Three interest-rate anchors
The framework compares two real interest rates that are usually discussed in separate literatures. Both are defined in the same units, which is what makes them directly comparable and what the whole measure rests on.
Monetary r-star
The real interest rate at which output sits at potential and inflation holds at target. This is the rate the economy gravitates to for macroeconomic balance, in the natural-rate tradition running from Wicksell to Laubach and Williams.
Fiscal r-star
The effective real interest rate on government debt that holds the debt-to-GDP ratio steady, given the primary balance, trend growth, the inflation target, and the debt level. This is the rate the fiscal position can afford.
The fiscal-monetary gap
Monetary r-star minus fiscal r-star. When it is positive, the rate the economy is pulling toward exceeds the rate the fiscal stance can sustain, and one of a small set of adjustments has to follow.
The law of motion of debt
Everything begins with the government budget constraint. Let \(D_t\) be the nominal debt stock and \(PB_t \equiv T_t - G_t\) the primary balance, the difference between tax revenue and non-interest spending. Debt carried into the period is repaid with interest at the effective nominal rate \(i_{t-1}\):
\[ D_t = (1 + i_{t-1}) D_{t-1} - PB_t \tag{A.5} \]Dividing through by nominal GDP and writing lower-case letters for ratios to GDP gives the same relationship in ratio form, where \(\pi_t\) and \(g_t\) are the growth rates of the GDP deflator and real GDP:
\[ d_t = \frac{1 + i_{t-1}}{(1 + \pi_t)(1 + g_t)}\, d_{t-1} - pb_t \tag{A.6} \]Defining the effective real interest rate on government debt as \(r_t \equiv i_{t-1} - \pi_t\), the change in the debt ratio can be written as the law of motion of debt:
\[ \Delta d_t = \frac{r_t - g_t}{1 + \pi_t + g_t}\, d_{t-1} - pb_t \tag{1} \]Read directly, debt dynamics improve with faster growth \(g_t\), higher inflation \(\pi_t\), or a larger primary balance \(pb_t\), and worsen when the real rate \(r_t\) rises. The term \(r_t - g_t\) is the familiar interest-growth differential: when \(r \lt g\), there is room to run primary deficits and still hold the debt ratio flat.
Deriving fiscal r-star
Under an active fiscal policy the primary balance does not adjust to stabilize debt, so its path can be taken as given. The natural question then becomes: what real interest rate would hold the debt ratio steady? Fiscal r-star is that rate. Formally, it is the effective real rate that stabilizes the debt ratio at a target level \(\bar d\), given the inflation target \(\bar\pi\), a constant primary balance \(\overline{pb}\), and trend growth \(\bar g\).
To obtain it, set \(\Delta d_t = 0\) in the law of motion (1), evaluated at those steady-state values:
\[ 0 = \frac{r_f^* - \bar g}{1 + \bar\pi + \bar g}\, \bar d - \overline{pb} \]Solving for \(r_f^*\) gives fiscal r-star:
\[ r_f^* = \bar g + (1 + \bar\pi + \bar g)\, \frac{\overline{pb}}{\bar d} \tag{6} \]The comparative statics fall straight out of (6). Fiscal r-star is increasing in trend growth, the inflation target, and the primary balance, and decreasing in the debt target. Intuitively, a stronger fiscal position, whether from a larger surplus, faster growth, more inflation to erode the debt, or a smaller debt stock to service, lets the economy carry a higher real rate before the debt ratio starts to climb. A useful convention note: if debt is scaled by real rather than nominal GDP, the \((1 + \bar\pi + \bar g)\) factor drops and fiscal r-star reduces to \(r_f^* = \bar g + \overline{pb} / \bar d\).
Substituting the definition back into the law of motion expresses debt accumulation against fiscal r-star:
\[ \frac{\Delta d_t}{\bar d} = \frac{r_t - r_f^*}{1 + \bar\pi + \bar g} - \frac{pb_t - \overline{pb}}{\bar d} \tag{7} \]This is the sustainability reading. While fiscal r-star sits above the actual real rate on the debt, there is room to run wider deficits without lifting the debt ratio. Once it falls below the actual real rate, the ratio rises unless the primary balance is raised above \(\overline{pb}\).
Monetary r-star
Monetary r-star is the natural rate from a standard New Keynesian block: the real interest rate at which output is at potential and inflation is at target. Combining an IS curve with a Phillips curve, and abstracting from cost-push shocks, inflation evolves as:
\[ \Delta \pi_t = \Phi_t - \phi\,(r_t^P - r_m^*) \tag{10} \]Here \(r_t^P\) is the real policy rate, \(\phi\) is the sensitivity of inflation to the gap between the policy rate and monetary r-star \(r_m^*\), and \(\Phi_t\) is an inflation-expectations term. When the policy rate is set at monetary r-star, inflation is stable. Because fiscal r-star is defined in this same real-rate space, the two anchors can be placed on one axis and differenced.
