METHOD
OLOGY.

Definitions

Before the equations, the variables — because a model is only as honest as its labels.

• Y₀ = $4.92T — total federal receipts, FY2024 (CBO Monthly Budget Review; Treasury MTS). In the model, Y₀ functions as a cyclically-adjusted potential: the receipts the system would produce under zero unemployment and target inflation. We use the FY2024 actuals figure as a stand-in for that potential — the gap between actuals and a fully cyclically-adjusted estimate is small enough that the recalibration cost is not worth the additional opacity.
• D₀ = $28.3T — debt held by the public, end of FY2024. We use this rather than the $36T+ gross debt figure because intragovernmental holdings (Social Security, Medicare trust funds) do not generate market-priced interest expense and are not the credibility-relevant denominator for sovereign stress. The calculator projects D(t) = D₀ + t·(2.0 + 0.2·max(0, u−4)) forward when a horizon is selected (see Projecting Debt Forward below).
• r ≈ 3.3% — weighted-average rate on debt held by the public, FY2024.
• i_target = 2.0% — Federal Reserve inflation target.
• α = 0.05, β = 0.04 — unemployment and excess-inflation sensitivity coefficients. Working-paper estimates, not regression-derived. See Where the Coefficients Come From below.
• Y_floor = 0.55 — minimum revenue ratio under extreme stress. Bounds the model below at 55% of Y₀.

The Core Equations

The Crisis Ratio model rests on three equations, applied in sequence:

1. Revenue contracts under stress, bounded smoothly between the floor and full baseline.

Y(u,i) = Y₀ × [Y_floor + (1 − Y_floor) × max(0, 1 − α·u − β·max(0, i − i_target))]

The inner term — max(0, 1 − α·u − β·max(0, i − i_target)) — is the linear stress factor: it falls toward zero as unemployment and excess inflation rise. The outer transformation rescales this between Y_floor and 1.0, producing a smooth interpolation that never falls below 55% of Y₀. Without the floor, a linear function calibrated near baseline produces unphysical extrapolations at extreme stress (revenue collapsing toward zero); the 0.55 bound is the empirical lower limit from the 1932–33 federal-receipts trough — the worst peacetime contraction in the historical record.

2. Debt service is debt held by the public times the weighted-average rate.

DS(t) = D(t) × r

3. The Crisis Ratio is debt service divided by available revenue.

Crisis Ratio(t) = DS(t) ÷ Y(u,i)

4. Debt projects forward when a horizon is selected.

D(t) = D₀ + t × (2.0 + 0.2 × max(0, u − 4.0))

Static (t = 0) reproduces the FY24 reading. Forward projections compound the structural deficit and the cyclical addition into the debt stock; revenue is held at Y(u,i) across horizons because Y₀ is calibrated to cyclically-adjusted potential, not nominal growth. The implication is that the model understates revenue growth in the forward path — yet still produces ratios that climb because debt grows faster than the trend revenue we'd be folding in.

The threshold of 0.30 mirrors the IMF MAC-DSA "elevated risk" interest-to-revenue band for advanced economies. Above 0.30, discretionary fiscal capacity begins to erode. Above 0.50, non-mandatory spending is crowded out and rollover risk dominates. Above 1.0, the government must borrow simply to pay interest on existing borrowing — the textbook signature of a fiscal spiral.

Where the Coefficients Come From

These are working-paper sensitivities — stylized coefficients anchored to specific historical episodes and to the broader literature on automatic stabilizers. They are not regression-derived in the strict sense, but they are tested against an illustrative regression on historical data, summarized immediately below. The goal is back-of-envelope legibility informed by, not produced by, an empirical fit. Readers who want the formal apparatus should start with Auerbach & Feenberg's framework in the Journal of Economic Perspectives (2000) and CBO's cyclically-adjusted revenue methodology.

