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Indivisibility and preference for collective provision
Regional
Science
and Urban
Indivisibility provision
Economics
22 (1992) 559-577.
and preference
North-Holland
for collective
John H.Y. Edwards* Tulane University, New Orleans LA, USA Received
February
1990, final version
received
March
1991
Collective goods are usually defined by comparing per-unit cost, net of congestion, under collective and individual provision. This paper examines preference for the form of provision of indivisible goods, arguing that the public/private or collective/individual dichotomy is an artifice of the assumption that shared goods are perfectly divisible. For indivisible goods, collective provision may be preferred when unit-cost is relatively higher and it may be undesired when unit-costs are lower than private; lower per-unit cost is neither a necessary, nor a sufficient condition. The model provides an explanation of Oates’ (Journal of Urban Economics, 1988, pp. 85-94) ‘zoo-effect’.
1. Introduction The literature on local public finance and the theory of clubs is built around two opposing effects of sharing which determine the net per-unit cost of collective provision. In these models congestion costs work against the reduction in pecuniary costs brought about by sharing. Collective goods, including public goods, are defined as the class of goods for which per-unit cost is lower for collective provision than for individual provision. This paper argues that the dichotomizing of goods into opposing private and public classes, classes distinguished by per-unit cost, is an artifice of the assumption that all goods are perfectly divisible. When indivisible goods are examined the &dings suggest that lower per-unit cost is neither a necessary nor a sufficient condition of preference for collective provision. In fact it is shown that collective provision may be preferred when the unit cost of collective provision is relatively higher than private per-unit costs and that collective provision may be undesired even when collective per-unit costs are
Correspondence Tilton Hall, New *I would like Kong-Pin Chen, helpful comments 01660462/92/$05.00
to: John H.Y. Edwards, Department of Economics, Tulane University, 206 Orleans, LA 70118, USA. to thank Todd Sandler, Marcus Berliant, Dan Rubinfeld, Nahid Aslanbeigui, Timothy Goodspeed, William Oakland, Wallace Oates and Jon Pritchett for on earlier drafts. Any errors are my responsibility. 0
1992-Elsevier
Science Publishers
B.V. All rights
reserved
560
J.H.Y.
Edwards,
Indivisibility
and preference
for
collective
provision
lower. These arguments provide a theoretical basis for the ‘zoo-effect’ described in Oates (1988). The sharp distinction between collective and private goods first arose in 1954 when Samuelson derived the now well-known conditions for efficient provision of pure public goods. Pure public goods are extreme in that ‘
. . . each individual’s consumption of such a good leads to no subtraction from any other individual’s consumption of that good, so that X,+j=X~+ j simultaneously for every ith individual and each collective consumption good’ [Samuelson (1954, pp. 388-389)].
Samuelson makes it clear that he intended, not to maintain the existence of these polar, or ‘pure’ cases, but to introduce this extreme assumption as a tool of analysis because ‘doctrinal history shows that theoretical insight often comes from considering strong or extreme cases’ [Samuelson (1955, p. 350)]. One consequence of this extreme assumption was the preference revelation dilemma that confounded the search for efficiency in the provision of collective output. By distinguishing local public goods and club goods from other collective goods, Tiebout (1956) and Buchanan (1965) both sought to rescue them from this condemnation to inefficient provision. Paradoxically, it is by re-introducing congestion costs that local public finance and clubtheory models seek market-like solutions. 1 The core of these models is the assumption that, while it reduces unit-price, sharing also causes congestion. These two effects of sharing offset one another in such a way as to produce U-shaped average cost curves for collective provision. Optimal group size and efficient provision have been identified with the minimum point on the average cost curves.’ These arguments have shown that it is not essential for shared goods to be of the ideal type conceived by Samuelson. However, much of the literature continues to operate on the purported public/private or collective/individual knife’s_edge3 dichotomy between immutable ‘classes’ of goods. In one of the best and most widely cited applications of the median voter model, Bergstrom and Goodman (1973) examined the congestion properties of police protection and parks. On the basis of high estimates of congestion they question the appropriateness of public provision (p. 293).4 Niskanen ‘A more complete discussion of the evolution of congestion as an economic concept can be found in Cornes and Sandier (1986). 2There are many discussions of the question of congestion and the existence of an optimal group size. Some good sources are Oakland (1972, 1983, Rubinfeld (1988), Cornes and Sandler (1986, especially chapters 5, IO, 11 and 16), Tiebout (1961) Mueller (1979) and Scotchmer (1985). ?3amuelson (1966) uses ‘knife’s_edge’ to describe the distinction. “Similar models with equivalent results can be found in Borcherding and Deacon (1972) Dudley (198 I), and Pommerehne and Frey (1976). More recently Bruckner (1981). Craig (1987) and, Edwards (1990) find the non-publicness result to be highly sensitive to congestion function specitication.
J.H.Y.
Edwards,
Indivisibility
and pre&wnce
for
collective
provision
561
(1971) and Borcherding et al. (1977) suspect there is evidence of voter exploitation in the fact that highly congested goods are provided publicly.5 It is my intention to show that such dichotomizing of goods into opposing private and public classes, distinguished by their cost per unit, is an artifice of the assumption that all goods are perfectly divisible. Piganiol (1969) contains some insightful comments on the ways private markets overcome indivisibilities. Among the mechanisms he lists are rentals, used goods, and credit. All of his solutions to the problem of indivisibility maintain the individual character of private goods markets. There also exist collective solutions to indivisibility that go beyond Piganiol’s list. Among these are clubs, public provision and informal sharing arrangements (roommates). In fact, it seems very likely that indivisibilities are present in most collectively provided goods. A recent study by Oates (1988) points in this direction. In discussing the empirical literature on congestion he suggests that the measurement of local good congestion may be upwardly biased because studies done to date disregard the effect of population size on the range of services provided. Most data groups expenditures into broad categories such as ‘Fire Protection’ and ‘Parks and Recreation’. He argues that this aggregation may cloak a greater diversity of goods in larger population areas, making it easy to confound high per-capita spending with high per-capita costs.6 Oates cites an early study by Schmandt and Stevens (1960) on the diversity of goods provided across local jurisdictions, adducing it as evidence that a critical population size might be needed to support certain types of collective goods. This is what Oates calls the ‘zoo effect’.’ The following sections examine how indivisibility and congestion affect the choice between collective and private provision. In so doing conditions are described under which the aggregation bias and zoo-effect occur. Because the context of this inquiry is one of consumer preference that abstracts from provision mechanisms and, for the most part also ignores exclusion, I have reverted to Samuelson’s earlier terminology ‘collective goods’ to refer to public, club and all other shared goods. Instead of ‘private provision’, “Individual provision’ will be used as the complement of ‘collective provision’ since provision forms such as clubs are both private and collective. Provision through a club is thus grouped with provision through government and contrasted to individual provision. While the terminology is both ‘Romer and Rosenthal (1979) develop an interesting variant of this argument in which voters face the choice between inadequate present expenditure and a proposal for excessive public spending. ‘Gramlich and Rubinfeld (1982, p. 546) also find that aggregation distorts public service demand estimates. ‘On a similar tack, McMillan et al. (1981) suggested that measurement problems may arise from the existence of a discontinuous optimal facility size for local public goods. Some empirical support for the zoo effect was found by McMillan (1988).
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J.H.Y. Edwards, Indivisibility and preference for collective provision
broader and more precise, the results can be interpreted as applying to the more usual dichtomy between public and private goods. The analytical part of the paper begins in section 2 by examining the decision to purchase an indivisible good. Section 3 defines the conditions necessary for U-shaped cost of collective provision. Section 4 examines the choice between individual and collective provision of indivisible goods with U-shaped average cost.
2. Indivisibility
and the collective
provision of goods
Two assumptions are commonly implicit in the general theory of consumer preferences. All goods are assumed to be infinitely divisible and the marginal rate of substitution between any pair of goods is assumed to approach infinity (zero) as consumption approaches zero (infinity). Together with convexity of preferences and well known relational axioms these assumptions help guarantee interior solutions, a result under which some of all goods will be consumed, regardless of the relation of unit price to income. In fact the infinite divisibility assumption makes the definition of units and of unit price completely arbitrary. If the context is appropriate these are innocuous simplifying assumptions; they permit valuable insights into the process of consumer choice with minimum technical clutter, are a sine-qua-non for the use of calculus and simplify specification of empirical models. They are not necessary for the anlysis of consumer choice, and are by no means derived from it [see Debreu (1983)].8 If the context is not appropriate, this simplification may be a misguiding one. Consider the choice between collective and individual procurement of goods such as swimming pools, housing, education, day-care, transportation, museums and zoos. The divisibility assumption is less than helpful when indivisibility may be the essential characteristic affecting consumer choice.
