501PS06

# 501PS06 - ECON501 Advanced Microeconomic Theory 1 Fall Semester 2007 Problem Set 7 The due date for this problem set is Monday November 19 1 Assume

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ECON501 Advanced Microeconomic Theory 1 Fall Semester 2007 Problem Set 7 The due date for this problem set is Monday November 19 1. Assume a f nite state space S = { s 1 ,...,s N } and a one-dimensional consequence space X = R + Suppose a person’s preference relation over the set of state-contingent vectors c =( c 1 ,...,c N ) R N + , admits a Subjective Expected Utility Representation. That is, there exists a probability vector ( π 1 ,...,π N ) and utility index u : R + R such that for any pair of acts c, c 0 : c < c 0 N X n =1 π n u ( c n ) N X n =1 π n u ( c 0 n ) Furthermore, assume that u ( · ) is a strictly concave and strictly increasing function and that π n > 0foreach n =1 ,...,N .S in c e X = R + ,w ecande f ne a convex combination of two contingent consumption vectors as follows: for any pair of contingent con- sumption vectors c , c 0 and any α in (0 , 1), c 00 = α c +(1 α ) c 00 is the contingent consumption vector for which c 00 n = αc n +(1 α ) c 0 n for every n =1 ,...,N. (a) Show that if for any pair of contingent consumption vectors c and c 0 c c 0 and c 6 = c 0 then c 00 = 1 2 c + 1 2 c 0 Â c (b) Explain why is this not a contradiction of the independence axiom. Notation: for any pair of contingent consumption vectors c and c 0 and any event E S ,let c E c 0 denote the contingent consumption vector c 00 for which c 00 n = ½ c n if s n E c 0 n if s n

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(c) Show that a preference relation that admits a subjective ex- pected utility representation satis f es the following property called the sure-thing principle. Sure-Thing Principle: Fo ranytwocon t ingen tcon sump t ion vectors c and c 0 ,andanyevent E S c % c 0 E c c E
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## This note was uploaded on 09/04/2010 for the course ECON 501 taught by Professor Grant during the Spring '10 term at Rice.

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501PS06 - ECON501 Advanced Microeconomic Theory 1 Fall Semester 2007 Problem Set 7 The due date for this problem set is Monday November 19 1 Assume

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