Critical value of the total det in view of the dets durations I.A. Molotov, N.A. Ryaova N.V.Pushov Institute of Terrestrial Magnetism, the Ionosphere and Radio Wave Propagation, Russian Academy of Sciences, IZMIRAN, Moscow, Troits 4290, Kaluzhsoe shosse, 4, Russia e-mail: iamolotov@yandex.ru, ryaova@izmiran.ru Yanuary 29, 206 Astract Parastatistic distriution of a total det owed to a large numer of creditors considered in relation to the duration of these dets. The process of det calculation depends on the fractal dimension of economic system in which this process taes place. Two actual variants of these dimensions are investigated. Critical values for these variants are determined. These critical values represent the levels after that orrower anruptcy occurs. The calculation of the critical value is performed y two independent methods: as the point where the entropy of the system reaches its maximum value, and as the point where the chemical potential is zero, which corresponds to the termination of payments on the det. Both methods lead to the same critical value. When the velocity of money circulation decrease, it is found for what dimensions critical det value is increased and for what it is decreased in the case when the velocity of money circulation is increased. Keywords: Critical det; duration; parastatistic distriiution; entropy; chemical potential; velocity of money circulation
Introduction The economic prolem of the distriution of dets (loans) for their duration is discussed. The critical (threshold) value of the total det is calculated. This value descries the oundary of qualitative change in the system defining a reach in economic security. The methods for calculating these critical values were developed y Maslov []. The parallels etween statistical and thermodynamic properties of systems of physical particles and economic systems [2, 3] play an important role in this wor. As in physics, the concept of entropy is also significant in economics. We are referring to Hartley s entropy [4], wherein all possile options for the det calculation are equally proale. Next, we deal with the values of individual dets s ( j,2,... j = ). It is assumed that the numer of dets is large. The value of each det is characterized y its due date, or duration, l. The ratio s l is the main characteristic of the det for the orrower (detor); it determines the effort the orrower has to mae to pay off the det. The distriution of dets is considered with the following limitations in mind: we will not tae into account the interest and the derivatives incurred y the dets, nor consider past due ("toxic") dets. The main purpose of this article is to calculate the value of a critical det. At the end of this paper the use of long and short loans is analyzed. 2 The amount of dets, including durations Let s say the dets have durations l j. We will arrange the dets y duration, in ascending order, where l < l 2 <... < l, and consider the normalized reverse durations l l ( j,2,... j = ). Following the normalization and an aritrary addition of virtual durations for which the value of dets s j = 0, we receive a sequence of normalized reverse durations. Evidently, it is possile to assume such sequence to e integer: 2
l l,..., l = = + j,... =. l l l j Let us consider the total det. It is convenient to deal with dimensionless expressions that do not depend on the choice of monetary units. Let s introduce the value of the arithmetic mean of the dets ( ŝ ), and then consider each det divided y the value of ŝ. Consider the dimensionless det total: s j to e = s j. () j= Similarly, we introduce the dimensionless quantity of money l E = s = + E E = j s l j ( ), j, (2) j= j j= which is required to pay off all dets, taing into account their duration. To determine the value of critical det, it is necessary to choose the appropriate statistics. In physics, specifies the numer of identical particles on a single quantum level. In contrast to Bose-Einstein statistics, where =, and Fermi-Dirac statistics, where =, parastatistics is characterized y a finite value of, which can e quite large and variale. As the numer of creditors (and dets) during det settlement can vary, the distriution of dets in relation to their durations should e considered a parastatistic distriution of identical elements (see [5]). The mathematical justification of parastatistic formulas was performed in [6]. When analyzing the physical systems it is important to indicate the fractal dimension of the system under consideration [7]. The economic system, in which det calculation taes place, also has a certain dimension from which the velocity of money circulation is dependent. However, the calculation of the dimension of the economic system presents certain difficulties. Information on dimensions of the currency time series gives a certain ideas aout dimensions of economic systems. The Hurst method [7, ch.8] permits to estimate dimensions of these series. For advanced countries with the stale currency the dimensions of currency series is found to e fractional with range in vicinity of d =.5 [8,9]. 3
