Biophysical Economics - MPP & MEPP

The Maximum Power Principle and 
                                 The Maximum Entropy Production Principle

Maximum Power Principle (MPP)

The Maximum Power Principle (MPP) is one of the most intriguing ideas I have come across in my readings of authors who call themselves Biophysical Economists.
This principle is, in my view, extremely important to a proper understanding of economic systems and their dynamics.  It explains why capitalism is more potent that other forms of economies, and it explains why capitalist systems evolve to consume mass and energy at ever higher rates.  Cheap throw-away but sophisticated goods consume energy and increase entropy at the highest rates that can be effectively managed.

The Maximum Power Principle in Wikipedia

Excerpted from Wikipedia September 2014


The maximum power principle has been proposed as the fourth principle of energetics in open system thermodynamics, where an example of an open system is a biological cell. According to Howard T. Odum (H. T. Odum 1995, p. 311).


"The maximum power principle can be stated: During self-organization, system designs develop and prevail that maximize power intake, energy transformation, and those uses that reinforce production and efficiency."


Further down it quotes H. T. Odum as saying (1970):


"Lotka provided the theory of natural selection as a maximum power organizer; under competitive conditions systems are selected which use their energies in various structural-developing actions so as to maximize their use of available energies. By this theory systems of cycles which drain less energy lose out in comparative development. However Leopold and Langbein have shown that streams in developing erosion profiles, meander systems, and tributary networks disperse their potential energies more slowly than if their channels were more direct. These two statements might be harmonized by an optimum efficiency maximum power principle (Odum and Pinkerton 1955), which indicates that energies which are converted too rapidly into heat are not made available to the systems own use because they are not fed back through storages into useful pumping, but instead do random stirring of the environment."

The Continuing Importance of Maximum Power

Charles A. S. Hall; Ecological Modelling 178 (2004) 107-113

In this article CAS Hall describes the origins of HT Odum's ideas on Maximum Power, and explains the concept.  this is a really good article to read, so I include it here.

Maximum Power Principle - Ed. by CAS Hall

C.A.S.Hall (1995) Maximum Power: The ideas and applications of H.T.Odum, Colorado University Press.


This book is just what the title says - a collection of articles by and about H. T. Odum and his ideas.  I found it to be very dense and difficult to read, as I have no training as a biologist or ecologist.

Presentation at CANSEE 2015

I attended CANSEE 2015 and made a presentation about the MPP.  The slide deck can be downloaded with this link:


151116 PPT MPP Presentation 30 Mins R15.pptx


In this presentation:

 - I formulated a re-statement of the MPP in terms that enabled me to build a model to test its operation;

 - Then, I explained the relevance of Atwood's machine;

 - I then described the model and its results, using a video produced using the model;

 - And I concluded with a few observations.


A few of the slides from the presentation, and the video, are shown below.

My restatement of the MPP is in three falsifiable hypotheses.  The terminology used might appear a bit arcane.  I am in the process of writing a paper that specifies the meaning of some of these words.  I felt this "restatement" was needed as I was unable to find a single clear statement of the MPP that was falsifiable, comprehensive, and fully general.  This is my attempt to meet those criteria, to enable me to make a serious attempt to model its action, and to test its veracity.  The good news is, I did make a model of a persistent autocatalytic system in which all energy transfer mechanisms were strictly concave, and it did evolve to function at maximum power, in agreement with hypothesis MPP#3.  (See video below.)

Hypothesis MPP#1 says it applies to all autocatalytic systems.  In the lingo of complexity science, I think this means all complex adaptive systems.  Odum strongly believed that this MPP applied to all persistent systems, including purely physical systems such as stars.  I am uncertain whether the language I use here extends that far.  However, I wanted to cover the territory for which I felt certain it applies, so my scope for the MPP might be a little less than Odum intended.  This slide shows a selection of applicable systems.

Hypothesis MPP#2 might seem a bit outrageous, but I believe it to be true.  In this slide I show a pictorial of a persistent energy pathway, consisting of energy stores and energy transfer mechanisms.  I use a formula that comes from my study of Atwood's machine, which goes with the curve described in Odum's paper of 1955, and shown at the bottom of the slide.


If this hypothesis is indeed true, then it applies to every energy transfer mechanism in every persistent metabolic, ecological or social process.  I also suspect that it applies to every persistent capital transfer process in all economies worldwide.  This would, I think, be an extension of Odum's views.

Looking a little more closely at the above formula, it has three parts.  Useful power is expressed as a function of efficiency (the Greek letter eta).  Useful power is determined by the energy still useful after the transfer is completed, and the time required to transfer the energy.  The function has two factors.  The first is a situational constant, or possibly a scaling factor.  In this case it is from Atwood's machine and was developed by applying Newtonian equations of mass and motion.  The second factor determines the shape of the strictly concave curve, shown to the right.  I think this is a VERY curious function.  You can read more on my page about Atwood's Machine, here.


