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The evolution and homogeneity of economies. (Attractors among economies) M.Caputo Texas A&M University, College Station 77843, Texas. The study of the evolution of economies is of interest particularly concerning the homogeneity of economies of the European Union (EU). Since the economies of the EU are non homogeneous because of different cultures, labour market, business intelligence, fiscal policy and administration, standard living, governance size, administrative structures and traditions is natural to assume that the evolution of these economies occurs with a different pace specially because to the different pace of their bureaucratic structures. This is generally considered as an internal friction which causes delays in the important processes of planning, of loaning, of decision making and of acting. This phenomenon has been already been considered in the economic literature and taken into account by introducing memory formalisms in the governing equation used in macroeconomy (Caputo and Kolari 2001, Caputo and Di Giorgio 2006). Concerning the economy Demaria stated : “a dynamic economy without memory is unthinkable” (Demaria 1978), Galbraith, mostly referring to economic crises, added: ”the number of people restrained by memory is bound to decrease in time” (Galbraith 1962). There could be more than one way to represent the memory in mathematical form in the various field of science, the form used in this note, as well in all the works quoted below and found in many treatises (the most recent and readily available are e.g. Podlubny 1999, Kilbas and Marzan 2005, Magin 2006, Mainardi 2010) is

(1)

called, perhaps improperly, fractional derivative. We consider a model for the evolutions of m > 2 economies yi(t) where we assume that their interaction is based on the differences of the values of their evolution status. Since the economies have structures which cause delays, we introduce in the equations a mathematical memory formalism represented by a derivative of fractional order which leads to a system of integro-differential equations.

(2)

where we assume that each economy is affected by a different memory, so the coefficients vi are different. We will prove that the asymptotic behaviour of the economies tends toward the economy which has the memory represented by a fractional derivative of order smaller than the others which acts as an attractor on the other economies. In order to study the solution of system (2’) we take its Laplace transform (LT) which is

(3)

where Yi is the LT of yi and p is the LT variable, and rewrite the system (3) as

or

2

(4)

Multiplying both members by p equation (4) is

(5)

It is clear that each diagonal element of the square matrix of the homogeneous system is the opposite of the sum of the other elements of its row. With application of the extreme value theorem (EVT) we obtain from equation (5)

(6)

Considering the homogeneous system (6) it is verified that = C = constant (i = 1, 2,3 …..m) is the solution. In other words the asymptotic values of yi(t) are all equal. In the following we will show that if all vi are different and z is the smallest order of fractional differentiation then yi (∞) = yz(0) for all values of i. That is all economic systems converge to the initial value of the system which is governed by the memory formalism represented by the fractional derivative of the smallest order in the system (2). To prove it we write the system (2) in expanded form

(7)

The incomplete matrix of the system with unknowns Yi is

(8)

The term independent from p of the determinant of (8) is obtained setting p = 0, it is given by the determinant of the following matrix

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(9)

We recall that each diagonal term of the matrix (8) is the opposite of the sum of the other term of the row. The determinant of (9) is then nil as is readily verified adding to the first column all the others which transforms the first column in a column of zeros. The determinant of (8), for instance obtained by development of the minors of the first row, gives a polynomial of rational powers of p where is missing the term independent of p and appears a power of p with the smallest value of the vi. Let z be that exponent which may be factorized in the determinant polynomial. Then for very small values of p we have N = Cpz where C is the determinant obtained from the incomplete matrix obtained eliminating the row and the column crossing in pz. We consider now the solutions of the system (7) obtained using Kramer’s method Yj = Mi,j / N (10) where Mij is the determinant of the following matrix (11) obtained from incomplete matrix (8) substituting with the column of the known terms of the system (7) with the column of the matrix where is Yj and N is the determinant of the incomplete matrix (8) (il cui primo termine é e la colonna diventa quella centrale nelle seguente matrice)

(11)

Then we extract the factor 1/p from the matrix (11) obtaining

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p-1 (12)

