Richard Melson

July 2006

Economics & Entropy


Borisas Cimbleris

Emeritus Professor, Federal University of Minas Gerais

An elementary definition of energy is "capacity of producing work". A rough definition of money is "the ability to make other people work". Money and its equivalents are the motive power of human action. This is admittedly cheap philosophy, but it works.

I believe in the usefulness of intuitive ideas in entering the tracks of precise concepts. In this paper I try to associate the ideas of classical, bona fide economists of the last two centuries, with the concepts of Thermodynamics.


In our material world we deal with fluxes of matter and energy, which are shaped by human action into economic phenomena. The economic phenomenon arises whenever we deal with finite resources. But what is infinite on this planet? And for the time being we have to deal only with this unique abode. The original endowment of Earth is in the form of reserves of matter and energy.

The transformable part of the reserves, duly identified, forms the resources. Part of these is transformable into consumable goods, which have a value. The value of things is conditioned by a complex of social, cultural and conceptual realities. It is beyond the scope of this review to consider the world as a whole.

All real energetic processes generate entropy. If this thermodynamic concept is applied by analogy to economic processes, an analog of entropy is generated. Let us remember that even social phenomena have associated physical fluxes. In all biological and social processes some organization is lost, and that is equivalent to generation of entropy, according to a Generalized Second Law of Thermodynamics. It is an extrapolation, of course, but it seems to work: all evidence is in favor of its existence, and there are no contradictions so far, even if it was necessary to invent a negative entropy for living beings.

(Schrödinger, Brillouin).

Every time available work is spent, something similar to entropy is generated. This "spent" work is transferred to the environment. Whatever this is, we are not able to extract more work from it. This anergy is lost for human use.

Perhaps I am insisting on the obvious, but some simple facts were, apparently, not taken into account by most of the economists, until the 60's. I was surprised by looking for "entropy" in a standard text (the highly respected Samuelson's book, of about 1000 pages).

There is not a chapter on economy of energy.

The first economist to deal with entropy is apparently Nicholas Georgescu-Roegen. His 1971 book is not generally available, and possibly not generally accepted. Other pioneers

are Schumpeter and Forrester, who emphasize environmental concerns.

An ABC of Production Economy

In short, Production Economy is about production, exchange and use of goods and services, and also the science of allocation of scarce resources. It does not deal with infinities.

A resource is scarce if it cannot be utilized or exchanged but for another scarce resource. The means of exchange for such transactions is generalized money, meaning coins, banknotes, financial "papers", contracts, leases - valuable items. Price is the quantity of money given in exchange for a unit of resource. For instance, an acre of land in a specified location; a gallon of treated water; a gun; a lease contract on a building; real estate in general. The price in a free market constitutes the economic measure of the value of an object. It may be a work of art, or even an unwritten idea.

When someone uses a resource, a potential user may be deprived of this resource. But it becomes difficult to evaluate if the users are a community. What is the market value of pure air, clean water, a green square for public use ? Such is the dilemma of "commons".

Now the value of a car is much more concrete, it can be technically evaluated, mainly due to the existence of specialists everywhere, with an exclusive interest in car sales. The value of a share or bond appears very precisely on the Exchange bulletin board, though it a strong function of time. Besides, it is a function of "local atmospheric conditions", of the general climate of a country. An investor came up with an interesting tautology: "The market factors are a function of the market factors".

A resource can be land, capital, energy, materials and combinations of such. Yet they have to be either tangible or measurable. The inborn resources of a human being are just body and soul. The soul is intangible. Gogol invented the business of selling dead souls, but nowadays there is little demand for these.

Production Processes

Any scarce resource has a price, which is its exchange value. A very simple scheme shows the basics:

In an open market, something enters into the process (inputs); something is transformed, and something utilizable comes out (products). There is also waste.

All these quantities are heterogeneous. In order to reduce them to a common denominator, we have to express everything as monetary values. Then it becomes possible to treat capital, labor, resources on an equal basis.

Land, labor, capital, feed materials and energy are production factors. An economic model classifies these factors by sectors and then permits a quantification of the correlations between them.

Economic exchanges

. The economic models of goods and services are fundamental processes of Economy. The simplest model for such exchanges considers rational agents, which try to attain a maximum usefulness, buying and selling goods and services in a free competitive market.

