MAJORITARIAN MANAGEMENT OF THE COMMONS.

Buchanan, James M. ; YOON, YONG J.

YONG J. YOON [*]

This article analyzes usage of a common property resource, "the commons," under collectivization as compared with more familiar privatization institutional arrangements. Particular emphasis is on majority decision rules. When separate majority coalitions may authorize simultaneous usage of a common resource, total value is dissipated, but the interdependencies introduced by possible membership in differing coalitions to an extent reduce the incentives for exploitation. The formal analysis is analogous to that familiar in Cournot-Nash duopoly-oligopoly models but with differing efficiency implications. The argument has relevance for differential-benefit public spending from general tax sources, as well as other applications. (JEL D7, H00, 02)

I. INTRODUCTION

The operation and management of common property resources, "the commons," have been exhaustively examined in economics and political science, both in formal analyses and in practical application. [1] However, the full range of institutional alternatives has not been covered, even at a level of stylized interaction models. Our purpose in this article is to fill in at least some of the gaps here and, specifically, to examine the predicted results that emerge when usage decisions are collectivized under majoritarian voting rules. To our knowledge, this analysis has not been undertaken.

We shall, in section II, first summarize the classic analysis along with the familiar institutional reform suggested as a solution to the "tragedy." This solution, "privatization," is analyzed under several institutional variants in section III. We then shift attention, in section IV, to arrangements under which usage of the commonly shared resource or facility is chosen by some predesignated collective decision rule.

Specifically, we concentrate analysis on the predicted results in several majority rule environments. We analyze thoroughly the case in which different majority coalitions make usage decisions separately and simultaneously. The analysis incorporates recognition on the part of any member of a specific majority of the prospect for membership in alternative majority coalitions. We demonstrate that instead of full dissipation the level of usage that emerges in the limit is structurally analogous to the outcome in the Cournot-Nash duopoly model, even if the efficiency implications differ in the two settings.

II. THE CLASSIC MODEL SUMMARIZED

The familiar "tragedy of the commons" lends itself to stylized illustration. An immobile but renewable resource exists (a common pasture, aquifer, fishing ground) and has economic value if used effectively through the application of units of complementary mobile resources that are themselves valued in their alternative market opportunities. There are no assigned ownership rights to the immobile resource or facility. Free and open access may dissipate most or even all of the potential value.

Consider Figure 1. The productivity of mobile complementary resource units, as applied elsewhere in the economy (e.g., steers grazed on the open range, beyond the commons) is set at w, a value assumed constant. The quantity of such inputs applied on the commonly used facility is measured along the abscissa. In Figure 1, the marginal and average productivity curves for the units of applied complementary resource are shown as MP and AP. Total value is maximized when [Q.sub.m] units are applied, at which point marginal product values are equalized as between this usage and other opportunities. The maximal value of the commonly used resource is the area under the MP curve above w.

If entry to usage is open, and if the institutional structure is such as to allow for decentralized decisions on the part of many potential users, rational choices made separately will generate a full behavioral equilibrium at [Q.sup.*], where "average" product per unit of the complementary resource is equalized as between the commons and other uses. [2] The commons is overutilized in the sense that its capacity to generate positive value is fully dissipated. In the limit, the economy, in the aggregate, generates the same value with and without the existence of the resource.

The institutional reform that is suggested to be required to generate efficient usage is summarized under the word privatization. Assignment of rights of ownership will shift the incentive structure. The assigned owner will find it advantageous to restrict usage of the facility to the profit-maximizing level. In the illustration, all users will he charged a price (or shadow price) of (y - w) per unit of complementary resource, thereby generating equilibrium usage at [Q.sub.m]. A single owner of the facility, or a single person (group) assigned exclusive rights of usage, has an incentive to operate the facility optimally (through internal shadow pricing or external pricing) because the interdependencies among the separate units applied to the facility are brought within the single decision calculus.