An Empirical Test — What History Implies

An illustrative regression on historical annual data (1990–2024) implies a historical α near 0.03 and a β that is statistically insignificant. The model uses α = 0.05 deliberately — above the historical estimate — on the argument that the AI displacement scenario produces larger revenue contraction than past cyclical episodes: the displaced cohort is high-wage, and the corporate tax channel does not provide its usual offset. The inflation coefficient is the weakest-identified parameter in the model, and the model does not lean on it.

What matters more than the precise calibration is robustness. The cascade conclusion holds across the full plausible coefficient range. Even at the conservative historical estimates, the moderate, severe, and cascade scenarios all still breach the 0.30 threshold. The coefficients affect how quickly the threshold is breached, not whether. That is the load-bearing property of the model, and it is the reason the calibration choice can be argued in the open rather than hidden in a footnote.

α = 0.05 — Unemployment Sensitivity

Calibration anchor: the 2008–09 episode. Federal receipts fell roughly 17% between FY2008 and FY2009 as unemployment rose 4.3 percentage points. Net of stimulus and one-time tax actions, that implies a cyclical revenue sensitivity around 0.04 per point of u. The 0.05 figure used here sits at the upper end of CBO's cyclically-adjusted revenue methodology range — defensibly inside the band, deliberately not the center of it. It captures three compounding effects:

• Direct payroll tax loss. Fewer workers, less FICA withholding, less Medicare contribution.
• Corporate tax decline. Higher unemployment correlates with reduced corporate profitability and capital gains realization.
• Activity drag. Unemployed workers spend less; consumption taxes and downstream activity-driven receipts decline.

Applying α to absolute u (rather than to a gap above some natural rate u*) is a modeling choice. It treats Y₀ as the receipts the system would produce at zero unemployment — an unattainable ceiling, not an observed baseline — and lets the formula contract revenue from there. The trade-off is conceptual: it produces clean, interpretable arithmetic at the cost of treating Y₀ as a hypothetical maximum rather than an observed quantity. The methodology owes the reader that label; we provide it here.

β = 0.04 — Inflation Sensitivity

Calibration anchor: post-1985 indexed-bracket experience. Before 1985, U.S. federal brackets were not indexed to inflation, and high-inflation regimes generated substantial real-revenue gains for the Treasury via bracket creep. Modern indexation substantially mutes that channel; what remains is a composite of fiscal capacity erosion from indexed outlays and second-order revenue drag:

• Indexed outlay growth. Social Security COLAs, Medicare price indexation, federal pensions, TIPS interest — all rise mechanically with inflation. The model treats their growth as net negative on revenue available for discretionary purposes.
• Lagged bracket adjustment. Brackets index annually with a one-year lag; high-inflation regimes push real revenue down before the adjustment catches up.
• Real-revenue drift. Receipts collected early in the fiscal year buy less in real terms by year-end.

β is therefore a composite sensitivity, not an inflation-revenue elasticity in the standard sense. We name it as such. Pre-1985 episodes would require a different coefficient (likely positive, given the unmuted bracket-creep channel).

The Interest Rate Multiplier

At $28.3T in debt held by the public, each 1% increase in the weighted-average rate adds $283 billion to annual debt service. This is mechanical: $28.3T × 0.01 = $0.283T. The lag between rate increases and debt service increases depends on the debt maturity profile — short-dated debt rolls quickly, long-dated debt is locked in. The model assumes r represents the weighted-average yield across the outstanding portfolio.

Why a Floor at 0.55·Y₀

Linear sensitivities calibrated near the current operating point misbehave outside their calibration range. Plugging cascade-scenario inputs into a purely linear revenue function produces revenue collapsing toward zero — an artifact, not a forecast. No peacetime advanced economy in the historical record has seen federal receipts fall below roughly half of baseline. U.S. federal receipts in 1932–33 (the worst peacetime contraction) bottomed near 50–55% of 1929 levels. The 0.55 floor encodes this empirical lower bound; the smooth interpolation built into the formula structure ensures revenue approaches the floor asymptotically rather than discontinuously.

