E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

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ANALYSIS

Inefficiency and common property regimes

Daniel Fuentes-Castro⁎

University of Zaragoza, Department of Economic Analysis, Faculty of Economic Sciences and Business Administration, Gran Vía, 2,50005 Zaragoza, Spain

A R T I C L E D A T A

⁎ Tel.: +34 976 76 18 19; fax: +34 976 76 19 96E-mail address: dfuentes@unizar.es.

0921-8009/$ – see front matter © 2008 Elsevdoi:10.1016/j.ecolecon.2008.11.005

A B S T R A C T

Article history:Received 5 April 2006Received in revised form8 November 2008Accepted 9 November 2008Available online 29 December 2008

Much of the literature on the commons focuses on the fact that many agents are assignedusage rights simultaneously, but less attention has been paid to the exercise of exclusionrights. The simultaneous exercise of one of the two rights by all the owners of a commoncauses a problem of overexploitation in the first case (competition “in use”) and underuse inthe second (competition “in exclusion”). The relevance of both inefficiencies stems from theway they illustrate the general conflict between individual and collective interests. Thispaper proposes a formal synthesis of the problems of inefficiency associated with theexploitation of resources in common property regimes. The synthesis takes into account thefollowing features: i) the importance of the consumer surplus for the analysis of the issue; ii)the attitude of economic agents in the face of a reciprocal externality linked to theexploitation of the common; and iii) the social and the private costs of exploitation.

© 2008 Elsevier B.V. All rights reserved.

Keywords:CommonsAnticommonsOverexploitationUnderutilizationNatural resourcesCommon property

JEL classification:Q20; Q30

1. Introduction

Though the problem of resource overexploitation was firstformalized by Gordon (1954), it was not until the publication ofa paper by Hardin (1968), in which the author coined theexpression “tragedy of the commons”, that thematter enteredthemainstream. Since then the economic literature has seen adeluge of works concerning the overexploitation of “commongoods”, which are now more commonly referred to asresources “in common property regimes” (Bromley, 1992,1995). Overexploitation is posited as a problem of efficiencythat is not intrinsically tied to the nature of the goods andusually arises when several economic agents exploit acommon resource and none is capable of excluding any ofthe others. In his famous example Hardin concluded that

.

ier B.V. All rights reserve

communal pastures were doomed to overexploitation unlessthere was an “extension in the morality” of the economicagents. Hardin's pessimistic predictions were based on theexistence of a reciprocal externality that could not becorrected by the market. The major weakness of thisargument was his failure to the costs inherent in theexploitation of the resource. When these costs are taken intoaccount, overexploitation loses its extreme character. Also, asthe theory of market failure indicates, intervention mechan-isms exist that can help the regulator correct overexploitationand safeguard resources from inevitable exhaustion (Baumoland Oates, 1975).

Three decades later Heller (1998, 1999) popularized theantagonistic expression, “tragedy of the anticommons”, whichfirst entered the economic literature in Michelman (1967).

d.

1741E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

What Heller (1998) had in mind was a scarce resource that isprone to underuse because “multiple owners each have aright to exclude others, and no one has an effective privilegeof use.” In Heller's argument, the “tragedy of the antic-ommons” is identified with the extreme and unfailing under-utilisation of a resource in a common property regime. Hellerused the difficulties observed in the processes of privatizationduring the transition to capitalism in the Eastern Europe toillustrate his reasoning. In a contemporary paper Heller andEisenberg (1998) also identified the problem of anticommonswith the inefficiency of the system of biomedical patents inthe United States, arguing that “more intellectual propertyrights may lead paradoxically to fewer useful products forimproving human health (Heller and Eisenberg, 1998)”.Buchanan and Yoon (2000) were the first to rise to thechallenge of formalizing the “tragedy of the anticommons”.They proposed an algebraic example that shows certainsymmetry between the tragedy of the commons and thetragedy of the anticommons. These authors identify theanticommons problem with the inefficiencies caused byoverlapping regulatory bureaucracies in residential construc-tion. In a similar context Sim, Lum and Malone-Lee (2002)identify a situation of anticommons in the Singaporean real-estate market. Coloma (2003) improved the Buchanan andYoon model and showed that the apparent formal symmetrybetween the commons and the anticommons does not in factobtain.

In the case of both the commons the anticommons, theowners of common resources are assigned both usage andexclusion rights. When no owner is able to exclude any otherfrom access to the resource overexploitationmay occur. In theopposite case, the resource will be prone to underuse if anyowner is able successfully to enforce the right of exclusion.The present work proposes a formal synthesis of both types ofinefficiencies.We present amodel that combines the essentialfeatures of underuse and overexploitation of resources incommon property regimes. In our opinion these problemsturn out to be associated not only with the reciprocalexternality indicated in Hardin's example, but also with issuesof market power.

