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American Finance & Banking Review; Vol. 4, No. 2; 2019 
ISSN 2576-1226    E-ISSN 2576-1234 

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

     11 
 

 

USA Income Distribution Counter-Business-Cyclical Trend 

(Estimating Lorenz curve using Continuous L1 norm estimation) 
 
 

Bijan Bidabad 

Professor  
Economics and Chief Economic Advisor  

Bank Melli Iran 

E-mail: bijan@bidabad.com 

 
Abstract 
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are 
defined. The continuous L1 norm estimation problems of linear one and two parameters models are solved. We proceed to use 
the functional form and parameters of the probability distribution function of income to exactly determine the L 1 norm 
approximation of the corresponding Lorenz curve of the statistical population under consideration. U.S. economic data used to 
estimate income distribution. An interesting finding of these calculations is that the distribution of income obeys counter-wise 
business cycles fluctuations. This finding is a new area for research in the realm of the theory and application of income 
distribution and business cycles interrelationship. 

Keywords: Income Distribution, Lorenz Curve, L1 norm statistics, Business Cycle  

 
JEL: C63 
 
1. Introduction 
The skewness of income distribution is persistently exhibited for different populations and at different times. It is discussed that 
Pearsonian family distributions are rival functions to explain income distribution. Lorenz curve is a method to analyze the skew 
distributions. There is a relation between the area under the Lorenz curve and the corresponding probability distribution 
function of the statistical population (see, Kendall and Stuart (1977)). That is, when the probability distribution function is 
known, we may find the corresponding Gini index as the measure of inequality. 
     Estimation of the Lorenz curve is confronted with some difficulties. For this estimation, we should define an appropriate 
functional form which can accept different curvatures (see, Bidabad and Bidabad (1989a,b)). There is another problem, that is , 
to create the necessary data set for estimating the corresponding parameters of the Lorenz curve, a large amount of computation 
on raw sample income data is inevitable. Obviously, these problems, despite their computational difficulties, make the 
significance of the estimated parameters poor (see, Bidabad and Bidabad (1989a,b)). To avoid this, we try to estimate the 
functional form of the Lorenz curve by using continuous information. In this paper, we use the probability density function of 
population income to estimate the Lorenz function parameters. The continuous L1 norm smoothing method, which will be 
developed for estimating the regression parameters, is used to solve this problem. However, we concentrate on two rival 
probability density functions of Pareto and log-normal. Since the former is simply integrable, there is no general problem to 
derive the corresponding Lorenz function, and the function is uniquely derived. But in the latter case, the log-normal density 
function (which has better performance for full income range) than Pareto distribution (which better fits to higher income 
range, (see, Cramer (1973), Singh and Maddala (1976), Salem and Mount (1974)), is not integrable and we can not determine 
its corresponding Lorenz function. In this regard, we should solve the problem by defining a general Lorenz curve functional 
form and applying the L1 norm smoothing to estimate the corresponding parameters. 
     In this paper, continuous L1 norm estimation is developed by using a similar method proposed in Bidabad (1987a,88a,89a,b) 
for the discrete case. Then the method is applied to the estimation of the Lorenz curve functional forms which have been 
proposed by Gupta (1984) and Bidabad and Bidabad (1989,92). In the end, we use our formulation to estimate Gini index and 
Kakwani length indices of inequality for the United States for the period of 1971-1990, based on the assumption that income is 
distributed log-normally. 
 
2. L1 norm of continuous functions 
Generally, Lp norm of a function f(x) (see, Rice and White (1964)) is defined by, 

         ||f(x)||p = ∫ xεI (|f(x)|pdx)1/p                       (1) 

mailto:bijan@bidabad.com


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Where, "I" is a closed bounded set.  The L1 norm of f(x) is simply written as, 

         ||f(x)||1 =  ∫ xεI |f(x)|dx                                  (2) 

Suppose that the non-stochastic function f(x,β) of "x", is combined with stochastic disturbance term "u" to form y(x) as 
follows, 

     y(x) = f(x, β) + u                                                                                     (3) 

Where, β is unknown parameters vector.  Rewriting u as the residual of y(x)-f(x,β), for L1 norm approximation of "β" we 

should find "β" vector such that the L1 norm of "u" is minimum.  That is, 

     Min: S=||u||1=||y(x)-f(x,β)||1=∫ xεI |y(x)-f(x,β)|dx                                (4) 

      β 
 
3. Linear one parameter L1 norm continuous smoothing 

Redefine f(x,β) as βx and y(x) as the following linear function, 

     y(x) = βx + u                                                                    (5) 

Where, "β" is a single (non-vector) parameter.  Expression (4) reduces to: 

     min: S = ||u||1 = ||y(x)- βx||1 = ∫ xεI |y(x)-f(x,β)|dx                                                                  (6) 

