Monetary Economics & Sovereign Risk
Cagan Seigniorage Revenue & Peak Inflation Rate Formula
Calculate steady-state real seigniorage (pi + g_Y) * m_0 * EXP(-alpha * pi), Cagan’s revenue-maximizing inflation rate pi* = 1/alpha - g_Y, and maximum monetizable deficit.
Pinpoint the exact monetization ceiling beyond which money printing triggers hyperinflation
Under Phillip Cagan’s (1956) semi-logarithmic money-demand schedule M / (P * Y) = m_0 * EXP(-alpha * pi) (where m_0 > 0 is zero-inflation base-money-to-GDP ratio and alpha > 0 is inflation semi-elasticity), steady-state real seigniorage revenue (% of GDP) at real GDP growth g_Y >= 0 and inflation pi is S(pi) = (pi + g_Y) * m_0 * EXP(-alpha * pi). Differentiating log seigniorage sets 1 / (pi* + g_Y) = alpha, yielding Cagan’s exact seigniorage-maximizing inflation rate pi_max* = (1 / alpha) - g_Y and maximum sustainable monetization ceiling S_max* = (m_0 / alpha) * EXP(-(1 - alpha * g_Y))!
Verify on base money m_0 = 0.25 (B2, 25% of GDP), semi-elasticity alpha = 2.0 (B3), real GDP growth g_Y = 0.02 (B4, 2.0%/yr), and actual inflation pi = 0.20 (B5, 20%/yr): current seigniorage is (0.20 + 0.02) * 0.25 * EXP(-0.40) = 0.0369 (3.69% of GDP), peak-revenue inflation sits at pi_max* = 0.50 - 0.02 = 0.4800 (48.00%/yr!), and maximum possible steady-state seigniorage is (0.25 / 2.0) * EXP(-0.96) = 0.0479 (4.79% of GDP — proving that attempting to money-finance a 6.0% fiscal deficit has NO steady-state solution and spirals into Cagan hyperinflation!).
| Cagan Monetization Metric (m_0=25%, alpha=2.0, g_Y=2%) | Formula | Verified Output |
|---|---|---|
| Current Seigniorage at pi = 20% | (pi + g_Y) * m_0 * EXP(-alpha * pi) | 0.0369 (3.69% of GDP) |
| Revenue-Maximizing Inflation (pi*) | (1 / alpha) - g_Y | 0.4800 (48.00%/yr) |
| Maximum Monetizable Deficit (S_max*) | (m_0 / alpha) * EXP(-(1 - alpha * g_Y)) | 0.0479 (4.79% of GDP) |
| Wrong-Side Seigniorage at pi = 100% | 1.02 * 0.25 * EXP(-2.0) | 0.0345 (< S(20%)!) |
Steady-State Real Seigniorage Revenue (% of GDP) Formula
Validate m_0 > 0, semi-elasticity alpha > 0, real growth g_Y >= 0, and non-negative nominal money growth pi + g_Y >= 0.
=LET(m_bs, B2, alp_c, B3, g_rl, B4, pi_in, B5, IF(OR(m_bs <= 0, alp_c <= 0, g_rl < 0, pi_in + g_rl < 0), "Invalid Cagan Input", ROUND((pi_in + g_rl) * m_bs * EXP(-alp_c * pi_in), 4)))Maximum Sustainable Seigniorage Ceiling S_max* Formula
Compare any proposed money-financed fiscal deficit against S_max* to detect hyperinflationary fiscal dominance.
=LET(m_bs, B2, alp_c, B3, g_rl, B4, IF(OR(m_bs <= 0, alp_c <= 0, g_rl < 0), "Invalid Cagan Input", ROUND((m_bs / alp_c) * EXP(-(1 - alp_c * g_rl)), 4)))Try the example
Use the original values or change them to test your understanding.