A Consistent Decomposition of the Redistributive Effect. An application to Taxes and Welfare Expenditures.

A Consistent Decomposition of the Redistributive Effect. An application to Taxes and Welfare Expenditures. Abstract The aim of this work is to solve the problem of non-additivity revealed by the works that calculate the redistributive effect of the budget or public policies made up of different instruments of income or public spending. We use Shapley’s value to do this. This technique allows us to decompose the redistributive effect and the vertical and horizontal effects consistently and not arbitrarily. The main result obtained for the case of taxes and social transfers in the US is that previous calculations undervalued the redistributive effects and their vertical and horizontal components for taxes and transfers. Undervaluation is more important for taxes. 1. Introduction Within public economy studies there is an important tradition in the analysis of the redistributive effects of taxes. One of the most used indicators to calculate redistributive effects is the difference in Gini indexes before and after taxes (Reynolds y Smolensky, 1977). Its versatility and simplicity has meant the extending of the measure to all kinds of public intervention instruments1 and even to the analysis of redistributive effect of the whole budget2. An additional refinement is in decomposing the ReynoldsSmolensky index to assess the effects on vertical equity and reranking, according to the method of Kakwani (1984), or also on horizontal inequality, according to the proposal of Aronson, Johnson and Lambert (1994). The problem of works that calculate the redistributive impacts of policies made up of various instruments or the whole budget is that they tend to present inconsistent results. Additive inconsistence is produced when the sum of the redistributive effects of each instrument does not coincide with the redistributive effect of the measures taken as a whole. This non-additivity is due to calculating the redistributive effects of each instrument only taking as a reference the original income and excluding the other instruments. That is, the interaction between instruments having an effect on the redistributive effect is not taken into account. The most elemental solution to take this interaction into account is to sequentially aggregate the different instruments to calculate their redistributive impacts. This was done by: Ferrarini and Nelson (2003) in a compared study of the financing and spending of the social insurance in ten countries, Keselman and Cheung (2004) to analyze the budget in Canada, Mahler and Jesuit (2006) for the budget in developed countries via the Luxembourg Income Study (LIS) data set, or Wolff and Zacharias (2007) for the budget in the US. The problem that this solution produces is that the redistributive effects calculated for the same policy are different according to the sequence chosen. That is, a problem of arbitrariness and judgment values arises. The aim that we set out for this work is to solve the problem of inconsistency of the decomposition of the redistributive effects without incurring in arbitrariness. To achieve this we propose additively decomposing the Reynolds-Smolensky index, and its disaggregation into vertical and horizontal effects in the manner of Kakwani (1984) or Aronson, Johnson and Lambert (1994), applying the Shapley´s value (1953). In broad terms, this technique considers the marginal effect on budget of eliminating each of the contributory factors in sequence, and then assigns to each factor the average of its marginal contributions in all possible elimination sequences. Shorrocks (1999), instigator of its use for the decomposing of inequality indicators, 3 shows that the sum of these effects is consistent and symmetrical. 1 For spending policies in the US compared with other countries we can see Mahler and Jesuit (2006), Marical, et. al (2008), Tanzi (2008), Warren (2008) or Prasad (2008). 2 Among the most recent we find those of Atkitson (2004), Smeeding (2005), Förster and Mira d’Ercole (2005) or Garfinkel, Rainwater and Smeeding (2006) in which the redistributive effects of the US Administration budget are compared with the results obtained in other countries. For other countries see Forteza and Rossi (2009) for Uruguay, Bargain and Callan (2010) for the EU, Clark and Leicester (2004) for the UK or Fuchs and Lietz (2007) for Austria. 