Tests2011.pdf - ECON 5113 Advanced Microeconomics Winter...

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ECON 5113 Advanced Microeconomics Winter 2011 Test 1 February 4, 2011 Answer ALL Questions Time Allowed: 1 hour 20 min Attention: 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. Re- member 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) Show that ∩ ∼ = . (c) Show that for every x R n + , ( x ) ( x ) ∪ ≺ ( x ) = R n + . 2. Let U : R n + R + be an increasing, twice di ff eren- tiable, and quasiconcave utility function. (a) Define the indirect utility function. Make sure you explain every variable you use. (b) State and prove Roy’s identity. 3. Suppose a consumer’s preference structure is repre- sented by the utility function U ( x ) = x α 1 x 1 α 2 , 0 < α < 1 . (1) (a) Set up the expenditure minimization problem. (b) Derive the Hicksian demand function h ( x , u ). (c) Find the expenditure function. (d) Find the indirect utility function. 4. Let U : R n + R + be an increasing, twice di ff eren- tiable, and quasiconcave utility function. (a) Define price elasticity ij . (b) Define income elasticity η i . (c) Prove that for i = 1 , . . . , n , n j =1 ij + η i = 0 . Hint: The ordinary demand function for good i is homogeneous of degree zero in ( p , y ). 5. Suppose a consumer’s utility function is given by Equation (1) with α = 1 / 2. Define the Laspeyres- Kon¨us cost of living index as P K ( p 0 , p 1 , u 0 ) = E ( p 1 , u 0 ) E ( p 0 , u 0 ) , where E is the expenditure function with the super- scripts 0 and 1 indicating the time periods. (a) The consumer’s income changes from y 0 to y 1 . Show that the consumer will be worse o ff in period 1 if y 1 /y 0 is less than P K ( p 0 , p 1 , u 0 ). (b) In period 0 the prices are p 0 = (1 , 1) and income is y 0 = 12. Suppose in period 1 p 1 = (2 , 1). Calculate the price index P K . Happy New Year!
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ECON 5113 Advanced Microeconomics Winter 2011 Test 2 March 4, 2011 Answer ALL Questions Time Allowed: 1 hour 20 min Attention: 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. Re- member to put your name on the front page.
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