Available online at www.sciencedirect.com

SciVerse ScienceDirect

Procedía - Social and Behavioral Sciences 55 (2012) 445 - 450

INTE -2012 Prague

International Conference on New Horizons in Education 2012 Prague, Czech Republic, 5-7 June 2012

Analytical Theory ofMonotone Commodity State Development with Inflexion

Tomás R. Zeithamer*

University of Economics, Faculty of Informatics and Statistics, Department of Mathematics, Ekonomická 957, 148 00 Prague 4, Czech Republic

Abstract

A method of modelling commodity depreciation, based on the methodology of theoretical physics, is used to derive a deterministic linear motion equation of the second order to describe the degressive and progressive development of the instantaneous relative depreciation and price of a commodity over time in a model of market structure with perfect competition. The same approach is used to derive a non-linear motion equation of the second order for instantaneous relative depreciation with degressive/progressive development over time.

©2012PublishedbyElsevierLtd.Selectionand/orpeer-review under responsibilityofTheAssociationofScience, EducationandTechnology

Keywords: Depreciation, differential equation, econophysics, equation of motion, perfect competition.

* Corresponding author. Tel.: +420 224 094 236; fax: +420 224 094 202. E-mail address: zeith@vse.cz.

1877-0428 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of The Association of Science, Education and Technology doi:10.1016/j.sbspro.2012.09.523

1. Introduction

Let us assume that instantaneous commodity depreciation w at every time t throughout the entire lifetime of the commodity is composed of the instantaneous commodity physical depreciation wPD and the instantaneous commodity external depreciation wED. The law of internal composition of both types of depreciation is designated as A, so the instantaneous commodity depreciation wisw = wPD A wED. The law of composition of magnitudes of instantaneous commodity physical and external depreciation is also designated by the symbol A, so that w(t) =wPD(t) A wED(t). In further considerations we presume (in linear approximation) that the law of composition of magnitudes of instantaneous commodity physical and external depreciation is algebraic addition, thus w(t) = wPD{t) + wED(t), so that the law of internal composition A of instantaneous commodity physical and external depreciation is designated „+" and so w = wPDA wED = wPD+wED. For simplicity, we assume that the law of internal composition A, or law of composition of magnitudes A respectively, does not change over time for both kinds of depreciation. We further assume in the linear approximation that the instantaneous commodity depreciation w, the instantaneous commodity physical depreciation wPD and the instantaneous commodity external depreciation wED are continuous real functions at interval (t0,te) (t0 is the initial time of monitoring of the instantaneous commodity state and te is the time at which we cease monitoring the instantaneous commodity state i.e. level of the instantaneous commodity depreciation). The instantaneous commodity physical depreciation wPD is defined as the permanent adverse change in the surface or dimensions of bodies of various states, induced by the interaction of functional surfaces or a functional surface and medium which causes wear (Posta, Vesely, & Dvorak, 2002). The instantaneous commodity external depreciation wED is defined as a supplement to the instantaneous commodity physical depreciation i.e. the instantaneous commodity external depreciation is the permanent adverse or favorable change in market value of a commodity, which is not caused by instantaneous commodity physical depreciation (damage).

In a market structure with perfect competition , the instantaneous commodity relative depreciation RD is defined by the magnitudes of instantaneous commodity relative depreciation in accordance with relation (Drozen, 2008; Zeithamer, 2011 b)

v J W(t0) 1 y

where w(t0) = w0 is the magnitude of instantaneous commodity depreciation at the initial time t0 and w(t) is the magnitude of instantaneous commodity depreciation at time t(t > t0). In addition to instantaneous commodity relative depreciation RD, the instantaneous commodity relative price RP is also defined under the condition of perfect competition by the magnitudes RP(t) at time t in accordance with the relationship (Drozen, 2008; Zeithamer, 2011b)

RP(t) = p(to)"pW (2)

v J p(t0) 1 y

where p(t0) = Po is the magnitude of instantaneous commodity price p at the initial time t0 of

monitoring the instantaneous commodity price on a select model market and p(t) is the magnitude

of instantaneous commodity price attime t > t0.

In the model of a market structure with perfect competition we assume the following conditions are met: a) in each market there are a large number of buyers and sellers, none of which are strong enough to influence the price or output of a sector; b) all goods are homogeneous; c) there is free entry to and exit from all markets; d) all manufacturers and consumers have perfect information about prices and quantities traded on the market; e) companies attempt to maximize profit and consumers attempt to maximize utility; f) companies have free access to information about technologies (Goodwin, Nelson, Ackerman & Weisskopf, 2009; Nicholson & Snyder, 2008).

