Globally and Universally Convergent Price Adjustment Processes

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Introduction
The simplest price adjustment process studied in general equilibrium theory is the Walrasian tâtonnement process.The famous examples of Scarf (1960) made it apparent that Walrasian tâtonnement may not converge to a competitive equilibrium.The work of Sonnenschein (1972Sonnenschein ( , 1973)), Mantel (1974) and Debreu (1974), basically claiming that any continuous function satisfying homogeneity of degree zero and Walras' law is the excess demand function of some economy, makes clear that the examples of Scarf (1960) do not correspond to exceptional cases and that it is possible to construct many examples where Walrasian tâtonnement does not converge and displays highly irregular dynamic behavior.The work of Saari and Simon (1978) and Saari (1985) points out that simple adaptations of the Walrasian tâtonnement process will not have better convergence properties.
Still, at least three universally convergent price adjustment processes are known in the literature: Smale's global Newton method introduced in Smale (1976), the process of van der Laan and Talman (1987), and the process proposed in Kamiya (1990).Such processes can also be used to compute equilibria, thereby enabling comparative statics exercises as well as policy recommendations.See also Judd (1997) and Eaves and Schmedders (1999) for further elaborations on this point.
The global Newton method of Smale adjusts prices in such a way that the vector of excess demands remains proportional to the vector of excess demands at the initial price system.The method does converge to a competitive equilibrium for almost any economy, but it does not do so for any initial price system.Only when the initial price system is chosen such that the prices of some commodities are sufficiently close to zero, convergence to a competitive equilibrium can be guaranteed.The work by Keenan (1981) makes clear that there may exist an open set of starting price systems for which Smale's process does not converge to a competitive equilibrium.
Another universally convergent price adjustment process has been presented in Kamiya (1990).Kamiya's process can be interpreted as a weighted average of Smale's global Newton method and Walrasian tâtonnement.At the initial price system, the process corresponds to Walrasian tâtonnement, whereas it approaches Smale's global Newton method when prices are close to a Walrasian equilibrium.Under rather weak conditions on the excess demand function, convergence to a competitive equilibrium price system is guaranteed for almost every starting price system.Although the boundary conditions of Kamiya (1990) are weak, they are still more demanding than boundary conditions that are derived from assumptions on the primitives of the model.
Yet another price adjustment process has been proposed in van der Laan and Talman (1987).In this process, prices of commodities with positive excess demand are kept relatively maximal, that is relative to the initial price system, whereas prices of commodi-ties with negative excess demand are kept relatively minimal.For this process universal convergence has been shown in Herings (1997).Under standard conditions on utility functions, consumption sets, and initial endowments, this price adjustment process converges to a Walrasian equilibrium price system for almost all economies and any starting price systems.
To show convergence of these processes, one first defines a set of price systems for which excess demands satisfy the properties of the price systems generated by the price adjustment process.For instance, for Smale's global Newton methods, this would be the set of price systems at which excess demand is proportional the excess demand at the initial price system.Next one shows that, for generic economies, this set consists of an arc connecting the initial price system and precisely one competitive equilibrium as boundary points, a finite number of arcs containing precisely two competitive equilibria, both being boundary points, and a finite number of loops that neither contain the initial price system nor any competitive equilibrium.The proofs for the van der Laan and Talman (1987) process have to deal with the complication that the path of prices generated by the process displays non-differentiabilities whenever one of the markets gets equilibrated during the process.To deal with such issues we need tools from differential topology like regular constraint sets and manifolds with generalized boundary.We explain these tools in detail as well as how they are applied in the convergence proof of the van der Laan and Talman (1987) process.
The price adjustment processes of Smale (1976), van der Laan and Talman (1987), and Kamiya (1990) are defined for exchange economies.The paper concludes with an overview of extensions of such processes to production economies and to processes where not only prices but also quantities may adjust.It also makes the connection to strategy adjustment processes in game theory.Although all processes discussed admit a natural economic or game-theoretic interpretation, they may also be considered as algorithms to compute an equilibrium.Finally, we briefly discuss the role of universally convergent price adjustment processes in experimental economics.
The organization of the paper is as follows.Section 2 provides the general set-up and Sections 3, 4, and 5 are devoted to the processes of Smale (1976), Kamiya (1990), and van der Laan and Talman (1987), respectively, which are defined by the properties of the orbits that they generate.Section 6 explains how these orbits can be obtained as the solutions to an appropriate system of differential equations.Section 7 presents an overview of the tools from differential topology that are needed for the convergence proofs and Section 8 explains how these tools are used in the convergence proof for the van der Laan and Talman (1987) process.Section 9 concludes the paper by discussing some further developments.

Price Adjustment Processes
We consider an economy E where L commodities are traded.Excess demand for the commodities in this economy as a function of prices is given by a function z : P → R L , where P denotes the set of possible price vectors.Typical choices for , the non-negative orthant of R L excluding the zero vector, or the set of strictly positive price vectors The excess demand function z satisfies homogeneity if, for every p ∈ P, for every λ ∈ R such that λp ∈ P, z(λp) = z(p).Because of homogeneity, it is possible to normalize prices.In some of the price adjustment processes studied in this paper, it is convenient to require the sum of the prices to be equal to one and take P equal to the unit simplex , or to consider prices whose squares sum up to one and take P equal to Apart from homogeneity, the excess demand function is assumed to satisfy Walras' law, the property that, for every p ∈ P, p • z(p) = 0.
