7. Method of meeting paths

The described algorithms are nonapplicable for the general linear exchange model, when the budgets of consumers are not fixed. In this case, the associating mapping G no longer has the property of potentiality. But, the complementarity approach makes possible to propose a modification of the proses [3]. We name it a method of meeting paths.

As mentioned earlier, on the current k-step of the process, we have a structure B<sup>k</sup> ∈ B. We consider two cells <sup>Ω</sup><sup>k</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ <sup>B</sup><sup>k</sup> ,Ξ<sup>k</sup> <sup>¼</sup> <sup>Ξ</sup>ð Þ <sup>B</sup><sup>k</sup> and two points pk <sup>∈</sup> <sup>Ω</sup>k, qk <sup>∈</sup>Ξk. Let Lk <sup>⊃</sup> <sup>Ω</sup>k, Mk <sup>⊃</sup>Ξ<sup>k</sup> be the affine hulls of these cells. For the points of their intersection Lk ∩ Mk, we obtain from (1), (3) the common system:

$$\frac{p\_k}{c\_k^i} = \frac{p\_j}{c\_j^i} \qquad (i,k), (i,j) \in B\_{k\prime} \tag{10}$$

We consider the situation when t

Figure 6. Illustration of a meeting paths step.

Under this condition, it holds t

be extended.

the process.

8. Generalizations

Nondegeneracy condition. Only one of the above two cases can occur.

zero except ~zi<sup>1</sup> , ~zi<sup>2</sup> , then the following two conditions are equivalent:

This condition will be satisfied if a bit to move the starting points p<sup>0</sup>, q0.

Lemma Let A be a nonnegative and indecomposed matrix, and x is its positive eigenvector, λ is the corresponding eigenvalue. If for a positive vector ~x the vector ~z ¼ λ~x � A~x has all components equal

Figure 6 illustrates one step of this method. In the figure, the point p tð Þ reaches the face of its cell earlier than the point q tð Þ does. For the next step the cell Ω will be reduced, the cell Ξ will

It should be noted that for the model with variable budgets, an iterative method was proposed [10] that uses the developed simple algorithm for Fisher's model in each step of

1. The models with upper bounds: the considered approach permits to develop the algorithms for deferent variations of the classical exchange model. The simplest of those models is the model in which the costs are limited for certain goods (the spending

x~i1 x~i2 ≥ xi1 xi2 :

~zi<sup>1</sup> ≥ 0 ()

Theorem 5. Under nondegeneracy condition, the process of meeting paths is always finite.

<sup>∗</sup> is limited by both above conditions as degenerate.

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models

http://dx.doi.org/10.5772/intechopen.77206

41

<sup>∗</sup> > 0. To justify this, the following lemma was proved [3].

$$\sum\_{j \in I\_v} p\_j = \sum\_{i \in I\_v} (p, w^i), \qquad \nu = 1, \dots, \tau. \tag{11}$$

where the sets Jν, J<sup>ν</sup> correspond to ν-th connected component of the graph Γð Þ B<sup>k</sup> .

Under some assumption about starting structure this system has rank ð Þ n � 1 and under additional condition P <sup>j</sup><sup>∈</sup> <sup>J</sup> pj <sup>¼</sup> 1 the system defines uniquely the solution <sup>r</sup><sup>k</sup> <sup>¼</sup> <sup>r</sup>ð Þ <sup>B</sup><sup>k</sup> . This is the intersection point of the affine hulls of the cells <sup>Ω</sup>ð Þ <sup>B</sup> and <sup>Ξ</sup>ð Þ <sup>B</sup> : It can be shown that <sup>r</sup><sup>k</sup> <sup>∈</sup>σ. If <sup>r</sup><sup>k</sup> <sup>∈</sup> <sup>Ω</sup>ð Þ <sup>B</sup><sup>k</sup> , r<sup>k</sup> <sup>∈</sup>Ξð Þ <sup>B</sup><sup>k</sup> , we have an equilibrium price vector.

