Advances in Systems Science and Applications (2014) Vol.14 No.2 183-189 Competitive Equilibrium of a Sequence of Incomplete Markets with a Continuum of Agents Guo-sheng Zhang The School of Economics, Trade and Event Management, Beijing International Studies University, Beijing, China, 100024 Abstract By introducing the Large Economical Ideas to multi-period financial market, we have constructed the multi-period economy with incomplete market and a contin- uum of agents. The competitive equilibrium has been proposed and the existence has been claimed. Our equilibrium definition is a development compared to that described by Radner, and our conclusion for equilibrium existence has generalized the related results obtained by Aumann and Zhang, if only the future contracts and goods are traded on security-spot markets. Keywords perfect competition, equilibrium, incomplete markets, correspondence integral 1 Introduction Consider a sequence of markets at successive dates, no one of which is complete in the Arrow Debreu Sense, i.e., at every date and for every commodity there will be some future dates and some events at those date for which the spot goods and future contracts contingent on those events are traded. For such economy, Rad- ner had first proposed the concept of common expectations that require traders to associate the same future prices to same future exogenous events[1]. An equi- librium is a set of prices at the first date, a set of common price expectations for the future, and a consistent set of individual plans for agents such that, given the current prices and price expectations, each individual agents plan is optimal for him, subject to an appropriate sequence of budget constraints. Radner’s com- mon expectation is a foundation for modern incomplete market theory. But a basic assumption of such model is that the current prices and price expectations for future Contracts be not affected by a single agents action, which needs the market be perfect competition. Otherwise a change in an individuals offer to buy or sell can easily upset the prevailing prices, so that the equilibrium will never be achieved, and price system is meaningless. As early as 1964, Aumann had suggested that the most natural mathematical model for a commodity market with such perfect competition is one in which there are a continuum of traders (like the continuum of points on a line)[2]. For decades, Aumann’s large economy has always been a vigorous field of economics. A deficiency of modern incomplete market theory is the lack of introduction of Aumann’s large economy ideas. As a novelty, Zhang first discussed security-spot markets with a measurable space of agents[3-4], where Aumann’s ideas have been 184 Guo-sheng Zhang: Competitive Equilibrium of a Sequence of Incomplete Markets with ... applied to security-spot market, particularly, to financial markets. But these re- search works are primary: the model is two-periods, and the existing discussion needs to proceed technically. In this paper, by combining Aumann’s ideas with Radner’s model, we discuss the economy with a sequence of incomplete markets and a continuum of agents. First, the concept of a competitive equilibrium for such economy is proposed, which has developed Radner’s definition. Then, we claim the existence of equi- librium. The sufficient conditions are all used by Aumann and Radner[5-6]. For the security-spot market with goods and future contracts, our conclusion is the generalization of related results proved by Aumann and Zhang[4,5]. 