Inequality Decomposition Background

Conventional approaches towards inequality measurement have contained in them assumptions regarding the type of social welfare function and the relationship between the inequality index and individual utility. These assumptions are either made explicit via specification of the social welfare axioms (Atkinson (1970)) or one can stipulate the properties or axioms a measure of inequality should have^^. The property of being invariant to a scaling up or down of population size or incomes (mean independence) is one such desirable property. Another very desirable property for our purposes of analysing how much national inequality comes from changes across regions is, the decomposability of an inequality measure into population subgroups. The class of measures that satisfy these properties are those which are ordinally equivalent to the single parameter, generahsed entropy family (Cowell (1980)). The single parameter can be assigned to any real value, the choice of which will determine whether the measure of inequality is sensitive to income changes in the upper tail of the distribution or those which are sensitive to the lower tail.

The decomposition property enables a breakdown of total inequality into absolute levels for each region plus a term representing inequality between regions. This cross regional income differential or “between group” tells us how large national inequality would be if each person in each region received the mean^° income for that region. The size of this between group component relative to total inequality can be interpreted as how important the regional partition is in explaining total inequality^’. When it is a small fraction of total inequahty, income differentials across regions explains or accounts for little. Extending this approach to incorporate multiple partitions, such as region and ethnic background gives a between group component which indicates how well the explanatory power of race and regional partitions are in explaining total inequality.

’^This implicitly imposes a type of social welfare functions. Or the regional representative-income, see Cowell (1994).

The most intuitive way to aggregate levels of inequality within each region is to take a weighted average. The weighted average gives an impression of the amount of total inequality that comes from the unequal distribution of earnings in each region with weights that reflect the relative population or income of each region. Across time, these weights will change. It is possible that this alone may lead to changes in inequahty. For example if there are only two regions and all the rich people move out of one region, inequality in that region would fall. The income of the mean worker of that region would also fall, causing a rise in the between group component of inequality. Mookherjee and Shorrocks (1982) have derived a way to separate out these effects , i.e. changes in inequality brought about by changes in the population structure across regions^^ are distinguished from pure changes in inequality within and across regions.

The non-conventional approach towards measuring inequality comes from use of the human capital model of wage determination. The use of log earnings and log wages in this literature provides a case for using the variance of logarithms as an indicator of dispersion, but as survey data is used, a robust measure of dispersion such as the log earning differential between the 90* and 10* percentile worker is preferred^^. The human capital literature asserts earnings differentials can be explained partly by differences in productivity that come about from differential investments in education or on-the-job training. The unexplained part comes from differences in other factors such as unobservable skill. In order to understand what the nature of earnings inequality is we consider earnings differentials before and after the effect of human capital variables have been taken out, i.e. we analyse both weekly and residual earnings differentials.

Results

Following the conventional axiomatic approach to measuring inequality for two measures, half the squared Coefficient of Variation and the Log Mean Deviation^ (LMD) is calculated.

^^For details see the Decomposition Appendix.

^^The reliability of the main results presented here have been tested against the In 84-16 wage differential. ^'‘Equivalent to the generalised entropy measure with parameter 2 and 0 respectively.

- c v ^ = -

2 2 [3.2.1]

LM D=—V lo g — y,-

[3.2.2]

where |X = mean income.

Table 3.1 shows, for both measures, the level of inequality in 1990 is almost double what it was in 1980, with a change of 0.105. The size of the between group component is small and its value as a proportion of the total is given in column 4 as the Rb-value. The Ry values indicate eliminating between group differences in the regional partition have very little effect (3% in 1980) on total inequality. Further 0.100 out of the 0.105 of the change in inequality comes from changes that have occurred within each region.

