The Econometrics Of Household Equivalence Scales

Paper to the Wellington Statistics Group (WSG), 11 February, 2004.

Keywords: Distributional Economics; Statistics;

1. Claudio Michelini
2. Household Equivalence Scales
3. Characterising Equivalence Scales
4. Available Scales
5. Which Scale Should We Use?
6. Which Scale Do We Use?
7. A Simple Econometric Procedure
8. The Michelini Scale
9. Conclusion

1. Claudio Michelini

I am grateful for the opportunity that this Wellington Statistics Group occasion gives to me honour Claudio Michelini, who died unexpectedly from a brain tumour just four years ago. I knew him professionally rather than socially, so there are others who could speak more knowledgeably of his personal qualities, but he was one of those wonderfully eccentric immigrants who have given this country so much character. Professionally I found him extremely competent and so recall with pleasure a number of lively sessions between him and myself and Suzie Ballantyne when he enthusiastically and cheerfully discussed the development of his work, the estimation of scientifically based equivalence scales based on the preference-consistent extended linear expenditure system.

Claudio, who came from Italy, had worked on the underlying theory as a part of his postgraduate work at the University of Bristol. But in those days the computing power was insufficient to cope with the non-linear estimation that (ironically) the linear theory required. He put the work aside but returned to it in the late 1990s, using the quasi-unit household data set provided by the Prince Albert Trust, which despite each observation being an average of three households (to guarantee respondents’ confidentiality), had been structured to maintain sufficient of the underlying empirical reality to enable the estimation of empirically based household equivalence scales.

What we did not know when he was visiting us in Wellington from Palmerston North, where he was a Senior Lecturer at Massey University, is that he was here to see a specialist about what proved to be a brain tumour. Alas it had advanced too far, and in March 2000 Claudio did not recover from the operation. At 59 he was in the prime of his life as a teacher, as a researcher and as a friend.

In many ways Claudio’s work was data intensive, computer intensive, and arcane and complex in its economic and estimation theory. My intention today is not just to go through this theory and estimation, but to show how it is also practically important, and that Claudio’s work is a major leap forward in the practice.

2. Household Equivalence Scales

If we want to make useful comparisons of the standard of living of different households, or to predict commonalities of their economic behaviour, 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,. 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.

Do more sophisticated adjustments matter? There are a surprising number of statements in the New Zealand literature which claims that the choice of the equivalence scale is not important. They are based on eyeing scales and suggesting that the differences are not great. Table 1 is an example in which a scale based on Claudio’s work is compared with Jensen88, the standard scale usually used. The convention is to set the scale level for a couple at 1.00. The figures for a single person means the Michelini scale says a single person needs 57 percent of the income of a married couple to attain the same standard of living, while the Jensen scale says that 65 percent is needed.

Table 1. Comparison of Two Equivalence Scales

Household Type Michelini Jensen88
Single Person 0.57 0.65
Single Adult, 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

The real test is not whether they look similar but whether they give similar outcomes. If we are adjusting the raw household income distribution (say, because we are interested in trends in inequality or poverty levels) we find the cumulative distributions are very steep in the ranges where we are interested, so quite small changes along the horizontal income axis (on which the equivalence scales operate) lead to much greater changes on the vertical axis, which gives the numbers of households involved.

Table 2: Percentage of People Below Given Equivalised Incomes By Equivalence Scale

Income Level:
Couple ($p.a.)
Michelini Jensen 88
12200 8.6 7.9
13200 10.6 9.7
14200 13.5 12.5
17.3 16.6
16200 22.2 21.3
17200 27.2 26.7
18200 32.4 31.5

Small differences in income scales – at last to some people as they eye Table 1 – leads to greater changes in the numbers of peoples involved as we see from Table 2. The Michelini scale says that 17.3 percent of New Zealanders were below the Royal Commission Benefit Datum Line in 1997 whereas the Jensen scale says there 16.6 percent. That difference represents about 28,000 people. Notice too, that the bias is systematic. Jensen88 gives lower estimates of poverty than Michelini for all poverty lines.

In many ways the exact level of poverty is not so important as the composition of the poor, because who is poor determines who social policy targets. Table 3 presents estimates of the percentages of household types below the RCSS BDL by Equivalence Scale. The totals correspond to those in Table 2 for the $15200 line. (Note the two adult proportions are the same (8.3 percent) because the nominal values of the actual poverty line is the same for each.)

