ONCE MORE ABOUT ECONOMIC ENTROPY

ONCE MORE ABOUT ECONOMIC ENTROPYAuthor(s): B. LUKÁCSSource: Acta Oeconomica, Vol. 41, No. 1/2 (1989), pp. 181-192Published by: Akadémiai KiadóStable URL: http://www.jstor.org/stable/40729377 .

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Ada Oeconomica, Vol. 41 (í-S), pp. ¡81-192 (1SS9)

ONCE MORE ABOUT ECONOMIC ENTROPY

B. LUKÁCS

The degree of structural similarity between economy and thermodynamics is investigated.. Some counterarguments are mentioned against describing the irreversibili ty of economic processes by a single entropy, although such an approximation might have been quite decent in the last century when free competition was a rule and state's redistribution activity was restricted.

Introduction

Some years ago an article [1], published in this journal, demonstrated a structural similarity between thermodynamics and the possible description of economy. Thermodynamics, as well as other fields of theoretical physics, has developed very powerful and refined matemathical techniques, whose application is an attractive possibility. However, first one should decide if there is structural isomorphy or similarity.

The above fine distinction seems to be mere hairsplitting, but it is not. This can be visualised by translating it into Greek, when we arrive at the famous distinction between homoousion or homoiousion, with obvious consequences. In the case of isomorphy the full arsenal of thermodynamics can be deployed; otherwise not algorithms, only ideas can be adopted.

If no convincing counterevidence is shown, then the demand for distinction is a mere overcautious commonplace, to be rejected by presumption of innocence, representing here the nonconstructivity of mere doubts. But the question is not. without importance; so here we try to collect some arguments for or against, ev- idences being out of reach in the present stage, still before detailed quantitative analyses.

Our claims are as follow. By semiquantitative arguments four statements seem to be verifiable:

1. For such long time intervals when technical evolution can already be felt, the tangible assets are insufficient to characterise the actual state in a thermodynamic type description; the same is true for such short periods, when seasonal changes are not averaged.

2. During inflation (no matter, creeping or gallopping) nominal prices, cap- itals, &c. cannot be used as fundamental variables. First the inflation is to be rescaled from them, which can be uniquely done ¿/isomorphy holds.

3. There cannot be a single economic entropy (by simpler words the Pfaf- fian form [2] differs from that of thermodynamics) for national economies having

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182 B. LUKÁCS: ONCE MORE ABOUT ECONOMIC ENTROPY

recourse to redistribution processes for correcting the results of pure economic evolution (which technnique is quite worldwide nowadays).

4. Therefore the only economy for which the isomorphy is clear is that of "free competition and minimal state" .

Unfortunately, [1] did not give an algorithm to construct the economic entropy function Z for an actual, observable economic system. Therefore first this problem is to be discussed.

The economic entropy

This Chapter is a rather brief recapitulation of the main results of [1] for the sake of clear argumentation. There an idealised model system was discussed; the results are valid for that system. Consider a productive economic subsystem of any size. Its economic state can be characterised by a sufficiently large set of its assets, as, e.g. its monetary founds (henceforth money), M, and tangible goods and services, N% (measured in natural units); the latter are n different data, n » 1, but the set can be reduced in practical cases. As a limiting case pure commerce is possible, but in the generic case there is production as well. The production possesses a preferred direction (one can always tell endproduct from raw material, although the direction may depend on the actual situation), therefore is irreversible. In contrast, ideal commerce is reversible: by subsequent buying and selling processes the initial state can be recovered. Fundamental experience demonstrates that pure commerce is not enough for development, therefore the irreversible processes drive the subsystem towards its "goal" (whatsoever be it).

So there are n + 1 independent extensives. Now, consider an initial state (Mi,7V"i), and a wanted final state with N^. This state can always be reached by pure commerce, buying the needed differences N[ - iVJ at market prices. Assume that there is no investment from outside (by the terminology of [1] the process is adiabaiic)] then M<i is fully determined. So, commerce is transitive on hypersurfaces of n dimensions in the n + 1 dimensional state space. An alternative formulation is that during pure commerce one function Z

Z = Z(M,Ni) (1) is constant. In the model pure commerce is reversible and vice versa, so the sub- space of reversibility is of n dimensions, by other words, its dimensionality is the (total)- 1.

Therefore for reversible processes

dZ s (l/T)dM - (mr/T)dNr = 0, (2) where we have introduced the shorthand notations

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B. LUKÁCS: ONCE MORE ABOUT ECONOMIC ENTROPY 183

Ì/T = SZ/6M; -mi/T = 6Z/6N{ (3)

together with the Einstein convention for indices [3]; there is an automatic summa- tion for indices occurring twice, above and below.

