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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

Theorem 7. A vector ~p is an equilibrium price vector if and only if ~p is a minimum point of the function

Theorem 8. A vector ~p is an equilibrium price vector if and only if ~p is a maximum point of the function

The finite algorithms developed for Fischer's model do not require any significant changes and

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

> X j ∈J pj xk <sup>j</sup> ! max

X j∈ J dk j xk <sup>j</sup> ≤ ζk,

xk

tions, 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>

c i

<sup>p</sup>; xi � � <sup>≤</sup> <sup>α</sup><sup>i</sup> <sup>þ</sup> <sup>X</sup>

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

x<sup>i</sup> ≥ 0:

<sup>j</sup> ≥ 0, j∈ J:

Let λkð Þp be the optimal value of this problem. The consumer i∈ I has the initial money stock

; xi � � ! max

k∈K

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.

θikλkð Þp ,

<sup>i</sup>∈<sup>I</sup> α<sup>i</sup> ¼ 1. The revenues of the firms are divided between consumers in some propor-

<sup>j</sup> indicate the resource cost per unit of product j.

<sup>k</sup>∈<sup>K</sup> θikλkð Þp .

ð Þ ln q . For these functions, the main results of classical case

ð Þ� p; ln p f pð Þ and ψð Þ¼ q f

44 Optimization Algorithms - Examples

following problem:

under the conditions

are applicable for this generalized model.

Here, ζ<sup>k</sup> is allowable resource and dk

Thus, the i th consumer has the following problem:

remain valid.

φ on σ<sup>∘</sup> .

ψ on σ<sup>∘</sup> .

αi, P ∗

Vadim I. Shmyrev1,2\*

