Measuring relative equality of concentration between different. income/wealth distributions

Transcription

1 Measuring relative equality of concentration between different income/wealth distributions Quentin L Burrell Isle of Man International Business School The Nunnery Old Castletown Road Douglas Isle of Man IM 1QB via United Kingdom q.burrell@ibs.ac.im Submitted to the Organising Committee of the International Conference to commemorate Gini and Lorenz, University of Siena, Italy, 3-6 May, 5.

2 Abstract In a recent paper Burrell (5a) introduced two new measures - both based on the Gini mean difference - for measuring the similarity of concentration of productivity between different informetric distributions. The first was derived from Dagum s (1987) notion of relative economic affluence (REA); the second in some ways analogous to the correlation coefficient is a new approach giving the so-called co-concentration coefficient (C-CC). Models and methods adopted in the field of informetrics very roughly, the metrics aspects of information systems - are very often based upon, or have direct analogues with, or at very least are inspired by, ones from econometrics and it is the purpose of this paper to suggest ways in which these new measures of similarity could be useful in, for instance, studies of income distributions (a) between different countries and (b) over different periods of time. The measures are illustrated using exponential, Pareto, Weibull and Singh-Maddala distributions.

3 1. Introduction One of the most intuitively reasonable requirements of a measure of concentration of income/wealth within a population is that it should be invariant under scale transformations the degree of inequality should be the same if incomes are measured in or US$. The situation is rather different if we are comparing inequalities between populations. For instance, suppose we have two populations whose unit income distributions are identical, but with one measured in, the other in US$. Then their within population concentration measures will be the same and yet there is clearly a difference between populations since if both were expressed in the same units, we would have two populations with different degrees of affluence. Dagum (1987) sought to address this problem by introducing a measure of relative economic affluence (REA) based upon the Gini mean difference, defined as the average absolute difference between incomes of (randomly chosen) members of the two populations. It turns out that the REAs of two income distributions are the same if and only if the means of the two distributions are the same. The first of the new measures proposed by Burrell (5a) is a simple adaptation of the REA; the second incorporates the Gini mean difference and the Gini coefficients of the two populations separately to give a normalised measure somewhat analogous to the correlation coefficient lying between and 1, with the upper value being achieved if and only if the two population income distributions (measured in the same units) are identical. Although we will talk in terms of income distributions, the notions clearly extend to other fields. 3

4 . Basic definitions We imagine a population of individuals and let denote the income of a randomly chosen individual. Suppose that the distribution of in the population is given by the (absolutely) continuous probability density function (pdf) f (x) defined on the nonnegative real line. (Restricting attention to the absolutely continuous case is done purely for simplicity of presentation.) Notation. (i) E[] mean of xf (x) dx (ii) Tail distribution function of (x) P( x) x f (y)dy At this early stage, let us recall that for a continuous non-negative random variable we have E [] (x) dx (1) (see, for instance, Stirzaker (1994, p38)). Without further comment we will always assume that the mean is finite. There are many different approaches to the measurement of concentration/inequality of (income) distributions and we refer the reader to Lambert (1) and Kleiber & Kotz (3) for further discussion. See also Egghe & Rousseau (199). Notwithstanding the opinion that the overemphasis bordering on obsession on the Gini coefficient as the measure of income inequality is an unhealthy and possibly misleading development (Kleiber & Kotz, 3, p3), the Gini coefficient is the cornerstone of our analysis. Definition 1. The Gini coefficient/index/ratio for is defined formally as 4

