Systems of variational inequalities related to economic equilibrium

36

Systems of variational inequalities related to economic equilibrium

by

Zdzis law Naniewicz ¹ and Magdalena Nockowska ²

1 Cardinal Stefan Wyszy´ nski University, Faculty of Mathematics and Science Dewajtis 5, 01-815 Warsaw, Poland

2 Technical University of L´ od´z, Institute of Mathematics W´ olcza´ nska 215, 93-005 L´ od´z, Poland, and

Technical University of L´ od´z, Center of Mathematics and Physics Al. Politechniki 11, 90-924 L´ od´z, Poland

e-mail: naniewicz@uksw.edu.pl, magdan@mail.p.lodz.pl Abstract: In the paper a new approach to the Walrasian gen- eral equilibrium model of economy is presented. The classical mar- ket clearing condition is replaced by suitably formulated variational inequality. It states that the market clears for a commodity if its equilibrium price is positive; otherwise, there may be an excess sup- ply of the commodity in equilibrium and then its price is zero. Such approach enables establishing new existence results without assump- tions which were fundamental for the currently used methods:

(i) Dis-utility functions are not required to be strictly convex and they may attain their minima in the consumption sets (the local nonsatiation of preferences is not required).

(ii) The boundary of the positive orthant is allowed for the price vector in equilibrium. It allows for investigation of certain new prob- lems, e.g. bankruptcy conditions.

Keywords: optimization problem, variational inequalities, du- ality, competitive equilibria.

1. Introduction

Let x j ∈ R ⁿ + be minimizers of a finite collection of convex objectives V j : R ⁿ + → R ∪ {+∞}, j = 1, . . . , m. The minimizers are assumed to fulfill unilateral con- straints

A j π, x j

 ≤ φ _j (π) determined by given nonnegative continuous func- tions φ j (·) depending on a vector π ∈ R ⁿ + . The problem is to find π ∈ R ⁿ + and (x j ) ∈ (R ⁿ + ) ^m which are linked together by a subdifferential relation of the form P m

j=1 A ^T _j x j ∈ ∂Φ + (π), Φ + being a convex function.

The main feature of the aforementioned problem is that the feasible set for the unknowns π, x j , j = 1, . . . , m, is nonconvex and, hence, the standard theory of variational inequalities (see Kinderlehrer and Stampacchia, 1980, Ekeland and Temam, 1976) cannot be applied directly to obtain solutions. The approach pre- sented here does not involve the notion of Pareto optimum or its generalizations (see Pallaschke and Rolewicz, 1997; Luc, 1989; Lee et al., 1998; Hadjisavvas and Schaible, 1998 and the references therein), but, roughly speaking, is based on the analysis of objectives’ parametrized constrained minima x j (π)

as func- tions of π. Some ideas from Naniewicz and Panagiotopoulos (1995) concerning the treatment of nonmonotone inequality problems are applied.

The considered problem has been first studied in Naniewicz (2002) under the hypothesis that φ j (τ ) ≥ δ j , ∀ τ ∈ R ⁿ + , δ j > 0. Now we examine the case in which the functions φ j , j = 1, . . . , m are positive homogeneous of degree 1 and Φ = P m

j=1 φ j . Sufficient conditions for existence of the solutions for the problem will be formulated.

The motivation for this work comes from mathematical economics (see, e.g., von Neumann, 1945-6; Nash, 1950; Arrow and Intrilligator, 1982; Arrow and Debreu, 1954; Aliprantis, Brown, Burkinshaw, 1989; Nagurney, 1999; Nagur- ney and Siokos, 1997; Panek, 2000, and the references quoted there). Under the aforementioned assumptions on φ j , j = 1, . . . , m, and Φ the problem to be studied is related to the general equilibrium model of an economy in finite dimensional commodity space, in particular to the models of Arrow-Debreu and Arrow-Debreu-McKenzie. There is a large literature on the general equilibrium model as given by both finite and infinite dimensional commodity spaces. For this issue we refer the reader to von Neumann (1945-46), Nash (1950), Arrow and Debreu (1954), Arrow and Intrilligator (1982), Aliprantis, Brown, Burkin- shaw (1989), Bulavsky (1994), Chichilnisky and Heal (1993, 1998), Chichilnisky (1993), Nagurney and Siokos (1997), Mas-Colell, Whinston, Green (1995), Mas- Colell (1986), Mas-Colell and Richard (1991), McKenzie (1959), Gale and Mas- Colell (1975), Negishi (1960), Scarf (1973), Eaves (1972), Hirsh and Smale (1979), Smale (1976), Aliprantis, Tourky, Yannelis (2001), Aliprantis, Mon- teriro, Tourky (2004), Aliprantis, Florenzano, Tourky (2005), and the references therein. The use of the homotopy methods for economic equilibria can be found in Eaves (1972), Hirsh and Smale (1979), Smale (1976) (see also Chichilnisky, 1993 and the references quoted there).

Recall that in the Arrow-Debreu-McKenzie model there are m consumers (indexed by j ∈ J = {1, . . . , m}), n firms (indexed by i ∈ I : = {1, . . . , n}), and s goods (indexed by l ∈ L : = {1, . . . , s}). In such economy, society’s initial endowments and technological possibilities (i.e., the firms) are owned by consumers. The initial endowment of j’s consumer is given by ω j ∈ R ⁿ + . In addition, we suppose that consumer j owns a share κ ji of firm i, where P

j∈J κ ji = 1. Denote by Y i ⊂ R ⁿ the production set associated with i’s firm.

Recall that allocation (x ^⋆ ₁ , . . . , x ^⋆ _m , y ^⋆ ₁ , . . . , y ^⋆ _n ), x ^⋆ _j ∈ R ⁿ + , j ∈ J, y ^⋆ _i ∈ R ⁿ , i ∈

I, and price vector p ^⋆ ∈ R ⁿ + constitute a competitive (or Walrasian) equilibrium

if the following conditions are satisfied (Mas-Colell, Whinston, Green, 1995):

Profit maximization: For each firm i ∈ I, y _i ^⋆ solves

y max

∈Y

p ^⋆ , y i

 ; (1)

Dis-utility minimization: For each consumer j ∈ J, x ^⋆ _j solves min n

V j (x j ) : p ^⋆ , x j

 ≤ p ^⋆ , ω j

 + X

i∈I

κ ij

p ^⋆ , y ^⋆ _i 

, x j ∈ R ⁿ +

o

; (2)

Market clearing:

X

j∈J

x ^⋆ _j = X

j∈J

ω j + X

i∈I

y ^⋆ _i . (3)

In the presented approach we introduce convex, nonnegative valued, positive homogeneous of degree 1 functions φ j (p) : =

p, ω j

 + P

i∈I κ ij sup _y

_∈Y

p, y i

 and

Φ(p) : = X

j∈J

φ j (p) = p, X

j∈J

ω j

 + X

i∈I

sup

y

∈Y

p, y i

 , p ∈ R ⁿ + ,

and instead of (3) the variational inequality, called the balance condition, will be considered

D

p − p ^⋆ , − X

j∈J

x ^⋆ _j E

+ Φ(p) − Φ(p ^⋆ ) ≥ 0, ∀ p ∈ R ⁿ + . (4)

It states that the market clears for a commodity if its equilibrium price is posi- tive. Otherwise, there may be an excess supply of the commodity in equilibrium and then its price is zero.

Finally, a more general problem can be stated:

Find {x ^⋆ _j } j∈J ⊂ R ^nm + and p ^⋆ ∈ R ⁿ + such that

Dis-utility minimization: For each consumer j ∈ J, x ^⋆ _j solves min n

V j (x j ) :

A j p ^⋆ , x j

 ≤ φ j (p ^⋆ ), x j ∈ R ⁿ +

o ; (5)

Balance condition:

D p − p ^⋆ , − X

j∈J

A ^T _j x ^⋆ _j E

+ Φ(p) − Φ(p ^⋆ ) ≥ 0, ∀ p ∈ R ⁿ + , (6)

which will be examined from the point of view of the existence issues.