Since z is the smallest value of vi expanding the determinant of the matrix (13) using the elements of the columns which contains only p and considering the values p <<1 we find Yz (p)= K y0z p-1+z/ Npz = K y0z/ pC, (13) where K is the determinant of the matrix obtained eliminating in the matrix (12) the column and the row crossing in pz. Then we apply to Yz the EVT by letting p => 0 finding (14)

Calculating the determinant of the matrix (11) by expanding with the minors of the j column and letting p => 0 we obtain the same determinant factor of y0z of the matrix (8) then

Finally since all asymptotic values are equal are equal we may conclude that they converge

to y0z . It is the economic entity with longer memory to dominate the pathway of all economic entities to the common asymptotic value. This property of a set of different economies seems acceptable also from the intuitive point of view: the economy less sensitive to the presence of the other economies could be the surviving one, the low sensitivity being represented by the long memory of its past. Obviously this will not happen in reality since those responsible or the economies have the opportunity to adjust the course of their economy acting on the appropriate structures in order to avoid the possible extreme event. This model mimics mathematically what Montesano (2013) defines what we understand as Pantaleoni’s first type of evolution which has as reference the static equilibrium of that economy which here has the longest memory or the slowest pace of evolution. 4. The problem of the homogeneity of the economies. The problem of the evolution of the economies of the European Union (EU) was considered by Caputo (2012) who showed that in a model based on a single memory formalisms the economies of a generic set would converge to single type of economy and when the economies are 2 they would converge to state of that with slowest pace of evolution. In the same note (Caputo 2012), is studied the homogeneity of 3 sets of economies of the EU and their per capita GDP indicated with y

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a) The set of 10 East European countries Bulgaria, Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Slovakia and Slovenia for which was estimate their mean per capita GDP (y) = 18.623 ( σ/y = 0.27). b) The set of 14 European countries Austria, Belgium, Cyprus, France, Germany, Greece, Ireland, Italy, Malta, The Netherland, Portugal, Slovakia, Slovenia, and Spain for which was estimated y = 30.865 (σ/y = 0.21). c) The set of 6 Western Continental European countries Austria, Belgium, France, Germany, Italy and The Netherland for which was estimated their y = 35.939 (σ/y = 0.11) where σ is the standard deviation of y. The comparison of the σ/y of the 3 sets was made assuming that the more evolved economies are at a later stage of the evolution of their economies and comparing their σ/y with those of the lesser evolved countries assumed in an earlier stage of their evolution. This procedure avoids the comparison of the evolution of an economy in time which, due to the instability of the world economy, could possibly impair the validity of the comparison. The comparisons at the same time, in fact, implies that all the economies considered have been exposed to the same world history. Caputo (2012) concluded that values of σ/y, assumed as measure of the homogeneity of the countries of the set, seem to decrease with the increase of the economic evolution of the countries which could indicate that there is more homogeneity, from the point of view of σ/y, in the groups of European countries with a greater economic evolution and that the economies evolve to a smaller σ/y, or to a greater homogeneity Also that, although the disjunctions between the 3 groups are still remarkable, the y seem to have a limited variation within each group which could indicate some degree of homogeneity in the European countries of each group from the point of view of y. In a subsequent note (Caputo 2013) it was theoretically shown that also in the case of a set of economies with different memories the set would converge to a single type of economy: that with slowest pace of evolution. The econometric study of the homogeneity was concentrated in the set of economies formed by: France, Germany, Italy, Spain and the UK, assuming the method already considered, but assuming as measure of the evolutions of the single country not only the per capita GDP, but a set of 29 parameters obtained from the tables of CIA World factbook . As measure of the disjunction between economies was assumed the classic algebraic distance in the Cartesian orthogonal system of the normalised 29 parameters considered, thus forming a metric space where, however, is not clear the econometric meaning of the triangle inequality. Caputo (2013) concluded that the economies considered may be considered somewhat homogenous. In principle the problem could be attacked with the method of cluster analysis however, also due to the limited number of economies considered, we preferred the way indicated in the work of Caputo (2012). 5. The probabilistic approach. The first approach for estimating the homogeneity of a set of economies was probabilistic. It made first normalising each parameter pj to the maximum value of its norm, obtaining a new set qj, then considering the set of the couples of difference of the parameters normalised and pj is substituted with

