Household Equivalence Scales

This is Chapter 6, written jointly with Suzie Carson, from The Health and Economic Status of Households. The appendices are not published, and the acknowledgement to Claudio Michelini has been published separately. Even so website presentational requirements have led to some changes – and perhaps infelicities. The full chapter is available from the authors.

Keywords: Distributional Economics; Statistics;

Introduction

The (disposable) income of a household has to be adjusted for the composition of the household, the numbers and ages of those who belong to it, if we want to make useful comparisons of the standard of living of different households, or to predict commonalities of their behaviour such as expenditure patterns. A simple adjustment might be to convert the income to a per capita basis, but that ignores the impact of household economies of scale and for the different characteristics of the inhabitants. It is not true that ‘two can live as cheaply as one’, but two living together are likely to spend less than if they live separately in order to attain the same standard of living. It also seems reasonable to postulate (and the research evidence supports) that different ages have different needs. Other relevant factors might be gender, employment status(for employed people may have outlays that the not-employed do not have, such as on transport to work), and marital status (since a couple may have expenditure savings relative to two singles living in the same house).

This suggests a more sophisticate adjustment to incomes than per capita. In practice the incomes are scaled back by a ‘household equivalence index’ so that instead of dividing the income of a household of two adults by 2.0 (to a per capita) basis, the divisor is, say, 1.8, the lower figure reflecting the notion that, in this case, a couple living together make a savings of, in this case, ten percent on their outlays to attain the same standard of living.

Table 6.1 shows a couple of Household Equivalence Scales for some common households. (Their derivation will be explained shortly.) Following standard NZ practice they are set so that the index for a household with a couple is unity.

Table 6.1: Two Household Equivalence Scales

Household Type Michileni (CM*) Jensen 88
Single Person 0.57 0.65
Single, one child 0.83 0.91
Couple 1.00 1.00
Couple, one child 1.22 1.21
Couple, two children 1.44 1.41
Couple, three children 1.65 1.58
Couple, four children 1.85 1.75
Three adults 1.38 1.29
Three adults, one child 1.59 1.47

While the two look rather similar, and others have eyeballed them and said they were, the extent to which the various scales are broadly the same is a major concern of this chapter.

Table 6.2, with a rearrangement of the Michelini Scale, draws attention to one of their key properties Notice how as the index moves to the right (increasing number of children) or down (increasing numbers of adults), the numerical level increases. This (weak) regularity arises because the more there are, the more it costs to run a household for a given standard of living.

Table 6.2: Michelini Scale(CM*) by Number of Children in the Household (across)

Number
of Adults
0 1 2 3 4
1 0.574 0.826 1.060 1.281 1.494
2 1.000 1.224 1.439 1.646 1.846
3 1.384 1.592 1.795 1.991 2.183
4 1.742 1.940 2.133 2.322 2.507

A Brief History of Equivalence Scales in New Zealand

The modern investigation of the income distribution in New Zealand can said to have been stimulated by the Royal Commission on Social Security, which sat between 1969 and 1972 when it reported. If so, the first New Zealander to address the issue of equivalence scales was Bill Sutch, who quoted a New York Standard to be described below. (1971:87). The Department of Social Welfare also consider the issue, again using the New York Standard, its submission being reproduced in an appendix to the Royal Commission’s report (1972:582-597). It, and the Royal Commission itself, also toyed with an implicit food based one, which will be also be discussed below.

However, the first New Zealander to use an equivalence scale for the sorts of purposes that it will be used in this study – that is to scale incomes to allow comparisons of living standards between households – was probably Peter Cuttance in his pioneering 1974 Masters thesis Poverty Among Large Families in New Zealand. Following Sutch and the DSW, he used a New York Scale, which was based on Community Council of Greater New York estimate of the cost of maintaining individuals in households. For each individual there was a detailed list of commodities that were required for this sustenance, and additionally there was an estimate of the cost of rent and common household requirements for different households. Adding together the amounts for each member of the household and the total household requirements gave an amount (in American dollars) which could be treated at the relativities of an equivalence scale.

Cuttance presumably used this because there was no other available excepting the food based scale which the Royal Commission on Social Security had rejected as unsatisfactory. Its essence was professional dieticians chose a suitable pattern of food consumption for each type of person (typically by age and gender), to value it and to use the quantum as proportional to an equivalence scale (although sometimes there would be a deduction for household groups reflecting some – apparently arbitrary – assessment of the degree of economies of scale in the household food purchasing). The underlying justification for this scale is the ‘Engels law’ that food is a constant proportion of a household budget. In fact the ‘law’ is not empirically robust. This can be seen by a comparison of the food quantums as a proportion of total spending in actual households or in the New York Community Council recommended budget. An even simpler demonstration is that since dieticians recommend a higher food intake for men than women, their expenditure of food is higher. If Engels law was exactly true, all other spending by men would be similarly proportionally higher to attain the same standard of living as for a woman. Thus because a man eats more the scale awards him a larger clothing allowance.

