The existence of equilibrium in a simple exchange model
© Maćkowiak; licensee Springer 2013
Received: 11 January 2013
Accepted: 5 April 2013
Published: 18 April 2013
This paper gives a new proof of the existence of equilibrium in a simple model of an exchange economy. We first formulate and prove a simple combinatorial lemma and then we use it to prove the existence of equilibrium. The combinatorial lemma allows us to derive an algorithm for the computation of equilibria. Though the existence theorem is formulated for functions defined on open simplices, it is equivalent to the Brouwer fixed point theorem.
MSC: Primary 91B02; secondary 91B50; 54H25.
Consider an economy with n goods populated with a finite number m of consumers whose preferences , defined on , are continuous, strictly monotone and strictly convex.a Suppose also that each consumer possesses a stock of goods and that the (total) supply is positive, . Suppose that at each positive price vector each consumer i wants to maximize his/her preferences among affordable bundles of goods, i.e., he/she plans to buy a bundle of goods such that its value is not greater than the value of the disposable stock and is the best among affordable bundles: , , implies and it is not true that . The monotonicity of preferences implies that . Hence, at the given prices p, it holds , where is the (total) demand for goods at prices p. Plans of all consumers can come into effect only if - again by the monotonicity assumption on preferences. Does there exist an equilibrium price vector, i.e., a positive price vector p such that ? It is well known that the answer to that question is positive; see  for a survey of the basic existence results. It is obvious that p is an equilibrium price vector if and only if the difference vanishes. If we allow p to vary over the positive orthant of , we obtain the function z; the excess demand function of the economy. One can show that z is homogeneous of degree zero, continuous on the set of positive prices, it satisfies Walras’ law and a boundary condition, and it is bounded from below [, Theorem 1.4.4]. One can also show that if a function f defined on the positive orthant of possesses the properties listed in the previous sentence, then there exists an economy whose excess demand function z is different from f only on a neighborhood of the boundary of in and the set of equilibrium prices for z coincides with the set of zeros of f . In this work, we are going to use the excess demand approach to prove the existence of equilibrium [, Section 3]: we just impose conditions a function should possess to be the excess demand function of an economy and then we prove that there exists an equilibrium price vector.b The novelty of our approach is that we are proving the existence of equilibrium (see the theorem in Section 4) in a new and constructive way.c It is important to emphasize that we do not rely on the Sperner lemma [, p.19] to prove the result. Instead of that, we introduce a combinatorial lemma (Lemma 1) formulated for a special triangulation of a closed simplex only. The particular triangulation decreases generality of the lemma but is computationally advantageous [, p.65].d
In the next section, we introduce notation. Section 3 presents necessary notions from combinatorial topology and ends with the combinatorial lemma (Lemma 1). In Section 4, we define the notions of excess demand function and equilibrium, and then we derive some properties of excess demand functions. Finally, we prove the existence theorem. Section 5 contains an algorithm for computation of equilibria. In Section 6, we clarify some differences between the boundary condition we use (see Definition 1(3)) and the standard boundary condition met in the literature. We also present a connection between fixed points of continuous functions and equilibria (zeros) of excess demand functions. At the end of Section 6, we pose a few open questions.
Let ℕ denote the set of positive integers and for any let denote the n-dimensional Euclidean space, and , . Moreover, is the i th unit vector of the standard basis of , where . In what follows, for the set , where is the set of nonnegative real numbers, is the standard -dimensional (closed) simplex and is its (relative) interior. For a set , denotes its boundary (or relative boundary of the closure of X if X is convex). For vectors their scalar product is . The Euclidean norm of is denoted by . For any set A, #A denotes its cardinality.
3 Definitions, facts and a combinatorial lemma
Let , , , be affinely independent. The set σ defined as is called a -simplex with vertices , . We write it briefly as or . If we know that σ is a -simplex, then the set of its vertices is denoted by . If , then the vector is called the vector of the barycentric coordinates of p in σ, if . For each , its barycentric coordinates in the simplex σ are uniquely determined.
If σ is a -simplex, then , where , is called a -face of σ.
