Frank Schmid · 13 October 2026 · The flow of the paper, section by section

Fiscal Capacity and Sovereign Debt Crises in the G7

Sovereign debt is priced as a fixed point: the yield carries a spread for a default hazard, and the hazard depends on how far the debt that the yield itself produces stands from a fiscal limit. Over a range of debt the pricing equation has three solutions, two of them stable, and beyond it only the crisis solution exists. Where that range lies is set by the fiscal capacity, which the paper takes from each economy's historical record, and the capacity is also where a resolution would come from.

The price of the debt

The yield is the risk-free rate plus expected inflation, an inflation risk premium, and a credit spread. The spread compensates for the hazard that the debt ratio produced by the yield implies, so the price is a fixed point, with a benign solution, a crisis solution, and a sunspot that selects between them where both exist.

The fiscal capacity

The fiscal limit is the present value of the largest primary surplus the government can sustain. The paper reads that surplus from history, as the largest 10-year average each economy has sustained, and asks what capacity, ceiling, or growth increment, announced or delivered, would remove the crisis solution.

The line of sight

the section belongs to the price of the debtthe section belongs to the fiscal capacitythe theme passes through

Introduction

Establishes: The setting: the Congressional Budget Office projects federal debt held by the public at 100.6 percent of GDP for fiscal year 2026 and 120.2 percent for 2036, and names the risk of a fiscal crisis among the consequences. A crisis in the issuer of the world's reserve asset differs from the episodes the literature has studied in three respects the model must carry: the convenience yield on Treasury debt, the U.S. long rate as the reference rate of the global financial cycle, and debt denominated in the issuer's own currency.
Leads to: In what model are those three features carried together?

Section 2 · The macroeconomic framework

2.1–2.3

Debt Dynamics, the Interest-Growth Differential, and the Fiscal Limit

Fiscal capacity

Establishes: The law of motion of the debt ratio, the two strands of literature on the interest-growth differential, and the fiscal limit as a present value: the largest sustainable primary surplus discounted at the real differential at a neutral policy rate, the financing rate carrying the spread being priced. The limit is unbounded while the real financing rate at that policy rate does not exceed trend real growth, and finite once it does.
Leads to: How does the market price debt that stands near such a limit?

2.4–2.5

The Price of Sovereign Debt, the Crisis Zone, and the Sunspot

Price of the debt

Establishes: The pricing fixed point. A root of the year's pricing equation is a crisis root where the default hazard it carries exceeds half of its bound. An economy lies below its crisis zone where no crisis root exists, inside it where a crisis root stands beside a benign one and the market can coordinate on it, and beyond it where the crisis root is the only one. An economy is in crisis in a year it spends on the crisis equilibrium by either route, coordination or sole survival. A closed-form corner condition places the boundary in proportion to the fiscal capacity; the operative boundary, the one the simulated paths obey, lies 19 through 60 percentage points of GDP above it for the economies without a ceiling.
Leads to: Once the market is on the crisis solution, what acts on the gap between the debt and the limit?

2.6–2.8

Monetary-Fiscal Regimes, Transmission, and Central-Bank Backstops

Price of the debtFiscal capacity

Establishes: Two courses a crisis can take, Leeper's active-money regime (regime M) and active-fiscal regime (regime F): the gap between the debt and the limit is left to the primary balance in the first and to the price level in the second. Output, inflation, asset prices, and the transmission of a U.S. crisis to the other six economies. Central banks enter twice: as holders, whose share of the stock is financed at the policy rate, and as a backstop, a ceiling on the sovereign spread modeled from the Eurosystem's Transmission Protection Instrument as a stylized experiment, available while the primary balance stood within a tolerance of its projected path and the debt did not exceed the fiscal limit in the year before, and so withdrawn the year after an economy is first priced on a crisis root.
Leads to: On what values does the model run?

2.9

Calibration, Estimates, and the Baseline

Fiscal capacity

Establishes: The capacity is taken from history rather than assumed: the largest 10-year average primary surplus each economy has sustained, in percent of GDP, with every receipt from abroad excluded; China's, with no positive 10-year average on its record, is scaled from revenue. Debt is read net of the government's financial assets. Parameters are of three kinds: values from the CBO and the IMF, estimates from the Jordà–Schularick–Taylor macrohistory database, and 30 structural assumptions, each listed with its basis and its treatment in the simulation. The baselines of France and the United Kingdom are themselves on the crisis equilibrium, the United Kingdom's in every year and France's from 2032.

