Incorporating energy into production functions

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In my last post on my Debtwatch blog, I finished by saying that the Physiocrats were the only School of economics to properly consider the role of energy in production. They ascribed it solely to agriculture exploiting the free energy of the Sun, and specifically to land, which absorbed this free energy and stored it in agricultural products. As Richard Cantillon put it in 1730:

The Land is the Source or Matter from whence all Wealth is produced. The Labour of man is the Form which produces it: and Wealth in itself is nothing but the Maintenance, Conveniencies, and Superfluities of Life. (Cantillon, Essai sur la Nature du Commerce in Général (Essay on the Nature of Trade in General)

Quesnay’s famous but neglected “Tableau Economique” therefore described the agricultural sector as “the productive sector” and manufacturing as “sterile”—see Figure 1.

Figure 1: Quesnay’s “Tableau Economique“, first drafted in 1759, two decades before Watt’s steam engine

This was a justified assertion at the time, given that the Physiocrats wrote before the Industrial Revolution—and in particular the widespread exploitation in manufacturing of stored solar energy in fossil fuels– and originated in France, which was then overwhelmingly a rural nation.

Smith, who was influenced by the Physiocrats and wrote in Britain when industry was starting to exploit fossil fuels (specifically coal) on a grand scale, could have corrected this oversight. But rather than following the Physiocrats’ lead on energy, Smith instead saw labour—not energy—as the font of wealth (which he described in the same terms as Cantillon: the “conveniencies of life”), and ascribed the increase in productivity over time to “the division of labour”:

The annual labour of every nation is the fund which originally supplies it with all the necessaries and conveniencies of life which it annually consumes, and which consist always either in the immediate produce of that labour, or in what is purchased with that produce from other nations…

The greatest improvement in the productive powers of labour, and the greater part of the skill, dexterity, and judgment with which it is anywhere directed, or applied, seem to have been the effects of the division of labour. (Smith 1776, An Inquiry into the Nature and Causes of the Wealth of Nations)

Economics thus lost the Physiocrats’ focus on energy, and instead descended first into the “Labour theory of value” and then into the Neoclassical (and Post Keynesian) notions of “production functions” in which energy played no role at all.

The abiding weakness of all schools of economics, ever since the Classicals—including today’s Neoclassical and Post Keynesian schools, which are normally at pains to point out how superior one is to the other—is this failure to acknowledge the key role of energy in production. In this brief note, I want to record some speculations about how modern mathematically-inclined economics, with its use of production functions, might be made as energy aware as the Physiocrats were two and a half centuries ago. For the sake of non-mathematical readers, I’ve put the equations in an appendix at the end of this post.

Post Keynesian models typically see Output (Y) as a function of Capital (K) with a fixed ratio (v) between Capital and Output, and a fixed ratio (a) between Output and Labour (L) (see Equation 1).

When they go further than “corn economy” models, they employ a so-called “Leontief production function”, in which the ratio between capital and labour is fixed in each industry, though it varies between industries.

Neoclassical economists criticise this approach because it ignores the substitutability of capital and labour in production, which they embody in their core concept of an “isoquant” which alleges that the same level of output can be produced by very different combinations of labour and capital. Post Keynesians normally respond that this substitutability is a chimera, and continue using this “fixed coefficients” model of production anyway.

Neoclassicals typically see output as a function of capital and labour where one can be smoothly substituted for the other. Their canonical model is the Cobb-Douglas Production Function with constant returns to scale (Equation 2).

Neoclassicals tout this model’s fit to empirical data as a strength; Post Keynesians note that this is simply the result of accounting identities, since this “production function” can be derived by manipulating the identity that Output equals Wages plus Profits under conditions of a relatively constant (or slowly varying) distribution of income (see Anwar Shaikh’s brilliant paper “The Humbug Production Function”).

In this “he said/she said” battle, both sides ignore the shared weakness that their models of production imply that output can be produced without using energy—or that energy can be treated as just a form of capital. Both statements are categorically false according to the Laws of Thermodynamics, which—in strong contrast to so-called “Economic Laws” like the “Law of One Price” and the “Law of Demand“—cannot be broken. As Sir Arthur Eddington once put it:

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. (Sir Arthur Stanley EddingtonThe Nature of the Physical World (1915), chapter 4)

Arguably therefore, the production functions used in economic theory—whether spouted by mainstream Neoclassical or non-orthodox Post Keynesians—deserve to “collapse in deepest humiliation”.

This unacceptable state of affairs has inspired a number of ecologically oriented economists to attempt to come up with production functions in which energy plays a crucial role. One such model is the LINEX (“LINear-Exponential”) or KLEC (“capital-labor-energy-creativity”) model used by Kummel, Lindenberger, Ayres and colleagues. At a basic level, this treats output as a function of labour, capital, energy and time—effectively adding Energy as a third input to the Cobb-Douglas model (Equation 3).

