When financial markets are unconstrained, redundant (financial) assets do not play a role in risk-sharing 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 short-selling constraints, credit limits, bid-ask 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 Cea-Echenique and Torres-Martinez (2014), among others. Carosi et al. (2009) describe portfolio constraints by restriction functions, which depend on first-period 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.
Cea-Echenique and Torres-Martinez (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 income-based portfolio constraints. In particular, attainable allocations are price-dependent. However, they impose a restrictive assumption that the set of price-dependent 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 is 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.
The paper considers an exchange economy with financial asset markets, extending over two periods. There are I agents and J financial assets. The uncertainty of the second period is described by a finite set S :＝{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 are L 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 consumption x_{i}(0) and invests portfolio θ_{i} with his endowments. In the second period, agent i makes consumptions (x_{i}(s))_{s∈S} with his endowments and payoffs of his portfolio. Hence, agent i chooses consumption bundle x_{i}:＝(x_{i}(0), x_{i}(1), ..., x_{i}(S)) in his consumption set X_{i}⊂R^{ℓ}, which contains his initial endowment e_{i} of commodities. Preferences over X_{i} are represented by a preference relation ≻i on X_{i}, which is irreflexive, complete, and transitive. The preference relation ≻i defines the preference correspondence P_{i}:X_{i} →. by P_{i}(x_{i}) :＝{ ∈X_{i} :≻_{i}x_{i}}, which is the set of consumption bundles that agent i prefers to x_{i}. Agent i is subject to portfolio constraints, as represented by correspondence Θ_{i} : R^{J} → 2^{RJ} of asset price q∈R^{J}. To finance his consumption in the second period, agent i chooses portfolio θ_{i}∈Θ_{i}(q) in the first period.
The payoff of asset j in state s∈S is denoted by r_{j}(s), and the payoff vector of asset j over S states by an S dimensional column vector r_{j}＝(r_{j}(s))_{s∈S}. Payoff vector in state s is denoted by a J dimensional row vector r(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 risk-sharing role without portfolio constraints because their payoffs can be replicated by those of the other assets. In contrast, redundant assets participate in risk-sharing 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 constraint p(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 constraint p(s)⋅x_{i}(s)≤p(s)⋅e_{i}(s)＋r(s)⋅θ_{i}, ∀s∈S, where p(s)∈R^{L} is a vector of commodity prices at state s∈S. Therefore, given price vector (p, q)∈R^{ℓ}× R^{J}, agent i maximizes his preference ≻i by choosing a pair (x_{i}, θ_{i}) of consumption and portfolio in his budget set:1
where
A pair (x_{i}, θ_{i})∈B̅_{i}(p, q) is optimal for agent i if [P_{i}(x_{i})×Θ_{i}(q)]⌒B̅_{i}(p, q)＝∅.
Definition 2.1: A competitive equilibrium of economy E is a profile (p^{*}, q^{*}, x^{*}, θ^{*})∈R^{ℓ}×R^{J}×(R^{ℓ})×(R^{J})^{I}, such that
We 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 convex-valued 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 short-selling constraints, bid-ask 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 non-empty 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).
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).9
Definition 3.1: Asset price q∈R^{J} is said to admit a constrained arbitrage for agent i if there is a portfolio θ_{i}∈C_{i}(q), such that W(q)⋅θ_{i}＞0. Asset price q∈R^{J} is said to admit no constrained arbitrage for economy E if it admits no constrained arbitrage for every agent 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 E. Let N_{i}(q) be the lineality space of Θ_{i}(q).10 We define N_{0}(q)＝Σ_{i∈I}(N_{i}(q)⌒V^{⊥}) and denote by N_{0}(q)^{⊥} its orthogonal complement in R^{J}. Let us define Q^{*}:＝{q∈Q: q∈N_{0}(q)^{⊥}}. The following results show what is appropriate for equilibrium asset prices.
Proposition 3.1: It holds that Q and Q^{*} are non-empty.
Proof : To show Q≠∅, suppose otherwise, that is, Q＝∅. Consider q＝λ ⋅R with λ ∈. Since Q＝∅, we see that q∉Q. Then there is some agent i with θ_{i}∈C_{i}(q) satisfying W(q)⋅θ_{i}＞0, which makes q⋅θ_{i}＝λ⋅R⋅θ_{i}≥0 necessary. If q⋅θ_{i}＞0, then q∈Q, then q∈Q, which is a contradiction. If q⋅θ_{i}＝λ ⋅R⋅θ_{i}＝0, then R⋅θ_{i}＝0. This implies that q∈Q, which is a contradiction. Hence, Q is non-empty.