The case, when the dimension of economic system is equal to d = 2, (3) is the simplest for analysis. Examples of currency series show that variants of the fractional dimensions < d < 2 are of interest. Therefore we egin from detail examination of the case (3). Then we will consider the variant < d 2 δ, 0 < δ <<. (4) The investigation of variant (4) rings to results of dets calculation which essentially differs from results for the case (3). Another values d > 2 (including fractional d ) are investigated analogously as in the case (3). Thus, we suppose that d = 2. It follows from parastatistic distriution that []: =, (5) j= exp( ( j + )) exp( ( j + )) E = j. (6) j= exp( ( j + )) exp( ( j + )) Here and are positive parastatistic parameters defined y (5) and (6). Parameter refers to the inverse velocity of money circulation in the system (see [, 3]). Parameter taes into account possile changes in the numer of the dets during settlement: in thermodynamics it corresponds to the chemical potential of the opposite sign. The volatility of the numer of dets maes it necessary to consider the overall parastatistic distriution. 3 Asymptotic relations Let us assume further that the quantities, E and are large, and the dets s j are of the same order. Then, the constant will e small [, 0]. Our next goal is deriving an asymptotic relationship etween the quantities E,,,, and. The ultimate goal 4
is to determine the critical values of 0 and E 0 for the total det and for the total money required to pay it off over duration. B The equation we see depends sustantially on the multiplication results of = and. It can e shown that with a small value of B, 0 is of no relevance. Therefore, we will further assume that this product is either greater than, or close to unity: B (7) Considering the ehavior of the terms of series (5), we can see that with an increase j, the terms in the series decrease when j + and do not contriute significantly to the value of the sum. This fact justifies replacing the difference ( ) exp j + (8) y the first term ( j + ) of the series expansion. The condition (7) indicates that we consider periods of time during which money turnover in the system is of the same order as the value of the total det, or lower. The condition (7) and the approximation of the difference (8) give us the opportunity to transform the sums (5) and (6). As a result, we get: = ( B( j+ )) e B ( j + ). ( B( j+ )) e ( j + ) (9) j= We will sum it up y means of the Euler Maclaurin formula, and then divide oth the numerator and the denominator of the integrand y exp B ( x + ) : B( x+ ) e B ( x + ) dx = B( x+ ) e ( x + ) = B( x+ ) B( x+ B ) ( x + ) e e B ( x+ e ) dx. x + (0) 5
Considering that, in view of (7), the value of exp B ( x + ), we can get rid (0) of the denominator: exp B x + ( ) ( ) + exp B x + After deploying () and replacing the sum x integration, we otain: + is small compared to. () + with a new variale of Bx 2Bx dx B Bx e Bx e ln e. + x (2) Similarly to (2): ( x ) Bx 2Bx E Bx e Bx e dx x + + Bx Bx e dx. (3) The evaluation of integrals in leading-order terms yields B B E = e e. B Together with equations (2) and (2), the last expression determines the amount necessary to repay the dets: B E = ( + + ) ln + e + e B. (4) Equation (2) in leading-order terms yields the equation B B ln + e (5) etween B, и, where the second term on the right serves for corrective purposes. Equation (5) simplifies the expression for the required amount: 6
