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West // Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms…

Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms

In response to a bunch of the points I made in Principles for Radical Tax Reform and Speech is Free, Distribution is Not about the distribution of power across network according to power laws, a friend pointed me to Geoffrey West’s Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life, in Organisms, Economies, and Companies.

Unsurprisingly, I love the scope / breadth / ambition of this book, all of which it does a pretty good job of living up to.

West looks at a wide range of phenomena in the natural and social worlds to try to derive a “grand unified theory of sustainability.” The key distinction West draws in the book is between the sublinear scaling of energy consumption in organisms (economies of scale / metabolic rate scales sublinearly with size, so as organisms increase in size the supply of energy cannot keep up with the maintenance demands of cells, leading to the eventual cessation of growth) with the superlinear scaling of energy consumption in cities and in economies (increasing returns / the social metabolic rate in cities scales superlinearly, so as cities grow the creation of social capital increasingly outpaces the demands of maintenance, leading to faster and faster open-ended growth).

His conclusion about the open-ended growth (exactly the kind of growth that is a fundamental premise of our current global economic regime of capitalism) is:

This kind of growth behavior is clearly unsustainable because it requires an unlimited, ever-increasing, and eventually infinite supply of energy and resources at some finite time in the future in order to maintain it. Left unchecked, the theory predicts that it triggers a transition to a phase that leads to stagnation and eventual collapse….

Some bits I highlighted…

It is interesting to me that Zipf named his book describing power distributions “Human Behavior and The Principle of Least Effort” — although Zipf does not provide an argument explaining the causality of power distributions, the title of his book suggests his instinctual place to look for the cause was consistent with phenomena in economics following the a similar power distribution as described by Pareto:

Zipf’s law owes its name to the Harvard linguist George Kingsley Zipf, who popularized it through his fascinating book Human Behavior and the Principle of Least Effort, published in 1949.17 He first enunciated his law in 1935, not for cities but for the frequency of word use in languages. As originally stated, the law says that the frequency of occurrence of any word in a corpus of written text such as all of Shakespeare’s plays, the Bible, or even this book is inversely proportional to its rank in the frequency table. Thus, the most frequent word occurs approximately twice as often as the second most frequent, three times as often as the third, and so on…


In economics, Zipf’s law actually predates Zipf. Much earlier it had been discovered by the influential Italian economist Vilfredo Pareto, who expressed it as a frequency distribution of incomes in a population rather than in terms of their ranking. This distribution, which is valid for many other economic metrics like income, wealth, and the size of companies, follows a simple power law with an exponent of approximately -2. When expressed in terms of rankings, this exponent corresponds to Zipf’s law. It quantifies the obvious economic fact that there are very few very wealthy people or large institutions but an enormous number of very poor or small ones. Pareto’s law, or the Pareto principle, has often been loosely stated in the form of the so-called 80/20 rule, in which the richest 20 percent of a population controls 80 percent of the total income, which is approximately true for the entire globe. Similarly, roughly 80 percent of a company’s profits come from 20 percent of its customers, as do 80 percent of its complaints. This asymmetry, in which there are only a very small number of very large entities but a huge number of very small ones, is characteristic of Zipf’s law. For instance, you need only about 20 percent of the dictionary to understand 80 percent of literature and roughly 80 percent of the population lives in the top 20 percent of the largest cities. Everything in between approximately follows the inverse proportionately according to the power law.

I like this pithy quote from Gandhi about greed and need (and West’s nuanced treatment of the tradeoffs of greed, which btw sound a little more pro-greed/growth in this passage than the book actually is overall):

In terms of the socioeconomic dynamics of cities we can likewise ask what, if anything, is being optimized in urban social networks. This is a tough question to answer definitively, and many scholars have obliquely tried to address it from multiple points of view. If we think of the city as the great facilitator of social interactions or as the great incubator for wealth creation and innovation, it is natural to speculate that its structure and dynamics evolved so as to maximize social capital by optimizing the connectivity between individuals. This suggests that social networks and the entire social fabric of cities and urban systems — that is, who is connected to whom, how much information flows between them, and the nature of their group structure — is ultimately determined by the insatiable drive of individuals, small businesses, and giant companies to always want more. Or, to put it in crass terms, that the socioeconomic machinery that we all participate in is primarily fueled by greed in both its negative and positive connotations as in the sense of the “desire for more.” Given the enormous disparities in income distributions that are observed in all cities across the globe, and the apparent drive of most of us to want more despite having plenty, it’s not hard to believe that greed in its various forms is an important contributor to the socioeconomic dynamics of cities. To quote Mahatma Gandhi: “The Earth provides enough to satisfy every man’s needs, but not every man’s greed.”