Now, obviously the - m 's are some prices [1] and for pure commerce they coincide with market prices (cf. eq. (2)); otherwise external prices are distinguished as ft* [1]. As a next step include some production: then dZ ^ 0, and due to irreversibility it must have a definite sign chosen to be positive. So, calculating the change of the capital C:

dC = dM- mrdNr = TdZ > 0 (4)

By the meaning of production irreversibility, so the increase of C and, synoni- mously, of Z, is connected with steps through more and more desirable (or at least advanced) stages of evolution, so Z somehow measures some utility, and then T is the inverse value of unit money from this viewpoint. Now we have indeed recapit- ulated the main results of [1], partly in inverse way. One to one correspondences (sometimes with inverse sign) have been obtained between words of thermodynamic and economic vocabularies as

energy < - ► money chemical components < - ► commodities and services

heat < - ► capital entropy < - ► entropy

chemical potentials < - ► internal prices temperature < - ► inverse utility of currency

Z plays the role of potential. Obviously it would be very useful to build up the entropy function Z(M , N%)

from observation: then one could predict the useful steps of advance, thus speeding up economic development. In the next Chapter we demonstrate that this is possible in principle; maybe the task is difficult in practice, and this might have been the reason that [1] suggested only the measurement of the second derivatives of Z.

How to determine ZI

Usual thermodynamics contains one more fundamental assumption, namely that the entropy is a homogeneous function of first order of its variables. This ensures the invariance of intensives when dividing the subsystem into parts. Since no serious counterexample seems to exist (market prices do not scale with size), we can accept this; then

Z(lMìlNi) = lZ(MìNi). (5)

Acta Oeconomica 41, 1989




Hence, by standard calculation, one gets

Z = (1/T)M - (mr/T)Nr = M z(n* = N*/M). (6)

Now, via eqs. (4), (6)

blnzjM = -m /(I - mrnr) = /¿(n*). (7) This system consists of n partial differential equations for the single z(nl); however the very existence of a single z function guarantees the integrability, whose mathematical conditions are

Sfi/6nk = 6fk/6n' (8) Now wè need functions /»(n*). Prices and commodity to money ratios can

be observed. We have seen that m's act as market prices in reversible commerce processes; so they can be observed in such ones. Prices and ratios change; we get their corresponding pairs. In principle then these tabulated functions can be interpolated without limit and a funcional dependence is obtained. We omit here the technical details; they are well elaborated in measurement theory [4].

The moral of this Chapter is that the economic entropy function Z = Mz(nl) can be calculated from observing the changing prices, commodities and money if

1. there exists indeed a single Z function in the system, and not many; 2. the extensives (now M and N%) can indeed be measured; and 3. the kinds of the independent extensives are indeed known. Then the only problem is technical: how to evaluate measurements [4]. Ob-

viously there are errors in the measurements, propagating into Z. Outside the domain of measured values n* and m¿ extrapolation is needed and the error of Z will rapidly increase. But this is simply the quantitative aspect of the well known problems of prediction.

At this point it may seem as if we were ready with the existence problem of Z' if the integrability Conds. (8) hold, then one can calculate Z, and then it exists and is unique (maybe up to some very restricted additive terms as in thermodynamics [5]), if they do not hold, then the m's and T cannot be obtained from a single Z. However, the first part of this statement is true but the second is not necessarily; there may be other reasons of the invalidity of (8) as well, when being checked. These possibilities are discussed in the next two Chapters.

Further independent data?

Are we indeed sure that the set of M and the chosen N% 's is complete? One may expect e.g. something representing the extension of the system too; this would

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be the analogon of the volume V in thermodynamics, and may e.g. be the number of population. Here we are not going to analyse actual data, so the question cannot be answered. But the way how to answer it is elaborated in thermodynamics. A direct check would be to observe market prices in pure commerce steps at different times but so that the commodity to money ratios n1 return again and again to the same values. If the prices are the same too, the set may be complete; if not, there must be new hidden variables responsible for changes. This direct check is difficult; however an equivalent way is to use the Gibbs-Duhem relation

Md(l/T) + Nrd(-mr/T) = 0. (9) This relation expresses the fact that there are n + 1 intensives but only n

extensive ratios on which they can depend. If (9) does not hold, that indicates the existence of more intensives, whose incorporation restores the balance [2].

