One of the most interesting emerging sciences today, in my opinion, is cliodynamics. Their practitioners attempt to come to with mathematical models of history to explain “big history” – things like the rise of empires, social discontent, civil wars, and state collapse. To the casual observer history may appear to be chaotic and fathomless, devoid of any overreaching pattern or logic, and consequently the future is even more so (because “the past is all we have”).

This state of affairs, however, is slowly ebbing away. Of course, from the earliest times, civilizational theorists like Ibn Khaldun, Oswald Spengler and Arnold Toynbee dreamed of rationalizing history, and their efforts were expounded upon by thinkers like Nikolai Kondratiev, Fernand Braudel, Joseph Schumpeter, and Heinz von Foerster. However, it is only with the newest crop of pioneers like Andrei Korotayev, Sergey Nefedov, and Peter Turchin that a true, rigorous mathematized history is coming into being – a discipline recently christened cliodynamics.

As an introduction to this fascinating area of research, I will summarize, review, and run an active commentary on one of the most comprehensive and theoretical books on cliodynamics: Introduction to Social Macrodynamics by Korotayev et al (it’s quite rare, as there’s only a single copy of it in the entire UC library system). The key insight is that world demographic / economic history can be modeled to a high degree of accuracy by just three basic trends: hyperbolic / exponential, cyclical, and stochastic.

Korotayev, Andrei & Artemy Malkov, Daria Khaltourina – Introduction to Social Macrodynamics: Secular Cycles and Millennial Trends (2006)

Category: cliodynamics, world systems; Rating: 5*/5

Summary: Andrei Korotayev (wiki); review @ cliodynamics.ru; a similar text на русском.

Introduction: Millennial Trends

Google Books has the first chapter Introduction: Millennial Trends.

In 1960, Heinz von Foerster showed that the world’s population at any given time between 1-1958 CE could be approximated by the simple equation below, where N is the population, t is time, C is a constant, and t(0) is a “doomsday” when the population becomes infinite (worked out to be 13 November, 2026).

(1) N(t) = C / ( t(0) – t )

According to Korotayev et al, this simple formula of hyperbolic explains 99%+ of the micro-variation in world population from 1000 to 1970. Furthermore, a quadratic-hyperbolic equation of the same type accurately represents the increase in the GDP. Why?

He discusses the work of Michael Kremer, who attempted to build a model by making the Malthusian assumption that “population is limited by the available technology, so that the growth rate of population is proportional to the growth rate of technology”, and the “Kuznetsian” assumption that “high population spurs technological change because it increases the number of potential inventors”.

(2) G = r*T*N^a

(3) dT/dt = b*N*T

Above, G is gross output, T is technology, N is population, and a, b, and r are parameters. Note that dT, change in technology, is dependent on both N (indicates potential number of inventors) and T (a wider technological base enabled more inventions to be made on its basis). Solving this system of equations results in hyperbolic population growth, illustrated by the following loop: population growth → more potential inventors → faster tech growth → faster growth of Earth’s carrying capacity → faster population growth.

Korotayev then counters arguments dismissing such theories as “demographic adventures of physicists” that have no validity because the world system was not integrated until relatively recently. However, that is only if you use Wallerstein’s “bulk-good” criterion. If one instead uses the softer “information-network” criterion, noting that there is evidence for the “systematic spread of major innovations… throughout the North African – Eurasian Oikumene for a few millennia BCE” – and bearing in mind that this emerging belt of cultures of similar technological complexity contained the vast majority of the global human population since the Neolithic Revolution – then this can be interpreted as “a tangible result of the World System’s functioning”.

Then Korotayev et al present their own model that describes not only the hyperbolic world population growth, but also the macrodynamics of global GDP in the world system until 1973.

(4) G = k1*T*N^a

(5) dN/dt = k2*S*N

(6) dT/dt = k3*N*T

Above, T is technology, N is population, S is surplus per person (and S = g – m, where g is production per person and m is the subsistence level required for zero population growth), and k1, k2, k3, and a are parameters. This can be simplified to:

(7) dN/dt = a*S*N

(8) dS/dt = b*N*S

(9) G = m*N + S*N

As S should be proportional to N in the long run, S = k*N. Replace.

(10) dN/dt = k*a*N^2

Recall that solving this differential equation gives us hyperbolic growth (1).

