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Explicit Solution to a government debt reduction problem: a stochastic control approach

Claudia Ceci, Luca Semerari

TL;DR

The paper tackles the problem of optimally reducing a debt-to-GDP ratio when fiscal policy simultaneously affects debt dynamics and GDP growth under uncertainty. It uses a stochastic control framework with a cost functional that accounts for both deficits and surpluses, and derives explicit solutions via the Hamilton-Jacobi-Bellman equation for the case of a linear GDP response. A key result is the emergence of a threshold-type policy when policy costs are present, with a constant deficit policy arising if the GDP impact is sufficiently strong (α>1) and a two-region, switching policy for 0<α<1, determined by a smooth-pasting condition. Numerical analyses illustrate the threshold’s behavior across economic regimes and parameter values, validating the analytical findings and highlighting implications for debt sustainability under uncertainty.

Abstract

We analyze the problem of optimal reduction of the debt-to-GDP ratio in a stochastic control setting. The debt-to-GDP dynamics are modeled through a stochastic differential equation in which fiscal policy simultaneously affects both debt accumulation and GDP growth. A key feature of the framework is the introduction of a cost functional that captures the disutility of fiscal surpluses and the perceived benefit of fiscal deficits, thus incorporating the macroeconomic trade-off between tighten and expansionary policies. By applying the Hamilton-Jacobi-Bellman approach, we provide explicit solutions in the case of linear GDP response to the fiscal policies. We rigorously analyze threshold-type fiscal strategies in the case of linear impact of the fiscal policy and provide closed-form solutions for the associated value function in relevant regimes. A sensitivity analysis is conducted by varying key model parameters, confirming the robustness of our theoretical findings. The application to debt reduction highlights how fiscal costs and benefits influence optimal interventions, offering valuable insights into sustainable public debt management under uncertainty.

Explicit Solution to a government debt reduction problem: a stochastic control approach

TL;DR

The paper tackles the problem of optimally reducing a debt-to-GDP ratio when fiscal policy simultaneously affects debt dynamics and GDP growth under uncertainty. It uses a stochastic control framework with a cost functional that accounts for both deficits and surpluses, and derives explicit solutions via the Hamilton-Jacobi-Bellman equation for the case of a linear GDP response. A key result is the emergence of a threshold-type policy when policy costs are present, with a constant deficit policy arising if the GDP impact is sufficiently strong (α>1) and a two-region, switching policy for 0<α<1, determined by a smooth-pasting condition. Numerical analyses illustrate the threshold’s behavior across economic regimes and parameter values, validating the analytical findings and highlighting implications for debt sustainability under uncertainty.

Abstract

We analyze the problem of optimal reduction of the debt-to-GDP ratio in a stochastic control setting. The debt-to-GDP dynamics are modeled through a stochastic differential equation in which fiscal policy simultaneously affects both debt accumulation and GDP growth. A key feature of the framework is the introduction of a cost functional that captures the disutility of fiscal surpluses and the perceived benefit of fiscal deficits, thus incorporating the macroeconomic trade-off between tighten and expansionary policies. By applying the Hamilton-Jacobi-Bellman approach, we provide explicit solutions in the case of linear GDP response to the fiscal policies. We rigorously analyze threshold-type fiscal strategies in the case of linear impact of the fiscal policy and provide closed-form solutions for the associated value function in relevant regimes. A sensitivity analysis is conducted by varying key model parameters, confirming the robustness of our theoretical findings. The application to debt reduction highlights how fiscal costs and benefits influence optimal interventions, offering valuable insights into sustainable public debt management under uncertainty.

Paper Structure

This paper contains 6 sections, 8 theorems, 90 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model Framework and Problem Formulation
  3. The Hamilton-Jacobi-Belman equation
  4. Numerics
  5. Conclusions
  6. Proofs of Lemmas 3.2, 3.4 and 3.5

Key Result

Proposition 2.5

For every admissible strategy $u \in \mathcal{U}$ and for any $x \in (0,+\infty)$ we have that $|J(x, u)| < +\infty$.

Figures (5)

  • Figure 1: Graphic determination of the threshold b
  • Figure 2: Value function $v(x)$ in the baseline case
  • Figure 3: Value function $v(x)$ comparison between Strong and Weak economies.
  • Figure 4: Value function $v(x)$ trajectories across scenarios with scenario-specific thresholds $b$.
  • Figure 5: Simulated sample paths of the optimal debt-to-GDP ratio process $\{X_t^{u^*}\}_{t\ge0}$ (blue lines) and the debt-to-GDP ratio process $\{X_t^{0}\}_{t\ge0}$ in the absence of government intervention (orange lines). The dotted line marks the threshold $b \approx 1.28$, separating expansionary ($u_t=-U_1$) and contractionary ($u_t=U_2$) regimes.

Theorems & Definitions (22)

  • Remark 2.1
  • Definition 2.2
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • ...and 12 more