Introduction

Equilibrium in incomplete markets where agents are subject to restricted participation was studied in seminal articles by [1], [2] and [3] or [4]. [1] guarantees equilibrium existence in real asset markets under exogenous short-sale constraints. [3] or [4] and [2] explore a more general framework in economies with nominal assets, considering restrictions given by closed and convex sets containing the zero vector.

Subsequent contributions to this literature have encompassed different types of constraints. Linear equality constraints were considered in [5] for nominal assets, and in [6] for real assets. Meanwhile, constraints given by quasi-concave inequalities are studied in [7]. Nevertheless, the most general approach remains to be the one in which restrictions are given by arbitrary closed and convex set containing the zero vector, as in [3] or [4], [2], [8], [9], [10] or [11].

Recently, the focus has been the study of restrictions that are affected by endogenous variables. [7] and [12] include consumption-price dependencies. Borrowing constraints depending on first-period consumption or wealth are considered, respectively, by [13] and [14]. A more general configuration is provided by [15], where the portfolio set depends on prices of commodities and assets.

In this paper, we analyze the existence of equilibria in incomplete financial markets when agents are subject to price-dependent trading constraints that affect the access to commodities and regulate financial trade. Two results of equilibrium existence are developed. First, we add investment constraints in [15], ensuring that a competitive equilibrium exists when individuals can fully hedge segmented assets payments by trading unsegmented contracts. Secondly, requiring weaker assumptions on preferences, we extend the model and the results of [13] allowing price-dependent trading constraints that affect the access to financial trade.

Model

We focus on a two-period economy with a finite set of agents and uncertainty about the realization of a state of nature at the second period. Let be the set of states of nature in the economy, where s = 0 denotes the unique state at the first period and S denotes the finite set of states that can be attained at the second period.

There is a finite number of commodities, which are perfectly divisible. Although previous results of the literature allow the transformation of commodities between periods, to simplify notation we focus on the case of perishable goods. However, our results can be easily extended to a framework with durable commodities. Commodity prices are denoted by , where is the set of commodities available for trade at each state of nature. Financial markets are composed by a finite set of contracts and each is characterized by price-dependent contingent promises (R_s,j(p))_s∈S which are continuous functions of prices and satisfy (R_s,j(p))_s∈S ≠ 0 for every p ≫ 0. Prices of financial contracts are denoted by . Let and be, respectively, the space of prices and the set of admissible consumption bundles and financial portfolios.

There is a finite set of consumers . Each agent i is characterized by a strictly increasing, strictly quasi-concave, and continuous utility function , physical endowments , and trading constraints determined by a correspondence . Hence, given , each agent i maximizes her utility function by choosing consumption and financial positions in her choice set

This model is compatible with extreme cases of financial market segmentation, excluding agents of the financial trade of some assets, i.e.,

Also, we can allow credit market exclusion, as some agents may not have access to liquidity through financial contracts, i.e.,

Definition 1. A competitive equilibrium for the economy is a vector of prices, allocations, and portfolios such that:

1. (i). Individual optimality: for each .
2. (ii). Market feasibility:

Under traditional assumptions on preferences and endowments, the difficulties that may appear to ensure equilibrium existence are associated with the effect of trading constraints on asset prices. Indeed, the restricted access to financial trade may increase the scarcity of instruments to transfer resources between periods. This situation may compromise the existence of upper bounds for asset prices and, therefore, our economy cannot be truncated to ensure equilibrium existence using standard fixed-point techniques.

These difficulties do not appear in the classical models without credit market segmentation. Indeed, when agents can always short-sale any financial contract, upper bounds on asset prices can be directly obtained by normalization of prices at the first-period. In our context, this normalization may compromise the well-behavior of choice set correspondences, inducing discontinuities on individual decisions. Indeed, since we do not require financial survival, i.e., that all agents have access to some amount of liquidity through any financial contract, after normalization of first-period prices, choice set correspondences may have an empty interior.

We conclude this section with some basic assumptions ensuring that the well-behavior of choice sets is not affected by trading constraints. The first of these hypotheses introduces some regularity conditions on trading constraints. To shorten notations, given , let be the plan composed by just one unit of investment on asset j.

Assumption A

1. (i). For any , is lower hemicontinuous with convex values and closed graph.
2. (ii). For any and , (0, 0) ∈ Φⁱ(p, q) and .
3. (iii). Given and , there is an agent such that

Notice that, in contrast to [15], we may allow for restrictions on investment and, therefore, the property does not necessarily holds.

Restrictions on trading constraints are also imposed through assumptions over the correspondence of attainable allocations , defined as the set-valued mapping that associates prices with market feasible allocations satisfying individuals’ budget and trading constraints, i.e.,

Notice that, any element of Ω(p, q) satisfies budget constraints with equality.

Assumption B

For any compact set is bounded.

Boundedness conditions for the set of admissible consumptions and portfolios are already present in the literature. For instance, non-redundancy of asset payoffs induces this kind of requirements. Nevertheless, in a more general setting it should be interesting to consider some redundancy that arises when agents are restricted to participate in financial markets. Thus, Assumption B allows redundancy in assets markets as is already discussed in the Proposition and Example 4 by [15]. Our Assumption B is slightly stronger than the one required by [15] (Assumption A3) and it can be replaced only in our first result (Theorem 1) by a weaker one.

