L15PartialDifferentiation

L15PartialDifferentiation - Introduction to Microeconomics...

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Introduction to Microeconomics Lecture 15 Partial Differentiation Ian Crawford
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Outline Partial differentiation Marginal products and elasticities Total differentiation Isoquants and indifference curves Monotonic transformations
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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 = + + = + In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant.
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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 = = ∂ ∂ = = ∂ ∂ 4 ( , ) 4 2 6 z x y x xy y = + +
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1 2 3 2 2 ( , , ,... ) N i j j i z x x x x z z x x x x = ∂ ∂ ∂ ∂ Partial Derivatives A function of N variables has N first order and N ( N-1 ) second order partials. Luckily, second-order partials are symmetric (Young’s Theorem) for well–behaved functions, (the order of partial differentiation is reversible) so there are really only ½ N ( N-1 )
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1 2 3 1 2 3 ( , , ,... , ) ( , , ,... , ) N i N i m p p p p u m x p p p p u p = Partial Derivatives Young’s Theorem comes in useful in the Slutsky Equation: It turns out that the partial derivative of the expenditure/cost function (which we met in Lecture 6) gives the Hicksian demands: This is a result called Shephard’s Lemma.
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1 2 3 2 1 2 3 ( , , ,... , ) ( , , ,... , ) i N i i N j i j m x p p p p u p x p p p p u m p p p = = ∂ ∂ Partial Derivatives If you take the partial derivative of the Hicksian demand function with respect to a price you are asking how the demand changes as you change a price whilst holding utility constant . This is the substitution effect.
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It is symmetric by Young’s Theorem: E.g. the substitution effect on leisure induced by a change in the price of goods, is the same as the substitution effect on goods induced by a change in the wage. ( 29 ( 29 ( 29 ( 29 2 2 p, p, p, p, j i j i j j i i x u x u m u m u p p p p p p = = = ∂ ∂ ∂ ∂ Partial Derivatives ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 p, p, p, p, p p, , i i j j j j i i i j j i x m x m x p m x m x m x x u p p x u p m = - = -
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Alternative Notation ( 29 2 2
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L15PartialDifferentiation - Introduction to Microeconomics...

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