Farm Wealth Inequality Within and Across States in the United States

Agricultural and Resource Economics Review,  Oct 2006  by Mishra, Ashok K,  Moss, Charles B,  Erickson, Kenneth W

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(6) ...

where D(p,q) is a measure of the difference (or distance) between the two probability measures, and p(x) and q(x) are probability measures defined on x. The discrete form of this measure becomes

(7) ...

In our applications we want to examine a uniform distribution of wealth against an empirical distribution of wealth:

(8) ...

This measure is sometimes referred to as the relative entropy in the signal.

Theil's measure of income inequality is an adaptation of the discrete form of the relative difference measure presented in Equation 7:

(9) ...

where I(p,q) is the measure of inequality (or dispersion) of income, p^sub i^ is the share of income1 in state or region i, and q^sub i^ is the share of the overall population in that state or region. Adapting this procedure to examine the inequality in farm wealth, we let p^sub i^ be the share of farm wealth in state i and q^sub i^ be the share of farmers in state i. I(p,q) is then defined as the dispersion of farm equity. If the share of the number of farms is close to the share of farm equity, then there is little additional information and the information inequality is small. A small inequality means that the distribution of farm wealth, per farm, is uniform across states, and thus the value of I(p,q) approaches 0. Conversely, an increase in the information inequality indicates divergence in farm wealth across the states.

An important aspect of the Theil measure of inequality is its decomposability. To develop this decomposability we divide the overall inequality in Equation 9 into two groups:

(10) ...

where G^sub 1^ and G^sub 2^ are mutually exclusive and exhaustive sets of individuals (states or regions). Defining

...

the equality represented in Equation 10 can be rewritten as

(11) ...

Defining the average inequality within each group, i.e., inequality between farms in each state, as

(12) ...

and the inequality between groups, i.e., inequality between states, as

(13) ...

we are left with the decomposition of the inequality measure in a given region as

(14) I = I + I^sub R^.

Letting p^sub 1^ equal firm-level wealth (instead of stateor regional-level wealth) and q^sub i^=1/N^sub 1^ if i ∈ G^sub 1^ or q^sub i^=1/N^sub 2^ if i ∈ G^sub 2^ (e.g., N^sub 1^ is the number of farms in group G^sub 1^ and N^sub 2^ is the number of farms in group G^sub 2^), the decomposability of the inequality measure in Equation 14 can be expanded to the firm level as

(15) ...

Thus, Q^sub 1^=N^sub 1^/(N^sub 1^+N^sub 2^) represents the share of farms in group 1 as depicted in Equation 13 and Q^sub 2^=N^sub 2^/(N^sub 1^+N^sub 2^) represents the share of farms in group 2, as depicted in Equation 13. Equation 15 states that total inequality in a given region is composed of inequality within farms in a state and between states.

As previously stated, a number of inequality measures have been proposed, including the coefficient of variation, Lorenz curves, and Gini coefficients. Given this diversity, it behooves the researcher to justify the choice of inequality measure. To justify our application of the Theil inequality measure, we rely on the axiomatic characteristics developed by Foster (1983), particularly those emphasizing the role of decomposability of inequality. Foster develops four criteria for measuring inequality: (i) the Pigou-Dalton transfer principle, (ii) symmetry, (iii) homogeneity, and (iv) the population principle.