# prelims_Micro Prelim ANSWERS Sept 2005 - MICROECONOMIC...

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M ICROECONOMIC T HEORY P RELIM F ALL 2005 A NSWER K EY Q UESTION 1 Throughout this question we assume that consumers have preferences over the quantities of two market goods, x 1 > 0 and x 2 > 0, as well as the size g > 0 of a park. A consumer has no control over g , and therefore takes the value g as given, but he or she spends all his of her (after-tax) wealth w , also given to him or her, buying x 1 and x 2 in the market at the given prices ( p 1 , p 2 ). In summary, in his or her Walrasian demand the consumer takes ) , , , ( 2 1 g w p p as given, and chooses x 1 and x 2 subject to the budget constraint w x p x p + 2 2 1 1 . Throughout this question we assume that wealth is always large enough so that at the chosen point the amount of x 2 is strictly positive. 1.1. Consumer Karl’s preference relation on 3 + can be represented by the utility function 2 1 2 1 3 ) 1 ( 2 ) , , ( ˆ : : ˆ x x g g x x u u + + = + + . 1.1(a). Compute Karl’s Walrasian demand functions ) , , , ( ~ 2 1 1 g w p p x and ) , , , ( ~ 2 1 2 g w p p x , and indirect utility function. Comment. A NSWER . Given p 1 , p 2 , w and g , choose x 1 and x 2 in order to maximize 2 1 ) 1 ( 2 x x g + + subject to p 1 x 1 + p 2 x 2 = w , or, equivalently, choose x 1 in order to maximize 2 1 1 1 ) 1 ( 2 p x p w x g + + , with FOC 2 1 1 1 p p x g = + , i.e., ) 1 ( ) , , , ( ~ 2 1 2 2 2 1 1 + = g p p g w p p x , and therefore ) 1 ( ) 1 ( ) , , , ( ~ ) , , , ( ~ 1 2 2 2 2 1 2 2 1 2 2 2 1 1 1 2 2 1 2 + = + = = g p p p w p g p p p p w p g w p p x p p w g w p p x . Indirect utility: ) 1 ( ) 1 ( ) 1 ( 2 1 2 2 2 1 2 2 + + + + g p p p w g p p g = 2 1 2 ) 1 ( p w p p g + + . 1.1(b). Let ( p 1 , p 2 ) = (1, 1), and let Karl’s initial wealth be 10. The government is considering a program that would improve g from 0 to 3, and would require Karl to pay a tax of 2. The prices ( p 1 , p 2 ) would not change as a result of the program. Would Karl benefit from the program? A NSWER . Karl’s indirect utility without the program = w w + = + + 1 1 1 1 ) 1 0 (. Karl’s indirect utility with the program = . 1 2 4 1 2 1 1 ) 1 3 ( w w w + > + = + + Thus, Karl benefits from the program (his willingness to pay for the improvement in g is 3, greater than the tax).

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2 1.2 . Karl has a cousin, Priscilla, whose preference relation on 3 + can be represented by the utility function 1 ) 1 ( 2 2 1 + + + g x x g . 1.2(a). Is Priscilla’s preference relation on 3 + identical to Karl’s? A NSWER . No way. Notice that for Karl the park is a good, whereas for Priscilla it is a bad (perhaps the park’s deer eat her vegetable garden). Note that Priscilla’s utility function is NOT an increasing transformation of Karl’s.
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prelims_Micro Prelim ANSWERS Sept 2005 - MICROECONOMIC...

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