2 A multiperiod model for a sequence of incomplete markets with a contin- uum of agents Consider an economy extending through a finite sequence of elementary dates 1, 2, ..., T , in an environment with a finite set S of alternative states. The set of events observable at date t will be represented by partition φt of S . It is assumed the sequence of partition, φt is monotonous, no decreasing in fineness, that is, φt+1 is as fine as φt 1 . Also, take φ1 = {S}. For each date, there is a finite set of commodities, numbered 1, 2, ..., l. Trade contract (such as future contract) at date t in event A, denoted by θhtu(A,B), specifies the number of units of commodity h that the trader will receive from the market at date u ≥ t in event B(θhtu(A,B)) < 0 means the delivery to market; u > t means a future trade and u = t means a spot trade). For each pair of dates t and u such that u > t , and each commodity h, there is a given family F h tu of events, which is either empty or is a partition of S. In the latter case, φu must be as fine as F h tu. Assume that . Assume further that if F h tu is not empty and t ≤ v ≤ u, F h vu is as fine as F h tu. In other word, if at date t, one can buy a contract for receipt at date u contingent on event B, then at a later date V , one can do the same. A portfolio plan, which was described as a trade plan by Radner, is an array (θhtu(A,B)), one for each combination(h,t,u,A,B) such that for A in φt , B in F h tu, B ⊆ A, t ≤ u. The security price paid at date t in event A for receipt of commodity h at date u in event B will be denoted by phtu(A,B) .When t = u, phuu(A) is spot price of commodity h. An array p = {phtu(A,B)} will be called a commodity price system. In the situation just described, there is for each event pair (t,A), with A ∈ φt, a market in contracts for current and future receipt, with cost to be made currently in units of account. To simplify the notation, let denote the set of all pair ( t,A) such that, t = 1, 2, ..., T , A ∈ φt. Endow order for m = (t,A) and n = (u,B) in 1φ is said to be as fine as partition φ′ if, for every A’ in φ ,either A ⊂ A′ or A’ ∩ A = ϕ Advances in Systems Science and Applications (2014) Vol.14 No.2 185 M : m ≤ n if and only if t ≤ u, A ⊇ B. We assume short sales have low bound L, −L ∈ R++. Then all of the portfolio plans can be denoted as Z = ×m∈MZm with Zm denoting the vector space of all arrays of number θhtu(A,B) ≥ L; h = 1, 2, ..., l; u = t, t+ 1, ..., T ; B ⊆ A,B ∈ F h tu for each m = (t, A) ∈ M . Thus a portfolio θ = (θm)m∈M,θm ∈ Zm, is a point in Z. For m = (t, A), n = (u,B), if u ≥ t and B ⊆ A, A ∈ φt, B ∈ F h tu, we denote θhmn = θhtu(A,B); or θmn = (θ1mn, θ 2 mn, · · · θlmn) = 0. The payment at m for portfolio plan θ = (θm)m∈M , θm = (θmn)n≥m ∈ Zm,given the commodity price system p = (pm)m∈M , is the inner product pm · θm. Let {A,F, u} be the measure space of agents. For any agent a ∈ A, we denote his preference as ≺a, which is a binary relation on (×m∈MRl +)× (×m∈MRl +). We assume the ≺a is complete, transitive, reflexive and satisfies a) closeness: the set {(x, y) ∈ (×m∈MRl +) × (×m∈MRl +)|x≺ay} is closed in (×m∈MRl +)× (×m∈MRl +) and b) monotony: if x,y are two points in ×m∈MRl + , such that x < y, then x≺ay. The set of all preference relations on ×m∈MRl + satisfying all of these assump- tions is denoted by β. We endow β with the topology of closed convergence on (×m∈MRl +) × (×m∈MRl +). For each a ∈ A, we require ε = (≺a, ea) : A → β × (×m∈MRl +) is measurable and ea is integrable, where ea ∈ ×m∈MRl + is the real endowment of agent α. We call ε large sequence