The Suntory and Toyota International Centres for Economics and Related Disciplines
Optimal Club Size: A Reply
Author(s): Yew-Kwang Ng
Source: Economica, New Series, Vol. 45, No. 180 (Nov., 1978), pp. 407-410
Published by: Wiley on behalf of The London School of Economics and Political Science and The
Suntory and Toyota International Centres for Economics and Related Disciplines
Stable URL: http://www.jstor.org/stable/2553455 .
Accessed: 01/02/2015 22:00
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
Wiley, The London School of Economics and Political Science, The Suntory and Toyota International Centres
for Economics and Related Disciplines are collaborating with JSTOR to digitize, preserve and extend access to
Economica.
http://www.jstor.org
This content downloaded from 155.69.24.171 on Sun, 1 Feb 2015 22:00:14 PM
All use subject to JSTOR Terms and Conditions
Economica, 45, 407-410
Optimal Club Size: A Reply
NG
By YEW-KWANG
Monash University
My analysis of clubs (Ng, 1973), aiming at Pareto optimality or maximizing total benefits of the whole population, has been misinterpreted as
maximizing total benefits of one club, which can be shown to be, in general,
not Pareto-optimal. In particular, Helpman and Hillman (1977) argue that
my conditions "cannot be relied upon to yield a Pareto-efficient club
population", and that "the clubs problems specified by Buchanan and Ng
are conceptually distinct". I shall first reply to their criticisms in a general
way (Section I) before going into the details (Section II). Section I may also
be taken as a reply to Berglas (1976).
I
Helpman and Hillman correctly note that there are two conceptually
distinct problems of clubs, one analysing a particular club and the other
analysing the formation of a large number of identical clubs in a big
population. However, I doubt that Buchanan has the second approach in
mind; in his paper no mention is made on a central requirement of the
second approach, namely that the number of individuals in the whole
economy must be very large (strictly speaking, infinite) in comparison to the
number of individuals in one club. Second, it is not true that "Ng's problem
is that of determining optimal club size when only one club may form for
each collective" (Helpman and Hillman, 1977, p. 293; italics added). In my
1973 paper I explicitly noted that, although my paper does not include an
explicit analysis of multi-club goods, "we can interpret a new club for the
'same' good as the consumption of a new good" (Ng, 1973, p. 298). With
this interpretation, the benefit to a "new member of changing clubs is the
difference in total benefit between joining the new club and staying at the
old club" (p. 298). Hence, my solution is not necessarily confined to one
club and does not lead to a Pareto-inferior situation.
It is true that, if the number of individuals is so large as to justify a large
number of clubs providing the same good, my optimality conditions, though
formally correct, are not very useful. Then a framework of analysis (which a
number of writers have adopted) with an infinite number of individuals of
identical tastes (I allowed for differences in tastes) becomes an acceptable
approximation. However, on a practical level there are hardly any collective
goods in respect of which consumers are indifferent as to which club they
join, since there is usually a difference in club location if not in other
aspects. Thus, my framework of analysis is also a more realistic one.
II
The conclusion of Helpman and Hillman that my conditions are not
Pareto-efficient is based on: (a) misunderstanding of my conditions; (b) an
This content downloaded from 155.69.24.171 on Sun, 1 Feb 2015 22:00:14 PM
All use subject to JSTOR Terms and Conditions
408
ECONOMICA
[NOVEMBER
inadequate distinction between the conditions for a welfare maximum
according to a specific social welfare function (SWF) and the conditions for
general Pareto efficiency.
Helpman and Hillman reproduce my condition for club membership as
E
i Cj
UNjIUKn=
where Cj is the club for the jth collective good Xj, and Xn is a numeraire
private good. They then argue that the condition "can be satisfied only if at
least one term in the summation is positive, whereas marginal congestion
implies UN <0 for all iE Cj". However, my condition is
ui-
where s is the number of individuals in the whole economy. The left-hand
side sums over all individuals and hence UNj is positive for the marginal
individual being included in the club. It is this increase in utility for this
individual that accounts for the term AU' in a later equation that puzzles
Helpman and Hillman. Thus, I did not fail "to allow for the effect of the
change in the range of summation". Neverthless, by analysing a, strictly
speaking, non-continuous variable Nj with differential calculus, I probably
have not made the point as clear as I should have. Hence, the misunderstanding of Helpman and Hillman is understandable and at least partly due
to the ambiguity of my notation.
