answer2 - Solutions to Problem Set 2 EC720.01 Math for...

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Solutions to 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 ) 0 , λ * 0 , 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 = 0 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. 1
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With this result in mind, return to the problem solved by a consumer who uses his or her income I to purchase c 1 units of good 1 at the price of p 1 per unit and c 2 units of good 2 at the price of p 2 per unit to maximize utility U ( c 1 , c 2 ) = a ln( c 1 ) + (1 - a ) ln( c 2 ) , with 0 < a < 1, subject to the budget constraint I p 1 c 1 + p 2 c 2 . a. The Lagrangian for the consumer’s problem is L ( c 1 , c 2 , λ ) = a ln( c 1 ) + (1 - a ) ln( c 2 ) + λ ( I - p 1 c 1 - p 2 c 2 ) . b. From question 3 of problem set 1, we know that c * 1 = aI p 1 , c * 2 = (1 - a ) I p 2 , and λ * = 1 I . c. Following the pattern shown in the theorem, the bordered Hessian matrix for this problem is H = 0 p 1 p 2 p 1 - a/c 2 1 0 p 2 0 - (1 - a ) /c 2 2 and therefore satisfies the second-order condition | H | = (1 - a ) p 2 1 c 2 2 + ap 2 2 c 2 1 > 0 confirming that the values of c * 1 and c * 2 do, in fact, solve the constrained maximization problem.
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