Mathematical Modeling of the Household Behavior in the Labor Market

We describe the economic behavior of the household. On the one hand, the household acts as a consumer that seeks to maximize the discounted consumptions in the imperfect lending and saving market. On the other hand, the household acts a worker in the labor market that receives salary and wants to enlarge its competencies to receive higher wages. In this work we present the model of the worker behavior that spends its salary on consumptions and on the investments in human capital. The investments in human capital helps to obtain new skills and increase the qualifications of the employee. This provides an opportunity to receive higher wages. The problem is formalized as an optimal control problem on the infinite time horizon. We introduce its solution in the form of the Pontryagin maximum principle, find the transversality conditions of the conjugate variables, and introduce the identification approach to reproduce the behavior of employees in different social layers based on the Russian Federation Household Budget Survey. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.