Mathematical model of labour population shift

The dynamic model of Tiebout's hypothesis is offered to study interregional migratory streams. This model allows to formalize the regulation of the population movement by redistribution of capital investments between regions. There are considered N identical regions having own economy and the main production resources. Let х(к) be the amount of people living in the region i at the step k. Write the following difference equation for dynamics of change of the population: х(к+1)= x,(k) + W(k) - Y,(k), i=1,...,n, k=0,1,..., (1) where Wi(k) is the amount of people which arrives to region i at the step k, Yi(k) is the amount of the population which leaves region i at the step k. Here, for simplicity, natural change (increasing or decreasing) of the population is omitted. It is connected with the fact that this process rather slow. It follows from this assumption that the total population in all regions at each time step remains constant. Then, Y _t (k) = J zj, (k)x (k) < x (k), W _t (k) = J z _v (k)xj (k), (2) j=1 j=1 where zj(k) is the population share of the region i moved to the region j. It is obvious that the following conditions have to be satisfied: n z _u(k) = 0, 0i(k) < 1, i = 1,...,n. j=1 Substituting (2) in (1), we obtain the equations x, (k +1) = f 1 - J Zj (k) 1 x, (k) + J z j (k) Xj (k), i=\,...,n, k=0,1,.... (3) j=1 V j= У The coefficients Zj(k) are defined by the region factors, which can be presented as follows: h. (k) = q. + , i = 1,...,n, k = 0,1,., x, (k) where q _t are constant factors (climate, cultural level, political stability, etc.), Z _t(k)/x _t(k) are economic factors: the ratio of capital investments Z,(k) into the region i to the amount of the people x,(k). According to Tiebout's hypothesis, there is a movement of the population from regions with the low total factor to regions with the high total factor proportionally to the difference of these factors. Mathematically it can be presented as Zji(k) = b, G{h/k)- h,(k)}, where fx, if x > 0, CT( x) = \ ' [0, if x < 0, bjj is the coefficient characterizing activity of inhabitants of the region i to remove to the region j. Note that in (3) it is possible a steady state when the migration between regions is lacking. It corresponds to a case when total factors of all regions are identical. The regulation of migration is carried out by redistribution of capital investments between regions. The task of optimum distribution of manpower resources between regions to maximize the total production efficiency is solved.

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