       β 
The discrete analog of (6) is solved by Bidabad (1987a,88a,89a,b). In these papers, we proposed applying discrete and regular 
derivatives to the discrete problem by using a slack variable "t" as a point to distinguish negative and positive residuals. A similar 
approach is used here to minimize (6).  To do so in this case, certain Lipschitz conditions are imposed on the functions involved 
(see, Usow (1967a)).  Rewrite (6) as follows, 

     Min: S = ∫ xεI |x||y(x)/x – β|dx                                                                                                (7) 
      β 
For convenience, define "I" as a closed interval [0,1]. The procedure may be applied to other intervals with no major problem 
(see, Usow (1967a), Hobby and Rice (1965), Kripke and Rivlin (1965)).  To minimize this function, we should first remove 
the absolute value sign of the expression after the integral sign. Since "x" belongs to closed interval "I", y(x) (which is a linear 
function of "x") and also y(x)/x are smooth and continuous. Thus, since y(x)/x is uniformly increasing or decreasing function 

of "x", a value of tεI can be found to have the following properties, 

     y(x)/x < β       if x < t 

     y(x)/x = β       if x = t                                                    (8) 

     y(x)/x > β       if x > t 
Value of the slack variable "t" actually is the border of negative and positive residuals. If the value of "t" were known, from (8) 

(middle equation), we could calculate the optimal value of "β" or inversely.  But nor "t" neither "β" are known.  To solve this 
problem, according to (8), we can rewrite (7) as two separate definite integrals with different upper and lower bounds. 

                      ⌠t                                ⌠1  

     min: S = - ⌡0 |x| (y(x)/x - β)dx +⌡t |x| (y(x)/x - β)dx                     (9) 

      β 
Decomposition of (7) into (8) has been done by use of the slack variable "t". Since both "β" and "t" are unknown, to solve (9), 

we partially differentiate it with respect to "t" and "β"  variables. 

       δS       ⌠t             ⌠1 

     ───  = ⌡0 |x|dx - ⌡t |x|dx = 0                                (10) 

       δβ 
and using Liebniz' rule to differentiate the integrals with respect to their variable bounds "t", yields, 

      δS                y(t)                 y(t) 

     ───  = -|t| [─── - β] - |t| [─── - β] = 0                                 (11) 

       δt                  t                      t 
Since "x" belongs to [0,1], equation (10) can be written as, 

     ⌠t           ⌠1 

     ⌡0 xdx - ⌡t xdx = 0                                   (12) 
or, 
     ½ t2 - ½ + ½t2 = 0                                                              (13) 
Which yields, 
     t = √2/2                                                            (14) 
Substitute for "t" in equation (11), yields, 
 



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             y(√2/2) 

     β = ─────                                                                               (15) 
               √2/2 

Remember that y(t) is function y(x) evaluated at x=t.  Value of "β" given by (15) is the optimal solution of (6). The above 
procedure actually is a generalization of Laplace weighted median for the continuous case. 
     Before applying this procedure to the Lorenz curve, let us develop the procedure for the two parameters linear model. 
 
4. Linear two parameters L1 norm continuous smoothing 
     Now, we try to apply the above technique to the linear two parameters model. Rewrite (4) as, 

     Min: S=||u||1=||y(x)-α-βx||1=∫ xεI |y(x)-α-βx|dx                             (16) 

     α,β 

Where, "α" and "β" are two single (non-vector) unknown parameters and y(x) and "x" are as before. According to Rice 

(1964c), let f(α*,β*,x) interpolates y(x) at the set of canonical points {xi;i=1,2}, if y(x) is such that y(x)-f(α*,β*,x) changes sign 

at these xi's and at no other points in [0,1], then f(α*,β*,x) is the best L1 norm approximation to y(x) (see also, Usow  (1967a)). 
With the help of this rule, if we denote these two points to t1 and t2 we can rewrite (16) for I=[0,1] as,  

            ⌠t1                           ⌠t2                           ⌠1 

     S = ⌡0  [y(x)-α-βx]dx - ⌡t1 [y(x)-α-βx]dx + ⌡t2 [y(x)-α-βx]dx                           (17) 

Since t1 and t2 are also unknowns, we should minimize S with respect to α, β, t1 and t2. Taking partial derivative of (17) using 
Liebniz' rule with respect to these variables and equating them to zero, we will have, 

       δS         ⌠t1        ⌠t2         ⌠t1 

     ─── = - ⌡0 dx + ⌡t1 dx - ⌡t2 dx = 0                                       (18) 

       δα 
       δS         ⌠t1        ⌠t2         ⌠t1 

     ─── = - ⌡0 dx + ⌡t1 dx - ⌡t2 dx = 0                              (19) 

       δβ 
      δS 

     ─── = 2[y(t1) -α-βt1] = 0                                                     (20) 

      δt1 

      δS 

      ─── = - 2[y(t2) -α - βt2] = 0                                                   (21) 