A collection , , of nonempty subsets of a -simplex , , is called a triangulation of S if it meets the following conditions:
is a -simplex, ,
if for , then is a common face of and ,
Two different -simplices , , , , in a triangulation of a -simplex S are adjacent if is a -face for both of them. Each -face of a simplex , , is a -face for exactly two different simplices in the triangulation, provided the -face is not contained in .
The K-triangulation of an -simplex with grid size , where m is a positive integer,e is the collection of all -simplices σ of the form , where vertices satisfy the following conditions: The K-triangulation of S with grid size is denoted by and the set of all vertices of simplices in is denoted by . Obviously, . For any and for a sufficiently large m, each simplex in has the diameter not greater than ε. Moreover, there exists exactly one simplex in such that is its vertex.f
each barycentric coordinate , , of in S is a nonnegative multiple of ,
, where is a permutation of , is the vector of the barycentric coordinates of , , .
A basic tool used in the proof of our main result is the following.
Then there exists a unique finite sequence of simplices , , such that and are adjacent for , , , , , and , .g
4 The existence of equilibrium
z is continuous on ,
Walras’ law holds, that is, for ,
the boundary condition holds: if , , and , , then ,
z is bounded from below: , .
Definition 2 Let be an excess demand function, . A point is called an equilibrium point for z, if .
The main goal of the paper is to give a new proof of the fact that for each excess demand function there exists an equilibrium point. First, we are going to characterize the behavior of z near the (relative) boundary of its domain, which is crucial for the theorem to follow. The intuition for the lemma below is as follows: if the price of a good i is low (in comparison to some other price - prices are standardized; they sum up to 1) then the demand significantly exceeds the supply of that good; if the price is (relatively) high - so all the other prices are low - then the demand for the i th good is considerably less than its supply.
Proof Suppose that the former implication is not true. Then there exist and a sequence , , , and , which contradicts the boundary condition. This implies that there exists for which the just considered implication is true and without loss of generality we can assume that . To prove the latter implication, observe that implies , , so the first implication guarantees that , . Now, from Walras’ law, we get , and is satisfied. □
is bounded from below: , ,
if , , and , , then ,
If , , converges to a point p with for some then diverges to +∞ and since the product is bounded from below it holds: . To prove that (5) is true it suffices to observe that for we have and to proceed as in the proof of Lemma 2 with in place of z. □
But H is not a homotopy since the domain of changes as t changes.
The important thing which Lemma 3 reveals is that at each fixed the function is an excess demand function defined on a simplex of dimension instead of .i
We can assume that the vectors , , are linearly independent; it suffices to take sufficiently small . The set is an -simplex with the vertices , . If , then , and if (i.e., ) then ; similarly, if (i.e. ) then . Moreover, if and then and if then (see Lemma 2). We are now in a position to prove the main result of the paper.
Theorem Let z be as in Lemma 3. For each there exists .
where is defined in (1). For , if , , and then it is clear that , since if , then we would obtain . Assume that and . Since , Lemma 3(3) ensures that for some - so, is well defined. Moreover, implies for some nonnegative integer k such that and, therefore, due to Lemma 3(5), it holds that for from which we obtain whenever . Therefore, the assumptions of the combinatorial Lemma 1 are satisfied. Hence, there exists a sequence of simplices in such that and are adjacent and , , , . There exists the first simplex in that sequence, call it , such that for all the last barycentric coordinate of all vertices of in are less than . Simplices are adjacent, i.e. they share an -face, and in other words, they differ by one vertex only. By the choice of all vertices satisfy , and there is a vertex such that . Now, the adjacency of and , the fact that all simplices in have diameters less than and the inequality entail that for , which implies for . Reasoning analogously, we get for the last simplex, , that it holds: , . By the choice of , all simplices , , are contained in . Moreover, their diameters are less than δ so , , implies , . Since is (arcwise) connected and and then by the continuity of there exists a simplex . Let . So , .Since for each i, there exists a vertex of such that (by the inclusion ), , . Further, , , since , if , due to Lemma 2. We have found a point , , which ends the proof. □
Corollary Let z be as in the above theorem. There exists an equilibrium point for z.
