Existence of Competitive Equilibria without Standard Boundary Behavior

We study the existence of competitive equilibria when the excess demand function fails to satisfy the standard boundary behavior. We introduce alternative boundary conditions and we examine their role in proving the existence of strictly positive solutions to a system of non-linear equations (competitive equilibium prices). In addition, we slightly generalize a well-known theorem on the existence of maximal elements, and we unveil the link between the hypothesis of our theorem and one of the boundary conditions introduced in this work.


Introduction
The purpose of this paper is twofold.In general, we provide a set of sufficient conditions in order for a system of N non-linear equations in N unknown to possess a strictly positive solution.In particular, since we deal with excess demand functions (vector fields) defined on suitable price-domains, from the standpoint of Economic Theory the natural interpretation of our results is the existence of a price-vector that clears every markets that are assumed perfectly competitive.In other words, the existence of a vector of strictly positive prices such that demand equals supply on every market.Such state of the economy is called competitive equilibrium.
Note that the literature about the existence of competitive equilibria is vast, and a survey of the numerous and remarkable contributions would hardly do justice to them.So, why yet another paper on the existence of competitive equilibria?To answer this question, first it is worth recalling briefly the established literature.

5) If
, where p ≠ 0 and p i = 0 for some i, then Recall that any finite-dimensional economy with continuous, strictly convex and strictly monotonic preferences, and with production sets that are closed, strictly convex, bounded above, and such that a strictly-positive aggregate consumption bundle is producible from the aggregate endowment, gives rise to an aggregate excess demand function enjoying the above properties (see, e.g., Aliprantis et al. [1], Arrow and Hahn [2], Mas-Colell et al. [3]).Notice that property is the standard boundary behavior.Clearly, a competitive equilibrium price vector is a 5) such that .
  = 0 Z p  Obviously, with constant returns-to-scale production, the production set is neither strictly convex nor bounded above.We borrow the formulation of the economy and the definition of competitive equilibrium from Geanakoplos [4].
Specifically, assume that preferences are continuous, strictly convex and strictly monotonic.Then, a constant returns-to scale economy can be formalized by an agg- such that is nonempty. 1Under these assumptions, 0 pY  Z still satisfies properties 1) through 5) above.A competitive equilibrium for a constant returns-to-scale production economy can now be defined as a price To summarize: if one is dealing with general production economies, then Z defined above is the (production-inclusive) aggregate excess demand function.If, rather, one is dealing with constant returns-to-scale economies, then Z is the aggregate net demand function that stems solely from consumers' preference maximization.In either cases, when preferences are continuous, strictly convex and strictly monotonic, Z satisfies properties 1) through 5) above.Now, let us turn to our contribution in this paper: we weaken the continuity of the excess demand function, we drop the standard boundary behavior (replacing it with alternative boundary conditions), and we prove a new mathematical theorem which is then utilized to study the existence of competitive equilibria.More precisely, following in Tian's footsteps [5], we do not assume that the aggregate excess demand function is lower semicontinuous, whereas in the literature the excess demand function is typically continuous (see above).
Moreover, we address hypothetical economies in which the standard boundary behavior of the aggregate excess demand function (property 5) above) is not necessarily satisfied, and we prove two theorems on the existence of competitive equilibria.Indeed, to motivate our work, in Section 3 we exhibit two model-economies: in the former, the standard boundary behavior fails.In the latter, the sufficient conditions for the standard boundary behavior are violated, and therefore the standard boundary behavior may or may not be satisfied.On the other hand, it is well-known that, whenever the excess demand function is defined on a relatively-open pricedomain2 (as it is the case in this paper), some sort of boundary conditions are needed to demonstrate the existence of a zero for such a function.In fact, loosely put, proper boundary conditions remedy the lack of compactness of the price-domain, and thus enable the application of specific fixed-point theorems.For these reasons, we introduce two alternative boundary conditions and we study how they are related to one another.Our alternative assumptions on the boundary behavior formalize inward-pointing conditions of the aggregate excess demand function.The former condition is formalized by means of the projection mapping (see Section 2), and the latter by means of convex combinations.This comes in handy because this method of modelling the boundary conditions enables us to retain the central idea of the first existence theorem in the proof of the second one, which thus becomes a variant of the first theorem.Hence, this approach offers a somewhat unified framework for two seemingly different problems Finally, in the context of Hilbert spaces, we prove a slight generalization of a theorem on the existence of maximal elements due to Yannelis and Prabhakar [6] (Theorem 5.1).Interestingly, one of the assumptions in our theorem lends itself to be interpreted in terms of boundary behavior 1 defined in Section 2. This strong analogy enables us to prove again the existence of a strictly positive equilibrium price vector as a short corollary of our new theorem.
Clearly, in this work we treat the excess demand function as the primitive of the economy at hand.This may be regarded by economists as an unconventional route to proving the existence of competitive equilibria.Nevertheless, the approach we follow, based directly upon the excess demand function, is well-suited to highlight the mathematical aspects of our contribution.Indeed, in Section 4 we shall develop a unifying treatment that can handle both general production economies and constant returns-to-scale economies.We shall detail the proof of the existence of competitive equilibria only for the former, since virtually the same method may be used to analyze constant returns-to-scale production economies as well.
The lay-out of the paper is as follows: In Section 2, we set our notation and we develop some background.Also, we introduce two alternative boundary conditions on the excess demand function.We also explain how our conditions relate to the literature we know of, and finally we state the main mathematical theorem that will be used in this paper.It is a celebrated selection theorem due to Michael [7].In Section 3, we construct two model economies for which the standard boundary behavior of the excess demand function is not necessarily satisfied.This justifies our interest in proving existence of competitive equilibria under alternative boundary conditions.In Section 4, we prove two theorems on the existence of competitive equilibria or, rather, on the existence of a strictly positive solution to a finite system of non-linear equations.In the process, we also compare our approach to the relevant literature.In subsection 4.1, we prove a theorem on the existence of maximal elements for correspondences 3 whose domain is different from the range, and domain and range are both subset of a Hilbert space.It is a natural generalization of Yannelis and Prabhakar [6], and thus it may be interesting in its own right.In subsection 4.2 we employ our new theorem to study the existence of competitive equilibria.In Section 5 we conclude and we outline a few directions for future research.

