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# lecture15kg - Introduction to Microeconomics Lecture 15...

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Introduction to Microeconomics Lecture 15 Partial Differentiation Kathryn Graddy

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Outline Partial differentiation Marginal products and elasticities Total differentiation Isoquants and indifference curves Monotonic transformations and Cobb Douglas Comparative Statics
Partial Derivatives 4 3 ( , ) 4 2 6 How does change when increases but doesn't change? 16 2 is called the partial derivative of with respect to z x y x xy y z x y z x y x z z x x = + + = +

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Partial Derivatives 3 First Order Partial Derivatives 16 2 2 6 z x y x z x y = + = + 2 2 2 2 2 Second-order partial derivatives 48 0 z x x x y = = 2 2 Cross- partial derivatives 2 2 z z y x y x z z x y x y = = ∂ ∂ = = ∂ ∂
2 2 2 2 2 2 2 2 A function ( , ) of two variables has four second-order partial derivatives: , , and but z x y z z z z x y y x x y z z x y y x ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ Note that the two cross-partial derivatives are the same: it doesn't matter whether we differentiate with respect to first and then with respect to , or vice-versa. This happens with all "well-beh x y aved" functions.

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Variables ( 29 1 2 1 2 2 2 2 2 2 1 3 1 1 2 2 3 1 1 3 , ,..., , ,..., , , ,... n
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lecture15kg - Introduction to Microeconomics Lecture 15...

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