Reynolds-Smolensky

Reynolds-Smolensky Index of Redistributive Effect

The Reynolds-Smolensky index is a summary number for how far payments towards health care, or any tax or transfer, make the distribution of income more or less equal. Health economists use it alongside the Kakwani index, which measures how progressive a payment is, because a payment can be strongly progressive and still redistribute very little if it raises little money. In equity analysis of health system financing it shows whether compulsory payments for health care narrow or widen income inequality, and by how much. The name is used in the literature for two closely related quantities, and the gap between them is the reranking of households that payments can cause.

Two quantities that carry the same name

The usual starting point is the redistributive effect. The World Bank guide to analysing health equity by O'Donnell, van Doorslaer, Wagstaff and Lindelow, following Lambert, defines the redistributive impact of a compulsory payment as the reduction in the Gini coefficient that the payment causes.

$$RE = G_X - G_{X-P}$$

where $RE$ is the redistributive effect, $G_X$ is the Gini coefficient of prepayment income (or another measure of ability to pay) and $G_{X-P}$ is the Gini coefficient of postpayment income, with households ranked by postpayment income.

Urban's 2009 review of the measurement literature distinguishes this from the index of redistributive effect proposed by Reynolds and Smolensky in 1977. Their index replaces the postpayment Gini coefficient with the concentration coefficient of postpayment income, for which households keep their prepayment order.

$$RS = G_X - C_{X-P}$$

where $RS$ is the Reynolds-Smolensky index and $C_{X-P}$ is the concentration index of postpayment income with households ranked by prepayment income. The concentration curve of postpayment income never lies below its Lorenz curve, so $C_{X-P}$ can never exceed $G_{X-P}$ and the index can never be smaller than the redistributive effect.

Applied health studies use the name for both. Sanwald and Theurl's 2015 study of out-of-pocket payments in Austria computes the index in one table from the Kakwani index and the payment share, and in another states that the Gini difference is captured by it, so published values need checking for the formula used.

Reranking and the link with the Kakwani index

The two quantities differ by the reranking term that Atkinson and Plotnick introduced. Urban notes that Atkinson saw the concentration coefficient of postpayment income as understating true postpayment inequality, so the measure using it overstates the redistributive effect whenever payments change the order of households.

$$RE = RS - R, \qquad R = G_{X-P} - C_{X-P}$$

where $R$ is the Atkinson-Plotnick reranking index, which is zero when no household changes position and positive otherwise, and the other symbols are as above.

The Reynolds-Smolensky index itself can be written in terms of progressivity and scale. Urban sets out the identity, which goes back to Kakwani, that the vertical effect of a payment equals its Kakwani index scaled by the average payment rate and equals the prepayment Gini coefficient minus the postpayment concentration coefficient.

$$RS = \frac{g}{1-g} K$$

where $g$ is the average payment rate, total payments divided by total prepayment income, and $K$ is the Kakwani index, the concentration index of payments minus $G_X$, with both ranked by prepayment income. A proportional payment has $K$ equal to zero and so leaves the index at zero whatever its size. A progressive payment gives a positive index and a regressive one a negative index, and with $K$ fixed the index grows slightly faster than the payment rate.

Decomposing redistribution in health care finance

Health financing studies often go one step further and split the reranking term. The World Bank guide sets out the decomposition by Aronson, Johnson and Lambert, which separates unequal treatment of equals from reranking.

$$RE = V - H - R$$

where $V$ is vertical redistribution, $H$ is horizontal inequity among households with the same prepayment income and $R$ is reranking. The guide defines $V$ as the $g/(1-g)$ scaling of the Kakwani index that would arise under horizontal equity. Urban shows that with exact groups of equals the reranking term is the same in both decompositions, so the Reynolds-Smolensky index equals $V$ minus $H$. The horizontal equity and vertical equity pages cover the two principles.

