# On the privatization of excludable public goods.

There is a growing interest among state and local government
officials in the possibility of privatization of the provision of public
goods. In the face of increasing opposition to tax increases and
government growth, the transfer of responsibility for the delivery and
pricing of services from the public sector to the private sector becomes
an attractive option. Assessment of the impact of the transfer from
public to private institutional arrangements upon provision of public
goods requires that properties of the private and public equilibria be
established.

In a series of provocative papers the possibility of private market provision of public goods has been explored. Building upon early contributions by Thompson [10] and Demsetz [5], both competitive [7] and monopoly [3; 2; 4] models have been developed in an attempt to characterize private market solutions to the public goods allocation problem under conditions of costless exclusion. A common conclusion of these recent papers is that private provision of excludable public goods is inefficient. The private pricing mechanism results in underallocation of resources to the production of public goods capacity and in the underutilization of the capacity which is produced. The implication is that, relative to the first-best Lindahl-type public equilibrium, the private equilibrium represents an inferior alternative.

Although the Lindahl solution provides an appropriate benchmark for pure theoretical inquiry, the actual process of privatization develops from a majority voting status quo in the public sector. In this paper we wish to extend the scope of comparative analysis to the case of private market versus majority voting public sector provision of excludable public goods. The concept of public sector equilibrium employed is that of median voter equilibrium. This concept has become a standard within the theoretical public goods literature [1] and has enjoyed some recent empirical support as well [6]. The particular research issue we will address is the identification of conditions under which a majority of the consumer/voters would prefer the private monopoly outcome to the public monopoly outcome. Given the increasing interest in the privatization of public service delivery, the framework of analysis contained in this paper would seem to be of significant potential public policy import. A nonexhaustive list of potential candidate services for monopoly privatization would include roads, waterways, parks, museums, zoos, and airports.

We employ consumer surplus analysis to determine the percentage of voters who prefer the private monopoly solution to the public outcome. Successful privatization requires majority support from an interesting coalition of low and high demanders of the public good. Low demanders can benefit from the voluntary purchase nature of the private regime. High demanders can secure surplus increases in situations where the private monopolist selects a larger capacity than that chosen under public provision. Identification of surplus gains and losses requires specific solutions for the equilibrium prices and quantities (capacities) which obtain under the public and private institutional arrangements. We first generate solutions within a linear demand framework introduced by Burns and Walsh [4]. In this case each individual's preference between monopoly and public provision and hence, the percentage of voters who support the privatization initiative can be determined analytically. The numerical analysis is employed for the cases involving non-uniform distributions of individuals and nonlinear demand functions.

For the linear demand distribution case, we prove that public sector provision always dominates monopoly provision under a majority voting ranking. The requisite coalition of high and low demands in support of the private outcome never materializes. The only support for privatization comes from the low demand group. The dominance result prove to be robust with respect to a variety of demand and distributional specifications.

The negative conclusions concerning privatization above follow from the significant increase in price and concomitant transfer of surplus which accompany the movement from public to pure private monopoly. These price and transfer effects can be ameliorated by considering partial privatization alternatives. We explore three mixed public/private arrangement: (1) price regulation, (2) quantity regulation, and (3) public auction. For the linear demand distribution case we demonstrate the existence of successful privatization proposals for each three mixed schemes. Parameter conditions amenable to passage of the privatization measures are also identified.

The paper is organized as follows. In section I we introduce the basic privatization issue and identify the motivation for coalition formation. In sections II and III we present the basic model structure and define the public and private monopoly solution concepts used in the comparative analysis. Section IV includes the presentation and discussion of the analytical and numerical results of equilibrium quantity and price and the two modes of provision, and the comparisons of majority voters' choice. In section V we describes and analyze the three partial privatization alternatives. Conclusions and suggestions for further research are found in section V.

I. A Simple Privatization Model

The basic issue raised in this paper can best be introduced within the context of a simple three person community model. Assume that each consumer possess a linear demand curve for the public good, q, of the form. (1) [q.sup.i] = [a.sup.i] + bp, i = 1, 2, 3. If we denote the total quantity of the public good available as q0, and if we allow individual consumption of the pure public service to be discretionary, then the constraint (2) [q.sup.i] [is less than or equal to] q0 must hold.

Under public sector provision the choice of capacity, qv, will be determined via majority voting. If we assume that the marginal cost of producing is a constant. C, and if we assume that a head tax is used to finance the production costs, then each individual faces a fixed tax price, pv, where (3) [P.sub.v] = C/3. Substituting into [1], each individual determines his most preferred output, [Mathematical Expression Omitted]. If we assume that [a.sup.3] > [a.sup.2] > [a.sup.1], then appeal to the median voter theorem yields [Mathematical Expression Omitted] as the public sector equilibrium output. Each consumer thus faces a tax bill of [p.sub.v] [q.sub.v] = [Cq.sub.v] /3. If the government then employs a zero user charge for access to the public good, individual consumption will equal min [[a.sup.i], [q.sub.v]].

Suppose now the possibility of provision of q under conditions of monopoly is introduced. Assume that it is both feasible and costless to monitor and to exclude individuals from consuming all of the monopolist's output. Assume further that the monopolist is constrained to charge a uniform price for each consumption unit of q sold. Let the monopolist's choice of price and output be denoted as [p.sub.m] and [q.sub.m]. Consider then the case of [p.sub.m] > [p.sub.v] and [q.sub.m] > [q.sub.v] illustrated in Figure 1 for our three person example. Consumer surplus for voter 1 will increase under monopoly if area B is larger than area A. Similarly for voter 3 consumer surplus will be greater under monopoly as long as area E is greater than the sum of areas A + C + D. Person 2, the median voter under public supply, is obviously made worse off under privatization (loses A + C). Thus the possibility exists for a majority coalition of low and high demand consumers to form in support of the private monopoly solution. Low demanders can gain since they need only for the units which they actually consume, [q.sup.1] ([P.sub.m]), rather than being forced to contribute to total capacity as was the case with public financing. High demanders can gain if the monopolist chooses a larger output than obtained under public equilibrium and thereby eliminates (or, in the general case, at least alleviates) the foregone surplus due to effective quantity rationing. The possibility of the existence of such gains and thus for the emergence of such privatization coalitions will be greater (i) the smaller the price differential ([p.sub.m] - [p.sub.v]) and (ii) the greater the capacity differential ([q.sub.m] - [q.sub.v]). The relevant differentials, of course, involve the profit-minimizing choice of p and q on the part of the monopolist. It is to the determination of these values which we now turn our attention.

