ECON10A_4 - REVIEW 3IMPORTANTUTILITY FUNCTIONS Linear...

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 1/4/2008 1 REVIEW –  3 IMPORTANT UTILITY  FUNCTIONS Linear: U(x 1 ,x 2 )=ax 1 +bx 2 Cobb-Douglass: U(x 1 ,x 2 )=ax 1 α x 2 β Leontief: U(x 1 ,x 2 )=min[x 1 /a, x 2 /b]
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 1/4/2008 2 Linear: U(x 1 ,x 2 )=ax 1 +bx 2 Perfect Substitutes MRS constant X 1 X 2 c/b c/a Slope=-a/b U(x 1 ,x 2 )=c
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 1/4/2008 3 Cobb-Douglass: U(x 1 ,x 2 )=ax 1 α x 2 β Diminishing MRS Willingness to trade depends on how much of each good is possessed X 1 X 2 U(x 1 ,x 2 )=c
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 1/4/2008 4 Leontief: U(x 1 ,x 2 )=min[x 1 /a, x 2 /b] Perfect Complements Goods must be used in fixed preportions MRS = 0 or undefined X T X M 1 4 (1,4) (2,8)
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 1/4/2008 5 MATH REVIEW Constrained Optimization
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 1/4/2008 6 Example 1 Unconstrained–one 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*
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 1/4/2008 7 Example 1a Unconstrained–one independent variable max x 2x-4x 2 X*= A. 4 B. 2 C. ¼ F.O.C.: 2-8x=0 X f(x) X*
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 1/4/2008 8 Example 2 Unconstrained–Two 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, 0 f x x f x x x x = =
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 1/4/2008 9 Example 2a Unconstrained–Two 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 - - +
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 1/4/2008 10 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 *
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 1/4/2008 11 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
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 1/4/2008 12 Some algebra F.O.C.s 2 2 1 1 2 1 2 2 1 1 2 1 ) , ( ) , ( ) , ( ) , ( x x x g x x x g x x x f x x x f = d c b a d c b a λ λ = = 2 * 6 1 * 6 2 1 4 2 2 1 = = c x x g x x x g x x x f
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