14.123 Homework Assignment 1 - B c W ∈ C such that for...

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I 1 14.123 Microeconomic Theory III Problem Set 1 The due date for this assignment is Thursday February 11. 1. Let P be the set of all lotteries p = ( p x , p y , p z ) on a set C = { x, y, z } of consequences. Below, you are given pairs of indi ff erence sets on P . For each pair, check whether the indi ff erence sets belong to a preference relation that has a Von-Neumann and Morgenstern representation (i.e. expected utility representation). If the answer is Yes, provide a Von-Neumann and Morgenstern utility function; otherwise show which Von-Neumann and Morgenstern axiom is violated. (In the fi gures below, setting p z = 1 p x p y , we describe P as a subset of R 2 .) (a) I 1 = { p | 1 / 2 p y 3 / 4 } and I 2 = { p | p y = 1 / 4 } : p x p y 1 1 I 2 1/4 1/2 3/4 (b) I 1 = { p | p y = p x } and I 2 = { p | p y = p x + 1 / 2 } : p x p y 1 1 I 1 I 2 1/2 2. For any preference relation º that satis fi es the Independence Axiom, show that the following are true. (a) For any p, q, r, r 0 P with r r 0 and any a (0 , 1] , ap + (1 a ) r º aq + (1 a ) r 0 ⇐⇒ p º q. (1) (b) For any p, q, r P and any real number a such that ap +(1 a ) r, aq +(1 a ) r P , if p q , then ap + (1 a ) r aq + (1 a ) r. (2) (c) For any p, q P with p  q and any a, b [0 , 1] with a > b , ap + (1 a ) q  bp + (1 b ) q. (3) 1
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(d) There exist c
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Unformatted text preview: B , c W ∈ C such that for any p ∈ P , c B º p º c W . (4) [Hint: use completeness and transitivity to fi nd c B , c W ∈ C with c B W for º c º c all c ∈ C ; then use induction on the number of consequences and the Independence Axiom.] 3. Let P be the set of probability distributions on C = { x, y, z } . Find a continuous preference relation º on P , such that the indi f erence sets are all straight lines, but º does not have a von Neumann-Morgenstern utility representation. 4. Let º ˙ be the "at least as likely as" relation de fi ned between events in Lecture 3. Show that º ˙ is a qualitative probability. 2...
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