
@+z;a~y
(19“)
1972
THE BERGSO?; SOCIAL WELFARE F U N C T I O S
52 1
As the social marginal rate of substitution is not independent of the
price ratio, it might be thought that (1Y) is not necessai'y for productive efficiency as such, because to maximize the production of one
product given the amounts of other products, we do not have to know
the relative prices. This, however, is shown to be incorrect in Appendix
B.
The Pareto-Barone-Cambridge Conditions or The Fundamc?ntal Value
Propositions of Individual Preference :
If the amounts of the varions commodities and types of work
were constant for all individuals in the community except any ith
individual, and if the i t h individual consumed the various commodities
and performed the various types of work in combinations which were
indifferent to him, welfare would be constant.
This proposition assumes away externality in consumption. Thus,
if externality is present, even if the ith individual is indifferent between two combinations of goods, some kth individual ( k # i ) may not
be indifferent between the two situations even if his oKn emsumption
and work stay the same.6
To account f o r externality in consumption, we use instead the
following proposition.
The Revised V d u e Propositions of Individual Prefcrence :
Social welfare is a positive function of individual prefuences, and
of indiT.idua1 preferences alone.
We have, thus,
E = E ( U ' , C 2 , . . . , li")
(20')
where U ' = C'(x,, yl, a;, b;, a:, b:, . . .
x,, y,,, a:, b;, a:, &, C'", DT,Cy,Dy)
(21')
or U' = U ' ( z , ,y l , a:, br. a:, b:, . . .
x,, Y",a;,b;, a;,
(22')
depending on whether we assume Cz etc. will or will not affect individual preferences.
By combining (5) and ( 6 ) , we get
dE dE
I,
=
i,
(23)
Using the Revised Value Propositions of Individual Preference, (5)
may be written as
dE
2~ aui
-=
-= I&*,
dXi
2-d U " d X i
Equations (6) through (10) can be similarly written. But one example
will suffice. Using (5'), equation (23) can be written as
6Rothenberg's interpretation of the proposition [7, p. 131 is, therefore, misleading.
522
T H E ECONOMIC RECOBD
DEC
.
aE
Unlike equation (24) of Bergson, we cannot, in general, delete -in (24’) to
a ul
, or the equality of the marginal rate of substitution
for all individuals and with the price ratio. Except by sheer chance, this
au* auj = 0 for all j # &. In this case, the goods
will be possible only if - =
8%
aY‘
are pure private goods without any external effect.
However, by assuming’ that there is at least one pure private good,
a Ui
Y,i.e. - 0 (j# i),
we have, from (6),
a yi
aE aE aui
= A, (i= 1, . . . , n).
-
ay*= S ’ a y ,
Using this equation, we can eliminate the aE/dUi’ in (24’) to gets
which is Evans’ equation [2, p. 811, In the case where x is a pure public
good proper, (24”) can further be reduced to
which is the Samuelson’s condition -[8, p. 3871 .O
T h e Paradox of Universal Externality
A problem arises if we drop the assumption of a pure private
good, as then we cannot just delete aE/aPin (24’) t o get the marginal
rate of substitution. aE/aUj can still be eliminated from the maximum
condition by an indirect method. Remembering that there are n
equations in (24’),
we may thus solve for the n aE/a’CTr in terms of
aUJ/aX, and aUJ/aY,, Substitute these back to (24’) t o eliminate
dE/aV. However, the maximum conditions thus obtained are, in
7 Evans 521 does not make this assumption explicit. He introduces a numeraire
which enters into the utility along with other goods, but the amount of the numCraire
held by i does not effect the ut~lityof j (i j). This of course is equivalent to the
assumption of private good.
aE auj 8~ auj
aE
Firat eubatitute
a Y , - a u L ay, into (24’) to get aLij*ax, auj*ay,
+
-
c, -1-
9 For derivation of (24”’) from (24”), see Evans [2, p. 81 f.]. See, however,
Ng [4] for the argument that this derivation cannot be applied to some public goods
Ng‘s argument is followed by Evans’ reply [3] which is followed by Ng‘s rejoinder
r51.
1972
THE BERGSON SOCLhL WELFARE FUNCTION
523
general, not in the simple terms of the (aggregate) marginal rates of
substitution. "hey are likely to be very complicated, especially in the
many-good, many-person case. We are thus faced with the followhg
di%iculties :
(1) In the cases of million of persons, it may not be feasible to
solve for dE/aUj. In these cases, the Pareto maximum conditions cannot be expressed to be free of interpemonal comparison of utility.
(2) Even if the solution is possible, the resultant Pareto conditions may be so complicated that the attainment of them is
made very dBcult, if not impossible.
It may seem paradoxical that the Pareto conditions cannot be
expressed in the simple terms of the (aggregate) marginal rate of
substitution. This paradox may be briefly explained.
Whenever there is external effect, the Pareto optimum point is
usually represented by the equivalence of marginal utility and
marginal disutility. To avoid interpersonal comparison of utility, these
marginal utility and &utility must be expressed in ternis of some
numiraire (usually money). But if this numiraire itself hila external
effect, equivalence of marginal utility and disutility measured in this
numiraire is, in general, no longer Pareto optimal. If every good in
the system has external effect, we are caught in a complete cycle; the
condition for each pair of goods has then to be expressed, in general,
in terms of the marginal utilities of all individuals for all goods.
The ansumption of the existence of a pure private good seems very
weak indeed. But the private good must be consumed by enough
individuals to cover the extensiveness of externality. If we have a
public good consumed by all individuals, e.g. the Presidtmt, or the
maintenance of law and order, then the private good must also be
consumed by all. To derive the optimality conditions for this public
good, we must sum the marginal utilities over all individuals. And to
transform this into the simple terms of the marginal rate of substitution without having to solve for the million aE/aUj we must express
these marginal utilities relative to some private good consumed by all
individuals.
