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utility - 6 Utility maximization CONTINUITY 6.1 Consumer...

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6 Utility maximization 6.1 Consumer Preferences It is assumed that preferences are: COMPLETE: x , y X, x º y or y º x or both . REFLEXIVE: x X, x º x . TRANSITIVE: x , y , z X, x º y and y º z = x º z .  and are de fi ned obviously. CONTINUITY: y X, sets { x : x º y } and { x : x ¹ y } are closed . WEAK MONOTONICITY: x y = x º y . STRONG MONOTONICITY: x y and x 6 = y = x  y . LOCAL NONSATIATION: x X, ε > 0 , y X, | x y | < ε that y  x . CONVEXITY and STRICT CONVEXITY: as usual.
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6.2 Existence of a utility function. Theorem: Suppose preferences are COMP, REFL, TRAN, CONT, and SMON. Then continuous u : R k + R , that represents these preferences, exists. Proof: x X de fi ne u ( x ) as x u ( x ) e . De fi ne: MRS (Marginal Rate of Substitution): ... NOTE: MRS is invariant to monotonic transforma- tions of utility functions: g ( x ) = v ( u ( x )) , where v 0 > 0 . 6.3 Consumer behavior Suppose p = ( p 1 , . . . , p n ) is vector of prices for goods, and m is money holdings of a consumer. De fi ne B = { x X : px m } . Consumer problem: max u ( x ) s.t. x X, px m. Solution: x ( p , m )– Marshallian demand , v ( p , m ) = u ( x ( p , m ))– indirect utility function . Brief analysis: Usual problems; also p i = 0 with pos- sibility of x i = .
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6.4 Indirect utility
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