To show Q^{*}≠∅, suppose otherwise, that is, Q^{*}＝∅. Take any q∈Q, and we have q∉N_{0}(q)^{⊥}. Then there exists v∈N_{0}(q) such that q⋅v＜0 without loss of generality. Since there exists v_{i}∈N_{i}(q)⌒V^{⊥}, ∀i∈I such that v＝Σ_{i∈I}v_{i}, it follow that q⋅v_{i}＜0 for some i. Noting that v_{i}∈C_{i}(q) and R⋅vi＝0, we see that v_{i} is a constrained arbitrage opportunity at q. Therefore, q∉Q_{i} and q∉Q, which is a contradiction. Hence, Q^{*} is nonempty.
Proposition 3.2: Under Assumption (A4), an equilibrium asset price q belongs to Q^{*}.
Proof: Let (p, q, x, θ) be an equilibrium of E. Suppose that q∈Q. Then there is some i∈I with v_{i}∈C_{i}(q) satisfying W(q)⋅v_{i}＞0. This implies that θ_{i}＋v_{i}∈Θ_{i}(q) and W(q)⋅θ_{i}W(q)⋅(θ_{i}＋v_{i}). Due to Assumption (A4), there exists a consumption bundle x_{i}’∈X_{i}, such that x_{i}’≻_{i}x_{i} and (x_{i}’, θ_{i}＋v_{i})∈B̅_{i}(p, q), which contradicts the optimality of (x_{i}, θ_{i} ) in B̅_{i}(p>, q). Hence, q∈Q.
We now show that q∈N_{0}(q)^{⊥}, that is, q⋅v＝0 for all v∈N_{0}(q). Suppose otherwise. Then there exists v∈N_{0}(q) such that q⋅v<0 without loss of generality. Since there exists v_{i}∈N_{i}(q)⌒V^{⊥}, ∀i∈I such that v＝Σ_{i∈I} v_{i}, it follow that q⋅v_{i}＜0 for some i. Noting that v_{i}∈C_{i}(q) and R⋅v_{i}＝0, we see that v_{i} is a constrained arbitrage opportunity at q. That is, q∉Q_{i}, and therefore q∉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 that Q and Q^{*} appear as cones with vertex zero under Assumption (A6). We observe that Q^{*} may not be convex. Therefore we consider Q̂ which is the convex hull of Q^{*}. Then Q̂ 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 risk-sharing in constrained asset markets, we need to examine feasible zero-income portfolios. We call portfolios in C_{i}(q)⌒V^{⊥} scale-free feasible zero-income portfolios for agent i in that, if v_{i}∈C_{i}(q)⌒V^{⊥}, we have λ v_{i}∈Θ_{i}(q) and R⋅(λv_{i})＝0 for every λ≥0. Particularly, if q∈cl(Q̂)\{0}, some agent i may have a portfolio v_{i}∈C_{i}(q)⌒V^{⊥} satisfying q⋅v_{i}≤0. Therefore, in the presence of scale-free feasible zero-income portfolios, some agent’s portfolio choices may be unbounded with his budget constraint satisfied. To prevent such negative effect of scale-free feasible zero-income 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 agent i has a scale-free feasible zero-income portfolio v_{i}∈C_{i}(q)⌒V^{⊥}, which is supported by the other agents because －v_{i}∈Σ_{j≠i}C_{j}(q)⌒V^{⊥}. Therefore, agent i can hold an indefinite amount of portfolios in the direction of v_{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 non-empty 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)].
Financial intermediaries prohibit short-selling 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} and b_{i}: R^{J}→R^{J} take the values of the short-selling limits or buying limits of agent i on securities, respectively. The portfolio constraints of agent i can therefore be described by
where 0∈(a_{i}(q), b_{i}(q)), ∀q∈R^{J}, a_{i}: R^{J}→R^{J}, and b_{i}: R^{J}→R^{J} are continuous functions and homogeneous of degree zero in q.
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 that
where a_{i}: R^{J}→R_{＋＋} is a continuous function and homogeneous of degree one in q and b_{i}: R^{J}→ is a continuous function and homogenous of degree zero in q.