of security-spot market. Each agent must select a consumption plan xa = (xam)m∈M ∈ ×m∈MRl +, and a portfolio plan θa = (θam)m∈M ∈ Z within his budget set. An pair (xa, θa) is called an assign if (xa, θa) : A → (×m∈MRl +)× Z is integrable (we endow Z with Bore -field). Give p = (pm)m∈M , agent a′s budget set is defined as xa(p) = {(x, θ) : x = (xm)m∈M ∈ ×m∈MRl +, θ = (θm)m∈M ∈ Z, such that for each m ∈ M,pm · θm ≤ 0, xm ≤ eam + ∑ j≤m θjm} The agent a′s demand correspondence is then defined as ξa(p) = {(x, θ) ∈ xa(p)|∀(x′, θ′) ∈ xa(p), x≻ax ′} Definition 2.1 An equilibrium of the economy ε is an assign (xa, θa) of consumption-portfolio plan and a price system p such that a · e, a ∈ A, (xa, θa) ∈ ξa(p) (1)∫ A θadu = 0, ∫ A xadu = ∫ A eadu (2) where ∫ A θadu = 0 means ∫ A θamdu = 0 for each m ∈ M . This definition has obviously generalized Radner’s definition for pure exchange 186 Guo-sheng Zhang: Competitive Equilibrium of a Sequence of Incomplete Markets with ... economy, that is, the finite market participants is replaced by infinite market participants. Note that (1) implies a.e., a ∈ A, (xa, θa) is the best consumption- portfolio in his budget set and (2) implies the market is clear at each data- event pair. Our definition is also a development for the equilibrium described by Debreu[7], which only considered spot-market with one-period. But compared with general incomplete market as that Geankoplos proposed[8], we only consider future contracts as securities here. Equilibrium Existence Our main result can be described as the following. Theorem 3.1 If {A,F, u} is atomless2 and endowment satisfies ∫ A eam ≫ 03, for any m ∈ M , then economy ε has equilibrium. For each m ∈ M , let P = ×m∈MPm, Pm be the set of all nonnegative vectors in Zm whose coordinates sum to 1.Denote the excess demand correspondence as ξ(p) = ∫ A ξa(p)du− ∫ A (ea, 0)du 4, here (ea, 0) ∈ (×m∈M )Rl +×Z. We will give some properties of ξ(p) restricted on . For the reason of shortening this paper, some simple proof processes similar to that used by Debreu are omitted here[9]. Proposition 3.1 Under conditions of theorem 3.1, ξ(p) is non-empty, com- pact, lower bounded and upper hemicontinuous at every p ≫ 0 in P . Proof. We only prove ξ(p) is upper hemicontinuous. We first prove xa(p) is continuous at each p ≫ 0 in P .The graph of corre- spondence xa(p) is obviously closed in P× (×m∈MRl +)×Z. Thus, xa(p) is upper hemicontinuous on P . To show that xa(p) is lower hemicontinuous at any point po ≫ 0 in P , we consider a pt sequence in P converging to po(t → ∞) and a point (xo, θo) ∈ xa(p o).Denote xo = (xom)m∈M , θo = (θom)m∈M , θom ∈ ZM . We will discuss the problem under two cases. i) pom · θom < 0 for any m ∈ M . Because of ptm → pom, We have ptm · θom < 0 for large enough t. So (xo, θo) ∈ xa(p t), which satisfies the condition appearing in the definition of lower hemicon- tinuity. ii) ∃m1,m2 · · · ,mg ∈ M , such that pomj · θomj = 0, j = 1, 2, . . . , gand pom · θom < 0, for m ̸= m1,m2 . . . ,mg. For any j, we select a point θ ′ mj ∈ Zmj satisfying pomj · θ′mj < 0. This is possible because −L > 0.Thus pomj ·θ′mj < pomj ·θomj , and for large enough t, the hyperplane {θmj ∈ Zmj |ptmj · θmj = 0} intersects the straight line through θ′mj and θomj in a unique point θ t mj . 2{A,F, u} is called atomless if, for any B ∈ F , u(B) > 0, there is C ∈ F such that 0 < u(C) < u(B). 3For x ∈ l, x ≫ 0 means all of its coordinates are strictly positive. 