400ps9-11-soln - ECON 400 Winter 2011 Answers for Problem...

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E C O N 4 0 0 H a r t m a n Winter 2011 Answers for Problem Set IX 1.a. The Marshallian demand functions are * (,, ) 2 M xp sM p and * 2 M yp s . The indirect utility function is 2 * 4 M Up ps . 1.b The Hicksian demand functions are 1/2 1/2 1/2 (,,) H x psu p s u and 1/2 1/2 1/2 H y . The expenditure function is * 1/2 1/2 1/2 (,,) 2 M psu . 1.c.i. Substitute 4 p , 9 s , and 216 M into the indirect utility function to see that the individual's initial utility index is 2 * (216) 46656 ( , , ) 324 4(4)(9) 144  . Alternatively, you can find the initial utility index by substituting 4 p , 9 s , and 216 M into the Marshallian demand functions to see that the individual consumes 27 x and 12 y , and then substituting these values into the utility function to get 324 U . Now, substitute 324 u , 9 p , and 9 s into the expenditure function to see that the individual needs 1/2 1/2 1/2 2(9) (9) (324) 2(3)(3)(18) 324 M to stay on the original indifference curve after the price change. (The numbers happen to work out so that the utility index and the required expenditure both equal 324; this does not happen in general.) 1.c.ii. The compensating variation is the expenditure in the original situation minus the expenditure required to stay on the original indifference curve after the price change: 1/2 1/2 1/2 216 2(9) (9) (324) 216 2(3)(3)(18) 216 324 108 CV   . This means that the individual would need an additional $108 to be as well off after the price increase as before. 1.c.iii. Substitute 9 p , 9 s , and 216 M into the indirect utility function to see that the individual's initial utility index would be 2 * (216) 46656 ( , , ) 144 4(9)(9) 324 if the price increase occurs. Alternatively, you can find this initial utility index by substituting 9 p , 9 s , and 216 M into the Marshallian demand functions to see that the individual would consume 12 x
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400ps9-11-soln - ECON 400 Winter 2011 Answers for Problem...

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