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
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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
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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.
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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.
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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.
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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? Because the present value of the tax is reduced by shifting it to the future, and this is accomplished by shifting some production to the future.
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