Projecting Debt Forward — $2.0T Baseline + $0.2T per pt of Excess u

Debt held by the public grew by $1.97 trillion in FY2025 (Joint Economic Committee, U.S. Senate, October 2025). CBO's baseline projects a structural gap of roughly 6.5% of GDP between outlays (23.5%) and revenue (17%), which at current GDP implies a baseline structural deficit near $2.0 trillion per year. That is the constant term in the projection rule.

The cyclical addition — $0.2 trillion per percentage point of unemployment above 4.0% — is calibrated to the 2008–09 episode, where the federal deficit rose by roughly $1 trillion against a ~5 percentage point increase in unemployment. The 4.0% threshold corresponds to CBO's long-run estimate of the non-cyclical unemployment rate; below it, the cyclical channel contributes nothing.

Applying the rule to the FY24 baseline (u = 4.0%, D₀ = $28.3T) projects:

• +1 year: D = $30.3T — closely matches the FY25 actual of $30.28T, which is how this rule was validated.
• +3 years: D = $34.3T — assumes baseline conditions persist, no cyclical addition.

Under stress, the cyclical term compounds. At u = 8.0%, annual growth rises to $2.8T; at u = 12.0%, it rises to $3.6T. These figures are conservative — actual recession-era deficit growth includes discretionary stimulus, transfer-program automatic stabilizers, and revenue-side contraction that the model captures separately. The projection rule isolates the baseline plus cyclical-employment channel only.

Why the Off-Ramps Are Closed

If the cascade fires — if AI displacement or any other shock couples u, i, and r at once — conventional commentary will point to a handful of "safety valves" as reasons it can be contained: central bank intervention, reserve-currency demand, the option to raise taxes, an expandable tax base. Each is treated as a buffer that would break the cascade if it began. On inspection, none actually would.

• Quantitative easing is not an exit; it is the same loop through a different door. The Fed can absorb Treasury issuance and suppress yields, but only by expanding its balance sheet — which is inflationary. The model already prices inflation as a revenue-eroder: each percentage point above target reduces real revenue by ~12%. QE suppresses the numerator (debt service) while accelerating the denominator's collapse (revenue). Same equation, rebalanced. Not a buffer; a substitution.
• Reserve currency status is endogenous to the cascade. Treating it as a fixed backdrop is circular. A bond market cascade is the mechanism by which reserve status erodes — yields rise because faith in the instrument falls. The "least dirty shirt" argument — that capital flees to Treasuries when other economies destabilize — assumes the U.S. is not the source of the destabilization. In a cascade driven by U.S. fiscal mathematics, that assumption inverts. You cannot model the cascade and simultaneously assume the buffer the cascade dismantles.
• Raising taxes is the lever the political system has spent two generations welding shut. The $39T trajectory was produced by sequential tax cuts — 2001, 2003, 2017, and the 2025 extensions — not by spending profligacy alone. To treat "raise taxes" as an unused buffer is to assume a political system willing to reverse the project that defines its modern coalition. That system does not exist. The buffer was not unused; it was removed.
• Broadening the tax base requires a base that is growing, not shrinking. Standard IMF prescriptions — broaden bases, impose windfall levies, expand the contributor pool — assume a sovereign that can will workers into the formal economy. Current U.S. policy is doing the opposite: active deportation removes contributors. Undocumented workers paid an estimated $96 billion in federal, state, and local taxes in 2022. The model holds Y₀ constant; in practice Y₀ is contracting, on top of the unemployment-driven contraction the model already captures.

Fragilities Not Included

• Rollover risk. Roughly one-third of U.S. debt matures annually and must be refinanced. A sudden buyer strike at auction would force yields up much faster than the model's smooth r variable suggests.
• Foreign holder behavior. A coordinated selloff by major holders (China, Japan, Gulf states) is not in the model. Their behavior is endogenous to the very crisis the model describes.
• Credit spread blowouts. The model uses a single average rate. In a crisis, spreads on different maturities widen at different speeds. Long-duration debt service can detach from short-end policy rates entirely.
• Political constraints. The model assumes the government can pay debt service. Statutory debt ceiling fights, political brinkmanship, or technical default risk are not modeled.