One of the authors used the New York Scale in his early poverty studies, but he also repriced the scale using New Zealand prices of the time. Easton (1973) The exercise taught a number of lessons:

First it indicated there was a considerable element of judgement in the choice of commodities in the budgets.

Second, the resulting New Zealand priced scale was quite different from the New York priced one, in part because housing was relatively cheaper (affecting the household economies of scale) as was health and education (affecting the relative cost of children). The lesson here is that scales from other countries were unlikely to be transferable. Indeed changes in relative prices over time within a country may affect the equivalence scale. As a general rule this is unlikely to be important in the case of inflation, but sometimes policy changes can change relative prices in a manner which impact on the relativities between households. These include the switch on housing assistance from subsidised rents to income supplements, increasing user charges in medicine, and rising tertiary students fees. That these happened in the 1990s means that scales estimated before then may no longer be valid. It is not clear how a Delphic approach discussed below should respond to changes in relative prices.

Third, the application of the scales to the measurement of poverty resulted in very different outcomes, with the numbers below the poverty line considerably greater with the New York scale (because there were lower economies of scale from the higher cost of housing, and children were more expensive). That means the household income distribution (and therefore the numbers in poverty) will be sensitive to the choice of scale.

These latter lessons – the non-transferability of foreign scales, the problems of major price changes, and the sensitivity of the household distribution to the choice of scale – have informed Easton’s subsequent work and critiques although they have hardly impinged on most of the other New Zealand work, some thirty years later. For that reason the subsequent discussion does not pay much attention to attempts to construct a New Zealand scale by drawing on overseas scales.

The first lesson led to an attempt to empirically estimate an equivalence scale using household budget data. The procedure can be seen as an earlier attempt to do what Claudio Michelini did with very much greater sophistication later, using an inferior data base, a more limited estimation method, and a cruder economic theory. (Easton 1980)

Meanwhile John Jensen, at the Department of Social Welfare, proposed a quite different scale, which he updated a decade later, suing a method quite divorced from empirical evidence, using a methodology which might be generously called ‘delphic’. (Jensen 1978, 1988) There was very little justification for the shift in parameters between 1978 and 1988, although as we shall see, there is a major impact on the resulting distribution.

Another delphic scale has been the so-called square root scale, used in overseas studies. Such scales have little empirical justification other than the wisdom of their proposers, and as we shall see – and as foreshadowed in Easton’s earlier work – the shape of the household distribution is sensitive to the one which is used. This is despite a common statement, based on eyeballing the scales, that the various scales are all much the same. (Rochford & Pudney 1984, Whiteford 1988 (which compares possible New Zealand scales with foreign ones without noting that relative prices may matter), Rutherford et al 1990)

Shortly after, Harry Smith at the Department of Statistics attempted to estimate equivalence scales using household data. (Smith 1989), and a year later, showing no awareness of this earlier work, two Treasury officials, Edna Brashares and Maryann Ainsley, reverted to implicit scales based on food budgets despite the defects discussed above and the method’s rejection 18 years earlier by the Royal Commission on Social Security. (Brashares & Ainsley 1990; Brashares 1993; Easton 1995)

In the 1990s the practice has been to use a Jensen scale in both official and non-official studies of the household income distribution. However sometimes users revert to the 1978 scale, and often it is not always clear which they are using. Arguably the Jensen 1988 scale has been used to justify the social security benefit structure. This is despite its lack of empirical underpinnings or attempt to validate it. The validation issue is discussed in the next section. More recently Statistics New Zealand has used the square root scale, again without any justification for the choice other than it is used overseas. (SNZ 1998)

In the late 1990s, quasi-unit records of households became available for a single year. Generally non-government researchers do not have ready access to government survey records because of the statutory requirements imposed on Statistics New Zealand to protect the confidentiality of its respondents. However, they agreed to allow data based on averages of three households to be placed in the public domain. The threes were not selected randomly, but stratified by income, household composition, and household tenure, which retained enough of their relevant economic structure to enable Michelini to estimate household equivalences scales for selected households, assuming their consumption behaviour reflected a particular economic theory (linear consumption expenditure). The method used was comparable to overseas estimation methods of household equivalence scales, although the resulting parameters were different. (Michelini 1998, 2000, 2001; Chatterjee and Michelini 1998) As the introduction reports, Michelini was unable to complete this work because of an untimely death.

Scale values were not estimated for every household type. As the appendix to this chapter explains, it was possible to generalise the scale values by fitting a function to the Michelini estimates, although the fit was not perfect. It is the generalised scale values which were shown in Table 6.2.

