Dynamic decomposition

What has been explained so far refers to the decomposition of the level of income inequality in any one year, this is the static decomposition. However, in order to address more directly the question of what causes changes in income inequality, one needs to decompose these changes rather than the levels. Several approaches have been used in the literature for dynamic decomposition of inequality. Probably the easiest way consists of decomposing the inequality index into within- and between- group components, as was shown earlier, and then calculating the proportional contribution of each factor by simply dividing the change in the between-group component by the change in overall inequality^. This would be as follows:

Cb= (4.5)

where Cb represents the proportional contribution of changes in explained inequality

(by the characteristic which yields to the partition of the population in question), to changes in overall inequality in the time interval ( t ,t +1). As a first stage towards the dynamic decomposition analysis, I will apply this approach in Chapter 8, since it gives an idea of the relevance of changes in the explanatory power of a specific characteristic for overall inequality changes.

However, Cb is only a summary measure of the change in explained inequality in a

period of time which does not allow us to identify the causes of this "explained" change that often counteract one another, such as changes in the population shares and relative mean incomes of the different sub-groups. Indeed, the impact of changes in the differences between sub-groups on overall inequality depends on changes in both population shares and relative mean incomes and, looking at equations (4.1) to (4.3), one realises that these changes not only impact on the between-group component of aggregate inequality, but also on the within-group component. Therefore, interpreting Cb as a summary measure of the full effect of changes between sub-groups on overall inequality according to the characteristic in question would be misleading, since one would be ignoring the impact of changes in both population shares and relative mean incomes on within-group inequalities*.

To deal with this problem one needs to decompose changes in aggregate inequality rather than calculating proportional changes in explained inequality. According to the decomposition equations (4.1) to (4.3), overall inequality values depend only on the relative mean incomes, population shares and within-group inequalities. Therefore, changes in these three factors will determine changes in overall inequality. So, what one needs is to measure, separately, the contributions of changes in these three factors to overall inequality changes, and what is relevant when trying to establish the channels through which economic changes might have affected income distribution during certain periods of time. Mookerjee and Shorrocks (1982) propose an approach based on the decomposition principle presented earlier, which takes account of the effects of changes in both population proportions over sub-groups and sub-group mean income relative to the population mean income. This approach yields an exact decomposition of changes in overall inequality that measures the effects mentioned above. However, due to the complicated arithmetic involved, and for computational purposes, these authors propose an approximate decomposition using the GEF(O) index which is more sensitive to changes at the lower end of the income distribution, as pointed out earlier. If equal weights across the distribution were required, then the Theil index (G EF(l)) would be the suitable choice, but in this case the arithmetic

involved by the approach of Mookerjee and Shorrocks (1982) is even more complicated and impracticable.

Coady and Wang (1997) put forward a less complicated formula yielding an exact decomposition of overall inequality changes that also attempt to measure, separately, the influences of changes in both population shares and relative incomes. They also applied this approach using GEF(O), although its application using the Theil index is manageable and brings the advantage of applying equal weights across the income distribution. Therefore, from the static decomposition equation proposed by Mookerjee and Shorrocks (1982), and based on the procedure proposed by Coady and W ang (1997), I have produced a decomposition equation for overall inequality changes using the Theil index as follows:

Bringing back the decomposition equation for the Theil index proposed by Mookherjee and Shorrocks (1982), shown earlier as equation (4.3), I have

// = (4.6)

where = — , being /J,k the mean income of sub-group k, and fi the mean income

of total population or sample; = — , being Uk the number of individuals in sub­ group k, and n the number of individuals in total population (or sample); while is the within-group inequality measured by the Theil index.

Thus, the general functional form for overall inequality measured by 7/ is

/, = I,

(v. I , I,

where v , À , and îj^ are all vectors of dimension k.

Differentiating (4.7) with respect to time I obtain:

I calculate the derivatives of 7/ with respect to Vk, JUk, and if^ , which are in equation (4.8), separately, applying equation (4.6). These are:

dii

In (I)

- + 1 ) w

dX^

= ( I I I )

Putting the results of the derivatives given in (1), (II), and (III) back into equation (4.8), shows that the dynamic decomposition of income inequality, using the Theil index (/;), can be expressed as follows:

A = + (TermA)

[^hv +

\

+ l)]A

+

(Term C)

01.9)

Thus, term A in equation (4.9) measures the effects of changes in population shares of sub-groups to changes in overall inequality (the "reallocation effect"); term B captures the contribution attributable to changes in the relative mean income of the sub-groups (the "relative income effect"); whilst term C represents the impact of

changes in the within sub-group inequalities due to changes in unobserved factors (the "pure inequality effect").

Following Jenkins (1995), to facilitate the analysis it is convenient to work with proportionate changes which is achieved by dividing Æy by the initial value of /;, and this implies dividing the terms A, B, and C in equation (4.9) by the initial value of /; as well. To illustrate the usefulness of this methodology let us assume that the population is partitioned by some specific characteristic, say gender. The contribution of changes in the variable gender to overall inequality changes might come through changes in either, the number of female-headed households relative to male-headed households, or in their relative mean incomes, or in both. Thus, if changes in the relative number of female-headed households and male-headed households were the most relevant causes of changes in overall inequality, then term A should be the largest relative to AIj/Ij . If, instead, changes in their relative mean incomes were the most important causes of changes in total income inequality, then term B should be the largest relative to A/y/Zy. Finally, if term C is the largest relative to A/y//y, then this means that gender does not have "explanatory power" of changes in overall income inequality. In Chapter 8 I apply this approach in order to determine causes of inequality changes and the possible channels through which economic changes might have affected the distribution of income in Venezuela during 1989-

1997. Also, by calculating the statistic Cb , I address, in the same chapter, the differences between both approaches and the risk of obtaining misleading results.