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ch08-multivariate-calculus

# ch08-multivariate-calculus - Multivariate Calculus...

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Unformatted text preview: Multivariate Calculus Professor Erkut Ozbay Economics 300 Multivariate calculus • Calculus with single variable (univariate) • Calculus with many variables (multivariate) 1 2 ( , ,..., ) n y f x x x = ( ) y f x = Partial derivatives • With single variable, derivative is change in y in response to an infinitesimal change in x • With many variables, partial derivative is change in y in response to an infinitesimal change in a single variable x i (hold all else fixed) • Total derivative is change in all variables at once Cobb-Douglas production function • How does production change in L? • Marginal product of labor (MPL) • Partial derivatives use ∂ instead of d 1 1 2 2 20 Q K L = 1 1 2 2 10 Q K L L- = Cobb-Douglas production function • How does production change in K? • Marginal product of capital (MPK) • Partial derivatives use ∂ instead of d 1 1 2 2 20 Q K L = 1 1 2 2 10 Q K L K- = Cobb-Douglas production function • Note • Produce more with more labor, holding capital fixed • Produce more with more capital, holding labor fixed 1 1 2 2 20 Q K L = 1 1 2 2 10 Q MPK K L K- = = 1 1 2 2 10 Q MPL K L L- = = Second-order partial derivatives • Differentiate first-order partial derivatives 3 1 2 2 3 1 2 2 2 2 2 2 5 5 Q Q K L L L L Q Q K L K K K-- = = - = = - 1 1 2 2 10 Q K L L- = 1 1 2 2 10 Q K L K- = Second-order partial derivatives...
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ch08-multivariate-calculus - Multivariate Calculus...

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