Since none of the scales are properly validated we cannot assess directly their relative merits. Burt we can show that the choice of equivalence scale matters.

Validating Equivalence Scales

To validate a household equivalence scale requires some demonstration that household types on the same equivalised income are on the same standard of living, preferably using evidence which was not used to construct the scales. There is a sense that scales derived by econometric methods are internally validated, although typically their construction involves an assumption (say linear consumer expenditure functions) which needs separate validation. Those built up from budget lists also have some internal validation, although the judgements of the composers need separate validation. Delphic scales have not even this possibility of internal validation. There has been hardly any attempt to validate scales.

A practical method of validation involves asking a variety of questions about a household’s living and consumption experiences, and statistically scaling them to give a measure of living standards. It is then possible to compare the equivalised incomes of different household compositions to see to what extent they are associated with similar levels on this measure. (Alternately, an equivalence scale could be constructed by aligning the measure for different household compositions.) By comparing the outcomes for different scales it would be possible to judge their merits, and to some extent validate them.

An preliminary attempt to do this was done in the estimate the economies of scale coefficient for the elderly based on the data in the living standards survey. It ended up with a wide and practically useless, confidence interval. (DSP 2001) The implication may not be merely that the sample size was too small, but possibly income is a poor correlates with living standards. If this is so, it involves a major reevaluation of the methodology of the measurement and interpretation of poverty. However, before that is done there is a need for more empirical investigation of the phenomenon.

The Survey of Living Standards does offer one other insight in regard to the validation problem. It suggests, almost as an aside, that the living standards of the elderly, on its measures, are on average higher than for the population as a whole. Future work will elaborate and probably confirm this finding. In the interim it offers one way of judging the merits of some equivalence scales.

Six Household Equivalence Scales

This section examines eight Household Equivalence Scales, six of which are usable. They can be broadly characterised by two parameters (ignoring a scaling parameter which sets the two adult household at unity):

ES = (A+αC)^(β)

Where
A = number of adults in household;
C = number of children in household;
α = the child adult equivalence parameter;
β = the household economies of scale parameter;

and where
ES is the Household Equivalence Scale which divides household disposable income (which may involve a number of further adjustments) to obtain equivalent income.

If α = 1, and β = 1, the equivalence scale is the per capita scale. However it is generally assumed both parameters are less than unity.

The seven scales we are looking at have the parameters shown in Table 6.3.

Table 6.3: Parameters for New Zealand Household Equivalence Scales

Scale Code Child/Adult
Equivalence α
Economies of
Scale β
Per Capita PC 1.000 1.000
Square Root SR 1.000 0.500
Easton (1973) E73 0.778 0.664
Jensen (1978) J78 0.737 0.781
Easton (1980) E80 0.916 0.606
Jensen (1988) J88 0.632 0.730
Smith (1989) SM 0.713 0.972
Michelini (1999) CM* 0.535 0.801

A brief summary of the characteristics of each table is as follows:

Per capita scale (PC)
The per capita scale is simply the number of people in the household. Equivalent income is calculated by dividing household income by the number of people in the household. No account is taken of the differing needs of adults and children or of economies of scale. As a result, we would expect households with children and larger households to be located nearer the bottom of the distribution when compared to scales that consider economies of scale and the lower needs of children.

Square root scale (SR)
The square root scale has been used internationally by Atkinson et al (1995:19) and by Statistics New Zealand (1998). The scale is the square root of the number of people in the household. The economies of scale coefficient (0.5). This scale has the strongest economies of scale coefficient and so it most raises the equivalised incomes of large households relative to small ones. The square root scale, as with the PC scale, assumes that adult and children have the same needs. The effect of this is to push smaller households with children further down the income distribution.

Easton 1973 (E73)
This is the scale based on the Greater New York Community Council budget requirements. It is shown here to give an indication of what such a scale might look like, even though it is not used in the subsequent analysis. It proves to be a very middling one as assessed by both parameters.

Jensen scales (J78, J88)
The 1978 and 1988 revised Jensen scales, in contrast to the SR and PC scales, allow for the differing needs of adults and children. The adult/child equivalence for the J78 scale (0.737) is higher than that of the 1988 scale (0.632), giving lower needs for children relative to adults in the 1988 scale. The scales use an economies of scale coefficient somewhere in the middle of the scales we looked at.

Easton 1980 (E80)
In contrast to the scales discussed so far, Easton used an econometric approach to estimating the equivalence scales. Using published data from the HES, a savings to income ratio, after tax income, household size and the age of the household head are used to estimate the household equivalence, it being assumed that the same savings ratio on average applied at the same equivalised household income. The Easton scale has the highest child/adult equivalence (0.916) with the exception of the PC and SR scales, which assume that adults and children have the same needs. The economies of scale coefficient is comparatively low at 0.606.