2.1. Indivisibility und individuul provision In what follows I will formally introduce indivisibilities into the theory of choice between collective and individual provision, then develop the implications for goods with U-shaped average cost of provision curves. The issue addressed here is not whether collective provision is more or less socially efficient than individual, nor will it be asked whether collective provision is feasible. The maintained hypothesis is that collective provision is feasible, with individual cost and access rights clearly specified. The central question ‘For other early Samuelson (1942).
discussions
of
these
assumptions
see
Georgescu-Roegen
(1969)
and
J.H.Y. Edwards, Indivisibility and preference for collective prouision
T%bkil Definition of symbols. income
563
--
of the representative consumar
a composite individual good with unitary pr& a normal, indivisible D-good, directly produced with purchased inputs the norm&lc-good the number of persons with whom Z* is shared a matrix of group member characteristics which rrffsctperceived crowding a vector of characteristics which, along with group size, determine tax share the price per unit of Z the individual’s tax share the individual’s price per unit of Z*, where P* s tPZ/Z* the smallest amount of Z availab& on the market the smallest amount of Z* available, where Z: =Z*(N, Z,) the maximum affordable Z, where Z=NY,/P the maximum affordable Z*, where z* = Y,/P*
asked is: if some goods are indivisible would a representative consumer prefer collective or individual provision? Table 1 contains a summary of symbolic notation for reference. A good Z will be called ‘indivisible’ if it is unavailable below some minimum amount ‘Z,’ but can be augmented in infinitesimal increments for Z>Z,. Consumers face a choice between bundles (Z,X) where X is an infinitely divisible good. For expository simplicity it will be assumed that units of X are chosen in a way that ensures a unitary price. The representative consumer with income Ye has the budget X+ PZZ Y,, where ‘P’ is the price of the minimum amount Z,.9 The indivisibility of Z is the basis of two definitions. Definition
I (D.1). If Z,>O, there exists a Y, : Y,/P=Z,. be called an ‘individually unaffordable good’.
For
Y < Y,, Z will
Definition 2 (0.2). An indivisible good is said to be individually undesired if it is not unaffordable in the sense of (D.l) but it is also true that (0, Y,)R[Z,, ( Yo- PZ,)], where R represents the binary relation of ‘weak preference’. lo
‘Since units of Z are being defined as multiples of Z, perhaps it would be clearer to use the notation P, to denote unit price. To simplify notation however the ‘m’ subscript is omitted on P and later on P*. “This definition and subsequently definition (D.4) will hold for any fixed level of Z. That is, it will be true that for any pair (Z,, Y,) there is a price which will make Z, an undesired level of provision. If the good is perfectly divisible a quantity smaller than Z, will be consumed. That some level of Z is undesired has little economic significance unless consumption is constrained to that level. The purpose of including Z, in the detinitions of undesired goods is that indivisibility is such a constraint on the level of consumption.
564
J.H.):
Edwards, Indivisibility
and preference
for collective provision
X
Fig. 1: (a) Unaffordable
The situations and l(b).
described
by
private
(D.l)
good; (b) undesired
and
(D.2)
are
private
good
portrayed
in figs.
l(a)
2.2. Indivisibility and collective provision Bradford et al. (1968) distinguished between goods directly provided by the government, or ‘D-goods’ and the goods that are of direct concern to the citizenry, or ‘C-goods’. An example of a D-good is police squad cars, for which the level of safety perceived by citizens is the corresponding C-good.” Modifying their framework somewhat, we may envision the choice of buying Z (the D-good) as an individual good or paying for a contract Z*(N, QZ), for joint use of Z with N other people who have a set of characteristics described by the matrix Q. The contract provides for unrestricted access to and use of Z for all N users and spells out a cost-sharing “This
is analogous
to the distinction
between
‘inputs’ and ‘outputs’
in Shoup
(1989).
J.H.Y. Edwards, Indivisibility and preference for collective provision
___-_-_-____-_--_ set o(
dnlmd
-N
565
t
ollunallm
-----
N
Fig. 2.
Un~~a&
collective good.
scheme t(N,q), where q is a vector of contractually relevant characteristics.12 Let P* be the price that must be paid by the representative group member for a contract which specifies the minimum collective purchase Zz(N, Q, Z,), such that P*Zz = tPZ,. Thus P* -tPZ,/Zz.13 Note that the group also faces an indivisibility constraint but, since N is variable, the group member price P* is also variable and may be less than P. This suggests two definitions for collectively provided goods that closely parallel (D.l) and (D.2).14 Definition 3 (0.3). If Zz(N, Q, Z,)>O, there exists a Y,: Y,/P* = Zz(N,Q,Z,). Z* is collectively unaffordable if Y < Y, for the representative group member. Definition 4 (0.4). An indivisible good is collectively undesired if for the representative group member (0, Y,)R[Z*, (Y, - tPZ,)]. Fig. 2 illustrates the effect on an individual budget constraint of joining a group that shares an uncongestable good Z. Let z be defined as the most Z that can be afforded by an income-homogeneous group of size N, i.e., “-The share price t(N, q) could be anonymous, or ‘q’ could be defined to discriminate between agents’ willingness to pay, making t a Lindahl price. Of course, if communities are completely homogeneous this distinction becomes trite. 13For example, if Z* is a pure (uncongestable) public good and with an equal cost-sharing scheme t(N) = l/iv, then P* = P/N. 14For simplicity it is assumed that sharing groups are homogeneous in income.
566
J.H.Y.
Edwards,
Indiuisihility
and preference
for collective
provision
Z = N Y,/P. Z(N) is shown in the first quadrant of fig. 2. For groups N < N,, Z is collectively unaffordable in the sense of (D.3).
of size
3. Collective provision and U-shaped average cost More can be said about the choice between collective and individual provision if we make some simplifying assumptions about the congestion function Z*(N, Q,Z) and the cost-sharing scheme. The first assumption qualifies the relationship between Z*(N, Z) and t(N) such that P* is Ushaped as assumed in those models which follow Tiebout (1956) and Buchanan (1965). Secondly, it will be assumed that congestion is ‘anonymous’ in the sense of Scotchmer and Wooders (1987). Z* can then be written in the general form Z* = F(Z, N). The tradeoff between cost sharing and congestion is at the center of collective good models that follow Buchanan (1965) and Tiebout (1956). It is often assumed that such a tradeoff guarantees a U-shaped average cost of provision and hence a unique minimum average cost. However, these are not sufficient conditions. This section attempts to clarify the conditions under which the average cost of collective provision will be U-shaped. In the process it is shown that if U-shapedness is assumed, then further assumptions about the cost-sharing rule restrict the elasticity of congestion. Let Z* represent the C-good, Z the directly produced output and N the number of persons sharing Z. If the directly produced goods (Z) are purchased in a competitive market, collective purchases will have no effect on price. In this case the U-shaped average cost curve referred to in the Tiebout and Buchanan models must be that of the C-good (Z*) and not that of the D-good. If Z is shared with ‘N’ others an anonymous ‘congestion function’ of the form Z* =f(Z, N) can be defined as the transformation over N of Z into Z*. If N = 0, Z is procured individually and Z enters the utility function directly as does any other individual good. Therefore, f(Z,O)-Z. Edwards (1990) lays out four desirable properties for congestion functions. These are all satisfied by the family of functions of the general form Z* = Zg(N), where g’(N) < 0 for sufficiently large N. l5 Expenditures on Z by a representative individual are tPZ. If Z is acquired individually, t = 1. Expenditures on the congestable good can be expressed in terms of either Z or Z*, that is p*z*
= tPZ,
“This functional form satisfies the four ‘desirable’ properties for a congestion out in Edwards (1990). It is equivalent to assuming Hicks-neutral congestion consumption.
(1) function laid technology of
J.H.K Edwards, Indivisibility and preference for collective prouision
P* is then implicitly
defined
P* E t(N)PZ/Z*(Z,
561
as N).