In this slide I create a family of strictly concave curves by the simple expedient of replacing various exponents in the above expression with variables A, B, C and D.  When C=0 the curve is not strictly concave, so I have excluded that value.  Otherwise, when the parameters are less than or equal to five, there are 750 members of this family.  You can see the first 50 here.


Now, I am NOT saying that every power-efficiency curve for every persistent energy transfer mechanism in all autocatalytic systems are members of this family.  


But, I can say that all three that I have managed to produce so far ARE members of this family.  That, I think, is a trend worth exploring.

The curve for MT constant is discussed in the paper by Odum and Pinkerton of 1955.  The curve for MH constant is discussed in the paper by Silvert in 1982, and both are referenced in Odum's rebuttal of 1983.

The curve for Jacobi's law reduces to the logistic map, the curve that is the basis of studies of chaotic behaviour, and by which Feigenbaum's constants were developed.  It is the source of the well-known bifurcation diagrams.

Hypothesis MPP#3 looks like the classic presentation of Odum's Maximum Power Principle (MPP).  If these three hypotheses are true, then I believe that the two inferences shown here are also true.  But, I also suspect that this third hypothesis, itself, is not logically independent of the previous two.  I think MPP#1 is self-evident.  I think that MPP#2 is not self-evident, but true.  I suspect that all else is a logical consequence of MPP#2.  And, IF that is true, then I wonder if an empirical test of the veracity of the MPP need only focus on MPP#2.  Just a thought.  With respect to inference b), the first two were discussed in Lotka's paper of 1922.  Item b-iii is an extension of Lotka's idea that I would propose for economic systems.

This is the main panel of my model testing the MPP.  A population of heterotrophs live in a garden of daisies and eat plants and one another, grow older, and reproduce (by fission) or die of old age or starvation or predation.  When they eat one another they form an energy transfer event governed by a strictly concave curve.  This curve has two parameters, being the values of two genes, one from each heterotroph.  At first, the average efficiency of the collection of current predations is randomly determined by the original population.  But, over time, the system evolves such that the average efficiency of all predations is 0.5, as predicted by MPP#3.

The following video requires some explanation.  I took the above MppLab application and adjusted the screen dramatically to produce the video.  You will not be able to produce exactly these graphs using the version available on this site, but, IT IS THE SAME MODEL.  The video shows you output from the first 2006 ticks of the standard default scenario of the model.  It is just a special display of the data.   (See the graphic to the left.)

Layout:  There are three histograms in the video, and several monitor windows along the bottom in which you can see a variety of numbers.  Here is a description of each.


Descriptions of Monitors: 

 - Ticks - The number of discrete time steps, starting at 1 and rising to 2006, shown here as tick 26.

 - Autotrophs - The number of plants (autotrophs, daisies), which is held close to 1,000 to ensure a constant supply of energy into the population of heterotrophs.  The AM-gene value of autotrophs is 128.  It does not change.  Autotrophs do not evolve.

 - Heterotrophs - The number of animals (heterotrophs), which can vary greatly.  The gene value of heterotrophs varies from 0 to 128.  A heterotroph can eat any organism with a gene value between 1 and 4 times its own.  Since the efficiency of the predation event is calculated as {prey gene value}/{predator gene value} it is always between 0.25 and 1.0.

 - OAMs - This is an acronym for Open Atwood's Machines - Each is formed when a predator seizes a prey and begins to devour it in a predation event.  So, the number of OAMs is the number of currently active predation events, during this tick.  Most such events may have a duration of several ticks.  They are called OAMs as a reminder to myself that I have used the energy transfer dynamics of Atwood's Machine as a model for the energy transfer dynamics of a typical predation and digestion event.  By this, I have, essentially, assumed that MPP#2 is true, and that one strictly concave curve is as good as another, when testing MPP#3.  For more information about Atwood's Machine, see here.  At tick number 26 (shown) the number of OAMs (number of currently active predation events) is 110.

 - 'A's as Prey - The count of OAMs in which a plant (Autotroph) is the prey.

 - 'H's as Prey - The count of OAMs in which an animal (Heterotroph) is the prey.  Note that 5+105=110.

 - Ave Efficiency per OAM - The average efficiency over all current predation events.  THIS IS THE NUMBER TO WATCH, TO TEST MPP#3.  This number converges to 0.5 within 1,000 ticks.


Description of Histograms:

Efficiency - The long wide histogram across the top is the most important one.  This is the histogram showing the number of current predation events during this tick, versus efficiency of the event.  A predation event is called an OAM, and occurs as a heterotroph devours another organism.  You will see that the efficiencies run from 0.25 up to 1.00 as the video progresses. 

AM-Gene - The other two histograms below that show the number of heterotrophs versus the size of the AM-gene that controls predation.  One histogram is an inset from the other.  Plants are to the far right.  Heterotrophs may eat organisms on their right, but not too far to the right.

Things to Watch For:

 - All animals start with AM-gene of 100.  All plants start with AM-gene of 128.  All initial herbivorous predations, therefore, start with an efficiency of 100/128 ~ 0.78125.

