DEFINITIONS AND WELFARE CONDITIONS
O F P U B L I C GOODS
Y. K . N G
The purpose of this paper is to show that there is a type of public goods
which is not covered by Evans’ general welfare conditions for private goods,
externalities. and public goods (Evans, 1970).
Public good was defined by Samuelson (1954, p. 387) as a good ‘ which
all enjoy in common in the sense that each individual’s consumption of such
goods leads to no subtraction from any other individual’s consumption of that
good ’. Lately, he has altered his definition to be any good ‘ with the property
of . . . entering into two or more persons’ preference functions simultaneously ’. (Samuelson 1969, p. 102.)
Evans complains, we think correctly, that, with the latter definition. the
whole range of consumption externality is classified as publicness. We therefore follow Evans in adopting the first definition.’
Evans’ welfare conditions are2:
-.
ulk
1=1
1
z - - -1
F
-
ur
f
( k = 1; . . . . . s ; r = 1 ; j = 1. . . . . . n)
...(I )
F,
The first condition states that, for an optimum, the sum, over all individuals i = 1, . . . . . s, of the marginal rates of substitution of consumption of
the good j by the individual k for consumption of the numeraire r by the
respective individuals i = 1, . . . . . s must be equal to the marginal rate of
transformation of the good j for the numeraire r. The second condition
states that the marginal social welfare of the numeraire is equal for all
individuals.
Evans then argues that the above conditions are applicable to pure
private goods, goods with external effects in consumption, and pure public
goods. In the case of pure private good. ufk=0 for i #k. hence the first
f
condition reduces to
-ukr
- -(k
= 1 . . . . . s; r = 1: j = 1,.
F,
. . . . n)
.. .(3)
See however footnote 5 below.
aui
Zu1=,where X,k stands for the consumption of the good j by the individual
; ax;
i, and uL stands for the utility function of individual i; F, = aF/aX,, where F is the
production function; and U, = dU/duf,where U is the soclal welfare function.
199
200
Y. K. N G
which states that the marginal rates of substitution of consumption of the
good j by each individual k must be equal to the marginal rate of transformation of the good j for the numeraire r.
In the case of consumption externality, the terms
(i + k) are not all
equal to zero and the welfare condition stands as it is. In the case of pure
public good, Evans argues that the ‘ good j must be consumed collectively
by all consumers. An increase in its provision to any single consumer must
be equivalent to the same increase in its provision to any other consumer.
Whether the good j is provided for himself or for any other person the
consumer must derive exactly the same marginal utility fromthe increase in
its provision ’ (1, p. 82.) Hence we have,
= u,: = u;
..
u;,= u:*= . .
Thus, equation ( I ) can be reduced to
a
uj‘
z-=l=lur(
Fj
.. .(5)
F,.
which is exactly Samuelson’s condition for public goods (3, p. 387).
It seems. therefore, that Evans has a general formula for the whole
spectrum of goods with the pure private good and the pure public good as
polar cases. However, there is at least one group of goods which cannot be
covered by Evans’ formula, as illustrated below.
Consider the building of a national museum. The size of the museum
can be varied, but overcrowding of visitors will not happen for any reasonable size. Hence, once built, the museum could be made available to all
individuals in the nation without additional cost. Assume also that the cost
of excluding someone from visiting the museum is negligible, so that the
museum need not be made available to all.
Now an individual may derive utility from the national museum in two
ways: (i) a sense of pride by the sheer existence of the museum; (ii) by
actually visiting the museum. Once the museum is provided to any individual,
it is clearly provided to all in the first sense Hence as far as this aspect of
the public good of the museum is concerned, Evans’ formula is applicable.
However, an individual does not derive any utility in the second sense from
the provisions of the good to other individuals. In this case, equation (4) is no
longer applicable and Samuelson’s condition (i.e. equation 3) cannot be
reduced from Evans’ formula. Nevertheless, Samuelson’s condition is still
the correct formula for Pareto optimality in this case, as it is still true that
‘ each individual’s consumption of such a good leads to no subtraction from
any other individual’s consumption of that good ’.s In deciding the socially
optimal supply of the good (the size of the museum), it is still true that
the marginal valuation curves of the various individuals have to be summed
vertically to arrive at the collective valuation curve, the intersection of which
Assuming that each individual has no special utility or disutility attached to the
number of people visiting the museum.
