Alan G. White

General equilibrium theory aims at studying the equilibrium price vector as a function of the parameters defining the economy; it describes those states in which the independent plans of many agents with conflicting interests are compatible such a state is called an equilibrium.

Ideally, it is desirable to have some general principles concerning the equilibrium concept. Naturally, the first question is existence and the existence theorem has been proved by Arrow-Debreu (1954). As equilibrium has no connections with well-being, welfare theorems are particularly interesting. The first and second theorems of welfare economics, which make explicit the connection between equilibrium and Pareto optimality are well understood. Uniqueness of equilibrium is desirable so that policy measures and outcomes can be equivocally compared. Finally, one would like to have a dynamic theory according to which equilibria are approached over time.

One can think of many examples of economies with an infinite number of equilibria, the two good, two agent Edgeworth box is a good example. It is an arduous task to provide conditions guaranteeing (global) uniqueness of equilibrium. Assumptions guaranteeing global uniqueness have been seen to be excessively strong.[1] Without results guaranteeing uniqueness of equilibrium comparative statics becomes devoid of meaning, and a dynamic theory becomes particularly difficult to develop.

The introduction of differential techniques into economics was brought about by the study of several basic questions stemming from models proporting equilibrium concepts, one such being the uniqueness issue. The theory of regular and singular economies can be seen as an attempt to progress general equilibrium theory in the absence of a definitive uniqueness result. Debreu (1970, p. 387) recognises the possibility of multiplicity of equilibria but requires them to be locally unique. He notes ...if the set of equilibria is compact (a common situation), local uniqueness is equivalent to finiteness. One is thus lead to investigate conditions under which an economy has a finite set of equilibria. The equilibria are well defined and are not destroyed by small perturbations in parameters. Indeed, the data of an economy (such as endowments) cannot be exactly observed. If the equilibrium correspondence W() is not continuous at the economy, a small observational error will yield entirely different sets of predicted equilibria: in this case the explanatory power of the model is limited. Thus the continuity principle is a desirable property. The differential approach to general equilibrium theory thus attempts to go beyond the often overstudied existence question, endowing the equilibrium set with a more regular structure and with differentiability assumptions it permits a greater examination of the properties of equilibria that are economically interesting (such as uniqueness).

The following exposition of the differential approach examines the nature and charac-teristics of the differential approach for a simple pure exchange economy the emphasis is not on intricacies of the mathematics per se. Section two outlines some of the mathematical concepts and definitions, while Section three presents the methodology of the differential approach. Section four considers some extensions that have been developed concurrently with the differential approach.


This section and the next draw heavily on Balasko (1988)

Consider a pure exchange economy where there are l commodities and m consumers where each consumer is endowed with an amount of each of the l commodities i l. Individual demand is fi and is assumed to be fixed.

The space of economies is denoted by = lm. Let r l , be the vector of total resources for the economy. Then the space (r) = { lm | i i = r } is the space of economies with totat resources r and S is the set of prices.

The equilibrium correspondence or the Walras correspondence (Balasko 1975a, p 907). W() associates with each economy the set of prices for which the economy is in equilibrium

W() = {p S | i fi (p,p.) = i i } S

The pair (p,) S x is said to be an equilibrium if supply is equated with demand at (p,). The set E S x denotes the set of equilibria. The natural projection (Balasko 1988) : E is the restriction to the equilibrium set E of the projection : S x : (p,) X Note that the preimage of the natural projection

-1() = {(p,) : (p,)} =

and thus -1() = W() x

It is easily seen from this that the study of the correspondence W is thus formally equivalent to the study of the function . Indeed the cardinal number of this set #-1 () is the number of equilibria associated with the economy . Thus #-1 () = 1 corresponds to a unique euqilibrium, and #-1 () = is the case of an infinte number of equilibria.

The program of study suggested by the above has become apparant. Firstly, the structure and properties of the equilibrium set E must be explored. Secondly, the relationship #-1 () = W () x indicates that studying the characteristics and behaviour of the natural projection will provide information on the equilibrium correspondance W.

The Nature of the Differential Approach

The standard axioms of preference (such as transitivity and convexity) do not guarantee the differentiability of the demand function. In order to use the differentiablility properties of demand functions it is necessary to strengthen the set of axioms without weakening their economic relevance. The solution to this lies in the mathematical concept of approximation, the best example of one such approximation technique being that depicted by the Stone - Weierstrass Theorem. One special case of this theorem states that any continuous function can be approximated arbitrarily close by a C polynomial (one whose derivatives exist up to any order exist and are continuous).