II. Model Specification

We assume a community of N individuals. Each individual is identified by a preference parameter x. We assume that x is distributed over a normalized compact parameter set [10,1] with a distribution function f(x), such that [Mathematical Expression Omitted]. The demand function for the public good of an individual with preference parameter x is specified as (4) [Mathematical Expression Omitted] Each demand function is assumed to be monotonic with respect to each argument so that there exist unique functions. (5) Q = Q (x, Q) P = P(x, Q) x = x(Q, P). It is assumed that demand curves do not interest, i.e., (6) Q(x, P) [is not equal to] Q(x', P), x [is not equal] x' for all P. For analytical convenience, and the comparability with Burns and Walsh, we normalize the demand function by (7) d(x, q, p) = 0 where [Mathematical Expression Omitted], and Q [bar] and P [bar] are defined by D(1, Q [bar], p]) = 0 and D(1, 0, P [bar]) = 0 respectively. Q [bar] and P [bar] are simply the intercepts of the quantity and price axes, respectively, of the demand curved of an individual who has the highest preference for the public good (x = 1). In the following discussion we simplify further by setting Q [bar] = 1. The normalized demand function also has the properties (5):

q = q(x, p) q [Epsilon] [0, 1]

p = p(x, q) p [Epsilon] [0, 1]

x = x(q, p) x [Epsilon] [0, 11]

The last equation is of significant value in the subsequent analysis. It allows us to identify by the preference parameter the type of individuals whose maximum quantity demanded is q for a given price p. In the following discussion we will refer to an individual of preference type x simply as individual x.

III. Price and Output under Alternative Modes of Provision

Monopoly Provision

Building upon foundations laid by Burns and Walsh, we structure our analysis of the monopoly firm utilizing the demand distribution approach. The demand distribution, n(q,p), defines the total demand for the qth unit of output when the joint good is available at a constant per-unit price of p. The characteristics of this construct will depend upon both the underlying structure of preferences for the public good and upon the distribution of those preferences across the population of consumers. Under uniform pricing, the number of individuals who demand the qth unit of output at a price p is given by (8) [Mathematical Expression Omitted] where [x.sub.q] = x((q, p), the preference parameter of an individual whose maximum demand is q at price p. The non-normalized total revenue from q units of output at price p is then obtained from (9) [Mathematical Expression Omitted] and the profit function follows (10) [Pi] (q, p) = TR(q, p) - Cq. Maximizing (10) with respect to p and q yields the profit-maximizing price and output pair, ([p.sub.m], [q.sub.m]) for the uniform-pricing monopolist.

The quantity demanded by an individual x at the uniform monopoly price [p.sub.m] is given by

q(x) = q(x, [p.sub.m]) if q(x, [p.sub.m]) < [q.sub.m]

= [q.sub.m] otherwise.

Public Provision

Employing the median voter theorem, we assume that the equilibrium quantity supplied under public provision is equal to the median of the distribution of individual optimal quantities for a given tax financing system. Within the context of the demand structure in our model, the median quantity will be quantity demanded by the individual whose preference parameter is [x.sub.v], where [x.sub.v] is determined by (11) [Mathematical Expression Omitted]

We first consider public provision under the assumption of head tax financing. The head tax represents the public analogue to uniform pricing, with the relevant tax price to each individual determined as (12) [P.sub.v] = C/NP [bar]. The quantity demanded by the median voter under head tax financing can then be determined by (13) [q.sub.v] = q([x.sub.v], [p.sub.v]).

As an alternative financing method one may consider a proportional tax system. A proportional tax on public goods demand intensity, x, will provide increased comparability with actual tax instruments such as income and property taxes. The a priori impact of switching from head tax to proportional tax financing upon the voting outcome is uncertain. The proportional scheme will decrease support for privatization among low demanders while increasing the potential support among the high demanders.

Under the proportional tax scheme [P.sub.t] (x) = P [bar] [p.sub.t] = P [bar] [p.sub.t] (x) = tx where [P.sub.t] is the non-normalized price and t is a fixed tax rate. The tax is set to recover the constant unit production cost C; [Mathematical Expression Omitted] where X [bar] is the mean of the distribution. The normalized price paid by individual x is then (14) [P.sub.t] (x) = (C/PX [bar]) x and hence, the quantity demanded by the median individual under the proportional tax financing arrangement is given by (15) [q.sub.t] = q([x.sub.v], [p.sub.t] ([x.sub.v])).

IV. Public versus Monopoly Provision

We now turn our attention to the central question of this paper: When will a majority of consumers prefer the private equilibrium outcome to the public equilibrium solution? We address this question by generating the individual consumer surplus rankings of the public and monopoly price-quantity pairs. The intuition expounded in section I indicates that the majority ranking will depend on two important factors: the distribution of individuals over the preference set and the shape of the demand curve. The distribution of individuals is important on a priori grounds because the number of individuals involved in the coalition of low and high demand consumers in support of the private solution will depend on the skewness and thickness n the tail of the distribution. If both tails are thick, the probability of a successful coalition will be higher. The shape of the demand curves may also be important because it will determined the magnitude of consumer surplus. To examine the effects of these two factors on the choice of a majority we posit the following general specifications.

Distribution of individuals: (16) [Mathematical Expression Omitted]

Demand function of individual x: (17) q(x, p) = (x - p) (1 - [Beta] p), -1 < [Beta] < 1.

The quadratic function f(x) takes various shapes depending on the values of the parameter [Delta] and [Theta]. f(x) has the extremum point at x = [Theta] and [Delta] is the value of the function at its extremum point. The distributions of particular interest for our purpose are when [Delta] = 0 and N with values [Theta] = 0, 0.5 and 1. The distribution function is uniform if [Delta] = N The value of [Theta] = 0.5 gives thick tails when [Delta] < N and thin tails when [Delta] > N. On the other hand, a value 0 or 1 for [Theta] gives a distribution skewed to one side. The alternative specifications are illustrated in Figure 2 for the case of [Delta] = 0. The demand function of individual x is strictly convex if [Beta] > 0, strictly concave if [Beta] < 0, and linear if [Beta] = 0.

We first examine the case of linear demand [Beta] = 0 and uniform distribution of individuals [Delta] = N. This case offers an immediate comparability with most of the extant literature and analytic solutions under these assumptions are relatively easy to obtain. The comparison of the consumer surplus of an individual under different modes of provision requires the equilibrium quantity and price. The following proposition presents the equilibria under monopoly and public provisions.

Proposition 1. Let the equilibrium output and price under monopoly be denoted as [qsub.m] and [P.sub.m], under provision with head tax financing as [q.sub.v] and [p.sub.v], and under public provision with proportional tax financing as [q.sub.t] and [p.sub.t] (x). Let [x.sub.v] denote the medium voter, k = C/NP [bar], [Tau] = [(1 + 12k).sup.1/2] and [Alpha] = (11 - 4 [(7).sub.1/2])/6 [is approximately equal to] 0.07. If the demand functions are linear and the distribution of individuals is uniform, then [x.sub.v] = 1/2 and (a) [q.sub.v] = [q.sub.t] = 0.5, - k if k [is less than or equal to] 1/2,

[p.sub.v] = k, [p.sub.t] (x) = 2kx, (b) [q.sub.m] = (4 - 2 [Tau])/3 if 0 [is less than or equal to] 1/4

= 0 otherwise

[p.sub.m] = (1 + [Tau])/6 if 0 [is less than or equal to] k [is less than or equal to] 1/4 (c) [Mathematical Expression Omitted] Proof. The proofs are presented in Appendix A.

