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.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

G. Anello 1 and F. Rania 2

Academic Editor:Naseer Shahzad

1, Department of Mathematics, University of Messina, Viale F. Stagno d'Alcontres 31, 98166 Messina, Italy

2, Department of Legal, Historical, Economic and Social Sciences, Magna Graecia University of Catanzaro, Campus loc. Germaneto, Viale Europa, 88100 Catanzaro, Italy

Received 30 December 2015; Revised 26 March 2016; Accepted 7 April 2016; 20 June 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 a-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 ( 1 ) show the properties of consequent demand (see the literature about regular economies, for instance) and ( 2 ) 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 _{Y j} of the j th producer will be a convex and compact set in ^{R l} containing the origin;

(iii): the utility function _{u i} of the i 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 3.2 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 G Q V I , 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 G Q V I , recalling, in particular, Theorem 3.2 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 n ∈ N , ^{R n} denotes the Euclidean space of the real n -vectors x = ( _{x 1} , ... , _{x n} ) equipped with the usual inner product [...] x , y [...] = ^{∑ i = 1 n} [...] _{x i y i} and norm _{( x ) n} = [...] x , x [...] , for any x , y ∈ ^{R n} .

The symbols ^{R + n} , ^{R 0 + n} , and ^{R [composite function] 0 + n} will indicate the cone of nonnegative, positive, and strongly positive vectors of ^{R n} , respectively. Furthermore, the set _{Δ n - 1} = ( x ∈ ^{R + n} : ^{∑ i = 1 n} [...] _{x i} = 1 ) indicates the unit simplex of ^{R + n} . We adopt the usual notation for vector inequalities, that is, for any x , y ∈ ^{R n} , one has x ≥ y if x - y ∈ ^{R + n} ; x > y if x - y ∈ ^{R 0 + n} ; and x >> y if x - y ∈ ^{R [composite function] 0 + n} . Let A ⊂ ^{R n} ; we will write int [...] ( A ) and A ¯ to indicate its interior and its closure, respectively. Open and closed balls of radius [straight epsilon] , centered at x ∈ ^{R n} , are denoted by _{B [straight epsilon]} ( x ) and _{B ¯ [straight epsilon]} ( x ) , respectively.

Let X be a subset of ^{R n} . A function f : X [arrow right] R is said to be quasiconcave iff for every r < _{s u p X} f , the set { x ∈ X : f ( x ) ≥ r } is convex. A function f : X [arrow right] R is said to be locally Lipschitz continuous near x ∈ X if there exist constants L > 0 and [straight epsilon] > 0 such that y ∈ _{B [straight epsilon]} ( x ) implies ( f ( y ) - f ( x ) ) <= L _{( y - x ) n} . L is called Lipschitz constant or r a n k of f .

Let f be a locally Lipschitz function near x ∈ X and let z ∈ ^{R n} . According to Clarke (see [19, page 25]), the generalized directional derivative of f at x in the direction z is defined by [figure omitted; refer to PDF] and the generalized gradient , or simply subdifferential , of f at x is the set-valued mapping, x [arrow right] ^{∂ [composite function]} f ( x ) , defined as follows: [figure omitted; refer to PDF]

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

Proposition 1 (Proposition 2.1 . 1 of [19]).

Let f be a locally Lipschitz function near x of r a n k L . Then the following hold:

(a) the function z [arrow right] ^{f [composite function]} ( x , z ) is finite, positively homogeneous, and subadditive on ^{R n} and satisfies [figure omitted; refer to PDF]

(b) ^{f [composite function]} ( x , - z ) = ( - f ^{) [composite function]} ( x , z ) .

Proposition 2 (Proposition 2.1 . 2 of [19]).

Let f be a locally Lipschitz function near x of r a n k L . Then the following hold:

(a) ^{∂ [composite function]} f ( x ) is a nonempty, convex, and compact set of ^{R n} ,

(b) for every z ∈ ^{R n} , one has [figure omitted; refer to PDF]

Proposition 3 (Proposition 2.1 . 5 of [19]).

Let f be a locally Lipschitz function near x of r a n k L . Then

(a) the multifunction ∂ f is closed,

(b) the multifunction ∂ f is upper semicontinuous at x .

Let X be a set; we write ^{2 X} for the family of all nonempty subsets of X . A correspondence or a multifunction between two sets X and Y is a function F : X [arrow right] ^{2 Y} . The graph of a multifunction F : X [arrow right] ^{2 Y} is the subset of X × X defined by g r ( F ) = { ( x , y ) ∈ X × Y : x ∈ X ⋀ y ∈ F ( X ) } . Let X be a subset of ^{R n} and let Γ : X [arrow right] ^{2 X} , Φ : X [arrow right] ^{2 R n} be two multifunctions. The classical generalized quasivariational inequality problem associated to X , Γ , Φ , denoted briefly by G Q V I ( X , Γ , Φ ) , is to find ( x ¯ , z ¯ ) ∈ X × ^{R n} such that [figure omitted; refer to PDF]

For the reader convenience, we report here the statement of Theorem 3.2 of [23], which the prove of our main result is based on.

