# mid10 - Question 3. Consider the function f ( x 1 ,x 2 ) =...

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Name Midterm Exam, Econ 210A, Fall 2010 Answer as many questions as you can. Put your answers on these sheets. Question 1. Let f ( x 1 ,x 2 ) = p max { x 1 ,x 2 } for all x 1 0, x 2 0. A) Is f a homogeneous function? If so, of what degree? Explain your answer. If not, show that it is not. B) Is f a concave function? Is f a quasi-concave function? Explain your answers. C) Is f a convex function? Is f a quasi-convex function? Explain your answers.

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Question 2. Ms Rodent has the utility function u ( x 1 ,x 2 ) = (min { x 1 ,x 2 } ) 2 . Wild Thing has the utility function u ( x 1 ,x 2 ) = max { x 1 ,x 2 } . A) Find Ms Rodent’s Marshallian demand function, her indirect utility function, her expenditure function, and her Hicksian demand function. B) Find Wild Thing’s Marshallian demand function, his indirect utility func- tion, his expenditure function, and his Hicksian demand function.
C) Find the Slutsky substitution matrix for Ms Rodent. What can we say about the income and substitution eﬀects of a change in prices on her demand?

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Unformatted text preview: Question 3. Consider the function f ( x 1 ,x 2 ) = x 1 + x 2 + ax 1 x 2-1 2 x 2 1-1 2 x 2 2 dened on the Euclidean plane, < 2 . A) What is the gradient of f ? Evaluate this gradient at the point x 1 = 1, x 2 = 2. B) Find the Hessian of f . C) For what values of a , if any, is f a concave function? For what values of a if any is f a convex function? Justify your answer. D) For what values of a does f have a global maximum? If f has a global maximum, nd this maximum. Question 4. A) State Eulers theorem on homogeneous functions. B) Suppose that the function f ( x 1 ,...,x n ) is homogeneous of degree k . Consider the function f i ( x 1 ,...,x n ) = f ( x 1 ,...,x n ) x i . Is this also a homogeneous function? If so, of what degree is it homogeneous? C) Extra Credit: Supply proofs for A and B....
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## This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.

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mid10 - Question 3. Consider the function f ( x 1 ,x 2 ) =...

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