A stochastic control approach to public debt management

We discuss a class of debt management problems in a stochastic environment model. We propose a model for the debt-to-GDP (Gross Domestic Product) ratio where the government interventions via fiscal policies affect the public debt and the GDP growth rate at the same time.We allow for stochastic interest rate and possible correlation with the GDP growth rate through the dependence of both the processes (interest rate and GDP growth rate) on a stochastic factor which may represent any relevant macroeconomic variable, such as the state of economy. We tackle the problem of a government whose goal is to determine the fiscal policy in order to minimize a general functional cost. We prove that the value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation and provide a Verification Theorem based on classical solutions. We investigate the form of the candidate optimal fiscal policy in many cases of interest, providing interesting policy insights. Finally, we discuss two applications to the debt reduction problem and debt smoothing, providing explicit expressions of the value function and the optimal policy in some special cases

A Stochastic Non-Zero-Sum Game of Controlling the Debt-to-GDP Ratio

We introduce a non-zero-sum game between a government and a legislative body to study the optimal level of debt. Each player, with different time preferences, can intervene on the stochastic dynamics of the debt-to-GDP ratio via singular stochastic controls, in view of minimizing non-continuously differentiable running costs. We completely characterise Nash equilibria in the class of Skorokhod-reflection-type policies. We highlight the importance of different time preferences resulting in qualitatively different type of equilibria. In particular, we show that, while it is always optimal for the government to devise an appropriate debt issuance policy, the legislator should optimally impose a debt ceiling only under relatively low discount rates and a laissez-faire policy can be optimal for high values of the legislator's discount rate

Debt management game and debt ceiling

We introduce a non zero-sum game between a government and a legislative body to study the optimal level of debt. We succeed in characterising Nash equilibria in the class of Skorokhodreflection policies which implies that the legislator imposes a debt ceiling. In addition, we highlight the importance of the time preferences in the magnitude of the optimal level of the statutory debt ceiling. In particular, we show that laissez-faire policy can be optimal for high values of the legislator’s discount rate

Optimal Control of Debt-to-GDP Ratio in an N-state Regime Switching Economy

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 regime-dependent 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 are characterised through 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.Comment: A previous version of this paper was circulating under the title "Optimal Management of Debt-to-GDP Ratio with Regime-Switching Interest Rate". In the new version we have improved exposition, generalized the setting and some of the results, and reorganized the structure of the paper. 27 pages, 3 figures. To appear on SIAM Journal on Control and Optimizatio