Research Article | Open Access
Existence for Competitive Equilibrium by Means of Generalized Quasivariational Inequalities
A competitive economic equilibrium model integrated with exchange, consumption, and production is considered. Our goal is to give an existence result when the utility functions are concave, proper, and upper semicontinuous. To this aim we are able to characterize the equilibrium by means of a suitable generalized quasi-variational inequality; then we give the existence of equilibrium by using the variational approach.
In this paper a competitive economic equilibrium model integrated with exchange, consumption, and production is considered. In 1874, Walras  laid the foundations for the study of the general equilibrium theory. However, the first rigorous result on the existence of general equilibrium is due to Wald; see, for example, . After this pioneering work, several authors (including [3–5]), stimulated by the advances in linear programming, activity analysis, and game theory, obtained equilibrium existence results. In particular, Arrow and Debreu considered the application of fixed point theory to equilibrium problems, establishing the existence of an equilibrium in an abstract economy that included production and consumption.
Recently, an alternative approach to the study of general economic equilibrium has been considered in terms of variational inequalities. Many economic equilibrium models, including the general equilibrium model of Arrow-Debreu, can be formulated as variational inequalities and/or complementarity problems. Variational theory was introduced in the early 1960s with the works of Fichera and Stampacchia; they study equilibrium problems arising from elastoplastic theory and from mechanics. Subsequently this theory was applied in different kinds of equilibrium problems and, now, represents a powerful tool for the study of a large class of equilibrium problems arising in mechanics, physics, optimization and control theory, operations research, and several branches of engineering sciences. For the state of art about this topic, we refer the reader to [6–14] and the bibliography therein.
In particular, Jofré et al. in  studied the competitive equilibrium by means of a variational inequality, which involves the Lagrange multipliers. Our purpose in this paper is to study the economic competitive equilibrium through a different variational method. In fact, we can characterize the equilibrium by means of a generalized quasi-variational inequality without using the method of Lagrange multipliers. A competitive equilibrium as solution of a suitable quasi-variational inequality has already been studied in [16–18], in which the authors consider a market of exchange and consumption. In this paper we consider weaker assumptions on the utility function with respect to those considered in the latter mentioned economic models. In particular, we consider upper semicontinuous utility functions, instead of ones and, consequently, the differentiability assumption is relaxed. This hypothesis leads to a quasi-variational inequality involving a multivalued map.
More precisely, an economic market with different goods, consumers, and producers will be here considered. Each consumer has a starting endowment of goods and receives a given fraction of the total production, determined by a system of fixed weights. To each commodity a nonnegative price not less than a fixed minimum price is associated. The purpose of the market is to determine a price, a production, and a consumption equilibrium: the production equilibrium maximizes the profit of producers and simultaneously the consumption equilibrium maximizes the preferences of consumers under a natural budget constraint. Furthermore, these chooses are made so that the total consumption does not exceed the total production plus the total endowment. Mathematically, the preferences are represented by a utility function, that is, assumed to be convex, proper, and upper semi-continuous, and not necessarily monotone and/or differentiable. We further suppose that the production set is closed, bounded, and convex; namely, producers, cannot have an infinite production.
Finally, we would like to stress that in the classical literature the so-called survivability assumption is required: each agent has, at the beginning, a positive quantity of each commodity (see, e.g., ). Thanks to the used variational approach, the novelty of the current paper is mainly to weaken the described survivability assumption. In fact, in this market there is the possibility that, at the beginning, some agent is not endowed with some goods. In this situation, it is required that at least one of the goods owned by the agent must have minimum price not zero; namely, this commodity cannot be a free good. In this way, for any current price the agent has always the opportunity to earn on the sale of its endowment; hence, he can survive in the market.
The plan of the paper is as follows. Firstly, for the reader's convenience, we recall some basic definitions and properties which will be useful in the sequel. After, we introduce the competitive economic equilibrium model integrated with exchange, consumption, and production and we reformulate it in terms of a generalized quasi-variational inequality. Finally, we use this approach to investigate the existence of equilibrium.
It is worth to mention that this model allows to consider a wide class of utility functions, which are only convex, proper, and upper semi-continuous.
In the whole section , , and are Banach spaces. We denote with the ball in centered at and of radius .
A multivalued map (multimap) is said to be (a)upper semicontinuous (u.s.c.) if for any and for any neighborhood of there exists such that for all , ; (b)lower semicontinuous (l.s.c.) if for any , for any sequence of elements , , and for any , there exists a sequence of elements , with , ; (c)closed if for any sequences , , if and , then ;(d)compact if its range is relatively compact in ; that is, is compact in ;(e)quasicompact if its restriction to any compact subset is compact.We recall two results useful to obtain upper semicontinuity and lower semicontinuity (see, e.g.,  Theorem and Corollary .).
Theorem 1. Let be a closed and quasicompact multimap. Then is u.s.c.
