Given a level of fiscal stimulus one can wonder how to allocate that optimally to reduce as much as possible inequalities in a population.
We can rewrite the Gini coefficient formula to make it simpler to analyse. We will now consider that the X suite is sorted in such a way that X is increasing (not necessarily strictly). Also, suppose that the first N terms of X are equals. (Since N can be equal to 1, that is if only the first term is equal to itself, the case we are considering is still the general case).
We have: and since ,
\sum_{i=1}^{n}\sum_{j=1}^{i-1}\left | x_j-x_i \right |=\sum_{i=1}^{n}\sum_{j=i+1}^{n}\left | x_j-x_i \right | we have :
G =\frac{\sum_{i=1}^{n}\sum_{j=i+1}^{n}\left | x_j-x_i \right |+\sum_{i=1}^{n}\sum_{j=1}^{i-1}\left | x_j-x_i \right |}{2*n*X } …show more content…
If we allocate to individuals who earn strictly more than the median earnings, we will increase the Gini coefficient.
From these computations, we can also deduce that the optimal way to allocate earnings is to allocate the maximum amount to the poorest, until his income level reaches the income level of the second poorest. At that point, we allocate most resources to the two equally poorest people until the income level of the third poorest is reached. We keep doing that until the level of fiscal payment reaches the total amount A that we fixed.
Comparing this level of Gini coefficient reduction to the reduction we obtained, can enable us to estimate the efficiency of the spending in terms of inequality
We can rewrite the Gini coefficient formula to make it simpler to analyse. We will now consider that the X suite is sorted in such a way that X is increasing (not necessarily strictly). Also, suppose that the first N terms of X are equals. (Since N can be equal to 1, that is if only the first term is equal to itself, the case we are considering is still the general case).
We have: and since ,
\sum_{i=1}^{n}\sum_{j=1}^{i-1}\left | x_j-x_i \right |=\sum_{i=1}^{n}\sum_{j=i+1}^{n}\left | x_j-x_i \right | we have :
G =\frac{\sum_{i=1}^{n}\sum_{j=i+1}^{n}\left | x_j-x_i \right |+\sum_{i=1}^{n}\sum_{j=1}^{i-1}\left | x_j-x_i \right |}{2*n*X } …show more content…
If we allocate to individuals who earn strictly more than the median earnings, we will increase the Gini coefficient.
From these computations, we can also deduce that the optimal way to allocate earnings is to allocate the maximum amount to the poorest, until his income level reaches the income level of the second poorest. At that point, we allocate most resources to the two equally poorest people until the income level of the third poorest is reached. We keep doing that until the level of fiscal payment reaches the total amount A that we fixed.
Comparing this level of Gini coefficient reduction to the reduction we obtained, can enable us to estimate the efficiency of the spending in terms of inequality