1.91United States, percent of GDP, federal basis, 1993 through 2002

1.43 · 1.06United Kingdom · France, percent of GDP

2.18 · 3.27Germany · Italy, percent of GDP

3.52 · 1.31Canada · Japan, percent of GDP

Reads as: On the macrohistory record the U.S. primary balance shows no response to lagged debt over 1871 through 2020; the interest-growth differential has been negative in 62 percent of years since 1950; and the response of the primary balance to debt, which the record separates from neither zero nor a positive value, is set to zero.
Exhibits: Table 1, initial conditions · Table 2 and Figure 1, the primary balance against lagged debt · Table 3, the response of the balance to debt · Table 4, the composition of the debt · Table 5, projection errors · Table 6, the differential and the pass-through · Tables 7 through 9, fiscal capacity by economy · Tables 10 through 12, the baseline beside the published projections
Leads to: Where does each economy stand against its crisis zone?

Section 3 · Results

3.1–3.2

The Crisis Zone, and the United States under the Two Regimes

Price of the debtFiscal capacity

Establishes: In 2027 the United States, France, and Japan are inside their crisis zones, the United Kingdom is beyond its own, and Germany, Italy, and Canada lie below theirs. The U.S. crisis begins in 2027 under either regime and does not end within the ten-year horizon under either; the regimes differ in where the cost falls. The second solution of the pricing equation exists because the fiscal limit is discounted at a rate that carries the spread being priced.

	Regime M: the primary balance adjusts	Regime F: the price level adjusts
Debt held by the public, 2036	195.1percent of GDP	121.2percent of GDP, 1.1 percentage points above its own baseline
Inflation by 2036	−0.8percent in 2036, fallen from the target	9.9percent
Sovereign spread, 2036	4.1percentage points; output gap −9.2 percent of potential	3.85percentage points

Exhibits: Figure 2, the pricing fixed point and the crisis zone · Tables 13 and 14, the limit formed without the spread · Table 15, the output loss against the record · Figure 3, U.S. paths under the baseline and the two regimes · Table 16, crisis outcomes by economy
Leads to: What reaches the other six economies?

3.3–3.4

Transmission, and the Interest-Growth Differential along the Paths

Price of the debt

Establishes: In the central scenarios France, the United Kingdom, and Japan are on the crisis equilibrium beside the United States in every year under both regimes, Germany and Italy under regime M alone, and Canada in no year. On the paths of regime M the Eurosystem's ceiling binds for France in 2027, for Italy in 2031 and 2032, and for Germany in 2034 and 2035, and is withdrawn the year after each is first priced on a crisis root; Italy is on the crisis equilibrium in 2032 through 2036 with the ceiling and in 2028 through 2036 without it. The differential that history shows negative in most years turns positive under regime M, the convenience yield that holds the U.S. rate below growth being lost in the coordination.
Exhibits: Table 17, the central-bank ceilings across the stochastic paths · Figure 4, the 10-year yield of each economy · Figure 5 and Table 18, the differential, the primary balance, and the decomposition of the change in the debt ratio
Leads to: How much of this depends on the central values?

3.5–3.6

The Stochastic Simulation and Sensitivity

Price of the debtFiscal capacity

Establishes: 10,000 paths that draw the capacity around its estimate, the other quantities the assumption tables mark as varied, and annual shocks. The share of paths on which each economy spends at least one year on the crisis equilibrium by 2036, with a bar on one scale from zero:

Economy	Percent of paths
United Kingdom	80
France	77
United States	54
Japan	44
Italy	40
Germany	24
Canada	10

Reads as: Each economy carries its own zone and the sunspot is common, so a crisis abroad does not require one in the United States. For the United States the market coordinates on 21 percent of paths and the benign equilibrium disappears on the rest of the 54; a credit event occurs on 27 percent. Conditional on a crisis, the median peak 10-year yield is 12.1 percent and the median debt ratio in 2036 is 141.7 percent of GDP. Trend growth is the largest single contributor to the variance of whether a crisis occurs; the fiscal capacity is the second. At a loss given default of 10 percent, the haircut of the illustration of Blanchard, Martínez-Bruera, and Huertas, in place of 40, the U.S. share is 32 percent and coordination occurs on 1. The shares are shares of simulated paths under stated ranges, not forecasts.
Exhibits: Figure 6, the distribution of the U.S. debt ratio and yield · Table 19, outcomes conditional on a U.S. crisis · Table 20, the variance between parameter draws and across shocks · Table 21, the level sensitivity · Table 22, the range of multiplicity against the dispersion of the limit · Table 23, deterministic variants · Figure 7, shares of variance
Leads to: Do faster growth or the central banks' announced portfolio paths change the reading?

3.7–3.8

The AI-Adoption Productivity Path, and the Announced Central-Bank Portfolio Paths

Price of the debtFiscal capacity

Establishes: Two variants. Adding the Penn Wharton Budget Model's projected contribution of generative AI to productivity growth to every trend path, with the fiscal capacity responding to growth, moves the share of paths carrying a U.S. crisis from 54 percent to 52. Carrying the announced run-off of the central banks' portfolios raises the crisis-state shares of Japan, France, Italy, Germany, and the United Kingdom and leaves the U.S. distribution where the central calibration puts it.