Approximation techniques are widespread in mathematics. Taylor expansions can be used to approximate more complex functions. Such techniques permit the original functions to be replaced by ones that are simpler to analyse, characterise and manipulate. The focus here is on the consumer and similar approximation results exist in the context of analysing preferences and consumer choice (see Mas-Colell, 1974). It can thus be assumed that functions in economics exhibit the property of differentiability.

The following definition provides the link with the differential approach.

Definition 1 Let M V W N be a function where M,N are finite dimensional real vector spaces. The function f is called a Cr diffeomorphism if

(a) V is open in M, W is open in N.

(b) V W is bijective

(c) f and f-1 are of class Cr, that is the derivatives up to the rth order exist and are continuous.

Theorem 1.

The demand fuction of consumer i, fi: S x l is a C diffeomorphism. Put simply, this means that fi is smooth and bijective and has a well defined smooth inverse gi where

gi : l S x : gi (xi) = (gradn ui (xi) , xi gradn ui (xi))

Structure of the Equilibrium Set

The desirable properties of discreteness of equilibria and continuity of the equilibrium correspondence can be examined once the theoretical framework has been established. The conditions for the solution of the problems posed are differentiability assumptions, and the main mathematical tools for the solution of the problem of discreteness of the set of equilibria have been provided by differential topology.

To appreciate the concept of smooth manifold it is necessary to have an understanding of the notion of a coordinate system which is used to parameterise (label) the elements belonging to a topological space. The search for structure then, is one of finding appropriate parameterisations.

Definition 2 Let y = (y1 , . . . . , yn ) be a sequence of real valued functions on an open subset V of a topological space. Then y is called an n-dimensional coordinate system on n with domain V if

X V y(V) n

is a homeomorphism of open V onto open y(V) in n. If z = (z1, . . . ,zn) is another n-dimensional coordinate system on X, then y is Cr compatible with z if F = z y-1 is a Cr diffeomorphism of y (V W) onto z(V W).

Coordinate systems and compatible coordinate systems are brought together in the manifold concept.

Definition 3 Let y = (y1 , . . . . , yn ) be a sequence of real valued functions on an open subset V of a topological space. Then y is called an n-dimensional co-ordinate system on X with domain V if

X V y(V) n

is a homeomorphism of open V onto open y(V) in n. If z = (z1 , . . . . ,zn ) is another n-dimensional co-ordinate system on X, then y is Cr compatible with z if F = z y-1 is a Cr diffeomorphism of y (V W) onto z(V W)

Definition 4 A topological space X is called a Cr manifold if a family Y, A of mutually Cr compatible coordinate systems is given, each Y has domain V and

As a first approximation a manifold is a topological space that can be identified with a Euclidean space, that is, for any point belonging to the manifold, there exists a neighbourhood of this space that is hoeomorphic to a Euclidean space. More precisely, and in line with the formal definition, the smooth manifold structure consists of a collection of local coordinate systems covering the whole space, where functions defining coordinate substitutions are of class C.

Theorem 2

The equilibrium set E is a smooth manifold of dimension lm embedded in S x and thus the equilibrium (p,) E can be parameterised by lm local coordinates.

The natural projection : E : (p,) is a mapping from the equilibrium set E S x into the space of economies. It has been proved that the equilibrium set E has the structure to a smooth manifold diffeomorphic to lm via a diffeomorphism. This enables one to use the notion of smooth mappings, that is mappings of class C, whose natural projection is smooth, then it puts at ones disposal many powerful tools of differential topology.

Figure 1. Regular and Singular Values

Regular and Singular Economics

Definition 5: Let f: X Y be a smooth mapping between two smooth manifolds X and Y. A point x X is said to be a critical point of f: X Y if the linear tangent mapping at x, Tx f is not surjective. The image f(x) is called a singular value. It is said that y Y is a regular value of f: X Y if y is not the image of a critical point.

The way out of the difficulty with uniqueness and continuity issues is provided by differential topology. A well-behaved economy (with locally unique equilibria and continuous dependence with respect to the parameters defining the economy) is called a regular economy such that

(a) there are not many non-regular economies

(b) each regular economy has a finite set of equilibria.

(c) the equilibrium set depends continuously on the economy.

