A case for private provision but collective ownership of public goods.

Introduction

Since Samuelson's two seminal papers (1954, 1955) on public goods, it generally is accepted that private provision of public goods under competitive conditions is Pareto inefficient. With perfect competition, underproduction of public goods occurs. This is due to free tiding behavior where individuals have an incentive to understate their true preferences for public goods.(1) Recent game theoretic laboratory experiments generally provide support for Samuelson's concern with free riding behavior. Despite a considerable variation in paradigms, experiments involving voluntary (or private) provision of public goods consistently reveal the presence of free riding behavior. See, for example, Marwell and Ames (1981); Kim and Walker (1984); Isaac and Walker (1988); Andreoni (1988); Isaac, Walker, and Williams (1994); Weimann (1994), and Andreoni (1995).

Given difficulties with private provision, it does not follow that government, by default, should provide public goods. Efficiency considerations are important here; too. It generally is understood that Pareto results occur under government provision if Lindahl (1958) prices (or tax shares) are assessed. The same free riding behavior associated with private provision, however, militates against derivation of Lindahl tax-shares required for effectuating this outcome.

More recent use of game theory models has not resolved this public good pricing dilemma, which Hurwicz (1972) describes as an incentive-compatibility problem. The difficulty is not one of constructing models which have Lindahl-compatible Nash equilibria. Mechanisms in the public good models of Hurwicz (1979) and Walker (1981), for example, have Nash equilibria with Lindahl tax-shares for consumers (or players). Incentive compatibility problems remain, however, in the form of Nash equilibria which accommodate free tiding behavior in the sense of agents playing from "disguised" utility functions. Bailey (1994) argues that this is endemic to game-theoretic models with public goods.

The positive externality which encourages free riding behavior derives from the non-exclusion feature of public goods. A strategy for generating Pareto results in a model with public goods is to establish an environment conducive to internalizing this externality. That is what the model presented here does.

The central institutional feature of the model is the separation of provision and ownership of public goods. This dichotomy has two important implications for modeling public goods provision. First, it obviates the condition that all economic agents truthfully reveal their preferences to the government (or to others). This cumbersome requirement is a common feature in public goods models from Lindahl's (1958) pioneering work to more recent demand-revealing models such as those by Groves and Ledyard (1977) and Groves (1979).

Second, it renders unnecessary another feature common to public good models: the calculation and assessment of a tax (or tax-price) to finance the public good. With private provision and collective ownership, prices (and quantities) of public goods are determined through the voluntary exchange activities of consumers. The focus here is not on optimal voluntary tax shares with government provision, but rather on a set of negotiated market prices which lead to optimal private provision of public goods.

Private Provision And Collective Ownership

Provision of a good is determined by who purchases the good. One purchased by individuals is privately provided; a good purchased by government, publicly provided. Here a two good economy is assumed: [X.sub.1] is a private good and [X.sub.2] a public good. Both goods are privately produced under conditions of perfect competition. Because both are purchased by individuals, there is private provision of the public good.

While provision is private, ownership of the public good is collective. This necessitates an institutional arrangement to facilitate collective payment for that good. Such an arrangement, established through voluntary association, includes both a Processing Agent and a set of rules describing the mechanics of purchases and sales.(2) The Processing Agent must have the authority to do the following: 1) collect funds from and disburse funds to economic agents to finance purchases and sales of the public good; and, 2) collect, hold, and transfer title to the public good.

Rules governing purchases and sales assume the following form. For purchases, all n consumers pay an equal share for each unit of the public good acquired. Purchase of a unit of [X.sub.2] by an individual consumer necessitates that each of the other n - 1 consumers remit (through the Processing Agent) a payment to the buyer in the amount of (l/n) X [P.sub.2]. The aggregate rebate for the purchaser is [(n - 1)/n] x [P.sub.2]. Thus, while the purchase price of [X.sub.2] is [P.sub.2], the post-rebate price for the purchaser of [X.sub.2] (and all other consumers) is (l/n) x [P.sub.2].

Collective ownership of [X.sub.2] implies that all n consumers have ownership rights. These ownership rights are such that individuals can sell [X.sub.2] at their own discretion - including units of [X.sub.2] which they did not directly purchase.(3) While liquidation of [X.sub.2] is undertaken by individuals, proceeds from liquidations are distributed in accordance with the consumer's share of the purchase price.

For equilibrium to occur, all individuals must choose to consume an identical amount of [X.sub.2]. Where differences occur, "veto power" resides with those wanting fewer units of the public good. If one individual, for example, purchases an additional unit of [X.sub.2], a second individual can negate that purchase by selling that unit of [X.sub.2].

Market transactions involving compensation are a means of resolving such differences. Individuals desiring more of the public good compensate those wanting less. Those receiving compensation agree not to negate the purchase of marginal units of the public good. Compensation occurs until an equilibrium amount of [X.sub.2] is determined, i.e., when all consuming units are satisfied with the amount of [X.sub.2] consumed.