In Section 1 of the paper we refer briefly to the state of theart. The model analysing the exploitation of resources incommon property regimes is formulated in Section 2. InSection 3 we solve the model for the commons, and theanticommons are analyzed in Section 4, which also providessome figures to illustrate the main results of the analyses. InSection 5 we summarize our key conclusions from the study.

1 “He generalized the Pigouvian model for situations where privateproduction results in additional social benefits. The solution to themarket failure (supplying less of the good than is socially desirable) is tosubsidize the producer so that his private marginal cost is reduced to thepoint where he is willing to supply the socially optimal amount of thegood.” Carlson, Zilberman and Miranowski (1993, p 224).2 “While the tragedy of the commons is a problem of definition o

property rights that tends to attenuate when the economic agents havemarket power, the tragedy of the anticommons is a problem of definitionof property rights that is exacerbated by the market power (Coloma2003, p. 8)”.

2. Property rights, commons and anticommons

The overexploitation of commons is commonly occurs incontexts where too many agents have a right to use theresource, but no one has a legal right to exclude any otherplayer from access. Much of the literature on the commonsfocuses on the fact thatmany agents are assigned usage rightssimultaneously. However, little attention has been paid to theexercise of exclusion rights, which may cause a different kindof inefficiency, consisting of the underuse of commons. Therelevance of both inefficiencies stems from the fact that they

illustrate the global conflict between individual and collectiveinterests.

Though itwasMeade (1952)who first introduced the conceptof “anticommons” in the sense of a positive externality, asCarlson, Zilberman andMiranowski (1993, p 224) note1, the firstexamplesuccessfully to illustrate this inefficiencywasproposedby Heller (1998), who analyzed why many Moscow storefrontsremained empty in the early 1990's while tiny bazaars wereimprovised on the streets. The problem was in fact thatnumerous different stakeholders, privatization agencies, localauthorities and federal governments possessed a right toexclude. No-one could open a business without collecting allpermits from each one of these players. A similar case isanalyzed by Sim, LumandMalone-Lee (2002), whodescribe howmany property owners in Singapore banded together “tocapitalize on the marriage value of the en bloc site to reap bigwindfalls as compared to individual sales”.

Buchanan and Yoon (2000) also identify the anticommonsproblem with the inefficiencies introduced by overlappingbureaucracies, even going so far as to suggest that the environ-mentalmovementhaspreventedvalue-enhancingdevelopmentby interposing additional authorities with the right to excludedevelopments. These authors were the first to formalize thetragedyof theanticommonsbut, beyond its innovativeness, theirmodel displays certain weaknesses that make it unsuitable forthe analysis of resource exploitation in common propertyregimes.Notonlydo they fail to consider the costs of exploitation(a factor that was also omitted by Hardin), but their model failscrucially to introduce the reciprocal externality typical ofsituations of overexploitation, and therefore it fails to accountfor the relationship between congestion, market power andoverexploitation. For these authors the monopoly (or soleownership) solution is optimal, since it maximizes the benefitsand reduces the consumption of the commonwith regard to theperfect competition equilibrium. In their argument, however,Buchanan and Yoon omit the loss of well-being associated withthe reduction of the consumer surplus caused by the soleownership solution. This paper focuses on all of these points.

Coloma (2003) proposes an improved version of theBuchanan and Yoon model, including a quadratic costfunction to demonstrate that the formal symmetry betweenthe commons and the anticommons does not in fact obtain2.Though Coloma's contribution successfully resolves some ofthe limitations of the Buchanan and Yoon model (especiallythose relating to market power) it is open to improvement ontwo points. First, Coloma does not distinguish betweensocially responsible behaviour and potential free-riding bythe owners of the resource. Going back to Hardin's classicexample, this distinction would appear substantial. In Colo-

f

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1742 E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

ma's model, however, “the only lack of efficiency is thenegative reciprocal externality that every firm generates toother firms, and this leads unequivocally to a solution inwhich there is overexploitation of the common resource(Coloma, 2003, p. 5)”. Second, the private cost should bedistinguished from the social cost linked to the exploitation ofthe resource. This matter is not addressed in Coloma's model,with important consequences as we shall see.

In the following sectionswe construct a formal synthesis ofthe problems of inefficiency associated with the exploitationof resources in common property regimes, emphasizing thoseaspects that in our opinion are not considered simultaneouslyor are not explained with sufficient clarity in the literature, asfollows: i) the importance of the consumer surplus for theanalysis of the issue, ii) the attitude of the economic agents inthe face of a reciprocal externality linked to the exploitation ofthe common, and iii) the introduction of the private cost and(negative) external effect of exploitation.

