Incomplete Markets with Endogenous Portfolio Constraints and Redundant Assets
 Author: Hahn Guangsug
 Publish: Seoul Journal of Economics Volume 27, Issue4, p469~488, Nov 2014

ABSTRACT
This paper shows that a competitive equilibrium exists in an exchange economy with incomplete financial markets where redundant assets are traded and the asset trading of each agent is subject to endogenous portfolio constraints. The set of budgetfeasible portfolios need not be bounded in the presence of redundant assets. To address this problem, we impose the positive semiindependence condition on individual portfolio constraints.

KEYWORD
Incomplete markets , Competitive equilibrium , Endogenous portfolio constraints , Redundant assets , Constrained arbitrage

I. Introduction
When financial markets are unconstrained, redundant (financial) assets do not play a role in risksharing and thus they are useless. Therefore, without loss of generality, we can assume that there is no redundant asset. In reality, however, redundant assets such as futures and options exist because financial markets are subject to portfolio constraints. In financial markets, agents usually face portfolio constraints when they trade financial assets. Portfolio constraints capture market frictions such as shortselling constraints, credit limits, bidask spreads, margin requirements, and proportional transaction costs. It is noted that many of portfolio constraints (
e .g ., margin requirements) depend on asset prices. It is important to investigate how redundant assets and endogenous portfolio constraints affect equilibrium asset prices in financial markets.The purpose of this paper is to show that there exists a competitive equilibrium in an exchange economy with incomplete financial markets, where agents are subject to portfolio constraints depending on asset prices. Aouani and Cornet (2011) and Hahn and Won (2014) among others demonstrate the existence of a competitive general equilibrium in an exchange economy with incomplete markets, where each agent’s asset trading is subject to exogenous portfolio constraints, which do not depend on endogenous variables. However, when agents participate in financial markets, they are often faced with endogenous portfolio constraints such as margin requirements, which depend on asset prices.
Several recent papers have studied this problem, including Carosi
et al . (2009) and CeaEchenique and TorresMartinez (2014), among others. Carosiet al . (2009) describe portfolio constraints by restriction functions, which depend on firstperiod consumption and commodity prices, as well as financial asset prices. They assume that portfolio restriction functions are continuously differentible in order to characterize the generic regularity of equilibrium. Thus such approach cannot cover cases in which portfolio constraints are represented by convex cones (e .g ., margin requirements). Moreover, by assuming that the payoff matrix has a column full rank, they exclude redundant assets such as financial derivatives, whose raison d'être is portfolio constraints.CeaEchenique and TorresMartinez (2014) employ endogenous trading constraints represented by correspondences that depend on both commodity and asset prices. Restrictions on consumption and portfolio choices are incorporated into a single trading constraint set. Trading constraints are so general and can therefore cover collateralized borrowing constraints and incomebased portfolio constraints. In particular, attainable allocations are pricedependent. However, they impose a restrictive assumption that the set of pricedependent attainable allocations is bounded. This assumption may not be fulfilled in constrained incomplete markets with redundant assets, in which asset demand correspondences are unbounded. Therefore, they
de facto exclude financial derivatives from incomplete financial markets.The rest of the paper is organized as follows: in Section II, we present the model of an exchange economy with incomplete markets where each agent
i s faced with endogenous portfolio constraints. In Section III, we define constrained arbitrage and provide additional assumptions for endogenous portfolio constraints. Section IV contains examples of endogenous portfolio constraints. In Section V, we show that a competitive equilibrium exists in the economy and present a numerical example of a competitive equilibrium. Section VI contains the concluding remarks.> II. The Model
The paper considers an exchange economy with financial asset markets, extending over two periods. There are
