3. Correspondences Daisuke Oyama Mathematics II April 10, 2020 - PowerPoint PPT Presentation

3. Correspondences Daisuke Oyama Mathematics II April 10, 2020

Correspondences Let X and Y be nonempty subsets of R N and R K , respectively. ▶ A correspondence F : X → Y is a rule that assigns a set F ( x ) ⊂ Y to every x ∈ X . ▶ “ F : X →→ Y ”, “ F : X ⇒ Y ”, and “ F : X ⇒ Y ” are also used. ▶ F is nonempty-valued if F ( x ) ̸ = ∅ for all x ∈ X . ▶ In Debreu, a correspondence is defined to be a nonempty-valued correspondence. ▶ F is compact-valued if F ( x ) is compact for all x ∈ X . ▶ F is convex-valued if F ( x ) is convex for all x ∈ X . ▶ F is closed-valued if F ( x ) is closed (relative to Y ) for all x ∈ X . ▶ F is singleton-valued if F ( x ) is a singleton set for all x ∈ X . 1 / 47

▶ The graph of F is the set Graph( F ) = { ( x, y ) ∈ X × Y | y ∈ F ( x ) } . ▶ F is locally bounded (or uniformly bounded ) near x ∈ X if there exists ε > 0 such that F ( B ε ( x ) ∩ X ) is bounded. F is locally bounded if for all x ∈ X , it is locally bounded near x . ▶ F ( A ) = { y ∈ Y | y ∈ F ( x ) for some x ∈ A } = ∪ x ∈ A F ( x ) · · · the image of A under F . 2 / 47

Examples ▶ Define B : R L ++ × R ++ → R L + by B ( p, w ) = { x ∈ R L + | p · x ≤ w } . B is a nonempty- and compact-valued correspondence. ▶ Given a function u : R L + → R , define the correspondence x : R L ++ × R ++ → R L + by x ( p, w ) = { x ∈ R L + | x ∈ B ( p, w ) and u ( x ) ≥ u ( y ) for all y ∈ B ( p, w ) } (the Walrasian demand correspondence). If u is continuous, then x is ▶ nonempty-valued by the Extreme Value Theorem, and ▶ compact-valued. —Why? 3 / 47

Continuous Correspondences: Notice ▶ Terminology: We use “upper/lower semi-continuous” instead of “upper/lower hemi-continuous”. ▶ Definition: We adopt general definitions using open sets. ▶ For lower semi-continuity, our definition is equivalent to that in MWG. ▶ For upper semi-continuity, under some additional assumption our definition is equivalent to that in MWG 4 / 47

Continuous Functions: Review ▶ For a function f : X → Y , the following conditions are equivalent: 1. For any open neighborhood V of f (¯ x ) (relative to Y ), there exists an open neighborhood U of ¯ x (relative to X ) such that f ( U ) ⊂ V . 2. For any sequence { x m } ⊂ X such that x m → ¯ x as m → ∞ , we have f ( x m ) → f (¯ x ) as m → ∞ . ▶ For correspondences , these are no longer equivalent. 1. Condition 1 will be used to define upper semi-continuity . 2. (A generalized version of) Condition 2 will be equivalent to lower semi-continuity . 5 / 47

▶ 1. An upper semi-continuous correspondence ▶ may have a “downward jump”, but ▶ may not have an “upward jump”. 2. A lower semi-continuous correspondence ▶ may have an “upward jump”, but ▶ may not have a “downward jump”. 6 / 47

Upper Semi-Continuity Let X and Y be nonempty subsets of R N and R K , respectively. Definition 3.1 ▶ A correspondence F : X → Y is upper semi-continuous at ¯ x ∈ X if for any open neighborhood V of F (¯ x ) (relative to Y ), there exists an open neighborhood U of ¯ x (relative to X ) such that F ( U ) ⊂ V . ▶ For A ⊂ X , F : X → Y is upper semi-continuous on A if it is upper semi-continuous at all ¯ x ∈ A . ▶ F : X → Y is upper semi-continuous if it is upper semi-continuous on X . ▶ F ( U ) = { y ∈ Y | y ∈ F ( x ) for some x ∈ U } · · · the image of U under F . 7 / 47

