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We solve an infinite time-horizon bounded-variation stochastic control problem with regime switching between N states. This is motivated by the problem of a government that wants to control the country's debt-to-GDP (gross domestic product) ratio. In our formulation, the debt-to-GDP ratio evolves stochastically in continuous time, and its drift - given by the interest rate on government debt, net of the growth rate of GDP - is affected by an exogenous macroeconomic risk process modelled by a continuous-time Markov chain with N states. The government can act on the public debt by increasing or decreasing its level, and it aims at minimising a net expected cost functional. Without relying on a guess-and-verify approach, but performing a direct probabilistic study, we show that it is optimal to keep the debt-to-GDP ratio in an interval, whose boundaries depend on the states of the risk process. These boundaries are given through a zero-sum optimal stopping game with regime switching with N states and we completely characterise them as solutions to a system of nonlinear algebraic equations with constraints. To the best of our knowledge, such a result appears here for the first time. Finally, we put in practice our methodology in a case study of a Markov chain with N = 2 states; we provide a thorough analysis and we complement our theoretical results by a detailed numerical study on the sensitivity of the optimal debt ratio management policy with respect to the problem's parameters.

We consider the problem of a government that wants to manage the country's debt-to- GDP (gross domestic product) ratio. The latter evolves stochastically in continuous time, and its drift is given by the interest rate on government debt, net of the growth rate of GDP. We further allow the interest rate to be affected by an exogenous macroeconomic risk process modelled by a continuous-time Markov chain with N states. The debt-to-GDP ratio level can be reduced by the government, e.g. through austerity policies in the form of spending cuts, or increased, e.g. by public investments. The aim of the government is to choose a policy which minimises the total expected cost of having debt plus the total expected cost of austerity policies, counterbalanced by the total expected gain from public investments. We model this as a bounded-variation stochastic control problem over an infinite time-horizon with regime switching, and we provide its explicit solution. To the best of our knowledge, such a problem has not been previously solved in the literature. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio in an interval, whose boundaries are depending on the states of the risk process, and are given through a zero-sum optimal stopping game with regime switching. We completely characterise these boundaries as solutions to a system of nonlinear algebraic equations with constraints. Finally, we put in practice our methodology in a case study of a Markov chain with N = 2 states; we provide a thorough analysis and we complement our theoretical results by a detailed numerical study on the sensitivity of the optimal debt ratio management policy with respect to the model's parameters.

Federal debt held by the public as a share of GDP: Figure 1: Federal debt held by public, as share of GDP (blue). Source: BEA, Treasury via FRED.

The empirical data on the effect of a high debt-to-GDP ratio on economic growth is conflicting. The article used the GDP Indicator to investigate the trajectory in the debt-to-GDP ratio in 45 nations in Sub-Saharan Africa during the COVID-19 Pandemic and to determine whether there is a point at which public debt becomes damaging to the region's economy. The major results demonstrated that as long as the economies of Sub-Saharan Africa continue to expand, a high debt-to-GDP ratio is not always negative; in fact, the majority of nations with a debt-to-GDP ratio greater than the 77 percent threshold had increased economic growth. Additionally, the paper discovered that countries in the region may face a high debt-to-GDP ratio as a result of excessive spending prompted by the Covid-19 pandemic, as well as an unpredictable slowdown in economic activity as a result of movement restriction policies imposed by subregional governments. Another intriguing conclusion is that Sub-Saharan Africa's average debt-to-GDP ratio of 56.6 percent in 2020 is much lower than the suggested limit of 77 percent. However, the report advised that governments reduce wasteful spending without jeopardizing the region's economic development rate. Finally, fiscal policies should be accompanied by monetary policies to promote effective public investment, simplified tax expenditures, increased public financial management, and better debt management and transparency.

This paper will argue that since the ratio of government debt to GDP cannot diverge to infinity, fiscal collapse is not possible. Using a macroeconomic model of a growing economy with a simple microeconomic foundation about consumers' behavior, with overlapping generations model in mind, we show the following results: 1) The budget deficit including interest payments on the government bonds equals an increase in the savings from a period to the next period. 2) If the savings in the first period is positive (unless the savings are made solely through stocks), we need budget deficit to maintain full employment under constant prices or inflation in the later periods in a growing economy. 3) Excess budget deficit induces inflation under full employment. 4) Under an appropriate assumption about the propensity to consume, the debt to GDP ratio converges to a finite value. It does not diverge to infinity. 5) In the case of balanced budget excluding interest payments, if an appropriate weak assumption about the propensity to consume holds, the debt to GDP ratio cannot diverge to infinity. 6) When the propensity to consume is small, we need budget deficit, not budget surplus, to prevent the debt to GDP ratio to diverge infinity.