(2)
‘Offsetting price and congestion effects’ are interpreted to mean t’(N) < 0 and 6Z*/6N 1, P* will be strictly increasing in N. Differentiating (2) and recalling that t is a function of group size N, we obtain
(3)
Recalling that P*(Z*/Z) =Pt, we can divide the left-hand first term and the right-hand side by Pt, obtaining 6P*
1
z
6N Pi Z* = Simplifying
side of (3) by the
1 P a 6N tP
(4)
(4) we see that
(5) Eq. (5) decomposes the percentage change in P* from changes in group size into its two effects: The percentage change in tax share (a price effect), and the percentage change in perceived quality/quantity of the good shared (a congestion effect). The price effect will always be negative, and SZ*/SN will be negative for large enough values of N, though it may be positive if there is a camaraderie effect, a sort of negative congestion [see Sandler and Tschirhardt (1980)]. When the negativity of the second term holds the two terms are offsetting. The necessary condition for U-shaped P* is
for small values of N, and that the inequality be reversed N = N, P* is minimized if SP*/dN =O, implying that
for N >m.
At
568
J.H.Y. Edwards, Indivisibility
and preference
for collective
t(R) =z*(R).
provision
(7)
Define the function R(N) as the absolute value of the ratio i/Z*. The Ushape of P* implies that R’(N)<0 and that there exists an fl such that R(R)= 1. U-shapedness requires that R(N) intersect R= 1 (P*=O) from above.“j Z is an individual good if R(1) < 1. It is interesting to note the interdependence between the tax scheme assumed and the congestion function. For example, the frequent equal-tax-shares assumption, t = l/N, restricts the congestion elasticity to equal - 1 when N =fi and implies that R(N) is the inverse of the congestion elasticity.
4. Indivisibility
and sharing with U-shaped average cost
A collective provision constraint is implied by the U-shape of P*. Sharing relaxes the constraint on consumer options imposed by indivisibility, but it does not completely eliminate the constraint. Note that the indivisibility of Z implies a minimum population size for collective provision. Zz only becomes collectively affordable at N, (see fig. 4), where Z* =Zz. Furthermore, congestion and the U-shape of P* impose a limit on the amount of Z* that can be obtained by increasing the size of the group that shares Z. Definition 5 (D.5). If P* is U-shaped, then associated price P*(n) is a maximum Z*(m) that can be collectively
with the minimum provided.
For uncongestable goods the collectively affordable amount of shared good is strictly increasing in N. Congestion drives a wedge between Z= N Y,/P and Z* = N Y,/P*. If P* is U-shaped, Z* will be shaped as an inverted U. Associated with the mininum P*(N) is a maximum Z*(N). This implies that increasing sharing group size above N will not be a feasible way of raising Z*. The function Z*(N) is shown in the first quadrant of fig. 3. Quadrant III shows the U-shaped P*(N), while quadrant II depicts the hyperbola Y,/P for O< P< co. Arrows in fig. 3 illustrate the derivation of Z* at the value N which minimizes P* (maximizes Z*). If all goods are perfectly divisible and exclusion is feasible congestion and tax-share alone determine preference for the form of provision. That is, P*
570
J.H.Y. Edwards, Indivisibility and preference for collective provision
x
Dewed
alternatives
budget
line for group
an
N
v,
Fig. 4. Collectively
affordable
and collectively
desired
congestable
goods.
condition for the existence of an indivisible goods market because Z* may be undesired (D.4) or even unaffordable (D.3) as a collective good. The existence of a collective market for any given group depends not on P* alone, but on the relation of P* to Y. Therefore if Z* is an undesired collective good for P* = P*(N,), Y= Y,. there is some Y, > Y, for which Z* is collectively desired. Conversely, if Y = Y, and P* is decreasing in N, there may be a sharing group of size N, >N,, P*(N,)
J.H.Y. Edwards, Indivisibility and preference for collective provision
571
Z, into Zz. The transformation Zz(N,Z,) is a deterioration in the perceived quality of Z, but also a ‘reduction’ in the minimum facility size. The ‘congestion wedge’, the distance between Z, and Zz(Z,,N), is therefore not all economic cost for it includes a range of facility size options which are not available to the individual (for example 11’in fig. 4). Proposition 1 suggests that as income and population grow with economic development, new goods will enter the public domain, as suggested by Schmandt and Stevens (1960) and Oates (1988). Note that if income for members in a group of size N, is less than Ye in fig. 4, Z* will be undesired. The public sector should in fact develop in a rather lumpy fashion if several Z-goods exist. With constant N, income growth would make Z* collectively affordable (D.4), then collectively desired (D.3), individually affordable (D.2) and individually desired (D.l). Even if Z-goods are individually desired, as long as P*(N,)P, one might suspect that group members were somehow coerced into collective consumption. The empirical finding Z* P) has been interpreted as evidence of voter exploitation by budget padding bureaucrats [for example, Niskanen (1971) and Borcherding (1977)]. If exclusion and replication are not possible group size may become ‘excessive’ for some goods in the sense that it causes the collective price to exceed the individual price for Z. This is a consequence of the collective provision constraint and inverted-U-shape of Z*. But should collective provision cease if P*(N) > P? Does continued collective provision then imply exploitation of group members as it might if Z were perfectly divisible? Proposition
2.
Suppose
that:
(a) preferences are complete, reflexive, transitive, continuous, monotonically increasing in X and Z*, and are convex over X and Z*; is positive, (b) P* is U-shaped and that the congestion function Zz(Z,,N) continuous and differentiable for N > 0, with 6Zz/6N ~0; (c) for income level Y0 and population N, there exists a Z: such that Z~(Z,,N)
Yo) and
CZ::(Y,-P*Z,*)IZCZ,,(Y,-PZ,)l.
If assumptions (a)-(c) hold, collective provision of an indivisible good Z will be preferred even if P*>P and Z is individually affordable and desired. Proof
Z,
is affordable
and individually
desired.
Proposition
2 requires
that
572
J.H.Y.
Edwards,
Indivisibility
and preference
for collective
provision
the indifference curve going through [Z,, ( Y0- PZ,)] intersect the collective budget constraint at a feasible level of provision. Preference for collective provision follows from the convexity of the indifference curves and the fact 0 that Z cannot be reduced below Z,,,.l’ Proposition 2 will hold for a broad class of well-behaved convex preferences. Proposition 2 suggests that when exclusion is not possible and collective goods are ‘over-congested’ (P* > P) there will be combinations of Y and N at which collective provision of indivisible goods is preferred. This is true even if individual provision is not undesired. Of course collective provision will also be preferred under the weaker condition that individual provision is undesired or unaffordable. Collective provision may be desired in spite of the higher unit price under over-congested sharing if such sharing lowers the minimum quantity constraint below what the individual faces. Even if the individual equivalent is affordable, a person may prefer access to a ‘smaller quantity’ of shared good though the unit price is higher. Proposition 2 highlights the fact that while unit-cost reduction is important, it is not the only reason for preferring collective provision. At levels of income that are low in relation to unit price ‘reducing’ the minimum facility size through some sharing arrangement will be the prevailing concern, A state satisfying Proposition 2 is illustrated by the lowest of the three indifference curves in fig. 5. Definition 6 (0.6). The upper collective provision threshold, if it exists, is defined as the population size N, below which Proposition 2 holds, but for which Z*(N, Z,) is collectively undesired for N 2 N,. Formally speaking, if Z is individually unaffordable with income Y,, N, from (D.6) is the least upper bound and N, which can be defined from (D.4) is the greatest lower bound of the set S= {no N : [Z*(n), Y, - P*Z*]R(O, Y,)} over which collective provision is preferred to none. The dependence of S on N and Y, suggests any classification of indivisible goods as collective or individual is impermanent. Consider the effect of income growth on a sharing group of fixed size. If communities are within S, they will only provide goods of type Z collectively, regardless of the level of congestion. Ceteris paribus, income growth causes vertical shifts in Z*, while “Debreu (1983, pp. 3637) has pointed out that to assume convex preferences is to assume they are continuous and therefore that ‘quantities of all commodities can be varied continuously’. No such assumption is made in (D.3) or (D.4). The concepts ‘unaffordable’ and ‘undesired’ do not depend on convexity. Proposition 2 does assume convex preferences and yet this does not contradict the assumption that Z is indivisible below Z,. Indivisibility arises as an imperfection, or technical constraint in the market for individual purchases. While convexity it is not empty of meaning over X and Z* as over X and Z may be hard to define for Z
J.H.Y
Edwards, Indivisibility and preference for collective provision
Fig. 5. Income
and choice between
collective
and private
513
provision
Z, (and therefore Zz) remain unchanged. Income growth broadens consumer options by increasing the range of feasible choices (the distance Z* -Zz). For sufficiently large changes in income, Z becomes affordable as an individual good. However, if communities are within the subset of S, G= {n E N : P*(n) < P}, goods of type Z will never be provided individually (regardless of the Y,/P ratio) because collective provision is cheaper. Proposition 3. If Proposition 2 holds for Y = YO, and Z is a normal good, then there exists an income level Y, > YO at which Z will cross the collective preference boundary. At this higher level of income, preference will revert from collective
to individual
provision.
