تجزیه و تحلیل تعادل عمومی داوری استراتژیک

We analyze a general equilibrium framework with Cournot arbitrageurs and with price-taking investors who are subjected to restricted participation constraints. Restricted participation may leave some arbitrage opportunities unexploited by investors. We show existence of Cournot–Walras equilibria with an endogenous number of arbitrageurs. The number of arbitrageurs is endogenous since they have to sink entry costs in order to arbitrage across the relevant markets. We characterize equilibria and analyze the effects on equilibrium prices and quantities of increased competition among arbitrageurs due to lower entry costs.

In this paper, we set out to describe a simple modeling framework for financial problems and to study the existence of equilibria as well as some equilibrium comparative statics. The model was motivated by the need to model the interaction between a large number of small investors and depositors on one hand, and an endogenous number of large strategic financial players, called arbitrageurs or financial intermediaries, on the other hand. As a stylized fact, individual investors have access to restricted investment opportunities compared to the universe of investments that the large financial institutions (e.g. investment banks, mutual funds, hedge funds, etc.) can trade and invest in. This advantage allows them to profit from the inefficiencies in the market by arbitraging across the various market places, or exchanges, which explains why we refer to them as arbitrageurs. Arbitrageurs are not endowed with any capital in order to guarantee that the unique source of their profits is the exploitation of arbitrage opportunities across exchanges. An exchange (or market place) is formally defined as a market where a certain subset of assets is traded. There is one auctioneer per exchange. This asymmetry of investment opportunities is captured in this framework by resorting to two assumptions. First, we restrict the participation of investors to one exchange, while the arbitrageurs can, upon sinking the required fixed investment cost, trade across all market places. This assumption is strong, but has proved useful in financial economics, and some of its applications are reviewed below where we discuss the related literature. Because investors on different exchanges cannot trade directly between themselves, they will trade indirectly via the arbitrageurs who thereby cash in the difference in the marginal willingnesses to pay in form of an assured arbitrage profit. This compensation is reminiscent of bid-ask spreads, so arbitrageurs can be viewed as intermediaries. The second assumption concerns the game form underlying the model: while investors are small and price-taking, arbitrageurs are large and play a Cournot game among themselves, taking the price impacts of their decisions into account. The equilibrium concept is therefore the Cournot–Walras equilibrium (CWE), introduced by Gabszewicz and Vial (1972) and extended and elaborated on in Hart, 1979 and Novshek and Sonnenschein, 1978 and Roberts (1980), and surveyed by Mas-Colell (1982). Here, we describe a general economy, with an arbitrary but finite number of exchanges, each one of which is inhabited by an arbitrary, but finite, number of price-taking investors that are restricted to trade on their own exchange only. The number of strategic and non-price-taking arbitrageurs à la Cournot, who can trade across all market places provided they have sunk the required fixed costs, is endogenous because of free (but costly) entry. This framework covers many restricted participation models encountered in the financial economics literature. The main raison d’être of this paper is therefore to set up a sufficiently general framework that encompasses and generalizes many useful models (some of which we shall review below), but that still admits equilibria and permits fairly explicit and qualitatively unambiguous comparative statics results that can be used to address explicit financial problems. This unfortunately requires some strong assumptions, both in view of known problems with the existence of equilibria, and due to the fact that we intend to stay within a class that allows transparent comparative statics properties. The most restrictive assumption we impose is downward sloping demand, which guarantees that the Walrasian correspondence is single-valued and, importantly, continuous. We therefore do not aim at the greatest level of generality. It is well-known that the existence of Cournot–Walras equilibria is a delicate issue, for instance as pointed out by the counterexamples in Dierker and Grodal (1986) (due to the multi-valuedness of the Walrasian correspondence and the non-existence of a continuous selection) or in Roberts and Sonnenschein (1977) (due to the non-quasi-concavity of the optimization problem). We show that equilibria exist, at least if demand is downward sloping and if entry costs are low enough so that competition