4. The main idea of the approach

The equilibrium problem can be considered in two different ways.

1∘ : The traditional point of view: supply–demand balance.

Given a price vector p, the economy reacts by supply and demand vectors:

$$p \Bigvee \{ \begin{array}{c} \text{demand } D(p) \\ \text{supply } S(p) \end{array} \Big\}$$

The goods' balance is the condition of equilibrium:

$$
\widehat{p}\text{ is equilibrium price vector}\iff S(\widehat{p}) = D(\widehat{p}).
$$

2<sup>∘</sup> : Another point of view.

The presented consideration is based on the new notion of consumption structure. Definition. A set B⊂ I � J is named a structure, if for each i∈ I there exists ið Þ ; j ∈B. Say that a consumption prescribed by x<sup>i</sup> � � is consistent with structure B if

$$(i,j)\notin \mathcal{B} \quad \implies \quad \mathfrak{x}\_j^i = 0.$$

This notion is analogous to the basic index set in linear programming. Two sets of the price vectors can be considered for each structure B. We name them zones:

$$
\mathcal{B} \bigvee \begin{matrix} \text{the } preference \text{ zone } \Xi(B) \\ \text{the } balance \text{ zone } \Omega(B) . \end{matrix}
$$

Ξð Þ B is the set of prices by which the consumers prefer the connections of the structure, ignoring the budget conditions and balances of goods. Ωð Þ B is the set of prices by which the budget conditions and balances of goods are possible when the connections of the structure are respected, but the participants' preferences are ignored.

Now, it is clear that

X i∈ I

<sup>σ</sup> <sup>¼</sup> <sup>p</sup> <sup>∈</sup>Rn

8 < :

The equilibrium problem can be considered in two different ways.

Given a price vector p, the economy reacts by supply and demand vectors:

p

The presented consideration is based on the new notion of consumption structure.

ð Þ <sup>i</sup>; <sup>j</sup> <sup>∉</sup><sup>B</sup> ¼) xi

<sup>B</sup> ↗ the preference zone <sup>Ξ</sup>ð Þ <sup>B</sup> ↘ the balance zone Ωð Þ B :

<sup>j</sup> ¼ 0:

Definition. A set B⊂ I � J is named a structure, if for each i∈ I there exists ið Þ ; j ∈B.

Say that a consumption prescribed by x<sup>i</sup> � � is consistent with structure B if

This notion is analogous to the basic index set in linear programming. Two sets of the price vectors can be considered for each structure B.

: The traditional point of view: supply–demand balance.

The goods' balance is the condition of equilibrium:

2<sup>∘</sup> : Another point of view.

We name them zones:

P i

1∘

in the unit simplex

30 Optimization Algorithms - Examples

4. The main idea of the approach

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

In what follows, we normalize the initial endowment of each commodity to 1, that is,

þj X j∈ J

For the sake of simplicity assume c<sup>i</sup> > 0, ∀i∈ I. It is sufficient for existence of equilibrium [13].

↗ demand D pð Þ ↘ supply S pð Þ :

<sup>b</sup><sup>p</sup> is equilibrium price vector () <sup>S</sup>ð Þ¼ <sup>b</sup><sup>p</sup> <sup>D</sup>ð Þ <sup>b</sup><sup>p</sup> :

pj ¼ 1

9 = ;:

<sup>w</sup><sup>i</sup> <sup>¼</sup> ð Þ <sup>1</sup>; …; <sup>1</sup> <sup>∈</sup> Rn. The sum of pj is also normalized to 1, restricting the price vector <sup>p</sup> to lie

p is an equlibrium price vector () ð Þ ∃B p∈ Ωð Þ B ∩Ξð Þ B :

We show that in this way the equilibrium problem is reduced to polyhedral complementarity one.

The question is as follows: What kind of the structures B ∈ B should be considered and what should be the collection B?

3∘ : The parametric transportation problem of the model.

Given a price vector p consider the following transportation problem of the model:

$$\sum\_{i \in I} \sum\_{j \in J} z\_{ij} \ln c\_j^i \to \max$$

under conditions

$$\{z\_{ij}\} \in Z(p) \left| \sum\_{i \in I} z\_{ij} = (p, w^i)\_{\prime} \quad \text{ $i \in I$ }, \quad \mathbf{i} \in I\_{\prime} \right|$$

$$\sum\_{i \in I} z\_{ij} = p\_{j\prime} \quad \quad \quad j \in I\_{\prime}$$

$$z\_{ij} \gtrless 0, \quad \quad \quad (i, j) \in I \times I.$$

The equations of this problem represent the financial balances for the consumers and commodities. The variables zij are introduced by zij ¼ pj xi j .

This is the classical transportation problem. The price vector p is a parameter of the problem. Under the assumption about w<sup>i</sup> � � this problem is solvable for each p∈σ.

