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The Equilibrium ManifoldPostmodern Developments in the Theory of General Economic Equilibrium$

Yves Balasko

Print publication date: 2009

Print ISBN-13: 9780262026543

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262026543.001.0001

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The Equilibrium Manifold and the Natural Projection

The Equilibrium Manifold and the Natural Projection

Chapter:
(p.20) (p.21) 2 The Equilibrium Manifold and the Natural Projection
Source:
The Equilibrium Manifold
Author(s):

Yves Balasko

Publisher:
The MIT Press
DOI:10.7551/mitpress/9780262026543.003.0039

Abstract and Keywords

This chapter introduces the concept of the equilibrium manifold as a tool for the study of properties that might not be satisfied by all equilibria. The equilibrium manifold leads to the related concept of natural projection, a map from the equilibrium manifold into the space of economies. It then shows that the natural projection is smooth and proper, which suffices for this map to define a structure that is known in mathematics as a ramified covering. The results of Debreu (1970) then become straightforward consequences of this structure.

Keywords:   equilibrium manifold, natural projection, economies, ramified covering

2.1 Introduction

In this chapter I introduce the concept of the equilibrium manifold as a tool for the study of properties that might not be satisfied by all equilibria. The equilibrium manifold leads to the related concept of natural projection, a map from the equilibrium manifold into the space of economies. I then show that the natural projection is smooth and proper, which suffices for this map to define a structure that is known in mathematics as a ramified covering. The results of Debreu (1970) then become straightforward consequences of this structure.

2.2 Equilibrium Manifold

2.2.1 Definition

Now that we know that a given property is not necessarily satisfied by all economies ω ∈ Ω, the issue is to characterize the subset of Ω that consists of the economies for which that property is satisfied. But properties of economies ω ∈ Ω are of two kinds: properties of each equilibrium taken in isolation, and properties of the equilibrium set as a whole, in other words, of all the equilibria of an economy. The relative perspective leads us to focus in a first stage on the properties that are satisfied by the pairs (p, ω) ∈ S × Ω, where the price vector pS is an equilibrium price vector of the economy ω ∈ Ω, i.e., pW(ω).

Walras Correspondence

We recognize in this setup the Walras correspondence and, more particularly, its graph, the set of pairs (p, ω) ∈ S × Ω such that pW(ω). This new importance of the graph of the Walras correspondence justifies an adjustment of the terminology. By definition, the pair (p, ω) ∈ S × Ω (p.22) is an equilibrium if it satisfies the equation z(p, ω) = O. This is equivalent to the price vector pS being an equilibrium price vector associated with the economy ω ∈ Ω, i.e., pW(ω).

The change in terminology from the Walras correspondence to the equilibrium manifold is not purely semantic. with the equilibrium manifold, the emphasis shifts from comparative statics, unquestionably an important subject, to the wider subject of the properties of equilibria regardless of whether these properties come under the heading of comparative statics.

Equilibrium Manifold

The equilibrium manifold E is the subset of S × Ω defined by the equation z(p, ω) = O. At this stage, the equilibrium manifold E has no more structure than just being a subset of the Cartesian product S × Ω. The name equilibrium manifold will be justified by the fact that E is indeed a smooth submanifold of the Cartesian product S × Ω, a property that is insufficiently obvious without proof. The following property is far easier to prove.

Proposition 2.2.1

The equilibrium manifold E is a closed subset of S × Ω.

Proof The equilibrium manifold E is closed in S × Ω as the preimage of the point (or vector) O ∈ IR by the continuous map (p, ω) → z(p, ω). ▪

Note that proposition 2.2.1 requires only the continuity of the aggregate demand function, itself a consequence of the continuity of the individual demand functions.

2.2.2 The Relative Perspective

Let 픅(p, ω) denote some property that may be satisfied at the equilibrium (p, ω) ∈ E. Let E(픅) be the subset of the equilibrium manifold E that consists of the equilibria for which the property 픅(p, ω) is true. The study of 픅(p, ω) then reduces to the study of the set E(픅) as a subset of the equilibrium manifold E.

This program requires a thorough understanding of the equilibrium manifold. An important issue in that regard is the existence of practical coordinate systems. Their existence would greatly simplify the study of the subsets that would now be defined by systems of equations and inequations. Hopefully, all the subsets we are interested in will belong to that category.

2.2.3 (p.23) Local Structure

The problem of the existence and nature of coordinate systems for the equilibrium manifold E has two versions. The local version deals with the structure of sufficiently small open sets of the equilibrium manifold. The most relevant mathematical property in that direction is provided by the structure of smooth manifold, where sufficiently small open sets are homeomorphic to Euclidean spaces. These open subsets can then be parameterized by a finite number of real numbers that act as local coordinates. with these local coordinates, it is possible to give sense to the concept of smooth maps. Differentiability makes it possible to apply the powerful methods of differential topology. Before undertaking the study of the smooth manifold structure of the equilibrium manifold, I insert a short mathematical parenthesis.

2.3 Mathematical Parenthesis: Smooth Maps and Related Concepts

2.3.1 Regular and Critical Points

Let ρ : XY be a smooth map (a map differentiable up to any order) between the smooth manifolds X and Y. Let the vector spaces TxX and Tρ(X) Y be the tangent spaces to the smooth manifolds X and Y at the points X and ρ(X), respectively.

The point xX is a critical point of the map ρ if the tangent map (also known as the derivative) ()x : TxXTρ(x) Y is not onto. We denote by Γ the set of critical points.

The point xX is regular if the tangent map ()x : TxXTρ(x) Y is onto.

Proposition 2.3.1

The set Γ of critical points of the smooth map ρ : XY is closed.