The parameter k summarizes all the characteristics of the community and may be considered as a community index. A small value of k indicates that the community size or the individual demand for the public good is large, and/or that the unit production cost is low. The range of this index in which the private monopoly produces a positive output is only half of that under public provision. Therefore, there is large set of communities where the private monopoly will not provide the public good, while a positive output is produced under public provision. Furthermore, result (c) in proposition 1 shows that the set of communities (the range of k) where the private monopoly output exceeds the public output is even smaller. Notice that when th monopoly output is positive, the difference between the monopoly price [p.sub.m] and the public price [p.sub.m] and the public prices [p.sub.v] and [p.sub.t] (x) is substantially higher than the difference in quantities, regardless of whether [q.sub.m] [is greater or less than [q.sub.v]. The private/public price differential ([p.sub.m] - [p.sub.v]) and quantity differential ([q.sub.m] - [q.sub.v]) are shown as a function of k in Figure 3. These differences will have significant implications for the majority ranking of the private and public options.

The source of the large price differential can be identified from examination of the first order conditions for the private monopoly optimization problem. For any given capacity, the profit-maximizing price is determined where marginal revenue equals zero. This condition follows from the assumed noncongestibility (zero marginal user cost) property of the public good. When evaluated at the profit-maximizing price, the elasticity of conditional demand (conditional on a given q) will be unity. When evaluated at the public price, however, the price elasticity is extremely low. For example, with [p.sub.v] = 0.01, the price elasticity evaluated at the monopoly capacity is only 0.015. Further, the conditional demand remains inelastic over a broad price range (until [p.sub.m] = 0.343 given [q.sub.m] = 0.6278). The inelasticity of aggregate conditional demand reflects the totally inelastic response to changes in price exhibited by the fraction of consumers who are being quantity rationed rather than price rationed. Since that subset of consumer is large when measured at the public price, the opportunity for sizeable profitable price increases by the private monopolist is apparent.

It is transparent that the majority will not prefer the monopoly provision when the public output exceeds the monopoly output regardless of the public financing method. With [q.sub.v] > [q.sub.m] > 0, the consumer surplus for the median and higher demand individuals will be lower under monopoly since [p.sub.m] is higher than [p.sub.v] and [p.sub.t] (x) for all [q.sub.m] > 0. When the monopoly output exceeds the public output, there is a possibility of a coalition of low and high demanders in support of the monopoly proposal: Low demanders whose consumer surplus is negative under public provision prefer non-participation in the public goods market under monopoly, and higher demand individuals may gain consumer surplus from a larger monopoly output. This is illustrated in Figure 4 for the case of head tax financing of public provision. The preference parameter of the individual whose consumer surplus is zero under the public provision(2) is indicated by [Mathematical Expression Omitted] in Figure 4. Individuals in the interval [Mathematical Expression Omitted] will prefer monopoly to public provision, while individuals in the interval [x.sub.v], [x.sub.1] have smaller consumer surpluses under monopoly provision, and hence, will prefer public to monopoly provision. The preference of individuals in the interval ([x.sub.1], 1) is determined by the difference in areas A and B: If A is greater than B, the individual prefers monopoly to public provision. The difference in consumer surpluses (A - B) is the largest for x = 1 among all individuals in the interval ([x.sub.1], 1]. The case of proportional tax financing is illustrated(3) in Figure 5. The price that each individual pays under public provision is determined by the price line [p.sub.t] (x) = 2kx. Area B is thus a decreasing function of x and area A is an increasing function of x for all x [Epsilon] ([x.sub.1], 1]. The difference in consumer surpluses (A - B) is again the largest for individual x = 1.

The preference of individual in intervals [Mathematical Expression Omitted is not so obvious in both cases. It is apparent, however, that some individuals in the interval ([x.sub.1]] must prefer monopoly provision to form a majority of the voters for the monopoly choice. The comparison of consumer surpluses of individuals in ([x.sub.1]] between the two modes of provision leads to a striking result as presented in the following proposition.

PROPOSITION 2. When the demand functions are linear and the distribution of individuals is uniform, a majority of the voters prefers public provision to monopoly provision under both the head tax and proportional tax financing methods of public provision.

Proof. It is sufficient to show that all individuals x [Epsilon] ([x.sub.1], 1] prefer the public provision when [q.sub.m] > [q.sub.v]. Since area A in Figure 4 and Figure 5 is the largest for individual x = 1, we only have to show that the highest demander prefers public provision.

Let [CS.sub.m], [CS.sub.v] and [CS.sub.t] be the consumer surplus of individual x = 1 under monopoly, under public provision with head tax, and under public provision with proportional tax, respectively. Then, [Mathematical Expression Omitted] which is a function of k. Using the results in Proposition 1, it is easy to show [Mathematical Expression Omitted] which imply [[Phi].sub.v]] (k) < 0 for all k [is less than or equal to] [Alpha]. Therefore, all individuals in the interval ([x.sub.1], 1] prefer public provision with head tax financing. The proof for the proportional tax case is similar, and thus omitted.

The necessary coalition of low and high demanders in support of the monopoly proposal never materializes in the case of linear demand and uniform distribution of individuals. This is due to the marked increase in the relative price of the public good under monopoly compared to that under public provision. A sharp increase in the relative eliminates the potential consumer surplus gains to the supramedian voters, leaving them with greater consumer surplus under the public regime. The only supporters of the monopoly proposal are those consumers [Mathematical Expression Omitted] who would prefer to buy zero from the monopolist than to oversubscribe to the public good under median voter rule.(4)

One might ask whether the strong negative conclusion concerning monopoly privatization is peculiar to the assumptions of linear demand and uniform distribution of individuals. In order to assess the robustness of the results, we generated a variety of cases by altering the demand function parameter [Beta] and distribution parameters [Delta] and [Theta]. Each individual's consumer surplus under different modes of provision and the percentage of voters who prefer the monopoly provision are then computed by using the procedure described in Appendix B. Since the qualitative results are similar, we report in Table I the results for the case of head tax financing under three alternative distribution functions. Linearity of the demand function is maintained for an easy comparison with the previous cases. The three distribution functions presented here are expected to be most favorable to the monopoly solution. In particular, the distribution with ([Delta] = 0, [Theta] = 0.5) is symmetric and bimodal with thick tails at both ends of the distribution (see Figure 2), and thus, the likelihood of the hypothesized coalition of low and high demand individuals forming a majority is expected to be highest. This distribution indeed generates the highest percentage of the population voting for privatization. However, it still fails to attract the support from the upper end individuals and hence fails to form a majority.