Table 3.1 - Ry is small

Decomposition of Total Inequality in each year

(1) (2) (3) (4) Total Inequality Inequality Between Inequality Within Rb= Between Region Regions Regions Inequality as a

Proportion of Total Inequality Mean Log Deviation

1980 0.130 0.002 0.128 0.015

1990 0.235 0.007 0.228 0.030

change 0.105 0.005 0.100

Half of the Squared Coefficient of Variation

1980 0.134 0.002 0.132 .015

1990 0.239 0.007 0.232 .029

change 0.105 0.005 0.100

Source: GHS.

To analyse the trend or change in total inequality over the decade, changes in the structure and movements of the population are crucial. The decomposition derived by Mookheijee and Shorrocks

which takes these effects into account is given in Table 3.2. In row 1 it is revealed that changes in the distribution of earners across regions accounts for 1.9% (0.002/0.105) of the total change. The contribution of changes in the population shares to the overall inequality change (column 4) is also very small. Changes in the levels of inequality within each region on the other hand account for 93.3%. Results of partitioning the population by region and age is given in the second row. To keep cell sizes reasonably large, 3 age categories are utilised, young (16-35) middle aged (36-50). and old (51-65).

Table 3.2 - Most o f the “Change” is W ithin Regions or Across Qualifications

Partitions (1) The Total Change in Inequality (2) Changes in Within Group Inequality (3) Changes in Between Group Inequality (4) Inequality Changes that arise from Changing Population Region 0.105 0.098 0.005 0.002 Region, Age^ 0.105 0.091 0.013 0.001 Region, Education^ 0.109 0.085 0.018 0.005 Region, Age, Education 0.100 0.077 0.022 0.001

Notes : 1. The age partition is defined as 16-35, 36-50, 51-65.

2. Four educational groups are used (no qualifications, low, middle and high). Source: GHS.

The effect is a rise in the between group inequality component, implying earnings differentials across age groups (or years of potential labour market experience) have risen between 1980 and 1990. The inclusion of an education partition, high qualifications (ATevels and above), low qualifications (OTevels, CSE’s or clerical qualifications) and no qualifications to the regional partition dramatically increases the size of the between group component. The rise in dispersion of earnings that arise from changes in earnings differentials across individuals with differing education is large. But even when both age and education partitions are used the contribution of

changes in inequality within regions dominates at 77%. This motivates study of changes within regional labour markets separately.

One point to note when analysing inequality trends at the regional level is some cells will be sm alP so that the likelihood for sampling variation is large. The only other data set that could provide bigger samples at the regional level is the New Earnings Survey (NES) micro dataset but this has no information on education. Given the sizeable attention on how education has contributed to the rise in wage differentials, the NES is unsuitable. Hence we continue to use GHS data but pay particular attention to sampling variation and the corresponding standard errors.

A further noteworthy point is the inequality trend measures the difference between the level of inequality in 1990 from the level in 1980. The inequality level in each year itself is the difference between the 90^ and 10**^ percentile worker. Thus, the inequality trend is obtained by taking the difference of a difference. It is worth stressing that when estimates from small samples are differenced (or in this case double differenced) they may be hable to very large sampling variations. Moreover if they are subject to any measurement error this is likely to be compounded at each differencing stage. In other words, even if the percentiles or levels of inequality for each region are quite accurate the changes may be inaccurate and cautions us against ranking regions in terms inequality trend. The only valid way in which this may be done is when the associated standard errors are accounted for. The simplest way to do this is via a simple t-test on the equality

of estimates for all pairs of regions^.

Table 3.3 documents the inequality trend for Great Britain and each region separately. The first column shows the change in the dispersion of earnings between 1980 and 1990, levels of dispersion in 1980 are subtracted from levels of dispersion in 1990. All numbers are positive meaning there has been a rise in dispersion in each and every region. The standard errors corresponding to each of

^^The smallest ones contain about 200 individuals and the largest over 1000.

^^The null hypothesis is, the change in the In 9010 wage diff in region i is different from that of region j. The t-statistic for this test is derived by taking the square root of the sum of the squared standard errors for each regional estimate.

