Exx5_Solution - x 2 ” x and x ” x for all x ∈ X 1 By...

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Intermediate Microeconomics Homework Set 1: Solution to Exercise 5 Exercise 5 Denise chooses an alternative in the set X = { x 1 ,...,x 20 } . Show that if her pref- erences are complete, transitive and reflexive, then there exists an alternative she prefers to all others: That is, there is x * X such that x * x for all x X . Answer In this exercise, you are asked to prove that there exists a most-preferred alterna- tive. It helps to think about this problem as a situation where you would have to find the heaviest person in a group of 20 people, only using a balance scale. Proof . Step 1 : Pick any x X . By completeness, any other alternative in X is either preferred to x or less-preferred. So, let us define X 1 = { x X : x x } and X 1 = { x X : x x } . Step 2 : Pick any x 2 X 1 . It follows from transitivity that x 2 x for all x X 1 , because
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Unformatted text preview: x 2 ” x and x ” x for all x ∈ X 1 . By completeness, any alternative in X 1 other than x 2 is either preferred to x 2 or less preferred. So, let us define X 2 = { x ∈ X 1 : x ” x 2 } and X 2 = { x ∈ X 2 : x „ x 2 } . Note that X 1 ⊂ X 2 by transitivity. Step 3 : Pick any x 3 ∈ X 2 . It follows from transitivity that x 3 ” x for all x ∈ X 2 , because x 3 ” x 2 and x 2 ” x for all x ∈ X 2 . By completeness, define X 3 and X 3 . Transitivity implies X 2 ⊂ X 3 ... . . . Step n : Since there is a finite number of alternatives, the process must end at some point. The alternative in X n is the most-preferred alternative (without further assumptions, there may be many). 1...
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This note was uploaded on 11/07/2010 for the course ECO 33358 taught by Professor Mathevet during the Fall '10 term at University of Texas.

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