3. The model

We consider an inverse linear demand function of the typeP=a−b·Q, where P is the price and Q the total quantity of agood exchanged in the market. For every unit of output theproducers consume one unit of a common input. Theparameters a and b in the demand function are both positive(the former is the consumer's reservation price and the lattermeasures the price increment associated with a marginalvariation in the quantity exchanged in the market). Theprivate cost for the firms exploiting the common is denotedby cQ, where c is positive. Note that the marginal cost isidentical for any firm (i.e., the firms in our model aresymmetric). In addition to the private cost there is a negativeexternal effect associated with the exploitation of thecommon. This external effect increases as the firms exploitthe common. We denote it by eQ, where e is positive, so thatthe social cost function (SC) involving private costs andexternal effects is written as follows:

SC Qð Þ = c � Q + e � Q ð1Þ

Themarginal social cost in the model takes the linear formMSC(Q)=c+e. In accordance with the example proposed byHardin (1968), Eq. (1) may be interpreted as affirming that theintroduction of an additional animal in themeadow generatesa private cost of cmonetary units (which is independent of thetotal number of cattle and is related, for example, withexpenditure on veterinary services, phytosanitary treatmentsand thenecessary conditions tomaintain every animal) and anexternal effect eQ, which is shared by all the animals grazing inthe meadow (because of the progressive impoverishment ofthe pasture in line with the total number of animals).

We consider a general function ofwell-beingwhich adds theconsumer surplus consumers to the producers' benefit and theexternal effect: W Qð Þ = RQ

0 P xð Þdx� c + eð ÞQ. Since the specifica-tion of the inverse demand function is known, the definition ofwelfare can be simplified to obtain the following expression:

W Qð Þ =Q � a� c� eð Þ � bQ2=2 ð2Þ

A simple calculation verifies that the quantity maximizingthis function (the social optimum) is

Q* =a� c� e

bð3Þ

Note that (a− c−e)N0 is required if Q4 is to be positive. Thisis a necessary condition for any result in the paper (otherwisecosts would be too high and the exploitation of the commonwould be unprofitable).

It is known that the consumption of a common goodinvolves rivalry but not exclusivity (Ostrom, 1990). In Hardin'spastures case this provides an incentive to keep adding moreheads of cattle to the meadow, which impoverishes thepastures and ends up harming the interests of the farmersby generating a reciprocal externality. Hardin (1968) concludedthat the common goods were doomed to overexploitation inthese circumstances, unless “a fundamental extension inmorality” exists. At this point the relevant issue is to includethe attitude adopted by agents in the face of damage to thecommon resource in the model. In the following sections weconsider two alternative cases. In the first the social cost isinternalized, and in the second none of the firms takes intoaccount the external effect on the environment (in any casethe exponent 0 refers to free-ridingwhile the exponent 1 refersto internalization).

4. Exploitation of resources in commonproperty regimes: competition in use

4.1. Basic findings

We refer to “competition in use” when the firms make use oftheir right to exploit the common resource (i.e. they decidewhether or not to use the resource and how it will be used intheir production processes). This is actually the usual mean-ing of competition. When there is internalization of the socialcost the maximization program for the i-th firm takes theform maxqi

Bi = P Qð Þ � qi � c + eð Þqi. Given the specification of thedemand function, the solution under Cournot competition(i.e., all the firms compete in use simultaneously) and in thepresence of identical firms verifies the quantities given below.Eq. (4) shows the Nash equilibriumwhen the firms themselvesabsorb the total external effect (QUse

1 ), while the Eq. (5) showsthe free-riding Nash equilibrium (QUse

0 ).

Q0Use =

n a� cð Þb n + 1ð Þ ð4Þ

Q1Use =

n a� c� eð Þb n + 1ð Þ ð5Þ

The literature on the exploitation of natural resourcesgenerally identifies the social optimum with the sole owner-ship equilibrium (Scott, 1955). The argument is that bycoordinating their interests, the n firms are obliged torecognize the total external effect as a proper cost. When then firms cooperate they have an interest in avoiding theoverexploitation of the resource, and so the social cost iseffectively internalized. As is well known, however, a mono-polist will exchange a small quantity of goods in the market

Fig. 1.

1743E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

than several firms competing under perfect conditions. This isbecause the monopolist can fix prices whereas each of the nfirms in perfect competition is a price acceptor. Themonopolyprovokes an irrecoverable loss of the consumer surplus. Thequantity exploited by themonopolist can be obtained from Eq.(5) when n=1:

Q1M =

a� c� e2b

ð6Þ

As may be seen from Eq. (5), the quantity representinginternalization of the social cost grows as market powerdiminishes (i.e. when the number of competitors increases)3.This means the consumer surplus can be increased at theexpense of the “extra” revenues of the n firms and the loss ofwell-being associated with market power can thus be pro-gressively recovered. Well-being reaches a maximum whenthe loss of the consumer surplus with the sole ownershipequilibrium is recovered. In other words, as n tends to infinityand market power diminishes, the quantity QUse

1 in Eq. (5)tends to the social optimum Q⁎ in Eq. (3). Meanwhile, profitdecreases with the number of firms (since it moves away fromthe monopolist equilibrium), while the welfare increases(approaching the social optimum).