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 risk-less 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_{＋}, and b_{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 risk-less bond, a_{i}≥2, and b_{i}∈R_{＋＋}. For example, assume that J＝2 and b_{i}＝0.12 Suppose that security 1 is risk-less bond and security 2 is a stock. Now suppose that agent i shorts one stock and maintains a margin account with mi proportion of the stock price in the bond. The portfolio constraint is therefore reduced to
which implies that m_{i}≥a_{i}/(a_{i}－1). In the case where a_{i}＝3, we have m_{i}≥3/2, that is, agent i 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 risk-less bond with q_{i}＝1, we modify his example to present the portfolio constraint set of agent i at q∈cl(Q)\{0} by
where a_{i}≥1, b_{ij}≥0, c_{ij}≥0, d_{i}＞0, and δ_{ij}＞0 are constants for every i and j 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 that
we have 0∈lnt(Θ_{i}(q)), ∀q∈cl(Q)\{0}. To see that Θ_{i} is convex-valued, we define continuous function : R^{J} × R^{J}→R by
The 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 of θ_{i}. Hence, we see that is a concave function function of θ_{i} and therefore Θ_{i} is convex-valued.
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 large n,
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.
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}, where
We observe that Δ is compact and convex.
̂Let X :＝Π_{i∈I} X_{i} and A_{X} :＝{(x_{1}, ..., x _{I})∈X: Σ_{i∈I} (x_{i}－e _{i})＝0}. We denote by X̂_{i} the projection of X_{i} onto A_{X} and let X̂: 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 that K_{n}⊂R^{ℓ} with X̂_{i}⊂int(K_{1}), and M_{n}⊂R^{J} with 0∈int(M_{1}) which satisfy ∪_{n}K_{n}＝R^{ℓ} and ∪_{n}M_{n}＝R^{J}. For each n and _{i∈I}, we define X_{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 correspondence P_{i}^{n} : X_{i}^{n} → 2X_{i}^{n} is defined by P_{i}^{n} (x_{i}) :＝P_{i}(x_{i})⌒X_{i}^{n}.
We denote by E^{n} the truncated economy <(X_{i}^{n}, P_{i}^{n}, e_{i}, Θ_{i}^{n})_{i∈I}>. We observe that each X_{i}^{n} is compact and each Θ_{i}^{n} is lower hemicontinuous with non-empty compact convex values and has a closed graph. Moreover, each P_{i}^{n} inherits the properties of P_{i}. We define function γ : Δ → R^{S＋1} by γ (p, q)＝(γ_{s}(p, q))_{s∈S0} with S_{0}＝S◡{0}, where
Let Ψ_{i}^{n}＝Mn, ∀ i∈I, and Ψ^{n}＝Π_{i∈I}, Ψ^{i}. For every i∈I and every n, we define correspondences B _{i}^{n}: Δ → 2X_{i}^{n}×Ψ_{i}^{n} and B̅_{i}^{n}: Δ → as follows:
Proposition 5.1: Under Assumptions (A1)-(A7), for each n, there is a profile (p^{n}, q^{n}, x^{n}θ^{n})∈Δ×X^{n}×Θ^{n}(q^{n}) such that
where z^{n}(s):= Σ_{i∈I}(x_{i}^{n}(s)-e_{i}(s)) for every s∈S_{0}.
Proof: See Appendix. ■
Lemma 5.1: Suppose that Assumption (A6) holds. Let {(q^{n}, θ_{i}^{n})} be a sequence in R^{J} × R^{J} with q^{n} → q^{*} and θ_{i}^{n}∈Θ_{i} (q^{n}). Suppose that {a_{n}} be a sequence in R_{＋}, such that a_{n} → 0. If sequence {a_{n} θ_{i}^{n} } converges to v_{i}, then v_{i}∈C>_{i}(q^{*}).
Proof: Apply 3.2 Lemma on p. 396 of Page (1987) to Θ_{i}. ■
From Proposition 5.1, we obtain an equilibrium existence theorem for economy E.
Theorem 5.1: Under Assumptions (A1)-(A8), economy E has a competitive equilibrium.
Proof: Take a sequence {(p^{n}, q^{n}, x^{n}θ^{n})} of profiles obtained in Propostion 5.1. Since each X_{i} is closed and bounded from below, X̂_{i} is compact and so is X̂. 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.
