# Econ100a spr07 hw1 answers - Econ 100A Spring 2007 Problem...

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Econ 100A, Spring 2007 Problem Set 1 Key A. Find the first and second derivatives for the following functions of x. (1) f(x) = a+bx+cx 2 +dx 3 dx c x f dx cx x x f 6 2 ) ( 3 2 ) ( 2 + = ! ! + + = ! (2) f(x) = ln(4x 3 ) 2 3 2 3 ) ( 3 4 12 ) ( x x f x x x x f ! = " " = = " B. Let z = 4x 4 y 3 -5x 2 y+xy 2 (1) Derive the first and second partial derivatives for z. What is true about the cross partials? y x y x x y y x z y x z x y x y y x z y x z y x y x y x y x z y x z y y x x y x z y x z xy x y x y y x z y x z y xy y x x y x z y x z yx yy xy xx y x 2 10 48 ) , ( ) , ( 2 24 ) , ( ) , ( 2 10 48 ) , ( ) , ( 10 48 ) , ( ) , ( 2 5 12 ) , ( ) , ( 10 16 ) , ( ) , ( 2 3 2 4 2 2 2 3 2 3 2 2 2 2 2 4 2 3 3 + ! = " " " = + = " " = + ! = " " " = ! = " " = + ! = " " = + ! = " " = Note that ) , ( ) , ( y x z y x z yx xy = (2) Derive the total differential of z dy xy x y x dx y xy y x dz d y x z d y x z dz y y x x ) 2 5 12 ( ) 10 16 ( ) , ( ) , ( 2 2 4 2 3 3 + ! + + ! = + =

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C. Calculate the slope and the elasticity of the following functions We examine the slope dx x df x f ) ( ) ( = ! and elasticity ) ( * ) ( x f x dx x df of a few different functions f( x) Slope Elasticity 3+4x 4 4* x x 4 3 + 12+5x 5 5* x x 5 12 + 2 10 x 20x 20x* 2 10 x x =2 3 2 x 2 6 x 3 2 2 6 x x x =3 Functions of the type f(x)=a+bx have constant slope, f’(x)=b, but not constant elasticity. Functions of the type f(x)=cx d have constant elasticity, ) ( * ) ( x f x dx x df =d. D. A consumer is asked to rank order the commodity bundles below: X 1 X 2 Bundle A 9 3 Bundle B 25 1 Bundle C 4 2 D1. For an economist to represent the consumer’s preferences with a utility function, what
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Econ100a spr07 hw1 answers - Econ 100A Spring 2007 Problem...

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