In my last post I defined the awkwardly termed "collectively auto-technological sets," a set of input and output goods in an economy denoted by nodes, or dots, with arrows leading respectively into and out of boxes representing production functions. In addition, some goods are production capacities carrying out specific production functions are are depicted by arrows from such nodes to the boxes whose production function they carry out.
I now present a fanciful example from 32,547 years ago in Peru. I am there. I wish to make pork stew. I take a pig bladder, well washed, line it with sweet leaves, fill it with water, chucks of pork. I add chunks of potatoes, native to my Peru, garlic, salt carried from the nearby shore, carrots and other good stuff. Then I build a fire, and I add small round granite stones to the fire. The stones become hot. I tie the ends of two sticks together to make a third-degree lever, and use them to pluck the hot stones from the fire, and plop them into the propped up pig bladder. The hot stones heat the water. When they cool, I replace them with more hot stones heated by the fire. Soon, I have cooked my pork stew.
Consider the logic of my enterprise: The pig bladder is a good and also performs a production function — it holds water, ingredients and hot stones, which are also goods. The fire heats the stones, another production function. The tied sticks, another good produced by three goods, two sticks and a thong to tie them, allow me to lift the stones into the pig bladder, another production step. The hot stones, cold stones transformed by the fire into hot stones, once plopped into the bladder, perform the production function of cooking my stew.
This is a tiny collectively auto-technological set, including my own labor.
Note next that this system has a kind of closure. It is sufficient unto itself. Suppose we wished to hang the bladder over the fire and cook the stew that way. Nope, the bladder would burn up, spilling half cooked stew into the fire. Thus, our little set is functionally isolated. It cannot easily expand to new goods and production functions. To cook over a fire, we had to invent something like fired pottery that could withstand being hung over a fire.
The next two ideas are slightly formal and then formal mathematically.
Fifty thousand years ago, the diversity of goods in the global economy was perhaps 10,000. Now it is about 10 billion. But old goods can be recombined to make new goods, some of which are also production technologies like the tied pair of sticks cited above. Now the number of unique pairs of N goods is N squared. Thus, as the diversity of goods in an economy increases, the diversity of adjacent possible goods — and thus production technologies, too — in an economy sharply increases.
But this is not sufficient, for any economy must always form a collectively auto-technological set, or it cannot expand its diversity of goods and production capacities and become supracritical.
Here, some wonderful mathematical results come in to help. They concern the connectivity properties of "erdos renyi" or random graphs. Consider N buttons on a wood floor. Use a spool of red thread. Break off pieces of red thread and pick a random pair of buttons and thie them with a piece of red thread. Keep doing this. Every now and then, lift up a button to see how many total buttons you lift. As the ratio of threads divided by buttons, T/B, increases from 0, at first the largest cluster of buttons you lift stays small, 1 or 2. Then triples of tied buttons form, then quadruples. Soon there are a modest number of mid-sized clusters of tied buttons. Then the magic happens: tie at random a few more pairs of buttons together and most, or all, of the mid-sized clusters of buttons get tied together into a giant cluster. This is called the giant component of a random erdos renyi graph.
This is, mathematically, a "first order phase transition." If the number N of buttons is large, the jump to the giant component is sudden as the ratio of threads divided by buttons, T/B, increases. At this point, mathematically, a collectively auto-technological set emerges!
There is another wonderful point to help us. Recall my question: name all the functionalities of a screwdriver for all possible purposes. We cannot do so. But as the diversity, N, of goods in the economy increases, the number of adjacent possible transformations of these to new combinatorial goods increases as N squared, (thinking only of uses of pairs of existing goods).
But each of these new combinations of, say, only pairs or triples of the N goods, cannot be produced in reality unless there is a production capacity — the analogue of the red thread in the erdos renyi random graph — that is a good that is a production capacity for that particular transformation, say of S + S + T -> L (two sticks, S, plus thong, T yield a third-degree lever. Here I am the production technology since I tie the sticks together.)
But as the number of candidate new-adjacent-possible-combinatorial goods increases, so too does the number of required production transformations to create them. Magically, the number of weird, unforeseen possible uses of screwdrivers, tied sticks, poles, pots, as novel production capacities that may be relevant to some of the required candidate production transformations, itself increases. Thus, as the diversity of goods, N, increases, the formation of collectively auto technological sets becomes ever easier.
Once formed, the giant component of a sufficiently large collectively auto-technological set can continue to expand to new goods and production capacities supracritically.
We do not know what "sufficiently large" is, because we do not know how the relevant possible production uses of screw drivers and other goods scales with N and the adjacent possible N squared new goods.
The pig bladder is a subcritical, functionally isolated collectively auto-technological set. A third-world nation is trapped in such sets, or is an export economy in a subcritical economy. Between the two, as N increases to an intermediate level, there is evidence that new but isolated auto-technological sets pop up. These may explain the bursts of growth in the second world. Sufficiently large collectively auto-technological sets are supracritical and, I believe, yield sustained growth.