2. Statement of the problem and preliminaries

First, basic notations are presented. Denote by R ⁿ the Euclidean vector space of all vectors x = 

x 1 , . . . , x n

 , x i ∈ R, i = 1, . . . , n, equipped with the inner product

· , · 

: R ⁿ × R ⁿ → R defined by π, x 

= P n

i=1 x i p i , x = 

x 1 , . . . , x n  , π = 

p 1 , . . . , p n 

∈ R ⁿ .

By R ^n×n we denote all n × n real valued matrices. Moreover, the following notations will be used:

R + = {α ∈ R: α ≥ 0}, R ⁿ + = 

x = 

x 1 , . . . , x n

 ∈ R ⁿ : x i ≥ 0, ∀ i = 1, . . . , n , R ^n×n + = 

A = (A ik ) ∈ R ^n×n : A ik ≥ 0, ∀ i, k = 1, . . . , n , R ⁿ − = 

x = 

x 1 , . . . , x n

 ∈ R ⁿ : x i ≤ 0, ∀ i = 1, . . . , n , Moreover, denote by ind K the indicator function of a set K i.e.

ind K (y) =

( 0 if y ∈ K, +∞ otherwise.

Throughout the paper it will be assumed that the functions

V j : R ⁿ → R ∪ {+∞}, j = 1, . . . , m, (7)

are convex, proper and lower semicontinuous functions and V j := V j + ind _R

.

Assume that the functions

φ j : R ⁿ + → R with φ j (τ ) ≥ 0, ∀ τ ∈ R ⁿ + , j = 1, . . . , m, (8) are continuous functions. Moreover, let the matrices A j ∈ R ^n×n + satisfy

Ker A j = {0}, j = 1, . . . , m, (9)

where Ker A j = {τ ∈ R ⁿ + : A j τ = 0}. Furthermore, let

Φ : R ⁿ → R ∪ {+∞} (10)

be a convex, proper, lower semicontinuous function.

Recall that if H is Hilbert space and ϕ : H → R∪{+∞} is a convex function, the subdifferential ∂ϕ : H → 2 ^H is defined by

∂ϕ(u) = {w ∈ H : ϕ(v) − ϕ(u) ≥

w, v − u 

, ∀v ∈ H},

provided that ϕ(u) < +∞ and ∂ϕ(u) = ∅, otherwise.

We are now in a position to formulate the main problem of the paper.

Problem (P): Find π ∈ R ⁿ + and x j ∈ R ⁿ + , j = 1, . . . , m, such as to satisfy the conditions:

(P M ) V j (x j ) = min 

V j (x) : A j π, x 

≤ φ j (π) and x ∈ R ⁿ +

, j = 1, . . . , m,

(P E) D

− X m j=1

A ^T _j x j , τ − π E

+ Φ(τ ) − Φ(π) ≥ 0, ∀ τ ∈ R ⁿ + .

If we assume that for any j = 1, . . . , m the matrix A j is the identity matrix, the function −V j describes the preferences of j-trader’s, the function φ j (π) =

π, ω j

 is a profit of j-trader’s and Φ = P m

j=1 φ j , then the problem (P ) with π > 0 (each component of π is positive) is equivalent to finding of an equilibrium in the Arrow-Debreu model of pure exchange. In general, (P E) states that the market clears for a commodity if its equilibrium price is positive. Otherwise, there may be an excess supply of the commodity in equilibrium and then its price will be zero.

Moreover, a solution of the problem (P ) can be treated as the problem of a competitive equilibrium of Arrow-Debreu-McKenzie model, when the functions φ j (π) =

π, ω j

 + P s

k=1 θ jk sup _y

_∈Y

π, y k

 describe a profit of j-trader’s and Φ = P m

j=1 φ j (see Aliprantis, Brown, Burkinshaw, 1989; Panek, 2000).

The case of φ j (τ ) ≥ δ j ∀ τ ∈ R ⁿ + , δ j > 0 is the starting point to an examina- tion of the case in which the functions φ j , j = 1, . . . , m are positive homogeneous of degree 1 and Φ = P m

j=1 φ j . Now we remind the main results for such a case.

The sufficient condition for (P M ) to have solutions reads as follows.

Theorem 1 (Theorem 1, p. 149, Naniewicz, 2002) Assume that for any j = 1, . . . , m, the hypotheses below hold:

(H 1 ) 0 ∈ cl(Dom ∂V j ), (R ⁿ ₋ \{0})∩B _R

(0, r j ) ⊂ Int Dom V _j ^⋆ for some r j > 0;

(H 2 ) 

x ∈ R ⁿ + : {hx ^⋆ , xi : x ^⋆ ∈ ∂V _j (x)} ∩ R − 6= ∅

⊂ B _R

(0, M j ) for some M j > 0;

(H 4 ) φ j (τ ) ≥ δ j ∀τ ∈ R ⁿ + for some δ j > 0.

Then for any π ∈ R ⁿ + \ {0} the optimization problem: Find x _j ∈ R ⁿ + such that V j (x j ) = min{V j (y) : ∀y ∈ R ⁿ ₊ with

A j π, y 

≤ φ j (π)} (11)

has at least one solution. Moreover, there exists α j ∈ Λ j (π), Λ j (π) being the set of all solutions of variational inequality

A j π, −∂V _j ^⋆ (−α j A j π) 

(t − α j ) + φ j (π)(t − α j ) ≥ 0, ∀ t ≥ 0, (12) with the property that

x j ∈ ∂V _j ^⋆ −α j A j π

. (13)

Additionally, Λ j : R ⁿ + \ {0} → 2 ^R

when extended to R ⁿ + by setting Λ j (0) := {0}

has nonempty, closed, convex and bounded values and it is an upper semicon- tinuous mapping from R ⁿ + into 2 ^R

.

Remark 1 From the proof of the Theorem 1 we get that

∂V _j ^⋆ (0) 6= ∅, j = 1, . . . , m.

Now the problem (P E) can be considered. Taking into account (13) we introduce a multivalued mapping R : R ⁿ + → 2 ^R

by setting

R(π) := − X m j=1

A ^T _j ∂V _j ^⋆ (−Λ j (π)A j π), π ∈ R ⁿ ₊ , (14)

(where y ∈ R(π) if and only if there exist α j ∈ Λ j (π), x j ∈ ∂V _j ^⋆ (−α j A j π) for any j = 1, . . . , m such that y = − P m

j=1 A ^T _j x j ).

It is easily seen that (P E) can be equivalently formulated as follows: Find π ∈ R ⁿ + and X ∈ R(π) such that

X, τ − π 

+ Φ(τ ) − Φ(π) ≥ 0, τ ∈ R ⁿ + . (15)

As far as R is concerned we have the result.

Proposition 1 (Proposition 4, p.150, Naniewicz, 2002) Under the hypotheses of Theorem 1, R given by (14) is a multivalued, upper semicontinuous mapping from R ⁿ + into 2 ^R

with nonempty, convex, closed and bounded values.

This allows the formulation of the following result.

Theorem 2 (Theorem 2, p.151, Naniewicz, 2002) Suppose that for any j = 1, . . . , m the hypotheses below hold:

(H 1 ) 0 ∈ cl(Dom ∂V j ), (R ⁿ − \{0})∩B _R

(0, r j ) ⊂ Int Dom V _j ^⋆ for some r j > 0;

(H 2 ) 

x ∈ R ⁿ + : {hx ^⋆ , xi : x ^⋆ ∈ ∂V j (x)} ∩ R − 6= ∅

⊂ B _R

(0, M j ) for some M j > 0;

(H 4 ) φ j (τ ) ≥ δ j ∀τ ∈ R ⁿ + for some δ j > 0;

(H ₅ ⁰ ) 

τ ∈ R ⁿ + : Φ(τ ) ≤ P m

j=1 φ j (τ ) + Φ(0)

⊂ B _R

(0, M ) for some M > 0;

(H 6 ) ∂Φ + (0) 6= ∅, where Φ + := Φ + ind _R

.