A Walrasian equilibrium is p * ∈ P such that z(p * ) = 0 L .We are interested in the question whether processes of price adjustment terminate in a Walrasian equilibrium, i.e., converge to a price vector at which excess demand of all commodities is equal to zero when starting from a price vector that does not correspond to a Walrasian equilibrium.
A standard way to define a price adjustment process is by means of a system of firstorder differential equations where p(t) ∈ P denotes the price vector reached at time t ∈ R + and f is a function from P into R L .The initial price vector p(0) is assumed to be given.The function f determines the way in which prices adjust.A common specification is that f is in proportion to excess demand at the price system p.This results in a price adjustment process that is known as Walrasian tâtonnement, described in Walras (1874), and formalized in Samuelson (1941).
Conditions for which the system of differential equations has a solution are well-known, see for instance Hirsch and Smale (1974) for details.The orbit γ(p(0)) is the set of price vectors that is generated by the system of first-order differential equations when the initial state is p(0), The closure of γ(p(0)) is denoted by γ(p(0)) and is called an orbit as well.
Rather than describing the three price adjustment processes of interest in this paper by a system of differential equations, it will turn out to be more convenient to describe them by the orbit that they generate.The next three sections describe this orbit as the solution to a particular system of equations that has one degree of freedom, where each section is devoted to one particular price adjustment process.

Smale's Global Newton Method
To describe Smale's global Newton method, we take P = ∆ L .The mean excess demand at We make the following assumption on the excess demand function z throughout this section.
Assumption 3.1: The excess demand function z : ∆ L → R L is twice continuously differentiable, satisfies Walras' law, and has the following boundary behavior: For every p ∈ ∆ L with at least one zero component, z(p) − z(p)1 L is not radially outward pointing, i.e., there is no Assumption 3.1 is weaker than the assumptions in Smale (1976), where a stronger condition on boundary behavior is stated.The assumption that for every p ∈ ∆ L with at least one zero component z(p)−z(p)1 L is not radially outward pointing is satisfied if, for every p ∈ ∆ L with at least one zero component, for some = 1, . . ., L with p = 0, it holds that z (p) > 0, a rather natural requirement.To see this, observe that by Walras' law there is with p > 0 and z (p) ≤ 0, so z (p) − z(p) > z (p) − z(p), whereas −1/L = p − 1/L < p − 1/L, which is easily seen to imply that z(p) − z(p) is not radially outward pointing at p. Smale (1976) discusses a number of variations when defining the global Newton method.Here we follow Herings (2002) and combine the approaches as suggested in Smale (1976), p. 117, andVarian (1977).Let a disk containing ∆ L in its interior, and let π : D L → ∆ L be the radial projection on ∆ L .For p ∈ ∆ L the radial projection of p on ∆ L is equal to p and for p ∈ D L \ ∆ L , the radial projection of p on ∆ L is given by the price system at which the line between p and (1/L)1 L intersects the relative boundary of ∆ L , so We apply Smale's global Newton method to the function z : The function z extends a function equal to a positive multiple of z − z1 L from ∆ L to D L .The vector z(p) is a weighted sum of the terms π(p) − p and z( π(p)) − z( π(p))1 L , with weight 1 on the former term and non-negative weight (1 − p − (1/L)1 L 2 ) on the latter term.Note that π(p) − p vanishes on ∆ L , while the contribution of the term (1 L is equal to zero on the relative boundary of D L , where it holds that p − (1/L)1 L 2 = 1.The function z is therefore radially inward pointing on the relative boundary of D L .
We argue next that the zero points of z and z coincide, so when a price adjustment process finds a zero point of z, it has found a Walrasian equilibrium.Let p belong the relative boundary of D L .The term 1 − p − (1/L)1 L 2 vanishes and the remaining term is π(p) − p, which is clearly not equal to zero.Let p ∈ D L \ ∆ L be such that it does not belong to the relative boundary of D L .Then z(p) = 0 L if and only if where the second equality follows from the definition of the radial projection at a price system in D L \ ∆ L .We find that z is radially outward pointing at π(p), a contradiction to Assumption 3.1.Finally, let p ∈ ∆ L .Then z(p) is a positive multiple of z(p) − z(p)1 L , so z(p) = 0 L if and only if z(p) = 0 L .Take a starting price system p 0 in the relative boundary of D L .Smale's process generates prices at which the excess demand is a non-negative multiple of the excess demand at p 0 .The orbit generated by Smale's process therefore belongs to the set The choice of θ = 1 shows that p 0 ∈ P and the choice of θ = 0 guarantees that p * ∈ P if p * is a Walrasian equilibrium.
In general, the set P consists of multiple components.Suppose the component of P containing p 0 is homeomorphic to an interval, containing p 0 and a unique Walrasian equilibrium as its boundary points.The price adjustment process then generates the prices of that component and terminates in a uniquely specified Walrasian equilibrium.Smale (1976) shows that, for a generic economy, the component of P containing p 0 is indeed homeomorphic to an interval and connects p 0 to a Walrasian equilibrium.
Instead of defining Smale's global Newton method by the orbit that it generates, we can also describe it by a system of differential equations.To do so, drop the last component of z and denoted the resulting function by z.Consider the following system of differential equations: where λ is an arbitrary scalar function of p such that so the price of commodity L is adjusted on the basis of the same principles as the prices of the other commodities.The equation 1 L dp dt = 0 makes sure that the sum of the prices is kept equal to one, so prices remain in D L .The fact that z is radially inward pointing on the relative boundary of D L makes sure that prices move towards the relative interior of D L .It is now immediate that the orbit of the system of differential equations belongs to the set P .