5 E[ ] 1 γ, where 1 and are independent copies of. (See the previous references, among many others, as well as the original presentation by Gini (1914).) The idea behind the definition is that we look at each pair of individuals within the population in turn, find the absolute difference between their incomes and then average out over all possible pairs. For purposes of calculation, the above definition is not very convenient. Of the many others available (see itzhaki, 1998), the one that best suits our purposes is given in the following: Proposition 1. γ 1 (x) dx () According to Kleiber & Kotz (3), this is originally due to Arnold & Laguna (1977) at least in the non-italian literature. It was independently rediscovered by Dorfman (1979) in economics and by Burrell (1991, 199a) in informetrics. The Gini coefficient is usually held to be one of the, if not the, best inequality measures in that it obeys all seven of the desirable properties proposed by Dalton (19) for such a measure, see Dagum (1983). (But note the previously quoted counter opinion of Kleiber & Kotz.) One of these properties is that it is invariant under scale, or is independent of the unit of measurement. This is clearly almost a necessary property in measuring inequality within a population. However, this property should not necessarily carry over to comparative studies of inequality between populations if different units of income are used. As an example, suppose that the income in two populations each follows an 5

6 exponential distribution, measured in the same units but with different means. Then clearly the general level of income is greater in the population having the greater mean. On the other hand, since the exponential is a scale-parameter family, the Gini coefficient for each will be the same. (Indeed, the Gini coefficient for an exponential distribution is ½, see, e.g. Burrell (199a).) Hence reliance on standard measurements of inequality such as the Gini coefficient is inappropriate for measuring relative inequality between populations. Instead, we follow Dagum (1987) to extend the Gini coefficient to become a measure of the overall inequality of income between two populations. Aside. Within the field of income/wealth distributions, the standard graphical representation of inequality is via the Lorenz curve where one arranges individuals in increasing order of income. In informetrics the focus is usually upon the most productive sources ( richest individuals ) in which case it is natural to arrange individuals in decreasing order of income so that what is plotted is the tail distribution function against the tail-moment distribution function. In the informetrics literature this graphical representation is sometimes known as the Leimkuhler curve, see Burrell (1991, 199a). The geometric relationship with the Lorenz curve is immediate, as is the geometric interpretation of the Gini coefficient via the Leimkuhler curve, see Burrell (1991). Let us denote the income of a randomly chosen individual from each population by,, respectively. The mean and tail distribution function are defined as before and denoted, respectively, and similarly for the population. The idea behind the 6

7 construction of the Gini ratio between the two populations is exactly analogous to that of the Gini coefficient for a single population, namely we look at pairs of individuals, but now one from each population, find the absolute difference between their incomes, measured in the same monetary units, and average this difference over all possible pairs. Thus we have the following: Definition. The Gini ratio between the two populations, denoted by G(,), is given by E[ ] G(, ), where, are independent. + (The numerator of the above expression is what is referred to as the Gini mean difference, Dagum (1987).) The analogy with Definition 1 is clear. Indeed, the Gini coefficient becomes a special case since if and have the same distribution then G(, ) G(, ) γ so that the (comparative) Gini ratio becomes the (single population) Gini coefficient. Note also that G(, ) < 1, where we can get G 1 as a limiting case, see Dagum (1987). Again, for purposes of calculation, the defining formula for the Gini ratio is not very convenient so that we make use of the following: Theorem 1. With the above notation: G(, ) 1 (x) + (x) dx (3) Proof. See Burrell (5a), reproduced here in Appendix 1. 7

8 3. Normalized measures. I: The relative concentration coefficient In the paper in which he introduced the Gini ratio, Dagum (1987) proposed the notion of relative economic affluence. We briefly recap Dagum s approach, but modifying some of his notation and terminology. As described earlier, the Gini ratio is derived from the average absolute difference in income between the -population and the -population. Dagum s relative measure results from splitting this difference into two components: the average excess income of members of the -population over less affluent -sources, which we denote by p(,), or p in Dagum s notation; and the average excess income of -sources over less affluent 1 -sources, denoted p(,), or Dagum s d 1. (These two components are in fact those considered in the proof of Theorem 1 in the Appendix.) In Burrell (5a) we suggested a (relative) concentration coefficient between the and populations defined as D(,) p(,)/p(,) assuming, without loss of generality, that E[] E[]. This is just one minus the relative economic affluence defined by Dagum (1987, Definition 7). An alternative representation of D(, ) is given in the following: Proposition. Assuming, without loss of generality, that E[] E[] D (, ) (1 (1 ) ) Proof. See Appendix. 8