Compensation payments change the consuming unit's effective price for the marginal unit of [X.sub.2]. Let [[Alpha].sub.1], [[Alpha].sub.2], .... [[Alpha].sub.n] represent the post-compensation shares of [P.sub.2] paid by consumers 1,2,3,...n for the marginal unit of [X.sub.2].(4) The effective price paid by the ith consumer for the marginal unit now is ([[Alpha].sub.i] x [P.sub.2]). Effective prices differ from (1/n) x [P.sub.2] by the amount of any compensation paid or received. For the jth consumer making compensation payments, [[Alpha].sub.j] x [P.sub.2] [greater than] (l/n) x [P.sub.2], while [[Alpha].sub.k] x [P.sub.2] [less than] (l/n) x [P.sub.2] for the kth consuming unit receiving compensation.

The motivation for individual consumers to participate in such an arrangement is twofold. The first is an understanding of the nature of the public good, i.e., that all benefit from the good no matter who buys it. The second is that all payments for the public good are voluntary. Each individual consumer has the option of disposing of undesired units of the public good.(5)

General Equilibrium 

Given such an environment with private provision but collective ownership of public goods, the ith optimizing consumer selects equilibrium quantities of [X.sub.1] and [X.sub.2] subject to its wealth constraint and the relative-share constraint for financing the public good. Because the consuming unit may not pay the same price for every unit of the public good, an average price is entered in the wealth constraint. Mathematically, the ith consuming unit maximizes the following Lagrangian function.

[Mathematical Expression Omitted] (1)

Given the likely outcome that [Alpha] is not identical for every consumer, the equilibrium value of the marginal rate of substitution of [X.sub.2] for [X.sub.1] is not the same for all consumers. For the ith consumer,

[Mathematical Expression Omitted] (2)

The nonexclusion feature of the public good implies one must sum the marginal rates of substitution to obtain the amount of [X.sub.1] that all consumers are willing to sacrifice in order to obtain the marginal unit of [X.sub.2]. With [Sigma] [[Alpha].sub.i] = 1 (where i = 1,2,...n),

[Sigma] [([MRS.sub.X2,X1]).sup.i] = [P.sub.2]/[P.sub.1]. (3)

i = 1,2,...n

In competitive production equilibrium, the marginal rate of transformation of [X.sub.1] into [X.sub.2] ([MRT.sub.X1,X2]) also is equal to [P.sub.2][/.sub.P].. Therefore,

[Sigma] [([MRS.sub.X2,X1]).sup.i] = [MRT.sub.X1,X2]. (4)

i = 1,2,...n

This establishes that an environment characterized by private provision and collective ownership of the public good leads to a Pareto optimal provision of that good. In addition, the price paid by each consumer for the marginal unit of the public good is a Lindahl price. This is confirmed as follows. The equilibrium value of [MRS.sub.X2,X1] for the ith consumer is equal to [Mathematical Expression Omitted]. Multiplying through by [P.sub.1], [[Alpha].sub.i] x [P.sub.2] = [([MRS.sub.X2,X1]).sup.i] x [P.sub.1]. That is, the effective price paid by the ith consumer for the marginal unit of [X.sub.2] is equal to what that consumer is just willing to pay for the marginal unit. For Lindahl (1958), a "just" distribution of the nominal price of a public good is one where shares are allocated in accordance with the marginal value of the public good to the individual consumer. That would require, in this case, that the ith consumer pay [[Alpha].sub.1] x [P.sub.2].(6)

Once an externality is internalized, there no longer is an incentive for free riding behavior. In the process of internalization, however, consumers may attempt to disguise their preferences in order to obtain a more favorable [Alpha]-value. Such behavior is not precluded in the bargaining process which leads to the establishment of equilibrium [Alpha]-values.

Consider the case where the ith consumer disguises its preferences in an effort to negotiate a lower [[Alpha].sub.i].(7) To do so, it must restrict consumption of [X.sub.2] and assume a non-maximizing position where [Mathematical Expression Omitted]. If no other consumer offers compensation for an expanded consumption of [X.sub.2], the ith consumer must abandon its strategy of understating its preference for that good. The reason is the utility maximizing assumption, which is not satisfied with inequality in the marginal utility-price ratios.

On the other hand, if another economic agent does respond in a manner which reduces [[Alpha].sub.i], this simply means that individual was paying less than it was willing to pay for the marginal unit of [X.sub.2]. That is, the situation was a disequilibrium one prior to the understatement of preferences by the ith unit.

Whether or not this negotiating strategy "works" in the sense of inducing additional compensation, the ith consumer cannot persist in attempting to disguise its preferences. Ultimately, this unit (like all other consumers) is obliged to move to a position where the marginal utility-price ratios are the same. Otherwise, there is a violation of the utility maximizing assumption.

Conclusions

Given an appropriate environment, the voluntary exchange activities of consumers result in a Pareto optimal provision of public goods. Private provision but collective ownership of public goods is one such environment. Under this arrangement, persistent attempts to free ride are not consistent with maximizing behavior which effectively internalizes the public good externality. Reliance upon "Coasian" market negotiations to generate Lindahl prices does raise the specter of transactions costs which are not addressed in the paper.