The fact that they have not adequately distinguished between the conditions for a welfare maximum and for general Pareto efficiency is also
understandable, as usually these conditions (except one) are the same. For
the case of two individuals and two private goods, either we maximize
W(U1, U2) or we maximize U1 +k (U2- U2) subject to F(X, Y) =0; we get
exactly the same condition for Pareto efficiency, i.e. U'/UQ= U2/ U=
FX/FY,where Ul -a Ullax', x1+X2= X, etc. There is an additional condition in each case:
Ul = (W2/ W1)U2
and
Ul =AkU2
respectively correspond to the requirement for a specific welfare maximum
and a specific Pareto-efficient point. However, for the case studied by
Helpman and Hillman, the condition derived (their equation 4) is not the
same as my condition (11), though they are consistent with each other. This
point, which may appear a little surprising at first, is in itself worthy of some
attention. The explanation lies in the fact that, for the case of clubs, there is
an inherent indivisibility: an individual is either in or out of a club. When an
individual is admitted to the club, the change in his consumption of the
facility is not infinitesimal. It is this indivisible change that necessitates a
significant reallocation of the private good to preserve the equal welfare
significance of marginal consumption. And it is this reallocation that makes
their equation (4) different from my condition, which refers purely to Pareto
efficiency and hence does not need the reallocation.
To illustrate the difference diagramatically, consider Figure 1, where
Curve A measures the marginal utility of the private good with no club
This content downloaded from 155.69.24.171 on Sun, 1 Feb 2015 22:00:14 PM
All use subject to JSTOR Terms and Conditions
1978]
OPrIMAL CLUB SIZE: A REPLY
409
J
0
C
D
amount of the
private good
Figure 1.
membership. While club membership must increase the total utility, it may
either increase or lower the marginal utility. For simplicity I shall just
consider the case of an increase. Hence, Curve B measures the marginal
utility with club membership. To achieve equality in marginal utility for all
individuals (this is for welfare maximization according to the particular
SWF, not necessary for Pareto efficiency as such), suppose all individuals in
the club are allocated OC amount of the private good and all individuals not
in the club are allocated OD. Thus, as an individual is being admitted into
the club he not only gains utility from the act of admission itself measured
by the area GHJI, he also gains the area DCFH owing to an additional
allocation (DC) of private good (transferred from all other individuals, each
sacrificing an infinitesimal unit; remembering Helpman-Hillman's model of
uncountable infinity of individuals). This additional allocation is costed, in
utility terms, at DCFG (corresponds to the term py(yin- yOUt) in their
equation 4). Hence the area GFJI (= GHJI+ DCFH- DCFG) has to be
compared with the disutility of congestion imposed on the existing club
members owing to this additional admission to see whether it is desirable to
admit him. This is Helpman-Hillman's equation (4) illustrated diagramatically.
On the other hand, if we are concerned only with Pareto efficiency, we do
not require any reallocation of the private good, Thus, the gain to an
individual of getting admission to the club depends on his initial income and
may be measured by GHJI, EFJI or some other measures, but does not
include DCFG. My condition (11) says that, for a Pareto optimum, any
individual not in/in the club must benefit from membership by an amount
(measured by GHIJ/EFJI) smaller/greater than the disutilities (appropriately weighted) imposed on other membership owing to his membership. If
this condition, or (11') is satisfied, it is not possible to make someone better
off without making some other person worse off. It does not necessarily
imply a welfare maximum according to a specific SWF, since welfare may be
increased by making some individuals better off and some worse off.
This content downloaded from 155.69.24.171 on Sun, 1 Feb 2015 22:00:14 PM
All use subject to JSTOR Terms and Conditions
410
ECONOMICA
[NOVEMBER
However, if welfare is maximized according to, say Helpman-Hillman's
SWF (which is Paretian), then it can be shown that my condition (11) is also
satisfied. Thus, suppose all individuals in the club are allocated OC and
others OD amount of private good and the marginal member imposes
congestion disutilities aggregated to precisely GFJI, satisfying HelpmanHillman's equation (4). Then all those in the club benefit from the membership (i.e. membership as such, not including the transfer of private good)
measured by EFJI, which is larger than GFJI, and all those not in the club
would benefit by GHJI only which is smaller than GFJI, by joining,
satisfying both parts of my condition (11). While a Pareto-efficient point
need not be welfare-maximal, a welfare-maximal point, provided the SWF is
Paretian, must be Pareto-efficient.
Since I am concerned with Pareto optimality conditions and not with a
specific welfare-maximal point, I conclude that the conditions I derive in my
1973 Economica paper are correct.
REFERENCES
BERGLAS,E. (1976). On the theory of clubs. American Economic Review, Papers and
Proceedings, 66, 116-121.
BUCHANAN,J. M. (1965). An economictheory of clubs. Economica,32, 1-14.
HELPMAN,
E. and HILLMAN,A. L. (1977). On optimalclub size. Economica,44, 293-296.
NG, Y-K. (1973). The economica theory of clubs: Pareto optimality conditions. Economica, 40,
291-298.
This content downloaded from 155.69.24.171 on Sun, 1 Feb 2015 22:00:14 PM
All use subject to JSTOR Terms and Conditions