B. Lukács

CRIP RMKI H-1525 Bp. 114. Pf. 49., Budapest, Hungary

Edited draft of the lecture at "Thermodynamic Afternoon",BME, Building H 655, Budapest, Sept. 22, 2000

Problems and answers arising at the lecture in questionsand argumentations are in bold italics


Sustainable development or sustainable consumption is a catchwordof the end of the century. Many people say that we should consumesustainably. Everybody thinks that the purpose or notion is clear;in Chapter 5 we shall be in the position to see in what extentit is clear.

There is a slogan telling that things do not go in such a directionthat we would have inherited Earth and her bounties; no, ratherit belongs to the next generations and we only keep the bountiesfor them. And in the same time the legal systems of all developedcountries are based in such a way that they recognise the rightsof present owners, give very few rights about property to thenext generations if they are already living, symbolic rights ornothing at all to "inheritor" in embryonic stage (nowin Hungary even not the right to be born), and absolutely no rightto persons in the future.

Popular wisdoms also do not go for virtual, future generations.There is a popular sentence in the Carpathian Basin which, onthe official klanguage of Hungary between 1000 and 1841, Latin,goes as "Utilior passer hodiernus quam tarda crastina!".Here "tarda" is Otis tarda, the Latin word is from thetime of Linneaus, and the animal was unknown for Romans, but alsofor British; but it is the biggest bird of Europe, so comparepasser and tarda.

Therefore Romans and other Latins could not have had this sentence;and also, Oscs of Southern Italy, once bigger and more culturedbrothers of Romans could not have had; they might have expressedthe notion somehow so: "Nesimú egmú faamat"i.e. The next need commands.

But if we do not know exactly, what is sustainable consumption,but anyways Roman Law did not know anything about the rights offuture generations, Modern Laws do not know either, wisdoms ofmany nations speak against, then is it possible at all?

Well, if Roman ius did not tell anything about, mos told something.It was a pious act to think about posteriority. Romans (and Atheniansand Spartans) were at least fanatic to establish the next generation;but we are no more fanatic.

Still, physics may help to formulate what can be a sustainableconsumption; and then everybody can decide if that is good tohim/her or not. And I will demonstrate that sustainable consumptionis not self-contradictory, so is possible, because was possible;although most of us would tell that the actual example would notbe good to him/her.

Now let us proceed.


Macroeconomy is a macroscopic theory. Thermodynamics is alsoa macroscopic theory.

Thermodynamics assumes that the state of the system can be characterizedby a finite set of macroscopic variables called extensives, XI,and its behaviour by a thermodynamic potential which is the functionof extensives, say entropy S, so S=S(XI).

The Bródy-Martinás-Sajó model [1] of economicthermodynamics assumes a set of variables M, the cash and relatives,and Ni, a finite set of quantities of products. We accept thisas starting point, although obviously the set should be appendedby population P or some related quantity.

The role of potential is then played by the utility function

U = U(M,Ni,...)    (1)

[1]. This quantity expresses the value of the given economic statefor the society. U(XI) is assumed to be homogeneous linear

U = Sum(R)XRdU/dXR    (2)

(where, for technical reasons, d stand even for partial derivation too) to ensure thermodynamic formalism.

It is possible to generalize thermodynamics beyond (2).However from the model [1] not the homogeneous linearity is thefirst thing to investigate, generalize &c. I am sure thatin proper variables U is pretty homogeneous linear. Introduceas extra variable population P. Such variable, or available labourforce L are often used in economic models. Then let us write

U(P,M,Ni) = Pu(m=M/P,ni=NiP;P)

Now, in a society in which individuality is strong enough,the global utility is a sum of individual utilities ("well-being"),i.e.

du(m,ni,P)/dP~ 0

and then homogeneous linearity holds. The omission of Pfrom the variables is irrelevant today because in Europe P isfairly constant.

As for further properties of utility U, it seems fairly entropy-like.I would mention a course held by Kirschner at ELTE about 1985,for U as a function of aggregated consumption I refer [2], [3]and citations therein. Utility is a slower than linear functionof consumption ("the second dish of soup is already lessvaluable"), and this seems to be an entropic property: entropy,while homogeneous linear in the extensives, is slower than linearin energy (this is one of the 4 Callen axiomes [4]). That is enoughfor a while to regard U as an analogon of entropy. Skepticismmust have a practical limit.

