Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem
From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part of) a competitive equilibrium. That will entail finding the prices that make that allocation an equilibrium. First, we had to change the definition of equilibrium to deal with budget constraints. ( ) Given an economy {X i, i, ω i } I i=1, {Y j } J j=1, an allocation x, y and a price vector p are a price equilibrium with transfers if there exists a vector of wealth levels I I J w R L with w i = p ω i + p yj i=1 such that: 1. For each j = 1,..., J: p y j p yj for all y j Y j ; 2. For each i = 1,..., I : xi i x i for all x i {x i X i : p x i w i } ; 3. I xi I ω i + J yj, with p l = 0 if the inequlity is strict for good l. i=1 i=1 j=1 i=1 j=1
Picture Need Interior Allocations
Quasi-Equilibrium We fix this problem with a small change to the definition of equilibrium. Definition Given an economy {X i, i } I i=1, {Y j } J j=1, ω, an allocation x, y and a price vector p constitute a quasi-equilibrium with transfers if there exists a vector of wealth levels such that: w = (w 1, w 2,..., w I ) with I w i = p ω + i=1 J j=1 p y j 1 For each j = 1,..., J: p y j p yj for all y j Y j. 2 For every i = 1,..., I : 3 if x i xi then p x w i I = I ω i + J i=1 Make sure you see why this deals with the previous problem. Any equilibrium with transfers is a quasi-equilibrium (why?). The maximzing condition for consumers is weaker. x i i=1 j=1 y j
Second Welfare Theorem: Preliminaries We have seen a few counterexamples to a possible second welfare theorem. This is what we need to avoid those issues. To show that for any Pareto optimal allocation one can find prices that make it into a competitive equilibrium requires a few assumptions We need to transfers to overcome the limitations imposed by private ownership. We need convexity of production sets. We need convexity and local non-satiation of preferences. We need to eliminate boundary issues.
Second Welfare Theorem Theorem (Second Fundamental Theorem of Welfare Economics) ) Consider an economy ({X i, i } Ii=1, {Y j } Jj=1, ω and assume that Y j is convex for all j = 1,..., J, and i is convex and locally non-satiated for all i = 1,..., I. Then, for each Pareto optimal allocation ˆx, ŷ there exists a price vector ˆp 0 such that (ˆx, ŷ, ˆp) form a quasi-equilibrium with transfers. The proof uses the separating hyperplane theorem. If an allocation is Pareto optimal there is an hyperplane that simultaneously supports the better-than sets of all consumers and all producers. That hyperplane yields a candidate equilibrium price vector. The proof is in three parts: aggregation, separation, and decentralization. We start with a Pareto optimal allocation and construct the corresponding quasi-equilibrium with transfers.
Proof of the Second Welfare Theorem: Aggregation First, we aggregate all consumers preferences when evaluating the Pareto effi cient consumption bundle x. Define the following sets: V i = {x i X i : x i ˆx i } R L and V = i V i V is the set of all bundles strictly preferred to ˆx by every consumer. Claim: V is convex. Take x,x V i (so both are strictly preferred to ˆx i ) and w.l.o.g. assume x i x. Since preferences are convex, for any λ [0, 1] λx + (1 λ) x i x By transitivity, we have λx + (1 λ) x i x i ˆx i Therefore, λx + (1 λ) x is an element of V i and therefore each V i is convex. V is convex because it is the sum of I convex sets.
Proof of the Second Welfare Theorem: Aggregation Second, we aggregate all firms and define the set of attainable consumption bundles. Define the aggregate production set as Y = Y j = y j R L : y 1 Y 1,..., y J Y J j j The set of consumption bundles that can be allocated to consumers is Y + {ω} This set is convex since it is the sum of J + 1 convex sets. Geometry of the Proof Draw V and Y + {ω}.
This is the set of consumption bundles strictly preferred to x * by all consumers V=Σ i V i = Σ i {x i in X i : x i >x i* } Y + ω Σ i x i * This is the set of attainable consumption bundles given the aggregate production set and the aggregate endowment x * is a Pareto optimal consumption bundle
Proof of the Second Welfare Theorem: Separation Next, we separate the sets V and Y + {ω}. Since (ˆx, ŷ) is a Pareto optimal allocation, V Y + {ω} =. If not, some consumer can obtain a consumption bundle preferred to what she gets in ˆx, contradicting the assumption that ˆx is Pareto optimal. Since V and Y + {ω} and two disjoint convex sets, one can apply the Separating Hyperplane Theorem. Separate V and Y + {ω} By the Separating Hyperplane Theorem, there exist a p R L with p 0 and an r R such that p z r for all z V, and p z r for all z Y + {ω}
Proof of the Second Welfare Theorem: Separation Next, we look at the implication of separation for consumers. By separation, p z r for all z V, Claim: if x i i ˆx i for all i, then p ( i x i ) r Take any x i i ˆx i for all i. remember this as we will use it later By local non-satiation, for each i there exists an ˇx i (near x i ) such that ˇx i i x i. Hence, ˇx i V i for all i, and i ˇx i V. So, p ( i ˇx i ) r (by separation); Take a sequence of ˇx i that goes to x i (check how this works): p ( i x i ) r. Applying this result to ˆx i i ˆx i, separation ( ) tells us that p ˆx i r Geometry of the Proof i We have shown that i {x i X i : x i i ˆx i } belongs to the closure of V which is contained in the half-space { z R L : p z r }.