Theorem 4 (Theorem 3.2 of Cubiotti [23]).

Let X be a closed convex subset of ^{R n} , K ⊆ X a nonempty compact set, and Φ : X [arrow right] ^{2 R n} and Γ : X [arrow right] ^{2 X} two multifunctions. Assume the following:

(i) the set Φ ( x ) is convex for each x ∈ K , with x ∈ Γ ( x ) ;

(ii) the set Φ ( x ) is nonempty and compact for each x ∈ X ;

(iii): for each y ∈ X - X , the set ( x ∈ X : _{i n f z ∈ Φ ( x )} [...] z , y [...] <= 0 ) is closed;

(iv) Γ is a lower semicontinuous multifunction (i.e., { x ∈ X : Γ ( x ) ∩ A ≠ ∅ } is open in X , for each open set A in X ) with closed graph and convex values.

Moreover, assume that there exists an increasing sequence _{[...] k} of positive real numbers, with X ∩ B ¯ ( 0 , _{[...] 1} ) ≠ ∅ and _{l i m k [arrow right] ∞ [...] k} = + ∞ such that if one puts _{D k} = B ¯ ( 0 , _{[...] k} ) , for each k ∈ N one has the following:

(v) Γ ( x ) ∩ _{D k} ≠ ∅ , for all x ∈ X ∩ _{D k} ;

(vi) for each x ∈ ( X ∩ _{D k} ) \ K , with x ∈ Γ ( x ) , [figure omitted; refer to PDF]

Then, there exists at least one solution to G Q V I ( X , Γ , Φ ) belonging to K × ^{R n} .

3. Economic Model

We consider a private ownership economy E a-la Arrow-Debreu (see [2, 3] for major details), where there are l commodities, m producers, and n consumers ( l , n , m ∈ N ). We index the commodities by the subscripts h = 1 , ... , l , the producers by the subscripts j = 1 , ... , m , and the consumers by the subscripts i = 1 , ... , n . We regard ^{R l} as the commodity space. By assuming the vector p = ( _{p 1} , ... , _{p l} ) ∈ ^{R + l} as a price system, the value of a commodity bundle a ∈ ^{R l} relative to the price p will be given by the inner product [...] p , a [...] = ^{∑ h = 1 l} [...] _{p h a h} .

3.1. Environments

The n consumers are labeled by i = 1 , ... , n . The consumption set _{X i} = ^{R + l} related to the consumer i is the set of all the l -uple of nonnegative real numbers. The preferences of the consumer i are expressed by utility functions _{u i} : _{X i} [arrow right] R , endowments by vectors _{e i} ∈ _{X i} , and shares of the profits of each firm by vectors _{θ i} ∈ [ 0,1 ^{] m} . The m producers are labeled by j = 1 , ... , m . The production set _{Y j} ⊆ ^{R l} related to consumer i is a subset of the set of all the l -tuple of real numbers. The aggregate endowment is e = ^{∑ i = 1 n} [...] _{e i} . We assume that the firms are owed by someone, or in other terms, 1 = ^{∑ i = 1 n} [...] _{θ i j} with _{θ i j} ∈ [ 0,1 ] for all j = 1 , ... , m . We summarized a private ownership economy by the tuple E = ( { _{X i} , _{e i} , _{u i} , _{θ i } i} , { _{Y j } j} , e ) .

3.2. Basic Restrictions

For a private ownership economy E we assume the restrictions listed as follows:

(A1) any price system p = ( ^{p ( 1 )} , ... , ^{p ( l )} ) ∈ ^{R + l} is normalized and bounded below by the vector q = ( ^{q ( 1 )} , ... , ^{q ( l )} ) ∈ ^{R + l} , called the minima prices, whose h t h component does not exceed the value of 1 / l , or equivalently [figure omitted; refer to PDF]

: is the set of the available price systems;

(A2) in any initial commodity bundle _{e i} ∈ ^{R + l} there exists at least one good h of positive quantity and positive minimum price: [figure omitted; refer to PDF]

(A3) for all j = 1 , ... , m , _{Y j} is convex and compact in ^{R l} such that _{0 ^{R l}} ∈ _{Y j} ;

(A4) there exists an open convex A ⊃ ^{R l} such that, for all i = 1 , ... , n , the following holds:

(a) _{u i} : A [arrow right] R is locally Lipschitz continuous and quasi-concave,

(b) _{0 ^{R l}} ∉ ^{∂ [composite function]} ( - _{u i} ) ( _{x i} ) for all _{x i} ∈ _{K i} ,

(c) ( - _{u i} ^{) [composite function]} ( _{x i} , _{i h} ) < 0 , for all _{x i} ∈ _{K i} , and h = 1 , ... , l such that ^{x i ( h )} = 0 ,

: where [figure omitted; refer to PDF]

: with M = _{m a x 1 <= j <= m m a x ( ^{y j ( 1 )} , ... , ^{y j ( l )} ) ∈ Y j} ^{∑ h = 1 l} [...] ( ^{y j ( h )} ) , and _{i h} is the unit vector of the h t h axis.

