Section 3
Low income indexes under alternative lines

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Headcounts under alternative low income lines
Other aggregate indexes under different low income lines
Different indexes under the same line

A low income line per se does not tell us how many individuals are in low income, how much their income shortfalls are, and how these shortfalls are distributed. To answer these questions, we examine a number of aggregate low income indexes. 

Headcounts under alternative low income lines

The most often used low income index is the "headcount", also referred to as the low income rate or low income incidence. The index simply tells us what proportion of individuals whose incomes are below a given threshold. Although a comparison of the headcounts under alternative low income lines is problematic due to the fact that different lines are subject to different assumptions and arbitrary choices, tracing their movements over time is meaningful.

The low income rate is a special case of the so-called FGT (Foster, Greer and Thorbeck, 1984) index. It can be written as,

Description

where y is a vector of income, z is the low income thresholds, N is the total number of individuals, q is the number of individuals whose incomes are below the low income threshold, and g_i = (z - y_i)/z is a measure of income shortfall for person i. The low income rate is obtained by setting a = 0,

Description

Figure 1 presents low income rates under alternative low income-lines for the 1976-2007 period. The after-tax low income lines and family after-tax income were employed to calculate the low income rates under Low Income Cut-Offs (LICO), variable and fixed Low Income Measures (LIMs), while family disposable income was used to obtain the low income rate under the Market Basket Measure (MBM) line.¹ To help visualize the historical variations, the estimated incidences were standardized to 1 using their corresponding values in the year 2000.² The unemployment rates for individuals 15 and older were also plotted in the figure (right axis scale).

Figure 1 Trend in low income incidence under alternative lines

Several observations can be made from Figure 1. Overall, low income incidences under different lines appeared to track each other well and they all tracked business cycles. From 1976 to 1981, the incidences declined under LICO, variable and fixed LIMs. They then increased briefly during the recession in the early 1980s, followed by six years of continuing decline until 1989.  Thereafter, they followed an upward trend during the next seven years to the 1996 peak and continued to decline up to 2007, the latest year in which income data were available.

Secondly, low income incidences under LICO, fixed LIM and MBM lines fluctuated over time and behaved in essentially the same way, while the incidence under the variable LIM line varied much less, particularly during relatively short periods. This means that LICO, fixed LIM and MBM may produce the same trend, although they measure low income from different angles. However, the variable LIM line seems to be able to generate its independent information, and thus it probably cannot be replaced by the other lines. This is not surprising, as we mentioned before, the variable LIM line is based on a contemporary income standard, while the other three lines are all based on some fixed standard and the year-over-year changes in their thresholds depends on changes in price.

Thirdly, in year-over-year comparisons, the direction of the changes in low income incidence can be different if different lines are employed. There were numerous occasions in which year-over-year comparisons on the change of low income incidence led to different conclusions when different lines were employed. For example, from 1976 to 1977, the variable LIM line indicated that low income incidence increased while the LICO and fixed LIM lines suggested that the incidence did not change. From 1990 to 1991, while LICO and fixed LIM indicated that the incidence increased, the incidence decreased under the variable LIM line. The differences can also be found over multi-year periods. From 1990 to 1993, the LICO and the fixed LIM line suggested that low income incidence increased, but according to the variable LIM line, the incidence did not change much. Similarly, from 2000 to 2004, the incidence increased slightly under variable LIM, while under the fixed LIM, LICO and MBM lines, the incidence decreased. 

Finally, even when the changes in the incidence are in the same direction, the magnitudes of the changes can be different under different low income lines. For example, from 1996 to 2000, low income incidence declined under LICO, fixed LIM and variable LIM. But the decline under the LICO and fixed LIM lines seemed to be much larger than that under the variable LIM line. Indeed, Appendix table 1 shows that, from 1996 to 2000, low income incidence under LICO and fixed LIM dropped by about three percentage points (from 15.7% to 12.5% under LICO, and from 12.7% to 9.9% under fixed LIM). But during the same period, the incidence under variable LIM declined by less than one percentage point (from 11.4% to 11.7%).