Proof Let , , be a sequence converging to 0. In view of the proof of the theorem, for each there exists a point such that , . The Bolzano-Weierstrass theorem and compactness of imply that there exists a convergent subsequence of , such that . From the continuity of z, it follows that , for . Since , , . Walras’ law ensures that . □
5 An algorithm for the computation of equilibrium
From the proof of the theorem, we can derive the following algorithm for computation of a point satisfying , , where is a given accuracy level. The algorithm below uses the function defined in (3) and we reasonably assume that .
Step 0: Determine , satisfying claim of Lemma 2 and Lemma 3(5), respectively. Fix accuracy level: . Find such that if , where is defined as in the proof of the theorem, and then and let be an integer for which all simplices in have diameter less than . Let be is as in the proof of Lemma 1 for , set , (see formula (2)) and go to step 1.
Step 1: Determine the only vertex such that and . Go to step 2.
Step 2: If , where is defined in Lemma 3, and STOP: v satisfies , . Otherwise, assign the only element of as the value of . Set and go to step 1.
Step 0 initializes the necessary parameters for correct course of the algorithm and in fact it is the most difficult part of the algorithm, unless we know some properties of the considered excess demand function (e.g., differentiability, its lower bound or if it is a Lipschitz function on compact subsets of ). It is easy to determine m if we know δ and ; it suffices to take , which is a consequence of the definition of the K-triangulation and the fact that the diameter of a simplex equals the maximum distance between its vertices. In Steps 1 and 2, set is a face of an element of such that . In Step 2, we check if currently considered simplex , where v is such a vertex in that is common -face of the currently considered simplex and its direct predecessor , is contained in , which implies that the value of l depends on function (see Lemma 3 and formula (3)). If it is the case, and in addition , then v is what we seek for. If not, we have to find the next adjacent simplex; to this goal, we have to decide which vertex should be removed from FaceVertices. To achieve this, we find the vertex , which bears the same value of l as v and we form the new set FaceVertices substituting v in place of and then we repeat the operations. The algorithm succeeds in finding approximate zero in a finite number of iterations due to Lemma 1, the theorem and its proof. It is worth to emphasize that at a given iteration of the algorithm (Step 1-Step 2) exactly one new value of l is computed and to proceed on with computations it is sufficient to know only the last simplex; there is no need to remember the earlier stages in the course of the algorithm. Moreover, the values of l need to be computed only at the vertices of the constructed sequence of simplices.
6 Final comments
6.1 The boundary condition
The standard form of the boundary condition imposed on/satisfied by an excess demand functions is:k , , , , implies . The difference is that we assume that if the (relative) price of a good i tends to 0, then the excess demand for the good i goes to +∞. The standard condition claims that if the (relative) price of a good i tends to 0, then the excess demand for some good, not necessarily i, goes to +∞. Our condition is satisfied if there is a consumer with Cobb-Douglas preferences and owns a positive quantity of each good. But even if z is an excess demand function that satisfies the standard boundary condition, we can approximate z (as close as we wish on compact subsets of ) with an excess demand function satisfying the version of the boundary condition used in the paper; see the below construction of the function and just put there z in place of g.
6.2 Fixed points of continuous functions defined on the standard simplex
Let and (taking a subsequence if necessary). If , then . If then and , so , but boundedness of g implies that , , is bounded. We obtain , which ensures that (see ). Hence, to find an approximate fixed point of f, we can apply the algorithm for , h sufficiently small.
The equivalence of the existence of equilibria for excess demand functions defined on the standard closed simplicesl and Brouwer’s theorem was shown in . The proofs of the equivalence for the excess demand functions considered in the current paper can be found in  or .
6.3 Open questions
Combinatorial Lemma 1 seems to be interesting for its own sake in spite of the fact that it is proved for a particular triangulation. We have seen that it implies the existence of equilibrium for an excess demand function. A slight modification of the proof of Theorem 7 in  allows to claim that the existence of equilibrium for an excess demand function is equivalent to the Brouwer fixed point theorem (see also ). The famous Sperner lemma, which is a combinatorial tool used to prove Brouwer’s fixed point theorem (and which is equivalent to it [, p.21]) has many implications (e.g., see [, pp.101-103]). What are other implications of Lemma 1? Does Lemma 1 generalize to any triangulation of the standard simplex? Is it equivalent to Sperner’s lemma? What about the behavior of the algorithm presented in the paper in comparison to the behavior of other computational methods for finding equilibria (e.g., methods presented in )? How to modify the algorithm to allow for the computation of (approximate) equilibria of excess demand mappings rather than functions?