is intense enough. The flavor of this result is reminiscent of the conditions for existence of equilibria in Novshek and Sonnenschein (1978) and Roberts (1980), as well as in the literature that these papers generated. There the high level of competition required to show existence is guaranteed by relying on various versions of the Debreu and Scarf replicating technique consisting of expanding the demand side relative to each firm’s production possibilities (see Debreu and Scarf, 1963). In our paper, no such replications are necessary, and the required level of competition is brought about by entry when entry costs are sufficiently small. However, while in the papers just mentioned existence of Cournot–Walras equilibria is shown indirectly when the Cournot–Walras equilibria are sufficiently close to the limiting Walrasian equilibrium, in which case they inherit existence from the Walrasian equilibrium, we show existence of equilibria directly. In order to show existence of pure strategy equilibria directly, we strengthen the assumptions by requiring that demand is downward sloping everywhere, while Novshek and Sonnenschein (1978) only require demand to be downward sloping at the limiting Walrasian equilibrium. This is sufficient in their case since they restrict each player’s strategy set by reducing the minimum efficient scale to an arbitrarily small number, while we do not exogenously restrict the players’ strategy sets. In the restricted participation framework of this paper, it turns out that the limiting equilibria may not be Walrasian equilibria (even with restricted participation), as shown in Zigrand (2001a), and we therefore do need a direct existence result. While Zigrand (2001a) does not study existence of equilibria per se, it does show that equilibria may not converge to Walrasian equilibria. The intuition for why limiting equilibria may not be Walrasian and why arbitrage profits may not vanish in the limit is as follows. As competition becomes more intense, each arbitrageur scales down his operations. In order to converge to a Walrasian equilibrium, aggregate arbitrage positions must grow so as to eliminate the arbitrages. There are situations, however, where each arbitrageur scales his positions down so as to leave the aggregate arbitrage positions (which includes the supplies of the new entrants) constant. The reason is that if they were to scale their own positions down less than that (as required if equilibria were to converge to Walrasian equilibria), the resulting higher equilibrium aggregate supplies would drive prices outside of the set of arbitrage prices. This discontinuity acts as a restraining device that prevents equilibria from converging to competitive equilibria. For a more complete characterization, the reader may refer to Zigrand (2001a). Like the papers by Codognato and Gabszewicz, 1991, Gabszewicz and Vial, 1972, Novshek and Sonnenschein, 1978, Roberts and Sonnenschein, 1977 and Roberts, 1980 and Yosha (1997), our paper therefore also touches, in conjunction with Zigrand (2001a), on the literature on the strategic foundations of competitive general equilibrium. For the sake of completeness, we mention that the convergence to competitive equilibria has also been studied in alternative frameworks, refer for instance to the strategic market games of Sahi and Yao, 1989 and Shapley and Shubik, 1977 or Codognato and Ghosal (2000) where all agents are strategic, or to the bargaining approach inspired from Gale (1987). 1.1. Related literature Very few general equilibrium models of economies with uncertainty and with imperfectly competitive financial intermediaries exist. Yosha (1997) and Bisin (1998) are notable exceptions. Yosha (1997) replicates a Cournot–Walras economy à la Debreu-Scarf in which individuals can invest in one risky project (all projects are independent and identically distributed) via one Cournot intermediary. The aim of his study is to analyze the trade-off between on one hand the beneficial effect of intermediaries’ size upon their ability to diversify and on the other hand the welfare costs arising from market power, and how this tension persists as the economy is replicated and the equilibrium converges to the competitive equilibrium without uncertainty. Yosha also retains the assumption of a downward sloping demand (of the unique asset) for the same reasons we do in this paper, namely to guarantee that the Walrasian correspondence is in fact single-valued. Bisin (1998) on the other hand studies an abstract model of security design. He is interested primarily in showing that, with nominal securities, the real indeterminacy of equilibrium allocations pointed out in the incomplete markets literature vanishes, because if assets are not exogenously given but optimally chosen, then intermediaries would choose to index asset payoffs. In his model, a fixed and exogenous number of strategic intermediaries choose payoff matrices and vectors of bid-ask spreads, exactly matching investors’ sell orders with other investors’ buy orders. This is in contrast to our model where intermediaries choose the quantities to trade across any number of markets with exogenous payoff matrices, thereby determining in which assets they make markets. Our model can therefore be interpreted as a model of security design as well, with the difference that costs in our setup are fixed, while they are variable in Bisin. Furthermore, in our model arbitrageurs may also choose to strategically hold portfolios whose payoffs do not sum to zero, and thereby influence prices to their advantage and enjoy utility from later consumption. Rather than assume downward sloping demand, Bisin circumvents the existence problems by opting to rely on the existence theorem by Simon and Zame (1990) in conjunction with a weaker equilibrium concept which only requires that there be some (rational) beliefs of intermediaries about equilibrium prices in the multivalued Walrasian correspondence which, when held, guarantee the non-emptiness of the set of (possibly mixed-strategy) Nash equilibria. In a slightly different spirit, Townsend (1983) derived many deep results about competitive, cooperative and non-cooperative intermediated general equilibrium structures with a complete set of contingent commodities. Yanelle (1996) analyzes two-sided competition among Bertrand intermediaries in an economy without uncertainty. Existence and regularity of equilibria with restricted participation were studied for instance by Balasko et al. (1990) and further by Polemarchakis and Siconolfi (1997). In financial economics the setup was mainly used to drive wedges between the marginal valuations of different market participants. Restricted participation has a long history in finance. In international finance, it is commonly assumed that markets are segmented, at least with respect to a subset of commodities or assets which are posited or argued to be non-tradable across countries, see for instance, Adler and Dumas (1983). This has also been found a necessary ingredient to understand the observed home-bias in investors’ asset allocations, as in Stockman and Dellas, 1989 and Tesar, 1993 or Serrat (2001). In fixed income theory, the market segmentation hypothesis of Culbertson (1957) and the preferred habitat theory of Modigliani and Sutch (1966) aim to provide explanations of the behavior of the term structure of interest rates. Here it is asserted that individuals have strong maturity preferences and that the different maturity markets are segmented. In the theory of futures markets, some authors have rationalized the behavior of basis risk in futures contracts (normal contango versus normal backwardation) by segmenting the market three ways, see for instance Keynes (1930). There is a clientèle which has only access to long futures positions, a clientèle which has only access to short futures positions, and speculators which can have any positions in futures, but have no access to the markets for the underlying assets. A similar segmentation is also the basis of many banking and financial intermediation papers in which the world is exogenously segmented into borrowers and lenders which cannot trade among themselves, so that transactions among them must pass through an intermediary. In the literature on security design, the wedges introduced by the restricted participation (or some related concept) is the basis for innovation since it provides innovators with a source of profit, as in Allen and Gale, 1988, Chen, 1995 and Hara, 1995 or Pesendorfer (1995). Alternatively, the segmentation is required to show existence of separating equilibria in an innovation game with adverse selection, as in Santos and Scheinkman (2001). Restricted participation has also been advanced as one of the possible explanations of various asset pricing puzzles, as in Allen and Gale, 1994, Allen and Gale, 1998 and Basak and Cuoco, 1998, or to generate instability as in Allen and Gale (2000), where the “domino effect” of contagion occurs only if not all institutions are linked by cross institutional stakes. Most of the models just mentioned assume that all agents are likewise restricted, which we believe is at best an approximative statement given the lure of profits to large institutions that are left on the table. Holden (1995) for instance describes a simple model of stock index arbitraging by strategic arbitrageurs. Similarly, the logic underlying the behavior of the basis in futures markets as explained by Keynes does not take arbitrage into account, i.e. it does not consider that the speculators have access to the underlying markets, as we allow in our model. If they had access, futures prices would be priced as in our model with reference to an endogenous no-arbitrage price. The futures price, the prices of the underlying assets and the distance of the futures price from the no-arbitrage price are determined endogenously and depend upon the competition among the speculators and the depth of the