The last two observations drive to the point that, when examining low income under different lines, some inconsistencies do occur. With potential controversy between different low income lines, one thing is probably clear: it is not the best practice to pick one line and disregard the others. More careful investigations are necessary. A first such step would be to test if the changes and the differences are statistically significant. After all, the estimated low income incidences are often based on a survey sample. A change in low income can reflect a fundamental change of the underlying poverty trend or it can be just due to a sampling error. Appendix 1 contains the standard error estimates for various low income indexes.³ They can be used to infer if a change in a low income index is significant or not.

As an example, Figure 2 presents the 95% confidence interval estimates for low income incidences under various lines. Using these estimates, some of the inconsistencies can be assessed. For example, the observed inconsistency in the declines of the incidence between 1996 and 2000 under different lines are likely due to sampling errors because the confidence interval estimates for the incidence for these two years under LICO and fixed LIM were not overlapping, and those under the variable LIM line overlapped only marginally.

Figure 2 Confidence interval (95%) estiamtes: low income rates (1976 to 2007)

Statistical analyses appeared to confirm our observation regarding the trend in low income incidence. In the long-run, the incidence moved in the same direction no matter which low income line was employed. But in the short-run, they could move in different directions or in the same direction with different magnitudes, depending on which measuring rod was employed. In particular, the low income incidences under the LICO, fixed LIM and the MBM lines mostly moved in close tandem, both in the long-run and in the short-run, while the incidence under variable LIM sometimes changed independently from the other lines in the short-run. Given the inconsistency, a sensible question to ask is, how do other low income statistics such as the gap ratio and severity indexes behave under different lines.⁴

Other aggregate indexes under different low income lines

The simplicity of the headcount ratio made it virtually the only low income statistic used in public debates for a long time. This has changed, at least in the academic world, since Sen's seminal work (Sen, 1976) that inspired a large literature on the axiomatic approach in measuring economic well-being. The FGT index (Equation 1) is one of the influential measures that satisfy a number of desirable axioms.⁵

When we set  a = 1 in Equation (1), the FGT index becomes,

Description

This can be referred to as the low income gap ratio or low income depth of a population. It shows on average, how far the incomes of low income individuals are away from the low income line. Foster, Greer and Thorbecke (1984) demonstrated that P₁ satisfies the monotonicity axiom, which states that, other things be equal, a reduction in income of a low income person must increase the overall low income gap ratio. Notice that P₁ is defined over the whole population, not the low income population alone.

Description

This statistic can be referred to as the low income severity index of a population. In addition to the monotonicity axiom, this index satisfies the transfer axiom which states that, other things being equal, a pure transfer of income from a low income person to anybody who has higher income must increase the severity index. With this index, individuals with large income shortfalls contribute more than individuals with small income shortfalls to low income severity, and hence inequality among low income persons is accounted for.

Several other low income indexes have also been developed. One is the average gap ratio among the low income population, , which is known as Sen's gap ratio. Another, as demonstrated by Osberg and Xu (2000), is the Sen-Shorrocks-Thon (SST) index,

Description

where G(g) is the Gini inequality index of the low income gap ratio gi = (z - y_i)/z in the population. The SST index is also referred to as a low income intensity measure. One advantage of the SST index is that it summarizes low income incidence, gap, and severity in a single statistic.  Heisz (2001) and Picot, Morissette and Myles (2003), among others, have employed the SST index to study low income intensity in Canada.

Figures 3 to 5 illustrate the trends in P₁, P₂, SST and under alternative low income lines. The results again suggest that LICO, fixed LIM, and MBM tracked each other well in the long-run. Like the incidence, P₁, P₂ and SST essentially followed a declining trend from the mid 1970s to 1989, increased from 1989 to 1997and decreased from 1997 to 2007. But in the short-run, different lines lead to different observations. In particular, the aggregate indexes under variable LIM moved differently in short time spans from those under the other lines. For example, between 1996 and 2000, low income depth, intensity and severity under LICO and fixed LIM declined markedly, while those under variable LIM changed little.

Figure 3 Trend in low income gap ratio (P₁) under different lines

Figure 4 Trend in low income severity (P₂) under different lines

Figure 5 Trend in low income intensity under different lines

Again, to make rigorous comparisons, we need to calculate the sampling standard errors for the indexes. As an example, Table 3 contains the testing result with statistics developed by Xu (1998). The table shows that the SST indexes based on all of the four low income lines dropped significantly between 1996 and 2007 and between 2000 and 2007. For example, the z-test statistics were -7.9, -2.4, -7.5 and -8.8 for testing the equality of the SST indexes between 2000 and 2007 under LICO, variable LIM, fixed LIM and MBM, respectively, led to the rejections of the null hypotheses at 1% of significance.