I would like to emphasize, which was not a central pointin [1], that the form of the function U(XI) cannot be determinedfrom natural laws &c. Being a utility function, U(XI) expressesthe preferences of the given society. Think about the differentdesirability of the same amount and variety of goods for an averageSicilian and a Dutch, to remain within Europe.


RR This short Chapter is the recapitulation of the results of [5]and of some statements of [6]. Details of the necessary formalismcan be found in [5] and citations therein.

In theories of thermodynamic type with irreversibility generallyappears an infinitesimal quantity dQ whose positivityexpresses the irreversibility. So under proper conditions (e.g.for isolated system) the process is permitted if

dQ>/= 0    (3)

(Observe structural analogy with light cone structure in relativity.)In thermodynamics it is called non-compensated heat. Generallythere is no function Q=Q(XI) which could be called heat, stillthe differential dQ is homogeneous linear. Thenwithout loss of generality it can be written as

dQ = Sum(R)qR(XI)dXR    (4)

where the coefficients qI are of homogeneous zeroth order.

Now let us introduce new, homogeneous linear, functions of theextensives so that (4) become "the simplest". One memberof the following sequence will be met:

dQ = dQ(XI)   K=1

dQ = T(XI)dS(XI)   K=2

dQ = T(XI)dS(XI) + dZ(XI)   K=3    (5)

dQ = T(XI)dS(XI) + V(XI)dZ(XI)   K=4

and so on

Usual physical thermodynamics is K=2. In economy K=1 was erroneouslyguessed for mid-1700 France by monetarists: the goal of theiractivities at the top of finances was to maximize the gold currency(cca. M) in treasury. The experiment ended with financial collapseand bloodshed so K was >1. For K>2 local irreversibility(3) does not prevent full accessibility [7], and hence perpetuamobilia of second kind are possible [5].

We know of economic processes resembling perpetua mobilia ofsecond kind. They are circular processes so that each individualstep seems to increase or at least not to decrease some "well-being"(cf. eq. (3)). In ancient times circular activity was common,but in the last 20 years of Central Europe I can cite the Austriannuclear power plant at Zwentendorf, built up and then closed downby a referendum, and the dam at Bös (Gabcikovo)-Nagymarosbuilt up to 95% and then partly demolished.

It seems that in reality K is not 2, the model [1] is an idealisation.However independently of this utility functions exist. Maybe notonly one and they are competing. Henceforth we remain at K=2.Even in that model a lot can be shown and understood.


Let us go back to the Bródy-Martinás-Sajómodel [1]. Take the utility function. From homogeneous linearity

U(M,Ni) = Mu(ni=Ni/M)    (6)

The equivalent of chemical potentials µi are internal pricesor market prices (the model is free enterprise), and then forthese prices

µi = -(du/dni)/(u-nrdu/dnr)    (7)


µi = µi(Nk/M)    (8)

Eq. (8) would prohibit e.g. explicit time dependence. Howeverexplicit time dependence does occur. An example is tomato prices.Tomato is very expensive out of season, because then productionis expensive. Production is expensive because off-season temperatureis low, so greenhouses must be heated. But note (8): where isthe place for cold weather?

In addition note that some conservation laws hold:

fK(M,Ni,...) = const.    (9)

R [8]. A lot of such relations exist. E.g.: steel and castiron bothcontain iron atoms. Reduced iron and iron ore also contain ironatoms. Processes reducing iron ore use up money M. And so on.Some of the constraints depend on technology, which is again time-dependent.

Therefore a complete description is possible if we introducenew extensives (as an example for the process see [9]). My guessis that, besides cash and population, at least 3 sets are needed:

A) Material goods {N(m)}, measured in natural units.

B) "Environmental" extensives {N(e)}. They may standfor conditions explicitly needing compensation (e.g. for greenhousesthe needed heating energy) or for the inutility of processes (e.g.air pollution in iron industry).

C) "Spiritual" goods {N(sp)}. They may be directly goodfor the population (a musical; a nice ritual), but may be indirectlyuseful as skill or technologic level.

