Existence of General Competitive Equilibria: A Variational Approach

Abstract

We study the existence of general competitive equilibria in economies with agents and goods in a finite number. We show that there exists a Walras competitive equilibrium in all ownership private economies such that, for all consumers, initial endowments do not contain free goods and utility functions are locally Lipschitz quasiconcave. The proof of the existence of competitive equilibria is based on variational methods by applying a theoretical existence result for Generalized Quasi Variational Inequalities.

1. Introduction

The proof of the existence of economic equilibria is certainly considered the first principal problem to be solved in General Equilibrium Theory (see [1]). Specifically, the question is to investigate what assumptions concerning environment and mechanism are able to guarantee the existence of one state where the aggregate demand does not excess the aggregate supply for all markets at prices endogenously determined.

In the model à-la Arrow-Debreu (see [2, 3], for major details) convexity and closure for the choice sets, price systems in the unit simplex, closure and convexity for the production plans, impossibility that two plans are able to cancel an other one, continuity and monotonicity for the utility functions, and nonsatiability and global survivability for the consumers are its main assumptions. Over the years, the assumptions’ successive refinements had to contextually () show the properties of consequent demand (see the literature about regular economies, for instance) and () adopt suitable mathematical tools able either to compute or only to show the existence of a feasible equilibrium. In favour of this thesis we report the results in [2–9], obtained by Kakutani’s fix point theorem in [10], by gradient’s algorithm, or by the techniques known as differential approach, because continuity, differentiability, and concavity are guaranteed for utility functions or for the preference relations.

Given the classicism of the problem and many excellent papers, as listed above, and books on the economic equilibrium (see, e.g., [11–13]), there is however the need to justify yet an other paper on the subject.

It is well known that variational analysis introduced by Stampacchia in [14, 15] became an extremely useful tool to solve the optimization problems and so to give a solution to the competitive equilibrium problem in economics (see, e.g., [5, 16, 17] and the references therein), too. Furthermore, the importance of concavity in economics for describing the increase of the consumer’s preferences and the consequential returns to scale (see, e.g., [7]) is well known. From convex analysis, clearly, strong concavity implies uniform concavity, uniform concavity implies strict concavity, strict concavity implies (weak) concavity, (weak) concavity implies quasiconcavity, and quasiconcavity does not imply differentiability. Therefore, combining all these facts, the purpose of this paper is to give a more extensive result on the existence of general competitive equilibrium by using nonsmooth analysis (see, e.g., [18, 19]) combined with variational analysis (see also [20]).

Our model describes a private ownership economy with two classes of agents, consumers and producers, and a finite number of no free goods at disposal. Consumers, as shareholders, will control the producers. The equilibrium will be realized when each consumer optimizes his utility under the budget set, each producer will realize the maximum profit according to his own production plan, and the sum of the total endowment plus the total consumption will not exceed the total production. In this economic scenery we will admit a list of assumptions which for now is the more generalized both from a mathematical and from an economic viewpoint. In detail, the assumptions are as follows:(i)the initial endowment of any consumer will consist of at least one type of goods of minima or greater price (absence of free goods in the initial endowment) (as in [21]);(ii)the production set of the th producer will be a convex and compact set in containing the origin;(iii)the utility function of the th consumer will be assumed quasiconcave and locally Lipschitz continuous.

The principal novelty consists in the assumptions on the utility function. As said above, because a quasiconcave function is not always differentiable, adding the locally Lipschitz condition we can drop gradient with generalized gradient (see [19, pages 25–28]) and so, differently to what it was proved in [6, 21], we are able to advance every possibility on the return to scale in satisfaction terms for the consumer. In other words, since the quasiconcave condition for utility function is weaker than weak concave and strong concave ones, or, also, it is the most general assumption in Consumer’s Theory (see [7, 9]), this fact allows us to express how marginal utility changes for any increase of consumption. Thus for the consumer’s progressive satiety we could attend all the feasible consequences which go from a possible reduction (exclusively for the strong concave utility) to a possible increase (typical for the quasiconcave utility) through the constant state (typical for the weak concave utility). From this fact, we can reasonably take into account a wide range of utility functions including also the typical economic ones, in generalized form, as Cobb Douglas’s and Constant Elasticity of Substitution (briefly CES) class, which, till now, have got involved in the proof of the existence of a general competitive equilibrium iff the utility is supposed to be concave (see [21]). With a more complicated type of generalized quasivariational inequality (see [20] for major details) and with the help of nonsmooth analysis (see, e.g., [18] for the continuity of set-valued map) we can yet treat the existence of a competitive economic equilibrium problem via variational method in generalized way with respect to [6, 21, 22]. The proof of the existence of competitive equilibria will be based on variational methods and, in particular, on an abstract existence result for Generalized Quasi Variational Inequalities due to Cubiotti (Theorem   in [23]).

