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# PS6 - Intermediate Microeconomics 2008 Problem Set No306...

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Intermediate Microeconomics, 2008 Problem Set No 6 due: Thursday, Jan 31; Wed, Jan 30 Problems 1) We considered preferences represented by the following utility function in the last homework: U ( q 1 ; q 2 ) = q 1 q 2 . Income is °xed at Y = 2 and the price of good two is °xed at p 2 = 1 . a) Suppose p 1 increases. Illustrate graphically how the change in demand can be decomposed into a substitution e/ect and an income e/ect. b) Suppose p 1 increases from 1 4 to 1 2 . Derive the total change of the demand for good one. Decompose that change into the change due to the substitution e/ect and the change due to the income e/ect. Note: The total change in demand is D 1 ° 1 2 ; Y ± ° D 1 ° 1 4 ; Y ± at Y = 2 . The change due to the substitution e/ect is H 1 ° 1 2 ; ° U 0 ± ° H 1 ° 1 4 ; ° U 0 ± . The change due to the income e/ect is the change in demand D 1 ° 1 2 ; Y ± when income changes from the hypothetical level Y = E ° 1 2 ; ° U 0 ± to Y = E ° 1 2 ; ° U N ± = 2 . ° U 0 and ° U N are the utility levels at the old and the new price, p 0 1 = 1 4 and p N 1 = 1 2 , respectively. c) Suppose p 1 increases from 1 4 to 4 . Again, decompose the change of demand into the change that is due to the substitution e/ect and the change that is due to the income e/ect. d) Calculate the elasticity of demand at p 1 = 1 4 . Decompose this elasticity into the Hicks elasticity " ° and the weighted income elasticity °± . 3) Let p 2 = 1 . Consider the following utility functions: U ( q 1 ; q 2 ) = min f q 1 ; q 2 g U ( q 1 ; q 2 ) = q 1 + q 2 U ( q 1 ; q 2 ) = q 2 + p q 1 a) Illustrate the indi/erence curves graphically and illustrate how a price change a/ects demand. Carefully illustrating the indi/erence curves will be helpful. You might want to check earlier problem sets when graphing the indif- ference curves. b) Illustrate how a change in income changes demand. 1

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c) Derive the expenditure function E ( p 1 ; U ) and the Hicks demand curve for good one, H 1 ( p 1 ; U ) for U = 4 and for general U .
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PS6 - Intermediate Microeconomics 2008 Problem Set No306...

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