701_prelim_2011_Spring final version-1_0

701_prelim_2011_Spring final version-1_0 - Microeconomic...

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Unformatted text preview: Microeconomic Theory I Preliminary Examination University of Pennsylvania June 6, 2011 Instructions This exam has 5 questions and a total of 100 points. You have two hours to complete it. Answer each question in a SEPARATE exam book. If you need to make additional assumptions, state them clearly. Be concise. Write clearly so that you might get partial credit. Good luck! 1. (20 pts) Suppose that u . & / is a continuous utility function representing a locally nonsatiated preference relation on the consumption set X D R L C : Write v . p ; w/ for the indirect utility function, where p 2 R L CC is the price vector and w 2 R C is the income. (a) (5 pts) Show that v is nonincreasing in p and strictly increasing in w . Point out where the property of locally nonsatiated preference is used in your argument. (b) (5 pts) Give an example of a utility function u that satisfies the previous properties, and the corresponding v does not strictly decrease in some p ‘ : (c) (5 pts) Show that v is quasi-convex in . p ; w/ : (d) (5 pts) Let p n ! p and w n ! w be sequences of prices and incomes, where p 2 R L CC and w 2 R C . Show that lim inf n v & p n ; w n ¡ ¡ v . p ; w/ : Point out where continuity and local nonsatiation are used in your argument. (Merely citing thePoint out where continuity and local nonsatiation are used in your argument....
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701_prelim_2011_Spring final version-1_0 - Microeconomic...

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