# Section 2 The LIM and proposed modifications

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## 2.1 The current LIM methodology

Currently, Statistics Canada produces three types of low income lines: the Low Income Cut-Off (LICO), the Low Income Measure (LIM) and the Market Basket Measure (MBM). Together, they help policy makers and researchers examine low-income from more than one perspective1. Since 1991, the LIM thresholds have been produced for three income concepts: market income, before-tax income and after-tax income.2 Regardless of the income concept, the calculation involves the following steps:

• For each economic family, calculate the "equivalent size" using its actual size and composition information and the equivalence scale.
• Divide total income of a family by its equivalent size to obtain the "equivalent income".
• Rank all economic families according to their equivalent incomes to obtain the median of the equivalent incomes for all families in the population.
• Divide the above median by 2 to obtain the "standard LIM threshold", which is equal to the LIM threshold for a single person.
• For families consisting of two or more individuals, multiply the standard LIM threshold by the equivalent sizes of these families to obtain their LIM thresholds.

The equivalence scales are employed to account for the economies of scales in consumption for different family compositions and sizes. A family of two persons needs more income than a single-person family, but not twice as much to maintain the same standard of living. Consequently, if the single-person family needs one unit of income, the two-person family needs more than one but less than two units of income. The equivalence scale system under LIM assigns a one to a single-person family, 1.4 to a two-person family (two adults or one adult and one child under 16 years of age), 1.7 to a three-person family consisting of two adults and one child, etc.3

Table 1 contains the after-tax LIM thresholds for the year 20064. Using data from the Survey of Labour and Income Dynamics (SLID), the estimated median of adjusted after-tax family income is \$30,358. Thus the standard LIM threshold is \$30,358 ÷ 2 = \$15,179. The LIM threshold for a single-person family is simply equal to the standard threshold since its equivalent size is unit. For a family of size 2, since its equivalence scale is 1.4, its LIM threshold would be \$15,179 x 1.4 = \$21,251.

## 2.2 Replacing economic family by household

This paper proposes three modifications to the existing LIM methodology. The first is to replace economic family by household as the basic accounting unit in which individuals pool income and enjoy economies of scale in consumption.

As discussed in the previous section, the current LIM methodology uses economic family as the basic accounting unit. The economic family is defined with two conditions: (1) they live in the same dwelling and (2) they are related by blood, marriage (including common-law marriage) and adoption. The vast majority of Canadians live within unique economic family households. In the year 2007, 94.9% of people lived in a household with only one economic family while the other 5.1% of the population was sharing a dwelling with another economic family. As such, only 2.5% of the population lived in secondary economic families within a household. This situation is fairly stable over the last 30 years.

The implicit assumption underlying the LIM methodology is that economies of scale in consumption for each individual are generated from within the economic family only. The assumption may appear reasonable because as just noted the vast majority of Canadians are living exclusively with those who are related to them by blood, marriage, or adoption. Nevertheless, we propose to replace economic family by household, based on two considerations:

First, economic family is not the only unit in which individuals share resources and enjoy the economies of scale in consumption. In some cases individuals from different economic families living in the same dwelling will take turns purchasing certain household consumption goods and services. More to the point, they are able to save on utilities and shelter expenses such as mortgage interest and expenses on heating, cooling, lighting, etc. This is especially important as housing costs represent a large proportion of the consumption of low-income families. For example, two couples, each being an independent economic family, may choose to live in a 2-bedroom apartment, sharing common spaces as well as appliances. These couples will need less income than if living in two one-bedroom apartments and sharing common areas. Under the existing LIM methodology, the above savings are excluded. The proposed modification would take that saving into consideration.

In low income measurement, it is important to take economies of scale into consideration since living arrangements are likely to be correlated with income. It is reasonable to assume that privacy is a normal good. Similar to all normal goods, an increase in income in an economic family will reduce its propensity to share a dwelling with other families or individuals, and a decrease in income will increase the possibility for them to live under one roof. Hence, low-income families or individuals are more likely than their high-income counterparts to trade-off privacy in order to maintain their standard of living. The current LIM methodology ignores the above savings and may thus over-estimate low-income incidence.

### Box 1 How Statistics Canada defines economic family?

Economic family is defined as a group of two or more persons who live in the same dwelling and are related to each other by blood, marriage, common-law or adoption. A person who lives alone is also counted as an economic family for statistical purposes of this report.

The economic family concept used by Statistics Canada is compatible with the definition of family within the household presented in the Principles and Recommendations for Population and Housing Censuses(United Nations, 1998) which defines a "family within a household" as "those members of the household who are related, to a specified degree, through blood, adoption or marriage".

