Physica A 395 (2014) 269–274

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Physica A

journal homepage: www.elsevier.com/locate/physa

A macro-physics model of depreciation rate ineconomic exchangeRui F. Marmont Lobo a,b,c,∗, Miguel Rocha de Sousa d,e

a Department of Physics, Faculty of Sciences and Technology, Universidade Nova de Lisboa, Monte da Caparica, 2829-516, Caparica,Portugalb GNCN-ICEMS, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516, Caparica, Portugalc ICEMS, Instituto Superior Técnico IST-Av. Rovisco Pais, 1000 Lisboa, Portugald Department of Economics, Universidade de Évora, Largo dos Colegiais, no 2, Évora, 7005-324 Évora, Portugale NICPRI-UE, Núcleo de Investigação em Ciência Política e Relações Internacionais, Universidade de Évora, Palácio do Vimioso,Largo Marquês de Marialva, 8, -7000-809 Évora, Portugal

a r t i c l e i n f o

Article history:Received 29 January 2013Received in revised form 2 September 2013Available online 9 October 2013

Keywords:Depreciation rateEcono-physics pure barter model (EPB)Edgeworth boxPure barterTheory of value

a b s t r a c t

This article aims at a new approach for a known fundamental result: barter or trade in-creases economic value. It successfully bridges the gap between the theory of value andthe exchange process attached to the transition fromendowments to the equilibrium in thecore and contract curve. First, we summarise the theory of value; in Section 2, we presentthe Edgeworth (1881) box and an axiomatic approach and in Section 3, we apply our pureexchange model. Finally (in Section 4), using our open econo-physics pure barter (EPB)model, we derive an improvement in value, which means that pure barter leads to a de-cline in depreciation rate.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Economics is a field of knowledge dealing with ‘wealth dynamics’ (production, distribution and consumption) and cer-tainly the most clearly value dependent among the social sciences. It attempts to determine what is valuable at a given timeby studying the relative exchange values of goods and services. Its conceptions and models are based on value systems andviews of human nature. In the economic models, the values can be quantified by being assigned monetary weightings. Thisemphasis on quantification gives economics the appearance of an exact natural science.

In its turn, physics is traditionally a science dealing with the quantification of the observable world, and thus is appropri-ate to develop a theoretical framework of economic dynamics based on suitable macro- and micro-models. The evolutionof physics has been based on a progressive microscopic interpretation of reality from a previous well-established macroframework. In this work, the authors also want to depart from a macro-econo-physics perspective, trying to find a verysimple model which is able to lead to a well-established economic result as an output.

This article contributes to the field of econo-physics, which has been steadily growing in the recent past, in particular bygiving a new formal-physics approach to the theory of value, and thus obtain a well-known result in the economic domain.The authors introduce and discuss a model of depreciation rate within economic exchange.

∗ Corresponding author at: Department of Physics, Faculty of Sciences and Technology, Universidade Nova de Lisboa, Monte da Caparica, 2829-516,Caparica, Portugal. Tel.: +351 212 948 549; fax: +351 212 948 549.

E-mail address: rfl@fct.unl.pt (R.F. Marmont Lobo).

0378-4371/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physa.2013.09.064

270 R.F. Marmont Lobo, M.R. de Sousa / Physica A 395 (2014) 269–274

Neoclassical economics gave birth to mathematical economics through the notion of economic core, a refinement of theEdgeworth [1] box, but has not forgotten that economics is also a social science. The formalisation of existence and numberof equilibria is a way to approach economics as a social engineering (Debreu [2], Scarf and Debreu [3], Aumann [4], Balasko[5] and more recently Balasko [6] using differential topology methods); nevertheless, Sen’s [7] approach, even though notbeing formal, extended also the economics domain as a social ethics domain.

First, we sum up some economic fundamentals. Our main aim will be to understand the theory of value; we start fromthe Edgeworth [1,8] box, in an economic model with no production, and then we create our general exchange model.

