On the topology of the set of critical equilibria

• Published date: 01 June 2016
• Date: 01 June 2016
• Author: Andrea Loi,Stefano Matta
• DOI: http://doi.org/10.1111/ijet.12084

doi: 10.1111/ijet.12084

On the topology of the set of critical equilibria

Andrea Loi∗and Stefano Matta†

In this study, we explore the connection between the dimension of a pure exchange smooth

economy and the topology of the set of the critical equilibria. In particular, using the dual

framework developed by Balasko, we show that the set of regular equilibria is path connectedif

two consumers (goods) and at least four goods (consumers) exist, or if the sum of the number of

goods and consumers is even. This result depends mainly on the fact that a real projectivespace

of dimension n,RPn,whichisembeddedinRPn+1, does not disconnect RPn+1unless n=1.

Key wor ds catastrophe, equilibrium manifold, regular economy

JEL classification C65, D50, D51, D52

Accepted 31 March2015

1 Introduction

In a ground-breaking study, Debreu (1970) showed that in a finite pure exchange smooth setting,

outside a null closed subset of the space of economies, every economy has a finite set of equilibria

and the equilibrium price set varies continuously if the endowments vary within the set of regular

economies. This closed and measure-zero set of singular economies plays a crucial role in determin-

ing the discontinuities of prices and in explaining important global properties of the equilibrium

manifold, such as path connectedness, and thus manystudies have aimed to understand its geometric

structure, particularly the important studies by Balasko (1978, 1979, 1992).

In Balasko (1978), the set of singular economies was shown to be a geometrical envelope. More-

over, in the two-consumer case, the set of regular economies with a unique equilibrium is path

connected and if suitable initial endowments at infinity are added, the set of economies with multi-

ple equilibria is path connected.

A different approachusing intersecting manifolds and affine spaces embedded in Euclidean spaces

(Balasko 1979) yielded different insights, by demonstrating the stratification of the set of critical

equilibria and providing alternative proofs of the results on the path connectedness of economies

with multiple equilibria.

Balasko (1992) showed that the set of critical equilibria is a closed subset of measure zero of the

equilibrium manifold; hence, the set of regular equilibria is open and dense.

∗Department of Mathematics, University of Cagliari, Cagliari, Italy. Financial support provided by Prin 2010/11, Italy,

under the project Var i et `a reali e complesse: geomet ria, topologia e analisi armonica, is gratefully acknowledged.

†Department of Economics, University of Cagliari, Cagliari, Italy.Email: smatta@unica.it

We would like to thank Yves Balasko, Andy McLennan, an anonymous referee, and the participants at the General

Equilibrium Days Workshop,2014, York, for helpful discussions and critical comments.

International Journal of Economic Theory 12 (2016) 107–126 © IAET 107

International Journal of Economic Theory

Topology of the critical equilibria Andrea Loi and Stefano Matta

According to the latter result, the probability of encountering a critical equilibrium is null.

However, if one considers the evolutionof equilibr ia, whichis represented by a continuous path on

the equilibrium manifold, the probability of encountering a critical equilibrium changes dramatically

and it becomes a certainty if the endpoints of the path belong to two different path-connected

components of the equilibrium manifold disconnected by the set of critical equilibria. Indeed, this

dynamic perspective has stimulated some recent studies (Balasko 2012; Loi and Matta 2009, 2011)

which have focused on the issue of catastrophe minimization, that is, the problem of minimizing

the number of intersections of continuous equilibrium paths with the codimension- one stratum of

the set of critical equilibria, which is the only one that can disconnect the equilibrium manifold (the

other strata are not as relevant because a path generally does not intersect strata with codimension

greater than 1).

In Loi and Matta (2009), the length between two regular equilibria was defined as the number

of intersection points on the evolution path connecting them with the set of critical equilibria.

Moreover,the existence of a minimal path according to this definition of distance was demonstrated

and it was given an upper bound based on this pseudo-distance. In Balasko (2012), by using the

real-algebraic nature of the set of critical equilibria in any fiber of the equilibrium manifold, it was

shown that a minimal equilibrium path joining two arbitrary equilibria contains at most two critical

equilibria.

In Loi and Matta (2011), byextending the result reported by Loi and Matta (2008), we constructed

a Riemannian metric such that a geodesic is arbitrarily close to a minimal path.

In the present study, we consider the geometry of the embedding of the set of critical equilibria

into the equilibrium manifold. This analysis demonstrates the existence of a connection between

the dimension of the economy and the embedding of the set of critical equilibria in the equilibrium

manifold. Our study helps us to understand the roles playedby the numbers of goods and consumers,

and the conditions under which the set of critical equilibria does not disconnect the equilibrium

manifold (see Theorem 8).

We employ the dual approach developed by Balasko (1979, 1988), which studies prices and

incomes that are consistent with equilibria. This duality theory shows that the set of critical equilibria

possesses the desirable topological structure of a disjoint union of closed smooth manifolds, which

belong to the equilibrium manifold (see Theorem 7). In particular, the codimension of one stratum

of critical equilibria PS1, which is the only one that can disconnect the equilibrium manifold PE,

is diffeomorphic to the product between Grassmannian manifolds and a Euclidean space. If PS1

is embedded in PE, in order to understand whether PS1disconnects PE, it is sufficient to work

on their diffeomorphic copies. Unfortunately, we lack a proof of an embedding of PS1in PE.In

fact, the only published proof Balasko (1979, theorem 3) is not correct, which we highlight at the

beginning of Section 4. We havenot found any counterexample to the existence of this embedding,

so throughout this study, we assume that PS1is embedded in PE, which is a reasonableassumption

if we consider the previously reported results described above.

We establish a connection between the dimension of the economy and the path connectedness

of the set of regular equilibria. In particular, in an economy of lgoods and mconsumers, our main

Theorem 8 shows that PSidoes not disconnect PE if two consumers (goods) and at least four goods

(consumers) exist, or if the sum of the consumers and goods is even.

We provide an intuitive explanation of this result. When l+mis even, PS1becomesanon-

orientable codimension-one submanifold of PE, which is a compact orientable manifold, and (see

Proposition 3) PS1does not disconnect PE. When m=2orl=2, the problem reduces to studying

whether an n-projective space RPnembedded in RPn+1disconnects RPn+1.InTheorem6,we

demonstrate the non-trivial fact that this never occurs unless n=1.

108 International Journal of Economic Theory 12 (2016) 107–126 © IAET