Notes-Chapter12-Optimization2

Notes-Chapter12-Optimization2 - Chapter 12 Optimization...

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Chapter 12 – Optimization With Equality Constraints Econ 130 Class Notes from Alpha Chiang, Fundamentals of Mathematical Economics, 3 rd Edition
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Introduction Previously, all choice variables were independent of each other. However if we are to observe the restriction Q 1 +Q 2 = 1000, the independence between choice variables is lost. The new optimum satisfying the production quota constitutes a constrained optimum .
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Effects of a constraint: 1 2 1 1 2 1 2 2 , U x x x U U U U x x = + are positive for all positive levels of x1 and x2. Budget constraint: 1 2 4 2 60 x x + = Such renders x1 and x2 mutually dependent. Problem: How to maximize U subject to the given constraint .
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Lagrange Multiplier Method: 1 2 1 1 2 2 (60 4 2 ) Z x x x x x λ = + + - - The symbol λ is called a Lagrange multiplier. It is treated as an additional variable: 1 2 1 2 1 2 1 2 1 2 60 4 2 0 2 4 0 3 equations, 3 unknowns 2 0 8, 14, 4, 128 Solution Z Z x x Z Z x x Z Z x x x x Z = - - = ÷ = + - = ÷ = - = ÷ = = = =
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Lagrange Multiplier Method: In general : Maximize or minimize ( , ) subject to ( , ) ( , ) [ ( , )] Necessary conditions: Z ( , ) 0 0 0 x x x y y y z f x y g x y c Z f x y c g x y c g x y Z f g Z f g λ = = = + - = - = = - = = - = dZ dc = λ measures the sensitivity of Z to changes in the constraint:
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n-Variables Case 1, 2 1, 2 1, 2 1, 2
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Notes-Chapter12-Optimization2 - Chapter 12 Optimization...

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