With this the utility fuction U seems possible.


Consider a minimally generalised one-sector Harrod-Domar model.Population is contant for simplicity, production Y (GDP, GNP oranything similar) is proportional to capital K, consumption Cis the difference of production and investment, sY. Then

dK/dt = s(t)Y(t) - aK(t)    (10)

Y(t) = g(t)K(t)    (11)

C(t) = (1-s(t))Y(t)    (12)

where a is the amortisation rate. Now, what is g(t); giving itwhat would be optimalised by society; and then what would be therpath?

It is, first, a general belief that constant savings/investmentrate is somehow "optimal". This belief was reexaminedby Kovács & Virág in 1981 [10], with the resultthat it is not so. Putting a=0 (not important) and g=go they foundsome changing investment rates superior.

In what sense? We should take first an U(C), but in this Chapterfor simplicity we choose U(C)C. The more serious problem is thatwe must take a functional of C(t) to be maximised.

The simplest functional is

F[C(t)] = Integral(0T)C(t)dt   R (13)

A discount in the integrand would make the principle more realistic,but let us continue. Principle (13) is maximal consumption intime T. The solution of the problem is a Dirac delta, i.e. investborrowed money at t=0 and then pay back and consume until T. Ifloans are ruled out, then invest all GDP until To and thenceforthconsume all Y; To can be calculated from a transcendent equation.

The discontinuous solution of the extremum principle is strange,but in this approximation the Lagrangian of the problem is degenerate.Namely let us introduce

y(t) = Integral(0t)gos(t')dt'    (14)


dy/dt = gos(t)    (15)


L = L(y,dy/dt) = Ko(1-go-1dy/dt)exp(y)    (16)

and then

d2L/d(dy/dt)2 = 0    (17)

Such a Lagrangian leads to nonexistent, distributional & [11].

Now let us keep the principle (13) but remove the degeneracy.The simplest way is to take

g = g(s), g(0) = g(1) = 0    (18)

[12]. Then we get continuous optimal investment paths. They aredescribed in [12], [13] and [14]. I do not go into details because1) for technical reasons I cannot give Figures here; 2) the formulaeare boringly clumsy; and 3) I am, by the order of the team leader,under (temporary?) boycott in OTKA T/29542 so just now I do notwant to help the other 4 members in their work (they must liftthe boycott first). But the qualitative behaviour of solutionsis simply formulable (most detailed curves can be found in [13]and a detailed analytic analysis in [14]):

Optimal paths start with high s(0) and monotonously decrease,but not to 0 at the end t=T.

They give higher total consumption than the best path with constants; hovever the optimum is sharp and even not too different pathsgive worse.

On optimal path even final K(T) can be greater than on the bests=so path, but that is sharp too.

With increasing time horizon T total consumption/time is growing.

Then the first conclusion would be long-range planning. But technologyis changing, so g(s) changes in time. For too long range the planneds(t) ceases to be optimal and then the total consumption willbe low.

Then let us plan for the longest T for which technologic predictionsare possible, say 15 years. That is possible, but the pathoptimal for 15 years surely will not be that of sustainable consumptionor development!

Optimal paths may be allowed for the near future too [15]. Itis another matter if the investment policy applies this tool ornot.

Now if we introduce an exponential discount (according to e.g.interest rates) and a sublinear consumption function [13] thens(t) changes slower on the optimal path but still the resultsremain qualitatively similar. In many cases one expects optimumfor even shorter periods (political cycles &c.).


Sustainable consumption is both a purpose (of somebody) or a(popular?) slogan. Let us cite a definition of sustainable consumptionfollowing Julia Haake & P. Jolivet [16]. According to it,World Comission on Environment and Development defined sustainabledevelopment as an activity meeting present needs "withoutcompromising the ability of future generations to meet [later]their own needs". The problem is that this principle (?)cannot be formulated as an extremum principle; and if we formulateit as two principles they very probably will be inconsistent.Obviously some combination is to be fulfilled, but which one?