Proof Let us use local coordinates for the manifolds X and Y at x and y = ρ(x). This means that we have open neighborhoods U of xX and V of y = ρ(x) ∈ Y that are homeomorphic to IRp and IRq, respectively. We can use the coordinates (ξl,…, ξp) and (ρl,…, ρq) of IRp and IRq to represent the elements of U and V, respectively. Then, the map ρ is represented by its coordinate functions ρll,…, ξp),…, ρql,…, ξp). The tangent map (dρ)x becomes in the local coordinate system the linear map represented by the Jacobian matrix of ρ. This matrix is defined by the first-order partial derivatives of the q coordinate functions with respect to the p variables. These derivatives are also smooth functions of the coordinates.

(p.24) The condition that the tangent map (dρ)x is not onto is equivalent to the property that the rank of the Jacobian matrix is less than the dimension q of the tangent space Tρ(X)Y. This is equivalent to having all the determinants of the square matrices of order q that can be extracted from the Jacobian matrix equal to zero. These determinants are polynomial functions of their coefficients, which are themselves smooth functions of the coordinates. The set of critical points of the smooth map ρ is therefore the set of zeros of a collection of continuous maps. Its complement, the set of regular points, is therefore open in U. Now, the manifold X can be made the countable union of open sets like U. The union of all the open sets made of regular points is therefore an open subset of X. Its complement, the set of critical points of the map ρ : XY, is therefore closed in X. ▪

2.3.2 Singular and Regular Values

By definition, the image ρ(x) of the critical point xX is a singular value of the map ρ : XY. Let ∑ denote the set of singular values. We then have ∑ = ρ(Γ).

By definition, the element yY is a regular value of the map ρ : XY if it does not belong to ∑. In other words, a regular value is not the image of any critical point. Let R be the set of regular values. We have R = Y\∑, the complement of ∑ in Y.

It follows from this definition that an element yY that does not belong to the image ρ(X) of the map ρ is a regular value, although it is not a value of that map. This observation underlines the importance of being accurate when specifying the domain and range of maps.

2.3.3 Regular Value Theorem

The following proposition is very useful for proving that a set defined by an equation system is actually a smooth manifold. It is in fact an extension to the setup of smooth manifolds of the implicit function theorem. It also illustrates the importance of the concept of regular value.

Proposition 2.3.2 (Regular value theorem)

The preimage ρ−1 (y) of the regular value yRY for the smooth map ρ : XY is a smooth submanifold of X whose dimension is equal to dim X − dim Y.

For mathematical references, see, for example, Guillemin and Pollack (1974, 21), Hirsch (1976, 22), and Milnor (1997, 11).

(p.25) 2.3.4 Sard’s Theorem

A particularly important property of smooth maps is Sard’s theorem. Since a smooth manifold is the union of a countable collection of open sets each diffeomorphic to an open set of a Euclidean space, it is possible to define sets of measure zero in smooth manifolds: a set has measure zero if its intersection with each open set of the collection has measure zero. Then, Sard’s theorem states the following:

Proposition 2.3.3 (Sard’s theorem)

The set ∑ of singular values of the smooth map ρ : XY has measure zero in Y.

An alternative formulation is to say that the complement of ∑, the set R of regular values, has full measure in Y. For proofs, see, for example, Dubrovkin, Fomenko, and Novikov (1985, 79), Guillemin and Pollack (1974, 39), Hirsch (1976, 69), or Milnor (1997, 10).

Exercises

  1. 2.1. Let E be a Euclidean space. Let (xn)n ∈ ℕ be a sequence in E that converges to the element x* ∈ E. By using the characterization of compact subsets of a Euclidean space as closed and bounded, show that the set k = ∪n ∈ ℕ {xn} ∪ {x*} is compact.

  2. 2.2. Let E be a metric space. Let (xn)n ∈ ℕ be a sequence in E that converges to the element x* ∈ E. Show that the set k = ∪n ∈ ℕ {xn} ∪ {x*} is compact. (Hint: Use the property that every open covering of a compact set has a finite subcovering.)

  3. 2.3. Let E and F be two metric spaces. Let f : EF be a proper map, i.e., a map such that the preimage f−1(K) of every compact set of F is compact in E. Prove that the direct image by f of every closed set of E is closed in F.

  4. 2.4. Let k be a set equipped with the discrete topology. What are the open sets for this topology? Show that k compact is equivalent to k finite.

  5. 2.5. Let E be a metric space. Let (xn)n ∈ ℕ be a sequence in E that converges to the element x* ∈ E. Assume x* ≠ xn for all n ∈ ℕ. The set k is equipped with the topology induced by the topology of E. Prove that the subset {x*} of k cannot be open in K. Can the induced topology on K be the discrete topology?

  6. 2.6. Let E and F be two metric spaces. The map f : EF is said to be continuous at the point xE if, for every sequence (xn)n ∈ ℕ converging to (p.26) x, the sequence (f(xn))n ∈ ℕ is also convergent and its limit is f(x). The map f : EF is said to be continuous if it is continuous for all xE.

    1. a. Show that the preimage of every closed subset of F is closed in E if f is continuous.

    2. b. Show that the preimage of every closed subset of F is closed in E if and only if the preimage of every open subset of F is open in E. (Hint: Compare the preimage of the complement F\U with the complement E\f−1(U).)

    3. c. Show that the map f : EF is continuous if the preimage of every open subset of F is open in E.

  7. 2.7. Let E and F be two metric spaces. Let f : EF be a continuous map. Show that the image of a connected set is connected. (Hint: Assume the contrary and get a contradiction.)

  8. 2.8. Let ℕ be the set of natural integers equipped with the discrete topology. Show that the only connected subsets are reduced to a point.