The convergence of Smale's global Newton method to a Walrasian equilibrium depends crucially of the choice of p 0 in the relative boundary of D L .Keenan (1981) shows that convergence may not hold for starting price systems in the relative interior of D L .This is problematic from both a computational and an economic point of view.From a computational point of view, it makes sense to take the initial price system such that it is expected to be close to a Walrasian equilibrium price system.There is no reason to expect that such a price system is located near the relative boundary of D L .From an economic point of view, it would be natural to take the equilibrium prices of the previous period as a starting point of the adjustment process to find the equilibrium prices of the current period.Again, there is no reason to expect the previous period's equilibrium prices to be close to the relative boundary of D L .
4 The Price Adjustment Process of Kamiya Kamiya (1990) introduces a price adjustment process that can be interpreted as a weighted average of Smale's global Newton method and Walrasian tâtonnement.This process addresses the problem that Smale's process may not converge when p(0) belongs to the relative interior of D L .For Kamiya's process it is convenient to normalize commodity prices by taking the sum of the squares of prices equal to one, so P = S L .
We make the following assumption on the excess demand function throughout this section.
Assumption 4.1: The function z : S L → R L is twice continuously differentiable, satisfies Walras' law, and has the following boundary behavior: For every p ∈ S L , for every = 1, . . ., L, p = 0 implies z (p) > 0.
We have argued in Section 3 that the boundary behavior on z imposed in Assumption 4.1 is more demanding than the boundary behavior required in Assumption 3.1.
Walras' law in combination with Assumption 4.1 implies that we may drop the last component of z in the search for an equilibrium.Also, it is possible to represent the price system by the first L − 1 components only, and take the price system in the set We can then replace the function z : S L → R L by the function z : ṠL−1 → R L−1 , where The function z is obtained by omitting the last component of z and making use of the price normalization.Take a starting price system p 0 in the interior of ṠL−1 .In Kamiya's process the prices are adjusted in such a way that z(p) is proportional to p − p 0 .The orbit generated by Kamiya's process therefore belongs to the set The choice of θ = 1 yields p 0 as the unique solution, so p 0 ∈ P .
Again, suppose the component of P containing p 0 is homeomorphic to an interval, containing p 0 and a unique Walrasian equilibrium as its boundary points.The price adjustment process then generates the prices of that component and terminates in a uniquely specified Walrasian equilibrium.Kamiya (1990) shows that, for a generic economy, the component of P containing p 0 is indeed homeomorphic to an interval and connects p 0 to a Walrasian equilibrium.The considerable advantage of Kamiya's price adjustment process over the global Newton method, both form a computational and an economic point of view, is that it converges for any initial price system.
Instead of defining Kamiya's price adjustment process by the orbit that it generates, we can also describe it by a system of differential equations.If we denote the (L − 1) × (L − 1) identity matrix by I, then at prices p different from p 0 such that z(p) = 0 L−1 , Kamiya's process is defined by where λ is an arbitrary scalar function of p such that At p = p 0 and at p ∈ ṠL−1 such that z(p) = 0 L−1 , the price adjustment process is defined by taking a limit.Kamiya's process can be seen to be a weighted average of Smale's global Newton method, ∂ z(p) dp dt = −λ(p) z(p), and Walrasian tâtonnement, dp dt = z(p).The weights depend on the norm of the excess demand and the distance between p and p 0 .The process is approximately equal to Walrasian tâtonnement at prices close to p 0 and it approaches Smale's global Newton method when prices are close to a Walrasian equilibrium.

The Price Adjustment Process of van der Laan and
Talman Although Kamiya's process displays generic convergence to a Walrasian equilibrium for any initial price system, the required boundary behavior, though fairly natural, does not follow from standard assumptions on primitives like consumptions sets, utility functions, and initial endowments.This section presents a price adjustment process proposed by van der Laan and Talman (1987), which displays generic global and universal convergence, i.e., convergence takes place for any initial price system and for excess demand functions that can be derived from standard assumptions on the primitives.
To describe the price adjustment process of van der Laan and Talman (1987), we normalize prices of commodities to belong to ∆L , i.e., the relative interior of the unit simplex.
We make the following assumption on the excess demand function throughout this section.
Assumption 5.1: The function z : ∆L → R L is twice continuously differentiable, satisfies Walras' law, and has the following boundary behavior: The boundary behavior postulated in Assumption 5.1 requires that excess demand explodes when one of the commodity prices converges to zero.Contrary to Assumption 4.1, it is possible to derive Assumption 5.1 from standard assumptions on consumption sets, utility functions, and initial endowments.
Take a starting price system p 0 in ∆L .The dynamics of the price adjustment process can be described as follows.First, the sign of the excess demand is evaluated at the starting price system p 0 .Typically it holds that for every commodity = 1, . . ., L, z (p 0 ) = 0. Initially, the price of a commodity with z (p 0 ) < 0 is decreased, while the price of a commodity with z (p 0 ) > 0 is increased.The ratio of prices of commodities for which there is a negative excess demand is kept constant among those commodities for which there is a negative excess demand, and similarly for the ratio of prices of commodities for which there is a positive excess demand.This generates a line segment of prices.The price system is adjusted in this way until a market, say the market for commodity 1 , attains an equilibrium.Typically, there is exactly one market for which this happens.
