Pure public goods, as defined by Paul Samuelson, are well understood. However, Burton Weisbrod argues that a substantial number of private goods have public good attributes. Among the goods mentioned by Weisbrod are visits to a park or hospital. These goods have two important attributes: uncertainty in consumption and substantial time and other costs associated with changing capacity to produce them. The question is: How do these public-type goods differ from Samuelsonian public goods? The analysis is based on a social welfare function whose arguments are the individual utilities of the members of society. Each member of society is assumed to gain utility from a private good for which he or she has an uncertain demand and from another private good. Individual endowments can be used to produce the private good or the capacity to enjoy the good for which demand is uncertain. Capacity, once produced, is available to all. Under these conditions, Weisbrodian public goods differ from Samuelsonian public goods and from pure private goods. It is shown, however, that a discriminating monopoly may provide the efficient level of capacity.
Pure public goods, as defined by Paul Samuelson (1954), are well understood, forming the very polar case to private goods. However, Burton Weisbrod (1964) argues that a number of services that appear to be purely private consumption goods in fact have collective consumption properties. Paraphrasing Weisbrod's position, the demand for the service is uncertain (a visit to a hospital or national park); consumption of the service is purely private, in the sense that one individual gets no utility from another individual's consumption of the service; and the capacity to provide the service is costly. Thus, for example, a consumer has an uncertain demand for a visit to a local hospital where the capacity to provide the service is either available or not. Therefore, during shortage situations, the consumer must either do without the service or possibly endure great danger or expense to obtain the service. Unlike Samuelson, Weisbrod argues by example, not an easy task, making comparisons with Samuelson's ideas difficult. In this article, I attempt to model Weisbrodian public goods in order to contrast them with private goods and Samuelsonian public goods.
Collective consumption goods, as described by Weisbrod, appear to include many goods and services provided in the private sector (e.g., theater seats, restaurant seats, hotel rooms, electric services, etc.), many goods and services provided in the public sector (e.g., police protection, fire protection, library services, park services, etc.), and some services provided in both the public sector and the private sector (e.g., hospital services).
Clearly, most public utilities would appear to have the attributes that Weisbrod associates with collective consumption goods. In fact, a substantial literature has developed concerning the lumpiness of capital investments in utilities and uncertain demand (Williamson 1966; Brown and Johnson 1969; Crew and Kleindorfer 1979). The main thrust of this literature relates to the problem of paying for capacity costs when marginal costs are low. The pricing problem related to this question is well understood but does not directly deal with the public good properties of service capacity. In fact, Samuelson's work and Weisbrod's work are not cited in this literature.
A separate body of literature concerning Weisbrodian public goods and Weisbrod's concept of option demand has also arisen. Option demand is the demand for the option to consume future services. Much of this literature associates option demand with the provision of local public services like museums, libraries, and fire protection (e.g., Huszar and Seckler 1974; Cowing and Holtmann 1976). At least one theoretical work suggests that Weisbrodian goods can be optimally provided by a perfectly discriminating monopolist (Zeckhauser 1970). Such a view focuses on the fact that a discriminating monopolist captures the entire consumer's surplus, eliminating all willingness to pay on the part of the consumer. Of course, a discriminating monopolist sets marginal cost equal to each consumer's marginal willingness to pay and charges take-it-or-leave-it prices that may pay for high fixed cost. Thus, one might argue that the discriminating monopolist offers a private sector solution to providing Weisbrodian public goods. Later, it is shown that a discriminating monopolist may, in fact, provide an efficient level of Weisbrodian capacity.
Analyzing Weisbrod's ideas at this time seems particularly important, since government agents are contemplating decreasing the government's subsidies to many of the types of services that Weisbrod discusses. For example, many cities are decreasing their commitment to certain local hospitals. Also, some government health plans, such as the British health system, are experiencing shortages (Lyall 1997).