    The following exercises assume some knowledge of the properties of the Lebesgue measure.

  9. 2.9. The set V has measure zero in the Euclidean space E = IRn if, for every ∈ 〉 0, there exists a countable set of cubes that cover V and such that the sum of the volumes (or Lebesgue measures) of the cubes that make up the covering is less than ∈. Prove that the Lebesgue measure of a point in a Euclidean space is zero.

  10. 2.10. Prove that the Lebesgue measure of any finite subset of a Euclidean space is zero.

  11. 2.11. Prove that the Lebesgue measure of the subset of a Euclidean space defined by a sequence of elements is equal to zero.

  12. 2.12. Show that the Lebesgue measure of the subset ℚ ∩ [0, 1] consisting of the rational numbers of the interval [0, 1] is equal to zero.

  13. 2.13. Show that the closure   [ 0, 1 ] ¯ of the set   [ 0, 1 ] is equal to [0, 1].

  14. 2.14. Let E = IRp+q = IRp × IRq and F = IRp × {0} with 0 ∈ IRq and q ≥ 1. Show that the Lebesgue measure in E of the subset F of E is equal to zero.

  15. 2.15. Let E be the Euclidean space IRn. Let V be a submanifold of E of dimension mn. Show that the Lebesgue measure of V in E is equal to zero.

  16. (p.27) 2.16. Let E be the Euclidean space IRn. Let W be the union of a finite number of smooth submanifolds of E of dimension strictly less than n. Show that the Lebesgue measure of W in E is equal to zero. (Hint: Apply exercise 2.15.)

  17. 2.17. Let E and F be two Euclidean spaces. Let f : EF be a smooth map. Show that the direct image of a set of measure zero by F has measure zero.

  18. 2.18. Let Y be the subset of Z = IR2 defined by equation x2 + y2 = 1. Let X be the subset of Z × IR equal to Y × IR. Let π : XZ be the map defined by formula π(x, y, z) = (x, y). Interpret geometrically the sets Y and X and the map π. What is the domain of the map n? Show that every point of the domain of π is critical. Show that Y is the image of the map π. Show that Y is also the set of singular values of the map π. Does this contradict Sard’s theorem?

  19. 2.19. Define the sets X, Y, and Z as in exercise 2.18. Consider the map π : XY by the same formula π(x, y, z) = (x, y) except that now the range is Y instead of Z. Show that the map π has no critical point.

  20. 2.20. Let X and Y be two smooth manifolds. Let α : XY and β ˜  : Y  X be two smooth maps. The composition α   β ˜  : Y  Y is the identity map idY.

    1. a. Let Z =  β ˜ ( Y ) be the image of the map β ˜ . Define β : Y → Z by β( y ) =  β ˜ ( y ) . Show that β : YZ is a bijection whose inverse is the map α|Z : ZY, the restriction of the map α to Z.

    2. b. Show that the maps α | Z : ZY and β : YZ are continuous for Z equipped with the topology induced by the topology of X.

    3. c. Let yY. Let x = β(y) ∈ Z. Let Ty(Y) and Tx(X) be the tangent spaces to Y at y and to X at x, respectively. Let d β ˜ y  :  T y ( Y )   T x ( X ) be the tangent (or derivative) map to β ˜  : Y  X . Show that the linear map d β ˜ y is an injection. (Hint: Use the fact that the relation α   β ˜  =  id Y implies for the tangent maps the relation d α x   d β ˜ y  = i d T y ( Y ) .)

    4. d. Conclude that the map β ˜  : Y  X is an immersion that defines a homeomorphism between its domain Y and its image Z =  β ˜ ( Y ) . (Such a map β ˜ is known as an embedding.)

  21. 2.21. Let the smooth map β : YX be an embedding, i.e., an immersion that is a homeomorphism between the domain Y and the image Z = β(Y). Show that the subset Z is a smooth submanifold of X diffeomorphic to Y.

(p.28) 2.4 Smooth Manifold Structure of the Equilibrium Manifold

The question is whether the equilibrium manifold E is actually a smooth manifold or, even better, a smooth submanifold of S × Ω. The answer turns out to be positive, as follows from proposition 2.4.1.

Proposition 2.4.1

The equilibrium manifold E is a smooth submanifold of dimension ℓm of S × Ω.

We are going to prove this property as a consequence of the regular value theorem. The idea is to apply the regular value theorem to the map (p, ω) → (p, ω) defined by the first ℓ − 1 coordinates of the aggregate excess demand z(p, ω). (Recall that the ℓ coordinates of the map (p, ω) → z(p, ω) are not independent, because of Walras’ law.)

2.4.1 Application of the Regular Value Theorem

The equilibrium manifold E is defined by equation (p, ω) = 0. It is therefore the preimage of 0 ∈ IRℓ−1 by the map : S × Ω → IRℓ−1. The regular value theorem tells us that a sufficient condition for E to be a smooth submanifold of S × Ω is that the element 0 ∈ IRℓ−1 is a regular value of the map . This is equivalent to the map having no critical point that is also an equilibrium. In fact, it will be shown that the map has no critical point, which is equivalent to showing that the Jacobian matrix of at (p, ω) ∈ S × Ω has rank ℓ − 1, this matrix having ℓ − 1 rows and mℓ + ℓ − 1 columns.