Remark 5.

Locally Lipschitz condition in an open convex set A ⊃ ^{R + l} , listed in (A4), will be needed to consider the Clark-subdifferential of _{u i} at each point of the closed set ^{R + l} . Quasiconcavity condition ( b ) in (A4) is weaker than the concavity condition and the strictly concave condition usually considered in the literature. Finally, the existence of the constant M comes from the compactness of the set ^{[...] j = 1 m} [...] _{Y j} (see assumption (A3)).

3.3. Existence Equilibrium Problem

Because consumer i owns two resources (initial endowment and profit share), indicated by [...] p , _{x i} [...] his expenditure and by [...] p , _{y j} [...] his profit derived from producer j , [figure omitted; refer to PDF] represents his budget set.

Set Y = ^{∏ j = 1 m} [...] _{Y j} , and M ( p , y ) = ^{∏ i = 1 n} [...] _{M i} ( p , y ) .

Definition 6.

An allocation is a couple ( x , y ) ∈ ^{R + n × l} × Y , where x = ( _{x 1} , ... , _{x n} ) represents the consumptions of all the consumers and y = ( _{y 1} , ... , _{y m} ) represents the productions of all the producers. In particular, an allocation ( x , y ) is said to be an individual allocation if x ∈ M ( p , y ) . An allocation is said to be a weakly balanced allocation if ^{∑ i = 1 n} [...] ( _{x i} - _{e i} ) - ^{∑ j = 1 m} [...] _{y j} <= 0 . 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 E is a triple ( p , x , y ) ∈ P × ^{R + n × l} × Y , where p is a price vector and ( x , y ) is an available allocation.

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

Problem 8.

Find ( p ¯ , x ¯ , y ¯ ) ∈ P × ^{R + n × l} × Y , with x ¯ ∈ M ( p ¯ , y ¯ ) satisfying [figure omitted; refer to PDF]

Remark 9.

A solution to Problem 8 is, then, a price vector p ¯ and an available allocation ( x ¯ , y ¯ ) such that x ¯ maximizes the consumers' utility, y ¯ maximizes the producers' profit, and ( x ¯ , y ¯ ) makes the market clear.

Definition 10.

A state ( p ¯ , x ¯ , y ¯ ) ∈ P × ^{R + n × l} × Y , with x ¯ ∈ M ( p ¯ , y ¯ ) , 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 E .

An equilibrium ( p ¯ , x ¯ , y ¯ ) ∈ P × M ( p ¯ , y ¯ ) × Y is said to be a Walras competitive equilibrium if in addition it satisfies the Walras' law: [figure omitted; refer to PDF]

Remark 11.

If there is no production (i.e., _{Y j} = _{0 ^{R + l}} for all j = 1 , ... , m ), then E 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 G Q V I problem as follows.

Problem 12.

Find ( p ¯ , x ¯ , y ¯ ) ∈ P × ^{R + n × l} × Y , with x ¯ ∈ M ( p ¯ , y ¯ ) , such that there exists T = ( _{T 1} , ... , _{T n} ) ∈ ^{∏ i = 1 n ∂ [composite function]} ( - _{u i} ) ( _{x ¯ i} ) satisfying [figure omitted; refer to PDF]

Let K = ^{∏ i = 1 n} [...] _{K i} , where _{K i} is as in condition (A4c).

Proposition 13.

Let ( p ¯ , x ¯ , y ¯ ) ∈ P × ^{R + n × l} × Y , with x ¯ ∈ M ( p ¯ , y ¯ ) , satisfying condition (11c) of Problem 8. Then x ¯ ∈ K .

Proof.

From (11c) of Problem 8 and the definition of the constant M , it promptly follows that [figure omitted; refer to PDF] for all i = 1 , ... , n . Now, suppose that x ¯ ∉ K . Then, [figure omitted; refer to PDF] for some i ∈ { 1 , ... , n } . This means that, for all h = 1 , ... , l , we should have [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] which is in contradiction with _{x ¯ i} ∈ _{M i} ( p ¯ , y ¯ ) . Thus, x ¯ ∈ K .

Proposition 14.

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

Proof.