3 Previous Works belong to Rongve (1995) and Chantreuil and Trannoy (1997). This methodology has been employed more profusely in the decomposing of poverty indicators, see as examples Kolenikov and Shorrocks (2005), Sami and Duclos (2008) or Deustsch and Silber (2008). To present analytical results we are going to take as a reference the recent work of Kim and Lambert (2009) in which the redistributive impact is analyzed for taxes and social spending in the USA for the years 1994, 1999 and 2004.4 Our calculations will allow the consistent and symmetrical decomposing of the total redistributive effect, the vertical effect, the horizontal inequality and the reranking that taxes and social benefits produce in the US and compare the results with those of the article cited. From this comparison we will be able to draw conclusions about the consequences of assessing redistributive effects that do not produce consistent results. This paper is organized as follows. The next section discuss the problems of traditional methodologies of redistributive decomposition and the relative merits of the Shapley decomposition. In section 3 we apply our proposal. The last section summarizes the results obtained and offers some concluding remarks. 2. Methodology One of the most usual ways of measuring the redistributive effect is the comparison of the Gini index of the initial income (market income) with the Gini index of the final income (post-tax income) (Reynolds and Smolensky 1977): TXXTX GRE −− −= G (1) At the same time, the most used ways of calculating the vertical and horizontal redistributive effects of the taxes are those devised by Kakwani (1984) and Aronson, Johnson and Lambert (1994).5 The first distinguishes between the vertical effect and reranking: TXTXTX RRE −−− −= V (2) And the second also identifies the horizontal inequality without an order change:6 TXTXTXTX RHRE −−−− −−= V (3) 4 Other recent works that are concerned with the redistributive effect of this type of policies are those of Verbist (2007) for taxes on work, pensions and unemployment benefits in the EU-15, Ferrario and Zanardi (2009) for the National Health of Italy or Biggs et al (2009) for the National Health in the USA. There is also a broad literature of health spending from the work of Wagstaff and van Doorslaer (1997)for the health spending and financing in Holland, which later Van Doorslaer et. al (1998) extended to an important group of countries. 5 There is another decomposing alternative put forward by Lerman and Yitzaki (1995), in which the order change is assessed first, and then V is resolved as the income change, but with the final order. 6 In spite of the terminology being undistinguishable, it must be pointed out that the original values of V and R are not coincident in both formulas. The work of Urban and Lambert (2008) recently put forward a reformulating of the values given by Aronson, Johnson and Lambert that identifies the relationship between both equations. The vertical effect is defined by the difference between pre-tax distribution inequality index and the inequality index of a fictitious income distribution obtained with the aid of taxation not generating either horizontal effects or reranking effects. This fictitious taxation leads to the pre-tax income of a household of rank in the distribution of pre-tax income being equal to its expected net income before taxation. Furthermore, the effect (V) measures the redistributive effect generated by the taxation system. This difference is positive if the taxation system is progressive and negative if it is regressive. The horizontal inequality (H), on the other hand, relates to unequal treatment of equals. A further concept of reranking (R) corresponds to the income scale rank-switching induced by the fiscal system. 