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ps2_101_2011_solution

ps2_101_2011_solution - Economics 101 Problem Set 2...

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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 Andrew’s 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 Andrew’s 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 Bruce’s preferences rank x 2 and x 4 ? Soln: 1
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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. Anthony’s 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 0 , we calculate U ( x ) and U ( x 0 ). Since we can always compare U ( x ) with U ( x 0 ), Anthony can always compare x and x 0 .
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