Such a model constitutes the laissez faire of the classical (liberal) economists such as Adam Smith. But it is only an ideal model. Any governmental control complicates things, not to speak of the imponderable human factors. Both cause the prices to oscillate and to make offer and demand to vary in an impredictable manner. In fact, they are social phenomena, more complex than the purely economical ones. This model also does not take into account economic growth nor the upgrading of technology. It is simple, closed and static.

Fig. 1. Static system for production and consumption

Consumers are paid for their services or resources provided; at the same time, they pay for other goods and services. Every payment must be counterbalanced by another. Thus, the workers are paid for their services by salaries; the owners of land or capital are paid for letting use their resources, in the form of interest and rent. Under simple hypotheses a price may be shown to exist which maximizes simultaneously the utility of all factors, both for buyers and sellers.

Stable equilibrium

Our simple model supposes a permanent equilibrium of exchanges. For such an equilibrium we need a regulating factor, which is the invested capital. This is also a condition for the creation of cycles.

Fig. 2. Model with a regulating factor.

The resources are known and fixed. Individual preferences are invariable and determinate; technology is invariable; economic variables have linear correlations. The reasons for this is just simplifying. The main thing that's lacking is the economic growth.

However, this model has permitted mathematical approaches such as those of Walras (1860) and Cassel.

It's time to introduce the fluxes of mass and energy. Capital and consumption implicate the use of materials and energy. Services also imply, in some way, the input of materials and energy - even if it be only mental energy.

However, real materials never vanish really. Consumption does not infringe the principle of matter conservation; the word is to be used advisedly. Materials just return to the " environment ", as waste. The economic cycle cannot really be closed. We come closer to the truth of the cycle the extraction and treatment of feed materials, as well as the "utilization" of waste, it is still not closed. We have to consider the global environment of the planet and even perhaps that of the solar system - what with the implicate universe. The global system is not, obviously, in equilibrium. A closed system is not in thermodynamic equilibrium, it must become inert, passive and devoid of matter and energy fluxes, if given enough time.

An approximation to economic equilibrium would the famous zero growth scenario, a state of null growth

Economic model with investment and technological change.

Now, for the first time, we consider explicitly matter and energy.

Such is the model due to von Neumann (1945) (who was not an economist),

and Piero Sraffa (1960).

Fig. 3. General equilibrium, considering fluxes of matter and energy

Thermodynamically we have an open system. Extraction here means taking from an initial endowment of the planet. A part of the waste goes back to the planet.

Earth Matter and energy extracted Economic cycle


Fig. 4. The essential cycle

The fluxes are proportional to quantities of matter and energy processed and transformed in material goods. The scale of use has something to do with the outcome.

However, the greater the NGP, the more garbage is generated.

Regretfully, we must note that the concept of equilibrium growth due to von Neumann is inconsistent with a closed system.

Environmental effects

On contemplating the model above, a disturbing question arises: is pollution proportional to the Gross National Product? Not necessarily. Many kinds of waste are not harmful or can be neutralized. Let us focus on the worst: pesticides, radioactive waste, toxic chemicals (such as heavy metals). They do not represent a very large mass, and may be substituted by less dangerous materials; they can be encapsulated and deposited in a safe manner, and in many cases, recycled. But such measures also demand the use of energy and matter.

Available energy

Energy is conserved. The magnitude wasted is available energy. On entering any process it is in fact consumed; any such transaction increase the global entropy or unavailable energy (thet of the environment). All real processes are irreversible, the Second Law is merciless. There is entropic degradation of the materials at any stage, from extraction to final use, although sometimes there may be a local decrease of entropy.

Production waste is mostly mass, not available energy. The wasted energy stays in the environment and may not be recovered, but mass can be recycled, at least in principle. But recycling demands more available energy, which is to be obtained from the initial endowment of the planet. And recycling also generates its own waste; in the best case that is waste heat . In the ideal case this can be reirradiated to space.

I shall present examples that will permit the engineers to retain the main idea:

1. Extraction of metals

The metal contains less energy than the ore. The processing generates entropy in the waste. Usually (almost compellingly) heat is utilized; combustion increases the environmental entropy through heat transfer, smoke, degradation of complex organic substances into simple gases.