various markets. This argument also applies to the term structure theories. If the term structure was priced according to the preferred habitat theory, it is more than likely that fixed income instruments would violate the no-arbitrage conditions. Again, arbitrageurs would trade along the yield curve and mitigate the preferred habitat features, possibly in interesting directions. This explicit modeling of how prices are determined locally (such as in dealer markets, or across exchanges as in the case of S&P index futures on the CME versus the underlying shares on the NYSE) and how arbitrage opportunities are affected by the arbitrage activity yields a more realistic picture than assuming that all asset prices are determined simultaneously by one centralized auctioneer. While restricted participation provides a motivation for innovation, we believe that a fuller picture may emerge when innovation is embedded into a model with arbitrageurs. For instance, assume that a put option on a stock is introduced on a derivatives exchange to overcome a short-selling constraint faced by retail investors. If the resulting put price differs substantially from the price of the replicating portfolio and the market is deep enough, arbitrageurs will mitigate the gains from innovation. Interestingly, as shown in this model, the arbitraging of the derivative has in turn repercussions on the prices of the assets in the replicating portfolio. As spelled out in Zigrand (2002), those repercussions can be surprising. And conversely, we can view arbitrageurs as the market makers of the innovated asset, or even as the innovators, see for example Rahi and Zigrand (2002) for a model of security design based upon the framework developed in the present paper. That model allows to address questions such as how the welfare maximizing innovation depends on the behavior of arbitrageurs, and thereby on the depth of the exchanges as well as on entry costs, and whether arbitrageurs would design the same securities as the social planner. 1.2. Structure of the paper In Section 2, we introduce the setup of the economy, we provide some preliminary definitions and we discuss the decision problem faced by the price-taking investors. In Section 3, we turn to the arbitrageur’s optimization problem. The main results on Cournot–Walras equilibria are studied in 4 and 5 concludes. Proofs are relegated to Appendix A.

In this paper, we presented a simple two-date general equilibrium model where only a subset of sophisticated traders have the opportunity to simultaneously trade all of the existing assets. This allows us to capture realistic aspects of financial markets, as for instance studied in Zigrand (2001a). In such economies arbitrage opportunities arise that only global and sophisticated traders can exploit. In standard models, the intertemporal marginal rate of substitution of any price-taking investor defines prices at equilibrium and thereby insures no-arbitrage across all assets because investors are thought to continually submit demand functions for all assets. This assumption is clearly unrealistic. In reality investors only infrequently submit asset demand functions, and then typically for small subsets of assets or mutual funds. The resulting degree of mispricing implicit in the pricing of the many assets across many exchanges is the outcome of a Cournot–Walras game played among the endogenous number of non-competitive arbitrageurs. In this paper we analyze the decision problem of arbitrageurs and provide a Slutsky-type decomposition of the individual reaction functions. The most important repercussion from increased competition is the market price impact of the additional trades as well as the effect this may have on the depth of the markets. Each arbitrageur is therefore led to scale back his own trades in an optimal way. We show existence of Cournot–Walras equilibria by direct means, as opposed to inducing existence from the neighbourhood of a Walrasian equilibrium. This is necessary for we show in Zigrand (2001a) that in the framework of this paper, limiting equilibria may not be Walrasian. We also feel that modeling technological progress (lower c) as the driver for competition is possibly more realistic for our purposes than a replication of the pool of investors, and that constant returns to scale in trading (at given prices) may also be more reasonable than decreasing returns to scale, at least up to a point. Finally, by relying on some ideas from non-smooth analysis, we are able to characterize the equilibrium comparative statics results by relying on extensions of the implicit-function theorem to such environments (as opposed to being able to provide limiting results only), in particular as to an exogenous change in the entry costs. We can predict to what extent lower entry costs attract more entrants, how this affects equilibrium arbitrage trades and asset prices across all exchanges. We also study the limiting equilibrium prices as entry costs go to zero.