4For integrals of correspondences and their properties, see Definition of Part , D. of [7]. Advances in Systems Science and Applications (2014) Vol.14 No.2 187 Define θt as θt = (θtm)m∈M , θtm =  θom, m ̸= m1,m2, · · · ,mg θ t m, m = mj and θ t mj is between θ′mj and θomj 0, others. It is easily checked that θtm → θom(t → ∞) for any m ∈ M , and ptm · θtm ≤ 0. Let xom = (xohm ), h = 1, 2, . . . , l, eam = (eahm ), h = 1, 2, . . . , l for each m ∈ M . For any h, if xohm < eahm + ∑ j≤m θohjm, we can take xthm such that xth < eahm + ∑ j≤m θthjm, and xthm → xohm (t → ∞). This is possible because xohm < eahm + ∑ j≤m θthjm for large enough t. If xohm = eahm + ∑ j≤m θohjm, then we take xthm = eahm + ∑ j≤m θthjm, which also implies xthm → xohm . Therefore, we can find xt = (xtm)m∈M → xo such that (xt, θtm) satisfies the con- dition for lower hemicontinuity of xa(p). Note that xa(p) is nonempty ((0, 0) ∈ xa(p)) and compact, According to Debreu[10],ξ(p) is upper hemicontinuous at each p ∈ P , p ≫ 0 . # Proposition 3.2 Under the conditions of theorem 3.1, ξ(p) satisfies Boundary Condition, that is, if pt ≫ 0 in converges to p0 in ∂P, then d(0, ξ(pt)) → (t → ∞). Proof. By the similar discussion to Debreu (1982, p. 729), the proposition holds if only for a.e.,a ∈ A d(0, ξ(pt)) → ∞. Suppose that the conclusion does not hold. Then there is a subsequence (pt ′ ) such that d(0, ξa(p t′)) is bounded. For each t′, one can select (ct ′ , θt ′ ) ∈ ξa(p t′) in such a way that sequence (ct ′ , θt ′ ) is bounded. Therefore, one can extract from (pt ′ , ct ′ , θt ′ ) a sequence (pt ′′ , ct ′′ , θt ′′ ) converging to (po, co, θo). By proposition 3.1 we have (co, θo) ∈ ξa(p o). Since po ∈ ∂P , there is mo ∈ M such that some coor- dinates of pomo is zero. Without loss of generality, we suppose the first coordinate of pomo is zero. Then by replacing the first coordinate of θomo with large one, we can get θ′ = (θ′m)m∈M ∈ Z, satisfy in pom · θ′m ≤ 0 and eau + ∑ j≤u θ′ju > eau + ∑ j≤u θoju. Thus, we can obtain x′ = (x′m)m∈M , such that (x′, θ′) ∈ xa(p o) and x′ > x0. This contradicts the monotony of preference ≺a.# Proposition 3.3 Under conditions of theorem 3.1, Walras Law holds, that is, for any p ∈ P , p ≫ 0 and (x, θ) ∈ ∫ A ξa(p)du, we have (pm · θm)m∈M = 0 and xm = ∫ A eamdu+ ∑ j≤m ∫ A θajmdu. Proof. For any a ∈ A, (x, θ) ∈ ξa(p), by the monotony of preference ≺a, it is easily proven that (pm · θam)m∈M = 0 and xam = eam+ ∑ j≤m θajm. By integrating the two sides of (pm · θam)m∈M = 0 and xam = eam + ∑ j≤m θajm, the conclusion holds. # 188 Guo-sheng Zhang: Competitive Equilibrium of a Sequence of Incomplete Markets with ... Proposition 3.4 Under conditions of Theorem 3.1, is convex-valued. Noticing the fact that ξa(p) is convex-valued for any a ∈ A, the proposition is the direct corollary of theorem 3 of Part I, D.II of [9]. Proof of theorem 3.1. For the real number b > 0, let b·Pm = {b·p : p ∈ pm}. Notice that if we replace P by P′ : P′ = ×m∈M (bm · Pm), all of the conclusions of proposition 3.1-proposition 3.4 are true. Assume the number of dimension- s of Pm is vm, and ∑ m∈M νm = ν. Let P ′ m = νm ν · Pm, P′ = ×m∈MP ′ m, and E = {p ∈ P′|p ≫ 0, there is(x, θ) ∈ ξ(p), such that ∑ n≥m,n,m∈M l∑ h=1 θhmn ≤ 0}. Since ξ(p) is bounded below, when p varies in E, the θ with (x, θ) ∈ ξ(p) remains bounded. So does χ. So, by the Boundary Conditions, there cant be in E a sequence pt converging to p0 in ∂P ′. Consequently, the distance from p ∈ E to ∂P ′ is bounded below by a strictly positive real number. Thus, there is a closed convex cone C with vertex 0 in ×m∈MRm, such that E ⊂ intC and C\0 ⊂ int(×m∈MR+ m), where Rm denotes the vector space of all arrays