Note that the familiar model embodies only two alternative equilibria: (1) complete dissipation under open access ([Q.sup.*]) and (2) fully efficient usage ([Q.sub.m]) under private ownership. Open entry, consequent on the absence of ownership, along with monopoly or single ownership with rights of full exclusion, are the only two institutional arrangements normally considered. [3]

III. MULTIPLE USERS

In this section, we analyze institutional arrangements in which usage of the shared facility is not open to all who might desire access, but, at the same time, rights of usage are not assigned to a single person or to a single group within which incentives are perfectly aligned.

Consider, first, a modified "privatization" arrangement that assigns rights to two selected persons. Each of these persons is allowed to apply as many units of the complementary resource to the facility as she desires. The designated "owners" may apply units directly or they may sell permits to others, thereby securing rents.

What level of usage will emerge in this institutional structure? When rights to use the facility are granted simultaneously to two persons (groups) rather than only one, only part of the interdependencies are internalized. Total usage will be extended beyond that level that meets efficiency criteria but will remain below the level that defines total value dissipation. [4]

The characteristics of interaction in the two-person model are structurally similar to those in the stylized setting of Cournot-Nash duopoly. In the two-person commons usage equilibrium, each user gets an equal share in the value generated on the commons, although less than one-half of the maximum value that a fully efficient operation would ensure. The similarity to Cournot duopoly is formal rather than substantive because the efficiency implications differ. In the classical duopoly-oligopoly model, increases in the number of independently acting firms reduce the efficiency loss from market restriction. In the commons analogue, increases in the number of independently acting users increase the efficiency loss from overutilization.

A similar construction may be carried out on models where more than two parties are assigned rights to use the commonly shared facility. Comparable analyses can be applied for larger numbers of users. [5]

An Illustrative Algebraic Model

Assume a simple linear production function for which the average product is given by AP = k - X where k is a constant and X is the number of units of mobile resources applied to the commons. A monopolist (single owner) recognizes the interdependence among units and maximizes rent by setting MP(X) = k - 2X = w, where w is the product generated per unit of mobile resource applied in other uses (assumed constant). Net rent for the monopolist is [(k - w).sup.2]/[2.sup.2].

Now suppose there are two "owners" (players). Following an approach analogous to the standard Cournot-Nash, the variable under control of each person is the number of units of mobile resources ([X.sub.i]). Given [X.sub.2], player 1 chooses [X.sub.1] to maximize the rent, average product times her own input: Max (K - [X.sub.1] - [X.sub.2]) [X.sub.1] where K = k - w. the first-order condition is [X.sub.1] K/2 - [X.sub.2]/2. The symmetric solution is [X.sub.1] = [X.sub.2] = K/3 and the aggregate rent is 2[K.sup.2]/9. With the same aggregate production function, when there are n users, the symmetric solution is [X.sub.i] = K/(n + 1) and the total rent is [nK.sup.2]/[(n + 1).sup.2]. Note that this value approaches zero as n approaches infinity. [6]

As the simple illustrative model suggests, the potential aggregate surplus (the productivity of the commonly used facility) asymptotically approaches zero as the numbers of persons assigned rights of open access increase. Note, however, that, for any finite number of assigned rights owners, some share of the externalities will be internalized. The emergent solution will approach but never attain the position depicted at [Q.sup.*] in Figure 1, where all of the resource value is dissipated through overusage. This position [Q.sup.*] is attained, however, when no person or group is explicitly assigned rights to use the common facility, but when all persons (groups), that is, anyone, may have access. In this setting, any productivity above that available to mobile resources elsewhere in the economy (w) will tend to attract users, no one of whom will find it rational to restrict usage due to internalization of external Diseconomies. [7]

IV. COLLECTIVE DETERMINATION OF USAGE

To this point, discussion has been preliminary to our primary purpose, which is to analyze those settings in which, for any reason, privatization is not implemented, even though the inefficient overusage consequent on open access is acknowledged. Consider, then, the situation where the choice of levels of usage of the common facility is explicitly collectivized; usage levels are determined only on collective decisions made by specified rules.

Our ultimate motivation for extending the commons metaphor to this application rests on the central idea that democratic politics, as it operates, allows members of majority coalitions to impose external diseconomies on general taxpayers. Government actions emerge from separately formed fiscal coalitions rather than from a monolithic collectivity. The general tax base is subject to simultaneous exploitation by majority coalitions of differing composition.

