An Alternative Single Parameter Functional Form for Lorenz Curve

Transcription

1 Crawford School of Public Policy Crawford School working aers An Alternative Single Parameter Functional Form for Lorenz Curve Crawford School Working Paer 7 Setember 07 Satya Paul Amrita University, India Sriram Shankar Australian National University Abstract This aer rooses a single arameter functional form for the Lorenz curve and comares its erformance with the existing single arameter functional forms using Australian income data for 0 years. The roosed arametric functional form erforms better than the existing Lorenz functional forms. The Gini based on the roosed functional form is closest to true Gini each year. Keywords: Gini coefficient; Lorenz curve; Parametric functional form JEL Classification: C80, D3 THE AUSTRALIAN NATIONAL UNIVERSITY

2 Suggested Citation: Paul, S and Shankar, S. (07) An Alternative Single Parameter Functional Form for Lorenz Curve, Crawford School working aer 7, Setember 07. Crawford School of Public Policy, The Australian National University. Address for Corresondence: Name: Sriram Shankar Position: Fellow Address:. Beryl Rawson Building, ANU Centre for Social Research and Methods, Australian National University Sriram.Shankar@anu.edu.au Crawford School of Public Policy College of Asia and the Pacific The Australian National University Canberra ACT 000 Australia The Crawford School of Public Policy is the Australian National University s ublic olicy school, serving and influencing Australia, Asia and the Pacific through advanced olicy research, graduate and executive education, and olicy imact. THE AUSTRALIAN NATIONAL UNIVERSITY

3 An Alternative Single Parameter Functional Form for Lorenz Curve Satya Paul Amrita University, India and Sriram Shankar Australian National University Abstract This aer rooses a single arameter functional form for the Lorenz curve and comares its erformance with the existing single arameter functional forms using Australian income data for 0 years. The roosed arametric functional form erforms better than the existing Lorenz functional forms. The Gini based on the roosed functional form is closest to true Gini each year. Keywords: Gini coefficient; Lorenz curve; Parametric functional form JEL Classification: C80, D3 Corresonding author:. Beryl Rawson Building, ANU Centre for Social Research and Methods, Australian National University, Sriram.Shankar@anu.edu.au. Contact details for Satya Paul are: Centre for Economics and Governance, Amrita University, Kerala, India. satyaaul@outlook.com.

4 . Introduction The Lorenz curve reresents a grahical relationshi between the cumulative roortion of oulation and the cumulative roortion of income and forms the backbone of several inequality measures including the widely used Gini coefficient. World Institute of Develoment Economics Research (WIDER) and World Bank ublish data on income shares by decile or quintile grous of oulation for a large number of countries. Based on grou data, Lorenz curve can be constructed (i) by interolation techniques (Gastwirth and Glauberman, 976), (ii) by assuming a statistical distribution of income and deriving the Lorenz curve (McDonald, 984), or (iii) by secifying a arametric functional form for the Lorenz curve. The interolation techniques assume the homogeneity of incomes within sub-grous, thereby leading to a downward bias in Gini estimate. The existing income distribution functions are known to be oorly fitting the Lorenz curve and resulting in inaccurate inequality estimates. The arametric Lorenz functional forms are directly estimated with the grou data without assuming homogeneity of incomes within sub-grous and thus are not downwardly biased. Various authors have suggested a variety of arametric functional forms to directly estimate Lorenz curve (Kakwani and Podder, 973; Aggarwal, 984; Guta, 984; Chotikaanich, 993; Rohde, 009). It would be useful and interesting to determine which functional form erforms best. This is articularly imortant when the aim is to construct inequality measures based on Lorenz functional form. This aer rooses an alternative single arameter Lorenz functional form and comares its erformance with the existing single arametric functional forms. The Australian data are used to comare the goodness-of-fit of the widely used arametric Lorenz secifications and the one suggested here. Alternative Functional Forms for the Lorenz Curve If (x) is the roortion of individuals that receive an income u to x and η is the roortion of total income received by the same units, then the Lorenz curve is defined as η= L((x)) () The regularity conditions for the function L() to describe the Lorenz curve are as follows. (i) L(0) = 0, (ii) L() =, (iii) dl 0 d and (iv) dl 0 d >. Also see Chotikaanich (993).