The price adjustment continues by keeping the market for commodity 1 in equilibrium, while the price p 1 is relatively increased (decreased) if there was a negative (positive) excess demand in the market for commodity 1 before attaining equilibrium.Other prices are kept relatively minimal in case of a negative excess demand and relatively maximal in case of a positive excess demand.In general this generates a 1-dimensional curve of prices.Two situations can occur.Either another market, say the market of a commodity 2 , attains an equilibrium.In this case the price system is adjusted in such a way that the markets of commodities 1 and 2 are kept in equilibrium, while the price of commodity 2 is relatively increased (decreased) when there was a negative (positive) excess demand on the market of commodity 2 before attaining equilibrium.Or the price on the market of commodity 1 becomes relatively minimal or maximal.In this case the market of commodity 1 is no longer kept in equilibrium but its excess demand is allowed to become negative or positive, respectively, while p 1 is kept relatively minimal or relatively maximal, respectively.The price adjustment process continues in this way until a Walrasian equilibrium is reached.
Formally, the orbit generated by van der Laan and Talman's process belongs to the set (5.1)For a price system to belong to P it should hold that the relative price of a commodity, i.e. the ratio of the price of a commodity and its initial price, is minimal if the commodity is in positive excess supply and it is maximal if the commodity is in positive excess demand.This is closely related to the ideas behind Walrasian tâtonnement, where prices of commodities in positive excess supply are decreased and those of commodities in positive excess demand are increased.The starting price system p 0 belongs to P as all relative prices are equal to one at p 0 , so the relative prices of all commodities are both minimal and maximal at the same time.A Walrasian equilibrium p * belongs to P , since if z(p * ) = 0 L then all implications in the definition of P are trivially met.Herings (1997) shows, for a generic economy, that the component of P containing p 0 is indeed homeomorphic to an interval and connects p 0 to a Walrasian equilibrium.Like Kamiya's process, there is convergence to a Walrasian equilibrium for any starting price system.There are two differences with Kamiya's process.First, standard assumptions on the primitives suffice to guarantee convergence to a Walrasian equilibrium.Second, whereas the orbit of Kamiya's process is generically a differentiable manifold with boundary, the orbit of van der Laan and Talman's process is generically a piecewise differentiable manifold with boundary.
It is also possible to present the price adjustment process of van der Laan and Talman (1987) as the solution to a system of differential equations.In fact, it is possible to do so for any piecewise differentiable orbit.The next section explains how to do this in general.

From Orbits to Systems of Differential Equations
After making some minor modifications to the formulations in Sections 3 and 4, it is possible to formulate the orbits of the processes of Smale (1976) and Kamiya (1990) as the zero points of a function f : [0, 1] × P → R L , where the set P is L-dimensional.The explicit construction for Smale's process is presented later in this section.The variable in the interval is denoted by λ and is equal to 1 − θ for the processes of Smale and Kamiya.The variable λ is not explicit in the formulation of van der Laan and Talman's process, but as demonstrated in Herings (2002) it can be taken equal to max k, ∈L (p k /p 0 k − p /p 0 ) and belongs to the interval [0, max =1,...,L 1/p 0 ].It is easy, though not essential, to renormalize prices to ensure that λ belongs to the interval [0, 1] as in the other two processes.
The orbits generated by the processes of Sections 3, 4, and 5 are well-behaved sets under suitable assumptions.We now ask the question whether it is possible to find a system of differential equations that generates a given orbit.Let (λ(t), p(t)) be the orbit corresponding to the zero points of f, parametrized by arc length t.The starting point of the price adjustment process equals p(0) and corresponds to parameter λ(0) = 0. Assume that zero is a regular value of both f and of the restriction of f to {0, 1} × P. Let J(λ, p) denote the Jacobian of f evaluated at (λ, p), a matrix of dimension L × (L + 1).The regularity assumption implies that the Jacobian has rank L. It also implies that there is a unique vector v(J(0, p(0))) ∈ R L+1 such that J(0, p(0)) • v(J(0, p(0))) = 0 L , v(J(0, p(0))) 2 = 1, and the first component of v(J(0, p(0))) is positive.For p ∈ P, let v(J(λ, p)) denote the unique vector such that J(λ, p) .
It can be shown that the orbit of zero points induced by f is generated by the system of differential equations see for instance Allgower and Georg (1983).This system of differential equations was first proposed by Davidenko (1953) and is also referred to as the system of Davidenko equations.We now illustrate that using the Davidenko equations immediately leads to Smale's process.The function f : [0, 1] × P → R L whose zero points correspond to the orbit of Smale's process is obtained by taking P = R L + \ {0 L } and defining, for (λ, p) It follows that The Davidenko equations specify that from which it follows that Substituting z(p) = (1 − λ) z(p 0 ) leads to the specification of Smale's process in Section 3.

Regular Constraint Sets and Manifolds with Generalized Boundary
In Sections 3, 4), and 5 it was asserted that the orbits generated by the various price adjustment processes are generically well-behaved sets.Section 6 required 0 L to be a regular value of f, the function whose zero points describe the orbit, and of the restriction of f to {0, 1} × P.This section presents the tools that can be used to give precise definitions of "well-behaved" and to demonstrate that certain sets are actually well-behaved.The exposition is based on Section 2.10 of Herings (1996).For further elaborations of this material, the reader is refererred to Milnor (1965), Golubitsky and Guillemin (1973), Jongen, Jonker, andTwilt (1983, 1986), and Mas-Colell (1985).
