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LN012810

# LN012810 - CB046/Starr LN012810 10:52 Lecture Notes for and...

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Unformatted text preview: CB046/Starr LN012810 January 21, 2010 10:52 Lecture Notes for January 28, 2010, and following: Households 12.1 The structure of household consumption sets and preferences Households are elements of the finite set H numbered 1 , 2 ,..., # H . A house- hold i ∈ H will be characterized by its possible consumption set X i ⊆ R N + , its preferences followsequal i , and its endowment r i ∈ R N + . 12.2 Consumption sets (C.I) X i is closed and nonempty. (C.II) X i ⊆ R N + . X i is unbounded above, that is, for any x ∈ X i there is y ∈ X i so that y > x , that is, for n = 1 , 2 ,...,N,y n ≥ x n and y negationslash = x . (C.III) X i is convex. X = ∑ i ∈ H X i . 12.2.1 Preferences Each household i ∈ H has a preference quasi-ordering on X i , denoted followsequal i . For typical x,y ∈ X i , “ x followsequal i y ” is read “ x is preferred or indifferent to y (according to i ).” We introduce the following terminology: If x followsequal i y and y followsequal i x then x ∼ i y (“ x is indifferent to y ”), If x followsequal i y but not y followsequal i x then x follows i y (“ x is strictly preferred to y ”). We will assume followsequal i to be complete on X i , that is, any two elements of X i are comparable under followsequal i . For all x,y ∈ X i , x followsequal i y , or y followsequal i x (or both). Since we take followsequal i to be a quasi-ordering, followsequal i is assumed to be transitive and reflexive. The conventional alternative to describing the quasi-ordering followsequal i is to as- sume the presence of a utility function u i ( x ) so that x followsequal i y if and only if 1 CB046/Starr LN012810 January 21, 2010 10:52 2 Lecture Notes for January 28, 2010, and following: Households u i ( x ) ≥ u i ( y ). We will show below that the utility function can be derived from the quasi-ordering. Readers who prefer the utility function formulation may use it at will. Just read u i ( x ) ≥ u i ( y ) wherever you see x followsequal i y . 12.2.2 Non-Satiation (C.IV) (Non-Satiation) Let x ∈ X i . Then there is y ∈ X i so that y follows i x . 12.2.3 Continuity We now introduce the principal technical assumption on preferences, the assumption of continuity. (C.V) (Continuity) For every x ◦ ∈ X i , the sets A i ( x ◦ ) = { x | x ∈ X i ,x followsequal i x ◦ } and G i ( x ◦ ) = { x | x ∈ X i ,x ◦ followsequal i x } are closed. Example 12.1 (Lexicographic preferences) The lexicographic (dictionary-like) ordering on R N (let’s denote it followsequal L ) is described in the following way. Let x = ( x 1 ,x 2 ,...,x N ) and y = ( y 1 ,y 2 ,...,y N ). x follows L y if x 1 > y 1 , or if x 1 = y 1 and x 2 > y 2 , or if x 1 = y 1 , x 2 = y 2 , and x 3 > y 3 , and so forth ... ....
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LN012810 - CB046/Starr LN012810 10:52 Lecture Notes for and...

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