World Development Vol. 39, No. 10, pp. 1888–1890, 2011� 2011 Elsevier Ltd. All rights reserved

0305-750X/$ - see front matter

www.elsevier.com/locate/worlddevdoi:10.1016/j.worlddev.2011.08.001

Comment: On the Interpretation (and Misinterpretation) of Inequality

Decompositions by Income Sources

AYAL KIMHI *

The Hebrew University of Jerusalem and Center for Agricultural Economic Research, Rehovot, Israel

Summary. — This paper discusses interpretations of different inequality decomposition rules when inequality is decomposed by incomesources. It argues that authors of a recent article based their conclusions on misinterpreted decomposition results. It also argues thatmarginal effects, derived as elasticities of inequality with respect to uniform increases in income from each source, are easily interpretedand can be compared across different decomposition rules.� 2011 Elsevier Ltd. All rights reserved.

Key words — income inequality, income sources, decomposition, interpretation, marginal effects

1. INTRODUCTION

In a recent article, Davis et al. (2010) examine, amongother things, the effect of rural nonfarm activities on incomeinequality in a number of developing and transition coun-tries, using a well-known methodology of inequality decom-position by income sources. Their results “. . .suggest thatnon-farm income induces greater income inequality in ruralareas. . .” The purpose of this note is to show that this con-clusion is based upon a common misinterpretation of thedecomposition results, and that a correct interpretation ofthe results could very well lead to the opposite conclusion.In doing this, this paper follows Podder and Chatterjee(2002), who stated that “. . .the disaggregation of inequalityby factor components . . . is probably the most misused andmisunderstood concept in the income inequality literature(p. 3).” Unfortunately, misinterpreted decomposition resultsare still common in the development literature and this iswhy this note is in place.

The next section describes the inequality decompositionmethodology, and the following section discusses the differentinterpretations of the decomposition results. The final sectionconcludes.

* This research was funded by the Center for Agricultural Economic Re-

search. Helpful comments and suggestions by Myoung-jae Lee and Shl-

omo Yitzhaki, as well as seminar participants at the Department of

Agricultural Economics and Management of the Hebrew University, the

School of Economics at Nagoya University, The Department of Eco-

nomics at Bar Ilan University, and the 2007 annual meeting of the Israel

Economic Association, are gratefully acknowledged. Final revision acce-pted: February 3, 2011.

2. INEQUALITY DECOMPOSITION BY INCOMESOURCES: GENERAL METHODOLOGY

Shorrocks (1982) was the first to offer a unified approach toinequality decomposition by income sources. Earlier, Fei,Ranis, and Kuo (1978) and Pyatt, Chen, and Fei (1980),among others, offered a decomposition of the Gini index ofinequality by income sources, but this happens to be a specialcase of Shorrocks’ (1982) approach. Specifically, Shorrocks(1982) suggested focusing on inequality measures that can bewritten as a weighted sum of incomes:

IðyÞ ¼ RiaiðyÞyi; ð1Þwhere ai are the weights, yi is the income of household i, and yis the vector of household incomes. These include as specialcases the Gini index as well as the class of Generalized Entro-py indices. If income is observed as the sum of incomes from kdifferent sources, yi = Rkyi

k, the inequality measure (1) can bewritten as the sum of source-specific components Sk:

1888

IðyÞ ¼ RiaiðyÞRkyki ¼ Rk½RiaiðyÞyk

i � � RkSk: ð2ÞDividing (2) through by I(y), one implicitly obtains the pro-

portional contribution of income source k to overall inequalityas:

sk ¼ RiaiðyÞyki =IðyÞ; ð3Þ

where Rksk ¼ 1.