Then the problem: Find π ∈ R ⁿ + and X ∈ R(π) such as to satisfy the variational inequality

X, τ − π 

+ Φ(τ ) − Φ(π) ≥ 0, ∀ τ ∈ R ⁿ + , (16)

has at least one solution.

Therefore, there exists π ∈ R ⁿ + , x j ∈ R ⁿ + , j = 1, . . . , m such that (P M ) V j (x j ) = min 

V j (x) : A j π, x 

≤ φ j (π) and x ∈ R ⁿ ₊

, j = 1, . . . , m,

(P E) D

− X m j=1

A ^T _j x j , τ − π E

+ Φ(τ ) − Φ(π) ≥ 0, ∀ τ ∈ R ⁿ + .

Equivalently, there exists π, (x j ), (α j )

∈ R ⁿ + × (R ⁿ + ) ^m × (R + ) ^m such that

−α j A j π ∈ ∂V j (x j ), A j π, x j

 − φ j (π) ∈ ∂ ind ≥0 (α j )

Φ(τ ) − Φ(π) ≥ D

τ − π, P m j=1 A ^T _j x j

E , ∀ τ ∈ R ⁿ +

 

 



 

 

. (17)

3. Approximation result

The multivalued mapping R defined in (14) is not upper semicontinuous in 0 and this mapping does not have bounded values for the functions φ j positive homogeneous of degree 1. In order to get the solution of the problem (P ) in this case we construct the appropriate approximation. We proceed in two steps. First, we shift the argument π in (17) to have boundary points of R ⁿ + ,

∂R ⁿ + := R ⁿ + \ Int R ⁿ + , out of the domain. This will allow for taking advantage of Theorem 2. Second, we extend the dimension and construct the family of extended dimensional problems parametrized by a small parameter ε > 0. The examination of the behaviour of the corresponding solutions when ε → 0 enables establishing of new existence results for the problem under consideration.

Step 1. In order to take advantage of Theorem 2 we choose π 0 ∈ R ⁿ + \ {0} in the way described below and for any ε > 0 consider the problem with shifted argument, namely

−α j A j (π + επ 0 ) ∈ ∂V j (x j ), A j (π + επ 0 ), x j

 − φ j (π + επ 0 ) ∈ ∂ ind ≥0 (α j ) Φ(τ ) − Φ(π) ≥

τ − π, P m j=1 A ^T _j x j

 , ∀ τ ∈ R ⁿ + .

 

 

 



(18)

This problem is a modification of (17) by shifting π 7→ π + επ 0 , π ∈ R ⁿ + , and replacing Φ(·) by Φ(· − επ 0 ). An element π 0 should be chosen in such a way that for each j = 1, . . . , m, and any ε > 0 we can find δ jε > 0 to fulfill the estimates

φ j (τ + επ 0 ) ≥ δ jε , ∀τ ∈ R ⁿ + . (19)

Then, (H 4 ) holds and the application of Theorem 2 is allowed. By reformulating (H ₅ ⁰ ) we are led to the following result.

Theorem 3 Let us assume that there exists π 0 ∈ R ⁿ + \ {0} such that (19) is satisfied for any ε > 0. Suppose that for each j = 1, . . . , m the hypotheses below hold:

(H 1 ) 0 ∈ cl(Dom ∂V j ), (R ⁿ ₋ \{0})∩B _R

(0, r j ) ⊂ Int Dom V _j ^⋆ for some r j > 0;

(H 2 ) 

x ∈ R ⁿ + : {hx ^⋆ , xi : x ^⋆ ∈ ∂V j (x)} ∩ R − 6= ∅

⊂ B _R

(0, M j ) for some M j > 0;

( ˆ H ₅ ^ε ) 

τ ∈ R ⁿ + : Φ(τ ) ≤ P m

j=1 φ j (τ + επ 0 ) + Φ(0)

⊂ B _R

(0, M ε ) for some M ε > 0;

(H 6 ) ∂Φ + (0) 6= ∅, where Φ + := Φ + ind _R

.

Then there exists (π ^ε , (x ^ε _j ), (α ^ε _j )) ∈ R ⁿ + × (R ⁿ + ) ^m × (R + ) ^m fulfilling (18).

Step 2. The idea is to introduce an additional dimension for the problem under consideration. For suitable modified data we shall formulate the approximation problem of (18)-type that makes possible to take advantage of Theorem 3.

Let us consider the data system (W _j ^ε , e φ ^ε _j , e Φ), where (i) W _j ^ε : R ⁿ + × R + → R ∪ {+∞}:

W _j ^ε (x, z) := V j (x) + ϕ ^ε _j (z), x ∈ R ⁿ , z ∈ R + , ε > 0, (20) with ϕ ^ε _j (z) := −εs j z ^1−ε + ind ≤s

(z), s j > 0, P m

j=1 s j = s < 1, W _j ^ε :=

W _j ^ε + ind _R

_×R

; (ii) e φ ^ε _j : R ⁿ + × R + → R + :

φ e ^ε _j (π, q) := φ ^ε _j (π) + qs j , π ∈ R ⁿ + , q ∈ R + , (21) with φ ^ε _j (τ ) := min{φ j (τ ), ¹ _ε };

(iii) e Φ : R ⁿ + × R + → R ∪ {+∞}:

Φ(π, q) = Φ(π) + q, e π ∈ R ⁿ ₊ , q ∈ R + . (22) The problem is to find (π ^ε , q ^ε ) ∈ R ⁿ + × R + , (x ^ε _j , z _j ^ε ) ∈ R ⁿ + × R + and α ^ε _j ∈ R +

such that

−α ^ε _j (A j π ^ε , q ^ε + εq 0 ) ∈ ∂W _j ^ε (x ^ε _j , z ^ε _j ), (A j π ^ε , q ^ε + εq 0 ), (x ^ε _j , z _j ^ε ) 

− e φ ^ε _j (π ^ε , q ^ε + εq 0 ) ∈ ∂ ind ≥0 (α ^ε _j ) Φ(τ, q) − e e Φ(π ^ε , q ^ε ) ≥

τ − π ^ε , P m

j=1 A ^T _j x ^ε _j 

+ P m j=1 z _j ^ε

(q − q ^ε ),

∀ (τ, q) ∈ R ⁿ + × R + .

 

 

 



 

 

  (23)

In the foregoing system the “π”-argument has been shifted by an element π 0 = (0, q 0 ) ∈ R ⁿ + × R + \ {(0, 0)} to deal with the admissible set of the form R ⁿ + × R + + ε(0, q 0 ) on which one can more clearly control the boundedness of the corresponding α ^ε _j ’s. Moreover, we have replaced φ j by φ ^ε _j for better handling the boundedness of the corresponding π ^ε ’s.

Now we are ready to state the hypotheses under which the existence of

solutions of(23) follows. From now on, for convenience we set q 0 = s j = _m ^s .

First, let us notice that from φ e ^ε _j (τ, q + ε _m ^s ) ≥ ε _m ^s
2

> 0, ∀ (τ, q) ∈ R ⁿ + × R + , it follows that (19) holds with δ jε = ε _m ^s
2

> 0.

Assume that (H ₂ ^ε ) 

x ∈ R ⁿ ₊ : {hx ^⋆ , xi : x ^⋆ ∈ ∂V j (x)} ∩ (R − + ε) 6= ∅

⊂ B _R

(0, M jε ) for some M jε > 0;

(H ₅ ^ε ) 

τ ∈ R ⁿ ₊ : Φ(τ ) ≤ P m

j=1 φ ^ε _j (τ ) + Φ(0) + ε _m ^s

⊂ B _R

(0, M ε ) for some M ε > 0.