7 In theory, for each of the m instrumentsaj,(j=1,…,m) that make up a policy M we can calculate its respective redistributive effects RE X+ajand the problem of inconsistency or asymmetry is given because REX+M≠REX+aJ J = 1 M ∑. The same takes place for V, H and R. As was cited above, the most elemental solution to the problem is to establish a sequence. To see the problem that it produces, we take the example of the work of Wolff and Zacharias (2007), in which the effects of taxes and spending for the years 1989 and 2000 in U.S. are analyzed. The authors first calculate the redistributive effects for the taxes (T), then for the transfers (Tr) and finally for public consumption (E) and the results fulfil the additive property 8. Now, why choose this sequence and not another (T+E+Tr, E+T+Tr, E+Tr+T, Tr+E+T o Tr+T+E)? If we did this, we would have obtained different redistributive impacts for each instrument according to the sequence used. That is, the order influences the result. To sum up, if we change the order, the redistributive effect values change, producing asymmetrical results. If this is so, what value do we take? Why must taxes precede public spending? It would not seem reasonable to us to consider that the administration first receives payments and then spends. Then, this sequence does not have the guarantee of certainty. Spending can be financed from the public deficit and in this case the spending is first and the taxes come afterwards. What happens when taxes fall back on transfers received? The reality is that modern economies are complex and it is not possible to find unquestionable sequences in the budget procedure. This is why to choose a sequence implies arbitrariness and the establishing of a value judgment that conditions the results. We can find the problem´s solution calculating the decomposition via the value of Shapley (1953). This methodology comes from game theory and has been applied in the study of inequality since the work of Shorrocks (1999).9 The applying of Shapley´s value allows an additive, symmetrical and exact decomposition for any index. 7 See decompostion of taxes in Creedy and Van Ven (2001), for Australia, Urban (2008) for Croacia, Lambert and Thoresen, T (2009) for Norway. 8 This can be checked in their Table 6 (Wolff and Zacharias, 2007, p.710). 9 See Chantreuil and Trannoy (1999); Sastre and Trannoy (2002), Deustch and Silber (2008). The system consists of quantifying the impact of a factor calculating the average of the marginal effects, determined by the elimination of the said factor in all the possible sequences. Indeed, let M be the set of instruments aj, { } mMj ,....2,1 = ∈ , that make up a public policy. Let G be a subset of M made up of g instruments. Let GX RE + be the value of RE when the instruments aj, MGj ⊆ ∉ have been eliminated. If all the sizes are equally probable, a determined size g will occur with a probability of g/1 . On the other hand, if we have a determined instrument aj that belongs to a subgroupG, the )1( −g instruments remaining of this subgroup can be chosen between the )1( −m instruments remaining in a number of sequences equal to: (m−1)! (m−1) −(g−1) [] !(g−1)! =(m − 1)! (m−g)!(g−1)! (4) The probability of one of these combinations being chosen is the reciprocal. Therefore, the probability of a group G containing the element aj is equal to the said reciprocal multiplied by m/1 . If we take into account the marginal contribution of aj ∈ G to the inequality of the subgroup G it is: REX+G j=REX+G−REX+(G−aj) (5) The contribution of the instrument aj to the total value of the index MX RE +is equal to the summation of all the groupings that can contain it, pondered by the probability of them occurring: RE X+M j=(m− g )!( g −1)! m! G⊆M j∈G ∑REX+G−REX+(G−aj) [ ] (6) Such that: REX+M=REX+M j j=1 m ∑ (7) This expression is generalizable to determine the contribution of each j a to the indices MX V+, MX H+ and MX R+ such that: 1 1 1 ∑ ∑ ∑ = + + = + + = + + = = = m j j MX MX m j j MX MX m j j MX MX VR HH VV (8) Therefore: ∑∑∑∑ = + = + = + ++++ = +−−=−−== m j j MX m j j MX m j j MX MXMXMX j MX m j MX RHVRHVRERE 1111 (9) With this methodology the sequence ceases to be determinant. All the possible combinations are considered and the interaction of the instruments that make up the policy is taken into account. To appreciate the consequences of its application, we are going to use the case of a policy with two instruments, for example taxes (T) and transfers (B). Such that for equation 7 must be met: RE X − T + B = RE X−T+B T+RE X−T+B B (10) Being: BXTXBTX B BTX TXBXBTX T BTX RERERERE RERERERE +−+− +− −++− +− +−= +−= 2 1 )( 2 12 1 )(. 