(11) N(t) = C / ( t(0) – t )

Furthermore, replacing N(t) above with S = k*N gives (12), allowing us to work out the “surplus world product” S*N (13).

(12) S = k*C / ( t(0) – t )

(13) S*N = k*C^2 / ( t(0) – t )^2

Hence in the long-run, this suggests that global GDP growth can be approximated by a quadratic hyperbola. Other indices that can be described by these or similar models include literacy, urbanization, etc.

One finding is that after 1973, there world GDP growth rate itself falls (rather than just a slowing of the growth of the GDP growth rate, as predicted by the original model): the explanation is, “the literate population is more inclined to direct a larger share of its GDP to resource restoration and to prefer resource economizing strategies than is the illiterate one, which, on the one hand, paves the way towards a sustainable-development society, but, on the other hand, slows down the economic growth rate”. To take this into account, they build a modified model, according to which, “the World System’s divergence from the blow-up regime would stabilize the world population, the world GDP… technological growth, however, will continue, though in exponential rather than hyperbolic form”.*

The consequences for the future are that though GDP growth will reach an asymptote, technological improvements will continue raising the standard of living due to the “Nordhaus effect” (e.g. combine Moore’s Law – exponentially cheapening computing power, with the growing penetration of ever more physical goods by IT).

“It appears important to stress that the present-day decrease of the World System’s growth rates differs radically from the decreases that inhered in oscillations of the past… it is a phase transition to a new development regime that differs radically from the ones typical of all previous history”. As evidence, unlike in all past eras, the slowing of the world population growth rate after the 1960’s did not occur against a backdrop of catastrophically falling living standards (famine, plague, wars, etc); to the contrary, the causes are the fall in fertility due to social security, more literacy, family planning, etc. Similarly, the decrease in the urbanization and literacy growth rates is not associated this time by the onset of Malthusian problems, but is set against continuing high economic growth and the “closeness of the saturation level”.

(AK: This rosy-tinged analysis is persuasive and somewhat rigorous, but there is a gaping hole – they used only “technology” as a proxy for the carrying capacity. However, as Limits to Growth teaches us, part of what technology did is open up a windfall of energy resources – high-grade oil, coal, and natural gas – that have been used to fuel much of the post-1800 growth in carrying capacity (disguised as “technology” in this model), yet whose gains are not permanent because of their unsustainable exploitation. Furthermore, the modern technological base is underpinned by the material base, and cannot survive without it – you can’t have semiconductor factories without reliable electricity supplies – and generally speaking, the more complex the technology, the greater the material base that is needed to sustain it (this may constitute an ultimate limit on technological expansion). This major factor is also neglected in Korotayev’s millennial model. As such, the conclusion that the world has truly and permanently reached a sustainable-development regime does not follow. This is not to say that it is without merit, however – it’s just that it needs to be integrated with the work done by the Limits to Growth / peak oil / climate modelers.)

Chapter 1: Secular Cycles

Korotayev et al conclude that these millennial models are only useful on the millennial scale (duh!), and that typical agrarian political-demographic cycles follow Malthusian dynamics because in the shorter term, population tends to growth much more rapidly than technology / carrying capacity, which led to a plateauing of the population, growing stress due to repeated perturbations, and an eventual tipping point over into collapse / dieoff.

The basic logic of these models is as follows. After the population reaches the ceiling of the carrying capacity of land, its growth rate declines toward near-zero values. The system experiences significant stress with decline in the living standards of the common population, increasing the severity of famines, growing rebellions, etc. As has been shown by Nefedov, most complex agrarian systems had considerable reserves for stability, however, within 50–150 years these reserves were usually exhausted and the system experienced a demographic collapse (a Malthusian catastrophe), when increasingly severe famines, epidemics, increasing internal warfare and other disasters led to a considerable decline of population. As a result of this collapse, free resources became available, per capita production and consumption considerably increased, the population growth resumed and a new sociodemographic cycle started.

He notes that newer models are far more complex and predict the dynamics of variables such as elite overproduction, class struggle, urbanization, and wealth inequality with a surprisingly high degree of accuracy (e.g. see A Model of Demographic Cycles in a Traditional Society: The Case of Ancient China by Nefedov). Korotayev et al then list three major approaches to modeling agrarian political-demographic cycles: Turchin (2003), Chu & Lee (1994), and Nefedov (1999-2004).

