Development Economics & Welfare Indices
Alkire-Foster Multidimensional Poverty (MPI) Formula
Implement Sabina Alkire and James Foster’s (2011) dual-cutoff censoring c_i(k) = IF(c_i >= k, c_i, 0) to calculate Multidimensional Headcount H, Deprivation Intensity A, and MPI = H * A.
Censor non-poor single deprivations before computing poverty intensity A and MPI = H * A!
Sabina Alkire and James Foster (2011) designed the Oxford/UNDP Multidimensional Poverty Index (MPI) around a rigorous Dual-Cutoff Identification & Aggregation rule! First, each household i accumulates a weighted Deprivation Score c_i = SUM_j(w_j * I_ij) in [0, 1] across health, schooling, and living-standard indicators (SUM(w_j) = 1). Second, applying Poverty Cutoff k in (0, 1] (UNDP standard k = 1/3 = 0.3333) identifies household i as multidimensionally poor iff c_i >= k, and CENSORS non-poor households’ scores to zero: c_i(k) = IF(c_i >= k, c_i, 0)! That yields Censored Headcount Ratio H = q / N (share of households with c_i >= k), Average Deprivation Intensity among the poor A = SUM(c_i(k)) / q, and Adjusted Headcount Ratio MPI = M_0 = H * A = AVERAGE(c_i(k)) — guaranteeing Dimensional Monotonicity (if a poor household suffers an ADDITIONAL deprivation, A rises so MPI rises, unlike crude headcount H!)!
Verify on a 5-household survey (Rows 2..6) with Poverty_Cutoff k = 0.3333 (G2) and Weighted Deprivation Scores c_i = [0.60, 0.50, 0.40, 0.20, 0.00]: Households 1..3 satisfy c_i >= 0.3333 (q = 3 poor out of N = 5 -> Headcount H = 3 / 5 = 0.6000), while Household 4’s isolated deprivation 0.20 < 0.3333 is censored to 0.00! Censored Scores c_i(k) = [0.60, 0.50, 0.40, 0.00, 0.00] yield Intensity among the Poor A = (0.60 + 0.50 + 0.40) / 3 = 0.5000 (50.0% average Simultaneous Deprivations), and exact MPI = H * A = 0.6000 * 0.5000 = 0.3000 (= AVERAGE(C2:C6) = 1.50 / 5)!
| Alkire-Foster Metric (c = [0.60, 0.50, 0.40, 0.20, 0.00], k = 0.3333) | Formula | Verified Output |
|---|---|---|
| Multidimensional Headcount Ratio (H) | COUNTIF(B2:B6, ">=" & G2) / ROWS(B2:B6) | 0.6000 (3 of 5 Poor) |
| Censored Score Sum SUM(c_i(k)) | SUM(C2:C6) = 0.60 + 0.50 + 0.40 | 1.5000 (0.20 Censored to 0) |
| Average Deprivation Intensity (A) | SUM(C2:C6) / Poor_Count | 0.5000 (50.0% Intensity) |
| Multidimensional Poverty Index (MPI) | AVERAGE(C2:C6) = H * A | 0.3000 |
Row-Level Dual-Cutoff Censored Deprivation Score c_i(k) Formula
Forgetting to censor c_i < k (computing uncensored average (0.60+0.50+0.40+0.20+0)/5 = 0.3400 instead of 0.3000) violates Alkire-Foster Poverty Focus!
=IF(B2 >= $G$2, B2, 0)Deprivation Intensity A & Adjusted Headcount MPI = H * A Formula
Test Dimensional Monotonicity: raise poor Household 1’s deprivation score B2 from 0.60 to 0.80 -> Headcount H stays 0.6000, but Intensity A rises to 0.5667 and MPI rises from 0.3000 to 0.3400!
=IF(OR(G2 <= 0, G2 > 1, MIN(B2:B6) < 0, MAX(B2:B6) > 1), "Invalid MPI Input", ROUND(AVERAGE(C2:C6), 4))Try the example
Use the original values or change them to test your understanding.