      δt2 

Equations (18) through (21) may be solved simultaneously for α, β, t1 and t2. Thus, we have the following system of equations, 
     2t2 - 2t1 - 1 = 0                                                               (22) 
     t2

2 - t1
2 - ½ = 0                                                                                                                      (23) 

     y(t1) - α - βt1 = 0                                                                                                                   (24) 

     y(t2) - α - βt2 = 0                                                                                                                   (25) 
The solutions are, 
     t1=1/4                                                                   (26) 
     t2=3/4                                                                            (27) 

     α = y(3/4)-(3/4)β = y(1/4)-(1/4)β                                              (28) 

     β = 2[y(3/4)-y(1/4)]                                                         (29) 
     This procedure, similar to that of multiple regression model for discrete case may be expanded to include "m" unknown 
parameters which is not discussed here. Some computational methods for solving the different cases of m parameters model are 
investigated by Ptak (1958), Rice and White (1964), Rice (1964a,b,c,69,85), Usow (1967a), Lazarski (1975a,b,c,77) (see also, 
Hobby and Rice (1965), Kripke and Rivlin (1965), Watson (1981)). Now, let us have a look at Lorenz curve and its proposed 
functional forms. 
 
5. Lorenz curve 
     The Lorenz curve for a random variable with probability density function f(v) may be defined as the ordered pair1,  
                        E(V|V ≤ v) 

     (P(V|V ≤ v), ──────)         vεR                              (30) 

                                                             
1 Taguchi (1972a,b,c,73,81,83,87,88) multiplies the second element of (30) by P(V|V≤v) which is not correct; his definition of (31) is 
equivalent to ours.    



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                           E(V) 
Where "P" and "E" stand for probability and expected value operators. For a continuous density function f(v), (30) can be 
written as, 
 

                           ⌠v 

      ⌠v                 ⌡-∞ wf(w)dw 

     (⌡-∞ f(w)dw, ────────) ≡ (x(v),y(x(v)))                                (31) 

                           ⌠+∞ 

                           ⌡-∞ wf(w)dw 
We denote (31) by (x(v),y(x(v))) where x(v) and y(x(v)) are its elements. Therefore, "x" is a function which maps "v" to x(v) 
and "y" is a function which maps x(v) to y(x(v)).  The function y(x(v)) is simply the Lorenz curve function. In recent years 
some functional forms for the Lorenz curve have been introduced. Among different proposed functions, we use the forms of 
Gupta (1984) and Bidabad and Bidabad (1989,92) which benefits from certain properties (see the papers for more 
explanations). Gupta (1984) proposed the functional form, 
     y=xAx-1      A>1                                                                (32) 
Bidabad and Bidabad (1989,92) suggest the following functional form: 
     y=xBAx-1       B≥ 1, A≥ 1   
                                                        (33) 
     To estimate the above functions by regular estimating method, we should gather discrete data from the statistical population, 
and manipulate them to construct relevant x and y vectors to estimate "A" of (32) or "A" and "B" of (33).  If the probability 
distribution of income is known, instead of gathering discrete observations, we can estimate the Lorenz curve by using the 
continuous L1 norm smoothing method for continuous functions.  In the following section, we proceed to apply this method to 
estimate the parameters "A" of (32) and "A" and "B" of (33) by using the information of probability density function of 
income. 
 
6. Continuous L1 norm smoothing of Lorenz curve 
     To estimate the Lorenz curve parameters when income probability density function is known, we cannot always take 
straightforward steps. When the probability density function is easily integrable, there is no major problem in advance. We can 
find the functional relationship between the two elements of (31) by simple mathematical derivation. But, when integrals of (31) 
are not obtainable, another procedure should be adopted. 
     Suppose that income of a society is distributed with probability density function f(w). This density function may be a 
skewed function such as Pareto or log-normal, as follows 

     f(w)=θkθw-θ-1,       w,k>0, θ>0                                                 (34) 

     f(w)=[1/wσ√(2π)]exp{-[ln(w)-μ]2/2σ2},   wε(0,∞), με(-∞,+∞), σ>0                            (35) 
These two distributions have been known as good candidates for presenting distribution of personal income. 
     In the case of Pareto density function of (34), we can simply derive the Lorenz curve function as follows. Let F(w) denote 
the Pareto distribution function: 

     F(w)=1-(k/w)θ                                                                    (36) 
with mean equal to, 

     E(w)= θk/(θ-1),  θ>1                                                                    (37) 
If we find the function y as stated by (31) as a function of x, the Lorenz function will be derived. Now, proceed as follows. 
Rearrange the terms of (31) as, 

                ⌠v 

     x(v) = ⌡-∞ f(w)dw                                                          (38) 

                                 ⌠ tv 

     y(x(v)) = [1/E(x)]⌡-∞  wf(w)dw                                                (39) 
Substitute Pareto distribution function, 

     x(v) = F(v) = 1-(k/v)θ                                                                 (40) 