 - Predations within a trophic level (e.g. omnivores eating omnivores, carnivores eating carnivores, apex carnivores eating apex carnivores) are characterized by similar AM-genes and efficiency above 0.5.

 - Predations between trophic levels (e.g. omnivores eating plants, carnivores eating omnivores, apex carnivores eating carnivores) are characterized by dissimilar AM-genes and efficiency between 0.25 and 0.5.

 - The distribution of efficiencies in the top histogram evolves to become a bi-modal histogram.

 - In the bottom histograms, heterotrophs to the left have more feeding opportunities, and so outcompete their near cousins to their right, which they can eat when given the chance.  So, there is evolutionary pressure on the population of heterotrophs to evolve ever smaller values on the AM-genes.

 - Gene values of 32 are a watershed dividing the carnivores on the left from the omnivores (i.e. those able to eat both plants and animals) on the right.  When the population of heterotrophs evolves below this critical value, a speciation event occurs as the populations bifurcates into two separate trophic levels.  Further bifurcations happen at 16, at 8 and at 4, although the definition of the line of demarcation becomes more fuzzy as the value gets further from the rock-steady values of 128 (4x32) and 32.  Up to five trophic levels can be seen towards the end of the video.  The apex predators are at the far left.

Note the three panels on the left of this slide.  In the topmost histogram, we see the distribution of efficiencies for all extant predations happening at the very instant, the moment, that this histogram was drawn.  They are, you might say, all over the map.  However, the second line graph is the history of the instantaneous average of all predations.  While the predations have varied efficiency, the average quickly converged on the value 0.5 and, with some temporary perturbations, remained there.  The bottom panel shows the formula that was used to emulate digestion, and we see that the system evolved to function at maximum power, in accordance with MPP#3.

Maximum Entropy Production Principle (MEPP)

The Maximum Entropy Production Principle (or MEPP) has been proposed by many scientists as an alternative candidate for the fourth law of thermodynamics.  However, based on the little bit of research I have done so far, the proposed statements of this principle are either very narrowly applicable to some special cases, or are not well-accepted by other scientists.  The situation here is in some ways reminiscent of the literature on the MPP, but more varied and less accepted.  In fact, in one case, in the work of Ilya Progogine, he proposed a principle of minimum entropy.  Strangely enough, although this appears to be blatantly in opposition to an MEPP, the difference is entirely a matter of perspective and semantics.


I, personally, believe that the Lotka/Odum MPP and the MEPP are two sides of the same coin.  When a complex adaptive system evolves to function at an average efficiency consistent with maximum rate of energy consumption, it will also have a maximal rate of entropy production (whatever that means).  I also believe that a careful examination of Prigogne's arguments will show that the self-organized part of the adaptive system will then exhibit a state of minimal exhibited local entropy (whatever that means).  Just so, is it possible to be in a state of maximum entropy production and minimal entropy at the same time?  So, I suspect that Lotka's, Odum's, and Prigogine's conceptions of adaptive systems are all consistent with each other.


I hope to explore these ideas in future modeling exercises.

The Economics of Diminishing Marginal Returns.

Having attempted to make a GRAND mathematical argument that the MPP is ubiquitous in all persistent economic practices, and failed, I look for another path up to this peak.  My mathematical skills were not up to the task to produce a formal mathematical proof.  I still think, though, if I could get a mathematician interested in the problem, that it is quite do-able.


So my fallback approach is to analyze a number of specific instances in which I think it can be more easily demonstrated.  So, the attached NTF is a shot at such an exercise.  In this "Note To File" I describe an idealized workshop in which there is a fixed demand for widgets, and a variable number of workers.  I am searching for that number of workers which provides the maximum power (maximum profits per day), and, not surprisingly, I get a hump-backed curve.  If we define T as the time (one day), C as the daily cost of production of the widgets, I as the daily income or revenue generated from sale of the widgets, and B as the daily profits where B = I - C, then the "power of the stream of benefits" is B/T, the efficiency is B/I, and the ROI is B/C.


To download the note to file (a diary note), click on this link.

151201 NTF DimMargRet CD-Shop R5.PDF


The two images below show two variations on the power-efficiency curve. 

 - The graph on the left shows a classic curve as we might expect from reading Odum's works, for which perfect efficiency is associated with zero power.  This is produced when the cost of materials and facilities are excluded from pricing decisions, and only quality of product is a factor. 

 - The graph on the right, however, is more complicated.  In that graph the cost of materials and facilities are included in pricing, and perfect efficiency is not possible.  We get a looped graph in which only the upper portion forms a simulacrum of the classic curve.

This third graph, derived from the weird graph above and to the right, is a scatter plot of ROI vs Power.  I note that power and ROI DO NOT PEAK AT THE SAME POINT.  This is a very curious phenomenon, and I wonder if it happens in real world businesses.  My understanding of the MPP would tell me that businesses are naturally selected to operate at maximum power, whereas most investors look for maximization of ROI.  This might explain why investors are so often disappointed, and why managers are so often confused by results.

Last updated:  December 2015