Otherwise, we consider for t ∈½ Þ 0; 1 two moving points:

$$p(t) = p^k + t(r^k - p^k), \quad q(t) = q^k + t(r^k - q^k)$$

It can be shown that in consequence of the assumption ci > 0, ∀i∈ I, there exists t <sup>∗</sup> <sup>¼</sup> max<sup>t</sup> under the conditions p tð Þ∈ Ωð Þ B<sup>k</sup> ,q tð Þ∈Ξð Þ B<sup>k</sup> :

It is the case when t <sup>∗</sup> < 1. The two variants may occur:

(i) t <sup>∗</sup> is limited by some of the inequalities zijð Þ p tð Þ <sup>≥</sup> <sup>0</sup>, ið Þ ; <sup>j</sup> <sup>∈</sup>Bk. Corresponding pairð Þ <sup>i</sup>; <sup>j</sup> should be removed from <sup>B</sup>k:B<sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>B</sup>k\ i f g ð Þ ; <sup>j</sup> .We accept <sup>q</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ, pkþ<sup>1</sup> <sup>¼</sup> p t<sup>∗</sup> ð Þ and pass to the next step.

(ii) t <sup>∗</sup> is limited by some of the inequalities (2) in description of the cell <sup>Ξ</sup>ð Þ <sup>B</sup><sup>k</sup> . Corresponding pair ð Þ <sup>i</sup>; <sup>l</sup> should be added to <sup>B</sup>k: <sup>B</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>B</sup>k∪f g ð Þ <sup>i</sup>; <sup>l</sup> . We accept <sup>q</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ, pkþ<sup>1</sup> <sup>¼</sup> p t<sup>∗</sup> ð Þ and pass to the next step.

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models http://dx.doi.org/10.5772/intechopen.77206 41

Figure 6. Illustration of a meeting paths step.

The optimal solutions of the consumer's problems are:

7. Method of meeting paths

40 Optimization Algorithms - Examples

(3) the common system:

additional condition P

It is the case when t

pass to the next step.

(i) t

(ii) t

Figure 5 shows the moving of the point q tð Þ to the equilibrium.

modification of the proses [3]. We name it a method of meeting paths.

pk ci k ¼ pj ci j

pj <sup>¼</sup> <sup>X</sup> i∈I<sup>ν</sup>

where the sets Jν, J<sup>ν</sup> correspond to ν-th connected component of the graph Γð Þ B<sup>k</sup> .

X j∈ J<sup>ν</sup>

If <sup>r</sup><sup>k</sup> <sup>∈</sup> <sup>Ω</sup>ð Þ <sup>B</sup><sup>k</sup> , r<sup>k</sup> <sup>∈</sup>Ξð Þ <sup>B</sup><sup>k</sup> , we have an equilibrium price vector.

Otherwise, we consider for t ∈½ Þ 0; 1 two moving points:

under the conditions p tð Þ∈ Ωð Þ B<sup>k</sup> ,q tð Þ∈Ξð Þ B<sup>k</sup> :

<sup>x</sup>~<sup>1</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup>:5; <sup>1</sup> , <sup>x</sup>~<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>0</sup>:5; <sup>0</sup> :

The described algorithms are nonapplicable for the general linear exchange model, when the budgets of consumers are not fixed. In this case, the associating mapping G no longer has the property of potentiality. But, the complementarity approach makes possible to propose a

As mentioned earlier, on the current k-step of the process, we have a structure B<sup>k</sup> ∈ B. We consider two cells <sup>Ω</sup><sup>k</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ <sup>B</sup><sup>k</sup> ,Ξ<sup>k</sup> <sup>¼</sup> <sup>Ξ</sup>ð Þ <sup>B</sup><sup>k</sup> and two points pk <sup>∈</sup> <sup>Ω</sup>k, qk <sup>∈</sup>Ξk. Let Lk <sup>⊃</sup> <sup>Ω</sup>k, Mk <sup>⊃</sup>Ξ<sup>k</sup> be the affine hulls of these cells. For the points of their intersection Lk ∩ Mk, we obtain from (1),

Under some assumption about starting structure this system has rank ð Þ n � 1 and under

the intersection point of the affine hulls of the cells <sup>Ω</sup>ð Þ <sup>B</sup> and <sup>Ξ</sup>ð Þ <sup>B</sup> : It can be shown that <sup>r</sup><sup>k</sup> <sup>∈</sup>σ.