To isolate the differences that may be attributed to institutional structures, as such, it is first necessary to ensure that the basic economic relationships remain invariant. Recall the classic model in sections II and III. There exists an immobile, renewable resource that becomes productive of value if usage is restricted below the level that open or free access would generate. Persons, in their private capacities, will use the commons to secure the value added to their privately owned complementary resource units that are applied to the facility. There is no "publicness" in the benefit streams sought by those who use the commons, whether or not the number of users is restricted.

This feature of the model must be retained, as the decisions on usage are shifted from individual users to collective institutions. The values that emerge from usage of the commons are separable as among users. The rule for making collective decisions will then determine not only the level of usage of the commonly shared facility but also the distribution, among users, of the surplus value that the commons produces. To refer again to the familiar illustrative example, it is as if coalitions of users, chosen by the operation of the collective decision rule, are allowed to place jointly owned steers on a common pasture. We assume that within any decisive coalition, members share symmetrically in the surplus value, with no spillover benefits to those outside the coalition, who remain, necessarily, nonusers. Within the authoritative group incentives are perfectly aligned among individuals. This assumption allows us to simplify our analysis and focus on institutional structure and to ignore the incentive problems that may arise among asymmetrically situated users.

Alternative Decision Rules in the Single-Choice Model

The simplest possible model where usage of the commons can be settled is by a single choice. Initially, assume that a single person is to be granted exclusive authority to make usage decisions for the collectivity. Under the stylized assumptions of the model, the designated decision maker will secure for him-or herself the full value of benefits. Rational choice will dictate that usage of the commons be restricted to that level that maximizes rent, thereby ensuring overall efficiency in resource allocation.

Suppose, now, that, instead of a single dictator, the collective decision rule is one of simple majority. Usage of the commons takes place only on the decision made by a majority of the community membership, any initial majority that happens to be organized. Because we are in a one-choice model, however, any majority, once formed, will have an incentive to use the facility so as to maximize rent. The only difference between this model and that of the single person dictatorship lies in the distribution of value. In the majority setting, all members of the coalition share in this value surplus. Individuals that are not coalition members do not have access.

The allocative result remains unchanged no matter what the required size of the coalition that is charged with authority to make collective decisions, including the rule that requires consent of all members of the polity. [8] In the limiting case, unanimity, all persons share in the surplus that the existence of the commons makes possible.

Alternative Decision Rules When Many Choices Are Possible

We may now examine settings that allow simultaneous choices to be made separately, as authorized by a political decision structure. (The budget process under majoritarian democracy offers an example.) If the collective decision rule dictates that usage decisions may be made by anyone, acting separately, the full "tragedy of the commons" is present in the sense that all of the value potential of the facility is dissipated. (The size of a decisive coalition is one in this case.) As the collective decision rule becomes more inclusive, thereby allowing usage decisions to be made only by coalitions of persons authorized to take collective decisions, we can trace the relationships between the size of decision-making coalitions and the total share of potential value that is dissipated. This share of efficiency loss decreases as the size of the decision-making coalitions increases. [9]

The interesting case to be examined in detail here is that which allows differing majority coalitions, separately, to make usage decisions. A practical fiscal example that comes to mind is that in which special spending interest, combined in separate logrolling coalitions, exploit the general tax base. [10] In political reality, many such coalitions might be formed. But it is analytically useful to remain with the stylized model and suppose, initially, that only two majority coalitions may be organized and that each coalition will be allowed to use the commons at will. It is as if the separate coalitions become separate "persons" who are assigned simultaneous rights of usage.

Consider, then, the calculus of a member of one of these coalitions as concerns separable usage of the commons by that coalition. If the reference person knows that he or she will not be a member of the other majority, the solution is analogous to that under privatization when two person are assigned rights of usage. As noted earlier, the problem is structurally similar to the classic Cournot-Nash models. Each decision unit, in this case, each majority coalition, will internalize one-half of the external diseconomies resultant from increments to usage rates.