2.4.2 The Rank Property

To prove the rank property, it suffices to extract from the Jacobian matrix a submatrix that still has rank ℓ − 1. Pick arbitrarily some consumer i. Let us look at the block made of the ℓ columns (and ℓ − 1 rows) made of the derivatives of with respect to the coordinates ω i 1 ,,  ω i of ωi, the endowment of consumer i. In the computation, we apply the chain rule. Given the fact that consumer i’s demand does not depend on consumer j’s wealth, with ji, this yields for the Jacobian matrix the rather simple expression (p.29)

[ z 1 ω i 1 z 1 ω i z 2 ω i 1 z 2 ω i z 1 ω i 1 z 1 ω i ] = [ p 1 f i 1 w i   1 p 2 f i 1 w i p 1 f i 1 w i f i 1 w i p 1 f i 2 w i p 1 f i 2 w i   1 p 1 f i 2 w i f i 2 w i p 1 f i 1 w i p 2 f i 1 w i p 1 f i 1 w i   1 f i 1 w i ].

In the right-hand matrix, multiply the last column by p1 and subtract from the first column, multiply again the last column by p2 and subtract from the second column, and so on until multiplication of the last column by pℓ − 1 and subtraction from the (ℓ − 1)th column. This yields the ℓ − 1 × ℓ matrix

[ 1 0 0 f i 1 w i 0 1 0 f i 2 w i 0 0 1 f i 1 w i ]

that has same rank. The rank of this new matrix is equal to ℓ − 1 because the block made of its first ℓ − 1 columns has obviously rank ℓ − 1.

(p.30) 2.4.3 Dimension of the Equilibrium Manifold

It also follows from the regular value theorem that the dimension of the equilibrium manifold E is equal to the dimension of S × Ω minus the dimension of IRℓ − 1, hence to ℓ − 1 + mℓ − (ℓ − 1) = mℓ.

2.4.4 Smoothness of the Embedding Map

The following consequence of E being a smooth submanifold of S × Ω is going to be particularly useful.

Proposition 2.4.2

The embedding map ES × Ω is smooth.

Proof The embedding map from E into S × Ω is the identity map from S × Ω restricted to the subset E. The proposition then follows readily from the definition of a smooth submanifold. ▪

The property conveyed by the proposition may seem to be of limited value. However, it is very convenient when it comes to proving the differentiability of maps that are the restrictions to the equilibrium manifold E of maps defined on the Cartesian product S × Ω. An immediate application of that observation is to the projection map S × Ω → Ω restricted to the equilibrium manifold E.

2.5 Natural Projection

The properties of the solutions of the equilibrium equation z(p, ω) = 0 when the parameter ω is varied in the parameter space Ω depend not only on the structure of the equilibrium manifold E but also on how this manifold is embedded in the Cartesian product S × Ω.

The latter aspect is conveyed by the map that is the restriction of the projection map S × Ω → Ω to the equilibrium manifold E. This map π : E → Ω, called from now on the natural projection, is related to the Walras correspondence by the equality π−1 (ω) = W(ω) × {ω}. The equilibrium manifold E is the domain of the natural projection.

We now establish two important properties of the natural projection, smoothness and properness. Smoothness enables us to apply to the natural projection important properties of differential topology. Properness is an additional property that also plays an important role. Properness is usually defined by the property that the preimage of every compact set is compact. This property then implies that the direct image of every closed set is closed.

(p.31) 2.5.1 Smoothness

Proposition 2.5.1

The natural projection π : E → Ω is smooth.

Proof The natural projection π : E → Ω is the composition of two maps: the natural embedding ES × Ω, which is smooth because E is a smooth submanifold of the Cartesian product S × Ω, and the projection map S × Ω → Ω, which is also smooth because the coordinate functions of this map are smooth. ▪

2.5.2 Properness

Proposition 2.5.2

The natural projection π : E → Ω is a proper map.

Proof A map is proper if the preimage of every compact set is compact. Let k be a compact subset of the parameter space Ω. Let H be the image of k by the continuous map ω = (ω1, ω2,…, ωm) → ωi + ω2 +⋯+ ωm from Ω into X. The set H is compact and therefore bounded from above: there exists some r* ∈ X such that ω1 + ω2 +⋯+ ωmr* for all ω ∈ K. It follows from the equilibrium equality

f 1 ( p, p   ω 1 ) +  +  f m ( p, p   ω m ) =  ω 1  +  +  ω m

that

f 1 ( p, p   ω 1 ) +  +  f m ( p, p   ω m )  r*  for ( p, ω )   π 1  ( K ).

Pick now some consumer i arbitrarily. It follows from the previous inequality combined with the fact that the consumption space is the strictly positive orthant X that

f i ( p, p   ω i )  r*  for ( p, ω )   π 1  ( K ).

Let Ki be the image of k by the projection map

ω = ( ω 1 ,,  ω i ,,  ω m )   ω i .

That map being continuous, its image Ki is a compact subset of X = ++ . There exists a lower bound A such that the coordinates ω i k satisfy the inequalities 0  A   ω i k for k = 1, 2,…, ℓ for ω ∈ K. Let ω i *  = ( A, A, , A ) .

The utility ui(fi (p, p ⋅ ωi)) is greater than or equal to uii), itself greater than or equal to u i ( ω i * ) . We therefore have that, for every (p.32) (p, ω) ∈ π−1(K), the demand xi = fi(p, p ⋅ ωi) is such that xir* and u i ( x i )   u i ( ω i * ) . We can assume without loss of generality the strict inequality ω i *   r* .

Lemma 2.5.3

The nonempty set L i  = { x i   X |  u i ( x i )   u i ( ω i * ) and  x i   r* } is compact.

Proof Let us show that Li is a closed and bounded subset of IR. Closedness in IR of the set { x i   X |  u i ( x i )   u i ( ω i * ) } follows from property (iv) of the utility functions considered in section 1.2.2. (See exercise 1.3 in chapter 1.) The set {xi ∈ IR | xir*} is also closed in IR. The intersection of these two sets is therefore closed in IR.