2. Live beings

Organisms achieve a state of low entropy in comparison to the environment, notwithstanding continuous dissipative processes. Live beings consume matter and energy, as food or photons with high available energy and are converted into mechanical work or biomass, with a low efficiency. Now the mention of efficiency obliges us to remember a few banalities of Thermodynamics.

Thermodynamic efficiencies

Q1 ///////////

Thermal engine W

Q2 \\\\\\\\\\\\\\\\\

In economic terms, the engine uses a "valuable" fuel and produces a "valuable" mechanical work.


--- >









Now "valuable", "expensive" and "worthless" are economic words. Usually it is not for the engineer to decide what is the "product" in each case. In a refrigerator, the waste heat is but a minor inconvenience. In a heat pump, which is essentially the same machine, this waste heat is the "product"; the input energy is declared "free".

In a heat engine, the usual efficiency is

e = W / Q1 ,

where W is the product, or whatever you gain or what is utilizable; Q1 is the energy you pay for, and Q2 is low quality heat, or "waste" heat, but not always. It depends on the design of the power-generating plant. We may declare that the heat waste is just that, or else utilize this waste heat in a lower temperature process, plus the electric energy generated. My point is that this decision is an economic sentence.

We have seen that in a heat pump the product is just the waste heat, whereas in a refrigerator the product is negative heat, or cold. In both cases we can estimate the performance of the engine as

kind of heat desired / mechanical energy bought

or similar fractions. The nature of the product is dictated by the social system. The engineer must be guided by a previous definition. The choice of the input energy shall be decided by the inputs available in the market.

There are an absolute limits in thermal processes. One limit is imposed by the conservation of energy (the First Law). Another limit is the transformability of energy.

Heat cannot be completely transformed into work. In this context an exergetic efficiency is preferable. Let us denote the available energy by A, synonymous with availability, maximum realizable work, being incorporated with every input, product, cycle or process. This general measure is conveniently subsumed under the word exergy. [Rank]. Then

h = Aout / Ain

In order to define availability, according to my teacher Keenan, let us introduce a few variables of the system, namely

internal energy








and some properties of the environment. such as To, Po and the chemical potentials in the environment, m io .

A = U + PoV - ToS - S i m io Ni

This formula is nearly identical to that for the Gibbs free energy (isobaric-isothermal potential), which is

G = U + PV - TS - S i m i Ni

The only difference is that A is a function of the properties of the system and those of the environment.

Carnot's limit

A reversible heat engine operated between the reservoir temperatures of T and To has the same thermodynamic and exergetic efficiencies:

e = h = 1 - T/ To

This is an absolute, ideal and unattainable limit for thermal processes, the utility of which is to determine an easy point of reference. The efficiencies of all real processes or heat engines are lower than that. In open cycles the efficiencies tend to be lower and the difference between thermal and exergetic efficiencies becomes much greater.

There is a vast literature on this subject. I shall give only one simple example, that of a boiler, an open system. A fossil fuel, therefore of low entropy, is burned at a high temperature, which may reach 2000 ° C. The product is steam, at much lower temperature; say, 120 ° C. The First Law efficiency may be of the order of 80 %, but h is only 25 %.

Thus, available energy has been lost. Hot flame heats up the operating fluid (water), producing steam and losing temperature in heat transfer through pipe walls. A heat pump would better utilize the available energy, but would demand a larger investment. A thermodynamic efficiency does not express the economic reality.

Let us present the concept of economic equilibrium due to Pareto. An ideal economic system would be a competitive free market, in which the agents are rational-minded and well informed consumers and producers, but which does not include "externalities" nor "indivisibilities". In this it reminds one of the perfect transformabilities of all energies but heat. In this system the resources are allocated, in a way as to leave no one better without making at least another one worse. Such is the Pareto Optimum, which may seem the essence of savage capitalism.

The existence of a Pareto optimum depends on the existence of a set of preferences of individuals, which in turn basically depend on the individual's income. However, people do not maximize a well defined utilitary function. Almost everybody subordinates his/hers

interests to the ideals of a group (family), of a nation, a religion, and the restrictions of tradition and laws. Altruistic behavior complicates the calculations. The Pareto optimum is at most a convenient abstraction. It may serve as a starting point for a notion of economic efficiency.

