We propose a simple single parameter functional form for the Lorenz curve. The underlying probability density function and cumulative density functions for the Lorenz curve are derived and are shown to have some useful properties. The proposed functional form is fitted to existing data sets and is shown to provide a better fit than existing single parameter Lorenz curves for the given data

Rohde, Nicholas C.

University of Queensland eSpace

AN ALTERNATIVE FUNCTIONAL FORM FOR 
ESTIMATING THE LORENZ CURVE 
 
Nicholas Rohde 
School of Economics 
The University of Queensland 
Brisbane, Australia 
July 2008 
 
    Email: n.rohde@uq.edu.au 
    Telephone: +61 042 229 7225  
 
We propose a simple single parameter functional form for the Lorenz 
curve.  The underlying probability density function and cumulative 
density functions for the Lorenz curve are derived and are shown to 
have some useful properties.  The proposed functional form is fitted to 
existing data sets and is shown to provide a better fit than existing 
single parameter Lorenz curves for the given data. 
 
 
Author key words: Gini coefficient; Parametric form; Lorenz curve 
 
JEL classification codes: C80, D31 
 
 
1. INTRODUCTION 
 
The Lorenz curve is an intuitive method for representing the distribution of income.  
Created by plotting cumulative income shares against cumulative population shares, the 
Lorenz curve forms the backbone of several inequality measures including the popular 
Gini coefficient.  Lorenz curves may be constructed from grouped data using 
interpolation techniques (Gastwirth, 1976) or may be presumed to follow a particular 
parametric form and fit to tabulated data (see Kakwani and Podder (1976), Rasche et al. 
(1980) Ortega et al. (1991)).  Parametric forms such as these are advantaged over Lorenz 
curves constructed directly from grouped data as they do not assume homogeneity of 
incomes within subgroups and thus are not downwardly biased (Lerman and Yitzaki, 
1989).   These techniques however face the disadvantage of imposing a rigid and in some 
cases unrealistic distribution upon the data and may result in poorly fitting Lorenz curves 
and inaccurate inequality estimates.   
 
In this paper we propose an alternative single parameter functional form for the Lorenz 
curve and derive the implicit probability density function (PDF) and cumulative 
distribution function (CDF).  We argue that these functions take appropriate shapes for 
modeling the distribution of income and demonstrate this by showing that the proposed 
method provides a better fit than other comparable Lorenz curves for our data set.  
2. FUNDAMENTALS OF THE LORENZ CURVE 
  
A Lorenz curve may be defined as 
 
         )(πη f=             (1) 
 
where  π is the cumulative population share of persons earning income equal to or 
  below income level x. 
  η is the cumulative income share of population subgroup π. 
 
A Lorenz curve must have the following properties: 
 
               0>π
η
d
d ,            02
2
>π
η
d
d ,          ( ) 00 =η ,         ( ) 11 =η  
 
and is defined on the domain 10 ≤≤ π  
 
The most popular single parameter Lorenz curves are the forms proposed by Kakwani 
and Podder (1973), Gupta (1984) Chotikapanich (1993) and a form implied by the Pareto 
distribution.  These are: 
 
Kakwani-Podder:     ( ) ( )πδππη −−= 1e ,   0>δ                       (2) 
 
Gupta:    ( ) 1−= πππη a ,   1>a             (3) 
 
Chotikapanich:  ( )
1
1
−
−= k
k
e
e ππη  ,    0>k            (4) 
 
Pareto:    ( ) ( )γππη 111 −−= ,     1>γ            (5) 
 
Here we propose the functional form 
 
    ( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
−
−= πβ
βππη 1 ,    1>β            (6) 
 
It is simple to verify that (6) passes through the coordinate points (0,0) and (1,1) and that 
the first and second derivatives are greater than zero. The derivatives are: 
 
  ( ) πβ
β
πβ
βππη −
−+⎟⎟⎠
⎞
⎜⎜⎝
⎛
−
−= 11)(' 2  >0 for 1>β ,  10 ≤≤ π         (7) 
 
  ( )
( )
( )23
1212)('' πβ
β
πβ
βππη −
−+⎟⎟⎠
⎞
⎜⎜⎝
⎛
−
−=  >0 for 1>β ,  10 ≤≤ π         (8) 
 
Using Kakwani’s (1980) result ( )μ
πη ')( =Fx  we can derive the implicit PDF and CDF 
for this Lorenz curve. 
 