3. INTERPRETING INEQUALITY DECOMPOSITIONRESULTS

Shorrocks (1982) noted that in principle, the weights ai(y)can be chosen in numerous ways, so that “. . .the contributionof any factor expressed as a proportion of total inequality canbe made to give any value between plus and minus infinity (p.202)!” In other words, the decomposition formula (3) yields aninfinite number of potential decomposition rules. In his empir-ical work, Shorrocks (1983) used three different decompositionrules, each based on a particular inequality index: (a) the Giniindex, with ai(y) = 2(i�(n + 1)/2)/(ln2), where i is the index ofobservation after sorting the observations from lowest to high-est income, n is the number of observations, and l is mean in-come; (b) the squared coefficient of variation, withai(y) = (yi � l)/(nl2); and (c) Theil’s T index with ai(y) = l-n (yi/l)/n. Morduch and Sicular (2002) and Kimhi (2007) alsocompared the results of these three decomposition rules, whileDavis et al. (2010) used only the rules based on the Gini andTheil’s T inequality indices. Paul (2004) did the same for theseand several other decomposition rules. All these authors foundthat the decomposition results vary across the decompositionrules, sometimes quite considerably.

INEQUALITY DECOMPOSITIONS BY INCOME SOURCES 1889

The question that emerges is, therefore, are we obtaining dif-ferent answers to the same question or to different questions?In other words, what is the meaning of the inequality decom-position results? Shorrocks (1983) answers this in part by not-ing that “This turns the calculation of inequality contributionsinto a meaningless exercise. . . (p. 315).” Shorrocks (1982) re-sponds to this challenge by adding intuitive restrictions onthe choice of decomposition rule, and comes up with a uniquedecomposition rule based on the squared coefficient of varia-tion inequality index. Fields (2003) reached the same conclu-sion in a different way. One of the restrictions imposed byShorrocks (1982) is that Rai(y) = 0, implying that “. . .the con-tribution of a factor to total income inequality is zero if allindividuals receive the same income from that source (p.201).” This is satisfied by the decomposition rules based onthe Gini and squared coefficient of variation inequality indices.Shorrocks (1982) himself is not completely happy with thisrestriction, noting (in a footnote) that it is “. . .perhaps ques-tionable, since we may feel that identical positive lump sumtransfers are an equalizing force and hence should be associ-ated with a negative contribution to inequality (p. 202).”

Morduch and Sicular (2002) took Shorrocks’ (1982) foot-note seriously. They defined the property of uniform additionsof an inequality index in the following way. An inequality in-dex is said to satisfy this property if adding a fixed amount ofincome across the entire population decreases inequality. It iseasy to agree that this is a desired property of inequality indi-ces. Morduch and Sicular (2002) then took a further step andadopted this definition to inequality decomposition methods,so that the property is satisfied if the contribution to inequalityof a positive equally-distributed income component is nega-tive. Paul (2004) makes the same argument and calls it the neg-ativity property.

The question whether the property of uniform additions is adesired property of inequality decomposition rules brings usback to the question of what is the meaning of the inequalitydecomposition results and how they should be interpreted.Shorrocks (1982) also emphasized the need “. . .to examine indetail the intuitive interpretations normally attached to state-ments of the form ‘factor X contributes Z percent of totalinequality’ (p. 203).” The answer may not be the same forall decomposition rules. Shorrocks (1982) has shown that forthe decomposition rule based on the squared coefficient of var-iation, the inequality contribution of an income source is equal

Table 1. Marginal effects of income sources

Country Crop Livestock Ag wage

Malawi �13.0 3.9 �6.5Madagascar �13.5 2.5 �2.5Bangladesh �9.4 �0.1 �20.1Nepal �13.5 �6.4 �12.4Ghana �29.2 �2.1 0.1Tajikistan 4.1 9.0 �7.6Vietnam �34.0 �10.9 �4.6Nigeria �34.0 �2.6 18.1Pakistan 1.0 �1.8 �7.5Nicaragua �12.8 �5.5 �10.0Indonesia �16.0 �0.3 �1.8Guatemala �19.5 �1.3 �8.5Albania �10.4 �18.7 1.5Ecuador 16.6 �1.0 3.6Bulgaria 1.6 0.5 3.0Panama �13.3 �0.7 �7.7

