In this paper, we consider a model of the evolution of the Lorenz curve describing the distribution of income between economic agents. In this model, the Pigou–Dalton transfer system generates a steady-state income distribution chosen by the welfare state. The model is described by a mixed problem whose main equation is a partial integro-differential equation. We prove that the steady-state solution of this equation is asymptotically stable. The proof involves studying the spectrum of the integral operator obtained by linearization of the integro-differential operator. We proceed to the boundary value problem on the eigenvalues for a linear non-autonomous system of first-order ODEs. It is shown that an explicit solution to this system can be derived. It appears that the study of the spectrum can be reduced to finding zeros of an integral as a function of a complex parameter. Sufficient conditions for the absence of zeros of the integral in the right half-plane are given.
G. S. Parastaev (Sat,) studied this question.