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ch08-multivariate-calculus

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

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Multivariate Calculus Professor Erkut Ozbay Economics 300

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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

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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 - =

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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 0 Q MPK K L K - = = 1 1 2 2 10 0 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 - =

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Second-order partial derivatives Note
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ch08-multivariate-calculus - Multivariate Calculus...

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