The probability density function in terms of income (x) with average income μ is: 
 
                 
x
x
xf α
αα
22
);( =    for             ( )22 1−≤≤ β
α
β
α x          (9) 
     where         ( )μββα 1−=         (10) 
 
The cumulative distribution function is: 
 
                  
x
xF αβ −=)(                  for        ( )22 1−≤≤ β
α
β
α x                             (11) 
 
The PDF of income defined in (9) has some unusual properties.  Ostensibly this function 
of income is manipulated by the single parameter α, which depends on mean income 
level μ and Lorenz curve parameter β.  Such a parameterization appears flawed as 
different combinations of β and μ can yield the same curve for equation (9).  This does 
not imply that density functions with the same value for parameter α will be identical 
however, as the PDF is only defined on the domain ( )maxmin xxx ≤≤  where the lower and 
upper bounds depend on both β and μ.  As such each combination of β and μ defines a 
unique PDF for x, which is typically downward sloping over the domain in a manner 
similar to an exponential decay function. 
 
Lorenz curves such as proposed in (6) that imply probability distribution functions that 
only exist on a subset of x are not uncommon, with the Chotikapanich, Gupta and 
Kakwani-Podder functional forms also exhibiting positive lower bounds and finite upper 
bounds.  Such restrictions on x need not be unrealistic as institutional structures such as 
social welfare systems, minimum wages and high marginal tax rates may effectively 
constrain incomes to lie within certain bounds.  
 
A further interesting property of the domain of equation (9) is that the distribution mean 
may be calculated directly from the upper and lower limits. The bounds on x are: 
 
2min β
α=x    ( )2max 1−= β
αx  
 
Solving these expressions with equation (10) gives the result  
 
     maxmin xx=μ          (12) 
 
-that the mean of the implicit PDF is equal to the geometric average of the highest and 
lowest incomes available under the distribution. 
 
3. A COMPARISSON OF SINGLE PARAMETER FUNCTIONAL FORMS 
 
In this section we estimate the single parameter Lorenz curves given in Section (2) using 
decile data from Chotikapanich et al. (2005).  The data covers incomes statistics eight 
Asian countries in 1988 and 1993 and is referred to in an abbreviated form in Table 1. 
The abbreviations are: Hong Kong-HK, Japan-JP, Korea-KR, Malaysia-ML, Philippines-
PH, Singapore-SG, Taiwan-TW and Thailand-TL.  Each Lorenz curve is fitted to every 
data set and we follow Sarabia et al. (2001) by measuring the goodness of fit with the 
Mean Squared Error. We calculate this as 
          
( )( )∑
=
−=
n
i
ii
n
MSE
1
2πηη
         (13) 
 
where iη is the cumulative income share of population group i calculated from raw data 
and ( )iπη  is the fitted value of the Lorenz curve at iπ . We exclude the Gupta Lorenz 
curve from this analysis as it can be shown to be functionally equivalent to the Kakwani-
Podder specification and thus gives identical goodness of fit statistics1. The results are 
presented in Table 1, with parameter estimates and goodness of fit statistics given for 
each Lorenz curve. 
 
Table 1. A comparison of functional forms for the Lorenz curve 
Data 
set 
Kakwani-Podder Chotikapanich Pareto Proposed 
δ MSE k MSE γ MSE β MSE 
HK88 2.312 0.00196 3.302 0.00165 2.719 0.00068 1.282 0.00044 
HK93 2.703 0.00270 3.737 0.00233 3.039 0.00058 1.230 0.00062 
JP88 1.031 0.00027 1.700 0.00021 1.721 0.00060 1.774 0.00009 
JP93 1.064 0.00036 1.747 0.00029 1.749 0.00052 1.743 0.00012 
KR88 1.378 0.00068 2.171 0.00056 1.987 0.00067 1.542 0.00023 
KR93 1.219 0.00025 1.957 0.00020 1.852 0.00101 1.637 0.00011 
ML88 2.206 0.00138 3.180 0.00112 2.620 0.00096 1.301 0.00024 
ML93 2.334 0.00148 3.326 0.00121 2.720 0.00101 1.280 0.00025 
PH88 1.216 0.00057 1.956 0.00046 1.865 0.00047 1.631 0.00015 
PH93 1.974 0.00115 2.910 0.00092 2.438 0.00090 1.347 0.00020 
SG88 1.117 0.00033 1.820 0.00025 1.785 0.00060 1.702 0.00008 
SG93 1.939 0.00098 2.868 0.00077 2.407 0.00104 1.356 0.00017 
TW88 1.136 0.00045 1.847 0.00036 1.804 0.00051 1.686 0.00015 
                                                 