To illustrate informally the idea of singular and regular values outlined above, consider a function f: such that f is of class C (Debreu 1976). The critical points are those where the linear tangent mapping is not surjective (at that point the derivative is zero). The three critical points are a,b, and c. Critical values are, by definition the images of critical points. Thus the critical values associated with the critical points a,b and c are respectively d,e and f. Sards theorem guarantees that, in some sense there are not many critical values.

These concepts can be extended to examples involving smooth manifolds and diffeomorphisms. The equilibrium set of an economy defined by finitely many parameters can be represented by a finite-dimensional smooth manifold. To show that economies are well-behaved in the sense of criteria (a), (b) and (c), introduce a C funtion F from a manifold L to M such that a regular economy is defined as a regular value of F satisfying (a), (b) and (c). Sards theorem shows that most economies are of this type.

To cast the issues in the framework that has been constructed, precise definitions of regular economies and singular economies are now introduced.

Definition 6. The economy is regular (resp. singular) if it is a regular value (resp. singular) value of the natural projection : E .. A singular value is the image of a critical point.

Denote by R, the set of regular economies and by the set of singular economies.[2] As these are the only two classifications of economy types that will be used, then = R . For a regular economy the projection of the tangent space of E at (p,) covers .

Sards theorem states that the set of singular values of a smooth mapping has Lebesgue measure zero. Thus the set of singular values of is small from a measure theoretic point of view. It is possible to attach a probabilistic interpretation to Lebesgue measure. Essentially, the probability that a randomly chosen economy will be singular is zero, in the sense that singular economies are quite exceptional. Of course, having measure zero is not equivalent to smallness from the topological point of view.

Balasko (1975b) derives a criterion for ascertaining the type of a particular economy.

Theorem 3 (Balasko)

The equilibrium (p,) E is critical if and only if

det J (p,) = 0

Where J (p,) is the Jacobian matrix of aggregate excess demand z*(p,), where z* is the vector of the first l-1 coordinates of z l .

This theorem gives an alternate description of a regular economy, and while less convenient from a diagrammatic point of view, is useful for computational purposes.

Uniqueness and Multiplicity of Equilibria

The desirable property of uniqueness of equilibria has been extensively surveyed: see Arrow & Hahn (1971). It is seen that global uniqueness, while desirable is a property which is excessively strong: the global uniqueness of equilibria can only be derived under highly restrictive assumptions. Clearly then, uniqueness of equilibria must be established by diluting or weakening some of the more restrictive assumptions

Debreus 1970 paper made the break. He showed that the number of equilibria for almost all exchange economies with continuously differentiable demand funtions was finite, and moreover, that the equilibria were locally unique. Furthermore, by introducing the notion of regular and singular economies he established that the set of economies with an infinite number of equilibria has a closure whose Lebesgue measure is zero.

To explore the issues associated with the uniqueness and multiplicity of equilibria. It is necessary to exploit fully the structure of the equilibrium manifold and its associated topological properties. The following outlines some of the ideas of connectedness which has already been mentioned briefly. It can be shown that uniqueness of equilibrium can be established for Pareto optimal economies. For an alternate proof of Debreus finiteness theorem see Dierker & Dierker (1972).

Extensions to the Differential Framework

It must be emphasised that only one member of a very large family of general equilibium models has been examined namely the simple pure exchange economy with a fixed number of l commodities and a fixed number of m consumers. Needless to say, there is ample room for further extensions. Rather, the initial intention was to introduce the nature of the differential approach, and give a characterisation of some of its features. Some of the aims and results of the differential approach have been outlined in a pure exchange setting. Indeed, any investigation of these issues would require several volumes - instead some of the more important developments and extensions will be mentioned.

There are several aspects of economics that are impossible to ignore. Production is one such feature and its inclusion in the general equilibrium model is so obvious as to obviate the need for justification. Arrow-Debreu (1954) included it in their model and its inclusion in the differential framework not only makes sense, but is entirely logical and consistent with earlier developments in the subject. Taking uncertainty into account is another natural extension. Debreus (Debreu 1954, Ch 7) concept of a contingent commodity embodies in its definition the state of nature, the realisation of which is necessary and sufficient for the contingent commodity to actually be delivered.

The Core of an Economy

One very important theme in the development of mathematical economics in recent years has been the study of economies with large numbers of agents. This has been helped by the introduction into economics of game theoretic concepts such as the core and by measure theory. The core was first named by Gillies and Shapley (Aumann 1964) who studied properties of the von Neumann-Morgentern solutions. The core consists of that set of allocations which can be improved upon by no coalition of agents.

