Public Economics & Income Distribution
Diamond-Saez Optimal Top Marginal Rate & Pareto Formula
Estimate the Pareto income tail index a = z_bar / (z_bar - z*) and compute the Diamond-Saez optimal top bracket rate tau* = (1 - g) / (1 - g + a * e).
Combine top-income Pareto thickness, behavioral elasticity, and welfare weights in one formula
Emmanuel Saez (2001) and Peter Diamond & Emmanuel Saez (2011) showed that above a top-bracket income threshold z*, upper-tail incomes closely follow a Pareto distribution with empirical thickness parameter a_par = z_bar / (z_bar - z*) > 1 (where z_bar is mean income among filers above z*; a smaller a_par means a fatter upper tail!). Given behavioral elasticity e_eti > 0 of reported income with respect to the net-of-levy retention rate (1 - tau) and marginal social welfare weight g_top in [0, 1) on top-bracket earners, the optimal asymptotic top marginal rate is tau* = (1 - g_top) / (1 - g_top + a_par * e_eti)!
Verify with top-bracket cutoff z* = 400,000 (B2), conditional mean income above cutoff z_bar = 1,000,000 (B3), elasticity of reported income e_eti = 0.25 (B4), and social welfare weight g_top = 0.10 (B5): empirical Pareto parameter is a_par = 1,000,000 / 600,000 = 1.6667 (5/3), Rawlsian / revenue-maximizing top rate (g_top = 0) is 1 / (1 + (5/3)*0.25) = 12 / 17 = 0.7059 (70.59%), and welfare-weighted optimal rate at g_top = 0.10 is 0.90 / (0.90 + 5/12) = 10.8 / 15.8 = 0.6835 (68.35%)! Double behavioral elasticity to e_eti = 0.50 and tau* drops sharply to 0.90 / (0.90 + 5/6) = 0.5192!
| Diamond-Saez Quantity (z*=400k, z_bar=1M, e=0.25) | Formula | Verified Output |
|---|---|---|
| Empirical Pareto Tail Index (a_par) | z_bar / (z_bar - z*) | 1.6667 (Thicker Tail < 2) |
| Revenue-Maximizing Rate (g_top = 0) | 1 / (1 + a_par * e_eti) | 0.7059 (70.59%) |
| Optimal Welfare-Weighted Rate (g = 0.10) | (1 - g_top) / (1 - g_top + a_par * e_eti) | 0.6835 (68.35%) |
Empirical Pareto Tail Index a_par = z_bar / (z_bar - z*) Formula
Validate conditional mean strictly above bracket threshold (z_bar > z* > 0).
=IF(OR(B2 <= 0, B3 <= B2), "Require z_bar > z* > 0", ROUND(B3 / (B3 - B2), 4))Diamond-Saez Optimal Top Marginal Rate tau* Formula
Evaluate (1 - g_top) / (1 - g_top + a_par * e_eti) using unrounded exact a_par = z_bar / (z_bar - z*) inside LET.
=LET(z_cut, B2, z_mn, B3, e_bh, B4, g_wt, B5, IF(OR(z_cut <= 0, z_mn <= z_cut, e_bh <= 0, g_wt < 0, g_wt >= 1), "Invalid Diamond-Saez Input", LET(a_pr, z_mn / (z_mn - z_cut), ROUND((1 - g_wt) / (1 - g_wt + a_pr * e_bh), 4))))Try the example
Use the original values or change them to test your understanding.