Bernard Cornet

When financial investors’ portfolio holdings are unconstrained, financial economies are assumed, w.l.o.g., to have no redundant assets. Indeed, eliminating redundant assets allows to replace the initial financial structure by an equivalent one, i.e., one that has the same consumption equilibria. Moreover, at the end of the process, absence of redundant assets guarantees that the set of admissible portfolio allocations is bounded, a fundamental property for the existence of equilibria. In the presence of institutional (exogenous) portfolio constraints, eliminating redundant assets is not innocuous anymore since bounded arbitrage may persist at equilibrium, the law of one price does not hold, and some zero-income portfolios may not be free. The goal of the paper is to replace the elimination of redundant assets by the elimination of useless portfolios, a process that eliminates in particular Werner useless portfolios, but needs to go beyond to obtain the boundedness of the set of admissible portfolio allocations at the end of the purification process. Moreover, the elimination process is carried out without affecting the set of consumption equilibria, hence replacing at each step the financial structure by an equivalent one.

The boundedness of the set of admissible allocations is a basic property in economic models that proved to be of fundamental use to show the existence of equilibria (Debreu 1959; Hurwicz and Reiter, Int. Econ. Rev. 14(3), 580–586, 1973). In the study of financial markets without portfolio constraints, this boundedness property is standardly derived from the absence of redundant assets, an assumption that can be made without loss of generality since redundant assets can be eliminated at no cost. However, there are no a priori grounds to do so when agents do face portfolio constraints, and the elimination of redundant assets should be replaced by the elimination of useless portfolios as shown by Aouani and Cornet (2014). The purpose of this paper is thus to characterize financial structures that are useless-free (i.e., without useless portfolios) when agents face portfolio constraints, and show that the absence of useless portfolios adequately extends the absence of redundant assets in the constrained case. Our main result will then show the equivalence between the absence of useless portfolios, a dual non-redundancy property, and the boundedness property of different sets of admissible portfolio allocations.

We study the convexity property of the set Q[subscript F] of arbitrage-free prices of a multi-period financial structure F. The set of arbitrage-free prices is shown to be a convex cone under conditions on the financial structure F that hold in particular for short-lived assets. Furthermore, we provide examples of equivalent financial structures F and F’ such that Q[subscript F] is a convex cone, but Q[subscript F’] is neither convex nor a cone.

We consider the two-date model of a financial exchange economy (E, F), with agents’ portfolionext term restrictions either represented by finitely many linear inequality constraints or satisfying Hart (1974)’s Weak No Market Arbitrage condition. The economy (E, F) is shown to have the same consumption equilibria as a reduced economy (E, T’), for which the set of admissible previous termportfolio allocations is bounded. Building upon the equilibrium existence result for reduced financial economies (E, !F’), (Aouani and Cornet, 2009), we then deduce the existence of equilibra of (E, F), under standard assumptions on the consumption side and under the aforementioned assumption on the financial side.

We consider a multiperiod financial exchange economy with nominal assets and restricted participation, where each agent’s portfolio choice is restricted to a closed, convex set containing zero, as in Siconolfi (Non-linear Dynamics in Economics and Social Sciences, 1989). Using an approach that dates back to Cass (CARESS Working Paper, 1984; J Math Econ 42:384-405, 2006) in the unconstrained case, we seek to isolate arbitrage-free asset prices that are also quasi-equilibrium or equilibrium asset prices. In the presence of such portfolio restrictions, we need to confine our attention to aggregate arbitrage-free asset prices, i.e., for which there is no arbitrage in the space of marketed portfolios. Our main result states that such asset prices are quasi-equilibrium prices under standard assumptions and then deduces that they are equilibrium prices under a suitable condition on the accessibility of payoffs by agents, i.e., every payoff that is attainable in the aggregate can be marketed through some agent’s portfolio set. This latter result extends previous work by Martins-da-Rocha and Triki (Working Paper, University of Paris 1, 2005).

We consider a two-date model of a financial exchange economy with finitely many agents having nonordered preferences and portfolio constraints. There is a market for physical commodities at any state today or tomorrow and financial transfers across time and across states are allowed by means of finitely many nominal assets or numéraire assets. We prove a general existence result of equilibria for such a financial exchange economy in which portfolios are defined by linear constraints, extending the framework of linear equality constraints by Balasko et al. (1990), and the existence results in the unconstrained case by Cass (1984, 2006), Werner (1985), Duffie (1987), and Geanakoplos and Polemarchakis (1986). Our main result is a consequence of an auxiliary result, also of interest for itself, in which agents’ portfolio constraints are defined by general closed convex sets and the financial structure is assumed to satisfy a “nonredundancy-type” assumption, weaker than the ones in Radner (1972) and Siconolfi (1989).

In a financial economy with asymmetric information and incomplete markets, we study how agents, having no model of how equilibrium prices are determined, may still refine their information by eliminating sequentially “arbitrage state(s)”, namely, the state(s) which would grant the agent an arbitrage, if realizable.