# On the Failure to Reach the Optimal Government Debt Ceiling

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## Abstract

**:**

## 1. Introduction

#### 1.1. Economic Motivations

#### 1.2. Contributions

## 2. The Bounded Government Intervention Model

**Remark**

**1.**

**Definition**

**1.**

**Problem**

**1.**

**Proposition**

**1.**

**Proof.**

## 3. The Value Function and the Verification Theorem

**Proposition**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. The Solution

#### 4.1. Construction of the Solution

**Definition**

**2.**

**Remark**

**2.**

**Lemma**

**1.**

- (i)
- $\xi \phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}0$,
- (ii)
- $\lambda \phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}\mu $,
- (iii)
- ${\gamma}_{2}\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}m+1$,
- (iv)
- ${c}_{3}\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}0$.

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 4.2. Verification of the Solution

**Theorem**

**2.**

**Proof.**

#### 4.3. The Debt Policy Generated by the Optimal Debt Ceiling

#### 4.4. Numerical Solutions

**Example**

**1.**

**Remark**

**3.**

## 5. Time to Reach the Debt Ceiling

#### 5.1. The Theoretical Result

**Proposition**

**3.**

- (i)
- If $\tilde{\mu}\le 0$ or, equivalently, $r-g\le {\sigma}^{2}/2$, then$$\begin{array}{c}\hfill {P}_{x}\left\{\tau <\infty \right\}=1.\end{array}$$
- (ii)
- If $\tilde{\mu}>0$ or, equivalently, $r-g>{\sigma}^{2}/2$, then$$\begin{array}{c}\hfill {P}_{x}\left\{\tau <\infty \right\}={\displaystyle \frac{\Gamma (\nu -1)-\Gamma (\nu -1,\beta /x)}{\Gamma (\nu -1)-\Gamma (\nu -1,\beta /c)}},\phantom{\rule{0.166667em}{0ex}}\end{array}$$$$\begin{array}{c}\hfill \Gamma (a,z):={\int}_{z}^{\infty}{t}^{(a-1)}{e}^{-t}dt,\phantom{\rule{1.em}{0ex}}z\ge 0,\phantom{\rule{0.277778em}{0ex}}a>0.\end{array}$$$$\begin{array}{c}\hfill \underset{\overline{U}\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\infty}{lim}{P}_{x}\{\tau <\infty \}=1.\end{array}$$

**Proof.**

#### 5.2. Application 1: Countries with Big Economic Growth

**Example**

**2.**

#### 5.3. Application 2: Countries with Moderate Economic Growth

**Example**

**3.**

#### 5.4. Application 3: Time to Reach the Optimal Debt Ceiling

## 6. Comparative Statics Analysis

- Compare the results of the debt policy associated with the optimal debt ceiling with the policy of non-government intervention.
- Compare the results of the debt policy associated with the optimal debt ceiling presented in this paper with the policy derived in the unbounded model of Cadenillas and Huamán-Aguilar (2016).
- Analyze the effects of some parameters on the optimal debt ceiling.

**Example**

**4.**

#### 6.1. The Debt Policy Associated with the Optimal Debt Ceiling versus the Non-Intervention Policy

#### 6.2. The Bounded Model versus the Unbounded Model

#### 6.3. The Effects of $\alpha $, g, and $\sigma $ on the Optimal Debt Ceiling

## 7. Summary of Analysis

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

**Proof.**

## Appendix B. Proof of Proposition 2

**Proof.**

## Appendix C. Proof of Lemma 2

**Proof.**

## Appendix D. Proof of Proposition 3

**Proof.**

## Appendix E. Mathematica Code

## References

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1 | That $60\%$ was simply the median of the debt ratios of those European countries. Although it was not binding, it was considered as a reference value. For more details, please see Footnote 3 of this paper. |

2 | Optimal debt is the debt ratio that arises as a result of welfare analysis considering that debt, on the one hand, smooths consumption and, on the other hand, has negative effects in wealth distribution; see, for example, Aiyagari and McGrattan (1998) or Barro (1999). Credit ceiling is the level of debt above which the country is not allowed to borrow in the financial markets; see, for example, Eaton and Gersovitz (1981). Debt limit is the debt level at which a debt crises takes place; see, for instance, Ostry et al. (2010). |

3 | Article 104c of the Maastricht Treaty Council of the European Communities (1992) says the following: “2. The Commission shall monitor the development of the budgetary situation and of the stock of government debt in the Member States with a view to identifying gross errors. In particular, it shall examine compliance with budgetary discipline on the basis of the following two criteria: (a) whether the ratio of the planned or actual government deficit to gross domestic product exceeds a reference value, unless either the ratio has declined substantially and continuously and reached a level that comes close to the reference value;—or, alternatively, the excess over the reference value is only exceptional and temporary and the ratio remains close to the reference value; (b) whether the ratio of government debt to gross domestic product exceeds a reference value, unless the ratio is sufficiently diminishing and approaching the reference value at a satisfactory pace. The reference values are specified in the protocol on the excessive deficit procedure annexed to this Treaty”. In the Maastricht Treaty, the “ratio of the planned or actual government deficit to gross domestic product” is the debt ratio, and the “reference value” is the debt ceiling (which was selected as 60%). Thus, the Maastricht Treaty does not mention explicitly any bound for the government interventions. Similarly, the debate in the USA Senate about the selection of the debt ceiling does not mention any bound for the government interventions. |