The set Li is bounded from above by r* and from below by 0. This proves the compactness of Li. ▪

Lemma 2.5.4

The image Dnui(Li) is a compact subset of the price set S.

Proof The normalized gradient map Dnui : XS is continuous. (See exercise 1.1 in chapter 1 for the definition of the normalized gradient of utility.) The set Li is compact. The image Dnui(Li) is then compact as the image of a compact set by a continuous map. ▪

It follows from the continuity of the natural projection that the restriction of π to Dnui(Li) × k is continuous. Therefore, π−1 (K) is a closed subset of the set Dnui(Li) × k by the continuity of π. The Cartesian product of the two compact sets k and Dnui(Li) is compact. The combination of the two properties implies that π−1(K) is compact, which proves the properness of the natural projection π : ∈ → Ω. ▪

2.6 Structure of the Equilibrium Manifold over the Set of Regular Economies

2.6.1 Critical and Regular Equilibria

Let ℜ(p, ω) denote the property for the equilibrium (p, ω) ∈ E to be a regular point of the natural projection π : ∈ → Ω. An equilibrium (p, ω) ∈ E that satisfies the property ℜ(p, ω) is known as a regular equilibrium. Let the set E(ℜ) consist of the regular equilibria. We denote by Γ the set of critical equilibria in E. The two sets E(ℜ) and Γ are complementary to each other.

It follows from proposition 2.3.1 that the set of critical equilibria Γ is closed in E. Its complement, the set of regular equilibria E(ℜ), is therefore open in E. (p.33)

Figure 2.1 The equilibrium manifold and the natural projection

Figure 2.1 The equilibrium manifold and the natural projection

2.6.2 Openness of the Set of Regular Economies R

Proposition 2.6.1

The set of regular economies R is open in Ω.

Proof The set of regular economies R is the complement in Ω of the set ∑, the set of singular values of the map π : E → Ω. By definition, singular values are images of critical points: ∑ = π(Γ). The set Γ of critical points of the map π : E → Ω is closed in E. In general, the direct image of a closed set by a continuous map is not closed. But the map π : E → Ω is also proper. The images of closed sets by proper maps are closed. This implies that ∑ is closed and its complement, the set of regular economies R, is therefore open in Ω (figure 2.l). ▪

2.6.3 Finiteness of the Equilibrium Set of Regular Economies

Proposition 2.6.2

For every ω ∈ R, the set π−l(ω) is finite.

Proof The proposition follows from the compactness and discreteness of the set π−l (ω).

Compactness The set π−l(ω) is compact as the preimage of the compact set {ω} by the proper map π : E → Ω.

Discreteness Let us show that the topological space π−l(ω) is discrete, which means that it is equipped with the discrete topology. The discrete topology is the topology where each subset is open. For a (p.34) topological space to be discrete, it suffices that each set consisting of one element is open.

Let {x} be the subset of π−l(ω) consisting of the unique element X. Clearly, x cannot be a critical point of the map π because ω is a regular value. Therefore, the tangent map (dπ)x is a bijection and, by the inverse function theorem, the natural projection π : E → Ω is a local diffeomorphism at the point xE. This means that there exist open neighborhoods U of ω and V of x such that the restriction π | V : VU is a diffeomorphism. This map being one-to-one, the intersection π−l(ω) ∩ V contains the element x and no other element. Therefore, we have {x} = π−l(ω) ∩ V, where V is open in E. It follows from the definition of the open sets of π−l(x) as the intersection with π−l(ω) of open sets of E that the subset {x} is open in π−l(x).

Open Covering of π−l(ω) The topological space π−l(ω) being equipped with the discrete topology, the subsets consisting of a unique elements of π−l(ω), the subsets {x} with x ∈ π−l(ω), are open. Their union is obviously equal to X. They define an open covering of the set π−l(ω).

Finiteness of π−l(ω) It follows from the compactness of π−l(ω) that the open covering of the set π−l(ω) by sets with a unique element (the sets defined by the elements of π−l(ω)) has a finite subcovering. The set π−l(ω) is therefore the union of only a finite number of its elements. This set is therefore finite. ▪

2.6.4 Finite Covering Property

The following proposition gives us a fairly accurate image of the equilibrium manifold E over the set of regular economies. This image is only partial, however, because the statement of the proposition is only local, i.e., it holds true only for sufficiently small open subsets of R.

Proposition 2.6.3

For every ω ∈ R, there exists an open neighborhood U of ω with U ⊂ R and the property that, if π−l(ω) is nonempty, the preimage π−l(U) is the union of a finite number of pairwise disjoint open sets V1, V2,…, Vk,…, Vn and the restriction πk : VkU of the map π is a diffeomorphism for k = l, 2,…, n.

Proof Let n ≥ l be the (finite) number of elements of π−l(ω).

Let xl,…, xn be all the elements of π−l(ω). Provided the open sets are small enough, it is always possible to consider open pairwise disjoint neighborhoods U 1 ,  U 2 ,,  U k ,,  U n in E of xl, x2,…, xk,…, xn (p.35) such that the restriction of π to U k is a diffeomorphism with U k  = π( U k ) .

The set E\( U 1    U k    U n ) is closed in E. Its image by the natural projection π is closed in Ω because π is proper. Let us define the set U as U = ( U 1    U 2    U k    U n )\π( E\( U 1    U k    U n ) ) , i.e., U is the intersection of the sets Uk for k varying from 1 to n and of the complement in Ω of the set π( E\( U 1    U k    U n ) ) . Clearly, U is open in Ω. Let us show that ω belongs to U. All we have to check is that ω does not belong to π( E\( U 1    U k    U n ) ) , which follows from the inclusion

π 1 ( ω )   U 1    U k    U n .

