In this paper we propose economic-mathematical model of labor potential optimal distribution of the region by economic sectors, which has been tested on statistical data for the Stavropol Territory. Research on the formation and use of labor potential of the region can use various methods: economic, social, mathematical, etc., that allow to evaluate the effects of the labor potential for economic development and to justify the ways of managing them [3, 4].

In the region there is allocated n sectors of the economy, which we denote conditional № 1, № 2, ..., № n. The region's economy may be in the m states: 1st, ..., m-e. We denote qij as income, which brings one employee to the i-th economic sector of the region, if the economy will be in the j-th state, i = 1, ..., n; j = 1, ..., m. Matrix {qij} is called effects matrix (in accordance with the common terminology). In addition, we denote by xi - the number of employees at the i-th sector, p - the minimum income per employee of the region. Then the value of income, that one employee of the i-th sector will bring to the region, if its economy is in the j-th state, is .

The model, that allows to distribute optimally the labor potential by economic sectors, obviously, would be:

, (1)

, (2)

, (3)

, (4)

, , ,,

where T - transpose operation.

Dividing the variables xi, i = 1, ..., n, by p and denoting , we go from the tasks (1) - (4) to the following linear programming task (which is more convenient to use in applied research than (1) - ( 4)):

, (5)

, (6)

. (7)

Note that the results of these research overlap with the results of the research set out in [1, 2].

We use the model (5) - (7) for the optimal distribution of labor potential of the Stavropol region on its economic sectors. According to [5] in the region there are the following economic sector (see Table 1.).

Table 1

Employed population distribution in the manufacturing process of the Stavropol region by economic activity in 2009

Table 2 shows the costs of the Stavropol Territory employers for the labor resources in 2009 in the various economic sectors.

Table 2

Costs of the Stavropol Territory employers for the labor resources in 2009 (thousand rubles per employee per year)

According to the data in Tables 1, 2 and the fact, that the cost per employee is 20% of the income that he brings to the sector in which he works, it is easy to calculate the value of p = 890142,9 rubles.

Effects matrix {qij} is calculated depending on the economy state: "bad", "satisfactory", "good" (i = 1, ..., 10; j = 1, 2, 3)

. (8)

We find the solution of the task (5) - (7) with m = 3, n = 10, and the found matrix (8). We have: s1 = 0,03; s2 = 0,03; s3 = 0,02; s4 = 0,02; s5 = 0,02; s6=0,01; s7 = 0,03; s8 = 0,05 ; s9 = 0,03; s10 = -0,00004. Since p = 890142,9 rubles., Then x1 = 27295, x2 = 22276, x3 = 21979, x4 = 20587, x5 = 17968, x6 = 11496, x7 = 30335, x8 = 44507, x9 = 25329, x10 = 0 .

As a result, with a minimum income of 890,142.9 rubles per employee of the Stavropol Territory, it is necessary to plan the labor resources by economic sectors in accordance with the above values ​​of xi, i = 1, ..., 10 (note that such planning Unemployment in the region disappears, x10 = 0).

2. Zaitseva, I.V. Mathematical model of the labor market self-organization for several economic sectors / I.V. Zaitseva, E.A. Semenchin / / Economics and Mathematical Methods, 2007. - Volume 43. - № 1. - Pp. 133 - 136.

3. Zaitseva, I.V. The development of the concept "working capacity" as a socio-economic category / I.V. Zaitseva, M.V. Popov, Y.V. Vorohobina / Management / economic systems: electronic scientific journal in 2013. - № 1. - Access mode: http://www.uecs.ru/index.php.

4. Zaitseva, I.V. A systematic approach to determining the structure of the regional labor potential / I.V. Zaitseva, M.V. Popova / / System Analysis and Information Technology (SAIT 2013): Proceedings of the 15th International Scientific and Technical Conference SAIT in 2013. - Kiev, the ESC "IASA" NTU "KPI", 2013. - 516 p. - Pp. 101-102.

5. Labor and Employment in the Stavropol region. 2012: The Stat.sb. / Stavropolstat. - Stavropol, 2012. – 137 p.