The results obtained so far (including Table I) are derived based upon the assumption that production costs are identical under both private and public provision. Proponents of privatization often argue, however, that public provision results in inefficient, non-cost-minimizing production behavior.(5) In order to assess the importance of potential cost advantages accruing from privatization, we calculated the cost differential required to secure majority support for monopoly proposals. The public/private cost ratio required to reverse the equal cost voting results ranged from 12/1 (if private C = 2) to 3/1 (if private C = 10). The necessary relative cost efficiencies which must obtain under private monopoly in order to secure majority support are quite substantial. [Tabular Data I Omitted]

V. Partial Privatization Alternatives

In the analysis above, the two choices available to voters were to provide the public good utilizing a collective choice allocation process or to turn the delivery of the public good over to a profit-maximizing monopoly firm. Complete privatization led to a marked increase in price relative to any increases in output, and thereby failed to garner majority support. In this section we explore three partial privatization alternatives designed to attenuate the price increases associated with the private monopoly solution. In the first case price is set by the public contractor and quantity is determined by the monopoly firm to maximize its conditional profit. The second case has the public authority setting capacity and the firm setting price so as to maximize profit. In the final case the right to produce and to sell a given capacity is allocated through a price auction. We investigate the properties of these three institutional arrangements under the assumption of linear demand functions and a uniform distribution of individuals.

When demand is linear and the distribution of individuals is uniform, monopoly's profit function is given by (see Appendix A) [Pi] (q,p) = n P [bar] p [(1 - p)q - [q.sup.2] / 2] - [C.sub.q].

The profit maximizing price [p.sub.s] at a given quantity [q.sub.s] under quantity regulation is found from [Alpha] [Pi] (q, p)/ [Alpha] p = 0: (18.a) [p.sub.s] = 0.5 - [q.sub.s] /4 if [q.sub.s] [is less than or equal to] 2/3, (18.b) [p.sub.s] = 1 - [q.sub.s] if [q.sub.s] > 2/3.

Similarly, from [Alpha] [Pi] (q, p) / [Alpha] q = 0, we find the profit maximizing quantity [q.sub.r] at a given price [p.sub.r] under price regulation: (19) [Mathematical Expression Omitted] where [p.sub.L] = (1 - [(1 - 4k).sup.1/2])/2 and [p.sub.U] = (1 + (1 - 4k)[.sup.1/2]) /2 for k [is less than or equal to] 0.25. Finally, the locus of prices and quantities that yields zero profit is defined by (20) [Mathematical Expression Omitted]

The functions (18)-(20) and the highest demand line are illustrated in Figure 6 for k = 0.02. Since the monopoly output without any constraint must satisfy both (18) and (19), [p.sub.m] and [q.sub.m] are determined at the intersection of these two curves.

The public contractor can induce the monopolist to provide a larger capacity than [q.sub.v at a low price by selecting a regulated price from the interval(6) ([p.sub.min], [(k).sup.1/2]), as shown as in shown in Figure 6. Note that, in the neighborhood of [P.sub.min], monopoly output is elastic to changes in the regulated price. A large public-private capacity differential can thus be obtained for a relatively small public-private price differential. The probability of high demander support for privatization that was absent in the nonregulated monopoly setting is, correspondingly, increased. We find a set of regulated prices for which such support indeed materializes. For example, when k = 0.01, a regulated price [P.sub.r] =0.125 and optimal monopoly output of 0.575 secures a majority (56%) of the votes when paired against the associated public solution. The support for the monopoly alternatives comes from both low end (20%) and upper end (36%) demanders.

If the public agent chooses to specify output when contracting with the private monopolist, the monopoly price is determined by the [P.sub.s] line if quantity is set below 2/3 and by the highest demand line (q = 1 - p) if quantity is set above 2/3. The public authority needs only to search for a quantity in the range of ([q.sub.m], [q [bar].sub.s]), where [q [bar].sub.s] is determined by the intersection point of the highest demand line and the zero profit ([Pi] = 0) curve.(7) If the quantity is set below [q.sub.m], the monopoly price [P.sub.s] is even higher than [P.sub.m] which makes the quantity-regulated monopoly even less attractive to the upper end demanders. When the quantity is set above [q [bar].sub.s], the monopoly profit is negative for all feasible prices and hence, the monopoly firm will not produce. It is apparent from Figure 6 that the monopoly firm can be induced to produce the largest output at the lowest price when the quantity is set at [q [bar].sub.s]. The price at this output is (1 - [q [bar].sub.s]) and the monopoly makes zero profits. When the regulatory authority sets the quantity at [q [bar].sub.s], we find a majority coalition of the low and upper demanders in favor of the monopoly provision: 64% when k = 0.01 and 59% when k = 0.02. A majority support for the monopoly is preserved even when the quantity is set slightly below [q [bar].sub.s], allowing positive for the firm. The percentage of the voters in support of the monopoly provision decreases as k increases, and the majority coalition ceases to exits when k becomes large.(8)

Finally we consider awarding the monopoly franchise on the basis of competitive price bidding. This may represent the most realistic partial privatization arrangement, as it mirrors the actual process of contracting out refuse collection services in a number of U.S cities [9]. The public agency again specifies the output, but now potential suppliers enter bids as to the user price to be charged. We will assume that the auction design results in a zero profit solution. From the analysis above we know that output [q [bar].sub.s] and price (1 - [q [bar].sub.s] represent a politically viable zero-profit solution (for low values of k). Unlike the case of quantity regulation, however, the zero-profit price auction strategy attracts the largest support for the monopoly when the quantity is set well below [q [bar].sub.s]. This follows from the positive slope of the zero profit locus over the relevant quantity range. The inframarginal consumer surplus gains due to the fall in price dominate the marginal surplus losses due to the reduction in capacity. The range of quantities and the range of values of k for which a majority of consumers can be made better off by privatizing via the auction approach is, therefore, quite wide.(9)

VI. Conclusions and Extensions

In order to assess the potential advantage afforded by the privatization of delivery of public goods, the properties of the private and public equilibria must be established. In this paper, we have identified the relevant equilibrium prices and quantities associated with public and private monopoly models of the provision of excludable, non-congestible public goods. For this class of commodities, our analysis suggests that unfettered private monopoly may not provide a politically viable alternative public provision, since the majority of our consumer-voters are made worse off (in the absence of sizeable cost differentials) under the pure monopoly solution. We are, however, able to identify mixed/private contracting arrangements which are preferred to the pure public situation by a majority of the voters. These results highlight the importance of evaluating privatization along a continuum from the pure public to the pure private poles. The variety of contracting arrangements and the associated variety of equilibria should provide a basis for continuing research in the public goods area. This research will provide an interesting complement to the current work on auction and contract design in the industrial organization literature [8].

