The methodologies of the studies - Institutional Constraints to Small Farmer Development in Southern Africa - page 301

In the context of poverty measurements, the concept of income inequality decomposition
was applied to calculate Gini Coefficients which allowed for the extent of inequality
between household categories to be assessed. Consideration was also given to the fact that
the Gini index satisfies the following five properties:
• Transfer sensitivity which allows the measure to respond to the relative positions of
the source and target of an income transfer - transferring income from a rich person
to a poorer person results in a more equal distribution and hence a lower Gini measure
than otherwise. This is also referred to as the 'transfer principle' (Shorrocks and Foster,
1987; Debraj, 1998).
• Symmetry - which ensures that the relative position of individuals in the income order
does not nothing to alter the absolute size of the inequality measure. Debraj (1998)
describes this as 'anonymity'. When individuals switch places in the income order
through some shuffling that did not affect the distribution, the inequality measure will
not be affected.
• Mean/Scale neutrality which ensures that when all incomes change by the same
proportion, there is no effect on the size of the measure of inequality.
• Population independence/homogeneity which ensures that the size of the population
does not affect the measure of inequality - it is the distribution of income across the
population that matters rather than the absolute size of the economy itself.
• Decomposability - the measure should be able to capture the contributions of sub-groups
and activities (income sources) to the total inequality (Foster, 1985; Adams, 1999).

Assuming n households engaged in k diverse economic activities for purposes of income
generation, the total household income of the i th household can be depicted as y ik . It is
also possible to represent the mean household income for the sample of households by
the standard notation µ, while the cumulative distribution of the total household income
would be shown as F(Y). A generalized function of the following form was defined:
G = ( 2 cov[ Y,F ( Y )] ) (A.6)

µ

where:
cov[ Y, F(Y) ] is the covariance of total income, Y, with mean µ and its cumulative
distribution, F. By decomposing the total household income into k sources to take account
of the contribution of each component of income to the overall inequality, equation A.6
can be rewritten and expanded so that the overall Gini Coefficient can be expressed as:

∑ ( cov[ Y k ,F ( Y )] )( 2 cov[ Y k ,F ( Y k )] )( µ k )

k

cov[ Y k ,F ( Y )] µ k µ

G=

(A.7)

k =1

which can be shortened further to:

G = Σ k k =1 G k R k S k (A.8)