While this is an improvement on the basic Cobb-Douglas model, it still implies logically that the contribution of energy to production could be nil: just set its exponent ? to zero. This is still in violation of the Laws of Thermodynamics: we need a production function in which energy plays an essential and irreducible role.

A potential way to achieve this is to accept that the whole idea of “labour” and “capital” without energy is a farce: labour without energy is a corpse, and capital without energy is a sculpture.

Why not acknowledge this by, as an initial abstraction, treating labour and capital as both being means to harness energy to do work, and treating output (Y) itself as work? Then we start from treating Labour and Capital as means to harness the energy they contain: EL for the energy flow that a worker can harness in a day, and EK for the energy flow that a machine can harness in a day (Equation 4).

The type and amount of energy that a worker or a machine can embody (as flows of energy at a point in time) are of course vastly different: the former is limited to food, and has a biophysical maximum (say, 5000 calories per day) whereas the latter can be the Sun itself directly, agricultural products, fossil fuels, or nuclear energy, and has risen from trivial levels before the Industrial Revolution to truly astronomical levels today.

Unpacking this further, we also need to acknowledge that not all the energy embodied in labour or capital can be used for work. Energy available to do work (these days called “exergy”) is the relevant factor, rather than the total energy embodied in labour or a machine; the efficiency with which that available energy is harnessed is also a vital ingredient.

So rather than simply showing the energy embodied in labour and capital, we need to multiply it by the ratio of available energy (exergy, with ExL for labour and ExK for machinery) to energy, and by the efficiency with which that exergy is harnessed . Then, using L to signify the number of workers and K (however imperfectly) to signify the number of machines, we get Equation 5, which treats output as a function of labour, capital and energy.

Rearranging Equation 5, we can derive the basic Cobb-Douglas formulation for labour and capital, times energy inputs. This is superficially like the Kummel/Ayres LINEX formulation, but it has the advantage that the energy contribution of either labour or capital cannot be set to zero without setting the contribution of the related “factor of production” also to zero, since they have the same exponents (see Equation 6). Energy therefore plays a crucial role in production using this formulation: if the energy input is zero, then so is output.

It may also transpire that the available energy embodied in machinery, and the efficiency of its exploitation, is the major explanation for the “Solow Residual”—the apparent paradox that, despite economists seeing output at any point in time as a function of labour and capital, the vast majority of the change in output over time comes not from an increase in the amount of Labour or Capital employed, but from the relatively unspecified A(t) term in the standard Cobb-Douglas function.

With this term replaced by two energy related terms—one of which has a definite maximum (since there is only so much exergy that a human body can exert in a day, and this can comfortably be treated as a constant), the other of which has gone from that of a water wheel in pre-Industrial times to that harnessed by the Spacex Falcon Heavy today—the “Solow Residual” may turn out to be the exponential increase in energy and exergy input into production over time (see Figure 2).

Figure 2: US Energy consumption over time

This energy-aware model of production is just a first step in properly integrating energy and the ecological effects of using energy into economics. It does not as yet consider the different types of energy resources, the impact of energy use upon energy resources and the ecology, or the mining (not “production”) costs of energy in terms of energy itself (the “Energy Return on Energy Invested” or EROEI). But it is necessary to have a model of production in which energy plays an essential role to be able to consider these issues about the viability of our energy usage properly.



About Steve Keen

I am Professor of Economics and Head of Economics, History and Politics at Kingston University London, and a long time critic of conventional economic thought. As well as attacking mainstream thought in Debunking Economics, I am also developing an alternative dynamic approach to economic modelling. The key issue I am tackling here is the prospect for a debt-deflation on the back of the enormous private debts accumulated globally, and our very low rate of inflation.
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63 Responses to Incorporating energy into production functions

  1. twowithinthreethatisone says:

    Just to follow up. I don’t tilt at windmills like pointing at fraudulent or unnecessary costs that there is often no way one can prior-ly perceive or effectively police except to make modern markets, which are inherently cost inflationary actually work and so true costs become more apparent. Rather I advocate everything that Dr. Keen has shown to be invalid about neo-liberal economic theory AND a new economic paradigm whose policies will enable the mere “this for that” current paradigm that apparently you and nearly everyone is still stuck in….to actually and stably work. That is innovative and integrative. Think a new thought. It’s healthy and stimulating, and as I’m sure Dr. Keen would agree is extremely hard for the orthodox to even countenance let alone actually do.

    Finally, iconoclasm is a great value, but good science is always open to new ideas and factors, and the signature of scientific breakthroughs has always been a valid integration of the scientific method and an aspect of consciousness like curiosity, the ability to visualize and/or the ethical impulse that actually arises from self awareness and hence the realization that others are essentially aware also and so worthy of every right and consideration as oneself. So let us visualize such a valid, workable and ethical economic philosophy and its reflective policies.