The hypothesis (H ₅ ^ε ) together with the assumption that s < 1 in (i), allows easily to ensure that

 (τ, q) ∈ R ⁿ + × R + : e Φ(τ, q) ≤ P m

j=1 f φ ^ε _j (τ, q + ε _m ^s ) + e Φ(0, 0)

is bounded. (24) Thus ( ˆ H ₅ ^ε ) holds. Moreover, from

∂W ^ε _j (x, z), (x, z) 

≤ 0, (x, z) ∈ R ⁿ + × R + , it easily follows that

∂V j (x), x 

≤ ε, and z ∈ (0, _m ^s ].

Consequently, (H ₂ ^ε ) yields |x| ≤ M jε which implies (H 2 ). Finally, we have checked that for the data system (W _j ^ε , e φ ^ε _j , e Φ) all the requirements of Theo- rem 3 are fulfilled. Therefore one allows to conclude the existence of a system

(π ^ε , q ^ε ), (x ^ε _j , z _j ^ε ), (α ^ε _j )

∈ (R ⁿ + × R ⁺ ) × R ⁿ + × R ⁺
m

× (R ⁺ ) ^m such that (23) is fulfilled. It can be written equivalently as

−α ^ε _j (A j π ^ε , q ^ε + ε _m ^s ) ∈ ∂V j (x ^ε _j ), ∂ϕ _j ^ε (z _j ^ε )
A j π ^ε , x ^ε _j 

+ (q ^ε + ε _m ^s )z _j ^ε − φ ^ε _j (π ^ε ) − (q ^ε + ε _m ^s ) _m ^s ∈ ∂ ind ≥0 (α ^ε _j ) Φ(τ ) − Φ(π ^ε ) ≥

τ − π ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀ τ ∈ R ⁿ + , q − q ^ε ≥ P m

j=1 z _j ^ε

(q − q ^ε ), ∀ q ∈ R ⁺ .

 

 

 

 

 

 (25)

We summarize the obtained result as follows.

Proposition 2 Suppose that for any ε > 0 the hypotheses (H 1 ), (H ₂ ^ε ), (H ₅ ^ε ) and (H 6 ) are assumed to hold. Then there exists a system π ^ε , (x ^ε _j ), (α ^ε _j )

∈ R ⁿ + × (R ⁿ + ) ^m × (R + ) ^m such that

−α ^ε _j A j π ^ε ∈ ∂V j (x ^ε _j ), (26)

z ^ε _j = (1 − ε)

_α

¹

+r

, r ^ε _j ≥ 0, _m ^s ≥ z _j ^ε , ( _m ^s − z _j ^ε )r ^ε _j = 0 (27) A j π ^ε , x ^ε _j 

= φ ^ε _j (π ^ε ) + ε _m ^s ( _m ^s − z _j ^ε ), (if α ^ε _j > 0), (28) A j π ^ε , x ^ε _j 

≤ φ ^ε _j (π ^ε ), (if α ^ε _j = 0), (29)

Φ(τ ) − Φ(π ^ε ) ≥

τ − π ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀ τ ∈ R ⁿ + . (30)

Proof. Let us notice that 0 < z ^ε _j ≤ _m ^s , j = 1, . . . , m, and P m

j=1 z ^ε _j ≤ P m j=1 s

m = s < 1. In view of (25) ₄ this means that q ^ε = 0. Therefore we can express (25) as (26)-(30).

Remark 2 From the proof of the Proposition 2 we get that

∂V _j ^⋆ (0) 6= ∅, j = 1, . . . , m.

4. Positive homogeneity of degree one

We shall investigate the problem (P ) under the hypotheses (H ₄ ³ ) φ j (tτ ) = tφ j (τ ) ∀τ ∈ R ⁿ + , ∀ t > 0;

(H ₆ ¹ ) Φ = P m

j=1 φ j is convex and Φ(τ ) ≥ γ |τ | ∀ τ ∈ R ⁿ + , γ > 0,

involved, for instance, in the models of Arrow-Debreu or Arrow-Debreu-McKen- zie.

Before the formulation of the next Theorem, recall that for a convex set K and x ∈ K, ∂ ind K (x) is called the normal cone to K at x and is denoted by N K (x) := ∂ ind K (x) (see Aubin, 1993). Moreover, recall that for a convex func- tion f : X → R ∪ {+∞}, X being a Banach space, the asymptotic generalized gradient of f at x, denoted ∂ ^∞ f (x), which is defined

∂ ^∞ f (x) := 

x ^⋆ ∈ X ^⋆ : (x ^⋆ , 0) ∈ N epi f (x, f (x)) ,

where epi f denotes the epigraph of f (see Clarke, 1983; Rockafellar and Wets, 1998).

Theorem 4 Suppose that for any j = 1, . . . , m the following hypotheses hold:

(H 1 ) 0 ∈ cl(Dom ∂V j ), (R ⁿ − \{0})∩B _R

(0, r j ) ⊂ Int Dom V _j ^⋆ for some r j > 0;

(H ₂ ¹ ) 

x ∈ R ⁿ + : { x ^⋆ , x 

: x ^⋆ ∈ ∂V j (x)} ∩ (R − + ε 0 ) 6= ∅

⊂ B _R

(0, M j ), for some M j > 0, ε 0 > 0;

(H ₄ ³ ) φ j (tτ ) = tφ j (τ ) ∀τ ∈ R ⁿ + , ∀ t > 0;

(H ₆ ¹ ) Φ = P m

j=1 φ j is convex and Φ(τ ) ≥ γ |τ | ∀ τ ∈ R ⁿ + , for some γ > 0;

(H 0 ) ∂V _j ^⋆ (0) is compact;

(H 9 ) A j τ , x j

 > φ j (τ ) for any τ ∈ R ⁿ + \ {0} and x j ∈ ∂V _j ^⋆ (0).

Moreover, for any j = 1, . . . , m assume that one of the conditions holds:

(H ₈ ⁰ ) Dom V j is closed or

(H ₈ ¹ ) if x ^k _j → x _j , p ^k → p,  p ^k 

 = 1, as k → +∞, α ^k _j > 0 such that −α ^k _j A j p ^k ∈

∂V j (x ^k _j ) and φ j (p) = 0, then lim inf k→∞ V _j ^⋆ (−α ^k _j A j p ^k ) > −∞.

Then there exists a system π, (x j ), (α j )

with π ∈ R ⁿ + \ {0}, x j ∈ R ⁿ + and α j ∈ R + ∪ {+∞} for j = 1, . . . , m, such that

−α _j A j π ∈ ∂V j (x j ), A j π, x j

 − φ _j (π) ∈ ∂ ind ≥0 (α j ), if α j ∈ R + ,

−A j π ∈ ∂ ^∞ V j (x j ), A j π, x j

 = φ j (π) = 0, if α j = +∞, Φ(τ ) − Φ(π) ≥

τ − π, P m j=1 A ^T _j x j

 , ∀ τ ∈ R ⁿ + ,

 



  ( e P )

Proof. The proof will be divided into two steps.

Step 1. We assume that

min{φ j (τ ) : τ ∈ R ⁿ + , |τ | = 1} = γ j , γ j > 0, j = 1, . . . , m. (31) Let us begin with checking the validity of the hypotheses of Proposition 2.

Notice that by (H ₄ ³ ) and (H ₆ ¹ ) the hypothesis (H ₅ ^ε ) holds. The remaining ones are satisfied immediately. Therefore one can assume the existence of

π ^ε , (x ^ε _j ), (α ^ε _j )

∈ R ⁿ + × (R ⁿ + ) ^m × (R + ) ^m , 0 < ε ≤ ε 0 , fulfilling (26)-(30).