Usually one starts from a convenient optimizable function. In the general case one has an extensive set of variables, such as materials, energy, rent and prices; capital, labor. As of now there is no general correlation between the thermodynamic and the economic efficiencies. I am convinced that the value of things derives, in an ultimate manner, from the existence of matter and energy. Hopefully, in the future we shall arrive at the concept of objective physical value of things. In that case it will be possible to estimate the value of a diamond, say, in kWh on the Universal Energetic Economic Scale, based ultimately on the intrinsic value of the available energy embedded in an object.

For that we should reach a consensus about value, by weighting advisedly all the variables. Some idea like that certainly has occurred as soon the Law of Conservation of Energy was established in 1870. I cannot perform a thorough search, but I would suggest a starting point in the writings of Stuart Mill, Spencer and Balfour Stewart.

Explicitly, a similar idea occurred to Frederick Soddy in 1922. He wrote that the price of a merchandise reflect, directly or not, the energy invested in its production. (Today, he would say available energy). The same idea was proposed by Howard Scott during the Great Depression of the 30's. And now it surfaces with the ecological movement since the 60's, while emphasizing the erroneous use of planet's resources. An indirect consequence of this idea is the analysis of industrial processes by net energy use, taking into account all the energy involved since the extraction of natural resources and the "primary" energy. This approach is gaining adepts.

Thermodynamic and economic optima do not coincide, except in very few cases. Berry et al. have shown that the two optima do coincide in the case of a free market in which the only "scarce" resource is available energy (or utilizable work). That means that labor, capital, inputs, etc. should all be for free.

Production factors

Among the interesting problems of Economy are the substitution and variation of different inputs. For instance,

Such questions have to do with margins or marginal costs; mathematically, with partial derivatives; physically of susceptibilities. Example: When the cost of a fertilizer goes up, production decreases; the market price then increases.

The main questions are mainly the following:

  1. If a certain input of a process increases or decreases by a small quantity, what happens to the product price?

  2. If the price of an input increases or decreases, how to maintain a constant production by manipulating the marginal variation of another input?

Question a) has to do with the allocation equilibrium, the distribution of marginal payments for the land, labor, capital, feed materials and energy. We may complete the questions above with

  1. For an incremental gain (profit), how should profit be distributed between the workers and the capitalists (owners of the capital and of the resources) ?

A good illustration of question b) is in the energy crises of 1974 and 1979. One of the inputs rose in price suddenly. How did the system adjust, what combinations of cheaper inputs endeavored to supply the former product ?

A natural reaction was to try to substitute energy for labor and capital. Those who assimilated the alternative energy idea thought mostly about solar, which would complement conventional energy, and thus a mixture of equilibrium factors.

A more general question is dynamic or evolutionary, namely, how will economic growth be affected by the relative variation of prices ?

The increase in the price of energy probably caused a slower growth of riches, the capital having kept constant. This effect was painfully felt by the Brazilian society (and probably others; but we still have no clear idea about the mechanism of change.

Another timely question is the effect of automation and informatization on the relations between labor and capital. These processes are obviously interrelated. Suffice it to remember that robots demand computers.

The decrease of the price and increase of variety and availability of microchips permits an ever increasing application of automatic controls in the industry. The historical ratio labor/ capital tends to diminish. A preliminary analysis showed that advances of Informatics would contribute to an overall economic growth. There would be some loss of jobs in the industry, but the creation of other, hitherto unknown, jobs elsewhere would compensate that. At the same time, leisure time would increase for the majority. However, the increase of unemployment did not permit complete translation of power to the capital side.

What is lost in exchange for such advantages ? An objective answer would require creation of quantitative models of production processes. This problem was clearly formulated by Wicksteed in 1894 as follows.

The production in an economic process is a function of the inputs, the production factors:

P = P (x1, x2,.....xn)

Now this is almost intuitive. It remains to define the xi as discrete or continuous variables. Are they really independent ? In order to use calculus, the variation must be continuous, and therefore there is a strong temptation to declare them continuous. But Georgescu-Roegen showed that one type of variable excludes the other type.

The econometrists promptly chose P to be differentiable, probably because then the problem becomes tractable. If P is continuous and homogeneous in the first degree, Euler's theorem applies:

P = x1 P / x1 + x2 P / x2 + ....