(15)

where ≤ 1 and define the differences

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xik,j = qij – qkj (16) Then Caputo (1955) we seek the density distribution of

(17)

The countries considered are A) France, Germany, Italy, Spain, UK and the 29 parameters used are all with positive values and listed in appendix A. It is readily verified that the variables xik,j for each couple i,k have bell-shaped density distribution which however due to the limited number of parameters is not so robust to entitle a reliable fitting to a Gaussian curve. This is instead obtained using the all set of couples i,k shown in the figure 1. The most probable value ε (Caputo 1955) is (18)

where σj is the standard deviation of the sets xik,j defined in equation (18). We first note that the selection of the order of the economies for computing the differences of parameters is immaterial to the results. Then we note that formula (4) gives the sum of the squares of the distances, however from it we may obtain the single distances as reported in the following table 1 Table 1

Table 1. Distances between the economies of the set computed with the probabilistic method in the year 2012. The TOTAL indicates the sum of the distances of the economies relative to the others, below is the relative standard deviation of these distances, RSD is the relative standard deviation. The probabilistic sum of the distances with 29 parameters is 21,06 and the RSD of all distances is 0,016. The sum of the distances from each economic system to the others has average 8,42 and the RSD is only 0,061. 6. The comparison with the geometric method. The probabilistic sum of the distances with 29 parameters is 21,06 and the RSD of all distances is 0,017. The sum of the distances from each economic system to the others has average 8,42 and the RSD is only 0,061 which indicates that the 5 economies may be considered relatively homogeneous. We will estimate the sums with the geometric method. In the theory (Caputo 2012) is measured the convergence simply using the variable y as a measure of the state of the economy, this method is here extended associating to each economy a set of n parameters, such as those usually considered to establish their financial conditions of an economy for instance inflation, unemployment, foreign debt etc., and then consider them as coordinates in an n dimensional space with a Cartesian orthogonal coordinate system. We may then assume the Cartesian distance as the disjunction between

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the economies; which would give a quantitative measure of the inhomogeneity of the economies which, in practice, is an abstract measure. The parameters pij identifying the each economy (in the matrix i indicates the column, the economy, and j indicates the raw) the parameters are normalised according to equation (15). From the definition (16) follows that

(19)

or

(20)

where the numerator of the left hand member of the equation (20) is the distance Dik between the economies i and k. The normalizing factor of Dij is obtained considering first the case when all parameters assume non negative values and that m is even: consider now that if the values of the parameters of a given subset of u < m of the m economies of the set are unity and all the others are zero the sum of all the m(m-1)/2 distances is n0,5u(m-u) whose maximum is obtained when u = m/2 which gives the distance m2 n0,5 /4. If one, or more than one, of the zero value parameters were to assume a positive value the sum of the distances would decrease. The same applies also to the values 1. The case when m is odd is obtained with the same procedure. When are present r parameters which may assume negative values we first remember that all qij subject to the limitation . Then the squared difference of two of them may

be 4 which implies the presence of the factor in formulae (5) taking into account that, having fixed the number of parameters, the presence of a parameter which may assume negative values implies one less parameters which may not assume negative values. It is seen that when all parameters assume non negative values the sum of the distances Dik has the maximum n0,5 m2/ 4 when m is even (21) n0,5(m2 - 1)/4 when m is odd Moreover taking into account the possible presence r parameters which may assume negative values, the values of qij are subject to the limit ≤ 2 and formulae (16) are modified according to U = Dik /[(n + 3r)0,5 m2/ 4] when m is even U = Dik /[(n + 3r)0,5 (m2 - 1)/4] when m is odd (22) where U ϵ [0,1] could tentatively be considered as a relative abstract measure of the inhomogeneity of the set of economies. The homogeneity of the set is then inversely proportional to the value of Dik. Obviously the distances obtained are only abstract tools and so far we may compare the different economies with the understanding that larger values of U imply relevant differences in the economies. The meaning of those measures would specially be in the comparison of the values of U obtained at subsequent times, these values may indicates if the members of the set of

