probset2

probset2 - Problem Set 2 EC720.01 Math for Economists Peter...

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Unformatted text preview: Problem Set 2 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, September 24 1. Utility Maximization - Second-Order Conditions The following result specializes Theorem 19.8 from Simon and Blume’s book to provide first and second-order conditions for a constrained optimization problem with two choice variables and a single constraint that is assumed to bind at the optimum. Theorem Let F : R 2 → R and G : R 2 → R be twice continuously differentiable functions, and consider the constrained optimization problem max x 1 ,x 2 F ( x 1 ,x 2 ) subject to c ≥ G ( x 1 ,x 2 ) , with parameter c ∈ R . Associated with this problem, define the Lagrangian L ( x 1 ,x 2 ,λ ) = F ( x 1 ,x 2 ) + λ [ c- G ( x 1 ,x 2 )] . Suppose there exist values x * 1 , x * 2 , and λ * of x 1 , x 2 , and λ that satisfy the first-order conditions L 1 ( x * 1 ,x * 2 ,λ * ) = F 1 ( x * 1 ,x * 2 )- λ * G 1 ( x * 1 ,x * 2 ) = 0 , L 2 ( x * 1 ,x * 2 ,λ * ) = F 2 ( x * 1 ,x * 2 )- λ * G 2 ( x * 1 ,x * 2 ) = 0 , L 3 ( x * 1 ,x * 2 ,λ * ) = c- G [( x * 1 ,x * 2 ) ≥ , λ * ≥ , and λ * [ c- G ( x * 1 ,x * 2 )] = 0 . Suppose also that c- G ( x * 1 ,x * 2 ), so that the constraint binds at the optimum, and that the “bordered Hessian” matrix H = G 1 ( x * 1 ,x * 2 ) G 2 ( x * 1 ,x * 2 ) G 1 ( x * 1 ,x * 2 ) L 11 ( x * 1 ,x * 2 ,λ * ) L 21 ( x * 1 ,x * 2 ,λ * ) G 2 ( x * 1 ,x * 2 ) L 12 ( x * 1 ,x * 2 ,λ * ) L 22 ( x * 1 ,x * 2 ,λ * ) satisfies the second-order condition that | H | > 0, so that the determinant of H is strictly positive. Then x * 1 and x * 2 are local maximizers of F ( x 1 ,x 2 ) subject to c ≥ G ( x 1 ,x 2 ). Note that this result provides sufficient conditions for a solution to the problem: it says that if the first and second-order conditions are satisfied, then the values of x * 1 and x * 2 constitute at least a local maximum....
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probset2 - Problem Set 2 EC720.01 Math for Economists Peter...

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