Economics & Market Microstructure
Kakwani Tax Progressivity & Reynolds-Smolensky Redistribution Index Formula
Decompose how much a tax system reduces income inequality using Nanak Kakwani’s Progressivity Index K = C_T - G_X and the Reynolds-Smolensky Net Redistribution identity RS = G_X - G_Y = (t / (1 - t)) * K - D_rerank.
A highly progressive tax with a tiny average tax rate reduces Gini inequality LESS than a moderately progressive broad tax!
Formulated by Nanak C. Kakwani (1977, Economic Journal) and Morgan Reynolds & Eugene Smolensky (1977), modern fiscal-incidence analysis separates Tax Progressivity (departure from proportionality) from Redistributive Capacity (actual reduction in Gini inequality)! If pre-tax income has Gini coefficient G_X and taxes ordered by pre-tax income have Concentration Index C_T, Kakwani’s Index is K = C_T - G_X: K > 0 indicates a progressive tax (taxes are more concentrated at the top than pre-tax income), K = 0 is flat/proportional, and K < 0 is regressive (such as an unrebated consumption excise)!
Crucially, the Reynolds-Smolensky net Gini reduction RS = G_X - G_Y obeys the exact Kakwani-Lambert decomposition: RS = (t / (1 - t)) * K - D_rerank, where t is the aggregate average tax rate (Total Tax / Total Pre-Tax Income) and D_rerank >= 0 is Atkinson-Plotnick horizontal reranking! At G_X = 0.42, C_T = 0.60 (K = +0.18 progressive), t = 0.25, and zero reranking, vertical equity reduces the post-tax Gini by (0.25 / 0.75) * 0.18 = 0.0600 (to G_Y = 0.3600)!
| Fiscal Incidence Regime (Pre-Tax Gini G_X = 0.4200) | Kakwani Progressivity K = C_T - G_X | Post-Tax Gini Drop RS = (t/(1-t))*K - D |
|---|---|---|
| Broad Progressive Income Tax (C_T=0.60, t=25%, D=0.002) | +0.1800 (Progressive) | +0.0580 Gini drop (G_Y = 0.3620) |
| Narrow Luxury Surtax (C_T=0.82, t=2%, D=0) | +0.4000 (Steeply Progressive) | +0.0082 Gini drop (Tiny tax base!) |
| Regressive Excise Tax (C_T=0.30, t=12%, D=0) | -0.1200 (Regressive!) | -0.0164 (Raises Gini to 0.4364!) |
Kakwani Tax Progressivity Index K = C_T - G_X
Subtracts pre-tax Gini G_X (B2) from tax concentration index C_T (B3).
=IF(OR(B2 < 0, B2 > 1, B3 < -1, B3 > 1), "Invalid Gini or Concentration Index", ROUND(B3 - B2, 4))Reynolds-Smolensky Net Gini Reduction RS = (t / (1 - t)) * (C_T - G_X) - D_rerank
Computes net Gini inequality reduction G_X - G_Y from Kakwani progressivity, average tax rate t (B4), and horizontal reranking D (B5).
=IF(OR(B2 < 0, B2 > 1, B3 < -1, B3 > 1, B4 <= 0, B4 >= 1, B5 < 0), "Invalid Fiscal Incidence Parameters", ROUND((B4 / (1 - B4)) * (B3 - B2) - B5, 4))Try the example
Use the original values or change them to test your understanding.