P U B L I C GOODS
20 1
with the marginal cost curve determines the optimal point.’ It seems, therefore, that Evans’ formula is not applicable in this case, and a separate
formula (Samuelson’s condition) is required after all.
From the above discussion, it is clear that there are two distinct types of
public goods. If we define a public good as one the consumption of which
by each individual leads to no subtraction from any other individual’s consumption, then there are two groups of public
The first is where
exclusion is impossible and thus the good, if produced, is provided for all
individuals, e.g. defence. The second is where exclusion is possible and the
good need not necessarily be provided for all, e.g. swimming ~ 0 0 1 Evans’
.~
formula is applicable to the first but not to the second type. Of course it may
be said that the cost of exclusion varies from case to case. Hence the two
types of public goods we speak of are actually a spectrum of public goods
with absolute non-exclusibility and zero cost of exclusion as polar cases. But
differences in quantity can have qualitative significance. Hence it may be
useful to distinguish two types of public goods. For those public goods where
no exclusion is possible or where exclusion costs are fantastically high, their
provision generally has to be made collectively. For those public goods where
exclusion costs are low, they could be provided in the market. But generally,
they are not efficiently supplied in the market. As the consumption of any
individual does not lead to any subtraction of the consumption of others,
the price has to be made zero to achieve Pareto optimality.
It has, however, been argued that this latter type of public goods ‘ need
not necessarily imply that competitive industry cannot produce the efficient
output ’ (Millward, 1970, p. 30). ‘ Consider cinemas in a large town . . . if
the town is populous and distances are small, free enterprise might well result
in optimal replication of cinema theatres, each operating at capacity
audiences, with fares set competitively at short and long-run marginal and
average costs. This is a case where an exclusion principle can, and should
operate ’ (Samuelson, 1969, p. 110).
It is quite true that, if each theatre is operating at capacity audience with
fare set competitively at short and long-run marginal and average costs,
which implies the lowest point on the average cost curve, Pareto optimality
4 Given the constraint that some individual will be excluded from entering the
museum (by a price or anything), the ‘second-best’ optimal size of the museum is
determined by the collective valuation curve which is a summation only of marginal
valuation curves of the non-excluded individuals.
5 If we strictly require ‘ common consumption for all ’ as the essential condition
for a public good, and interpret this to mean X,’= X, (i= 1, . . . . s), as did Samuelscm
(1954, p. 387), the second groups of public goods mentioned in the test will be
excluded. But this would leave this type of goods without a name. It is certainly not
private goods and the relevant welfare conditions are similar to the ‘public goods
proper ’ (i.e. the first type).
6 This second type of public good is quite distinct from Evans’ ‘ local public good ’.
The latter is a good which, once produced is automatically available to all individuals
in that locality, but not to individuals in other areas. The former is a good which, once
produced, could be made without additional cost, available to all individuals (in the
locality, the whole nation or the whole world, as the case may be), but need not
necessarily be made available to any individual.
6
202
Y. K. N G
has been attained. But it seems to me that, once this is so, the good in
question is no longer a public good. If there is any increase in demand,
either new theatres have to be built or existing theatres have to operate
beyond the point of minimum average costs. An increase in someone's consumption must lead to decrease in others' consumption or to increase costs of
production.
University of New England,
New South Wales
REFERENCES
EVANS,A. W. (1970). Private Good, Externality, Public Good. Scottish Journal of
Political Economy (February).
MILLWARD,
R. (1970). Exclusion Costs, External Economies, and Market Failure.
Oxford Economic Papers (March).
SAMUBLSON, P. A. (1954). The Pure Theory of Public Expenditure. Review of Economic
Statistics (November).
SAMUELSON,
P. A. (1969). The Pure Theory of Public Expenditure and Taxation. In
Public Economics (ed., J. Margolis and H. Guitton). London : Macmillan.