EC487 Advanced Microeconomics, Part I: Lecture 5 Leonardo Felli - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 5 Leonardo Felli 32L.LG.04 27 October, 2017

Pareto Efficient Allocation Recall the following result: Result An allocation x ∗ is Pareto-efficient if and only if there exists a vector of weights λ = ( λ 1 , . . . , λ I ) , λ i ≥ 0 , for all i ≤ I and λ h > 0 for at least one h ≤ I, such that x ∗ solves the following problem: I λ i u i ( x i ) � max x 1 ,..., x I i =1 (1) I x i ≤ ¯ � s.t ω i =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 2 / 53

Pareto Efficient Allocation (cont’d) Recall we proved the only if statement. Proof: If: If x ∗ is Pareto-efficient then there exist λ such that x ∗ solves (2). To prove this implication we need the Second Welfare Theorem and the following Result. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 3 / 53

Pareto Efficient Allocation (cont’d) Result Let U : R L + → R be continuously differentiable, concave and monotonic. Consider the following problem: max U ( x ) s.t. p x ≤ p ω x ∈ R N + Then there exists a µ > 0 such that ∂ U ( x ) = µ p ℓ ∀ ℓ = 1 , . . . , L ∂ x ℓ Proof: By Kuhn-Tucker Theorem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 4 / 53

Pareto Efficient Allocation (cont’d) We can now re-state the if statement above as follows. Result ( If:) Assume that x ∗ is a Pareto-efficient allocation with x ∗ , i > 0 for all i ≤ I, and that u i ( · ) are monotonic, concave and continuously differentiable. Then there exists an I-tuple λ 1 , . . . , λ I > 0 such that x ∗ solves the planner’s problem (2). Moreover, λ i is the inverse of the marginal utility of income. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 5 / 53

Pareto Efficient Allocation (cont’d) Proof: By Second Welfare Theorem, since x ∗ is Pareto-efficient it is a Walrasian Equilibrium for endowments x ∗ , i = ω i and a price vector p ∗ > 0. Therefore for a given p ∗ consumers maximize their utility subject to budget constraint by choosing x ∗ , i . In other words, by the result above, there exists a I -tuple: γ 1 , . . . , γ I > 0 such that: ∂ u i ( x ∗ , i ) = γ i p ∗ ∀ i , ∀ ℓ ℓ ∂ x i ℓ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 6 / 53

Pareto Efficient Allocation (cont’d) Consider now the central planner’s Problem: I λ i u i ( x i ) � max x 1 ,..., x I i =1 (2) I x i ≤ ¯ � s.t ω i =1 It is a concave problem: concave objective function and linear constraint, therefore x ∗ solves it if we can find an L -tuple α 1 , . . . , α L > 0 such that: � I � λ i u i ( x ∗ , i ) � ∂ i =1 = α ℓ ∀ i , ∀ ℓ ∂ x i ℓ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 7 / 53

Pareto Efficient Allocation (cont’d) or λ i ∂ u i ( x ∗ , i ) = α ℓ ∀ i , ∀ ℓ ∂ x i ℓ Choosing now λ i = 1 α ℓ = p ∗ ℓ γ i and noticing that γ i is the marginal utility of income concludes the proof. Notice that α ℓ are the shadow prices of the feasibility conditions , and according to the result above correspond to the Walrasian equilibrium prices. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 8 / 53

Production Economy Consider now an economy with I consumers and J producers characterized by their production possibility set Y j . Define a production economy with private ownership as follows: � � ( ω 1 , . . . , ω I ); u i ( · ); Y j ; θ j E = i , ∀ i ≤ I , ∀ j ≤ J where θ j i is the share owned by consumer i of firm j , of course: I � θ j i = 1 ∀ j ≤ J i =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 9 / 53

Walrasian Equilibrium with Production: Definition Define a Walrasian equilibrium for a production economy E as: � � p ∗ ; ( x ∗ , 1 , . . . , x ∗ , I ); ( y ∗ , 1 , . . . , y ∗ , J ) such that: x ∗ , i solves for every i ≤ I : J p ∗ x i ≤ p ∗ ω i + � θ j p ∗ y ∗ , j � u i ( x i ) � max s.t. i x i j =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 10 / 53

Walrasian Equilibrium with Production: Definition (cont’d) y ∗ , j solves for every j ≤ J : y j ∈ Y j y j p ∗ y j max s.t. and market clearing conditions are satisfied: I J I x ∗ , i − y ∗ , j − ω j ≤ 0 � � � i =1 j =1 i =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 11 / 53