Proof. From the definition of normality and the assumptions on preferences in Proposition 2 there exists a budget constraint defined by the income level Y, with an indifference curve tangent at bundle [Z,,( Ye- PZ,)]. If P* > P convexity requires that this indifference curve be above any that can be reached through collective consumption. 0 Propositions
2 and
3 are illustrated
in fig. 5. Setting
N=N,,
P*(N,)
>P
514
J.H.Y. Edwards, Indivisibility and preference for collectiue provision
and Y= Y,, Z is individually desired, but collective provision is preferred. At a higher income Yi, if Z is a normal good preference reverts to individual provision though P/P* remains unchanged. These propositions imply that there is a range of incomes for the subset H= {n : n ES and n q!G) over which collective provision is preferred even though P* BP, but that at sufficiently high levels of income preference will revert to individual provision. A sort of natural ‘sunset law’ emerges from examining preference for the provision of indivisible goods. The preference for collective (including public) over individual (including private) provision may cease in two ways. The first way is if the sharing group size grows above the collective provision threshold (D.6). For the second way to operate goods must be over congested and yet preferred as collective goods. At sufficiently high income levels the minimum size constraint will not be binding for individual choice and preference will switch to individual provision.
5. Conclusions Is there a class of goods that should always be collectively provided and another that should always be private? It is certainly true that some goods are most often found in the private sector and some are seldom (if ever) shared. Defense is nearly always collectively provided and sneakers nearly never. Yet there are wealthy people who hire guards (armies in the Philippines) and it is not unthinkable among the world’s poorest for friends or family members to share footwear. The foregoing examination of consumer preference indicates that the presence of indivisibility in congestable goods has three important implications: (1) The effect of population size and income on consumer preference between collective and individual provision of indivisible goods is such that even with uniform tastes we should not expect at one point in time to find all local governments, clubs, or other sharing groups such as households and cooperatives, providing the same kinds of goods. (2) Collective provision is not a permanent feature. As time passes, changes in income and sharing-group size will alter the individual/collective preference boundaries drawn by household, group, club or community members. The fact that a decision was once made to provide a good collectively does not mean it will always be preferable to continue doing so. (3) If indivisibility is of paramount importance in determining individual preference for collective provision the consequences for empirical work need to be carefully studied. Local public good demand estimates, for one, need to be reconsidered, since demand may not be a simple monotonic and continuous function of income, price and population.
J.H.Y
Edwards, Indivisibility and preference for collective provision
575
All this calls into question the unequivocal categorization of goods as collective or ‘private’. Individual preference for the form of provision will not depend only on the relation between individual and collective unit prices. Preference for the form of provision depends on a complex relation between prices, individual income, group income and sharing group size. As the relation between these variables changes preference for the form of provision may also change. Indivisibility imposes a constraint on minimum unit size and therefore a minimum expenditure. For finite income some indivisible goods will simply not be affordable. Sharing is one way to overcome the indivisibility constraint. The result that individual preference between individual and collective provision is impermanent suggests some interesting and testable hypotheses. For instance, among groups of the same size, richer groups should provide a different private/public mix than poor groups. The suggestion that wealthier groups might provide more of each good is not news. If some goods are indivisible, it is also not too surprising that wealthy groups might provide a greater diversity of goods. The new implication here is that some goods provided by poor groups may not be provided by wealthy ones. In fact, finding that groups with very high income provide fewer goods collectively would not be inconsistent with arguments developed here. Private clubs and local communities are more obvious candidates for examining the effect of indivisibility on collective choice. In the local public sector we might expect to see the wealthy leaving town as growth in their income makes the local goods mix (not only level) unappealing. In their thought-provoking test of the median voter and Tiebout hypotheses Gramlich and Rubinfeld (1992) compare results from regressions which use micro survey data with the more usual macro specification. They find that ‘(. . .) the macro income elasticity of about 0.4 is confirmed in the micro regressions. But virtually all of the positive elasticity is due to community income, with individual family income having a coefficient that is very close to zero in all three equations. Our results do suggest a positive income elasticity of demand for public spending, but the increased demand is seen to come in a very special form. As higherincome individuals within a community are surveyed, they do not appear to have any greater taste for public spending’ (p. 544). Their finding is consistent with the arguments presented in this paper. Though formal proof would require relaxing the assumed homogeneity of incomes and tastes, it seems intuitively clear in the indivisible goods framework that the most important determinant of individual preferences for collective provision is group, and not individual income. McMillan et al. (1981) and Oates (1988) have also suggested that indivisibility plays an important role in the mechanisms of collective good
516
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Edwards, Indivisibility and preference for collective provision
provision. Both the zoo effect and optimal facility size arguments that they put forth imply the issues of indivisibility and congestion are linked. To ignore indivisibility is to ignore an essential determinant of collective choice.
References Bergstrom, Theodore C. and Robert P. Goodman, 1973, Private demands for public goods, American Economic Review 63, 28G296. Borcherding, Thomas E., 1977, ed., Budgets and bureaucrats (Duke University Press, Durham, NC). Borcherding, Thomas E., and Robert T. Deacon, 1972, The demand for the services of nonfederal governments, American Economic Review 12, 891-901. Borcherding, T.E., W.C. Bush and R.M. Spann, 1977, The effects on public spending of the divisibility of public outputs in consumption, bureaucratic power, and the size of the taxsharing group, in: Borcherding (1977) 21 l-228. Bradford, D.F., R.A. Malt and W.E. Oates, 1968, The rising cost of local public services: Some evidence and reflections, National Tax Journal 22, 1855202. Brueckner, Jan K., 1981, Congested public goods: The case of tire protection, Journal of Public Economics 15,45-X Buchanan, James M., 1965, An economic theory of clubs, Economica 32, l-14. Buchanan, James M., 1975, The limits of liberty: Between anarchy and Leviathan, Chicago (University of Chicago Press, Chicago, IL). Cornes, R. and T. Sandier, 1986, The theory of externalities, public goods and club goods (Cambridge University Press, New York). Craig, Steven G., 1987, The impact of congestion on local public good production, Journal of Public Economics 32, 331-353. Debreu, Gerard, 1983, The coefficient of resource utilization in: Mathematical economics; twenty papers of Gerard Debreu (Cambridge University Press, New York). Dudley, Leonard, 1981, The demand for military expenditures: An international comparison, Public Choice 37, 5531. Edwards, John H.Y., 1986, A note on the publicness of local public goods: Evidence from New York municipalities, Canadian Journal of Economics 14, 5688573. Edwards, John H.Y., 1990, Congestion function specitication and the publicness of local public goods, Journal of Urban Economics 27, 8&96. Geogescu-Roegen, N., 1969, Revisiting Marshall’s constancy of marginal utility of money, Southern Economic Journal, 176181. Gramhch, Edward M., 1982, Models of excessive government spending: Do the facts support the theories?, in: Robert Haveman, ed., Public finance and public employment (Wayne State University Press, Detroit, MI). Gramlich, Edward M. and Dan L. Rubinfeld, 1982, Micro estimates of public spending demand functions and tests of the Tiebout and median voter hypotheses, Journal of Political Economy 90, 536560. McMillan, Melville L., 1988, On measuring congestion of local public goods, Journal of Urban Economics, forthcoming. McMillan. Melville L., W.R. Wilson and L.M. Arthur, 1981, The publicness of local public goods: Evidence from Ontario municipalities, Canadian Journal of Economics 14, 396433 Mueller. Dennis C., 1979. Public choice (Cambridge Universitv Press, New York). Niskanen, W.A., Jr., 1971. Bureaucracy and representative government (Adeline-Atherton, Chicago, IL). Oakland, William H., 1972, Congestion, public goods and welfare, Journal of Public Economics I, 229-375. Oakland, William H., 1985, Club goods in handbook of public economics, A.J. Auerbach and Martin Feldstein, eds. (North-Holland, New York).