Appendix A: Proof of Proposition 1

In a series of provocative papers the possibility of private market provision of public goods has been explored. Building upon early contributions by Thompson [10] and Demsetz [5], both competitive [7] and monopoly [3; 2; 4] models have been developed in an attempt to characterize private market solutions to the public goods allocation problem under conditions of costless exclusion. A common conclusion of these recent papers is that private provision of excludable public goods is inefficient. The private pricing mechanism results in underallocation of resources to the production of public goods capacity and in the underutilization of the capacity which is produced. The implication is that, relative to the first-best Lindahl-type public equilibrium, the private equilibrium represents an inferior alternative.

Although the Lindahl solution provides an appropriate benchmark for pure theoretical inquiry, the actual process of privatization develops from a majority voting status quo in the public sector. In this paper we wish to extend the scope of comparative analysis to the case of private market versus majority voting public sector provision of excludable public goods. The concept of public sector equilibrium employed is that of median voter equilibrium. This concept has become a standard within the theoretical public goods literature [1] and has enjoyed some recent empirical support as well [6]. The particular research issue we will address is the identification of conditions under which a majority of the consumer/voters would prefer the private monopoly outcome to the public monopoly outcome. Given the increasing interest in the privatization of public service delivery, the framework of analysis contained in this paper would seem to be of significant potential public policy import. A nonexhaustive list of potential candidate services for monopoly privatization would include roads, waterways, parks, museums, zoos, and airports.

We employ consumer surplus analysis to determine the percentage of voters who prefer the private monopoly solution to the public outcome. Successful privatization requires majority support from an interesting coalition of low and high demanders of the public good. Low demanders can benefit from the voluntary purchase nature of the private regime. High demanders can secure surplus increases in situations where the private monopolist selects a larger capacity than that chosen under public provision. Identification of surplus gains and losses requires specific solutions for the equilibrium prices and quantities (capacities) which obtain under the public and private institutional arrangements. We first generate solutions within a linear demand framework introduced by Burns and Walsh [4]. In this case each individual's preference between monopoly and public provision and hence, the percentage of voters who support the privatization initiative can be determined analytically. The numerical analysis is employed for the cases involving non-uniform distributions of individuals and nonlinear demand functions.

For the linear demand distribution case, we prove that public sector provision always dominates monopoly provision under a majority voting ranking. The requisite coalition of high and low demands in support of the private outcome never materializes. The only support for privatization comes from the low demand group. The dominance result prove to be robust with respect to a variety of demand and distributional specifications.

The negative conclusions concerning privatization above follow from the significant increase in price and concomitant transfer of surplus which accompany the movement from public to pure private monopoly. These price and transfer effects can be ameliorated by considering partial privatization alternatives. We explore three mixed public/private arrangement: (1) price regulation, (2) quantity regulation, and (3) public auction. For the linear demand distribution case we demonstrate the existence of successful privatization proposals for each three mixed schemes. Parameter conditions amenable to passage of the privatization measures are also identified.

The paper is organized as follows. In section I we introduce the basic privatization issue and identify the motivation for coalition formation. In sections II and III we present the basic model structure and define the public and private monopoly solution concepts used in the comparative analysis. Section IV includes the presentation and discussion of the analytical and numerical results of equilibrium quantity and price and the two modes of provision, and the comparisons of majority voters' choice. In section V we describes and analyze the three partial privatization alternatives. Conclusions and suggestions for further research are found in section V.

I. A Simple Privatization Model

The basic issue raised in this paper can best be introduced within the context of a simple three person community model. Assume that each consumer possess a linear demand curve for the public good, q, of the form. (1) [q.sup.i] = [a.sup.i] + bp, i = 1, 2, 3. If we denote the total quantity of the public good available as q0, and if we allow individual consumption of the pure public service to be discretionary, then the constraint (2) [q.sup.i] [is less than or equal to] q0 must hold.

Under public sector provision the choice of capacity, qv, will be determined via majority voting. If we assume that the marginal cost of producing is a constant. C, and if we assume that a head tax is used to finance the production costs, then each individual faces a fixed tax price, pv, where (3) [P.sub.v] = C/3. Substituting into [1], each individual determines his most preferred output, [Mathematical Expression Omitted]. If we assume that [a.sup.3] > [a.sup.2] > [a.sup.1], then appeal to the median voter theorem yields [Mathematical Expression Omitted] as the public sector equilibrium output. Each consumer thus faces a tax bill of [p.sub.v] [q.sub.v] = [Cq.sub.v] /3. If the government then employs a zero user charge for access to the public good, individual consumption will equal min [[a.sup.i], [q.sub.v]].

Suppose now the possibility of provision of q under conditions of monopoly is introduced. Assume that it is both feasible and costless to monitor and to exclude individuals from consuming all of the monopolist's output. Assume further that the monopolist is constrained to charge a uniform price for each consumption unit of q sold. Let the monopolist's choice of price and output be denoted as [p.sub.m] and [q.sub.m]. Consider then the case of [p.sub.m] > [p.sub.v] and [q.sub.m] > [q.sub.v] illustrated in Figure 1 for our three person example. Consumer surplus for voter 1 will increase under monopoly if area B is larger than area A. Similarly for voter 3 consumer surplus will be greater under monopoly as long as area E is greater than the sum of areas A + C + D. Person 2, the median voter under public supply, is obviously made worse off under privatization (loses A + C). Thus the possibility exists for a majority coalition of low and high demand consumers to form in support of the private monopoly solution. Low demanders can gain since they need only for the units which they actually consume, [q.sup.1] ([P.sub.m]), rather than being forced to contribute to total capacity as was the case with public financing. High demanders can gain if the monopolist chooses a larger output than obtained under public equilibrium and thereby eliminates (or, in the general case, at least alleviates) the foregone surplus due to effective quantity rationing. The possibility of the existence of such gains and thus for the emergence of such privatization coalitions will be greater (i) the smaller the price differential ([p.sub.m] - [p.sub.v]) and (ii) the greater the capacity differential ([q.sub.m] - [q.sub.v]). The relevant differentials, of course, involve the profit-minimizing choice of p and q on the part of the monopolist. It is to the determination of these values which we now turn our attention.