2. Linear motion equation of commodity state without inflexion

Instantaneous commodity depreciation w is a real composite function of time, i.e. w(t) = w(p(t)), where w(p) is the continuous decreasing real function of instantaneous commodity price p and instantaneous commodity price p is a continuous decreasing real function of time t. If we monitor the development of instantaneous commodity depreciation at time interval (t0, te), then for the first derivation of functions w(p) and p(t) it holds that ^ (p) < 0 for p £ <p(te), p(t0)) and ^r (t) < 0 for t £ (t0, te).

It directly follows from these relationships that for the interval (t0,te), ^ (t) = (p(0)- (0 > 0-

This means that instantaneous commodity depreciation w is a continuous increasing real function of time t, which corresponds to trends for common commodities over time. Then, instantaneous commodity relative depreciation RD is also a continuous real function at interval (t0,te) and ~j~(t') > 0 f°r every time t £ (t0, te).

The magnitude of instantaneous commodity relative depreciation RD over time t increases with acceleration and the acceleration of instantaneous commodity relative depreciation increases in direct proportion to the instantaneous speed of change of instantaneous commodity relative depreciation at time t. The motion equation of instantaneous commodity relative depreciation is thus (Zeithamer, 2011 b)

^co-B^rn. ((

where B is the constant of proportionality, B > 0. In addition, let initial conditions be met where RD(t0) = RD0 > 0, (t0) = v0 > 0, so that the solution of differential equation (3) at interval (t0, te) is then

RD(t) = RD0-^ + (4)

From here it directly follows that instantaneous commodity relative depreciation RD is a purely convex function at interval (t0,te). This means that the increase in instantaneous commodity relative depreciation at interval (t0, te) is progressive.

Instantaneous commodity relative depreciation RD increases with acceleration at time t again and the acceleration of instantaneous commodity relative depreciation increases in direct proportion to the speed of change of relative depreciation at time t while the constant of proportionality is negative. The motion equation of instantaneous commodity relative depreciation is then (Zeithamer, 2011 a; Zeithamer, 2011 b)

^« = -0«. (( where (—B) is the constant of proportionality, B > O.In addition, let initial conditions be met where RD(t0) = RD0 : interval (t0, te) is then

where RD(t0) = RD0 > 0, ^^ (t0) = v0 > 0, so that the solution of the differential equation (5) at

RD(t) = RD0+^ -^e-^-^. (6)

From here it directly follows that instantaneous commodity relative depreciation RD is a purely concave function at interval (t0,te). This means that the increase in instantaneous commodity relative depreciation at interval (t0, te) is degressive. The progressive increase of instantaneous commodity relative depreciation is characteristic, for example, of certain types of food goods, while degressive increase of relative depreciation may be seen in certain commodities in the automotive industry.

Specific types of commodities are not listed here as the breakdown of all commodities under the condition of perfect competition into individual disjoint classes of commodities is the subject of a separate investigation.

Motion equations (3) and (5) for instantaneous commodity relative depreciation RD yield a deterministic differential equation for instantaneous commodity price p while a commodity is an element of one of the disjoint classes of the set of all commodities. For each commodity class found, it will be necessary to determine the functional relationship between instantaneous commodity depreciation w and the instantaneous commodity price p at interval (t0,te). Assume that we have selected a single specific class of commodity from the set of all commodities. For each commodity of this particular class

let w(t) = D(p(t0) - p(t)), so that, in accordance with equation (1), RD(t) = [0(P°-P(t))~w°] at interval

(t0, te). A constant D (D > 0) is given in such units to ensure that the same units are found on both sides of the equation w(t) = D(p(t0) — p(t)). Directly following from deterministic differential equation (3) for instantaneous commodity relative depreciation RD is the deterministic differential equation for

instantaneous commodity price p at interval (t0, te) , which is (t) = B ^ (t) with initial conditions

p(t„) = Po > 0> ^ (t0) = 7"o < 0> where (0 < 0 for t £ (t0, te). The solution of this differential equation for a purely concave drop in the instantaneous commodity price may be written as p(t) = p0 (1 - eB^t~t°')). Deterministic differential equation (5) for instantaneous commodity relative depreciation RD yields a deterministic differential equation for instantaneous commodity price p at interval (t0, te) which is ^f (t) = —B ^ (t) with initial conditions p(t0) = Po > 0> ^¡T (fo) = ro < 0> where ^ (t) < 0 for t £ (t0, te). The solution of this differential equation for a purely convex drop in the instantaneous commodity price may be written as p(t) = p0 + ^ (1- e~B^t~t°')).