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Indiuisihility
and preference
for collective
provision
571
Oates, Wallace E., 1988, On the measurement of congestion in the provision of local public goods, Journal of Urban Economics 24, 85594. Piganiol, B., 1969, Consommation, epargne et biens durables (Dunod, Paris). Pommerehne, W.W. and B.S. Frey, 1976, Two approaches to measuring public expenditures, Public Finance Quarterly 4, 3955407. Romer, T. and H. Rosenthal, 1979, Bureaucrats versus voters: On the political economy of resource allocation by direct democracy, The Quarterly Journal of Economics 94, 563-587. Rubinfeld, Daniel L., 1988, The economics of the local public sector: in: A.J. Auerbach and M. Feldstein, eds., Handbook of public economics (North-Holland, New York). Samuelson, Paul A. 1942, Constancy of the marginal utility of income, in: 0. Lange, F. McIntyre and T.O. Ynmeta, eds., Studies in mathematical economics and econometrics (University of Chicago Press, Chicago, IL). Samuelson, Paul A., 1954, The pure theory of public expenditure, Review of Economics and Statistics 36, 387-389. Samuelson, Paul A., 1955, Diagrammatic exposition of a theory of public expenditure, Review of Economics and Statistics 37, 35G356. Samuelson, Paul A., 1966, Pure theory of public expenditure and taxation, Paper presented at the meetings of the International Economics Association at Biarritz, Mimeo. Sandier, Todd and John T. Tschirhart, 1980, The economic theory of clubs: An evaluative survey, Journal of Economic Literature 18, 1481-1519. Schmandt, H.J. and G.R. Stephens, 1960, Measuring municipal output, National Tax Journal 13, 369-375. Scotchmer, Suzanne, 1985, Profit maximizing clubs, Journal of Public Economics 27, 2545. Scotchmer, Suzanne and M. Holtz Wooders, 1987, Competitive equilibrium and the core in club economies with anonymous crowding, Journal of Public Economics 34, 159-173. Shoup, Carl S., 1989, Rules for distributing a free government service among areas of a city, National Tax Journal 42, 1033122. Tiebout, Charles M., 1956, A pure theory of local expenditures, Journal of Political Economy 64, 416424. Tiebout, Charles M., 1961, An economic theory of fiscal decentralization, in: Public finances: Needs, sources and utilization, 79-96 (NBER, Princeton, NJ).
Science
and Urban
Indivisibility provision
Economics
22 (1992) 559-577.
and preference
North-Holland
for collective
John H.Y. Edwards* Tulane University, New Orleans LA, USA Received
February
1990, final version
received
March
1991
Collective goods are usually defined by comparing per-unit cost, net of congestion, under collective and individual provision. This paper examines preference for the form of provision of indivisible goods, arguing that the public/private or collective/individual dichotomy is an artifice of the assumption that shared goods are perfectly divisible. For indivisible goods, collective provision may be preferred when unit-cost is relatively higher and it may be undesired when unit-costs are lower than private; lower per-unit cost is neither a necessary, nor a sufficient condition. The model provides an explanation of Oates’ (Journal of Urban Economics, 1988, pp. 85-94) ‘zoo-effect’.
1. Introduction The literature on local public finance and the theory of clubs is built around two opposing effects of sharing which determine the net per-unit cost of collective provision. In these models congestion costs work against the reduction in pecuniary costs brought about by sharing. Collective goods, including public goods, are defined as the class of goods for which per-unit cost is lower for collective provision than for individual provision. This paper argues that the dichotomizing of goods into opposing private and public classes, classes distinguished by per-unit cost, is an artifice of the assumption that all goods are perfectly divisible. When indivisible goods are examined the &dings suggest that lower per-unit cost is neither a necessary nor a sufficient condition of preference for collective provision. In fact it is shown that collective provision may be preferred when the unit cost of collective provision is relatively higher than private per-unit costs and that collective provision may be undesired even when collective per-unit costs are
Correspondence Tilton Hall, New *I would like Kong-Pin Chen, helpful comments 01660462/92/$05.00
to: John H.Y. Edwards, Department of Economics, Tulane University, 206 Orleans, LA 70118, USA. to thank Todd Sandler, Marcus Berliant, Dan Rubinfeld, Nahid Aslanbeigui, Timothy Goodspeed, William Oakland, Wallace Oates and Jon Pritchett for on earlier drafts. Any errors are my responsibility. 0
1992-Elsevier
Science Publishers
B.V. All rights
reserved
560
J.H.Y.
Edwards,
Indivisibility
and preference
for
collective
provision
lower. These arguments provide a theoretical basis for the ‘zoo-effect’ described in Oates (1988). The sharp distinction between collective and private goods first arose in 1954 when Samuelson derived the now well-known conditions for efficient provision of pure public goods. Pure public goods are extreme in that ‘
. . . each individual’s consumption of such a good leads to no subtraction from any other individual’s consumption of that good, so that X,+j=X~+ j simultaneously for every ith individual and each collective consumption good’ [Samuelson (1954, pp. 388-389)].
Samuelson makes it clear that he intended, not to maintain the existence of these polar, or ‘pure’ cases, but to introduce this extreme assumption as a tool of analysis because ‘doctrinal history shows that theoretical insight often comes from considering strong or extreme cases’ [Samuelson (1955, p. 350)]. One consequence of this extreme assumption was the preference revelation dilemma that confounded the search for efficiency in the provision of collective output. By distinguishing local public goods and club goods from other collective goods, Tiebout (1956) and Buchanan (1965) both sought to rescue them from this condemnation to inefficient provision. Paradoxically, it is by re-introducing congestion costs that local public finance and clubtheory models seek market-like solutions. 1 The core of these models is the assumption that, while it reduces unit-price, sharing also causes congestion. These two effects of sharing offset one another in such a way as to produce U-shaped average cost curves for collective provision. Optimal group size and efficient provision have been identified with the minimum point on the average cost curves.’ These arguments have shown that it is not essential for shared goods to be of the ideal type conceived by Samuelson. However, much of the literature continues to operate on the purported public/private or collective/individual knife’s_edge3 dichotomy between immutable ‘classes’ of goods. In one of the best and most widely cited applications of the median voter model, Bergstrom and Goodman (1973) examined the congestion properties of police protection and parks. On the basis of high estimates of congestion they question the appropriateness of public provision (p. 293).4 Niskanen ‘A more complete discussion of the evolution of congestion as an economic concept can be found in Cornes and Sandier (1986). 2There are many discussions of the question of congestion and the existence of an optimal group size. Some good sources are Oakland (1972, 1983, Rubinfeld (1988), Cornes and Sandler (1986, especially chapters 5, IO, 11 and 16), Tiebout (1961) Mueller (1979) and Scotchmer (1985). ?3amuelson (1966) uses ‘knife’s_edge’ to describe the distinction. “Similar models with equivalent results can be found in Borcherding and Deacon (1972) Dudley (198 I), and Pommerehne and Frey (1976). More recently Bruckner (1981). Craig (1987) and, Edwards (1990) find the non-publicness result to be highly sensitive to congestion function specitication.
J.H.Y.
