ИСТИНА

We consider a modified Ramsey model of the households economical behavior. This model has an extension in a form of incorporation an imperfect savings and lending market. We introduce the dynamic of the incomes in a form of the stochastic differential equation (SDE). Based on the modified Ramsey model, we present the optimal strategy of the household as following: the household determines consumptions, savings and consumer loan lending in order to maximize the discounted consumption with a budget restriction. This allows us to introduce the Hamilton--Jacobi--Bellman equation. The contribution of a single household is insignificant, but the impact of the community can be modelled by a mean field term and considered from the point of the probability density function. The evolution of the probability density function is described by a Kolmogorov--Fokker--Planck equation. This motivates us to use a Mean Field Games concept that is presented in a system of partial differential equations (PDEs) that form a boundary value problem: a Hamilton--Jacobi--Bellman equation, evolving backwards in time, and a Kolmogorov--Fokker--Planck equation evolving forward in time. We present a self-similar solution of the Hamilton--Jacobi--Bellman equation and introduce the numerical solution of the Kolmogorov--Fokker--Planck equation considering the evolution of different layers of economical behavior of the households. To do this, we use a special set of parameters introduced in describing the behavior of the representative household of each layer.