GPT 6 Astra Ultra attempt on Econ-153
economicsEconomics cataloguecontributionexplorationClaim: exploration
Reported by the author as unresolved.
Manuscript
Contents
- Safe debt as a tail constraint: exact and ambiguity-robust benchmarks
- Abstract
- Short target and a deliberately restricted bond
- The exact law and atoms
- What a mean restriction does and does not buy
- A finite-sample safety certificate
- Safety and welfare are different optimizations
- References
- Completion Estimate
Safe debt as a tail constraint: exact and ambiguity-robust benchmarks
Status: Unresolved. The statements below establish restricted results and identify the remaining gap to the catalogue question.Abstract
In a one-period real repayment model, safe debt capacity is determined by a lower-tail resource threshold, with an explicit strict-inequality convention for atoms. We solve the robust capacity problem under a mean and an upper resource bound, and show that a mean alone gives no positive robust capacity. A finite candidate-grid certificate accounts for sampling uncertainty. These are credit-risk benchmarks, not a solution of endogenous sovereign-debt equilibrium and pathwise price risk.Short target and a deliberately restricted bond
The catalogue asks how much sovereign debt remains safe at a specified horizon and how this interacts with fiscal feasibility and welfare. Here total real resources available for repayment are an exogenous nonnegative random variable $R$. Issue $b>0$ units of an indexed one-period claim, sharing resources pro rata. Real repayment per unit is $\min\{1,R/b\}$, so loss is $L(b)=(1-R/b)_+$. There is no intermediate trading, inflation exposure, strategic default, or endogenous refinancing. At $b=0$ set loss to zero. Fix a tolerance $0\le\ell<1$ and probability bound $0<\alpha<1$. Safety means ${\mathbb{P}}(L(b)>\ell)\le\alpha$.The exact law and atoms
Define $F_<(x)={\mathbb{P}}(R<x)$ and \[ q_\alpha=\sup\{x\ge0:F_<(x)\le\alpha\}. \] Allow $q_\alpha=+\infty$.Theorem (Tail capacity): For $b>0$, safety is equivalent to $F_<((1-\ell)b)\le\alpha$. Consequently the supremum safe debt is $b^*=q_\alpha/(1-\ell)$. If $q_\alpha<\infty$, it is itself safe and the safe set is $[0,b^*]$. If the policy separately imposes a deterministic issuance cap $\bar b$, capacity becomes $\min\{b^*,\bar b\}$.
Proof: For $\ell<1$, the loss event is exactly $R<(1-\ell)b$. The safe thresholds form a downward-closed set containing zero. For a finite supremum $q_\alpha$, choose a sequence of safe thresholds increasing to it. The events $\{R<x_k\}$ increase to $\{R<q_\alpha\}$, so continuity from below of probability gives $F_<(q_\alpha)\le\alpha$. Every smaller nonnegative threshold is safe by monotonicity. Positive scaling proves the result, and intersection with an issuance interval gives the last assertion. ∎
For example, if resources equal zero with probability $\alpha$ and 10 otherwise, then $q_\alpha=10$. Replacing $F_<$ by ${\mathbb{P}}(R\le x)$ at the boundary would incorrectly declare the threshold 10 unsafe. This example is purely a discrete mathematical illustration. If $\ell\ge1$ all finite debt passes this credit-loss criterion, showing why the threshold restriction is necessary for an informative benchmark. What a mean restriction does and does not buy
Proposition (Mean-only ambiguity): Fix $\mu>0$ and let the ambiguity class contain all nonnegative resource laws with mean $\mu$. For every $x>0$, $\sup_P{\mathbb{P}}_P(R<x)=1$. Hence robust safe debt capacity is zero for every $\alpha<1$ and $\ell<1$.
Proof: For any $p<1$ sufficiently close to one, put probability $p$ at zero and $1-p$ at $\mu/(1-p)>x$. The mean is $\mu$ and the loss-threshold probability is $p$. Its supremum is one. Every positive debt has a positive threshold, so fails robust safety; zero debt is safe by convention. ∎
Now impose $0\le R\le M$ almost surely and ${\mathbb{E}} R=\mu$, with $0<\mu<M$. Theorem (Exact bounded-support ambiguity frontier): For $0<x\le\mu$, \[
\sup_P{\mathbb{P}}_P(R<x)={M-\mu\over M-x},
\] where the supremum need not be attained. For $x>\mu$ it equals one, and for $x=0$ it equals zero. The robust safe-debt capacity is \[
b_{\rm rob}^*={1\over1-\ell}
\max\left\{0,\ M-{M-\mu\over\alpha}\right\}.
