EC487 Advanced Microeconomics, Part I: Lecture 4 Leonardo Felli 32L.LG.04 20 October, 2017
Marshallian Demands as Correspondences We consider now the case in which consumers’ preferences are strictly monotonic and weakly convex . In this case Marshallian demands may be correspondence . Assume they are. Definition A correspondence is defined as a mapping F : X ⇒ Y such that F ( x ) ⊂ Y for all x ∈ X . Definition The graph of a correspondence F : X ⇒ Y is the set: G ( F ) = { ( x , y ) ∈ X × Y | y ∈ F ( x ) } Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 2 / 45
Fixed Point Theorem Definition A fixed point for a correspondence F : X ⇒ Y is a vector x ∗ such that x ∗ ∈ F ( x ∗ ) Theorem (Kakutani’s Fixed Point Theorem) Let X be a compact, convex and non-empty set in R N . Let F : X ⇒ X be a correspondence. Assume that G ( F ) is closed and that F ( x ) is convex for every x ∈ X then F has a fixed point. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 3 / 45
Kakutani’s Fixed Point Theorem Here is an alternative statement of Kakutani’s Fixed Point Theorem (equivalent). Theorem (Kakutani’s Fixed Point Theorem) Let X be a compact, convex and non-empty set in R N . Let F : X ⇒ X be a correspondence such that: ◮ F is non-empty; ◮ F is convex valued; ◮ F is upper-hemi-continuous. Then there exists a vector x ∗ ∈ X such that: x ∗ ∈ F ( x ∗ ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 4 / 45
Upper-hemi-continuity Definition A correspondence F : X ⇒ Y where X and Y are compact, convex subsets of Euclidean space is upper-hemi-continuous if and only if given the two converging sequences { x n } ⊂ X and { y n } ⊂ Y such that: { x n } ∞ { y n } ∞ n =1 → x ∈ X ; n =1 → y ∈ Y and: y n ∈ F ( x n ) ∀ n it is the case that y ∈ F ( x ) . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 5 / 45
Upper-hemi-continuity (cont’d) Notice that Result Upper-hemi-continuity of the correspondence F ( · ) is equivalent to F ( · ) having a closed graph only if Y is a compact set. Notice that if we consider a degenerate correspondence y = F ( x ) upper-hemi-continuity of F ( x ) is equivalent to continuity of this function and implies that its graph is closed. However, a closed graph does not imply that the function is continuous: example of a discontinuous function with a closed graph is: � 1 if x � = 0 F ( x ) = x 3 if x = 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 6 / 45
Existence of Walrasian Equilibrium Theorem (Existence Theorem of Walrasian Equilibrium) Let the excess demand function Z ( p ) be such that: 1. Z ( p ) is upper-hemi-continuous; 2. Z ( p ) is convex-valued; 3. Z ( p ) is bounded; 4. Z ( p ) homogeneous of degree 0 ; 5. Z ( p ) satisfies Walras Law: p Z ( p ) = 0 ; then there exist a vector of prices p ∗ and an induced allocation x ∗ such that: Z ( p ∗ ) ≤ 0 . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 7 / 45
Existence of Walrasian Equilibrium (cont’d) Proof: Once again we shall start from a normalization of prices: p ∈ ∆ L . In this way the excess demand function will be such that: Z : ∆ L ⇒ R L We then consider the following correspondence: p ⇒ Z ( p ) ⇒ p ′ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 8 / 45
Existence of Walrasian Equilibrium (cont’d) Where we define p ′ so that p ′ Z ( p ) is maximized. In other words, denote Z a vector in R L and m ( Z ) the following set of price vectors ˆ p : m ( Z ) = arg max p Z ˆ ˆ (1) p p ∈ ∆ L s.t. ˆ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 9 / 45
Existence of Walrasian Equilibrium (cont’d) Claim The set m ( Z ) is convex. Proof: Consider p ∈ ∆ L and p ′ ∈ ∆ L that solves (1). Then necessarily: p Z = p ′ Z and for every λ ∈ [0 , 1]: [ λ p + (1 − λ ) p ′ ] Z = p Z = p ′ Z . Therefore: [ λ p + (1 − λ ) p ′ ] ∈ m ( Z ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 10 / 45
Existence of Walrasian Equilibrium (cont’d) Claim The set m ( Z ) is upper-hemi-continuous. In other words, consider the following two sequences: { Z n } → Z ∗ and { p n } → p ∗ such that p n ∈ m ( Z n ) ∀ n then p ∗ ∈ m ( Z ∗ ) . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 11 / 45
