The methodologies of the studies - Institutional Constraints to Small Farmer Development in Southern Africa - page 302

where:
S k is the share of income source k on overall income (i.e. S k =µ k /µ));
G k is the Gini coefficient measuring the inequality in the distribution of income from
activity k within the group, and R k is the Gini correlation of income from source k with
total income which can be expressed separately as:

cov[ Y k ,F ( Y )]

cov[ Y k ,F ( Y k )]

(A.9)

R k =

Contribution of source k to overall income inequality (P k ) is given by the product of G k,
R k and S k

P k = G k R k S k / G (A.10)

Adams (1999) pointed out that the larger the product of G k R k S k , the greater the
contribution of income from source k to overall income inequality. S k is the fraction of
income from source k to total income, it should be noted that it is always positive and less
than one, G k is always positive and may exceed one, and R k is between -1 and 1.

Source elasticity on total inequality P k measures the response of inequality to changes in
income from source k . It assesses whether an income source is inequality-increasing or
inequality-decreasing on the basis of whether or not marginal increase on share of that
income source leads to an increase or decrease in overall inequality. This can only be valid
under the assumption that additional increments in income are distributed in the same
manner as the original units since anything that disrupts the pattern of increase also affects
overall inequality (Van de Berg and Kumbi, 2006).

P k = S k ( R k G k - G) / G (A.11)

The above procedures were used to establish the extent of rural poverty, the pattern of
rural activity choices, and extent of inequality which have been reported in Chapter 4 of
this topic.

A.3 Models for the component studies assessing institutional constraints

The generalized choice model was employed to estimate the probability that a smallholder
farmer in the enumerated localities faced or did not face a specific constraint. Since only
two options are available, namely 'all produce sold' or 'not all produce sold , a binary model
is set up which defines Y = 1 for situations where the farmer sold all produce, and Y =
0 for situations where some or all produce was not sold. Assuming that x is a vector of
explanatory variables and ρ is the probability that Y = 1, it is valid to proceed in the same
way as suggested by Equations A.3 - A.5 above.