Claim 1: Σ_{i∈I}(x_{i}^{*}－e_{i})＝0 and Σ_{i∈I}θ_{i}^{*}＝0, where (x_{i}^{*}, θ_{i}^{*})∈X _{i} × Θ_{i}(q^{*}) for each i∈I.
Proof: From (d) of Proposition 5.1, it is immediate that Σ_{i∈I} (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 each n, we set a_{n}＝(1＋Σ_{i∈I} ‖θ_{i}^{n}‖)^{－1}, which converges to 0. We see that anθ_{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 to v_{i} for each _{i∈I}. Since Σ_{i∈I} a_{n}θ_{i}^{n}＝0 for all n, it holds that Σ_{i∈I} v_{i}＝0 and Σ_{i∈I}‖v_{i}‖＝1, which implies that v_{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 all n and s∈S. By multiplying both sides of the inequalities by a_{n} and passing to the limit, we obtain R⋅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 that v_{i}∈C_{i}(q^{*})⌒V^{⊥}. Since Σ_{i∈I} v_{i}＝0, by Assumption (A8), we obtain v_{i}＝0 for all i∈I, which leads to a contradiction.
Therefore, {θ_{i}^{n}} is bounded for each i∈I. Without loss of generality, we may assume that {θ_{i}^{n}} converges to θ_{i}^{*}. From Assumption (A6) and (d) of Proposition 5.1, it follows that θ_{i}^{*}∈Θ_{i}(q^{*}) and Σ_{i∈I}θ_{i}^{*}＝0. □
Claim 2: γ (p^{*}, q^{*}) = 0.
Proof: This immediately follows from (e) of Proposition 5.1. □
Claim 3: (x_{i}^{*}, θ_{i}^{*})∈B̅_{i}(p^{*}, q^{*})
Proof: This directly follows from (a) of Proposition 5.1 and Claims 1 and 2. □
Claim 4: p^{*}(0) ≠ 0.
Proof: If p^{*}(0)＝0, agent i has x_{i}∈X_{i}such that x_{i}≻_{i}x_{i}^{*} and (x_{i}, θ_{i}^{*})∈B̅_{i}(p^{*}, q^{*}) in view of Assumption (A4) and Claim 3. Since p^{*}(s)≠0, ∀s∈S due to Claim 2, by Assumption (A5), there is x_{i}°∈int(X_{i}) such that p^{*}(s)⋅x_{i}°(s)≪p^{*}(s)⋅e_{i}(s), ∀s∈S. Since ‖q^{*}‖＝1 by Claim 2, Assumption (A7) ensures that there exists θ_{i}°∈int(Θ_{i}(q^{*})) such that q^{*}⋅θ_{i}°＜0. Then, for t∈(0, 1) sufficiently close to 1, we see that tx_{i}＋(1－t) x_{i}°≻_{i}x_{i}^{*} and p^{*}□ [tx_{i}＋(1－t)x_{i}°－e_{i}]≪W(q^{*})⋅[tθ_{i}^{*}＋(1－t) θ_{i}°] with tθ_{i}^{*}＋(1－t) θ_{i}°∈Θ_{i}(q^{*}). Since Θ_{i} is lower hemicontinuous, there exists a sequence {θ̂_{i}^{n}} converging to t(θ_{i}^{*}＋(1－t) θ_{i}° such that θ̂_{i}^{n}∈Θ_{i}(q^{n}), ∀n. Therefore, for sufficiently large n, we have tx_{i}＋(1－t) x_{i}°≻_{i}x_{i}^{n} and p^{n}□ [tx_{i}＋(1－t)x_{i}°－e_{i}] ≪W(q^{n})⋅θ̂_{i}^{n} with tx_{i}＋(1－t)x_{i}°∈X_{i}^{n} and θ̂_{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 that p^{*}(0)≠0. □
Claim 5: q^{*}∈Q^{*}.
