Economy of the household

It is somewhat amusing, that the very word economy is derived from the Greek word oikos (οικος), meaning household; yet the household is rarely, if ever, subject to economic analysis.

It is not the case that the household couldn't be grasped by such an attempt; quite the opposite, as I will prove by example.  Such analysis would also be very beneficial, in my opinion.  The only reason that seems possible to me is a simple unwillingness to do this work; not so much due to laziness, but because many people wouldn't want to see the results.

Household, not family

Since family is nowadays taken to mean nuclear family, I will refer to households throughout the analysis.  Another factor is that family is taken to necessarily imply some kind of close genetic relationship; again, this is completely unnecessary for the economic analysis (though there are reasons for there to be a high relatedness).  On the other hand, it's good to recognize that the extended family is a perfect example of a large household in the sense that I use the word.

The first and probably most important observation we can make is that housework scales well below linear as a function of the number of members.  We can also assume that everything that appears as housework ("chores") can scale at most as a linear function, because for something to scale at a higher power, it must necessarily be a form of "communication", a member-to-member exchange.  Thus such factors are not expressable in terms of "who does the chores", and I will simply treat them as a limit on the number of people in the household (for reasons of cohesion).

The second observation is that necessary the monetary expenditures (rent or equivalent, utility bills, etc.) taken together also scale below linear.  Again, it should be obvious that no component can scale at a power above linear; and for low N, the significantly sublinear rent dominates the expenses.

Building the household

Here I will make the assumption that the household should be able to raise fully dependent children.  This requirement means that for a significant part of the childrens' 0-6 years age range—and there would be multiple, overlapping ranges—at least one adult must stay at home.  I think this is a completely reasonable assumption.

However, caring for children does not constantly occupy the person, but includes frequent periods where only availability is needed.  For instance, a sleeping infant only requires that the caretaker be within hearing range.  This means that the caretaker can also do some portion of the housework.

Looking at the other side of the issue, we can now ask: how many income-earning adults do we need to cover the expenses?  In view of what we have so far fixed (at least one person staying at home and caring for the children), I think we can say that a single income probably does not cut it (unless it is unusually high).  Thus the household cannot be a nuclear family, since at least three adults are involved.  It should be noted that I only explore the economic issues, not emotional ones, since my normal approach is useless here.

Among the possible arrengements are newly retired grandparents helping with raising their grandchildren (this assumes a generation gap of 30 years or more), which naturally transitions into the now-adolescent grandchildren being able to contribute care for their grandparants.  Further, this arrangement can be seen as "self-propagating", since young adults could be expected to stay (or return to) home well into their 20s (to help their highly dependent elders), rather than founding their own families; also, at this point their own parents are nearing retirement.

If we want to shorten the generation gap from the 30+ years of the previously outlined attempt, we might try to substitute a different relative in stead of the grandparents.  I couldn't come up with any that was a good fit—generation age gaps didn't work out (no, 20 is not acceptable); their existence was dubious at best (with more than 2 children per couple, (grand)xN-parents are "diluted"), and generally the setup couldn't be replicated in following generations.

Going bigger

After exhausting the possibility-space of three-adult households, we should countinue by exploring still larger ones.  In these cases, we are not limited to a single pair of similar-age people and an individual from a different generation, so I'm not going to explore all possibilities in detail.

In general, it stays true that both the time-related costs and monetary costs per person continue to shrink as the number of members grows, so I'm willing to guess that e.g. a five- or six-adult setup (with two staying at home) and the rest earning a fairly average wage would be stable in that nobody would have to do too much work (either as wage-work plus housework, or as housework), and the disposable income per person after substracting cost of living (rent and utility bills) is comparable to that seen elsewhere.

Working out specific setups is left as a exercise to the reader; there must be loads of solutions to the above constraints.

Cohesion

I only mentioned the issue of cohesion as a "communication cost" growing as the square of members (per capita as a linear function).  I also passingly mentioned that a high relatedness of members is beneficial.  This is a result of inclusive fitness modifying the (iterated) prisoners' dilemma in such a way that the payoff from defection is lowered relative to that of cooperation.

Going beyond households

Due to the problems of cohesion, it might not be feasible to expand the size of the household much beyond what I already described.  However, we can still harness the scaling effects described above in ways where multiple households cooperate.  An already common example is family childcare or childminding, whereby multiple families' children are cared for by usually a single person.

I don't know whether such ideas of cohousing and cooperative housing arrangements will flourish, but they should.