Sankhy¯ a : The Indian Journal of Statistics 1995, Volume 57, Series B, Pt. 3, pp.428–432

ON THE RELATIONSHIP BETWEEN THE GINI COEFFICIENT AND INCOME ELASTICITY By NRIPESH PODDER The University of New South Wales SUMMARY. The area between the Lorenz curve of income and the concentration curve of expenditure on some item is often used as an overall weighted index of income elasticity minus one. This note shows that the area is in fact a linear transform of the deviation of elasticity from unity estimated at the mean of income, and thus establishes the exact relationship between the area between the curves and elasticity. This facilitates an accurate estimation of the elasticity itself, instead of an index of its deviation, from the values of the concentration index and the Gini index.

1.

Introduction

In economics the deviation from proportioality of one variable with respect to another variable is measured by elasticity. Thus, if expenditure on a certain item is proportional to income (or total expenditure) then the income elasticity of demand, better known as Engel elasticity, is unity. On the other hand, if expenditure on the item rises more (less) than proportionately with income, Engel elasticity is greater (less) than one. Roy, Chakravarty and Laha (1959), and Mahalanobis (1960) were the first to show that the situation can be graphically represented by the positions of the Lorenz curve of income (total expenditure) and the concentration curve of expenditure on the item. Iyenger (1960) estimated the elasticities from the same curves by assuming lognormality of the distribution of total expenditure. While the concept of the Lorenz curve is well known, concentration curves are yet to gain popularity. When the cumulative proportion of expenditure (on an item) is shown against the cumulative proportion of population, the population having been arranged in order of income (total expenditure), we call it the concentration curve. In the case of unit elasticity, the Lorenz curve and the concentration curve coincide. On the other hand, if the concentration curve is below (above) the Lorenz curve, elasticity is greater (less) than unity. Therefore, it is natural that the area (or twice the area) between the concentration curve and the Lorenz curve is taken to be an index of the deviation of overall elasticity from unity. Let the Gini coefficient of income be denoted by Gx and the concentration index of expenditure on the

relationship between the gini coefficient and income elasticity 429

Paper Paper received. AMS (1980) Subject Classification 62P20 Key words and phrases

index of the deviation of overall elasticity from unity. Let the Gini coefficient of income be denoted by Gx and the concentration index of expenditure on the item be denoted by Cy . Then the index of the deviation of elasticity from unity is given by Iη−1 = Cy − Gx

. . . (1)

This formula has been used by a number of authors [see, e.g., Kakwani (1977, 1978)] to compute an index of Engel elasticity. When C represents the concentration index of taxes, the same formula is also widely used as a global measure of tax progressivity [for a full survey of the literature see Pfaler ˙ (1989)]. The first observation to be made is that the range of the above index is between −2 and 1, whereas the Engel elasticity, and therefore the deviation of elasticity from unity, ranges between −∞ and ∞. Since both of them are continuous, we can asume that they are related in a continuous manner. Second, intuitively it can be felt, since the index is the representation of the overall elasticity situation, there must be some weights attached at different levels of income in obtaining the aggreagate, or it may be elasticity at some specific value of income. The analysis in the next section, under same a priori assumption will establish the exact relationship between the relevant concentration indexes and the elasticity. 2.

Elasticity and the concentration ratio

Suppose X is a variable which is the sum of n components. It is welll established that when X = X1 + X2 + · · · + Xn XGx = X 1 C1 + X 2 C2 + · · · + X n Cn

. . . (2)

where Ci is the concentration index of Xi , and the bar sign over a variable represents its arithmatic mean. This result was first derived by Rao (1967). Since any proportional change in any variable leaves its Gini index or concentration index unchanged, it can be stated that Gx = Gax or Cx = Cax . For a linear function of X, such as Y = aX + b, it is easy to prove that Y Gy = a X Cx = aXCx

. . . (3)

when the observed values of the variables are arranged in ascending order of Y values. On the other hand if they are arranged by ascending order of X values we have Y Cy = a X Gx = aXGx . . . (4)

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nripesh poddar

since either the Gini index or the concentration index of a constant, such as b is zero. The elasticity of any given function, g(X) at any value of X is defined as: η=

Xg 0 (X) . g(X)

. . . (5)

When Y = g(X) = aX + b, (5) can be written as η=

Xg 0 (X) aX = . g(X) aX + b

. . . (6)

Suppose this elasticity is evaluated at the average value of X. Then the elasticity may be written as aX ηX = . . . (7) Y which, in view of the fact that Y Cy = aXGx , becomes ηX =

aX Cy = . Gx Y

. . . (8)