We use the notation N = {0, 1, . ..},N + = {1, 2, . ..},N * = N ∪ {∞}, and N * + = N + ∪ {∞}.Let X and Y be topological spaces.The set of continuous functions from X into Y is denoted by C 0 (X, Y ).Let U be an open subset of R m and V a non-empty subset of R n .For r ∈ N + , C r (U, V ) is defined as the set of r times continuously differentiable functions from A k-dimensional topological manifold, where k ∈ N, is a set which is locally like R k .More formally, we have the following definition.
Definition 7.1 For k ∈ N, a subset X of R m is a k-dimensional topological manifold if for every element x of X there exists an open subset U of X containing x, an open subset V of R k , and an injective and surjective function ϕ : U → V such that ϕ ∈ C 0 (U, V ) and ϕ −1 ∈ C 0 (V, U ), i.e., ϕ : U → V is a homeomorphism.
Every neighborhood of a point in a k-dimensional topological manifold can be continuously deformed via a function ϕ into a set which looks like R k and vice versa.Since only continuity is required, such a manifold can have kinks.For the special case where k = 0, the definition implies that a 0-dimensional topological manifold is a discrete set of points.The function ϕ is called a coordinate system for X around x.When the coordinate systems ϕ are required to be C r diffeomorphisms instead of homeomorphisms, we obtain the definition of a C r manifold.

and an injective and surjective function
It follows from Definition 7.2 that the empty set is a k-dimensional C ∞ manifold for every k ∈ N. When the coordinate systems are required to be differentiable as in Definition 7.2, it is not possible for the manifold to have kinks.
We next generalize the concept of a C r manifold to a piecewise C r manifold.
In Definition 7.3 it is allowed that the dimension of some of the C r manifolds whose union is equal to X is less than k.A piecewise C r manifolds can have kinks, but these kinks are restricted to occur at a finite number of lower-dimensional manifolds.
For k ∈ N, for r ∈ N * + , a characterization of a k-dimensional C r manifold is given in the following theorem.Some authors use this characterization as a definition of a kdimensional C r manifold.A function ϕ : To study globally and universally convergent price adjustment processes, the notion of a manifold is too restrictive.Typical choices for P like R L + \ {0 L } or ∆ L are not even topological manifolds because they have a boundary.The same is true for the various sets P defined in Sections 3, 4, and 5.For the set ∆ L it holds that even the boundary has a boundary when L ≥ 3. It turns out that the formulation of a manifold presented in Theorem 7.4 can be easily generalized to deal with manifolds that have boundaries, and where the boundaries may have boundaries themselves, and so on.
The neighborhood of an element x in a k-dimensional MGB X looks like the set R For k ∈ N, for r ∈ N * + , let the set X be a k-dimensional C r MGB.For j ∈ {0, . . ., k}, define the set B j (X) by For every j ∈ {0, . . ., k}, a path-component of B j (X) is called a stratum of X.A stratum in B j (X) of a C r MGB is a (k − j)-dimensional C r manifold.The collection of strata of X forms a partition of X.The set B 0 (X) is called the relative interior of X and the set X \B 0 (X) is called the relative boundary of X.
, then X is called a manifold with boundary.The following theorem reveals that compact 1-dimensional MGB's have a particularly nice structure.
Theorem 7.6 For r ∈ N * + , let the set X be a compact 1-dimensional C r MGB.Then X has a finite number of components, each being C r diffeomorphic to either the unit circle or the unit interval.
A set which is homeomorphic to the unit circle is called a loop and a set homeomorphic to the unit interval is called an arc.Theorem 7.6 therefore implies that a compact 1dimensional C r manifold with generalized boundary consists of a finite number of arcs and loops.
Many sets that play a role in economic theory are so-called regular constraint sets.Let U be an open set of R m and let, for some n 1 ∈ N, for every i ∈ {1, . . ., n 1 }, g i : U → R, and, for some n 2 ∈ N, for every i ∈ {1, . . ., n 2 }, h i : Notice that n 1 and n 2 are allowed to be zero in the definition above.For every element x of U, define the set The next result shows the relation between an RCS and an MGB.
Theorem 7.8 For r ∈ N * + , let the subset X of R m be a C r RCS and let (g, h) be a C r regular constraint system such that M [g, h] = X.If g has n 1 components, then X is an (m − n 1 )-dimensional C r MGB and, for every x ∈ X, (x) = #I 0 (x).
Theorem 7.8 is very convenient to show that a certain set is an MGB.Let us illustrate this for the unit simplex ∆ L .Define the functions g : R L → R and, for every i ∈ {1, . . ., L}, h i : R L → R by It is easy to verify that (g, h) is a C ∞ regular constraint system.Since M [g, h] = ∆ L , it holds that the unit simplex ∆ L is a C ∞ RCS and by Theorem 7.8 a C ∞ MGB.
Next, the tangent space and the tangent cone of a manifold with generalized boundary are defined.
Definition 7.9 For k ∈ N, for r ∈ N * + , let X be a k-dimensional C r MGB.Let x be an element of X and let ϕ be a C r coordinate system for R m around x.The tangent space of X at x, denoted by T x X, is the set ∂ϕ It can be shown that both the tangent space T x X and the tangent cone C x X as defined in Definition 7.9 do not depend on the choice of the coordinate system.It is easily verified that a tangent cone is indeed a cone.Since ϕ −1 is a C r diffeomorphism, it holds that ∂ϕ −1 (0 m ) is an invertible matrix and therefore T x X is a k-dimensional vector space.The tangent cone C x X is a k-dimensional MGB.
In case a set X is an RCS, the following theorem gives an easy way to determine the tangent space of X at an element x of X.
