Signature
C = 1 - sum_(t=1)^T [(p_t - p_(t-1)) * (L_t + L_(t-1))]
| Inputs | Definition | Unit |
|---|---|---|
p_t | Cumulative share of the population up to and including group t, with groups ranked from poorest to richest, so that p_T equals 1 | proportion |
p_(t-1) | Cumulative share of the population up to and including group t minus 1, with p_0 equal to 0 | proportion |
L_t | Cumulative share of the payment, or of ability to pay for a Lorenz curve, held by groups 1 to t, so that L_T equals 1 | proportion |
L_(t-1) | Cumulative share of the payment, or of ability to pay, held by groups 1 to t minus 1, with L_0 equal to 0 | proportion |
C | Concentration index of a payment, or Gini coefficient of ability to pay when L_t is a Lorenz curve | index without unit, from minus 1 to 1 |
|---|
TNumber of groups, for example 5 for quintiles or 10 for deciles (count)
Function
Vertical equity in health financing: progressivity and redistribution function
Maps the distribution of health care payments and of ability to pay, across households ranked from poorest to richest, to measures of vertical equity in financing: the Kakwani index of progressivity for each source of finance, the revenue-weighted index for the whole financing mix, and the vertical redistributive effect of compulsory payments on the Gini coefficient of income. Progressivity is measured as a departure from proportionality between payments and ability to pay. Vertical equity in delivery is judged against a need norm that the analyst chooses and defends, so it has no single formula and is not covered here.
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Implementations
Excel
Grouped-data concentration index by trapezoids in one cell
For five groups, with p_0 to p_5 in A2:A7 and L_0 to L_5 in B2:B7, row 2 holding the zeros for the origin, SUMPRODUCT multiplies each group width by the sum of its two curve heights. For T groups the ranges run to row T plus 2.
=1-SUMPRODUCT(A3:A7-A2:A6,B3:B7+B2:B6)
Assumptions
Groups ranked by ability to pay with cumulative shares from zero
Groups are ordered from poorest to richest by ability to pay, the shares are cumulative proportions, and the sum starts from the origin with p_0 and L_0 equal to 0 and ends at p_T and L_T equal to 1.
Straight-line curve within each ability to pay group
The curve is a straight line between group points, as if everyone within a group held the same share. Inequality within groups therefore does not enter the index, and results from different numbers of groups are not strictly comparable.
Worked examples
Gini coefficient of income from quintile shares in the vertical equity example
Income shares of 6, 11, 16, 23 and 44 per cent give cumulative shares of 0.06, 0.17, 0.33, 0.56 and 1.00. The pair sums add to 3.24 and each quintile has width 0.2, so the Gini coefficient is one minus 0.648, or 0.352.
T = 5; p_t = [0.2,0.4,0.6,0.8,1]; p_(t-1) = [0,0.2,0.4,0.6,0.8]; L_t = [0.06,0.17,0.33,0.56,1]; L_(t-1) = [0,0.06,0.17,0.33,0.56]; C = 0.352
Concentration index of illustrative tax payments by income quintile
Tax shares of 3, 8, 14, 23 and 52 per cent give cumulative shares of 0.03, 0.11, 0.25, 0.48 and 1.00. The pair sums add to 2.74, so the concentration index is one minus 0.548, or 0.452.
T = 5; p_t = [0.2,0.4,0.6,0.8,1]; p_(t-1) = [0,0.2,0.4,0.6,0.8]; L_t = [0.03,0.11,0.25,0.48,1]; L_(t-1) = [0,0.03,0.11,0.25,0.48]; C = 0.452
Concentration index of illustrative out-of-pocket payments by income quintile
Out-of-pocket shares of 12, 15, 19, 23 and 31 per cent give cumulative shares of 0.12, 0.27, 0.46, 0.69 and 1.00. The pair sums add to 4.08, so the concentration index is one minus 0.816, or 0.184.
T = 5; p_t = [0.2,0.4,0.6,0.8,1]; p_(t-1) = [0,0.2,0.4,0.6,0.8]; L_t = [0.12,0.27,0.46,0.69,1]; L_(t-1) = [0,0.12,0.27,0.46,0.69]; C = 0.184
Common errors
Leaving out the first trapezoid from the origin
Starting the sum at the first group point instead of the origin drops the trapezoid for the poorest group. For the article's income data the pair sums then add to 3.18 and the Gini coefficient comes out as 0.364 instead of 0.352.
Entering group shares where cumulative shares are required
Using each group's own share of income as L_t, instead of the running total, gives a curve that never reaches 1. For the article's income data the result is 0.688, nearly twice the correct Gini coefficient of 0.352.
Sources
World Bank guide on the grouped-data concentration index
O'Donnell O, van Doorslaer E, Wagstaff A, Lindelow M. Analyzing Health Equity Using Household Survey Data: A Guide to Techniques and Their Implementation. Washington, DC: World Bank; 2008. Chapter 8, pages 95 to 98: the concentration index as twice the area between the concentration curve and the line of equality (equation 8.1) and the grouped-data formula of Fuller and Lury (equation 8.4), which gives the same result as the trapezoid sum for a curve joined by straight lines.
Canonical Identity
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