9 Corollary. < D(, ) 1 and D(,) 1 if and only if E[] E[]. The proof is immediate, but see Burrell (5a) for details. Thus D(, ) is normalized in that it lies between and 1 and the upper bound is achieved if and only if the two means are the same. However, if the means are not the same then the upper bound is given by the ratio of the means and this leads us to make the following Definition 3. The (modified) relative concentration coefficient is D * (, ) D(, ) where wlog E[] > E[]. ( )/ ( )/ 1 (4) 1 II: The coefficient of co-concentration Note that although the Gini ratio already gives some sort of measure of the degree of similarity/dissimilarity between two income distributions so far as their concentration/inequality is concerned, it is not very informative on its own. One problem is that the ratio is minimised when the two distributions are the same whereas we would like a comparative measure to be maximised in this situation. This is easily resolved if we make the following: 9

10 Definition 4. θ and 1 γ (x) dx coefficient of equality within the distribution of (5) H(, ) 1 G(, ) (x) + (x) dx (6) Note that both equality ratio between the distributions of and. θ and H(,) lie between and 1 and that θ 1 corresponds to the case where all individuals have the same income. If all individuals across both populations have equal income, then H(,) 1. Also, zero values can only be achieved via a limiting process so that in practice both may be taken as being strictly greater than zero. We can now construct a new measure that focuses on the degree of equality rather than inequality of concentration between the two populations. Definition 5. The coefficient of co-concentration or co-concentration coefficient (C-CC) is given by H(,) (1 G(,)) Q(,) θ θ (1 γ )(1 γ ) (7) (8) + ( )( ) where the representation (8) follows from (7) together with (5) and (6). The following shows that this is a standardised measure and is (joint) scale invariant: 1

11 Theorem. (i) < Q(, ) 1 (ii) Q(,) 1 if and only if the two distributions are the same. (iii) Q(k, k) Q(, ) for any constant k. Proof. See Appendix 3. Note. The equality of distributions required for the co-concentration coefficient to achieve its upper bound is a much stronger condition than the equality of means required in the case of the relative concentration coefficient and, we would argue, a more natural requirement. 4. Some theoretical examples In Burrell (5a), simple examples for Exponential and Pareto distributions were considered. Here we look at two rather more substantial cases. (a) Weibull distribution Suppose that ~ Wei(, β), i.e. has a Weibull distribution with index and scale parameter β, so that the tail distribution function of is given by (x) P( > x) exp[ (x / β) ] The mean of the distribution is well known to be 1 E[] βγ 1+. This results from, or can be viewed as providing, the useful identity 11

12 1 exp[ (x / β) ]dx βγ 1+ (9) Noting that (x) exp[ (x / β) ] exp[ (x / λ) ] 1/, where λ β /, we can use the identity (9) to straight away write 1/ 1 ( β/ ) Γ 1+ (x) dx. It then follows that the Gini coefficient, using (), is γ 1. 1/ This is, of course, a well-known result; see e.g. Kleiber& Kotz (3, p177) for an alternative derivation. Similarly, if ~ Wei(, β 1 ) and ~ Wei(, β ) then (x) (x) exp [ ] 1 1 (x / β ) (x / β ) exp x + exp[ (x / λ) ] 1 β 1 β, where now β β 1 λ 1/ ( β + β ) 1. It then follows from the identity (9) that x) (x)dx β β 1 ( / 1 ( β + β ) 1 1 Γ 1+ Of course, the above derivation can be much simplified if we recall that the Weibull parameter β is a scale parameter and that the measures we are considering are (joint) scale invariant, see Theorem (iii). For instance, notice how λ in the above depends only on the ratio of the two β-values. Hence there is no loss in assuming that, say, β 1 β and β 1 throughout. With this assumption we find the equality ratio (Definition 4) as H (,) (x) + (x)dx (1 β 1 Γ 1+ 1/ + β ) β 1 (1 + β)(1 + β (1 + β) Γ 1+ ) 1/ 1