For this we need T. But from eq. (3) one gets

l/T=-nr6z/6nr (10) and z has been calculated from the prices.

Now it is easy to show an example when (9) does not hold even for a handmade model system. Construct a Z function with explicit time dependence, Z = Z{MiNi't). Then the right hand side is not 0 but -{ôZ/ôtfdt, and any formalism of thermodynamics breaks down [2]. A clear signal is if the prices depend not only on n*'s but on time too. And definitely this happens with any prices in a long time evolution via technologic <5¿c. changes, and with vegetable and fuel prices between summer and winter.

It is not quite clear how to solve this problem, but in thermodynamics simi- lar problems can be solved by eliminating the time dependence by means of extra extensives representing the changes [6]. The long term dependence is caused by development of knowledge; now, different kinds of knowledge are intangible goods (sometimes people are ready to pay real price for them, e.g. when learning lan- guage), their development is as necessary for describing the actual state of economy as the quantity of castiron. The short term dependence is from the change of external (e.g. weather) conditions; the changing fuel need in glasshouses demonstrates that the wanted status of natural environment is not a cost- free godsend, something which we are just learning from pollution effects. So the moral of this Chapter is that nontangible goods may be as important (and so as real) as tangible ones.

Inflation

Now assume that by some precognition we know that our set of extensives is complete, there cannot be anything else important quantity, and still something is

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wrong, e.g. prices continuously go up. This is just the case of inflation. Has the thermodynamic structure broken down or not?

This paper is not the detailed analysis of inflation. So here we make only a classification. It seems that two subcases of inflation exist, according to the relation (8). Namely, if it holds, then the function Z exists. Then some intrinsic property of the Z function is responsible for the increasing prices. (Note that the N%/Nk ratios may be constant, but N%/M is continuously decreasing, so the prices do not have to be constant.) Maybe this is the case when inflation occurs in (or near) an absolute shortage (e.g. just after wars). But a worldwide experience from the last two decades is the possibility of serious tendencious peacetime price increases. This may be something else.

Assume that one has such a long sequence of observed prices and commodity to money ratios that the derivatives in the integrability condition (8) can be calculated (for technical problems see again [4]), and the result is that (8) does not hold. Then there is no such entropy function which could produce the observed prices; however, the whole increase may still be an artefact. Namely, consider the hypothetical case when authorities are continuously renaming the numbers on the banknotes. Then the observed M is a completely arbitrary quantity, whose pres- ence smuggles an explicit time dependence into the Z function. But this is just which should be avoided; so the changing currency unit is to be rescaled before using it in the formalism.

Now, if prices are continuously going up, but commodity ratios and price ratios remain roughly constant, and (8) does not hold, then one can suspect that something (market, central bank, &c.) is changing indeed the currency unit, and he may hope that the scale factor could be empirically found out and removed. This might happen by assuming an m(t) function for money multiplication, then

M - > M/m(t)] m, - ► mi/m(t) (11)

and with the new prices and ratios (8) can again be checked. The correct m(t) function is that one, with which (8) (fairly) holds if ever. The technical problems may be serious, but this is a regular fitting procedure [4].

The problem is caused by the fact that the Nl>s do possess natural units, but M does not. However, if the above construction can be made, we have fixed the units of the currency (up to a constant factor or unit).

More complicated Pfaffian forms?

Now, the previous two Chapters dealt with serious technical problems, but did not necessarily touch the structural isomorphy (homoousion) of economy and thermodynamics. Now we have arrived at the fundamental question. [1] states that

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the existence of the integrating factor 1/T and so that of the entropy Z too can be obtained from some simple assumptions. One is the set of independent extensives, discussed in a former Chapter. The other is the irreversibility formulated for C as dC > 0, for which good arguments exist. The third was formulated as Proposition 26 in [1] stating that any particular state possesses such neighbourhood which cannot be reached without irreversibility.1 However, this third assumption is not sufficiently strong to get dC = TdZ.

We do not want to make here a detailed mathematical argumentation with nice corollaries and proofs, but restrict ourselves to transparent constructions. First we note that what is really needed for the existence of T and Z is a foliation in the state space: that the infinitesimal neighbourhood of any point separate into 3 parts: one, which is accessible by reversible adiabatic process, but whence the original point is not; another, which is inaccessible, but thence the original one is accessible, and a third exceptional part separating the first two [2, 7]. For the last case dC = 0; here Z =const., downwards dZ < 0, upwards dZ > 0. Now, this foliation is a consequence of the simple inaccessibility (with some unimpor- tant mathematical assumptions for continuity, convexity &c.) if the dimension of transitivity of reversible processes is n, i.e. only by 1 dimension smaller than the total space.2 If this assumption does not hold, then the existence of irreversibility cannot lead to dC = TdZ.

