Week10 - ECON 117 Mathematical Economics Spring 2008 Week...

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ECON 117 Spring 2008 Mathematical Economics Week 10 1.2 Multiple variable case: () , : n yf f =→ x \\ Def) f x has a local max at if 0 x 00 ( ) ( ) , small 0 n ff h h >+ Δ Δ ∈∀ xxx x 0 Theorem) f x has a local max at if 0 x () ( ) 0 0 12 , ,, 0 , 0 0 n n f f xx x ⎛⎞ ∂∂ ≡= ⎜⎟ ⎝⎠ Dx x x x …… \ And 22 11 1 2 0 1 n nn f ⎡⎤ ⎢⎥ = ⎣⎦ " # " n # is neg. def. Page 1 of 9
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ECON 117 Spring 2008 Mathematical Economics Q) 12 2 1 2 (, ) 2 x x fxx x ex e e =+⋅− Show that ( 1 2 0, ) 2 is a local max. Q) 32 123 1 1 3 2 2 3 (, , ) 3 2 3 fxxx x xx x x x =− + + Find a local max. Page 2 of 9
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ECON 117 Spring 2008 Mathematical Economics Page 3 of 9
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ECON 117 Spring 2008 Mathematical Economics Theorem) ( ), , , i.e., : nn yf y f =∈ xx \\ f is continuously differentiable concave neg. semi-def. strictly concave neg. def. ( ) is iff is convex pos. semi-def. strictly convex pos. def. f f ⎛⎞ ⎜⎟ ⎝⎠ xH Page 4 of 9
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ECON 117 Spring 2008 Mathematical Economics 1.3 Profit maximization Ex) • A firm has two production input 12 (, ) x x . • The production function yf x x = . • Input prices, , are given. ww • The output price p is also given. Q) Define the firm’s decision problem.
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Week10 - ECON 117 Mathematical Economics Spring 2008 Week...

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