The answer on the question about B reads: B is the collection of all dual feasible basic index sets of the transportation problem and of all their subsets being structures.

4∘ : Polyhedral complexes of the model.

For B ∈ B, we obtain the description of zones Ωð Þ B and Ξð Þ B in the following way.

$$\mathcal{B}\in\mathfrak{B}\Rightarrow\begin{array}{l} \text{(a)}\quad\Omega(\mathcal{B})\subset\sigma\text{ is the balance zone of the structure :}\\ \Omega(\mathcal{B})=\left\{p\in\sigma\,|\,\exists z\in\mathcal{Z}(p),z\_{\mathcal{Y}}=0,\,(i,j)\notin\mathcal{B}\right\};\\ b)\quad\Xi(\mathcal{B})\subset\sigma^{\*}\text{ is the preference zone of the structure :}\\ \Xi(\mathcal{B})=\left\{q\in\sigma^{\*}\,\middle|\,\max\_{k}\frac{c\_{k}^{i}}{q\_{k}}=\frac{c\_{j}^{i}}{q\_{j}},\,\,\forall(i,j)\in\mathcal{B}\right\}.\end{array}$$

Here, σ<sup>∘</sup> is the relative interior of σ.

It is easy to give these descriptions in more detail.

For q∈Ξð Þ B , we have the linear system

$$\frac{q\_k}{c\_k^i} = \frac{q\_j}{c\_j^i} \qquad (i,k) \in \mathcal{B}, (i,j) \in \mathcal{B}. \tag{1}$$

<sup>c</sup><sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>2</sup>; <sup>3</sup> , w<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup>=2; <sup>1</sup>=2; <sup>1</sup>=<sup>2</sup> ,

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models

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

33

<sup>2</sup> <sup>¼</sup> ð Þ <sup>3</sup>; <sup>2</sup>; <sup>1</sup> , w<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup>=2; <sup>1</sup>=2; <sup>1</sup>=<sup>2</sup> :

<sup>1</sup> � ð Þ <sup>1</sup>=6; <sup>2</sup>=6; <sup>3</sup>=<sup>6</sup> , c<sup>2</sup> � ð Þ <sup>3</sup>=6; <sup>2</sup>=6; <sup>1</sup>=<sup>6</sup> :

Figure 2 illustrates the polyhedral complexes of the model. Each of both complexes has 17 cells. Figure 3 illustrates the arising complementarity problem. The point c<sup>12</sup> is its solution: c<sup>12</sup> ∈ Ω12. Thus, the corresponding vector <sup>p</sup><sup>∗</sup> <sup>¼</sup> ð Þ <sup>3</sup>=8; <sup>2</sup>=8; <sup>3</sup>=<sup>8</sup> is the equilibrium price vector of the model.

c

We need c<sup>1</sup> and c<sup>2</sup> only up to positive multipliers:

Figure 2. Polyhedral complexes in exchange model.

Figure 3. Complementarity problem: c<sup>12</sup> is the solution.

c

Thus, c<sup>1</sup> and c<sup>2</sup> can be considered as points of the unit price simplex σ.

$$\frac{q\_l}{c\_l^i} \gg \frac{q\_j}{c\_j^i} \qquad (i, l) \notin \mathcal{B}, (i, j) \in \mathcal{B}. \tag{2}$$

Thus, <sup>Ξ</sup>ð Þ <sup>B</sup> is the intersection of a polyhedron with <sup>σ</sup><sup>∘</sup> .

To obtain the description of Ωð Þ B , we should use the well-known tools of transportation problems theory. Given B∈ B, introduce a graph Γð Þ B with the set of vertices V ¼ f g 1; 2; …; m þ n and the set of edges f g ð Þj i; m þ j ð Þ i; j ∈B . Let τ be the number of components of this graph, let V<sup>ν</sup> be the set of vertices of ν th component, I<sup>ν</sup> ¼ I ∩ V<sup>ν</sup> and J<sup>ν</sup> ¼ f g j∈Jjð Þ m þ j ∈ V<sup>ν</sup> :. It is not difficult to show that the following system of linear equations must hold for p∈ Ωð Þ B :

$$\sum\_{j \in I\_r} p\_j = \sum\_{i \in I\_r} (p, w^i), \qquad \nu = 1, \ldots, \tau. \tag{3}$$

Under these conditions, the values zij can be obtained from the conditions z ∈Z pð Þ and

$$z\_{ij} = 0, \qquad (i, j) \notin B\_{\prime}$$

presenting linear functions of p: zij ¼ zijð Þp . Now, for p ∈ Ωð Þ B , we have in addition the system of linear inequalities

$$z\_{\vec{\eta}}(p) \gg 0, \qquad (i, j) \in \mathcal{B}.$$

Thus, Ωð Þ B is described by a linear system of equalities and inequalities. Therefore, it is also a polyhedron.