  2. twowithinthreethatisone says:

    As Minsky and Dr. Keen say: “The fundamental direction of capitalism is up.” What that means is businesses will try to price whatever the market will bear….and ALSO that the lower bound of cost, especially in an increasingly FIXED capital intensive modern economy, will dynamically be upward as well…because the cost accounting convention that all costs must go into price is valid and always in force.

  3. twowithinthreethatisone says:

    Actually, I don’t need to reference any data, especially data that would probably only be endlessly and obsessively contended anyway, all I really have to do is cite a convention because a convention….is a valid and universally applied rule.

  4. Tim Ward says:

    Some references, books and article collections:
    Economists and the Powerful: Convenient Theories, Distorted Facts, Ample Rewards
    by Norbert Häring, Niall Douglas
    ISBN-10: 0857284592
    ISBN-13: 978-0857284594

    Killing the Host: How Financial Parasites and Debt Bondage Destroy the Global Economy
    by Michael Hudson
    ISBN-10: 3981484282
    ISBN-13: 978-3981484281

    The Best Way to Rob a Bank is to Own One: How Corporate Executives and Politicians Looted the S&L Industry
    by William K. Black
    ISBN-10: 0292754183
    ISBN-13: 978-0292754188

    Direct quote from twowithinthreethatisone “Actually, I don’t need to reference any data, especially data that would probably only be endlessly and obsessively contended anyway, all I really have to do is cite a convention because a convention….is a valid and universally applied rule.”

    No scientist, mathematician, economist, engineer, or professional from various fields, would ever accept that, nor would reputable people behave in such a manner. Twowithinthreethatisone pretends to have a new paradigm, and will not show any data for it, and uses false statements attempting to support it. No one would ever accept that. Twowithinthreethatisone is not providing a real reference, (citation) to a refereed journal, recognized textbook or anything, for the alleged convention. Twowithinthreethatisone asserts that the alleged convention is universal, and never not in effect, but it demonstrably breaks down under various circumstances, and therefore cannot be true, because it is not universally true, nor is it ‘never not in effect’. It is false. There is no such convention. No recognized reference exists for it.

    Direct quote from twowithinthreethatisone: “As Minsky and Dr. Keen say: “The fundamental direction of capitalism is up.” What that means is businesses will try to price whatever the market will bear….and ALSO that the lower bound of cost, especially in an increasingly FIXED capital intensive modern economy, will dynamically be upward as well…because the cost accounting convention that all costs must go into price is valid and always in force.”

    Professors Minsky and Keen have also written about deflation, stagnation, debt deflation, and economic crashes. From Professor Keen’s article Oct 4, 2016 in Forbes, quotes: “This is not a complete model either of course: it omits government spending and bankruptcy for starters—which is why the collapse into a crisis, once it begins, does not slow down” and “Minsky’s key conclusion about capitalism that it is “inherently unstable””. Under various of the conditions written about by professors Minsky, Keen and others, the allegations of twowithinthreethatisone do not apply. Twowithinthreethatisone also fails to point out the difference between price takers and price makers in the economy. Most economic agents, price takers, have no ability to force prices into the economy. Twowithinthreethatisone fails to point out how costs can fall under various conditions, deflation and crashes for example. Twowithinthreethatisone is using an appeal to authority (the quote), to support deceptive misleading reasoning including false logic and a false statement, (the alleged accounting convention) to draw false conclusions in an apparent attempt to direct attention away from destabilizing inflationary pressures, that are the subject of the false statement by twowithinthreethatisone farther above, direct quote: “I don’t tilt at windmills like pointing at fraudulent or unnecessary costs that there is often no way one can prior-ly perceive or effectively police”. It appears that Twowithinthreethatisone is attempting to direct attention away from unnecessary costs, (monopoly fees, evaluation fraud, rent extraction, spurious fees tolls charges penalties, etc), and down play their significance, when in fact they are major problems destabilizing the economy, as detailed in the books and articles by professors Hudson and Black referred to above. The statement is false, because economists know very well how to effectively perceive and police unnecessary costs, as those authors detail. According to twowithinthreethatisone, pointing to those unnecessary costs is tilting at windmills, which professors Hudson and Black disprove. This quote from twowithinthreethatisone “businesses will try to price whatever the market will bear “ is not where the problems come from. Directing attention to real problems is tilting at windmills, according to twowithinthreethatisone

  5. twowithinthreethatisone says:

    Actually a lot of heat but no light. The cost accounting convention I cite that all costs must go into price is an actual one. Go to any accountant and he will confirm it that it is indeed an accounting rule. Understanding its enforcing effect and the fact that in an economy with continually increasing fixed capital costs along with depreciation and waste, the rate of flow of total costs will exceed the rate of flow total individual incomes alone should bring insight.