Constant Correspondences ▶ Any correspondence F with F ( x ) = F ( x ′ ) for all x, x ′ ∈ X is upper semi-continuous according to our definition. 8 / 47

Upper Semi-Continuity + Compact-Valuedness Proposition 3.1 F : X → Y is upper semi-continuous at ¯ x and F (¯ x ) is compact if and only if for any sequence { x m } ⊂ X such that x m → ¯ x , any sequence { y m } ⊂ Y such that y m ∈ F ( x m ) for all m ∈ N has a convergent subsequence whose limit is in F (¯ x ) . Proposition 3.2 If F : X → Y is upper semi-continuous and compact-valued, then F ( A ) is compact for any compact set A ⊂ X . ▶ F ( A ) = { y ∈ Y | y ∈ F ( x ) for some x ∈ A } · · · the image of A under F . 9 / 47

Closed Graph Definition 3.2 F : X → Y has a closed graph if its graph, Graph( F ) = { ( x, y ) ∈ X × Y | y ∈ F ( x ) } , is closed (relative to X × Y ). Definition 3.3 ▶ F : X → Y is closed at ¯ x if x m → ¯ x, y m ∈ F ( x m ) for all m ∈ N , and y m → y ⇒ y ∈ F (¯ x ) . ▶ F : X → Y is closed if it is closed at all ¯ x ∈ X . Proposition 3.3 F : X → Y has a closed graph if and only if it is closed. 10 / 47

Upper Semi-Continuity + Closed-Valuedness Proposition 3.4 If F is upper semi-continuous and closed-valued, then it has a closed graph. 11 / 47

Proof ▶ Let y m ∈ F ( x m ) for all m ∈ N and ( x m , y m ) → (¯ x, ¯ y ) ∈ X × Y . ▶ Take any ε > 0 . ▶ B ε ( F (¯ x )) being an open neighborhood of F (¯ x ) , there exists an open neighborhood U of ¯ x such that F ( U ) ⊂ B ε ( F (¯ x )) by the upper semi-continuity of F at ¯ x . ▶ Since x m → ¯ x , there exists M such that for all m ≥ M , x m ∈ U and hence y m ∈ F ( U ) ⊂ B ε ( F (¯ x )) . y ∈ ¯ Therefore, we have ¯ B ε ( F (¯ x )) . ▶ Since ε > 0 has been taken arbitrarily and since F (¯ x ) is closed, we have ¯ y ∈ F (¯ x ) (by Proposition 2.8). 12 / 47

Upper Semi-Continuity + Compact-Valuedness Proposition 3.5 For correspondences F : X → Y and G : X → Y , define the correspondence F ∩ G : X → Y by ( F ∩ G )( x ) = F ( x ) ∩ G ( x ) for all x ∈ X . If 1. F has a closed graph, and 2. G is upper semi-continuous and compact-valued, then F ∩ G is upper semi-continuous and compact-valued. 13 / 47

Proof x ∈ X , and consider any sequence { x m } ⊂ X such ▶ Take any ¯ that x m → ¯ x . Let { y m } be any sequence such that y m ∈ ( F ∩ G )( x m ) = F ( x m ) ∩ G ( x m ) for all m . ▶ Since y m ∈ G ( x m ) for all m , and by the upper semi-continuity of G at ¯ x and the compactness of G (¯ x ) , there exist x ) such that y m ( k ) → ¯ a subsequence { y m ( k ) } and ¯ y ∈ G (¯ y . ▶ Since y m ∈ F ( x m ) for all m , we thus have a sequence { ( x m ( k ) , y m ( k ) ) } ⊂ Graph( F ) that converges to (¯ x, ¯ y ) . By the closedness of Graph( F ) , we have (¯ x, ¯ y ) ∈ Graph( F ) , y ∈ F (¯ i.e., ¯ x ) . ▶ Hence, we have ¯ y ∈ ( F ∩ G )(¯ x ) . The conclusion therefore follows from Proposition 3.1. 14 / 47

