sol_set6 - k using the boundary condition ( T ) = a . This...

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Solutions to Problem Set 6 ARE 261 November 8, 2001 Question 2 1. For the monopolist the current value Hamiltonian for the exhautible re- source problem is H = aq ¡ bq 2 ¡ c 2 q 2 ¡ ¸q (1) The necessary conditions are a ¡ 2 bq ¡ cq ¡ ¸ = 0 (2) ¡ 2 b ¡ c · 0 (3) _ ¸ = ±¸ (4) _ R = ¡ q (5) We know that the stock is exhausted at T so the only variable that is free is T . We need a transversality condition for this which is given by H ( T ) = 0 . 2. We know that the resource is exhausted at T and so q ( T ) = 0 . From the …rst necessary condition we know that ¸ ( T ) = a . Furthermore the third necessary condition implies that ¸ ( t ) = k exp ( ±t ) where k is a constant. We can determine
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Unformatted text preview: k using the boundary condition ( T ) = a . This implies that k = ae T and so ( t ) = ae ( t T ) . We can use the rst order condition and the expression for ( t ) to write the optimal policy function q ( t ) as a function of T q ( t ) = a 2 b + c (1 e ( t T ) ) (6) The only unknown is T . 3. The resource is exhausted at T and so the condition that determines T is Z T t =0 q ( t ) dt = R (7) where R is the initial reserve of the stock. The Planners problem is solved in the same manner. 1...
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This note was uploaded on 08/01/2008 for the course ARE 263 taught by Professor Karp during the Fall '06 term at University of California, Berkeley.

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