Theorem 2. Assume that(i) is a continuous map such that is affine for every ; (ii) and are l.s.c. multimaps with closed and convex values; (iii)for every there exists such that ;
Then the multimap defined by is l.s.c. with closed and convex values.
Definition 3. Let be a convex map. For an arbitrary the set of all continuous linear functionals on with
is called the subdifferential of at . A continuous linear functional is called a subgradient of at .
Here it follows some properties of the subdifferential.
Proposition 4 (see, e.g.,  Propositions 2.1.2. and 2.1.5). Let be a convex and l.s.c. map. Then (1)for , is a nonempty convex and compact set; (2) is an u.s.c. multimap.To prove our main theorem we will rely on the following existence result for generalized quasi-variational inequalities.
Theorem 5 (see, e.g.,  Corollary 3.1). Let and be multimaps from into itself. Suppose that there exists a nonempty, compact, and convex set such that (i);(ii) is a nonempty, contractible, compact valued, and upper semicontinuous multimap on ;(iii) is a nonempty, continuous, and convex valued multimap on .
Then there exist and a vector such that
3. Equilibrium Model
We consider a marketplace consisting of two types of agents: consumers, indexed by , and producers, indexed by . We denote with , and , respectively, sets of consumers, producers, and goods. We denote by and the nonnegative quantities of commodity , respectively, owned and consumed by agent . The vectors and represent, respectively, the initial endowment and the consumption of agent and represents the consumption of the market. For each we denote by the set of indexes corresponding to initial holdings, namely, and we assume that he is endowed with at least one positive commodity, then . We denote by the quantity of commodity produced by producer . We note that the commodity can also assume negative values: the positive quantity represents the commodity offered in the market by producer , the negative quantity represents the demand required by the market but not satisfied by producer , and equals zero which means that the producer does not produce the commodity . The vector and the matrix represent the productions, respectively, of producer and of the market. To each commodity is a fixed minimum price such that . More precisely, each commodity has a positive price , which we suppose for all . We denote by the price vector and we suppose that prices belong to the set We observe that as usual in the economic literature, in order to survive in the market, it is required that each agent is endowed with each commodity , that is, . Thanks to our variational approach and by introduction of the minima prices, we can weaken this assumption by requiring the following weak survivability assumption: “for all agents there exists such that and .”
Each agent is endowed with at least one commodity with minimum price greater than zero, namely, . From an economic point of view this means that even if the agent is not endowed with some goods, he can be active in the market. In fact each agent has always the opportunity to earn on the sale of its endowment .
We denote by the production set of producer , where is assumed to be, as usual, a closed, bounded, convex set of with . The boundedness of means that producers cannot have an infinite production and includes the possibility that there is no activity for producer . We indicate with the total market production. The total production of commodity is shared between consumers: each consumer receives the given fraction , determined by a system of fixed weights such that for all . Hence, each consumer , relative to commodity , has at command the quantity . We note that can assume negative values, then it is possible that, the holdings of consumer is negative. Given a price vector , the inner product represents the value of the consumption plan , represents the value of the production plan , and represents the wealth of consumer at the going prices.
We notice that in the considered model, when the required demand of a commodity is not satisfied, the consumer does not suffer any loss. In this market producers act to maximize their profit , while consumers act to maximize their preferences described by a utility function , subject to budget constraints: the value of the consumption plan of agent cannot exceed the agent's wealth at the going prices. The multimap defined as represents the budget constraint set. Previous arguments lead to the following definition of competitive equilibrium.
Definition 6. Letting , and , we say that is a competitive equilibrium if and only if
We consider the equilibrium problem under the following assumptions:(H1)for all is a closed, bounded, convex set of with ;(H2)for all is concave, proper, and upper semi-continuous.
Denoting with the multimap defined as , we give now a characterization of the competitive equilibrium in terms of the following generalized quasi-variational inequality:
“find and , with :
Theorem 7. Let assumptions (H1)-(H2) be satisfied. A solution to (9), , such that is a competitive equilibrium, where is the vector of minima prices.
Proof. We divide the proof in several steps.
Step and are solutions to the generalized quasi-variational inequality (9) if and only if, for all , is a solution to for all , and are solutions to and is a solution to This is easily seen by testing (9), respectively, with for such that with for such that and with for all . Vice versa, let , and satisfy (11), (12), and (13), then (9) is verified.
Step 2. The maximization problem (6) is equivalent to the variational inequality (11).
It follows directly from the definition of the variational inequality (11).
Step 3. For all , is a solution to maximization problem if and only if there exists such that It is well known that from (H2) is locally Lipschitz continuous. Hence, there exists the generalized directional derivative, , of at on the direction and it holds Furthermore, since is convex, there exists the directional derivative ; then by (18) we have Thus, if is a minimal point of , it follows that then there exists such that (17) holds.
Conversely, let be such that (17) holds, then From the convexity of it follows that is an increasing function, implying and Hence, we get for , for all . Consequently, is a maximum point of in .