1 Setting αδ ln=  allows the Kakwani-Podder specification given in equation (2) to be written in terms of 
the Gupta specification in equation (3).  
TW93 1.149 0.00040 1.864 0.00032 1.811 0.00058 1.678 0.00012 
TL88 2.023 0.00150 2.968 0.00123 2.486 0.00068 1.334 0.00031 
TL93 2.298 0.00176 3.286 0.00145 2.700 0.00075 1.285 0.00032 
 
 
The results demonstrate the capability of the proposed functional form to closely model a 
variety of different data sets.  The new Lorenz curve provides the best fit (as measured by 
MSE) of the four single parameter forms in 15 of the 16 considered cases (indicated on 
the table in bold) with the Pareto Lorenz curve being slightly superior for Hong Kong 
data in 1993.  The Chotikapanich and Pareto specifications appear roughly equivalent at 
fitting the given data while the Kakwani-Podder functional form was the poorest 
performer. 
 
4. CONCLUSION 
 
The proposed functional form appears to be a worthy addition to the existing class of 
single parameter Lorenz curves.  The new specification is shown to meet the required 
regularity conditions for a Lorenz curve and demonstrates a strong capacity for modeling 
income data.  The ability for this Lorenz curve to effectively model data is likely to be 
due a similarity of shapes behind the underlying PDF and typical income distributions. 
 
 
REFERENCES 
 
 
Chotikapanich, D, 1993 “A Comparison of Alternative Functional Forms for the Lorenz 
Curve” Economics Letters 41, pp. 21-29. 
 
Chotikapanich, D, Griffiths, B, and Rao, D.S, 2007 “Estimating and Combining National 
Income Distributions Using Limited Data” Journal of Business and Economic Statistics 
25, pp. 97-109. 
 
Gastwirth, J and Glauberman, M, 1976 “The Interpolation of the Lorenz Curve and Gini 
Index from Grouped Data” Econometrica 44, pp. 479-483. 
 
Gupta, M.R, 1984 “Functional Form for Estimating the Lorenz Curve” Econometrica 52, 
pp. 1313-1314. 
 
Kakwani, N.C, 1980 “Income Inequality and Poverty: Methods of Estimation and Policy 
Applications” Oxford University Press, 1980. 
 
Kakwani, N.C, and Podder, N, 1973 “On the Estimation of Lorenz Curves from Grouped 
Observations” International Economic Review 14, pp. 278-292. 
 
Kakwani, N.C, and Podder, N, 1976 “Efficient estimation of the Loren curve and 
associated inequality measures from grouped observations” Econometrica 44 pp.137–
148. 
 
Lerman, R and  Yitzhaki, S, 1989 “Improving the accuracy of estimates of Gini 
coefficients”  Journal of Econometrics 42, pp. 43-47. 
 
Ortega, P.  Martín, A.  Fernández, A.  Ladoux, M. and García, A, 1991 “ A new 
functional form for estimating Lorenz curves” Review of Income and Wealth 37, pp. 447-
452. 
 
Rasche, R. Gaffney, J.  Koo, AYC and Obst, N, 1980  “Functional Forms for Estimating 
the Lorenz Curve” Econometrica 48, pp.1061-1062. 
 
Sarabia, J. M,  Castillo, E. and Slottje, D.J, 1999 “An ordered family of Lorenz curves” 
Journal of Econometrics 91, pp. 43–60. 
 
 
 
 
 
 
 
 
 
 


An alternative functional form for estimating the Lorenz Curve

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An alternative functional form for estimating the Lorenz Curve

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