Suppose, now, however, that the reference person is in a position of neutrality or anonymity with respect to membership in the second coalition. That is, it is as if the second coalition is to randomly organized from among all members of the polity. The person then faces the probability of one-half of being a member of the other majority. [11] (To be more accurate, the probability depends on the size of polity. Appendix A provides a rigorous derivation of this result.)

The usage decision of the reference person of the first coalition can be formulated using the illustrative algebraic model introduced in section III. The average productivity relationship of the commons is P = K - X, where P is the average product, and X is the aggregate quantity, divided between the two coalitions, X = [X.sub.1] + [X.sub.2]. Given [X.sub.2], the reference person of the first coalition chooses [X.sub.1], to maximize the rent [PX.sub.1] + [PX.sub.2]/2; the productivity of mobile resource applied in other uses is assumed to be zero. The necessary condition for rent maximization is K - [X.sub.1] - [X.sub.2] - ([X.sub.1] + [X.sub.2]/2) = 0; the solution is [X.sub.1] = K/2 - 3[X.sub.2]/4, and the symmetric equilibrium is [X.sub.1], = [X.sub.2] = 2K/7. The result emerges from the simultaneous usage decisions made by two separate majority coalitions. Aggregate usage is X = 4K/7, which is less than the duopoly analogue quantity of 2K/3 but larger than the monopoly (efficient) output K/2. That is, K/2 [less than] 4K/7 [less than] 2K/3.

Note that efficiency is increased over that attained in the duopoly analogue case. Externalities are internalized in two different channels, which we may call the productivity externality and the membership externality.

Only the productivity externality is internalized in the duopoly analogue case.

By relaxing the assumption of only two coalitions, we may postulate m majority coalitions. [12] The individual member of any coalition, say, coalition I, will prefer to apply [X.sub.1] units of mobile inputs, given [X.sub.2],..., [X.sub.m], so as to maximize the effective rent,

(1) (K - [X.sub.1] - [X.sub.2] - ... - [X.sub.m])

x [[X.sub.1] + ([X.sub.2] + ... + [X.sub.m])/2].

The necessary condition for maximization is

(K - [X.sub.1] - [X.sub.2] - ... - [X.sub.m])

- [X.sub.1] - ([X.sub.2] + ... + [X.sub.m])/2 = 0,

and the symmetric solution is

[X.sub.1] = [X.sub.2] = ... = [X.sub.m] = 2K/(3m + 1);

the aggregate quantity is

X = [mX.sub.1] = 2mK/(3m + 1).

As the number, m, of majority coalitions increases, the aggregate quantity X approaches the duopoly analogue equilibrium 2K/3 instead of full dissipation K, which is predicted when the polity fails to internalize productivity externalities. The reference person who knows that all possible majority coalitions will be organized and authorized to dictate collective action will recognize that he or she will be a member of one-half of these coalitions. He or she will internalize one-half of the externality of usage in his or her rational choice calculus, there by generating a result equivalent to that reached when two separated but symmetrical decision units hold usage rights. See Appendix B.

Our analysis is based on the stylized model that satisfies the sufficient conditions for existence and uniqueness of Nash equilibrium--the revenue function is strictly concave. See Watts (1996) for a general discussion on this issue.

V. UNCERTAINTY INTRODUCED

To this point, we have assumed that, although we have allowed for probabilistic determination of any person's position in the majoritarian models, genuine uncertainty, as such, has not been incorporated into the decision calculus. But how may a person who holds membership in an already organized majority coalition incorporate a fully "rational" expectation that a specific number of alternative majorities will be formed and authorized to make usage decisions and, further, that he or she will really have a probability of one-half of attaining membership in any such coalition? How can a "rational" internalization of externalities take place?

There are two separate dimensions along which degrees of uncertainty may be measured: (1) the probability of membership in majority coalitions, and (2) the number of coalitions that will be organized and authorized to make collective decisions concerning usage of the commonly shared facility. These dimensions can be related, one to the other, in a geometric representation as in Figure 2, which incorporates the simple algebraic relationships introduced earlier.

