Copyright © CC-BY-NC 2019, CRIBFB | AMFBR 

American Finance & Banking Review; Vol. 4, No. 1; 2019 
ISSN 2576-1226    E-ISSN 2576-1234 

Published by Centre for Research on Islamic Banking & Finance and Business, USA 

 

     17 
 

 

Functional Form for Estimating the Lorenz Curve 
 
 

Bijan Bidabad  
B.A., M.Sc., Ph.D., Post-Doc.  

Professor  
Economics and Chief Islamic Banking Advisor  

Bank Melli, Iran 
E-mail:bijan@bidabad.com 

 
Behrouz Bidabad 

Faculty of Mathematics 
Polytechnics University, Hafez Ave 

 Tehran, 15914, Iran 
 E-mail:bidabad@aut.ac.ir 

 

 
 
Abstract 
A flexible Lorenz curve which offers different curvatures allowed by the theory of income distribution is introduced. The 
intrinsically autoregressive nature of the errors in cumulative data of the Lorenz curve is also under consideration. 
 
Keywords: Lorenz Curve, Functional Form, Estimation, Income Inequality, Distribution 
 

1. Introduction 
Income distribution is often portrayed on a Lorenz curve. In recent years some of its functional forms have been introduced. 
These forms should satisfy some definitional properties, and also make estimation of the function parameters by the known 
estimating methods simple. This note emphasizes on two other characteristics of the Lorenz curve which have been neglected. 
First, the Lorenz curve could be non-symmetric with respect to the line y=1-x, for 0≤x≤1. This enables that different Lorenz 
curves cross the others which are the same in functional form and different in parameters for 0<x<1 (see E. E. Hagen(2)).  
Figure 1 shows this phenomenon by two different Lorenz curves of A and B. 

y=1-x

y=x

x
100%

100%

y

B

A

 
Figure 1. 

mailto:bijan@bidabad.com
mailto:bidabad@aut.ac.ir


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18 
                         
 

M. R. Gupta (1) proposed the following definitional properties. The function y=f(x) represents the Lorenz curve, if: 

(i)     f(0) = 0 

(ii)    f(1) = 1 

(iii)   f’(x) ≥ 0     for    0≤x≤1 

(iv)   f”(x) ≥ 0    for    0≤x≤1 

(v)    f(x) ≤ x      for    0<x<1 

(vi)   0 ≤ ∫ 𝑓(𝑥). 𝑑𝑥
1

0
≤

1

2
  

It is obvious that (vi) is redundant; when (i) to (v) are satisfied. Because by manipulating (i) to (v) we have: 0 ≤ f(x) ≤ x. By 
integrating this inequality, (vi) is derived: 

∫ 0 𝑑𝑥
1

0
≤ ∫ 𝑓(𝑥). 𝑑𝑥

1

0
≤ ∫ 𝑥. 𝑑𝑥

1

0
  

Or: 

0 ≤ ∫ 𝑓(𝑥). 𝑑𝑥
1

0

≤
1

2
 

Therefore property (vi) is always satisfied, and we need no more to test (vi) for any function which satisfies (i) to (v).  

Let’s review the proposed functional forms: 

Kakwani et al. (4):  

𝑀 = 𝑎. 𝑁𝑙 (2 1/2 − 𝑁)𝑐         where  𝑀 =
𝑥−𝑦

21/2;         𝑁 =
𝑥−𝑦

21/2;       a≥ 0;   0 ≤ l ≤ 1;     0≤ c ≤1 .  

This form does not satisfy all the properties. 

Rasche et al. (5): 

𝑦 =  [1 − (1 − 𝑥)𝑎]1/𝑙     where 0≤ a ≤1; 0 ≤ l≤ 1.  

This form makes the estimation of the parameters by the least squares method difficult. 

Gupta (1): 

𝑦 = 𝑥. 𝐴𝑥−1     where A>1 

This form satisfies definitional properties and simply can be estimated by ordinary least squares method; but by changing the 
parameter A (from Ai to Aj), the resulted functions (yi and yj) will never intersect for 0 < x < 1. To prove this, we can solve the 
following system: 



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19 
                         
 

{
𝑦 = 𝑥. 𝐴𝑖

𝑥−1

𝑦 = 𝑥. 𝐴𝑗
𝑥−1 

Solutions are x = y = 0 and x = y =1 which are not in the domain 0 < x < 1. 

2. Proposition 
This note suggests the following functional form which satisfies the definitional properties (i) to (v); and by changing its 
parameters, the resulting curves may cross each other: 

𝑦 = 𝑥𝐵 . 𝐴𝑥−1     where    B≥1;   A≥1      for   0 < x ≤ 1 

Definitional properties satisfy as follows: 

(i)     f(0) = 0 

(ii)    f(1) = 1 

(iii)   f’(x) = 𝑥𝐵−1. 𝐴𝑥−1 (B + x.logA) > 0     for    0<x<1  

(iv)   f’’(x) = 𝑥𝐵−2. 𝐴𝑥−1 [(B +  x. logA)2 – B] ≥ 0     for    0≤ x ≤1 

(v)    f(x) = 𝑥𝐵 . 𝐴𝑥−1 = 
𝑥𝐵

𝐴1−𝑥 
 ≤ 𝑥      for 0 < x < 1  

Different shapes of the function as y = 𝑥𝐵𝑖 . 𝐴𝑖
𝑥−1 and y = 𝑥𝐵𝑗 . 𝐴𝑗

𝑥−1 may have intersection for 0 <  x < 1. By solving the 

following system: 

{
𝑥𝐵𝑖 . 𝐴𝑖

𝑥−1

𝑥𝐵𝑗 . 𝐴𝑗
𝑥−1                  (1) 