Thus, we have proved the suficiency part of the following theorem: Theorem. For a function Y = g(X), the elasticity of Y with respect to X evaluated at the mean of X can be estimated as the ratio of the concentration index of Y and the Gini index of X, if and only if the function g(X) is linear. Proof. To prove the necessity part of the theorem, suppose the sample values of the variable X are arranged in ascending order as, x1 , x2 , ...., xn and the corresponding values of Y are y1 , y2 , . . . , yn . In that case, a variant of the formula for the Gini index and the concentration index are respectively. n X

Gx =

n X

(2i − n − 1)xi

i=1

n2 X

and Cy =

(2i − n − 1)yi

i=1

n2 Y

.

Suppose, that for any given g(X), linear or nonlinear, the following equality holds. n X (2i − n − 1)yi /n2 Y 0 Cy g (X)X = i=1 = . . . (9) n X Gx g(X) 2 (2i − n − 1)xi /n X i=1

relationship between the gini coefficient and income elasticity 431 Notice that the left hand side of (9) is the definition of elasticity of g(X) at the mean value of X. Simplifying the above condition and using the fact that g(X) ≡ Y , we get n n X g(X) X (2i − n − 1)g(xi ) − g 0 (X) (2i − n − 1)xi = 0. g(X) i=1 i=1

. . . (10)

This can be further simplified as n X

(2i − n − 1)[αg(xi ) − βxi ] = 0,

. . . (11)

i=1

where α = g(X)/g(X) and β = g 0 (X). In view of the fact that n X

(2i − n − 1) = 0.

. . . (12)

i=1

Condition (11) will always be satisfied if the expression within the square bracket in (11), namely, αg(xi ) − βxi is either a zero or a constant for all i. In either of these cases g(X) must be a linear function which we have already proved. Now, suppose, g is a nonlinear function that generates all the sample value of Y except the jth sample value. Then (11), in conjuction with (12), will necessarily imply X αg(xj ) − βxj =

(2i − n − 1)g(xi )

i6=j

2j − n − 1

.

. . . (13)

Equation (13) tells us that the function g must also depend on the sample size as well as the rank of the sample value of X. Since the function g solely depends on X, nonlinearity of g is inadmissible. This completes the proof of the necessity part of the theorem. The theorem has an important corollary. Corollary. The sample statistic a = Y Cy /XGx is the concentration of the coefficient of a simple linear regression.

3.

Conclusion

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nripesh poddar

It then follows that the commonly used index of the deviation of Engel elasticity from unity, given by Equation (1), is obtained by multiplying the deviation of Engel elasticity by the Gini index of income (total expenditure). Iη−1 = Cy − Gx = Gx (ηX − 1)

. . . (14)

Thus, it is obvious that the ratio of the area under the concentration curve of expenditure on an item and the area under the Lorenz curve of income (total expenditure) as the Engel elasticity at the mean income is more informative and formula (9) rather than (14) should be used in empirical applications. It is proved that this elasticity is in fact based on a linear Engel function. The widely used tax progressivity measure is also nothing but the index of elasticity of the taxes as in (14), based on a linear tax function. Acknowledgement. The author gratefully acknowledges some helpful comments by an anonymous refree on an earlier version of this paper. Thanks are also due to Binh Tran-Nam and Pundarikaksha Mukhopadhyay for their help in revising the paper. The usual disclaimer, however, applies.

References

Iyenger, N. S. (1960). On a method of computing engel elasticities from concentration curves, Econometrica, 28, 881 - 893. Kakwani, N. (1977). On the estimation of engel elasticities from grouped observations with applications to indonesian data, Journal of Econometrics, 6, 1-17. Kakwani, N. (1978). A new method of estimating engel elasticities, Journal of Econometrics, 8, 103-110. Mahalanobis, P. C. (1960). A method of fractile graphical analysis, Econometrica, 28, 325-352. ´ ler, W. Redistributive effects of tax progressivity: evaluating a general class of aggregate Pfa measures, Public Finance / Finance Publiques, 37, 1-31. Rao, V. M. (1967). Two decompositions of concentration ratio, Journal of the Royal Statist. Soc., Ser. A, 132, 418-425. Roy, J., Chakravarty, I. M. and Laha, R. G. (1959). A study of concentration curves as a description of consumption patterns, In: Studies in Consumer Behaviour, Indian Statistical Institute, Calcutta.

The University of New South Wales U.S.A.