Patryk Pagacz. Equilibrium theory in infinite dimensional Arrow-Debreu s model. Uniwersytet Jagielloński

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1 Patryk Pagacz Uniwersytet Jagielloński Equilibrium theory in infinite dimensional Arrow-Debreu s model Praca semestralna nr 1 (semestr zimowy 2010/11) Opiekun pracy: Marek Kosiek

2 Uniwersytet Jagielloński Wydział Matematyki i Informatyki Instytut Matematyki mgr Patryk Pagacz Equilibrium theory in infinite dimensional Arrow-Debreu s model. Praca semestralna w ramach Środowiskowych Studiów Doktoranckich Opiekun pracy semestralnej: dr hab. Marek Kosiek Kraków 2010

3 EQUILIBRIUM THEORY IN INFINITE DIMENSIONAL ARROW-DEBREU S MODEL. PATRYK PAGACZ Abstract. This work explains Arrow-Debreu model. The general purpose is to get some review of quasiequilibrium existence theorem in ecomonies with infinite-dimensional commodity spaces. We focus on pure exchange economy. 1. The Arrow-Debreu model In this section we recall the finite-dimensional Arrow-Debreu model. In our market there are k commodities, n consumers and m producers. Let I = {1, 2, 3,..., n} denote a set of consumers and J = {1, 2, 3,..., m} a set of producers (firms). Every commodity is represented by a real number. Thus R k is a commodity space. Each producer j J is described by a production set Y j R k, which is a set of all possible production plans. Positive value of any component of vector from Y j shows how many commodity j-th firm can produce and negative how many it needs. A consumer i I has some consumption plans x i = (x 1 i, x 2 i,..., x k i ). The collection of all plans is said to be a consumption set X i a subset of commodity space R k. It has also a initial endowment ω i X i. Consumers make choices between consumption plans. So to characterize an ith consumer we need a preference relation i X i X i, where x i x means x is no less desirable than x. Moreover each consumer has a share of profit of producers, so is also characterized by a vector θ i of shares θ j i of the profit of j-th firm. So we can say that a model of economy is 4-tuple: E = ((X i, i, ω i, θ i ) i I, (Y j ) j J ) Usually there are some natural assumptions as below. Assumption 1. For each producer j J: (1) 0 Y j, (Possibility of Inaction) (2) Y j is closed and convex, (3) Y j Y j = {0}, (4) Y R k +, where Y = m Y j is the total production set. j=1 Assumption 2. For each consumer i I: 1

4 2 PATRYK PAGACZ (1) X i has a lower bound, (2) ω i X i, (3) X i is convex, (4) X i is closed. Assumption 3. (1) For all j J n θ j i = 1. (2) i is preorder, for all i. (3) For every x i X i a set {x X i x i x i } is closed and convex. (4) If t (0; 1) and x i, x i X i are such that x i i x i then tx i + (1 t)x i i x i. Where x i is strictly preferred then x i (x i i x i) if x i i x i and not x i i x i. We also use notation P i (x i ) = {x X i x i x i }. All consumers can exchange thier commodity bundles with respect to some global price. Where price is some vector p from R k and price of consumer bundle x is standard inner product < p, x >= p(x). The goal of each producer is have to the biggest benefit, wich is also inner product p and thier production plan. We do not need consider all consumer and producer plans. For us the most important thing is set of all feasible allocations. It means { } n m n n m A(E) = (x, y) X i Y j x i = ω i + y j Moreover A X (E) denotes the projection of A(E) on n X i. j=1 For example, let it be two consumers Tom and John. At the beginning Tom has 300 tonnes of beet and John has three cars. Moreover, Tom works in John s beet-sugar factory and he earns one third factory s benefit. So there are three commodities: beets, sugar and cars. Thus we can write ω 1 = (300, 0, 0), ω 2 = (0, 0, 3) and θ 1 = 1 3, θ 2 = 2 3. In some kind of possible situation it is a price p = (0, 9; 4; 100) and Tom spend all beets. Then he gets 270 zl and factory makes 75 tonnes of sugar. Therefore Tom buys two cars form John. After that they can buy 20 and 55 tonnes of sugar, because the factory s benefit is equal to 30 zl. So we can write this situation as y = ( 300, 75, 0), x 1 = (0, 20, 2), x 2 = (0, 55, 1). We see that x 1 + x 2 = y + ω 1 + ω 2. j=1 2. Basic definition for infinite dimensional model In this work, we are concerned on equilibrium existence theorems in economics, where a commodity space is infinite dimensional. From now we make some global assumptions.

