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slide17-sept30 - PREFERENCE AND REVEALED PREFERENCE Siyang...

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Unformatted text preview: PREFERENCE AND REVEALED PREFERENCE Siyang Xiong Rice University September 29, 2011 Xiong (Rice University) ECON 401 September 29, 2011 1 / 21 consumer preferences Suppose we are given a choice set X . Let x ; y ; z denote elements in X , i.e., x ; y ; z 2 X . I have a binary relation (more speci&cally, preference) on elements in X . For example, x is strictly preferred y : x & y : I am indi/erent between x and y : x ¡ y : x is weakly preferred y : x ¢ y : Xiong (Rice University) ECON 401 September 29, 2011 2 ¡ 21 binary relation Let B denote the binary relation (i.e., & , ¡ or ¢ ): xBy Some property of binary relation: Relexive : xBx for all x 2 X ; Irrelexive : : xBx for all x 2 X ; Symmetric : xBy implies yBx for all x ; y 2 X ; Asymmetric : xBy implies : yBx for all x ; y 2 X ; Transitive : xBy and yBz implies xBz for all x ; y ; z 2 X ; Negatively Transitive : : xBy and : yBz implies : xBz for all x ; y ; z 2 X ; Complete : either xBy or yBx for all x ; y 2 X ; Acyclic : x 1 Bx 2 Bx 3 ::: x n & 1 Bx n implies x 1 6 = x 2 . Xiong (Rice University) ECON 401 September 29, 2011 3 / 21 de&ne preferences: method #1 start with " & ", then " ¡ " is de&ned as follows. x ¡ y i/ both x & y and y & x : " ¢ " is de&ned as follows. x ¢ y i/ x & y and : y & x : Xiong (Rice University) ECON 401 September 29, 2011 4 ¡ 21 de&ne preferences: method #2 start with " & ", then " ¡ " is de&ned as follows. x ¡ y i/ : y & x : " ¢ " is de&ned as follows. x ¢ y i/ both : x & y and : y & x : We will prove that the two methods are equivalent! Xiong (Rice University) ECON 401 September 29, 2011 5 ¡ 21 assumptions on preference: method #1 suppose we start with " & ", consider the following preoperties of " & ". complete : for any x ; y 2 X , either x & y and y & x ; re&exive : for any x 2 X , we have x & x ; transitive : for any x ; y ; z 2 X , we have & x & y and y & z ¡ ) x & z ....
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slide17-sept30 - PREFERENCE AND REVEALED PREFERENCE Siyang...

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