2 1) (11) Taking these results, we can construct Table 1, in which are included the solution obtained via applying Shapley´s value and the two possible sequences. These latter produce consistent results but they are not symmetrical. Table 1. Decomposition of the redistributive effect of a public policy made up of two instruments, taxes and transfers. Sequence 1 (-T+B) Sequence 2 (B-T) Shapley Solution Taxes TX RE − (I) BXBTX RERE ++− − (III) TXBXBTX T BTX RERERERE −++− +− +−= 2 1 )( 2 1) (V) Transfers TXBTX RERE −+− − (II) BX RE + (IV) BXTXBTX B BTX RERERERE +−+− +− +−= 2 1 )( 2 1 (VI) Total BTX RE +− (VII) BTX RE +− (VII) BTX RE +− (VII) Source: own elaboration. The problem of inconsistency that tends to appear in works of this kind, such as happens in Kim and Lambert (2009), originates in calculating the total effect, VII and the effects I and IV. The three effects are estimated as the marginal effect on the initial income, but in VII the interaction of both instruments is taken into account, while in I and IV it is not. This inconsistency is also produced in the case of our calculating the partial effects III and II (calculating the redistributive effect of each policy, eliminating it in the final income instead of aggregating it to the initial income), as can be observed in Inmervoll et. al (2006). For their part, in the works that adopt the sequential solution, as happens in the work of Wolff and Zacharias (2007), sequence 1 is opted for (values I and II), which fulfils the additive property, but incurs in arbitrariness. Sequence 2 could be taken and the result would also be consistent, but with different values (values III and IV).10 Shapley´s solution consists of calculating the average of the marginal effects of each instrument in each sequence. It fulfills the properties of additivity and symmetry (V+VI=VII) and takes into account any interaction between instruments. In the same way in which we proceed with RE, the vertical, horizontal and reordering effects V, H and R can be decomposed. 3. Data and Results As was pointed out in the introduction, to contrast the efficiency of the methodology proposed and its consequences, we employ the data of the work of Kim and Lambert (2009), which is the most recent study for the United States, and we will analyze the consequences that derive from the new estimation. In this work, data of personal taxes and transfers for social benefits are taken in the United States for the years 1994, 1999 and 2004 proceeding from the U.S. Current Population Survey (CPS). The different indexes that we use are computed with the relation specified in equation (3). In Table 2 the data offered by the authors is reproduced and the new results with value´s Shapley for the three years considered is included.11 As can be observed, taxes and transfers reduce the income inequality of the market around 30%. The greatest inequality after taxes and transfers is in 2004, as a consequence of the market income being divided more unequally. Taxes have their greatest redistributive effect and their greatest relative weight, close to 30%, in 1999. Later they lose importance in favour of transfers which are responsible for 76.29% of the redistribution in 2004. Likewise, taxes and transfers lose their capacity to correct the vertical inequality of the 1999 market income. Nevertheless, the increase of the vertical equality of the transfers means that in 2004 taxes and transfers together generate the greatest horizontal inequality correction. In comparative terms we can see that the redistributive effects of taxes and transfers separately are greater than those previously calculated and explain the whole redistributive effect. To explain this, we take taxes as an example and with Shapley value we calculate the average of the two redistributive effects on the original income and on the income after social transfers (I and III of Table 1). When taxes are applied after the social transfers, these have already reduced the inequality and consequently with the same quantity of resources a greater equality can be attained. The most relevant effect for the taxes is that their redistributive effect is broader: they explain between 10 As was pointed out above, in the work of Wolff and Zacharias the marginal effect on the initial income of the transfers is also calculated, which is why it can be seen that II and IV do not coincide. 