to the average of two quantities: the inequality that would beobserved if this income source was the only source of inequal-ity, and the amount by which inequality would fall if inequal-ity in this income source was eliminated. Hence, thedecomposition results reflect the variability in each incomesource. Lerman and Yitzhaki (1985) have shown that in thecase of the decomposition rule based on the Gini inequality in-dex, the relative contribution of each income source can bewritten as sk = 2cov(yk, F)/l, where cov(yk, F) is the covari-ance of income source k with the cumulative distribution of in-come, F. Consider a multiplicative mean-preserving spread inyk that does not change the cumulative distribution F, i.e.,a(yk � lk) where a is a small positive scalar. This clearly in-creases the absolute value of the covariance and of sk by a fac-tor of a. Hence, the Gini-based decomposition results alsoreflect the variability in each income source.

This leads to the conclusion that inequality decompositionresults reflect changes in the variability of income sources, atleast for these two decomposition rules. Hence, it makes per-fect sense that the contributions to inequality of incomesources with zero variability are zero. Interestingly, the litera-ture does not offer similar intuitive results for the inequalitydecomposition rule based on Theil’s T inequality index, whichsatisfies the property of uniform additions. This comes as nosurprise if one examines carefully the logic that underlies thisproperty. This was based on the idea that an increase in anequally-distributed income source reduces inequality. But suchan increase in not mean-preserving, and this is why it reducesoverall inequality. Perhaps, then, the results of inequalitydecomposition rules that satisfy the property of uniform addi-tions do not reflect mean-preserving changes in income source.If results based on different decomposition rules reflect differ-ent questions, it comes as no surprise that these results couldbe quite different from each other, as was obtained by Mor-duch and Sicular (2002), Paul (2004) and Kimhi (2007), as wellas Davis et al. (2010).

In their conclusions, Morduch and Sicular (2002) noted that“the aggregate Gini coefficient falls if an income source is in-creased by a constant amount for all members of the popula-tion, but none of the components of the standarddecomposition of the Gini are affected (p. 104)” and thus con-clude that “it is of limited use in describing causes of inequality(p. 105).” In fact, increasing income by a constant amount hasthe same effect on inequality regardless of the particular in-

derived from Davis et al. (2010) results

NonAg wage Self-emp Transfers Other

14.4 6.6 �5.2 0.0�4.2 21.6 �3.6 �0.31.3 20.3 12.7 �4.715.3 6.3 8.1 2.65.0 32.7 �6.0 �0.40.1 2.7 �8.9 0.5�3.2 57.6 �4.7 �0.117.8 2.3 �1.2 �0.3�8.5 13.8 �3.5 6.513.4 17.2 �2.2 �0.121.5 12.2 �14.8 �0.922.6 12.7 �7.2 1.010.9 31.9 �15.0 �0.2�20.1 4.9 �7.3 3.420.3 5.2 �30.2 �0.327.9 �0.8 �5.7 0.4

1890 WORLD DEVELOPMENT

come source that is increased. The logic of the property of uni-form additions may lead to the absurd conclusion that allinequality contributions should be negative!

The question that Morduch and Sicular (2002) perhaps hadin mind is what happens to inequality as a result of a uniformincrease in a particular income source. Shorrocks (1983) hasnoted that comparing sk and lk/l, the share of income fromsource k in total income, is useful for knowing whether thekth income source is equalizing or disequalizing. In the caseof the Gini decomposition rule, sk = 0 if k stands for anequally-distributed income component. Hence, it follows thatthe effect of a uniform increase in this income component onthe Gini index is unambiguously negative. Lerman andYitzhaki (1985) have shown that the elasticity of the Giniinequality index with respect a uniform percentage change inyk is sk � lk/l, which supports the logic of Shorrocks (1983)for the case of the Gini decomposition rule. Paul (2004) de-rived equivalent elasticities for other decomposition rules.These “marginal effects” are more informative than the pro-portional contributions to inequality sk when one wants toknow whether a particular income source is equalizing ordisequalizing (Podder, 1993). In fact, both Paul (2004) andKimhi (2007) found that the marginal effects obtained fromdifferent decomposition rules are much more similar thanthe decomposition results.