Background, Notation, and Boundary Conditions
 be the aggregate excess demand function of a general production economy.The question we are after in this paper is: Problem: Does there exist some To set the stage for the subsequent analysis, let , where 1 is the dimensional , and let .


, let Clearly, since we are searching for a , by virtue of property 2) above we can restrict the domain of ( ) = 0 Z p  Z to Int .We choose this normalization over other admissible ones (for example, one might have Z defined on the intersection of the unit sphere with N   ) because convexity is very handy in our proofs.
Recall that the standard boundary behavior (property 5)) plays a crucial role in proving the existence of a strictly positive vector, , such that (see, e.g., Aliprantis et al. [1], and Mas-Colell et al. [3]).When production exhibits constant returns-to-scale, the standard boundary behavior of the aggregate net demand still comes into play to prove the existence of competitive equilibria.For details see, for example, Geanakoplos [4].
For our purposes it will be convenient to use a formulation of the standard boundary behavior which does not involve asymptotic conditions.To this end, the following result is a straightforward consequence of the standard boundary behavior of the excess demand function (and of property 4).It is not difficult to prove: Proposition 2.1: Let : N Z Int   be a map satisfying properties 4) and 5) listed above.Assume that is such that , with As the examples in Section 3 demonstrate, one can conceive of an economy for which the standard boundary behavior may fail.
Therefore, we still wish to provide an affirmative answer to Problem above, but we have to drop the standard boundary behavior hypothesis.To this end, we shall now introduce two alternative boundary conditions for the aggregate excess demand function, but first we need to provide some mathematical background.
For any > 0  , define the restriction of the (metric) projection mapping to , that is To the best of our knowledge, the projection mapping was introduced in Economics by Todd [9] in a general equilibrium model of production with activity analysis.It was used also by Kehoe [10].In this paper we utilize the projection function in a different manner.Basically, boundary behavior 1 formalizes the assumption that the excess demand function is "inwardpointing" on   .A different "inward-pointing" condition on the excess demand function was introduced by Neuefeind [11], and Husseinov [12].We stress, however, that Neuefeind works with continuous excess demand functions, whereas in the next section we are able to address Problem above without assuming continuity of the excess demand function.
Boundary behavior 2: There exists a > 0  Begin by noticing that, by 3), for each p    )) we have that .We claim that for all .Indeed, suppose, by contradiction, that there exists a such that . By assumption, there exists 0 < < 1  such that . Hence, by definition of projection mapping, the latter inequality implies that 0   , which is impossible.The proof is complete.
In the proofs presented in Section 4 we shall invoke the following selection theorem due to Michael [7], (Theorem 3.1'''): 5Theorem 2.2 (Michael): Let X be a perfectly normal . Illustrative Examples e now present two examples of economies whose ple 1: Consider a competitive economy with one re 3 W excess demand function does not necessarily satisfy property 5).The former example is very simple, admittedly artificial, but its virtue is to convey the main ideas.As for the latter, we refer the reader to Impicciatore et al. [13].The key ingredient in both models is that not all of the goods and services traded affect consumers preferences, while agents are endowed with strictly-monotonic preferences over a subset of commodities and services.
Exam presentative consumer and one representative firm.There are two commodities.A consumption good, denoted by c, which is produced by the firm with linear technology and consumed by the consumer.We let p be the price of the consumption good.The second commodity, denoted by x , is a production input, owned by the consumer, which is available in fixed and limited quantity, say x .The production input is not produced and is supplied by the consumer to the firm.We let w be the price of the input production services.T consumer is endowed with he x units of the production input, but she is not endowed ith the consumption good.Consumer's preferences are represented by the utility function : u     which is a function only of the consumpt nd is assumed to be strictly increasing.The production technology is such that w ion good a x units of the production input yield ax units of th consumption good, with > 0 a .Thus, the firm's profit-ization p e maxim roblem is axi- , and the consumer's preference m can be described as x x It is very easy to see that, if a competitive equilibrium pr ice vector ( , )  p w exists, then we must have ( , ) 0 p w    .So, in equilibrium prices must be strictly is why in Section 4 we will be concerned with the existence of strictly positive equilibrium price vectors.It is routine matter to check that the consumer's for this economy the standard e net demand function does not ho Now we show that boundary behavior of th ld.This is why in Section 4 we shall put forward a method for proving existence of competitive equilibria under alternative boundary conditions of the aggregate net demand functions.To see that the standard boundary behavior fails, note that in view of Proposition 2.1 above it will suffice to exhibit a price vector ˆ( , ) 0 p w  such that, for each > 0


