Macroeconomics & Sovereign Fiscal Analysis
Debt Sustainability (r - g) & Stabilizing Primary Balance
Project public debt-to-GDP d_t = ((1 + r)/(1 + g)) * d_{t-1} - pb_t and calculate the exact debt-stabilizing primary balance pb* = ((r - g)/(1 + g)) * d_{t-1}.
Audit IMF / Blanchard debt snowball dynamics using exact discrete (1 + r) / (1 + g)
In sovereign Debt Sustainability Analysis (DSA), prior-year public debt-to-GDP ratio d_{t-1} evolves as d_t = ((1 + r) / (1 + g)) * d_{t-1} - pb_t, where r is the effective interest rate on public debt, g is GDP growth (both nominal OR both real — never mix nominal r with real g!), and pb_t is the primary fiscal balance (% of GDP, revenue minus non-interest spending). Setting d_t = d_{t-1} solves the exact Debt-Stabilizing Primary Balance: pb* = ((r - g) / (1 + g)) * d_{t-1}! Notice why Continuous-Time approximations (r - g) * d_{t-1} miss the (1 + g) denominator and misstate required fiscal consolidation when nominal growth g is high!
Verify on initial debt d_{t-1} = 0.90 (B2, 90% of GDP), effective nominal bond rate r = 0.045 (B3, 4.5%), nominal GDP growth g = 0.025 (B4, 2.5%), and actual primary surplus pb_t = 0.005 (B5, +0.50% of GDP): stabilizing debt at 90% requires pb* = ((0.045 - 0.025) / 1.025) * 0.90 = 0.0176 (+1.76% of GDP!). Because actual pb_t = +0.50% falls 1.26 pp short, debt rises to (1.045 / 1.025) * 0.90 - 0.005 = 0.9126 (91.26% of GDP). Flip growth to g = 0.055 > r = 0.045 (Blanchard’s negative r - g regime!) and pb* drops to -0.0085 — stabilizing debt even with a -0.85% primary deficit!
| DSA Regime (d_{t-1} = 90%, r = 4.5%, pb_t = +0.50%) | Stabilizing Primary Balance pb* | Next-Year Debt d_t |
|---|---|---|
| Positive Snowball (g = 2.5%, r > g) | +0.0176 (+1.76% Surplus Needed) | 0.9126 (91.26%, Rising) |
| Neutral Growth (g = 4.5%, r = g) | 0.0000 (Balanced Primary) | 0.8950 (89.50%, Falling) |
| Favorable Growth (g = 5.5%, r < g) | -0.0085 (-0.85% Deficit Allowed) | 0.8865 (88.65%, Falling) |
Exact Debt-Stabilizing Primary Balance ((r - g) / (1 + g)) * d_{t-1} Formula
Validate d_{t-1} >= 0, r > -1, and GDP growth g > -1.
=LET(d_pr, B2, r_eff, B3, g_gdp, B4, IF(OR(d_pr < 0, r_eff <= -1, g_gdp <= -1), "Invalid DSA Input", ROUND(((r_eff - g_gdp) / (1 + g_gdp)) * d_pr, 4)))Next-Year Debt-to-GDP Ratio d_t and Fiscal Effort Gap (pb* - pb_t) Formula
Project exact next-year debt ratio d_t and required fiscal adjustment pb* - pb_t.
=LET(d_pr, B2, r_eff, B3, g_gdp, B4, pb_ac, B5, IF(OR(d_pr < 0, r_eff <= -1, g_gdp <= -1), "Invalid DSA Input", ROUND(((1 + r_eff) / (1 + g_gdp)) * d_pr - pb_ac, 4)))Try the example
Use the original values or change them to test your understanding.