3. Non-linear motion equation of commodity state with inflexion

In this section of our work we again presume the following conditions to be met: (1) the commodity is on one of the markets of the model of market structure with perfect competition at initial time t0; (2) at time t0 the commodity is found in its initial state, which is uniquely determined by the magnitude of instantaneouscommodity depreciation w(t0) = w0.

Let the acceleration of of the instantaneous commodity relative decreciation be the sum of two components, i.e.

d2RD _ id2RD\ , (d2RD\

dt2 ~ I~dtr)1 I dt2 )2'

The first component of acceleration is a consequence of physical and chemical processes (including also social/psychological processes in physico-chemical approximation), which cause the first component of the instantaneous acceleration to increase in direct proportion to the magnitudes of rate of change of the instantaneous commodity relative depreciation, i.e.

GUto),-»™®- ®

where B is the proportionality constant, B > 0 and t £ (t0,te). The second component of acceleration results from physical and chemical processes (including also social/psychological processes in physico-chemical approximation), which cause the second component of the instantaneous acceleration to be directly proportional to the product of the magnitude of rate of change of the instantaneous commodity relative depreciation (t) and the magnitude of instantaneous commodity relative depreciation RD(t), while the proportionality constant is negative, thus

where (—.A) is the proportionality constant, A > 0, t £ (t0, te).

By substituting relations (8) and (9) into equation (7), we obtain the following motion equation for the acceleration of instantaneous commodity relative depreciation

= (io)

where A > 0,B>0,te (t0, te).

One of the subsets of the set of solutions for motion equation (10) is given by

1+eVD(t+C2) '

where for constants D,y1,y2,C2 it follows that D =B¿+2AC1 , yí =- , y2 =

2 i t ¿y -. _ b+VD __ _ B-VD

A ' J ' A

0 < \y2 \ < yi.y2 <0. _< < = (l^^l) _ ip• va^ue instantaneous

commodity relative depreciation is zero. The given subset of the solutions of motion equation (10) shows the progressive - degressive increase of instantaneous commodity relative depreciation with an inflexion point at time t = —C2 and a limit at limt^+œ RD(t) = y±.

4. Conclusions

Assuming that the market value of the commodity at time t is fully determined exclusively by the value of the instantaneous commodity price p(t), methodological procedures taken from theoretical physics were used to construct motion equations for instantaneous commodity relative depreciation RD. Motion equations (3) and (5) for the progressive and degressive increase of instantaneous commodity relative depreciation are linear differential equations of the second order with constant coefficients assuming market structures with perfect competition. Motion equation (10) of instantaneous commodity relative depreciation for the progressive/degressive growth of depreciation is a non-linear differential equation of the second order with constant coefficients. Motion equation (10) was also derived for instantaneous commodity relative depreciation on a market with perfect competition. In the solutions set for motion equation (10), there is the subset of solutions which model progressive/degressive growth of the magnitudes of instantaneous commodity relative depreciation with a single inflexion point.

Acknowledgement

The author is grateful to Mrs. Pavla Jara and the National Technical Library for their great effort and excellent work, which was indispensable in the completion of a large portion of this work. This paper is dedicated to Mrs. Vera Ruml - Zeithamer and Mrs. Anna Ruml and Mr. Frantisek Ruml.

References

Drozen, F. (2008). Modelling of price dynamics and appreciation. Ekonomicky casopis (Journal of Economics), Vol. 56, (No. 10), pp. 1033-1044, ISSN 0013-3055.

Goodwin, N., Nelson, J. A., Ackerman, F., & Weisskopf, T. (2009). Microeconomics in context. 2nd ed., Armonk, New York: M. E. Sharpe, Inc., ISBN 978-0-7656-2301-0.

Nicholson, W., & Snyder, Ch. (2008). Microeconomic Theory-Basic principles and extensions, 10th ed., South - Western College Pub., ISBN 978-0324-42162-0.

Posta, J., Vesely, P., & Dvorak, M. (2002). Degradace strojnich soucasti, (Degradation of machine parts). 1st ed., Praha: CZU, ISBN 80-213-0967-9 (in Czech).

Zeithamer, R. T. (2011 a). The Approach of Physics to Economic Phenomena. 10th International conference Aplimat 2011, Bratislava-. Proceedings, pp. 1303 - 1308. ISBN 978-80-89313-51-8.

Zeithamer, R., T. (2011 b). On the Possibility of Econophysical Approach to Commodity Valuation Theory. 7. Konference o matematice a fyzice na vysokych skolach technickych, Proceedings, Brno, pp. 142 - 150, Brno, ISBN 978-80-7231-816-2.