Notes-Chapter 11 - Optimization

Notes-Chapter 11 - Optimization - Chapter 11-...

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Unformatted text preview: Chapter 11- Optimization: More Than One Choice Variable Alpha Chiang, Fundamentals of Mathematical Economics, 3 rd Edition One variable case 2 2 ( ) necessary but not sufficient '( ) for either maximum or minimum Max: Min: z f x dz dz f x dx d z d z = = = < Function of 2 variables 2 2 2 2 2 ( , ) 0 for dx and dy not both zero , first partials ( ) second order partials ( ) x y x y x y xx x yy y xy z f x y dz f dx f dy dz f f z z f f x y z z f f or x x x x z z f f or x y y y z z f or x y = = + = ⇒ = = ∂ ∂ ≡ = ∂ ∂ ∂ ∂ ∂ ∂ ≡ ≡ ÷ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ≡ ≡ ÷ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ≡ ÷ ∂ ∂ ∂ 2 Young's theorem: = yx xy yx x y z z f or y x y x f f ∂ ∂ ∂ ∂ ≡ ÷ ∂ ∂ ∂ ∂ Examples: 1. 3 2 5 z x xy y = +- 2 3 5 5 2 6 2 5 x y xx yy xy f x y f x y f x f f = + =- = = - = Examples: 2. 2 2 2 2 2 2 1 y y y x xx y xy y y y yy z x e f xe f xe f xe f x e f x e------ = = = = - = - = 2nd Order Total Differential 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 x y x y x y xx yx xy yy xx yx xy yy xx xy yy dz f dx f dy d z d dz dz dx dz dy x y f dx f dy dx f dx f dy dy x y f dx f dy dx f dx f dy dy f dx f dydx f dxdy f dy f dx f dxdy f dy = + ∂ ∂ = = + ∂ ∂ ∂ ∂ = + + + ∂ ∂ = + + + = + + + = + + Examples: 3. 3 2 2 5 Find , z x xy y dz d z = +- ( 29 ( 29 2 2 2 2 3 5 5 2 6 2(5) 2 x y dz f dx f dy x y dx x y dy d z xdx dxdy dy = + = + +- = +- Second Order Condition: 2 2 2 2 2 2 ( , ) for maximum, arbitrary values of dx and dy for minimum iff 0, 0, iff 0, 0, xx yy xx yy xy xx xy yx yy xx yy xx yy xy z f x y d z d z d z f f f f f f f f f d z f f f f f = < < < < Conditions for Relative Extremum x y f f = = x y f f = = 2 0, xx yy xx yy xy f f f f f < < 2 0, xx yy xx yy xy f f f f f CONDITIONS FOR RELATIVE EXTREMUM Maximum Minimum FOC - first order conditions SOC - 2nd order conditions Objective Functions With More Than Two Variables Objective Functions With More Than Two Variables 3 variables:...
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This note was uploaded on 11/09/2011 for the course ECON 101 taught by Professor Richards during the Spring '11 term at Cambrian College.

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Notes-Chapter 11 - Optimization - Chapter 11-...

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