Public Economics & Actuarial Pensions
Aaron-Samuelson PAYGO vs Funded Pension Formula
Compare Pay-As-You-Go social security’s Samuelson biological rate of return g_bio = (1 + n)*(1 + g_w) - 1 against funded capital return r and compute equilibrium replacement rates.
Understand Henry Aaron’s (1966) Social Insurance Paradox and demographic aging!
Paul Samuelson (1958) and Henry Aaron (1966) proved the foundational theorem of pension economics in Overlapping-Generations (OLG) models! Under an unfunded Pay-As-You-Go (PAYGO) pension collecting payroll contribution rate tau from L_t = (1 + n) * L_{t-1} active workers earning real wage w_t = (1 + g_w) * w_{t-1} to pay L_{t-1} retirees, each generation’s contributions tau * w_{t-1} * L_{t-1} return Retiree Benefit tau * w_t * L_t = tau * w_{t-1} * L_{t-1} * (1 + n) * (1 + g_w) — earning Samuelson’s Biological Rate of Return g_bio = (1 + n) * (1 + g_w) - 1 (compound labor-force growth plus real wage growth)! Steady-state PAYGO beats a Fully Funded Capital Account earning real return r iff g_bio > r (dynamic inefficiency), whereas in dynamically efficient aging economies (r > g_bio), PAYGO delivers lower steady-state benefits by factor (1 + g_bio) / (1 + r) and achieves Equilibrium Replacement Rate (Benefit / Current_Wage) = tau * (1 + n) (= tau / Old_Age_Dependency_Ratio)!
Verify over a 30-year generational horizon: Generation_Labor_Growth n = 0.10 (+10% cohort growth, B2 -> Old-Age Dependency Ratio 1 / 1.10 = 0.9091), Generation_Real_Wage_Growth g_w = 0.50 (+50% over 30 yr, B3), Generation_Funded_Capital_Return r = 1.20 (+120% real compounding over 30 yr, B4), and Contribution_Rate tau = 0.15 (15.0%, B5): Samuelson Biological Return is (1.10 * 1.50) - 1 = 0.6500 (+65.00% over 30 yr); Funded-to-PAYGO Benefit Multiplier is (1 + 1.20) / (1 + 0.65) = 2.20 / 1.65 = 1.3333x (33.33% higher steady-state benefit if pre-funded, ignoring transition windfall debt!); and Current-Wage PAYGO Replacement Rate is 0.15 * 1.10 = 0.1650 (16.50% of active wage)!
| OLG Pension Metric (n=+10%, g_w=+50%, r=+120%, tau=15%) | Formula | Verified Output |
|---|---|---|
| Samuelson Biological Rate of Return (g_bio) | (1 + n) * (1 + g_w) - 1 | 0.6500 (+65.0% / generation) |
| PAYGO Steady-State Replacement Rate | tau * (1 + n) | 0.1650 (16.5% of Current Wage) |
| Funded-Over-PAYGO Benefit Ratio | (1 + r) / ((1 + n) * (1 + g_w)) | 1.3333x (r > g_bio) |
| Aaron Condition Regime Verdict | IF(r > g_bio, "Funded Dominates...", ...) | Dynamically Efficient (r > g_bio) |
Samuelson (1958) Biological Rate of Return g_bio Formula
Include interaction term n * g_w via exact product (1 + n) * (1 + g_w) - 1 — over 30-year generations, linear n + g_w misses +0.0500 (500 bps)!
=IF(OR(B2 <= -1, B3 <= -1), "Invalid OLG Input", ROUND((1 + B2) * (1 + B3) - 1, 4))Aaron (1966) Funded-to-PAYGO Steady-State Benefit Ratio Formula
Validate 1 + n > 0, 1 + g_w > 0, and 1 + r > 0.
=LET(n_lb, B2, gw_r, B3, r_cp, B4, IF(OR(n_lb <= -1, gw_r <= -1, r_cp <= -1), "Invalid OLG Input", ROUND((1 + r_cp) / ((1 + n_lb) * (1 + gw_r)), 4)))Try the example
Use the original values or change them to test your understanding.