## Friday, December 19, 2014

### Peak pyramids: the way to ruin is rapid

Originally published on Cassandra's Legacy on Friday, December 19, 2014
The graph above is a little exercise in cliodynamics, the attempt of quantitatively modeling historical data. Here, the size of the great Egyptian pyramids is plotted as a function of their approximate building date, taken as the last year of the reign of the Pharaoh associated to them. The data are fitted with a simple Gaussian, which approximates the cycle of the Hubbert model of resource depletion.

The great Egyptian pyramids built during the 3rd millennium BCE are the embodiment of the power and of the wealth of the Egyptian civilization of the time. But why did the Egyptians stop building them? Not lack of interest, apparently, since they kept building pyramids for a long time. But they never built again pyramids on such a giant scale.

Probably, we will never have sufficient data to understand the economics of the Egyptian pyramid building cycle of the 3rd and 4th Egyptian dynasties. But we can try at least to examine the quantitative data we have. So, I went toWikipedia and I found data for the size of pyramids and their approximate dates. The result is the graph above. Here, I show only the data for completed pyramids as a function of the last year of the reign of the Pharaoh associated for each one.

As you can see, it is possible to fit the data with a Gaussian curve, which approximates the Hubbert curve, known to describe the depletion of a limited, non renewable resource. This suggests that the Egyptians had run out of resources, possibly in the form of the fertile soil necessary to sustain the large workforce needed to build pyramids. Or, perhaps, in an age of increasing warring activity, they were forced to funnel more and more resources into the military sector, taking them away from pyramid building.

Another phenomenon we can note in the graph is the rapid collapse of the size of the pyramids at the end of the cycle. The last pyramid of this cycle, the one associated to Pharaoh Menkaure, is even smaller than the first one of the cycle, the "stepped pyramid" of Pharaoh Djoser. Perhaps, this rapid decline is a manifestation of the "Seneca Effect", a term that I coined to describe economic cycles in which decline is faster than growth. Unfortunately, however, the data are too scattered and uncertain to be sure on this point. But surely there was no "plateau" nor a slow decline after the construction of the largest pyramids andit is suggestive to think that even pyramid building may be described with Seneca's words "increases are of sluggish growth, but the way to ruin is rapid."

## Monday, December 15, 2014

### Seneca cliffs of the third kind: how technological progress can generate a faster collapse

Originally published on Cassandra's Legacy on Monday, December 15, 2014

The image above (from Wikipedia) shows the collapse of the North Atlantic cod stocks. The fishery disaster of the early 1990s was the result of a combination of greed, incompetence, and government support for both. Unfortunately, it is just one of the many examples of how human beings tend to worsen the problems they try to solve. The philosopher Lucius Anneus Seneca had understood this problem already some 2000 years ago, when he said, "It would be some consolation for the feebleness of our selves and our works if all things should perish as slowly as they come into being; but as it is, increases are of sluggish growth, but the way to ruin is rapid."

The collapse of the North Atlantic cod fishery industry gives us a good example of the abrupt collapse in the production of resources - even resources which are theoretically renewable. The shape of the production curve landings shows some similarity with the "Seneca curve", a general term that I proposed to apply to all cases in which we observe a rapid decline of the production of a non renewable, or slowly renewable, resource. Here is the typical shape of the Seneca Curve:

The similarity with the cod landings curve is only approximate, but clearly, in both cases we have a very rapid decline after a slow growth that, for the cod fishery, had lasted for more than a century. What caused this behavior?

The Seneca curve is a special case of the "Hubbert Curve" which describes the exploitation of a non renewable (or slowly renewable) resource in a free market environment. The Hubbert curve is "bell shaped" and symmetric (and it is the origin of the well known concept of "peak oil). The Seneca curve is similar, but it is skewed forward. In general, the forward skewness can be explained in terms of the attempt of producers to keep producing at all costs a disappearing resource.

There are several mechanisms which can affect the curve. In my first note on this subject, I noted how the Seneca behavior could be generated by growing pollution and, later on, how it could be the result of the application of more capital resources to production as a consequence of increasing market prices. However, in the case of the cod fishery, neither factor seems to be fundamental. Pollution in the form of climate change may have played a role, but it doesn't explain the upward spike of the 1960s in fish landings. Also, we have no evidence of cod prices increasing sharply during this phase of the production cycle. Instead, there is clear evidence that the spike and the subsequent collapse was generated by technological improvements.

The effect of new and better fishing technologies is clearly described by Hamilton et al. (2003)

Fishing changed as new technology for catching cod and shrimp developed, and boats became larger. A handful of fishermen shifted to trawling or “dragger” gear. The federal government played a decisive role introducing newtechnology and providing financial resources to fishermen who were willing to take the risk of investing in new gear and larger boats.
...

