Depletion1

Depletion1 - Optimal Depletion: Concepts and Derivations...

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Optimal Depletion: Concepts and Derivations
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Outlines 1. Basic Concepts and an Informal Derivation 2. Formal Derivation and the Welfare Theorem 3. Assumptions Underlying the Welfare Theorem 4. Concluding Remarks
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Basic Concepts and an Informal Derivation How is an exhaustible resource different from other goods or resources? It is limited in supply and not producible.
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Basic Concepts and an Informal Derivation What are the implications for efficient use or market equilibrium? Extracting and consuming a unit today carries an opportunity cost, beyond the cost of the inputs used in extracting the resource: the value that might have been obtained at some future date. This opportunity cost must be considered in deciding how to allocate the resource over time.
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Instead of the usual efficiency or equilibrium condition, price equals marginal cost, we have: Price equals marginal cost plus marginal opportunity cost. This implies that resource owners will wish to extract less today than they would if the resource were producible, as shown in Figure 1, where q* is the optimal level of extraction and q’ is the level at which price equals marginal cost.
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Figure 1 p 0 q* q Marginal Cost + Marginal Opportunity Cost Marginal Cost Demand q’
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What we have identified as the marginal opportunity cost is known by a number of names in the resource-economics literature: royalty, user cost, net price, marginal profit, and scarcity rent. We’ll stick with royalty, probably the most common term.
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Another equilibrium or efficiency condition describes the behavior of the royalty over time. Consider the decision on whether to extract a unit of the resource. What is the net benefit to the owner from extracting today? The royalty, as we have defined it. But: that same unit would also yield a royalty if extracted next year (for now, let’s assume just two periods, this year and next). So when should it be extracted to yield the largest net benefit?
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There is a simple and elegant answer to this question. To see it, let’s consider an example: For an individual producer, considering whether to extract the marginal unit of the resource this period or next, let marginal cost MC = 2, price p 0 = 5, and expected future price p 1 = 6. Then, this year’s royalty on the unit = 3, and next year’s = 4. But the present value of next year’s royalty = 4/(1 + r) , where r is the discount rate. Let r = 0.1, so 4/(1 + r) = 4/1.1 = 3.64. Extraction is optimally deferred to next year even with discounting.
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Now suppose this calculation is made by many independent producers. How does the market adjust? The current price will rise, reflecting the fall in current supply. The expected future price might fall, based on what
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Depletion1 - Optimal Depletion: Concepts and Derivations...

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