Proposition 1. When n→∞ firms without market powercompete in use and internalize the total external effect, thesocial optimum is reached. As market power increasesconsumption of the common resource moves away from thesocial optimum and changes into underutilisation. At the soleownership equilibrium, benefits are maximized but there is aloss of well-being because of the underutilisation of theresource.

4.2. Underexploitation versus overexploitation undercompetition in use

The maximization of well-being under Cournot competitionwith identical agents requires two conditions: i) the n firmsmust internalize the social cost associated with the exploita-tion of the common; and ii) the market power of each of thefirms must be minimal. Let us now consider what happenswhen none of the n firms internalizes the social cost.

From Eq. (4) we know that the consumption of the commonincreases as new competitors join the market when firmsbehave as free-riders4. At the perfect competition equilibrium(market power disappears when n→∞) the quantity QUse

0 wouldbe

QPC =a� cb

ð7Þ

It is evident that the common is overexploited at theperfect competition equilibrium, since the social optimum Q⁎

in Eq. (3) is lower than QPC (it should be remembered here thateN0). Unlike in the case of full internalization, the absence ofany market power turns out now to be harmful to the

3AQ1

UseAn = a�c�e

b 1 + nð Þ2 N0f a� c� eð ÞN04 AQ0

UseAn = a�c

b 1 + nð Þ2 N0f a� cð ÞN0 which is verified because (a – c - e)N0.

.

collective interest: overexploitation is maximum when ntends to be infinite. Nevertheless, this does not mean thereis always overexploitation when firms compete in use but donot internalize the social cost.

FromEqs. (3) and (4),wemaydeduce thatQUse0 bQ⁎ if and only

if ebe0= (a−c)/3 and nbn0= (a−c−e)/e (the first is a necessary andthe second a sufficient condition)5. This means that nbn0 firmscompeting in use and behaving as free-riders will not overusethe common provided the external effect does not surpass thecritical value e0 in the model. Note that the critical value n0increaseswith the external effect. Hence, this result hasmore todo with the weakness of the external effect than with thenumber of firms. Note also that when the above conditions arebounded, QUse

0 =Q⁎ is obtained. Eq. (8) shows the order betweenthequantities in the casesmentioned.As Fig. 1 shows, the orderis fixed except for QUse

0 (overlooking the QExc quantities for themoment), which is placed before or after Q⁎ depending on themagnitude of the external effect and the number of firms in themarket:

QMbQ1UsebQ*bQPC 8nz2

Q1UsebQ

0UsebQPC 8nz2

If ebe0 = a� cð Þ=3; Q0UsebQ⁎fnbn0 = a� c� eð Þ=e

If eNe0 = a� cð Þ=3; Q0UseNQ⁎

ð8Þ

5 The proof is in Appendix A.

Fig. 2.

1744 E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

The conclusionswemay draw about profits andwelfare areintuitive. As mentioned above, profits are non-negative in anycaseprovided that (a− c−e)N0.Welfaredecreaseswithquantityonly if there is overuse. Otherwisewelfare increases6. FromEq.(8)we know thatQUse

0 NQ⁎necessarily if eNe0 (the samehappenswhen ebe0 provided that nNn0). Since QUse

0 increases with n, itmight be supposed that the welfare could be negative if thenumber of firms were high enough. This issue is illustrated inFig. 2. From the Eqs. (2) and (4), we can obtain the conditionsunder which welfare is negative7:

If eVe1 = a� cð Þ=2; W0Usez0

If eNe1 = a� cð Þ=2;W0Useb; = ; N0fnN; = ;bn1 = 2 a� c� eð Þ= �a + c + 2eð Þ

ð9Þ

Since WUse0 is a continuous function on n and since

0bWMbW⁎, Eqs. (8) and (9) ensure that if e is significant thena value of n exists such that WUse

0 =WM. For this particularvalue of n the firms generate the same profits and well-beingas in the sole owner solution, as Fig. 1 illustrates. This is due tothe conjunction of two contrary effects: on the one hand, areduction in market power increases the consumer surplusand recovers part of thewell-being lost by themonopolist; and

6 From Eq. (2) we know that ∂W(Q)/∂Q= (a–c–e) -bQ.7 The proof is in Appendix B.

on the other, the increasing number of firms generates anincreasing external effect.

An interesting result is obtained from Eqs. (8) and (9). Therecan be overuse and positive welfare simultaneously (indeedwelfare is not maximal, as we can be seen in Fig. 1). This isreflected in Eq. (10).

