Modeling the process of establishing equilibrium on the labor market

The article is devoted to studying the process of establishing equilibrium in the labor market. For this purpose, a synergistic model is proposed that describes the redistribution process between the employed and the unemployed agents of the system. The model is based on a nonlinear first-order differential equation that relates the change in the number of employed agents to the current number of employed and unemployed agents. The equation also includes parameters that describe the probabilities of transitions between different states of the system. Situations are considered when the probabilities of transitions are constant (do not change when the state of the system changes), and when they depend linearly on the number of employed and unemployed agents. In all of the above cases, the question of the dynamics of the system and the possibility of its transition to a stationary state (an equilibrium state) is studied. The paper shows that both in the case of constant transition probabilities and under the condition that these probabilities depend on the number of employed and unemployed, the system has a stable stationary state, to which it transitions regardless of the initial conditions. To demonstrate the stability of the system, the dependence of transition probabilities on the number of employed and unemployed agents is considered, and based on the resulting nonlinear differential equation, a stationary solution is obtained, after which this solution is analyzed for stability. Exact (analytical) solutions are found for the corresponding differential equation, and the stability of the stationary state is proven. The results of the study can be useful in studying the processes of establishing equilibrium in the labor market and in analyzing the stability of the corresponding systems.