Fishermen in open boats and some long-liners continued to fish cod, lobster and seal inshore. Meanwhile draggers  and other long-liners moved onto the open ocean, pursuing cod and shrimp nearly year round. At the height of the boom, dragger captains made \$350,000–600,000 a year from cod alone. ... The federal government helped finance boat improvements, providing grants covering 30–40% of their cost.
....
By the late 1980s, some fishermen recognized signs of decline. Open boats and long-liners could rarely reach their quotas. To find the remaining cod, fishermen traveled farther north, deployed more gear and intensified their efforts. A few began shifting to alternative species such as crab. Cheating fisheries regulation—by selling unreported catches at night, lining nets with small mesh and dumping bycatch at sea—was said to be commonplace. Large illegal catches on top of too-high legal quotas drew down the resource. Some say they saw trouble coming, but felt powerless to halt it.

So, we don't really need complicated models (but see below) to understand how human greed and incompetence - and help from the government - generated the cod disaster. Cods were killed faster than they could reproduce and the result was their destruction. Note also that in the case of whaling in the 19th century, the collapse of the fishery was not so abrupt as it was for cods, most likely because, in the 19th century, fishing technology could not "progress" could not be so radical as it was in the 20th century.

The Seneca collapse of the Atlantic cod fishery is just one of the many cases in which humans "push the levers in the wrong directions", directly generating the problem they try to avoid. If there is some hope that, someday, the cod fishery may recover, the situation is even clearer with fully non-renewable resources, such as oil and most minerals. Also here, technological progress is touted as the way to solve the depletion problems. Nobody seems to worry about the fact that the faster you extract it, the faster you deplete it: that's the whole concept of the Seneca curve.

So take care: there is a Seneca cliff ahead also for oil!

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## A simple dynamic model to describe how technological progress can generate the collapse of the production of a slowly renewable resource; such as in the case of fisheries.

by Ugo Bardi

Note: this is not a formal academic paper, just a short note to sketch how a dynamic model describing overfishing can be built. See also a similar modeldescribing the effect of prices on the production of a non renewable resource

The basics of a system dynamics model describing the exploitation of a non renewable resource in a free market are described in detail in a 2009 paper byBardi and Lavacchi. According to the model developed in that paper, it is assumed that the non renewable resource (R) exists in the form of an initial stock of fixed extent. The resource stock is gradually transformed into a stock of capital (C) which in turn gradually declines. The behavior of the two stocks as a function of time is described by two coupled differential equations.
R' = - k1*C*R C' = k2*C*R - k3*C,
where R' and C' indicate the flow of the stocks as a function of time (R' is what we call "production"), while the "ks" are constants. This is a "bare bones" model which nevertheless can reproduce the "bell shaped" Hubbert curve and fit some historical cases. Adding a third stock (pollution) to the system, generates the "Seneca Curve", that is a skewed forward production curve, with decline faster than growth.

The two stock system (i.e. without taking pollution into account) can also produce a Seneca curve if the equations above are slightly modified. In particular, we can write:
R' = - k1*k3*C*R C' = ko*k2*C*R - (k3+k4)*C.
Here, "k3" explicitly indicates the fraction of capital reinvested in production, while k4 which is proportional to capital depreciation (or any other non productive use). Then, we assume that production is proportional to the amount of capital invested, that is to k3*C. Note how the ratio of R' to the flow of capital into resource creation describes the net energy production (EROI), which turns out to be equal to k1*R. Note also that "ko" is a factor that defines the efficiency of the transformation of resources into capital; it can be seen as related to technological efficiency.
The model described above is valid for a completely non-renewable resource. Dealing with a fishery, which is theoretically renewable, we should add a growth factor to R', in the form of k5*R. Here is the model as implemented using the Vensim (TM) software for system dynamics. The "ks" have been given explicit names. I am also using the convention of "mind sized models" with higher free energy stocks appearing above lower free energy stocks

If the constants remain constant during the run, the model is the same as the well known "Lotka-Volterra" one. If the reproduction rate is set at zero, the model generates the symmetric Hubbert curve.

In order to simulate technological progress, the "production efficiency" constant is supposed to double stepwise around mid-cycle. A possible result is the following, which qualitatively reproduces the behavior of the North Atlantic cod fishery.

Among other things, this result confirms the conclusions of an early paper of mine (2003) on this subject, based on a different method of modeling.