From (H 9 ) it follows that π ^ε 6= 0. Moreover, by the hypothesis (H 9 ) com- bined with (29) and the fact that φ ^ε _j (π ^ε ) ≤ φ j (π ^ε ) it follows that α ^ε _j > 0 for each j = 1, . . . , m. Thus from (28) we get

A j π ^ε , x ^ε _j 

= φ ^ε _j (π ^ε ) + ε _m ^s ( _m ^s − z _j ^ε ), j = 1, . . . , m. (32) Hence

P m j=1

A j π ^ε , x ^ε _j 

= P m

j=1 φ ^ε _j (π ^ε ) + ε _m ^s s − P m j=1 z ^ε _j

, s − P m

j=1 z _j ^ε ≥ 0. (33) By means of (30) it follows that

Φ(π ^ε ) = π ^ε , P m

j=1 A ^T _j x ^ε _j  , P m

j=1 A ^T _j x ^ε _j ∈ W, (34)

because Φ is positively homogeneous of degree 1. Therefore P m

j=1 φ j (π ^ε ) = Φ(π ^ε ) = P m j=1

A j π ^ε , x ^ε _j 

= P m

j=1 φ ^ε _j (π ^ε ) + ε _m ^s s − P m j=1 z ^ε _j

. (35) Now let us suppose that lim sup _ε→0 |π ^ε | < +∞. Then, for sufficiently small ε > 0, φ ^ε _j (π ^ε ) = φ j (π ^ε ), j = 1, . . . , m which, by (35), yields

s − P m

j=1 z _j ^ε = 0.

But such an equality can happen only when _m ^s = z _j ^ε for each j = 1, . . . , m. This means that π ^ε , (x ^ε _j ), (α ^ε _j )

is a solution of ( e P ) whenever ε > 0 is small enough.

Now, suppose that lim sup _ε→0 |π ^ε | = +∞. From (35) we get

ε→0 lim φ j (π ^ε ) − φ ^ε _j (π ^ε )

= 0, j = 1, . . . , m. (36)

First, we show that {α ^ε _j |π ^ε |} ε≤ε

is bounded. Suppose, on the contrary, that α ^ε _j |π ^ε | → +∞, as ε → 0 (by choosing a subsequence, if necessary). Taking into account −α ^ε _j A j π ^ε ∈ ∂V j (x ^ε _j ) we get

−α ^ε _j A j π ^ε , x ^ε _j 

= V j (x ^ε _j ) + V _j ^⋆ (−α ^ε _j A j π ^ε ).

From ∂V _j ^⋆ (0) 6= ∅ we obtain that there exists c j ∈ R that for any y ∈ Dom V j , V j (y) ≥ −c j . Using (32) and the definition of the Fenchel conjugate function we get the following estimate

α ^ε _j φ j (π ^ε )+α ^ε _j (φ ^ε _j (π ^ε )−φ j (π ^ε )) ≤ c j +V j (y)+hα ^ε _j A j π ^ε , yi, ∀y ∈ Dom V _j . (37) Dividing (37) by α j ε |π ^ε |, letting ε → 0 and using the fact that min{φ j (τ ) : τ ∈ R ⁿ + , |τ | = 1} = γ j , γ j > 0 we get

0 < γ j ≤ |A _j | |y| , ∀y ∈ Dom V _j ,

which contradicts the assumption 0 ∈ cl(Dom ∂V j ).

Therefore in this case it can be assumed that p ^ε := _|π ^π

| → p, x ^ε _j → x j , α ^ε _j |π ^ε | → e

α j , as ε → 0, for some p ∈ R ⁿ + , |p| = 1, x j ∈ R ⁿ + , e α j ∈ R + (by passing to subsequence, if necessary). Using positive homogeneity of degree one the functions φ j , j = 1, . . . , m, Φ, the conditions (26), (28), (30) obtain the following form

−α ^ε _j |π ^ε | A j p ^ε ∈ ∂V j (x ^ε _j ), A j p ^ε , x ^ε _j 

− φ j (p ^ε ) = _|π ¹

| φ ^ε _j (π ^ε ) − φ j (π ^ε )
+ _|π ^ε

|

s m

s m − z _j ^ε
Φ(τ ) − Φ(p ^ε ) ≥ D

τ − p ^ε , P m

j=1 A ^T _j x ^ε _j E

, ∀ τ ∈ R ⁿ + ,

 

 

 



(substituting _|π ^τ

| into τ ). By letting ε → 0 we obtain

−e α j A j p ∈ ∂V j (x j ), A j p, x j

 = φ j (p) Φ(τ ) − Φ(p) ≥ D

τ − p, P m j=1 A ^T _j x j

E

, ∀ τ ∈ R ⁿ + .

 

 

 



This means that p, (x j ), (e α j )

is a solution of ( e P ).

Step 2. Now we assume that for some j 0 ∈ 1, . . . , m min{φ j

(τ ) : τ ∈ R ⁿ + , |τ | = 1} = 0.

We apply the results from Step 1 to the system V j (·), φ j (·) + ε |·| , Φ(·) + mε |·|
for sufficiently small ε > 0. First, we notice the conditions (H 0 ), (H 9 ) imply that there exists e ε ≤ ε 0 such that for any j = 1, . . . , m we get

A j τ , x j

 > φ j (τ ) + e ε |τ | , ∀τ ∈ R ⁿ + \ {0}, ∀x j ∈ ∂V _j ^⋆ (0). (38)

Therefore, we obtain that for any ε ≤ e ε there exist π ^ε ∈ R ⁿ + , x ^ε _j ∈ R ⁿ + , α ^ε _j ∈ R + , j = 1, . . . , m such that

−α ^ε _j A j π ^ε ∈ ∂V j (x ^ε _j ), A j π ^ε , x ^ε _j 

− φ _j (π ^ε ) − ε |π ^ε | ∈ ∂ ind ≥0 (α ^ε _j ), Φ(τ ) − Φ(π ^ε ) + εm(|τ | − |π ^ε |) ≥

τ − π ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀ τ ∈ R ⁿ + .

 

 

 

 (39)

From (38), (39) it follows that π ^ε 6= 0,  x ^ε _j 

 ≤ M j , j = 1, . . . , m. As in Step 1 we get the estimate

α ^ε _j φ j (π ^ε ) ≤ c j + V j (y) +

α ^ε _j A j π ^ε , y 

, ∀y ∈ Dom V j . (40)

Let p ^ε = _|π ^π

| , ε > 0. Therefore, there exist p ∈ R ⁿ + , |p| = 1, x j ∈ R ⁿ + , j = 1, . . . , m such that p ^ε → p, x ^ε _j → x j , j = 1, . . . , m as ε → 0 (by passing to a subsequence, if necessary).

Let j ∈ {1, . . . , m}. We consider two cases:

Case 1 φ j (p) > 0. Then, analogously as in Step 1, from (40) we get that {α ^ε _j |π ^ε |} ε≤ε

is bounded. Hence there exists e α j ∈ R (by passing to a subse- quence if necessary) such that α ^ε _j → e α j . From positive homogeneity of degree one of the function φ j , letting ε → 0 in (39) ₁ , (39) ₂ we have

−e α j A j p ∈ ∂V j (x j ), A j p, x j

 − φ j (p) ∈ ∂ ind ≥0 (e α j ).

Case 2. φ j (p) = 0. From (39) ₂ we get A j p, x j

 = φ j (p), φ j (p) = 0. (41)

Moreover, if (H ₈ ⁰ ) holds, then it is easily seen that x j ∈ Dom V j . If (H ₈ ¹ ) holds, then from the condition −α ^ε _j |π ^ε | A j p ^ε ∈ ∂V j (x ^ε _j ) we get

0 ≥ V j (x j ) + lim inf

ε→0 V _j ^⋆ (−α ^ε _j |π ^ε | A j p ^ε ), which implies x j ∈ Dom V j .

If lim inf ε→0 α j |π ^ε | = e α j ∈ R + , then letting ε → 0 in (39) ₁ we get −e α j A j p ∈

∂V j (x j ). It means that

−e α j A j p ∈ ∂V j (x j ), A j p, x j

 − φ j (p) ∈ ∂ ind ≥0 (e α j ).

If lim inf ε→0 α ^ε _j |π ^ε | = +∞ then (41) means that (−A j p, 0) ∈ ∂ ind _{epi V}

(x j , V j (x j )), which can be written equivalently as

−A j p ∈ ∂ ^∞ V j (x j ).

From (39) ₃ it follows that

Φ(τ ) − Φ(p ^ε ) + εm(|τ | − |p ^ε |) ≥

τ − p ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀ τ ∈ R ⁿ + ,

(substituting _|π ^τ

| into τ ). Thus by letting ε → 0 we are led to Φ(τ ) − Φ(p) ≥

τ − p, P m

j=1 A ^T _j x j 

, ∀ τ ∈ R ⁿ + , as desired. This completes the proof.

Remark 3 For the function Φ + , which is convex, lower semicontinuous, pos- itively homogeneous of degree 1, there exists a nonempty, convex, closed set W ⊂ R ⁿ such that Φ + (τ ) = sup _y∈W

τ, y 

, τ ∈ R ⁿ (see Aubin, 1993). There- fore, problem ( e P ) from Theorem 4 can be formulated equivalently as to find

π, (x j ), (α j )

such that

−α j A j π ∈ ∂V j (x j ), A j π, x j

 − φ j (π) ∈ ∂ ind ≥0 (α j ), if α j ∈ R + ,

−A j π ∈ ∂ ^∞ V j (x j ), A j π, x j

 = φ j (π), φ j (π) = 0, if α j = +∞, Φ(π) =

π, P m j=1 A ^T _j x j

 , P m

j=1 A ^T _j x j ∈ W.

 



  ( e P ^′ )

Remark 4 Note that from the proof of Theorem 1 it follows that ∂V _j ^⋆ (0) 6= ∅.

In order to obtain the result similar to that of Theorem 4 for V j ’s which may not attain global minimum on R ⁿ + , it is sufficient to assume that the set W ∩ R ⁿ +

in Remark 3 is bounded. Then we can apply Theorem 4 for the functions e V j :=

V j + ind _B

_(0,K)∩R

, j = 1, . . . , m, with suitably chosen constant K > 0. For such functions (H 1 ) reduces to the requirement that 0 ∈ cl(Dom ∂V j ) and (H 2 ) is redundant (note that then Dom( e V _j ^⋆ ) = R ⁿ ). Therefore, for K sufficiently large and W ∩ R ⁿ + bounded, any solution of the modified problem becomes a solution of the initial problem ( e P ).

Remark 5 In order to find a solution of the problem ( e P ) in the case when

∂V _j ^⋆ (0) is not a compact set, it is sufficient to replace the assumption (H 9 ) by the following one

(H ₉ ^′ ) there exists e ε > 0 for any j = 1, . . . , m, such that A j τ , x j

 > φ j (τ ) + e ε |τ | , ∀τ ∈ R ⁿ + \ {0}, x j ∈ ∂V _j ^⋆ (0).

The main result reads as follows:

Theorem 5 Suppose that for any j = 1, . . . , m the following hypotheses hold:

(H 1 ) 0 ∈ cl(Dom ∂V j ), (R ⁿ − \{0})∩B _R

(0, r j ) ⊂ Int Dom V _j ^⋆ for some r j > 0;

(H ₂ ¹ ) 

x ∈ R ⁿ + : { x ^⋆ , x 

: x ^⋆ ∈ ∂V j (x)} ∩ (R − + ε 0 ) 6= ∅

⊂ B _R

(0, M j ), for some M j > 0, ε 0 > 0;

(H ₄ ³ ) φ j (tτ ) = tφ j (τ ) ∀τ ∈ R ⁿ + , ∀ t > 0;

(H ₆ ¹ ) Φ = P m

j=1 φ j is convex and Φ(τ ) ≥ γ |τ | ∀ τ ∈ R ⁿ + , γ > 0;

(H ₇ ¹ ) 0 / ∈ ∂V j (0).

Moreover, for any j = 1, . . . , m, assume that one of the following conditions holds:

(H ₈ ⁰ ) Dom V j is closed or

(H ₈ ¹ ) if x ^k _j → x _j , p ^k → p,  p ^k 

 = 1, as k → +∞, α ^k _j > 0 such that −α ^k _j A j p ^k ∈

∂V j (x ^k _j ) and φ j (p) = 0, then lim inf k→∞ V _j ^⋆ (−α ^k _j A j p ^k ) > −∞.

Then there exist 0 < r ≤ 1 and π, (x j ), (α j )

, π ∈ R ⁿ + \ {0}, x j ∈ R ⁿ + and α j ∈ R ⁺ ∪ {+∞} for j = 1, . . . , m such that

−α j A j π ∈ ∂V j (x j ), A j π, x j

 − φ j (π) ∈ ∂ ind ≥0 (α j ), if α j ∈ R ⁺ ,

−A _j π ∈ ∂ ^∞ V j (x j ), A j π, x j

 = φ j (π) = 0, if α j = +∞, Φ(τ ) − Φ(π) ≥

τ − π, ¹ _r P m j=1 A ^T _j x j

 , ∀ τ ∈ R ⁿ + .

 



  (P Q)

Proof. The proof will be divided into two steps.

Step 1. We assume that

min{φ j (τ ) : τ ∈ R ⁿ + , |τ | = 1} = γ j , γ j > 0, j = 1, . . . , m. (42) Let 0 < δ ≤ ε 0 . Let us make the following replacement in Proposition 2:

Φ 7→ Φ ^δ , 0 < δ ≤ ε 0 where Φ ^δ (τ ) := Φ(τ )
1+δ

, τ ∈ R ⁿ + .

We have to check the validity of hypotheses stated in Proposition 2 in case of such replacement.

First it will be shown that (H ₅ ¹ ) holds, i.e.

 τ ∈ R ⁿ + : Φ(τ )
1+δ

≤ P m

j=1 φ ^ε _j (τ ) + _m ^ε

⊂ B _R

(0, f M )

for some f M > 0 and any 0 < ε ≤ δ. Indeed, taking into account (H ₆ ¹ ) and φ ^ε _j (τ ) ≤ φ j (τ ), j = 1, . . . , m, we get the estimate for Φ(τ ) ≥ 1 and τ ∈ 

τ ∈ R ⁿ + : Φ(τ )
1+δ

≤ P m

j=1 φ ^ε _j (τ ) + _m ^ε Φ(τ ) ≤ (1 + _m ^δ ) ^{1 δ} ≤ e ^{m 1} ≤ e.

Having in mind that Φ(τ ) ≥ γ |τ |, γ > 0, we finally arrive at

|τ | ≤ ( _e

γ if Φ(τ ) ≥ 1,

1

γ if Φ(τ ) ≤ 1. (43)

Hence the assumption (H ₅ ¹ ) follows with f M = _γ ^e . Further, it is not difficult to

verify that ∂Φ ^ε ₊ (0) ∩ R ⁿ ₊ = {0}, which yields (H 6 ) from Theorem 2.