Theorem 7.10 For k ∈ N, for r ∈ N * + , let the subset X of R m be a k-dimensional C r RCS, let x be an element of X, and let the pair of functions (g, h) be a regular constraint system such that For r ∈ N * + , let the subsets X of R m and Y of R n be C r manifolds, let x be an element of X, and let f be a function of C r (X, Y ).Let U be an open set of R m such that X ⊂ U and let the function g ∈ C r (U, R n ) be such that, for every x ∈ X, g(x) = f (x).It can be shown that ∂g(x) |T x X is a function from T x X into T f (x) Y.Moreover, the function ∂g(x) |T x X does not depend on the choice of the function g.The derivative of f at x, denoted by ∂f (x), is defined by ∂f Otherwise x is called a critical point of f.Let an element y of Y be given.The element y is called a critical value of f if it is the image of a critical point of f.Otherwise y is called a regular value of f.Notice that every element y of Y \ f (X) is a regular value of f.
For r ∈ N * + \ {1}, for m ∈ N + \ {1}, let the subset X of R m be an m-dimensional C r manifold with boundary.Let x be an element of B 1 (X).To each element x we associate the vector g(x) of R m satisfying ∀x ∈ T x B 1 (X), g(x) • x = 0, ∀x ∈ C x X, g(x) • x ≤ 0, and g(x) 2 = 1.It is easily verified that the vector g(x) is uniquely determined.Obviously, it belongs to the set ), obtained by associating with every element x of B 1 (X) the vector g(x), is called the Gauss map of B 1 (X).The function g is continuously differentiable and, for every x ∈ B 1 (X), ∂ g(x) is a function from T x B 1 (X) into T x B 1 (X).For every x ∈ B 1 (X), the determinant of the linear function ∂ g(x), is called the Gaussian curvature of B 1 (X) at x.
Let C 1 manifolds X, Y, and Z, Z being a subset of Y, an element x of X, and a function The function f is said to intersect Z transversally if f £ Z at every x ∈ X.The following theorem follows almost immediately from the definition of transversality.
The following result is complementary to Theorem 7.11 and applies to the case where The following result is an easy corollary to Theorems 7.11 and 7.12.
To state the last result in this section, we need the notion of Lebesgue measure zero of a set in a manifold X.For k ∈ N, for r ∈ N * + , let the subset X of R m be a k-dimensional C r manifold and let S be a subset of X.Then the set S is said to have Lebesgue measure zero in X if there exists a countable cover {U n | n ∈ N} of S and, for all n ∈ N, a coordinate system ϕ n with domain U n such that ϕ n (U n ∩ S) has Lebesgue measure zero.In case X ⊂ R m is an m-dimensional C r manifold, then the notions of Lebesgue measure zero and Lebesgue measure zero in X coincide.

Convergence Proof
So far, an economy E was characterized by an excess demand function z : P → R L .We now go back to the primitives and derive the excess demand function from the consumption choices of a finite set M of consumers that face a budget constraint.
For every consumer i ∈ M, the consumption set X i and the preference relation i are assumed to be given in this section as is the starting price system p 0 ∈ ∆L .After choosing, for every i ∈ M, initial endowments ω i ∈ X i we obtain an economy E = ((X i , i , ω i ) i∈M ).To show that, generically in initial endowments, a price adjustment process is globally and universally stable, a number of standard assumptions on consumption sets and preference relations are made.

A1. For every consumer
Consider some consumer i ∈ M. The preference relation i satisfies the boundary condition if, for every x i ∈ X i , the closure of the set Let i be complete, transitive, continuous, monotonic, i.e., a consumption bundle containing more of every commodity than another consumption bundle is preferred to this other consumption bundle by consumer i, and of the class C r .For every x i ∈ X i , we define the sets as the sets of consumption bundles indifferent to x i and weakly preferred to x i , respectively, by consumer i.It holds that P i (x i ) is an L-dimensional C r manifold with boundary, with boundary equal to the (L − 1)-dimensional C r manifold I i (x i ).The real number c i (x i ) is defined as the Gaussian curvature of P i (x i ) at x i .The preference relation i is said to have non-zero Gaussian curvature if, for every x i ∈ X i , c i (x i ) = 0. Intuitively, i has non-zero Gaussian curvature if indifference curves are nowhere flat.A2.For every consumer i ∈ M, the preference relation i is complete, transitive, continuous, strongly monotonic, strongly convex, of the class C 3 , satisfies the boundary condition, and has non-zero Gaussian curvature.
If the economy E = ((X i , i , ω i ) i∈M ) satisfies Assumptions A1 and A2 and for every consumer i ∈ M it holds that ω i ∈ X i , then it can be shown that the total excess demand function z : R L ++ → R L of the economy E belongs to C 2 (R L ++ , R L ).Let (X i , i ) i∈M satisfy Assumptions A1-A2 and let p 0 ∈ ∆L be the starting price system.The set Ω of initial endowments is defined by Ω = i∈M R L ++ .The next definition introduces the notion of regularity for initial endowments ω ∈ Ω, which means that the set P ω as defined in equation (5.1) of Section 5 has a nice structure.Definition 8.1 Let (X i , i ) i∈M satisfy Assumptions A1 and A2 and let p 0 ∈ ∆L be the starting price system.The set of regular initial endowments Ω * is the set of initial endowments ω of Ω for which the components of the set P ω as defined in (5.1) for the economy E = ((X i , i , ω i ) i∈M ) with starting price system p 0 are given by 1. an arc containing p 0 and precisely one Walrasian equilibrium price system as boundary points, 2. a finite number of arcs containing precisely two Walrasian equilibrium price systems both being boundary points, 3. a finite number of loops containing neither p 0 nor any Walrasian equilibrium price system.