bHomogeneity of degree zero is among these conditions: we can restrict our considerations to excess demand functions defined on the open standard simplex and not on the whole positive orthant of - see Definition 1 in Section 4.
cConstructive in the sense that it allows to derive a (simplicial) algorithm for computation of an approximate equilibrium.
dWe find  by Yang as a comprehensive source of information on computation of equilibria and fixed points. Since the primary goal of this paper is to derive the existence of equilibria in a novel way without referring to Brouwer’s fixed point theorem and not to construct algorithm for computation of equilibria, the algorithm presented below should be treated as a by-product which is important, as we believe, but whose properties should be examined in the future.
eOur K-triangulation is called the -triangulation in [, p.64].
fWe could not have found a reference for this statement but it is proof is elementary.
gFor simplicity: if we know that σ is a simplex we write instead of - formally correct way - . Notice, that the codomain of the function l could be easily changed to in place of , but we do not do that to discern the ’top’ of a simplex from its ’bottom’ - see Figure 1.
hThe method of construction of the sequence is similar to the one used in the proof of the correctness of the Scarf algorithm - see [, p.68].
iThe idea for the definition of comes from the proof of Theorem 1 in  as it comes as a loose suggestion for the proof of our main theorem below.
jThe idea for l is closely related to the notion of the standard integer labeling rule [, p.63].
lThis assumption eliminates both boundary conditions presented above.
I would like to thank participants of the Seminar of Department of Mathematical Economics (Poznań University of Economics), Nonlinear Analysis Seminar at Faculty of Mathematics and Computer Science (Adam Mickiewicz University in Poznań), Seminar of the Game and Decision Theory at Institute of Computer Science (Polish Academy of Science, Warsaw) for helpful comments and criticism. I also thank the referees for their comments and remarks that improved the paper. All remaining errors are mine. This work was financially supported by the Polish National Science Centre, grant no. UMO-2011/01/B/HS4/02219.
- Debreu G: Existence of competitive equilibrium. 2. In Handbook of Mathematical Economics. Edited by: Arrow KJ, Intriligator MD. North-Holland, Amsterdam; 1982:697–743.Google Scholar
- Aliprantis C, Brown D, Burkinshaw O: Existence and Optimality of Competitive Equilibria. Springer, Berlin; 1990.View ArticleGoogle Scholar
- Mas-Colell A: On the equilibrium price set of an exchange economy. J. Math. Econ. 1977, 4: 117–126. 10.1016/0304-4068(77)90002-7MathSciNetView ArticleGoogle Scholar
- Yang Z: Computing Equilibria and Fixed Points. Kluwer, Boston; 1999.View ArticleGoogle Scholar
- Scarf H: The computation of equilibrium prices: an exposition. 2. In Handbook of Mathematical Economics. Edited by: Arrow KJ, Intriligator MD. North-Holland, Amsterdam; 1982:1006–1061.Google Scholar
- Maćkowiak P: The existence of equilibrium without fixed-point arguments. J. Math. Econ. 2010, 46(6):1194–1199. 10.1016/j.jmateco.2010.09.003View ArticleGoogle Scholar
- Uzawa H: Walras’ existence theorem and Brouwer’s fixed-point theorem. Econ. Stud. Q. 1962, 13(1):59–62.Google Scholar
- Maćkowiak P: Some equivalents of Brouwer’s fixed point theorem and the existence of economic equilibrium. Scientific Books 222. In Quantitative Methods in Economics. Edited by: Matłoka M. Poznań University of Economics Press, Poznań; 2012:164–171.Google Scholar
- Toda M: Approximation of excess demand on the boundary and equilibrium price set. Adv. Math. Econ. 2006, 9: 99–107. 10.1007/4-431-34342-3_6MathSciNetView ArticleGoogle Scholar
- Dugundji J, Granas A: Fixed Point Theory. Springer, New York; 2003.Google Scholar
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