Edwards,
Indivisibility
and pre&wnce
for
collective
provision
561
(1971) and Borcherding et al. (1977) suspect there is evidence of voter exploitation in the fact that highly congested goods are provided publicly.5 It is my intention to show that such dichotomizing of goods into opposing private and public classes, distinguished by their cost per unit, is an artifice of the assumption that all goods are perfectly divisible. Piganiol (1969) contains some insightful comments on the ways private markets overcome indivisibilities. Among the mechanisms he lists are rentals, used goods, and credit. All of his solutions to the problem of indivisibility maintain the individual character of private goods markets. There also exist collective solutions to indivisibility that go beyond Piganiol’s list. Among these are clubs, public provision and informal sharing arrangements (roommates). In fact, it seems very likely that indivisibilities are present in most collectively provided goods. A recent study by Oates (1988) points in this direction. In discussing the empirical literature on congestion he suggests that the measurement of local good congestion may be upwardly biased because studies done to date disregard the effect of population size on the range of services provided. Most data groups expenditures into broad categories such as ‘Fire Protection’ and ‘Parks and Recreation’. He argues that this aggregation may cloak a greater diversity of goods in larger population areas, making it easy to confound high per-capita spending with high per-capita costs.6 Oates cites an early study by Schmandt and Stevens (1960) on the diversity of goods provided across local jurisdictions, adducing it as evidence that a critical population size might be needed to support certain types of collective goods. This is what Oates calls the ‘zoo effect’.’ The following sections examine how indivisibility and congestion affect the choice between collective and private provision. In so doing conditions are described under which the aggregation bias and zoo-effect occur. Because the context of this inquiry is one of consumer preference that abstracts from provision mechanisms and, for the most part also ignores exclusion, I have reverted to Samuelson’s earlier terminology ‘collective goods’ to refer to public, club and all other shared goods. Instead of ‘private provision’, “Individual provision’ will be used as the complement of ‘collective provision’ since provision forms such as clubs are both private and collective. Provision through a club is thus grouped with provision through government and contrasted to individual provision. While the terminology is both ‘Romer and Rosenthal (1979) develop an interesting variant of this argument in which voters face the choice between inadequate present expenditure and a proposal for excessive public spending. ‘Gramlich and Rubinfeld (1982, p. 546) also find that aggregation distorts public service demand estimates. ‘On a similar tack, McMillan et al. (1981) suggested that measurement problems may arise from the existence of a discontinuous optimal facility size for local public goods. Some empirical support for the zoo effect was found by McMillan (1988).
562
J.H.Y. Edwards, Indivisibility and preference for collective provision
broader and more precise, the results can be interpreted as applying to the more usual dichtomy between public and private goods. The analytical part of the paper begins in section 2 by examining the decision to purchase an indivisible good. Section 3 defines the conditions necessary for U-shaped cost of collective provision. Section 4 examines the choice between individual and collective provision of indivisible goods with U-shaped average cost.
2. Indivisibility
and the collective
provision of goods
Two assumptions are commonly implicit in the general theory of consumer preferences. All goods are assumed to be infinitely divisible and the marginal rate of substitution between any pair of goods is assumed to approach infinity (zero) as consumption approaches zero (infinity). Together with convexity of preferences and well known relational axioms these assumptions help guarantee interior solutions, a result under which some of all goods will be consumed, regardless of the relation of unit price to income. In fact the infinite divisibility assumption makes the definition of units and of unit price completely arbitrary. If the context is appropriate these are innocuous simplifying assumptions; they permit valuable insights into the process of consumer choice with minimum technical clutter, are a sine-qua-non for the use of calculus and simplify specification of empirical models. They are not necessary for the anlysis of consumer choice, and are by no means derived from it [see Debreu (1983)].8 If the context is not appropriate, this simplification may be a misguiding one. Consider the choice between collective and individual procurement of goods such as swimming pools, housing, education, day-care, transportation, museums and zoos. The divisibility assumption is less than helpful when indivisibility may be the essential characteristic affecting consumer choice.
2.1. Indivisibility und individuul provision In what follows I will formally introduce indivisibilities into the theory of choice between collective and individual provision, then develop the implications for goods with U-shaped average cost of provision curves. The issue addressed here is not whether collective provision is more or less socially efficient than individual, nor will it be asked whether collective provision is feasible. The maintained hypothesis is that collective provision is feasible, with individual cost and access rights clearly specified. The central question ‘For other early Samuelson (1942).
discussions
of
these
assumptions
see
Georgescu-Roegen
(1969)
and
J.H.Y. Edwards, Indivisibility and preference for collective prouision
T%bkil Definition of symbols. income
563
--
of the representative consumar
a composite individual good with unitary pr& a normal, indivisible D-good, directly produced with purchased inputs the norm&lc-good the number of persons with whom Z* is shared a matrix of group member characteristics which rrffsctperceived crowding a vector of characteristics which, along with group size, determine tax share the price per unit of Z the individual’s tax share the individual’s price per unit of Z*, where P* s tPZ/Z* the smallest amount of Z availab& on the market the smallest amount of Z* available, where Z: =Z*(N, Z,) the maximum affordable Z, where Z=NY,/P the maximum affordable Z*, where z* = Y,/P*
asked is: if some goods are indivisible would a representative consumer prefer collective or individual provision? Table 1 contains a summary of symbolic notation for reference. A good Z will be called ‘indivisible’ if it is unavailable below some minimum amount ‘Z,’ but can be augmented in infinitesimal increments for Z>Z,. Consumers face a choice between bundles (Z,X) where X is an infinitely divisible good. For expository simplicity it will be assumed that units of X are chosen in a way that ensures a unitary price. The representative consumer with income Ye has the budget X+ PZZ Y,, where ‘P’ is the price of the minimum amount Z,.9 The indivisibility of Z is the basis of two definitions. Definition
I (D.1). If Z,>O, there exists a Y, : Y,/P=Z,. be called an ‘individually unaffordable good’.
For
Y < Y,, Z will
Definition 2 (0.2). An indivisible good is said to be individually undesired if it is not unaffordable in the sense of (D.l) but it is also true that (0, Y,)R[Z,, ( Yo- PZ,)], where R represents the binary relation of ‘weak preference’. lo
‘Since units of Z are being defined as multiples of Z, perhaps it would be clearer to use the notation P, to denote unit price. To simplify notation however the ‘m’ subscript is omitted on P and later on P*. “This definition and subsequently definition (D.4) will hold for any fixed level of Z. That is, it will be true that for any pair (Z,, Y,) there is a price which will make Z, an undesired level of provision. If the good is perfectly divisible a quantity smaller than Z, will be consumed. That some level of Z is undesired has little economic significance unless consumption is constrained to that level. The purpose of including Z, in the detinitions of undesired goods is that indivisibility is such a constraint on the level of consumption.
564
J.H.):
Edwards, Indivisibility
and preference
for collective provision
X
Fig. 1: (a) Unaffordable
The situations and l(b).
described
by
private
(D.l)
good; (b) undesired
and
(D.2)
are
private
good
portrayed
in figs.
l(a)
2.2. Indivisibility and collective provision Bradford et al. (1968) distinguished between goods directly provided by the government, or ‘D-goods’ and the goods that are of direct concern to the citizenry, or ‘C-goods’. An example of a D-good is police squad cars, for which the level of safety perceived by citizens is the corresponding C-good.” Modifying their framework somewhat, we may envision the choice of buying Z (the D-good) as an individual good or paying for a contract Z*(N, QZ), for joint use of Z with N other people who have a set of characteristics described by the matrix Q. The contract provides for unrestricted access to and use of Z for all N users and spells out a cost-sharing “This
is analogous
to the distinction
between
‘inputs’ and ‘outputs’
in Shoup
(1989).
J.H.Y. Edwards, Indivisibility and preference for collective provision
___-_-_-____-_--_ set o(
dnlmd
-N
565
t
ollunallm
-----
N
Fig. 2.
Un~~a&
collective good.
scheme t(N,q), where q is a vector of contractually relevant characteristics.12 Let P* be the price that must be paid by the representative group member for a contract which specifies the minimum collective purchase Zz(N, Q, Z,), such that P*Zz = tPZ,. Thus P* -tPZ,/Zz.13 Note that the group also faces an indivisibility constraint but, since N is variable, the group member price P* is also variable and may be less than P. This suggests two definitions for collectively provided goods that closely parallel (D.l) and (D.2).14 Definition 3 (0.3). If Zz(N, Q, Z,)>O, there exists a Y,: Y,/P* = Zz(N,Q,Z,). Z* is collectively unaffordable if Y < Y, for the representative group member. Definition 4 (0.4). An indivisible good is collectively undesired if for the representative group member (0, Y,)R[Z*, (Y, - tPZ,)]. Fig. 2 illustrates the effect on an individual budget constraint of joining a group that shares an uncongestable good Z. Let z be defined as the most Z that can be afforded by an income-homogeneous group of size N, i.e., “-The share price t(N, q) could be anonymous, or ‘q’ could be defined to discriminate between agents’ willingness to pay, making t a Lindahl price. Of course, if communities are completely homogeneous this distinction becomes trite. 13For example, if Z* is a pure (uncongestable) public good and with an equal cost-sharing scheme t(N) = l/iv, then P* = P/N. 14For simplicity it is assumed that sharing groups are homogeneous in income.
566
J.H.Y.
