The Suntory and Toyota International Centres for Economics and Related Disciplines
The Economic Theory of Clubs: Pareto Optimality Conditions
Author(s): Yew-Kwang Ng
Source: Economica, New Series, Vol. 40, No. 159 (Aug., 1973), pp. 291-298
Published by: Wiley on behalf of The London School of Economics and Political Science and The
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1973]
The Economic Theory of Clubs: Pareto Optimality
Conditions'
NG
By YEW-KWANG
Following its conclusion in Readingsin Microeconomics[1], Professor
Buchanan's seminal paper on the economic theory of clubs [2] can now
be regarded as a standard reference.2 The purpose of this paper is to
provide an alternative analysis of the theory of clubs which, it is
believed, will overcome some of the shortcomings of Buchanan's
analysis.
I. BUCHANAN's ANALYSIS
The salient feature of Buchanan's analysis is the explicit specification
of the numbers of individuals consuming collective goods. The effect of
these numbers on the satisfaction of each individual is the most
important innovation of his analysis. I shall follow Buchanan in this
important step.
In deriving his Pareto-optimality conditions, Buchanan maximizes
the utility function of a single individual subject to his own feasibility
constraint. As elements of the utility function are included Ni, the
numbers of individuals consuming the various collective goods which
are also consumed by the individual in question. Hence, one set of the
necessary conditions is [Buchanan's Eq. (6)]
(1)
UN/ Ur=ANfi,
which states "that the marginal rate of substitution 'in consumption'
between the size of the group sharing in the use of good Xj, and the
numeraire good Xr, must be equal to the marginal rate of substitution
'in production"' ([2], p. 5).
It seems to me that the conditions thus obtained are the equilibrium
conditions for an individual, given his market opportunities and assuming that he can choose his preferredNI. For Pareto optimality, we have
to maximize the utility of an individual, subject to the constraints that
the level of utility of each other individual is held constant and that
society's production possibilities are given. This is the approach used in
the following model.3
1 I am grateful to a referee and to my colleagues Mendel Weisser and Michael
Berry for helpful comments on the first draft of this paper.
2
Referencesin square brackets are listed on p. 298.
3 Alternatively but equivalently, we may maximize a social welfare function
W= W(Ul, U2,..., US).This is an equivalent method since the various aW/lUl
are not assumedto be known.
291
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292
[AUGUST
ECONOMICA
II. AN ALTERNATIVE
ANALYSIS
Suppose that there are s individuals, m collective goods and (n -im)
private goods. We have the following utility functions:
(2)
Ui Ui(X1, N1, Di,.... Xn, N, Din,,
xi +1,.... Xi),
where Ui is the utility function of the ith individual, Xj, Nj are, respectively, the amount of and the number of individuals consuming the jth
collective good, Xi is the amount of the jth private good consumed by
the ith individual, and Dij = 1 if individual i belongs to club j (i.e.
consuming Xj) and =0 if he does not. Obviously, if Dij=O,aU0
/X-=
,aUiaNjO-0. To facilitate the use of calculus we also assume divisibility
in the quantity of all goods and that the number of consumers in each
club is large so that Nj is approximately a continuous variable.
We have the production function
s
(3)
F(X1,
...X x
z
s
...
XMi+1
X) = O.
E
To derive the necessary conditions for a Pareto optimum, we may
maximize, without loss of generality, U' subject to Ui = Bi (i= 2, ...., s),
where CI is the given level of utility for individual i, and to the production constraint (3). The relevant Lagrangean function is
(4)
L = Ul
+ E A\i(Ui i-2
i)- OF.
The following equations are obtained by setting the respective partial
derivatives of L equal to zero:'
s
(5)
(6)
, ..
(j- ,m)l
EAiUij=OFxj
i=l
AiUxlj= OFxj
(j-m+
1, ... ., n, i= 1...,s)
s
(7)
E AiUiNi=
(j-l
i.,n).
Apart from the conventional marginal equality for private goods
n), obtained from
s; j, k=m+1,...,
UJ/UiXk=FXj/Fxk, (i=1,...,
(6), we have, by combining (5), (6) and (7), and using Xn as a numeraire,
s
(8)
ya UixilUXn =FXJIFXn
j=j
(j= 1. . n, )
s
(8)
UN=l
n
m).
(1=1,.