Smith scale (SM)
Smith also uses an econometric approach and uses unit record HES data for the analysis. The scale categorises households by the number of adults and children and the age of the household head. Smith uses two methods to estimate the scales, one based on utility theory and the other on Engel’s law. The scale used in this paper is based on utility theory and uses household disposable income. The adult/child equivalence (0.713) is similar to that of J78. The economies of scale coefficient at 0.972 is far higher than other scales developed in New Zealand and is similar to that of the PC scale which takes no account of economies of scale. The effect of this would be to put larger households, and in particular, those with adults, lower down the distribution. The Smith scale was only calculated for some households, so it cannot be used for comparative purposes.

Michelini scale (CM*)
Like Easton80 and Smith, Michelini uses econometric techniques to estimate his household equivalences. This scale is generalised from the last estimates he made, as described in the appendix to this chapter. The adult/child equivalence for the five household types is the lowest of the seven scales at 0.585. The economies of scale coefficient (0.801) is slightly higher than some of the scales. The scale is labeled CM*, the asterisk to note it has been derived from Michelini’s estimates.

Comparing Household Equivalence Scales

This section has the primary purpose to show that claims that all the household equivalence scales give much the same answers is misleading both in research terms and for policy purposes. To do this we are going to use the method elaborated in the next chapter in which the incomes of households are equivalised, and the characteristics of the resulting distribution are explored.

Comparing the effects of scale is not easy. The method used here is to look at the distribution of the social group relative to the overall distribution. Consider the distribution by deciles. If a group had exactly the same distribution as the population as a whole, it’s distribution would appear as 10 percent in each decile. In practice this rarely happens. If the share of the distribution of the group tends to be in the lower deciles it can be said to be poorer than average, if in the upper ones it can be said to be richer than average. What we shall see is that the shares vary according to the choice of Equivalence Scale, sometimes markedly.

To do this we need to summarise the distribution of equivalised household incomes. The summary measure locates of the median of the social group relative to the percentile of the overall distribution. If the two distributions were identical the median of the social group would be located at the 50th percentile of the overall group. Again in practice this rarely occurs. Subject to various caveats if the median of a group is associated with a higher percentile of the population the group might be said to be richer than the population as a whole, if with a lower percentile it is poorer.

An important caveat is that the distribution of the social group may be wider or narrower in the distribution than the population distribution as a whole. An indication of this is given by the lower and upper quartiles of the social group being located relative to the percentile of the overall distribution. Because there tends to be more interest in the poorer part of the distribution, only the lower quartile is shown.

As will be apparent from the next chapter, we have derived the group medians (and quartiles) for a wide range of social groups. For this chapter’s purposes we look at only those by household type and by age.

The Distribution of Household Type by Equivalence Scale

Table 6.4M: Median Income of Household Type: Relative to Population Decile By Equivalence Scale

Household Type PC SR J78 E80 J88 CM*
Adult not in
Labour Force
58.9 19.9 38.9 53.4 28.7 39.4
Adult in
Labour Force
89.6 65.5 71.0 85.0 72.3 81.4
2 Adults neither
in Labour Force
46.8 33.6 37.9 42.3 39.0 34.5
2 Adults, 1+
in Labour Force
83.6 76.5 79.3 80.7 77.7 79.7
1 Adult with
1+ Children
18.2 13.3 15.8 20.2 15.4 19.3
2 Adults with
1 Child
49.3 50.4 52.6 50.8 52.5 54.2
2 Adults with
2 Children
41.9 52.0 51.1 50.5 52.1 52.6
2 Adults with
3+ Children
20.6 35.7 29.9 23.8 26.5 33.8
3 Adults, no
Children
66.1 71.0 66.3 62.1 67.3 54.3
3 Adults with
Children
30.5 48.9 39.7 31.4 45.2 38.7

Table 6.4L: Lower Quartile Income of Household Type: Relative to Population Decile By Equivalence Scale

Household Type PC SR J78 E80 J88 CM*
Adult not in
Labour Force
53.1 14.0 32.7 44.6 22.7 33.0
Adult in
Labour Force
77.5 49.7 61.1 70.3 53.3 65.7
2 Adults neither
in Labour Force
37.5 24.8 23.5 33.0 23.6 24.8
2 Adults, 1+
in Labour Force
68.1 55.1 59.1 69.1 56.3 58.7
1 Adult with
1+ Children
09.2 06.6 09.4 11.5 08.6 11.3
2 Adults with
1 Child
29.6 28.7 28.8 28.8 28.8 28.9
2 Adults with
2 Children
24.0 29.2 26.3 25.3 30.2 28.2
2 Adults with
3+ Children
11.3 16.6 13.3 13.1 15.4 14.4
3 Adults, no
Children
43.1 46.8 37.3 37.6 44.2 39.1
3 Adults with
Children
12.4 26.2 16.3 13.1 18.1 14.7

There is considerable variation in the location of households types in the household (equivalised) income distribution. To illustrate the point, consider the impact of Jensen’s apparently minor change in his parameters between his 1978 and 1988 scale. Consider households whose sole member is not in the labour force (mainly the elderly). The change in scale reduced the median from the 38.9 percentile to the 28.7 percentile, and the lower quartile from the 32.7 percentile to the 22.7 percentile (in the latter case from above the population quartile to below it).

More generally, the Jensen 1988 and the Square Root scales tend to lower the relative incomes of smaller households, with the policy implication that they should be paid more attention.

Age

Table 6.5M: Median Income of Age Group: Relative to Population Decile By Equivalence Scale

Age Group PC SR J78 E80 J88 CM*
0-14 27.1 37.6 33.4 31.6 37.5 36.3
15-24 54.1 58.2 55.4 51.3 56.2 53.8
25-44 52.6 56.8 56.5 54.6 56.7 56.6
45-64 69.2 63.1 64.5 66.2 63.4 61.7
65+ 55.6 44.7 42.9 42.0 36.9 41.1

Table 6.5L: Lower Quartile Income of Age Group: Relative to Population Decile By Equivalence Scale

Age Group PC SR J78 E80 J88 CM*
0-14 13.3 16.6 14.5 15.0 15.6 16.3
15-24 26.8 32.6 26.9 32.9 30.3 24.1
25-44 25.8 30.6 26.4 26.2 30.6 28.5
45-64 33.2 34.6 28.0 39.4 34.8 35.1
65+ 45.0 22.4 37.1 32.3 40.3 30.8

The same pattern is evident with the impact on age, since generally younger people tend to live in larger households. The scales used by Statistics New Zealand and the Department of Social Welfare and its successors tend to raise the relative incomes at the younger end of the population and to lower them at the older end, relative to the other scales.

So the choice of scale does affect the shape of the household income distribution for research purposes, and it does have implications for policy making involving the income distribution. It is inconsistent with the evidence to argue that the scales are all broadly the same. The choice of scale matters, and implicitly or explicitly the scale so chosen will influence the conclusions.

The next section discusses the choice of equivalence scale. However there is a lemma from the conclusion in this section, of some significance when using an equivalence scale for other purposes. It cannot be assumed that the resulting index will be age independent, since we know that scales are age sensitive. This means that , for instance, the New Zealand Deprivation Scale, which uses Jensen 1988 to equivalise household incomes may have an age bias in it. (Crampton and Salmon 2000) This is a matter for those who have constructed the scale to investigate. Until they have it should be used with caution wherever age has some relevance to the problem being investigated. (This example is chosen not because it is unique, but because epidemiologists go to considerable trouble to adjust mortality rates for age composition. Clearly it could be misleading to associate the age adjusted mortality figures with the potentially age biased deprivation scale.)

The Impact of a Choice of Scale on Estimates of the Numbers in Poverty

Equivalence scales are commonly used in the estimation of the numbers in poverty where an income threshold is used. The most commonly used poverty line – the income below which a household’s equivalised income falls its members are deemed to be in poverty – is the Royal Commission’s Benefit Datum Line (BDL), set in real terms in 1972, as the minimum practical benefit level for a couple for which it was necessary to enable them to participate in and belong to their community. (Easton (1997) notes that an alternative proposed by Stephens et al (1998) is very close to the Royal Commission level.) In the prices at the time of the surveys that this analysis is based upon, that amounted to $15,200 p.a. In order to get the poverty lines for other household types this amount is adjusted according by an equivalence scale, the intention being that reflected the same standard of living.

That means, subject to a caveat to be explained, that while the numbers of couples in households of two adults will be the same for all scales, the numbers will differ for other household types because the poverty levels will vary according to the choice of household. Thus the numbers of poor will vary according to the choice of equivalence scale. Table 6.4 gives an indication of the degree of variation not only for the BDL but for other possible poverty levels.