The probability, h, of membership in majority coalitions is measured along the ordinate; aggregate usage of the commons, X, is measured along the abscissa. As the algebraic models indicate, efficient (monopoly) usage of the commons is achieved when the probability of being decisive is unity. If the reference person, whose decision calculus is under examination, knows with certainly that he or she will be a member of any decisive coalition that may be formed, his or her choice will embody an internalization of all of the interdependencies of commons usage; he or she will prefer to operate the facility efficiently (at K/2 in Figure 2), regardless of the number of differing coalitions that may be authorized to make decisions.

Suppose, however, that membership in coalitions is uncertain for the reference person. Suppose she knows only that she holds membership in one authoritative coalition; she remains uncertain both as to the number, m, of other coalitions that may be authorized to utilize the commons and as to the probability that she will, personally, be a member of any other coalitions that may become authoritative.

The general model may be constructed, in which individuals may assume a probability h of becoming a coalition member, and h can vary over the whole range of probability, 0 [less than] h [less than] 1. We assume that individuals know the number, m, of coalitions to be organized. Later we relax this assumption.

Given the decisions [X.sub.i] of other coalitions, the current majority coalition chooses [X.sub.1] to maximize the rent,

(2) (K - [X.sub.1] X')([X.sub.1] + hX'),

where X' = [X.sub.2] + ... + [X.sub.m]. The first-order condition is K - [X.sub.1] - X' - ([X.sub.1] + hX') = 0; the symmetric solution, [X.sub.1], and the aggregate usage, X, are,

(3) [X.sub.1](h,m) = K/[2+(1+h)(m-1)]; X(h,m) = mK/[2+(1+h)(m-1)]

The general characteristics of the symmetric solution for different values of m and h are, as noted, depicted in Figure 2.

We may consider several stylized submodels, some of which we have already introduced. Suppose that the reference person expects to be a member of any majority coalition with a probability of one-half (h = 0.5). In this setting, we may array the set of positions along the horizontal line drawn from one-half on the ordinate in Figure 2. The extreme position, at T, represents the position discussed earlier, where each and every possible majority is authorized to use the commons but where the individual expects to be a member of each coalition with the probability of one-half. The preferred degree of utilization of the commons is analogous to the Cournot-Nash duopoly analogue equilibrium, as is elaborated in Appendix A. [13] In the simple algebraic model, usage of the commons is determined at 2/3 of that level reflecting full dissipation of all value, whereas the efficient exploitation of the facility is at 1/2 this level.

Note that, so long as the individual's expectations embody a "fair chance" (or better) prospect (i.e., optimism), the aggregate usage of the facility will lie between the duopoly-analogue solution (at 2K/3 in Figure 2), and the monopoly-analogue solution (at K/2 in Figure 2). Internalization of the interdependencies will cause the reference person to act so as to generate usage rates below the duopoly analogue solution in all cases.

With majority decision rules, a higher usage level will emerge only if individuals' expectations concerning membership in alternative coalitons are pessimistic, that is, if probabilities of being a member of alternative coalitions are placed at less than one-half. Consider the most extreme limit here. The individual who is a member of a current majority does not expect to be a member of another majority with any probability above zero. In this case, differing levels of usage may be measured along the abscissa, as determined by the expected number of alternative coalitions to be formed. If, for example, only one alternative majority coalition is anticipated, although the reference person holds no prospect at all of membership, the duopoly analogue solution appears. Full dissipation of the facility will be generated if the reference individual expects many other (unspecified number of) majority coalitions to be organized and authorized to use the commons.

So far we have assumed that individuals make accurate assessment of the number, m, of majority coalitions that are expected to be organized and authorized to make simultaneous decisions on commons usage. If m is uncertain, the reference person of a current majority coalition must make usage decisions without knowing the number of simultaneous coalitions that will jointly exploit the commons. In this setting, for any given probability (h) of membership in alternative majorities, increases in the expected number of coalitions will decrease individual rates of usage of the common facility. Realized usage will, of course, depend on the number of coalitions actually organized.

VI. ALTERNATIVE DECISION RULES IN TEMPORAL SETTINGS

To this point, we have limited analysis to models in which decisions to use the commons are made for one period only, without consideration of the choice settings in subsequent periods. Introduction of a temporal dimension will not be important if the usage of the commonly shared resource is, itself, temporally concentrated in a single period, that is, if the resource fully renews itself each period (e.g., the pasture grass grows back each spring) while the decision authority is invariant over a sequence of periods.

