Macroeconomics & Household Finance
Modigliani Life-Cycle Saving & Wealth-Income Formula
Calculate Franco Modigliani’s Life-Cycle Hypothesis (LCH) smoothed annual consumption C = (W_0 + N*Y)/(N + M), active-worker saving rate, and retirement peak wealth-to-income ratio.
Understand hump-shaped wealth accumulation across working and retirement years!
Franco Modigliani and Richard Brumberg (1954) reconciled short-run Keynesian consumption slopes with long-run constant saving shares through the Life-Cycle Hypothesis (LCH)! An agent entering the planning horizon with Initial_Net_Wealth W_0, Constant_Real_Annual_Labor_Income Y earned over Remaining_Working_Years N, and Expected_Retirement_Span M (Total Remaining Lifespan L = N + M, under zero real interest / constant consumption smoothing) commands Lifetime Resources Omega = W_0 + N * Y and smooths Annual Consumption at C = (W_0 + N * Y) / (N + M) = (1 / L) * W_0 + (N / L) * Y!
Verify on Initial_Net_Wealth W_0 = 30,000 (B2), Annual_Labor_Income Y = 80,000 (B3), Remaining_Working_Years N = 30 (B4), and Expected_Retirement_Years M = 15 (B5 -> Lifespan L = 45 years): Lifetime Resources equal 30,000 + 30 * 80,000 = 2,430,000; Smoothed Annual Consumption is 2,430,000 / 45 = 54,000; Active-Worker Annual Saving is Y - C = 80,000 - 54,000 = 26,000 (Active Saving Rate s = 26,000 / 80,000 = 0.3250, or 32.50% — below the zero-initial-wealth benchmark M / (N + M) = 15 / 45 = 33.33% because existing wealth W_0 finances 30,000 / 45 = 666.67/yr of consumption!); and Hump-Peak Retirement Wealth W_ret = M * C = 15 * 54,000 = 810,000 (Wealth-to-Income Ratio W_ret / Y = 10.1250x)! Extend longevity to M = 20 retirement years (L = 50): Consumption drops to 48,600 and Active Saving Rate rises to 0.3925 (39.25%)!
| Modigliani LCH Metric (W_0=30k, Y=80k, N=30, M=15) | Formula | Verified Output |
|---|---|---|
| Smoothed Annual Consumption (C) | (W_0 + N * Y) / (N + M) | 54,000.00 / yr |
| Marginal Propensity to Consume Income (MPC_Y) | N / (N + M) | 0.6667 (MPC_W = 0.0222) |
| Active-Worker Saving Rate (s = (Y - C) / Y) | (Y - C) / Y | 0.3250 (32.50% of Income) |
| Peak Retirement Wealth-to-Income (W_ret / Y) | (M * C) / Y | 10.1250x Annual Income |
Modigliani-Brumberg Smoothed Annual Consumption Formula
Notice why temporary windfalls (Delta W_0) have tiny annual MPC_W = 1 / (N + M), whereas permanent salary raises (Delta Y) move consumption by N / (N + M)! Before accepting a peer’s suggested retirement-gap edit in a shared Arte Sheets household financial plan, test M = 20 in cell B5 to verify longevity sensitivity.
=LET(w_in, B2, y_lb, B3, n_wk, B4, m_rt, B5, IF(OR(y_lb <= 0, n_wk <= 0, m_rt < 0, w_in + n_wk * y_lb <= 0), "Invalid Life-Cycle Input", ROUND((w_in + n_wk * y_lb) / (n_wk + m_rt), 2)))Active-Worker Saving Rate & Peak Retirement Wealth Ratio Formula
Validate Y > 0, N > 0, M >= 0, and positive lifetime resources W_0 + N * Y > 0.
=IF(ISTEXT(E2), E2, ROUND((B3 - E2) / B3, 4))Try the example
Use the original values or change them to test your understanding.