Let V k  =  U k    π 1  ( U ). . The restriction π k  = π |  V k defines a diffeomorphism between Vk, and π(Vk).

Let us check that π−1(U) is the union of V1 ∪ ⋯ ∪ Vk ∪ ⋯ ∪ Vn. Let xn−1(U). Assume that x does not belong to any Vk. Then it belongs to the set E\( U 1    U k    U n ) , which implies

ω = π( x )  π( E\( U 1    U k    U n ) ).

Therefore, ω belongs to the open set U, hence a contradiction. This ends the proof of the proposition. (See figure 2.2.) ▪

The mathematical structure described by proposition 2.6.3 is known as the property that the restriction of the map π to π−1 (R) is an open finite covering of the set of regular economies R.

Selections of Equilibrium Prices

Proposition 2.6.4

Let ω ∈ R. There exists an open neighborhood U of ω with UR and a finite number n of smooth maps sk : US such that the union 1  k n s k ( ω ) is identical to the set W(ω′) of equilibrium price vectors associated with every ω′ ∈ U.

In other words,

π 1 ( ω ) =  1  k  n { ( s k ( ω ),  ω ) }   for  ω   U.

Proof To prove this proposition, let us go back to proposition 2.6.3. It suffices to compose the map π k 1  : U   V k with the projection S × Ω → S to define the map sk : US, which ends the proof of the proposition. ▪ (p.36)

Figure 2.2 Structure of the equilibrium manifold over the neighborhood U of the regular economy ω ∈ R

Figure 2.2 Structure of the equilibrium manifold over the neighborhood U of the regular economy ω ∈ R

Proposition 2.6.4 enables us to express the equilibrium prices associated with a regular economy as a function of the parameter ω describing these economies.

This property also implies the lower hemicontinuity of the Walras correspondence over the set of regular economies R.

Remark Smooth selections of equilibria are defined only for open neighborhoods of regular economies. It is important for both practical and theoretical reasons to have some more information about the domains of these selection maps. In particular, does there exist some largest domain for the smooth selection maps instead of these open sets?

A complete treatment of this question is beyond the scope of this chapter. This problem is easier in the context of chapter 5 in the general case of an arbitrary number of goods and consumers.

Local Constancy of the Number of Equilibria at Regular Economies

Proposition 2.6.5

Let ω ∈ R be a regular economy. There exists an open neighborhood U of ω such that UR and the number of equilibria is constant all over U.

(p.37) Proof Let ω ∈ R be a regular economy. It then suffices to pick U as in proposition 2.6.4 to prove the proposition. ▪

Proposition 2.6.5 is often stated as the function number of equilibria N : ω → #π−1(ω) being locally constant.

Constancy of the Number of Equilibria over the Connected Components of R

By definition, the connected component of a point in a topological space is the largest connected set containing that point (see, e.g., Dieudonné 1960, sec. 3.1.9). The connected components of a set are the various connected components of the points of this set. It follows from this definition that the set of regular economies R is partitioned into its connected components. By partitioned is meant that the various connected components are pairwise disjoint and that their union is equal to the full set R.

Proposition 2.6.6

The number of equilibria is constant over each connected component of the set of regular economies R.

Proof Let N(ω) = #π−1(ω) denote the number of equilibria of the regular economy ω ∈ R. This defines a function π : R → ℕ, where ℕ is the set of natural integers. Let us equip this set with the discrete topology, the topology where each subset is open (and also closed).

Proposition 2.6.5 tells us that, for ω ∈ R, there exists an open neighborhood U of ω contained in R over which the number of equilibria N(ω) is constant. The function N is said to be locally constant.

Let us show that a locally constant function is necessarily constant on every connected component of its domain. First, let us show that the function number of equilibria N is continuous.

Continuity is established if we can show that the preimage by the function N of every open subset of ℕ is open in R. Since the topology of ℕ is discrete, open subsets of ℕ are the union of open subsets reduced to just one point. The preimage of a union of sets being also the union of the preimages of the sets, and the union of open sets being open, it therefore suffices to show that the preimage N−1(k) of the set reduced to the element k by the map N : R → ℕ is open. This set consists of the economies ω ∈ R that have k equilibria. If this set is empty (i.e., there are no such economies), then it is open, because an empty set is open by definition. If N−1(k) is nonempty, let ω ∈ N−1(k). It follows from Proposition 2.6.5 that there exists an open set U where the (p.38) number of equilibria is equal to k. This is equivalent to the inclusion UN−l(k). This shows that the set N−l(k) is an open neighborhood of each of its elements. This property is characteristic of an open set and implies that N−l(k) is open. This proves the continuity of the map N : R → ℕ.

Let C be a connected component of R. The image of a connected set by a continuous map is connected. Therefore, the image N(C) is a connected subset of the set ℕ (equipped with the discrete topology). It follows from the definition of a connected set combined with the definition of the discrete topology that the only connected sets of ℕ equipped with the discrete topology are the sets that consist of a unique element. This implies that the set N(C) consists of just one element, which is another way of saying that the map ℕ is constant on C. This ends the proof that the number of equilibria is constant on every connected component of the set of regular economies. ▪

2.7 Genericity of Regular Economies

We have seen that regular economies and their equilibria enjoy relatively nice properties. It is therefore important to have some information about the size of the set of regular economies R.

2.7.1 Full Measure

We already know that the set R is open. The following proposition tells us that this set is really large in the sense that its complement has Lebesgue measure zero in Ω.

Proposition 2.7.1

The set of regular economies R is open with full measure in Ω.