Let ( p ¯ , x ¯ , y ¯ ) ∈ P × M ( p ¯ , y ¯ ) × Y be a competitive equilibrium and fix i ∈ { 1 , ... , n } . From Proposition 13, we have _{x ¯ i} ∈ _{K i} . So, by assumption (A4b) and condition (11a) of Problem 8 it cannot be that _{x ¯ i} ∈ int [...] ( _{M i} ( p ¯ , y ¯ ) ) .

Moreover, assumption (A4c) and again condition (11a) of Problem 8 imply that ^{x i ( h )} > 0 for all h = 1 , ... , l . Therefore, (12) is verified, which means that ( p ¯ , x ¯ , y ¯ ) 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 i ∈ { 1 , ... , n } and let _{x i} , _{z i} ∈ ^{R + l} be such that _{u i} ( _{x i} ) < _{u i} ( _{z i} ) . Then, ( - _{u i} ^{) [composite function]} ( _{x i} , _{z i} - _{x i} ) <= 0 .

Proof.

Let { ( _{y n} , _{t n} ) } be a sequence in A × ( 0,1 ) (the set A is as in (A4a)) such that ( _{y n} , _{t n} ) [arrow right] ( _{x i} , 0 ) . In force of locally Lipschitz continuity (and thus continuity) and _{u i} ( _{x i} ) < _{u i} ( _{z i} ) , we can suppose that _{u i} ( _{y n} ) < _{u i} ( _{z i} ) for all n ∈ N . In force of quasiconcavity, one has _{u i} ( _{y n} ) <= _{u i} ( _{y n} + _{t n} ( _{z i} - _{y n} ) ) for all n ∈ N . Consequently, [figure omitted; refer to PDF] Taking into account the arbitrariness of the sequence { ( _{y n} , _{t n} ) _{} n ∈ N} , 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 ( p ¯ , x ¯ , y ¯ ) ∈ P × ^{R + l} × Y , with x ¯ ∈ M ( p ¯ , y ¯ ) . Assume that ( p ¯ , x ¯ , y ¯ ) is a solution to Problem 12. Then, ( p ¯ , x ¯ , y ¯ ) is a solution to Problem 8.

Proof.

Let ( p ¯ , x ¯ , y ¯ ) be a solution to Problem 12, and let T ∈ ^{∏ i = 1 n} [...] ^{∂ [composite function]} ( - _{u i} ) ( _{x ¯ i} ) satisfying inequality ( G Q V I ) .

Testing ( G Q V I ) with ( p , x ¯ , y ¯ ) , p ∈ P , one has [figure omitted; refer to PDF] Moreover, from x ¯ ∈ M ( p ¯ , y ¯ ) , we promptly obtain [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] Now, choosing p = ( 0 , ... , 0,1 , 0 , ... , 0 ) ∈ P (the fact is possible in force of assumption (A1)), with 1 at the h t h position, we obtain condition (11c) of Problem 8. Furthermore, for each fixed j ∈ { 1 , ... , m } , testing ( G Q V I ) with ( p ¯ , x ¯ , ( _{y ¯ 1} , ... , _{y ¯ j - 1} , _{y j} , _{y ¯ j + 1} , ... , _{y ¯ m} ) ) , _{y j} ∈ _{Y j} , we obtain [...] p ¯ , _{y j} - _{y ¯ j} [...] <= 0 , for all _{y j} ∈ _{Y j} , which is condition (11b) of Problem 8.

