The study of the dynamic model of the distribution of labor resources by region in conflict interaction

The introduced models turn out to be the development of forces and resources allocation in municipal economy branches if a lot of interests appear in a competitive case. The models combine the reasonable level of adequacy with the practical application in solving the problems. The plans of labour force allocation can be complied on their basis.


Introduction
The process when two competitive companies (participants in economic process) are aimed at solidifying at several interfacing economic regions (market sectors) is considered. To do this they locate their liquid asserts, industrial facilities, labour force and other resources in some regions. The capital productivity ratio from every participant investing labour force in every region is set by marketed law and normally depends on joint allocation of labour force in all the interfacing regions. The process is supposed to be uncapped and sampled time. Every participant is aimed to maximize mean income received at every process step or to maximize its resource share taking part in the regional activity.

Game-theoretic model of labour force allocation
We introduce an extend dynamic model (EDM) ) , , ( are compact metric spaces given for every x ,  -function: We interpret X as a state space of some process  . Let consider U as a control set of this process. The process is controlled by two agents  and  (but, in any case, there can be any finite quantity), -agents control sets at every condition x of the process  , they are assumed metric spaces, correspondently, Suppose that We define metric  in space U induced by   (  ,  :  ) , Let us define the dynamics of the process  by recurrence formula. Suppose that t is a time parameter. The process initial state  (1) and etc. The sequence of space points   ,... , ,... , is called the process path if the following point is built out of the previous one by function  .
We consider the game  which dynamics is defined by the process  .
The spaces  S 2 ,  S 2 are considered as the sets of all the spaces  S ,  S . Let us make them metrical introducing the metrics by the following. If , then the distance between them is defined as : It is possible to consider the task of Nash equilibrium situations finding for the game  .
The complexity of the decisions in these conflicts is stipulated by the fact that the income functions as the function 1 : lose the continuity property if they are reduced to the normal game. In case of the income function is approximated by the consequence of finite games at the transition from the finite game to the infinite one, different peculiarities that, in general, do not allow considering the equilibrium property appear.
Partially, this problem is solved by introducing some additional conditions to the consequence of equilibrium situations in finite games that approximate the initial game. The researches on this problem allow receiving the following result.
Further we need the following result (also see [1], [2], [3], [5], [8]). The question about the existence of mathematical decision for the game in its normal form is followed by the urgent question concerning the solution ampleness if the parameters defining the following task are altered.
Suppose that the space  is generally topological and its point is the game in its normal form  .
The situation is mainly the following. If there is the open and dense set in the space Let us consider as earlier that where I -the set of players, i S -the set of the player's strategy, i -the compact metric space. 1 : is the real continuous income function of the player i . Let us consider the metric space ( )   , (for such games: The metrics  is allocated in the following:

 
and . The process is analyzed by the simplicial subdivision of two-dimensional sphere. Suppose that the sphere is divided into N simplexes. The process dynamics is assumed by the following function: where  takes the form:  As the result of numerical experiments the following exposition is received.
One and only one of the following expositions is always true for the function  : The exposition A -function  has three critical points: the 1st point ( ) The exposition C is the same as for the exposition A , except the following: the properties of the 3rd critical point from the exposition A proceed to the 2nd critical point and the properties of the 2nd critical point proceed to the 3rd critical point.

Numerical example for model of labour force allocation in regions with infinite time
Nowadays the labour market mostly follows the model of monopolistic competition [9] and has the following structure. Between two poles, the employee, on the one hand, and the consumer (the employer), on the other hand, the chain of mediators has been built: -"big wholesalers" (in other words, dealers) that are, as a rule, government organizations supplying vacant jobs; -"small wholesales" are the companies dealing with recruitment. The employee interacts with other market participant only through "big wholesalers". The employer, in its turn, also interacts through them or seldom through "small wholesalers".
Let's consider the labour market per the criterion "price group". There are four main groups: -the 1st group includes employees of low quality that are situated on the lower range of retail prices (from 0.10 to 0.25 c.u. per hour), -the 2nd group includes employees of middle quality and correspond to the second-from-thebottom range of retail prices (from 0.25 to 0.50 c.u. per hour), the 3rd group includes employees of high quality that characterize by the high level of retail prices (from 0.50 to 0.90 c.u. per hour), -the 4th group is presented by employees of high consumer quality imported to Russia (as a rule, of elite quality) and with the highest level of retail prices (from 0.90 to 1.50 c.u. per hour).
The fractions of the labour market occupied by every analyzing price group are in the following correlation: the 1st price group complies 48%, the 2nd -27%, the 3rd -22%, the 4th -7% from the total labour force volume.
Let us confine the further analysis to the competition among one type of labour market participants, i.e. the employees. Herein, there are two lines of competitive interaction: the 1st is between the foreign and Russian employees, the 2nd is among Russian employees.
Suppose that employees are the supports of two competitive technologies (   and ). Let's analyze the multistep process where employees are considered to be goods. Every process step is presented as a manufacturing cycle including allocation of capital, manufacture of goods, its disposal and repayment.
Let's introduce agents' production functions and suppose that all the factors except current assets are unaltered. Then, the agents' production functions will be one inner functions: Suppose that the production function allows linear approximation and can be presented as , where j i c , is the reciprocal variable of cost-per-unit. After agents having allocated their capitals the full volume of product output in every four labour market fractions will be defined. Let's correlate the total supply to the vector ( ) in every four market fractions. Then, using the abovementioned production functions, we define the total supply in i -fraction by the function: where   ithe capital provided by the employee  ,   ithe capital provided by the employee  , for production and distribution of the goods in market i -fraction. Suppose that other price-forming factors are constant, then all the volume of goods will be sold by prices in accordance with the demand curve [9], implicitly defined by functionals: where, i is the index identifying the labour market fraction, v is the earlier introduced supply volume vector, is the vector variable presenting retail prices in the1sr, the2ng, the 3rd and the 4th labour market fractions, correspondently.
It's appropriate to suppose that the marginal utility for every group of goods is the monotonic decreasing function per the argument "quantity of purchased goods". Let's assume as the utility function the functions: where, k is the index identifying the group of goods,  a is the coefficient of exponential growth,  b is the factor, v is the number of goods in this group purchased by the customer.
Let's define with  , , , p p p p are, as earlier, the prices of the 1st, the 2nd, the 3rd and the 4th labour market fractions. Using economic terms [6], the employer forms his consumer basket so that the marginal utilities of all the groups of goods included into the basket are equal. In mathematical terms, the consumer is finding the constrained maximum: As the objective functional is a convex function and the acceptable region is also convex, this problem can be changed into the equivalent problem of solving simultaneous equations: of capital units into the production of the following price group (the 2nd market fraction) and etc [12][13]. Thus, the vector variables: