Abstract
The main contribution of this paper is the analysis of a patch model which includes migration effects and interactions between two different economies. The migration coefficients are driven by differences between salaries. The dynamics of each economy is described through a generalized Solow model which combines together a convex-concave production function and logistic population dynamics. Numerical simulations show the long-run behavior of these systems.
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Notes
- 1.
The author showed the explosion of the birth rate when income increases and then the increase of mortality because of competition on the relatively scarce output of productive land
- 2.
(i) lim k→0 f′(k)=+∞, (ii) lim k→∞ f′(k)=0, (iii) f (0)=0 (see [1])
- 3.
In order to generate a simulation for which the system reaches an equilibrium in set B of Equation (8), we consider our functions g1 and g2. We calculate that g1(x)=0.00634726x+0.00000242x2=0 at x=0 and x ≈ −2623, while g2(x)=0.10548241x − 0.00000776x2=0 at x=0 and x ≈ 13593. This means that there are no (necessarily positive) equilibria in set B. When we use the initial conditions L 01=1, L 02=27888, K 01=1.5 * 103.3 /1000, and K 02=103.3/1000, at time t=200, we reach (L 1 , L 2)=(1, 13593). But it turns out that (Y 1)L 1=(Y 2)L 2=0.62247, Y 1=δ 1 K 1=0.6875, and Y 2=δ 2 K 2=9337, so we have reached an equilibrium in set A
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Capasso, V., Kunze, H.E., La Torre, D. (2012). Population dynamics in a patch growth model with S-shaped production functions and migration effects. In: Perna, C., Sibillo, M. (eds) Mathematical and Statistical Methods for Actuarial Sciences and Finance. Springer, Milano. https://doi.org/10.1007/978-88-470-2342-0_9
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DOI: https://doi.org/10.1007/978-88-470-2342-0_9
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