On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Abstract

Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an infinite time-horizon. In a general not necessarily Markovian framework, we first show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such link to characterise the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein-Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation-dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem.Public Debt Control 2 of a crisis. This negative effect on economic growth from high debt levels has been observed in 18 different advanced economies (see [8]).In this paper we propose a continuous-time stochastic model for the control of the debtto-GDP ratio. The problem we have in mind is that of a government aiming to answer the question: How much is too much? 1 Following classical macroeconomic theory (see, e.g., [5]), in any given period the debt ratio stock grows by the existing debt stock multiplied by the difference between real interest rate and GDP growth, less the primary budget balance 2 . We assume that the government can reduce the level of the debt-to-GDP ratio by adjusting the primary budget balance, e.g. through fiscal interventions like raising taxes or reducing expenses. We therefore interpret the cumulative interventions on the debt ratio as the government's control variable, and we model it as a nonnegative and nondecreasing stochastic process. Uncertainty in our model comes through the GDP growth rate of the country, and its inflation and nominal interest rate which, by Fisher law [21], directly affect the growth rate of the debt ratio. We first assume that they are general not necessarily Markovian stochastic processes whose dynamics is not under government control. Indeed, the level of these macroeconomic variables is usually regulated by an autonomous Central Bank, whose action, however, is not modelled in this paper (see, e.g., [10] and [25] for problems related to the optimal control of inflation).Since high debt-to-GDP ratios can constrain economic growth making it more difficult to break the burden of the debt, we assume that debt ratio generates an instantaneous cost/penalty. This is a general nonnegative convex function of the debt ratio level, that the government would like to k...

On the Failure to Reach the Optimal Government Debt Ceiling

“…We next present a verification theorem that will be used to obtain the solution. Other verification theorems for classical stochastic control problems can be found, for instance, in Fleming and Soner (2006).…”

Continuity of the optimal stopping boundary for two-dimensional diffusions

“…This is motivated by the fact that in recent years several specific problems of this kind have appeared in the literature (cf. [6], [8], [10], [12], [16], [17]) and further ones are on their way. Often these papers are motivated by real-world applications where dimension two (or higher) plays a crucial role (cf.…”

“…Moreover, the known uniqueness arguments [22,, originally established in [19] and further refined in [11], extend to this fully two-dimensional case as well (cf. [6], [8], [12], [16], [17]) and this enables one to characterise the optimal stopping boundary as the unique solution to a nonlinear integral equation in the class of continuous functions (with the value function being expressed in terms of the optimal stopping boundary itself). This characterisation of the optimal stopping boundary is known to be sufficient for practical purposes of optimal stopping (using Picard iteration for numerics for instance) and higher degrees of regularity can then be studied subsequently as/if needed.…”

Determination of Optimal Public Debt Ceiling for Kenya Using Stochastic Control

“…In our study, the dynamic programming equation (HJB) is a second order linear inhomogeneous second ordinary differential equation (rather than a PDE). This means that in the inaction region the value function depends only parametrically on the variable associated to the purely controlled state [11].…”

“…In this model, it is prudent for government to enforce a fiscal policy that maintains the debt/GDP ratio under an inflation dependent ceiling. However, the Markovian formulation of the singular control problem is fully two-dimensional [11]. We however, leave the multidimensional singular control approach for future research.…”