At this point, condition (11a) of Problem 8 remains the only one to be proved. Fix i ∈ { 1 , ... , n } . Testing ( G Q V I ) with ( p ¯ , ( _{x ¯ 1} , ... , _{x ¯ i - 1} , _{x i} , _{x ¯ i + 1} , ... , _{x ¯ n} ) , y ¯ ) , _{x i} ∈ _{M i} ( p ¯ , y ¯ ) , we obtain [figure omitted; refer to PDF] Testing, for each h ∈ { 1 , ... , l } , inequality (21) with [figure omitted; refer to PDF] one has [figure omitted; refer to PDF] which clearly implies [...] _{T i} , _{x ¯ i} [...] <= 0 . We claim that [figure omitted; refer to PDF] Indeed, assume, on the contrary, that [figure omitted; refer to PDF] Then, for each h = 1 , ... , l , one has ^{x i ( h )} > 0 . Indeed, if ^{x i ( h )} = 0 for some h ∈ { 1 , ... , l } , by condition (A4c), we should have [figure omitted; refer to PDF] Now, from assumption (A2), there exists h ∈ { 1 , ... , l } such that ^{p ¯ ( h )} > 0 and ^{e i ( h )} > 0 . So, in particular, [...] _{e i} , p ¯ [...] > 0 . Then, if we fix ρ > 0 small enough, we have _{x ρ} [: =] ( 0,0 , ... , ρ , 0 , ... , 0 ) ∈ _{M i} ( p ¯ , y ¯ ) , with ρ at the h t h position. Consequently, testing (21) with _{x i} = _{x ρ} and taking (25) into account, we have [figure omitted; refer to PDF] which is in contradiction with (26). Therefore, ^{x i ( h )} > 0 , for each h = 1 , ... , l . In view of (23) and (25) this implies _{T i} = _{0 ^{R l}} , which is again in contradiction with (26). Then, strict inequality (24) holds. At this point, let _{z i} ∈ _{M i} ( p ¯ , y ¯ ) . By observing that _{z i} / 2 , _{x ¯ i} / 2 ∈ _{M i} ( p ¯ , y ¯ ) , if we put ^{y i θ , 1} = ( 1 - θ ) ( _{x ¯ i} / 2 ) + θ ( _{z i} / 2 ) and ^{y i θ , 2} = ( 1 - θ ) _{x ¯ i} + θ ( _{x ¯ i} / 2 ) , for all θ ∈ ( 0,1 ) , by the convexity of _{M i} ( p ¯ , y ¯ ) , we infer ^{y i θ , 1} , ^{y i θ , 2} ∈ _{M i} ( p ¯ , y ¯ ) . So, taking (21) into account, one has [figure omitted; refer to PDF] Moreover, in view of (24), one also has [figure omitted; refer to PDF] for all θ ∈ ( 0,1 ) . Adding side to side the above inequalities, we obtain [figure omitted; refer to PDF] From the above inequality and Proposition 15, it follows that _{u i} ( _{x ¯ i} ) ≥ _{u i} ( θ _{z i} ) , for all θ ∈ ( 0,1 ) . By the continuity of _{u i} , we then obtain _{u i} ( _{x ¯ i} ) ≥ _{u i} ( _{z i} ) . From the arbitrariness of _{z i} ∈ _{M i} ( p ¯ , y ¯ ) , condition (11a) of Problem 8 follows.

Remark 17.

Under the assumptions of Theorem 16, condition (11c) actually holds as equality. Indeed, fix i ∈ { 1 , ... , n } and define [figure omitted; refer to PDF] We claim that _{g i} ( _{x ¯ i} ) = 0 . Indeed, if not, taking in mind that _{x ¯ i} ∈ _{M i} ( p ¯ , y ¯ ) , it should be _{g i} ( _{x ¯ i} ) > 0 . Then, for each h = 1 , ... , l , there exists _{ρ h} > 0 such that [figure omitted; refer to PDF] Testing ( G Q V I ) with ( p ¯ , x ¯ ( ρ ) , y ¯ ) , we obtain [figure omitted; refer to PDF] Thus, in view of (A4c), it must be ^{x ¯ i h} > 0 for all h = 1 , ... , l . This fact, together with _{g i} ( _{x ¯ i} ) > 0 , implies _{x ¯ i} ∈ int [...] ( _{M i} ( p ¯ , y ¯ ) ) . Testing ( G Q V I ) with ( p ¯ , x , y ¯ ) , where x = ( _{x ¯ 1} , ... , _{x ¯ i - 1} , _{x i} , _{x ¯ i + 1} , ... , _{x ¯ n} ) , with _{x i} being arbitrarily chosen in _{M i} ( p ¯ , y ¯ ) , we obtain [...] _{T i} , _{x i} - _{x ¯ i} [...] ≥ 0 for all _{x i} ∈ _{M i} ( p ¯ , y ¯ ) . Since _{x ¯ i} ∈ int [...] ( _{M i} ( p ¯ , y ¯ ) ) , from this inequality it follows that _{T i} = _{0 ^{R l}} 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 i = 1 , ... , n , the map ^{∂ [composite function]} ( - _{u i} ) : ^{R + l} [arrow right] ^{2 R l} has closed graph.

Proof.

If, for each i = 1 , ... , n the utility _{u i} is locally Lipschitz on ^{R + l} , then according to (b) of Proposition 1, to (a) of Proposition 2, and to (b) of Proposition 3, for each i = 1 , ... , n , the subdifferential of - _{u i} has closed graph.

Now, put X = P × ^{R + n × l} × Y and define [figure omitted; refer to PDF] where ^{∂ [composite function]} ( - u ) ( x ) = ( ^{∂ [composite function]} ( - _{u 1} ) ( _{x 1} ) , ... , ^{∂ [composite function]} ( - _{u n} ) ( _{x n} ) ) , for all x = ( _{x 1} , ... , _{x n} ) ∈ ^{R + n × l} .

