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Econ-348 · economics · Economics

State-contingent sovereign borrowing

LLM status Open unverified

Statement

Original subject group: Fiscal policy and sovereign debt

Audit classification: Explicit published open question. Limited indexation is expressly a puzzle with existing explanations; this contract design is editorial. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
Fix a sovereign with stochastic true output $y_t$, observable output report $\tilde y_t$, and an ability to influence reporting at a real cost. A debt contract promises repayment $g(\tilde y_t)$, possibly subject to a maximum payment. Investors have a declared stochastic discount factor, ambiguity set for output and reporting, and a liquidity cost depending on contract standardization. The sovereign’s preferences and default technology are given. Let $\mathcal G$ be a finite-dimensional, publicly stated family of repayment schedules. Determine whether a schedule $g\in\mathcal G$ yields a welfare-improving equilibrium relative to noncontingent debt while satisfying lender break-even and truthful-reporting incentives. In particular, require \[ E[m_t g(\tilde y_t)]-\text{liquidity and ambiguity costs}\ge \text{issue price}, \] where $m_t$ is the investor discount factor, together with a specified sovereign incentive-compatibility condition for every feasible false report. Quantify how much insurance survives pricing premia, manipulation, and reduced market liquidity; explain conditions under which noncontingent issuance is an equilibrium despite gross risk-sharing benefits. Assess contract standardization or independent verification as feasible remedies. The problem concerns this explicitly modeled adoption puzzle, not whether state-contingent debt can provide insurance in an ideal frictionless model, a benefit already established.

For a one-period contract let the issue price be $Q(g)$, actual repayment after default be $\widehat g(\tilde y,y)\le g(\tilde y)$, and the lender's actual pricing functional be $\inf_{P\in\mathcal P}E_P[m\widehat g]-c_{\rm liq}(g)$. Competitive zero-profit pricing equates that functional to $Q(g)$, rather than treating the issue price as a free parameter. The report incentive requires $u(y-g(y))\ge u(y-g(r))-k(y,r)$ for every feasible false report $r$, in the no-default one-period benchmark with $k$ in utility units; the full default model replaces this by continuation values and actual repayments. Contracts must keep consumption positive or specify treatment at zero. A welfare comparison holds the required resources raised fixed, or optimizes borrowing in both regimes; comparing different issue proceeds can confound insurance with financing volume. Require a strictly positive lender discount factor with finite discounted repayments and define all liquidity costs in the issue-price numeraire.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Verified GDP indexation. An independent statistical authority observes output without manipulation and the lender has one known law. Compare optimal smooth repayments with noncontingent debt, keeping funding proceeds and default technology identical.
  2. Ambiguous indexation and standardization. Allow an explicitly bounded ambiguity set and a liquidity cost depending on contract features. Compare continuous schedules with threshold warrants and assess whether standardization lowers costs enough to reverse the ranking.

References and source audit

  • Leonardo Martinez; Francisco Roch; Francisco Roldán; Jeromin Zettelmeyer. Sovereign Debt. 2022. Locator: Section10, State-contingent debt, printed pp.25--26. Primary text checked. https://fqroldan.github.io/resources/MRRZ.pdf.
Limited indexation is expressly a puzzle with existing explanations; this contract design is editorial.
  1. Leonardo Martinez; Francisco Roch; Francisco Roldán; Jeromin Zettelmeyer. 2022. Sovereign Debt. Section10, State-contingent debt, printed pp.25--26. https://fqroldan.github.io/resources/MRRZ.pdf. DOI: https://doi.org/10.5089/9798400213250.001.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-348-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-348

State-contingent sovereign borrowing: a research attempt

Target and outcome

Status: unresolved. The question is whether state-contingent sovereign borrowing survives pricing premia and truthful-reporting constraints when compared at the same funding proceeds. This attempt solves a two-state, one-period, no-default contract family with log utility, costly false reports and robust lender pricing. It gives an exact adoption condition and an explicit surviving-insurance bound. General dynamic default and endogenous statistical institutions remain unresolved.

Martinez, Roch, Roldán and Zettelmeyer [survey] discuss state-contingent debt and its limited use. Their survey supplies background, not the parameter values or the particular contract family below. The derivation is a conditional model calculation rather than a new empirical explanation of all observed nonindexation.

Fixed funding and the pricing functional

True output is $y_L<y_H$, with probabilities $p\in(0,1)$ and $q=1-p$. Required date-zero proceeds are $B\in(0,y_L)$. The sovereign values date-one consumption by log utility under the true law. Lenders have discount factor one and consider low-state probabilities in $[p-\epsilon,p+\epsilon]\subset[0,1]$. A contract promises nonnegative payments $g_L,g_H$, with spread $d=g_H-g_L\geq0$. Its liquidity cost in issue-price units is $\ell d$, where $\ell\geq0$.

Restrict attention to contracts that are enforceable without default and keep both truthful-state consumptions strictly positive. This can be implemented by an exogenous default technology that rules out default in this family. It is a substantive restriction: no promised payment is substituted for an endogenous default-adjusted payment without justification.

Worst-case lender repayment is attained at low-state probability $p+\epsilon$, because $g_H\geq g_L$. Consequently zero-profit issue pricing is \[ Q(g)=p g_L+qg_H-(\epsilon+\ell)d=B. \] Writing $\kappa=\epsilon+\ell$, every fixed-funding contract in the family has \[ g_L=B-(q-\kappa)d,\qquad g_H=B+(p+\kappa)d. \] Thus greater insurance raises the expected payment burden when $\kappa>0$. The comparison does not conceal an increase in issue proceeds.