The guide illustrates the decomposition with van Doorslaer and colleagues' 1999 study of twelve OECD countries. In the Netherlands in 1992, compulsory public payments for health care took 8.21 per cent of income with a Kakwani index, assuming horizontal equity, of minus 0.0999, giving vertical redistribution of 0.0821 divided by 0.9179, multiplied by minus 0.0999, or minus 0.0089. The total redistributive effect was minus 0.0096, so public finance widened income inequality, and the guide reports that horizontal inequity and reranking added a further 6.6 per cent to that regressive redistribution. The abstract of the 1999 study reports that private sources generally had larger negative redistributive effects, caused to a substantial degree by differential treatment.

Estimating the index from household survey data

The World Bank guide estimates every Gini coefficient and concentration index from household records by the covariance method, using each household's weighted fractional rank. The reranking term needs a concentration index with households ranked by prepayment group and then by postpayment income within each group. For taxes, the guide calls for only the part of taxation that funds health care to be counted, by scaling all tax payments by tax-funded health spending as a proportion of total general government spending on goods and services.

The measure of ability to pay matters too. In the Austrian study, deducting the benefit level of the means-tested income maintenance programme from household net income doubled the Gini coefficient and made the regressive effect of out-of-pocket payments much larger.

Worked example with five households and one reranking

All figures in this example are illustrative. Five equally weighted households have prepayment incomes of GBP 10,000, GBP 15,000, GBP 16,000, GBP 30,000 and GBP 49,000, with mean GBP 24,000 and total GBP 120,000. A compulsory health contribution takes 3, 3, 10, 8 and 10 per cent of income, so payments are GBP 300, GBP 450, GBP 1,600, GBP 2,400 and GBP 4,900, totalling GBP 9,650 with mean GBP 1,930. The third household's higher rate stands for a contribution charged on a different base.

1. Rank the households. With equal weights the fractional ranks are 0.1, 0.3, 0.5, 0.7 and 0.9, so the deviations from their mean of 0.5 are minus 0.4, minus 0.2, 0, 0.2 and 0.4.

2. Gini coefficient of prepayment income. Income deviations from the mean are minus 14,000, minus 9,000, minus 8,000, 6,000 and 25,000. Multiplied by the rank deviations they give 5,600, 1,800, 0, 1,200 and 10,000, which sum to 18,600, so the covariance is 3,720.

$$G_X = \frac{2 \times 3{,}720}{24{,}000} = 0.3100$$

where $G_X$ is the Gini coefficient of prepayment income.

3. Kakwani index of the payments. Payment deviations are minus 1,630, minus 1,480, minus 330, 470 and 2,970, and the products with the rank deviations are 652, 296, 0, 94 and 1,188, which sum to 2,230, a covariance of 446. The concentration index of payments is 892 divided by 1,930, or 0.4622.

$$K = 0.4622 - 0.3100 = 0.1522$$

where $K$ is the Kakwani index, positive because payments are more concentrated among richer households than income is.

4. Reynolds-Smolensky index. The average payment rate is 9,650 divided by 120,000, or 0.0804, so the identity gives the index.

$$RS = \frac{0.0804}{0.9196} \times 0.1522 = 0.0133$$

where $RS$ is the Reynolds-Smolensky index. As a check, postpayment incomes are GBP 9,700, GBP 14,550, GBP 14,400, GBP 27,600 and GBP 44,100, with mean GBP 22,070. Kept in prepayment order, their deviations multiplied by the rank deviations give 4,948, 1,504, 0, 1,106 and 8,812, which sum to 16,370, a covariance of 3,274, so the concentration index is 6,548 divided by 22,070, or 0.2967, and 0.3100 minus 0.2967 is 0.0133.

5. Redistributive effect and reranking. After payment the third household, at GBP 14,400, falls below the second, at GBP 14,550. Re-ranked by postpayment income, the third household takes rank deviation minus 0.2 and the second takes 0, so the products become 4,948, 1,534, 0, 1,106 and 8,812, which sum to 16,400. The covariance is 3,280 and the postpayment Gini coefficient is 6,560 divided by 22,070, or 0.2972.

$$RE = 0.3100 - 0.2972 = 0.0128$$

where $RE$ is the redistributive effect. Reranking is 0.2972 minus 0.2967, or 0.0005, and 0.0133 minus 0.0005 gives the same 0.0128. About 4 per cent of the redistribution implied by progressivity and scale is lost because one household overtakes another.