One can directly see this on an example. The simplest nontrivial case is a state space of 3 dimensions, spanned by M , Nl and N2. If the reversible motions span 2 dimensions, then C will not change on the 2- surfaces Z =const., so dC ~ dZ. But what happens, if the reversible processes span only one dimension?

The obvious answer is that then there are lines in the 3 dimensional space on which dC = 0. Such lines cannot produce foliation. A line is two relations between three extensives, i.e. say

5 = S(M,N' N2) = const, Z = Z(M}N'N2) = const. (12)

Then C changes with both 5 and Z, and one gets

dC = TdS + WdZ (13) In thermodynamics the difference of the dimensions of the state space and reversible hypersurfaces is either 1 or there is no change in the extra extensives (thus the

1The exact form of this proposition in [l] is formulated as: "Close to every point in the phase space there is another point which can be reached only by a regular economic process in the sense of Proposition 2a." (i.e. by dC > 0.)

2H&d [1] stated in Proposition 2a the reversibility of pure commerce too, the single entropy would be a mathematically correct consequence in the model system. Then there would remain the question of the validity of the model.

Acta O económica 41, 1989




state space is effectively reduced), from reasons partly immaterial here partly to be mentioned later. However, in economy the question is to be discussed.

A possibility is the existence of extra extensives which are not objects of commerce as e.g. the number of population. For such quantities one cannot a priori know if their changes can be reversible, and if not, the dimensionality of the reversible hy persurf aces is n, but it is < (total) - 1. If there are no extra extensives, and the complete set is (M, A^'), then the dimensions of space and pure commercial motion are n + 1 and n, respectively. However, not necessarily all pure commercial processes are reversible. Indeed, all developed countries apply some income taxation of serious degree. Now, the usual way is to apply taxes when buying goods of consumption, but not for goods of investment. (The technical details of achieving the desired purpose may be various and complicated.) Therefore if one sells machines, buys whisky and again back, he may lose some important percent of capital. Then commerce is still reversible within two disjoint classes of goods, but not between classes. This is at least one dimension loss. Therefore for contemporary developed countries one may expect eq. (13). To be sure, [1] assumed no external source of money, and tax office could be regarded as a negative source. However, taxes belong to the rules of the actual economy. Tax is not a (positive or negative) investment arbitrarily decided by the economic units but strict law (although parlamentai, not natural). Therefore one should not define adiabatic processes without the prescribed tax.

This is an important structural difference between this kind of economy and thermodynamics. First, m's, therefore prices, cannot be obtained from a single potential. This will be reflected in the fact that the integrability condition (8) will not hold even after rescaling the value of currency, a fact which is now predicted and could be experimentally observed. Second, the irreversibility will be produced by the increase of two different entropies. From dC > 0 neither dS > 0 nor dZ > 0 follow automatically. So, for such an economy accumulation is not necessarily synonimous with increasing Z [1].

One may contemplate about the most important consequences of this structural homoiousion instead of homoousion. But it is very frequently told that contemporary national economies try to follow two different paths in the same time: one is the preference of market, the other the preference of social welfare or justice. This is just the expected behaviour if there are two quantities with tendency of increase as "natural" laws, instead of a simple Second Law of Thermodynamics.

An oversimplified but quantitative example

This last but one Chapter is a mere illustration by means of the most prim- itive possible quantitative example. But first observe that it is not easy to decide

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if market activities are truely reversible or not. One either has some internal information about this, or he is an external observer assuming (as long as possible) that market prices belong to reversible acts, so they are the - ra's. Consider a very simple model with money M, investment goods X = Mx (price is -mx) and food Y = My (price is - my).

Now, our observer goes to the market, observes the changes of prices and quantities for a long time, and in the changes believes to see the price functions

- mx = a/x' -my = b/y. (14)

Prices go up, if goods are rarer. Then he constructs the /'s of eq. (7), and finds that the integrability conditions (8) hold. The economic entropy and temperature can be calculated:

Z = cMxa/(1+a+6'y6/(1+fl+i';r = (1 + a + b)M/Z. (15)

The constant c is arbitrary, reflecting the definition of the currency unii. Eq. (14) shows the possibility of a "real inflation": if y/x is fixed but x goes down, prices will uniformly increase.