B E = ( + + ) B + e. (6) 4 Entropy for d=2 Lie in physics, the entropy in economics is defined y a logarithm of the numer of possile implementation options in the system under study. Let s proceed to the computation of the entropy S. The point of maximum entropy viewed as a function of determines the critical value of 0, see []. The calculation of entropy can e performed y taing into account the properties of parastatistic distriution, as it was done in [0]. Instead of this cumersome way, we will use the analogy of the physical and economic variales and determine S y means of the formula E S = K ( ), (7) which was estalished in [0, ] for physical systems with fractal dimension d 2. Here, K is the proportionality factor, which depends only on the system s dimension. It is shown for the dimension value d = 2 that K = 2 6, (8) see [0]. Another independent method of calculating 0 is presented in the next section. The coincidence of these two calculations justifies the use of formula (7) in econophysics and thus provides an additional confirmation for the analogy aove. Let s turn to calculating the entropy on the asis of equations (6) and (7). We determine the derivative E ( ) E ( V ) = ( V = - velocity of money circulation in the system). This derivative is calculated for fixed and, accounting for the dependence of on V. This dependence is determined y (5), from which we can otain that Let us rewrite (6) in this form: V V ( V ) ln B. (9) 7
E = ( + + ) V + V exp V. (20) Differentiating (20) with respect to V in view of the dependence ( V ), we find that ( ) E = + B + e + B B + B V 2 B 2 2 2. (2) As a result, according to (7), (8), and after discarding lower-order terms, we otain the following expression for the entropy E S = 2 6 2 6 + B e e B V 2 B B 2 2. (22) The last step of calculating the critical value of 0 is setting to zero the derivative S (or, equivalently, S B ). The differentiation is performed for a fixed, ut with consideration for the dependence of on B, ( B) =. Assuming the derivative S B equals zero results in 2 2 B e B = + 2. Transforming this relation y means of equation (5) in the higher-order, we find that Thus, the maximum entropy S is reached at 0 = ln. (23) ln = = 0 V ln. (24) Here V means the velocity of money circulation in the system. If the velocity of money circulation is unnown, then y using the formulas (2) and (4) one can express the critical value of 0 via det parameters of E and : 0 ln = ( E ). (25) Equation (25) allows to express the velocity of money circulation in the system in an explicit form: 8
E V = =. (26) 5 Another method to calculate the critical value of 0 The idea of another method of calculating the critical value of 0 is suggested in []. The critical value of the total det corresponds to a moment of det settlement, in which the numer of dets is no longer changing. At this point of calculation the value of equals zero. Taing into account expression (2), one can determine that = 0 Bx 2Bx dx Bx e O ( e ), (27) x = + where B = B0 = 0. By means of (27) and (5) we otain 0 = ln O + 2, (28) which is consistent with formula (23). The matching results otained from formulas (23) and (28) justify oth the use of formula (7) and the applicaility of the concept of entropy as a whole in economic prolems. In (28), parameter does not depend on a large numer - the numer of creditors. V = is small, ut it Thus, the value of the critical det in the case of dimension (3) for a large numer of creditors is determined y the expressions (23) or (28). Formulas (23) and (28) show that when > 0, the orrower is forced to declare anruptcy, ut when < 0, the orrower is ale to repay the det. 6 Changes in a det s parameters during the repayment process Let s consider the det parameters at a critical point. All parameters at that point are tagged with a zero in formulae elow. It has een already noted that the critical det 9
total 0 is determined via formulas (23) and (28). It follows from these formulas that the critical numer of dets 0 is determined y ln The critical value of the chemical potential is 0 = 0. (29) V 0 0 = 0. (30) At the critical point equation (5) transforms into the following equation: 0 0 0 = V0 ( ln 0 ln0 ) V0 exp V0. (3) A joint consideration of (29) and (3) shows that these equations can e satisfied only with simultaneous approach to zero at the critical point of the parameters V and ln. We determine that when approaching the critical point, the three values, V, and ln tend toward zero. Equality V 0 = 0 means that money circulation related to paying off the dets stops at the critical point, which is consistent with the equality (30). As a result, in the process of repayment, the values V and decrease to zero, and the det total and the numer of creditors decrease to 0 and 0, respectively. 7 Critical value of the total det for dimension (4) Now, we start to investigate of the total det dynamics in the case of dimension, which satisfies to condition (4). The parastatistic equation instead of (5) now has following form where α = d 2., (32) = α α ( j ) ( j ) j j + + = e e We performed a set of transformations which are similar to transformations at the case d = 2. With neglect of small terms we have: 0