6. Halve the contribution. If every rate were halved, payments would total GBP 4,825 and the concentration index of payments would be unchanged, since it does not depend on scale, so the Kakwani index stays at 0.1522. The average rate falls to 0.0402, and the index falls to 0.0402 divided by 0.9598, multiplied by 0.1522, or 0.0064. Postpayment incomes of GBP 14,775 and GBP 15,200 for the second and third households keep their order, so there is no reranking and the redistributive effect is also 0.0064.

7. Interpretation. The contribution reduces the income Gini coefficient by about 0.013 points. Equally progressive payments at half the scale redistribute about half as much, so a comparison of financing systems that uses the index alone mixes structure and size.

Reading the index in health financing analysis

The sign shows the direction of redistribution and the size shows its extent in Gini points. The identity implies small values whenever payments are a modest share of income. Austrian out-of-pocket payments of 5.4 per cent of net income, with a Kakwani index of minus 0.107, gave a Reynolds-Smolensky index of minus 0.006 in Sanwald and Theurl's grouped analysis, which agrees with 0.054 divided by 0.946, multiplied by minus 0.107.

Their ungrouped decomposition shows why the choice of formula matters. For total out-of-pocket payments the redistributive effect was minus 0.0121, made up of vertical redistribution of minus 0.0060, horizontal inequity of 0.0021 and reranking of 0.0040, so the vertical effect explained only half of the regressive impact. Their grouped Reynolds-Smolensky index of minus 0.006, computed in the scaled Kakwani form, was about half the redistributive effect of minus 0.0121.

The World Bank guide argues that voluntary payments made in return for care have no redistributive effect on economic welfare, since they buy the care, while accepting that how voluntary out-of-pocket payments are is debatable. For out-of-pocket costs the index therefore describes the change in income left for other goods, not a welfare transfer.

How the index differs from nearby measures

Several equity measures use the same household data on income and payments but answer different questions. The table sets the Reynolds-Smolensky index against its neighbours.

In health financing studies both Gini-based measures cover the payment side only. Who receives the care that payments finance is a separate question, answered with the concentration index of use or benefit incidence analysis.

Limitations and common misreadings

The index is a single number for the whole distribution and does not show where redistribution happens. Urban reports that Atkinson preferred transition matrices or Lorenz and concentration curves to single summary statistics, whose assumptions he thought could be little more than arbitrary.

The treatment of reranking is contested. In the Kakwani and Aronson, Johnson and Lambert framework reranking reduces redistribution, which invites the reading that removing it would raise redistribution without extra revenue. Urban argues that Atkinson and Plotnick saw reranking as a by-product that does not itself change post-tax inequality, and notes that Lerman and Yitzhaki built a decomposition, ranked by post-tax income, in which reranking adds to the fall in inequality.

Results also depend on the analyst's incidence assumptions about who bears each payment, the ability-to-pay measure, the equivalence scale and, for the horizontal and reranking terms, the width of the income intervals used to define equals, which the World Bank guide notes changes the split between them but not their sum.

Sources

• 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 17, Redistributive effect of health finance. https://openknowledge.worldbank.org/handle/10986/6896
• Reynolds M, Smolensky E. Public Expenditures, Taxes, and the Distribution of Income: The United States, 1950, 1961, 1970. New York: Academic Press; 1977.
• Sanwald A, Theurl E. Out-of-pocket payments in the Austrian healthcare system: a distributional analysis. International Journal for Equity in Health. 2015;14:94. https://doi.org/10.1186/s12939-015-0230-7
• Urban I. Kakwani decomposition of redistributive effect: origins, critics and upgrades. ECINEQ Working Paper 2009-148. Society for the Study of Economic Inequality; 2009. https://www.ecineq.org/milano/WP/ECINEQ2009-148.PDF
• van Doorslaer E, Wagstaff A, van der Burg H, Christiansen T, Citoni G, Di Biase R, et al. The redistributive effect of health care finance in twelve OECD countries. Journal of Health Economics. 1999;18(3):291-313. https://doi.org/10.1016/S0167-6296(98)00043-5