Secondly and equally if not more important, household is the international standard in comparative statistical surveys of income and well-being while the economic family concept is rarely employed by other countries.5 Indeed, the United Nations' Principles and Recommendations for Population and Housing Censuses (1998), while defining family as "those members of the household who are related, to a specified degree, through blood, adoption or marriage", states that the "degree of relationship used in determining the limits of the family in this sense is dependent upon the uses to which the data are to be put and so cannot be established for worldwide use". Similarly, the Canberra Expert Group on Household Income Statistics in its Final Report and Recommendations (2001) considers household as the preferred unit of analysis on income distribution, and this "preference was driven by to a high degree by the relationship of households to both micro (survey) and macro (System of National Accounts (SNA)) data uses". Hence, household rather than the economic family has the maximum international comparability.6

### Box 2 How Statistics Canada defines dwelling and household?

A dwelling is defined as a set of living quarters. Two types of dwellings are identified in the census: collective dwellings and private dwellings. The former pertains to dwellings which are institutional, communal or commercial in nature. The latter, private dwellings refers to a separate set of living quarters with a private entrance either from outside the building or from a common hall, lobby, vestibule or stairway inside the building.

A household is defined as being composed of a person or group of persons who co-reside in, or occupy the same dwelling and do not have a usual place of residence elsewhere in Canada or abroad. The dwelling may be either a collective dwelling or a private dwelling. The household may consist of a single economic family, or it may contain two or more families sharing a dwelling. It is also possible for a group of unrelated persons living together or a person living alone to constitute a household.

An interesting feature of the modification is that such a change creates a further conceptual difference with two other low-income lines produced by Statistics Canada. However, this difference does not mean we can't compare low-income statistics produced under different lines. In fact, the three low-income lines measure the same phenomena of economic well-being using different assumptions and perspectives. They are all subject to their own set of arbitrary choices and implicit assumptions, and the levels of low-income under different low-income lines are, therefore, not directly comparable with or without the modification. The three low-income lines are designed to complement each other. By allowing LIM to pick up the economies of scale not measured by other lines, LIM becomes a better complement to the LICO and MBM lines.

Under the proposed modification, an individual will be defined as in low-income if the household as a whole is in low-income which in turn will generate different low-income statistics. Under the existing methodology, the economies of scale in consumption are likely under-captured, with respect to shelter costs. As a result, one may end up with upward bias in low-income statistics. On the other hand, since not everything is shared between economic families within a household, the revised LIM may exaggerate the economies of scale. This implies that, other things being equal, low-income statistics under the modification may contain downward bias relative to the existing LIM. Empirically, it is unclear which factor dominates: the upward bias under the existing method or the downward bias under the new method.

It is not expected that the modification will make significant changes in low-income statistics since, as noted earlier, the number of households containing multiple families is not large and is fairly constant over the past 30 years. But it would also appear that those individuals that choose to form multiple economic family households do in fact achieve a household-like distribution of income. As can be seen in Figure 1, persons in households with multiple economic families have a distribution of economic family income that is very similar to that of unattached individuals living in single economic family households. However, the distribution of household income for those same people closely approximates the shape of the distribution for households with one economic family and two or more persons.

Another important point is the fact that the large majority of multiple economic family (EF) households have relatively small household sizes, implying a conscious choice to form a new household. About 44% of individuals living in multiple EF's live in a two-person household with a further 32% living in three-person households. That is, in three quarters of the cases, two unattached individuals are choosing to live together or an unattached individual is living with a two person economic family.

Figure 2 shows the low-income rates, measured at person level, under the existing LIM methodology and those for which household is adopted as the basic accounting unit, with the existing equivalence scale and weighting scheme. This modification reduces the estimated low-income rates slightly, but does not change the low-income trend in any noticeable way.

## 2.3 Adopting the squared-root equivalence scale

One of the key ingredients under the LIM methodology is to choose the equivalence scale.7 In essence, the equivalence scale measures how the consumption of an individual will have to change when her/his family status changes such that her/his level of well-being is maintained. For example, a woman lives alone and consumes a basket of goods and services for given prices and attains a certain level of utility. The problem in identifying the equivalence scale for her is to ask how much she would save if she were to live with somebody else, attaining the same utility level as before. Since a person cannot be living alone and together with somebody at the same time, it is generally impossible to identify the equivalence scale for each individual.

Nevertheless, income/resources pooling and sharing do occur within a family or household and economies of scale in joint consumption exist. For example, if two families, each of size two, were to decide to form a new family of size four, the new family would not need as many cars, stoves and refrigerators as when they were living separately to attain their previous levels of satisfaction. They may also be able to take advantage of bulk pricing and volume discounts. Thus, in practice, the equivalence scale is primarily employed to account for savings accrued in consumption expenditures for people who live together. But the problem is that there is no agreement about the degree and extent of the saving, and hence various equivalence scales have been proposed and employed.