II. Model Specification

We assume a community of N individuals. Each individual is identified by a preference parameter x. We assume that x is distributed over a normalized compact parameter set [10,1] with a distribution function f(x), such that [Mathematical Expression Omitted]. The demand function for the public good of an individual with preference parameter x is specified as (4) [Mathematical Expression Omitted] Each demand function is assumed to be monotonic with respect to each argument so that there exist unique functions. (5) Q = Q (x, Q) P = P(x, Q) x = x(Q, P). It is assumed that demand curves do not interest, i.e., (6) Q(x, P) [is not equal to] Q(x', P), x [is not equal] x' for all P. For analytical convenience, and the comparability with Burns and Walsh, we normalize the demand function by (7) d(x, q, p) = 0 where [Mathematical Expression Omitted], and Q [bar] and P [bar] are defined by D(1, Q [bar], p]) = 0 and D(1, 0, P [bar]) = 0 respectively. Q [bar] and P [bar] are simply the intercepts of the quantity and price axes, respectively, of the demand curved of an individual who has the highest preference for the public good (x = 1). In the following discussion we simplify further by setting Q [bar] = 1. The normalized demand function also has the properties (5):

q = q(x, p) q [Epsilon] [0, 1]

p = p(x, q) p [Epsilon] [0, 1]

x = x(q, p) x [Epsilon] [0, 11]

The last equation is of significant value in the subsequent analysis. It allows us to identify by the preference parameter the type of individuals whose maximum quantity demanded is q for a given price p. In the following discussion we will refer to an individual of preference type x simply as individual x.

III. Price and Output under Alternative Modes of Provision

Monopoly Provision

Building upon foundations laid by Burns and Walsh, we structure our analysis of the monopoly firm utilizing the demand distribution approach. The demand distribution, n(q,p), defines the total demand for the qth unit of output when the joint good is available at a constant per-unit price of p. The characteristics of this construct will depend upon both the underlying structure of preferences for the public good and upon the distribution of those preferences across the population of consumers. Under uniform pricing, the number of individuals who demand the qth unit of output at a price p is given by (8) [Mathematical Expression Omitted] where [x.sub.q] = x((q, p), the preference parameter of an individual whose maximum demand is q at price p. The non-normalized total revenue from q units of output at price p is then obtained from (9) [Mathematical Expression Omitted] and the profit function follows (10) [Pi] (q, p) = TR(q, p) - Cq. Maximizing (10) with respect to p and q yields the profit-maximizing price and output pair, ([p.sub.m], [q.sub.m]) for the uniform-pricing monopolist.

The quantity demanded by an individual x at the uniform monopoly price [p.sub.m] is given by

q(x) = q(x, [p.sub.m]) if q(x, [p.sub.m]) < [q.sub.m]

= [q.sub.m] otherwise.

Public Provision

Employing the median voter theorem, we assume that the equilibrium quantity supplied under public provision is equal to the median of the distribution of individual optimal quantities for a given tax financing system. Within the context of the demand structure in our model, the median quantity will be quantity demanded by the individual whose preference parameter is [x.sub.v], where [x.sub.v] is determined by (11) [Mathematical Expression Omitted]

We first consider public provision under the assumption of head tax financing. The head tax represents the public analogue to uniform pricing, with the relevant tax price to each individual determined as (12) [P.sub.v] = C/NP [bar]. The quantity demanded by the median voter under head tax financing can then be determined by (13) [q.sub.v] = q([x.sub.v], [p.sub.v]).

As an alternative financing method one may consider a proportional tax system. A proportional tax on public goods demand intensity, x, will provide increased comparability with actual tax instruments such as income and property taxes. The a priori impact of switching from head tax to proportional tax financing upon the voting outcome is uncertain. The proportional scheme will decrease support for privatization among low demanders while increasing the potential support among the high demanders.

Under the proportional tax scheme [P.sub.t] (x) = P [bar] [p.sub.t] = P [bar] [p.sub.t] (x) = tx where [P.sub.t] is the non-normalized price and t is a fixed tax rate. The tax is set to recover the constant unit production cost C; [Mathematical Expression Omitted] where X [bar] is the mean of the distribution. The normalized price paid by individual x is then (14) [P.sub.t] (x) = (C/PX [bar]) x and hence, the quantity demanded by the median individual under the proportional tax financing arrangement is given by (15) [q.sub.t] = q([x.sub.v], [p.sub.t] ([x.sub.v])).

IV. Public versus Monopoly Provision

We now turn our attention to the central question of this paper: When will a majority of consumers prefer the private equilibrium outcome to the public equilibrium solution? We address this question by generating the individual consumer surplus rankings of the public and monopoly price-quantity pairs. The intuition expounded in section I indicates that the majority ranking will depend on two important factors: the distribution of individuals over the preference set and the shape of the demand curve. The distribution of individuals is important on a priori grounds because the number of individuals involved in the coalition of low and high demand consumers in support of the private solution will depend on the skewness and thickness n the tail of the distribution. If both tails are thick, the probability of a successful coalition will be higher. The shape of the demand curves may also be important because it will determined the magnitude of consumer surplus. To examine the effects of these two factors on the choice of a majority we posit the following general specifications.

Distribution of individuals: (16) [Mathematical Expression Omitted]

Demand function of individual x: (17) q(x, p) = (x - p) (1 - [Beta] p), -1 < [Beta] < 1.

The quadratic function f(x) takes various shapes depending on the values of the parameter [Delta] and [Theta]. f(x) has the extremum point at x = [Theta] and [Delta] is the value of the function at its extremum point. The distributions of particular interest for our purpose are when [Delta] = 0 and N with values [Theta] = 0, 0.5 and 1. The distribution function is uniform if [Delta] = N The value of [Theta] = 0.5 gives thick tails when [Delta] < N and thin tails when [Delta] > N. On the other hand, a value 0 or 1 for [Theta] gives a distribution skewed to one side. The alternative specifications are illustrated in Figure 2 for the case of [Delta] = 0. The demand function of individual x is strictly convex if [Beta] > 0, strictly concave if [Beta] < 0, and linear if [Beta] = 0.

We first examine the case of linear demand [Beta] = 0 and uniform distribution of individuals [Delta] = N. This case offers an immediate comparability with most of the extant literature and analytic solutions under these assumptions are relatively easy to obtain. The comparison of the consumer surplus of an individual under different modes of provision requires the equilibrium quantity and price. The following proposition presents the equilibria under monopoly and public provisions.

Proposition 1. Let the equilibrium output and price under monopoly be denoted as [qsub.m] and [P.sub.m], under provision with head tax financing as [q.sub.v] and [p.sub.v], and under public provision with proportional tax financing as [q.sub.t] and [p.sub.t] (x). Let [x.sub.v] denote the medium voter, k = C/NP [bar], [Tau] = [(1 + 12k).sup.1/2] and [Alpha] = (11 - 4 [(7).sub.1/2])/6 [is approximately equal to] 0.07. If the demand functions are linear and the distribution of individuals is uniform, then [x.sub.v] = 1/2 and (a) [q.sub.v] = [q.sub.t] = 0.5, - k if k [is less than or equal to] 1/2,

[p.sub.v] = k, [p.sub.t] (x) = 2kx, (b) [q.sub.m] = (4 - 2 [Tau])/3 if 0 [is less than or equal to] 1/4

= 0 otherwise

[p.sub.m] = (1 + [Tau])/6 if 0 [is less than or equal to] k [is less than or equal to] 1/4 (c) [Mathematical Expression Omitted] Proof. The proofs are presented in Appendix A.