Proof Sard’s theorem, proposition 2.3.3, tells us that the set of singular values of a smooth map between two smooth manifolds has Lebesgue measure zero. Let us apply this to the set ∑ = π(Γ), the set of singular values of the natural projection π : E → Ω. The set ∑ has therefore measure zero in Ω and its complement R = Ω\∑ has full measure. ▪

2.7.2 Density

The full measure property implies an interesting topological property, density, as follows from corollary 2.7.2.

(p.39) Corollary 2.7.2

The set of regular economies R is open and dense in Ω.

Proof All we have to show is that R is dense. Assume the contrary. Then there exists a nonempty open cube U such that the intersection RU is empty. This means that the nonempty open set U is contained in ∑. Therefore, the measure of ∑ must be larger than, or at least equal to, the measure of U. But the measure of the nonempty cube U is the product of the lengths of its sides and, as such, is strictly positive. This yields a contradiction. ▪

Remark Density is considered to mean “large” from a topological perspective. Note, however, that the set of rational numbers ℚ is dense in the set of real numbers IR. Nevertheless, its Lebesgue measure is equal to zero as the measure of a countable set. (Remember, the set of rational numbers is countable.) Here, we have much more than just density because the set R is also open.

Exercises

  1. 2.22. The correspondence W : Ω → S is upper hemicontinuous (u.h.c.) at ω* ∈ Ω if W(ω*) ≠ ø and if for every neighborhood U of W(ω*) there exists a neighborhood V of ω* such that W(ω) ⊂ U for every ω ∈ V. The correspondence W is u.h.c. if it is u.h.c. at every ω* ∈ Ω (Hildenbrand 1974, 21, def. 1). Show that properness of the projection map π : E → Ω implies that the Walras correspondence W : Ω → S is u.h.c. (Hint: See Hildenbrand 1974, 24, theorem 1.)

  2. 2.23. The correspondence W : US (where U is an open subset of Ω) is lower hemicontinuous (l.h.c.) at ω* ∈ U if W(ω*) ≠ ø and if, for every open set G of S with W(ω*) ∩ G ≠ ø, there exists a neighborhood V of ω* such that W(ω) ∩ G ≠ ø for every ω ∈ V. The correspondence is l.h.c. if it is l.h.c. at every ω* ∈ U (Hildenbrand 1974, 26, def. 3). Show that the restriction of the Walras correspondence to the set of regular economies R is 1.h.c. (Hint: See Hildenbrand 1974, 27, prop. 7.)

2.8 Economies with a Large Number of Equilibria

I conclude this chapter with another property that requires nothing more than the smoothness and properness of the natural projection, a property that complements nicely the fact that the set of economies with an infinite number of equilibria is contained in a closed set with measure zero.

(p.40) The only reservation I have about including it at this early stage is that its proof is mathematically demanding in terms of covering maps and Riemannian geometry. The proof needs mathematical concepts that are not used anywhere else in the book. The reader is therefore encouraged to focus on the general ideas, which are quite simple, and to skip the technical details in a first reading.

2.8.1 Upper Bound of the Measure of the Set of Economies with More Than a Given Number of Equilibria

Proposition 2.6.2 gives no upper bound on the number of equilibria. In fact, DMS tells us that there is no upper bound. Nevertheless, there exists an upper bound to the size of the set of economies with more than a given number of equilibria.

Let us consider a compact subset k of Ω. Let Ωn(K) denote the set of economies ω ∈ k having at least n equilibria, and let μ(Ωn(K)) denote the Lebesgue measure of this set.

Proposition 2.8.1

There exists a constant c(K) such that the inequality μ(Ωn(K)) ≤ c(K)/n is satisfied for every n ≥ l.

2.8.2 Sketch of the Proof

Assume to fix ideas that Ω is two-dimensional and S one-dimensional. Then, S × Ω can be identified with the (strictly positive orthant of the) ordinary Euclidean space IR3, and the equilibrium manifold E is a surface in that space. This surface comes with a notion of area (e.g., the definition of the area of the sphere). The area in Ω coincides with the Lebesgue measure of IR2.

The next important step is to observe that the area does not increase through orthogonal projection. More specifically, let H be some subset of the surface E. Then the area of H is bigger than the area of its orthogonal projection π(H) in the plane Ω. This property is just the generalization to surfaces of a well-known property of solid geometry, namely, that the area of the orthogonal projection of a triangle is less than or equal to the area of the triangle that is projected (figure 2.3).

Let now k be some compact subset of Ω. The preimage π−l(K) is compact in the surface E. It has therefore a finite area c−l(K)).

The set of singular economies S having measure zero, the intersection Ωn(K) ∩ R has the same measure as Ωn(K). (p.41)

Figure 2.3 Area of an orthogonal projection

Figure 2.3 Area of an orthogonal projection

Let now U be some subset of Ωn(K) ∩ R such that π−l (U) consists of a finite number of layers diffeomorphic to U. Not only is the number of layers ≥ n but the area of each layer is ≥ to the area of U onto which each layer projects orthogonally. Therefore, the area of π−l(U) is ≥ ℕ times the area of U. It is shown in a following section that the set Ωn(K) ∩ R can be partitioned into a countable collection of such open sets U. The inequalities for each open set U then add up to the inequality

c( π 1 ( K ) )  area( π 1 ( Ω n ( K ) ) n × area( Ω n ( K ) ) ,

which ends the proof of the proposition. Let us now develop the more technical aspects of this proof.

Area Element for the Equilibrium Manifold

The equilibrium manifold E is embedded in S × Ω, which is itself an open subset of the Euclidean space IRℓ − 1 × IRm. The restriction of the scalar product to the tangent spaces to E defines a Riemannian structure on E. (See, e.g., Hicks l965 for these elementary notions of Riemannian geometry.) This Riemannian structure leads to a concept of ℓm dimensional area on E, which is the extension to arbitrary dimensions of the usual area concept for two-dimensional surfaces.