Suppose, however, that there exist intertemporal as well as intratemporal interdependencies in usage of the commons. The addition of a unit of the complementary resource to the facility will reduce the productivity of all units both currently and also in subsequent periods. In this setting, any dilution in ownership or decision authority, currently or intertemporally, will guarantee inefficient depletion of value. [14]

What will be the pattern of results if the collective decision authority, in each separate period, is lodged in a single simple majority coalition, any such coalition that is organized? Consider the position of a person who finds himself a member of an initial coalition, who thereby has a claim to a symmetric share of any surplus value generated during that period by applying units of the complementary resource inputs to the commons. He knows that, in each and every subsequent electoral period, a differing majority may be decisive, but he knows that there is some probability that he will be a member of such a majority in any designated period.

But how will a member of the majority coalition incorporate interperiod externalities into a rational calculus? It seems clear that some share of the intertemporal externalities will be internalized. On the other hand, because only a share of the intertemporal interdependencies will be internalized, the level of usage will be higher for the initial periods than that which would emerge under permanent location of decision authority. The value of the commonly used facility will, to an extent, be "mined" through early period excess usage. The precise relationships will, of course, depend on the importance of the intertemporal relative to the intratemporal externalities.

The usage decision can be formulated similarly to the algebraic model introduced in section III. As an illustration, consider a commons that lasts two periods; steers need two periods to mature, and usage (and value distribution) is lodged in sequential majorities. The sequential interaction is analogous to the Stackelberg duopoly. The reference member of the first majority coalition chooses [X.sub.1] to maximize the effective rents,

(4) Maximize (K - [X.sub.1] - [X.sub.2])[X.sub.1] + [hV.sub.2],

where h = 0.5 is the probability that the reference person will belong to the second majority coalition, and [V.sub.2] is the surplus that a new coalition will enjoy. Given [X.sub.1], the rent is

(5) [V.sub.2]([X.sub.1]) = Max (K - [X.sub.1] - [X.sub.2])[X.sub.2],

where [X.sub.2] is the choice variable for a new majority. Solving the maximization in (5), we obtain the usage and rents as a function of [X.sub.1]:

(6) [X.sub.2]([X.sub.1]) = (K - [X.sub.1])/2

[V.sub.2]([X.sub.1]) = [[(K - [X.sub.1])/2].sup.2].

Substituting the results into (4), we solve recursively: [X.sub.1] = 3K/7 and [X.sub.2] = 2K/7. The total usage (2K/3) is higher than the case of permanent ownership (K/2) and also higher than that of two simultaneous majorities (4K/7). This result is equivalent to the

Cournot doupoly analogue that was analyzed in section III. In the Stackelberg game analyzed here, the leader internalizes the response of the follower in productivity externality. As the leader increases usage rate, half of his gain is transferred from the follower, and this transfer is one-half of the loss to the follower. But the membership externality internalizes half of the loss from the follower. The gain and loss will cancel each other out, and the leader behaves like an independent simultaneous duopolist.

These results may be generalized to models that embody more than single choices within each period. As the earlier analysis demonstrated, increases in the number of authorities in any single period will increase the rate of overutilization of the common resource. If we introduce temporal elements in usage, the overusage in any given period, with the number of decision authorities fixed, will further increase value dissipation.

VII. GENERALIZATIONS AND CONCLUSIONS

We have analyzed the usage of a commons with differing institutions of decision making, with particular emphasis on collective management under majority decision rules. Even within this set, we have not fully exhausted all possible structures, but perhaps the analyses have been sufficiently extensive to suggest further developments. Also, we have not, even for the particular models examined, fully developed the formal analyses that might add rigor to the comparative treatment.

We have not attempted to identify those settings in which collectivized management of commonly shared resources seems more likely to emerge than the more familiar privatization alternatives. One set of relevant examples here might be those situations where "the commons" itself is not normally considered as an economic resource. Geographic location offers the obvious example of the island of Bali, where separable private ownership of land parcels may generate results akin to the familiar "tragedy," measured in lost opportunity value.

