Lecture Notes, May 20, 2011Fundamental Theorems of Welfare EconomicsPareto EfficiencyDefinition: An allocation , is attainable if there is (notexi,iHyjYj,jFchange in notation) so that . (The inequalities hold0iHxijFyjiHrico-ordinatewise.) Definition: Consider two assignments of bundles to consumers, vi, wiXi, i H.vi is said to be Pareto superior to wiif for each iH, ui(vi) ui(wi) and for somehH, uh(vh) uh(wh) . Note that Pareto preferability is an incomplete ordering. There are many allocationpairs that are Pareto incomparable. Definition: An attainable assignment of bundles to consumers, , is saidwi,iHto be Pareto efficient (or Pareto optimal) if there is no other attainable assignmentso that vi is Pareto superior to wi. viDefinition: , x0iRN, y0j RN, is said top0,x0i,y0j,p0RN,iH,jFbe a competitive equilibrium in a private ownership economy if (i)y0jYjandp0yojp0yfor allyYj, for alljF(ii)x0iXi,Mip0 p0rijFijp0y0jp0x0iMip0and ui() ui(x) for all with for all , andx0ixXip0xMip0iH(iii)0iHx0ijFy0jiHri(co-ordinatewise) with = 0 for co-ordinates k so that the strict inequality holds.pk0This definition is sufficiently general to include the equilibrium developed in eachof Theorems 14.1, 18.1, and 24.7. Properties (i) and (ii) embodydecentralization. Property (iii) is market clearing.Economics 113Spring 2011University of California, San Diego Prof. R. Starr, Ms. S. FriedMay 20, 20111
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