Table 3: Percentage of Household Type Below RCSS BDL By Equivalence Scale

Household Type Michelini Jensen88
Single Adult 7.0 12.2
Single Adult, 1 child 17.0 33.5
Single Adult, 2+ children 47.9 60.5
Two Adults 8.3 8.3
Two Adults, 1 child 15.9 15.6
Two Adults, 2 children 17.2 16.0
Two Adults, 3 children 24.7 21.8
Two Adults, 4+ children 34.6 26.8
Three adults 10.1 7.9
Three adults, children 27.7 23.9
Other household types 16.9 11.9
All children 20.8 20.1
All parents 19.2 17.9
Other adults 10.8 11.0
ALL 17.3 16.6

Compared to the Michelini scale, the Jensen88 scale identifies relativitly more small households as impoverished compared to large ones, and relativity more one parent households compared to two parent households. Thus it is less likely to identify children and their parents as poor. Thus the Jensen88 scale focuses us on the single elderly, solo parent families and small families compared to the Michelini scale. Even so, in the late 1990s children and their parents were the largest group in poverty, a result which is independent of the choice equivalence scale.

To give an indication of the fiscal significance, the payment structure to New Zealand superannuants follows the Jensen Scale, giving the single elderly 65 percent of the rate of the married couple. If the Michelini scale ratio had been used instead, the single rate would be only 57 percent of the couple rate – about $30 a week less. That cost difference to the government budget comes to around 5 percent of the outlay on New Zealand superannuation, perhaps a net fiscal cost of $225m a year.

My point is not that we should reduce the rate to the single elderly by $30 a week. One could equally argue that if the Michelini scale was the more accurate, the married rate should be increased by around $26 a week to each partner so that they had a material standard of living similar to the single elderly. The critical point is that the parameter matters, and deserves to be measured accurately, since the fiscal cost of error could be large.

(I would mention that my intuition is the Jensen scale is high. A way of thinking about it is that a couple in receipt of $100, each spend $35 on themselves and use $30 to purchase goods which they jointly consume at exactly the same level as if they were by themselves. That means that when one leaves or dies, the remaining partner spends $35+$30 = $65 to maintain the same material standard of living. $30 out of $100 seems a bit high. The comparable figure for the Michelini scale is $14 (where $43 + $14 = $57 and $57 +$43 = $100) , which seems more plausible.)

3. Characterising Equivalence Scales

Thus far I have illustrated that the choice of scale matters. But how to choose a scale? Here are some examples of those used in New Zealand. But first, a mathematical way of characterising scales.

They can be broadly characterised by two parameters (ignoring a scaling parameter which sets the two adult household at unity):

ES = (A+αC)^(ß)

A = number of adults in household;
C = number of children in household;
α = the child adult equivalence parameter;
ß = 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 ß = 1, the equivalence scale is the per capita scale, although it is generally assumed both parameters are less than unity, that is children cost less than adults and there are household economies of scale.

4. Available Scales

A variety of scales have been used in New Zealand. Dismissing overseas generated ones – I’ll explain why shortly – here are the main New Zealand ones grouped by the method by which each is derived.

Mathematical Scales

Some equivalence scales are purely a priori mathematical, with no justification given for them other than some intuitive elegance.

Per Capita Scale
ES = (A+C), has already been mentioned.

The Square Root Scale
The square root scale is ES = (A+C)^(.5) is the square root of the number of people in the household. It has been used internationally by Atkinson et al (1995:19) in the Luxembourg Income Study and by Statistics New Zealand (1998). The square root scale, as with the PC scale, assumes that adult and children have the same needs, and has strong economies of scale.

Mathematical Judgement Scales.

In New Zealand the two main scales are due to John Jensen, in which he took the mathematical function used above, and put in parameters based upon his personal judgement of what was the relativity between a one and two adult household, and that between a two adult and a two adult and four child household. There appears to be no systematic empirical evidence for those judgements and
the 1978 parameters were changed in 1988, for no clear reason.

In 1978 John Jensen proposed ES = (A+.737C)^(.781).

In 1988 Jensen changed his mind and proposed instead ES = (A+.632C)^(.730).

The Jensen88 equivalence scale is the one used by the Ministry of Social Development and many poverty researchers, which is why I used it early to compare it with the Michelini scale. The change may seem minor, but it reduces the numbers to below the RCS-BDL poverty line from 20.3 percent of the population to 16.6 percent of the population, or by about 150,000 people. The cynic might observe that Jensen’s 1988 paper has done more to reduce poverty than any other single policy measure, with no impact whatsoever on the material state of the poor.

Expenditure Judgement Scales

A quite different approach but still requiring judgement, was to use a set of judged expenditures to calculate relative needs by households and so infer an expenditure scale.

Food Based Scales

The most primitive, still used in the US, is to calculate some assessment of the minimum cost of a healthy diet, by various household compositions. The procedure than multiplies by some integer to get a poverty line, but implicitly the costs of the baskets of food map to equivalence scales.