I agents andJ financial assets. The uncertainty of the second period is described by a finite setS :＝{1, ...,S } of states of nature. In the first period, no agent knows which state will be realized in the second period. The payoffs of asset j∈J :＝{1, 2, ...,J } are realized depending on the state in the second period. There areL commodities in each state s＝S _{0} :＝S ◡{0} where the first period is regarded as state s＝0. Therefore, the commodity space is equal to Rℓ where ℓ:＝L (S ＋1).In the first period, agent
i ∈I :＝{1, 2, ...,I } makes consumptionx_{i} (0) and invests portfolioθ _{i} with his endowments. In the second period, agenti makes consumptions (x _{i}(s ))_{s∈S} with his endowments and payoffs of his portfolio. Hence, agenti chooses consumption bundlex _{i}:＝(x _{i}(0),x _{i}(1), ...,x _{i}(S)) in his consumption setX _{i}⊂R^{ℓ}, which contains his initial endowmente _{i} of commodities. Preferences overX _{i} are represented by a preference relation ≻i onX _{i}, which is irreflexive, complete, and transitive. The preference relation ≻i defines the preference correspondenceP _{i}:X _{i} →. by P_{i}(x_{i}) :＝{ ∈X_{i} :≻_{i}x_{i}}, which is the set of consumption bundles that agenti prefers tox_{i} . Agenti is subject to portfolio constraints, as represented by correspondence Θ_{i} : R^{J} → 2^{RJ} of asset priceq ∈R^{J}. To finance his consumption in the second period, agenti chooses portfolioθ _{i}∈Θ_{i}(q ) in the first period.The payoff of asset
j in states ∈ is denoted by r_{j}(S s ), and the payoff vector of assetj overS states by anS dimensional column vector r_{j}＝(r_{j}(s ))_{s∈S}. Payoff vector in state s is denoted by aJ dimensional row vectorr (s )＝(r _{j}(s ))_{j∈J}. We denote the asset payoffs by an (S ×J ) payoff matrix R＝[(r _{j})_{j∈J}]. An asset is called redundant if its payoffs can be replicated by those of the other assets. We allow redundant assets,i.e. , V^{⊥}≠{0} where V^{⊥}＝{θ ∈R^{J}:R ⋅θ ＝0}. We note that redundant assets do not play a risksharing role without portfolio constraints because their payoffs can be replicated by those of the other assets. In contrast, redundant assets participate in risksharing under portfolio constraints, which may prevent the replication of redundant assets. We represent this economy by ＝<(E X _{i}, ≻_{i},e _{i}, Θ_{i})_{i∈I};R >.In the first period, agent
i is subject to budget constraintp (0)⋅x _{i}(0)＋q ⋅θ _{i}≤p (0)⋅e _{i}(0), where (p (0),q )∈R^{L}×R^{J} is a vector of commodity and asset prices in the first period. In the second period, he is subject to budget constraintp (s )⋅x _{i}(s )≤p (s )⋅e _{i}(s )＋r (s )⋅θ _{i}, ∀s∈S , wherep (s )∈R^{L} is a vector of commodity prices at state s∈S . Therefore, given price vector (p ,q )∈R^{ℓ}× R^{J}, agenti maximizes his preference ≻i by choosing a pair (x _{i},θ _{i}) of consumption and portfolio in his budget set:1where
A pair (
x _{i},θ _{i})∈B̅ _{i}(p ,q ) is optimal for agenti if [P _{i}(x _{i})×Θ_{i}(q )]⌒B̅ _{i}(p ,q )＝∅.Definition 2.1 : Acompetitive equilibrium of economy is a profile (E p ^{*},q ^{*},x ^{*},θ ^{*})∈R^{ℓ}×R^{J}×(R^{ℓ})×(R^{J})^{I}, such thatWe now provide the list of basic assumptions for every agent
i ∈I , which are necessary for our main results.Note that Assumptions (A1)(A5) are standard assumptions. Assumption (A6) states that the portfolio constraint of agent
i is represented by a convexvalued correspondence that has a closed graph. Moreover, Assumption A6 requires that portfolio constraints ‘nicely’ depend on asset prices. This assumption can cover market frictions such as shortselling constraints, bidask spreads, margin requirements, and proportional transaction costs.8 Moreover, Assumption (A6) states that portfolio choice sets depend solely on the relative price of assets.1Let v and v’ be vectors in a Euclidean space. Then v≥v’ implies that v－v’∈ Rℓ＋; v＞v’ implies that v≥v’ and v≠v’; and v≫v’ implies that v－v’∈Rℓ＋＋. 2Note that B̅i is a correspondence from Rℓ×RJ to Rℓ×RJ. 3The preference relation ≻i is continuous if Pi(xi) and Pi－1(xi) :＝{xi’∈Xi : xi≻ixi’} are open for every xi∈Xi, and is convex if Pi(xi) is convex for every xi∈Xi. 4For each xi∈Xi and for each s∈S there exists xi’(s)∈Xi(s) such that (xi’(s), xi(－s))≻ixi, where xi(−s) = (xi(0),…, xi(s − 1), xi(s + 1),…, xi(S)). 5Let A be a nonempty subset of a Euclidean space. The closure of A is denoted by cl(A) and the interior of A is denoted by int(A). 6Let X and Y be subsets of Euclidean space. A correspondence ϕ : X→2Y is lower hemicontinuous if {x∈X: ϕ(x)⌒V≠∅} is open for every open set V⊂Y and has a closed graph if Gϕ:＝{(x, y)∈X×Y: y∈ϕ(x)} is closed. 7The homogeneity of degree zero for constrained choice sets can be also found in and Page and Wooders (1999) and et al Carosi et al. (2009). 8See Heath and Jarrow (1987), Luttmer (1996), and Elsinger and Summer (2001).