By means of these notations, we can rewrite the variational inequality ( G Q V I ) as follows: find ( p ¯ , x ¯ , y ¯ ) ∈ P × ^{R + n × l} × Y , with ( p ¯ , x ¯ , y ¯ ) ∈ Γ ( p ¯ , x ¯ , y ¯ ) , and ( z ^ , T , w ^ ) ∈ Φ ( p ¯ , x ¯ , y ¯ ) such that [figure omitted; refer to PDF]

Theorem 19.

Assume that conditions (A3), (A4a), and (A2) hold. Then, Problem 12 admits at least a solution in P × C × Y , where [figure omitted; refer to PDF] and M > 0 is as in ( A 4 c ) .

Proof.

First, note the following:

(i) the set X is nonempty closed and convex in ^{R l} × ^{R n × l} × ^{R m × l} ;

(ii) the set K [: =] P × C × Y ⊂ X is nonempty and compact in ^{R l} × ^{R n × l} × ^{R m × l} ;

(iii): Γ ( p , x , y ) is a nonempty convex subset of X , for all ( p , x , y ) ∈ X .

Moreover, recalling that ^{∂ [composite function]} ( - _{u i} ) ( _{x i} ) is (nonempty) convex and compact in ^{R l} , for all i = 1 , ... , n and for all _{x i} ∈ ^{R + l} , we also have the following:

(iv) Φ ( p , x , y ) is a nonempty convex and compact subset of ^{R n} × ^{R n × l} × ^{R m × l} , for all ( p , x , y ) ∈ X .

Thus, to satisfy all the assumptions of Theorem 3.2 of [23], it remains to check that the following further conditions hold:

( _{a 1} ) : the set [figure omitted; refer to PDF]

: is closed, for each ( ρ , τ , ω ) ∈ X - X ;

( _{a 2} ) : the map Γ : X [arrow right] ^{2 X} is lower semicontinuous with closed graph;

( _{a 3} ) : there exists _{R 0} such that if for each R ∈ [ _{R 0} , ∞ [ we denote by _{B R} the closed ball in ^{R n} × ^{R n × l} × ^{R m × l} centered at 0 with radius R , one has _{B R} ∩ X ≠ ∅ , and

(i) Γ ( p , x , y ) ∩ _{B R} ≠ ∅ , for all ( p , x , y ) ∈ X ∩ _{B R} ;

(ii) _{s u p ( ^{p [variant prime]} , ^{x [variant prime]} , ^{y [variant prime]} ) ∈ Γ ( p , x , y ) ∩ B R i n f ( z , T , w ) ∈ Φ ( p , x , y )} [...] ( z , T , w ) , ( p , x , y ) - ( ^{p [variant prime]} , ^{x [variant prime]} , ^{y [variant prime]} ) [...] > 0 , for all ( p , x , y ) ∈ X ∩ _{B R} \ K , with ( p , x , y ) ∈ Γ ( p , x , y ) .

At the end we divide the proof in several steps.

Step 1. To prove that condition ( _{a 1} ) holds true, fix ( ρ , τ , ω ) ∈ X - X and let { ( _{p k} , ^{x k} , ^{y k} ) } be a sequence in Λ ( ρ , τ , ω ) such that ( _{p k} , ^{x k} , ^{y k} ) [arrow right] ( _{p [low *]} , ^{x [low *]} , ^{y [low *]} ) as k [arrow right] ∞ . Let us show that ( _{p [low *]} , ^{x [low *]} , ^{y [low *]} ) ∈ Λ ( ρ , τ , ω ) . At first observe that since X is closed, one has ( _{p [low *]} , ^{x [low *]} , ^{y [low *]} ) ∈ X . Moreover, since Φ ( _{p k} , ^{x k} , ^{y k} ) is compact for each k ∈ N , and the function [figure omitted; refer to PDF] is continuous in ^{R n} × ^{R n × l} × ^{R m × l} , then, for each k ∈ N , we can find ( ^{z k} , ^{T k} , ^{w k} ) ∈ Φ ( _{p k} , ^{x k} , ^{y k} ) such that [figure omitted; refer to PDF] Note that, from the definition of Φ , for each k ∈ N , one has [figure omitted; refer to PDF] Moreover, recalling that _{u i} is locally Lipschitz continuous in ^{R + l} for all i = 1 , ... , n , then, for each k ∈ N , there exist an open neighborhood _{A k} of ^{x k} in ^{R n × l} and a constant _{L k} ≥ 0 such that [figure omitted; refer to PDF] Furthermore, there exist an open neighborhood _{A 0} of ^{x [low *]} in ^{R n × l} and a constant _{L 0} ≥ 0 such that [figure omitted; refer to PDF] At this point, observe that the family of open sets { _{A k } k ∈ N ∪ { 0 }} is a covering of the compact set { ^{x k} _{} k ∈ N} ∪ { ^{x [low *]} } . Therefore, from (43) and (44), we infer that there exists a constant L ≥ 0 such that [figure omitted; refer to PDF] Consequently, from (42), up to a subsequence, we can suppose that the sequence { ^{T k} } converges to some ^{T [low *]} ∈ ^{R n × l} . Now, from (40) and (41), we infer that [figure omitted; refer to PDF] and, from Proposition 18 and (42), we also infer that [figure omitted; refer to PDF] Furthermore, from (39), passing to the limit as k [arrow right] ∞ , one has [figure omitted; refer to PDF] At this point, observe that considering together conditions (46) and (47) means that ( ^{z [low *]} , ^{T [low *]} , ^{w [low *]} ) ∈ Φ ( _{p [low *]} , ^{x [low *]} , ^{y [low *]} ) and this latter condition, together with (48), gives ( _{p [low *]} , ^{x [low *]} , ^{y [low *]} ) ∈ Λ ( ρ , τ , ω ) . Therefore, condition ( _{a 1} ) is proved.