**Figure 2.**The value function (for the bounded-intervention model) versus the total cost of the non-intervention policy.

c | $\phantom{\rule{2.em}{0ex}}\mathbf{Country}\mathbf{A}\phantom{\rule{2.em}{0ex}}$ | $\phantom{\rule{2.em}{0ex}}\mathbf{Country}\mathbf{B}\phantom{\rule{2.em}{0ex}}$ | |
---|---|---|---|

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.010\phantom{\rule{1.em}{0ex}}$ | 0.60 | 0.01538 | 0.00000 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.025\phantom{\rule{1.em}{0ex}}$ | 0.60 | 0.19876 | 0.00000 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.050\phantom{\rule{1.em}{0ex}}$ | 0.60 | 0.99542 | 0.00215 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.100\phantom{\rule{1.em}{0ex}}$ | 0.60 | 1.00000 | 0.89679 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.150\phantom{\rule{1.em}{0ex}}$ | 0.60 | 1.00000 | 0.99998 |

b | $\mathbf{Country}\mathbf{A}$ | $\mathbf{Country}\mathbf{B}$ | |
---|---|---|---|

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.010\phantom{\rule{1.em}{0ex}}$ | 0.60982 | 0.02356 | 0.00000 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.025\phantom{\rule{1.em}{0ex}}$ | 0.62576 | 0.29127 | 0.00000 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.050\phantom{\rule{1.em}{0ex}}$ | 0.64324 | 0.99598 | 0.00215 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.100\phantom{\rule{1.em}{0ex}}$ | 0.65495 | 1.00000 | 0.89679 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.150\phantom{\rule{1.em}{0ex}}$ | 0.65808 | 1.00000 | 0.99998 |

$\overline{\mathit{U}}=0.001$ | $\overline{\mathit{U}}=0.01$ | $\overline{\mathit{U}}=1$ | $\overline{\mathit{U}}=5$ | $\overline{\mathit{U}}=\mathit{\infty}$ | |
---|---|---|---|---|---|

b | 0.300024 | 0.310426 | 0.331213 | 0.331325 | 0.331352 |

$\phantom{\rule{1.em}{0ex}}\mathit{\alpha}=\mathbf{0.5}\phantom{\rule{1.em}{0ex}}$ | $\phantom{\rule{1.em}{0ex}}\mathit{\alpha}=\mathbf{1}\phantom{\rule{1.em}{0ex}}$ | $\phantom{\rule{1.em}{0ex}}\mathit{\alpha}=\mathbf{1.3}\phantom{\rule{1.em}{0ex}}$ | |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.01\phantom{\rule{1.em}{0ex}}$ | 0.609820 | 0.310426 | 0.241035 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=2.00\phantom{\rule{1.em}{0ex}}$ | 0.662425 | 0.331283 | 0.254845 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=\infty \phantom{\rule{2.em}{0ex}}$ | 0.662704 | 0.331352 | 0.254886 |

$\mathit{\mu}=\mathbf{0.05}$ | $\mathit{\mu}=\mathbf{0.10}$ | $\mathit{\mu}=\mathbf{0.14}$ | |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.01\phantom{\rule{1.em}{0ex}}$ | 0.310426 | 0.263990 | 0.253090 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=2.00\phantom{\rule{1.em}{0ex}}$ | 0.331283 | 0.303408 | 0.282346 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=\infty \phantom{\rule{2.em}{0ex}}$ | 0.331352 | 0.303466 | 0.282397 |

$\mathit{\sigma}=\mathbf{0.05}$ | $\mathit{\sigma}=\mathbf{0.13}$ | $\mathit{\sigma}=\mathbf{0.17}$ | |

$\phantom{\rule{1.em}{0ex}}\overline{U}=0.01\phantom{\rule{1.em}{0ex}}$ | 0.310426 | 0.301363 | 0.295110 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=2.00\phantom{\rule{1.em}{0ex}}$ | 0.331283 | 0.349697 | 0.359007 |

$\phantom{\rule{1.em}{0ex}}\overline{U}=\infty \phantom{\rule{2.em}{0ex}}$ | 0.331352 | 0.350220 | 0.359954 |

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Cadenillas, A.; Huamán-Aguilar, R.
On the Failure to Reach the Optimal Government Debt Ceiling. *Risks* **2018**, *6*, 138.
https://doi.org/10.3390/risks6040138

**AMA Style**

Cadenillas A, Huamán-Aguilar R.
On the Failure to Reach the Optimal Government Debt Ceiling. *Risks*. 2018; 6(4):138.
https://doi.org/10.3390/risks6040138

**Chicago/Turabian Style**

Cadenillas, Abel, and Ricardo Huamán-Aguilar.
2018. "On the Failure to Reach the Optimal Government Debt Ceiling" *Risks* 6, no. 4: 138.
https://doi.org/10.3390/risks6040138