Depletion2

# Depletion2 - Optimal Depletion Some Examples 1 Base Case p...

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Optimal Depletion: Some Examples

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1. Base Case Given q t = 10 p t t = 0,1 MC t = 2 q 0 + q 1 = 10 r = .1 p 10 2 0 8 10 q
Equilibrium Condition From demand equation p 0 MC 0 = p 1 MC 1 1 + r 10 q 0 MC 0 = 10 q 1 MC 1 1 + r

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Substituting for MC and r From stock constraint 10 q 0 2 = 10 q 1 2 1.1 8 q 0 = 8 q 1 1.1 ) 10 ( 8 1 . 1 8 . 8 0 0 q q = q 0 = 5.14
q 1 = 10 q 0 = 4.86 from stock constraint p 0 = 10 q 0 = 4.86 from demand equation p 1 = 10 q 1 = 5.14 from demand equation.

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2. Suppose we ignore the stock constraint. Then, But this is impossible since p 0 = MC 0 10 q 0 = 2 q 0 = 8 p 1 = MC 1 10 q 1 = 2 q 1 = 8. q 0 + q 1 = 16 > 10.
3. Suppose we ignore discounting. Then, and we get egalitarian allocation of the resource over time or, in other words, price and extraction time paths are flat. 10 q 0 2 = 10 q 1 2 8 q 0 = 8 q 1 8 q 0 = 8 10 q 0 ( ) 2 q 0 = 10 q 0 = 5 q 1 = 5

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4. Suppose 0 < r < .1, say, r = .02. Then, 10 q 0 2 = 10 q 1 2 1.02 8 q 0 = 8 10 q 0 ( ) 1.02 q 0 = 5.03 q 1 = 4.97 p 0 = 4.97 p 1 = 5.03.
So price and extraction paths are flatter than in the case where r = .1, but not perfectly flat, as in the case where r = 0. In general, for r > r’ , q r r r t t r p

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5. Suppose we add a severance tax, = \$1/unit extracted. p 0 MC 0 1 = p 1 MC 1 1 1 + r 10 q 0 MC 0 1 = 10 q 1 MC 1 1 1 + r 7 q 0 = 7 10 q 0 ( ) 1.1 q 0 = 5.10 q 1 = 4.90 p 0 = 4.90 p 1 = 5.10.
So price rises more slowly than without the tax (though the royalty continues to rise at rate r ), and production is reduced in the present, shifted to the future. In other words, the severance tax leads to conservation of the resource. Why?

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## This note was uploaded on 09/11/2011 for the course ECON 102 taught by Professor Sunding during the Spring '07 term at Berkeley.

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Depletion2 - Optimal Depletion Some Examples 1 Base Case p...

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