Table 6.6: Percentage of People Below Given Equivalised Incomes By Equivalence Scale

Income Level of
Two Adults ($p.a.)
PC SR J78 E80 J88 CM*
12200 22.0 07.9 09.9 11.0 07.9 08.6
13200 26.2 10.4 13.0 14.0 09.7 10.6
14200 30.2 13.5 16.4 17.7 12.5 13.5
15200
(RCSS BDL)
34.0 17.8 20.3 21.7 16.6 17.3
16200 37.9 22.8 25.4 25.8 21.3 22.2
17200 41.8 27.1 30.9 29.8 26.7 27.2
18200 44.9 31.3 35.7 34.6 31.5 32.4

The table confirms the conclusion of the previous section. Outcomes are sensitive to choice of equivalence scale. The per capita scale gives higher poverty numbers than others, while the Jensen 1988, the most used in ‘official’ circles gives the lowest levels of poverty numbers for a given poverty line.

A comparison of the Jensen 1978 to the Jensen 1988 equivalence scale is instructive. It reduces the numbers below the poverty line by between 25.4 and 11.8 percent, depending on the poverty line. The difference between them at the Royal Commission BDL amounts to over 130,000 people. Another way of presenting the same point is that the poverty line of $15200 p.a. for a couple using the Jensen 1988 scale is equivalent to $14300 using the Jensen 1978 scale, a reduction in the poverty line of 6 percent.(Given the substantial difference in poverty numbers, it may seem surprising that the difference in effective poverty levels is small. This is because the cumulative frequency distribution of household incomes is steep in this region (or the frequency distribution is dense).

Who among the poor are affected by the different equivalence scales? Table 6.6 shows the figures by household type and the proportions of adults and children for the RCSS BDL. (Easton 1980 gives proportions different from the others in the case of a two person household because it is includes an age of head of household/reference person effect. The remainder are the same because the poverty line is calibrated on an average two person household.)

Table 6.6: Percentage of Household Type Below RCSS BDL By Equivalence Scale

Household Type PC SR J78 E80 J88 CM*
One Adult 04.6 32.5 08.3 05.1 12.2 07.0
1 adult &
1 child
48.1 48.1 33.8 17.9 33.5 17.0
1 adult &
2+ children
93.9 70.2 69.6 60.3 60.5 47.9
Two adults 08.3 08.3 08.3 07.2 08.3 08.3
2 adults &
1 child
29.6 15.9 18.7 19.1 15.6 15.9
2 adults &
2 children
43.2 16.0 21.2 22.6 16.0 17.2
2 adults &
3 children
65.9 21.5 31.1 33.9 21.8 24.7
2 adults &
4+ children
83.5 23.7 46.9 55.8 26.8 34.6
Three adults 14.1 06.7 09.5 11.3 07.9 10.1
Three adults
& children
53.2 20.1 29.6 36.0 23.9 27.7
Other 31.2 08.4 15.3 23.2 11.9 16.9
All children 46.4 20.6 26.2 22.8 20.1 20.8
All Parents 39.3 17.6 22.5 24.7 17.9 19.2
Other Adults 13.2 15.1 10.9 10.9 11.0 10.8

In summary the Jensen 1988 and the Square Root scales tend to favour small households, and thereby underestimate the numbers (relative to the other scales) of children in two adult families, and children and their parents generally. (The Michelini results tend to be towards the lower numbers of poor, because they allow for age of children, and households income tends to increase with older children, as conjectured in Easton (1977).) Nevertheless, all the equivalence scales point the same way. Poverty in New Zealand is centrally a problem of children and their parents. The largest group is in two parent families.

Choosing Between Scales

What is the researcher (or policy maker) to do if he or she wants to avoid the pitfalls that this sensitivity generates? There are a number of options, until systematic validation identifies one equivalence scale as superior to the others.

Use All Scales
This is the strategy followed for some of the underlying analysis of this chapter and the next. We found the difficulty was the massive quantities of data generated, and the resulting enormous task of interpreting the output. Moreover, while some conclusions are robust to choice of scale (e.g. children are always disproportionately among the poor), others which are important are not and these are often very important research and policy questions (i.e. the location in the distribution of the elderly).

Try to Avoid Using Equivalence Scales.
The uncertainties suggest that wherever possible the use of equivalence scales should be avoided. For instance Carson(2000) treated the one and two person households of the elderly separately, rather than combining them by using an equivalences scale. While this is a counsel of perfection, the outcome may be very cumbersome or important questions may be unanswerable – as in the other extreme of using every available equivalence scale.

Select a Scale on a Systematic Criterion
There might be three sorts of selection criteria:

(1) Sophistication of the estimating procedure. This would eliminate delphic generated scales. E73 involves various judgements as to the content of the household expenditure. E80 and Smith are both less sophisticated examples of the generalised econometric estimating which Michelini uses. On this basis CM* is the preferred choice.

(2) Comparisons of parameters. The economies of scale parameters would eliminate the Per Capita scale and Smith, while the child equivalence for Per Capita, Square Root and E80 seems to be too close to that of an adult. (The child equivalence for CM* appears low. This arises because it represents the equivalence for a young child (less than aged 4) rather than the average child who is about 8 years old.)