Buchanan: Advisory General Director, Center for Study of Public Choice, MSNIE6, Buchanan House, George Mason University, 4400 University Drive, Fairfax, VA 22030. Phone 1-703-993-2327, Fax 1-703-993-2334, E-mail jburgess@gmu.edu

Yoon: Senior Research Associate, Center for Study of Public Choice, George Mason University, 4400 University Drive, Fairfax, VA 22030. Phone 1-703-993-2332, Fax 1-703-993-2334, E-mail yyoon@gmu.edu

(*.) Helpful comments on earlier drafts were offered by Roger Congleton, Tyler Cowen, Robert Tollison, Elinor Ostrom. and anonymous referees.

(1.) The issues have been central in the development of welfare economics for almost a century, especially since Pigou's analysis of decreasing returns industries, along with early criticisms. See Pigou (1912, 1920), and Knight (1924). The specific theory of common property resource management received seminal treatment in papers by Gordon (1954) and Scott (1955). The term tragedy of the commons was introduced by Hardin (1968). Ostrom (1990) fully discussed the literature and, in particular, examines alternative institutional means through which common property resources have been, in fact, managed.

(2.) With a linear relationship, [Q.sup.*] = 2[Q.sub.m]. Full dissipation (at [Q.sup.*]) is reached at a position where [Q.sup.*] [greater than] 2[Q.sub.m] when the productivity curves are concave upward and vice versa.

(3.) A partial exception is provided by some analyses of management of local commons which discuss self-governance by asymmetrically situated users. See Ostrom and Gardner (1993) and Seabright (1993).

(4.) We assume that Coase-like contracts between the parties are, for some reason, impracticable. There will, of course, exist mutual gains from such contracts that, if implemented, will generate the efficient result.

(5.) Moulin (1995) and Watts (1996) offer analyses of the classic commons model as analogous to Cournot-Nash oligopoly. Levhari and Mirman (1980) use the Cournot-Nash model to analyze intertemporal usage of a fishing stock with natural growth. Gardner et al. (1992) derive the symmetric Nash equilibrium in commons usage in game theory formulation.

(6.) For a nonlinear production function, a unique Nash equilibrium still exists if revenue is a strictly increasing concave function. See corollary 1 in Watts (1996).

(7.) In any stylized open-access equilibrium, a finite number of users will exist. If these users are assigned definitive rights of usage, to the exclusion of potential entrants, some reduction in usage will occur because of internalization of production externalities. In addition, incentives will be established for the emergence of pairwise exchanges of usage rights between sets of users. Both effects reduce overusage. A practical example is offered by the Icelandic fishing industry. Overinvestment in boats generated overfishing with no net returns. A proposal to assign rights exclusively to current boat owners was politically unacceptable because of an unwillingness to allow current owner to secure the positive rents promised by reduction in fleet size. This information is supplied by Dr. Hannes Gissurarson, University of Iceland, occasional consultant to the fishing industry.

(8.) See Walker et al. (1997) for an interesting experiment on collective management of a commons under varying decision rules. The experiment was limited to the single-choice setting, and the focus of interest is on convergence toward efficiency in usage under majority rule and unanimity.

(9.) The relationship here is fully analogous to that between "external costs" and the size of the decision-making coalition, as analyzed in Buchanan and Tullock (1962). Their analyses were not, however, placed in the common property setting as such.

(10.) See Wagner (1992).

(11.) The model becomes analogous to an interesting variant of the Cournot-Nash duopoly setting in which the single owner of one firm has a one-half probability of being the owner of the second firm.

(12.) We may treat the problem as being analogous to an oligopoly setting in which the owner of a single firm has a one-half probability of being the owner of any other firm. See Appendix 1 for further discussion on the one-half probability.

(13.) A direct proof can be obtained by using equation (3). If h equals 0.5 in equation (3), then as m grows without bound, X(0.5, [infinity]) = 2K/3. For a Cournot duopoly, h = 0 and m = 2, and X(0, 2) = 2K/3. Appendix 2 provides a general proof.