                                   ⌠v 

     y(x(v)) = [(θ-1)/θk]⌡k wθkθw-θ-1dw                                               (41) 
or, 

     y(x(v)) = 1-(k/v)θ-1                                                               (42) 
Now, by solving (40) for "v" and substituting in (42), the Lorenz curve for Pareto distribution is derived as, 

     y = 1-(1-x)(θ-1)/θ                                                                 (43) 



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     As it was shown in the case of Pareto distribution, formula of Lorenz curve is easily obtained. But, if we select the log-normal 
density function (35), the procedure may not be the same. Because the integral of log-normal function has not been derived yet. 
In the following pages, the L1 norm smoothing technique will be developed to estimate the parameters of given functional forms 
(32) and (33) by using the continuous probability density function. 
     According to (30) and (31) independent and dependent variables of (32) and (33) may be written as, 

                ⌠v 

     x(v) = ⌡0 f(w)dw                                                        (44) 

                                  ⌠v 

     y(x(v)) = [1/E(x)] ⌡0 wf(w)dw                                               (45) 
Substitute (44) and (45) inside (32) and define random error term u as, 

                                                                ⌠v 

                   ⌠v                    ⌠v                ⌡0 f(w)dw-1     

     [1/E(w)]⌡0 wf(w)dw = ⌡0 f(w)dw.A                     . eu                                                        (46) 
or briefly, 
     y(x)=xAx-1eu                                                                                                                         (47) 
Similarly for the model (35), 

                                                                    ⌠v 

                   ⌠v                    ⌠v              B    ⌡0 f(w)dw-1   

     [1/E(w)]⌡0 wf(w)dw={⌡0 f(w)dw}  . A                    . eu                                                    (48) 
or briefly, 
     y(x)=xBAx-1eu                                                                                                                       (49) 
Taking natural logarithm of (47) and (49), gives, 
     ln y(x)=ln x + (x-1)ln A + u                                                                                                 (50) 
     ln y(x)=B.ln x + (x-1)ln A + u                                                                                             (51) 
With respect to properties of Lorenz curve and probability density function of f(w) and equations (46) to (49), it is obvious 
that x belongs to the interval [0,1]. Thus the L1 norm objective function for minimizing (50) or (51) is given by, 

                    ⌠1 

     min: S = ⌡0 |u|dx                                                                                                                 (52) 
Now, let us deal with L1 norm estimation of "A" of Lorenz curve functional form (32) (redefined by (50)). The corresponding 
L1 norm objective function will be, 

                    ⌠1 

     min: S = ⌡0 |ln y(x) - ln x - (x-1) ln A|dx                                                                            (53) 
      A 
or, 

                    ⌠1 

     min: S = ⌡0 |x-1||[ln y(x)-ln x]/(x-1) - ln A|dx                                                                    (54) 
      A 
By a similar technique used by (9), we can rewrite (54) as,  

                    ⌠t                                                          ⌠1 

     min: S = ⌡0 |x-1|{[ln y(x)-ln x]/(x-1)-ln A}dx - ⌡t |x-1|{[ln y(x)-ln x]/(x-1)-ln A}dx       (55) 
      A 
since, 0≤ x ≤ 1 we have, 

                      ⌠t                                                   ⌠1 

     min: S = - ⌡0 [ln y(x) - ln x - (x-1) ln A]dx +⌡t [ln y(x) - ln x - (x-1) ln A]dx                   (56) 
      A 
Differentiate (56) partially with respect to "t" and "A" and equate them to zero; 
 

      δS            ⌠t               ⌠1 

     −−−− = + ⌡0 [(x-1)/A]dx - ut [(x-1)/A]dx = 0                                                 (57) 

      δA 

      δS 

     −−−− = - 2[ln y(t) - ln t - (t-1)ln A] = 0                                                                               (58) 

      δt 
 



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From equation (57), we have, 
     t = 1±√2/2                                                                                                   (59) 
Since "t" should belong to the interval [0,1], we accept, 
     t = 1-√2/2                                                                                                                             (60) 
Substitute (60) in (58), and solve for "A", gives the L1 norm estimation for "A" equal to, 
               1-√2/2          

     A = [−−−−−−−−]√2                                                                                                               (61) 
               y(1-√2/2)  
Now, let us apply this procedure to another Lorenz curve functional form of (33) (redefined by (51)).  Rewrite L1 norm 
objective function (52) for the model (51), 

                    ⌠1 

     min: S = ⌡0 |ln y(x) - B ln x - (x-1) ln A|dx                                                                        (62) 
     A,B 
or, 

                  ⌠1 

     min: S=⌡0 |x-1||[lny(x)]/(x-1)-(lnx)/(x-1)-lnA|dx                                                               (63) 
     A,B 
The objective function (63) - by some changing on variables - is similar to (16). Thus, by a similar procedure to those of (17) 
through (29) we can write "S" as, 