If initial endowments ω are regular, then the price adjustment process converges to a uniquely determined Walrasian equilibrium.Moreover, since one Walrasian equilibrium is connected to the starting price system, the other Walrasian equilibria come in pairs, and all Walrasian equilibria belong to P ω , the number of Walrasian equilibria is odd.
The result that we are after is that the set of non-regular initial endowments is small, both in a topological and in a measure theoretic sense.This is the content of Theorem 8.2, stated as Theorem 2.3 in Herings (1997).
Theorem 8.2 Let (X i , i ) i∈M satisfy Assumptions A1 and A2 and let p 0 ∈ ∆L be the starting price system.Then the set of non-regular initial endowments Ω \ Ω * has a closure in Ω with Lebesgue measure zero.
The proof of Theorem 8.2 conveys that the set of initial endowments ω ∈ Ω for which the set P ω is a compact 1-dimensional piecewise C 2 manifold, i.e., a 1-dimensional topological manifold that is a finite union of C 2 manifolds, some possibly of lower dimension, is such that its complement in Ω has a closure with Lebesgue measure zero.All these initial endowments are regular, as the conditions required for P ω in Definition 8.1 are clearly satisfied.
Theorem 8.2 confirms the well-known result of Dierker (1972) that, generically, an economy has an odd number of Walrasian equilibria.
For every ω ∈ Ω, for every s ∈ S, we define the set P ω (s) of price systems by
To show Theorem 8.2 it is convenient to define, for every s ∈ S, for every ω ∈ Ω, the set Q ω (s) by Let s ∈ S. Clearly, for every ω ∈ Ω, P ω (s) ⊂ Q ω (s), the difference between these two sets being that no inequality constraints are taken into account in the definition of Q ω (s).In Lemma 8.3 it is shown that there exists a subset Ω of Ω such that Ω \ Ω has Lebesgue measure zero and for every ω ∈ Ω the set Q ω (s) is a 1-dimensional C 2 manifold.Hence, it can be shown to consist of a number of disjoint sets that are diffeomorphic to either the unit circle or the open unit interval.
The proof of Lemma 8.4 involves similar techniques as the proof of Lemma 8.3.
Notice that the condition 0 (s) ≤ L − 3 is crucial since for an admissible sign vector s with 0 (s) = L − 2 a corresponding set Q ω (s, 1 , 2 ) is equal to the set of Walrasian equilibrium price systems in P ω (s) of the economy E = ((X i , i , ω i ) i∈M ).Clearly, it cannot be shown that this set is empty for almost every ω ∈ Ω.
The proof of Lemma 8.6 uses similar techniques as the proof of Lemma 8.3.
With this preliminary work out of the way, the proof of Theorem 8.2 consists of three parts.In the first part, which combines Lemmas 8.3,8.4,8.5,and 8.6, it is shown that, for almost every ω ∈ Ω, for every s ∈ S, the set P ω (s) is a compact 1-dimensional C 2 manifold with boundary.In this part of the proof, Lemma 8.5 is needed to show that, generically, at most one of the inequalities in (8.5)-(8.9)can be binding at the same time.In the second part, such a generic ω ∈ Ω is fixed, and the sets P ω (s) corresponding to the various sign vectors s ∈ S are linked together.Here Lemma 8.5 is crucial again, since it ensures that a boundary point of P ω (s) is also a boundary point of P ω (s ) for a uniquely determined sign vector s ∈ S \ {s}, unless the boundary point is equal to the starting price system p 0 or a Walrasian equilibrium price system p * .It then follows that for almost every ω ∈ Ω the set P ω consists of a finite number of arcs and loops.The argument corresponds to a non-linear version of the door-in door-out principle.There is a unique arc having the starting price system p 0 and a unique Walrasian equilibrium price system as boundary points.The other arcs have two Walrasian equilibrium price systems as boundary points, whereas the loops contain no Walrasian equilibrium price systems.Therefore, the set Ω \ Ω * has Lebesgue measure zero.In the third part of the proof it is shown that the closure in Ω of the set Ω \ Ω * has Lebesgue measure zero.For details of these three parts, the reader is referred to Herings (1997).

Further Developments
The price adjustment processes of Smale (1976), van der Laan and Talman (1987), and Kamiya (1990) have an appealing economic interpretation where the adjustment of prices is related to the values of excess demands.Another price adjustment process which admits such an interpretation was introduced in Joosten and Talman (1998), where prices of commodities with the highest excess demand are kept above their initial values, the prices of commodities with the lowest excess demand are allowed to be lower than their initial values, and the prices of other commodities are equal to their initial values.The literature has also considered various price adjustment processes for production economies.Economies with linear or constant returns to scale production were studied by van den Elzen (1993Elzen ( , 1997) ) and van den Elzen, van der Laan, and Talman (1994) and economies with non-convex production technologies by van den Elzen and Kremers (2006).Bénassy (1975), Drèze (1975), andYounès (1975) studied exchange economies where price adjustment is subject to rigidities.At a fix-price equilibrium, equality of supply and demand is achieved by adjusting quantities rather than prices.Chapter 11 of Herings (1996) presents a globally and universally converging quantity adjustment process that ends up at a fix-price equilibrium.Herings, van der Laan, Talman, and Venniker (1997), Herings, van der Laan, and Venniker (1998), and Herings, van der Laan, and Talman (1999) study processes where both prices and quantities adjust and eventually a Walrasian equilibrium is reached.
