3. Classical linear exchange model

We demonstrate the main idea of the approach on the classical linear exchange model in the well-known description [13].

Consider a model with n commodities (goods) and m consumers. Let J ¼ f g 1;…; n and I ¼ f g 1;…; m be the index sets of commodities and consumers.

Each consumer i ∈I possesses a vector of initial endowments w<sup>i</sup> ∈R<sup>n</sup> þ. The exchange of commodities is realized with respect to some nonnegative prices pj , forming a price vector p ∈R<sup>n</sup> þ.

The consumer i∈I has to choose a consumption vector x<sup>i</sup> ∈ Rn <sup>þ</sup> maximizing his linear utility function ci ; xi under budget restriction:

$$\begin{array}{c} \left(c^i, x^i\right) \to \max, \\ \left(p, x^i\right) \leqslant \left(p, w^i\right), \newline \Longrightarrow \text{The problem of consumer } i. \\ \left.x^i \geqslant 0. \end{array}$$

Let x~<sup>i</sup> be a vector xi that solves this program.

A price vector <sup>~</sup><sup>p</sup> 6¼ 0 is an equilibrium price vector if there exist solutions <sup>x</sup>~<sup>i</sup> , i ¼ 1, …, m, for the individual optimization problems such that

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

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

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models

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

31

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

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

zij ln c i <sup>j</sup> ! max

zij ¼ pj

The equations of this problem represent the financial balances for the consumers and com-

This is the classical transportation problem. The price vector p is a parameter of the problem.

The answer on the question about B reads: B is the collection of all dual feasible basic index sets of

aÞ Ωð Þ B ⊂σ is the balance zone of the structure :

k

<sup>b</sup><sup>Þ</sup> <sup>Ξ</sup>ð Þ <sup>B</sup> <sup>⊂</sup> <sup>σ</sup><sup>∘</sup> is the preferance zone of the structure :

ci k qk ¼ ci j qj

; ∀ð Þ i; j ∈B

) :

<sup>Ω</sup>ð Þ¼ <sup>B</sup> <sup>p</sup> <sup>∈</sup>σj∃z<sup>∈</sup> Z pð Þ; zij <sup>¼</sup> <sup>0</sup>;ð Þ <sup>i</sup>; <sup>j</sup> <sup>∉</sup> <sup>B</sup> � �;

� � � � �

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

<sup>Ξ</sup>ð Þ¼ <sup>B</sup> <sup>q</sup>∈σ<sup>∘</sup> max

(

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

, j ∈J,

zij ⩾0, ið Þ ; j ∈I � J:

xi j .

Given a price vector p consider the following transportation problem of the model: X i ∈I

X j∈ J

X j∈ J

� � � � � � � � � � �

Under the assumption about w<sup>i</sup> � � this problem is solvable for each p∈σ.

the transportation problem and of all their subsets being structures.

X i ∈I

respected, but the participants' preferences are ignored.

: The parametric transportation problem of the model.

zij � �<sup>∈</sup> Z pð Þ

modities. The variables zij are introduced by zij ¼ pj

: Polyhedral complexes of the model.

� � � � � � � � � � � �

B ∈ B )

Now, it is clear that

be the collection B?

under conditions

one.

3∘

4∘

In what follows, we normalize the initial endowment of each commodity to 1, that is, P i <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 in the unit simplex

$$\sigma = \left\{ p \in \mathbb{R}\_+^n \, | \, \sum\_{j \in J} p\_j = 1 \right\}.$$

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