1. Turchin has constructed an elegant “fiscal-demographic” model, in which the state plays a positive role by by a) maintaining armed order against banditry and lawlessness, and b) doing works such as roads, canals, irrigations systems, flood control, etc, – both of which increase the effective carrying capacity. However, as demographic growth brings the population to the carrying capacity of the land (in practice, the population plateaus somewhat below it due to elite predation), surpluses diminish. So do the state’s revenues, since the state taxes surpluses; meanwhile, expenditures keep on rising (because of the reasons identified by Tainter). Eventually, there sets in a fiscal crisis and the state must tax the future to pay for the present by drawing down the surpluses accumulated in better days; when those surpluses run out, the state can no longer function and collapses, which leads to a radical decline of the carrying capacity and population as the land falls into anarchy and irrigation and transport infrastructure decays.

2. The Chu and Lee model consists of rulers (including soldiers), peasants (grow food), and bandits (steal food). The peasants support the rulers to fight the bandits, while there is a constant flux between the peasants and bandits whose magnitude depends on the caloric & survivability payoffs to belonging in each respective class. However, it’s not a fully-formed model as its main function is to fill in the gaps in the historical record, by plugging in already-known historical data on warfare and climatic factors; they neglected to associate crop production with climatic variability (colder winters result in lesser crop yields) and the role of the state in food distribution (which staved off collapse for some time and was historically significant in China).

3. Nefedov has integrated stochasticity into his models, in which random climatic effects produce different year-to-year crop yields. One result is that as carrying capacity is reached, surpluses vanish and the effects of good and bad years play an increasingly important role – i.e. a closed system under stress suffers increasingly from perturbations. One bad year can lead to a critical number of people leaving the farms for the cities or banditry, initiating a cascading collapse. However, he neglects the “direct role of rebellion and internal warfare on cycle behavior”, so as the model is purely economic, each demographic collapse is, implausibly, immediately followed by a new rise.

The ultimate aim of Korotayev et al is to integrate the positive features of all three models (Chapter 3), but for now the take a closer look at the political-demographic history of China, the pre-industrial civilization that maintained the best records.

Chapter 2: Historical Population Dynamics in China – Some Observations

Below is a graph of China’s population on a millennial scale. Note the magnitude and cyclical nature of its demographic collapses. Note also that such cycles are far from unique to Chinese civilization (see collapse of the Roman Empire), and reflect for a minute, even, on the profound difference between the modern world of permanent growth, and the pre-industrial, “Malthusian” world.

Since it would be futile to repeat the fine details of every political-demographic cycle in China’s, I will instead just list the main points.

The cycles tend to be ones of a fast rise in population, when surpluses are high and people are prosperous. It plateaus and stagnates when the population reaches the carrying capacity, when there is overpopulation, much lowered consumption, increasing debilitation of state power, and rising social inequality and urbanization.

Sometimes, such as in the middle Sung period, population stress did not lead to a collapse, but instead to a “radical rise of the carrying capacity of the land” through administrative and technological innovations. This increased the permanent ceiling of Chinese carrying capacity from 60mn to around 120mn souls, and in doing so alleviated the population stress until the early 12th century ( AK : e.g. in Early Modern Britain, the problem of deforestation was solved by coal). At that point, China may have once again solved its problems, even escaping from its Malthusian trap ( AK : some historians have noted that it had many of the prerequisites for an industrial revolution). That was not to be, as “the Sung cycle was interrupted quite artificially by exogenous forces, namely, by the Jurchen and finally Mongol conquests”.

: e.g. in Early Modern Britain, the problem of deforestation was solved by coal). At that point, China may have once again solved its problems, even escaping from its Malthusian trap ( : some historians have noted that it had many of the prerequisites for an industrial revolution). That was not to be, as “the Sung cycle was interrupted quite artificially by exogenous forces, namely, by the Jurchen and finally Mongol conquests”. The Yuan dynasty would not reach the highs of the Sung because of the general bleakness of the 14th century – the end of the Medieval Warm Period, unprecedented floods and droughts in China, etc, which lowered the carrying capacity to a critical level. The resulting famines and rebellions led to the demographic collapse of the 1350’s, as well as the de facto collapse of the state, as China transitioned to warlordism.