Existence of Walrasian Equilibrium (cont’d) Proof: Suppose this is not true then p ∗ �∈ m ( Z ∗ ) p � = p ∗ such that in other words there exists ¯ p Z ∗ > p ∗ Z ∗ ¯ (2) Since { Z n } → Z ∗ and { p n } → p ∗ we get that: p Z n → ¯ p Z ∗ ¯ and p n Z n → p ∗ Z ∗ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 12 / 45
Existence of Walrasian Equilibrium (cont’d) Choose now n large enough or such that: p Z n − ¯ p Z ∗ | | ¯ < ( ε/ 2) (3) | p n Z n − p ∗ Z ∗ | < ( ε/ 2) Conditions (2) and (3) imply p Z n > p ∗ Z ∗ + ε p ∗ Z ∗ + ε 2 > p n Z n ¯ and 2 so that p Z n > p n Z n ¯ a contradiction of the assumption: p n ∈ m ( Z n ). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 13 / 45
Existence of Walrasian Equilibrium (cont’d) Let g : ∆ L ⇒ ∆ L be defined as: g ( p ) = m ( Z ( p )) . Result The composition of two upper-hemi continuous and convex-valued correspondences is itself upper-hemi-continuous and convex-valued. Therefore if Z ( p ) and m ( Z ) are upper-hemi-continuous and convex-valued then g ( p ) = m ( Z ( p )) is upper-hemi-continuous and convex-valued. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 14 / 45
Existence of Walrasian Equilibrium (cont’d) By Kakutani’s Fixed Point Theorem there exists p ∗ such that p ∗ ∈ g ( p ∗ ) . We still need to check that this price vector p ∗ is indeed a Walrasian equilibrium price vector. By definition of g ( p ) and the fact that p ∗ ∈ g ( p ∗ ) we know that p ∗ ∈ arg max p Z ( p ∗ ) p Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 15 / 45
Existence of Walrasian Equilibrium (cont’d) In other words p ∗ Z ( p ∗ ) ≥ p Z ( p ∗ ) ∀ p ∈ ∆ L . (4) By Walras Law we know that: p ∗ Z ( p ∗ ) = 0 From (4) we can then prove the following. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 16 / 45
Existence of Walrasian Equilibrium (cont’d) Result The price vector p ∗ is such that: Z ( p ∗ ) ≤ 0 Proof: Assume by way of contradiction that there exists an ℓ ≤ L such that Z ℓ ( p ∗ ) > 0. Choose then ˆ p = (0 , . . . , 0 , 1 , 0 , . . . , 0) where the digit 1 is in the ℓ -th position. p ∈ ∆ L and: We then obtain that ˆ p Z ( p ∗ ) > 0 = p ∗ Z ( p ∗ ) ˆ which contradicts (4). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 17 / 45
Existence of Walrasian Equilibrium (cont’d) Therefore, we have proved that: ◮ there exists a vector of prices p ∗ such that Z ( p ∗ ) ≤ 0 ◮ or that p ∗ is a Walrasian equilibrium price vector. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 18 / 45
Properties of Walrasian Equilibrium: Definitions Recall that x = { x 1 , . . . , x I } denotes an allocation. Definition An allocation x Pareto dominates an alternative allocation ¯ x if and only if: u i ( x i ) ≥ u i (¯ x i ) ∀ i ∈ { 1 , . . . , I } and for some i : u i ( x i ) > u i (¯ x i ) . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 19 / 45
Properties of Walrasian Equilibrium: Definitions (cont’d) In other words, the allocation x makes no one worse-off and someone strictly better-off. Definition An allocation x is feasible in a pure exchange economy if and only if: I � x i ℓ ≤ ¯ ω ℓ ∀ ℓ ∈ { 1 , . . . , L } . i =1 Definition An allocation x is Pareto efficient if and only if it is feasible and there does not exist an other feasible allocation that Pareto-dominates x . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 20 / 45
Properties of Walrasian Equilibrium: Definitions (cont’d) x 2 ✛ ✻ 2 x 1 1 x p x q q ¯ x q x 2 1 ✲ ❄ x 1 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 21 / 45
Benevolent Central Planner A standard way to identify a Pareto-efficient allocation is to introduce a benevolent central planner that has the authority to re-allocate resources across consumers so as to exhaust any gains-from-trade available. 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 = 1 , . . . , 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 (5) I x i ≤ ¯ � s.t ω i =1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 20 October, 2017 22 / 45
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