III. Constrained Arbitrage and Additional Assumptions
When no portfolio constraints are present in incomplete markets, no arbitrage opportunity is admitted and therefore the law of one price holds in equilibrium. However, the law of one price does not hold in incomplete markets with portfolio constraints, and it is not appropriate to apply the notion of arbitrage used for unconstrained incomplete markets to constrained incomplete markets. The notion of constrained arbitrage is employed in Jouini and Kallal (1999) and Luttmer (1996), which study incomplete markets with exogenous portfolio constraints. To introduce an appropriate notion of arbitrage for incomplete markets with endogenous portfolio constraints, let
C _{i}(q ) denote the recession cone Γ(Θ_{i}(q )) of Θ_{i}(q ).9Definition 3.1 : Asset priceq ∈R^{J} is said to admit a constrained arbitrage for agenti if there is a portfolioθ _{i}∈C _{i}(q ), such thatW (q )⋅θ _{i}＞0. Asset priceq ∈R^{J} is said to admit no constrained arbitrage for economy if it admits no constrained arbitrage for every agentE i ∈ .I No constrained arbitrage is equivalent to no arbitrage in unconstrained incomplete markets. Let Q_{i} denote the set of asset prices that admit no constrained arbitrage for agent
i . Then,Q : ∩_{i∈I}Q _{i} is the set of asset prices that admit no constrained arbitrage for . LetE N _{i}(q ) be the lineality space of Θ_{i}(q ).10 We defineN _{0}(q )＝Σ_{i∈I}(N _{i}(q )⌒V^{⊥}) and denote byN _{0}(q )^{⊥} its orthogonal complement in R^{J}. Let us defineQ ^{*}:＝{q ∈Q :q ∈N _{0}(q )^{⊥}}. The following results show what is appropriate for equilibrium asset prices. : It holds thatProposition 3.1Q andQ ^{*} are nonempty. : To showProof Q ≠∅, suppose otherwise, that is,Q ＝∅. Considerq ＝λ ⋅R withλ ∈. SinceQ ＝∅, we see thatq ∉Q . Then there is some agenti withθ _{i}∈C _{i}(q ) satisfyingW (q )⋅θ _{i}＞0, which makesq ⋅θ _{i}＝λ ⋅R ⋅θ _{i}≥0 necessary. Ifq ⋅θ _{i}＞0, thenq ∈Q , thenq ∈Q , which is a contradiction. Ifq ⋅θ _{i}＝λ ⋅R ⋅θ _{i}＝0, thenR ⋅θ _{i}＝0. This implies thatq ∈Q , which is a contradiction. Hence,Q is nonempty.To show Q^{*}≠∅, suppose otherwise, that is,
Q ^{*}＝∅. Take anyq ∈Q , and we haveq ∉N _{0}(q )^{⊥}. Then there existsv ∈N _{0}(q ) such thatq ⋅v ＜0 without loss of generality. Since there existsv _{i}∈N _{i}(q )⌒V^{⊥}, ∀i ∈I such thatv ＝Σ_{i∈I}v _{i}, it follow thatq ⋅v _{i}＜0 for somei . Noting thatv _{i}∈C _{i}(q ) andR ⋅vi ＝0, we see thatv _{i} is a constrained arbitrage opportunity atq . Therefore,q ∉Q _{i} andq ∉Q , which is a contradiction. Hence,Q ^{*} is nonempty. : Under Assumption (A4), an equilibrium asset priceProposition 3.2q belongs toQ ^{*}. : Let (Proof p ,q ,x ,θ ) be an equilibrium of . Suppose thatE q ∈Q . Then there is somei ∈I withv _{i}∈C _{i}(q ) satisfyingW (q )⋅v _{i}＞0. This implies thatθ _{i}＋v _{i}∈Θ _{i}(q ) andW (q )⋅θ _{i}W (q )⋅(θ _{i}＋v _{i}). Due to Assumption (A4), there exists a consumption bundlex _{i}’∈X _{i}, such thatx _{i}’≻_{i}x _{i} and (x _{i}’,θ _{i}＋v _{i})∈B̅ _{i}(p ,q ), which contradicts the optimality of (x _{i},θ _{i} ) inB̅ _{i}(p >,q ). Hence,q ∈Q .We now show that
q ∈N _{0}(q )^{⊥}, that is,q ⋅v ＝0 for allv ∈N _{0}(q ). Suppose otherwise. Then there existsv ∈N _{0}(q ) such thatq ⋅v <0 without loss of generality. Since there existsv _{i}∈N _{i}(q )⌒V^{⊥}, ∀i ∈I such thatv ＝Σ_{i∈I}v _{i}, it follow thatq ⋅v _{i}＜0 for somei . Noting thatv _{i}∈C _{i}(q ) andR ⋅v _{i}＝0, we see thatv _{i} is a constrained arbitrage opportunity atq . That is,q ∉Q _{i}, and thereforeq ∉Q , which is a contradiction. Hence,q ∈N _{0}(q )^{⊥}. ■From Proposition 3.2, we see that
Q ^{*} is an appropriate set of equilibrium asset prices and thatQ andQ ^{*} appear as cones with vertex zero under Assumption (A6). We observe thatQ ^{*} may not be convex. Therefore we considerQ̂ which is the convex hull ofQ ^{*}. ThenQ̂ is a nonempty convex cone.We now impose a portfolio survival condition, which states that there is no constraint on trading for sufficiently small amount of portfolios.