On the other hand the economies of scale for the Square Root Easton 1980 and Jensen 1988 seem too strong. For instance the last is set so that a single adult household is 65 percent of the two adult household. That implies that the additional adult increases the household spending by about a half (53.8 percent) to maintain the same standard of living.

(3) Judgements based on the impact on location of households in the income distribution. We might think of the scales all being attempts to get at a true scale. In that case one of the middle scales may be close to it. An average would not be appropriate since some of the scales might be thought of as extreme (e.g. per capita). A scale near the middle of all the scales would seem to be more appropriate, where the scales are applied to social sub groups of the population. So we may ask which scale is most commonly in the middle of the seven at each decile and by the three quantiles. This is an ad hoc test, but the results are revealing.

Table 6.7: Frequency in which Each Scale Generate an Estimate at the Centre (Percent)

Scale Measure 1 Measure 2
Per Capita 11.0 06.7
Square Root 05.1 12.0
Jensen (1978) 19.8 22.7
Easton (1980) 07.6 09.8
Jensen (1988) 08.1 15.8
Smith (1989) 08.7 04.3
Michelini* (1999) 21.4 27.8

It would appear that two – Jensen 1978 and the generalised Michelini – are most commonly in the middle. The Michelini scale is ahead of the Jensen 1978 but a slightly different criteria brings them closer. None of the other scales have nearly the same performance, including the widely used Jensen 1988 and Square Root scale. On this measure both would be judged extremist.

On the basis of these three criteria, the generalised Michelini seems to be the most convincing of all the available scales, although were it not delphic, perhaps Jensen 1978 might have some attractions. The ideal would be a scale which had been properly validated.

Allow the Econometrics to choose the Scale

This approach is elaborated in the next section.

The Econometric Approach: With Dummies

Suppose one was estimating the following simple equation

Log X(i)/EQ(i) = α + β Log (Y(i)/EQ(i))

Where X was household expenditure, Y household disposable income and, EQ the equivalence scale value for the household.

Suppose we are not sure what the value of EQ is. We could estimate

Log X(i) = α + β Log Y(i) + (1-β) Log EQ(i)
or
Log X(i) = α + β Log Y(i )+ Σγ(j) D(ij)

where D(ij) is a set of dummy variables which represents the various housing compositions. If the household is of composition j then D(ij) = 1, otherwise it is zero. This means that

Log EQ(i) = + γ(i)/(1-β)

so the econometric equation estimates an equivalence scale.

This is a simplified version of what Michelini was doing, and has the advantage that very little is being imposed upon the equivalence scale structure. However we still require the regularity of a larger household having a larger equivalence scale.

To illustrate the method, consider the estimated equation from Chapter 5.

Table 6.8: Estimates from an Econometric Equation by Number of Children in the Household

Number
of Adults
0 1 2 3
1 0.693 1.065 1.468+ n.a.
2 1.000 1.036 1.177 1.432+
3 1.460 1.765+ n.a. n.a.
4+ 1.882 n.a. n.a. n.a.

Estimated from the coefficients from the Total Medical Spending Equation (See Chapter 5)
+ indicates ‘X or more’ children., 4+ indicates 4 or more adults.

The estimated equivalence scales are horizontally weakly regular: the more children in a household for a given number of adults, the higher the scale value. However, the figures are not vertically regular,: that is for households with children the equivalence scale values fall – rather than rise – from when an extra adult is added to a single adult household.

But the estimates are subject to statistical error, and while it has not been possible to calculate the precise confidence intervals for the equivalence scale values, the irregularity could be explained from this source.

There is another explanation. One could conceive of requiring a different equivalence scale for expenditure from income. The income scaling would give an indication of the general standard of living, while the expenditure scaling would reflect the particularities of the expenditure. The mathematics now becomes

Log X(i)/EQX(i) = α + β Log (Y(i)/EQY(i))

Where X was household expenditure, Y household disposable income and, EQX is the equivalence scale value for the household’s expenditure and EQY for its income. This time we estimate

Log X(i) = α + β Log Y(i) + (Log EQX(i)- β Log EQY(i))
or
Log Xi = α + β Log Y(i) + Σγ(j) D(ij)

where D(ji) is a set of dummy variables which represents the housing composition which the equivalence scale is representing. If the household is of composition j then D(ij) = 1, otherwise it is unity. This means that

γ(i) = (Log EQX(i) – β Log EQY(i)).

This time, however, the two equivalence scales are not identified (that is, they can not be separately measured). The important thing however is while each component may be regular in some circumstances the observed γ(i) will not be.

Thus regularity of the coefficient dummies is not a necessary requirement in this method.