The assumption of a pure common private good still Beems very
weak. However, in this world of extensive external effects, the assumption of no common private good seems equally weak, if not
weaker. What is needed to make it impossible, in general, to express
all the Pareto conditions in the simple terms of the marginal rate
of substitution is the following weak condition of universal externality:
For each common good, the consumption of at least one individual
enters into the utility function of at least another individual.
While weak universal externality renders some Pareto conditions
incapable of being expressed in the simple terms of marginal rate of
substitution, complete universal externality renders all Pareto conditions incapable of being so expressed. T l e condition needed for the
524
THE E C O S O X I C RECORD
DEC
existence of complete unzversal externality is : the consumption of
each individual of each good enters into the utility function of at least
another individual. Since each good going to each individual enters
into the utility function of at least tlvo individuals, v e cannot eliminate
dE/dI“ to ayoid interpersonal comparison of utility without solring
first for these a E / d P .
Where nniversal externality is estensiye, we may be forced to
admit of interpersonal utility comparison just t o achieve
Pareto optimality.’O Otherwise, n-e hare to face the difficulty
of defining. let alone attaining, Pareto optimality. The paradox may
thus prove to be a serious blow t o the practicability of Pdreto
optimality.
The paradox of universal externality seems t o suggest some interesting problems for further study. For example, there is the
empirical question of the degree and extent to x\hich universal esternality is present in the economy. Secondly, there are the theoretical
and practical problems of the possibility of solving for the millions of
dE ‘aC*. Failing this, there is the methodological issue of the acceptability and practicability of interpersonal comparison of utility
Finally, there is the policy matter as t o the feasibility of any ‘shortcut’ o r ‘optimal feasible’ solntion in the presence of universal e s ternality.
YE\V--K\VASG
KC
I*nicPrsitg of Sex EnglaTicl
REFERENCES
[ 11 Bergson, Abran, ‘-4 Reformulation of Certain .Aspects oi LVeliare Economics‘:
Quarterly Journal of Economics, 1938, as reprinted in Kenneth J Arrow ana
Tibor Scitovsky, eds., Readings irt Welfare Economics, (London : George
.\Hen and Unwin, 1969).
[ 2 ] Evans, Alan W., ‘Private Good, Externality, Public Good‘, Scotfish Jourrtal of
Political Economy, February 1970, 79-89.
p, ‘Definitions and SVelfare Conditions oi Public Goods: X Reply’.
Scottish Joirriml of Political Economy, June 1971, 203-08.
[4] Ng, Yew-Kwang, ‘Definitions and Welfare Conditions of Public Goods’.
Scottish Journal of Political Economy, June 1971, 199-202.
[5]
, ‘Definitions and Welfare Conditions of Public Goods: h Rejoinder’,
Scottish Journal o f Political Economy, Oct. 1971, pp. 347-9.
[5] ___ , ‘L$7eliare Economics, X-alue Judgement, and Policy Recommendation.
Second Conference of Australasian Economists. riug. 1971. .A revised versiori
appears in The Econoi~icJoirmal, September 1972.
[7] Rothenberg, Jerome, The Mensirrcment of Social Welfare, (Englewood Cliffs :
Prentice-Hall 1961).
[XI Sarnuelson, Paul A,. ‘The Pure Theory oi Puhlic Expenditure‘, R c i l m r j
Economic Stafistics, November 1954, 387-89
[$I
~
1 0 1 have argued elsewhere [6j that interpersonal comparisons oi utility are
not value judgements but subjective judgements of fact, and that economists are
more qualified in making those subjective judgements of fact that are closely related
to h e i r field of study
1972
THE BERGSON SOCIAL WELFARE FUNCTION
APPENDIX A
Derivation of the Necessary Conditions for a Mazimum We.lfure.
Form the Lagrange function:
Similarly,
and
525
526
THE ECONOMIC RECORD
DEC.,
1972
APPENDIX B
Equality of Social Marginal Rate of Substitution Necessary for
Productive Eficrency
For simplicity, assume
X
Y
E:
X(C", D". Q, DY)
(B-1)
= Y(Cy,DY, C",D")
C"
+
c y
=c
= D.
(B-2)
Dx + D'
To maximize X given Y and subject to (B-2), we form the Lagrangean
equation below :
L = X(C", D", C'. DY)
+1[9Y ( C y ,DY, C", D")] + &(C
C" - Cy) + $3
( D Dx D')
-
-
ax
@
+ 61
ay
=
61-
- -
(B-3)
Similarly,
Now by maximizing X given Y , we are also simultaneously maximizing
Y given X. Hence, we also have,
ay
B
Y
ax
+ ++By
= .&.
(B-5)
Similarly,
-& and d3 we, respectively, the "marginal values" of C and D in terms of X .
4send d6 are those in t e r n of Y , hence +Jqbl = (bslCs. Hence by combining (B-3)
through (B-6), we get
~
ax + h
ay - ay
acx - acp +
ax
aDx +
ay - a y
$
ax
ax
4
~
9
(B-7)
+ 4 4 -a
01 is the marginal value of Y in terms of 2 , and 0, is the marginal value of X
in terms of Y.Hence (B-7) is seen to be equivalent to (19").
It should be noted that, for productive efficiency as such, the 'marginal value'
must not be interpreted as the valuation of the consumers or society. Rather, it is
just the transformation ratio, or the marginal rate of transformation. With this
perspective, the fact that (19") is necessary for productive efficiency and the
irrelevance of relative subjective valuation for productive efficiency can easily be
reconciled.
$1 ~x
m
y