5 EQUILIBRIUM THEORY IN INFINITE DIMENSIONAL ARROW-DEBREU S MODEL.3 A model of economy is 6-tuple: Assumption 1. (SA) E = ((L, τ), (X i, P i, ω i, θ i ) i I, (Y j ) j J ) (1) L is a Riesz space endowed with a Hausdorff locally convex linear topology τ; (2) L + is a closed cone in the τ-topology of L; (3) L is a sublattice of the order dual of L. Assumption 2. (C) (1) X i L is convex, τ-closed and ω i X i ; (2) P 1 i (x i ) = {y i X i x i P i (y i )} is σ(l, L )-open in X i, for all i I and (x i ) i I A X (E) (3) P i (x i ) is convex and x i P i (x i ) (4) P i (x i ) is τ-open in X i Assumption 3. (P) n (1) θ j i = 1. (2) Y j L is convex, τ-closed and 0 Y j. (3) A(E) is (σ(l, L )) n+m -compact Let us introduce some basic definitions. Definition 1. A feasible allocation (x, y) is Pareto optimal, if there is no feasible allocation (x, y ) that x i P i (x i ) at least one i. Definition 2. A feasible allocation (x, y) is weakly Pareto optimal, if there is no feasible allocation (x, y ) that x i P i (x i ) for all i I. Definition 3. A triple (x, y, p) is a quasiequilibrium of E iff (x, y) A(E), p (L, τ), p 0 and (1) i I : px i = p(ω i + m θ j i y j) and x P i (x i ) : px px i ; j=1 (2) j J y Y j : py py j ; Moreover, if for some i I: inf{p(z i ) z i X i } < p(ω i ), then (x, y, p) is a nontrivial quasiequilibrium. A quasiequilibrium such that z i P i (x i ) implies p(z i ) > p(x i ) is a Walrasian equilibrium or short equilibrium. If (x, y, p) is a nontrivial quasiequilibrium, then under some additional model condition is actually an equilibrium (see [1]). 3. Production economies Now we want to present existence theorem for production economies with E-properness.

6 4 PATRYK PAGACZ Definition 4. An element x of A(E) is said to be dominated (blocked) by a nonempty coalition S I iff there exists x S X i such that i S i S(x S i ω i ) ω j i Y j. i S j J Definition 5. A nonzero vector t = (t 1, t 2, t 3,..., t n ) [0; 1] n is said to dominate(block) an allocation x A(E) iff there exists x t X i such i I that t i (x S i ω i ) t i ω j i Y j. and i {i I t i > 0} : x t i P i (x i ) i I i I j J A set of all feasible allocations which cannot be dominated by any nonzero vector t is denoted by C f (E) and it is called a fuzzy core of the economy E. From our global assumption about economies C f (E) is nonempty. Definition 6. A subset A of L is said to be radial at point x A if for each y L there exists a real number λ > 0, such that (1 λ)x+λy A for every λ with 0 λ λ. Let K be some order ideal of L. Definition 7. A preference relation P i : X i 2 X i is said to be E- proper relative to K at x i X i if there exists a τ-open convex subset V of L, a lattice Z K verifying Z + K Z and some subset A of L, radial at x i such that x i V Z, = V Z A P i (x i ) and P i (x i ) A V (Z + L + ). Definition 8. A set Y i is said to be E-proper relative to K at y i Y i if there exists a τ-open convex subset V of L, a lattice Z K verifying Z K Z and some subset A of L, radial at y i, such that y i V Z, V Z A Y i and Y i A V (Z L + ). Definition 9. An economy E is said to be E-proper relative to K at (x, y) X i Y j if each preference relation and each production i I j J set are E-proper relative to K at the corresponding component of (x, y), with for each i, ω i Z i and for each j, 0 Z j (where Z i and Z j are taken from definition of properness P i and Y j ). Let L u be order ideal generated by u = x i + y j + ω i. i I j J i I Definition 10. A production economy E, E-proper at (x, y) relative to K, is said to be nontrivially E-proper if, in the previous definition, the set (Z i L K u ) j L i I j J(Z K u ) is a radial at ω subset of L K u, where L K u denotes the principal ideal generated in K by u. Theorem 11. Let (x, y) C f (E). If E is nontrivially E-proper relative to L u at (x, y) for some u L, then there exists p L such that (x, y, p) is a quasiequilibrium of E. More about this theorem is in [4].