One should take into account the fact that dynamics, namely the use of diffusion processes (Wiener processes), havebeen in the economic literature now for quite a while. For instance in macroeconomics, researchers include Turnovsky [9];Dowry, Pitchfork and Turnovsky [10]; Ljunqvist and Sargent [11]; and Stokey, Lucas and Prescott [12]. Recently, for eco-nomic geography, Fujita, Krugman and Venables [13] discussed the diffusion properties of economic physical goods; andmore recently endogenous growth models with human capital, the diffusion of technology and human capital spillovershave become important—see Aghion and Howitt [14] and Barro and Sala-I-Martin [15]. Stoneman [16] has abridged the roleof diffusion in economic systems from not only a physical but also an economic vision.

Regarding the next sections, in Section 2, we present the Edgeworth [1,8] box and our axiomatic approach; in Section 3,we present the transition from endowments (model without production) to equilibrium and simple core and contract curve.

Our model provides an intuitive approach which is common knowledge to economics, that is exchange increases upvalue (a fundamental theorem in economics), but in an extended framework: our basic econo-physics model. It is nowquite common in finance and economics to have diffusion processes of goods and information. The idea of diffusion hasbeen ‘imported’ from physics and mathematics [9–16], from concepts such as Brownian motion and Wiener processes, toevaluate option pricing (Black–Scholes) and international macroeconomics exchange rate risk pricing [9]. The main noveltyis that of the processes of economic diffusion, namely barter or trade between two goods that are blended together andthat can have implications on social welfare/economic valuation. What does happen in the barter exchange between two(at least) different goods? The diffusion process has physical properties which can be applied to economics. In the exchangeprocess (the mixing diffusion econo-physics), we conclude that in a macroscopic scale (in our economic model, for a societyor class or group of consumers) there is a decrease in depreciation (thus degradation). The conclusions might seem obvious,but the framework henceforth used is new and has been taken from the idea of diffusion in physics.

2. Axiomatic approach

We consider the following hypothesis for our first simple model:

1. We have non-perishable goods (A, B; A ≡ corn, B ≡ wheat).2. We consider a time interval in which there is no increase in production (constant production, P).3. We have two consumers (1, 2) and two goods (A, B) in the Edgeworth box [1,8], with quantities NA and NB, respectively;

this is what is called (2 × 2) model.4. The economic system is closed (there are no exchanges with the exterior).5. The social welfare function of this economy is given by V , the value of utility in consumption (Ut ) plus the production

(P):

V = Ut + P (2.1)

in which Ut is utility, which is satisfaction obtained from consumption of goods.

Marginal Utility (U ′=

∂U∂Ni

, in which i stands for each good A, B) is decreasing at an increasing rate; thus, U ′<0; U ′′>0.P is production, a kind of ‘manna’ from heaven, which is a standard hypothesis in simple economic models. A social

welfare function is just some function of the individual utility functions; it gives a way to rank different allocations thatdepends on the individual preferences, and it is an increasing function of each individual utility. The utility function is thedegree of satisfaction obtained from consumption of goods.

To pursue the goal of this article, we explain the transition process from initial endowments (ω) to the final equilibrium(E*) located in the core (optimal Pareto points in the contract line on the Edgeworth box)—see Fig. 1. The Edgeworth boxis a convenient graphical tool to analyse the exchange of two goods between two people, as a way of representing variousdistributions of resources between two agents. It is frequently used to illustrate general equilibrium, aiding in representingthe competition and efficient outcomes [8]. The width of the box measures the total amount of good A available in theeconomy and the height measures the total amount of good B. Agent 1’s consumption choices and utility are measuredfrom the left-hand lower corner, while agent 2’s variables are measured from the upper right. Indifference curves are utilityparametric curve levels which represent ordinal increasing satisfaction in consumption, thus increase in welfare. Paretopoints or efficient loci are allocations with which any consumer cannot improve his welfare without reducing the welfareof the other consumer and vice versa. The Pareto point is represented by the tangency of indifference curves (utility curves)in the Edgeworth box. The contract curve is the set of all Pareto allocations, and the core is the Pareto set of points obtainedfrom trading starting at the initial allocation, which naturally includes all viable equilibria. Thus in Fig. 1, from a given initialendowment (ω) for two goods (A, B), and two given consumers (1, 2), we have respectively a closed box in which we haveinitial utilities (U0