To analyze the effects of redundant assets on risksharing in constrained asset markets, we need to examine feasible zeroincome portfolios. We call portfolios in
C _{i}(q )⌒V^{⊥}scalefree feasible zeroincome portfolios for agenti in that, ifv _{i}∈C _{i}(q )⌒V^{⊥}, we haveλ v _{i}∈Θ_{i}(q ) andR ⋅(λ v _{i})＝0 for everyλ ≥0. Particularly, ifq ∈cl (Q̂ )\{0}, some agenti may have a portfoliov _{i}∈C _{i}(q )⌒V^{⊥} satisfyingq ⋅v _{i}≤0. Therefore, in the presence of scalefree feasible zeroincome portfolios, some agent’s portfolio choices may be unbounded with his budget constraint satisfied. To prevent such negative effect of scalefree feasible zeroincome portfolios at the aggregate level, we need the following assumption:If Assumption (A8) does not hold, there is an asset price
q ∈cl (Q̂ )\{0}, such that some agenti has a scalefree feasible zeroincome portfoliov _{i}∈C _{i}(q )⌒V^{⊥}, which is supported by the other agents because －v _{i}∈Σ_{j≠i}C _{j}(q )⌒V^{⊥}. Therefore, agenti can hold an indefinite amount of portfolios in the direction ofv _{i} such that the budget constraints of all agents and the market clearing condition are not violated. This possibility is eliminated by Assumption (A8).9Let A be a nonempty convex subset of Euclidean space X. The recession cone of A is the set Γ(A)＝{v∈E: A＋v⊂A}. When A is closed, Γ(A) is also closed and can be expressed as Γ(A)={v∈X:∃{xn} in A and {an} in R. with an → 0 such that anxn → v} 10The lineality space L(A) is the maximal subspace in A, that is, L(A)＝Γ(A)⌒[－Γ(A)].
IV. Examples of Endogenous Portfolio Constraints
Financial intermediaries prohibit shortselling above specific limits, which can depend on asset prices. Financial regulation prohibits the purchase of some securities above given limits, which may also depend on asset prices. Let continuous functions
a _{i}: R^{J}→R^{J} andb _{i}: R^{J}→R^{J} take the values of the shortselling limits or buying limits of agenti on securities, respectively. The portfolio constraints of agenti can therefore be described bywhere 0∈(
a _{i}(q ),b _{i}(q )), ∀q ∈R^{J},a _{i}: R^{J}→R^{J}, andb _{i}: R^{J}→R^{J} are continuous functions and homogeneous of degree zero inq .Financial intermediaries can provide credit to agents with limits that depend on asset prices. In this case, the trading strategies of agent
i are restricted such thatwhere
a _{i}: R^{J}→R_{＋＋} is a continuous function and homogeneous of degree one inq andb _{i}: R^{J}→ is a continuous function and homogenous of degree zero inq .Financial assets such as collateralized debt obligation (CDO) are used as debt instruments and should be backed by a pool of other financial assets. Supposing that security 1 is a riskless bond, we can express portfolio constraints in the following form:11
where
θ _{i}^{－}＝(－min{0,θ _{ij}})^{J}_{j}＝1,θ _{i}^{＋}＝(max{0,θ _{ij}})^{J}_{j}＝1,a _{i}∈R_{＋}, andb _{i}∈R_{＋＋}. It is obvious that the portfolio correspondences of the above examples satisfy Assumptions (A6) and (A7).As in Heath and Jarrow (1987), portfolio constraints that involve margin requirements can be described as
where security 1 is riskless bond,
a _{i}≥2, andb _{i}∈R_{＋＋}. For example, assume thatJ ＝2 andb _{i}＝0.12 Suppose that security 1 is riskless bond and security 2 is a stock. Now suppose that agenti shorts one stock and maintains a margin account with mi proportion of the stock price in the bond. The portfolio constraint is therefore reduced towhich implies that