Notes

1. Samuelson's interpretation of free riding behavior as the understatement of one's true preferences for public goods is the one employed in this study. This interpretation is referred to by some as the "weak" version of free riding behavior. It is contrasted with a more restrictive "strong" version which requires that consumers demand zero units of a public good. Marwell and Ames (1981), for example, make this distinction.

2. The possibility also exists for this arrangement to occur through the medium of government.

3. The institutional arrangement must make provision for an individual to sell the public good prior to its consumption. That is, consumption cannot occur prior to collective payment for the good. To successfully liquidate a unit of [X.sub.2], an individual must do one of two things: 1) sell the good back to a vendor; or, 2) sell to others who are not within the institutional arrangement governing the purchase and sale of this public good.

4. With payment for the marginal unit fully distributed among all n consumers, [Sigma] [[Alpha].sub.i] = 1.

5. Refer to p. 92 below for a discussion of the implications of attempts to free-ride by the optimizing consumer. Footnote 7 considers the limiting case of non-participation.

6. In Lindahl's analysis, this price would also be paid on intramarginal units. The same is true for the case examined here if the consumer does not pay or receive compensation. Otherwise, the effective price paid for the marginal unit can differ from prices paid for intramarginal units.

7. This case is the limiting one if [[Alpha].sub.i] [approaches] 0, i.e., there is (asymptotic) non-participation by consumer i. The consumer can still consume [X.sub.2] but pays nothing (in the limit) for it. If ([Delta]U/[Delta][X.sub.1])/[P.sub.1] = ([Delta]U/[Delta][X.sub.2])/([[infinity].sub.1] x [P.sub.2]), then (asymptotic) non-participation is optimal for that consumer. The same is true when ([Delta]U/[Delta][X.sub.1])/[P.sub.1] [greater than] ([Delta]U/[Delta][X.sub.2])/([[infinity].sub.i] x [P.sub.2]), where (in the limit) all wealth is spent on [X.sub.1]. On the other hand, if ([Delta]U/[Delta][X.sub.1])/[P.sub.1] [less than] ([Delta]U/[Delta][X.sub.2])/([[infinity].sub.i] x [P.sub.2]), non-participation is inconsistent with utility maximization. In maximizing, then, consumer i will choose to not participate (asymptotically) only when doing so is demand-revealing, i.e., in the first two situations above.

References

1. Andreoni, J. 1988. "Why Free Ride?: Strategies and Learning in Public Goods Experiments," Journal of Public Economics 37: 291-304.

2. Andreoni, J. 1995. "Warm Glow Versus Cold Pickle: The Effects of Positive and Negative Framing on Cooperation in Experiments," Quarterly Journal of Economics 110: 3-21.

3. Bailey, M.J. 1994. "Lindahl Mechanisms and Free Riders," Public Choice 80: 35-39.

4. Groves, T. 1979. "Efficient Collective Choice When Compensation is Possible," Review of Economic Studies 46: 227-241.

5. Groves, T. and J. Ledyard. 1977. "Optimal Allocation of Public Goods: A Solution to the "Free-Rider" Problem," Econometrica 45: 783-809.

6. Hurwicz, L. 1972. "On Informationally Decentralized Systems," pp. 297-336 in Decision and Organization, edited by C.B. McGuire and R. Radner. Amsterdam: North-Holland.

7. Hurwicz, L. 1979. "Outcome Functions Yielding Walrasian and Lindahl Allocations at Nash Equilibrium Points," Review of Economic Studies 46: 217-225.

8. Isaac, R.M. and J.M. Walker. 1988. "Group Size Effects in Public Goods Provision: The Voluntary Contributions Mechanism," Quarterly Journal of Economics 103: 179-200.

9. Isaac, R.M.; Walker, J.M., and A.W. Williams. 1994. "Group Size an Voluntary Provision of Public Goods: Experimental Evidence Using Large Groups," Journal of Public Economics 54: 1-36.

10. Kim, O. and M. Walker. 1984. "The Free Rider Problem: Experimental Evidence," Public Choice 43: 3-24.

11. Lindahl, E. 1958. "Just Taxation - a Positive Solution," pp. 168-76 in Classics in the Theory of Public Finance, edited by R.A. Musgrave and A.T. Peacock. London: Macmillan.

12. Marwell, G. and R.E. Ames. 1981. "Economists Free Ride, Does Anyone Else?" Journal of Public Economics 15:295-310.

13. Samuelson, P.A. 1954. "The Pure Theory of Public Expenditure," Review of Economics and Statistics 36: 387-389.

14. Samuelson, P.A. 1955. "Diagrammatic Exposition of a Theory of Public Expenditure," Review of Economics and Statistics 37: 350-356.

15. Walker, M. 1981. A Simple Incentive Compatible Scheme for Attaining Lindahl Allocations," Econometrica 49: 65-71.

16. Weimann, I. 1994. "Individual Behavior in a Free Riding Experiment," Journal of Public Economics 54:185-200.

William D. Gerdes is Professor of Economics, North Dakota State University, Fargo, North Dakota 58105-5075