Let me stress again that this is not an academic paper. I am just showing the results of tests performed with simple assumptions for the constants. Nevertheless, these calculations show that the Seneca cliff is a general behavior that occurs when producers stretch out their system allocating increasing fractions of capital to production. Should someone volunteer to give me a hand to make better models, I'd be happy to collaborate!

## Originally published on "Cassandra's Legacy" on Sunday, December 7, 2014

"It would be some consolation for the feebleness of our selves and our works if all things should perish as slowly as they come into being; but as it is, increases are of sluggish growth, but the way to ruin is rapid." Lucius Anneaus Seneca, Letters to Lucilius, n. 91

This observation by Seneca seems to be valid for many modern cases, including the production of a nonrenewable resource such as crude oil. Are we on the edge of the "Seneca cliff?"

It is a well known tenet of people working in system dynamics that there exist plenty of cases of solutions worsening the problem. Often, people appear to be perfectly able to understand what the problem is, but, just as often, they tend to act on it in the wrong way. It is a concept also expressed as "pushing the lever in the wrong direction."

With fossil fuels, we all understand that we have a depletion problem, but the solution, so far, has been to drill more, to drill deeper, and to keep drilling. Squeezing out some fuel by all possible sources, no matter how difficult and expensive, could offset the decline of conventional fields and keep production growing for the past few years. But is it a real solution? That is, won't we pay the present growth with a faster decline in the future?

This question can be described in terms of the "Seneca Cliff", a concept that I proposed a few years ago to describe how the production of a non renewable resource may show a rapid decline after passing its production peak. A behavior that can be shown graphically as follows:

It is not just a theoretical model: there are several historical cases where the production of a resource collapsed after having reached a peak. For instance, here are the data for the Caspian sturgeon, a case that I termed "peak caviar".

Do we risk to see something like this in the case of the world production of oil and gas? In my opinion, yes. There are some similarities; both fossil fuels and caviar are non-replaceable resources; and in both cases prices went rapidly up at and after the peak. So, if Caspian sturgeon showed such a clear Seneca cliff, oil and gas could do the same. But let me go into some details.

In the first version of my Seneca model, the fast decline of production was interpreted in terms of growing pollution that places an extra burden on the productive system and reduces the amount of resources available for the development of new resources. However, I found that the Seneca behavior is rather robust in these systems and it appears every time people try to "stretch out" a system to force it to produce more and faster than it would naturally do.

So, in the case of the Caspian sturgeon, above, growing pollution is unlikely to be the cause of the rapid collapse of production (even though it may have contributed to the problem). Rather, the main factor in the collapse is likely to have been the effect of the growing prices of a rare and non replaceable resource (caviar). High prices enticed producers to invest more and more resources in raking out of the sea as much fish as possible. It worked, for a while, but, in the end, you can't fish sturgeon which isn't there. It ended up in disaster: a classic case of a Seneca Cliff.

Can this phenomenon be modeled? Yes. Below, I describe the model for this case in some detail. The essence of the idea is that producers need to reinvest a fraction of their profits in developing new resources in order to keep producing. However, the yield of the new investments declines as time goes by because the most profitable resources (e.g. oilfields) are exploited first. As a result, less and less capital is available for new investments. Eventually production reaches a maximum, then it declines. If we assume that companies re-invest a constant fraction of their profits in new resources, the model leads to the symmetric bell shaped curve known as the "Hubbert Curve."

However, as I describe in detail below, decline can be postponed if high prices provide extra capital for new productivedevelopments. Unfortunately, growth is obtained at the cost of a fast burning out of capital resources. The final result is not any more the symmetric Hubbert curve, but a classic Seneca curve: decline is more rapid than growth.

Is this what we are facing for fossil fuels? Of course, we are only dealing with qualitative models, but, on the other hand, qualitative models are often robust and give us an idea of what to expect, even though they can't tell us much in terms of predicting events on a precise time scale. The ongoing collapse of oil prices may be a symptom that we are running out of the capital resources necessary to keep developing new fields. So, what we can say is that there are some good chances of rough times ahead - actually very rough. The Seneca cliff may well be part of our near term future.

_______________________________________________________

## The Seneca curve as the result of increasing fractions of profits allocated to the production of a non renewable resource

by Ugo Bardi - 07 Dec 2014

Note: this is not a formal scientific paper; it is more a rough "back of the envelope" calculation designed to show how increasing capex fractions can affect the production rate of a non renewable resource. If someone could give me a hand to make a more refined and publishable study, I would be happy to collaborate!

