Sunday, March 27, 2011

Tainter's law: where is the physics?

Joseph Tainter's interpretation of the cause of the collapse of civilisations is that social structures generate negative returns when they become too complex; as shown above (fromTainter's 1996 paper at We could call this relationship as "Tainter's law". But what is exactly that generates this behavior? In this post, I'll try to make a simple model that explains the law.

Joseph Tainter has written a fascinating interpretation of the collapse of human civilisations in his book "The Collapse of Complex Societies" (1988) (see also his 1996 paper) Collapse is a common event: it is the stuff history books are made of. The mighty empires of the past; from Sumeria to the Soviet Union, have all collapsed at some point. Yet, we don't seem to be able to understand the reasons why collapse is so common.

In his book, Tainter examines previous studies and lists at least eleven causes (or "concauses") of collapse that have been proposed by historians. Resource depletion, catastrophes, intruders, social conflict, and others. But is there a single cause of collapse? Or are there several? Tainter looks for a single, common root of the problem and finds it in what he calls "the decreasing returns of complexity".

Starting from a well known concept in economic theory, that of diminishing returns, Tainter builds his case on historical examples. It is clear that several societies have continued to build up and maintain complex and expensive structures even in conditions where it was very difficult to find the necessary resources. An example is that of the fortifications protecting the Western Roman Empire, that must have been such a burden that we may consider them to be of the factors that brought down the Empire. And, in general, we do see that societies, including ours, build up hypertrophic and complex bureaucracies which appear totally useless; an increase of complexity that generates only a waste of resources.

The idea of decreasing returns to complexity looks consistent and reasonable. But, why do societies behave in this way? Tainter does not provide a real explanation; on this point, he seems to follow the tradition of historians to describe rather than interpret. But, if you happen to have a more physics-oriented point of view, then describing what happens is not enough. You want to know what are the inner mechanisms that make civilisations evolve towards higher complexity. What is the physics of collapse?

So, let's see if we can build a model of civilisation growth and collapse. The simplest one that I have been able to put together is the following. It is a "toy model, if you like:

The model is based on the conventions of system dynamics. The rectangles indicate stocks of something. You could say that the box on the left contains fossil fuels, whereas the box on the right contains carbon dioxide. The central box contains all the stuff the economy is made of and that is created from the availability of energy from fossil fuels: people, machinery, building, facilities, you name it.

The fossil fuel stock is processed by the economy and eventually transformed into waste, as indicated by the double edged arrows which show the direction of the flux of matter. The single edged arrows indicate how the amounts stored in the stocks affect the flow; that is also influenced by two constants: how fast the economy can extract resources and how fast resources are transformed into waste.

There are a few more points about the model; the first is that the resource stock is assumed to be finite - that is "non renewable". This is an approximation, but it is a good one and not only for our society. Ancient civilizations were based on agriculture, which is supposed to be a renewable resource. But agriculture is not necessarily renewable; it is more often a way to transform fertile land into a desert by mining a non renewable resource: fertile soil.

Finally, note also that the model assumes a feedback relation between resources and the size of the economy. That is, the more resources there are, the faster they are exploited and - also - the bigger the economy, the faster it exploits resources. These assumptions imply a "positive feedback" between resources and the economy; which is a reasonable assumption. A similar relation holds for the waste and the economy.

Now, let's go on and "solve" the model. That is, let's see how the size of the stocks change as time goes by. Here are the results (obtained using the Vensim software for system dynamics)

As you see, the stock of resources gets depleted while the economy grows. At some point, however, the flow from the resource stock has been so much reduced that the economy can't keep growing and it starts declining. In the end, all the stock of resources has been transferred to the "waste" stock.

Note that the model describes a closed system in terms of mass. There is no flux of matter from or to the outside. And, indeed, mass is conserved in the results: the sum of the mass contained in the three stocks is constant. But the system does exchange energy with the surroundings. Burning fossil fuels generates heat, which is dispersed outside as we may assume that all three boxes maintain at the same average temperature.