It is easy to see that each face of the polyhedron Ωð Þ B is also a polyhedron Ω B <sup>0</sup> � � with <sup>B</sup> 0 ⊂B .

Therefore, we have on the simplex σ a polyhedral complex ω ¼ f g Ωð Þj B B ∈ B . The polyhedrons Ξð Þ B form on σ another polyhedral complex ξ ¼ f g Ξð Þj B B ∈ B . It is clear that

$$
\Omega(\mathcal{B}\_1) \subset \Omega(\mathcal{B}\_2) \implies \quad \Xi(\mathcal{B}\_1) \supset \Xi(\mathcal{B}\_2).
$$

Thus, the complexes ω, ξ are in duality, and we obtain the reduction of the equilibrium problem to a polyhedral complementarity one.

Example. In the model, there are 3 commodities and 2 consumers:

$$c^1 = (1,2,3)\_\prime \qquad w^1 = (1/2,1/2,1/2)\_\prime$$

$$c^2 = (3,2,1)\_\prime \qquad w^2 = (1/2,1/2,1/2)\_\prime$$

We need c<sup>1</sup> and c<sup>2</sup> only up to positive multipliers:

Here, σ<sup>∘</sup> is the relative interior of σ.

32 Optimization Algorithms - Examples

For q∈Ξð Þ B , we have the linear system

tions must hold for p∈ Ωð Þ B :

of linear inequalities

polyhedron.

It is easy to give these descriptions in more detail.

Thus, <sup>Ξ</sup>ð Þ <sup>B</sup> is the intersection of a polyhedron with <sup>σ</sup><sup>∘</sup> .

X j∈J<sup>ν</sup>

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

Under these conditions, the values zij can be obtained from the conditions z ∈Z pð Þ and

zij ¼ 0, ið Þ ; j ∉B,

presenting linear functions of p: zij ¼ zijð Þp . Now, for p ∈ Ωð Þ B , we have in addition the system

zijð Þp ⩾ 0, ið Þ ; j ∈B:

Thus, Ωð Þ B is described by a linear system of equalities and inequalities. Therefore, it is also a

Therefore, we have on the simplex σ a polyhedral complex ω ¼ f g Ωð Þj B B ∈ B . The polyhedrons

<sup>Ω</sup>ð Þ <sup>B</sup><sup>1</sup> <sup>⊂</sup> <sup>Ω</sup>ð Þ <sup>B</sup><sup>2</sup> ¼) <sup>Ξ</sup>ð Þ <sup>B</sup><sup>1</sup> <sup>⊃</sup>Ξð Þ <sup>B</sup><sup>2</sup> :

Thus, the complexes ω, ξ are in duality, and we obtain the reduction of the equilibrium

It is easy to see that each face of the polyhedron Ωð Þ B is also a polyhedron Ω B

Ξð Þ B form on σ another polyhedral complex ξ ¼ f g Ξð Þj B B ∈ B . It is clear that

Example. In the model, there are 3 commodities and 2 consumers:

problem to a polyhedral complementarity one.

qk ci k ¼ qj ci j

ql ci l ⩾ qj ci j

To obtain the description of Ωð Þ B , we should use the well-known tools of transportation problems theory. Given B∈ B, introduce a graph Γð Þ B with the set of vertices V ¼ f g 1; 2; …; m þ n and the set of edges f g ð Þj i; m þ j ð Þ i; j ∈B . Let τ be the number of components of this graph, let V<sup>ν</sup> be the set of vertices of ν th component, I<sup>ν</sup> ¼ I ∩ V<sup>ν</sup> and J<sup>ν</sup> ¼ f g j∈Jjð Þ m þ j ∈ V<sup>ν</sup> :. It is not difficult to show that the following system of linear equa-

ð Þ i; k ∈ B, ið Þ ; j ∈B, (1)

ð Þ i; l ∉B, ið Þ ; j ∈ B: (2)

<sup>p</sup>; <sup>w</sup><sup>i</sup> � �, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>, …, <sup>τ</sup>: (3)

<sup>0</sup> � �

with B 0 ⊂B .

$$c^1 \sim (1/6, 2/6, 3/6), \quad c^2 \sim (3/6, 2/6, 1/6).$$

Thus, c<sup>1</sup> and c<sup>2</sup> can be considered as points of the unit price simplex σ.

Figure 2 illustrates the polyhedral complexes of the model. Each of both complexes has 17 cells. Figure 3 illustrates the arising complementarity problem. The point c<sup>12</sup> is its solution: c<sup>12</sup> ∈ Ω12. Thus, the corresponding vector <sup>p</sup><sup>∗</sup> <sup>¼</sup> ð Þ <sup>3</sup>=8; <sup>2</sup>=8; <sup>3</sup>=<sup>8</sup> is the equilibrium price vector of the model.

Figure 2. Polyhedral complexes in exchange model.

Figure 3. Complementarity problem: c<sup>12</sup> is the solution.
