Lecture15 Partial Differentiation

Lecture15 Partial Differentiation - Introduction to...

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1 Introduction to Microeconomics Lecture 15 Partial Differentiation Ian Crawford Outline z Partial differentiation z Marginal products and elasticities z Total differentiation z Isoquants and indifference curves z Monotonic transformations Partial Derivatives 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. 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 zxy x x y y z xy x z zx x =++ =+ Partial Derivatives 3 First Order Partial Derivatives 16 2 z x z 2 2 2 2 Second-order partial derivatives 48 z x x = 4 2 6 x x y y = ++ 26 x y = + 2 0 x y = 2 2 Cross- partial derivatives 2 2 zz yx y x x y ∂∂ ⎛⎞ == ⎜⎟ ∂ ∂ ⎝⎠ 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) 123 22 (, , , . . . ) N ij ji zx x x x x xx x = for well–behaved functions, (the order of partial differentiation is reversible) so there are really only ½ N ( N-1 ) 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: ( , , ,... , ) ( , , ,... , ) N iN i mp p p p u m xppp pu p = This is a result called Shephard’s Lemma.
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2 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 . 123 2 (, , , . . . ,) ( , , ,... , ) iN i j ij m xppp p u p m p pp = = ∂∂ This is the substitution effect. It is symmetric by Young’s Theorem: ( ) ( ) () 22 p, p, p, p, j i ji j j i i x u xu mu mu p p === Partial Derivatives 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. ( ) ( ) ( ) p, p, p, p, p p, , ii j j jj i i i j j i xm x pm x xu p p p m =− = Alternative Notation First-order partial derivatives can be written: or and or xy ff 22 2 and the second-order derivatives are: or and or and or For a function , , ,... we sometimes xx yy xy ff f f xyx y fxxx ∂∂ ∂ ∂∂∂ 121 11 3 2 1 3 write: for , for , for , for etc.
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.

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Lecture15 Partial Differentiation - Introduction to...

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