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economies is becoming less or more homogeneous, that, is converge to a unique state when all parameters are theoretically equal. In the following this method is used for studying the homogeneity of 5 of the most advanced European economies: France, Germany, Italy, Spain and the UK and assuming as coordinates the 29 parameters obtained from the tables of World factbook for the year 2012 listed in the appendix, which have positive values. After normalising the data we obtain the results presented in the following table 2 Table 2

Table 2. Distances between the economies of the set computed with the geometric method. The total indicates the sum of the distances of the economy relative to the others. RSD is the relative standard deviation. The geometric sum of the distances with 29 parameters is 21,85 (21,06) which gives U = 0,474 (0,391) and the RSD of all distances is 0,016. The sum of the distances from each economic system to the others has average 8,74 (8.42) and the RSD is only 0,067 which are in excellent agreement with the values obtained with the probabilistic method indicated in brackets. . We note also that the differences between the single distances in tables 1 and 2 are relatively small: of the order of few percent. Moreover the differences of the sums of the distances from each economy to the others are within the respective standard deviations and that the same is valid also for the total distances. In fact the sum of the distances found and shown in table 3 is 21,85 while that obtained with probabilistic method gives 21,06 that is with only 3,6% difference, which, taking into account the difference of the two approaches, could be considered more than encouraging. We finally note that if we change the order of the terms when computing xik,j then each value of the set changes sign, the density distribution becomes symmetric relative to the preceding one thus conserving its presumed Gaussian density distribution and the standard deviation are the same. The probabilistic method, therefore, may be certainly be used as a check but also considered as alternative method to estimate the homogeneity of a sets of economies or the convergence of different sets.

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Figure 1. Density distribution and fitting to a Gaussian curve (squares) of the set formed with all 290 values of xik,j (diamonds) obtained from the sets of 29 positive parameters of the economies of France, Italy, Germany, Spain and UK, defined by formulae (13) and (18).. We may finally note that the density distribution of the xik,j is certainly bell shaped with a a satisfactory chi, which confirms the very good matching of the distances obtained with probabilistic and the geometric methods. 7. The time evolution of EU economies, the geometric approach. In order to explore the evolution of the homogeneity of the EU economies we now consider two sets of economies: A) France, Germany, Italy, Spain the UK B) Bulgaria, Czech Republic, Hungary, Poland, Romania, which, for historic reasons, have different economies and, obviously, should have different economic evolutions and, perhaps, different homogeneity. The study of the evolution and homogeneity of the economies in each set is carried out examining the variations of the state of the 5 sets in time. For this purpose was selected a set of 14 parameters for the years 2010, 2005 and 2010 taken from EUROSTAT and listed in the appendix A to be used as Cartesian orthogonal coordinates in a space with 14 dimensions. The results concerning the 2 sets are shown in the following tables 3 and 4. Table 3. Set A.

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where m is the number of economies of the set, n is the number of parameters used, r is the number of parameters which may assume negative values and U [0,1] is the normalized measure of inhomogeneity. The "sum" is the sum of the distances of each economy from the others of the set, the "SUM" is the sum distances between all the economies of the set. Table 4. Set B.