Q0UseNQ⁎ and W0

UseN0if ebe0 and nNn0if ea e0; e1ð Þ 8nif eNe1 and nbn1

8<: ð10Þ

The above result may explain why some commonresources, such as the environment, are overexploited evenwhen the economic agents are aware of what they are doing.Sometimes the reason is that welfare is positive in spite ofeverything. If QUse

0 NQ⁎ would necessarily imply that WUse0 b0

then there would be less arguments for overuse (in fact, therewould be a conflict between positive profits and negativewelfare). Based on the result in Eq. (10) overexploitation maycoexist with positive welfare in the following cases: i) whenthe external effect is low (ebe0) but the number of firms is high(nNn0); ii) when the external effect is high (eNe1) but thenumber of firms is low (nNn1); iii) when the external effect is inthe interval e ∈ (e0, e1), regardless of the number of firmsexploiting the common.

In this light, we may affirm that free-rider competitiondoes not unfailingly lead the resource to a tragic end. Thisoutcome in fact depends on the magnitude of the externaleffects linked to the exploitation of the common and on thepressure to which it is subject8. Hardin (1993) himself admitsthat “under special circumstances even an unmanagedcommons may work well. […] Thus, the ratio of supply todemand is of critical importance. The scale of the commons(the number of people using it) is also important.” When theexternal effect is weak, it is possible that the resource will beunderused as a result of free-riding, or it may approach thesocial optimum, though a significant number of firms can be acause of overexploitation. Even when there is overuse thisdoes not necessarily mean welfare is negative.

Proposition 2. There is underuse of the common in thepresence of free-rider firms competing in use if not only theexternal effect but also the number of firms does not exceedthe critical values e0 and n0 in themodel, respectively. At thesecritical values, competition between the free-rider firms leadsto the social optimum. Otherwise there is overexploitation,which increases with the number of firms. If the externaleffect is significant there exists a value of n such that free-rider competition provides the same well-being as the soleownership solution.

Pollution is a case in point.When the number of polluters islower than n0 the “environmental quality” resource is alwaysunderused. In this case the behaviour of the firms is irrelevantfrom a conservation standpoint and, what is more, the well-

8 Carlson, Zilberman y Miranowski (1993, p. 228) draw the sameconclusion: “Whether there will be overgrazing depends upon theregeneration rate of the resource: in this case, the growth rate ofthe grass relative to E0 or E⁎ levels of grazing effort. Overgrazingcan occur at both points, at either, or at neither point.”

1745E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

being is higher when the polluters behave as free-riders(because they increase the consumer surplus, approaching thesocial optimum more closely). When the number of pollutersis higher than n0 the resource is underused if they internalizethe social cost, and it is overexploited if they do not. Bothscenarios evolve in the opposite direction as new pollutersjoin the market. Thus, a market made up of responsible firmscomes closer to the social optimum, and a market formed byfree-rider firms moves away from it (the reduction of marketpower is desirable only in the first case).

9 The proof is in Appendix C.

5. Exploitation of resources in common propertyregimes: competition in exclusion

5.1. Initial results

As explained above, Heller (1998) identifies a problem ofunderutilization when property rights are exercised not onlyfor the purpose of exploiting the resource but also to obtainrents. In the example proposed by Hardin (1968), there wouldbe competition “in exclusion” if each of the farmers were tocharge other farmers a right of entry to the common pastures.On payment of this royalty, each agent allows the remainingagents to exploit the common (since no owner has the right ofexclusive use of the common, every owner exercises theproperty right by collecting rental income). Costs are shared byall of the agents exploiting the resource. In fact, behind thiskind of competition lies common management of theresource. First a cooperative gathering of all the agents isformed; then every agent names his price; and finally theresource is exploited in common. However, it is not hard to seethat the main difficulty of such individual entry rights is thatconsumers would eventually have to pay the price.

Let us denote the right of entry granted by the i-th agent bypi. Since the common cannot be partitioned (otherwise everyagent could exercise the right of exclusive use only on its ownpart of the same), the total right of entry paid by every agent isthe sum of the individual prices. In these conditions themaximization program of the i-th agent takes the followingform:

maxpi

Bi = pi � Q Pð Þ � c � Q Pð Þn

� e � Q Pð Þn

ð11Þ

Given the specification of the demand function, thesolution under Cournot competition in prices (i.e., all thefirms compete in exclusion simultaneously) and in thepresence of identical firms verifies the optimal quantitiesbelow.

Q1Exc =

a� c� eb n + 1ð Þ ð12Þ

Q0Exc =

a� cb n + 1ð Þ ð13Þ

Eq. (12) shows the Nash equilibrium when the firms absorbtotal external effect (QExc

1 ), while Eq. (13) shows the free-riderNash equilibrium (QExc

0 ). Note that the sole owner solution inEq. (6) is found when n=1 in Eq. (12). Since quantity decreaseswith the number of firms, the solution when the firms

compete in exclusion and internalize the social cost isnecessarily lower than the sole owner solution for any nN1:QExc1 bQM. In this context the number of agents plays the

opposite role to that of competition in use (see precedingsection). Thus, the higher the number of firms, the higher themarket power of each one will be, since their requirementshave to be satisfied by all of the other firms. This may explainwhy many common inheritances, not to mention commonresources such as those mentioned in Section 1, are sold at aloss even when the inheritors are aware of the inefficiency(after all, the welfare is positive in spite of everything). In theextreme case, when n tends to infinity, the market power ofevery firm is so significant that it is not possible to reachagreement on the total price of access to the resource.Underutilization thus increases with market power, andaccess to the resource can be blocked when the number offirms is significant.

