Economics 201B Second Half Lecture 6 4 1 10 The Second Welfare Theorem in the Arrow Debreu Economy Theorem 1 Second Welfare Theorem Pure Exchange Case If x is Pareto Optimal in a pure exchange economy with strongly monotone continuous convex preferences there exists a price vector p and an income transfer T such that p x T is a Walrasian Equilibrium with Transfers Outline of Proof Let Ai x i x i x i i x i A I i 1 Ai a1 aI ai Ai Then 0 A if it were we d have a Pareto improvement By Minkowski s Theorem nd p 0 such that inf p A 0 Show RL 0 Ai and hence p 0 Show inf p Ai 0 for each i De ne T to make x i a ordable at p Ti p x i p i Show I i 1 Ti 0 and x i Qi p T 1 Use strong monotonicity to show that p 0 Show p 0 Qi p T Di p T Now for the details Let Ai x i x i x i i x i A I i 1 Ai a1 aI ai Ai Claim 0 A If 0 A there exists ai Ai such that I i 1 ai 0 Let x i x i ai Since x i x i ai Ai we have x i i x i I i 1 x i I i 1 I i 1 I i 1 x i ai x i I i 1 ai x i Therefore x is a feasible allocation x Pareto improves x so x is not Pareto Optimal contradiction Therefore 0 A 2 p 0 inf p A 0 Ai is convex so A is convex easy exercise By Minkowski s Theorem there exists p 0 such that 0 p 0 inf p A I i 1 The fact that inf p A I i 1 inf p Ai inf p Ai is an exercise once you gure out what you have to prove it is obvious We claim that p 0 Suppose not so p 0 for some WLOG p 1 0 Let x i x i 1 0 0 p1 By strong monotonicity x i i x i so 1 0 0 Ai p1 So inf p Ai 1 p 0 0 p1 1 0 inf p A I i 1 inf p Ai I 0 a contradiction that shows p 0 We claim that inf p Ai 0 for each i Suppose 0 By strong monotonicity x i i x i 3 so Ai so inf p Ai p Since is an arbitrary positive number inf p Ai is less than every positive number so inf p Ai 0 Since I i 1 inf p Ai 0 inf p Ai 0 i 1 I De ne T to make x i a ordable at p We claim that T is an income transfer and x i Qi p T Let Ti p x i p i I i 1 Ti I i 1 p x i p i p I i 1 x i I i 1 i p 0 so T is an income transfer p x i p i x i i p i p x i i p i Ti 4 so x i Bi p T If x i i x i then x i x i Ai so p x i p x i x i x i p x i p x i x i p x i inf p Ai p x i p i Ti so x i Q p T Use strong monotonicity to show that p 0 Lemma 2 If i is continuous and complete and x i y then there exists 0 such that B x Xi i y Proof If not we can nd xn x xn Xi xn i y by completeness we have y Since i is continuous y i x so x i y a contradiction which proves the lemma Since p 0 and p 0 p 0 since in addition 0 p 0 so p i Ti 0 for some i If p 0 for some WLOG 1 let x i x i 1 0 0 5 i xn for each n By strong monotonicity x i i x i p x i p x i p i Ti 0 Find WLOG 2 such that p 0 x 2i 0 Since x i i x i let 0 be chosen to satisfy the conclusion of the Lemma If necessary we may make smaller to ensure that 2x 2i Let x i x i 0 2 0 0 Since Xi RL x i Xi so by the Lemma x i i x i But p x i p x i p i Ti which shows that x i Qi p T a contradiction which proves that p 0 Show p 0 Qi p T Di p T Case 1 p i Ti 0 Since p 0 Bi p T 0 so Qi p T Di p T 0 Case 2 p i Ti 0 Suppose x Qi p T but x Di p T Then there exists z i x such that z Bi p T hence p z p i Ti Since x Qi p T p z p i Ti so p z p i Ti 0 By Lemma 2 there exists 0 such that z z z RL z x 6 Let z z 1 2 z Since z RL z RL z z z 2 z 2 so z x p z p z 1 2 z p i Ti 1 2 z p i Ti which contradicts the assumption that x Qi p T This shows Qi p T Di p T since clearly Di p T Qi p T Qi p T Di p T What if preferences are not convex Second Welfare Theorem may fail if preferences are nonconvex Diagram gives an economy with two goods and two agents and a Pareto optimum x so that so that the utility levels of x cannot be approximated by a Walrasian Equilibrium with Transfers If p is the price which locally supports x and T is the income transfer which makes x a ordable with respect to the prices p there is a unique Walrasian equilibrium with transfers z q T z is much more favorable to agent I and much less favorable to agent II than x is 7 This is the worst that can happen under standard assumptions on preferences Given a Pareto optimum x there is a Walrasian quasiequilibrium with transfers z p T such that all but L people are indi erent between x and z Those L people are treated quite harshly they get zero consumption One could be less harsh and give these L people carefully chosen consumption bundles in the convex hull of their quasidemand sets but one would then have to forbid them from trading a prohibition that would in practice be di cult to enforce 8