ECON100A_4

ECON100A_4 - 1/4/2008 1 MATH REVIEW Constrained...

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Unformatted text preview: 1/4/2008 1 MATH REVIEW Constrained Optimization 1/4/2008 2 Example 1 Unconstrainedone independent variable max x f(x) We assume nice properties (continuity, second order conditions) F.O.C. for interior solution: A. f(x)=0 B. f(x)=0 C. f(x)=0 X f(x) X* 1/4/2008 3 Example 1a Unconstrainedone independent variable max x 2x-4x 2 X*= A. 4 B. 2 C. F.O.C.: 2-8x=0 X f(x) X* 1/4/2008 4 Example 2 UnconstrainedTwo independent variables We assume nice properties (continuity, second order conditions) F.O.C. s for interior solution: A. B. f(x)=o X f(x) ) , ( max 2 1 , 2 1 x x f x x 1 2 1 2 1 2 ( , ) ( , ) 0, f x x f x x x x = = 1/4/2008 5 Example 2a UnconstrainedTwo independent variables F.O.C. s for interior solution: 8+y-2x=0 x-2y=0 X f(x) 2 2 , 8 max 2 1 y x xy x x x-- + 1/4/2008 6 Example 3 - Constrained Optimization Optimum is where these are equal: slope of level curve of function x 1 x 2 Slope of constraint x 1 +x 2 =10 Also must be on line x 1 +x 2 =10 10 . . max 2 1 2 1 , 2 1 = + x x t s x x x x X 1 X 2 10 10 x 1 +x 2 =10 X 1 * X 2 * 1/4/2008 7 More General Form Given nice assumptions, optimum is x 1 * , x 2 * that equate: slope of level curve of f at x 1 * , x 2 * slope of constraint g at x 1 * , x 2 * c x x g t s x x f x x = ) , ( . . ) , ( max 2 1 2 1 , 2 1 2 2 1 1 2 1 ) , ( ) , ( x x x f x x x f 2 2 1 1 2 1 ) , ( ) , ( x x x g x x x g X 1 X 2 g(x 1 ,x 2 )=c Level curves of f 1/4/2008 8 Some algebra F.O.C.s 2 2 1 1 2 1 2 2 1 1 2 1 ) , ( ) , ( ) , (...
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ECON100A_4 - 1/4/2008 1 MATH REVIEW Constrained...

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