Dalton index

The Dalton Index is a measure of inequality (used to measure relative concentration ) named after the British politician Baron E. Hugh J. N. Dalton .

definition

Dalton assumes for each individual that marginal welfare (marginal welfare) decreases as income increases. This is called the income welfare function :

${\displaystyle U_{i}=U_{i}\left(x_{i}\right);\;i=1,2,\ldots ,N\,,}$

where represents the benevolence of a person receiving an income . This is monotonous and concave , which is expressed mathematically as follows: ${\displaystyle U_{i}}$ ${\displaystyle i}$ ${\displaystyle x_{i}}$

${\displaystyle {\frac {\partial U_{i}}{\partial x_{i}}}>0}$ such as ${\displaystyle {\frac {\partial ^{2}U_{i}}{\partial x_{i}^{2}}}<0}$

applies. Dalton also assumes that the welfare of different people are additively linked. Hence the welfare in his model is a simple summation (aggregation) of personal welfare. In other words, the following applies to welfare : ${\displaystyle W}$

${\displaystyle W=\sum \limits _{i=1}^{N}U_{i}\left(x_{i}\right)\,.}$

He also assumes that the income to welfare ratio is the same for everyone. This is:

${\displaystyle U_{i}=U\left(x_{i}\right);\;i=1,2,\ldots ,N\,.}$

In this case the relation can be noted as follows: ${\displaystyle W}$

${\displaystyle W=\sum \limits _{i=1}^{N}U\left(x_{i}\right)\,,}$

which makes it clear whoever gains welfare, the addition to public welfare is the same. For any given level of welfare, any distribution among members of society is permissible. However, it must be remembered that the relationship between individual incomes and their welfare is concave. Therefore, the redistribution of income from to does not become a symmetrical change in the welfare of those involved in the transaction. The result is an impact on the level of social welfare. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle W}$

Representation of the Dalton index

From the graph, one can see and compare the situation when two individuals who have the same relationship, two different income levels, and the situation when they have the same income (or the same income difference). It should be noted that the welfare sum of person 1 (BB ​​') and person 2 (DD') is less than double that of CC ', which would be the welfare level of both persons if they received the same income. It is easy to see that the loss (D'E) experienced by Person 2 is overcompensated for by the gain of Person 1 (C'F).

This demonstrates that under Dalton's assumptions, an equal distribution is preferable to an unequal distribution for a given total income from a welfare point of view. Indeed, for a given total income, the economic welfare of a society will reach its maximum when all incomes are equal. The disparity of any given distribution can therefore be called

${\displaystyle {\frac {\sum \limits _{i=1}^{N}U(\mu )}{\sum \limits _{i=1}^{N}U\left(x_{i}\right)}}={\frac {NU(\mu )}{\sum \limits _{i=1}^{N}U\left(x_{i}\right)}}}$

can be defined, which is equal to 1 for even distribution and greater than 1 for uneven distribution. Here is the mean income that everyone would receive if they were equally distributed. Therefore, one could define the Dalton Index as a number lying in the range of numbers as follows: ${\displaystyle \mu :={\frac {1}{N}}\sum _{i=1}^{N}x_{i}}$ ${\displaystyle [1,\infty )}$

${\displaystyle D_{2}={\frac {NU(\mu )}{\sum \limits _{i=1}^{N}U\left(x_{i}\right)}}\,,}$

which is obviously equal to 1 for a uniform distribution. Later authors transformed the Dalton index to a number from the interval [0,1] as follows, so that the uniform distribution corresponds to 0:

${\displaystyle D=1-{\frac {\sum \limits _{i=1}^{N}U\left(x_{i}\right)}{NU(\mu )}}=1-{\frac {\bar {U}}{U(\mu )}}\,.}$

The index makes it appear as if it has the interval limits. But there are many valid, concave functions where this is not the case. For example, if we have, then there is . Assuming the fact that holds, the index becomes negative. And can be less than one (1) if you can measure in each unit. The same also applies: ${\displaystyle [0,1]\,.}$ ${\displaystyle U\left(x_{i}\right)=\log x_{i}\,,}$ ${\displaystyle D=1-\left\{{\tfrac {\log {\hat {\mu }}}{\log \mu }}\right\}}$ ${\displaystyle {\hat {\mu }}<\mu }$ ${\displaystyle \mu <1}$ ${\displaystyle \mu }$ ${\displaystyle x}$ ${\displaystyle U\left(x_{i}\right)={\tfrac {1}{x_{i}}}\,.}$

However, in order to get the numeric value, it is not necessary to define the index. In 1920 Dalton showed that disparity must be measured in terms of income, although the index is defined in terms of economic welfare. Then there is no uniform measure of inequality. This depends on the assumed functional relationship. Dalton's intention of these two functions is to illustrate: The first relates to Bernoulli's hypothesis. It maintains that proportional injections of income (more than the simple subsistence / subsistence poverty line required) lead to equal increases in individual welfare. This means:

${\displaystyle \mathrm {d} U_{i}={\frac {\mathrm {d} x_{i}}{x_{i}}}}$ or ${\displaystyle U_{i}=\log x_{i}+c_{i}\,.}$

On the premise that every person has the same functional relationships / ratios, the Dalton Index can be noted as follows:

${\displaystyle D=1-{\frac {\log {\hat {\mu }}+c}{\log \mu +c}}}$

with harmonious means of personal income. The other phrase it deals with is defined as: ${\displaystyle {\hat {\mu }}=}$

${\displaystyle \mathrm {d} U_{i}={\frac {\mathrm {d} x_{i}}{x_{i}^{2}}}}$ or ${\displaystyle U_{i}=c-{\frac {1}{x_{i}}}\,,}$

where represents the maximum welfare that a person can achieve, if applicable. In this case, the Dalton index changes to the following: ${\displaystyle c}$ ${\displaystyle \!\ x\to \infty }$

${\displaystyle D=1-{\frac {c-{\frac {1}{\tilde {\mu }}}}{C-{\frac {1}{\mu }}}}}$

with harmonic mean. ${\displaystyle {\tilde {\mu }}=}$

literature

• Hugh Dalton: Some aspects of the inequality of incomes in modern communities. George Routledge and Sons Ltd., London 1920. 368 pp.

Web links

• Pramod Kumar Chaubey (eGyanKosh, IGNOU / Indira Gandhi National Open University ): Unit 11: Measures of Inequality - PD / public domain (PDF file; 3.34 MB).