The equivalence scales under LIM were chosen as a rough mid-point of several scales embodied in the various series of LICOs and administrative/legislative scales implied by the municipal budget guides and provincial social assistance levels. As Table 2 shows, they fall in between the Old Organisation for Economic Co-operation and Development (OECD) scales (also known as the Oxford scales) and scales derived by Poulin (1988) from Statistics Canada's Income Satisfaction Surveys. These equivalence scales have been employed by Statistics Canada to produce the LIM thresholds since 1991, as well as those extended versions to earlier years. LIM's equivalence scales are also employed by the MBM line.8

It is worth noting that all equivalence scales are subject to certain assumptions and some arbitrary choices, as accurately summarized by Jäntti and Danziger (2008, P319): "(t)here is no optimal method for deriving an equivalence scale. Indeed, without additional assumptions, there is no way of selecting the basis for choosing an equivalence scale, let alone the correct equivalence scale, out of the multitude that have been suggested". Yet the choice of equivalence scales does evolve over time. This is most evident at OECD. The "Old OECD" scales contained in Table 2 were popular in the 1980s. By the early 1990s, a set of "OECD-modified" scales were adopted by the Statistical Office of the European Union. These modified scales assign a value of 1.0 to the household head, 0.5 to each additional adult and 0.2 to each child. More recently, OECD publications on poverty and inequality typically employ the square root of household size to adjust household income.9

We propose to adopt the square root equivalence scale scheme to adjust household income. It is consistent with international practices and it is simple. One aspect of that simplicity is that it is monotonic with respect to family/household size, whereas, other scales are not.

Figure 3 shows the average equivalence factor of each additional family member. Under the LIM scale and the Square Root scale the second person is always counted as .4. However, the square root has declining factors for each subsequernt member while the LIM scale does not, and thus flattens out after the third member10. Furthermore, under the Square Root scale one needs only consider how many people are in the family whereas using the LIM scale one needs to keep in mind both the age of family members as well as whether the family is a single parent family.

This simplicity of function adds to the transparancy and interpretability of results. When comparing countries one needs not be concerned for how the differing family composition affects the low-income rate, as people are treated as individuals.

Some recent empirical research appears to indicate that the square root of household size is a good approximation of the true economies of scale.11 Under this modification, we expect the low-income rate to be lower than that under the existing equivalence scale since the new scale allows for more saving as can be seen from Figure 3. But again, we do not expect this modification to change low-income statistics in any significant way because the new scheme only differs meaningfully from the existing one when family/household size is large.

As demonstrated in Table 3, when family/household size is four or smaller, the two equivalence scales are practically identical. Indeed, the observed low-income statistics under the new equivalence scale are just as expected. Figure 4 shows the low-income rates under the current LIM methodology, except that the equivalence scale is now replaced by the square root of family size. The modification reduces the low-income rates slightly but the differences were not significant, and the low-income trends under the two schemes are the same.

## 2.4 The weighting factor

The economic well-being of individuals is the ultimate concern for policy makers and researchers concerned with welfare economics. For example, Statistics Canada routinely produces statistics on the number of elderly persons or children with low income and government income support payments are made to individuals. However, as noted above, the largest determinant of an individual's economic well-being is the income of the family or household in which they live.

As such, a choice arises as to whether to calculate the median income based on ordering households, economic families or persons. As mentioned earlier, the current LIM methodology ranks economic families and is largely consistent with the LICO in that regard12. However, the international norm is to follow the second approach. For example, the Canberra handbook provides the following steps in determining a threshold:

"Once equivalence scale adjustments have been applied to household income so that household income no longer directly reflects the size of the household, household income weights can be multiplied by the number of people in each unit to derive 'person weights'. By the application of these derived 'person weights' to equivalised household income, estimates of the distribution of income amongst all persons can be made. Thus a six person unit 'counts' six times as much as a one person unit."(Canberra, 2001, P41)13

Person weighting produces an estimate of the overall distribution of income among individuals in the population, assuming that all household or family incomes are pooled. This procedure is particularly appropriate for comparisons of populations across geography or time where there are large differences in family size or where family size is correlated with income. The individual distribution reflects the assumption that household or family income is shared equally between all members, and does not reflect the direct receipt of income by each individual. Because many household members receive no income, e.g., younger children, this assumption is hard to dismiss in practice pending further advancement in research in this area14.