5 3 Note that (i) and (ii) imly that the Lorenz curve is defined over the domain 0 and (iii) and (iv) suggest that the sloe of Lorenz curve is non-negative and monotonically increasing. We roose the following functional form for the Lorenz curve ( e ) e L( ) =, where γ > 0 () ( e e γγ ) It is easy to verify that Eq. () asses through the coordinate oints (0,0) and (,) and that the first and second derivatives are greater than zero. That is, L(0) = 0, L() = and γ( e ) ( ) γ e + e e L ( ) = + γ 0 (3) γ( e) γ( e) e e γ( e ) + γ( e ) + e e L ( ) = γ + γ ( ) ( γe + ) > 0 for γ > 0, 0 (4) γ e γ( e) e e This functional form is comared with the existing widely used single arameter functional forms roosed by Kakwani and Podder (973), Chotikaanich (993) and Rohde (009) 3 and a form imlied by Pareto distribution. Pareto: ( ) ( ) Kakwani-Podder: ( ) L = α, α > (5) ( ) L = e δ, δ > 0 (6) Chotikaanich: L( ) κ e =, κ > κ 0 e (7) Rohde: ( ) β L =, β > β The main motivation for fitting a Lorenz curve is to facilitate the estimation of inequality measures such as the Gini coefficient. This widely used index is defined as one minus twice the area under the Lorenz curve. Based on the roosed (new) and existing functional forms, Gini coefficients are exressed as follows: (8) 3 We do not consider here two other single arameter functional forms given by Aggrawal (984) and Guta (984). As ointed out in Sarabia et al (00), their functional forms do not add substantive value to the Lorenz curve comarison, as Rohde (009) is a rearameterization of Aggarwal (984), and Guta (984) that of Kakwani and Podder (973).

6 4 Proosed: ( ) γ e e G = d (9) 0 γ ( e) e G = α d 0 (0) Pareto: ( ) Kakwani-Podder: 0 δ ( ) G = e d () Chotikaanich: κ ( κ )e + ( κ+ ) G = κ κ(e ) () Rohde: G β ( β ) β = ln + β (3) 3. Performance of Alternative Functional Forms of Lorenz Curve Table resents the Australian income data by decile grous for Based on these data, the roosed and other four functional forms of Lorenz curve (equations and 5 to 8) are estimated using the non-linear least squares. All the arameter estimates are statistically significant at the % level (Table ). A comarison of these estimated functional forms is done based on two statistics. (i) Information Inaccuracy Measure (I) = N qiln(q ˆ i / q i) where i= qi and ˆq i denote actual and redicted income shares. The estimated function with smaller value of I is better than those with larger values. N criterion (ii) Mean Square Error (MSE) = η i L(, ˆ i θ) N. N = i It is always non-negative, and values closer to zero are better. Both the statistics are measures of goodness-of-fit. 4 These grou data were constructed using the individual income data from the first ten waves (00-00) of Household, Income and Labour Dynamics Australia (HILDA) Survey.

7 5 In terms of measure I, the roosed functional form erforms best followed by Kakwani- Podder, Chotikaanich, Rohde and Pareto resectively in each year (Table 3). In terms of MSE, the roosed functional form erforms best in seven years, 00 to 006 and 009, whereas Kakwani-Podder functional form erforms best in the other 3 years (Table 4). In 007 and 00, the Chotikaanich functional form erforms the second best. The worst erformance is shown again by the Pareto functional form. Table : Actual Income Shares by Decide Grous, Decide Grous Source: These decide grous are constructed using the data on individual income from first 0 waves of HILDA Survey. Table : Estimates of Lorenz Parameters (00-00) Proosed Rohde Chotikaanich Pareto Kakwani-Podder Year γ SE β SE κ SE α SE δ SE

