Lorenz Curve
The Lorenz curve, discovered by Max 0 Lorenz, famous economic statistician, is a graphic method of studying the dispersion. This curve was used by him to study the distribution of wages, profits, turnover, etc. However, still the most common use of this curve is in the study of the degree of inequality in the distribution of income and wealth between countries or between different time-period, it is a cumulative percentage curve in which the percentage of items is combined with the percentage of other things as profits, wealth, turnover etc.
While drawing the Lorenz curve the following procedure is taken:
(i) The size of items (i.e. variable values) and frequencies are both cumulated. By taking grand for each as 100 and take the percent of cumulative frequencies.
(ii) On the X-axis start from 0 to 100, and then take the percent of cumulative frequencies.
(iii) On the Y-axis start from 0 to 100 and take the per cent of the cumulated values of the variable.
(iv) Draw a diagonal line joining 0 (0, 0) with the point p (100, 100) as shown in the below diagram. The line OP will make an angle of 45. With the Y-axis and is known as the line of equal distribution. Any point on this diagonal represents that same percent on X as on Y.
(v) Provide the percentages of the cumulated values of the variable (Y) against the percentages of the corresponding cumulated frequencies (x) for the given distribution and join these points with never cross the line of equal distribution OP, it will always lie below OP unless the distribution is uniform in which it will coincide with OP. The greater the variability, the greater is the distance of the curve from OP.
Illustration: In the following table , the number of companies belonging to 2 areas A & B according to the amount of profits earned by them. Draw the Lorenz curves and interpret them.
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Profits earned $ '000
|
No. of companies
|
|
|
Area A Area B
|
|
6
|
6 2
|
|
25
|
11 38
|
|
60
|
13 52
|
|
84
|
14 28
|
|
105
|
15 38
|
|
150
|
17 26
|
|
170
|
10 12
|
|
400
|
14 4
|
Solution: calculations for drawing the Lorenz Curve:
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$ '000
|
Cumulative
profit
|
No. of companies
|
No. of companies
|
Cumulative number
|
Cumulative percentage
|
No. of companies
|
Cumulative number
|
Cumulative percentage
|
|
6
|
6
|
0.6
|
6
|
6
|
6
|
2
|
2
|
1
|
|
25
|
31
|
3.1
|
11
|
17
|
17
|
38
|
40
|
20
|
|
60
|
91
|
9.1
|
13
|
30
|
30
|
52
|
92
|
46
|
|
84
|
175
|
17.5
|
14
|
44
|
44
|
28
|
120
|
60
|
|
105
|
280
|
28.0
|
15
|
59
|
59
|
38
|
158
|
79
|
|
150
|
430
|
43.0
|
17
|
76
|
76
|
26
|
184
|
92
|
|
170
|
600
|
60.0
|
10
|
86
|
86
|
12
|
196
|
98
|
|
400
|
1000
|
100.0
|
14
|
100
|
100
|
4
|
200
|
100
|