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ch10-extreme-values-multivariate (1)

ch10-extreme-values-multivariate (1) - Extreme Values of...

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Extreme Values of Multivariate Functions Professor Erkut Ozbay Economics 300
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Extreme values of multivariate functions In economics many problems reflect a need to choose among multiple alternatives Consumers decide on consumption bundles Producers choose a set of inputs Policy-makers may choose several instruments to motivate behavior generalizes the univariate techniques
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Stationary points and tangent planes of bivariate functions 2 2 1 1 2 2 6 16 4 g x x x x = - + - 2 2 1 2 1 2 1 2 4 2 16 h x x x x x x = + - - +
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Slices of a bivariate function 2 2 1 1 2 2 6 16 4 g x x x x = - + - 1 1 6 2 0 g x = - = 2 2 16 8 0 g x = - =
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Multivariate first-order condition If is differentiable with respect to each of its arguments and reaches a maximum or a minimum at the stationary point, then each of the partial derivatives evaluated at that point equals to zero, i.e. 1 2 ( , ,..., ) n f x x x * * 1 ( ,..., ) n x x * * 1 1 * * 1 ( ,..., ) 0 ... ... ... ( ,..., ) 0 n n n f x x f x x = =
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Second-order condition in the bivariate case First total differential 1 2 1 1 2 1 2 1 2 2 1 1 2 2 ( , ) ( , ) ( , ) i.e. y f x x dy f x x dx f x x dx dy f dx f dx = = + = + 1 2 ( , ) f x x
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Second-order condition in the bivariate case Second total differential 2 1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 2
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