Step 2. Now, let us show that the map Γ is lower semicontinuous in X . To this end, it is sufficient to prove that, for every ( _{p 0} , _{x 0} , _{y 0} ) ∈ X , every ( p , x , y ) ∈ Γ ( _{p 0} , _{x 0} , _{y 0} ) , and every sequence { ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) } in X such that ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) [arrow right] ( _{p 0} , _{x 0} , _{y 0} ) as k [arrow right] + ∞ , there exists a sequence { ( _{p k} , ^{x k} , ^{y k} ) } in X such that ( _{p k} , ^{x k} , ^{y k} ) ∈ Γ ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) for all k ∈ N and ( _{p k} , ^{x k} , ^{y k} ) [arrow right] ( p , x , y ) as k [arrow right] + ∞ (see [18] at page 39, for instance).

So, let ( _{p 0} , _{x 0} , _{y 0} ) , ( p , x , y ) , and { ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) } be as above. For each i = 1 , ... , n , using the fact that ( p , x , y ) ∈ Γ ( _{p 0} , _{x 0} , _{y 0} ) , we have the following two situations:

either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Suppose that (49) holds. Then, since ( _{p ^ k} , ^{y ^ k} ) [arrow right] ( _{p 0} , _{y 0} ) , there exists _{k 0} ∈ N such that [figure omitted; refer to PDF] So, in this case, if we put ^{x i k} = _{x i} for k ≥ _{k 0} and ^{x i k} = 0 for k = 1 , ... , _{k 0} - 1 , it is easy to check that ^{x i k} ∈ _{M i} ( _{p ^ k} , ^{y ^ k} ) , for all k ∈ N . Moreover, it is clear that ^{x i k} [arrow right] _{x i} as k [arrow right] + ∞ .

Suppose that (50) holds. Then, from the survivability condition (A2), we have [figure omitted; refer to PDF] Consequently, since ( _{p ^ k} , _{x ^ k} , ^{y ^ k} ) [arrow right] ( _{p 0} , _{x 0} , _{y 0} ) , we have [figure omitted; refer to PDF] Therefore, if we put [figure omitted; refer to PDF] it is easy to check that ^{x i k} ∈ _{M i} ( _{p ^ k} , ^{y ^ k} ) , for all k ∈ N . Indeed, if _{a k} > 0 , then [figure omitted; refer to PDF] and so ^{x i k} ∈ _{M i} ( _{p ^ k} , ^{y ^ k} ) , while, if _{a k} = 0 , then ^{x i k} = _{x i} and [figure omitted; refer to PDF] from which we again obtain ^{x i k} ∈ _{M i} ( _{p ^ k} , ^{y ^ k} ) . Finally, observe that ^{x i k} [arrow right] _{x i} as k [arrow right] + ∞ .

So, for each i = 1 , ... , n , in both cases (49) and (50), we can find a sequence ^{x i k} which converges to _{x i} and such that ^{x i k} ∈ _{M i} ( _{p ^ k} , ^{y ^ k} ) , for all k ∈ N . Consequently, if we put ^{x k} = ( ^{x 1 k} , ... , ^{x n k} ) , for all k ∈ N , the sequence { ( _{p k} , ^{x k} , ^{y k} ) } , where _{p k} = p and ^{y k} = y , for all k ∈ N , satisfies ( _{p k} , ^{x k} , ^{y k} ) ∈ Γ ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) for all k ∈ N and ( _{p k} , ^{x k} , ^{y k} ) [arrow right] ( p , x , y ) as k [arrow right] + ∞ , as desired. Therefore, Γ is lower semicontinuous in X .