13 so that the co- Also, the product of the coefficients of equality is θ concentration coefficient is / θ Q (,) H(, ) θ + β β + β 1+ 1/ β (1 + β) 1/ θ (1 )(1 ) 1 + β 1/ Note the particular case where 1 gives the C-CC for the exponential distribution as 4β Q(, ), see Burrell (5a). (1 + β) Remark. In practice we have found that the graph of Q(, ) is fairly flat and so we recommend using its square for both illustrative and analytic purposes. (This is analogous to using the R measure, or coefficient of variation, rather than the basic correlation coefficient in correlation studies.) The graph of Q as a function of the scaling ratio β is given in Figure 1 for various values of. Note how in each case the peak, where Q 1, occurs when the scale ratio β 1, which corresponds to the two distributions coinciding. ******************* Insert Figure 1 about here **************************** For the modified relative concentration coefficient, using the formula (4) and the results above, routine algebra gives 1/ (1 + β ) 1 D *(,) if β 1 1/ (1 + β ) β (1 + β (1 + β ) ) 1/ 1/ β 1 if β > 1. 13

14 This is illustrated in Figure over the same range and for the same values of. Note the severely peaked nature of the graphs around β 1 for > 1. The differences between the general forms of the graphs reflect the different emphases of the two measures in assessing differences/similarities in concentrations. ************************* Insert Figure about here ********************* (b) Singh-Maddala distribution The Singh-Maddala (1975, 1976) distribution is a very flexible three-parameter income distribution model. (See Kleiber & Kotz (3) for a concise treatment of its various attributes.) For our purposes, an attractive feature is the simple form of its tail distribution function. Indeed, adopting the Kleiber & Kotz notation, if ~ SM(a, b, q) then q a x (x) 1 + and hence, as before, b q a x bγ(1 + 1/ a) Γ(q 1/ a) E[] (x)dx 1 + dx (1) b Γ(q) The final equality of (1) provides our useful identity. (Note that we have merely quoted the expression for the mean of the distribution, see Kleiber & Kotz (3, p1).) Clearly, then a x (x) dx 1 + b q bγ(1 + 1/ a) Γ(q 1/ a) dx Γ(q) using (1), so that the coefficient of equality is θ Γ(q) Γ(q 1/ a) Γ(q 1/ a) Γ(q) 14

15 Also if ~ SM(a, b, q) and ~ SM(a, b, p) then, using the same identity in (1) a x (x) (x)dx 1 + b and the equality ratio is then (q+ p) bγ(1 + 1/ a) Γ(q + p 1/ a) dx Γ(q + p) H(, ) (x) + (x)dx Γ(q + p 1/ a) Γ(q 1/ a) Γ(p 1/ a + Γ(q + p) Γ(q) Γ(p) 1 Note that both the coefficient of equality and the equality ratio do not involve the parameter b, as should be expected since it is a scale parameter for the SM distribution. From the above, it is clearly straightforward to derive a general expression for the C-CC although it is rather cumbersome and not too enlightening. However, certain special cases simplify matters greatly. For instance, if we take a 1 then we find after a little algebra that Q(, ) (q 1)(q 1)(p 1)(p 1) (q + p 1)(q + p ) It is now straightforward to plot this as a function of p > 1 for any value of q > 1 (to ensure finite means). This is illustrated in Figure 3, again using the squared form of the function. Notice that here we get Q 1 when p q, again corresponding to the two distributions coinciding. ***************************** Insert Figure 3 about here ********************* 15