11 As we did not have at our disposal the intermediate of the Gini indices of the different arrangement sequences, we will limit ourselves to disaggregating the redistributive effect of transfers and taxes, without continuing the disaggregation of them into the different subelements that they make up. 29.7% (1999) and 23.7% (2004) of the respective total effects. The new calculations also show a greater weight in the vertical effects of the taxes. This situation is coherent with its greater total redistributive effect. It is also observed that, as Kim and Lambert pointed out, there is a certain stability of equality loss due to reranking, between 30 and 35% of the redistributive effect. Now, with the new calculations we also moreover identify the contribution of each instrument to the reranking. Thus, in 1994, 76% of the reranking was produced by transfers and 24% by taxes. In 2004 the taxes hardly contributed 17.5% of the reranking, which means that the transfers have a more unequal distribution in terms of horizontal equality. Finally we have to highlight, also in the tax ambit, the sign change of the H values. In the original calculations the taxes produced inequality as a consequence of unequal treatment of equals. The new calculations point out that taxes and transfers follow a similar pattern. Inequality due to horizontal inequality is not produced, but rather this operates in favour of redistribution. Table 2. Redistributive Effect of taxes and transfers. Non -additive decomposition vs Shapley decomposition 1994 Non additive decomposition* Shapley decomposition Absolute value In percentage of RE Absolute value In percentage of RE In percentage of effect of taxes and public transfers Taxes (T) RE 0,03312 0,040595 26,28% VK 0,03961 119,60% 0,053275 131,24% 25,65% V 0,03962 119,64% 0,05317 130,96% 25,73% H 0,00001 0,02% -0,000115 -0,28% 10,65% RK 0,0065 19,62% 0,012685 31,24% 23,83% Public Transfers (B) RE 0,10638 0,1138505 73,72% VK 0,14075 0,154415 74,35% V 0,1399 131,51% 0,15345 134,78% 74,27% H -0,00085 -0,80% -0,000965 -0,85% 89,35% RK 0,03437 32,31% 0,040555 35,62% 76,17% Total (N=T-B) RE 0,15445 0,1544455 100,00% VL 0,20769 0,20769 100,00% T by VL decomposition 0,03126 0,053275 B by VL decomposition 0,17643 0,154415 V 0,20662 134% 0,20662 133,78% 100,00% H -0,00107 -0,69% -0,00108 -0,70% 100,00% RK 0,05324 34,47% 0,05324 34,47% 100,00% 1999 Non additive decomposition* Shapley decomposition Absolute value In percentage of RE Absolute value In percentage of RE In percentage of effect of taxes and public transfers Taxes (T) RE 0,03492 0,04296 29,70% VK 0,03847 110,17% 0,051685 120,31% 27,38% V 0,03847 110,17% 0,05163 120,18% 27,43% H 0,00001 0,02% -0,00005 -0,12% 9,26% RK 0,00354 10,15% 0,008725 20,31% 19,79% Public Transfers (B) RE 0,09367 0,10171 70,30% VK 0,12385 0,137065 72,62% V 0,12342 131,76% 0,13658 134,28% 72,57% H -0,00043 -0,46% -0,00049 -0,48% 90,74% RK 0,03018 32,22% 0,035365 34,77% 80,21% Total (N=T-B) RE 0,14467 0,14467 100,00% VL 0,18875 0,18875 100,00% T by VL decomposition 0,03144 0,051685 B by VL decomposition 0,15731 0,137065 V 0,18821 130% 0,18821 130,10% 100,00% H -0,00054 -0,38% -0,00054 -0,37% 100,00% RK 0,04409 30,48% 0,04409 30,48% 100,00% 2004 Non additive decomposition* Shapley decomposition Absolute value In percentage of RE Absolute value In percentage of RE In percentage of effect of taxes and public transfers Taxes (T) RE 0,02954 0,03766 23,71% VK 0,03323 112,49% 0,04754 126,23% 22,08% V 0,03326 112,57% 0,04747 126,05% 22,14% H 0,00002 0,08% -0,000075 -0,20% 8,24% RK 0,00369 12,49% 0,00988 26,23% 17,49% Public Transfers (B) RE 0,11307 0,12119 76,29% VK 0,1535 0,16781 77,92% V 0,15276 135,10% 0,16697 137,78% 77,86% H -0,00074 -0,66% -0,000835 -0,69% 91,76% RK 0,04043 35,76% 0,04662 38,47% 82,51% Total (N=T-B) RE 0,15885 0,15885 100,00% VL 0,21535 0,21535 100,00% T by VL decomposition 0,02613 0,04754 B by VL decomposition 0,1892 0,16781 V 0,21444 135% 0,21444 135,00% 100,00% H -0,00091 -0,57% -0,00091 -0,57% 100,00% RK 0,0565 35,57% 0,0565 35,57% 100,00% * Data obtained from Table 3 from the work of Kim, Lambert (2009). Source:own elaboration.