                   ⌠t1 

     min: S = ⌡0 |x-1|{[lny(x)]/(x-1)-(lnx)/(x-1)-lnA}dx 
     A,B 

              ⌠t2 

            - ⌡t1|x-1|{[lny(x)]/(x-1)-(lnx)/(x-1)-lnA}dx 
 

               ⌠1 

            + ⌡t1|x-1|{[lny(x)]/(x-1)-(lnx)/(x-1)-lnA}dx                                                                (64) 
Since  0≤x≤1, then (64) reduces to, 

                      ⌠t1                                                       ⌠t2   

     min: S = - ⌡0  [ln y(x) - B ln x - (x-1) ln A]dx + ⌡t1 [ln y(x) - B ln x - (x-1) ln A]dx 
     A,B 

                ⌠1 

              - ⌡t2 [ln y(x) - B ln x - (x-1) ln A]dx                                                                          (65) 
Differentiate "S" partially with respect to "A", "B", t1 and t2 and equate them to zero, 

      δS      1    ⌠t1               ⌠t2                  ⌠1         

     −−− = −  [ ⌡0 (x-1)dx -⌡t1 (x-1)dx +  ⌡t2 (x-1)dx  ]  = 0                                                    (66) 

      δA     A           

      δS        ⌠t1                 ⌠t2                  ⌠1 

     −−−− = ⌡0  ln(x)dx - ⌡t1 ln(x)dx +  ⌡t2  ln(x)dx = 0                                                           (67) 

      δB               

      δS 

     −−−− = -2{ln[y(t1)] - Bln(t1) - (t1-1)ln(A)} = 0                                                                   (68) 

      δt1 
 

      δS 

     −−−− = 2{ln[y(t2)] - Bln(t2) - (t2-1)ln(A)} = 0                                                                    (69) 

      δt2 
The above system of simultaneous equations can be solved for the unknowns t1, t2, "A" and "B".  Equation (66) is reduced to, 
     t1

2-t2
2-2(t1-t2)-1/2 = 0                                                                                                           (70) 

Equation (67) can be written as, 
     t1(ln t1-1) - t2(ln t2-1) – 1/2 = 0                                                                                             (71) 
Calculate t1 from (70) as, 
     t1 = 1 ±√ q (t2

2-2t2+3/2)                                                                                                        (72) 
Since 0st1s1, we accept, 



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     t1 = 1 - √ (t2
2-2t2+3/2)                                                                                                          (73) 

Substitute t1 from (73) into (71), and rearrange the terms, gives; 
                           [1-√ (t2

2-2t2+3/2)] 
         [1-√ (t2

2-2t2+3/2)] 

     ln −−−−−−−−−−−−−−−−−−−−− + t2-3/2+√(t2
2-2t2+3/2) = 0                                             (74) 

                        t2
t2 

The root of equation (74) may be computed by a suitable numerical algorithm. However, it has been computed and rounded for 
five digits decimal point as, 
     t2 = 0.40442                                                                                                                          (75) 
Value of t1 is derived by substituting t2 into (73); 
     t1 = 0.07549                                                                                                                          (76) 
Values of "B" and "A" are computed from (68) and (69) using t2 and t1 given by (75) and (76). Thus, 
          (t2-1)lny(t1) - (t1-1)lny(t2) 

     B = −−−−−−−−−−−−−−−−−−                                                                                              (77) 
          (t2-1)ln(t1) - (t1-1)ln(t2) 
or, 
     B = -0.84857ln[y(0.07549)] + 1.31722ln[y(0.40442)]                                                        (78) 
and, 
     A = [y(0.07549)]1.28986[y(0.40442)]-3.68126                                                                           (79) 
Now, let us describe how equation (61) for the model (32) and equations (78) and (79) for the model (33) can be used to 
estimate the parameters of the Lorenz curve when the probability distribution function is known.  In the model (32) we should 
solve (44) for x(v)=1-√2/2.  On the other hand, we should find value of "v" such that, 

                ⌠v 

     x(v) = ⌡0 f(w)dw = 1-√2/2                                                                                                  (80) 
By substituting this value of "v" into (45), value of y(1-√2/2) is computed. The value y(1-√2/2) is used to compute the 
parameter "A" given by (61) for model (32). 
     The procedure for the model (33) is also similar, with the difference that two values of "v" should be computed.  Once two 
different values of "v" are computed as follow, 

                ⌠v 

     x(v) = ⌡0 f(w)dw = 0.07549                                                                                                (81) 

                ⌠v 

     x(v) = ⌡0 f(w)dw = 0.40442                                                                                                (82) 
Values of "v" are substituted in (45) to find y(0.07549) and y(0.40442). These values of "y" are used to compute the 
parameters of the model (33) by substituting them into (78) and (79). 
     The only problem remains is computation of related definite integrals of x(v) defined by (80), (81) and (82) which can be 
done by appropriate numerical methods such as the enclosed sample computer program coded for MathCAD 11 for a complete 
example. 
 
