A thousand years ago, books were generally written by hand, on parchment made from sheep skin. I don't have a good source on how long it took a person to transcribe a typical book, so for the purpose of this post let's just call it 30 days. I do know that a typical book required the skins of about 12 sheep (source: Braudel).

We can represent this via two production constraints:

$N_{b o o k s} \leq \frac{1}{30} N_{t r a n s c r i p t i o n D a y s}$

$N_{b o o k s} \leq \frac{1}{12} N_{s h e e p}$

... and of course we could add more constraints to reflect all the other inputs to a book. We write it like this, rather than just saying "1 book = 12 sheep + 30 transcriptionDays", to highlight that each input is an independent limit on the number of books produced. If we only have 15 sheep on hand, then we can make at most 1 book, no matter how many bored transcriptionists are sitting around.

Another reason why writing out the constraints is useful: it offers a natural way to introduce technology changes.

Let's consider two possible technology changes:

switching from parchment to paper
switching from transcriptionists to a printing press

How do these modify the constraints? Well, paper eliminates the sheep constraint and replaces it with a paper constraint (of the form $N_{b o o k s} \leq C N_{p a p e r}$ for some $C$ ) - yet the transcription constraint remains exactly the same. Conversely, a press eliminates the transcription constraint - yet the sheep constraint remains exactly the same. The constraint representation is modular with respect to technology changes: introduction of new technology removes/modifies some constraints, while leaving most of them unaltered.

With a little creativity, this representation can be extended to other kinds of technology changes as well:

Before the invention of television, we had the constraint $N_{T V} \leq 0$ . The invention of television replaced this constraint with a bunch of television production constraints, like $N_{T V} \leq N_{v a c u u m T u b e s}$
Fixed-cost capital goods, e.g. a printing press, add a constraint that we need at least one of the capital good, independent of the number of things produced: $1 \leq N_{p r e s s}$
A more efficient sheep-skin processing technique might replace $N_{b o o k s} \leq \frac{1}{12} N_{s h e e p}$ with $N_{b o o k s} \leq \frac{1}{10} N_{s h e e p}$

... etc.

Conjugacy

One of the main lessons of optimization theory - be it linear programming, convex analysis, what have you - is that every constraint has a conjugate ...

Gears Which Turn The World

Gears Which Turn The World