1 ,U02 ). Therefore, after a trading or barter processwe have a final equilibrium E*, which cannot be improved

R.F. Marmont Lobo, M.R. de Sousa / Physica A 395 (2014) 269–274 271

Fig. 1. The Edgeworth Box with pure exchange in a (2 × 2) model.Source: Adapted from Mas-Collell et al. [1995: pp 520]-[8].

upon by any further exchange,and which yields an increase of utilities in both consumers. Thus, we have a final equilibriumallocation with prices, pA and pB, equal to the ratio of marginal utilities:

p∗= −

pAPB

= −∂U1

∂NB

∂U1

∂NA= −

∂U2

∂NB

∂U2

∂NA(2.2)

where U = U* in equilibrium.This transition process can be described as an exchange process from initial endowment (ω) to final equilibrium (E ∗).To do this, we construct the net welfare gain function:

χ = V ∗− (V 0

1 + V 02 ), (2.3)

which reveals the (net) gain in aggregate value from point ω (initial endowment) to E∗ of final equilibrium.Will it be possible that when the system reaches E∗, one has convergence to equilibrium and χ >0?Actually, there was an increase in the aggregate value due to the exchange between the two consumers; thus χ >0. This

is a standard result in economics since Edgeworth [1,8].Aumman [4] has generalised this result and Scarf and Debreu [3] proved its existence in different settings. Our contribu-

tion is next.

3. Econo-physics pure barter model

We have a closed system formed by a system of N goods of the same type (homogeneous good). We consider V to be thetotal value of the (economic) system, Ut the utility from consumption of the good and P the production.

We can increase the value of the system by applying labour to the economic system or by giving more utility to thesystem (benefited from more consumption):

dV = dU t + dP (3.1)

where P represents the total (human and mechanical) energy used to produce the system of goods.If the number of consumers increases, then (total) utility increases.Consider N ’, the number of consumers and R, the resistance to consumption (that is, a market neighbourhood) which

measures the conversion from potential consumers to effective consumers. Thus,

Ut =

N ′f

N ′i

R · dN ′= R.∆N = R ·

N ′

f − N ′

i

. (3.2)

Since it is not possible to convert 100% of production into (consumption) utility, one can introduce a depreciation factor ξ ,obeying

dξ >dPφ

(3.3)

272 R.F. Marmont Lobo, M.R. de Sousa / Physica A 395 (2014) 269–274

where φ is considered the productive capacity and ξ stands for degradation in the economic system. The well-knowndepreciation rate, which accounts for the temporal variation of the good’s value, is caused by a reduction in the remainingvalue of its future services [15]. Standard neoclassical growth theory uses this traditional approach (vide the capital intensitylaw of motion in Ref. [15]). With our approach, the depreciation factor is defined by (3.3), which is a simpler physics conceptapplied to the same economic idea.

Assuming that there is no change in the nature of goods, that the physical/economic system is closed, we can state thefollowing condition:

dV = dUt + dP = RdN ′+ φdξ . (3.4)

In an ideal equilibrium situation of ‘well-behaved’ markets, one can assume that demand equals supply; thus

R · N ′= N · φ. (3.5)

The left-hand side of (3.5) stands for consumer demand: R denotes consumer resistance and N ′ the number of consumers;the right-hand side of (3.5) stands for production, in which N is the number of goods and φ is the productive potential.