Furthermore, an other novel result is that every competitive economic equilibrium is also a Walras competitive equilibrium (see Proposition 14).

Finally, we point out that our main result contains, as special cases, some recent existence results of competitive equilibria for pure exchange economics established in [24, 25]. Probably, our techniques also work if we consider the dynamic version of pure exchange economics introduced in [16, 17] (see also [26, 27]) and so the existing results of these papers could be improved by replacing the concavity condition with the quasiconcavity condition.

The organization of the remainder of this paper is as follows.

In Section 2 for the sake of convenience, we will recall the main notations, definitions and results that will be used in the sequel for our analysis.

In Section 3 we will describe the economic model with its environments and related restrictions and its internal mechanisms, defining the general economic equilibrium problem through constrained maximization problems system.

In Section 4 we will point out the connections between the competitive economic equilibrium problem and a suitable , proving also that in a suitable compact set any competitive equilibrium is a Walras competitive equilibrium.

In Section 5 we will prove the existence of a solution to , recalling, in particular, Theorem   of Cubiotti in [23] and, thus, concluding with the existence of at least one (Walras) competitive equilibrium for the private ownership economy considered.

In Section 6 we shall show how two well-known functions, Cobb-Douglas and CES, could be employed as utility functions in economics under the locally Lipschitz and quasiconcavity assumption.

2. Preliminaries

Throughout this paper, for each , denotes the Euclidean space of the real -vectors equipped with the usual inner product and norm , for any .

The symbols , , and will indicate the cone of nonnegative, positive, and strongly positive vectors of , respectively. Furthermore, the set indicates the unit simplex of . We adopt the usual notation for vector inequalities, that is, for any , one has if ; if ; and if . Let ; we will write and to indicate its interior and its closure, respectively. Open and closed balls of radius , centered at , are denoted by and , respectively.

Let be a subset of . A function is said to be quasiconcave iff for every , the set is convex. A function is said to be locally Lipschitz continuous near if there exist constants and such that implies . is called Lipschitz constant or of .

Let be a locally Lipschitz function near and let . According to Clarke (see [19, page 25]), the generalized directional derivative of at in the direction is defined by and the generalized gradient, or simply subdifferential, of at is the set-valued mapping, , defined as follows:

For the sequel, we recall below three results on generalized derivative and subdifferential, here, for convenience, rewritten when is a subset of .

Proposition 1 (Proposition of [19]). Let be a locally Lipschitz function near of . Then the following hold: (a)the function is finite, positively homogeneous, and subadditive on and satisfies (b).

Proposition 2 (Proposition of [19]). Let be a locally Lipschitz function near of . Then the following hold:(a) is a nonempty, convex, and compact set of ,(b)for every , one has

Proposition 3 (Proposition   of [19]). Let be a locally Lipschitz function near of . Then(a)the multifunction is closed,(b)the multifunction is upper semicontinuous at .

Let be a set; we write for the family of all nonempty subsets of . A correspondence or a multifunction between two sets and is a function . The graph of a multifunction is the subset of defined by . Let be a subset of and let , be two multifunctions. The classical generalized quasivariational inequality problem associated to , , , denoted briefly by , is to find such that

For the reader convenience, we report here the statement of Theorem   of [23], which the prove of our main result is based on.