8 6 Table 3 Information Inaccuracy Measure Year Proosed Rohde Chotikaanich Pareto Kakwani-Podder Table 4 Mean Squared Errors (MSE) Year Proosed Rohde Chotikaanich Pareto Kakwani-Podder The true Ginis and the estimated Ginis based on alternative functional forms of Lorenz curve are statistically significant at the % level. (Table 5). For each year, the Gini based on the roosed Lorenz curve secification is closest to true Gini. The Gini based on the Kakwani- Podder functional form is second closest to true Gini. The Ginis based on the Chotikaanich, Rohde and Pareto Lorenz curve functional forms rank resectively third, fourth and fifth in terms of their closeness to true Gini.

9 7 Year TRUE SE Proosed SE Rohde SE Table 5: Estimates of Gini (00-00) Chotikaanich SE Pareto SE Kakwani- Podder Note: True Ginis are calculated using the data on individual income from the first 0 waves of HILDA Survey. Standard errors for true Ginis are calculated based on a method rovided in Giles (004). The exressions for the variances for Ginis corresonding to the alternative arametric functional forms of Lorenz curve are resented in Aendix A. SE 4. Concluding Remarks The Australian data show the sueriority of the roosed Lorenz curve functional form over other functional forms. In terms of information inaccuracy measure, the roosed form outerforms all the four functional forms in all the 0 years, However, in terms of MSE, the roosed Lorenz functional form erforms best in 7 years whereas the Kakwani-Podder functional form erforms best in the other 3. The Gini based on our Lorenz curve secification is closest to true Gini each year.

10 8 Aendix A The standard error for Gini coefficient is calculated using the asymtotic aroximation: ˆ dg var(g) = Vθ, where V θ is the asymtotic variance of the single arameter θ estimated dθ using non-linear least squares for the various functional forms of Lorenz curve. Chotikaanich: ( ( ) ) κ e e κ + Var ( G) = κ ( κ ( e ) ) Var ( κ ). Rohde: dg Var ( G) = Var( β ) dβ dg β = ln + dβ β.,where ( β ) Kakwani-Podder: dg Var ( G) = Var( δ ) dδ dg dδ, where = ( ) δ ( ) e d. 0 Pareto: ( ) dg Var ( G) = Var α dα dg = ln dα α, where ( )( ) α d. 0 Proosed: dg Var ( G) = Var( γ ) dγ. where γ ( e ) dg e e e = d 0 γ ( e ) γ( e) γ( e) dγ +. e e e Acknowledgements This article uses unit record data from the HILDA Survey. The HILDA Project was initiated and is funded by the Australian Government Deartment of Families, Housing, Community Services and Indigenous Affairs (FaHCSIA) and is managed by the Melbourne Institute of Alied Economic and Social Research (Melbourne Institute). The findings and views reorted in this article, however, are those of the authors and should not be attributed to either FaHCSIA or the Melbourne Institute.

11 9 References Aggarwal, V., 984. On otimum aggregation of income distribution data. Sankhya B, 46, Chotikaanich, D., 993. A comarison of alternative functional forms for the Lorenz curve. Economics Letters, 4, 9. Gastwirth, J., Glauberman, M., 976. The interolation of the Lorenz curve and Gini index from groued data. Econometrica, 44, Giles, D. E. A., 004. Calculating a standard error for the Gini coefficient: some further results. Oxford Bulletin of Economics and Statistics, 66, Guta, M.R., 984. Functional form for estimating the Lorenz curve. Econometrica, 5, Kakwani, N.C., Podder, N., 973. On the estimation of Lorenz curves from groued observations. International Economic Review, 4, McDonald, J. B., 984. Some generalized functions for the size distribution of income. Econometrica, 5, Rohde, N., 009. An alternative functional form for estimating the Lorenz curve. Economics Letters, 05, Sarabia, J.M., Prieto, F., Sarabia, M., 00. Revisiting a functional form for the Lorenz curve. Economics Letters, 07, 49-5.