Lueck Miceli-Property Lato investment in assets The models used in this section are fundamental to the later sections of the chapter that examine specific issues and legal doctrine As we noted above, there first systematic analysis of property rights began with the Enlightenment writers(.g, Blackstone, Hume, Locke, Smith), but the formal modeling of the economics of property rights began with Frank Knight's(1924)analysis of public and private roads. Knight showed that a public road with no charge for access would be overused compared to the private road because users would not face the full cost of their actions. Gordon(1954 further developed Knight's preliminary model-establishing the now famous"average product rule' for input use --in the context of an open ocean fishery where no one could be excluded Gordons model was completed with Cheungs(1970) paper, which fully characterized the Nash equilibrium for an open access resource Our analysis of various property rights regimes will use a common set of notation in which a fixed asset(e.g, plot of land )is used in conjunction with a variable input()in order to produce a market output (Y=f(x)). If the input is available at a market wage of w, then the first-best use of the input(x*()must maximize r=f(x)-wx and satisfy the first-order necessary condition flx)=w. The first-best value of the land is thus*=R*(r* t)edt, where r is the discount rate.We start with sS, or a complete lack of rights, and th examine private property rights, common property, and mixed property rights regimes 2.1 Assume there are n individuals who have unrestricted assess to a resource such as a piece of land, and that output from the land(e.g, beef from grazing animals)is given by Y=fir) where x, is the effort of the i h individual, f(>0 andf(<0, and the opportunity cost of effort is the market wage, w. 9 Each person's objective is to maximize his own rent subject to the constraint of open access, which means that each user can only capture(and own) the output proportion to his share or error This means each person must solve the following constrained maximization problem maxR=f(x)-wx subject to f=|x/∑:x/∑mx) Scott(1955) similarly shows the dissipation under open access and the private property solution In law, since Roman times, open access resources have been called res nullius, or things unowned his production function captures the effect of competing users of the open access asset and is standard in the literature. Also, note that while ownership of the land is absent each person is assumed to have perfect ownership themselves, their labor, and the product derived from the open access asset This is a standard assumption but might be modified to explicitly distinguish use effort from violence effort
Lueck & Miceli – Property Law 5 investment in assets. The models used in this section are fundamental to the later sections of the chapter that examine specific issues and legal doctrine. As we noted above, there first systematic analysis of property rights began with the Enlightenment writers (e.g., Blackstone, Hume, Locke, Smith), but the formal modeling of the economics of property rights began with Frank Knight’s (1924) analysis of public and private roads. Knight showed that a public road with no charge for access would be overused compared to the private road because users would not face the full cost of their actions. Gordon (1954) further developed Knight’s preliminary model – establishing the now famous ‘average product rule’ for input use -- in the context of an open ocean fishery where no one could be excluded.16 Gordon’s model was completed with Cheung’s (1970) paper, which fully characterized the Nash equilibrium for an open access resource. Our analysis of various property rights regimes will use a common set of notation in which a fixed asset (e.g., plot of land) is used in conjunction with a variable input (x) in order to produce a market output (Y= f(x)). If the input is available at a market wage of w, then the first-best use of the input (x*(w)) must maximize R = f(x) – wx and satisfy the first-order necessary condition f’(x) = w. The first-best value of the land is thus 0 * *( *, ) rt V R x t e dt ∞ − = ∫ , where r is the discount rate.17 We start with open access, or a complete lack of property rights, and then, in turn, examine private property rights, common property, and mixed property rights regimes.18 2.1. Open Access Assume there are n individuals who have unrestricted assess to a resource such as a piece of land, and that output from the land (e.g., beef from grazing animals) is given by 1 ( ) n i i Yf x = = ∑ where xi is the effort of the i th individual, f’(⋅) > 0 and f’’(⋅) < 0, and the opportunity cost of effort is the market wage, wi. 19 Each person’s objective is to maximize his own rent subject to the constraint of open access, which means that each user can only capture (and own) the output in proportion to his share of effort.20 This means each person must solve the following constrained maximization problem: max ( ) i i i i ii x R = − f x wx (2.1) subject to 1 1 ( ) n n i ii i i i f x xf x = = = ⎡ ⎤ ⎣ ⎦ ∑ ∑ 16 Scott (1955) similarly shows the dissipation under open access and the private property solution. 17 Each period’s rent can be viewed as a steady state outcome. 18 In law, since Roman times, open access resources have been called res nullius, or things unowned. 19 This production function captures the effect of competing users of the open access asset and is standard in the literature. Also, note that while ownership of the land is absent each person is assumed to have perfect ownership of themselves, their labor, and the product derived from the open access asset. 20 This is a standard assumption but might be modified to explicitly distinguish use effort from violence effort