Summing up, the hypotheses of Proposition 2 are fulfilled for the data system (V j ), (φ j ), Φ ^δ

. Thus, for any 0 < ε ≤ δ there exists a system π ^ε , (x ^ε _j ), (α ^ε _j )

∈ R ⁿ + × (R ⁿ + ) ^m × (R + ) ^m with the properties that

−α ^ε _j A j π ^ε ∈ ∂V j (x ^ε _j ), (44)

z ^ε _j = (1 − ε)

_α

¹

+r

, r ^ε _j ≥ 0, _m ^s ≥ z ^ε _j , ( _m ^s − z _j ^ε )r ^ε _j = 0 (45) A j π ^ε , x ^ε _j 

= φ ^ε _j (π ^ε ) + ε _m ^s ( _m ^s − z _j ^ε ), (if α ^ε _j > 0), (46) A j π ^ε , x ^ε _j 

≤ φ ^ε _j (π ^ε ), (if α ^ε _j = 0), (47)

Φ(τ )
1+δ

− Φ(π ^ε )
1+δ

≥

τ − π ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀ τ ∈ R ⁿ + . (48) From (H ₇ ¹ ), (43), (44) we easily establish the estimate

|π ^ε | ≤ _γ ^e , lim inf

ε→0 |π ^ε | > 0,  x ^ε _j 

 ≤ M j , j = 1, . . . , m. (49) From (49) we obtain that for sufficiently small ε, φ ^ε _j (π ^ε ) = φ j (π ^ε ), for j = 1, . . . , m. Hence from (44), (46) we get the estimate for α ^ε _j > 0

α ^ε _j φ j (π ^ε ) ≤ c j + V j (y) +

α ^ε _j A j π ^ε , y 

, ∀y ∈ Dom V j ,

where the constant c j ∈ R is such that V j (y) ≥ −c j , for any y ∈ Dom V j . Analysis similar to that in the proof of Theorem 4 shows that {α ^ε _j |π ^ε |} ε≤ε

is bounded. Therefore, from (49) in this case it can be assumed that π ^ε → π, x ^ε _j → x j and α ^ε _j → e α j for some π ∈ R ⁿ + , π 6= 0, x j ∈ R ⁿ + , e α j ∈ R + (by passing to subsequence, if necessary) such that

−e α j A j π ∈ ∂V j (x j ), A j π, x j

 − φ j (π) ∈ ∂ ind ≥0 (e α j ), (Φ(τ )) ^1+δ − (Φ(π)) ^1+δ ≥

τ − π, P m j=1 A ^T _j x j

 , ∀ τ ∈ R ⁿ + .

 



  (50)

Taking into account (50), we get for any 0 < ε ≤ ε 0 that there exist π ^ε , (x ^ε _j ), (α ^ε _j )
such that

−α ^ε _j A j π ^ε ∈ ∂V _j (x ^ε _j ), A j π ^ε , x ^ε _j 

− φ j (π ^ε ) ∈ ∂ ind ≥0 (α ^ε _j ), (Φ(τ )) ^1+ε − (Φ(π ^ε )) ^1+ε ≥

τ − π ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀ τ ∈ R ⁿ + .

 



  (51)

Moreover

0 < |π ^ε | ≤ _m ^e ,  x ^ε _j 

 ≤ M j , j = 1, . . . , m. (52)

Analysis similar to that in the proof of Theorem 4 shows that {α ^ε _j |π ^ε |} ε≤ε

,

j = 1, . . . , m are bounded. Therefore, there exist r ∈ [0, 1], p ∈ R ⁿ ₊ , |p| = 1,

x j ∈ R ⁿ + , e α j ∈ R, j = 1, . . . , m such that p ^ε → p (p ^ε = _|π ^π

| ), |π ^ε | ^ε → r, x ^ε _j → x j , α ^ε _j |π ^ε | → e α j . From positive homogeneity of degree one of the functions φ j , letting ε → 0 in (51) ₁ , (51) ₂ we have

−e α j A j p ∈ ∂V j (x j ), A j p, x j

 − φ j (p) ∈ ∂ ind ≥0 (e α j ).

Using positive homogeneity of degree one of the function φ j the condition (51) ₃ gets the equivalent form

|π ^ε | ^ε (Φ(τ )) ^1+ε − (Φ(p ^ε )) ^1+ε

≥

τ − p ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀τ ∈ R ⁿ +

(substituting _|π ^τ

| into τ ). Thus by letting ε → 0 we arrive at r(Φ(τ ) − Φ(p)) ≥

τ − p, P m j=1 A ^T _j x j

 , ∀τ ∈ R ⁿ + .

From the hypotheses (H ₇ ¹ ) we easily get that r > 0.

Therefore there exist r ∈ (0, 1] and the system (p, (x j ), (e α j )), |p| = 1 such that

−e α j A j π ∈ ∂V j (x j ), A j π, x j

 − φ j (π) ∈ ∂ ind ≥0 (e α j ), Φ(τ ) − Φ(p) ≥

τ − π, ¹ _r P m j=1 A ^T _j x j

 , ∀ τ ∈ R ⁿ + .

 



 

Step 2. Now we assume that for some j 0 ∈ {1, . . . , m}

min{φ j

(τ ) : τ ∈ R ⁿ + , |τ | = 1} = 0.

We apply the results from Step 1 to the system V j (·), φ j (·) + ε |·| , Φ(·) + mε |·|
for ε > 0. For any 0 < ε ≤ 1 there exist r ^ε ∈ (0, 1], p ^ε ∈ R ⁿ + , |p ^ε | = 1, x ^ε _j ∈ R ⁿ + , α ^ε _j ∈ R + , j = 1, . . . , m such that

−α ^ε _j A j p ^ε ∈ ∂V _j (x ^ε _j ), A j p, x ^ε _j 

− φ j (p ^ε ) − ε ∈ ∂ ind ≥0 (α ^ε _j ), r ^ε Φ(τ ) − Φ(p ^ε ) + εm(|τ | − 1)

≥

τ − p ^ε , P m

j=1 A ^T _j x ^ε _j 

, ∀ τ ∈ R ⁿ + .

 

 

 

 (53)

From (53) ₁ we get that  x ^ε _j 

 ≤ M j . Moreover as in Step 1 we get the estimate α ^ε _j φ j (p ^ε ) ≤ c j + V j (y) +

α ^ε _j A j p ^ε , y 

, ∀y ∈ Dom V j . (54)

From the fact that |p ^ε | = 1 and r ^ε ∈ (0, 1] we get that there exist p ∈ R ⁿ + ,

|p| = 1, r ∈ [0, 1], x j ∈ R ⁿ + , j = 1, . . . , m such that r ^ε → r, p ^ε → p, x ^ε _j → x j , as ε → 0 (by passing to a subsequence, if necessary).

Let j ∈ {1, . . . , m}. We consider two cases:

Case 1. φ j (p) > 0. Then, analogously as in Step 1, from (54) we get that {α ^ε _j } ε≤1 is bounded. Hence there exists e α j ∈ R (by passing to a subsequence if necessary) such that α ^ε _j → e α j . Letting ε → 0 in (53) ₁ , (53) ₂ we obtain that

−e α j A j p ∈ ∂V j (x j ), A j p, x j

 − φ j (p) ∈ ∂ ind ≥0 (e α j ). (55)

Case 2. φ j (p) = 0. From (53) ₂ we have that A j p, x j

 = φ j (p), φ j (p) = 0.

Similarly to the proof of the Theorem 4 we get that x j ∈ Dom V j . Moreover, if lim inf ε→0 α ^ε _j |π ^ε | = e α j ∈ R ⁺ , then (55) holds.

If lim inf ε→0 α ^ε _j |π ^ε | = e α j = +∞, then the conditions A j p, x j

 = φ j (p), φ j (p) = 0 mean

−A j p ∈ ∂ ^∞ V j (x j ).

Letting ε → 0 in (53) ₃ we get rΦ(τ ) − rΦ(p) ≥

τ − p, P m j=1 A ^T _j x j

 , ∀ τ ∈ R ⁿ + . (56)

Moreover, from the assumption (H ₆ ¹ ) we deduce that there exists j ^′ ∈ {1, . . . , m}

such that φ j

(p) > 0. Note that then x j

6= 0. Suppose, on the contrary, that x j

= 0. The condition

A j

p, x j

 − φ j

(p) ∈ ∂ ind ≥0 ( f α j

) means e α j

= 0 and 0 ∈ ∂V j

(0), contrary to (H ₇ ¹ ). Hence P m

j=1 A ^T _j x j 6= 0, which implies that that 0 < r ≤ 1.

Summing up, there exist 0 < r ≤ 1, p ∈ R ⁿ + , |p| = 1, x j ∈ R ⁿ + , f α j ∈ R + ∪ {+∞}, j = 1, . . . , m, such that

−e α j A j p ∈ ∂V j (x j ), A j π, x j

 − φ j (π) ∈ ∂ ind ≥0 (e α j ), if e α j ∈ R ⁺ ,

−A j p ∈ ∂ ^∞ V j (x j ), A j p, x j

 = φ j (p) = 0, if e α j = +∞, Φ(τ ) − Φ(p) ≥

τ − p, ¹ _r P m j=1 A ^T _j x j

 , ∀ τ ∈ R ⁿ + .