Now, even if we cannot do anything with the sociological definition,qualitatively two disjoint ways exist to satisfy something similarto sustainable consumption. One solution is a stationary economybased on renewable sources. And then, if we take the limes T,then (generally) we get a stationary path. That may be sustainableforever; but trivial and consumption will be low on it, so it"will not meet the needs", either of present or of future.The other solution uses up the sources but always finds new sources/methods.As an example for the second solution, our fuel sources are goingdeeper and deeper in the crust because near-surface deposits arebeing exhausted. However technical development keeps pace withthe deeper and deeper mines and it is a matter of argumentationif mining becomes more expensive or not. Just now when I am preparingthis text it is an open question if the present increase of oilprices is real or reflects an ingenious policy of OPEC; and thatwill continue, stop or go back. While the history of the lastdecades demonstrate that up to now the recent path of developmentwas sustainable, there is no guarantee that this path will besustainable in the future. However, since I cannot predict myfuture position in OTKA T/29542 (Futurology by means of toolsof physics), now we turn to past, to see if "sustainableconsumption" existed in past ituations. If the answer isYes, then we can learn something and maybe can formulate necessaryconditions.


If we go back before Technological Revolution, we see a few otherevents which opened up new frontiers and so promised a seeminglylimitless growth. Going toward past, the first such event wasthe Great Discoveries, opening up the Americas and Australia.For now, however, the Americas fairly have filled up, and Australiais the smallest continent.

The previous event was Urban Revolution about 4000 BC. Howeverabout 60 AD Rome became a city of million, the population of presentmetropolises.

There was the Neolitic Revolution about 7000 BC introducing foodproduction by means of artificial environment. That path seemsto have been finished in the last centuries. In developed countriesboth the extension of agricultural land or the weight of agricultureare decreasing.

War, in a sense of systematic manslaught between permanentlyidentifiable groups, a consequence of overpopulation, insufficientsources &c., can be proven from the end of the Würm glacial.The first certain documents are the rock paintings in CastellónProvince, Catalonia.

However, not before. In the Würm material remnants no specificweapons for manslaughter can be discovered (although, of course,hunting tools could be used against men too), and the multitudinouscave paintings never show organised manslaughter. Also, no traditionspeaks about such; albeit, I must admit, traditions certainlyrooted in Würm are rather scarce. African Würm, whichmay have differed qualitatively from European, is much less known.

In Europe it seems that the economic system was rather stationaryat the last third of Würm, maybe as back as 40000 AD.

Now, palaeoeconomy is on one hand only an analogy for futurologyor present-day economy; on the other I just started the researchwith Nóra Fáy and want to publish the results elsewhere(we just have started to write down the programme of research[17]), so it would be improper to go into details here. HoweverI am going to elucidate the connections with variational principles.

The end-Würmian leading way of existence was the so-called"specialized hunting" [18]. The community chooses oneprey species and hunts this overwhelmingly; other animals areonly exotics. The advantage of this economy is that the communitylearns a lot about the prey animal, so the necessary hunting willtake minimal time and effort.

In Western Europe most communities chose reindeer, a few horse.However in the Carpathian Basin and in Ukraina chosen preys ofherds showed great variety. I mention here only Staroselje wherethe hunters killed donkeys but not horses [19] (although theyare almost sibling species). The habit must have started before30000 Bc, because in the Bükk Mts., NE Carpathian Basin,the specialised hunting of Capra ibex was done by Homo neanderthalis[19].

It is rather hard to guess the ratio of hunted and hunter. Howeverfrom the size of hordes (several dozen persons) and from the scarcityof sites one can guess that prey were in relative abundance.

This way of economy could indeed have been stationary, and consumptioncould have been sustainable. Namely, in the language of the Bródy-Martinás-Sajómodel we can tell the following about the extensives.

(Population P, not in the model. Far from saturation, thereforewe do not have to analyze. Anyway, childbirth may have been highto modern standard; we do not know too much about infant mortalityand maybe epidemics were rare. However really old skeletons arerare too. Climate and diet suggests that the majority may havedied in his/her prime, in many circulatory catastrophes as strokeor heart attacks, without extended illness. Since medical carewas probably not too efficient, the population may have remainedlow; and it did.)

Money M. While it did not exist in present form, some desirable,rare and portable goods may have played this role as e.g. shellsor amber. Their quantity would have been rather hard to exhaust.

