Tests2009.pdf - ECON 5113 Advanced Microeconomics Winter...

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ECON 5113 Advanced Microeconomics Winter 2009 Test 1 February 11, 2009 Answer ALL Questions Time Allowed: 1 hour 20 minutes Instruction: Please write your answers on the answer book provided. Use the right-side pages for formal an- swers and the left-side pages for your rough work. Also, start each question on a new page. Read the questions carefully and provide answers to what you are asked only. Do not spend time on what you are not asked to do. Remember to put your name on the front page. 1. Let be a preference relation on R n + . (a) Define the relations and induced by . (b) Suppose is complete and transitive. Show whether or not and induced by are complete and transitive. 2. Suppose that U : R n + R + is a continuous, strictly increasing, and quasi-concave utility function which represents a preference relation . Show that satisfies (a) transitive, (b) locally nonsatiation, (c) convex. 3. Suppose that a consumer’s preference structure is represented by the utility function U ( x 1 , x 2 ) = a log x 1 + b log x 2 , where a, b > 0 and a + b = 1. (a) Find the Marshallian demand functions. (b) Find the indirect utility function. 4. Let U : R n + R + be a continuous, increasing, and strictly quasiconcave utility function. (a) Define the expenditure function association with U . (b) Prove that the expenditure function is homo- geneous of degree one in prices. (c) Define the Hicksian (compensated) demand function. (d) State and prove Shephard’s lemma. (e) Show that if U ( x ) is a homothetic function then the corresponding expenditure function can be written as: E ( p , u ) = e ( p ) ψ ( u ) . 5. The Slutsky matrix below belongs to a consumer with a regular utility function of three goods, with market price p = (1 , 2 , 6) T : 10 ? ? ? 4 ? 3 ? ? . (a) Supply the missing numbers. (b) Prove the Slutsky matrix must be negative semidefinite. 6. Let the observed consumption bundles of a con- sumer in period 0 and period 1 be x 0 and x 1 , given the price vectors p 0 and p 1 respectively.
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