Walrasian foundations for equilibria in segmented markets

While market fragmentation has accelerated in recent years, it has of course always been an important feature of the economic landscape. Allais [1] argued for a more realistic “economy of markets” in lieu of a “market economy”. In his Nobel speech he says: “\(\ldots \) I was led to discard the Walrasian general model of the market economy, characterized at any time, whether there be equilibrium or not, by a single price system, the same for all the operators,—a completely unrealistic hypothesis,—and to establish the theory of economic evolution and general equilibrium, of maximum efficiency, and of the foundations of economic calculus, on entirely new bases resting on \(\ldots \) a new model, the model of the economy of markets (in the plural)”.

We do not restrict state prices to be nonnegative, since we will have occasion to consider economies with no arbitrageurs later in the paper. In such economies there may be unexploited arbitrage opportunities, and hence negative state prices, in equilibrium. See Example 5.1.

Notice that each of the \(S\) terms that are summed up in the inner product must be less than or equal to zero, due to \(\mathbf{C1}\) and \(\mathbf{C2}\). Hence all of these terms must in fact be zero.

By uniqueness we mean that the equilibrium allocation and pricing functional on each exchange are unique. There may, of course, be multiple state-price deflators that induce the same equilibrium pricing functional.

Formally, \(p^A_s\) is the Lagrange multiplier associated with the arbitrageur’s no-default constraint in state \(s\).

In view of (16), Eq. (18) holds if and only if each of the \(S\) terms that are summed up in the inner product is zero.

Exactly the same proof goes through if we assume from the start that \(P^1=I\). Thus \(\bar{p}^{RP}\) is the (unique) WERP state-price deflator of the economy in which asset payoffs are the same as in the original economy except that markets are complete on exchange 1.

If markets are complete, \(p^{CM}\) is the unique solution to (22). If markets are incomplete, so that the common span \(\langle R\rangle \) is a strict subset of \(\mathbb R^S\), \(p^{CM}\) is still a solution, but it is not the only one. All solutions are of course equivalent to \(p^{CM}\).