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2000 Romania Hungary Poland Bulgaria Czech R Romania   1,90   1,59   1,58   1,55  Hungary 1,90     1,93   1,31   1,53  Poland 1,59   1,59     1,66   1,14  Bulgaria 1,58   1,58   1,31     2,14  Czech R. 1,55   1,55   1,66   2,14   Sum 6,62   6,67   6,32   6,69   3,36  rel st d 0,024806   0,044950   0,05189   0,05168   0,06484  0,660 = U n=14 m=5 r=1  SUM 16,34   rel st de 0,01808                       2005         Romania Hungary Poland Bulgaria Czech R Romania   1,7   2,07   1,53   1,84  Hungary 1,7     1,81   1,58   1,18  Poland 2,07   1,81     2,48   1,73  Bulgaria 1,53   1,58   2,48     1,6  Czech R. 1,84   1,18   1,73   1,6    Sum 7,14   6,27   8,09   7,19   6,35  rel st d 0,031989   0,04384   0,041752   0,063415   0,045486  0,653   = U m=5 n=14 r=2  SUM 17,52   rel st d 0,019                       2010         Romania Hungary Poland Bulgaria Czech R Romania   1,83   1,83   1,23   2  Hungary 1,83     1,78   1,11   1,58  Poland 1,83   1,78     1,45 1,36  Bulgaria 1,23   1,11   1,45     1,54  Czech R. 2   1,58   1,36   1,54    Sum 6,89   6,3   6,42   5,33   6,48  rel st d 0,049053   0,052108   0,036563   0,037032   0,04179  U =0,585   m= 5 n=14 r = 2  SUM 15,71   rel st d 0,018      

The results of the table 3 and table 4 with data set A and set B indicate that: 1) The SUM of distances between the economies of set A is first increasing then decreasing while that of set B is always decreasing as it is shown in the figure 2 which indicates that the economies of the set B are steadily proceeding to more homogenization while those of set A are oscillating but at the end in 2010 are more homogeneous than in 2000.. 2) The sum of distances of the single economies of set A from the others in set, at all times, is almost 50% larger than that of set B. 3) The relative standard deviations of the two sets are about of the same order of magnitude of few percent.

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Remembering that the homogeneity of a set is inversely proportional to the average separation in the set we see that in the year 2000 the homogeneity of set B is greater than that of set A, in 2005 (the middle points) the level of homogeneity is inversed relative to 2000 and in 2010 they have the same level of homogeneity. That is the 2 sets, after an oscillating level of homogeneity, seem to converge.

Figure 2. Inhomogeneity U of the two sets of economies analyzed. Squares are for set B and diamonds are for set A. The set A approaches the set B with an oscillation.

Figure 3. Evolution of two economic entities from Caputo (2012). Note that the initial value of one of the economy (diamonds), that with longer memory, is the attractor of the other economy (squares); both economies asymptotically converge to that initial value. The economy attractor is that whose evolution is modelled with the fractional derivative of smaller order. We note that there is some similarity in the figures 3 and 2 and one may be inspired to suggest that the economies of set A be similar to these represented by diamonds in figure 3 and the economies of set B be similar to these represented by squares in figure 2. This could imply that the set A could possibly be modelled with a derivative of order smaller than that to use for the economies of set B. If we were to compare the initial and final values of the inhomogeneity of the two sets, which is what we are concerned, we note that the initial value of the set A is far below that