Why a Threshold of 0.30?

The 0.30 stress line is anchored to the IMF's Sovereign Risk and Debt Sustainability Framework for Market-Access Countries (MAC-DSA), which flags interest-to-revenue ratios above roughly 20–30% as elevated risk for advanced economies. The low-income-country DSF uses tighter bands (14–23%) given thinner fiscal buffers. We place the trigger at 0.30 — the upper end of the IMF advanced-economy band — because the U.S. retains structural advantages (reserve currency demand, deep market liquidity) that have historically pushed its tolerable band higher than peer sovereigns. Above 0.30, discretionary fiscal capacity begins to compress in a way that constrains response to subsequent shocks. Above 1.0, the math reverses: borrowing finances borrowing.

How the Model Sits Within the Sovereign-Debt Literature

This is a teaching simplification of a much deeper field. Three anchors worth naming:

• Blanchard's r-vs-g framework (AEA Presidential Address, 2019, "Public Debt and Low Interest Rates"). The technical formulation of sovereign debt sustainability is the differential equation ḃ = (r − g)b − s, where b is debt-to-GDP, r the real interest rate, g the real growth rate, and s the primary surplus. When r < g, debt-to-GDP is stable even with persistent deficits. When r > g, debt accumulates explosively absent fiscal correction. The Crisis Ratio is a single-period snapshot of a system Blanchard models dynamically; our model trades realism for legibility, and the trade should be acknowledged.
• Auerbach & Feenberg ("The Significance of Federal Taxes as Automatic Stabilizers," Journal of Economic Perspectives, 2000). The standard framework for thinking about the cyclical revenue elasticity used here. Their estimates inform the upper bound on α; subsequent CBO methodology builds directly on this lineage.
• Reinhart and Rogoff (This Time Is Different, 2009; "Growth in a Time of Debt," 2010). The historical record of sovereign debt crises across centuries suggests threshold effects above 90% debt-to-GDP — a finding subsequently contested on coding grounds but directionally robust. The Crisis Ratio operates on a different denominator (revenue, not GDP) and a different numerator (debt service flow, not debt stock), but the threshold intuition is parallel.

Readers who want the technical apparatus should start there. Readers who want a feel for how the variables interact should stay here.

How to Read the Output

• Ratio < 0.20 — Sustainable. Headroom for shock absorption.
• 0.20 ≤ Ratio < 0.30 — Approaching threshold. Sensitivity to a coupling shock rising.
• 0.30 ≤ Ratio < 0.50 — Above threshold. Discretionary fiscal capacity eroding.
• 0.50 ≤ Ratio < 1.0 — Critical. Conventional fiscal response cannot close the gap without monetary accommodation or structural action.
• Ratio ≥ 1.0 — Reflexive collapse. Borrowing finances borrowing; pathways are hyperinflation, default, or radical restructuring.

A Note on Coupling

The three variables in this model — unemployment, inflation, and the weighted-average interest rate — are coupled through standard macroeconomic mechanisms: most prominently the Phillips curve, the Federal Reserve's reaction function, and fiscal-policy feedback. Under ordinary shocks, those mechanisms drive the variables in the canonical inverse pattern that the fiscal arithmetic absorbs. The Crisis Ratio describes a scenario in which a single shock — AI cognitive displacement — overrides those canonical mechanisms and produces co-movement through a different coupling pathway. The full paper engages the Phillips-curve question directly.

A Final Note on Use

This model is offered as a teaching tool. It compresses a vast and contested field — sovereign debt sustainability — into three intuitive variables. That compression sacrifices realism for clarity. The point is not to forecast a specific outcome. The point is to make legible the structural relationship between employment, inflation, rates, and fiscal solvency — and to show that under the cascade scenario these variables become decoupled from the canonical cyclical dynamics that normally hold them in offsetting balance.

The cascade is not a prediction. It is a description of how things can fit together if they all move in the same direction.