The fiscal-monetary gap
The gap is monetary r-star minus fiscal r-star. Combining the fiscal block (7) with the monetary block (10) and rearranging yields the gap in full, where \(\tau_t^* \equiv r_t - r_t^P\) is the spread between the real effective rate on government debt and the real policy rate:
\[ r_m^* - r_f^* = \frac{\Delta d_t}{\bar d}(1 + \bar\pi + \bar g) + \frac{1}{\phi}(\Delta\pi_t - \Phi_t) + \frac{pb_t - \overline{pb}}{\bar d}(1 + \bar\pi + \bar g) - \tau_t^* \tag{11} \]When the gap is zero, the two authorities can hit their targets at once: the rate that stabilizes inflation is also the rate that stabilizes debt. When monetary r-star rises above fiscal r-star, a positive gap opens, and equation (11) enumerates the only ways it can resolve, in practice some combination of the four:
- Debt rises. If neither authority changes stance, the debt ratio grows (first term).
- Inflation overshoots. If the central bank holds the real rate below monetary r-star to ease the fiscal burden, inflation runs above target (second term).
- Fiscal consolidation. Raising the primary balance above \(\overline{pb}\) closes the gap from the fiscal side (third term).
- The spread compresses. Lowering \(\tau_t^*\), by shortening debt maturity or through financial repression, eases the burden without either authority reaching its target (fourth term).
A phase-diagram view
The same logic reads cleanly off a phase diagram in primary-balance and real-rate space. The upward-sloping schedule traces the combinations that hold debt steady, so its height at any primary balance is fiscal r-star; it slopes up because a larger primary balance supports a higher debt-stabilizing rate. The flat schedule is monetary r-star. Their crossing is the debt-stabilizing primary balance \(pb^{DS}\). To the left of it, at an active primary balance \(\overline{pb}\), fiscal r-star sits below monetary r-star, and the vertical distance between the two schedules is the fiscal-monetary gap.
How we estimate it
Computing fiscal r-star from equation (6) requires four inputs for each country and year. The working paper estimates them as follows, over a sample of 16 advanced economies spanning 1878 to 2020, more than 140 years:
The macro-financial data come from the Jorda-Schularick-Taylor database, with fiscal series from Public Finances in Modern History (Mauro et al.). A second, forward-looking variant swaps the historical filters for five-year-ahead World Economic Outlook projections of growth, the primary balance, debt, and the policy rate. That version extends coverage to emerging markets and sidesteps the endpoint problem that filters suffer near the present, which makes it the natural template for estimates that are refreshed each data vintage.
Extensions
Open economy and foreign-currency debt
For an economy that issues part of its debt in foreign currency, fiscal r-star picks up the foreign real rate and the foreign-currency debt share \(\alpha\), with \(\zeta \equiv 1 / (1 - \alpha)\) and \(r^f\) the foreign real rate:
\[ r_f^* = \zeta \bar g + \zeta (1 + \bar g)\frac{\overline{pb}}{\bar d} - \zeta \alpha r^f \tag{A.3} \]Fiscal r-star now falls as the foreign rate rises, and by more the larger the foreign-currency share. When \(\alpha = 0\), this collapses back to equation (6). This is the version relevant to extending the measure across emerging markets.
A financial-repression proxy
The paper also notes that the same objects yield a proxy for financial repression, \((r_m^* - r^P)/(r_m^* - r_f^*)\), the position of the policy rate within the gap. When the policy rate sits closer to fiscal r-star under an elevated monetary r-star, the central bank is leaning toward debt sustainability rather than price stability.
Interpretation and limitations
The gap is a single summary statistic for fiscal-monetary tension, measured in real-rate units and comparable across countries and across the full historical sample. In the paper, larger gaps have tended to be followed by rising debt, higher inflation, currency depreciation, financial repression, and elevated crisis risk. A few limitations are worth stating plainly, and the paper is explicit about each:
- Monetary r-star is treated as exogenous for exposition, but it plausibly depends on the fiscal stance, since debt and deficits can shift the neutral rate. The assumption can be relaxed.
- The historical estimates lean on filters (HP filters and moving averages), which are noisy near the endpoints. The forward-looking projection variant is the intended remedy.
- The bubble term is set to zero in the debt-valuation equation via a transversality condition. That need not hold when \(r \lt g\).
- Fiscal r-star assumes a constant, exogenous primary balance (active policy). Under passive policy, where the primary balance adjusts to stabilize debt, the debt-stabilizing primary balance is the more natural object.