The relationship between the set of Walras (competitive) allocations W() and the set of core allocations C() has been the subject of much research. The usual rough statement is that the core approaches the set of equilibrium allocations as the number of agents tends to infinity. The two sets satisify the mathematical trivial but economically important relationship (Debreu & Scarf 1963)

W() C()

Thus a state of an economy decentralised by an equilibrium price system cannot be improved upon by co-operation of individual agents. This strengthens the well known first theorem of welfare economics which states that every equilibrium allocation is Pareto optimal.

The equivalence of the set of Walras allocations and the core has been established by Debreu & Scarf (1963) who consider m types of each of r consumers - the equivalence is proved as r tends to infinity (m fixed) given the equal treatment property that allocation in the core assign the same consumption to all consumer of the same type.

The notion of perfect competition is fundamental in the study of economic equilibrium. A mathematical model appropriate to the intuitive notion of perfect competition is one which has a continuum of agents. Aumann (1964) notes that the reason for adopting this model is that one can integrate over a continuum, and that is, the actions of a single individual agent are negligible. Aumann introduces a atomless measure space of agents[3] and the chief mathematical tools used are those of Lebesgue measure and integration, but only their most elementary properties are needed. Aumann proves that for a continuum of traders

W() C()

Basing his proof on one given by McKenzie (1959), he proves the existence of equilibrium for a market with a continuum of traders, thus establishing the non-emptiness of the core (Aumann 1966). Here, however, no convexity assumptions are needed to prove the existence in some sense the individually insignificant traders have a convexifying effect on the aggregate.

The mathematical notion that an economy becomes more competitive as the number of agents increases has led to questions concerning the rate of this convergence. Debreu (1975) considers an economy , whose n-replica n has each agent repeated n times. Cn is the core of En and W are the Walras allocations of and therefore of En. Debreu proves

Theorem 4 (Debreu)

For a regular economy , as n tends to infinty the Hausdorff distance

(Cn,W) 0 ( 1/n )

That is to say that n(Cn, W) is bounded. In other words, the distance between the core of n , and its set of Walras allocations converges to zero at least as fast as then inverse of the number of agents. Aumann (1979) notes that the generic nature of Debreus convergence theorem leaves open the possibility of cores converging as slowly as one wishes.

Global Analysis: Smales approach

The approach adopted by Smale to the study of general economic equilibrium was to put the main results of classical general equilibrium theory in a global calculus context. There has been a movement away from calculus during the 1950s recall that Debreu (1959) in his preface to Theory of Value set out to free mathematical economics from the traditions of differential calculus. One limitation of subsequent developments in the theory was in the inadequate treatment of how equilibrium was reached. Smale (1976, p 290) notes that one step in the liberation from the static point of view would be to use a mathematics of a different kind. The liberation that Smale speaks of is the liberation from the mathematics of fixed point theorems and such like. Interestingly, Debreu returned to calculus foundations in his 1970 paper on the finiteness of the number of equilibria. (Debreu 1970)

Smale (1976) enumerates a number of reasons for this reversion to calculus foundations. Dynamic questions are more accessible via calculus formulations. Comparative statics are integrated in the framework in a natural way as they depend on first derivatives. Mathematical approximations by differentiable (C) functions gives further justification to the use of calculus. It must be remembered that this particular approach is closest to the oldest traditions of the subject found in Hicks (1939) and Samuelson (1947). The goal of Smale was thus to approach equilibrium theory with mathematics with dynamic and algorithimic overtones.

Computation of Equilibrium

Although the first proof of existence is now close to sixty years old it is only in the past thirty years that attempts have been made to develop algorithims for the computation of Walras equilibria. Scarf (1967 & 1973) showed that how to compute an approximate Walras equilibrium and proposed a general algorithm for the calculation of a fixed point of a correspondence. This algorithm has been surprisingly efficient, though does not permit a gradual improvement in the degree of approximation of the solution. This problem was overcome by Eaves (1974)


The results of the differential approach have been extended and modified in different directions: the above represents only a small sample of the research that has been conducted in equilibrium analysis. Demand functions, which have been assumed to be fixed have been weakened and the number of agents has been increased. The computation of approximate economic equilibria, and the introduction of dynamics into the theory have progressed the mathematical development of the theoryof general equilibrium, providing a richer theoretical foundation for examining issues concerning the properties of equilibria.


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