The method was rejected by the 1972 Royal Commission on Social Security, but it popped up again in 1989 when the Treasury employed a visiting American. Like many overseas experts, the visitor does not appear to look at the local literature, nor the international literature outside the US. Her results seem to have been used as a part of the benefit cuts of 1991.

I have, elsewhere, pointed out all sorts of bizarre features of her method. One obvious example will do. The average man requires more food than the average woman. If we take equivalence based food scales seriously, then the level for men will be higher than the level for women, with the consequence that a man requires more income than a woman to attain the same standard of living. This income would not be spent only on food, and the import would be that a man needs to spend more on clothes than a woman too.

Total Expenditure Based Scales

Instead of basing the equivalence scale only on food expenditure, it might be extended to cover all items. Two submissions to the 1972 Royal Commission used a Community Council of Greater New York estimate of the cost of maintaining individuals in households. Thirty years ago that was all that was available, other than scales based on mathematical elegance. I used the New York Scale in my early poverty studies, but I also repriced the scale using New Zealand prices of the time on the New York expenditure weights. The exercise taught me a number of lessons:

First it indicated there was a considerable element of judgement in the choice of commodities in the budgets. For instance, the New York scale provided for a working woman two nighties a year, but only one for a woman who stayed at home.

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 because of the public provision in New Zealand (affecting the relative cost of children). The lesson here is that scales from other countries were unlikely to be transferable. Moreover, changes in relative prices over time within a country may affect the equivalence scale. This may not be important in the case of inflation, but sometimes policy changes can be important, including the switch on housing assistance from subsidised rents to income supplements, increasing user charges in medicine, and rising tertiary students fees. Not only does this paper pay no attention to attempts to construct a New Zealand scale by drawing on overseas scales, but it warns that the domestic scales may have changed over time, in ways that judgmental scales cannot respond. .

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 was my first indication that 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 my subsequent work and critiques although they have hardly impinged on most of the other New Zealand work, some thirty years later.

Econometric Estimates

This led me to seek an econometric method to estimate a New Zealand equivalence scale. In 1980 I published a scale which, I confess with hindsight, proves to be only of antiquarian interest. I shall shortly describe a generalisation of the estimation method. (The data base was the Household Survey, whose income statistics were still inadequate at that time, and in any case I had to used was based upon group averages rather than unit records.)

Harry Smith at the Department of Statistics had another go in 1989. Even though he had access to unit records the results were also not very satisfactory.

A decade later, Claudio turned his mind to the task using quasi-unit records, both alone and with Srikanta Chatterjee. I shall shortly report his method, and also mention some work done subsequently by Suzie Ballantyne and myself. But before doing so, I want to raise the issue of validation of equivalence scales. How do we know which one to use?

5. Which Scale Should We Use?

In the last few paragraphs I have mentioned something like a dozen different equivalence scales. In a longer paper I have discussed how to decide which one to use. It is an important question in social research, and it is even more important in social policy, since – as we have seen – the choice of scale affects the choice of issues that the policy addresses, the cost of the policy, and in the incomes and welfare of those who are affected by the policy. I take it we want to avoid the delphic method of selection, even if we knew which oracle to choose.

It turns out that there appears to be no scientific method which identifies one scale as unquestionably superior to others. Underlying all of them is a series of assumptions – for even the econometrician has to chose the underlying economic theory and the equation form. In the longer paper I suggest the following research strategies:

1. Use All Scales and only come to a conclusion which seems robust to the choice of scale.

2. Try to Avoid Using Equivalence Scales.

3. Select a Scale on a Systematic Criterion such as :
(i) Sophistication of the estimating procedure.
(ii) Comparisons of parameters.
(iii) 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.

On the basis of these three criteria, the Michelini scale seems to be the most convincing, although were it not delphic, Jensen78 would also have some attractions.

4. Allow the Econometrics to choose the Scale, which I shall shortly describe.

6. Which Scale Do We Use?

Given that the choice of scale is important, given that by the late 1980s we had the data base to use scientific/statistical criterion to select a scale, one might have expected that some effort would have been put into an econometric search to identify one. Recall that the social policy focus, the cost of social policy, and the material welfare of individuals all depend upon this choice.

However the policy process works in a mysterious ways. As I have reported it generally uses the Jensen88, there is no obvious merit in this scale, compared with the others I have looked at. (On the other hand, Jensen78 did come up by my selection criterion.)

This conclusion does not reflect upon John Jensen. What he did was sensible given the ignorance at the time. The point is that the social policy process – indeed even much of the research process – was unable to progress beyond his work.

