Square root of people

How do you infer the economic well-being of individuals from household income? At one extreme, you could just divide household income by the number of people in the household. This is naive because there are some economies of scale. It doesn’t take twice as much energy to heat a house for two people as a house for one person, for example.

The other extreme is the two-can-live-as-cheaply-as-one philosophy. All that matters is household income; the number of people in the house is irrelevant. That’s too simplistic because while there are fixed costs to running a home, there are marginal costs per person.

So if you have a household of n people do you divide by the total income by n or by 1? Said another way, do you divide by n1 or n0? Some economists split the difference and use an exponent of 0.5, i.e. divide by the square root of n. This would say, for example, that the members of a family of four with a total income of $100,000 have roughly the same standard of living as a single person with an income of $50,000.

Dividing by √n is a crude solution. For one thing, it makes no distinction between adults and children. But for something so simple, it makes some sense. It makes more sense than dividing by n or not taking n into account.

Source: Burkhauser on the Middle Class

9 thoughts on “Square root of people

  1. Perhaps this could be somehow linked to the ages of the people too, for example by summing their years and dividing by some nominal number of years. That’s too simple a suggestion but something more complicated along the same lines might work.

  2. Seems like the economists could use data and statistics: fit a power curve to expenditures as a function of family size. You might find that a different exponent (like 0.7) fits the data better.

  3. I think that a priori all one can really say without measurement is that
    you need to divide by f(n) where f(n) is increasing, asymptotically linear and 1 <= f(n) =1 and f(1)=1. I would expect that e.g. the
    form an + b n^0.5 + c where the coefficents are nonnegative and
    a+b+c = 1 would be a fairly reasonable model, but that’s just a
    stab at it.

  4. Maybe using an exponent is the wrong approach altogether. It seems to me that this is a case where fixed and variable costs should be used. For example:

    For a household with A adults and C children, let Va be the incremental cost of an adult, and Vc be the incremental cost for a child, while F is the fixed cost of maintaining the household. (We could make this model even simpler by not adjusting for adult/child status, but I leave it here to illustrate that this is an easy adjustment.)

    A household like mine, with 3 adults and 2 children would have costs of F+3Va+2Vc

    If you wanted to simplify things further, you could define a “single adult equivalent” (SAE), which would be some value equal to F+Va, we would then define Va and Vc to be some fraction of an SAE. For example, if we define Va = 0.7 SAE and Vc = 0.3 SAE, then for the family described above, we get costs of:

    1 SAE + (3-1)*(0.7 SAE) + 2*(0.3 SAE) = 3 SAE

    Now, you can divide income by SAE and have an idea of relative economic well-being – with fewer of the odd assumptions that roots bring in.

  5. Ben: I think the problem with a more sophisticated model is that the ratio of fixed costs to variable costs can vary greatly depending on your circumstances. In the context of Burkhauser’s research, sqrt(n) sounds reasonable. If you were interested in a narrower segment of the population, a linear model like you suggest would make good sense.

  6. John,

    I would make the exact same argument (does not account for greatly varying circumstances) for using the fractional exponent, and I would add that there were bad assumptions made too!

    I’d clearly need to know more about how Burkhauser is using the numbers, but I just have a hard time buying the argument that adding additional people to a household gets cheaper as you keep adding them. I don’t buy the argument that the second person adds .41 units of cost but the 5th only adds .2 units. I’ve spend some time quoting parts and unit price asymptotically approaches the variable cost, it doesn’t keep decreasing forever. Ultimately you can’t keep a person alive on $0. Many people have a hard time seeing this because they don’t see that

    Quantity 10 = $135.0 each
    Quantity 50 = $ 75.0 each
    Quantity 100 = $ 67.5 each

    is actually a linear function, with a fixed cost of $750 and a variable cost of $60. That decrease doesn’t go on forever.

    I would argue that if this model is slightly more sophisticated (we must arbitrarily assign the value of fixed and variable costs, or SAEs), the underlying assumptions are much more accurate than assuming that costs grow by some (equally arbitrary) fractional exponent.

    Also keep in mind that the cost of an additional person is related to the SAE by a fixed ratio – all it assumes is that the first person costs more than additional people – and if we are looking to divide income by this number to determine “economic well-being” then we are really studying the “circumstances” that you mention.


  7. Ben: I doubt there are many households with more than, say, 8 people, so you don’t have to worry about large values of n. Burkhauser is looking at averages over the entire US, and large households won’t impact the averages much. Also, larger households tend to have lots of children, and the cost per child does tend to go down with more children. At least while they’re young. (But then they go to college … :)

  8. You know, I’m rethinking the college thing. I think we see what it gets you: Arguing IN FAVOR of a linear model with a statistics professor who you’ve never met. ;-)

    In all seriousness, if you are examining economic well-being (and defining it as EWB = income/cost), then I would argue that you really do need to look closely at what the actual incremental costs of adding a generic person to a household is. They are really low: food, water, clothing, and shelter (I’m inserting any incremental utilities costs here) Everything else is a bonus that should be reflected in that EWB measurement – spending more than the minimum amount of money on baby clothes (or teenager clothes) indicates that you have more money to spend and few high-priority unmet needs, therefore your EWB number should be higher.

    The other side of this is the justification for using the exponential model – if we are truly not concerned about very large households, we are really only dealing with a few discrete numbers: 1, 2, 3, 4, 5, 6, and possibly 7 and 8. This doesn’t give us many points to fit data to, and assuming that each data point is going to have some uncertainty associated with it, choosing an exponential model over a linear model seems unwarranted. Also, it would be duplicitous to suggest that the exponential model has fewer variables – a full equation would be y = a * (n – b)^c +d, we’ve just decided that a = b = d = 0 (I respectfully refer you back to your post about cleverly applied simple models vs simplistically applied clever models.) In fact, we could simply select discrete values for our cost function and not have many more variables to deal with – but that would make the math a bit more tricky.

    But let’s be honest; Economists tend to be smart folks, and I would like to think that this one picked n^0.5 for some other reason than sheer mathematical convenience, but as someone who spent several years worrying about costs and economies of scale for several years (I spent a while building cost models and quoting for a manufacturing job shop), I just don’t see how this model could be better than the linear one.

  9. In some cases one can circumvent the problem all together by analyzing different statistics rather than accumulating them with some arbitrary normalization. If you define one wealth statistic for single-person household, one for couples, one for couples with one-child and so on, one can still deduce things like regional differences in wealth. Like you said, there is a limit to how many people live in a household, so even if one allow for different age groups in the categories one can probably end up with a manageable number of statistics to analyze. One can of course not compare how household-size effect wealth, but to that you would need a justification for your normalization anyway. Moreover, breaking the analysis up in this way can serve to validate the n^.5 model for analysis where you need to compare statistics between different household sizes.

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