 



 

This completes the proof.

Corollary 1 Assume the hypotheses of Theorem 5. Then there exist π ∈ R ⁿ + , π 6= 0, x j ∈ R ⁿ + , α j ∈ R ⁺ ∪ {+∞}, j = 1, . . . , m, and 0 < r ≤ 1 such that

V j (x j ) = min{V j (y) : A j π, y 

≤ φ j (π)}, if α j ∈ R + , (57)

−A j π ∈ ∂ ^∞ V j (x j ), A j π, x j

 = φ j (π) = 0, if α j = +∞, (58) Φ(π) = P m

j=1 φ j (π), Φ(π) =

π, ¹ _r P m j=1 A ^T _j x j

 , ¹ _r P m

j=1 A ^T _j x j ∈ W. (59) If one the inequalities

A j π, x j

 ≤ φ j (π) happens to be strict, then r < 1.

Remark 6 The results obtained lead to certain implications concerning the equilibrium price vector π. First, if φ j (π) > 0, i.e. the income of j’s consumer- producer at prices π is positive, then he maximizes his utility over the set of all admissible commodity bundles {y ∈ R ⁿ + :

π, y 

≤ φ j (π)}, as expected. But, if φ j (π) = 0, i.e. the income of j’s consumer-producer at prices π vanishes, it is not possible to determine whether x j maximizes his utility function −V j over the set {y ∈ R ⁿ + :

π, y 

= 0} of all zero-valued commodity bundles (at prices π), what has quite reasonable explanation. Note that all of those zero-valued commodity bundles are equivalent in the sense of economy: for the customer they are equally worthless.

Moreover, if for at last one j the strict inequality π, x j

 < φ j (π) occurs, i.e.

the value of the optimal commodity bundle of j’s consumer does not reach its budget line, then r < 1. Thus, 1 − r can be referred to as the measure of the difference between the value of the total endowment and aggregate demand at equilibrium prices π. This may happen when due to the preferences which are not bound to be strictly monotone, the Walras’ law fails.

References

Aliprantis, C. D., Brown, D. J. and Burkinshaw, O. (1989) Existence and Optimality of Competitive Equilibria. Springer Verlag.

Aliprantis, C. D., Florenzano, M. and Tourky, R. (2005) Linear and non-linear price decentralization. J. Econom. Theory 121, 51–74.

Aliprantis, C. D., Monteriro, P. K. and Tourky, R. (2004) Non-mark- eted options, non-existence of equilibria, and nonlinear prices. J. Econom.

Theory 114, 345–357.

Aliprantis, C. D., Tourky, R. and Yannelis, N. C. (2001) A theory of value with nonlinear prices. J. Econom. Theory 100, 22–72.

Arrow, K. and Debreu, G. (1954) Existence of an equilibrium for a com- petitive economy. Econometrica 22, 264–290.

Arrow, K. J. and Intrilligator, M. D. (1982) Handbook of Mathematical Economics, vol. II. North Holland.

Aubin, J. P. (1993) Optima and Equilibria. Springer Verlag.

Borwein, J. M. and Jofr´ e, A. (1998) A nonconvex separation property in Banach spaces. Math. Methods Oper. Res. 48, 169–179.

Bulavsky, W. A. (1994) On a solution for a class of equilibrium problems.

Siberian Math. J. 35, 990–999, in Russian.

Chichilnisky, G. (1993) Intersecting families of sets and the topology on cones in economics. Bull. A.M.S. (New Series) 29, 189–207.

Chichilnisky, G. (1999) A unified perspective on resource allocation: Lim-

ited arbitrage is necessary and sufficient for the existence of a competitive

equilibrium, the core and social choice. In: G. Chichilnisky, ed., Topology

and Markets, American Mathematical Society and the Fields Institute for

Research in Mathematical Sciences, 1999.

Chichilnisky, G. and Heal, G. (1993) Competitive equilibrium in Sobolev spaces without bounds on short sales. J. Economic Theory 59, 364–384.

Chichilnisky, G. and Heal, G. (1998) A unified treatment of finite and in- finite economies: Limited arbitrage is necessary and sufficient for the ex- istence of equilibrium and the core. Economic Theory 12, 163–176.

Clarke, F. H. (1983) Optimization and Nonsmooth Analysis. John Wiley &

Sons.

Eaves, C. (1972) Homotopies for computation of fixed points. Math. Pro- graming 3.

Ekeland, I. and Temam, R. (1976) Convex Analysis and Variational Prob- lems. North-Holland.

Gale, D. and Mas-Colell, A. (1975) An equilibrium existence theorem for a general model without ordered preferences. J. Math. Economics 2, 9–15.

Hadjisavvas, N. and Schaible, S. (1998) From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96, 297–

309.

Hirsh, M. and Smale, S. (1979) On algorithms for solving f (x) = 0. Comm.

Pure Appl. Math. 32, 281–312.

Kinderlehrer, D. and Stampacchia, G. (1980) An Introduction to Varia- tional Inequalities and their Applications. Academic Press.

Lee, G. M., Kim, D. S., Lee, B. S. and Yen, N. D. (1998) Vector varia- tional inequality as a tool for studying vector optimization problems. Non- linear Analysis 34, 745–765.

Luc, D. T. (1989) Theory of Vector Optimization. Lecture Notes in Eco- nomics and Mathematical Systems 319, Springer-Verlag.

Mas-Colell, A. (1986) The price equilibrium existence problem in topolog- ical vector lattices. Econometrica 54, 1039–1053.

Mas-Colell, A. and Richard, S. F. (1991) A new approach to the exis- tence of equilibria in vector lattices. J. Econom. Theory 53, 1–11.

Mas-Colell, A., Whinston, M. D. and Green, J. R. (1995) Microecono- mic Theory. Oxford University Press.

McKenzie, L. (1959) On the existence of general equilibrium for a competitive market. Econometrica 27, 54–71.

Mordukhovich, B. S. (2001) The extremal principle and its applications to optimization and economics. In: A. Rubinov and B. Glover, eds., Optimization and Related Topics, Kluwer Academic Publishers, 343–369.

Nagurney, A. (1999) Network Economics: A Variational Inequality Approach.

Second and revised edition, Kluwer Academic Publishers.

Nagurney, A. and Siokos, S. (1997) Financial Networks: Statics and Dy- namics. Springer-Verlag.

Naniewicz, Z. (2002) On some optimization problem related to economic equilibrium. Control and Cybernetics 31, 141–165.

Naniewicz, Z. and Panagiotopoulos, P. D. (1995) Mathematical Theory

of Hemivariational Inequalities and Applications. Marcel Dekker.

Nash, J. F. (1950) Equilibrium points in n-person games. Proc. Nat. Acad.

Sci. U.S.A. 36, 48–49.

Negishi, T. (1960) Welfare economics and existence of an equilibrium for a competitive economy. Metroeconomica 12, 92–97.

von Neumann, J. (1945-46) A model of general economic equilibrium. Re- view Econom. Stud. XIII, 1–9, (G. Morgenstern, transl.).

Pallaschke, D. and Rolewicz, S. (1997) Foundation of Mathematical Op- timization. Kluwer Academic Publishers.

Panek, E. (2000) Ekonomia Matematyczna. Akademia Ekonomiczna w Poz- naniu, in Polish.

Rockafellar, R. T. and Wets, R. J-B. (1998) Variational Analysis. Sprin- ger-Verlag.

Scarf, H. E. (1973) The Computation of Economic Equilibria. Yale Univ.

Press New Haven, CT.

Smale, S. (1976) Exchange processes with price adjustment. J. Math. Econom.

3, 211–226.