p tðÞ¼ <sup>p</sup><sup>k</sup> <sup>þ</sup> t r<sup>k</sup> � pk � �, qtðÞ¼ <sup>q</sup><sup>k</sup> <sup>þ</sup> t r<sup>k</sup> � <sup>q</sup><sup>k</sup> � �

<sup>∗</sup> is limited by some of the inequalities zijð Þ p tð Þ <sup>≥</sup> <sup>0</sup>, ið Þ ; <sup>j</sup> <sup>∈</sup>Bk. Corresponding pairð Þ <sup>i</sup>; <sup>j</sup> should be removed from <sup>B</sup>k:B<sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>B</sup>k\ i f g ð Þ ; <sup>j</sup> .We accept <sup>q</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ, pkþ<sup>1</sup> <sup>¼</sup> p t<sup>∗</sup> ð Þ and pass to the next step.

<sup>∗</sup> is limited by some of the inequalities (2) in description of the cell <sup>Ξ</sup>ð Þ <sup>B</sup><sup>k</sup> . Corresponding pair ð Þ <sup>i</sup>; <sup>l</sup> should be added to <sup>B</sup>k: <sup>B</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>B</sup>k∪f g ð Þ <sup>i</sup>; <sup>l</sup> . We accept <sup>q</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ, pkþ<sup>1</sup> <sup>¼</sup> p t<sup>∗</sup> ð Þ and

It can be shown that in consequence of the assumption ci > 0, ∀i∈ I, there exists t

<sup>∗</sup> < 1. The two variants may occur:

ð Þ i; k , ið Þ ; j ∈ Bk, (10)

<sup>∗</sup> <sup>¼</sup> max<sup>t</sup>

<sup>p</sup>; wi � �, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>, …, <sup>τ</sup>, (11)

<sup>j</sup><sup>∈</sup> <sup>J</sup> pj <sup>¼</sup> 1 the system defines uniquely the solution <sup>r</sup><sup>k</sup> <sup>¼</sup> <sup>r</sup>ð Þ <sup>B</sup><sup>k</sup> . This is

We consider the situation when t <sup>∗</sup> is limited by both above conditions as degenerate.

Nondegeneracy condition. Only one of the above two cases can occur.

This condition will be satisfied if a bit to move the starting points p<sup>0</sup>, q0.

Under this condition, it holds t <sup>∗</sup> > 0. To justify this, the following lemma was proved [3].

Lemma Let A be a nonnegative and indecomposed matrix, and x is its positive eigenvector, λ is the corresponding eigenvalue. If for a positive vector ~x the vector ~z ¼ λ~x � A~x has all components equal zero except ~zi<sup>1</sup> , ~zi<sup>2</sup> , then the following two conditions are equivalent:

$$
\tilde{\omega}\_{i\_1} \ge \mathbf{0} \iff \frac{\tilde{\mathcal{X}}\_{i\_1}}{\tilde{\mathcal{X}}\_{i\_2}} \ge \frac{\mathcal{X}\_{i\_1}}{\mathcal{X}\_{i\_2}} \dots
$$

Theorem 5. Under nondegeneracy condition, the process of meeting paths is always finite.

Figure 6 illustrates one step of this method. In the figure, the point p tð Þ reaches the face of its cell earlier than the point q tð Þ does. For the next step the cell Ω will be reduced, the cell Ξ will be extended.

It should be noted that for the model with variable budgets, an iterative method was proposed [10] that uses the developed simple algorithm for Fisher's model in each step of the process.

#### 8. Generalizations

1. The models with upper bounds: the considered approach permits to develop the algorithms for deferent variations of the classical exchange model. The simplest of those models is the model in which the costs are limited for certain goods (the spending constraints model [7]): pj xi <sup>j</sup> ≤ βij. In this case, the mappings G associated with the arising polyhedral complementarity problem are potential too. Some modifications of the developed algorithms are needed. More difficult is the model with upper on the purchase volumes of goods. In this case, the mappings G are not potential and algorithm becomes more complicated. Such a model arises if the functions of participants are not linear, but piecewise linear concave separable [5].