Proposition 3. The common resource is always underusedwhen firms compete in exclusion and internalize the socialcost. In this context, the sole owner solution is a second bestbecause the loss of well-being growswith the number of firms.

5.2. Underexploitation versus overexploitation undercompetition in exclusion

Let us focus now on the non-internalization issue when firmscompete in exclusion. The quantity in this case also decreaseswithn, so thatQExc

0 tends to zerowhen thenumber of firms is veryhigh (note that in any case QExc

1 bQExc0 ). Unlike the internalization

case, however, it is now not possible to state that QExc0 is always

lower than the sole owner solution. What is more QExc0 is not

necessarily lower than the optimal quantityQ⁎; it depends on themagnitude of the external effect and on the number of firms inthe market. This is illustrated by Fig. 1 (underuse in exclusion)and Fig. 2 (overuse in exclusion). From Eqs. (3) and (13), we obtainthat QExc

0 NQ⁎ if and only if ebe2=2(a−c)/3 and nbn2=e/(a−c−e) (thefirst is a necessary and the second a sufficient condition)9. Thismeans that n firms competing in exclusion and behaving as free-riders overuse the common if both the external effect and thenumber of firms do not exceed the critical values e2 and n2 in themodel. Note that the critical value n2 increases with the externaleffect. When the preceding conditions are bounded, competitionin exclusion without internalization provides the optimal well-being. Otherwise the resource is underused.

Q1ExcbQMbQ*; 8nz2

Q1ExcbQ

0Exc; 8nz2

If eNe2 = 2 a� cð Þ=3; Q0ExcNQ⁎fnbn2 = e= a� c� eð Þ

If ebe2 = 2 a� cð Þ=3; Q0ExcbQ⁎

ð14Þ

Aswe havementioned before thewelfare is positive for anyQbQ⁎. Since QExc

0 does not always verify this condition, it isnecessary to analyze when the welfare provided by n firmscompeting in exclusion and behaving as free-riders isnegative. Since QExc

0 decreases with n, it might be supposedthat welfare would be negative if the number of firms were

1746 E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

significantly low. From Eqs. (2) and (13), we may obtain theconditions under which welfare is negative10:

If eVe3 = 5 a� cð Þ=6; W0Excz0

If eNe3 = 5 a� cð Þ=6; W0Excb; = ;N0fnb; = ;Nn3 = �a + c + 2eð Þ= a� c� 2eð Þ

ð15Þ

Welfare is negative when the external effect is significantlyhigh and the number of firms low. From Eqs. (14) and (15), itcan be deduced that overuse and positive welfare may occursimultaneously, as in the competition in use case:

Q0ExcNQ⁎ and W0

ExcN0 if ea e2; e3ð Þ and nbn2 ð16Þ

This might be the case of overcrowded coasts, for example,where too many beachfront facilities are built becauseenvironmental standards depend only on local by-laws (i.e.the number of excluders is too low) and environmentaldamage is not extreme (i.e. development results neither in anatural paradise nor in a lifeless concrete desert).

12 “The environmental movement, emergent since the 1960s inparticular, has had the effect of interposing additional authoritieswith the right to exclude development of facilities. While these

6. Synthesis of findings

In this section we order all of the solutions obtained in theprevious sections in order to present the inefficiency problemsmentioned in perspective. Since welfare increases withquantity for any QbQ⁎, the order of welfare when there isinternalization of social cost can be found from Eqs. (8) and(14):

W1ExcbWMbW1

UsebW⁎ 8nz2 ð17Þ

The order including the free-rider solutions is not, in fact,intuitive. In this respect, we may make the followingcomments:

When firms compete in use and do not internalize socialcosts, welfare can be found at any point in the interval(WUse

1 ,W⁎) if QbQ⁎ depending on the magnitude of theexternal effects and on the number of firms (so thatWUse

0 NWMNWExc1 necessarily). If QNQ⁎ welfare, WUse

0 , islower than the optimum, decreasing with the number offirms, and it may turn negative if Eq. (9) is satisfied.When firms compete in exclusion and do not internalizesocial costs, welfare may be found at any point in theinterval (WExc

1 ,W⁎) if QbQ⁎, depending on the magnitude ofthe external effects and on the number of firms (thisincludes the caseWExc

0 NWMNWExc1 ). If QNQ⁎, then welfare is

lower than the optimum, and itmay turn negative if Eq. (15)is satisfied.In any case WUse

1 NWExc1 . Nevertheless, competition in use

in the presence of free-rider firms is preferable tocompetition in exclusion if and only if the external effectis significantly lower: W0

UseNW0Exc f eV a� cð Þ=2 . Other-

wise, competition in exclusion provides higher welfare11.