m _{i}≥a _{i}/(a _{i}－1). In the case wherea _{i}＝3, we havem _{i}≥3/2, that is, agenti should put the money from shorting the stock and an additional fifty percent of the stock price in his margin account.Won (2003) provides a more generalized form of the example in Heath and Jarrow (1987). Assuming that security 1 is a riskless bond with
q _{i}＝1, we modify his example to present the portfolio constraint set of agenti atq ∈cl (Q )\{0} bywhere
a _{i}≥1,b _{ij}≥0,c _{ij}≥0,d _{i}＞0, and δ_{ij}＞0 are constants for everyi andj and δ_{i}＝(δ_{ij}). It is obvious that Θ_{i} has a closed graph and satisfies the homogeneity of degree zero. If we assume thatwe have 0∈
lnt (Θ_{i}(q )), ∀q ∈cl (Q )\{0}. To see that Θ_{i} is convexvalued, we define continuous function : R^{J} × R^{J}→R byThe portfolio constraint correspondence is then given by
We can observe that max_{j}{⋅} is a convex function on R^{J} and ⋅ is a convex function on R, which implies that －max_{j∈J} {
c _{ij}q _{j}(θ _{ij}＋δ_{ij})} is a concave function ofθ _{i}. Hence, we see that is a concave function function ofθ _{i} and therefore Θ_{i} is convexvalued.To show that Θ_{i} is lower hemicontinuous, we define correspondence Θ_{i}°: R^{J} →2^{RJ} by
Suppose
θ _{i}^{*}∈Θ_{i}°(q ^{*}), that is, (q ^{*},θ _{i}^{*})＞0. Take a sequence {(q ^{n},θ _{i}^{n})} converging to (q ^{*},θ _{i}^{*}). Since is continuous, for sufficiently largen ,which implies that
θ _{i}^{n}∈Θ_{i}°(q ^{n}). Thus we see that Θ_{i}° is lower hemicontinuous. Noting that Θ_{i}(q )＝cl (Θ_{i}°(q )), we see that Θ_{i} is lower hemicontinuous. Hence, Θ_{i} satisfies Assumptions (A6) and (A7). □11This example is adapted from Elsinger and Summer (2001). 12To be precise, bi should be sufficiently close to zero.
V. Existence of a Competitive Equilibrium
In this section, we will show that there exists a competitive equilibrium of economy
E . We define the sets of normalized prices by Δ＝Δ_{0}×Δ_{1}, whereWe observe that Δ is compact and convex.
̂Let
X :＝Π_{i∈I}X _{i} andA _{X} :＝{(x _{1}, ...,x _{I})∈X : Σ_{i∈I} (x _{i}－e _{i})＝0}. We denote byX̂ _{i} the projection ofX _{i} ontoA _{X} and letX̂ : Π_{i∈I}X̂ _{i}. To consider a sequence of truncated economies, we take an increasing sequence {(K _{n},M _{n})} of compact convex cube pairs with center 0 such thatK _{n}⊂R^{ℓ} withX̂ _{i}⊂int (K_{1}), andM _{n}⊂R^{J} with 0∈int (M _{1}) which satisfy ∪_{n}K _{n}＝R^{ℓ} and ∪_{n}M _{n}＝R^{J}. For eachn and _{i∈I}, we defineX _{i}^{n} :＝X _{i}⌒K _{n}, Θ_{i}^{n}(q ) :＝Θ_{i}(q )⌒M _{n},X _{n} :＝Π_{i∈I}X _{i}^{n}, and Θ^{n}(q ) :＝Π_{i∈I} Θ_{i}^{n}(q ). Moreover, the preference correspondenceP_{i}^{n} :X _{i}^{n} → 2X _{i}^{n} is defined byP _{i}^{n} (x _{i}) :＝P _{i}(x _{i})⌒X _{i}^{n}.We denote by
the truncated economy <(E ^{n}X _{i}^{n},P _{i}^{n},e _{i}, Θ_{i}^{n})_{i∈I}>. We observe that eachX_{i}^{n} is compact and each Θ_{i}^{n} is lower hemicontinuous with nonempty compact convex values and has a closed graph. Moreover, eachP _{i}^{n} inherits the properties ofP _{i}. We define function γ : Δ → R^{S＋1} by γ (p ,q )＝(γ_{s}(p ,q ))_{s∈S0} with _{0}＝S ◡{0}, whereS Let Ψ_{i}^{n}＝Mn, ∀