The main force behind the transformation is energy potential, in this case the chemical potential of fossil fuels. In other words, the left box (resources) has a thermodynamic potential higher than the right box (waste). As we know from the second principle of thermodynamics, the transformation occurs with the creation of entropy. The economy is a grand machine for creating entropy - it could not be anything else.

If you like to use the term "exergy" (the fraction of energy able to do useful work) you can say that the "waste" stock contains much less exergy than the "resources" stock; while the "Economy" stock has an intermediate exergy content. There is no direct system dynamics convention to express stocks in terms of exergy. It could be taken into account in the model, but let's not go into that - let's keep this model as a "toy" one. The important thing is understanding what makes it move.

Now, let's go back to Tainter's interpretation of collapse. What could we take as "complexity" in the model? There is not an explicit parameter describing that but, as a first approximation, the size of an economy determines its complexity. That has been the rule for all known history and we see it happening even today. With the economic crisis, some structures we could once afford - say, mass instruction, public health care - must shrink and disappear. Society loses complexity in times of decline and gains it in times of growth.

So the "bell shaped" curve that describes the cycle of the economy should also describe its complexity. Now, let's walk one further step in quantifying Tainter's intuition. What can be the meaning of "benefits of complexity"? Well, it is clear from what Tainter says that the benefit of complexity have to do with the ability of society to solve problems. In our toy model, the only problem for the economy is to produce as much as possible in terms of resources. So we can define benefits of complexity as proportional to production, that is to the rate of exploitation of the natural resources stock.

Now we can replot Tainter's idea from the data of the model, that is, plot production ("benefits") as a function of the size of the economy ("complexity"). And the result is something that looks very much like Tainter's law! Here it is. (note that in the full plot the curve is a complete loop that goes back to zero at the end of the cycle):

To compare, here is again Tainter's original plot: the two graphs are not identical, but the similarity is evident.

Now, of course what we have been doing here is a "toy model" of the economy. When I present this kind of models at conferences, usually there is someone in the audience who stands up and says, "It is too simple; it is not realistic!". The idea seems to be that I am modelling societies using a "spherical cow model" - a term used to disparage the tendency of physicists to oversimplify their model.

This is a perfectly understandable criticism, but it can be answered noting that more detailed models of the same kind provide similar results. For instance, the "world3" model of "The Limits to Growth" study leads to curves that are very similar in shape to the ones shown here.

But I think that is not the point, you can make models simple or detailed, it depends on what is their purpose. The toy model presented here is not meant to describe how real societies behave. It is meant to be "mind sized", that is able to help us understand how physical factors affect the historical cycle of civilizations. It stresses that civilizations must obey the laws of thermodynamics; just as they must obey the law of gravity.

Some consequences of the model are obvious. It tells us that as long as we base our existence on non-renewable resources, we must eventually run out of them. But it gives us also some non-obvious hint on the path we are going to follow in this cycle. In particular, the model tells us that we will likely keep increasing the size and complexity of our society even with a diminishing flux of resources into the economy. In this sense, it confirms Tainter's intuition, but it tells us something more; that is it extends Tainter's curve beyond the limit of the plot shown in his 1996 paper. It says that after the phase of increasing complexity and reduced returns, the curve will loop back and, eventually, both complexity and production will go to zero as is the economy completes its cycle based on non-renewable resources. Here is the complete plot:

But the main point is that, eventually, Tainter's law derives from thermodynamics. As we know (or should know) thermodynamics is not only a good idea, it is the law!


Tainter's 1996 paper "Complexity, Problem Solving and Sustainable Societies"

A post of mine on Tainter's view of collapse

A paper of mine on modelling resource exploitation


Ugo Bardi is a member of the Club of Rome, faculty member of the University of Florence, and the author of "Extracted" (Chelsea Green 2014), "The Seneca Effect" (Springer 2017), and Before the Collapse (Springer 2019)