We get: 

𝑥−1

𝑙𝑜𝑔𝑥
 = 

𝐵𝑗− 𝐵𝑖

𝐿𝑜𝑔(𝐴𝑖 𝐴𝑗⁄ )
           (2) 

It is obvious when (2) is satisfied; there is an intersection between two curves of (1). So if we solve the following system  

{
𝑥 − 1 = 𝐵𝑗 −  𝐵𝑖

𝑙𝑜𝑔𝑥 = log (𝐴𝑖 𝐴𝑗⁄ )
             (3) 

We can find a relation in terms of 𝐴𝑖  , 𝐴𝑗  , 𝐵𝑖 and 𝐵𝑗 which satisfies (2). Therefore: 

𝐴𝑖 𝐴𝑗  ⁄ − 1 =  𝐵𝑗 − 𝐵𝑖             (4) 

Hence, when 𝐴𝑖 , 𝐴𝑗  , 𝐸𝑖 and 𝐵𝑗 can satisfy equation (4), there is a solution (or intersection) for (1). But the intersection is inside 

the domain 0 > x > 1 when: 

By using (3) and (4) 



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{
0 < 𝑥 =  𝐵𝑗 − 𝐵𝑖 + 1 < 1

0 < 𝑥 = (𝐴𝑖 𝐴𝑗⁄ ) < 1
 

Or: 

{
0 <  𝐵𝑗 −  𝐵𝑖 < 1

0 <  𝐴𝑖 <  𝐴𝑗
 

The second thing that has been ignored is the autoregressive nature of the errors in the Lorenz curve data. On the other hand, 

when there is an error in the (𝑡 − 1)th percent of income earners, this error completely will transfer to the next cumulative 

percent (t). This is because of using cumulative data to estimate the Lorenz curve. So if we define 𝑢𝑡 as disturbance term of the 
tth observation (cumulative percent), autoregressive specification of the error would be: 

𝑢𝑡 =  𝑢𝑡−1 +  𝑉𝑡   

with 𝑉𝑡  obeying classical assumptions of regression. Therefore the stochastic form of our suggested functional form could be as 
follow: 

𝑦𝑡 =  𝑥𝑡
𝐵 . 𝐴𝑥𝑡−1. 𝑒𝑢𝑡                        (5) 

or: 

𝑦𝑡−1 =  𝑥𝑡−1
𝐵 . 𝐴𝑥

𝑡−1−1 . 𝑒𝑢𝑡−1             (6) 

Dividing (5) by (6) and taking natural logarithm: 

Log(
𝑦𝑡

𝑦𝑡−1
) = 𝐵. log (

𝑥𝑡

𝑥𝑡−1
) + 𝑙𝑜𝑔𝐴. (𝑥𝑡 − 𝑥𝑡−1) + 𝑢𝑡 − 𝑢𝑡−1          (7) 

Since 𝑢𝑡 − 𝑢𝑡−1 = 𝑣𝑡 and E(𝑣𝑡 , 𝑣𝑡−1) = 0 the problem pf autoregression has been discarded and (7) can be estimated by 
Ordinary Least Squares easily. 

References  
Gupta, M.R. “Functional Forms For Estimating the Lorenz Curve”, Econometrica. 52(1984), 1313-1314. 
Hagen, E.E. “The Economics of Development”. Richard D. Irwin, Inc. Homewood, Illinois. 1975, 216-217. 
Kakwani, N.C. “Functional Form for Estimating the Lorenz Curve: A Reply”. Econometrica, 48, (1980). 1063-1064. 
Kakwani, N.C.;  N. Podder, “Efficient Estimation of the Lorenz Curve and Associated Inequality Measures from Grouped 

Observations.” Econometrica. 44(1976), 137-148. 
Rasche, R.H., J. Gaffney, A.Y.C. Koo, Anan. Ofst: “Functional Forms for Estimating the Lorenz Curve,” Econometrica, 

48(1980), 1061-1062. 
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Bidabad, Bijan, Estimating Lorenz curve for Iran by using continuous L1 norm estimation, Economics and Management 
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http://www.bidabad.com/doc/iraninc-l1.pdf  
Bidabad, Bijan, USA Income distribution counter-business-cyclical trend (Estimating Lorenz curve using continuous L1 norm 

estimation). First meeting of the Society for the Study of Economic Inequality (ECINEQ), Palma de Mallorca, 
Spain, July 20-22, 2005. 

http://www.uib.es/congres/ecopub/ecineq/general.html 
http://www.uib.es/congres/ecopub/ecineq/papers/039Bidabab.pdf  
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http://www.bidabad.com/doc/l1-article4.pdf
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Bidabad, Bijan, Hamid Shahrestani. An implied inequality index using L1 norm estimation of Lorenz curve. Global Conference 
on Business and Finance Proceedings. Mercedes Jalbert, managing editor, ISSN 1931-0285 CD, ISSN 1941-9589 
Online, Volume 3, Number 2, 2008, The Institute for Business and Finance Research, Ramada Plaza Herradura, San 
Jose, Costa Rica, May 28-31, 2008, pp. 148-163. Global Journal of Business Research, Vol. 4, No. 1, 2010, pp.29-
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http://www.bidabad.com/doc/L1-Implied-inequality-index-4.pdf 
http://www.theibfr.com/archive/ISSN-1941-9589-V3-N2-2008.pdf 
http://www.bidabad.com/doc/SSRN-id1631861.pdf 
 
 
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http://www.bidabad.com/doc/L1-Implied-inequality-index-4.pdf
http://www.theibfr.com/archive/ISSN-1941-9589-V3-N2-2008.pdf
http://www.bidabad.com/doc/SSRN-id1631861.pdf