i ∈ , and Ψ^{n}＝Π_{i∈I}, Ψ^{i}. For everyI i ∈ and everyI n , we define correspondencesB _{i}^{n}: Δ → 2X _{i}^{n}×Ψ_{i}^{n} andB̅ _{i}^{n}: Δ → as follows: : Under Assumptions (A1)(A7), for eachProposition 5.1n , there is a profile (p ^{n},q ^{n},x^{n} θ ^{n})∈Δ×X ^{n}×Θ^{n}(q ^{n}) such thatwhere
z ^{n}(s ):= Σ_{i∈I}(x _{i}^{n}(s )e _{i}(s )) for everys ∈ _{0}.S : See Appendix. ■Proof : Suppose that Assumption (A6) holds. Let {(Lemma 5.1q ^{n},θ _{i}^{n})} be a sequence in R^{J} × R^{J} withq ^{n} →q ^{*} andθ _{i}^{n}∈Θ_{i} (q ^{n}). Suppose that {a _{n}} be a sequence in R_{＋}, such thata _{n} → 0. If sequence {a _{n}θ _{i}^{n} } converges tov _{i}, thenv _{i}∈C >_{i}(q ^{*}). : Apply 3.2 Lemma on p. 396 of Page (1987) to Θ_{i}. ■Proof From Proposition 5.1, we obtain an equilibrium existence theorem for economy
.E : Under Assumptions (A1)(A8), economyTheorem 5.1 has a competitive equilibrium.E : Take a sequence {(Proof p ^{n},q ^{n},x^{n} θ ^{n})} of profiles obtained in Propostion 5.1. Since eachX _{i} is closed and bounded from below,X̂ _{i} is compact and so isX̂ . Noting that {(p ^{n},q ^{n},x^{n} )}∈Δ×X̂ , without loss of generality, we may assume that {(p ^{n},q ^{n},x^{n} )} converges to (p ^{*},q ^{*},x ^{*})}∈Δ×X̂ . : Σ_{i∈I}(Claim 1x _{i}^{*}－e _{i})＝0 and Σ_{i∈I}θ _{i}^{*}＝0, where (x _{i}^{*},θ _{i}^{*})∈X _{i} × Θ_{i}(q ^{*}) for eachi ∈I . : From (d) of Proposition 5.1, it is immediate that Σ_{i∈I} (Proof x _{i}^{*}－e _{i})＝0. To show Σ_{i∈I}θi*＝0, we claim that sequences {θ _{i}^{n}} for each _{i∈I} are bounded. Suppose otherwise. For eachn , we seta_{n} ＝(1＋Σ_{i∈I} ‖θ _{i}^{n}‖)^{－1}, which converges to 0. We see thatan θ _{i}^{n}∈Θ_{i}(q ^{n}) and sequence {a_{n} θ _{i}^{n}} for each _{i∈I} are bounded. Thus, without loss of generality, it converges tov _{i} for each _{i∈I}. Since Σ_{i∈I}a _{n}θ _{i}^{n}＝0 for alln , it holds that Σ_{i∈I}v _{i}＝0 and Σ_{i∈I}‖v _{i}‖＝1, which implies thatv _{i}≠0 for some _{i∈I}.Using Lemma 5.1, we see that
v _{i}∈C _{i}(q ^{*}). On the other hand,p ^{n}(s )⋅(x _{i}^{n}(s )－e _{i}(s ))≤r (s )⋅θ _{i}^{n}＋γ_{s} (p ^{n},q ^{n} ) for alln ands ∈ . By multiplying both sides of the inequalities byS a_{n} and passing to the limit, we obtainR ⋅v _{i}≥0. In view of Σ_{i∈I}v _{i}＝0, we obtain R⋅v _{i}＝0, that is,v _{i}∈V^{⊥}. This implies thatv _{i}∈C _{i}(q ^{*})⌒V^{⊥}. Since Σ_{i∈I}v _{i}＝0, by Assumption (A8), we obtainv _{i}＝0 for alli ∈I , which leads to a contradiction.Therefore, {
θ _{i}^{n}} is bounded for eachi ∈I . Without loss of generality, we may assume that {θ _{i}^{n}} converges toθ _{i}^{*}. From Assumption (A6) and (d) of Proposition 5.1, it follows thatθ _{i}^{*}∈Θ_{i}(q ^{*}) and Σ_{i∈I}θ _{i}^{*}＝0. □ : γ (Claim 2p ^{*},q ^{*}) = 0. : This immediately follows from (e) of Proposition 5.1. □Proof : (Claim 3x _{i}^{*},θ _{i}^{*})∈B̅ _{i}(p ^{*},q ^{*}) : This directly follows from (a) of Proposition 5.1 and Claims 1 and 2. □Proof :Claim 4p ^{*}(0) ≠ 0.Proof : Ifp ^{*}(0)＝0, agenti hasx _{i}∈X _{i}such thatx _{i}≻_{i}x _{i}^{*} and (x _{i},θ _{i}^{*})∈B̅ _{i}(p ^{*},q ^{*}) in view of Assumption (A4) and Claim 3. Sincep ^{*}(s )≠0, ∀s ∈ due to Claim 2, by Assumption (A5), there isS x _{i}°∈int (X _{i}) such thatp ^{*}(s )⋅x _{i}°(s )≪p ^{*}(s )⋅e _{i}(s ), ∀s ∈ . Since ‖S q ^{*}‖＝1 by Claim 2, Assumption (A7) ensures that there existsθ _{i}°∈int (Θ_{i}(q ^{*})) such thatq ^{*}⋅θ _{i}°＜0. Then, fort ∈(0, 1) sufficiently close to 1, we see thattx _{i}＋(1－t )x _{i}°≻_{i}x _{i}^{*} andp ^{*}□ [tx _{i}＋(1－t )x _{i}°－e _{i}]≪W (q ^{*})⋅[t θ _{i}^{*}＋(1－t )θ _{i}°] witht θ _{i}^{*}＋(1－t )θ _{i}°∈Θ_{i}(q ^{*}). Since Θ_{i} is lower hemicontinuous, there exists a sequence {θ̂ _{i}^{n}} converging tot (θ _{i}^{*}＋(1－t )θ _{i}° such thatθ̂ _{i}^{n}∈Θ_{i}(q ^{n}), ∀n . Therefore, for sufficiently largen , we havetx _{i}＋(1－t )x _{i}°≻_{i}x _{i}^{n} andp ^{n}□ [tx _{i}＋(1－t )x _{i}°－e _{i}] ≪W (q ^{n})⋅θ̂ _{i}^{n} withtx _{i}＋(1－t )x _{i}°∈X _{i}^{n} andθ̂ _{i}^{n}∈Θ_{i}^{n} (q ^{n}). This is a contradiction in view of of Proposition (b) and (e) of Proposition 5.1. Hence, it follows thatp ^{*}(0)≠0. □Claim 5 :q ^{*}∈Q ^{*}. : First, we show thatProof q ^{*}∈Q . Suppose otherwise. Then there is some agenti who has a portfolioθ _{i} ∈C _{i}(q ^{*}) satisfyingW (q ^{*})⋅θ _{i}＞0. Assumption (A4) ensures that there exists δ∈R^{ℓ} such thatx _{i}^{*}＋δ≻_{i}x _{i}^{*} andp ^{*}□δ＜W (q ^{*})⋅θ _{i}. Claim 3 implies thatp ^{*}□(x _{i}^{*}＋δ－e _{i})＜W (q ^{*})⋅(θ _{i}*＋θ _{i}) withθ _{i}*＋θ _{i}∈Θ_{i}(q ^{*}). Note that Claims 2 and 4 imply thatp * (s )≠0, ∀s ∈ _{0}. Assumption (A5) allows us to takeS x _{i}°∈int (Xi), such thatp ^{*}□x _{i}°≪p ^{*}□e _{i}. Therefore, fort ∈(0, 1) sufficiently close to 1, we obtaint (x _{i}^{*}＋δ)＋(1－t )x _{i}°≻_{i}x _{i}^{*} andp ^{*}□[t (x _{i}^{*}＋δ )＋(1－t )x _{i}°－e _{i}]≪W (q ^{*})⋅[t (θ _{i}*＋θ _{i})]. Sincet (θ _{i}*＋θ _{i})∈Θ_{i}(q ^{*}) and Θ_{i} is lower hemicontinuous, there exists a sequence {θ̂ _{i}^{n}}converging tot (θ _{i}*＋θ _{i}) withθ̂ _{i}^{n}∈Θ_{i}(q ^{n}). For sufficiently largen ,with
t (x _{i}^{n}＋δ)＋(1－t )x _{i}°∈X _{i}^{n} andθ̂ _{i}^{n}∈Θ_{i}^{n}(q ^{n}). This is a contradiction in view of (b) and (e) of Proposition 5.1. Hence,q ^{*}∈Q .We now show that
q ∈N _{0}(q ^{*})^{⊥}, that is,q ^{*}⋅v ＝0 for allv ∈N _{0}(q ^{*}). Suppose otherwise. Then we have somev ∈N _{0}(q ^{*}) such thatq ^{*}⋅v ＜0 without loss of generality. Since there existsv _{i}∈N _{i}(q ^{*})⌒V ^{⊥}, ∀i ∈ such thatI v ∈Σi ∈I v _{i}, it follow thatq ^{*}⋅v _{i}＜0 for somei . Noting thatv _{i}∈C _{i}(q ^{*}) andR ⋅v _{i}＝0, we know thatv _{i} is a constrained arbitrage opportunity atq ^{*}. Applying the same arguments presented in the previous paragraph, we arrive at a contradiction. Hence,q ^{*}∈N _{0}(q ^{*})^{⊥} and thereforeq ^{*}∈Q⌒N _{0}(q ^{*})^{⊥}, that is,q ^{*}∈Q ^{*}. □Let us now define the open budget set of agent
i by : [Claim 6P _{i}(x _{i}^{*})×Θ_{i}(q ^{*})]∩B _{i}(p ^{*},q ^{*}) = ∅. : Suppose that the claim does not hold. Then there is someProof i ∈ with (I x̂ _{i},θ̂ _{i})∈ [P _{i}(x _{i}^{*})×Θ_{i}(q ^{*})]⌒B _{i}(p ^{*},q ^{*}). Noting thatP _{i}^{－1} is openvalued by Assumption (A3), we see thatP _{i}^{－1} is lower hemicontinuous. SinceP _{i} and Θ_{i} are lower hemicontinuous andB _{i} has an open graph, the correspondence (p ,q ,x _{i}) ↦ [P _{i}(x_{i} )×Θ_{i}(q )]⌒B _{i}(p ,q ) is lower hemicontinuous. Therefore there is a sequence {(x̂ _{i}^{n} ,θ̂ _{i} ^{n})} converging to (x̂ _{i},θ̂ _{i}) such that (x̂ _{i}^{n},θ̂ _{i}^{n} )∈[P _{i}(x _{i}^{n})×Θ_{i}(q ^{n})]⌒B _{i}(p ^{n},q ^{n}). For eacht ∈(0, 1) and eachn ∈N, we sety _{i}^{n}(t ) :＝(t x̂ _{i}^{n}＋(1－t )x _{i}^{n},t θ̂ _{i}^{n} ＋(1－t )θ _{i}^{n}). Observe that, for sufficiently largen , we obtainy _{i}^{n}(t )∈x _{i}^{n} × Θ_{i}^{n}(q ^{n}) and thusy _{i}^{n}(t )∈B _{i}^{n} (p ^{n},q ^{n}) by aim 2. Therefore, for sufficiently largen , we havey _{i}^{n}(t )∈[P _{i}^{n} (x _{i}^{n})×Θ_{i}^{n}(q ^{n})]⌒B _{i}^{n} (p ^{n},q ^{n}), which contradicts (b) of Proposition 5.1. Hence, [P _{i}(x _{i}^{*})×Θ_{i}(q ^{*})]⌒B _{i}(p ^{*},q ^{*})＝∅. □ : For everyClaim 7i ∈ , [I P _{i}(x _{i}^{*})×Θ_{i}(q ^{*})]∩B̅ _{i}(p ^{*},q ^{*}) = ∅. : Suppose that the claim does not hold. Then there is someProof i ∈ with (I x _{i},θ _{i})∈[P _{i}(x _{i}^{*})×Θ_{i}(q ^{*})]⌒B̅ _{i}(p ^{*},q ^{*}). Since Claims 2 and 4 imply thatp ^{*}(s )≠0, ∀s ∈S _{0}, by Assumption (A5), we can take (x _{i}’,θ _{i}’)∈B _{i}(p ^{*},q ^{*})≠∅. Assumption (A3) implies that for t∈(0, 1) sufficiently close to 1,t (x _{i},θ _{i})＋(1－t )(x _{i}’,θ _{i}’)∈[P _{i}(x _{i}^{*})×Θ_{i}(q ^{*})]⌒B _{i}(p ^{*},q ^{*}), which contradicts Claim 6. Hence, [P _{i}(x _{i}^{*})×Θ_{i}(q ^{*})]⌒B̅ _{i}(p ^{*},q ^{*})＝∅. □By Claims 1, 3, and 7, we prove that (
p ^{*},q ^{*},x ^{*},θ ^{*}) is a competitive equilibrium for economy . ■E : We consider an exchange economy withExample 5.1I ＝2,L ＝1,J ＝3, andS ＝3. The utility functions and initial endowments of agents are provided as follows:Let
X _{t}＝R^{4}_{＋}, ∀i ＝1, 2 and consider the commodity as a numéraire. Payoff matrix is given byThis allows us to restrict noarbitrage asset prices to R^{3}_{＋＋}. Note that
V ^{⊥}＝{v ∈R^{3}:v ＝λ (1, 1, －1),λ ∈R}. Portfolio constraints for agents are described by:The recession cones of these constraints are:
Define R_{＋} :＝{
θ ∈R^{3}: R⋅θ ＞0}. SinceC _{i}(q )⌒R_{＋}＝R^{3}＋ for alli andq ∈R^{3}_{＋＋}, we find thatQ ＝R^{3}_{＋＋}, which is a nonempty open convex cone. We denote a competitive equilibrium of the economy by (q ^{*}, (x _{1}^{*},x _{2}^{*}), (θ _{1}^{*},θ _{3}^{*}))∈R^{2}＋×(R^{4})^{2} × (R^{3})^{2}. Then it follows thatSince
C _{i}(q ^{*})⌒V ^{⊥}＝{0} for alli ＝1, 2, we see that Assumption (A8) is trivially holds. Note that the law of one price does not hold and that thefirst inequality constraint of agent 1 is binding at the equilibrium. □13‖⋅‖ is the Euclidean norm.
VI. Concluding Remarks
It is shown that there exists a competitive equilibrium in a twoperiod exchange economy with incomplete markets where redundant assets are present and portfolio constraints are represented by a lower hemicontinuous correspondence of asset prices. Most of general equilibrium models, which study incomplete markets with endogenous portfolio constraints, either express portfolio constraints in terms of differentiable restriction functions that describe the boundary of constraints, or
de facto exclude redundant assets. The present paper not only models endogenous portfolio constraints via correspondences of asset prices, but also considers the risksharing role of redundant assets in incomplete markets. Assumption (A8) plays a key role of excluding the unboundedness of scalefree zeroincome portfolios, which arises due to redundant assets. Future possible directions of research include weakening Assumption (A8) for more general results and extending the results of this paper to economies with multiperiod incomplete markets.

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