Exhaustible - Exhaustible Resources Contents: Two-Period...

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Exhaustible Resources Contents: Two-Period Nonrenewable Resource Model with Extraction Costs Two-Period Nonrenewable Resource Model with Open Access Policies to Correct Open Access Market Failures Two-Period Nonrenewable Resource Model with Monopoly Two-Period Nonrenewable Resource Model with Changing Demand Two-Period Nonrenewable Resource Model with a Backstop Technology Sketch of An "n-Period" Model of Nonrenewable Resources Summary of Nonrenewable Resource Model Results Two-Period Nonrenewable Resource Model with Extraction Costs Utility Maximization Assume that we are concerned with a nonrenewable resource under unsatiated demand in two periods, t=0 and t=1. B ( X t ) is the gross benefit associated with using X t amount of the resource in period t. Now, let c denote marginal extraction cost of the resource. Hence, the Net Benefit for period t becomes: B(X t ) cX t We can maximize social welfare by solving: max X 0 ,X 1 NPV SW(X 0 ,X 1 ) [ ] = B(X 0 ) c X 0 + 1 1 + r B(X 1 ) c X 1 [ ] subject to the constraint of the total available resource, S: S 0 = X 0 + X 1 . The Lagrangian equation for this problem is: L = B(X 0 ) c X 0 + 1 1 + r B(X 1 ) c X 1 [ ] + λ (S 0 X 0 X 1 ) The F.O.C.'s are: (1) L X 0 = B X (X 0 ) c = 0 . (2) L X 1 = B X (X 1 ) c 1 + r = 0. (3) L = S 0 X 0 X 1 = 0. where B X (X t ) = MB of using X t amount of the resource in period t, and c = MC. Understanding the Relationships Equation (1) states that the price of the mineral resource, P 0 [which equals B X (X 0 ) ] equals marginal mining cost, c, plus the shadow cost of the resource constraint, or: P 0 = c + . The shadow cost of the resource constraint is also called the user cost in dynamic problems. It is the opportunity cost of not being able to use the marginal unit of the resource in the future if you use it today. Stated in another way, the shadow price is the benefit one could gain from increasing the constraint of initial resource by one unit.
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-2- Notice from Equation (2) that higher interest rates reduce the user cost, as before. A reduction in user cost implies that one should use more of the resource today and less in the future. If one uses more of the resource today and less in the future, then the resource will be plentiful today and scarce in the future, so the price of the resource will be lower today and higher in the future. Hence, an increase in the interest rate shifts a larger share of consumption from the future to the present, lowering the price today, but raising the price in the future. Impacts of Extraction Costs It is clear from both Equation (1) and Equation (2) that higher extraction cost reduces the net marginal benefit associated with using the resource in either time period, which in turn: reduces resource use and raises prices in both periods. reduces the user cost, increasing the incentive to shift some consumption from the future to today.
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This note was uploaded on 04/02/2008 for the course ECON 100B taught by Professor Wood during the Fall '08 term at University of California, Berkeley.

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Exhaustible - Exhaustible Resources Contents: Two-Period...

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