X i ∈I

> X i∈ I

Definition. A set B⊂ S � J is named a structure, if for each s∈ S there exists sð Þ ; j ∈B.

From this follows that we have to suppose P

The structure notion is generalized:

As before, we suppose that all vectors cs

problem of the model is changed and becomes a net problem: X i∈ I

X j∈ J

� X j∈ J

It can be shown that this problem is solvable for all p∈σ.

X i ∈I

p; xi � � ≤ λi. But for the λi, i ∈I and λk, k ∈K, we obtain the condition P

zij � <sup>X</sup> k∈ K

X j ∈J

zij ln c i <sup>j</sup> � <sup>X</sup> k∈ K

P

<sup>j</sup> <sup>∈</sup><sup>J</sup> pj <sup>¼</sup> 1 and <sup>P</sup>

are discussed.

polyhedral complexes.

only if ~p ∈ Ωð Þ B ∩ Ξð Þ B for some B∈ B.

<sup>x</sup>~<sup>i</sup> <sup>¼</sup> <sup>X</sup> k∈ K <sup>x</sup>~<sup>k</sup> <sup>þ</sup><sup>X</sup> i ∈I wi :

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models

<sup>i</sup> <sup>∈</sup><sup>I</sup> <sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>P</sup>

<sup>i</sup> <sup>∈</sup><sup>I</sup> <sup>w</sup><sup>i</sup> <sup>¼</sup> <sup>e</sup> with <sup>e</sup> <sup>¼</sup> ð Þ <sup>1</sup>; …; <sup>1</sup> . Thus, in the equilibrium, we have

<sup>x</sup>~<sup>i</sup> <sup>¼</sup> <sup>X</sup> k∈K

The polyhedral complementarity approach can be used for this generalized model as well. The main results remain valid [4], but the consideration becomes more complicated. Some features

> X j ∈J

zij ¼ �α<sup>i</sup> � <sup>p</sup>; <sup>w</sup><sup>i</sup> � �, i <sup>∈</sup>I,

zkj ¼ pj

zkj ¼ λk, k ∈K,

zij ≥ 0, zkj ≥ 0, i∈I, j∈ J, k∈ K:

As mentioned before, we consider the family B of structures B: it is the collection of all dual feasible basic index sets of the transportation problem and of all their subsets being structures.

For each B∈ B, we define the balance zone Ωð Þ B and the preference zone Ξð Þ B . The description of these sets is quite similar to those of the classical case. Thus, in this way, we again obtain two

Theorem 6. A vector ~p ∈σ<sup>∘</sup> is an equilibrium price vector of generalized linear exchange model if and

The generalized model can be considered with fixed budgets, and in this way, we obtain the generalization of the Fisher's model. The budget condition of the consumer i remains the same:

zkj ln c k <sup>j</sup> ! max

, j∈ J,

<sup>k</sup><sup>∈</sup> <sup>K</sup> λk. As before, we suppose also that

http://dx.doi.org/10.5772/intechopen.77206

43

<sup>x</sup>~<sup>k</sup> <sup>þ</sup> <sup>e</sup>: (12)

, s ∈S are positive. The parametric transportation

<sup>i</sup> <sup>∈</sup><sup>I</sup> <sup>λ</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>P</sup>

<sup>k</sup>∈<sup>K</sup> λk.

2. The generalized linear exchange model: the polyhedral complementarity approach is applicable to models with the production sector too. Some firms are added, those supply goods to the market. Describe more in detail one of those models.

The model with n products, m participants-consumers, and l participants-firms is considered. Let J ¼ f g 1; …; n , I ¼ f g 1;…; m , and K ¼ f g m þ 1;…; m þ l be the sets of the numbers of products, consumers, and firms. Thus, S ¼ I∪K is the set of numbers of all participants.