Upper Semi-Continuity + Compact-Valuedness Proposition 3.6 For a correspondence F : X → Y , consider the following conditions: 1. F is upper semi-continuous and compact-valued. 2. F has a closed graph and the images of compact sets are compact. 3. F has a closed graph and the images of compact sets are bounded. 4. F has a closed graph and is locally bounded. We have the following: ▶ 1 ⇔ 2 ⇒ 3 ⇔ 4. ▶ If Y is closed, 3 ⇒ 2 (hence these conditions are equivalent). 15 / 47

▶ Thus, if Y is closed, then our definition is equivalent to that in MWG (condition 3) for compact-valued correspondences. 16 / 47

Proof ▶ 1 ⇒ 2: By Propositions 3.2 and 3.4. ▶ 2 ⇒ 1: Take any sequence { x m } ⊂ X such that x m → ¯ x ∈ X and any sequence { y m } ⊂ Y such that y m ∈ F ( x m ) for all m ∈ N . Since A = { x m | m ∈ N } ∪ { ¯ x } is compact, { y m } ⊂ F ( A ) has a convergent subsequence with a limit ¯ y ∈ F ( A ) by the compactness of F ( A ) , where ¯ y ∈ F (¯ x ) by the closedness of the graph. Therefore, the conclusion follows by Proposition 3.1. ▶ 2 ⇒ 3: Immediate. 17 / 47

Proof ▶ 3 ⇒ 4: Suppose that F is not locally bounded, x ∈ X such that F ( B ε (¯ x ) ∩ X ) is not i.e., there exists some ¯ bounded for every ε > 0 . For each m ∈ N , let y m ∈ F ( B 1 /m (¯ x ) ∩ X ) be such that ∥ y m ∥ > m , and let x m ∈ B 1 /m (¯ x ) ∩ X be such that y m ∈ F ( x m ) . By construction, x m → ¯ x . Thus we have found a compact set { x m | m ∈ N } ∪ { ¯ x } whose image is not bounded. 18 / 47

Proof ▶ 4 ⇒ 3: Suppose that there exists a compact set A ⊂ X such that F ( A ) is not bounded. For each m ∈ N , let y m ∈ F ( A ) be such that ∥ y m ∥ > m , and let x m ∈ A be such that y m ∈ F ( x m ) . By the compactness of A , { x m } has a convergent subsequence { x m ( k ) } with a limit ¯ x ∈ A . x ) ∩ X ) contains { y m ( k ) } k ≥ K for For any ε > 0 , F ( B ε (¯ some K , which is unbounded. 19 / 47

Proof ▶ 3 ⇒ 2 under the closedness of Y : Let A ⊂ X be a compact set. Take any { y m } ⊂ F ( A ) , and let { x m } ⊂ A be such that y m ∈ F ( x m ) for all m ∈ N . By the compactness of A and the boundedness of F ( A ) , { ( x m , y m ) } has a convergent subsequence { ( x m ( k ) , y m ( k ) ) } y ) ∈ A × R K . with a limit (¯ x, ¯ By the closedness of Y , ¯ y ∈ Y , and therefore, by the closedness of the graph of F , ¯ y ∈ F (¯ x ) ⊂ F ( A ) . This implies that F ( A ) is compact. 20 / 47

Upper Semi-Continuity + Compact-Valuedness Corollary 3.7 Suppose that Y is compact. F : X → Y is upper semi-continuous and compact-valued if and only if it has a closed graph. ▶ Thus, if Y is compact, then our definition is equivalent to that in Debreu for compact-valued correspondences. 21 / 47