Let $x_L=y_L-B>0$, $x_H=y_H-B>0$, $a=q-\kappa$ and $b=p+\kappa$. Truthful consumption is \[ c_L=x_L+ad,\qquad c_H=x_H-bd. \] False reporting of the low state when output is high costs $k_H\geq0$ in utility units; false reporting of the high state when output is low has nonnegative cost and yields a weakly larger payment, so is never advantageous when affordable. An unaffordable false report is infeasible.

Incentive compatibility and the adoption threshold

Proposition (Surviving indexation): High-state truth-telling is equivalent to \[ 0\leq d\leq d_{IC}:= \frac{(e^{k_H}-1)x_H}{1+(e^{k_H}-1)b}. \] If $\kappa<q$, let $D$ be the maximum feasible spread after imposing this bound, nonnegative payments and any additional exogenous repayment caps. Suppose these constraints give a closed interval $[0,D]$ on which consumption is positive. Then expected welfare \[ V(d)=p\log(x_L+ad)+q\log(x_H-bd) \] is strictly concave. If $D>0$, some indexed contract improves on noncontingent debt if and only if \[ \frac{p(q-\kappa)}{y_L-B}> \frac{q(p+\kappa)}{y_H-B}. \] Its optimal spread is \[ d^*=\min\left\{D,\max\left(0, \frac{p a x_H-q b x_L}{ab}\right)\right\}. \] If $\kappa\geq q$, no positive spread improves welfare in this family.
Proof: Under high output, falsely reporting low would yield consumption $c_H+d$. Truth-telling requires $\log c_H\geq\log(c_H+d)-k_H$. Exponentiation and rearrangement give $d\leq(e^{k_H}-1)c_H$. Substituting $c_H=x_H-bd$ gives the claimed bound, with a positive denominator. It also shows that zero manipulation cost forces $d=0$.

For $\kappa<q$, $a,b>0$. The first and second derivatives are \[ V'(d)=\frac{pa}{x_L+ad}-\frac{qb}{x_H-bd},\qquad V''(d)=-\frac{pa^2}{(x_L+ad)^2}-\frac{qb^2}{(x_H-bd)^2}<0. \] Strict concavity means an improvement from zero exists exactly when $V'(0)>0$ and a positive spread is feasible. Solving $V'(d)=0$ gives the displayed unconstrained optimum because $p+q=1$, and clipping to $[0,D]$ gives the constrained optimum. If $\kappa\geq q$, increasing $d$ weakly lowers low-state consumption and strictly lowers high-state consumption; welfare therefore strictly decreases. ∎

The compactness premise is easy to implement by imposing minimum consumption $\underline c>0$ below both baseline consumptions, along with finite payment caps. Alternatively, if the upper feasible endpoint makes a consumption approach zero, log utility tends to minus infinity there, so the optimum is attained away from that endpoint; the interior formula still applies with the appropriate feasible restriction. These cases should not be conflated with admitting zero-consumption log utility as finite.

A numerical contract check

Take $p=q=1/2$, $(y_L,y_H)=(3,7)$, $B=1$, $\kappa=0.1$ and $k_H=\log2$. Then $a=0.4$, $b=0.6$, $(x_L,x_H)=(2,6)$ and $d_{IC}=6/1.6=3.75$. Nonnegative low-state payment requires $d\leq2.5$. The unconstrained welfare optimum is also $2.5$, so the optimal contract has \[ (g_L,g_H)=(0,2.5),\qquad(c_L,c_H)=(3,4.5). \] The issue price is $(0+2.5)/2-0.1(2.5)=1$, exactly the baseline funding. Truth-telling is satisfied because $2.5\leq3.75$. Expected log utility rises from $\tfrac12\log12$ to $\tfrac12\log13.5$. If verification is weaker, for example $k_H=\log1.1$, the incentive bound shrinks to $0.6/1.06\simeq0.566$, sharply limiting the insurance despite unchanged output risk.

These values illustrate the formulas only. They are not estimates of GDP-report manipulation costs or sovereign liquidity premia.

Remedies, extensions and the remaining gap

Improved verification increases $k_H$, expanding the incentive-feasible interval while leaving zero funding fixed. Standardization that reduces $\ell$ lowers $\kappa$ and can turn the strict adoption inequality from false to true. But a verification agency's real resource cost must be included before calling that remedy welfare improving. Similarly, moving ambiguity or liquidity costs outside the lender's actual pricing rule would incorrectly treat insurance as cheaper than the assumed market permits.

The benchmark already explains noncontingent issuance in three precise cases: no credible reporting cost, a nonpositive marginal insurance gain after pricing, or external caps that leave no positive spread feasible. It does not show which case explains a particular sovereign. Dynamic default, endogenous debt volumes, investor risk aversion, correlated discount factors, political reporting incentives and multiple competing contracts require a different equilibrium. The broad adoption and policy-evaluation target remains unresolved, with the two-state criterion available as a check on richer models.

Completion Estimate

49%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves a two state fixed funding contract family with truthful reporting, ambiguity and liquidity pricing, yielding exact adoption and surviving insurance conditions. The full sovereign contract target still requires endogenous default and continuation incentives, richer investor pricing, feasible verification costs, and a justified equilibrium adoption comparison.

References

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