7 EQUILIBRIUM THEORY IN INFINITE DIMENSIONAL ARROW-DEBREU S MODEL.5 4. Pure exchange economy In this section we consider an easier case. We assume that there is no producer, it means that we have E = (I, (L, τ), (X i, P i, ω i ) i I ). Additionally let for all i I be X i = L +. Definition 12. Let v L be such that v > 0. A correspondence P : L + L + is said to be pointwise v-proper at x L +, if there is an open convex cone Γ x with vertex x such that x v Γ x and P (x) Γ x =. A correspondence P : L + L + is pointwise proper at x L +, if there is some v > 0 in L such that P is pointwise v-proper at x. Definition 13. Let v L be such that v > 0. A correspondence P : L + L + is said to be v-proper at x L +, if there is a convex set P (x) such that x + v is a τ-interior point of P (x) and P (x) L + = P (x). The correspondence P : L + L + is proper at x L +, if there is some v > 0 in L + such that is v-proper at x L +, if there is some v such that P is v-proper at x. A correspondence P : L + L + which is v-proper at x L + is also pointwise v-proper at x. Indeed as in definition of v-properness, let U be a τ-open neighborhood of v such that x + U is a subset of P (x). Let Γ x := {µx + νu u U, µ, ν R + }. Γ x is an open convex cone with vertex x and x v Γ x. If Γ x P (x) then (from x P (x) and convexity P (x)) there exists u U such that x u P (x). Hence x = x+u+x u P (x). This is contradiction, so Γ 2 x P (x) =. Definition 14. Let v L be such that v > 0. A utility function u : L + R is said to be pointwise v-proper at x L +, if the correspondence defined by P (z) := {y L + u(y) > u(z)} is pointwise v-proper at x L +. Definition 15. An economy E = (X i, P i, θ i ) i I is said to be ω-proper economy if satisfies the following properties: (1) X i = L + for each consumer i, (2) for each i I and every weakly Pareto optimal allocation x we have x i P i (x i ) and: P i (x i ) is τ-open in L + or P i (x i ) = {y i L + u(y i ) > u(x i )} for some concave function u : L + R; there is a convex set P i (x i ) such that the vector x i + ω is τ-interior point of P i (x i ) and P i (x i ) L + = P i (x i ). Fact 16. Let L be a Riesz space with a locally convex topology τ such that its positive cone is closed. Fix ω L + \ {0}. Then subspace L ω := n N n[ ω; ω] is Archimedean and endowed with order normable topology.