The consumer i∈ I has the initial endowments w<sup>i</sup> ∈ Rn <sup>þ</sup> and also the initial money stock <sup>α</sup>i. His total budget after selling the initial endowments is equal to <sup>α</sup><sup>i</sup> <sup>þ</sup> <sup>p</sup>; <sup>w</sup><sup>i</sup> . Thus, the <sup>i</sup> th consumer will choose the purchase vector x<sup>i</sup> looking for an optimal solution to the following problem:

$$\left(c^i, x^i\right) \to \max$$

under the conditions

$$(p, \boldsymbol{x}^i) \le \alpha\_i + (p, w^i)\_{\boldsymbol{\cdot}},$$

$$\boldsymbol{x}^i \ge \mathbf{0}.$$

The firm k ∈K plans to deliver to the market the products to a total sum of at least λk. If xk <sup>¼</sup> <sup>x</sup><sup>k</sup> <sup>1</sup>;…; xk n denotes a plan of <sup>k</sup> th firm then the total cost of such a supply at the prices pj equals p; xk . The quality of the plan is estimated by the firm in tending to minimize the function c<sup>k</sup> ; ; <sup>x</sup><sup>k</sup> . Here, ck <sup>¼</sup> ck <sup>1</sup>;…; ck n is a fixed nonnegative vector whose components determine a comparative scale of the "undesirability" of various products for the firm (e.g., their relative production costs).

Thus, the k th firm makes its choice according to a solution of the optimization problem:

$$(c^k, x^k) \to \min$$

under the conditions

$$\begin{aligned} &(p, x^k) \geq \lambda\_{k'}\\ &x^k \geq 0. \end{aligned}$$

An equilibrium is defined by a price vector ~p and a collection of vectors x~<sup>i</sup> and x~<sup>k</sup> i∈ I and k ∈K, representing some solutions to optimization problems of the participants for p ¼ ~p and satisfying the balance of products:

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models http://dx.doi.org/10.5772/intechopen.77206 43

$$\sum\_{i \in I} \tilde{\mathbf{x}}^i = \sum\_{k \in K} \tilde{\mathbf{x}}^k + \sum\_{i \in I} w^i.$$

From this follows that we have to suppose P <sup>i</sup> <sup>∈</sup><sup>I</sup> <sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>P</sup> <sup>k</sup><sup>∈</sup> <sup>K</sup> λk. As before, we suppose also that P <sup>j</sup> <sup>∈</sup><sup>J</sup> pj <sup>¼</sup> 1 and <sup>P</sup> <sup>i</sup> <sup>∈</sup><sup>I</sup> <sup>w</sup><sup>i</sup> <sup>¼</sup> <sup>e</sup> with <sup>e</sup> <sup>¼</sup> ð Þ <sup>1</sup>; …; <sup>1</sup> . Thus, in the equilibrium, we have

$$\sum\_{i \in I} \tilde{\mathbf{x}}^i = \sum\_{k \in K} \tilde{\mathbf{x}}^k + e. \tag{12}$$

The polyhedral complementarity approach can be used for this generalized model as well. The main results remain valid [4], but the consideration becomes more complicated. Some features are discussed.

The structure notion is generalized:

constraints model [7]): pj

42 Optimization Algorithms - Examples

under the conditions

<sup>1</sup>;…; xk n

under the conditions

; ; <sup>x</sup><sup>k</sup> . Here, ck <sup>¼</sup> ck

their relative production costs).

satisfying the balance of products:

xk <sup>¼</sup> <sup>x</sup><sup>k</sup>

function c<sup>k</sup>

piecewise linear concave separable [5].

The consumer i∈ I has the initial endowments w<sup>i</sup> ∈ Rn

xi

<sup>j</sup> ≤ βij. In this case, the mappings G associated with the arising

<sup>þ</sup> and also the initial money stock <sup>α</sup>i. His

polyhedral complementarity problem are potential too. Some modifications of the developed algorithms are needed. More difficult is the model with upper on the purchase volumes of goods. In this case, the mappings G are not potential and algorithm becomes more complicated. Such a model arises if the functions of participants are not linear, but

2. The generalized linear exchange model: the polyhedral complementarity approach is applicable to models with the production sector too. Some firms are added, those supply

