Economics & Market Microstructure
Foster-Greer-Thorbecke (FGT_0, FGT_1, FGT_2) Poverty Headcount, Gap & Severity Formula
Calculate the World Bank’s unified Foster-Greer-Thorbecke (1984) poverty family P_alpha = (1/N) * SUM(MAX(0, (z - y_i)/z)^alpha): Headcount Ratio P_0, Poverty Gap Index P_1, and Distribution-Sensitive Squared Severity P_2.
Why transferring $1 from the poorest household to someone just below the poverty line lowers Headcount P_0—but RAISES Squared Poverty Severity P_2!
Formulated by James Foster, Joel Greer, and Erik Thorbecke (1984, Econometrica), the FGT class P_alpha = (1 / N) * SUM_{i=1..N} (MAX(0, (z - y_i) / z))^alpha is the global standard for World Bank and UN poverty measurement because it satisfies Subgroup Additivity (national poverty equals the population-weighted average of regional poverty rates) across three complementary aversion parameters alpha!
At alpha = 0, P_0 is the Poverty Headcount Ratio q / N (fraction below poverty line z—insensitive to how deep poverty is!). At alpha = 1, P_1 is the Poverty Gap Index (average normalized shortfall (z - y_i)/z across the ENTIRE population N, so N * z * P_1 equals the exact minimum fiscal transfer budget needed to lift everyone to z!). At alpha = 2, Squared Poverty Severity P_2 weights each shortfall by itself—satisfying Amartya Sen’s Pigou-Dalton Transfer Axiom so regressive transfers among the poor strictly INCREASE P_2!
| FGT Parameter alpha (Poverty Line z = $20, y = [8, 14, 20, 28, 45]) | Exact FGT Formula | Value & Economic Meaning |
|---|---|---|
| Headcount Ratio P_0 (alpha = 0) | COUNTIF(B2:B6, "<" & E2) / COUNT(B2:B6) | 0.4000 (2 of 5 = 40.0% Below Line z) |
| Poverty Gap Index P_1 (alpha = 1) | Average of [(20-8)/20, (20-14)/20, 0, 0, 0] | 0.1800 ((0.60 + 0.30) / 5 -> $18 Total Gap) |
| Squared Poverty Severity P_2 (alpha = 2) | Average of [0.60^2, 0.30^2, 0, 0, 0] | 0.0900 ((0.36 + 0.09) / 5) |
Poverty Gap Index FGT P_1 = SUMIFS(z - y_i) / (N * z) = (q * z - SUM_poor) / (N * z)
Computes exact FGT_1 without helper columns using COUNTIF and SUMIF against poverty line E2.
=IF(OR(E2 <= 0, COUNT(B2:B6) = 0, MIN(B2:B6) < 0), "Invalid Poverty Line or Income Vector", ROUND((COUNTIF(B2:B6, "<" & E2) * E2 - SUMIF(B2:B6, "<" & E2)) / (COUNT(B2:B6) * E2), 4))General Foster-Greer-Thorbecke Index P_alpha Across Normalized Shortfalls C2:C6
Evaluates P_alpha = AVERAGE(shortfall^alpha) for any aversion parameter alpha >= 0 in E3 (alpha=0, 1, 2, ...).
=IF(OR(E2 <= 0, E3 < 0, COUNT(B2:B6) = 0), "Invalid FGT Inputs", IF(E3 = 0, ROUND(COUNTIF(B2:B6, "<" & E2) / COUNT(B2:B6), 4), ROUND(SUMPRODUCT(C2:C6 ^ E3) / COUNT(B2:B6), 4)))Try the example
Use the original values or change them to test your understanding.