7. Income distribution in the United States of America 
In order to compute the Lorenz curve for the United States, we try to apply the above procedure for both (32) and (33) 
propositions and using log-normal distribution function assumption.  The source of data is "the U.S. economic report of the 
president to parliament, different years". Median income and disposable personal income per family report by table 1. The 

amount of mean and median of income were used to derive the log-normal density function parameters μ and δ. The explained 
procedure of estimation then applied to the series of data for 1977-2002, and corresponding results are reported in next table 2. 
The results of Slottje (1989), which are based on quintile data calculations, confirm our finding figures partially. Comparisons 
show the high compatibility of both procedures. An interesting finding of these calculations is that the distribution of income 
obeys counter-wise business cycles fluctuations. This finding is a new area for research in the realm of the theory and application 
of income distribution and business cycles interrelationship.  
     A sample computer program is also enclosed at the end of these pages. 
 
 
 
 
 
 



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18 
                         
 

Table 1. 

Year Population 

millions 

No. 

of 

famili

es 

millio

ns 

Disposable 

personal income,  

billions of 

current $ 

Per 

capita 

disposabl

e income 

$ 

Per 

family 

disposabl

e income 

$ 

Family 

median 

Income 

current $ 

Gross 

domestic 

product 

billions of 

$ 

Real gross 

domestic 

product billions 

of chained 

(2000) $ 

1977  220.3 57.2 1435.7 6,517 25,098 16009.0 2,030.9 4,750.5 

1978  222.6 57.8 1608.3 7,224 27,825 17639.9 2,294.7 5,015.0 

1979  225.1 59.6 1793.5 7,967 30,091 19587.2 2,563.3 5,173.4 

1980  227.7 60.3 2009.0 8,822 33,317 21023.2 2,789.5 5,161.7 

1981  230.0 61.0 2246.1 9,765 36,820 22387.8 3,128.4 5,291.7 

1982  232.2 61.4 2421.2 10,426 39,432 23433.3 3,255.0 5,189.3 

1983  234.3 62.0 2608.4 11,131 42,070 24673.9 3,536.7 5,423.8 

1984  236.4 62.7 2912.0 12,319 46,446 26433.1 3,933.2 5,813.6 

1985  238.5 63.6 3109.3 13,037 48,890 27735.2 4,220.3 6,053.7 

1986  240.7 64.5 3285.1 13,649 50,932 29458.2 4,462.8 6,263.6 

1987  242.8 65.2 3458.3 14,241 53,042 30970.2 4,739.5 6,475.1 

1988  245.1 65.8 3748.7 15,297 56,971 32191.0 5,103.8 6,742.7 

1989  247.4 66.1 4021.7 16,257 60,844 34213.1 5,484.4 6,981.4 

1990  250.2 66.3 4285.8 17,131 64,643 35353.3 5,803.1 7,112.5 

1991  253.5 67.2 4464.3 17,609 66,435 35938.7 5,995.9 7,100.5 

1992  256.9 68.2 4751.4 18,494 69,670 36573.1 6,337.7 7,336.6 

1993  260.3 68.5 4911.9 18,872 71,709 36929.5 6,657.4 7,532.7 

1994  263.5 69.3 5151.8 19,555 74,341 38781.9 7,072.2 7,835.5 

1995  266.6 69.6 5408.2 20,287 77,705 40610.6 7,397.7 8,031.7 

1996  269.7 70.2 5688.5 21,091 81,033 42300.2 7,816.9 8,328.9 

1997  273.0 70.9 5988.8 21,940 84,467 44568.2 8,304.3 8,703.5 

1998  276.2 71.6 6395.9 23,161 89,330 46736.8 8,747.0 9,066.9 

1999  279.3 73.2 6695.0 23,968 91,461 48789.3 9,268.4 9,470.3 

2000   282.5 73.8 7194.0 25,467 97,478 50731.7 9,817.0 9,817.0 

2001   285.6 74.3 7469.4 26,156 100,531 51407.4 10,100.8 9,866.6 

2002   288.6 75.6 7857.2 27,223 103,932 51680.0 10,480.8 10,083.0 

2003 290.5  8039.2 27,675   10,735.8 10,210.4 

http://www.gpoaccess.gov/eop/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

10,000

30,000

50,000

70,000

90,000

110,000

$
 

Year 

Family Median and Mean Income 

Per family disposable income $

Family median Income current $

http://www.gpoaccess.gov/eop/


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19 
                         
 

Table 2  

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Year Gupta Model Bidabad model Slottje figures 

