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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 CobbDouglas 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 = CobbDouglas 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 = CobbDouglas 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 = = Secondorder partial derivatives Differentiate firstorder 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 = Secondorder partial derivatives...
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This note was uploaded on 12/04/2011 for the course ECON 300 taught by Professor Cramton during the Fall '08 term at Maryland.
 Fall '08
 cramton
 Economics

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