    As I have mentioned a couple of times I agree with most if not all of Dr. Keen’s research so far as the disequilibrated nature of modern economies as well as his de-bunking of neo-liberal economics. I agree with Michael Hudson’s critiques of finance. So I cite their researches…..which simply are re-discoveries of C. H. Douglas’s theory Social Credit. So there, if you require references and citations, I cite them to confirm most of what I say.

  6. Tim Ward says:

    Apologies to all. I apologize to you twowithinthreethatisone, for my multiple errors. I apologize to twowithinthreethatisone, to professor Keen, and all readers (who understandably avoid my posts now) for for my mistakes and the tremendous nuisance I’ve been.

    I didn’t ask an accountant, nor consult a text. I realized I don’t know what cost and price are, they are not what I think.

    And I don’t understand this quote from twowithinthreethatisone five posts above: “Rather I advocate everything that Dr. Keen has shown to be invalid about neo-liberal economic theory ” To me this means twowithinthreethatisone advocates policies shown to be invalid. Now I’m thinking nothing I read means what I think.

  7. twowithinthreethatisone says:

    The problem is we hear nothing but a bunch of tight assed conservativism, chilly technocratic Austrian/libertarianism and tired old liberalism. These mindsets by themselves all amount to a lot of authoritarian, non-integrative and orthodox bull shit. Try taking the particles of truth in each, combining/integrating them and throwing out the culturally ingrained falsehoods they also all contain, add even a modicum of grace while coming into a new mental unit of time and the power of humans to self actualize could swiftly begin a cultural quantum leap toward sanity and civility. Integrate this integrative approach into an equally integrative and open minded scientific method and you’ll have the signature of scientific breakthrough. Wisdom is the integrative process itself. The world has a crying need for Wisdom. Let us have it.

  8. twowithinthreethatisone says:

    Yes, that should have been re-worded as I advocate Dr. Keen’s de-bunking of neo-liberal economics. Having said the same thing several times here and a couple of times on this thread I would have thought it would have been properly understood anyway, but score one for sarcasm and irrelevant to the discussion grammatical critique.

    Seriously though, you should ask an accountant whether all costs having to go into price is a convention or not. He may not understand the economic significance of that convention just as most macro-economic theorists who Dr. Keen himself has declared could get their degree without so much as taking an elementary course in accounting probably won’t either, especially an apparently “bean counting” one like cost accounting. But sometimes the little things are actually big and the abstract trip over them.
    like cost accounting

  9. Tim Ward says:

    There’s no mystery here. I learned twowithinthreethatisone‘s phrase ‘convention that all cost must go into price’ ages ago, worded completely differently. So differently that twowithinthreethatisone‘s phrase is an alien construct. It is obvious why 90% complain or disagree with twowithinthreethatisone‘s posts, think it’s propaganda or are bored with them. Far from accepting twowithinthreethatisone‘s economics, I fundamentally disagree with them, now that I really know what twowithinthreethatisone is writing about, and I’m neither an Austrian, nor neo-liberal, nor technocratic. So apologies for the confusion.

  10. twowithinthreethatisone says:

    Okay. How was it worded?

    “It is obvious why 90% complain or disagree with twowithinthreethatisone‘s posts, think it’s propaganda or are bored with them. ”

    Obvious? Did you take a poll to get that conclusion, or is it simply a smear tactic meant to ostracize and induce a reaction by an understandably busy Dr. Keen? Is this a discussion or an attempt at invalidation and long distance internet psycho-analysis…which is the only thing less accurate than neo-liberal economic theory?

    Let us have a discussion. So do you agree with what Dr. Keen has said about DSGE? Then we agree on much. Do you agree that a debt jubilee is a gift and so is aligned with grace as in gifting which I assert is the concept upon which a new economic philosophy and aligned policies needs to be based?

  11. Tim Ward says:

    I found my old notes, delivered by a finance prof. Should have looked there first. Good examples (instead of my uninformative ones) and a direct statement that effectively states advanced capitalism cannot operate with the rule twowithinthreethatisone states. All costs do not have to be in price.

  12. twowithinthreethatisone says:

    You’re right. Just ask Bill Black the guy who took down Charles Keating and invented the word “control fraud” which is dishonest accounting that rips off vendors and investors and pays upper management incredible profits. Actually many, many small businesses commit accounting fraud….because its the only way they can stay in business in an economy that is chronically short of spendable individual income.

    All costs must go into price if you’re honest. Oh, and by the way your statement “….advanced capitalism cannot operate with the rule twowithinthreethatisone states” is completely correct. That’s why monetary grace which is a costless gift is the answer to advanced finance capitalism’s dilemma.

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