Edwards,
Indiuisihility
and preference
for collective
provision
Z = N Y,/P. Z(N) is shown in the first quadrant of fig. 2. For groups N < N,, Z is collectively unaffordable in the sense of (D.3).
of size
3. Collective provision and U-shaped average cost More can be said about the choice between collective and individual provision if we make some simplifying assumptions about the congestion function Z*(N, Q,Z) and the cost-sharing scheme. The first assumption qualifies the relationship between Z*(N, Z) and t(N) such that P* is Ushaped as assumed in those models which follow Tiebout (1956) and Buchanan (1965). Secondly, it will be assumed that congestion is ‘anonymous’ in the sense of Scotchmer and Wooders (1987). Z* can then be written in the general form Z* = F(Z, N). The tradeoff between cost sharing and congestion is at the center of collective good models that follow Buchanan (1965) and Tiebout (1956). It is often assumed that such a tradeoff guarantees a U-shaped average cost of provision and hence a unique minimum average cost. However, these are not sufficient conditions. This section attempts to clarify the conditions under which the average cost of collective provision will be U-shaped. In the process it is shown that if U-shapedness is assumed, then further assumptions about the cost-sharing rule restrict the elasticity of congestion. Let Z* represent the C-good, Z the directly produced output and N the number of persons sharing Z. If the directly produced goods (Z) are purchased in a competitive market, collective purchases will have no effect on price. In this case the U-shaped average cost curve referred to in the Tiebout and Buchanan models must be that of the C-good (Z*) and not that of the D-good. If Z is shared with ‘N’ others an anonymous ‘congestion function’ of the form Z* =f(Z, N) can be defined as the transformation over N of Z into Z*. If N = 0, Z is procured individually and Z enters the utility function directly as does any other individual good. Therefore, f(Z,O)-Z. Edwards (1990) lays out four desirable properties for congestion functions. These are all satisfied by the family of functions of the general form Z* = Zg(N), where g’(N) < 0 for sufficiently large N. l5 Expenditures on Z by a representative individual are tPZ. If Z is acquired individually, t = 1. Expenditures on the congestable good can be expressed in terms of either Z or Z*, that is p*z*
= tPZ,
“This functional form satisfies the four ‘desirable’ properties for a congestion out in Edwards (1990). It is equivalent to assuming Hicks-neutral congestion consumption.
(1) function laid technology of
J.H.K Edwards, Indivisibility and preference for collective prouision
P* is then implicitly
defined
P* E t(N)PZ/Z*(Z,
561
as N).
(2)
‘Offsetting price and congestion effects’ are interpreted to mean t’(N) < 0 and 6Z*/6N
(3)
Recalling that P*(Z*/Z) =Pt, we can divide the left-hand first term and the right-hand side by Pt, obtaining 6P*
1
z
6N Pi Z* = Simplifying
side of (3) by the
1 P a 6N tP
(4)
(4) we see that
(5) Eq. (5) decomposes the percentage change in P* from changes in group size into its two effects: The percentage change in tax share (a price effect), and the percentage change in perceived quality/quantity of the good shared (a congestion effect). The price effect will always be negative, and SZ*/SN will be negative for large enough values of N, though it may be positive if there is a camaraderie effect, a sort of negative congestion [see Sandler and Tschirhardt (1980)]. When the negativity of the second term holds the two terms are offsetting. The necessary condition for U-shaped P* is
for small values of N, and that the inequality be reversed N = N, P* is minimized if SP*/dN =O, implying that
for N >m.
At
568
J.H.Y. Edwards, Indivisibility
and preference
for collective
t(R) =z*(R).
provision
(7)
Define the function R(N) as the absolute value of the ratio i/Z*. The Ushape of P* implies that R’(N)<0 and that there exists an fl such that R(R)= 1. U-shapedness requires that R(N) intersect R= 1 (P*=O) from above.“j Z is an individual good if R(1) < 1. It is interesting to note the interdependence between the tax scheme assumed and the congestion function. For example, the frequent equal-tax-shares assumption, t = l/N, restricts the congestion elasticity to equal - 1 when N =fi and implies that R(N) is the inverse of the congestion elasticity.
4. Indivisibility
and sharing with U-shaped average cost
A collective provision constraint is implied by the U-shape of P*. Sharing relaxes the constraint on consumer options imposed by indivisibility, but it does not completely eliminate the constraint. Note that the indivisibility of Z implies a minimum population size for collective provision. Zz only becomes collectively affordable at N, (see fig. 4), where Z* =Zz. Furthermore, congestion and the U-shape of P* impose a limit on the amount of Z* that can be obtained by increasing the size of the group that shares Z. Definition 5 (D.5). If P* is U-shaped, then associated price P*(n) is a maximum Z*(m) that can be collectively
with the minimum provided.
For uncongestable goods the collectively affordable amount of shared good is strictly increasing in N. Congestion drives a wedge between Z= N Y,/P and Z* = N Y,/P*. If P* is U-shaped, Z* will be shaped as an inverted U. Associated with the mininum P*(N) is a maximum Z*(N). This implies that increasing sharing group size above N will not be a feasible way of raising Z*. The function Z*(N) is shown in the first quadrant of fig. 3. Quadrant III shows the U-shaped P*(N), while quadrant II depicts the hyperbola Y,/P for O< P< co. Arrows in fig. 3 illustrate the derivation of Z* at the value N which minimizes P* (maximizes Z*). If all goods are perfectly divisible and exclusion is feasible congestion and tax-share alone determine preference for the form of provision. That is, P*
P) is the necessary and sufficient condition to prefer collective (private) provision. In fact it is this idea that more of the good can be had ibIt can also be seen that the requirement for Samuelson pure public goods is not Z*=Z, for all N. as usually stated. but that R(N)> 1 for all N. The distinction is that R(N) may be downward sloping (Z congestable) as long as it approaches R = 1 asymptotically. In the- real world where oonulation size N, is finite, it is sufftcient for fi > N,. More complete discussions of the relation between N, and ailocative efficiency can be found in” Mueller (1979) and Scotchmer (1985).
J.H.Y
Edwards, Indivisibility and preference for collective provision
569
Fii. 3. Collective provision constraint.
one way than the other which justifies classifying goods as collective or private. The following propositions jointly argue for a more flexible classification scheme. Proposition 1 shows that gains from sharing (tax cost falls ‘more’ than congestion costs) are insuffkient by themselves to guarantee collective provision. Proposition 2 considers the case P* > P and shows that ‘losses’ from sharing (tax savings are less than congestion costs) are insufficient by themselves to guarantee private provision. Proposition 3 states that if there is over-crowding preference for sharing will cease at sufliciently high income. Proposition 1. (i) Even if P*
growth
on
The proof follows from (D.3) and (D.4).
Indivisibility imposes a minimum consumption constraint, namely Z* 2 Zz, even when goods are shared. Even though sharing ‘reduces’ the minimum size constraint, in the sense that Zz < Z,, P*
570
J.H.Y. Edwards, Indivisibility and preference for collective provision
x
Dewed
alternatives
budget
line for group
an
N
v,
Fig. 4. Collectively
affordable
and collectively
desired
congestable
goods.
condition for the existence of an indivisible goods market because Z* may be undesired (D.4) or even unaffordable (D.3) as a collective good. The existence of a collective market for any given group depends not on P* alone, but on the relation of P* to Y. Therefore if Z* is an undesired collective good for P* = P*(N,), Y= Y,. there is some Y, > Y, for which Z* is collectively desired. Conversely, if Y = Y, and P* is decreasing in N, there may be a sharing group of size N, >N,, P*(N,)
J.H.Y. Edwards, Indivisibility and preference for collective provision
571
Z, into Zz. The transformation Zz(N,Z,) is a deterioration in the perceived quality of Z, but also a ‘reduction’ in the minimum facility size. The ‘congestion wedge’, the distance between Z, and Zz(Z,,N), is therefore not all economic cost for it includes a range of facility size options which are not available to the individual (for example 11’in fig. 4). Proposition 1 suggests that as income and population grow with economic development, new goods will enter the public domain, as suggested by Schmandt and Stevens (1960) and Oates (1988). Note that if income for members in a group of size N, is less than Ye in fig. 4, Z* will be undesired. The public sector should in fact develop in a rather lumpy fashion if several Z-goods exist. With constant N, income growth would make Z* collectively affordable (D.4), then collectively desired (D.3), individually affordable (D.2) and individually desired (D.l). Even if Z-goods are individually desired, as long as P*(N,)
2.