16 For the modified relative concentration coefficient, rather than give the general form let us just stay with the special case where a 1 as considered above. Routine calculation leads, for a given value of q, to D* p/q if p < q, D* q/p if p > q. Thus the graph of D* is linear in p for p < q and is proportional to 1/p for p > q. See Figure 4 and again compare with the corresponding Figure 3 for the Q measure. Our view is that, once again, D * is rather harsh in distinguishing between the distributions, which is not surprising given that it hinges on the mean rather than the overall distribution. ****************** Insert Figure 4 about here ******************************* Remark. The reason that the Singh-Maddala example works so neatly in the calculation of the C-C coefficient above is that the tail distribution function is of the form (x) g(x) where is the sole parameter of interest. This then gives the identity g ( ), say, and then g ( ). Also if and belong to the same parametric family, with parameter values, β respectively, then g +β ( + β) Substituting into (8) then gives the C-C coefficient as Q(, ) ( + β) ( ) + ( β) ( ) ( β) () (β) Use of this formula allows us to straight away write down the C-CC for such as the exponential and Pareto distributions as well as the Singh-Maddala considered here. 16

17 5. Concluding remarks. In this paper we have merely defined and given some simple examples of the Q and D * measures and have not made any investigation of their statistical properties, although we hope to have convinced the reader of the superiority of Q as the more subtle measure. Nor have we considered empirical applications, though there are several possibilities, including: Within informetrics, examples of comparative studies over several data sets have been given in Burrell (5b), leading to a so-called co-concentration matrix. The analogous treatments of income/wealth distributions for different countries or for the same country during different years, maybe in real terms, are obvious applications. It would seem that it could also be used in investigative studies to assess the effects of (proposed) taxation changes or degrees of inflation. Again from informetrics, much use is made of time-dependent stochastic models in which case the entire distributional shape changes as the length of the time period increases. This means that concentration, as measured by the Gini index or illustrated via the Lorenz curve, also changes, see Burrell (199a, b). An investigation of the behaviour of the Q measure in such circumstances is currently in hand (Burrell, 5c). Are there similar models appropriate for income/wealth distributions? After all, if we double the period of observation, we (roughly) double the average income so how does the income distribution change? 17

18 Any conclusions regarding the efficacy of the Q measure must be tentative at this stage anything definitive requires further experience of its application and interpretation - but we are hopeful that it might be a useful additional tool in comparative studies. References Arnold, B. C. & Laguna, L. (1977). On generalized Pareto distributions with applications to income data. International Studies in Economics No. 1, Department of Economics, Iowa State University, Ames, Iowa. Burrell, Q. L. (1991). The Bradford distribution and the Gini index. Scientometrics, 1, Burrell, Q. L. (199a). The Gini index and the Leimkuhler curve for bibliometric processes. Information Processing and Management, 8, Burrell, Q. L. (199b). The dynamic nature of bibliometric processes: a case study. In I. K. Ravichandra Rao (Ed.), Informetrics 91: selected papers from the Third International Conference on Informetrics (pp ), Bangalore: Ranganathan Endowment. Burrell, Q. L. (5a). Measuring similarity of concentration between different informetric distributions: Two new approaches. Journal of the American Society for Information Science and Technology. (To appear.) Burrell, Q. L. (5b). Some empirical studies of the measurement of similarity of concentration between different informetric distributions. (Submitted for publication.) 18

19 Burrell, Q. L. (5c). Time-dependent aspects of the co-concentration coefficient. (In preparation.) Dagum, C. (198). Inequality measures between income distributions. Econometrica, 48, Dagum, C. (1983). Income inequality measures. In S. Kotz & N. S. Johnson (Eds.), Encyclopaedia of Statistical Sciences, Volume 4 (pp. 34-4), New ork: Wiley. Dagum, C. (1987). Measuring the economic affluence between populations of income receivers. Journal of Business and Economic Statistics, 5, Dalton, H. (19). The measurement of inequality of incomes. Economic Journal, 3, Dorfman, R. (1979). A formula for the Gini coefficient. Review of Economics and Statistics, 61, Egghe, L. & Rousseau, R. (199). Elements of concentration theory. In L. Egghe & R. Rousseau (Eds.), Informetrics 89/9: Selection of papers submitted for the Second International Conference on Bibliometrics, Scientometrics and Informetrics (pp ), Amsterdam: Elsevier. Gini, C. (1914). Sulla misura della concentrazione e della variabilità dei caratteri. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti, 73, Kleiber, C. & Kotz, S. (3). Statistical size distributions in economics and actuarial sciences. New Jersey: Wiley. Lambert, P. J. (1). The distribution and redistribution of income. 3 rd edition. Manchester: Manchester University Press. 19