The parameter k summarizes all the characteristics of the community and may be considered as a community index. A small value of k indicates that the community size or the individual demand for the public good is large, and/or that the unit production cost is low. The range of this index in which the private monopoly produces a positive output is only half of that under public provision. Therefore, there is large set of communities where the private monopoly will not provide the public good, while a positive output is produced under public provision. Furthermore, result (c) in proposition 1 shows that the set of communities (the range of k) where the private monopoly output exceeds the public output is even smaller. Notice that when th monopoly output is positive, the difference between the monopoly price [p.sub.m] and the public price [p.sub.m] and the public prices [p.sub.v] and [p.sub.t] (x) is substantially higher than the difference in quantities, regardless of whether [q.sub.m] [is greater or less than [q.sub.v]. The private/public price differential ([p.sub.m] - [p.sub.v]) and quantity differential ([q.sub.m] - [q.sub.v]) are shown as a function of k in Figure 3. These differences will have significant implications for the majority ranking of the private and public options.

The source of the large price differential can be identified from examination of the first order conditions for the private monopoly optimization problem. For any given capacity, the profit-maximizing price is determined where marginal revenue equals zero. This condition follows from the assumed noncongestibility (zero marginal user cost) property of the public good. When evaluated at the profit-maximizing price, the elasticity of conditional demand (conditional on a given q) will be unity. When evaluated at the public price, however, the price elasticity is extremely low. For example, with [p.sub.v] = 0.01, the price elasticity evaluated at the monopoly capacity is only 0.015. Further, the conditional demand remains inelastic over a broad price range (until [p.sub.m] = 0.343 given [q.sub.m] = 0.6278). The inelasticity of aggregate conditional demand reflects the totally inelastic response to changes in price exhibited by the fraction of consumers who are being quantity rationed rather than price rationed. Since that subset of consumer is large when measured at the public price, the opportunity for sizeable profitable price increases by the private monopolist is apparent.

It is transparent that the majority will not prefer the monopoly provision when the public output exceeds the monopoly output regardless of the public financing method. With [q.sub.v] > [q.sub.m] > 0, the consumer surplus for the median and higher demand individuals will be lower under monopoly since [p.sub.m] is higher than [p.sub.v] and [p.sub.t] (x) for all [q.sub.m] > 0. When the monopoly output exceeds the public output, there is a possibility of a coalition of low and high demanders in support of the monopoly proposal: Low demanders whose consumer surplus is negative under public provision prefer non-participation in the public goods market under monopoly, and higher demand individuals may gain consumer surplus from a larger monopoly output. This is illustrated in Figure 4 for the case of head tax financing of public provision. The preference parameter of the individual whose consumer surplus is zero under the public provision(2) is indicated by [Mathematical Expression Omitted] in Figure 4. Individuals in the interval [Mathematical Expression Omitted] will prefer monopoly to public provision, while individuals in the interval [x.sub.v], [x.sub.1] have smaller consumer surpluses under monopoly provision, and hence, will prefer public to monopoly provision. The preference of individuals in the interval ([x.sub.1], 1) is determined by the difference in areas A and B: If A is greater than B, the individual prefers monopoly to public provision. The difference in consumer surpluses (A - B) is the largest for x = 1 among all individuals in the interval ([x.sub.1], 1]. The case of proportional tax financing is illustrated(3) in Figure 5. The price that each individual pays under public provision is determined by the price line [p.sub.t] (x) = 2kx. Area B is thus a decreasing function of x and area A is an increasing function of x for all x [Epsilon] ([x.sub.1], 1]. The difference in consumer surpluses (A - B) is again the largest for individual x = 1.

The preference of individual in intervals [Mathematical Expression Omitted is not so obvious in both cases. It is apparent, however, that some individuals in the interval ([x.sub.1]] must prefer monopoly provision to form a majority of the voters for the monopoly choice. The comparison of consumer surpluses of individuals in ([x.sub.1]] between the two modes of provision leads to a striking result as presented in the following proposition.

PROPOSITION 2. When the demand functions are linear and the distribution of individuals is uniform, a majority of the voters prefers public provision to monopoly provision under both the head tax and proportional tax financing methods of public provision.

Proof. It is sufficient to show that all individuals x [Epsilon] ([x.sub.1], 1] prefer the public provision when [q.sub.m] > [q.sub.v]. Since area A in Figure 4 and Figure 5 is the largest for individual x = 1, we only have to show that the highest demander prefers public provision.

Let [CS.sub.m], [CS.sub.v] and [CS.sub.t] be the consumer surplus of individual x = 1 under monopoly, under public provision with head tax, and under public provision with proportional tax, respectively. Then, [Mathematical Expression Omitted] which is a function of k. Using the results in Proposition 1, it is easy to show [Mathematical Expression Omitted] which imply [[Phi].sub.v]] (k) < 0 for all k [is less than or equal to] [Alpha]. Therefore, all individuals in the interval ([x.sub.1], 1] prefer public provision with head tax financing. The proof for the proportional tax case is similar, and thus omitted.

The necessary coalition of low and high demanders in support of the monopoly proposal never materializes in the case of linear demand and uniform distribution of individuals. This is due to the marked increase in the relative price of the public good under monopoly compared to that under public provision. A sharp increase in the relative eliminates the potential consumer surplus gains to the supramedian voters, leaving them with greater consumer surplus under the public regime. The only supporters of the monopoly proposal are those consumers [Mathematical Expression Omitted] who would prefer to buy zero from the monopolist than to oversubscribe to the public good under median voter rule.(4)

One might ask whether the strong negative conclusion concerning monopoly privatization is peculiar to the assumptions of linear demand and uniform distribution of individuals. In order to assess the robustness of the results, we generated a variety of cases by altering the demand function parameter [Beta] and distribution parameters [Delta] and [Theta]. Each individual's consumer surplus under different modes of provision and the percentage of voters who prefer the monopoly provision are then computed by using the procedure described in Appendix B. Since the qualitative results are similar, we report in Table I the results for the case of head tax financing under three alternative distribution functions. Linearity of the demand function is maintained for an easy comparison with the previous cases. The three distribution functions presented here are expected to be most favorable to the monopoly solution. In particular, the distribution with ([Delta] = 0, [Theta] = 0.5) is symmetric and bimodal with thick tails at both ends of the distribution (see Figure 2), and thus, the likelihood of the hypothesized coalition of low and high demand individuals forming a majority is expected to be highest. This distribution indeed generates the highest percentage of the population voting for privatization. However, it still fails to attract the support from the upper end individuals and hence fails to form a majority.

