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 effortLueck & 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