Lecture_6 - NR Econ Lecture 6 • Today’s two topics 1...

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Unformatted text preview: NR Econ Lecture 6 • Today’s two topics 1. Math tool: constrained optimization 2. Module III: Water Use Constrained optimization (the method of Lagrange) • Calculus technique for solving constrained optimization problems – Utilize basic rules of differentiation – Constrained maximization or constrained minimization • What are constraints? – Income constraint in the consumer problem – Public sector budget constraint – Natural resource constraints • Water supply constraint • Constraint on “# of endangered species protected” • Fossil fuel supply constraint • Constraint on “Accumulated greenhouse gas in the atmosphere” • Motivitation: the consumer’s problem General approach of constrained optimization • Three components of the problem – Objective function [function to optimize by the decision- maker] f(x 1 , x 2 ,…,x n ) – Constraint g(x 1 , x 2 ,…,x n ) = c [ c = a constant ] – Choice variable(s) Three-step approach • Step 1. Form a new function with all of the information L = f(x 1 , x 2 ,…,x n ) + λ ( c - g(x 1 , x 2 ,…,x n )) where L = Lagrangian function [ L = L(x 1 , x 2 , …,x n , λ ) ] λ = Lagrange multiplier-- The Lagrange multiplier is a new element Three-step approach (continued) • Step 2. Apply approach of establishing the first- order conditions (FOC); standard method of setting partial derivatives equal to 0 ) ,..., , ( ) ,..., , ( 2 2 1 2 2 1 2 = ∂ ∂- ∂ ∂ = ∂ ∂ x x x x g x x x x f x L n n λ ) ,..., , ( ) ,..., , ( 2 1 2 1 = ∂ ∂- ∂ ∂ = ∂ ∂ n n n n n x x x x g x x x x f x L λ...
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This note was uploaded on 04/03/2008 for the course ECON 472 taught by Professor Moore during the Winter '08 term at University of Michigan.

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Lecture_6 - NR Econ Lecture 6 • Today’s two topics 1...

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