Economics & Market Microstructure
Watts Poverty Index Formula & Morduch Average Time-to-Exit Poverty Calculator
Calculate Harold Watts’s (1968) distribution-sensitive logarithmic poverty index W = (1 / N) * SUM_{y_i < z} LN(z / y_i) and Jonathan Morduch’s (1998) expected years to exit poverty t_exit = W / g.
Why Watts (1968) was the first axiomatically coherent, transfer-sensitive poverty measure — with an exact exit-time meaning!
Proposed by Harold W. Watts (1968), the Watts Poverty Index W = (1 / N) * SUM_{i: y_i < z} LN(z / y_i) satisfies monotonicity, subgroup decomposability, and the Pigou-Dalton transfer axiom (because LN(z / y_i) is strictly convex in poor incomes y_i > 0, transferring $1 from a poor household at $900 to a destitute household at $600 strictly reduces W!).
Jonathan Morduch (1998) revealed a striking policy interpretation: if all poor incomes grow continuously at annual real rate g > 0, household i needs t_i = LN(z / y_i) / g years to cross poverty line z! Thus W / g equals the population-weighted average poverty exit time, while W / (H * g) (dividing by headcount ratio H = q / N) gives the average years needed among currently poor households to escape poverty!
| Watts & Morduch Metric (Incomes [600, 750, 900, 1200, 1500], z = 1000, g = 4%/yr) | Evaluated Value | Policy Interpretation |
|---|---|---|
| Watts Poverty Index W | 0.1808 | Mean log poverty shortfall across N = 5 households |
| Population-Weighted Exit Time W / g | 4.52 years | Per-capita person-years of poverty remaining at 4% growth |
| Mean Exit Time Among Poor Households | 7.53 years | Average years for the 3 poor households to reach $1,000 |
Watts Logarithmic Poverty Index W = (1 / N) * SUM_{y_i < z} LN(z / y_i)
Evaluates LN(z / y_i) for each poor household below poverty line D2 (requiring strictly positive incomes y_i > 0) and divides by total sample size N = 5.
=IF(OR(MIN(B2:B6) <= 0, D2 <= 0), "Requires Positive Incomes & Poverty Line", ROUND((IF(B2 < D2, LN(D2 / B2), 0) + IF(B3 < D2, LN(D2 / B3), 0) + IF(B4 < D2, LN(D2 / B4), 0) + IF(B5 < D2, LN(D2 / B5), 0) + IF(B6 < D2, LN(D2 / B6), 0)) / COUNT(B2:B6), 4))Morduch Population-Weighted Poverty Exit Time t_exit = W / g (Years)
Divides the Watts Index W by continuous annual real income growth rate g > 0 in D3.
=IF(OR(MIN(B2:B6) <= 0, D2 <= 0, D3 <= 0), "Requires Positive Incomes, Line z & Growth g", ROUND(((IF(B2 < D2, LN(D2 / B2), 0) + IF(B3 < D2, LN(D2 / B3), 0) + IF(B4 < D2, LN(D2 / B4), 0) + IF(B5 < D2, LN(D2 / B5), 0) + IF(B6 < D2, LN(D2 / B6), 0)) / COUNT(B2:B6)) / D3, 2))Try the example
Use the original values or change them to test your understanding.