with ˆ> p  and ˆ> w  , there exists a vector     Indeed, simple calculations reveal that, if we take   One might ask what goes wrong in this model, with regard to the standard boundary behavior.Basically, in this example what causes the standard boundary behavior of the net demand function to fail is the presence of a non-reproducible input available in fixed quantity.Also, consumer's preferences over both goods are convex, but not strictly-convex, and monotonic, but not strictlymonotonic (see Section 1).Example 2: Another example one might think of, deals with a mainstream reformulation of the original Walras' theory of savings and capital accumulation.We refer the reader to Impicciatore et al. [13] for the details.Here we just sketch informally a few elements of the model.
Time-horizon is finite with two periods, = 0,1 t .In each period there are C consumption goods, and J labor/leisure services; there are M capital goods, as as a consistent number of capital goods' production services.
There exists a complete array of Arrow-Debreu forward markets open at = 0 t .Consumers purchase capital well ods i ec goods produced at = 0 t in order to sell their production services at = 1 t .We assume that consumers have to store capital go n order to supply their services to the production s tor in the next period.
There is a finite number H of consumers, indexed by = 1, , h H  .We assume that capital goods are not consumed, nor do they affect agents' preferences.Hence, consumers' preferences are defined on the consumption set . Preferences on h 2(C X are continuous, strictly increasing and strictly quasi-concave.At = 0 t each consumer is endowed with labor/leisure services and capital goods.Similarly, at = r takes prices as given, and choos p iz the storage capacity constraint and the budget constraint. The authors then define the notion of virtual aggregate net demand function, which is instrumental in proving existenc is model the virtual aggregate net demand function may fail to satisfy the standard boundary behavior.To see this, note that we may well think of each consumer as being equipped with monotonic, and convex preferences defined over every goods and services traded in the economy.On the other hand, we know from Section 1 that sufficient conditions for the standard boundary behavior are strictly convex and strictly monotonic preferences.In other words, the sufficient conditions for the standard boundary behavior are violated.Furthermore, suppose we are given an arbitrary sequence   π n of strictly positive prices that converges to π 0  , where π belongs to the boundary of N   .By the capacity constraint on storage, the demand for capital goo s is bounded above, and one can prove that ast one nsumers' income is finite and positive.In a nutshell, these are the reasons why the virtual aggregate net demand function does not necessarily satisfy the standard boundary behavior.Therefore, as we pointed out above, we seek a method to prove existence of equilibria that does not hinge on the standard boundary behavior.[5] and Ewald's approach to proving the basic Ky-Fan theorem (see Ewald [14], Lemma 3.6.1,and Theorem 3.6.5).Our proof, though, departs from Ewald's in two significant ways.First of all, the correspondences defined in [14] are assumed to have open lower sections.In contrast, we posit the assumption of lower hemicontinuity (see Assumption 4.1 above) because, in general, it is weaker.Moreover, we assume lower hemicontinuity to facilitate a comparison with the approach followed by Tian [5], and because we believe it is a more natural assumptions when dealing with Economic models.