+ B ( x ) exp( B ( x ) ) dx α + ( +, ) α = + + x x and as efore B. form: As aove, to determine critical det 0 we write the value for = 0 in the α α 0 = ( B0 x exp( B0 x) ) dx ( 0 02 ) = +, (33) 2 α x B 0 = 0. The principal term in (33) is equal The correction term in leading order is = α α. 0 and exp( ) α B = exp 2 2 α 02 0 α 0 exp O α = + O α α + α. As a result, instead of formula (24) we have:. 0 α V (34) α The factors in the right hand of (34) are not independent. The decreasing of the dimension corresponds to the increasing of the velocity V [2]. The quantities α and V are connected y the low of energy conservation ( α ) + E = f V α, (35)
where function f ( α ) α 2 ( α ) ζ ( α ) = Γ + ( Γ is the gamma-function, ζ is the Riemann zeta function) monotonically increases for all α <.7 [0,2]. It is follows from here that the factor V + α monotonically decreases in spite of index + α α growth. The function V ( α ) is decreasing even more. Meanwhile the factor α (34) within the interval (4) is quicly increasing (from to ). Therefore expression 0 in (34) grows together with α, ut is diminishing together with V. Thus, the dependence of 0 on V in the case (4) is radically different from this dependence in the case (3). Simultaneously the dependence of 0 on the numer of creditors is found to e inessential in the case (4). in 8 Short and long dets Suppose we have m short dets s with the duration L and n long dets 2 L duration 2 the dets s and 2 s with the >> L. Oviously, m + n =, sˆ = ms + ns2. As efore, we assume s divided y ŝ. Now, the dimensionless det total is: and the dimensionless total payoff value = m s + n s, (36) 2 L E = m s + n s. (37) 2 2 L We assume that the values of the dets s and s 2 are of the same order and the numers m and n are finite and are also of the same order. On the asis of formulas (23) and (28) one can determine that the critical det is 0 = V ln ( m + n). (38) 2
Let s consider the cases of a dominance either in long or short-term dets (loans). The case n (3) are applicale. >> m does not yield fundamentally new results: the general formulas (28) - More interesting is the case m using (26), (36) and (37) one can find that >> n of the predominance of short-term dets. By V L s 2, (39) L L 2 0 s ln m. (40) L We can see that in this case the turnover rate is high; also high is the critical value of 0, which in turn means an increase in orrowing capacity. While the enefits of this case are ovious, the formula (40) provides a quantitative estimate of these enefits. 7 Conclusion The critical values 0 of the total det are calculated for a large numer of creditors and in view of dets duration. Thus, it is otained the threshold after that anrupt of orrower occurs. The conducted analysis assumes that totality of dets is suordinated to parastatistic distriution. The form of the parastatistic relations depends on the fractal dimension of economic system in which det calculation tae place. Two most actual cases of dimensions are investigated: case (3) and case (4). The necessity of the separate study of these cases is determined y the fact that oundary dimension d = 2 is a singular λ -point for parastatistic system [5,0,3]. This singularity determines specific ehavior of the most important thermodynamic characteristic: the entropy in λ -point has point of inflection, the second derivative of chemical potential with respect to the velocity of money circulation (to the temperature) has jump in this point. 3
The critical values calculation are realized here y two ways: as the point of the entropy maximum value and as the point zero for the chemical potential. Both ways give one and the same critical value. The dimension variants (3) and (4) lead to essentially different results. In the case (3) formulas (23) or (28) are otained for critical values. In this case critical measurement 0 linearly depend on the velocity of money circulation in the system (which identically connected y the law (35) with dimension) and logarithmically related with the numer of creditors. In the case (4) formula (34) is otained. Taing in an account relation (35) it estalished that value 0 increase with α and decrease with V. The dependence on the numer of creditors is inessential. The coincidence of formulas (23) and (28), otained y independent methods, proves the use the concept of entropy and chemical potential in economic prolem. Also it is shown that the use of short loans permits to increase the critical value 0 and thus gain orrowing capacity. 4
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