7. A Simple Econometric Procedure

To begin with an intuitive idea of how an equivalence scale might be derived via econometrics, I describe the procedure that Suzie and I used. We were estimating a consumption function for a particular group of items (medical goods and services) using quasi-unit records. At a simple level the required equation was for each household i.

(1)…..Log (X(i)/EQ(i)) = α + ß Log (Y(i)/EQ(i)) + ξ(i)

X was household expenditure on the group of items,
Y household disposable income and,
EQ the equivalence scale value for the household.

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

(2)…..Log X(i) = α + ß Log Y(i) + (1-ß) Log EQ(i) + ξ(i)
(3)…..Log X(i) = α + ß Log Y(i )+ Σδ(j) H(ij) + ξ(i)

where H(ij) is a set of dummy variables which represents the housing composition which the equivalence scale is measuring. If the household is of composition j then H(ij) = 1, otherwise it is zero. (Setting α = 0 in order to maintain equation rank.) This means that

(4)…..Log EQ(i) = + δ(i)/(1-ß).

(Other than a scaling factor, settled by setting EQ to unity for a household of two adults.)

Since Equation (3) is linear in the unknowns, the unknown parameters of β and δ(ii) can be estimated, and hence so can the EQ(i) from Equation (4).

8. The Michelini Scale

This is a simplified version of what Michelini was doing. The difficulty with the above equation is that the aggregation of each expenditure component in the equation will not sum to a sensible aggregate function, together with it being unclear where Equation (1) comes from.

So Claudio used an Extended Linear Expenditure System which is derived from a particular utility function (Kelin Rubin), modified in order to incorporate scale effects in household consumption. There is not the time to derive the system here – Claudio would have enthusiastically done so – but I can report that around 25 equations later, he would come to a Quadratic Almost Ideal Demand System (QAID) based on a Pricing Scale. (PS-QAID) which he further expanded with Demand Shifters to a (EPS-QAID) system. The difficulty with the system is that it is non-linear in its unknown parameters, so the Maximum Likelihood Estimates involve non-linear estimation. Claudio once remarked to me that, even on the high speed computer he was using, the convergence of the estimation procedure could be very slow.

Claudio estimated half a dozen different forms of his equations (and also applied his work to Italy). It is an enormous corpus of results, and I can but report here the favoured equivalence scale.

Table 4: Michelini’s Estimates of Equivalence Scale
Numbers of Adults X Numbers of Children

Numbers of
of adults
1 .573 n.a. n.a. n.a. n.a.
2 1.000 1.214 1.448 1.638 n.a.
3 n.a. n.a. n.a. n.a. n.a.
Numbers of
of adults
1 .574 .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

There is another problem in his raw scale. Look at the row for two adult families. It says that if an adult couple need $100 for a particular standard of living, they need an extra $21.40 for their first child, an extra $23.40 for their second child and an extra $19.00 for their third child. Now in the order of things the second child should be no more expensive than the first, and less so, given the economies of scale. (Note this happens with the generalised scale: $22.40, $21.50, $20.70.)When I discussed this irregularity with Claudio we agreed it could be within the standard errors of the estimates. However because the non-linearity of the system we dont know what they are.

Claudio would have been intrigued by the results from the much less sophisticated estimates which Suzie and I derived. We found a very similar irregularity. Now this could be due to a mis-specification. (It is not obvious that the equivalence scale for an expenditure group should be the same as for income as a whole). Even so, two different data bases and estimation periods generated similar irregularities, which suggests the problem may be structural. Claudio and I discussed whether the effect arose from not allowing for children by age. It is not impossible two children families on average have children who are on average older than one child families. It is less obvious how that could be simply parameterised and estimated, given it would add to the non-linearities and hence the computing problems.

In any case Claudio is not here to progress his work.

9. Conclusion

Whether we like it or not we have to use equivalence scales on a wide range of research and policy issues. The issue of the choice of scale is not an insignificant one, particularly where it has a practical impact on the policy focus, on the fiscal costs of policy, and on the incomes of individuals. Given this importance – given that the issue has been around for over thirty years – it is extraordinary how little systematic effort has been made to get a good quality scale.

A notable exception to this lacuna, was Claudio Michelini who, when the data became available, used the enhanced computer power to estimate equivalence scales based on actual human behaviour rather than based on assumptions and introspection. He did it because he was passionate about the research problem, and was always slightly diffident when I talked about the relevance of his work to policy. Yet he has, in my judgement, produced the best available equivalence scale for New Zealand. But it was not the best possible scale, and Claudio looked forward to improving his last estimates. Unfortunately he never had a chance.

The danger is with Claudio gone, we will lapse back into the torpor of opinion. A better tribute to Claudio’s work would be to continue to progress it – and to incorporate it into social policy in an intelligent way.

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