In this paper, the differentiable approach is emphasized.Herings (2002) presents an alternative approach towards the understanding of the universal convergence of price adjustment processes that does not rely on differentiability and points out that convergence can be understood from fixed-point theory, more precisely the fixed point theorem of Browder (1960).Methods from differential topology can be used to show that for a generic economy, where the primitives satisfy suitable differentiability assumptions, the initial price system is connected by exactly one path to a Walrasian equilibrium.Browder's theorem can be used to show that for every economy, where the primitives satisfy suitable continuity assumptions, the initial price system is always connected to a Walrasian equilibrium, though not necessarily by a uniquely specified path.Herings (2002) also extends these insights to strategy adjustment processes used in non-cooperative game theory like the tracing procedure of Harsanyi (1975) and the equilibrium selection procedure of McKelvey and Palfrey (1995).
In the tracing procedure, the players initially choose a best response against a given common prior distribution.Next, they lower the weight on the prior and give some weight to the initial best responses to form a new prior against which a best response is played.This process of updating the initial prior on the basis of best responses continues until the weight on the prior is zero and a Nash equilibrium is reached.Herings and Peeters (2001) show that the tracing procedure converges to a Nash equilibrium for a generic finite normal-form game.Herings and Peeters (2004) extend the tracing procedure to the class of stochastic games and show convergence to subgame perfect equilibrium in stationary strategies for a generic stochastic game.Jackson and Wolinsky (1996) studies network formation games and introduces the widely used concept of pairwise stability.A network is pairwise stable if it is robust against unilateral link deletion and bilateral link creation.Most of the literature considers unweighted networks, but recently, Bich and Morhaim (2020) study a weighted version, where the strength of each link is measured by a continuous variable.Agents can unilaterally decide to decrease the link strength, whereas it requires the consent of both agents to increase it.Bich and Morhaim (2020) prove that pairwise stable networks exist if all agents have quasi-concave and continuous utility functions.Herings and Zhan (2022) reformulate the network formation problem as a non-cooperative game played by the links and adapt the linear tracing procedure of Harsanyi (1975) to this problem.They show that for a generic network formation problem, a uniquely defined pairwise stable network is selected by the corresponding strategy adjustment process.
McKelvey and Palfrey (1995) study finite normal-form games and assume that each player's payoff is subject to random error.Their concept of quantal response equilibrium is consistent in the sense that all players maximize their utility given the choices made by the other players, and the utility maximizing behavior of a player, together with the error structure, leads to the mixed strategy against which the other players optimize.Quantal response equilibria are quite successful in describing the behavior of participants in experiments.McKelvey and Palfrey (1995) also consider a procedure similar to the tracing procedure to select a Nash equilibrium.Start with the quantal response equilibrium where choices are completely determined by the error terms and follow the path of quantal response equilibria that results when the error terms vanish.McKelvey and Palfrey (1995) show that for almost all games, a unique Nash equilibrium is selected in this way.
Among other frameworks, Tyson (2021) considers finite normal-form games where players are assumed to be exponential satisficers.For a satisficing player, there is a positive probability that the perceived payoffs of different strategies are equal, even when actual payoffs are different.The exponential satisficing model depends on a preference resolution parameter γ ≥ 0, where γ = 0 implies that all pure strategies are perceived indifferent and higher values of γ correspond to a higher probability that perceived preferences are in line with actual preferences.Tyson also studies the limit, when γ tends to infinity, of exponential satisficing equilibria, as a way to select Nash equilibria.
We have considered price adjustment processes with a natural economic interpretation and briefly discussed strategy adjustment processes with an appealing game-theoretic description.These processes can also be seen as algorithms to compute an economic or game-theoretic equilibrium and thereby contribute to the vast literature on equilibrium computation as pioneered by Scarf (1967), Kuhn (1968), andEaves (1972).To numerically follow the price adjustment processes presented in this paper, one can either resort to piecewise linear approximations of the problem of interest and then follow the exact price adjustment process for the piecewise linear approximation, or use predictor-corrector methods to approximately follow the price adjustment process for the exact problem.For numerical details, we refer the reader to Garcia and Zangwill (1981) and Allgower and Georg (2012).
There is some experimental support for price adjustment processes like the global Newton method.Asparouhova and Bossaerts (2009) performed experiments where a large pool of subjects trade in securities via a double auction mechanism.It turns out that security prices respond to excess demand of all securities, so not only to their own excess demand as in the Walrasian tâtonnement process.In fact, the authors provide evidence that the resulting price adjustment process coincides with the global Newton method.
Still, in other experimental designs, price adjustment seems to be described fairly well by Walrasian tâtonnement.For instance, in an economic environment based on Scarf's example, Anderson, Plott, Shimomura, and Granat (2004) find strong support for the hypothesis that price dynamics are mainly driven by a market's excess demand, and therefore prices follow the path predicted by Walrasian tâtonnement.This finding is corroborated in the experimental work by Gillen, Hirota, Hsu, Plott, and Rogers (2021), who find that price changes for a good are largely determined by excess demand in its own market with only second-order influences of excess demands in other markets.These findings raise the question whether globally and universally convergent price adjustment processes can be used to stabilize markets.The experimental work by Goeree and Lindsay (2016) suggests that this is the case.They also construct an economic environment based on Scarf's example, use the global Newton method to adjust prices, and observe convergence to a competitive equilibrium.