Proof: First, we show that q^{*}∈Q. Suppose otherwise. Then there is some agent i who has a portfolio θ_{i} ∈C_{i}(q^{*}) satisfying W(q^{*})⋅θ_{i}＞0. Assumption (A4) ensures that there exists δ∈R^{ℓ} such that x_{i}^{*}＋δ≻_{i}x_{i}^{*} and p^{*}□δ＜W(q^{*})⋅θ_{i}. Claim 3 implies that p^{*}□(x_{i}^{*}＋δ－e_{i})＜W(q^{*})⋅( θ_{i}*＋θ_{i}) with θ_{i}*＋θ_{i}∈Θ_{i}(q^{*}). Note that Claims 2 and 4 imply that p*(s)≠0, ∀s∈S_{0}. Assumption (A5) allows us to take x_{i}°∈int(Xi), such that p^{*}□ x_{i}°≪p^{*}□e_{i}. Therefore, for t∈(0, 1) sufficiently close to 1, we obtain t(x_{i}^{*}＋δ)＋(1－t)x_{i}°≻_{i}x_{i}^{*} and p^{*}□[t(x_{i}^{*}＋δ )＋(1－t)x_{i}°－e_{i}]≪W(q^{*})⋅[t(θ_{i}*＋θ_{i})]. Since t(θ_{i}*＋θ_{i})∈Θ_{i}(q^{*}) and Θ_{i} is lower hemicontinuous, there exists a sequence {θ̂_{i}^{n}}converging to t(θ_{i}*＋θ_{i}) with θ̂_{i}^{n}∈Θ_{i}(q^{n}). For sufficiently large n,
with t(x_{i}^{n}＋δ)＋(1－t)x_{i}°∈X_{i}^{n} and θ̂_{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 all v∈N_{0}(q^{*}). Suppose otherwise. Then we have some v∈N_{0}(q^{*}) such that q^{*}⋅v＜0 without loss of generality. Since there exists v_{i}∈N_{i}(q^{*})⌒V^{⊥}, ∀i∈I such that v∈Σi∈Iv_{i}, it follow that q^{*}⋅v_{i}＜0 for some i. Noting that v_{i}∈C_{i}(q^{*}) and R⋅v_{i}＝0, we know that v_{i} is a constrained arbitrage opportunity at q^{*}. Applying the same arguments presented in the previous paragraph, we arrive at a contradiction. Hence, q^{*}∈N_{0}(q^{*})^{⊥} and therefore q^{*}∈Q⌒N_{0}(q^{*})^{⊥}, that is, q^{*}∈Q^{*}. □
Let us now define the open budget set of agent i by
Claim 6: [P_{i}(x_{i}^{*})×Θ_{i}(q^{*})]∩B_{i}(p^{*}, q^{*}) = ∅.
Proof: Suppose that the claim does not hold. Then there is some i∈I with (x̂_{i}, θ̂_{i})∈ [P_{i}(x_{i}^{*})×Θ_{i}(q^{*})]⌒B_{i}(p^{*}, q^{*}). Noting that P_{i}^{－1} is open-valued by Assumption (A3), we see that P_{i}^{－1} is lower hemicontinuous. Since P_{i} and Θ_{i} are lower hemicontinuous and B_{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 each t∈(0, 1) and each n∈N, we set y_{i}^{n}(t) :＝(tx̂_{i}^{n}＋(1－t)x_{i}^{n}, tθ̂_{i}^{n} ＋(1－t) θ_{i}^{n}). Observe that, for sufficiently large n, we obtain y_{i}^{n}(t)∈x_{i}^{n} × Θ_{i}^{n}(q^{n}) and thus y_{i}^{n}(t)∈B_{i}^{n}(p^{n}, q^{n}) by aim 2. Therefore, for sufficiently large n, we have y_{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^{*})＝∅. □
Claim 7: For every i∈I, [P_{i}(x_{i}^{*})×Θ_{i}(q^{*})]∩B̅_{i}(p^{*}, q^{*}) = ∅.
Proof: Suppose that the claim does not hold. Then there is some i∈I with (x _{i}, θ_{i})∈[P_{i}(x_{i}^{*})×Θ_{i}(q^{*})]⌒B̅_{i}(p^{*}, q^{*}). Since Claims 2 and 4 imply that p^{*}(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. ■
Example 5.1: We consider an exchange economy with I＝2, L＝1, J＝3, and S＝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 by
This allows us to restrict no-arbitrage 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}. Since C_{i}(q)⌒R_{＋}＝R^{3}＋ for all i and q∈R^{3}_{＋＋}, we find that Q＝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 that
Since C_{i}(q^{*})⌒V^{⊥}＝{0} for all i＝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.
It is shown that there exists a competitive equilibrium in a two-period 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 risk-sharing role of redundant assets in incomplete markets. Assumption (A8) plays a key role of excluding the unboundedness of scale-free zero-income 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.