Examples of trading constraints

In this section we show that our general approach to incomplete markets with trading constraints is compatible with the existence of security exchanges or assets backed by financial collateral.

Existence of Equilibrium

When traditional fixed-point techniques are used to prove the existence of a competitive equilibrium, one of the main steps is to ensure that endogenous variables can be bounded without adding frictions on the model. Since Assumption B induces endogenous bounds on market feasible allocations, we only need to find upper bounds for asset prices.

With this objective in mind, some authors impose financial survival conditions, assuming that every agent has access to resources by short-selling any financial contract (see [8], [10], [17] and [11]). Thus, commodity and asset prices can be normalized without compromising the lower hemicontinuity of individuals’ choice sets.

Notwithstanding, as we include financial market segmentation, we need to follow an alternative approach to establish bounds for asset prices. For this reason, we identify the set of assets that always give access to liquidity: we refer to a financial contract j as unsegmented when for every there is δ > 0 such that . For notation convenience, let be the (possibly empty) maximal subset of composed of contracts that are unsegmented.

Since any agent can short-sale unsegmented contracts, their prices can be normalized without affecting the continuity of individual demands. For this reason, we focus on hypotheses that allow us to find bounds for segmented asset prices: the super-replication of its deliveries by unsegmented contract promises or the compensation of losses on future consumption through the increment of current demand.

Concluding Remarks

We provide a general framework for two-period economies with uncertainty where restricted access to markets is considered. We extend the literature by allowing a general form of constraints in consumption and portfolios that we called trading constraints in the spirit of [15]. In the one hand, we include price dependence in trading contraints generalizing configurations where restrictions are exogenous to the model. In the other hand, we include frictions to investment opportunities that were not present before in the context of endogenous trading constraints.

Our results combine two different alternatives to ensure equilibrium existence. The first one, requiring perfect hedging of the financial structure that is segmented. The second, by assuming a kind of impatience condition in agents’ preferences.

Some advantages of our approach are the generality of the configuration that includes several types of financial contracts, e.g. wealth-dependent access, borrowing constraints, commodity options, or financial collateral, and to dispense of differentiability assumptions over trading restrictions.

This general configuration is interesting, in particular, because if they are complemented with weaker conditions over the access to the market, then segmentation and/or exclusion of markets are present in the model.

As a matter of future research, there are different directions to take. A first one that study the relation between inter-temporal transfers and endogenous bounds for prices imposed by super-replication or impatience in preferences. A second one exploring time variations of the trading constraints, for instance, in order to consider credit contractions in the spirit of [20]. A third way may include structures justifying the formation of exchanges in terms of networks or group formation.

Appendix A: Proof of the Theorem 2

For each , let where

Given N > 0, define where

Notice that, Assumption C guarantees that is finite.

Let be the correspondence given by where

It follows from identical arguments to those given in [15] (Lemmata 1 and 2) that for each (M, N) ≫ 0 the correspondence Ψ_M,N has a non-empty set of fixed points. Therefore, our objective is to ensure that, for M and N large enough the fixed points of Ψ_M,N are competitive equilibria for our economy. Hence, we need to determine upper bounds for prices of segmented assets.

Lemma 1. Under the Assumptions of Theorem 2, let be a fixed point of Ψ_M,N satisfying Then, there is such that, for N large enough,

Proof. For any , let ρⁱ ∈ (0, 1) such that . Hence, property (i) of Theorem 2 imply that

Fix and satisfying part (ii) of Theorem 2. Then, there is zⁱ ≤ 0 such that and . Since Φⁱ has convex values and , it follows that . Also, the continuity of payoffs ensures that there is εⁱ ∈ (0, 1) such that,

In addition, it follows from Assumption A(ii) that, for each

Consequently, as is an optimal choice for agent i in and zⁱ ≤ 0, it follows that which implies that Since i = i(j) was fixed, we can consider

Lemma 2. Under the Assumptions of Theorem 2, fix Then, each fixed point of Ψ_M,N is a competitive equilibrium of our economy.

This result follows from identical arguments to those made in the proof of Lemma 4 in [15].

Appendix B: Proof of the Corollary

For each agent , let be the function defined by where is the commodity that satisfies the condition of the Corollary, , and . As customary, denotes the super-differential of at point x. Notice that, as , the monotonicity and concavity of ensure that is a non-empty subset of .

Consider the economy obtained by replacing with . Then, part (i) of Theorem 2 holds. Hence, Theorem 2 guarantees that there exists a competitive equilibrium for this auxiliary economy. To conclude the proof it is sufficient to ensure that satisfies .

Suppose, by contradiction, that there exists and such that . The market feasibility of consumption allocations ensures that, for every , . Therefore, and there exists λ ∈ (0, 1) such that .

Since , we conclude that,

This contradicts the optimality of for agent i in the auxiliary economy.

Acknowledgments

We are grateful to Antonela Racca for several suggestions and comments. The authors acknowledge financial support from Conicyt-Chile through Fondecyt project 1150207. Cea-Echenique is grateful to Conicyt-Chile for financial support given by the Ph.D. fellowship 21110938.