,
Since Ux = UNj= 0 for those individuals not consuming Xj, we may
rewrite equation j of the set (8) in the following form if we arrange our
individuals in the order of their consumption of Xj, i.e. for each jth
1 Defining Al= 1, and denoting a partial derivativeby an appropriatesubscript,
n+ n), FXJ= aFIaXJ,
m); = a Ut/XJ (j= m + 1,
m+n).
Xi= E Xi,(j=m+l,...
= aUlIaX1 (j= 1, . *. ,
e.g. UkxJ
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1973]
THE ECONOMICTHEORYOF CLUBS
293
collective good, all those consuming it are put before those not consuming it:
Nj
(10)
E Ui! Uixn=F1/FXn
(1=1 ..
. m)
Alternatively, we may define Sj as the set of indices corresponding to
individuals in club j. Then (8) becomes
(10')
im).
E UilUixn=FxjlFXn
(j=1,...,
iesi
In our derivation of (9) we have ignored the fact that N1 is not
strictly a continuous but, rather, a discrete variable, since it can only
vary over integers. A better appropriation to the conduction determining the number of individuals in club j is obtained by defining A ui = Ui
(when Dij= 1)- Ui (when Dij = 0). Then (9) can be replaced by
2 0 if i E-Si; < 0 if i 0 Si.
Or, in terms of the marginal rates of substitution,
All
AUj + I
k#ji
Uixn(UNj
U'Xn)
fi (Ui
< 0 if i 0 Si.
UxiU)dXj+I Ux/UUNn 2 0 if i e S;
This says that, for each collective good, any individual in the club must
derive a total benefit from the consumption of that good in excess of (or
at least equal to) the aggregate marginal disutility imposed on all other
consumers in the club, and that the reverse must hold for any individual
not in the club. On the other hand, (10') says that, given the set of
consumers, the aggregate marginal valuation must equal the marginal
cost. This is obviously the Samuelson condition [3]; the only difference
is that the set of consumers need not coincide with the set of individuals
in the whole society, or that Nj need not equal s.
In the case of Samuelsonian pure public goods, the addition of a new
consumer does not affect the utility levels of the existing consumers.
Therefore the summation term in (11) or (11') vanishes. [Hereafter(10)
and (11) are taken to include (10') and (11') respectively.] But as long
as all consumers place positive valuations on the good, the first term
of (11) is positive. Hence the second inequality in (11) cannot be
satisfied even if Nj=s, i.e. if all individuals in the society are included.
This weakness of (11) can be overcome by placing a feasibility constraint Nj<s (j=1,...,m)
in our maximization problem. I have
decided against this for reason of notational simplicity.
Now compare (11) with Buchanan's condition, i.e. Eq. (1). Our
analysis shows that, once the conventional equality for private goods
and conditions (10) and (11) are satisfied, a Pareto optimum has been
achieved, assuming that the second-order conditions are also satisfied.
These conditions, however, do not imply Buchanan's condition. In
other words, (1) is not necessary in order to define a Pareto-optimal
position. This is so because each Nj enters into the utility function of a
number of consumers simultaneously, and each consumer cannot vary
Nj at will. Hence the relevant condition for each Nj is the aggregate
marginal valuation rather than the individual marginal valuation.
(11')
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294
ECONOMICA
[AUGUST
III. THE GEOMETRY
We turn now to Buchanan's geometry. Our Figure 1 is a simplified
version of his Figure 1, where the total benefit and total cost curves to
each individual, B and C, are drawn for a given quantity of a collective
good. For simplicity, Buchanan assumes identical tastes and equal costsharing for all individuals. With these assumptions, the curve C is a
C
0
c,
X
0.
0~
en
H-I
,Z0
Number of Persons
FIGURE 1
rectangular hyperbola. The shape of B is explained by Buchanan as
follows: "As more persons are allowed to share in the enjoyment of the
facility, of given size, the benefit evaluation that the individual places
on the good will, after some point, decline. There may, of course, be
both an increasing and a constant range of the total benefit function, but
at some point congestion will set in, and his evaluation of the good will
fall" ([2], pp. 7-8).