Greed is the pejorative image of this insatiable desire for more, but it also has an extremely important, positive flip side. Metaphorically, it is the social analog of the evolutionary biological drive of animals, including us, to maximize their metabolic power relative to their size. As was discussed in chapter 3, this can be thought of as derivative from the principle of natural selection and underlies the allometric scaling laws that permeate biology. The extension of the concept of the survival of the fittest to the social and political domain has led many thinkers to the controversial concept of Social Darwinism, whose roots go back to Malthus. Regardless of its validity, this idea has been sadly misrepresented, abused, and misused by politicians and social thinkers, sometimes with devastating consequences, to support all sorts of extreme views ranging from eugenics and racism to rampant laissez-faire capitalism.

The desire for more can apply to many things beyond wealth and material assets. It is a hugely powerful force in society that poses enormous moral, spiritual, and psychological challenges at both the individual and collective levels. The desire to succeed, whether in sports, business, or academia — to run the fastest, have the most creative company, or generate the most profound and insightful idea — has been a major underlying societal dynamic that has been instrumental in bringing us the extraordinary standard of living and quality of life many of us are privileged to enjoy. At the same time we have tempered our rampant materialistic greed by evolving altruistic and philanthropic behavior that has been integrated into our sociopolitical structures to protect us from excesses.

With the invention of the city and its powerful combination of economies of scale coupled to innovation and wealth creation came the great divisions of society. Our present social network structures barely existed in their present form until urban communities evolved. Hunter-gatherers were significantly less hierarchical, more egalitarian and community oriented than we are. The struggle and tension between unbridled individual self-enhancement and the care and concern for the less fortunate has been a major thread running throughout human history, especially over the past two hundred years. Nevertheless, it seems that without the motive of self-interest our entrepreneurial free market economy would collapse. The system we have evolved critically relies on people continually wanting new cars and new cell phones, new widgets and gadgets, new clothes and new washing machines, new thrills, new entertainment, and pretty much new everything, even when they already have enough of “everything.” It may not be a pretty picture and it doesn’t work for everyone, but so far, it’s worked remarkably well for most of us, and apparently most of us seem to want it to continue. Whether it can is a topic I’ll return to in the last chapter.

From “The Myth of Open Ended Growth”


A crucial aspect of the scaling of companies is that many of their key metrics scale sublinearly like organisms rather than superlinearly like cities. This suggests that companies are more like organisms than cities and are dominated by a version of economies of scale rather than by increasing returns and innovation. This has profound implications for their life history and in particular for their growth and mortality. As we saw in chapter 4, sublinear scaling in biology leads to bounded growth and a finite life span, whereas in chapter 8 we saw that the superlinear scaling of cities (and of economies) leads to open-ended growth.

Their sublinear scaling therefore suggests that companies also eventually stop growing and ultimately die, hardly the image that many CEOs would cherish. It’s actually not quite as simple as that because the prediction for the growth of companies is more subtle than just a simple extrapolation from biology. To explain this I am going to present a simplified version of how the general theory applies to companies focusing on the essential features that determine their growth and mortality.

The sustained growth of a company is ultimately fueled by its profits (or net revenue), where these are defined as the difference between sales (or total income) and total expenses; expenses include salaries, costs, interest payments, and so on. To continue growth over a prolonged period, companies must eventually return a profit, part of which is sometimes used to pay dividends to shareholders. Together with other investors, they in turn may buy additional stocks and bonds to help support the future health and growth of the company. However, to understand their generic behavior it is more transparent to ignore dividends and investments, which are primarily important for smaller, younger companies, and concentrate on profits, which are the dominant driver of growth for larger ones.