So far so good. But what if he believed to see the slightly different functions

-mx=a/x' -my=b/y + q? (16)

His first idea, of course, is that he observed reversible acts, som = m*. Can it be? The integrability conditions (8) do not hold. What is the explanation?

1. Maybe there is no clear irreversibility in this economy: some people perform dC < 0 productions. This is a possible, but too crude conclusion.

2. Maybe there is a third important commodity W1 overlooked first. (The nominator of /'s contains all the commodities.) Cond. (8) is restored if it can be found in the market with the price

- mw = k/w - qy/w. (17)

3. But for n different commodities the number of the independent components of (8) is n(n - l)/2, growing rapidly. If W is not found, or its price is not appropriate, then maybe (16) is a result of an inappropriate fitting to the observed changes, because there is an inflation, and the fitting should have to be done according to (11).

4. We cannot follow the observer hither, not possessing his data. But if he fails again, then there is no Z function whence such m's could be obtained. Maybe these are not "real" prices developed from reversible market acts; e.g. it is possible that b/y + q is the price when buying, but q goes to the treasury. Then there is some Z generating the original prices (cf. eq. (15)), but certain commerce steps are irreversible, so not all the irreversibility comes from Z.

Acta Oeconomica 41, 1989




Prices (16) seem very harmless; still they do not fit into a structure isomorphic with thermodynamics. The above example is of course a handmade illustration. But who knows what prices can be produced by a resolute tax system? Economic systems may be very intricate and complex.

Conclusions

Here the thermodynamics type structure of economy proposed in [1] has been discussed. Our results can be summarised as follows. Generally one cannot expect the same structure for economy as for thermodynamics: the irreversible change of capital C will not be proportional to the change of one entropy, but rather a set of entropies will appear. (Another possibility is to use one entropy as potential function, but two disjoint kinds of money (MlyM2). Then the equivalence of entropie and energetic conventions might break down, but this question is still partly obscure even in thermodynamics.) The simplest reason of this phenomenon is when there are irreversibilities even in pure buying and selling; taxation systems singling out some goods of commerce very probably generate such irreversibilities. An even more pronounciated effect arises in the case of "labelled money", very familiar in Hungary, when certain amounts of money may be spent only within a prescribed family of goods.

Some analogies are known in physics. Certain systems may possess two different temperatures simultaneously, each for a given kind of internal energy, with negligible cross exchange. The most striking example is the crystallic LiF, at a low positive temperature of atomic motion with a possibility for a negative one for its nuclear spins [8]. However, such systems are more or less exceptional in physics; generally different kinds of disordered motions can very effectively excite each other, and the energy is fastly being rearranged between them. In economy this may be forbidden.

Our guess is that the simple thermo dynamic Pfaffian form dC = TdZ was valid only for last century free competition economies with "minimal state" . It is tempting to think that the manifestation of the more complicated contemporary Pfaffian forms is the frequent competition and compromise of "economic" and "social" preferences and guiding principles. The existence of a single entropy would lead to serious constraints among the changes of prices and amounts of various commodities; in principle this could be checked from observed data, but the author does not feel himself sufficiently experienced to perform this check. If indeed the dimensionality of reversible processes differs with more than one dimensions from that of the total state space, then the usual formalisms of termodynamics should be generalised before applying to economy.

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Acknowledgement

The author acknowledges that years ago he was present at a discussion of the authors of [1], where some doubts arose for the case of "labelled money". The present paper was triggered by an article of Beatrix Paály published in Hungarian, which questioned the equivalence of energetic and entropie versions of First Law.

References

1. Bródy, A. -Martinas, K.-Sajó, K.: An essay in macroeconomics. Acta Otconomica, Vol.ir>. Nos 3-4 (1985) pp. 337-343.

2. Landsberg, P.T.: Thermodynamics. Interscience, N.Y. 1961. 3. Eisenhardt, L.P.: Riemannian geometry. Princeton University Press, Princeton 1950. 4. Jánossy, L.: Theory and practice of the evaluating of measurements. Oxford University Press,

Oxford 1965. 5. Lukács, B.-Martinas, K.: Phys. Lett. 114A, 306 (1986) 6. Diósi, L.-Lukács, B.-Martinas, K.-Paál, G.: Astroph. Space Science 122, 371 (1986) 7. Lukács, B.-Martinas, K.-Paál, G.: in: Relativity Today, World Scientific, Singapore, 1988,

p. 247. 8. Purcell, E.M.-Pound, R. V.: Phys. Rev. 81, 279 (1951)

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