Theorem 4 (Theorem   of Cubiotti [23]). Let be a closed convex subset of  , a nonempty compact set, and and two multifunctions. Assume the following: (i)the set is convex for each , with ;(ii)the set is nonempty and compact for each ;(iii)for each , the set is closed;(iv) is a lower semicontinuous multifunction (i.e., is open in , for each open set in ) with closed graph and convex values.Moreover, assume that there exists an increasing sequence of positive real numbers, with and such that if one puts , for each one has the following: (v), for all ;(vi)for each , with , Then, there exists at least one solution to belonging to .

3. Economic Model

We consider a private ownership economy à-la Arrow-Debreu (see [2, 3] for major details), where there are commodities, producers, and consumers (). We index the commodities by the subscripts , the producers by the subscripts , and the consumers by the subscripts . We regard as the commodity space. By assuming the vector as a price system, the value of a commodity bundle relative to the price will be given by the inner product .

3.1. Environments

The consumers are labeled by . The consumption set related to the consumer is the set of all the -uple of nonnegative real numbers. The preferences of the consumer are expressed by utility functions , endowments by vectors , and shares of the profits of each firm by vectors . The producers are labeled by . The production set related to consumer is a subset of the set of all the -tuple of real numbers. The aggregate endowment is . We assume that the firms are owed by someone, or in other terms, with for all . We summarized a private ownership economy by the tuple .

3.2. Basic Restrictions

For a private ownership economy we assume the restrictions listed as follows:(A1)any price system is normalized and bounded below by the vector , called the minima prices, whose component does not exceed the value of , or equivalently  is the set of the available price systems;(A2)in any initial commodity bundle there exists at least one good of positive quantity and positive minimum price: (A3)for all , is convex and compact in such that ; (A4)there exists an open convex such that, for all , the following holds:(a) is locally Lipschitz continuous and quasi-concave,(b) for all ,(c), for all , and such that , where  with , and is the unit vector of the axis.

Remark 5. Locally Lipschitz condition in an open convex set , listed in (A4), will be needed to consider the Clark-subdifferential of at each point of the closed set . Quasiconcavity condition in (A4) is weaker than the concavity condition and the strictly concave condition usually considered in the literature. Finally, the existence of the constant comes from the compactness of the set (see assumption (A3)).

3.3. Existence Equilibrium Problem

Because consumer owns two resources (initial endowment and profit share), indicated by his expenditure and by his profit derived from producer , represents his budget set.

Set , and .

Definition 6. An allocation is a couple , where represents the consumptions of all the consumers and represents the productions of all the producers. In particular, an allocation is said to be an individual allocation if . An allocation is said to be a weakly balanced allocation if . Finally, an allocation is said to be an available allocation if it is both individual and weakly balanced.

Definition 7. A state of the economy is a triple , where is a price vector and is an available allocation.

Considering consumers and producers as price-takers, the above model leads to the following general economic problem.

Problem 8. Find , with satisfying

Remark 9. A solution to Problem 8 is, then, a price vector and an available allocation such that maximizes the consumers’ utility, maximizes the producers’ profit, and makes the market clear.

Definition 10. A state , with , satisfying conditions (11a), (11b), and (11c) of Problem 8 is said to be a competitive equilibrium or free disposal-equilibrium for the private ownership economy .
An equilibrium is said to be a Walras competitive equilibrium if in addition it satisfies the Walras’ law:

Remark 11. If there is no production (i.e., for all ), then becomes a pure exchange economy. In this case Problem 8 assumes the form of the one in [24, 25].

4. Variational Method

Now, we establish a problem as follows.

Problem 12. Find , with , such that there exists satisfying

Let , where is as in condition (A4c).

Proposition 13. Let , with , satisfying condition (11c) of Problem 8. Then .

Proof. From (11c) of Problem 8 and the definition of the constant , it promptly follows that for all . Now, suppose that . Then, for some . This means that, for all , we should have Hence, which is in contradiction with . Thus, .

Proposition 14. Let assumption (A4) be entirely satisfied. Then, any competitive equilibrium is a Walras competitive equilibrium.