As we’ve seen, growth in both organisms and cities is fueled by the difference between metabolism and maintenance. Using that language, the total income (or sales) of a company can be thought of as its “metabolism” while expenses can be thought of as its “maintenance” costs. In biology, metabolic rate scales sublinearly with size, so as organisms increase in size the supply of energy cannot keep up with the maintenance demands of cells, leading to the eventual cessation of growth. On the other hand, the social metabolic rate in cities scales superlinearly, so as cities grow the creation of social capital increasingly outpaces the demands of maintenance, leading to faster and faster open-ended growth.

So how does this dynamic play out in companies? Intriguingly, companies manifest yet another variation on this general theme by following a path that sits at the cusp between organisms and cities. Their effective metabolic rate is neither sub- nor superlinear but falls right in the middle by being linear. This is illustrated in Figures 63 and 64, where sales are plotted logarithmically against the number of employees showing a best fit with a slope very close to one. Expenses, on the other hand, scale in a more complicated fashion: they start out sublinearly but, as companies become larger, eventually transition to becoming approximately linear. Consequently, the difference between sales and expenses, which is the driver of growth, also eventually scales approximately linearly.


The fact that companies scale sublinearly, rather than superlinearly like cities, suggests that they epitomize the triumph of economies of scale over innovation and idea creation. Companies typically operate as highly constrained top-down organizations that strive to increase efficiency of production and minimize operational costs so as to maximize profits. In contrast, cities embody the triumph of innovation over the hegemony of economies of scale. Cities aren’t, of course, driven by a profit motive and have the luxury of being able to balance their books by raising taxes. They operate in a much more distributed fashion, with power spread across multiple organizational structures from mayors and councils to businesses and citizen action groups. No single group has absolute control. As such, they exude an almost laissez-faire, freewheeling ambience relative to companies, taking advantage of the innovative benefits of social interactions whether good, bad, or ugly. Despite their apparent bumbling inefficiencies, cities are places of action and agents of change relative to companies, which by and large usually project an image of stasis unless they are young.

To achieve greater efficiency in the pursuit of greater market share and increased profits, companies stereotypically add more rules, regulations, protocols, and procedures at increasingly finer levels of organization, resulting in the increased bureaucratic control that is typically needed to administer, manage, and oversee their execution. This is often accomplished at the expense of innovation and R&D (research and development), which should be major components of a company’s insurance policy for its long-term future and survivability. It’s difficult to obtain meaningful data on “innovation” in companies because it’s not straightforward to quantify. Innovation is not necessarily synonymous with R&D, especially as there are significant tax advantages in labeling all sorts of extraneous activities as R&D expenses. Nevertheless, from analyzing the Compustat data set we found that the relative amount allocated to R&D systematically decreases as company size increases, suggesting that support for innovation does not keep up with bureaucratic and administrative expenses as companies expand.

The increasing accumulation of rules and constraints is often accompanied by stagnating relationships with consumers and suppliers that lead companies to become less agile and more rigid and therefore less able to respond to significant change. In cities we saw that one very important hallmark is that they become ever more diverse as they grow. Their spectrum of business and economic activity is incessantly expanding as new sectors develop and new opportunities present themselves. In this sense cities are prototypically multidimensional, and this is strongly correlated with their superlinear scaling, open-ended growth, and expanding social networks — and a crucial component of their resilience, sustainability, and seeming immortality.

While the dimensionality of cities is continually expanding, the dimensionality of companies typically contracts from birth through adolescence, eventually stagnating or even further contracting as they mature and move into old age. When still young and competing for a place in the market, there is a youthful excitement and enthusiasm as new products are developed and ideas bubble up, some may be crazy and unrealistic and some grandiose and visionary. But market forces are at work so that only a few of these are successful as the company gains a foothold and an identity. As it grows, the feedback mechanisms inherent in the market lead to a narrowing of its product space and inevitably to greater specialization. The great challenge for companies is how to balance the positive feedback from market forces, which strongly encourage staying with “tried and true” products versus the long-term strategic need to develop new areas and commodities that may be risky and won’t give immediate return.