\] This capacity is itself robustly safe.
Proof: Let $p={\mathbb{P}}(R<x)$. Since $R\le x$ on that event and $R\le M$ everywhere, $\mu\le px+(1-p)M$, giving $p\le(M-\mu)/(M-x)$ for $x\le\mu<M$. To approach equality choose $r<x$ with $r\ge0$, put mass $(M-\mu)/(M-r)$ at $r$ and the rest at $M$, and let $r\uparrow x$. This also covers $x=\mu$, where the supremum is one. For $x>\mu$, the constant law $R=\mu$ attains probability one. The value at zero follows from nonnegativity.
The expression is a consequence of the supplied support and mean, not of estimated fiscal capacity or observed bond prices. In particular, robust capacity changes sharply when the support restriction is removed. For any positive safe threshold necessarily $x<\mu$ and $(M-\mu)/(M-x)\le\alpha$, equivalently $x\le M-(M-\mu)/\alpha$. Positive thresholds exist precisely when the right side is positive. Otherwise only zero is safe. If it is positive, it is below $\mu$ and the worst-case supremum equals $\alpha$ at the boundary. A supremum bound controls every law even if no law attains it, proving boundary safety and the formula after rescaling. ∎
A finite-sample safety certificate
Suppose independent samples $R_1,\ldots,R_N$ come from one fixed resource law. Fix $0<\delta<1$ and, in advance, a finite list $b_1,\ldots,b_J>0$, and put \[ \widehat p_j={1\over N}\sum_{i=1}^N {\mathbf{1}}\{R_i<(1-\ell)b_j\},\qquad e_N=\sqrt{\log(J/\delta)/(2N)}. \] Declare candidate $j$ certified if $\widehat p_j+e_N\le\alpha$, and always allow $b=0$.Proposition (Simultaneous certification): With probability at least $1-\delta$, every certified candidate is safe, including a candidate chosen after seeing the sample. The guarantee is uniform over all fixed resource laws. It does not make a previously unknown ambiguity class valid.
Proof: For each $j$, the indicators are independent and bounded in $[0,1]$. The one-sided bounded-variable exponential inequality gives ${\mathbb{P}}(p_j>\widehat p_j+e_N)\le e^{-2Ne_N^2}=\delta/J$. A union bound makes every upper bound valid simultaneously. On that event the certification inequality implies $p_j\le\alpha$ for every selected candidate. The bound contains no law-specific constants. ∎
If no positive candidate is certified, this procedure issues no positive safety certificate; it does not prove that true capacity is zero. A continuum of data-selected thresholds requires a uniform empirical-distribution argument or a fixed grid refinement theorem. Safety and welfare are different optimizations
Proposition (A transparent capped welfare example): If a separately supplied net convenience benefit is $B(b)=a\log(1+b)-cb$ with $a,c>0$, and the only feasibility restriction is $0\le b\le\bar b<\infty$, its unique maximizer is $\min\{\bar b,\max\{0,a/c-1\}\}$.
Proof: $B'(b)=a/(1+b)-c$ is strictly decreasing and $B''(b)<0$. The unique unconstrained nonnegative maximum is $\max\{0,a/c-1\}$, and truncation at the cap gives the constrained optimum. ∎
One may insert a proved safety capacity for $\bar b$, but the benefit function must be justified separately. Neither convenience yield nor a high mean repayment resource establishes a safety bound. The remaining catalogue task requires endogenous fiscal rules, policy-dependent resource distributions, equilibrium selection, and horizon-specific market-price losses. None is provided by this one-period calculation. References
- [MT] K. J. Mitchener and C. Trebesch, Sovereign Debt in the 21st Century, CESifo Working Paper 8959 (original March 2021; September 2022 revision). https://www.ifo.de/DocDL/cesifo1_wp8959.pdf. Survey background and concluding research agenda. The bond loss and ambiguity classes here are explicit restricted modeling choices, not a debt-capacity estimate from the survey.