In the case illustrated in Fig. 1, positive welfare alwaysexists. Nevertheless, in this figure there is overexploitation

10 The proof is in Appendix D.11 The proof is in Appendix E.

when the firms compete in use and do not internalize thesocial cost, and there is underuse when the firms competein exclusion independently of the behaviour of the firms:QExc1 bQExc

0 bQ⁎. In theparticular case shown inFig. 1, thewelfareprovided by the free-rider firms competing in use is identical tothe sole owner solution, even though the free-riders overexploitthe resource. In the case illustrated in Fig. 2 overexploitationalways exists. However, the welfare provided by free-ridercompetition inuse isnegative,while thewelfarewhen the firmscompete inexclusion is positive (in this case, it is identical to thesole owner solution).

Proposition 4. Let us consider n firms competing in exclusion.When the external effect is significantly higher and thenumber of firms lower (eN e2 and nbn2 in the model,respectively) the resource is underused only if the firmsinternalize the social cost, but it is overexploited if they do not.Otherwise the resource is underused. Although both scenariosevolve in the same direction as new firms join the market, thewelfare provided by the responsible firms directly approachesthe rent dissipation point, while the market formed by free-rider firms progressively approaches (and exceeds) W⁎, WUse

1 ,WM and finally the rent dissipation point when the number offirms is significant.

Proposition 5. Competition in use is preferable to competitionin exclusion when firms internalize the social cost. In thepresence of free-rider firms, however, competition in use ispreferable if and only if the external effect is significantlylower; ebe1 in the model (otherwise the competition inexclusion provides a higher welfare).

The result described in the above proposition qualifiesBuchanan's and Yoon's (2000) argumentswith reference to theenvironmental movement12. In fact, when the external effectis significantly higher and the owners of a common do notinternalize the social cost, the exercise of exclusive rightsamong the owners of a common improves welfare. Since theanticommons is a model of “veto power” in the words of Parisiet al. (2005), it is hardly surprising that institutions apply forexclusion in the presence of significant environmentaldamage. Nevertheless, as Buchanan and Yoon (2000) suggest,the unanimity rule often prevents value-enhancing develop-ment. International agreements on climate change failbecause countries negotiate following a competition in-exclusion strategy. The common in this case is the globalagreement. No country can implement the agreement by itself(i.e. no country has the exclusive right to use the common), butany country can demand compensation in exchange forsigning the agreement. If this compensation (i.e. a right ofentry to the common) is too high, or if the countries involved

authorities have, without question, prevented some value redu-cing development, they have also prevented value-enhancingdevelopment.” (Buchanan and Yoon, 2000; p.12)

1747E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

in the negotiation are very numerous, the treaty will probablynot be approved. The difficulties encountered in the ratifica-tion of the Kyoto Protocol illustrate this issue.

Proposition 6. In any case positive welfare will exist when acommon is underused. Furthermore, the overexploitation ofthe resource does not necessarily result in negative welfare.This explains why some open-access resources are over-exploited or underused evenwhen economic agents are awareof these inefficiencies.

7. Conclusions

This paper proposes a formal synthesis of the problems ofinefficiency associated with the exploitation of resources incommon property regimes. In our opinion the situations ofunderutilization and overexploitation turn out to be linked notonly to the attitude of the economic agents facing a reciprocalexternality but also to market power issues.

We distinguish two alternative scenarios in which theeconomic agents do or do not internalize the social cost linkedto the exploitation of a common resource. In the first scenariothe agents compete for the use of the resource and internalizethe social cost. In this case the optimum is reached at theperfect competition equilibrium. As market power increases,the exploitation of the common moves away from theoptimum and the resource is necessarily underused. In theextreme situation of sole ownership, the loss of well-being ismaximal and the consumption of the resource is minimal. Inthe second scenario the agents behave as free-riders and donot internalize the social cost. Here, perfect competitioncauses maximum overexploitation of the resource. Never-theless, as market power increases the inefficiencydiminishes. When the increase of the market power reachesa threshold level, the resource will be underused. Beyond thisthreshold the behavior of firms is irrelevant from the point ofview of the conservation of the resource, and well-being isactually higher when firms behave as free-riders. The twoscenarios evolve in opposite directions with regard to thenumber of competitors. While a market made up of respon-sible firms comes closer to the social optimum, a marketformed by free-rider firms move away. The reduction ofmarket power is desirable only in the first case. This evolutionin opposite directions means the sole ownership solution willprovide the same well-being as a given number of free-riderfirms competing in use.

In addition to the usual competition for the use ofresources, we focus on the exercise of exclusion rights. Inthis context the common is always underused when theagents internalize the social cost. Since the loss of well-beinggrows with the number of competitors the sole ownershipsolution provides a second-best equilibrium.