The results obtained so far (including Table I) are derived based upon the assumption that production costs are identical under both private and public provision. Proponents of privatization often argue, however, that public provision results in inefficient, non-cost-minimizing production behavior.(5) In order to assess the importance of potential cost advantages accruing from privatization, we calculated the cost differential required to secure majority support for monopoly proposals. The public/private cost ratio required to reverse the equal cost voting results ranged from 12/1 (if private C = 2) to 3/1 (if private C = 10). The necessary relative cost efficiencies which must obtain under private monopoly in order to secure majority support are quite substantial. [Tabular Data I Omitted]

V. Partial Privatization Alternatives

In the analysis above, the two choices available to voters were to provide the public good utilizing a collective choice allocation process or to turn the delivery of the public good over to a profit-maximizing monopoly firm. Complete privatization led to a marked increase in price relative to any increases in output, and thereby failed to garner majority support. In this section we explore three partial privatization alternatives designed to attenuate the price increases associated with the private monopoly solution. In the first case price is set by the public contractor and quantity is determined by the monopoly firm to maximize its conditional profit. The second case has the public authority setting capacity and the firm setting price so as to maximize profit. In the final case the right to produce and to sell a given capacity is allocated through a price auction. We investigate the properties of these three institutional arrangements under the assumption of linear demand functions and a uniform distribution of individuals.

When demand is linear and the distribution of individuals is uniform, monopoly's profit function is given by (see Appendix A) [Pi] (q,p) = n P [bar] p [(1 - p)q - [q.sup.2] / 2] - [C.sub.q].

The profit maximizing price [p.sub.s] at a given quantity [q.sub.s] under quantity regulation is found from [Alpha] [Pi] (q, p)/ [Alpha] p = 0: (18.a) [p.sub.s] = 0.5 - [q.sub.s] /4 if [q.sub.s] [is less than or equal to] 2/3, (18.b) [p.sub.s] = 1 - [q.sub.s] if [q.sub.s] > 2/3.

Similarly, from [Alpha] [Pi] (q, p) / [Alpha] q = 0, we find the profit maximizing quantity [q.sub.r] at a given price [p.sub.r] under price regulation: (19) [Mathematical Expression Omitted] where [p.sub.L] = (1 - [(1 - 4k).sup.1/2])/2 and [p.sub.U] = (1 + (1 - 4k)[.sup.1/2]) /2 for k [is less than or equal to] 0.25. Finally, the locus of prices and quantities that yields zero profit is defined by (20) [Mathematical Expression Omitted]

The functions (18)-(20) and the highest demand line are illustrated in Figure 6 for k = 0.02. Since the monopoly output without any constraint must satisfy both (18) and (19), [p.sub.m] and [q.sub.m] are determined at the intersection of these two curves.

The public contractor can induce the monopolist to provide a larger capacity than [q.sub.v at a low price by selecting a regulated price from the interval(6) ([p.sub.min], [(k).sup.1/2]), as shown as in shown in Figure 6. Note that, in the neighborhood of [P.sub.min], monopoly output is elastic to changes in the regulated price. A large public-private capacity differential can thus be obtained for a relatively small public-private price differential. The probability of high demander support for privatization that was absent in the nonregulated monopoly setting is, correspondingly, increased. We find a set of regulated prices for which such support indeed materializes. For example, when k = 0.01, a regulated price [P.sub.r] =0.125 and optimal monopoly output of 0.575 secures a majority (56%) of the votes when paired against the associated public solution. The support for the monopoly alternatives comes from both low end (20%) and upper end (36%) demanders.

If the public agent chooses to specify output when contracting with the private monopolist, the monopoly price is determined by the [P.sub.s] line if quantity is set below 2/3 and by the highest demand line (q = 1 - p) if quantity is set above 2/3. The public authority needs only to search for a quantity in the range of ([q.sub.m], [q [bar].sub.s]), where [q [bar].sub.s] is determined by the intersection point of the highest demand line and the zero profit ([Pi] = 0) curve.(7) If the quantity is set below [q.sub.m], the monopoly price [P.sub.s] is even higher than [P.sub.m] which makes the quantity-regulated monopoly even less attractive to the upper end demanders. When the quantity is set above [q [bar].sub.s], the monopoly profit is negative for all feasible prices and hence, the monopoly firm will not produce. It is apparent from Figure 6 that the monopoly firm can be induced to produce the largest output at the lowest price when the quantity is set at [q [bar].sub.s]. The price at this output is (1 - [q [bar].sub.s]) and the monopoly makes zero profits. When the regulatory authority sets the quantity at [q [bar].sub.s], we find a majority coalition of the low and upper demanders in favor of the monopoly provision: 64% when k = 0.01 and 59% when k = 0.02. A majority support for the monopoly is preserved even when the quantity is set slightly below [q [bar].sub.s], allowing positive for the firm. The percentage of the voters in support of the monopoly provision decreases as k increases, and the majority coalition ceases to exits when k becomes large.(8)

Finally we consider awarding the monopoly franchise on the basis of competitive price bidding. This may represent the most realistic partial privatization arrangement, as it mirrors the actual process of contracting out refuse collection services in a number of U.S cities [9]. The public agency again specifies the output, but now potential suppliers enter bids as to the user price to be charged. We will assume that the auction design results in a zero profit solution. From the analysis above we know that output [q [bar].sub.s] and price (1 - [q [bar].sub.s] represent a politically viable zero-profit solution (for low values of k). Unlike the case of quantity regulation, however, the zero-profit price auction strategy attracts the largest support for the monopoly when the quantity is set well below [q [bar].sub.s]. This follows from the positive slope of the zero profit locus over the relevant quantity range. The inframarginal consumer surplus gains due to the fall in price dominate the marginal surplus losses due to the reduction in capacity. The range of quantities and the range of values of k for which a majority of consumers can be made better off by privatizing via the auction approach is, therefore, quite wide.(9)

VI. Conclusions and Extensions

In order to assess the potential advantage afforded by the privatization of delivery of public goods, the properties of the private and public equilibria must be established. In this paper, we have identified the relevant equilibrium prices and quantities associated with public and private monopoly models of the provision of excludable, non-congestible public goods. For this class of commodities, our analysis suggests that unfettered private monopoly may not provide a politically viable alternative public provision, since the majority of our consumer-voters are made worse off (in the absence of sizeable cost differentials) under the pure monopoly solution. We are, however, able to identify mixed/private contracting arrangements which are preferred to the pure public situation by a majority of the voters. These results highlight the importance of evaluating privatization along a continuum from the pure public to the pure private poles. The variety of contracting arrangements and the associated variety of equilibria should provide a basis for continuing research in the public goods area. This research will provide an interesting complement to the current work on auction and contract design in the industrial organization literature [8].

Appendix A: Proof of Proposition 1

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Author: | Hwang Hae-Shin |
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Publication: | Southern Economic Journal |

Date: | Apr 1, 1992 |

Words: | 5563 |

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