Second Welfare Theorem

1 Second Welfare TheoremEcon 2100 Fall 2018 Lecture 18, October 31 (boo)Outline1 Second Welfare TheoremFrom Last ClassWe want to state a prove a Theorem that says that any Pareto optimalallocation is (part of) a competitive will entail nding the prices that make that allocation an , we had to change the de nition of equilibrium to deal with an economy fXi;%i; !igIi=1;fYjgJj=1 , an allocationx ;y and a pricevectorp are a price equilibrium with transfers if there exists a vector ofwealth levelsw2 RLwithIXi=1wi=p IXi=1!i+JXj=1p y jsuch that:1. For eachj=1; :::;J:p yj p y jfor allyj2Yj;2. For eachi=1; :::;I:x i%ixifor allxi2fxi2Xi:p xi wig; i IPi=1!i+JPj=1y j,withpl=0 if the inequlity is strict for x the boundary issues, we make another small nitionGiven an economyfXi;%igIi=1;fYjgJj=1; !, an allocationx ;y and a price vectorp constitute a quasi-equilibrium with transfers if there exists a vector of wealthlevelsw= (w1;w2; :::;wI)withIXi=1wi=p !

2 +JXj=1p y jsuch that:1 For eachj=1; :::;J:p yj p y jfor everyi=1; :::;I:ifx ix ithenp x wi3 IPi=1x i IPi=1!i+JPj=1y jwithpl=0 if the inequlity is strict for sure you see why this deals with the previous equilibrium with transfers is a quasi-equilibrium (why?).The maximzing condition for consumers is Welfare Theorem : PreliminariesWe have seen a few counterexamples to a possible Second Welfare Theorem ,and ways in which we can deal with is a summary of what we have learned so show that for any Pareto optimal allocation one can nd prices thatmake it into a competitive equilibrium requires a few assumptionsWe need transfers to overcome the limitations imposed by private need convexity of production need convexity and local non-satiation of need to eliminate boundary Welfare TheoremTheorem ( Second Fundamental Theorem of Welfare Economics)Consider an economy fXi;%igIi=1;fYjgJj=1; !

3 And assume that Yjis convex forall j=1; :::;J, and%iis convex and locally non-satiated for all i=1; :::; , for each Pareto optimal allocation^x;^y there exists a price vector^p6=0suchthat(^x;^y;^p)form a quasi-equilibrium with proof uses the separating hyperplane an allocation is Pareto optimal there is an hyperplane that simultaneouslysupports the better-than sets of all consumers and all hyperplane yields a candidate equilibrium price proof is in three parts: aggregation, separation, and start with a Pareto optimal allocation and construct the correspondingquasi-equilibrium with of the Second Welfare Theorem : AggregationFirst, we aggregate all consumers preferences when evaluating the Paretoe cient consumption bundle^ ne the following sets:Vi=fxi2Xi:xi ^xig RLandV=XiViVis the set of all bundles strictly preferred to^xby every : V is x0,x002Vi(so both are strictly preferred to^xi) and assumex0% preferences are convex, for any 2[0.]

4 1] x0+ (1 )x00%ix00By transitivity, we have x0+ (1 )x00%ix00 i^xiTherefore, x0+ (1 )x00is an element of Viand therefore each Viis is convex because it is the sum of I convex of the Second Welfare Theorem : AggregationSecond, we aggregate all rms and de ne the set of attainable ne the aggregate production set asY=XjYj=8<:Xjyj2RL:y12Y1; :::;yJ2YJ9=;The set of consumption bundles that can be allocated to consumers isY+f!gThis set is convex since it is the sum ofJ+1 convex of the ProofDrawVandY+f! iVi= i{xiin Xi: xi>xi*}Y + This is the set ofconsumption bundlesstrictly preferred tox*by all consumersThis is the set of attainable consumption bundles giventhe aggregate production setand the aggregate endowmentx*is a Pareto optimalconsumption bundle ixi*Proof of the Second Welfare Theorem : SeparationNext, we separate the setsVandY+f!