Table 3.3- Regions o f the North Experienced Larger Rises in Earnings Dispersion

Changes in the 90-10 log wage differential

Region (1) (2) (3)

Change in the Change in the Change in the Dispersion of Dispersion of Dispersion of

Earnings Residual Earnings Residual Earnings from a Human from a Full

Capital Specification Specification Great Britain 0.341 0.238 0.180 (.068) (.038) (.040) North 0.898 0.481 0.549 (0.230) (0.102) (0.105) Yorkshire Humberside 0.167 0.180 0.199 (0.081) (0.066) (0.061) North West 0.749 0.180 0.304 (0.122) (0.066) (0.060) E. Midlands 0.273 0.350 0.417 (0.082) (0.077) (0.075) W. Midlands 0.148 0.165 0.140 (0.074) (0.062) (0.057) E.Anglia 0.307 0.397 0.331 (0.132) (0.128) (0.111) London 0.128 0.238 0.218 (0.071) (0.062) (0.059) South East 0.226 0.224 0.141 (0.063) (0.049) (0.045) South West 0.212 0.165 0.150 (0.098) (0.078) (0.074) Wales 0.555 0.373 0.410 (0.136) (0.102) (0.102) Scotland 0.280 0.159 0.155 (0.091) (0.069) (0.066)

Notes: 1. Standard errors in parenthesis.

2. Residual wages are derived from a basic human capital wage equation. This equation includes age, age^ race and 10 dummies on levels of education.

3. The full specification extends the human capital equation with 12 occupation and 9 industry dummies.

these estimate is reported below each number^^. Results from the simple t-tests tell us that, the North and North West and are not significantly different from each other (or Wales) but are statistically bigger than all other regions. The inequality trend in Wales is significantly larger than all but regions of the East Midlands, East Anglia, South West, and Scotland. Noting that the samples for each region are drawn from independent samples, the large standard errors leave us with the result that the change in inequality estimates for the remaining regions are not significantly different from each other.

The estimates in the second and third columns are of the changes in inequality derived from residual earnings. The residual earnings come from wage equations for each year (a human capital wage equation and another which includes dummy variables for industry and occupation). In Appendix A we estimate earnings equations for 1980 and 1990. One can see that by 1990 the returns to skill have risen. These changes may be distinct across regions. It may also be that skills differ across industry and industry differ across regions and changes in the structure of industry over the last decade can explain why there has been a change in inequality. The estimates of column (2) and (3) in Table 3.3 can be interpreted as the inequality trend that is present once effects of levels and returns to personal and job characteristics are removed. The change in the residuals of the regional earnings equations maintain the pattern drawn out in the raw change (column 1). For regions with the significantly big changes in inequality, the North, the North West and Wales, characteristics seems to account for a lot of the differences. Testing the equality of estimates across pairs of regions (as before) reveal the residual inequality trend of the North and

^^The variance of the change in inequality (ci) between t, t+1 when data from the two periods comes from independent surveys is given by:

^ ci “ ^ / +1 ^ ~ 90. / +1 ^ 10, r +1 ^ 10,90, f +1 ) 90, f ^ 10, t ^ 10,90, t ) where G JO and G 90 are the variances of the 10“* and 90^** percentiles of the distribution and G fo 90 the covariance of the two percentiles. The method for deriving the variances and covariances of two quantités 0, and 0, is

2 8 , ( 1 - 8 , )

given by G = for i <j where//0, 7and/f0, ] are the densities at each quantité and T is " r / [ e , . ] / [ 0 . ]

North West is only significantly different than Yorkshire Humberside, the South West, West Midlands and Wales. To be explicit, total earnings inequality in the North, North West and Wales was significantly bigger than the other regions, yet human capital residual inequality in these regions is only bigger than four other regions. The implication is some of the regional variation in inequality comes from variation in observable characteristics across regions, removing these removes some of the observed inequality across regions.