The efficient and optimal use of natural resources. chapter 14. Objectives. Develop a simple economic model, built around a production function in which natural resources are inputs into the production process;
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Q = Q(K, R) (14.1)
0 < <
Figure 14.1 Substitution possibilities and the shapes of production function isoquants
Social welfare function (SWF)
In general form: (14.5)
Assume that the SWF is utilitarian in form:
Two constraints that must be satisfied by any optimal solution.
Constraint 1: Resource stock-flow constraint
(14.9 and 14.10)
QK (= ∂Q/∂K) and QR (= ∂Q/∂R) are the partial derivatives of output with respect to capital and the non-renewable resource. (i.e. the marginal products of capital and the resource).
Time subscripts are attached to these marginal products to make explicit the fact that their values will vary over time in the optimal solution.
The terms Pt and ωtare the shadow prices of the two productive inputs, the natural resource and capital. These two variables carry time subscripts because the shadow prices will vary over time. The solution values of Pt and ωt, for t = 0, 1, . . ., , define optimal time paths for the prices of the natural resource and capital.
The quantity being maximised in equation 14.8 is a sum of (discounted) units of utility. Hence the shadow prices are measured in utility, not consumption (or money income), units.
Equation 14.14a: In each period, the marginal utility of consumption UC,t must be equal to the shadow price of capital ωt . An efficient outcome will be one in which the marginal net benefit of using one unit of output for consumption is equal to its marginal net benefit when it is added to the capital stock.
Equation 14.14b: The value of the marginal product of the natural resource must be equal to the marginal value (or shadow price) of the natural resource stock, Pt. The value of the marginal product of the resource is the marginal product in units of output (QR,t) multiplied by the value of one unit of output(, ωt).
The static efficiency conditions
Equation 14.14c: Dividing each side by Ptheexpression states that the growth rate of the shadow price of the natural resource (that is, its own rate of return) should equal the social utility discount rate.
Equation 14.14d: Dividing both sides of 14.14d by ω, we obtain an expression
which states that the return to physical capital (its capital appreciation plus its marginal productivity) must equal the social discount rate.
The relations imply that along an optimal path:
PtB = P0Bet
PtA = P0Aet
Modelling extraction costs
Figure 14.4 Three possible examples of the relationship between extraction costs and remaining stock for a fixed level of resource extraction, R
t (for given value of Rt = )
Remaining resource stock, St
ρP = PGS (14.23)
ρ = GS (14.24)
Figure 14.5 The relationship between the resource stock size, S, and the growth of the resource, G
MSY = GMAX
The following situations are likely: