Tests2015.pdf - ECON 5113 Microeconomic Theory Winter 2015...

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ECON 5113 Microeconomic Theory Winter 2015 Test 1 January 30, 2015 Answer ALL Questions Time Allowed: 1 hour 20 minutes Instruction: This is a closed-book exam. No mobile phones or calculators are allowed. Please write your an- swers on the answer book provided. Use the right-side pages for formal answers and the left-side pages for your rough work. Answers should be provided in complete and readable essay form, not just in mathematical for- mulae and notations. Remember to put your name on the front page. You can keep the question sheet after the test. 1. Suppose that a consumer’s preference relation % on a consumption set X R n + is complete and transitive. (a) Define the following induced relations on X : (a) “is strictly preferred to”, , (b) “is indi erent to”, . (b) Explain if is complete, reflexive, transitive, circular, symmetric, asymmetric, or antisym- metric. (c) Show that is an equivalence relation. 2. Suppose that a consumer’s preference relation satis- fies A1, A2, A3, A4, and A5’ (but not A5, see ap- pendix A). (a) Explain if the solution to the utility maximiza- tion problem still exists. (b) If yes, is the choice for the optimal bundle unique? (c) Illustrate the answers above with a diagram with a two-good case. 3. Aminata buys two goods, xigua and yam every week. Her utility function is given by U ( x, y ) = 2 x + log( y + 4) . The market prices for xigua and yam are $2 and $1 per kg respectively. Aminata’s weekly budget for food is $5. (a) Set up Aminata’s grocery shopping as a Kuhn- Tucker optimization problem. (b) Find the optimal bundle. (c) State the condition of the marginal rate of sub- stitution at Aminata’s optimal bundle in rela- tion with the market prices. 4. A consumer’s utility function is given by U ( x 1 , x 2 ) = Ax 1 x 1 - 2 , A > 0 , 0 1 . (a) Set up the expenditure minimization problem and find the expenditure function E ( p , u ). (b) A cost of living index is defined as I ( p 0 , p 1 , u 0 ) = E ( p 1 , u 0 ) E ( p 0 , u 0 ) , where p 0 and p 1 are prices in period 0 and 1 respectively, and u 0 is the utility level in period 0. Show that the index for our consumer is in- dependent of the utility level. 5. Consider the Slutsky equation @ d i ( p , y ) @ p j = @ h i ( p , u ) @ p j - d j ( p , y ) @ d i ( p , y ) @ y . State and prove the law of demand. You can assume the properties of the expenditure function listed in appendix B.
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Appendix A: Axioms of Consumer Choice For all a , b , c in the consumption set X , the relation % satisfies the following axioms: A1 Completeness: Either a % b or b % a . A2 Transitivity: If a % b and b % c , then a % c . A3 Continuity: The upper contour set % ( a ) and the lower contour set - ( a ) are closed. A4’ Local Non-Satiation: For any > 0, there exists an x 2 B ( a ) \ X such that x a .
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