It is possible to assume the following ‘econometric’ potential, L:

dL = −N ′

· dR + ξ · dφ. (3.6)

Thus, from (3.6) taking partial derivatives

ξ = −

∂L∂φ

R

(3.7)

N ′= −

∂L∂R

φ

. (3.8)

For ideal markets, where N ′= N · φ/R, there is no variation of φ and R in the barter process. Thus, introducing a unitary

econometric potential:

η =LN

=

dLdR

R,φ

. (3.9)

One gets from (3.5) and (3.8):

dη =1NdL = −∅dR/R (3.10)

and

η = −∅(ln R + k) (3.11)

where k is the integration constant.In the barter process,we need to dealwith open systems. If we have in the very start, two types of goods in awell-behaved

perfect market to constant φ, we are in autarky; and then we remove the barriers to economic barter trade.For an open system:

dL = −NdR − ξdφ +

i

ηi · dNi (3.12)

in which i stands for the number of goods and where ηi =

∂L∂Ni

R,φ,Nj=i

is the ‘econometric unitary potential’ of type i goods.

As a consequence of exchanging two type of goods A and B, it is possible to use (3.5) to write a set of related equations:

R = RA + RB (3.13)N = NA + NB (3.14)RA,B = YA,BR (3.15)

YA,B =NA,B

NA + NB(3.16)

ln RA,B = ln R + ln YA,B. (3.17)

We have ex-ante barter (autarky) for two goods A and B

Li = NA · ηAi + NB · ηBi (3.18)

R.F. Marmont Lobo, M.R. de Sousa / Physica A 395 (2014) 269–274 273

with

ηAi = −∅ · (ln R + k1) (3.19)

ηBi = −∅ · (ln R + k2). (3.20)

After barter, the two goods are traded and ‘blend’ together and the final ‘econometric’ potential is

Lf = NA · ηAf + NB · ηBf (3.21)

with

ηAf = −∅ · (ln RA + k1) = −∅ · (ln R + ln Y A +k1) (3.22)

ηBf = −∅ · (ln RB + k2) = −∅ · (ln R + ln Y B +k2) (3.23)

where k1 and k2 have the same value in the initial and final states as they are exclusive functions of ∅.Thus, the increase in the ‘econometric’ potential in an open econo-physics system due to barter is

∆Lbarter = Lf − Li = −φ (NA · ln YA + NB · ln YB) = −Nφ (YA · ln YA + YB · ln YB) . (3.24)

Finally, considering ex-post barter in an open econo-physics model, then if we derive (3.24) in order to depreciation(degradation), we end up with

∆ξ barter = −

∂∆Lbarter

∂φ

R,NA,NB

= N (YA · ln YA + YB · ln YB) . (3.25)

As YA and YB are both smaller than 1, we end up with a negative result for (3.25):

∆ξ barter < 0. (3.26)

Result: In our extended econo-physics framework, barter or trade in an open system leads to a decrease in depreciation.One should note that the function of Section 2 simply predicts, in the conventional model, that barter or trade in the

Edgeworth [1,8] box leads to socioeconomic welfare improvement. Thus, the result χ >0 of Section 2 is also positive for ourecono-physics model even without introducing prices, because trade in our econo-physics model increases socioeconomicvalue.

One can illustrate the real behaviour at the market from the two goods (A = corn; B = wheat): on the Edgeworth boxof Section 2, we have shown that in the conventional economics approach there is an increase in social net welfare due topure barter or trade (see Fig. 1), as in (2.3). In our econo-physics pure barter (EPB) model, which has N ′ agents and N goods(N ′

× N model) one can give a practical example as follows:If population increases (e.g., due to immigration), the productive capacity in equilibrium will increase for the same

resistance: N ′

2 = 2N ′

1 and N2 = N1 = 2 (two goods, corn and wheat); then, φ = 2R.Thus, the classical idea that scale in markets tends to compensate costs is here replicated. Therefore, we have constant

returns to scale.Another important idea is the fact that increasing the number of goods from, for instance, two real goods to four

goods (corn, wheat, barley and oat) leads to a decrease in the productive capacity, which identifies the opportunity costof producing more variety of goods.