138.5conditional median U.S. debt ratio in 2036 under the AI path, percent of GDP, against 141.7

5.2 through 17.2percentage points added to the share of paths carrying the crisis state, across Japan, France, Italy, Germany, and the United Kingdom, under the run-off

Exhibits: Table 24, the AI-adoption sensitivity · Table 25, the Bank of England's gilt holdings under its announcement · Table 26, the central banks' holdings on their announced paths
Leads to: What would end the crisis?

Section 4 · The path to resolution

The Path to Resolution

Fiscal capacity

Establishes: The paper solves for the capacity that removes the crisis equilibrium, rather than reading a requirement off the crisis paths, for every economy that reaches the crisis equilibrium in a central scenario. No growth path in the record makes the limit unbounded. The requirement is one figure for any announcement no later than the onset and rises with every announcement after it.

2.75 · 2.67United States, percent of GDP, announced in 2027, regime M and regime F, against a historical 1.91

5.06 · 3.40the same, announced from 2033

2.93median U.S. requirement under regime M across the drawn parameter sets, percent of GDP, a set that requires no capacity entering at its drawn capacity

6.63 · 4.28United Kingdom, regime M and regime F, against a historical 1.43

Exhibits: Table 27, the resolving capacity by announcement year · Figure 8 and Table 28, the resolving capacity over the drawn parameter sets · Figure 9, the years in which the crisis equilibrium exists against the announced capacity
Leads to: Is the capacity the only instrument that reaches a resolution?

4.1

The Resolution Map: Three Instruments

Price of the debtFiscal capacity

Establishes: The same question solved for an announced capacity, an unconditional ceiling on the sovereign spread, and a permanent increment to trend growth. A ceiling removes the crisis equilibrium only where the baseline keeps a benign one, and removes none for the United Kingdom and France; a ceiling that removes it can still be reached, the purchases carrying a spread the hazard prices above it while no crisis root exists, as 18 of the 23 clearing ceilings are. Every announced instrument must hold beyond the horizon. A surplus actually delivered, rather than announced, trades size for time.

1.32ceiling on the spread, percentage points, announced in 2027, that removes the U.S. crisis equilibrium under regime M

0.61increment to trend growth, percentage points, that clears for the United States under regime M

1.08 through 4.97the increment, percentage points, for the United Kingdom and France

1.0 or belowceiling, percentage points, under which the U.S. historical capacity suffices

Exhibits: Table 29, the resolution map · Table 30 and Figure 10, the two instruments together · Table 31, the delivered primary surplus · Table 32, the swing against the revenue and expenditure base
Leads to: Has a consolidation of that size been sustained before?

4.2–4.3

The Requirement against the Historical Record, and the Prior Suppliers of the Safe Asset

Fiscal capacity

Establishes: The requirement set against the record of large primary surpluses, and against the fiscal history of the two economies that supplied the world's safe asset before the United States, each of which financed itself at full fiscal backing after losing that position.
Exhibits: Table 33, the requirement against the record of large primary surpluses · Table 34, fiscal backing of the three dominant suppliers of the world's safe asset
Leads to: What does the model not carry?

Sections 5 and 6, the addendum, and the appendix

Limitations

Price of the debtFiscal capacity

Establishes: The results are subject to 14 limitations, each stated with the reading it qualifies.
Leads to: What, then, does the paper conclude?

Conclusion

Price of the debtFiscal capacity

Establishes: The fiscal limit is proportional to the capacity. At the U.S. onset the coordination raises the limit's financing rate from below trend growth to above it, the spread carrying the larger part of the change and the loss of the convenience yield the smaller. Which policy authority adjusts decides the outcome of the crisis but, at the capacities the record shows, not whether it occurs: under active monetary policy the cost falls on output and on the debt ratio, and under active fiscal policy on the price level. The capacity is where a resolution would come from, and it is not the only instrument that reaches one. The multiplicity is conditional on two quantities the record does not pin for a G7 sovereign, the dispersion of the fiscal limit and the loss given default.
Leads to: Where does China stand, its spread administered rather than set by a marginal investor?

Addendum

China

Establishes: China is carried in the computation and stands outside the set of economies the paper studies. Its sovereign spread is administered by a state-directed holder base under capital controls rather than set by a marginal investor, so the coordination the paper measures does not price for it. Priced as a market issuer it is on the crisis equilibrium for 6 of the ten years under regime M.
Exhibits: Table 35, China under three readings
Appendix: A, the assumptions of the model: Tables A.1 through A.13, each parameter with its value, unit, basis, and treatment in the stochastic simulation