Carrying-capacity innovations under the Ming did not, eventually, outrun population growth, and it collapsed during the turmoil of the transition to the Qing dynasty. The innovations accelerated throughout the 18th century (e.g. New World crops, land reclamation, intensification of farming). Indications of subsistence stress as China entered the 19th century were a) declining life expectancies, b) rising staple prices, and c) a huge increase in female infanticide rates in the first half of the 18th century. By 1850, China was again under very severe subsistence stress and the state grew impotent just as Europeans began to encroach on the Celestial Empire.

Huang 2002: 528-9, worthy of quotation in extenso. “Recent research in Chinese legal history suggests that the same subsistence pressures behind female infanticide led to widespread selling of women and girls… Another related social phenomenon was the rise of an unmarried “rogue male” population, a result of both poverty (because the men could not afford to get married) and of the imbalance in sex ratios that followed from female infanticide. Recent research shows that this symptom of the mounting social crisis led, among other things, to large changes in Qing legislation vis-à-vis illicit sex… Even more telling, perhaps, is the host of new legislation targeting specifically the ‘baresticks’ single males (guanggun) and related ‘criminal sticks’ of bandits (guntu, feitu), clearly a major social problem in the eyes of the authorities of the time”. See Diagram V.13. ( AK : Interestingly, China’s one-child policy, by artificially restricting fertility in order to ward off a “Maoist dynasty” Malthusian crisis, has led to many of the same problems in the past two decades).

: Interestingly, China’s one-child policy, by artificially restricting fertility in order to ward off a “Maoist dynasty” Malthusian crisis, has led to many of the same problems in the past two decades). Speaking of which… China had further dips in its population after during perturbations in the 1850’s (the millenarian Taiping Rebellion), the 1930’s (Japanese occupation), and 1959-62 (the Great Leap Forward), each progressively smaller than the last in its relative magnitude. For instance, the latter just formed a short plateau.

Korotayev et al conclude the chapter by running statistical tests on China’s historical population figures from 57-2003. In contrast to linear regression (R^2 = 0.398) and exponential regression (R^2 = 0.685), the simple hyperbolic growth model described in “Introduction: Millennial Trends” produces an almost perfect fit with the observed data (R^2 = 0.968). So in the very, very long-term, the effects of China’s secular cycles are swamped by the millennial trend of hyperbolic growth.

Finally, the authors describe in-depth the general pre-industrial Chinese demographic cycle. Below is a functional scheme I’ve reproduced from the book (click to enlarge).

The main points are:

Fast population growth until it nears the carrying capacity, then a long period (100 years+) of a very slow and unsteady growth rate, accompanied by increasingly significant, but non-critical fluctuations in annual population growth due to climatic stochasticity (positive growth in good years, negative growth – along with dearth, minor epidemics, uprisings, etc – in bad years). These fluctuations get worse with time as the state’s counter-crisis potential degrades due to the drawdown of previously accumulated surpluses.

According to Nefedov’s model and historical evidence, the fastest growth of cities occurred during the last phases of demographic cycles, as peasants were driven off the land and there appeared greater demand for city-made goods from the increasingly affluent landowners (who could charge exorbitant rates on their tenants). Furthermore, some peasants are drawn into debt bondage because the landowner had previously given them food at a time of dearth. Other peasants turn to banditry.

Re-“elite overproduction → over-staffing of the state apparatus → decreasing ability of the state to provide relief during famines”. The system of state relief had been very effective earlier, e.g. in 1743-44 a state effort to prevent starvation in the drought-stricken North China core was successful. However: “By Chia-ch’ing times (1796-1820) this vast grain administration had been corrupted by the accumulation of superfluous personnel at all levels, and by the customary fees payable every time grain changed hands or passed an inspection point… The grain transport stations served as one of the focal points for patronage in official circles. Hundreds of expectant officials clustered at these points, salaried as deputies (ch’ai-wei or ts’ao-wei) of the central government. As the numbers of personnel in the grain tribute administration grew and as costs rose through the 18th century, the fees payable for each grain junk increased [from 130-200 taels per boat in 1732, to 300 taels in 1800, and to 700-800 taels by 1821]”. Similarly, the Yellow River Conservancy, whose task it was to prevent floods, degenerated into hedonistic corruption in the early 19th century; only 10% of its earmarked funds being spent legitimately.

