This paper shows that, in markets with transaction costs, even if a redundant security does not even save individual investors' total costs for their security trading, the prices of the other securities may well be different were it to not be available for trade, resulting in a different equilibrium consumption allocation. In this sense, a redundant security may give rise to the divergence of individual and social relevance in markets with transaction costs. We then show that this divergence may also be a robust phenomenon with respect to perturbations in utility functions, initial endowments, and transaction costs.
In this paper, we investigate the risk sharing role of a redundant security in markets with transaction costs. We use a fairly simple model of security markets with transaction costs. We assume: an exchange economy under uncertainty, with a single consumption good and two periods; securities in zero net supply; transaction costs incurred in the first period and proportional to transaction volumes, with no fixed costs; and symmetric information.
The first risk sharing role of a security is the spanning role: a security provides consumers with the risk-hedging opportunities that they could not enjoy by using any portfolio of the other securities. A security without this role is called redundant.
A second role emerges in the presence of transaction costs. It is referred to as the cost saving role: even when the same risk-hedging opportunity is provided by some portfolio of the other securities, such a portfolio may be more costly for consumers than the redundant security. Note that this discrepancy in prices need not result in arbitrage opportunities in the presence of transaction costs.
The first purpose of this paper is to show by means of an example that even if a redundant security does not have the cost saving role for individual consumers, the prices of the other securities may well be different were it to be unavailable for trade, resulting in a different equilibrium consumption allocation. In short, the redundant security is individually irrelevant but socially relevant. The possibility of this pecuniary externality by a redundant security on the others should not surprise us once we understood the role of transaction costs. Nevertheless, it is a significant effect, one that should not be forgotten, and it is worthwhile to present a work-out example in a simple model. An immediate, technical implication of this result is that there is no straightforward extension of the no-arbitrage equilibrium presented in Magill and Shafer (1991), which is an analytically simple tool to study frictionless incomplete security markets, to markets with frictions, because whether a budget set can arise at equilibrium depends now on which securities span it in the presence of transaction costs. The pecuniary externality should thus not be ignored in the abstract price functional approach to derivative asset pricing with transaction costs taken by Bensaid et al. (1992) and Jouini and Kallal (1995a); Jouini and Kallal (1995b); Jouini and Kallal (1995c); we shall elaborate on this point in the next subsection.
The second purpose of this paper is to show that this divergence of individual and social relevance (DISR) is a robust phenomenon. More specifically, starting from a security market equilibrium at which a redundant security gives rise to the divergence, even if the consumers' utility functions and initial endowments and the proportional transaction costs of the securities are perturbed, there is a new equilibrium near the original equilibrium at which the same redundant security gives rise to the divergence.
This robustness result is not an immediate consequence of the standard regularity analysis in general equilibrium theory initiated by Debreu (1970). To see this, note that when the divergence is present, some consumers' must have multiple security demands, so that the aggregate excess demand correspondence must necessarily be multi-valued at the equilibrium. This is because consumers can enjoy the same consumption with and without the redundant security. On the other hand, Debreu (1970) defined the regularity of an equilibrium as the invertibility of the Jacobian of the excess demand function at the equilibrium, which implies via the implicit function theorem the persistence of the equilibrium relative to perturbations in the utility functions and the initial endowments. As we do not have an excess demand function, let alone a differentiable one, establishing the robustness is not a straightforward application of these existing results. The key idea to establish the robustness is to use the Negishi mapping, which, roughly speaking, measures how much each consumer is better or worse off than the indirect utility level evaluated by the supporting price vector of each efficient security allocation. This makes considerable sense because the use of the Negishi mapping allows us to bypass the problems arising from the multi-valuedness of demand functions.
We provided the concept of the DISR and presented an example of an economy at whose unique equilibrium the DISR is present. We also proved that, under some conditions, the DISR is robust with respect to perturbations in utility functions, initial endowments, and transaction costs.
An interesting direction of future research is to extend the present analysis to a multi-period setting with incomplete security structures. In such a setting, the first welfare theorem no longer holds, and thus we cannot use our notion of regularity of an equilibrium, which we used for the robustness theorem and turned out to be nothing but the regularity of the Negishi mapping. A new line of proof will be necessary to carry out this extension. It will also be interesting to explore a more general notion of regular equilibria in security markets with transaction costs without using the direct sum condition, for which the genericity can be obtained.
