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Unformatted text preview: 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 Consider a continuous time version of the natural resource depletion problem that you solved previously, in discrete time, using the Kuhn-Tucker theorem in problem set 6. Let c ( t ) denote society’s consumption of an exhaustible resource during each period t ∈ [0 , ∞ ), and suppose that a representative consumer gets utility from this resource as described by Z ∞ e- ρt ln( c ( t )) d t, (1) where the discount rate satisfies ρ > 0. Let s ( t ) denote the stock of the resource that remains at each date t ∈ [0 , ∞ ). Since the resource is nonrenewable, this stock evolves according to- c ( t ) ≥ ˙ s ( t ) (2) for t ∈ [0 , ∞ ), which just indicates that consumption during period t subtracts from the stock that remains to be consumed from that period forward. Now the social planner’s problem can be stated as: choose continuously differentiable functions c ( t ) and s ( t ) for t ∈ [0 , ∞ ) to maximize (1) subject to s (0) = s given and (2) for all t ∈ [0 , ∞ ). a. Define the Hamiltonian for this problem, using π ( t ) to denote the multiplier correspond- ing to the constraint (2) for period t . b. Next, write down the first-order condition for c ( t ) and the pair of differential equations for π ( t ) and s ( t ) that, according to the maximum principle, help characterize the solution to the social planner’s problem....
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- Fall '09
- Economics, Boundary value problem, Stock and flow, Capital accumulation