The basics of a system dynamics model describing the exploitation of a non renewable resource in a free market are described in detail in a 2009 paper byBardi and Lavacchi. According to the model developed in that paper, it is assumed that the non renewable resource (R) exists in the form of an initial stock of fixed extent. The resource stock is gradually transformed into a stock of capital (C) which in turn gradually declines. The behavior of the two stocks as a function of time is described by two coupled differential equations.

R' = - k1*C*R
C' = k2*C*R - k3*C,

where R' and C' indicate the flow of the stocks as a function of time (R' is what we call "production"), while the "ks" are constants. This is a "bare bones" model which nevertheless can reproduce the "bell shaped" Hubbert curve and fit some historical cases. Adding a third stock (pollution) to the system, generates the "Seneca Curve", that is a skewed forward production curve, with decline faster than growth.

The two stock system (i.e. without taking pollution into account) can also produce a Seneca curve if the equations above are slightly modified. In particular, we can write:

R' = - k1*k3*C*R
C' = ko*k2*C*R - (k3+k4)*C.

Here, "k3" explicitly indicates the fraction of capital reinvested in production, while k4 which is proportional to capital depreciation (or any other non productive use). Then, we assume that production is proportional to the amount of capital invested, that is to k3*C. Note how the ratio of R' to the flow of capital into resource creation describes the net energy production (EROI), which turns out to be equal to k1*R. Note also that "ko" is a factor that defines the efficiency of the transformation of resources into capital; it can be seen as related to technological efficiency. These points will not be examined in detail here.

Here is the model as implemented using the Vensim (TM) software for system dynamics. The "ks" have been given explicit names. I am also using the convention of "mind sized models" with higher free energy stocks appearing above lower free energy stocks

If the k's are kept constant over the production cycle, the shape of the curves generated by this model is exactly the same as with the simplified version, that isa symmetric, bell shaped production curve. Here are the results of a typical run:

Things change if we allow "k3" to vary over the simulation cycle. The characteristic that makes "k3" (productive investment fraction) somewhat different than the other parameters of the model, is that it is wholly dependent on human choice. That is, while the other ks are constrained by physical and technological factors, the fraction of the available capital re-invested into production can be chosen almost at will (of course, there remains the limit of the total amount of available capital!).

Higher prices will lead to higher profits for producers and to the tendency to increase the fraction reinvested in new developments. It is also known that in the region near the production peak prices tend to be higher - as in the historical cases of whale oil and caviar and whale oil. In the case of caviar, the price rise was nearly exponential, in the case of whale oil, more like a logistic curve. Assuming that the fraction of reinvested capital varies in proportion to prices, some modeling may be attempted. Let me show here the results obtained for an exponential increase of the fraction of reinvested Capex.

I have also tried other functions for the rising trend of k3. The results are qualitatively the same for a linear increase and for a logistic one: the Seneca behavior appears to be robust, as long as we assume a significant increase of the fraction of the reinvested capex

Let me stress once more that these are not supposed to be complete results. These are just tests performed with arbitrary assumptions for the constants. Nevertheless, these calculations show that the Seneca cliff is a general behavior that occurs when producers stretch out their system allocating increasing fractions of capital to production.

## Saturday, April 19, 2014

### The Seneca effect in the Easter egg hunt

Originally published on Cassandra's legacy on Saturday, April 19, 2014
For the Easter of 2014, let me repropose my post on the dynamic modelling of the Easter egg hunt which, for some reason, has been the most successful post ever to be published (in 2012) on the former "Cassandra's Legacy" blog, now known as "Resource Crisis." -Happy Easter, everyone!

Here is a little Easter post where I try to model the Easter Egg hunt as if it were the production of a mineral resource. A simple model based on system dynamics turns out to be equivalent to the Hubbert model of oil production. We can have "peak eggs" and the curve may also take the asymmetric shape of the "Seneca Peak." So, even this simple model confirms what the Roman Philosopher told us long ago: that ruin is much faster than fortune. (Image fromuptownupdate)

For those of you who may not be familiar with the Easter Bunny tradition, let me say that, in the US, bunnies lay eggs and not just that: for Easter, they lay brightly colored eggs. The tradition is that the Easter Bunny spreads a number of these eggs in the garden and then it is up to the children to find them. It is a game that children usually love and that can last quite some time if the garden is big and the bunny has been a little mean in hiding the eggs in difficult places.

A curious facet of the Easter Egg hunt is that it looks a little like mineral prospecting. With minerals, just as for eggs, you need to search for hidden treasures and, once you have discovered the easy minerals (or eggs), finding the well hidden ones may take a lot of work. So much that some eggs usually remain undiscovered; just as some minerals will never be extracted.

