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 agoLueck & 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