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.

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 *} i*s* 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* f*i (*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

-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*.

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.*

*g*i* : l * *S *x : *g*i (*x*i) = (*grad*n*
u*i* *(*x*i) , *x*i *grad*n *u*i (*x*i))

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 = (y*1* , . . . . , y*n* ) 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 = (z*1*,
. . . ,z*n*) 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 = (y*1* , . . . . , y*n* ) 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 = (z*1*
, . . . . ,z*n* ) 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.

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

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**.

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

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.

*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.

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).

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 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.

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.

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