Material goods N(m). In Würm reindeer economy they belongedfundamentally to 3 kinds. 1) The hunted reindeer, meat and tallowas food, skin as clothes and canvas, bones and horns for moredelicate tools. Reindeer is renewed yearly. 2) Vegetables, collecteddaily by women. Renewed. The tricky business was Vitamin C sourcein winter, but maybe that was solved by moss or lichen. 3) Tools.For bone & horn tools there was a renewed source. For stonetools the source was in principle limited, but practically infinite.From sources N(m)'s can have been produced via labour force (present)and skill (see later).

Environmental goods N(e). As told, they are the equivalents ofbad environmental conditions. I know about only one such, winterheating of caves. It was done by means of wood, renewable. Tobe connected directly to entropy (and maybe to ektropy) so centralfor some in OTKA T/29542 I note that, in contrast to present situations,during Würm animal energy (heat) production was much higherthan human; human was higher than that of artificial heating (probably;reconstructions would differ, e.g. because of the argumentationswhether fires were made in or before the caves, but anyways observethe Esquimaux in their igloo), and the additional CO2 productionvia economic activity (broiling & heating) would have beennegligible even if accumulated during whole Würm. However,I think, methodical reconstructions are not yet done.

Spiritual goods N(sp). They can be divided into 2 subgroups. Firstis skills. Hunting and gathering skills were probably widely spread.Toolmaking, clothsewing &c. skills may have been rarer, butwhat is important, there are practically no conservation lawsfor such goods. The second subgroup is spiritual goods for spiritualwell-being. We see products of painting and sculpturing, suspectabout rites and do not know anything about verbal art (I shallhave to return to this point later). For painting & sculpturinga small quantity of raw material is needed, but that classifiestogether with stone for tools.

This economy may have remained stable until environmental conditonsremained roughly the same, provided the utility function obeyedsome conditions. Namely if U(M,N(m),N(e),N(sp)) was not too steep inM and after a threshold it increased very sublinearly in N(m), thenthe population was no danger for the renewable goods. For N(sp)there were no conservation laws. If the society liked arts, rites,tales &c., they could produce it practically without limit.

Now, the increase of U in food and substantial clothes may havebeen strongly sublinear, although modern examples exist for expensiveeating and clothing fashions, but also for the opposite. Howeverwe can conclude that the necessary condition for the stable glacialeconomy was this very sublinear dependence of U(M,Ni) in all Ni'sconnected with conservation laws.

There remains the question how and why the stable and stationaryglacial economy collapsed, why Man switched over another way ofoperation. At the end of Würm there were obviously largeperturbations but the system may have returned to the originalpath. However it has not. I can list only negative statements.

The present population is the same on species level as at, say,20000 BC. As far as subspecies are defined for Homo sapiens, someof them also lived through the end of Würm. As far as wecan guess, fundamental needs and fundamental mental processesof modern Man are the same as that of glacial Homo sapiens.

Also, the cca. 7 °C temperature increase in itself cannotbe in itself the reason. During Würm there was a substantialpopulation, for example, in latter Sahara, which then migratedaway. Now Würm pre-Sahara might have been similar for averagetemperature to neothermal mediterranean Europe. True, annual variationswere different and this is something we plan with NóraFáy to investigate; but these details are surely neitherThermodynamics nor Economics in the sense of the present workshop.

There is one possibility to mention, not as if I believed init. There are some opinions (in remote minority) that speech itselfis the product of the last millenia. Of course, in earlier centuriesthis was the universal idea, but then the history of Man itselfdid not exceed say 10 millenia. In modern times the founder ofthis school was maybe van Ginneken [20]. However he put the originof speech to 4000 BC, well after Neolithic. So this fundamentalchange could not be responsible for the switch at 10000 BC.

In some sense his follower is Jóhannesson [21] who triedto find stems in the Indo-European and Semitic languages whichstill would belong to primary words. Following this line, Halloran,in a study which I saw only on Internet [22] tries to interpretSumerian or proto-Sumerian language as not only artificial butthe first speech of that population. What is important, he putsthe transition to speech in proto-Sumerian population to cca.8000 BC, to the beginning of Neolithic. If speech had startedwith Neolithic, the change in social behaviour would have beenexplained; but the theory is not widely accepted.