Let λ denote the measure defined by the Riemannian structure on the equilibrium manifold E. The orthogonal projection does not increase the measure. More specifically, let μ denote the Lebesgue (p.42) measure of Ω. We then have μ(π(V)) ≤ λ(V) for every measurable subset V of E.

Finite Area of π−1(K)

The set k is compact and the natural projection π proper. The set π−l(K) is therefore compact, hence measurable, i.e., with a finite measure C(K) = λ(π−l (K)).

Reduction to Ωn(K) ∩ R

The set of regular economies R having full measure, we have

μ( Ω n ( K ) ) = μ( Ω n ( K )  R ).

Existence of a Suitable Countable Partition

Every open subset of a Euclidean space can be decomposed into a countable union of pairwise disjoint open cubes and a set of measure zero. (For a proof, see, e.g., Rudin l966, 52.) We apply this property to the open set R and neglect from now on the set of measure zero.

A cube is connected and simply connected. The connectedness property implies that each cube of the decomposition of R is contained in one connected component of R. Therefore, the number of equilibria of economies belonging to the same cube is constant.

Simple connectedness implies that the covering of each cube of the collection by the map π is trivial. (See, e.g., Dieudonné l973, sec. l6.28.6.) This means that the preimage is the disjoint union of a finite number of open sets all diffeomorphic to the cube through the projection map π, a map that we identify with the orthogonal projection.

Let us consider in the countable collection of cubes those that have at least n equilibria. Let Uj be the collection of these cubes. This collection is at most countable. In addition, each economy ω ∈ Uj has at least n equilibria. The set Ωn(K) is equal, up to a set of measure zero, to the union of those cubes Uj. The set π−l(Uj) consists of at least n layers, each one having a λ-measure at least equal to μ(Uj). Adding up all these inequalities yields

nμ( U j )  λ( π 1 ( U j ) ).

Summing up these inequalities over the cubes Uj gives

nμ( Ω n ( K ) )  λ( π 1 ( Ω n ( K ) ) )  λ( π 1 ( K ) ) = c( K ).   

(p.43) Proposition 2.8.1 tells us that the probability of observing economies with more than n equilibria, although not zero, tends to zero as 1/n and is therefore small for n large.

Proposition 2.8.1 is an asymptotic version of the property that the set of economies with an infinite number of equilibria has Lebesgue measure zero in Ω.

2.9 Conclusion

The natural projection π : E → Ω is therefore an open finite covering of its set of regular values R. In addition, the complement Ω\R, the set of singular values ∑, is closed with measure zero. Such a map π : E → Ω is known in mathematics as defining a ramified covering, the ramifications taking place over the set of singular values ∑. The genericity of regular values follows directly from Sard’s theorem. The other properties in Debreu (1970) are just reformulations of the finite covering property of the set of regular values R.

A first outcome of the relative postmodern perspective is therefore to put the Debreu (1970) results into the wider setup of the properties of the solutions of an equation system that depends on some parameter. The equation system is here the equilibrium equation, the parameter the vector of individual endowments, with variable total resources. Note that the results depend on the choice of the parameter space and would not necessarily hold with parameter spaces different from Ω = Xm.

The two key properties for the results of this chapter are the smoothness and properness of the projection map. Smoothness is a fairly general property in this kind of approach. It is in fact satisfied for a large class of equations in addition to the equilibrium equation of the Arrow-Debreu model. Properness is more specific to the economic model. Nevertheless, all it requires is that there be at least one consumer whose demand tends to infinity when (normalized) prices tend to the boundary of the price set, i.e., either to zero or to infinity. Interestingly, properness is the mathematical version of a property that economists have discussed for a long time in relation to the existence of free goods.

Smoothness and properness of the natural projection require only a small fraction of the assumptions made on utility functions. The next chapters are therefore devoted to the properties of the Arrow-Debreu (p.44) model that follow from a more intensive exploitation of these assumptions.

2.10 Notes and Comments

The proof that the equilibrium manifold is indeed a smooth manifold for the parameter space Ω = Xm , i.e., for total resources that are variable, is given by Delbaen (1971). A set closely related to the equilibrium manifold is also considered by Smale (1974a).

The name catastrophe theory is often given to the application of the theory of singularities of smooth maps to the study of the solutions of finite equation systems. Most models of catastrophe theory involve smooth maps similar to the natural projection. The main differences between these models are in the interpretation of the singularities and their importance. In the natural projection, the regular values (regular economies) are as important as the singular values (singular economies) or the critical equilibria. This is not the case for all the models of catastrophe theory. The most remarkable feature of catastrophe theory is its ability to generate discontinuities in otherwise differentiable models. Structural stability is a concept also introduced by catastrophe theory, and it corresponds to the lack of discontinuities. Typically, structural stability is observed at regular values.

Thom (1975) and Zeeman (1977) tend to reserve the name catastrophe theory to the methods that consist in deriving mathematical models of real-world phenomena through the identification of the phenomenon’s qualitative features with appropriate singularities of smooth maps. That method makes sense only when the mathematical model is unknown, which is not the case for economic theory with its Arrow-Debreu model. For various applications of catastrophe theory, see Arnold (1992), Thom (1975), and Zeeman (1977).

The results of sections 2.6 and 2.7 are in Debreu (1970). The introduction of the natural projection π : E → Ω and the proof of the ramified finite covering property of the natural projection are due to Balasko (1975a). It is the natural projection approach that Debreu (1976) presented in an address to the American Economic Association meeting of 1975, devoted to the theory of regular economies.

The bound on the size of the set of economies with at least n equilibria is given by Balasko (1979c).