Lueck miceli Assuming that all users are homogeneous (w;=wi, for all i*j), the Nash open access equilibrium is x=x(n, wl,..., w,, which must satisfy the first-order necessary condition ()f Equation(2.2), as Cheung shows, is indeed identical to Gordons asserted average product equilibrium, but only in the limiting case of an infinite number of users with unrestricted access. Thus, in the limit as n>o, (2.2) becomes =1 I X which states that the open access equilibrium level of effort occurs where the average product equals the wage. More importantly, this limiting case also implies that rents are completel dissipated;or that,2iR-Eialr'(xa)-wxa]=0. Similarly, the present value of the asset is also zero; that is, voa= R(xo, De"dt=0 In this framework, the absence of property rights leads to overuse of the asset and complete dissipation of its value. Complete dissipation is a limiting result, however, of the assumption of homogeneous users. If users are heterogeneous, dissipation under open access will be incomplete, and infra-marginal (low cost)users will earn rents(Libecap 1989). The presence rent under open access may be an important factor in preventing the establishment of rights to the open access resource because those earning rents will have incentives to maintain the open access regime 2. 2 Private Property rights Private ownership, as Knight first noted is the straightforward solution to the open access problem. Under the conditions of the Coase Theorem, the owner faces the full value and opportunity cost of asset use, he chooses the first-best level of use(x*<x), and generates V*> y=0. The Coase Theorem also implies that, as long as property rights are well defined the organization of the asset s use will not matter: the owner may use the land himself, he may hire inputs owned by others, input owners may hire(or rent) the asset, or there may be a sharing arrangement between the asset owner and the input owners. In fact, under the conditions of zero transaction costs, any property regime(e.g, common property, state ownership) would generate the first -best use of the asset This has been the starting point with Knight, Gordon and Cheung quation (2.2)is actually a weighted average of average and marginal products. Brooks et al. (1999)show that Cheung's(1970)equilibrium holds in a dynamic setting 2Hardin's(1968)famously named"tragedy of the commons"is a popularized version of this literature This was well understood by Hobbs, Bentham, Locke and Blackstone long ago
Lueck & Miceli – Property Law 6 Assuming that all users are homogeneous21 (wi = wj, for all i ≠ j), the Nash open access equilibrium is x = xoa(n, w1, …, wn), which must satisfy the first-order necessary condition ( ) ( ) 1 , 1 1 ( ) 1 1 '( ) 1,..., . n n i i n i i i i i f x n f x wi n n n x = = = ⎛ ⎞ − ⎜ ⎟ = == ⎝ ⎠ ∑ ∑ ∑ (2.2) Equation (2.2), as Cheung shows, is indeed identical to Gordon’s asserted average product equilibrium, but only in the limiting case of an infinite number of users with unrestricted access.22 Thus, in the limit as n→ ∞, (2.2) becomes 1 1 ( ) , n i i n i i f x w x = = ⎛ ⎞ ⎜ ⎟ = ⎝ ⎠ ∑ ∑ (2.3) which states that the open access equilibrium level of effort occurs where the average product equals the wage. More importantly, this limiting case also implies that rents are completely dissipated; or that, 1 1 () 0 n n i oa oa i i i R f x wx = = = −= ⎡ ⎤ ∑ ∑ ⎣ ⎦ . Similarly, the present value of the asset is also zero; that is, 0 ( , ) 0. oa oa rt V R x t e dt ∞ − = = ∫ In this framework, the absence of property rights leads to overuse of the asset and complete dissipation of its value.23 Complete dissipation is a limiting result, however, of the assumption of homogeneous users. If users are heterogeneous, dissipation under open access will be incomplete, and infra-marginal (low cost) users will earn rents (Libecap 1989). The presence of rent under open access may be an important factor in preventing the establishment of rights to the open access resource because those earning rents will have incentives to maintain the open access regime. 2.2 Private Property Rights Private ownership, as Knight first noted is the straightforward solution to the open access problem.24 Under the conditions of the Coase Theorem, the owner faces the full value and opportunity cost of asset use, he chooses the first-best level of use (x* < xoa), and generates V* > Voa =0. The Coase Theorem also implies that, as long as property rights are well defined the organization of the asset’s use will not matter: the owner may use the land himself, he may hire inputs owned by others, input owners may hire (or rent) the asset, or there may be a sharing arrangement between the asset owner and the input owners. In fact, under the conditions of zero transaction costs, any property regime (e.g., common property, state ownership) would generate the first-best use of the asset. 21 This has been the starting point with Knight, Gordon and Cheung, 22 Equation (2.2) is actually a weighted average of average and marginal products. Brooks et al. (1999) show that Cheung’s (1970) equilibrium holds in a dynamic setting. 23 Hardin’s (1968) famously named “tragedy of the commons” is a popularized version of this literature. 24 This was well understood by Hobbs, Bentham, Locke and Blackstone long ago
