This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 KuhnTucker 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 firstorder 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....
View
Full Document
 Fall '09
 IRELAND
 Economics, Boundary value problem, Stock and flow, Capital accumulation

Click to edit the document details