Monday, 25 January 2016

Family Roots: Equivalising Households

Hey gurrl, want to form an economy of scale?
How to compare your income, spending, and wealth against households of different sizes and composition.

TLDR: Divide household income, spending, or wealth by the square root of household size to estimate what a single would have to earn, spend, or own, at the same living standard.

Personal finance statistics are often averaged per household. However, your household may not be average. You may be single. You may have five children. Even being coupled with no children is far from the norm. Conventional wisdom says that a household of one needs less money than a family of four to achieve the same standard of living.

How much less? Enter equivalisation, (click for soporific definition). Some methods assign different weights to additional family by age. This can be difficult when age data is not available. Furthermore, weights vary by country as do spending habits.

The easiest method to apply is the OECD square root of household size.

Let's take it for a spin.


A couple needs √2 or 1.41 times the income of a single person to achieve the same standard of living. A family with one child needs √3 (1.73) times. A family of four needs double.

Let's say you're single and earning $50K a year. You'll be able to afford a better lifestyle than your couple friends earning a combined $70K.
[$70 ÷ √2 < $50]
The progression is not, as one might assume, linear. That is because of economies of scale. Both a single and a family of four need but one roof over their heads.

Income, to me, is not as important as expenditure to determine standard of living, so we'll leave it here.


The Australian Bureau of Statistics applies its own equivalisation method to wealth figures. It's fairlyhard to see how it fits actual data

Mean household net worth (2011-2012)
  • Lone person aged under 35 - $160K
  • Couple only, reference person aged under 35 - $259K
  • Lone person aged 65 and over - $623K
  • Couple only, reference person aged 65 and over - $1189K
For the square root method to be accurate, you would expect - all other variables like spending habits being the same - couples' mean wealth to be roughly 1.41 times that of singles. Yet it's not. It's 1.6 and 1.9 times for the under-35 and over-65 age ranges respectively. (Errors of +14% and +34%.) The ABS' method better fits Australian data, with the equivalence factor of 1.5 between singles and couples.

Age is a big factor in wealth, and my guess is that the differences in average age between couples and lone persons within the age ranges is a large part of the discrepancy.

Remember also that equivalisation is about producing a figure representative of a certain 'standard of living'. We assume that couples aim for the same living standards as singles. If large groups are more or less thrifty, then equivalised data may not match averages in reality.

Still, if you are a 42-year old couple with an $800K net worth, you're arguably 'wealthier' on an equivalised basis than the average 45-54y.o. household with its average of 3 members and $872K.
[($800 ÷ √2) > ($872 ÷ √3)]


The real test of how much more larger households need to spend is how much more they actually do spend.

Australian singles under 35 spent on average $869 per week in 2012 while their coupled counterparts spent $1429. Square root equivalisation applied to single spending predicts $1225, an error of 16%.

Nevertheless, if you were in a family of three, you could rest assured that your weekly spend of $1500 would not be extravagant by average standards.
[$869 x √3 = $1505]
In fact, Australian couples with eldest child under 5 spent close to the estimate: an average of $1489 per week.

Japan's Statistics Bureau maintains better figures on monthly expenditure by household size. (Also, I have a personal interest in measuring my household against other Japanese.)

2014 Monthly  Expenditure by Household

Number of household members1人2人3人4人5人6人+
Average Expenditure¥189,353¥294,024¥347,169¥381,970¥408,923¥442,325
√(Number of household members)1.001.411.732.002.242.45
Equivalised Expenditure¥189,353¥267,786¥327,969¥378,706¥423,406¥463,818

Square root equivalisation fits better for spending than for wealth but it is not perfect. It underestimates spending up to 4-person families, then overestimates for larger households.

This can be explained by families of different sizes having different standards of living. Childcare, for example, will be a more important expense for 3-person households than for 5-person households, as many 5-person households include grandparents who provide (free) childcare.


My remarks sound definitive, but are only so for brevity. Application of theoretical equivalisation results in theoretical figures.

A couple maintaining their living standards with a child spends only 23% more than they did before parenthood.
[√3 ÷ √2 = 1.23 ]
A linear scale would incorrectly assume a 50% jump in expenditure. Sure, children cost money to raise, but so do adults. Even so, kids are not fiscal suicide.

This is a great argument for cohabitation. Two young workers moving in together maintain the same income, vastly increase net wealth, but can safely drop 29% of their expenses.
[√2 ÷ 2 = 0.71]
I wonder which came first, marriage or the math?

Large families appear to be extremely efficient resource allocators. That said, I don't know how well the formula scales up to extremely large (8+) households.

On the flipside, divorce devastates finances. Two families of two (assuming a couple shares 50-50 custody of two kids) requires 41% more income to maintain the same standard of living as a family of four.
[2 x √2 ÷ √4 = 1.41]
At the same time, assuming a 50-50 split in matrimonial assets, the two parents' access to capital for deposits or spending decreases by 29%.
[1/2 ÷ (1÷√2) = 0.71]
Wealth, income, and spending vary with individuals and life stages. Equivalising by square root of household size is not precise. That said, its curve matches reality better than a straight linear function. Keeping in mind its limitations around varying circumstances, and assumptions about living standards, it remains a useful tool for comparison.

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