Lueck 8 Miceli-Property Lato Not only does private ownership create incentives for optimal resource use, it also creates incentives for optimal asset maintenance and investment. With open access, no user has any incentive to use inputs that have a future payoff. To see the effect on investment, consider a slightly modified version of the model above. Let future output be Yr+/=f(xv), where xr is current investment, available at a market wage of w, and the interest rate is r:. The first-best use of the input(x,)must maximize R=f(xv/(1+ry-wa, and satisfy the first-order necessary conditionf'lxy/(+r)=wr. This outcome is generated under perfect private ownership. Now let r be the probability of expropriation(because of imperfect property rights)of the future output so that(1-rm) is the probability the investor's output remains intact. The solution to the investment problem(x, )is now to maximize R=f(xv/(1-77(+r)/-wxr which must satisfy(xv/( T)/(1+r)=wr. This clearly implies less investment(x, <x'). Pure open access means that no investor could claim future output(T=1), so x =0, and the rent from investment also equals zero In a recent article, Heller( 1998)identifies a situation in which a large number of uncoordinated individuals have the right to exclude users, thus creating a regime in which assets are under-used because each rights holder can exercise a"veto'over use. Because of the incentive to under use rather than over use the asset, Heller labeled this the anti-commons'and argues that many of the development problems in post-communist Europe are plagued with this problem of too many owners.Buchanan and Yoon(2000) formalize Heller's idea and give it additional application cases where competing bureaucracies can stifle development by exercising veto rights. De Sotos(2000) documentation of the difficulties of operating in an economy heavily laden with overlapping bureaucracies is a similar application(as discussed in Section 5. 1. 2). In a similar application, Anderson and Lueck(1992) study 'fractionated ownership of land on American Indian reservations. They found that divided ownership of agricultural landamong large numbers of heirs to the original owner )led to dramatic reductions in the value of agricultural itput It is not clear that the anti-commons phenomenon can usefully be regarded as a distinct property regime. The lack of investment incentives does not stem fromto much ownership, but simply from severely divided interests in which unanimous agreement is required for decision. In this sense, the anti-commons is like open access too many people have access and thus no one can gain from optimal use. The difference seems to be that the decisions considered are investment decisions, so the anti-commons can be viewed as open investment problem. The same land or apartments governed by too many will be overused while investment will be suboptimal 2 Writing before Adam Smith, Wm. Blackstone( Book ll, Chapter 1, 1765)recognized this and wrote: 'And the art of agriculture, by a regular connexion and consequence, introduced and established the idea of a more permanent property in the soil, than had hitherto been received and adopted. It was clear that the earth would not produce he I fruits in sufficient quantities, without the assistance of tillage: but who would be at the pains of tilling it, if another might watch an opportunity to seize upon and enjoy the product of his industry, art, and labor? This is based on the detailed analysis of Bohn and Deacon(2000) Heller and Eisenberg(1998)also call open access a problem of too many owners, whereas the economic models xamined above characterize this as a lack of ownership
Lueck & Miceli – Property Law 7 Not only does private ownership create incentives for optimal resource use, it also creates incentives for optimal asset maintenance and investment. With open access, no user has any incentive to use inputs that have a future payoff.25 To see the effect on investment, consider a slightly modified version of the model above.26 Let future output be Yt+1 = f(xt), where xt is current investment, available at a market wage of w, and the interest rate is r. The first-best use of the input ( * t x ) must maximize R = f(xt)/(1+r) – wtxt and satisfy the first-order necessary condition f’(xt)/(1+r) = wt. This outcome is generated under perfect private ownership. Now let π be the probability of expropriation (because of imperfect property rights) of the future output, so that (1-π) is the probability the investor’s output remains intact. The solution to the investment problem ( t x π ) is now to maximize R = f(xt) [(1-π)/(1+r)] – wtxt which must satisfy f’(xt) [(1- π)/(1+r)] = wt. This clearly implies less investment ( * t t x x π < ). Pure open access means that no investor could claim future output (π = 1), so oa t x = 0, and the rent from investment also equals zero. In a recent article, Heller (1998) identifies a situation in which a large number of uncoordinated individuals have the right to exclude users, thus creating a regime in which assets are under-used because each rights holder can exercise a ‘veto’ over use. Because of the incentive to under use rather than over use the asset, Heller labeled this the ‘anti-commons’ and argues that many of the development problems in post-communist Europe are plagued with this problem of ‘too many owners.’ Buchanan and Yoon (2000) formalize Heller’s idea and give it additional application in cases where competing bureaucracies can stifle development by exercising veto rights. De Soto’s (2000) documentation of the difficulties of operating in an economy heavily laden with overlapping bureaucracies is a similar application (as discussed in Section 5.1.2). In a similar application, Anderson and Lueck (1992) study ‘fractionated’ownership of land on American Indian reservations. They found that divided ownership of agricultural land (among large numbers of heirs to the original owner) led to dramatic reductions in the value of agricultural output. It is not clear that the anti-commons phenomenon can usefully be regarded as a distinct property regime. The lack of investment incentives does not stem from ‘too much ownership’, but simply from severely divided interests in which unanimous agreement is required for decision.27 In this sense, the anti-commons is like open access: too many people have access and thus no one can gain from optimal use. The difference seems to be that the decisions considered are investment decisions, so the anti-commons can be viewed as open access investment problem. The same land or apartments governed by ‘too many’ will be overused while investment will be suboptimal. 25 Writing before Adam Smith, Wm. Blackstone (Book II, Chapter 1, 1765) recognized this and wrote: ‘And the art of agriculture, by a regular connexion and consequence, introduced and established the idea of a more permanent property in the soil, than had hitherto been received and adopted. It was clear that the earth would not produce her 1`fruits in sufficient quantities, without the assistance of tillage: but who would be at the pains of tilling it, if another might watch an opportunity to seize upon and enjoy the product of his industry, art, and labor?’ 26 This is based on the detailed analysis of Bohn and Deacon (2000). 27 Heller and Eisenberg (1998) also call open access a problem of ‘too many owners,’ whereas the economic models examined above characterize this as a lack of ownership
Lueck 8 Miceli-Property Lato The empirical literature on private property rights is of two types. First, there is a literature that private property. This rather small literature is dominated by studies on natural resources and attempts to measure the dissipation from open access and to compare resource use to that und especially of fisheries where open access regimes have been common(e.g, Agnello and Donnelly 1975, Bottomley 1963). These studies have estimated the deadweight losses from open access use and compared levels of asset use in open access regimes with those of private property and other limited access regimes. Second there is a recent and growing literature on the effects of property rights security on resource use and investment. Much of this literature has focused on the investment effects of differences in legal title to land. In his survey article Besley (1998 )notes that the econometric evidence for positive investment effects of more secure rights in developing countries is quite limited. These studies suffer, however, from data limitations(on oth measures of investment and measures of property rights security)and from potential property rights endogeneity. We expect more investment with better defined rights, but as we discuss in section 4, the choice of property rights regime can itself be influenced by investment levels or other correlated variables. Thus the econometric issue is how to find an instrument for property rights variables to isolate the effect of rights on investment. Still there is some compelling evidence for the importance of property rights in the cases of natural resource use (Bohn and Deacon 2000), American Indian reservation agriculture(Anderson and Lueck 1992), and urban residential land(Miceli et al. 2002) 2.3. Common Property rights In modern social science the term commons common property originated in the analysis of what is now called open access. Yet, in law and custom common property has long meant, in stark contrast to open access, exclusive ownership by a group. Common property regimes have been well documented, especially for natural resource stocks in less developed economies (Bailey 1992, McKay and Acheson 1987, Ostrom 1990), and their details have been studied in many settings(e.g, Acheson 1988, Dahlman 1980, Eggertson 1992, Stevenson 1991). Ma writers on common property have noted the gains from group enforcement of rights to the resource(Ellickson 1993, McKay and Acheson 1987, Ostrom 1990, and Stevenson 1991), and we examine common ownership to take this empirical feature into account Common property is best viewed as an intermediate case between open access and private ownership. Common property may arise out of explicit private contracting(e. g, unitized oil reservoirs, groundwater districts)or out of custom(e.g, common pastures and forests; it may have legal(e. g, riparian water rights)or regulatory(e. g, hunting and fishing regulations) bases that have implicit contractual origins. Contracting to form common property effectively creates a group that has exclusive rights to the resource( eggertsson 1992, Lueck 1994). Acting together There is also a property rights literature that focuses on differences between private and regulated firms(e.g bublic utilities)that we do not discuss here We return to this issue in the context of the choice of title system in section 5.1.1 Indeed Hardins(1968)famous paper incorrectly characterizes the common pastures of English villages as op access resources when the historical record shows clearly that they were common property (e.g, Dahlman 1980, Smith 2000) Further evidence that common property regimes are productive is seen from the disasters that have occurred when they have been dismantled by the state(effectively creating open access )as in the forests of Nepal and Thailand Ostrom 1990)