5 (^x;^y)is a Pareto optimal allocation,V\Y+f!g=;.If not, some consumer can obtain a consumption bundle preferred to what shegets in^x, contradicting the assumption that^xis Pareto +f!gare two disjoint convex sets, one can apply theSeparating Hyperplane +f!gBy the Separating Hyperplane Theorem , there exist a^p2 RLwith^p6=0 and anr2 Rsuch that^p z rfor allz2V,and^p z rfor allz2Y+f!gProof of the Second Welfare Theorem : SeparationNext, we look at the implication of separation for consumers. By separation,^p z rfor allz2V,Claim: if xi%i^xifor all i, then^p (Pixi) rremember this aswe will use it laterTake any xi%i^xifor all local non-satiation, for each i there exists an xi(near xi) such that xi , xi2 Vifor all i, andPi xi2V .So,^p (Pi xi) r (by separation);Take a sequence of xithat goes to xi(check how this works):^p (Pixi) this result to^xi%i^xi, separation tells us that^p Xi^xi!

6 RGeometry of the ProofWe have shown thatPifxi2Xi:xi%i^xigbelongs to the closure ofVwhichis contained in the half-space z2RL: ^p z r .VY + This is the set ofconsumption bundlesstrictly preferred tox*by all consumersThis is the set of attainable consumption bundles giventhe aggregate production setand the aggregate endowmentx*is a Pareto optimalconsumption bundle ixi*pProof of the Second Welfare Theorem : SeparationNext, use the implication of separation for rms. By separation^p z rfor allz2Y+f!gChoosingz=Pj^yj+!2Y+f!gone gets^p (Xj^yj+!) rNext, put together the implications of separation for consumersand Pareto optimal allocation is feasible, and therefore:Xi^xi Xj^yj+!2Y+f!gHence we have^p Xi^xi! rPutting together this inequality and the opposite one from the previous slide:^p Xi^xi!=rGeometry of the ProofPi^xibelongs toY+f!

7 Gand it lies in the half-space z2RL: ^p z r .VY + This is the set ofconsumption bundlesstrictly preferred tox*by all consumersThis is the set of attainable consumption bundles giventhe aggregate production setand the aggregate endowmentx*is a Pareto optimalconsumption bundle ixi* jyj*+ pSecond Welfare Theorem Proof: DecentralizationWe have shown the following holds^p Xi^xi!= ^p 0@!+Xj^yj1A=rClaim:^x satis es the consumers condition in a quasi-equilibrium with transfers atprices^ some consumer i, take an x such that x i^xi. We need to show that^p x wifor some shown previously,^p 0@x+Xn6=i^xn1A|{z}this satis es xi%i^xifor all i r= ^p 0@^xi+Xn6=i^xn1 AHence:^p x ^p ^xiSet wi= ^p ^xiso that we have^p x wias Welfare Theorem Proof: DecentralizationWe have shown the following holds^p Xi^xi!= ^p 0@!+Xj^yj1A=rClaim:^y maximizes pro ts at prices^ any rm j and any yj2Yj, we haveyj+Xk6=j^yk2 YHence, by separation and the equation above we have^p 0@!

8 +yj+Xk6=j^yk1A r=^p 0@!+ ^yj+Xk6=j^yk1 AHence,^p yj ^p ^yjTherefore,^yjmaximizes pro ts at prices^ of the Second Welfare Theorem : EndSummaryWe have shown that^xsatis es the consumers condition in a quasi-equilibriumwith transfers at prices^pand incomewi= ^p ^ have also shown that^yjmaximizes pro ts at prices^ have shown that the Pareto optimal allocation(^x;^yj)and the prices^p form aquasi-equilibrium with equilibirum prices are given by an hyperplane that simultaneously supportsall consumers better-than set and the aggregate production of the Second Welfare Theorem : CodaThe last step is to show that a quasi-equilibrium with transfers is also anequilibrium with will prove this in Problem Set , you have to show that, under local non satiation, if there is a consumptionbundle cheaper than a consumer s wealth, condition 2.

9 Of a quasi-equilibriumwith transfers is equivalent to condition 2. of an equilibrium with , add strict monotonicity, and show that a quasi equilibrium with transferswhich has strictly positive wealth for all consumers is an equilibrium Are The Welfare Theorems About?The rst Welfare Theorem says a competitive equilibrium is Pareto e cient:markets can yield e cient Second Welfare Theorem says that any Pareto e cient allocation can beobtained as an equilibrium provided one makes the right adjustment other words, any outcome that maximizes social Welfare can be obtained byredistributing income correctly across are many caveats to these resultsBoth theorems rule out theorems need local non the Second Welfare Theorem , even assuming all assumptions are satis ed, weneed someone to decide what are the is not practical as this someone would need to know everyone s preferences,and all production sets, to gure those transfers out.

10 How do you know those?In public economics, and optimal taxation theory, one asks if there is a way toget the consumers to reveal their preferences to the WeekEquilibrium characterization in the di erentiable theorems in the di erentiable