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Unformatted text preview: Economics 101 Problem Set 2 Solutions January 25, 2011 Lingwen Huang 1. Andrew consumes positive amounts of goods 1 and 2. He thinks that 3 units of good 1 is always a perfect substitute for 1 unit of good 2. Which of the following utility functions represent Andrews prefer ences? (a) U ( x 1 ,x 2 ) = min { x 1 , 3 x 2 } . (b) U ( x 1 ,x 2 ) = 9 x 2 1 + 6 x 1 x 2 + x 2 2 . (c) U ( x 1 ,x 2 ) = 5 x 1 + 15 x 2 + 100. (d) U ( x 1 ,x 2 ) = x 1 + 3 x 2 + 500. Soln: The preference represented by the utility function in 1(a) is perfect complementary. The utility function in 1(b) represents the same pref erence as a utility function W ( x 1 ,x 2 ) = 3 x 1 + x 2 (because W = U ), so 3 units of good 1 is always a perfect substitute for 9 units of good 2. Both the utility functions in 1(c) and 1(d) represent the same pref erence as the utility function W ( x 1 ,x 2 ) = x 1 + 3 x 2 (make sure you understand why). Since with utility function W ( x 1 ,x 2 ) = x 1 + 3 x 2 , 3 units of good 1 is always a perfect substitute for 1 unit of good 2 (you can calculate that MRS 21 = U/x 2 U/x 1 = 3 ), utility functions 1(c) and 1(d) represent Andrews preferences. 2. Bruce has preferences, , that are complete and transitive. Suppose there are four bundles x 1 , x 2 , x 3 , and x 4 , with the property that (a) x 1 v x 2 , (b) x 3 v x 4 , and (c) x 1 x 3 . How do Bruces preferences rank x 2 and x 4 ? Soln: 1 From 2(a) and 2(c), transitivity implies that x 2 v x 1 x 3 . Then, from 2(b), we have x 2 x 3 v x 4 . Therefore, x 2 x 4 . 3. Anthonys utility function for good 1 and good 2 is given as the fol lowing. Graph his indifference curves for different utility functions in the commodity space, R 2 + , and discuss whether these preferences are complete, transitive, monotonic and convex. Justify your answer. (a) U ( x 1 ,x 2 ) = (2 x 1 + 5 x 2 ) 2 50. (b) U ( x 1 ,x 2 ) = ( x 1 2) 2 ( x 2 3) 2 + 10. (c) U ( x 1 ,x 2 ) = min { 2 x 1 x 2 , 2 x 2 x 1 } . Soln: We first argue that all these utility functions represent preferences, which are complete and transitive (in fact, if a preference has util ity representation, it must be complete and transitive). For any two bundles x and x , we calculate U...
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This note was uploaded on 03/02/2011 for the course ECON 101 taught by Professor Dannicatambay during the Spring '08 term at UPenn.
 Spring '08
 DANNICATAMBAY
 Microeconomics, Utility

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