The derivatives above are the marginal products of the various factors and xi are the costs of those factors. The terms xi P / xi . Each parcel of production is paid for according to the degree of utilization of the xi (such as capital, land, resources). The homogeneity of the function has an interesting consequence. If all the inputs be augmented by a factor z, the increase of the product is proportional to z, being z P / xi. We call it constant return in scale. This increase is certain for small increments, but there may be increasing economies of scale. It is well known that in power plants, beginning from a certain value of the generated power, a thermal energy plant burns the fuel in a more efficient way, providing more energy per mass unit (more J/kg and also more W/ kg). The production process thus becomes financially and thermally more efficient with the increase of production scale.

In fact, the scale changes represent changes in production technology. For a given state of the art there exists a maximum product, consistent with the technology state at the time; the product grows in time.

Presently the economists admit that a first-degree homogeneous function is right only for the global economy, and only if they are able to express the various production factors in strictly monetary terms. But such global analysis is incompatible with some specific processes. We must distinguish the macroscale from the microscales.

An important historical instance of a homogeneous production function is the Cobb-Douglas function,

P ( x1, x2,.....) = x1a 1 x2a 2........ xna n

which reminds us the dimensional formula of a physical magnitude, expressed as a power product. By identifying the production factors as capital, C, labor, L, resources, R, the exponents may be considered as "economic dimensions". There we have the makings of an economic dimensional analysis, only we don't know what these dimensions are. An additional condition for a homogeneous function is

S a i = 1.

This function is beautiful, but incompatible with the balances of mass and energy. This is easy to see by introducing mass and energy directly into the production function. In this case the production function P obtains that is not zero, even if one of the inputs be zero, provided there is enough of the other inputs. In that case, the feed materials may become null, if only we provide enough capital and labor; the produced goods would still be the same. This is manifestly absurd. Without matter, no powers on earth can produce a single thread of textile, nor a crumb of bread.

Fig. 4. A model with a constant substitution elasticity

The fact is that there is always some mass and energy embedded in the inputs and in the final product. One cannot create mass or energy just by providing capital and manpower. In order to make iron you need ore or scrap; to make cheese, milk or soy. The substitutability of mass and energy for capital and labor is only possible from a physical threshold, as shown in Figure 5.

This paradox can be overcome if we limit the use of production functions in economic problems which involve only marginal substitutions near the operating point in market equilibrium.

Fig, 5. Transformation curves of C + L into M + E.

Capital and labor cannot, by themselves, substitute for natural resources beyond a certain limit, lest we admit a motus continuus.

Recurring to a sectorial equilibrium model (Walras) is also inconsistent with the real world. Exogenous available energy sources are indispensable.

The Leontief matrix is basically an energy balance supplemented by a matter balance. If we succeed in representing this matrix as a function of time, a real production function shall result. This only depends on a thorough collection of economic data, computer-aided. I believe that many of the difficulties of writing a Brazilian energetic matrix is due to the reluctance of releasing data by government-owned firms.

Some fascinating topics were left out in this survey of the parallelism between economic and thermodynamic concepts. Respectively, they are the connections between information and the economic variables, and the exhaustion of natural resources.

As a closing note, I must emphasize that this work is merely a translation of an unpublished paper written in the 80's. (I don't remember when and why). I'm sorry that I wasn't able to update it.

REFERENCES [old and few]

Samuelson, Paul. Economics, 9/e. Tokyo, McGraw-Kogakusha, 1973

Leroy-Beaulieu, Paul. Précis d'Economie Politique. 23/e. Paris, Delagrave, 1929

The pioneer papers

Wicksteed, Philip. An Essay on the Coordination of the Laws of Distribution. London,

Macmillan, 1894.

Soddy, Frederick. Cartesian Economics. London, Henderson, 1922.

Neumann, John von. Rev. Econ. Studies 13 (1), 1945-6/

Georgescu-Roegen, Nicholas. The Entropy Law and Economic Process.

Cambridge,Mass. Harvard University Press, 1971.

Berry, R.S., G.Heal, P.Salamon. Resources and Energy 1: 125, 1978.

Entropy & Economics

July 10, 2006