Most companies tend to be shortsighted, conservative, and not very supportive of innovative or risky ideas, happy to stay almost entirely with their major successes while the going is good because these “guarantee” short-term returns. Consequently, they tend toward becoming more and more unidimensional. This reduction in diversity coupled with the predicament described earlier in which companies sit near a critical point is a classic indicator of reduced resilience and a recipe for eventual disaster. By the time a company realizes its condition it is often too late. Reconfiguring and reinventing become increasingly difficult and expensive. So when a large enough unanticipated fluctuation, perturbation, or shock comes along the company becomes seriously at risk and ripe for a takeover, buyout, or simply going belly-up. In a word, it is, as the Mafiosi put it, il bacio della morte — the kiss of death.

On impending collapse of open-ended growth (and his comments on Malthus):

In biology, the network principles underlying economies of scale and sublinear scaling have two profound consequences. They constrain the pace of life — big animals live longer, evolve more slowly, and have slower heart rates, all to the same degree — and limit growth. In contrast, cities and economies are driven by social interactions whose feedback mechanisms lead to the opposite behavior. The pace of life systematically increases with population size: diseases spread faster, businesses are born and die more often, and people even walk faster in larger cities, all by approximately the same 15 percent rule. Moreover, the social network dynamic underlying superlinear scaling leads to open-ended growth, which is the primary assumption upon which modern cities and economies are based. Continuous adaptation, not equilibrium, is the rule.

This is a wonderfully consistent picture: the same conceptual framework based on underlying network dynamics and geometry with the same mathematical structure leads to quite different outcomes in these two very different cases, and both are strongly supported by a plethora of diverse data and observations. However, there is a big catch with potentially huge consequences. Even though the growth of organisms, cities, and economies follows essentially identical mathematical equations, their resulting solutions have subtle but crucial differences arising from one being driven by sublinear scaling (the economies of scale of organisms) and the other by superlinear scaling (the increasing returns to scale of cities and economies): in the superlinear case, the general solution exhibits an unexpectedly curious property technically known as a finite time singularity, which is a signal of inevitable change, and possibly of potential trouble ahead.

A finite time singularity simply means that the mathematical solution to the growth equation governing whatever is being considered — the population, the GDP, the number of patents, et cetera — becomes infinitely large at some finite time, as illustrated in Figure 76. This is obviously impossible, and that’s why something has to change.

Before addressing some of the consequences of this phenomenon, let me first elaborate on some of its salient features. Simple power laws and exponentials are continuously increasing functions that also eventually become infinitely large, but they take an infinite time to do so. Another way of saying this is that in these cases the “singularity” has been pushed off to an infinite time into the future, thereby rendering it “harmless” relative to the potential impact of a finite time singularity. In the case of growth driven by superlinear scaling, the approach to the finite time singularity, represented by the solid line in Figure 76, is faster than exponential. This is often referred to as superexponential, a term I’ve already used earlier when discussing the growth of cities.


This kind of growth behavior is clearly unsustainable because it requires an unlimited, ever-increasing, and eventually infinite supply of energy and resources at some finite time in the future in order to maintain it. Left unchecked, the theory predicts that it triggers a transition to a phase that leads to stagnation and eventual collapse, as illustrated in Figure 77. This scenario sounds just like a rehash of the standard Malthusian argument that has been summarily dismissed by generations of economists: namely, that we won’t be able to keep up with demand and that open-ended growth will eventually lead to catastrophe.

Which brings us to the crux of the matter. Because of the presence of a finite time singularity resulting from superlinear scaling, this scenario is categorically different from that of Malthus. If growth were purely exponential as assumed by Malthusians, neo-Malthusians, their followers, and critics, then the production of energy, resources, and food could at least in principle keep up with exponential expansion because all of the relevant characteristics of the economy or city remain finite, even if they continue to increase in size and become very large.

