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Unformatted text preview: Solutions to Problem Set 9 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, November 12 1. Natural Resource Depletion A social planner chooses continuously differentiable functions c ( t ) and s ( t ) for t ∈ [0 , ∞ ) to maximize Z ∞ e- ρt ln( c ( t )) d t, (1) subject to s (0) = s given and- c ( t ) ≥ ˙ s ( t ) (2) for all t ∈ [0 , ∞ ). a. The Hamiltonian for this problem can be defined as H ( s ( t ) ,π ( t ); t ) = max c ( t ) e- ρt ln( c ( t ))- π ( t ) c ( t ) . b. According to the maximum principle, the solution to the social planner’s problem is characterized by the first-order condition e- ρt c ( t )- π ( t ) = 0 and the pair of differential equations ˙ π ( t ) =- H s ( s ( t ) ,π ( t ); t ) = 0 and ˙ s ( t ) = H π ( s ( t ) ,π ( t ); t ) =- c ( t ) . c. Probably the easiest way to solve the pair of differential equations shown above is to use a guess-and-verify method. Accordingly, guess that those differential equations have solutions for the form π ( t ) = π and s ( t ) = 1 πρ e- ρt + k, where π and k are two constants that remain to be determined. The proposed solution for π ( t ) implies that...
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This note was uploaded on 02/19/2010 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.
- Fall '09