In this sense, the statements of Davis et al. (2010) that posi-tive proportional contributions indicate inequality-increasingeffects are not correct. Consequently, their empirical conclu-sion, that nonfarm income induces greater income inequality

in rural areas, is not necessarily true. In fact, it is easy to derivethe marginal effects of income sources on inequality based onthe Gini decomposition rule as sk � lk/l, using lk/l fromTable 2 and sk from Table 5 in Davis et al. (2010). The resultsare shown in Table 1. In essence, the conclusions of Daviset al. (2010) are found to be mostly correct, in the sense thatnonagricultural activities have positive marginal effects oninequality in most cases. However, several specific results arefound to be wrong. For example, crop income in Pakistan isnot inequality increasing but rather has a negligible effect oninequality, while nonagricultural wage income is an equalizingincome source. This is not surprising, given that the conclu-sions of Davis et al. (2010) are based on misinterpretation ofthe decomposition results.

4. SUMMARY AND CONCLUSIONS

This paper critically reviewed interpretations of inequalitydecompositions by income sources. Different decompositionrules give different results simply because they do not measurethe same things. Therefore, there is no meaning in derivingconclusions from these results. In contrast, marginal effectsof income sources on inequality are easily computed and easilyinterpreted, and, therefore, should be used to determinewhether particular income sources are equalizing or disequal-izing. Despite this, the major conclusions of Davis et al. (2010)are not changed after computing the marginal effects. How-ever, several particular results are completely different.

REFERENCES

Davis, B., Winters, P., Carletto, G., Covarrubias, K., Quinones, E. J.,Zezza, A., et al. (2010). A cross-country comparison of rural incomegenerating activities. World Development, 38, 48–63.

Fei, J. C. H., Ranis, G., & Kuo, S. W. Y. (1978). Growth and the familydistribution of income by factor components. Quarterly Journal ofEconomics, 92, 17–53.

Fields, G. (2003). Accounting for income inequality and its change: A newmethod, with application to the distribution of earnings in the UnitedStates. Research in Labor Economics, 22, 1–38.

Kimhi, A. (2007), Regression-based inequality decomposition: A criticalreview and application to farm-household income data, Discussionpaper No. 16.07, The Center for Agricultural Economic Research,Rehovot, Israel. Available from: http://departments.agri.huji.ac.il/eco-nomics/en/publications/discussion_papers/2007/kimhi-regression.pdf.

Lerman, R. I., & Yitzhaki, S. (1985). Income inequality effects by incomesource. A new approach and applications to the United States. Reviewof Economics and Statistics, 67, 151–156.

Morduch, J., & Sicular, T. (2002). Rethinking inequality decomposition,with evidence from rural China. The Economic Journal, 112, 93–106.

Paul, S. (2004). Income sources effects on inequality. Journal of Develop-ment Economics, 73, 435–451.

Podder, N. (1993). The disaggregation of the Gini coefficient by factorcomponents and its applications to Australia. Review of Income andWealth, 39, 51–61.

Podder, N., & Chatterjee, S. (2002). Sharing the national cake in postreform New Zealand: Income inequality trends in terms of incomesources. Journal of Public Economics, 86, 1–27.

Pyatt, G., Chen, C., & Fei, J. (1980). The distribution of income by factorcomponents. Quarterly Journal of Economics, 94, 451–474.

Shorrocks, A. F. (1982). Inequality decomposition by factor components.Econometrica, 50, 193–211.

Shorrocks, A. F. (1983). The impact of income components on thedistribution of family incomes. Quarterly Journal of Economics, 98,311–326.