p Y + ω Σ i x i * This is the set of consumption bundles strictly preferred to x * by all consumers V This is the set of attainable consumption bundles given the aggregate production set and the aggregate endowment x * is a Pareto optimal consumption bundle
Proof of the Second Welfare Theorem: Separation Next, use the implication of separation for firms. By separation p z r for all z Y + {ω} Choosing z = j ŷj + ω Y + {ω} one gets p ( ŷ j + ω) r j Next, put together the implications of separation for consumers and firms. The Pareto optimal allocation is feasible, and therefore: ˆx i ŷ j + ω Y + {ω} i j Hence we have ( ) p ˆx i r i Putting together this inequality and ( the opposite ) one from the previous slide: p ˆx i = r Geometry of the Proof i ˆx i belongs to Y + {ω} and it lies in the half-space { z R L : p z r }. i
p Y + ω Σ j y j* + ω Σ i x i * This is the set of consumption bundles strictly preferred to x * by all consumers V This is the set of attainable consumption bundles given the aggregate production set and the aggregate endowment x * is a Pareto optimal consumption bundle
Second Welfare Theorem Proof: Decentralization We have shown the following holds ( ) p ˆx i = p ω + i j ŷ j = r Claim: ˆx satisfies the consumers condition in a quasi-equilibrium with transfers at prices ˆp = p. For some consumer i, take an x such that x i ˆx i. We need to show that ˆp x w i for some w i. As shown previously, p x + ˆx n r = p ˆx i + ˆx n n i n i }{{} this satisfies x i i ˆx i for all i Hence: ˆp x ˆp ˆx i Set w i = ˆp ˆx i so that we have ˆp x w i as desired.
Second Welfare Theorem Proof: Decentralization We have shown the following holds ( ) p ˆx i = p ω + i j ŷ j = r Claim: ŷ maximizes profits at prices ˆp. For any firm j and any y j Y j, we have y j + ŷ k Y k j Hence, by separation and the equation above we have ˆp ω + y j + k j ŷ k r = ˆp ω + ŷ j + k j ŷ k Hence, ˆp y j ˆp ŷ j Therefore, ŷ j maximizes profits at prices ˆp.
Proof of the Second Welfare Theorem: End Summary We have shown that ˆx satisfies the consumers condition in a quasi-equilibrium with transfers at prices ˆp and income w i = ˆp ˆx i. We have also shown that ŷ j maximizes profits at prices ˆp. Therefore We have shown that the Pareto optimal allocation (ˆx, ŷ j ) and the prices ˆp form a quasi-equilibrium with transfers. The equilibirum prices are given by an hyperplane that simultaneously supports all consumers better-than set and the aggregate production set.
Proof of the Second Welfare Theorem: Coda The last step is to show that a quasi-equilibrium with transfers is also an equilibrium with transfers. You will prove this in Problem Set 9. First, you have to show that, under local non satiation, if there is a consumption bundle cheaper than a consumer s wealth, condition 2. of a quasi-equilibrium with transfers is equivalent to condition 2. of an equilibrium with transfers (there is nothing strictly cheaper than ω i in our counterexample from last class). Then, add strict monotonicity (something else violated by that counterexample) and show that a quasi equilibrium with transfers which has strictly positive wealth for all consumers is an equilibrium with transfers.
What Are The Welfare Theorems About? The first welfare theorem says a competitive equilibrium is Pareto effi cient: markets can yield effi cient allocations. The second welfare theorem says that any Pareto effi cient allocation can be obtained as an equilibrium provided one makes the right adjustment to income. In other words, any outcome that maximizes social welfare can be obtained by redistributing income correctly across consumers. There are many caveats to these results Both theorems rule out externalities. Both theorems need local non satiation. For the second welfare theorem, even assuming all assumptions are satisfied, we need someone to decide what are the transfers. This is not practical as this someone would need to know everyone s preferences, and all production sets, to figure those transfers out. How do you know those? In public economics, and optimal taxation theory, one asks if there is a way to get the consumers to reveal their preferences to the planner.
Next Week Equilibrium characterization in the differentiable case. Welfare theorems in the differentiable case. Existence.
Welfare Theorems in the Differentiable Case Question What is the relationship between the first order conditions that correspond to a competitive equilibrium and those that give Pareto optimality? Assumptions needed for differentiability Consumers Let X i = R L + and assume there exist u i (x) representing i that satisfy strong monotonicity and convexity for each i. Normalize things so that u i (0) = 0. Assume each u i (x) is twice continuously differentiable, with u i (x) 0 for any x, and also assume that u i (x) is quasi-concave. Producers Production sets are Y j = { y R L : F j (y) 0 }, where F j (y) = 0 defines the transformation frontier. Assume each F j (y) is convex, twice continuously differentiable, with F j (y) 0 for any y, and also assume that F j (0) 0.
Welfare Theorems in the Differentiable Case Remark Given these assumptions, Pareto effi ciency solves the planners problem. An allocation is Pareto optimal if and only if it is a solution to the following: subject to max u 1 (x 11, x 21,..., x L1 ) (x,y ) R LI R LJ u i (x 1i, x 2i,..., x Li ) u i i = 2, 3,..., I x li i ω l + y lj j l = 1, 2,..., L F j (y 1j,..., y Lj ) 0 j = 1, 2,..., J
Welfare and Equilibirum In the Differentiable Case (x, y ) is Pareto optimal if and only if it solves the following max u 1 (x 1 ) (x,y ) R L(I +J) s.t. u i (x i ) ū i F j (y j ) 0 i x li ω l + j y lj i = 2, 3,..., I j = 1, 2,..., J l = 1, 2,..., L We can also write the maximization problems that must be solved by a competitive equilibrium. (x, y, p ) is a competitive equilibrium with transfers if x, y solves: and max u i (x i ) s.t. p x i w i i = 2, 3,..., I x i 0 max y j p y j s.t. F j (y j ) 0 j = 1, 2,..., J What is the connection between the first order conditions of these two optimization problems?