LN042511 - CB046/Starr LN042511 April 17, 2011 17:17...

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Unformatted text preview: CB046/Starr LN042511 April 17, 2011 17:17 Lecture Notes for April 25 & 27, 2011: 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 + . We will soon move to using a utility function u i ( ) to represent followsequal i . 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. 1 CB046/Starr LN042511 April 17, 2011 17:17 2 Lecture Notes for April 25 & 27, 2011: Households 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 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. Note that this assumption represents precisely what we would expect from a continuous utility function: that the inverse images of the closed sets [0 ,a ] and [ a, + ) are closed, where a = u i ( x ). The following example represents an otherwise well-behaved preference ordering that is not continuous....
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LN042511 - CB046/Starr LN042511 April 17, 2011 17:17...

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