Consistent with Eq. (1), Buchanan argues that the optimal size
of a club is determined at the point where the slopes of the two curves B
and C are equal, shown at S in Figure 1. However, it can be seen
clearly that the number of persons, S, maximizes the net benefit per
person, but not necessarily the total net benefit. If we increase the number of persons by one, the net benefit per person will be decreased
slightly, but the increase in benefit to this new consumer may outweigh
this aggregate decrease. Hence optimality is achieved only when the
increase is just equal to the decrease, assuming perfect divisibility and
continuity. But as the number of persons can only be varied by an integer,
our condition (11) holds. Buchanan ends up by maximizing average net
benefit rather than total net benefit precisely because of his maximization of the utility of an individual without subjecting it to the constraints of the constancy of the utility levels of other individuals.'
1 Buchanan'sconditions, ratherthan being the "Pareto optimality conditions",
are more appropriate as equilibrium conditions assuming that club members
maximizethe averagebenefitand that there aresome othersimplifyingassumptions.
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1973]
THE ECONOMICTHEORYOF CLUBS
295
The geometrical illustration of our conditions turns out to be much
more simple and clear-cut. In Figure 2 the vertical axis measures the
total cost and aggregate total benefit. For a given quantity (or quality)
of a collective good, an aggregate total benefit curve A' can be derived
by aggregating the B curve in Figure 1 over the relevant ranges of the
number of consumers. Since the quantity of the good is fixed, total
cost is an horizontal straight line,1 shown as D. Subtracting D from
A', we have the aggregatenet benefit curve A. For this given quantity of
the collective good, the optimal size of the club occurs at T, where both
A' and A attain their maxima.
coj
70
DD
CD,
0
f
Numberof Persons
FIGURE
2
This analysis assumes a fixed quantity of the collective good. To
determine the optimal quantity and optimal number of consumers
simultaneouslywe turn to Figure 3, which is similar to Figure 2 but has
a number of A curves, each one corresponding to a given quantity of
the good. The larger the quantity, the higher is the aggregate benefit,
but also the higher the total cost. Hence the aggregate net benefit may
increase or decrease with the increase in quantity. For each quantity
of the good, we need to focus only on the point of maximum aggregate
net benefit. From these maxima, we choose the maximum maximorum,
P, which determines both the optimal quantity represented by A3 and
the optimal size of the club S.
I hope to produce another paper comparing these equilibrium conditions with
Pareto conditions and discussing the problem of optimal intervention. After
relaxingthe simplifyingassumptions,this problemis much more complicatedthan
it might seem at first sight.
1 Our construction can also be generalizedto cover the case of an increasing
total cost curve where the marginal cost of extending services to additional
consumers is not zero. In the diagrammaticaltreatment, this can be achieved by
definingtotal cost to be the cost of "production"only and subtractingthe cost of
extending services from the aggregatebenefit curve.
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296
ECONOMICA
[AUGUST
01)
Ei~
v~~~~~~~~
R
0)
0
Number of Persons
FIGURE 3
co
0
M
XCi)
00)
cOD
O
2,
MC
XE
<
MB
0<
0o
Quantityof Good
FIGURE 4
The point P clearly satisfies condition (11). That it also satisfies (10)
may be seen more clearly in Figure 4, where the horizontal axis measures
the quantity of the good. The marginal cost of producing the good (MC)
is drawn. The marginal aggregate benefit of increasing the quantity of
the good (MB) is obtained by comparing, successively, the points of
maximum aggregate total benefit associated with different quantities of
the good. The marginal aggregate net benefit curve (MNB) is similarly
obtained by comparing those of the maximum aggregate net benefit,
i.e. the points M, Q, P, R in Figure 3. Since P is the maximum maximorum, marginal aggregate net benefit must be zero. With MNB=O,
MB=MC by definition. And this last equality is what is specified in
(10).
The Samuelson pure public good can be seen to be a limiting case
of our analysis. As the number of consumers does not affect the
satisfaction of any consumer, the aggregate total benefits (A' in Figure
5) is an ever increasingfunction of the number of consumers. If we have
identical individuals, both A' and A are straight lines. But this is not
necessary for the analysis. The essential point is that A' and A are
increasing (assuming all individuals place positive valuations on the
good). Hence, the optimal size of the club occurs only when all individ-
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1973]
THE ECONOMICTHEORYOF CLUBS
297
A
0)
z
<o
Numberof Persons
FIGURE5
uals are included. We may again determine quantity and size simultaneously by drawing in the various A curves, and choosing the one that
intersectsthe vertical line ZZ' at the highest level, where Z is the number
of all individuals.