8 6 PATRYK PAGACZ Proof. Archimedean follows form that positive cone is close. So a map: : L ω x inf{λ λω x λω} R + is well define norm. Its topology is locally convex and each open set absorbs order intervals. So the norm topology is coaster then the order topology. On the other hand if V is an order open set then there exists λ > 0 such that [ ω; ω] λv, so B ω (0, 1 ) ( 1 ω, 1 ω) V. Therefore these λ λ λ topologies are equivalent. Proposition 17. Consider an ω-proper economy. Assume that x is a weakly Pareto optimal allocation and that, for some p L, (x, p) is a quasiequilibrium. Then (x, p) is nontrivial if and only if p(ω) > 0. Proof. Firstly let the quasiequilibrium (x, p) be nontrivial. For the contradiction, we assume that p(ω) = 0. Then there exist l I and z l L + such that p(z l ) < p(x l ). So from definition 15 there exist ɛ > 0 such that x l + ω + ɛ(z l x l ) P l (x l ), but for ɛ < 1 we have x l + ω + ɛ(z l x l ) L +. Hence x l + ω + ɛ(z l x l ) P l (x l ), and consequently p(x l +ω+ɛ(z l x l )) p(x l ). It means 0 ɛp(z l x l ) < 0. Now let p(ω) > 0. The space (L ω, ω ) has ω as an order unit and ω, 1 m ω are interior points of (L ω) +. Assume now that for every i I and for every z L + we have p(z) p(x i ). Fix z [ ω, ω] and then choose ɛ > 0 such that 1 m ω + ɛz (L ω) +. Hence for every i I we have p( 1 ω + ɛz) p(x m i). Summing over i, we obtain mɛp(z) 0. So, p(z) 0, for all z [ ω, ω], and therefore p = 0, but p(ω) > 0. Let us introduce the main theorem of this paragraph: If f is m-tuple of functionals (f 1, f 2,..., f m ), and x L + then n n R f (x) := sup{ f i (x i ) x i = x; x i L +, for all i} In this situation we say that m-tuple (f 1, f 2,..., f m ) is exact at point x with respect to (x 1, x 2,..., x m ) if and only if R f (x) = m f i(x i ). Theorem 18. Let L satisfy (SA), and let ω L +. The following condition are equivalent: (1) For each ω-proper economy E = (P i, ω i ) i with total endowment = ω > 0, every weakly Pareto optimal allocation is a i ω i nontrivial quasiequilibrium for some τ-continuous price. (2) For any list of continuous linear functionals f = (f 1, f 2, f 3,..., f m ) such that f i (ω) > 0 for each i and R f is exact at ω with respect to some x = (x 1, x 2, x 3,..., x m ) A m ω, there exists some (λ 1, λ 2,..., λ m ) 0 such that the Riesz-Kantorovich formula of the m-tuple of continuous linear functionals (λ 1 f 1, λ 2 f 2,..., λ m f m ) is exact at ω with respect to x and pointwise ω-proper at ω. We will make needed arrangements for proof of Theorem 18.

9 EQUILIBRIUM THEORY IN INFINITE DIMENSIONAL ARROW-DEBREU S MODEL.7 For functions g i : L R, where i = 1, 2, 3,..., m we define: n n ( m g i )(x) := sup{ g i (x i ) x i = x}. For f L we have a new function: { f(x), if x L+ f(x) :=, otherwise Now we see that for x L +, and m-tuple functionals (f 1, f 2,..., f m ) = f it is : R f (x) = ( m f i )(x). Thanks that we shall use: Theorem 19. (Moreau 1967) Assume that X, X is an arbitrary dual system. For each i = 1, 2,..., m let g i : X [, ] be a non identically equal to function. If m ĝ i is exact at x with respect to (x 1, x 2,..., x m ) that satisfies x = m x i then ( m g i )(x) = m g i (x i ). Lemma 20. (Podczeck) Let (L, τ) be an ordered topological vector space, M be a vector subspace of L(endowed with the induced order), Y be an open and convex subset of L such that Y M + and let y Y M +. If p is a linear functional on M satisfying p(y) p(z), for all z Y M +, then there exists some π L such that π M p and p(y) = π(y) π(z), for all z Y. Definition 21. Let E be an economy. The preference P is said to be local nonsatiation at x if and only if for each U τ-neighborhood of x there exists y U such that y P (x). Proposition 22. Let E be an ω-economy. If x = (x 1, x 2, x 3,..., x m ) is a weakly Pareto optimal such that for each i P i (x i ) has nonempty interior and P i is local nonsatiation at x i, than (x, p) is quasi-valuation equilibrium for some nonzero p L. Proof. Let Z := m P i (x i ), so Z is a convex set with nonempty interior and since x is weakly Pareto optimal, ω Z. Since from Hahna- Banach theorem there exists nonzero p L such that p(ω) p(z) for all z Z. Now fix arbitrary i 0 and y P i0 (x i0 ). From local nonsatiation for each i i 0 there exists a sequence {x n i } n=1 converting to x i such that