This cannot be achieved if you are growing superexponentially and approaching a finite time singularity. In this scenario demand gets progressively larger and larger, eventually becoming infinite within a finite period of time. It is simply not possible to supply an infinite amount of energy, resources, and food in a finite time. So if nothing else changes, this inextricably leads to stagnation and collapse, as illustrated in Figure 77. An extensive analysis carried out in 2001 by Didier Sornette and Anders Johansen, then at UCLA, showed that data on population growth and the growth of financial and economic indicators strongly support the theoretical predictions that we have been growing superexponentially and are indeed headed toward such a singularity.

I want to emphasize that this situation is qualitatively quite different from classic Malthusian dynamics, where this is no such singularity. The existence of a singularity signifies that there has to be a transition from one phase of the system to another having very different characteristics, analogous to the way the condensation of steam to water which subsequently freezes to ice epitomizes transitions between different phases of the same system, each having quite different physical properties. And indeed, underlying such familiar phase transitions are singularities in the thermodynamic variables characterizing the system (water) but in terms of temperature rather than time (0°C for freezing and 100°C for boiling). Unfortunately, for cities and socioeconomic systems the phase transition stimulated by the finite time singularity is from superexponential growth to stagnation and collapse, and this could lead to potentially devastating consequences.

So how can such a collapse be avoided, and can it be achieved while still ensuring open-ended growth? The first point to appreciate is that these predictions assume that the parameters of the growth equation do not change. So one clear strategy for forestalling a potential catastrophe is to intervene before reaching the singularity by “resetting” the parameters. Moreover, to maintain open-ended growth with these new settings requires that the driving term in the equation — the “social metabolism” — needs to remain superlinear, meaning that the new dynamic must still be driven by the positive feedback forces of social interaction responsible for innovation, and for wealth and knowledge creation. Such an “intervention” is none other than what is usually referred to as an innovation. A major innovation effectively resets the clock by changing the conditions under which the system has been operating and growth occurring. Thus, to avoid collapse a new innovation must be initiated that resets the clock, allowing growth to continue and the impending singularity to be avoided.

Major innovations can therefore be viewed as mechanisms for ensuring a soft transition to a new phase by circumnavigating the potentially disastrous discontinuity inherent in the black hole of a finite time singularity. Having made the transition and “reset the clock” to avoid stagnation and collapse, the process begins all over again with the continuation of superexponential growth, eventually leading to a new finite time singularity which likewise has to be circumvented. The entire sequence is continually repeated, thereby pushing potential collapse as far into the future as the creativity, inventiveness, and resourcefulness of human beings allow. This can be restated as a sort of “theorem”: to sustain open-ended growth in light of resource limitation requires continuous cycles of paradigm-shifting innovations, as illustrated in Figure 78.

West concludes not with a specific solution to this predicament, only a call for the necessity of “a scientifically predictable, quantitative framework” for the dynamics, growth, and evolution as a prerequisite.

This increasingly rapid rate of change induces serious stress on all facets of urban life. This is surely not sustainable, and, if nothing changes, we are heading for a major crash and a potential collapse of the entire socioeconomic fabric. The challenges are clear: Can we return to an analog of a more “ecological” phase from which we evolved and be satisfied with some version of sublinear scaling and its attendant natural limiting, or no-growth, stable configuration? Is this even possible? Can we have the kind of vibrant, innovative, creative society driven by ideas and wealth creation as manifested by the best of our world’s cities and social organizations, or are we destined to a planet of urban slums and the ultimate specter of devastation raised by Cormac McCarthy’s novel The Road? Given the special, unique role of cities as the originators of many of our present problems and their continuing role as the superexponential driver toward potential disaster, understanding their dynamics, growth, and evolution in a scientifically predictable, quantitative framework is crucial to achieving long-term sustainability on the planet. Perhaps of even greater importance for the immediate future is to develop such a theory within the context of a grand unified theory of sustainability by bringing together the multiple studies, simulations, databases, models, theories, and speculations concerning global warming, the environment, financial markets, risk, economies, health care, social conflict, and the myriad other characteristics of man as a social being interacting with his environment.

Geoffrey West’s Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life, in Organisms, Economies, and Companies is a very important book, very worth your time…