Suppose
that:
(a) preferences are complete, reflexive, transitive, continuous, monotonically increasing in X and Z*, and are convex over X and Z*; is positive, (b) P* is U-shaped and that the congestion function Zz(Z,,N) continuous and differentiable for N > 0, with 6Zz/6N ~0; (c) for income level Y0 and population N, there exists a Z: such that Z~(Z,,N)
Yo) and
CZ::(Y,-P*Z,*)IZCZ,,(Y,-PZ,)l.
If assumptions (a)-(c) hold, collective provision of an indivisible good Z will be preferred even if P*>P and Z is individually affordable and desired. Proof
Z,
is affordable
and individually
desired.
Proposition
2 requires
that
572
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Indivisibility
and preference
for collective
provision
the indifference curve going through [Z,, ( Y0- PZ,)] intersect the collective budget constraint at a feasible level of provision. Preference for collective provision follows from the convexity of the indifference curves and the fact 0 that Z cannot be reduced below Z,,,.l’ Proposition 2 will hold for a broad class of well-behaved convex preferences. Proposition 2 suggests that when exclusion is not possible and collective goods are ‘over-congested’ (P* > P) there will be combinations of Y and N at which collective provision of indivisible goods is preferred. This is true even if individual provision is not undesired. Of course collective provision will also be preferred under the weaker condition that individual provision is undesired or unaffordable. Collective provision may be desired in spite of the higher unit price under over-congested sharing if such sharing lowers the minimum quantity constraint below what the individual faces. Even if the individual equivalent is affordable, a person may prefer access to a ‘smaller quantity’ of shared good though the unit price is higher. Proposition 2 highlights the fact that while unit-cost reduction is important, it is not the only reason for preferring collective provision. At levels of income that are low in relation to unit price ‘reducing’ the minimum facility size through some sharing arrangement will be the prevailing concern, A state satisfying Proposition 2 is illustrated by the lowest of the three indifference curves in fig. 5. Definition 6 (0.6). The upper collective provision threshold, if it exists, is defined as the population size N, below which Proposition 2 holds, but for which Z*(N, Z,) is collectively undesired for N 2 N,. Formally speaking, if Z is individually unaffordable with income Y,, N, from (D.6) is the least upper bound and N, which can be defined from (D.4) is the greatest lower bound of the set S= {no N : [Z*(n), Y, - P*Z*]R(O, Y,)} over which collective provision is preferred to none. The dependence of S on N and Y, suggests any classification of indivisible goods as collective or individual is impermanent. Consider the effect of income growth on a sharing group of fixed size. If communities are within S, they will only provide goods of type Z collectively, regardless of the level of congestion. Ceteris paribus, income growth causes vertical shifts in Z*, while “Debreu (1983, pp. 3637) has pointed out that to assume convex preferences is to assume they are continuous and therefore that ‘quantities of all commodities can be varied continuously’. No such assumption is made in (D.3) or (D.4). The concepts ‘unaffordable’ and ‘undesired’ do not depend on convexity. Proposition 2 does assume convex preferences and yet this does not contradict the assumption that Z is indivisible below Z,. Indivisibility arises as an imperfection, or technical constraint in the market for individual purchases. While convexity it is not empty of meaning over X and Z* as over X and Z may be hard to define for Z
J.H.Y
Edwards, Indivisibility and preference for collective provision
Fig. 5. Income
and choice between
collective
and private
513
provision
Z, (and therefore Zz) remain unchanged. Income growth broadens consumer options by increasing the range of feasible choices (the distance Z* -Zz). For sufficiently large changes in income, Z becomes affordable as an individual good. However, if communities are within the subset of S, G= {n E N : P*(n) < P}, goods of type Z will never be provided individually (regardless of the Y,/P ratio) because collective provision is cheaper. Proposition 3. If Proposition 2 holds for Y = YO, and Z is a normal good, then there exists an income level Y, > YO at which Z will cross the collective preference boundary. At this higher level of income, preference will revert from collective
to individual
provision.
Proof. From the definition of normality and the assumptions on preferences in Proposition 2 there exists a budget constraint defined by the income level Y, with an indifference curve tangent at bundle [Z,,( Ye- PZ,)]. If P* > P convexity requires that this indifference curve be above any that can be reached through collective consumption. 0 Propositions
2 and
3 are illustrated
in fig. 5. Setting
N=N,,
P*(N,)
>P
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J.H.Y. Edwards, Indivisibility and preference for collectiue provision
and Y= Y,, Z is individually desired, but collective provision is preferred. At a higher income Yi, if Z is a normal good preference reverts to individual provision though P/P* remains unchanged. These propositions imply that there is a range of incomes for the subset H= {n : n ES and n q!G) over which collective provision is preferred even though P* BP, but that at sufficiently high levels of income preference will revert to individual provision. A sort of natural ‘sunset law’ emerges from examining preference for the provision of indivisible goods. The preference for collective (including public) over individual (including private) provision may cease in two ways. The first way is if the sharing group size grows above the collective provision threshold (D.6). For the second way to operate goods must be over congested and yet preferred as collective goods. At sufficiently high income levels the minimum size constraint will not be binding for individual choice and preference will switch to individual provision.
5. Conclusions Is there a class of goods that should always be collectively provided and another that should always be private? It is certainly true that some goods are most often found in the private sector and some are seldom (if ever) shared. Defense is nearly always collectively provided and sneakers nearly never. Yet there are wealthy people who hire guards (armies in the Philippines) and it is not unthinkable among the world’s poorest for friends or family members to share footwear. The foregoing examination of consumer preference indicates that the presence of indivisibility in congestable goods has three important implications: (1) The effect of population size and income on consumer preference between collective and individual provision of indivisible goods is such that even with uniform tastes we should not expect at one point in time to find all local governments, clubs, or other sharing groups such as households and cooperatives, providing the same kinds of goods. (2) Collective provision is not a permanent feature. As time passes, changes in income and sharing-group size will alter the individual/collective preference boundaries drawn by household, group, club or community members. The fact that a decision was once made to provide a good collectively does not mean it will always be preferable to continue doing so. (3) If indivisibility is of paramount importance in determining individual preference for collective provision the consequences for empirical work need to be carefully studied. Local public good demand estimates, for one, need to be reconsidered, since demand may not be a simple monotonic and continuous function of income, price and population.
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575
All this calls into question the unequivocal categorization of goods as collective or ‘private’. Individual preference for the form of provision will not depend only on the relation between individual and collective unit prices. Preference for the form of provision depends on a complex relation between prices, individual income, group income and sharing group size. As the relation between these variables changes preference for the form of provision may also change. Indivisibility imposes a constraint on minimum unit size and therefore a minimum expenditure. For finite income some indivisible goods will simply not be affordable. Sharing is one way to overcome the indivisibility constraint. The result that individual preference between individual and collective provision is impermanent suggests some interesting and testable hypotheses. For instance, among groups of the same size, richer groups should provide a different private/public mix than poor groups. The suggestion that wealthier groups might provide more of each good is not news. If some goods are indivisible, it is also not too surprising that wealthy groups might provide a greater diversity of goods. The new implication here is that some goods provided by poor groups may not be provided by wealthy ones. In fact, finding that groups with very high income provide fewer goods collectively would not be inconsistent with arguments developed here. Private clubs and local communities are more obvious candidates for examining the effect of indivisibility on collective choice. In the local public sector we might expect to see the wealthy leaving town as growth in their income makes the local goods mix (not only level) unappealing. In their thought-provoking test of the median voter and Tiebout hypotheses Gramlich and Rubinfeld (1992) compare results from regressions which use micro survey data with the more usual macro specification. They find that ‘(. . .) the macro income elasticity of about 0.4 is confirmed in the micro regressions. But virtually all of the positive elasticity is due to community income, with individual family income having a coefficient that is very close to zero in all three equations. Our results do suggest a positive income elasticity of demand for public spending, but the increased demand is seen to come in a very special form. As higherincome individuals within a community are surveyed, they do not appear to have any greater taste for public spending’ (p. 544). Their finding is consistent with the arguments presented in this paper. Though formal proof would require relaxing the assumed homogeneity of incomes and tastes, it seems intuitively clear in the indivisible goods framework that the most important determinant of individual preferences for collective provision is group, and not individual income. McMillan et al. (1981) and Oates (1988) have also suggested that indivisibility plays an important role in the mechanisms of collective good
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provision. Both the zoo effect and optimal facility size arguments that they put forth imply the issues of indivisibility and congestion are linked. To ignore indivisibility is to ignore an essential determinant of collective choice.
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