Lueck & Miceli – Property Law 8 The empirical literature on private property rights is of two types.28 First, there is a literature that attempts to measure the dissipation from open access and to compare resource use to that under private property. This rather small literature is dominated by studies on natural resources and especially of fisheries where open access regimes have been common (e.g., Agnello and Donnelly 1975, Bottomley 1963). These studies have estimated the deadweight losses from open access use and compared levels of asset use in open access regimes with those of private property and other limited access regimes. Second, there is a recent and growing literature on the effects of property rights security on resource use and investment. Much of this literature has focused on the investment effects of differences in legal title to land. In his survey article Besley (1998) notes that the econometric evidence for positive investment effects of more secure rights in developing countries is quite limited. These studies suffer, however, from data limitations (on both measures of investment and measures of property rights security) and from potential property rights endogeneity.29 We expect more investment with better defined rights, but as we discuss in section 4, the choice of property rights regime can itself be influenced by investment levels or other correlated variables. Thus the econometric issue is how to find an instrument for property rights variables to isolate the effect of rights on investment. Still there is some compelling evidence for the importance of property rights in the cases of natural resource use (Bohn and Deacon 2000), American Indian reservation agriculture (Anderson and Lueck 1992), and urban residential land (Miceli et al. 2002). 2.3. Common Property Rights In modern social science the term ‘commons’ or ‘common property’ originated in the analysis of what is now called open access. Yet, in law and custom common property has long meant, in stark contrast to open access, exclusive ownership by a group.30 Common property regimes have been well documented, especially for natural resource stocks in less developed economies (Bailey 1992, McKay and Acheson 1987, Ostrom 1990), and their details have been studied in many settings (e.g., Acheson 1988, Dahlman 1980, Eggertson 1992, Stevenson 1991). Many writers on common property have noted the gains from group enforcement of rights to the resource (Ellickson 1993, McKay and Acheson 1987, Ostrom 1990, and Stevenson 1991), and we examine common ownership to take this empirical feature into account.31 Common property is best viewed as an intermediate case between open access and private ownership. Common property may arise out of explicit private contracting (e.g., unitized oil reservoirs, groundwater districts) or out of custom (e.g., common pastures and forests); it may have legal (e.g., riparian water rights) or regulatory (e.g., hunting and fishing regulations) bases that have implicit contractual origins. Contracting to form common property effectively creates a group that has exclusive rights to the resource (Eggertsson 1992, Lueck 1994). Acting together 28 There is also a property rights literature that focuses on differences between private and regulated firms (e.g., public utilities) that we do not discuss here. 29 We return to this issue in the context of the choice of title system in section 5.1.1. 30 Indeed Hardin’s (1968) famous paper incorrectly characterizes the common pastures of English villages as open access resources when the historical record shows clearly that they were common property (e.g., Dahlman 1980, Smith 2000). 31 Further evidence that common property regimes are productive is seen from the disasters that have occurred when they have been dismantled by the state (effectively creating open access) as in the forests of Nepal and Thailand (Ostrom 1990)