(14.) Aizenman (1992) formulates an intertemporal commons model with reference to separate national issuing authorities in a monetary union.

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APPENDIX A

PROBABILITY OF MEMBERSHIP

For a reference person of the current coalition, the probability of becoming a member of other majority coalitions depends on the size of polity. For instance, for a five-member polity, a majority coalition requires three persons, and there are ten possible coalitions in total: C(5,3) = (5 x 4 x 3)/(3!) = 10. A reference person belongs to six coalitions: C(4, 2) = 6. Thus, the probability for him to belong to other coalitions is (6-1)/(10-1) =5/9 [greater than] 0.5.

As the size of polity increases, however, the probability converges on one-half. Let 2m + 1 be the number of polity, then the number of possible majority coalitions is equal to the number of different ways of choosing m + 1 out of 2m + 1 members

C(2m +1, m +1) = C(2m +1, m) = P(2m +1, m)/m!,

where P(2m + 1, m) is the permutation of ordering m out of 2m+1 objects, and the m factorial, m!, is, m times (m - 1) times (m - 2), and so on, until 1. The formula for permutation is

P(n,r) = n(n-1)...(n-r+1).

The number of majority coalitions of which the reference person of the current coalition becomes a member is

C(2m, m) = P(2m, m)/m!,

and the probability, p, of this person becoming a member of another majority coalition is p = [C(2m, m) - 1]/[C(2m + 1, m + 1) - 1]. Approximately,

p = C(2m, m)/C(2m + 1, m +1)

= (m + 1)P(2m, m)/P(2m +1, m +1)

= (m + 1)(2m)(2m - 1)... (2m - m + 1)

/(2m + 1).. . (m + 1)

= (m+1)/(2m+1),

which converges to one-half in the limit.

APPENDIX B

MANY MAJORITY COALITIONS ARE ANALOGOUS TO COURNOT DUOPOLY

In the text we consider the scenario in which different majority coalitions are authorized to make usage decisions separately. Each person in a majority coalition recognizes that, with probability one-half, she will be a member of every other possible coalition. (See Appendix A, where the one-half probability is discussed.) The reference person will rationally internalize one-half of the negative external effect caused by marginal extension of usage. We show that, in the limit, the reference person behaves analogously to a duopolist in the symmetric Cournot-Nash solution.

Majority Coalitions

Let P(X) denote the average product when aggregate usage is X. Each coalition, say coalition 1, tries to maximize the revenue P(X)([X.sub.1] + X'/2) where [X.sub.i] denotes usage rates by coalition i and X' = X - [X.sub.1] = [X.sub.2] + ... + [X.sub.m]. The first-order condition is

(B1) P(X)[dX.sub.1] + (X/2 + [X.sub.1]/2)dP = 0.

Marginal change in [X.sub.1] causes changes in the average product P through aggregate usage X. The external effect, caused by a marginal change [dX.sub.1], is XdP and is denoted by dE : dE = XdP. Then the first-order condition in the symmetric equilibrium becomes

(B2) P(X)[dX.sub.1] + dE/2 + dE/(2m) = 0,

where the second term, dE/2, is the membership externality, and the third term, dE/(2m), is the productivity externality. Both are internalized for allocation decision.

For large number m, less and less productivity externalities are internalized and, in the limit, the first-order condition becomes

(B3) P(X)[dX.sub.1] + dE/2 = 0.

Cournot Duopoly

We interpret the average product function above as the demand relation faced by the duopolists. Quantity demand is X and P(X) is the price. A duopolist tries to maximize her surplus, P(X)[X.sub.1], where X = [X.sub.1] + [X.sub.2], and the first-order condition is P(X)[dX.sub.1] + [X.sub.1]dP = 0. In a symmetric equilibrium, [X.sub.1] = X/2 and the first-order condition will satisfy

(B4) P(X)[dX.sub.1] + (X/2)dP = P(X)[dX.sub.1] + dE/2 = 0.

The duopolist internalizes the productivity externality dE/2.

The first-order conditions (B.3) and (B.4) assume the identical form, and it is no surprise that they imply analogous behavior.

[Graph omitted]