 A Gini Kakwani A B Gini Kakwani Gini Kakwani 

1977 7.938 0.442 0.172 5.798 1.214 0.438 0.170 0.426 0.109 

1978 8.080 0.444 0.173 5.899 1.214 0.441 0.172 0.427 0.108 

1979 7.484 0.434 0.166 5.475 1.212 0.430 0.164 0.427 0.111 

1980 8.189 0.446 0.175 5.978 1.215 0.442 0.173 0.428 0.112 

1981 9.095 0.456 0.185 6.631 1.218 0.546 0.183 0.435 0.114 

1982 9.693 0.467 0.191 7.064 1.220 0.464 0.190 0.447 0.118 

1983 10.051 0.471 0.194 7.324 1.221 0.469 0.193 0.447 0.120 

1984 10.909 0.481 0.202 7.952 1.222 0.479 0.201 0.449 0.121 

1985 11.004 0.482 0.203 8.021 1.223 0.480 0.202   

1986 10.442 0.476 0.198 7.609 1.222 0.473 0.197   

1987 10.175 0.473 0.196 7.416 1.221 0.470 0.194   

1988 11.123 0.483 0.204 8.110 1.223 0.481 0.203   

1989 11.269 0.485 0.205 8.216 1.223 0.482 0.204   

1990 12.137 0.493 0.212 8.858 1.224 0.491 0.211   

1991 12.493 0.496 0.215 9.122 1.225 0.494 0.214   

1992 13.518 0.505 0.222 9.886 1.226 0.503 0.221   

1993 14.207 0.510 0.226 10.403 1.226 0.509 0.226   

1994 13.741 0.507 0.223 10.052 1.226 0.505 0.223   

1995 13.676 0.506 0.223 10.004 1.223 0.504 0.222   

1996 13.717 0.507 0.223 10.034 1.226 0.505 0.222   

1997 13.339 0.504 0.221 9.751 1.226 0.502 0.220   

1998 13.637 0.506 0.223 9.973 1.226 0.504 0.222   

1999 12.962 0.500 0.218 9.472 1.225 0.499 0.217   

2000 13.825 0.507 0.224 10.115 1.226 0.506 0.223   

2001 14.470 0.512 0.229 10.600 1.226 0.511 0.227   

2002 15.759 0.521 0.235 11.573 1.227 0.520 0.235   

0.42

0.44

0.46

0.48

0.5

0.52

0.54

G
in

i 
In

d
e

x
 

Year 

Comparision of Gini Indices 

Gupta

Bidabad



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The following graph compares the calculated Gini index with real GDP for the period of 1977-2002.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Dependent Variable: GINI  

Method: Least Squares  

Date: 06/23/19   Time: 17:44  

Sample (adjusted): 1979 2002  

Included observations: 24 after adjustments 

          
Variable Coefficient Std. Error t-Statistic Prob.   

          
C 0.472788 0.009894 47.78343 0.0000 

@TREND 0.002299 0.000535 4.298753 0.0003 

GDPGROWTH(-1) -0.432267 0.191001 -2.263167 0.0343 

          
R-squared 0.521663     Mean dependent var 0.490375 

Adjusted R-squared 0.476107     S.D. dependent var 0.025039 

S.E. of regression 0.018123     Akaike info criterion -

5.066782 

Sum squared resid 0.006897     Schwarz criterion -

4.919525 

Log likelihood 63.80138     Hannan-Quinn criter. -

5.027714 

F-statistic 11.45105     Durbin-Watson stat 2.298751 

Prob(F-statistic) 0.000434    

          
 
 
As the countercyclical movement of Gini index and GDP is understandable from the above graph, the above simple regression 
between Gini index and the growth of GDP of USA with one lag also proves this phenomenon. The parameters are meaningful 
and the t statistics and other statistics are all significant.  
 
 
  

0.42

0.44

0.46

0.48

0.50

0.52

2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000

10,000
11,000

1
9
7

7

1
9
7

9

1
9
8

1

1
9
8

3

1
9
8

5

1
9
8

7

1
9
8

9

1
9
9

1

1
9
9

3

1
9
9

5

1
9
9

7

1
9
9

9

2
0
0

1

2
0
0

3

$
 

Year 

Income Distribution and GDP 

GDP

Gini



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www.cribfb.com/journal/index.php/amfbr                             American Finance & Banking Review                               Vol. 4, No. 2; 2019 
 

21 
                         
 

 
 

 
 
  

 (Using Sample Mean and Median) 

Calculations for 2002 USA data 

CONTINUOUS L NORM ESTIMATION OF LORENZ CURVE 

Bijan BIDABAD 

This program has been coded for MathCAD 11 

Mean = Sample mean of income distribution:  

Med = Sample median of income distribution:  

 

Calculation of Log-Normal density function parameters m and s according to sample mean and median 

 

  

 

Log-Normal Probability_Density Function 

 

Selective range for_Log-Normal plot, values of_increment and upper bound_may be changed  

Log-Normal plot 

 

Precision Tolerance level  

TOL value should be_ changed for more_ accurate solutions,_(less TOL = higher precision) 

 
(45) 

 
(44) 