Aggregate Excess Demand Define aggregate excess demand for this economy as: I J I � � � x i ( p ) − y j ( p ) − ω i Z ( p ) = i =1 j =1 i =1 where x i ( p ) is consumer i ’s Marshallian demand and y j ( p ) is firm j ’s net supply: y j ℓ ( p ) > 0 is the supply of commodity ℓ while y j κ ( p ) < 0 is minus unconditional factor demand of commodity κ . Notice that Z ( p ) is homogeneous of degree zero in p . Both x i ( p ) and y j ( p ) are homogeneous of degree zero in p . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 12 / 53

Walras Law Walras Law: p Z ( p ) = 0 . This is obtained once again by summing each consumer’s budget constraint: J p x i ( p ) − p ω i − � θ j p y j ( p ) � � = 0 i j =1 That is: I I I J p ω i − � � � � θ j p x i ( p ) − � p y j ( p ) � = 0 i i =1 i =1 i =1 j =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 13 / 53

Walras Law (cont’d) In other words: � I I I J � p ω i − � � � � p x i ( p ) − θ j p y j ( p ) � � = 0 i i =1 i =1 j =1 i =1 or   I I J ω i − � � � x i ( p ) − y j ( p )  = 0 p  i =1 i =1 j =1 We can now state the three main Theorems we have proved in a pure exchange economy for the production economy E . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 14 / 53

Existence Theorem Theorem (Existence Theorem) Consider a Z ( p ) that satisfies the following conditions: 1. Z ( p ) is non-empty; 2. Z ( p ) is convex valued; 3. Z ( p ) is upper-hemi-continuous; 4. Z ( p ) is homogeneous of degree 0; 5. Z ( p ) satisfies Walras Law; 6. Z ( p ) is bounded; then there exists p ∗ such that Z ( p ∗ ) ≤ 0 . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 15 / 53

Properties of Walrasian Equilibrium We now move to the properties of Walrasian equilibrium in the production economy E . Definition An allocation is feasible for a production economy E if and only if there exists a production plan y j for every firm j ≤ J such that y j ∈ Y j ∀ j ≤ J and I I J x i ≤ ω i + � � � y j i =1 i =1 j =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 16 / 53

Properties of Walrasian Equilibrium (cont’d) Definition An allocation is Pareto-efficient for a production economy E if and only if ◮ it is feasible ◮ and there does not exists an alternative feasible allocation ˆ x that Pareto-dominates it . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 17 / 53

First Welfare Theorem Theorem (First Fundamental Theorem of Welfare Economics) Let E be a production economy with consumer preferences satisfying weak monotonicity. p ∗ ; ( x ∗ , 1 , . . . , x ∗ , I ); ( y ∗ , 1 , . . . , y ∗ , J ) � � Let be a Walrasian equilibrium for E . Then x ∗ is a Pareto-efficient allocation. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 18 / 53

Second Welfare Theorem Theorem (Second Welfare Theorem) Assume that x ∗ , such that x ∗ , i > 0 , is Pareto-efficient and that ◮ preferences u i ( · ) are strongly monotonic; ◮ preferences are convex: u i ( · ) are quasi-concave; ◮ technology Y j is convex for every j ≤ J and 0 ∈ Y j . ω i for i ≤ I and Then there exists a redistribution of endowments ¯ a vector of shares ¯ θ j i , for i ≤ I and j ≤ J such that � � p ∗ ; ( x ∗ , 1 , . . . , x ∗ , I ); ( y ∗ , 1 , . . . , y ∗ , J ) is a Walrasian equilibrium for the economy ¯ E . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 19 / 53

Second Welfare Theorem: Lemma Lemma Assume that x ∗ , such that x ∗ , i > 0 , is Pareto-efficient and that: ◮ preferences are strongly monotonic; ◮ preferences are convex; ◮ Y j are convex for every j ≤ J; ◮ 0 ∈ Y j for every j ≤ J. Then there exists a y ∗ , j for every firm j ≤ J and a vector p ∗ such that: I J I x ∗ , i = y ∗ , j + � � � ω i ; a) i =1 j =1 i =1 b) u i ( x ′ i ) > u i ( x ∗ , i ) implies p ∗ x ′ i > p ∗ x ∗ , i ; c) p ∗ y ∗ , j ≥ p ∗ y j for every y j ∈ Y j and every j ≤ J. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 20 / 53

Second Welfare Theorem: Proof Proof: Given this Lemma we just need to find the re-distribution i , ∀ i , which at p ∗ gives exactly ω i , ¯ θ j � p ∗ x ∗ , i � ¯ to every consumer. I Let V ∗ = p ∗ x ∗ , i and � i =1 V ∗ i = p ∗ x ∗ , i . Let also s i = V ∗ i V ∗ be i ’s share of the total value. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 2017 21 / 53