Debt-Stabilizing Primary Balance & (r - g) Formula

Understand why (r - g) < 0 lets debt/GDP shrink even with small primary deficits — and flips violently when r > g!

Sovereign debt sustainability hinges on the exact Law of Motion of the Debt-to-GDP Ratio d_t = D_t / Y_t! Given Nominal Effective Interest Rate r (or Real Rate r_real) and Nominal GDP Growth Rate g (or Real Growth g_real, provided BOTH use the same nominal or real basis because inflation (1 + pi) cancels identically in (1 + i) / (1 + g_nom)!), Prior Debt Ratio d_{t-1}, and Primary Budget Balance Ratio pb_t = (Revenues - NonInterest_Spending) / GDP: Next-Year Debt-to-GDP is d_t = ((1 + r) / (1 + g)) * d_{t-1} - pb_t! Subtracting d_{t-1} isolates the Automatic Interest-Growth Snowball Term ((r - g) / (1 + g)) * d_{t-1} and Debt-Stabilizing Primary Balance pb* = ((r - g) / (1 + g)) * d_{t-1}!

Verify on Prior_Debt_GDP d_0 = 0.9000 (90.0% of GDP, B2), Nominal_Effective_Interest_Rate r = 0.0450 (4.50%, B3), Nominal_GDP_Growth g = 0.0250 (2.50%, B4), and Actual_Primary_Balance pb_t = -0.0100 (-1.00% primary deficit, B5): Automatic Snowball / Required Debt-Stabilizing Primary Surplus is ((0.045 - 0.025) / 1.025) * 0.90 = (0.02 / 1.025) * 0.90 = +0.0176 (+1.76% of GDP surplus needed just to hold debt flat at 90%!), and Actual End-of-Year Debt-to-GDP climbs to (1.045 / 1.025) * 0.90 - (-0.0100) = 0.9176 + 0.0100 = 0.9276 (92.76% of GDP, +2.76 percentage points jump)! Flip growth above interest (g = 0.0550 > r = 0.0450): pb* turns negative (-0.0085), stabilizing debt even at a 0.85% primary deficit!

Exact IMF Debt-Stabilizing Primary Balance pb* Formula

Never drop denominator (1 + g): during high nominal growth or inflation spikes, linearized (r - g) * d_0 overstates the snowball by factor (1 + g)!

=LET(d_pr, B2, r_ef, B3, g_ym, B4, IF(OR(d_pr < 0, r_ef <= -1, g_ym <= -1), "Invalid Debt-Dynamics Input", ROUND(((r_ef - g_ym) / (1 + g_ym)) * d_pr, 4)))

Next-Period Public Debt-to-GDP Ratio d_t Formula

Remember sign convention: pb_t > 0 is a primary surplus (reduces debt); pb_t < 0 is a primary deficit (adds to debt).

=LET(d_pr, B2, r_ef, B3, g_ym, B4, pb_t, B5, IF(OR(d_pr < 0, r_ef <= -1, g_ym <= -1), "Invalid Debt-Dynamics Input", ROUND(((1 + r_ef) / (1 + g_ym)) * d_pr - pb_t, 4)))

Further reading

• Arte Sheets Lesson — Harrod-Domar ICOR Growth & Financing Gap Formula
• Arte Sheets Lesson — Krugman Currency Crisis & Shadow Exchange Rate Formula