Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Vertical equity in health financing: progressivity and redistribution function

f(L_X, L_P, w_j, g) = (pi_K, pi_total, V)

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.

  • Kakwani index of progressivity for a health financing source

    pi_K = C_P - G_X

    Measures how far one source of health finance departs from proportionality to ability to pay, as the concentration index of the payment minus the Gini coefficient of ability to pay. It equals twice the area between the Lorenz curve of ability to pay and the concentration curve of the payment. A positive value indicates a progressive source, a negative value a regressive one, and zero is consistent with proportionality.

  • Grouped-data concentration index by trapezoids for vertical equity analysis

    C = 1 - sum_(t=1)^T [(p_t - p_(t-1)) * (L_t + L_(t-1))]

    Computes a concentration index, or a Gini coefficient when the curve is a Lorenz curve, from T groups ranked from poorest to richest. The curve joins the group points with straight lines, so the area under it is a sum of trapezoids and the index is one minus twice that area. The result is the same as the grouped-data formula of Fuller and Lury given in the World Bank guide.

  • Revenue-weighted Kakwani index for a health financing mix

    pi_total = sum_(j=1)^J [w_j * pi_j]

    Measures the progressivity of a whole health financing system as the average of the Kakwani indices of its sources, each weighted by that source's share of total health payments. Overall progressivity therefore depends both on how progressive each source is and on how much revenue it raises.

  • Vertical redistributive effect of compulsory health payments

    V = g / (1 - g) * K_E

    Gives the fall in the Gini coefficient of income that a compulsory health payment would produce through its progressivity alone, as the Kakwani index scaled by the average payment rate. It is the vertical term V in the decomposition of the redistributive effect into vertical redistribution, horizontal inequity and reranking. A larger payment rate magnifies the redistribution from a given degree of progressivity.