So what you have is an increasingly exploited peasantry, a growing (and volatile) urban artisan class – e.g., the sans-culottes of the French Revolution, and more banditry. The bandits create a climate of fear in the countryside and force more outmigration into the cities, and the abandonment of some lands. At the same time, state power – military and administrative – is on the wane, displaced by corruption. The effects of perturbations are magnified due to the system’s loss of resiliency. There eventually comes a critical tipping point after which there is a cascading collapse that involves a population dieoff, the fall of centralized power, and a prolonged period of internal warfare.

Fast population growth does not resume immediately after collapse because things first need to settle down.

In my Facebook Note, Musings on the decline and fall of civilizations, I draw a link between the fast population increase / abundance of the “rise” period, and the concept of the “Golden Age” common to all civilizations. Also ventures a theory as to why cities (hedonism, conspicuous consumption, etc) have such a poor reputation as a harbinger of collapse… because they are, it’s just that the anti-poshlost preachers haven’t identified the right cause (i.e. overpopulation, not “moral decadence” per se).

Furthermore, a tentative explanation of the reason for differential Chinese – European technological growth rates (compare and contrast with Jared Diamond’s explanation):

Incidentally, a possible reason why Western Europe emerged as the world’s economic hegemon by the 19th century, instead of China, a civilization that at prior times had been significantly more advanced. But in China, the depth of the Malthusian collapses was deeper and more regular (once every 300 years, typically) than in W. Europe… Once the Yangtze / Yellow River irrigation systems failed, tens of millions of peasants were doomed; nothing on an equivalent scale in Europe, which is geographically and politically fragmented into many chunks and nowhere has anywhere near the same reliance on vulnerable hydraulic works for the maintenance of complex civilization (control over water was at the heart of “Oriental despotism” (Wittfogel); the Chinese word “zhi” means both “to regulate water” and “to rule”).

This theory that the reason China began to lag behind Western Europe technologically was because of its more frequent collapses / destructions of knowledge should be explored further.

Finally, about the nature of perturbations in a closed system under increasing stress… That is our world in the coming decades: even as Limits to Growth manifest themselves, there will be more (and greater) shocks of a climatic, terrorist, and military nature. The stochasticity will increase in amplitude even as the System becomes more fragile. As a result, polities will increase the level of legitimization and coercion, i.e. they will become more authoritarian.)

Chapter 3: A New Model of Pre-Industrial Political-Demographic Cycles

To address the shortcomings of other models and taking into account what happens in typical pre-industrial demographic cycles, Korotayev with Natalia Komarova construct their own model that includes the following three main elements:

(1) The Malthusian-type economic model, with elements of the state as tax collector (and counter-famine reservoir sponsor), and fluctuating annual harvest yields; this describes the logistic shape of population growth. It explains well the upward curve in the demographic cycle and saturation when the carrying capacity of land is reached. (2) Banditry and the rise of internal warfare in time of need are the main mechanism of demographic collapse. Personal decisions of peasants to leave their land and become warriors / bandits / rebels are influenced by economic factors. (3) The inertia of warfare (which manifests itself in the fear factor and the destruction of infrastructure) is responsible for a slow initial growth and the phenomenon of the “intercycle”.

Reproducing the model in detail will take up too much space, so just the main conclusions: “the main parameters affecting the period of the cycle are a) the annual proportions of resources accumulated for counter-famine reserves, b) the peasant-bandit transformation rate, and c) the magnitude of the climatic fluctuations. Hence, the lengths of cycles – and this is historically corroborated – is increased along with the growth of the counter-famine (more reserves) and law-enforcement (repress banditry) subsystems.

Chapter 4: Secular Cycles & Millennial Trends

Full version of Chapter 4: Secular Cycles and Millennial Trends.

The chapter begins by modeling the role of warfare, and challenges recent anthropological findings that denser populations do not necessarily lead to more warfare.

First, this is explained by the fact that it’s not a simple relation, but more of a predator-prey cycle described by a Lotka–Volterra equation. When warfare breaks out in a time of stress it leads to the immediate reduction of the carrying capacity and demographic collapse; however, warfare simmers on well into the post-collapse phase because groups continue to retaliate against each other.

Second, the methodology is flawed because it treats all wars the same, whereas in fact they tend to be far less devastating for bigger polities than for small ones. This is because bigger polities have armies that are more professional, and the length of their “bleeding borders” relative to total territory, is much smaller than for territorially small chiefdoms, for whom even low-intensity wars are demographically devastating. As such, more politically complex polities fight wars more frequently more frequently than smaller ones, but tend to be far less damaged by them.

