Investigation of a Mathematical Model Linking GDP Growth with Changes in National Debt

Abstract

The article proposes a mathematical model linking GDP growth with changes in national debt. The model is based on a system of two linear ordinary differential equations. The rate of GDP growth is defined in the model as the difference between aggregate revenues and aggregate expenses, including debt service expenses. Possible types of GDP and national debt dynamics are investigated as a function of parameters. External investments increasing national debt may accelerate economic growth, whereas without new external loans GDP either does not grow or grows slowly. An equation describing the variation of relative external debt has been derived and fully investigated. It is shown that the external debt is never fully repaid, which is consistent with other models. Conditions of stable economic growth are derived, when GDP grows faster than or at the same rate as national debt, whereas the relative debt approaches a constant value. We investigate conditions that preclude a Ponzi game, so that a country cannot use new external debt to build a financial pyramid to repay old debt. The model features are demonstrated in application to statistical data from a number of countries. The model parameters are determined and growth trajectories are calculated for GDP and national debt.