The model with n products, m participants-consumers, and l participants-firms is considered. Let J ¼ f g 1; …; n , I ¼ f g 1;…; m , and K ¼ f g m þ 1;…; m þ l be the sets of the numbers of prod-

total budget after selling the initial endowments is equal to <sup>α</sup><sup>i</sup> <sup>þ</sup> <sup>p</sup>; <sup>w</sup><sup>i</sup> . Thus, the <sup>i</sup> th consumer will choose the purchase vector x<sup>i</sup> looking for an optimal solution to the following problem:

; xi ! max

<sup>p</sup>; xi <sup>≤</sup> <sup>α</sup><sup>i</sup> <sup>þ</sup> <sup>p</sup>; <sup>w</sup><sup>i</sup> ,

The firm k ∈K plans to deliver to the market the products to a total sum of at least λk. If

determine a comparative scale of the "undesirability" of various products for the firm (e.g.,

; <sup>x</sup><sup>k</sup> ! min

p; xk ≥ λk, xk ≥ 0:

An equilibrium is defined by a price vector ~p and a collection of vectors x~<sup>i</sup> and x~<sup>k</sup> i∈ I and k ∈K, representing some solutions to optimization problems of the participants for p ¼ ~p and

Thus, the k th firm makes its choice according to a solution of the optimization problem:

c k

 denotes a plan of <sup>k</sup> th firm then the total cost of such a supply at the prices pj equals p; xk . The quality of the plan is estimated by the firm in tending to minimize the

is a fixed nonnegative vector whose components

xi ≥ 0:

goods to the market. Describe more in detail one of those models.

ucts, consumers, and firms. Thus, S ¼ I∪K is the set of numbers of all participants.

c i

<sup>1</sup>;…; ck n Definition. A set B⊂ S � J is named a structure, if for each s∈ S there exists sð Þ ; j ∈B.

As before, we suppose that all vectors cs , s ∈S are positive. The parametric transportation problem of the model is changed and becomes a net problem:

$$\sum\_{i \in I} \sum\_{j \in I} z\_{ij} \ln c\_j^i - \sum\_{k \in K} \sum\_{j \in I} z\_{kj} \ln c\_j^k \quad \to \text{ max}$$

$$- \sum\_{j \in I} z\_{ij} = -\alpha\_i - (p, w^i), \quad i \in I,$$

$$\sum\_{i \in I} z\_{ij} - \sum\_{k \in K} z\_{kj} = p\_{j'} \quad \quad j \in I,$$

$$\sum\_{j \in J} z\_{kj} = \lambda\_{k'} \quad k \in K,$$

$$z\_{ij} \ge 0, \qquad z\_{kj} \ge 0, \qquad i \in I, \quad j \in I, \quad k \in K.$$

It can be shown that this problem is solvable for all p∈σ.

As mentioned before, we consider the family B of structures B: it is the collection of all dual feasible basic index sets of the transportation problem and of all their subsets being structures.

For each B∈ B, we define the balance zone Ωð Þ B and the preference zone Ξð Þ B . The description of these sets is quite similar to those of the classical case. Thus, in this way, we again obtain two polyhedral complexes.

Theorem 6. A vector ~p ∈σ<sup>∘</sup> is an equilibrium price vector of generalized linear exchange model if and only if ~p ∈ Ωð Þ B ∩ Ξð Þ B for some B∈ B.

The generalized model can be considered with fixed budgets, and in this way, we obtain the generalization of the Fisher's model. The budget condition of the consumer i remains the same: p; xi � � ≤ λi. But for the λi, i ∈I and λk, k ∈K, we obtain the condition P <sup>i</sup> <sup>∈</sup><sup>I</sup> <sup>λ</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>P</sup> <sup>k</sup>∈<sup>K</sup> λk.

For this variant of model, we have the reduction to optimization problems as well. To do this, we consider the function f pð Þ, which gives the optimal value of the transportation problem by given price vector p. Having this function, we introduce as before the functions φð Þ¼ p ð Þ� p; ln p f pð Þ and ψð Þ¼ q f ∗ ð Þ ln q . For these functions, the main results of classical case remain valid.

