PS1 KEY SP092

PS1 KEY SP092 - Econ 100A, Dr. Famulari, Spring 2010...

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Econ 100A, Dr. Famulari, Spring 2010 Practice Problems: Week 1 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 b 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 = . This illustrates the mathematical result that (under quite general conditions) the order in which partial differentiation is conducted does not matter when evaluating second order partial derivatives – sometimes called “Young’s Theorem” (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 + - + + - = + = 1
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C. Calculate the slope and the elasticity of the following functions Slope is dx x df x f ) ( ) ( = while elasticity is ) ( * ) ( x f x dx x df 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. Solve the following : 2 2 max x yx x - (1) What is the choice variable? X (2) What is the solution function? Differentiating the objective function with respect to X we get the first order condition, y-4X=0. Solving the F.O.C. we have the solution function: X*=Y/4 (3) What is the optimal value function (the maximal value of the function)? The optimal value function is 8 ) 4 ( 2 ) 4 ( *) ( 2 * 2 2 2 y y y y x yx = - = - E. A consumer is asked to rank order the commodity bundles below:
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PS1 KEY SP092 - Econ 100A, Dr. Famulari, Spring 2010...

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