The r vs g Sustainability Framework
The Rule
“The expected level of nominal interest rates relative to nominal growth rates tells us how debt and debt service are likely to grow or shrink.” — a-00078
r > g → debt/income expands automatically (even with zero deficit) r < g → debt/income can stabilize (if primary balance managed) r = g, primary balance = 0 → debt/income exactly flat
The Math
“If the primary deficit is zero (i.e., current expenses before interest = current revenue), debts will stay flat if the interest rate is equal to the revenue growth rate. If the primary deficit is 5% of the current debt level, interest rates would need to be 5% below the revenue growth rate.” — a-00079
Formula: For debt stability: r ≤ g - (primary deficit / debt level)
Example: If debt/revenue = 500% and primary deficit = 10% of revenue:
- Primary deficit / debt = 10% / 500% = 2%
- Required: r ≤ g - 2%
- At g = 4%, need r ≤ 2%
With current US interest rates at 3.5% and g at 3.9%, the margin is ~0.4% — razor thin.
The Compounding Effect
“When interest rates are higher than income growth rates, the existing debt grows relative to incomes because the debt compounds faster than incomes grow.” — a-00081
At r > g, every year the debt:income ratio increases automatically. This is mechanical compounding — it doesn’t require any new borrowing. Just existing debt + higher rate + lower growth.
US Sensitivity Table (Approximate)
| Scenario | r | g | Primary Deficit | Debt/Revenue in 10Y |
|---|---|---|---|---|
| Base case | 3.5% | 3.9% | -6% | 648% |
| Rate shock (+1%) | 4.5% | 3.9% | -6% | ~750% |
| Growth shock (-1%) | 3.5% | 2.9% | -6% | ~750% |
| Policy fix | 2.0% | 3.9% | -3% | ~500% |
The Interest Rate Spiral Risk
“Let’s say nominal income is growing at 3.9% a year, interest rates are 3.5%, and debt levels start at 580% of government income… in this example the government spends 32% more than it collects.” — a-00082
At these numbers, the debt trajectory is explosive unless private holders are willing to hold at low rates. Once private demand falls:
- Rates must rise to attract buyers
- Higher rates increase debt service
- Higher debt service increases the deficit
- Larger deficit requires more bond issuance
- More issuance pushes rates higher → spiral
The break point: when r crosses g with a large primary deficit, the spiral becomes self-sustaining unless CB intervenes (monetizes).