10 8 PATRYK PAGACZ x n i 0 P i (x i ) for all n. We see that y + x i Z, so p(y) + p(x i ) = i i 0 i i 0 p(y + x i ) p(ω) = p(x i ) for all n. Hence we obtain p(y) p(x i ) i i 0 i from continuity of p. Proof. (Theorem 18) (1) (2) Pick m continuous linear functionals (f 1, f 2,..., f m ), and let x = (x 1, x 2,..., x m ) be as in (1). We can define a ω-proper economy E = (P i, x i ) m with total endowment ω > 0, where P i (z) := {x f i (x) > f i (z)}. Thanks that R f (ω) = m f i (x i ), x is a weakly Pareto optimal. So according to our hypothesis, there exists some f L that supports the Pareto allocation x as a nontrivial quasiequilibrium. In other words thanks Proposition 17, f (ω) > 0 and for each i the bundle x i minimizes f (y i ) under the constraint f i (x i ) f i (y i ). Hence, for each i there exists λ i 0 such that f (λ i fi )(x i ). Since f 0, then λ i > 0 for each i. Let g = (λ 1 f 1, λ 2 f 2,..., λ m f m ). If m y i = ω with y i 0 for each i, then since f (λ i fi )(x i ), we get m λ i f i (y i ) m λ i f i (x i )+ m f (y i x i ) = m λ i f i (x i ) + f ( m y i x i ) = m λ i f i (x i ). So R g (ω) = m λ i f i (x i ). Moreover from Theorem 19, we obtain f ( m λ i fi )(ω). This implies f (v) f (ω) R g (v) R g (ω) for every v L +. Consequently, as f (ω ω) = 0 < f (ω), letting Γ ω = {v L f (v) < f (ω)}, we see that Γ ω satisfies the required properties for R g to be pointwise ω-proper at ω. (2) (1) Assume that x = (x 1, x 2,..., x m ) is a weakly Pareto optimal allocation for an ω-proper economy E with m consumers and total endowment ω > 0. Thanks L L for all f L there exists λ > 0 such that λf 1 ( 1; 1) [ ω; ω]. So [ ω, ω] is weak τ-bounded thus order intervals are τ-bounded. Therefore, the topology of (L ω, ω ) is stronger then topology induced by τ on L ω. So we have the ω-proper economy E Lω = ((L ω ) +, ω i, P i Lω ) m, where x is also Pareto optimal of E Lω. Moreover preferred set on L ω at x i has nonempty interior. Indeed let V be a τ-neighborhood of zero such that x i + ω + V P i (x i ), then there exists 1 > ɛ > 0 such that B ω (0, ɛ) V, but x i + ω + B ω (0, ɛ) x i + [0, 2ω], thus x i + ω + B ω (0, ɛ) (L ω ) + P i (x i ) = P i (x i ) L ω. For each 0 < λ < 1 we have λx i + (1 λ)ω P i (x i ) L ω, thus by Proposition 22 there exists non-zero p L ω := (L ω, ω ) such that

