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My sister and her partner are starting up a small farm in the US. They have almost everything they need at this point…except an affordable place to live. What is the lowest cost possible? This is a distinctly mathematical question for which the answer could shape their future.
People with financial experience might confidently navigate the shortest path to a suboptimal but stable solution. But how about my sister, a self-defined millennial with millennial-scale resources (a high grit-to-actual-money ratio)? Not only that, but a millennial with a deeply ingrained aversion to maths?
Maths Avengers Assemble.
How might we go about modelling this problem?
Four hundred years ago, someone starting a farm in that region could build themselves a turf house to stay in – a comfy hole in the ground that could be built relatively quickly from the land itself without purchasing materials. They could upgrade their homestead gradually if they succeeded – from log cabin all the way to fancy frame house. In some sense, house-building was a continuous function.
Since that time, building codes have been invented, geared towards urban housing. It is no longer permissible, in most places, to start up a farm using a turf house as a base of operations. What’s the new minimum in that region? Is it within reach? How would you begin to find out? Here is the answer according to my sister.
First, look into whether you’d be allowed to build a turf house, or the equivalent (live on your land in a glorified tent, shed, etc. for a few years to save up). Nope. You can’t. Done.
Next, see if you can afford to buy anything in the nearby town. Most of it is too big, built for commuters with jobs in nearby cities. A few dilapidated fixer-uppers could be affordable, but the repairs needed to be able to resell the house later and get on to the farm could push the cost up.
So maybe you want to build. Mobile homes are out, because they only depreciate; there’s no way to build equity in a mobile home. The next step up is modular, factory-built housing. At first glance, these homes seem like the cheapest. They are built in an assembly-line fashion and in theory offer the economy of scale. On paper this looks perfect! Medium-sized modular homes are out of your price range, but cheaper than traditionally-built houses, old or new. If smaller ones scale down to below 750 ft2 (70m2), they could even be affordable. However, the catalogues don’t list these smaller models.
The trouble is, the manufacturers don’t seem to want you to scale down too easily. But how small can they build? (What can you order from the secret menu?) After some research it would seem that there are three major factories used by all of the companies selling state-regulation-compliant homes in the US. One builds to floor plans with modules of 14ft (4.3m) on a side, one builds to 15ft (4.6m), and one to 16ft (4.9m). Two of the 14ft modules should come in under the threshold for affordability, if the pricing scale is consistent.
Of course, the pricing scale for modular housing does not turn out to be consistent (or at least, not linear). Very small homes end up being more expensive per square metre because the base cost of taking up time on the factory floor is higher relative to the cost of materials and transportation for small orders.
The final step is to request quotes from builders. The lowest for a sub-750ft2 (70m2) house is right on the line you were hoping to get under, and you know that costs can overrun on the estimate very easily.
Here is all this summarised in graphical form. A region of values is shown for the self-built data instead of a line because a very wide range of estimates was available which somewhat overlapped the estimates provided for cheap on-site builders. Building at one end of this range or the other would affect important decisions about materials, techniques and time.
What do you notice? What does this graph reveal that might support the decision-making process? What relevant contextual details does it miss? On the expensive end of things, why might there start to be a wider discrepancy between cheap and standard builder rates? Between builders and modular homes?
In these particular circumstances, by hook or by crook, there really does seem to be a local minimum. This may mark the first time my sister has modelled one, and she understands perfectly and viscerally what it is – and what it means for people who can’t meet it. The best hope they have for coming in under the line and still having time and energy to start a farm might well be renting nearby while slowly accumulating materials so that they can build as much as possible themselves – or supporting new legislation which would enable tiny house exceptions for building codes.
Students of maths become students of life, and it’s amazing what they can model when it matters!
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