Lueck miceli individuals can realize economies of enforcing exclusive rights to the asset. Equation (2.2) mplies that waste can be reduced simply by restricting access to the asset A contracting model can illustrate how common property can limit waste from the rule of capture.Contracting to form common property effectively creates a group that has exclu only to the group's size and the joint effort to exclude outsiders. In this setting, individuals ns rights to the resource. We assume that(contractual)agreement among group members pertains acting together can realize economies of enforcing exclusive rights to the asset, so we also assume the costs of excluding(or policing) non-members can be represented as p(n), where p'(n) <0andp"(m)>0. a simple and customary method of allocating use of common property is a rule that grants equal access to all members of the group(Ostrom 1990). Equal sharing of the asset avoids the explicit costs of measuring and enforcing individual effort(or use) but still creates an incentive for over use.Effort is not explicitly part of the common property contract'so each member chooses his own effort(xi) as he captures his share of the assets output(Y=f(xv again)in competition with other group members. The size of the group is chosen to maximize the wealth of the group subject to the constraint of aggregate effort(X )by members operating in a common property regime, and in recognition of the costs of excluding outsiders. Optimal group size is a tradeoff between increased resource use with a larger group and increased enforcement costs associated with a smaller group. Formally the problem is maxR=/∑mx(n)-∑m(wx(m)-p(m, where x f is the individuals solution to the problem in equation(2. 1). The optimal group size, no determines total effort and must satisfy the first-order necessary condition B0-∑((∑x)-)∠ dnp(n)=0 Equation (2.5)states that the gain from an additional member in terms of a marginal reduction in policing costs must equal the marginal reduction in aggregate rent from overuse of the resource The net present value of the common property resource is thus V=L R(x, ne"dt>0,where vs>y>v=0. While the value of an asset governed by common property is less than its first-best value, it could have greater value than private property depending on the magnitude of the policing cost and overuse effects The model is based on Lueck(1994). Also see Caputo and lueck(1994)and wagner(1995). Others us evolutionary game theory models(e.g. Sethi and Somanathan 1996) Common property might also be viewed as an output sharing contract with moral hazard. In this framework group members shirk as in a principal-agent model(see Lueck 1994). Evidence of both types of common property asset sharing(e. g, share access to a pasture)and output sharing(e.g share the cheese produced from cattle on the common pasture)-are found in the empirical literature Total effort is given by X =xne
Lueck & Miceli – Property Law 9 individuals can realize economies of enforcing exclusive rights to the asset. Equation (2.2) implies that waste can be reduced simply by restricting access to the asset. A contracting model can illustrate how common property can limit waste from the rule of capture.32 Contracting to form common property effectively creates a group that has exclusive rights to the resource. We assume that (contractual) agreement among group members pertains only to the group's size and the joint effort to exclude outsiders. In this setting, individuals acting together can realize economies of enforcing exclusive rights to the asset, so we also assume the costs of excluding (or policing) non-members can be represented as p(n), where p'(n) < 0 and p''(n) > 0. A simple and customary method of allocating use of common property is a rule that grants equal access to all members of the group (Ostrom 1990). Equal sharing of the asset avoids the explicit costs of measuring and enforcing individual effort (or use) but still creates an incentive for over use.33 Effort is not explicitly part of the common property ‘contract’ so each member chooses his own effort (xi) as he captures his share of the asset’s output (Y = f(Σxi) again) in competition with other group members. The size of the group is chosen to maximize the wealth of the group subject to the constraint of aggregate effort (Xc ) by members operating in a common property regime, and in recognition of the costs of excluding outsiders. Optimal group size is a tradeoff between increased resource use with a larger group and increased enforcement costs associated with a smaller group. Formally the problem is 1 1 max ( ( )) ( ( )) ( ) c n n c i i n i i i R f x n wx n p n = = =−− ∑ ∑ , (2.4) where c i x is the individual’s solution to the problem in equation (2.1). The optimal group size, nc determines total effort34 and must satisfy the first-order necessary condition 1 1 ( ( ) ) ' '( ) 0. c n n i i i i i R dx f x w pn n dn = = ∂ ⎡ ⎤ = − −= ∂ ∑ ∑ ⎢ ⎥ ⎣ ⎦ (2.5) Equation (2.5) states that the gain from an additional member in terms of a marginal reduction in policing costs must equal the marginal reduction in aggregate rent from overuse of the resource. The net present value of the common property resource is thus 0 ( , ) 0, c c rt V R x t e dt ∞ − = > ∫ where V* > Vc > Voa = 0. While the value of an asset governed by common property is less than its first-best value, it could have greater value than private property depending on the magnitude of the policing cost and overuse effects. 32 The model is based on Lueck (1994). Also see Caputo and Lueck (1994) and Wagner (1995). Others use evolutionary game theory models (e.g. Sethi and Somanathan 1996). 33 Common property might also be viewed as an output sharing contract with moral hazard. In this framework group members shirk as in a principal-agent model (see Lueck 1994). Evidence of both types of common property – asset sharing (e.g., share access to a pasture) and output sharing (e.g. share the cheese produced from cattle on the common pasture) – are found in the empirical literature. 34 Total effort is given by 1 c n c c i i X x = = ∑