## Abstract

We use club theory for the first time to provide a model of securities exchange (SX) formation. We think of a SX as a local public good that allows its traders to diversify risk by trading their securities with other SX members. In our two-stage equilibrium setting, traders evaluate SXs depending on their risk-sharing possibilities and, given these evaluations, choose the SX they want to join. Security prices can differ among SXs and traders may value SX memberships differently. We establish continuity properties in both stages and show that equilibrium exists for a generic set of economies.

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## Notes

For notational convenience, we use boldface letters to denote a set, except when we refer to a subset of traders, or exchange, here denoted by

*S*.As stated in the Hong Kong Exchange (HKEx) rules, any trader intending to operate a brokerage business for products available on HKEx, using the trading facilities of the stock exchange and/or futures exchange, must be admitted and registered as an exchange participant of that exchange, and pay the corresponding membership fee. The acquisition of an exchange membership usually involves a commitment for trading in the exchange for a long period of time—usually, the exchange participant stays in the exchange since it enters. See, for example, the MICEX list of participants: http://www.micex.com/markets/stock/members/list.

Alternatively, we could assign to each exchange

*S*a subset of*assets*(in the broad sense) \(\mathbf {J}(S)\subseteq \mathbf {J}\) that traders in exchange*S*agree to issue for later trading. We leave this possibility for future research.For simplicity, we do not model trading fees, which have become substantially less important since implementation of the Mifid regulation (see Colliard and Foucault 2012 for a survey of this literature and for an analysis of the effect of trading fees on the efficiency of the markets); nor do we consider transaction fees (e.g., stamp duty). We focus instead on the pure trading aspects of security markets.

See the HKEx security trading infrastructure at http://www.hkex.com.hk/eng/market/sec_tradinfra/CMTradInfra.htm.

Allouch and Wooders (2008) demonstrate that, given communication costs, for all sufficiently large economies, the core is non-empty and the set of price-taking equilibrium outcomes is equivalent to the core.

In a more sophisticated model with additional benefits of exchange membership (other than risk diversification, e.g., information acquisition or increasing gains from a larger exchange), we should modify the domain of the trader’s utility function as follows: \(u^{i}(x_{0},F[i;\mathbf {I} ],x_{1},x(1),\ldots , x(\Xi )).\) This extension is left for future research.

An example is a set of parameterized Cobb–Douglas utility functions, where the set of parameters is defined in a finite dimensional space.

Observe that AW (2008, pp. 271–272) only requires assumption (f) to be satisfied for a consumption \(x_{0}^{i}\) bounded above by the aggregate endowments plus some \(\varepsilon >0\).

Anecdotal evidence shows that this was a common belief until 2006. See

*The Economist*, March 25, 2006 (http://www.economist.com/node/6978712): “*Liquidity and technology will inevitably make trading a natural monopoly*”.Cole and Prescott (1997), Ellickson et al. (2001), Luque (2013), and Wooders (1980,1997) require bounded club sizes. In contrast, Wooders (1989) and Allouch and Wooders (2008) permit both unbounded club sizes and ever-increasing gains from larger clubs. The main difference between the two cases is the extent to which approximation or coalition formation costs are required to obtain existence of equilibrium. Konishi et al. (1998) allow for an arbitrary number of jurisdictions, but take as given the size of the population, rather than looking at “large” economies. See Luque (2014) for a review of the different approaches to the presence of equilibrium in local public good economies.

In Faias and Luque (2015a), we provide an equilibrium existence proof for a general equilibrium economy with

*exogenous*exchange structures, cross-listings, and multiple memberships. It remains an open question as to how to modify club models to allow for an agent’s utility function to depend on the whole structure of clubs.Notice that the equilibrium correspondence is defined in the finite set of exchange structures; therefore, a continuous measurable selector is not needed. Continuous selectors are in general used to construct continuous objective functions. Thus, they fit only if the correspondence is defined in a continuum set.

Here,

*q*stands for the security price vector in trader*i*’s exchange.A set is a continuously differentiable function if all of its elements are continuously differentiable functions.

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We thank comments and suggestions from Nizar Allouch, Antonio Cabrales, Jean-Eduard Colliard, Gionvani Nicolo, Mario Pascoa, Erwan Quintin, Mark Ready, Tim Riddiough, Marzena Rostek, Myrna Wooders, Randy Wright, and participants at the annual meetings of the APET 2010, EWGE 2011, SAET 2011, SEA 2011, and PEA 2012, and seminars at Indiana University, University of Miami, Purdue University, and University of Wisconsin–Madison. Luque gratefully acknowledges the Spanish Ministry of Education and Science for financial support in 2010 under Grant SEJ2008-03516. Faias gratefully acknowledges that this work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through UID/MAT/00297/2013 (CMA).

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Faias, M., Luque, J. Endogenous formation of security exchanges.
*Econ Theory* **64**, 331–355 (2017). https://doi.org/10.1007/s00199-016-0989-9

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DOI: https://doi.org/10.1007/s00199-016-0989-9

### Keywords

- Endogenous securities exchange structure
- Security prices
- Risk sharing
- Membership prices
- Equilibrium
- Club theory