Mean 103932

Med 51680

 2 ln
Mean

Med











 1.18209

 ln M ed( )  10.85283

f w( )
1

w  2 









exp
ln w( )  2

2 
2















w 10
5 M ean

200
 2 M ean

f w( )

w

TOL 0.00001

y v( )
1

M ean







 0

v

ww f w( )




d

x v( )

0.00001

v

wf w( )




d



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22 
                         
 

  

Calculation for Gupta model 

Initial guess for v. This value should be changed for faster convergence and less iterations 

 

 

(60) 

Calculating v for (80)  

Calculated v  

y(t )_ 0   

 

(61), estimated A:  

 
(53) 

Sum of absolute residuals  

Range variable for plotting the Lorenz curves 

 

 
Gupta Lorenz curve: 

Calculation of Gini index 
 

 

 

Calculation of Kakwani length of Lorenz curve 

 

v 20000

t
0

1
2

2


v root x v( ) t
0

 v 

v 27136.6437

y v( ) 0.04208 z
0

y v( )

A

t
0

z
0











2



A 15.54768

S

0

1

xln z
0  ln t

0  t
0

1  ln A( )




d

S 0

X 0 0.005 1

Y X( ) X A
X 1



Y X( )

X

X

Gini 1 2

0

1

XY X( )




d

Gini 0.51967

Length

0

1

X1 A
X 1

1 X ln A( )( ) 
2








d



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23 
                         
 

  Length of Lorenz curve 

 

 

Kakwani index of length  

Calculation For Bidabad Model 

(76)  

Initial guess for v. This value should be changed for faster convergence and less iterations 

 

Calculating v for (81)  

Calculated v  

y(0.07549)   

(75)  

Initial guess for v. This value should be changed for faster convergence and less iterations 

 

Calculatig v for (82)  

Calculated v  

y(0.40442)   

 
(79) 

(78)  

Estimated A and B:   

 
(62) 

Sum of absolute residuals  

Range variable for plotting the Lorenz curves 

 

 Bidabad Lorenz curve 

Length 1.5515

Kakwani
Length 2

2 2


Kakwani 0.23437

t
1

0.07549

v 8000

v root x v( ) t
1

 v 

v 9464.04318

y v( ) 0.00442 z
1

y v( )

t
2

0.40442

v 27000

v root x v( ) t
2

 v 

v 38826.25803

y v( ) 0.07722 z
2

y v( )

A z
1 

1.28986
z
2 

3.68126


B 0.84857 ln z
1  1.31722ln z

2 

A 11.41481 B 1.22709

S

0

1

xln z
1  B ln t

1  t
1

1  ln A( )




d

S 0.00002

X 0 0.005 1

Y X( ) X
B

A
X 1





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24 
                         
 

 

 

 

 

 
 
 
 

 
 

 
 
 

 
 
 

 
References 
Bidabad Bijan (1987a) Least absolute error estimation I, The 1st International Conference on Statistical Data Analysis Based 

on the L1 Norm and Related Methods. Neuchatel, Switzerland, 1987 . 
http://www.bidabad.com/doc/lae-I.pdf  

Bidabad Bijan (1987b) Least absolute error estimation II, The 1st International Conference on Statistical Data Analysis Based 

on the L1 Norm and Related Methods. Neuchatel, Switzerland, 1987 . 
http://www.bidabad.com/doc/lae-II.pdf  

Bidabad Bijan (1988a) A proposed algorithm for least absolute error estimation. Proceedings of the Third Seminar of 
Mathematical Analysis. Shiraz University, 24-34, Shiraz, Iran. 

Bidabad Bijan (1988b) A proposed algorithm for least absolute error estimation, part II. Proceedings of the Third Seminar of 
Mathematical Analysis, Shiraz University, 35-50, Shiraz, Iran. 

Bidabad Bijan (1989a) Discrete and continuous L1 norm regressions, proposition of discrete approximation algorithms and 
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Bidabad, Bijan (2019) L1 norm based computational algorithms. Bangladesh Journal of Multidisciplinary Scientific Research, 
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  http://www.bidabad.com/doc/l1-article6.pdf  
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Bidabad, Bijan (2019) L1 Norm Based Data Analysis and Related Methods, (1632-1989). Australian Finance & Banking 

Review, 3(1), 43-81, 2019.  

 
Calculation of Gini Index 

 

 

Calculation of Kakwani length of Lorenz curve 

 

Length of Lorenz curve 

 

 

Kakwani index of length 
 

Y X( )

X

X

Gini 1 2

0

1

XY X( )




d

Gini 0.51834

Length

0

1

X1 A
X 1

X
B 1

 B X ln A( )( ) 
2








d

Length 1.55118

Kakwani
Length 2

2 2


Kakwani 0.23381

http://www.bidabad.com/doc/lae-I.pdf
http://www.bidabad.com/doc/lae-II.pdf
http://www.bidabad.com/doc/l1-article6.pdf
http://www.bidabad.com/doc/l1-article5.pdf


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25 
                         
 

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