Ami Pro - LN021610

Ami Pro - LN021610 - Economics 113 University of...

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Lecture Notes, February 16, 2010 Fundamental Theorems of Welfare Economics Pareto Efficiency Definition : An allocation , is attainable if there is (note x i , i H y j Y j , j F change in notation) so that . (The inequalities hold 0 i H x i j F y j i H r i co-ordinatewise.) Definition : Consider two assignments of bundles to consumers, v i , w i X i , i H. v i is said to be Pareto superior to w i if for each i H, u i (v i ) u i (w i ) and for some h H, u h (v h ) u h (w h ) . Note that Pareto preferability is an incomplete ordering. There are many allocation pairs that are Pareto incomparable. Definition : An attainable assignment of bundles to consumers, , is said w i , i H to be Pareto efficient (or Pareto optimal) if there is no other attainable assignment so that v i is Pareto superior to w i . v i Definition : , x 0i R N , y 0j R N , is said to p 0 , x 0 i , y 0 j , p 0 R N , i H , j F be a competitive equilibrium in a private ownership economy if (i) y 0 j Y j and p 0 y oj p 0 y for all y Y j ,fo ra l l j F (ii) x 0 i X i , M i p 0   p 0 r i j F ij p 0 y 0 j p 0 x 0 i M i p 0 and u i u i (x) for all with for all , and x 0 i x X i p 0 x M i p 0 i H (iii) 0 i H x 0 i j F y 0 j i H r i (co-ordinatewise) with = 0 for co-ordinates k so that the strict inequality holds. p k 0 This definition is sufficiently general to include the equilibrium developed in each of Theorems 14.1, 18.1, and 24.7. Properties (i) and (ii) embody decentralization. Property (iii) is market clearing. Economics 113 Winter 2010 University of California, San Diego Prof. R. Starr, Mr. Troy Kravitz February 16, 2010 1
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First Fundamental Theorem of Welfare Economics (1FTWE) Every competitive equilibrium is Pareto efficient (CE PE). This result does not require convexity of tastes or technology (though attaining a CE may need convexity). Theorem 19.1 (First Fundamental Theorem of Welfare Economics) : Assume C.II, C.IV and C.VI(C) or C.II, and C.IV** (weak desirabilty: For each x X i there is x' X i so that x' >> x (x' is greater than x in every co-ordinate) and x' is strictly preferred by i to x). Let
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Ami Pro - LN021610 - Economics 113 University of...

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