Main Existence Theorems
Secondly, since Ewald deals with correspondences with open lower sections, he finds it natural to employ the finite-dimensio eorem.In contrast, we work directly with a lower hemicontinuous correspondence, U defined in (4.1), and therefore we shall employ Michael's selection theorem (Theorem 2.2 above).Incidentally, Theorem 2.2 above is a generalization of the finite-dimensional version of Browder's selection theorem used by Ewald.
Theorem 4.1: If Assumption 4.1 holds, then there exists a π easy to see that we are done.So, assume, generality, that any open neighborhood of  in  , say . Since U is lowe continuous, there e ts an open neighb , that is nuous selection, t . 7N n Th above.Th , admits a contihat is ther exists a c nuous function ext, consider the restrictio of U to W | : Clearly,  is convex and compact valued.We wish to prove that  is upper hemicontinu .To this end, d graph theorem it will suffice to show that ous by the close and open in π W : W is   (see above), there is a h that or all n N  .Thus, for all n N  we have that π take the composition of  with P ( P is the projection function de-  rk assumption assumes that the excess demand function is defined on the whole  , and demonstrates the existence of a q    such that ( ) 0 Z q   .Neuefeind [11] formalizes a condition that the excess demand function is "inward-po ting" close to the boundary of the p .His assum oes not require the excess demand function to satisfy property 5) above.However, Neuefeind assumes that the excess demand function is continuous.We dispense with the standard boundary condition on the excess demand function too, but unlike Clearly,  is convex and compact valued.Furthermore,  Theorem 4.  , as was to be proven.as open graph.In the context of Hilbert spaces, we can now prove a theorem which is a natural generalization of Theorem 5.1 in Yannelis and Prabhakar [6].It is a generalization in that we do not require the domain and range of the correspondence at hand to be the same.Moreover, in place of the first condition in Theorem 5.1 of Yannelis and Prabhakar [6], we posit a more general assumption (see assumption 1) below).It is more general in the sense that it collapses to Yannelis and Prabhakar's condition whenever the domain and range of the correspondence coincide.Interestingly, our assumption 1) below bears a natural economic interpretation in terms of boundary behavior 1 defined in Section 2. Consequently, we shall show, in Section 4.2, that our Theorem 4.1.2can be employed to provide another short proof of Theorem 4.1.

We
Let  be a Hilbert space, and let X and Y be non-empty, convex and compact subsets of  , with

4 )
There exists a s > 0 such that ( )i Z p   regate net demand function-technology pair   , and convex cone that allows for free disposal.Clearly, we must restrict attention to the set ofN Y   N p   such that 0 pY  , i.e., 0 py  for all y Y  .One can assume, without much loss in generality, that the set of Since P is continuous and  is upper hemicontinuous, P   is upper hemicontinuous.By construction, en follows from Kaku ni's fixed point theorem that there exists a see the proof of Now, we will prove that  is inward pointing.8T this end, pick any π o logical to investigate the relationship between boundary behavior 1 and boundary behavior 2. The next theorem, due to Iryna Topolyan 4 , serves this purpose: Theorem 2.1 (Topolyan): If : N Z Int   satisfies properties 3) above, and boundary behavior 2, then it also satisfies boundary behavior 1. Proof: Assume that Z satisfies boundary behavior 2.    and preceding inequality contradicts boundary stan oint of applied mathematics, we believe at this work is self-contained.From the perspective of be improved and ex-th economic theory, this paper can nded.Let us outline what it would be worth under-D.Aliprantis, D. J. Brown and O. Burkinshaw, "Existence and Optimality of Competitive Equilibria," Spri 990.doi:10.1007/978-3-642-61521-4te taking for future research.First of all, one should investigate the relationship between the standard boundary behavior and our boundary behavior 1.Secondly, if it turns out that neither of them implies the other, or that the standard boundary behavior implies our boundary behavior 1, then it would be interesting to construct relevant economic models for which the standard boundary behavior does not hold, but our boundary conditions are satisfied by the excess demand function of the model itself. the