Imperial expansions in territory coincide with periods of fast population growth and high per capita surpluses; later on, shrinking surpluses decimate the tax base and even defense proves increasingly hard (“imperial overstretch”). This correlation is very strong.

Now Korotayev et al combine their model from the last chapter with Kremer’s equation for technological growth (see the Introduction):

dT/dt = a*N*T

They also model a “Boserupian” effect, in which “relative overpopulation creates additional stimuli to generate and apply carrying-capacity-of-land-raising innovations”.

Indeed, if land shortage is absent, such stimuli are relatively weak, whereas in conditions of relative overpopulation the introduction of such innovations becomes literally a “question of life and death” for a major part of the population, and the intensity of the generation and diffusion of the carrying capacity enhancing innovations significantly increases.

Finally, they make the size of the harvest dependent not only on climatic fluctuations, but also on the level of technology.

Harvest i = H 0*random number i*T i.

Running this model with some reasonable parameters produces the following diagram, which reproduces not only the cyclical, but also the hyperbolic macrodynamics.

Furthermore,

Note that it also describes the lengthening of growth phases detected in Chapter 2 for historical population dynamics in China, which was not described by our simple cyclical model. The mechanism that produces this lengthening in the model (and apparently in reality) is as follows: the later cycles are characterized by a higher technology, and, thus, higher carrying capacity and population, which, according to Kremer’s technological development equation embedded into our model, produces higher rates of technological (and, thus, carrying capacity) growth. Thus, with every new cycle it takes the population more and more time to approach the carrying capacity ceiling to a critical extent; finally it “fails” to do so, the technological growth rates begin to exceed systematically the population growth rates, and population escapes from the “Malthusian trap” (see Diagram 4.26):

The cycles lengthen, and then cease:

AK: some confirmation for my rough explanation of why Chinese technological growth rate fell below Europe’s prior to the Industrial Revolution (see end of Chapter 2 in this post).

Of special importance is that our numerical investigation indicates that with shorter average period of cycles a system experiences a slower technological growth, and it takes a system longer to escape from the “Malthusian trap” than with a longer average cycle period.

Finally, they also add in an equation for literacy:

l i+1 = l i*b*dF i*l i*(1 – l i)

Which has the following effect on population growth:

N i+1 = N i*(1 + α × dF’)*(1 – l) – dR i – rob*N i*R i

And all added together, it produces the following stunning reproduction of China’s population dynamics from ancient past to today.

And concludes:

Of course, these models can be only regarded as first steps towards the development of effective models describing both secular cycles and millennial upward trend dynamics.

The Meaning of Cliodynamics

Turchin, Peter & Sergey Nefedov – Secular Cycles (2008)

Category: cliodynamics, world systems; Rating: 5/5

Summary: Read the whole book (PDF) or in chapters

This is a free online, quasi-popular book about eight different pre-industrial secular cycles (including Tudor England, the Roman Empire, Muscovy, and the Romanov Empire). Knowing the facts of history and the proximate causes of Revolutions – Lenin’s charisma, Tsarist incompetence, the collapse of morale and of the railway system, etc – is all well and good, but an entirely different perspective is opened up when looking at late Tsarist Russia through a social macrodynamic prism. The interpretation shifts to one of how late imperial Russia was under a panoply of Malthusian pressure, and of how the additional stresses and perturbations of WW1 “tipped” the system over into a state of collapse.

Finally, my reply to someone who sent me a message suggesting that cliodynamics may “make old school idiographic history redundant”.

I don’t think these trends will make idiographic history redundant, because there are many elements that are irreducible to mathematical analysis; furthermore, a major and inevitable weakness of cliodynamics is our lack of numbers for much of pre-mass literacy history. To the contrary, I think cliodynamics will end up complementing the “old school” rather than displacing it.

Footnotes

* Ray Kurzweil, one of the high priest of the singularitarian movement, extends Moore’s observations to also model technological growth (computing power, to be precise) as doubly exponential, or even hyperbolic. See Appendix: The Law of Accelerating Returns Revisited,

On the other hand, Joseph Tainter noted that in many areas the rate of technological innovation is actually slowing down. This is an argument that Kremer’s assumption that the rate of technological growth is linearly dependent on the product of the population and the size of the already-existing technological base is too simplistic.