To show that condition ( _{a 2} ) hold true, it remains to prove that Γ has closed graph. To this end, let { ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) } and { ( _{p k} , ^{x k} , ^{y k} ) } be two sequences in X such that ( _{p k} , ^{x k} , ^{y k} ) ∈ Γ ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) , for all k ∈ N , and suppose that ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) [arrow right] ( _{p 0} , _{x 0} , _{y 0} ) , ( _{p k} , ^{x k} , ^{y k} ) [arrow right] ( p , x , y ) , as k [arrow right] ∞ . Let us show that ( p , x , y ) ∈ Γ ( _{p 0} , _{x 0} , _{y 0} ) .

Since P and Y are closed, one has p ∈ P and y ∈ Y . Moreover, with ( _{p k} , ^{x k} , ^{y k} ) ∈ Γ ( _{p ^ k} , ^{x ^ k} , ^{y ^ k} ) for all k ∈ N , one has [figure omitted; refer to PDF] Passing to the limit as k [arrow right] ∞ , we obtain [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] which, together with p ∈ P and y ∈ Y , gives ( p , x , y ) ∈ Γ ( _{p 0} , _{x 0} , _{y 0} ) , as desired.

Step 3. To finish the proof of our Theorem, it remains to show that condition ( _{a 3} ) holds true as well.

Let _{R 0} > 0 be such that the closed ball _{B R 0} ⊂ ^{R n} × ^{R n × l} × ^{R m × l} contains the compact set K . Then, for each R ∈ [ _{R 0} , ∞ [ , one has K ⊂ X ∩ _{B R} and P × { 0 } × Y ⊂ Γ ( p , x , y ) ∩ _{B R} , for all ( p , x , y ) ∈ X . Therefore, condition ( i ) of ( _{a 3} ) holds. Suppose that condition ( i i ) of ( _{a 3} ) does not hold. Then, there should exist ( p ¯ , x ¯ , y ¯ ) ∈ X ∩ _{B R} \ K , with ( p ¯ , x ¯ , y ¯ ) ∈ Γ ( p ¯ , x ¯ , y ¯ ) , such that [figure omitted; refer to PDF] Now, let us put _{p [low *]} [: =] ( 1 / l , ... , 1 / l [...] l -times ) ∈ P . Then, ( _{p [low *]} , x ¯ , y ¯ ) ∈ _{B R} ∩ X . Moreover, from ( p ¯ , x ¯ , y ¯ ) ∈ Γ ( p ¯ , x ¯ , y ¯ ) , it trivially follows that ( _{p [low *]} , x ¯ , y ¯ ) ∈ Γ ( p ¯ , x ¯ , y ¯ ) . Thus, we can test (60) with ( ^{p [variant prime]} , ^{x [variant prime]} , ^{y [variant prime]} ) = ( _{p [low *]} , x ¯ , y ¯ ) . Doing so, we get [figure omitted; refer to PDF] Therefore, with Φ ( p ¯ , x ¯ , y ¯ ) being a compact set, there should exist ( z ¯ , T ¯ , w ¯ ) ∈ Φ ( p ¯ , x ¯ , y ¯ ) such that [figure omitted; refer to PDF] From the definition of Φ , the previous inequality is equivalent to [figure omitted; refer to PDF] which, taking in mind that x ¯ ∈ M ( p ¯ , y ¯ ) , implies [figure omitted; refer to PDF] Consequently, [figure omitted; refer to PDF] Therefore, x ¯ ∈ C . But this contradicts the fact that ( p ¯ , x ¯ , y ¯ ) ∈ X ∩ _{B R} \ K = ( P × ^{R + n × l} × Y ) ∩ _{B R} \ ( P × C × Y ) .

The proof is now complete.

6. Applications

The well-known Cobb-Douglas utility function _{u i} : ^{R + l} [arrow right] _{R +} defined by [figure omitted; refer to PDF] where _{λ h} > 0 for all h = 1 , ... , l , is concave if ^{∑ h = 1 l} [...] _{λ h} < 1 , but it is only quasiconcave (and not necessarily concave) regardless of the sum of exponents _{λ h} . 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 _{u i} given by (66) for arbitrary positive exponents _{λ h} . The same result also applies to CES utility function defined by [figure omitted; refer to PDF] where A , _{α h} > 0 , for all h = 1 , ... , l , and ρ <= 1 .

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 [figure omitted; refer to PDF] where _{g j} : _{R +} [arrow right] _{R +} are strictly increasing concave or convex functions, not necessarily of class ^{C 1} .

7. Conclusions

In [6, 21] under concavity assumption, a demand function _{x i} ( p ) 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 G Q V I 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 G Q V I 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 [figure omitted; refer to PDF]

: where A , _{λ h} > 0 , for all h = 1 , ... , l ;

(ii) the CES type utility so defined by [figure omitted; refer to PDF]

: where A , _{λ h} > 0 , for all h = 1 , ... , l and ρ <= 1 .

Acknowledgments

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

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