If we have N2 = 4 and keeping N1 = 2, then we end up with: φ = (N ′

1 · R)/4, leading to a decline in the productivecapacity, which pinpoints the opportunity cost of producing more variety of goods.

Therefore, the model replicates conventional and standard economic theory with a simpler framework.

4. Discussion and conclusions

Summarising:Pure trade or barter between at least two goods and two agents leads to a quantitative decline in depreciation rate, thus

to an economic welfare gain. In addition, this result was generalised in Section 3 of the model for N agents and two goods.This result is in line with economic theory but offers a new and simpler approach, from a physical point of view, which

can yield an improved framework for scientific debate around the issue of valuation of goods, and even pure trade of goods.Our quantitative and quite objective model does not depend on measuring empirical or aprioristic parameters to producethe well-known outcomes of real economy leading to concrete applications.

In economics, real markets are as markets in general for goods without monetary transactions or in goods wheremonetary transactions occur but of which money growth is deflated. Thus, our approach is valid for all the real markets,that is without having growing inflation, generated by that hypothesis.

If we refer to concrete or rather physical (other idea of real) goods, we can think of trading fruits, cattle, agricultureproduction and even some industrial goods. Even though for services the idea would be a bit more farfetched, eventuallywe can think of trading services by in-between vouchers; but this could again lead to monetising the economy but again ifwe discount by the growth of vouchers’ loss of value we would end up again with our viable model.

274 R.F. Marmont Lobo, M.R. de Sousa / Physica A 395 (2014) 269–274

Despite our approach being limited by our hypothesis framework, it reveals potential for further developmentsnamely, the discussion around the idea of perfect markets and the relationship between the macroscopic scale model andmicroscopic economic variables.

Economic intuition has shown, for centuries, that barter or trade leads to an economic improvement.Wehave shownwitha very simplistic physicsmodel and in a framework of an open-ended economic system that, on a first rough approximation,and even without introducing prices, we can have a decrease of depreciation due to pure barter or trade. This result holdsfor a macroscopic scale, which means, in economic terms, it has the potential to be applied in macroeconomics for groupsof countries or even households.

It is well known in economics that the work of Stiglitz, Spence and Akerlof has severely contested the idea of perfectinformation and perfect markets. Our model could eventually be extended to imperfect information or incompleteness.However, more development on this should come from the microscopic scale.

Our EPB model is simpler and intuitive, as opposed to some economic approaches, namely the standard general equi-librium stochastic dynamic models based upon the work of Stokey et al. [12] and Ljunqvist and Sargent [11], which usesimulation methods to obtain more elaborate results. Nevertheless, our approach is an innovation because it allows us toqualify the trend in depreciation, and without using prices, one can conclude that pure barter increases social welfare.

As it is known in economics, the choices of consumers are purely ordinal, and not cardinal. Nevertheless, our modelpresents the idea that just because consumers trade, they are left better in the ex-post trade, and so wemeasure a decline indepreciation, when we introduce production (at least energy to convert production in consumption). Thus, as in economics,production is made cardinal, which is measured accordingly. This interaction between consumption and production in ourmodel is captured by a decline in depreciation rate. Thus, this is our new approach to an economic result.

As any scientific model, ours is limited by the initial axiomatic hypothesis. Nevertheless, this is one of the main noveltiesof the article; even though limited by the idea of working with two goods/two agents; the general case can be furtheranalysed in other extensions. The most important fact is that we find a decrease in depreciation just because we have twoagents trading two goods.

This is new in terms of the physics result, and reinforces the idea in economics that trade leads to social welfare increase.Therefore, we have a new physics approach to a well-known fundamental result in economics.

Thus, as Occam’s razor pinpoints, we can have two competing explanations for the same result, but the simplest one tendsto become better.We believe that ourmacro-physics approach to economics presents a newand simpler formal background.

This model is a starting milestone for further developments already under focus connecting the measurablemacroeconomic parameters introduced here and a microscopic statistical point of view.

References

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