Table 3 Test statistics for changes in low income intensity

Notice that the finding that different lines might lead to different observations in the short-run was for the general population. If we examine , that is, if we focus on the income gap among low income individuals, it can be seen (Figure 6) that, different low income lines pointed in the same direction in both the long- and short-run, and no matter which low income line was employed, we would draw the same conclusion: the income shortfalls among low income individuals were relatively low from the mid 1980s to the late 1990s, and in the past 30 years, they changed little.

Figure 6 Trend in Sen's gap ratio under alternative low income lines

Different indexes under the same line

In order to characterize the four low income lines, we examined several aggregate indexes, one at a time, under these lines. The similarities in low income trend under different lines appeared to overshadow the differences between various aggregate indexes. To answer questions such as whether a single index contains all information on low income, this subsection compares different indexes under the same line. 

The analyses can be conducted by examining the relevant plots of various indexes across Figures 1 to 6. As an example, we extracted the plots of the LICO-based headcount, gap ratio, severity indexes and Sen's gap and put them together to obtain a new graph (Figure 7).⁶ An immediate observation was that higher order indexes generally varied more than lower order index: the severity index varied more than the gap ratio, which in turn varied more than the headcount. This reflects the conceptual differences between the indexes: the severity index is equal to the square of the gap ratio, and hence, any change in the gap ratio would be amplified in the severity index. 

Secondly, different indexes generally moved in the same direction in the long-run, although year over year or in multiple-year periods, they might vary in opposite directions or in the same direction with different magnitudes. From Figure 7, we can easily see that, overall, different indexes from the FGT family tracked each other well. And during the last ten 10 years, the trend under these indexes was virtually identical. But in short time spans, different results might be obtained. For instance, from 1976 to 1977, while the incidence did not change, the low income depth and the severity indexes both increased. On the other hand, from 1992 to 1994, the severity index dropped slightly while the incidence and gap ratio increased.

Figure 7 Comparing different indexes under the same line (after-tax LICO)

But the trend in Sen's gap, the income shortfalls among low income individuals seemed to be different than those implied by the other indexes. Overall, there was not much change in Sen's gap over time: it decreased from the mid 1970s to the end of 1980s, and increased slowly up to 2007. Over periods of several years, Sen's gap also behaved differently than the FGT family of indexes. For example, from1996 to 2007, all of the three FGT indexes dropped significantly. But in the same period, Sen's gap followed a slightly upward trend.

The same exercise was conducted for the variable LIM, fixed LIM and MBM based indexes. The observations were essentially same as those for the LICO based indexes. For instance, under the fixed LIM line (not shown here but can be seen by combining the relevant indexes from Figures 1 to 6), the incidence declined by about 5% between 1983 and 1984, but the corresponding low income gap and severity indexes increased by 3% and 6%, respectively, suggesting that even though considerable number of individuals escaped low income between 1983 and 1984, the well-being of those who remained under the fixed LIM line deteriorated.
 

The result that various low income indexes moved in the same direction in the long-run, and they even moved in the same direction most of the time in the short-run implies that  using a single, simple index such as the headcount may not necessarily be harmful in public discourse. But the finding that they might move in opposite directions or in the same direction with different magnitudes means that it is necessary to look at various indexes at the same time for policy development, and given that various indexes can be produced with little cost, the best practice would be to use multiple indexes, in addition to multiple lines.

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Notes

1. See the methodology appendix for the definition of disposable income.
2. The estimated low income rates under different low income lines can be found in Appendix Table A4.
3. For the period from 1976 to 1995, the standard errors for the incidence, gap ratio and severity indexes are based on analytical method based on linearized Taylor series approximation. For the period from 1996 to 2007 in which 1000 bootstrap weights are available, the standard errors are based on the bootstrap weights which take sampling design (clustering and stratification) into consideration.
4. In addition, stochastic dominance test, which is out of the scope of the current study, may also be pursued. For a recent example, see Chen (2008).
5. SeeHagenaars (1987) for a summary of relevant axioms.
6. TheSST index behaved much like the gap ratio and the severity index, it is thus ignored in Figure 7.