Frank Schmid · 7 November 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 includes 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 roots, two of them stable, and beyond it only the crisis root 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 sovereign 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 root, a crisis root, and a sunspot, a random event on which investors coordinate, that selects between the two 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 announced capacity, ceiling, or growth increment, or what delivered primary balance, would remove the crisis equilibrium.

The line of sight

the section belongs to the price of the debtthe section belongs to the fiscal capacitythe theme passes through
1

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 among the consequences “the risk of a fiscal crisis.” 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 represent: the convenience yield on Treasury debt, U.S. monetary policy as a driver of the global financial cycle, and debt denominated in the issuer's own currency.
Leads to
In what model are those three features represented together?
Section 2 · The Macroeconomic Framework
2.1–2.3

2.1 Debt Dynamics
2.2 The Interest-Growth Differential and the Stationarity of the Debt Ratio
2.3 The Fiscal Limit as a Present Value

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 including 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

2.4 The Price of Sovereign Debt
2.5 Multiplicity, 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 at that root 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 first year of the paths obeys, 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 equilibrium, what acts on the gap between the debt and the limit?
2.6–2.8

2.6 Monetary-Fiscal Regimes
2.7 Output, Inflation, Asset Prices, and Transmission
2.8 Central-Bank Backstops: Mandates and Record

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; its level and its conditions are assumptions. The ceiling is available while, in the year before, the primary balance, net of the automatic stabilizers, stood no more than a tolerance below its projected path and the debt did not exceed the fiscal limit, so it is 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 that a source identifies 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's and the IMF's projections and from published estimates, estimates made on other records, among them the pass-through of U.S. long rates on the OECD series and the fiscal capacities, and assumptions where no record identifies a value; each of the 30 structural parameters is 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
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

3.1 The Fixed Point and the Crisis Zone
3.2 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 10-year horizon under either; the regimes differ in where the cost falls. The crisis equilibrium exists because the fiscal limit is discounted at a rate that includes the spread being priced.
195.1 / 121.2U.S. debt held by the public in 2036, percent of GDP, regime M (the primary balance adjusts) / regime F (the price level adjusts), the second 1.1 percentage points above its own baseline
−0.8 / 9.9U.S. inflation in 2036, percent, regime M, fallen from the target / regime F
4.1 / 3.9U.S. sovereign spread in 2036, percentage points, regime M / regime F
−9.2 / −7.9lowest U.S. output gap, percent of potential, regime M / regime F
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

3.3 Transmission
3.4 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 are on it under regime M alone, in some years, and Canada in no year. On the deterministic 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 U.S. interest-growth differential on the stock turns positive at the onset under regime M.
Exhibits
Table 17, the central-bank ceilings across the simulated 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

3.5 The Stochastic Simulation
3.6 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:
EconomyPercent of paths
United Kingdom80
France77
United States54
Japan44
Italy40
Germany24
Canada10
Each economy has its own zone, and a crisis abroad does not require one in the United States: the crisis years of the United Kingdom and France rest on their own baselines and on the growth draw common to the eight economies as well as on the common sunspot. The United States spends at least one year on the crisis equilibrium on 54 percent of paths, the market coordinating on 21 percent and the crisis equilibrium arriving as the only root on the remaining 33, in 2027 on the paths that begin beyond the zone and later on the rest; a credit event occurs on 27 percent of paths. 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 an illustration the paper cites, in place of 40, the U.S. share of paths with a crisis year is 32 percent and the share on which the market coordinates 1 percent. 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

3.7 The AI-Adoption Productivity Path
3.8 Variant: The Announced Central-Bank Portfolio Paths

Price of the debtFiscal capacity
Establishes
A sensitivity and a variant. 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 with a U.S. crisis from 54 percent to 52. Applying the announced run-off of the central banks' portfolios raises the crisis shares of Japan, France, Italy, Germany, and the United Kingdom and leaves the U.S. shares within a percentage point of the central calibration's.
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 crisis share, 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 remove the crisis equilibrium?
Section 4 · Removing the Crisis Equilibrium
4

Removing the Crisis Equilibrium

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. For the United States, no trend growth path of the central calibration makes the limit unbounded at the maximal crisis yield of the corner condition. 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
3.74median U.S. requirement under regime M over the drawn parameter sets that require a capacity, percent of GDP
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. At the central calibration a ceiling removes the crisis equilibrium only where the baseline keeps a benign one, and removes none for the United Kingdom and France, while across the drawn parameter sets a ceiling removes it under regime M in 197 of the United Kingdom's 888 sets with a crisis and in 229 of France's 901. At the central calibration 18 of the 23 ceilings that remove a crisis equilibrium are still reached, the purchases absorbing a spread the hazard prices above them while no crisis root exists; every announced instrument must hold beyond the horizon; and a primary surplus delivered rather than announced can be smaller the later the year from which the crisis equilibrium is to be absent.
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, announced in 2027, that removes the U.S. crisis equilibrium under regime M
1.08 through 4.97the increment, percentage points, for the United Kingdom and France across the announcement years and regimes
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 · Table 33, the map over the drawn parameter sets
Leads to
Has a consolidation of that size been sustained before?
4.2–4.3

4.2 The Requirement against the Historical Record
4.3 The Loss of the Safe-Asset Position and the Record of the Prior Suppliers

Fiscal capacity
Establishes
The requirement set against the record of large primary surpluses, and against the fiscal history of the two economies that were dominant suppliers of a safe asset before the United States, each of which held a debt within its estimated fiscal backing after losing that position.
Exhibits
Table 34, the requirement against the record of large primary surpluses · Table 35, fiscal backing of the three dominant suppliers of a safe asset
Leads to
What limits the reading?
Sections 5 and 6, the Addendum, and the Appendices
5

Limitations

Price of the debtFiscal capacity
Establishes
The reading of the results is subject to 14 limitations.
Leads to
What, then, does the paper conclude?
6

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 accounting for the larger part of the change and the loss of the convenience yield the smaller. Which policy authority adjusts decides the outcome of the U.S. crisis but, at the capacities the record shows, not whether it occurs, and for Germany and Italy it decides the occurrence as well: the cost falls on output under both, and in addition on the debt ratio under regime M and on the price level under regime F. 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 modeled 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 10 years under regime M.
Exhibits
Table 36, China under three readings
Appendices
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 · Appendix B, supplementary results: Table B.1, the central scenarios against the product of the loss given default and the bound on the default hazard; Table B.2, spreads and yields at the start of the horizon; Table B.3, the crisis outcomes of Table 19 on the crisis paths with no drawn U.S. credit event