When the external effect is significantly higher and thenumber of firms lower, the resource is underused only if thefirms internalize the social cost, but it is overexploited if they donot do so. Otherwise the resource is underused. Although bothscenarios evolve in the same direction as new competitors jointhe market, the welfare provided by the responsible firmsdirectly approaches the rent dissipation point, while themarket

formed by free-rider firms approaches the optimal solutioninitially and then, if thenumberof competitors is significant, therent dissipation point.

When the two kinds of competition (use and exclusion) arecompared, we find that competing for the use of commons ispreferable to the exclusion case if the agents internalize thesocial cost. The contrary, however, does not hold. In thepresence of free-riding competition in use is preferable if andonly if the external effect is significantly lower (otherwisecompetition in exclusion provides higher welfare).

Finally, it may be affirmed in any case that there is positivewelfare when the common is underused. Furthermore theoverexploitation of the resource does not necessarily result innegative welfare. This may explain why some open accessresources are overexploited or underused even when eco-nomic agents are aware of the inefficiency.

Appendix A. QUse0 bQ*

Let Q0Use =

n a�cð Þb n + 1ð Þ and Q* = a�c�e

b . Then QUse0 bQ⁎ if and only if the

following equation is verified:

n a� cð Þb n + 1ð Þ b

a� c� eb

fnbn0 =a� c� e

eðA1Þ

The result in Eq. (A1) must satisfy n≥2 (otherwise we findthe monopoly equilibrium), so that n0≥2 and

a� c� eb

z2feVe0 =a� c3

ðA2Þ

Note that n0b2 is validated if eNe0. Hence, from (A1) it maybe affirmed that

If eNe0 = a� cð Þ=3; Q0UseNQ⁎ ðA3Þ

Appendix B. W0Useb0

Let us consider the welfare function in Eq. (2). Welfare ispositive for any Q in the interval (0, 2Q4), where Q4 maximizesthe welfare function. Otherwise welfare is negative. LetQ0

Use =n a�cð Þb n + 1ð Þ and Q⁎ = a�c�e

b . ThenW0

USEb; = ;N0fn a�cð Þb n + 1ð ÞN; = ;b 2 a�c�eð Þ

b fnN; = ;bn1 = 2 a� c� eð Þ= �a + c + 2eð Þ. As�a + c + 2eð Þb0feNe1 = a�c

2 , Eq. (9) follows from the aboveexpression.

Appendix C. QExc0 NQ⁎

Let Q0Exc =

a�cb n + 1ð Þ and Q⁎ = a�c�e

b . Then QExc0 NQ4 if and only if the

following equation is verified:

a� cb n + 1ð Þ N

a� c� eb

fnbn2 =e

a� c� eðC1Þ

The result in Eq. (C1) must satisfy n≥2 (otherwise we findthe monopoly equilibrium), so that n2≥2 and

ea� c� e

z2feze2 =2 a� cð Þ

3ðC2Þ

1748 E C O L O G I C A L E C O N O M I C S 6 8 ( 2 0 0 9 ) 1 7 4 0 – 1 7 4 8

Note that n2b2 is validated if ebe2. Hence, from (C1) it maybe affirmed that

If ebe2 = 2 a� cð Þ=3; Q0ExcbQ⁎ ðC3Þ

Appendix D. WExc0 b0

Let us consider thewelfare function inEq. (2).Welfare is positivefor anyQ in the interval (0, 2Q⁎), whereQ⁎maximizes thewelfarefunction. Otherwise welfare is negative. Let Q0

Exc =a�c

b n + 1ð Þ andQ* = a�c�e

b . ThenW0

Excb; = ; N0f a�cb n + 1ð Þ N; = ; b 2 a�c�eð Þ

b fnN; = ; bn3 = �a + c + 2e2 a�c�eð Þ .

Since �a + c + 2e2 a�c�eð Þ z2feze3 = 5 a�cð Þ

6 the Eq. (15) follows from theabove expression.

Appendix E. WUse0 NWExc

0

Let us consider the welfare function in Eq. (2):W(Q)=Q·(a−c−e)–b Q2/2. Then WUse

0 (Q)NWExc0 (Q) if and only if the following

equation is satisfied:

Q0Use a� c� eð Þ � b Q0

Use� �2

=2NQ0Exc a� c� eð Þ � b Q0

Exc� �2

=2 ðE1Þ

Eq. (E1) can be simplified to obtain the following expression:

Q0Use +Q0

Exc =2 a� c� eð Þ

bðE2Þ

Since Q0Exc =

a�cb n + 1ð Þ and Q0

Use =n a�cð Þb n + 1ð Þ, the condition in Eq. (E2)

is satisfied if and only if eb (a−c)/2. Therefore W0Use Qð ÞN

W0Exc Qð Þfeb a� cð Þ=2.

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