Now we are almost ready. For today a lot of raw materials arenot renewing (most fuels, ores, gems, for example). We saw thatnonstationary processes are superior for relatively short timehorizons T, and there is nothing present in modern societies preferringlong T. (In ancient Greek society, e.g., it was not absurd toplan for several generations, and household was continuous whilegenerations changed, see e.g. [23], but this was so because thepsykhe of the deceased lived in Underworld and descendants ingood circumstances were needed for rites. Today such an argumentwould be unheard of in economic planning.)

On the other hand it is not true that present economicscould not permit "preindustrial" behaviours. If thereare communities whose utility functions increase slowly aftera threshold, then in such a community activity drops after thethreshold. And if in U(M,Ni) after the threshold U grows stronglybut only with "spiritual" goods, then the communitywill produce mainly spiritual goods whose consumption seems sustainable.Not the mathematical and physical formalism is wrong, if expertsforget this possibility. The Bródy-Martinás-Sajómodel may have shortcomings, but I am sure that its utility functionhas a good structure.

But, as told, in the present situation, although I am a fullmember of the group OTKA T/29542 Futurology by Means of Toolsof Physics, it is not obvious that I should do direct futurology,so I stop here, when I have demonstrated that I could.


In the topics of this lecture I acknowledge collaboratios with:

M. Banai, ELTE, HTSzoft Ltd. 1982-

Nóra Fáy, ELTE, CRIP 1998-

J. Kovács, KgI 1985-1997 (died)

K. Martinás, ELTE 1983- (with breaks)

Ildikó Virág KgI 1985-1997.

I did not work with 3 other members of OTKA T/29542 becausemy attempts for contact were unsuccessful up to now. With NóraFáy we started the Palaeoeconomy research in 2000.


[1] A. Bródy, K. Martinás & K. Sajó:Acta Oec. 35, 337 (1985)

[2] S. M. Goldmann: The Review of Econ. Studies 35, 145(1968)

[3] F. R. Chang: Econometrica 56, 147 (1988)

[4] H. B. Callen: Thermodynamics. J. Wiley, New York, 1960

[5] B. Lukács & G. Paál: Acta Phys. Hung. 66,321 (1989)

[6] B. Lukács: Acta Oec. 41, 181 (1989)

[7] P. T. Landsberg: Thermodynamics. Interscience, New York,1961

[8] B. Lukács: Proc. ERÖFI II, 1995, KFKI-1995-11,p. 175

[9] L. Diósi, B. Lukács, K. Martinás &G. Paál: Astroph. Space Sci. 122, 371 (1986)

[10] Kovács J. & Virág Ildikó: Közg.Szemle 1981/6

[11] Banai M. & Lukács B.: Közg. Szemle 1987/4

[12] Banai M. & Lukács B.: Közg. Szemle 1988/11

[13] Banai M. & Lukács B.: KFKI-1989-68

[14] Banai M. & Lukács B.: Mat. Lapok 34, 307(1991)

[15] M. Banai & B. Lukács: Attempt of Closing Up byLong Range Regulators. in: Technological Lag and IntellectualBackground: Problems of Transition in East Central Europe, ed.J. Kovács, Darthmouth, Aldershot, 1995, p. 311

[16] Julia Hake & P. Jolivet: Towards Sustainable Consumption... on Internet,

[17] Fáy Nóra & Lukács B.: TKKE Hírlevél2000. május, 5. old.

[18] G. Clark: World Prehistory. Cambridge University Press, Cambridge,1969

[19] Gáboriné Csánk Vera: Az ôsemberMagyarországon. Gondolat, Budapest, 1980.

[20] J. van Ginneken: La Reconstruction Typologique des LanguesArchaiques de l'Humanité. Amsterdam 1939.

[21] A. Jóhannesson: The Third Stage in the Creation ofHuman Language. Rejkjavík & Oxford, 1963

[22] J. A. Halloran: The Proto-Sumerian Language Invention Process,on Internet,

[23] Aristotle: Oeconomica. 1343b and especially the deuterocanonictext.

My HomePage, with some other studies, if you are curious.