Acknowledgements

Author details

References

ics. 1976;3(2):197-204

Vadim I. Shmyrev1,2\*

\*Address all correspondence to: shmyrev.vadim@mail.ru

models. Soviet Mathematics - Doklady. 1983;27(1):230-233

vol. 9869. Heidelberg, Germany: Springer; 2016. pp. 61-72

Annals of Mathematical Statistics. 1959;30(1):165-168

Siberian Mathematical Journal. 1985;26:288-300

Mathematics. 2008;2(1):125-142

1 Novosibirsk State University, Novosibirsk, Russia

This chapter was supported by the Russian Foundation for Basic Research, project 16-01-00108 À.

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models

http://dx.doi.org/10.5772/intechopen.77206

45

2 Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia

[1] Eaves BC. A finite algorithm for linear exchange model. Journal of Mathematical Econom-

[2] Shmyrev VI. On an approach to the determination of equilibrium in elementary exchange

[3] Shmyrev VI. An algorithm for the search of equilibrium in the linear exchange model.

[4] Shmyrev VI. A generalized linear exchange model. Journal of Applied and Industrial

[5] Shmyrev VI. An iterative approach for searching an equilibrium in piecewise linear exchange model. In: Kochetov Yu. et al., editors. Lecture Notes in Computer Sciences.

[6] Eisenberg E, Gale D. Consensus of subjective probabilities: The pari-mutuel method. The

[7] Devanur NR, Papadimitriou CH, Saberi A, Vazirani VV. Market equilibrium via a primal– dual algorithm for a convex program. Journal of the ACM (JACM). 2008;55(5):22

[8] Shmyrev VI. An algorithmic approach for searching an equilibrium in fixed budget exchange models. In: Driessen TS et al., editors. Russian Contributions to Game Theory

[9] Shmyrev VI. An algorithm for finding equilibrium in the linear exchange model with fixed

and Equilibrium Theory. Berlin, Germany: Springer; 2006. pp. 217-235

budgets. Journal of Applied and Industrial Mathematics. 2009;3(4):505-518

Theorem 7. A vector ~p is an equilibrium price vector if and only if ~p is a minimum point of the function φ on σ<sup>∘</sup> .

Theorem 8. A vector ~p is an equilibrium price vector if and only if ~p is a maximum point of the function ψ on σ<sup>∘</sup> .

The finite algorithms developed for Fischer's model do not require any significant changes and are applicable for this generalized model.

3. The production-exchange models Arrow-Debreu type: these are modifications of previous model. Describe the simplest variant of the model. On the market, there is one unit of each good. The firms produce additional goods, spending some resource that is limited and seek to maximize revenue from the sale of manufactured goods. Thus, the k th firm solves the following problem:

$$\sum\_{j \in \mathcal{J}} p\_j x\_j^k \to \max$$

$$\sum\_{j \in \mathcal{J}} d\_j^k x\_j^k \le \zeta\_k$$

$$x\_j^k \ge 0, \qquad j \in \mathcal{J}.$$

Here, ζ<sup>k</sup> is allowable resource and dk <sup>j</sup> indicate the resource cost per unit of product j.

Let λkð Þp be the optimal value of this problem. The consumer i∈ I has the initial money stock αi, P <sup>i</sup>∈<sup>I</sup> α<sup>i</sup> ¼ 1. The revenues of the firms are divided between consumers in some proportions, those are given by <sup>θ</sup>ik. The total budget of <sup>i</sup> th consumers becomes <sup>α</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup> <sup>k</sup>∈<sup>K</sup> θikλkð Þp . Thus, the i th consumer has the following problem:

$$(c^i, x^i) \to \max$$

under the conditions

$$\begin{aligned} (p, x^i) &\le \alpha\_i + \sum\_{k \in K} \theta\_{ik} \lambda\_k(p), \\ x^i &\ge 0. \end{aligned}$$

The condition of good balances in equilibrium is given as before by the equality (12).

The polyhedral complementarity approach is applicable for this model too, but the consideration becomes much more complicated. An iterative method can be developed that uses the abovementioned generalized linear exchange model as an auxiliary in each step of the process.