Proof. Let be a competitive equilibrium and fix . From Proposition 13, we have . So, by assumption (A4b) and condition (11a) of Problem 8 it cannot be that .
Moreover, assumption (A4c) and again condition (11a) of Problem 8 imply that for all . Therefore, (12) is verified, which means that is a Walras competitive equilibrium.

The next proposition will be needed for the main theorem of this section.

Proposition 15. Let assumption (A4a) be satisfied. Let and let be such that . Then, .

Proof. Let be a sequence in (the set is as in (A4a)) such that . In force of locally Lipschitz continuity (and thus continuity) and , we can suppose that for all . In force of quasiconcavity, one has for all . Consequently, Taking into account the arbitrariness of the sequence , conclusion follows.

The next theorem will state that under the above assumptions any competitive equilibrium is a solution to Problem 12.

Theorem 16. Let assumptions (A1), (A2), (A3), and (A4c) be satisfied. Moreover, let , with . Assume that is a solution to Problem 12. Then, is a solution to Problem 8.

Proof. Let be a solution to Problem 12, and let satisfying inequality .
Testing with , , one has Moreover, from , we promptly obtain Hence, Now, choosing (the fact is possible in force of assumption (A1)), with at the position, we obtain condition (11c) of Problem 8. Furthermore, for each fixed , testing with , , we obtain , for all , which is condition (11b) of Problem 8.
At this point, condition (11a) of Problem 8 remains the only one to be proved. Fix . Testing with , , we obtainTesting, for each , inequality (21) with one haswhich clearly implies . We claim thatIndeed, assume, on the contrary, thatThen, for each , one has . Indeed, if for some , by condition (A4c), we should haveNow, from assumption (A2), there exists such that and . So, in particular, . Then, if we fix small enough, we have , with at the position. Consequently, testing (21) with and taking (25) into account, we have which is in contradiction with (26). Therefore, , for each . In view of (23) and (25) this implies , which is again in contradiction with (26). Then, strict inequality (24) holds. At this point, let . By observing that , if we put and , for all , by the convexity of , we infer . So, taking (21) into account, one has Moreover, in view of (24), one also has for all . Adding side to side the above inequalities, we obtain From the above inequality and Proposition 15, it follows that , for all . By the continuity of , we then obtain . From the arbitrariness of , condition (11a) of Problem 8 follows.

Remark 17. Under the assumptions of Theorem 16, condition (11c) actually holds as equality. Indeed, fix and define We claim that . Indeed, if not, taking in mind that , it should be . Then, for each , there exists such that Testing with , we obtain Thus, in view of (A4c), it must be for all . This fact, together with , implies . Testing with , where , with being arbitrarily chosen in , we obtain for all . Since , from this inequality it follows that in contradiction with assumption (A4b).

5. Main Result

At this point, it remains to prove that Problem 12 admits at least a solution.

First, we need the following proposition.

Proposition 18. For each , the map has closed graph.

Proof. If, for each the utility is locally Lipschitz on , then according to (b) of Proposition 1, to (a) of Proposition 2, and to (b) of Proposition 3, for each , the subdifferential of has closed graph.

Now, put and define where , for all .

By means of these notations, we can rewrite the variational inequality as follows: find , with , and such that

Theorem 19. Assume that conditions (A3), (A4a), and (A2) hold. Then, Problem 12 admits at least a solution in , where and is as in .