In summary, using dummies in a suitable econometric equation is a means of avoiding deciding on a particular scale. However, the meaningfulness of the dummy coefficients needs to be checked. And as we shall see in the next chapter, equivalence scales are used in contexts where econometric equations are not immediately relevant.

The Michelini Scale(s)

The previous section used a simple ad hoc demand function. Claudio Michelini’s work derived the demand function from an assumed household utility function. His results are therefore more rigorous, and avoid inconsistencies between parameters, such as the individual expenditure items not adding to total expenditure. However, the assumed utility function may be invalid, of course.

While in Table 6.3 we gave a Michelini type scale derived from these assumptions, as explained in the appendix the actual scale he produced was somewhat more limited. with estimates for only four household types. They are estimated separately, and therefore independent of one another.

Table 6.9: Michelini Estimates of Equivalence Scale by Number of Children in the Household

Number
of Adults
0 1 2 3
1 0.573 n.a. n.a. n.a.
2 1.000 1.214 1.448 1.638
3 n.a n.a. n.a. n.a.

The scale meets the criterion of regularity but the sequence for two adult households do not follow a simple function, so it is not possible to interpolate simply the missing values for other household types. The appendix shows how we interpolated by fitting an econometric function across the five observations. This gives different values for the estimated scale values (See appendix table 6.A.2). The estimates of the generalised scale appear to be within the standard errors of the individual estimates.

We can report that Michelini was aware of the problem. Shortly before his death we were discussing how he could remedy the deficiency by a simultaneous estimate of all the scale points together, having them to conform to the equation we used in the Appendix. He was apprehensive about the addition of further non-linear conditions, because that would slow down the convergence of the estimation procedure (possibly so much there was no practical convergence). In the interim he was working on the costs of children, the paper of which was posthumously published

The discussion on Michelini (2001) has not been included.

Summary

The Need for A Scale
For many purposes household income has to be adjusted for household composition. A means of doing this is a household equivalence scale which divides the income by an index which reflects the relative housing expenditure needs. The simplest would be the per capita scale, but it is generally accepted that there are economies of scale so that larger households use less of some resources to attain a given standard of living. It is also accepted that children have a different (lower) relativity to adults, and that while it seems likely that different age groups have different relativities, there is little agreement as to the exact level. Such empirical research there is supports both hypotheses.

Constructing a Scale
There are a number of ways of constructing household equivalence scales.
- Perhaps the least satisfactory method is the delphic where the scale is chosen by judgement with little empirical input.
- Using overseas constructed scales involves the assumption of international comparability. However different relative prices, especially for housing, and different public provision, especially for health and education, means that it is unlikely that foreign scales are particularly relevant to New Zealand. At minimum their usefulness is unproved. An instructive example of international differences is those based on Engles Law of the quantity of food expenditures in total household expenditure. Aside from various weaknesses – such that men consume more food than women – the proportion of food expenditure in household budgets varies internationally, in part because of different price relativities.
- Scales have also been constructed on the basis of household budgets consisting of quantities of consumption items (or just food) but these have a judgmental element to them.
- The most rigorous way to construct a household equivalence scale is to use an econometric method, the more sophisticated of which pursued in New Zealand being Claudio Michelini’s explicitly based on utility theory. However Michelini’s work is not complete.

The Sensitivity of Results to Scales
The choice of equivalence scale matters. It is simply not true that they are all broadly the same when actual effects are compared. The text illustrates a number cases, but one example here will illustrate the problem. While the Jensen 1978 scale estimates that 20.3 percent of the population are below the poverty line (the Royal Commission BDL), the Jensen 1988 scale estimate is 16.6 percent, a difference of 130,000 people. Not only is the numbers of poor affected by the choice of equivalence scale but so is the composition, although the conclusion that the poor are children and their parents remains robust to the choice of scale.

Choosing a Scale
How then to choose a scale? Until there is a validated one the use of an equivalence scale should be avoided unless it is absolutely necessary. One option where scale use cannot be avoided is to use a plausible set of them, and report only results which are robust to the choice of scale. Where that is not possible, then the sensitivity of the conclusion to scale choice needs to be included in the commentary.

Another procedure which avoids the arbitrary use of a scale is to use dummy variables in regression equations, but that is not always possible.

Where it is not, the generalised Michelini scale seems the best available. The only other with merit might be the Jensen 1978 scale. All others seem unsatisfactory, including the Jensen 1988 and the Square Root scales which are most used for official statistical purposes.

Conclusion

While the generalised Michelini Scale may be the best available, given the importance of an equivalence scale in research and policy, there is a strong case for more effort to improve its estimation, possibly along the lines that Michelini contemplated before his premature death.

There is also a desperate need to validate any chosen scale, that is to use evidence external to their construction to demonstrate they are doing what the claim to do.