finsols10

finsols10 - Solutions to Final Exam EC720.01 - Math for...

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Solutions to Final Exam EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2010 Due Tuesday, December 14, 2010 at 11:00am 1. Growth and Pollution, Part I The social planner’s problem is max z ( Akz ) 1 σ 1 1 σ ° B γ ± ( Akz β ) γ subject to 1 z. a. The Lagrangian for the social planner’s problem can be written as L ( z,λ )= ( Akz ) 1 σ 1 1 σ ° B γ ± ( Akz β ) γ + λ (1 z ) . b. According to the Kuhn-Tucker theorem, if z is the value of z that solves the social planner’s problem, then there exists an associated value λ of λ such that, together, z and λ satisfy the ±rst-order condition L 1 ( z )=( Ak ) 1 σ ( z ) σ βB ( Ak ) γ ( z ) βγ 1 λ =0 , the constraint L 2 ( z )=1 z 0 , the nonnegativity condition λ 0 , and the complementary slackness condition λ (1 z )=0 . c. Let’s look ±rst for a solution with a nonbinding constraint, that is, with 1 >z .B y the complementary slackness condition, this solution must have λ = 0. Hence, the ±rst-order condition implies that z = ° 1 βB ± 1 βγ + σ 1 ° 1 Ak ± γ + σ 1 βγ + σ 1 For this solution to apply, it must be that z < 1o r ,inl igh to ftheexp re s s ionju s t derived, that Ak >
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If, on the other hand, the constraint is binding, then z =1andtheFrst-ordercond it ion implies that λ =( Ak ) 1 σ βB ( Ak ) γ . ±or this solution to apply, it must be that λ 0o r ,inl igh to ftheexp re s s ionju s t derived, that ° 1 βB ± 1 γ + σ 1 Ak. Putting these results together, the optimal choice for z is given by z = 1i f ² 1 βB ³ 1 γ + σ 1 Ak ² 1 βB ³ 1 βγ + σ 1 ´ 1 Ak µ γ + σ 1 βγ + σ 1 if
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finsols10 - Solutions to Final Exam EC720.01 - Math for...

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