20 Singh, M. & Maddala, G. S. (1975). A stochastic process for income distributions and tests for income distribution functions. ASA Proceedings of the Business and Economic Statistics Section, Singh, M. & Maddala, G. S. (1976). A Function for the size distribution of incomes. Econometrica, 44, Stirzaker, D. (1994). Elementary Probability. Cambridge: Cambridge University Press. Stuart, A., & Ord, J. K. (1987). Kendall s Advanced Theory of Statistics. Volume 1: Distribution Theory (5th edition). London: Griffin. itzhaki, S. (1998). More than a dozen alternative ways of spelling Gini. Research on Income Inequality, 8, 13-3.

21 Appendix 1. Proof of Theorem 1. Although it is straightforward to give a general proof, either using Lebesgue-Stieltjes integration or via the expectation operator (see Dagum, 1987), we prefer to use elementary methods and restrict attention to the (absolutely) continuous case. Suppose that and are independent copies of the variables. Then E[ ] x y f (x)f (y)dxdy Splitting the region of integration into {x>y} and {y>x}, the former yields, let us say p(,) x> y x ( x y)f (x)f (y)dxdy (x y)f (y)dy f (x) dx x x xf (y)dy yf (y)dy f (x)dx x xf (x) yf (y) x F (y)dy f (x)dx x x y F (y)dy f f f x (x)dx (x)f (y)dy dx (x)f (y)dx dy ( y)f (y)dy (y)( 1 (y))dy Interchanging the roles of x and y in the above leads straight to 1

22 p(,) y> x y x)f (x)f (y) dxdy ( y)f (y)dy (y)( 1 (y))dy ( Adding these two expressions gives E[ ] p(, ) + p(, ) x ( y) ( 1 (y))dy + y f (x)f (y)dxdy y) ( 1 (y)) dy ( ( (y) + (y) (y) (y))dy and the result follows. + + (A1). Proof of Proposition 5. Eliminating p(,) from (6) and (7), and rearranging, gives p(, ) and similarly p(, ) (E[ ] + (E[ ] ( Hence if >, ) / )) / D(, ) E[ ] ( E[ ] + ( ) ) Now substitute from (A1) and then dividing numerator and denominator by gives the result Proof of Theorem.

23 (i) H(, ) 1 G(, ) H(, ) Now (x) + (x) dx ( ) so that 4 (A) + ( + ) ( ) ( )( ) (A3) by the Cauchy-Schwarz inequality, variants of which can be found in most introductory texts on analysis, see also Stuart & Ord (1987, p. 65). Also ( (A4) + ) ( ) Combining (A3) and (A4) then gives, from (A) H(, ) ( ) ( )( ) 4 ( + ) and the result follows. θ (ii) For Q(,) 1, both of the above inequalities (A3) and (A4) must be equalities. For the second, trivially the equality holds if and only if. The Cauchy-Schwarz inequality reduces to an equality if and only if there is a constant c such that, for all x, ( x) c Proof of Theorem 1 above, this leads to c c θ (x). Then using the note at the end of the end of the. Having established the requirement that the two means must be the same, this implies that c 1 and hence the two distributions are the same. 3

24 Figure 1 : Q-squared for the Weibull distribution Q-squared.6 alpha 1/ alpha alpha Scale ratio, beta Figure. D-star for the Weibull distribution D-star.6.4 alpha 1/ alpha alpha Scale ratio, beta 4

25 Figure 3: Q-squared for the Singh-Maddala distribution Q-squared.6 q 1.5 q q Parameter value, p Figure 4: D-star for the Singh-Maddala distribution D-star.6 q 1.5 q q Parameter value, p 5