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slides2_1 - Consumers Want to prove similar theorems...

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Consumers Want to prove similar theorems. Outline: Preferences, utility Demand Slutsky equation Revealed preferences Slutksy equation with differentiability 1

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Binary relations For a given (abstract) set X , a binary relation P on X is defined as a subset P X × X , where ( x, y ) P if x and y are in relation P . We will also write xPy . Here are some properties a relation might (or might not) have reflexive: xPx for all x P symmetric: xPy implies yPx transitive: xPy and yPz implies xPz complete: xPy or yPx for all x, y X 2
Preferences Write f instead of P . We write x f y if x is weakly preferred to y , i.e. ( x,y ) P . We say x y if x f y but not y f x and say that x is strictly preferred to y . Finally we write x y if x f y and y f x and say x and y are indifferent. We usually want to assume that our preference relation is complete and transitive. 3

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Utility We say a function u : X R is a utility function representing a preference relation f on X if u ( x ) u ( y ) x f y Note that for a given preference relation, there is never a unique utility function representing it. For any strictly increasing f : R R , if u ( . ) represents f , so does f ( u ( . )) 4
Utility and preferences A preference relation can be represented by a utility function, only if it is complete and transitive. This is enough is choice set is finite. Here we are interested in X R L + . Need additional condition to guarantee exis- tence of utility function. 5

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Lexicographic preferences Suppose there are two goods. Suppose for x, y R 2 + , x f y if x 1 > y 1 or x 1 = y 1 and x 2 y 2 . This is known as lexicographic preferences. 6
The preference relation f on X is continuous if for any sequence ( x n , y n ) in X × X , with x n f y n for all n , we have that if x n x and y n y , then x f y . Suppose that preferences are complete, transitive and continuous. Then there is a continuous utility function that represents them. 7

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slides2_1 - Consumers Want to prove similar theorems...

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