11 EQUILIBRIUM THEORY IN INFINITE DIMENSIONAL ARROW-DEBREU S MODEL.9 (x, p) is quasi-valuation equilibrium of E Lω. A pair (x, p) is also nontrivial quasi-valuation equilibrium. To see this, let ɛ > 0 be such that x i + ω + B ω (0, ɛ) P i (x i ) L ω, thus p(x i + ω + z) p(x i ) for each z B ω (0, ɛ). Hence p = 0 or p(ω) > 0. Applying Lemma 20 with M = L ω, Y = int P i (x i ), y = x i, and p the price, we obtain for each i a τ-continuous linear functional π i on L such that π i Lω p and p(x i ) = π i (x i ) π i (y) for all y P i (x i ). Like before, since ω + x i int P i (x i ), one deduces that π i (ω) > 0. For y i L + such that m y i = ω we obtain m m π i (y i ) = π(ω) p(ω) = p( x i ) = m π(x i ). So it means that R π is exact at ω with respect to x = (x 1, x 2,..., x m ), where π = (π 1, π 2,..., π m ) (L ) m. R π (ω) > 0, so R π (2ω) > R π (ω). From our assumption, there exist (λ 1, λ 2,..., λ m ) 0 such that R g is exact at ω with respect to x and pointwise ω-proper at ω, where g = (λ 1 π 1, λ 2 π 2,..., λ m π m ). It means that {x : R g (x) > R g (ω)} = P (ω) L +, for some convex set P (ω) with nonempty interior. So thanks Hahn-Banach theorem there exists a continuous price p such that p(x) p(ω) for x P (ω). Let ω L + be such that R g (ω ) = R g (ω) then R g (λω ) > R g (ω ) for λ > 1 so p(λω ) p(ω). Hence p(ω ) p(ω). Now we can observe that for ω = y + P j (x j ) with y P i (x i ) it is j i R g (ω ) R g (ω). Hence p(y) p(x i ). So (x, p) is nontrivial quasivaluation equilibrium. At the end we shall present some other equilibrium existence theorem. Theorem 23. Let E = (P i, x i ) m be an economy with total endowment ω > 0, where the interval [ ω, ω] is absorbing (ω is order unit) and [0, v] is compact in σ(l, L )-topology. Moreover there is a consumer i such that P i (x i ) + L + P i (x i ), for all x i L +. Then E has a quasi-equilibrium. Proof. Let F be the set of all finite-dimensional subsets F of L spanned by sets of vectors in L + and containing {ω 1, ω 2,..., ω m }. For each F F, we have an economy E F = (P i F, ω i ) m over F, where F + = F L +. By standard results on equilibria in economics, we obtain that E F has a quasi-equilibria (x F, p F ). We can assume that p F (v) 1 for all v F [ ω, ω]. Thanks to Hahn-Banach theorem we can extends p F to linear functional p F such that p F (v) 1 for v [ ω, ω]. Now by Banach-Alaoglu theorem the set B := {p L p F (v) 1

12 10 PATRYK PAGACZ for v [ ω, ω]} is compact in the σ(l, L)-topology and by assumption [0, ω] is compact in the σ(l, L )-topology. Thanks that the net {(x F, p F )} F F has an accumulation point (x 0, p 0 ) in [0, ω] m B. Now we must show that (x 0, p 0 ) is a quasi-equilibrium. Let {(x G, p G )} G be a subnet of {(x F, p F )} F F converting to (x 0, p 0 ). Fix i {1, 2, 3,..., m}. Since that x G i ω for each G, we obtain (p 0 p G ) (ω) (p 0 p G )(x G i ) (p 0 p G ) + (ω). Since (p 0 p G ) + (ω) 0 and (p 0 p G ) (ω) 0, then (p 0 p G )(x G i ) 0. But p 0 (x G i ) p 0 (x 0 i ), so p G (x G i ) p 0 (x 0 i ). Parallel p G (ω i ) p 0 (ω i ). Therefore by p G (x G i ) = p G (ω i ) we have p 0 (x 0 i ) = p 0 (ω i ). Now let x P i (x 0 i ), so x 0 i P 1 i (x) and by openness we can assume that x G i P 1 (x) for all G, thus x P (x G i ) so p G (x) p G (x G i ). Therefore like before we get p 0 (x) p 0 (x 0 i ). References [1] C.D. Aliprantis, M. Florenzano and R. Tourky, General equilibrium analysis in ordered topological vector spaces, Journal of Mathematical Economics, 40, [2] G. Debreu, Valuation equilibrium and Pareto optimum, Proceedings of the National Academy of Sciences, 40, [3] G. Debreu, Theory of Value, John Wiley New York [4] M. Florenzano and V. M. Marakulin, Production equilibria in vector lattices, Econom. Theory, 17, No. 3, [5] H. Keiding, Topological vector spaces admissible in economic equilibrium theory, Journal of Mathematical Analysis and Applications, 351, [6] V. M. Marakulin, Equilibrium analysis in Kantorovich spaces, Journal of Mathematical Sciences, 133, No. 4, [7] Walter Rudin, Analiza funkcjonalna, PWN Warszawa [8] H.H. Schaefer, Topological Vector Spaces, Springer Verlag, New York, Heildelberg 1971.