of real number ghmn ∈ R for any n = (u,B) ≥ m,B ∈ F h mn and h = 1, 2, ..., l, and R+ m is the positive cone of Rm. Let ξ′(p) = {θ ∈ Z : thereis(x, θ) ∈ ξ(p)}. If we restrict ξ′(p) on P ′, it is easily proved that ξ′(p) is convex-valued, bounded below and satisfies Walras Law. We further claim ξ′(p) is upper hemicontinuous at each p ≫ 0, p ∈ P′. Suppose pt in P′ converge to po ∈ P′, po ≫ 0 and θt ∈ ξ′(pt) converge to θo, which is a portfolio plan. By the definition of ξ′(p) , there is (xt, θt) ∈ ξ(pt). Let U be a compact neighborhood of p0 contained in the relative interior of P′. Then when t large enough, (xt, θt) ∈ ξ(pt) is uniformly bounded. Thus we can take a subsequence (xt ′ , θt ′ ) → (xo, θo) with xo ∈ ×m∈MR+ m. By proposition 3.1, we have (xo, θo) ∈ ξ(po), which yields θo ∈ ξ′(po). This shows that ξ′(p) is upper hemicontinuous at any po ∈ P′, po ≫ 0. According to Debreu[9], there is a point p∗ ∈ C ∩P′ such that ξ′(p∗)∩Co ̸= ϕ. Let θ∗ ∈ ξ′(p∗) ∩ Co. By Walras’ Law, the point ( 1 v , 1 v , · · · , 1 v ) in P′ belongs to E, hence to C. Therefore, ∑ n≤m,n,m∈M l∑ h=1 θ∗hmn ≤ 0. Consequently,p∗ ∈ E, hence p∗ ∈ intC. Moreover, by another application of the Walras’ Law, we has p∗ · θ∗ = 0. This equality together with p∗ ∈ intC and θ∗ ∈ Co implies θ∗ ∈ 0. Then (x∗, θ∗) is a equilibrium for price P ∗.# References [1] Aumann, R. J. (1964), “Space cannot be cut: Why self-identity naturally includes neighbourhood”, Integrated Psychological Behaviour, Vol.32, pp.39- 50. Advances in Systems Science and Applications (2014) Vol.14 No.2 189 [2] Aumann, R. J. (1966), “Existence of competitive equilibria in market with a continuum of traders”, Econometrica, Vol.34, pp.1-17. [3] Yun, X. (2007), “Optimal portfolio on optimal control”, Advances in System Science and Applications, Vol.7, No.1, pp.66-71. [4] Arrow, K., Debreu, G. (1954), “Existence of an equilibrium for a competitive economy”, Econometrica, Vol.22, pp.265-290. [5] Debreu, G. (1982), “Existence of competitive equilibrium”, Handbook of Mathematical Economics, Vol. II, Chapter 15, pp.697-743, North Holland, New York.http://www.calphysics.org/articles/chown2007.html [6] Radner, R. (1972), “Existence of equilibrium of plans, prices, and price ex- pectations in a sequence of market”, Econometrica, Vol.40, No.2, pp.289-302. [7] Hildenbrand, W. (1974), Core and equilibria of a large economy, Princeton University Press. [8] Geankoplos, J. (1990), “An introduction to general equilibrium with incom- plete Asset Market”, Journal of Mathematical Economics, Vol.19, pp.1-38. [9] Jifeng, Z. and Xuemou, Wu. (2002), “Pansystems or principles to society- economy systems”, Advances in System Science and Applications, Vol.2, No.1, pp.24-31. [10] Hart, O. D. (1975), “On the optimality of equilibrium when the market structure is incomplete”, Journal of Economic Theory, Vol.11, pp.418-443. [11] Bingan, J. etc.(2007), “Equilibrium of manufactures’ R&D decision-making in defense procurement”, Advances in System Science and Applications, Vol.7, No.1, pp.117-120. [12] Guosheng Zhang (1998), On large finacial market, Mathematics in Economy (in Chinese), Vol.15, No.3, pp.1-6. [13] Guosheng Zhang (2000), “Spot-Financial Market with large characteristics”, Systems Engineering-Theory & paractice (in Chinese), Vol.20, No.3, pp.39- 45. [14] Guosheng Zhang (2007), “Competitive equilibrium of Large Security-Spot Market with incomplete asset structure”, Journal of Systems Science and Complexity, Vol.20, No.3, pp.386-396. Corresponding Author Author can be contacted at: zhangguosheng@bisu.edu.cn.