THE MODEL
Although the arguments are perfectly general, for ease of understanding assume that there are two individuals in society, each with a quasi-linear utility function of the form U(q,x) = V(q) + x, where V(q) is an increasing concave function of q (e.g., visits to the hospital or public park) and x is another private good, which can be viewed as money. Furthermore, assume that expected individual utility is a suitable index of individual satisfaction, that all consumers face the same cumulative probability distribution function F(q) for all q is an element of (0, infinity), and that social welfare is measured by the sum of the individual expected utility functions. Suppose that individuals have some given endowment, w, of the private good x, and they use this endowment to trade x for the capacity q[sup *] to provide the service q such that w - q[sup *] = x. Then, an individual's constrained expected utility possibilities can be expressed as
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The first term on the right-hand side of this equation is the expected satisfaction that the individual gains from the good when the number of units demanded is less than the available capacity, and the second term is the expected utility gained when the number of units demanded is equal to or greater than the capacity available. The last two terms reflect the utility from the other private good.
Of course, the model is exceedingly simple. For example, there is no cost to consuming good q once you know you desire to have it. In the case of the national park, you can enter free of charge. In many cases, there is a cost associated with delivering the service ex post that would influence the demand for capacity by changing the cumulative probability function of demand. Although exogenous changes in the distribution function are considered, including service costs in the model makes the analysis more complex without contributing to the essence of the model. The important point is that all decisions about allocating resources between capacity and x must be made before the uncertainty is resolved. Also, once capacity q[sup *] is fixed, it is available to all, although shortages and surpluses may develop. One may wish to spend a week in the national park and find that only 3 days are available. Or, one may wish to spend a week in the hospital for maternity recovery and find that only 3 days are available. Conversely, one may wish to spend 1 day in the park, when a visit of several weeks would be possible. With two consumers, total capacity is the sum of the two individual contributions to capacity: Q[sup *] = q[sup *, sub 1] + q[sup *, sub 2]. Thus, capacity is a public good--it enters each consumer's utility function.
Social welfare, which is the sum of the constrained expected utilities of the individuals, can be written as
(1) [Multiple line equation(s) cannot be represented in ASCII text]
The first- and second-order conditions for a maximum are
(2) dW/dQ[sup *] = [dV[sub 1]/dQ[sup *] + dV[sub 2]/dQ[sup *]] [1 -F(Q[sup *])] - Q[sup *], i = 2
and
(3) d[sup 2]W/dQ[sup *2] = [d[sup 2]V[sub 1]/dQ[sup *2] + d[sup 2]V[sub 2]/dQ[sup *2]] [1 - F(Q[sup *])] - f(Q[sup *]) [dV[sub 1]/dQ[sup *] + dV[sub 2]/dQ[sup *]] < 0.
The marginal cost of an extra unit of capacity Q[sup *] is 1. Therefore, equation (2) suggests that sum of the marginal utilities of capacity multiplied by the probability of a shortage should equal the marginal cost of the public good in terms of private goods. If there was no uncertainty in this model and the Vs were a function of a pure public good, the sum of the marginal utilities of the public good would equal the marginal cost (Varian 1992, 419). Thus, if public goods were optimally provided, Weisbrodian public goods would have fewer resources devoted to them than if they were Samuelsonian public goods. If equation (2) is rearranged so that the sum of the marginal utilities of the public good is on one side of the equation and all else is on the other, it is easier to understand this result: Sigma[dV[sub i]/dQ[sup *]] = 1 + {Sigma[dV[sub i]/dQ[sup *]]}F(Q[sup *]). In this form, it is clear that the sum of the marginal utilities of the public good equals the marginal cost of providing another unit of capacity plus the sum of the marginal utilities of capacity multiplied by the probability of a surplus. That is, the extra capacity has no value when there is a surplus of capacity, and this must be considered when determining the amount of capacity. Of course, the assumption that everyone faces the same cumulative probability distribution function F(q) is a simplifying assumption that really has not changed the essence of the analysis to this stage. However, this assumption allows further analysis.
Suppose that the cumulative distribution function changed to H(q), such that H(q) > F(q) universal quantifier q. Then, the mean for the F distribution is greater than the mean for the H distribution. Furthermore, if Q[sup *] is the optimal capacity given the H distribution, the optimal capacity for the F distribution is greater than that for the H distribution, according to equation (2). That is, shifting the cumulative distribution to the right is similar to decreasing the marginal cost of the capacity. Also, welfare is greater for the F distribution than for the H distribution. To see this, subtract the expected welfare in equation (1) for the H distribution from the expected welfare in (1) for the F distribution for any given Q[sup *]. Then, integrating the first integral in (1) by parts for each consumer and simply integrating the second integral for each consumer gives
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This last result holds for any Q[sup *], and although the welfare planner would choose a larger Q[sup *] for the F distribution than for the H distribution, the planner could choose the same Q[sup *] for both distributions when the F distribution prevails, so the best Q* under the F distribution must result in greater social welfare.
