### Foster-Greer-Thorbecke

The **Foster-Greer-Thorbecke** (sometimes referred to as **FGT**) metric is a generalized measure of poverty within an economy.^{[1]} It measures the outfall from the povertyline and is weighted by $\backslash alpha$. Therefore it is also considering the inequality among the poor but as the proper amount for $\backslash alpha$ is not pre defined (and a normative question) we cannot say that the Gini is part of the FGT.
%Also P0 is not part of FGT. The only measure that is combining P0, P1 and Gini is the Sen Index.
FGT measure was developed by Professor Erik Thorbecke, his former student Professor Joel Greer, and another graduate student at Cornell University at the time, Professor James Foster.

The formula for the FGT is given by:

$FGT\_\backslash alpha=\backslash frac\; \{1\}\; \{N\}\; \backslash sum\_\{i=1\}^H\; (\backslash frac\; \{z-y\_i\}\; \{z\})^\backslash alpha$

where $z$ is an agreed upon poverty line (1.25$ or 2$ per day adjusted for purchasing power parity are the two most common poverty lines used by the World Bank. Developed countries usually have much higher poverty lines), $N$ is the number of people in an economy, $H$ is the number of poor (those with incomes at or below $z$), $y\_i$ are individual incomes and $\backslash alpha$ is a "sensitivity" parameter. If $\backslash alpha$ is low then the FGT metric weights all the individuals with incomes below z roughly the same. If $\backslash alpha$ is high, those with the lowest incomes (farthest below $z$) are given more weight in the measure. The higher the FGT statistic, the more poverty there is in an economy.

The FGT measure corresponds to other measures of poverty for particular values of $\backslash alpha$. For $\backslash alpha=0$, the formula reduces to

$FGT\_0=\backslash frac\; \{H\}\; \{N\}$

which is the Headcount ratio, or the fraction of the population which lives below the poverty line. If $\backslash alpha=1$ then the formula is

$FGT\_1=\backslash frac\; \{1\}\; \{N\}\; \backslash sum\_\{i=1\}^H\; (\backslash frac\; \{z-y\_i\}\; \{z\})$

which is the average poverty gap, or the amount of income necessary to bring everyone in poverty right up to the poverty line, divided by total population. This can be thought of as the amount that an average person in the economy would have to contribute in order for poverty to be just barely eliminated.

While the two above versions are widely reported, a good deal of technical literature on poverty uses the $\backslash alpha=2$ version of the metric:

$FGT\_2=\backslash frac\; \{1\}\; \{N\}\; \backslash sum\_\{i=1\}^H\; (\backslash frac\; \{z-y\_i\}\; \{z\})^2$

as in this form, the index combines information on both poverty and income inequality among the poor. Specifically in this instance the FGT can be rewritten as:

$FGT\_2=H\; \backslash mu^2\; +\; (1-\backslash mu^2)\; C\_v^2$

where $C\_v$ is the coefficient of variation among those with incomes less than $z$, $H$ is the total number of the poor as above, and $\backslash mu$ is given by

$\backslash mu=\backslash frac\; \{1\}\; \{H\}\backslash sum\_\{i=1\}^H\; (\backslash frac\; \{z-y\_i\}\; \{z\})$.

Other decompositions of the index are also possible.^{[2]}

The $\backslash alpha\; =\; 2$ version of the index is officially part of the Mexican Constitution as a basis for all government measurements of poverty.**
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