The two approaches would generate different low-income statistics if family or household size is correlated with income. As an example, consider a population consists of 11 families and 36 individuals. The numbers of person(s) in those families are 1, 2, 2, 2, 3, 3, 4, 4, 4, 5 and 6, respectively:

Assuming that their adjusted incomes are, respectively,

{15,000; 20,000; 30,000; 40,000; 45,000; 50,000; 60,000; 65,000; 70,000; 80,000; 85,000}

If we rank families according to their adjusted incomes, the estimated median would be 50,000, and half of the median is 25,000. In this case, two out of 11 families are in low- income. Since there are only 3 individuals from these two low-income families, the low-income rate measured at individual level is 8.3% in this population. On the other hand, if we rank individuals as suggested by the Canberra group, the estimated median would be 65,000, implying a low-income threshold of 32,500. Under this scenario, 5 persons from 3 families would be in low-income, resulting a 13.9% of low-income rate at individual level.

More generally, if families or households at the left tail of the income distribution are large in size, while families or households at the right tail are small, the median of the income distribution based on individual weighting would be lower than the median of the distribution based on family or household weighting. On the other hand, when families or households at the left tail are small, while families or households at the right tail are large, individual based median would be higher than family or household based median. The latter case is demonstrated by Figure 5, in which individual-weighted median equals \$33,311, while economic family weighted median equals \$31,363.

Notice that the vertical axis in Figure 5 indicates both the proportion of economic families in low income (family weighted line) and the proportion of persons in low income based on the income of their economic family (individual weighted line). Both make use of economic family income. The 18.8% low-income rate at the economic family level is higher than that at individual level. The lower rate for individuals is because the sizes of low-income families are smaller than those of non-low-income families. Indeed, the example from Figure 5 implies a low-income rate of 16.8%.

When we compare the impact of this revised weighting on low-income rates over time, as can be seen from Figure 6, two findings are immediate: On the one hand, the overall low-income trends during the past 30 years were almost identical, no matter which weighting scheme is employed. On the other hand, the estimated low-income rates under the two schemes were similar in the 1970s but they started to diverge in the 1980s, with low-income rates under the new scheme becoming higher over time than those under the existing scheme. In the mid 1980s, the differences were generally less than one percentage point, but by 2007, the difference was approximately two percentage points.

Since the only difference leading to the above result is the weighting factor, a plausible explanation is that the relative family sizes of high and low income families have changed over time. This turns out to have been the case during the past 30 years. Figure 7 shows that in the mid 1970s, there was no clear relation between income and family size. A positive relation would be rejected because families from the top quartile on average had the smallest average family size. A negative correlation would also be rejected because on average, families from the second and the third quartiles were smaller than those from the bottom quartile. But from the mid 1980s, a positive relationship appeared to form between family size and income. Then according to the simple model we discussed at the beginning of this section, when family size and income are positively correlated, the median based on ranking individuals would be higher than that based on family, and a higher standard low-income threshold would be obtained.

## Notes

1. It should be kept in mind that while using three low income lines gives a more complete picture of low-income, low-income itself remains only one facet of economic well being and poverty.
2. Market income is defined as the sum of earnings from employment, investment income, retirement income, support payment received and other incomes such as alimony, severance pay, scholarships, lump-sum payments from pensions and deferred profit-sharing plans received when leaving a plan and supplementary unemployment benefits. Income before-tax, also refered to as total income, is the sum of market income and government transfers, while after-tax income is defined as the above total income minus federal and provincial income taxes.
3. This is discussed in detail in section 2.3.
4. Because the aim of this paper is not to present the most recent low-income statistics using the LIM, the reader will find a number of different reference years used throughout.
5. Canada and the U.S. use similar economic family concepts for low income analysis, however, the U.S. refers to them as census families.
6. According to the Canberra Group (2001), the only difference in defining household among many countries is that some countries impose an "eating together" restriction on household members.
7. See Nelson (1993) for a discussion of the conceptual history of equivalence scales.
8. Under the MBM methodology, the low income thresholds were estimated for the standard family, consisting of two adults and two children, in the 48 regions. The thresholds for other family types in each region are derived by multiplying the threshold of the standard family in that region by the LIM equivalence scales of these families.
9. OECD (2008).
10. The LIM scale falls somewhere between .3(child) and .4(adult) which reflects the differing proportion of adults and children for a given family size in the population.
11. See for example, Fleurbaey and Gaulier (2007).
12. Wolfson and Evans (1989) point out that while LICO is based on an analysis of households on the Family Expenditure Survey, the thresholds are applied to economic families.
13. This point is also made in the Canadian context by Skuterund, Frenette and Poon, 2004.
14. There are some interesting empirical studies examining the intra-household distribution of consumption however these efforts are not sufficiently advanced so as to provide a robust basis for alternatives to the equal sharing assumption.

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