Continuing with the analysis of the influence of a change in the distribution function on welfare, consider two distributions with the same mean, but where
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This implies that the G distribution is riskier than the F distribution and that it has a greater variance (Rothschild and Stiglitz 1970). It is shown above that
[Multiple line equation(s) cannot be represented in ASCII text]
but because V(q) is an increasing concave function of q for all consumers,
[Multiple line equation(s) cannot be represented in ASCII text]
Thus, social welfare is always greater under the less risky distribution, although it is not clear whether optimal capacity is less or greater for the less risky distribution compared to the more risky distribution.
PRIVATE PROVISION OF CAPACITY
To continue with the comparison between Samuelsonian and Weisbrodian public goods, Weisbrod, in contrast to Samuelson, seems to suggest that a private market might sometimes offer the optimal level of capacity or provide options at no cost. Although this may be the case in some models, it does not hold for the model developed here.
In the model developed here, individuals could determine the amount of capacity to be provided by a purely private decisionmaking process, which allows one to contrast the welfare implications of the private decision-making equilibrium with that achieved by the social planner. Suppose that each consumer believes the other consumer's contribution to financing capacity is given. Then, for example, individual one would choose q[sup *, sub 1], given q[sup *, sub 2], to maximize
[Multiple line equation(s) cannot be represented in ASCII text]
such that w[sub 1] = x[sub 1] + q[sup *, sub 1], Q[sup *] = q[sub 1] + q[sub 2], and q[sup *, sub 1] >/= 0. That is, the individual's choice cannot decrease the amount of capacity available, and the total contribution to capacity is the sum of the two individual contributions. The solution to this problem requires that
[Multiple line equation(s) cannot be represented in ASCII text]
with equality holding when q[sup *, sub 1] > 0. The Nash equilibrium, such that each individual is contributing the optimal amount of capacity given the other individual's contribution (Varian 1992, 421), requires that this last inequality hold for both individuals simultaneously. Furthermore, if Q[sup *] is positive, at least one of the inequalities must be an equality. Therefore, if Differential V[sub 1]/differential Q[sup *] > Differential V[sub 2]/Differential Q[sup *], only individual one contributes to capacity and individual two is a free rider. Only when the individuals have identical tastes will both contribute to the capacity, but their contributions are not socially optimal. If only individual one contributes to the public good, we can write the equilibrium condition as Differential V[sub 1]/Differential Q[sup *] = 1 + [Differential V[sub 1](Q[sup *])/ Differential V[sub 1](Q[sup *])]F(Q[sup *]), which is different from equation (2). Thus, private actions do not result in the optimal level of Weisbrodian public goods, which, of course, is also true for Samuelsonian public goods.
Now, consider first-degree price discrimination by a monopoly producer (Varian 1992, 243) of Weisbrodian goods. Assume the discriminating monopolist can charge each consumer a different take-it-or-leave-it price ex anti for capacity and can produce capacity at a constant cost of 1 per unit. Then, in our two-consumer model, the monopolist wishes to maximize the profits function Pi = R[sub 1](q[sub 1]) + R[sub 2](q[sub 2]) - [sup *], such that R[sub 1] SUMMARY AND DISCUSSION
As indicated earlier, Weisbrodian collective consumption goods are provided in both private and public settings. It has been shown that private competitive markets do not lead to optimal amounts of capacity being provided, and there is no mason to believe that government decisions lead to optimal provision. Under certain conditions, monopoly provision of capacity is efficient, but all the consumer's surplus is absorbed by the monopolist. It is not surprising, then, that society has experimented with various means of providing some of these services, including purely private provision, government-regulated private provision, nonprofit provision, and government provision of services. Appropriate policy in this area is crucial to social well-being, and economists should continue to provide advice on which services have more private good properties and which have more public good properties.
AUTHOR'S NOTE: Brett Katzman, Tom Holtmann, Josh Ederington, and two reviewers made helpful comments on an earlier draft of this article.
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