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Unformatted text preview: Some Old Prelim Questions 1. Mr. Green consumes two goods, X and Y . His utility function is U ( x,y ) = ln( x + 2 y y 2 2 ) where x is his consumption of X and y is his consumption of Y . Let Good X be the numeraire with a price of 1 and let p be the price of Good Y . Let w be Mr. Green’s income and assume that w > 1. A Ms. Blue’s preferences over bundles of X and Y are represented by the utility function U ( x,y ) = x + 2 y y 2 2 . How do Ms Blue’s preference compare with Mr. Green’s? Explain. B Define homothetic preferences. Does Mr. Green have homothetic prefer ences? If so, prove it. If not, show that he does not. C Define convex preferences. Show that Ms. Blue has convex preferences. (In your proof feel free to use the fact that f ( y ) = 2 y y 2 is a concave function.) Does Mr. Green have convex preferences? D Find Mr. Green’s Walrasian demand functions x ( p,w ) and y ( p,w ) for the two goods X and Y when the price of X is 1, the price of Y is p and income is w . Be careful to specify demand at priceincome combinations that lead to a “corner solution” where Mr. Green buys no Y . Note: We have made life easier for you by assuming that w > 1. This ensures that there is no corner solution where Mr. Green buys no X . ( For extra credit if you have extra time, you could show that if w > 1 , then x ( p,w ) > for all p > . ) E Find Mr. Green’s indirect utility function v ( p,w ). (Be careful to specify the indirect utility function for priceincome combinations that lead to “corner solutions” as well those that lead to as interior solutions.) F What does Roy’s law tell us about the relation between the indirect util ity function and the Walrasian demand function for Y . Verify by direct calculation that Roy’s law is satisfied by Mr. Green’s demand function for Y . G Find Mr. Green’s expenditure function e ( p,u ) where the price of X is 1 and the price of Y is p ....
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This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.
 Spring '10
 fallahi
 Utility

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