The public good may be not quite pure in the sense that, although
the good is freely available to all and the number of consumers does not
enter any utility function, some individuals nevertheless place negative
valuations on the good. In other words, the good is "impure" not so
much in its "publicness" but in its "goodness". In this case, if we rank
consumers according to their degree of preference for the good (as
we should do to satisfy the second-order condition), the A curves will
turn downward after some point before Z. In this case, there are two
sub-cases. First, the good may be "non-rejectable" in the sense that,
once provided, no individual can refuse it. Then the relevant point of
comparison is along the line ZZ', and the Samuelson condition applies,
as does Eq. (8). Second, the good may be "rejectable", and hence the
optimal point will occur before Z; our conditions (10) and (11) apply,
but the Samuelson condition does not. As Buchanan points out, the
theory of clubs applies in the strict sense only if "exclusion" is possible
([2], p. 13). If exclusion is possible, the good is usually rejectable.
Hence for the theory of clubs, the second sub-case is the more relevant.
IV. A
COMPLICATION-THE NUMBER OF CLUBS
The foregoing analysis and that of Buchanan both assume implicitly
that the number of clubs for each collective good is unity, or, at least,
they do not include an explicit analysis of the number of clubs itself.
To show the implication of this matter, consider the following example
which I owe to Weisser.
In an economy of 20 individuals, the size of club for a particulargood
maximizes averagebenefit at 4 and maximizes total benefit at 5. It seems
that 5 clubs each with 4 members will maximize both average and total
benefits for the 20 individuals. Hence it appears that Buchanan's
analysis is valid if the number of clubs is taken into consideration.
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298
ECONOMICA
[AUGUST
However, this appears to be so only because of the specific total number
of individuals chosen and the implicit assumption of identical tastes.
What happens if the number of individuals is 21 ? If we adhere to the
yardstick of the maximization of average benefit, one individual has
to be excluded from enjoying the benefit of the good, even though the
"congestion cost" he imposes on others is less than the benefit to him
(so that he would be willing to compensate them, and everyone could be
made better off).
It may appear, therefore, that we have to include another set of
variables, the numbers of clubs, into our maximization problem.
This, however, is not really necessary, as we can interpret a new club
for the "same" good as the consumption of a new good.' Thus interpreted, it can easily be seen that our analysis is applicable even if the
number of clubs exceeds one. For example, in the previous example of
21 individuals, our analysis indicates that the "last" individual has
to be included in one of the clubs, provided that the benefit is larger
than the cost of congestion. Once everyone is included in one or other
of the clubs, our condition (11) is also satisfied, since a further increase
in the membership of a particular club has then to occur by attracting a
member from another club. Hence the benefit to this new member of
changing clubs is the differencein total benefit between joining the new
club and staying in the old club. Thus, even in the original case of 20
individuals and identical tastes, our condition is satisfied at 5 clubs
each with 4 members. The net benefit to a member of transferringfrom
one club to another is zero or actually negative, taking account of
indivisibility. Therefore the aggregate marginal benefit of extending the
membership from 4 to 5 is negative, thereby satisfying our condition
(11). While Buchanan's condition is satisfied only for certain numbers
of individuals, our analysis is applicable to any number of individuals,
whether with identical or with different tastes, and even to cases where
some individuals may choose to belong to two or more clubs.
If it is objected that our condition is also satisfied at 4 clubs each
with 5 members, we may reply that we have only been concerned with
first-order necessary conditions, and not with second-order and/or
total conditions, or with the problem of local-versus-global maxima.
Universityof New England, Australia
and London School of Economics.
REFERENCES
[1] Breit, W. and H. M. Hochman, Readings in Microeconomics,2nd ed., New
York, 1971.
[2] Buchanan,J. M., "An Economic Theory of Clubs", Economica,vol. 32 (1965),
pp. 1-14.
[3] Samuelson, P. A., "The Pure Theory of Public Expenditure", Review of
EconomicStatistics, vol. 36 (1954), pp. 387-9.
1 This is a more realistic point of view. There are hardly any collective goods in
respect of which consumers are indifferent as to which club (apart from its
size) they join, since there is usually at least a differencein club locations if not in
other aspects of the goods,
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