Proof. First, note the following: (i)the set is nonempty closed and convex in ;(ii)the set is nonempty and compact in ;(iii) is a nonempty convex subset of , for all . Moreover, recalling that is (nonempty) convex and compact in , for all and for all , we also have the following: (iv) is a nonempty convex and compact subset of , for all .Thus, to satisfy all the assumptions of Theorem   of [23], it remains to check that the following further conditions hold: the set  is closed, for each ;the map is lower semicontinuous with closed graph;there exists such that if for each we denote by the closed ball in centered at with radius , one has , and(i), for all ;(ii), , for all , with .At the end we divide the proof in several steps.
Step  1. To prove that condition holds true, fix and let be a sequence in such that as . Let us show that . At first observe that since is closed, one has . Moreover, since is compact for each , and the function is continuous in , then, for each , we can find such thatNote that, from the definition of , for each , one has Moreover, recalling that is locally Lipschitz continuous in for all , then, for each , there exist an open neighborhood of in and a constant such thatFurthermore, there exist an open neighborhood of in and a constant such thatAt this point, observe that the family of open sets is a covering of the compact set . Therefore, from (43) and (44), we infer that there exists a constant such that Consequently, from (42), up to a subsequence, we can suppose that the sequence converges to some . Now, from (40) and (41), we infer thatand, from Proposition 18 and (42), we also infer thatFurthermore, from (39), passing to the limit as , one hasAt this point, observe that considering together conditions (46) and (47) means that and this latter condition, together with (48), gives . Therefore, condition is proved.
Step  2. Now, let us show that the map is lower semicontinuous in . To this end, it is sufficient to prove that, for every , every , and every sequence in such that as , there exists a sequence in such that for all and as (see [18] at page 39, for instance).
So, let , , and be as above. For each , using the fact that , we have the following two situations:
either orSuppose that (49) holds. Then, since , there exists such that So, in this case, if we put for and for , it is easy to check that , for all . Moreover, it is clear that as .
Suppose that (50) holds. Then, from the survivability condition (A2), we have Consequently, since , we have Therefore, if we put it is easy to check that , for all . Indeed, if , then and so , while, if , then and from which we again obtain . Finally, observe that as .
So, for each , in both cases (49) and (50), we can find a sequence which converges to and such that , for all . Consequently, if we put , for all , the sequence , where and , for all , satisfies for all and as , as desired. Therefore, is lower semicontinuous in .
To show that condition hold true, it remains to prove that has closed graph. To this end, let and be two sequences in such that , for all , and suppose that , , as . Let us show that .
Since and are closed, one has and . Moreover, with for all , one has Passing to the limit as , we obtain Thus, which, together with and , gives , as desired.
Step  3. To finish the proof of our Theorem, it remains to show that condition holds true as well.
Let be such that the closed ball contains the compact set . Then, for each , one has and , for all . Therefore, condition of holds. Suppose that condition of does not hold. Then, there should exist , with , such thatNow, let us put . Then, . Moreover, from , it trivially follows that . Thus, we can test (60) with . Doing so, we get Therefore, with being a compact set, there should exist such that From the definition of , the previous inequality is equivalent to which, taking in mind that , implies Consequently, Therefore, . But this contradicts the fact that .
The proof is now complete.

6. Applications

The well-known Cobb-Douglas utility function defined bywhere for all , is concave if , but it is only quasiconcave (and not necessarily concave) regardless of the sum of exponents . Following the same arguments at page 178 of [25], Theorems 16 and 19 can be applied to prove the existence of competitive equilibrium for Problem 8 with given by (66) for arbitrary positive exponents . The same result also applies to CES utility function defined by where , for all , and .

It is also to be noted that, as we require only a local Lipschitz condition on the utility functions, our main theorems can be applied to generalized Cobb-Douglas utility functions of the type where are strictly increasing concave or convex functions, not necessarily of class .

7. Conclusions

In [6, 21] under concavity assumption, a demand function was defined and a competitive economic equilibrium was characterized as a solution of generalized quasivariational inequalities or variational inequality involving the Lagrange multipliers, respectively. In this paper we have weakened the concavity assumptions by requiring only the quasiconcavity on the utility functions. Moreover, we dot not require the differentiability of the utility functions but only a local Lipschitz condition.

In Theorem 16 we have shown that any solution of particular problem is a competitive equilibrium. In Theorem 19, using an abstract result due to Cubiotti ([23]), we have proved the existence of solution to this problem. Under the same assumptions, we finally have shown that any competitive equilibrium is a Walras competitive equilibrium too (Proposition 14).

As examples of utility functions to which our main results apply we have exhibited two typical well-known utility functions as follows:(i)the Cobb-Douglas type utility so defined by  where , for all ;(ii)the CES type utility so defined by  where , for all and .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions and comments that improved the paper.