Now, if searching for minerals is similar to searching for Easter Eggs, perhaps we could learn something very general if we try a little exercise in model building. We can use system dynamics to make a model that turns out to be able to describe both the Easter Eggs search and the common "Hubbert" behavior of mineral production. The exercise can also tell us something on how system dynamics can be used to make "mind sized" models (to use an expression coined by Seymour Papert). So, let's try.

System dynamics models are based on "stocks"; that is amounts of the things you are interested in (in this case, eggs). Stocks will not stay fixed (otherwise it would be a very uninteresting model) but will change with time. We say that stocks (eggs) "flow" from one to another. In this case, eggs start all in the stock that we call "hidden eggs" and flow into the stock that we call "found eggs". Then, we also need to consider another stock: the number of children engaged in the search.

To make a model, we need to make some assumptions. We could say that the number of eggs found per unit time is proportional to the number of children, which we might take as constant. Then, we could also say that it becomes more difficult to find eggs as there are less of them left. That's about all we need for a very basic version of the model.

Those are all conditions that we could write in the form of equations, but here we can use a well known method in system dynamics which builds the equations starting from a graphical version of the model. Traditionally, stocks are shown as boxes and flows as double edged arrows. Single edged arrows relate stocks and flows to each other. In this case, I used a program called "Vensim" by Ventana systems (free for personal and academic use). So, here is the simplest possible version of the Easter Egg Hunt model:

As you see, there are three "boxes," all labeled with what they contain. The two-sided arrow shows how the same kind of stock (eggs) flows from one box to the other. The little butterfly-like thing is the "valve" that regulates the flow. Production depends on three parameters: 1) the ability of the children to find eggs, 2) the number of children (here taken as constant) and 3) the number of remaining hidden eggs.

The model produces an output that depends on the values of the parameters. Below, there are the results for the production flow for a run that has 50 starting eggs, 10 children and an ability parameter of 0.006. Note that the number of eggs is assumed to be a continuous function. There are other methods of modeling that assume discrete numbers, but this is the way that system dynamics works.

Here, production goes down to nearly zero, as the children deplete their egg reservoir. In this version of the model, we have robot-children who continue searching forever and, eventually, they'll always find all the eggs. In practice, at some moment real children will stop searching when they become tired. But this model may still be an approximate description of an actual egg hunt when there is a fixed number of children - as it is often the case when the number of children is small.

But can we make a more general model? Suppose that there are many children and that not all of them get tired at the same time. We may assume that they drop out of the hunt simply at random. Then, can we assume that the game becomes so interesting that more children will be drawn in as more eggs are found? That, too can be simulated. A simple way of doing it is to assume that the number of children joining the search is proportional to the number of eggs found (egg production). Here is a model with these assumptions. (note the little clouds: they mean that we don't care about the size of the stocks where the children go or come from)

This model is a little more complex but not so much. Note that there are two new constants "k1" and "k2" used to "tune" the sensitivity of the children stock to the rest of the model. The results for egg production are the following:

Now egg production shows a very nice, bell shaped peak. This shape is a robust feature of the model. You can play with the constants as you like, but what you get, normally, is this kind of symmetric peak. As you probably know, this is the basic characteristic of the Hubbert model of oil production, where the peak is normally called "Hubbert  peak". Actually, this simple egg hunt model is equivalent to the one that I used, together with my coworker Alessandro Lavacchi, to describe real historical cases of the production of non renewable resources. (see this article published in "Energies" and here for a summary)

We can play a little more with the model. How about supposing that the children can learn how to find eggs faster, as the search goes on? That can be simulated by assuming that the "ability" parameter increases with time. We could say that it ramps up of a notch for every egg found. The results? Well, here is an example:

We still have a peak, but now it has become asymmetric. It is not any more the Hubbert peak but something that I have termed the "Seneca peak" from the words of the Roman philosopher Seneca who noted that ruin is usually much faster than fortune. In this example, ruin comes so fast precisely because people try to do their best to avoid it! It is a classic case of "pulling the levers in the wrong direction", as Donella Meadows told us some time ago. It is counter-intuitive but, when exploiting a non renewable resource, becoming more efficient is not a good idea.

There are many ways to skin a rabbit, so to say. So, this model can be modified in many ways, but let's stop here. I think this is a good illustration of how to play with "mind sized" models based on system dynamics and how even very simple models may give us some hint of how the real world works. This said, happy Easter, everyone!

(BTW, the model shown here is rather abstract and not thought to describe an actual Easter Egg hunt. But, who knows? It would be nice to compare the results of the model with some real world data. My children are grown-ups by now, but maybe someone would be able to collect actual data this Easter!)