Step 4. The equilibrium condition (8) holds.
Indeed, by Step 1, variational inequality (13) holds. We pose We suppose by contradiction that . We observe that is nonempty; in fact, if , since , we have , namely, there exists at least one index such that , then but this contradicts (10), then . We consider where . We have for all that and then . If we replace in (13), it results in But this is false because is a solution to variational inequality (13). Hence, we have the equilibrium condition (8).
Then we can conclude that is a competitive equilibrium.
Proof. By equilibrium condition (8), since for all , we have that for all , it results in Then is a solution to the variational inequality (13). Moreover, by Steps 2 and 3 of Theorem 7 it follows that, for all and , and are, respectively, solutions to variational inequalities (11) and (12). Hence, is a solution to the variational problem (9).
4. Existence Theorems
This section concerns the study of existence of solutions to the generalized quasi-variational inequality (9). Firstly we achieve an existence result for the following generalized quasi-variational inequality:
“find and with for all: where , with such that for all (the existence of is ensured by the boundedness of for all ).
Theorem 9. Let assumptions (H1)-(H2) be satisfied. Then there exists a solution of (29).
Proof. We prove that all the hypotheses of the existence result of Theorem 5 are satisfied obtaining the claimed result.
Step The multimap defined as with for any , is u.s.c. with compact and convex values.
Since, from (H2), the map is a convex and l.s.c. map, from Proposition 4, it follows that the subdifferential is u.s.c. with compact convex values. Furthermore, the other two components of are single valued continuous maps, thus we can conclude that the multimap is u.s.c. with compact convex values.
Step 2. The multimap defined as , with , is l.s.c.
Indeed the map defined in (5) is l.s.c. To prove it we apply Theorem 2 with , , , , .
To apply the mentioned theorem we only need to verify that for any there exists such that This follows easily from the fact that where the last inequality is due to the fact that, since there exists such that and for any , we have for any Finally, the map is l.s.c. as cartesian product of l.s.c. multimaps and trivially it has convex values.
Step 3. .
Follows directly from the definition of .
Step 4. The map is u.s.c.
First of all we prove that is a closed multimap. Indeed let , , and , with for any , such that , , and , one has . In fact, since we have passing to the limit and we have so .
Hence, is a closed multimap as intersection of closed multimaps and is a closed multimap as cartesian product of closed multimaps.
Moreover, from Step 3, is a compact multimap, thus, by Theorem 1 it is u.s.c.
Hence, we can apply Theorem 5, obtaining a solution of the quasi-variational inequality (29).
Proof. Fix , we prove that and are a solution to the generalized variational inequality
We suppose that there exists such that . Since , , and are a convex set, then , for all . It results in
Finally if , by (10) we can choose
Hence, . Then we have that there exists such that , but this contradicts solution to
and by Step 1 in Theorem 7 this contradicts the fact that is a solution to (29).
Then, we can conclude that is a solution to (11), is a solution to (36), and is a solution to (13); namely, is a solution to (9).
Remark 12. We observe that if each agent is endowed with each commodity , that is, , then it is possible to assume for all and (10) is verified for solution to (29).
In fact, for all it results in , then Moreover, since is a solution to (12), for all , namely, Summing for it follows that In conclusion, being , (10) holds.
Remark 13. If each agent is endowed with all commodities there exists a competitive equilibrium .
5. Summary and Conclusions
The main result of this paper has been to obtain the existence of a competitive economic equilibrium for a model integrated with exchange, production, and consumption. In order to obtain a wide applicability in the economic framework, care was taken to keep a level of generality on the assumptions of the market. In particular utility functions which are proper, convex, and upper semi-continuous have been considered, hence without assuming any differentiability. These assumptions allow us to consider a wide range of utility functions frequently used in the economic literature.
A class of utility functions most widely used in economics consists of Cobb-Douglas utility functions: where indicates the importance which agent gives to the commodity . This class of utility functions is very much appreciated in economics, thanks to its analytical tractability (see, e.g.,  for an historical overview on Cobb-Douglas utility functions).
Another very interesting class of utility functions is represented by Constant Elasticity of Substitution (CES) utility functions: where coefficients are distribution parameters. These functions are characterized by constant elasticity of substitution between any two differentiated goods. The CES utility functions were originally introduced by Kenneth Arrow, as a generalization of Cobb-Douglas utility functions.
The functions, before mentioned, satisfy the assumptions we have taken in this paper; in fact they are proper, convex, and upper semi-continuous. However, these functions are not always differentiable on all their domain. They are defined and continuous on all of , but are only differentiable on the interior of : in particular the Cobb-Douglas utility function is not differentiable when while the CES utility function is never differentiable.
To conclude we want to stress that, in our opinion, by using the variational approach, the generalized quasi-variational inequalities are especially suitable to handle equilibrium problems for a market of exchange, consumption, and production, for it allows to take into account a wide class of models.
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