Note to Self: Malthusian Growth Epoch Differential Equations—Analytic:

Principal Reference: Mark Koyama: Could Rome Have Had an Industrial Revolution?

## Quantifying "Two Heads Are Better than One"

Let us take four cases as possible rough descriptions of the agrarian age from 5000 BCE to 1500 CE. Note that all of these apply only during the agrarian age Malthusian epoch. They are irrelevant and do not apply during the previous pre-agriculture, pre-herding, pre-writing gatherer-hunter age. They are irrelevant and do not apply during the subsequent commercial age, when it did or whenever it would have started. They are:

1. Two heads are better than one, and twice as good as one: the rate of technological and organizational progress might be proportional to the global population.

2. While two heads are better than one, each of us has only a limited amount of time to listen; The rate of technological and organizational progress might be proportional to the square root of the global population.

3. While two heads are better than one, we really are far from being a true anthology intelligence: each doubling of the population only does as much to advance inmeaning that the rate of technological and organizational progress might be proportional to the logarithm of the global population.

4. Two heads are not better than one after all: technological and organizational progress does not fall from the air, but rather drops from the air at a near-constant rate.

We know that there were about 5 million people on the earth in 5000 BCE, when the agrarian age more or less began. We know that there were about 50 million people on earth in 1000 BCE. What do those two benchmarks imply about when there would be enough people on the earth thinking and communicating enough about better ways to perform tasks and organize societies to break us out of Malthusian poverty?

We could write down our four possible rought description cases in math, using "c" rather than "t" for our time index because there is no real point to working in units of less than a century, and to remind us that that is what we are doing:

(1) $\frac{dP_{c}}{dc} = γ{P_{c}}^{2}$

(2) $\frac{dP_{c}}{dc} = γ{P_{c}}^{(3/2)}$

(3) $\frac{dP_{c}}{dc} = P_{c}\left(γ{\ln{P_{c}}}\right)$

(4) $\frac{dP_{c}}{dc} = γP_{c}$

And then use http://wolframalpha.com to get us analytic solutions:

(1') $\frac{dP_{c}}{dc} = γ{P_{c}}^{2} → P_{c} = \frac{1}{A - γc}$

(2') $\frac{dP_{c}}{dc} = γ{P_{c}}^{(3/2)} → P_{c} = \frac{4}{{(A + γc)}^2}$

(3') $\frac{dP_{c}}{dc} = P_{c}\left(γ{\ln{P_{c}}}\right) → P_{c} = li^{^{(-1)}}\left(A + γc\right)$

(4') $\frac{dP_{c}}{dc} = γP_{c} → P_{c} = Ae^{(γc)}$

(Yes, I had forgotten that the "log integral" function $li( )$ existed, let alone that its inverse $li^{^{(-1)}}()$ was a thing.)

The fourth of these equations is, of course, simple exponential growth.

If we calibrate the first to the initial 5-million-people-in-5000-BCE start of the agarian age, and if we calibrate time in centuries we get:

(5) $5 = \frac{1}{A + 50γ}$

(6) $250γ + 5A = 1$

(7) $A = 0.2 - 50γ$

And this equation has an vertical asymptote at:

(8) $0.2 - 50γ = γc_{_{asymp}}$

(9) $c_{_{asymp}} = -50 + \frac{0.2}{γ}$ (in centuries around the year 1)

The second, calibrated, gives us:

(10) $5 = \frac{1}{(A - 50γ)^{^{2}}}$

(11) $±\sqrt{5} = \frac{1}{A - 50γ}$

(12) $-50(±\sqrt{5})γ + ±\sqrt{5}A = 1$

(13) $A = \frac{1}{±\sqrt{5}} - 50γ$

And this equation has an vertical asymptote at:

(14) $\frac{1}{±\sqrt{5}} - 50γ = -γc_{_{asymp}}$

(15) $c_{_{asymp}} = -50 + \frac{1}{γ\sqrt{5}}$ (in centuries around the year 1)

Slow growth after the year 1 is a sign rather than growth tending rapidly toward a vertical asymptote is a sign that there was indeed a great anomaly here: the Malthusian regime lasted signficantly longer than it "should" have...

## And So We Finally Get to Mark Koyama...

Mark:

First, there those who tend to think that market expansion is sufficient for sustained economic growth.... Second, there are those who argue that colonial empires or natural resources like coal were crucial for modern economic growth.... Third, there are those who argue that ultimately only innovation can explain the transition to modern economic growth... divided between those who seek to explain the increase in innovation in purely economic terms... and those who... that the answer has to be sought... in something... broadly defined as culture....

Adherents of... the view that trade, commerce, and market development were a sufficient condition for modern economic growth should find the Roman Industrial Revolution counterfactual highly appealing.... Similarly... the Roman empire was a coherent, capitalist, “world system”... colonies... periphery... based around the Mediterranean economy rather than the Atlantic world, but there seems little intrinsic reason why it should have been less successful than the early modern world system in generating economic growth....

The demand for slaves soared after 200 BCE, and... their ready supply encouraged landlords to practice commercial agriculture on a vast scale. Such an economy was ill-suited for modern economic growth....

The arguments of McCloskey and Mokyr suggest greater skepticism towards the counterfactual we have outlined. Mokyr argues that made 18th century Britain distinctive was a “culture of growth”... the importance of a competitive Republic of Science.... Similarly, I am not aware of evidence of the kind of rhetorical change in attitudes towards commerce in the Rome world that McCloskey documents in the 17th century Dutch Republic or 18th century England—no new-found respect for traders and merchants, over and above soldiers and adventurers, and no evidence of lessening distain for commerce or business. If these cultural attitudes were the binding constraint in late medieval and early modern Europe, then they were equally binding in antiquity. I’ve speculated... on the ways in which slavery and other Roman institutions reinforced a cultural ethos that was hostile to trade-based economic betterment.... But I would be eager to read counter evidence....

All of this suggests that a better understanding of why sustained or modern economic growth did not occur during earlier “efflorescences” can help us better understand which factors were important in the explaining the transition that did take place after 1800.

The "Two Heads Are Better than One" approach thus focuses our attention on what was, for it, a profound anomaly. The world in 600 CE—whether in China, the Roman Empire or ex-Empire, or in between—fell far short of what its Whig progressive narrative would have expected to have been created by then.