0,5

0,55

0,6

0,65

0,7

0,75

0,8

2000 2002 2004 2006 2008 2010

U

year

2,4  

2,6  

2,8  

3  

3,2  

3,4  

3,6  

3,8  

4  

-­‐4   -­‐3   -­‐2   -­‐1   0   1   2   3  

level  of  evolution  

log  time  

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of the set B while at the end the values of the two sets are almost coinciding. The abnormal initial grow of the set A in figure 2 is yet unexplained possibly due to exogenous factors, it is also to be noted that the set A in spite of the oscillations is finally an attractor for set B.. 8. Comparison of the results obtained using different methods or different parameters applied to set A. [using 14 parameters in 2010 and the geometric method with the results using 29 parameters in 2012 and the probabilistic method]. For sake of checking the stability of the results when changing the parameterswe or the method of analysis we may now tentatively compare the results obtained for set A using different methods and different parameters. Remark 1. Concerning the method we verify that in the analysis of set A the probabilistic and the geometric method, using the the 29 parameters gave practically tha same results. Concerning the use of different parameters we consider the geometric method analysis of the set A for the year 2010 shown in table 3 made using 14 parameters (listed in appendix A) giving the SUM = 21.85 and U= 0,406 shown in table 3 compared with the analysis for the year 2012 shown in table 1, using 29 parameters (listed in the appendix B) giving SUM = 15,71 and U = 0,351. But the SUM = 21,85 results from 29 parameters and in order to compare it with the corresponding SUM = 14,97 from table 3 for the year 2010 we could normalize it with the factor (14/29)0,5 = 0,695 (which takes into account that the distance generally increases as the square root of the number of parameters used) finding SUM = 15,18, not far from the other values 14.97. However if we compare the corresponding values of U= 0,406 from table 1 and U = 351 from table 3, which takes into account that among the 14 parameters of the year 2010 there are 2 which may assume negative values, then the normalizing factor is (20/29)0,5= 0,830 which gives SUM = 18,14 with a discrepancy with the value 15.71 resulting from the use of 14 parameters. But we may not rule out that this difference be due to the fact that the parameters are related to different years (2010 and 2012) and to the use of different parameters. Analogous results referring to the use of different parameters related to the same years support the following remarks: Remark 2. In the comparison of sets of economies the use of different sets of parameters one may obtain unreliable conclusions. Remark 3. Since the selection of parameters which assume non negative values allows a simpler discussion of the results the use of this type of parameters would be preferable. 8. Conclusions. A possible interpretation of the results of this note is that it may model the bureaucracy, as an energy dissipation internal friction mechanism, as a delaying factor represented by the mathematical memory formalism (Caputo 2012) which eventually takes into account, in an abstract and somewhat imaginative form, also the second principle of thermodynamics, which takes all systems of nature to the same energy level. However more comprehensive interpretations of qualified economitsts would be desirable.

A limit to the method used in this note is represented by the fact that, although it gives an

abstract measure of the disjunction of the economies, calling the attention to the fact that some economies are disjoined from the system were they are imbedded, which, in practice, is a relative comparison, and shows their evolution, it would not directly say about the internal equilibrium of the economies of the set which however is not the scope of the

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present note. In spite of this it may give information identifying the parameters responsible for the inhomogeneity of the economies (Caputo 2013). The results of this note could be of practical interest since those responsible of the economies would have the possibilities to correct the course of the evolution of their economies by changing the structures of the economies and affecting course of the evolution of the economies themselves. The methods used in order to estimate the homogeneity of the economies may need further discussion concerning its consistency and meaning which is postponed to another paper. We must note that the measure used depends on the parameters selected but also on the number of economies in the sets and therefore would rigorously be comparable only the results of the analysis of sets with the same number of economies and parameters or, in a comparison of results obtained with different number of parameters and /or number of economies, one must apply the necessary normalizations. The cause is that changing the parameters may change the values of the normalised parameters. We note that an appropriate level of inhomogeneity could possibly be useful for a healthy evolution of the systems of economies; the problem would then be to investigate which could be the physiological level. We note also that the law expressed by equation (2’) of Caputo (2013) and (2) of Caputo (2012), which mimics the law of attraction of elastic type and the measure of the distances used may lead us to laws of evolution of economies systems which we will consider in another study yet short of possibility of verification and practical use. Finally we cannot restrain ourselves to comment that the theories aiming to equilibrium are still very useful in spite of the fact that, in practice, the equilibrium is not existing. References. Barro R.J, A Cross-Country study of Growth, Savings and Government, NBER Working Paper 2855, 1989. Barro R.J., Economic Growth in a Cross-Country Section of Countries, Quarterly Journal of Economics, 51, 407-443, 1991. Barro, R.J. - X. Sala i Martin, Economic growth. McGraw-Hill, Boston, Mass., 1995. Caputo M., A linear model for population growth (with a tentative test), Atti Accademia Scienze Ferrara, 59, 1-20, 1981. Caputo M., Evolutionary population growth with memory in a limited habitat, Proceedings Meeting Urban Ecosystems, Atti Accad. Naz. Lincei, 182, 883 -916, 2001. Caputo M., Population self - growth with memory, Proceedings Meeting: Phenomena of selfregulation in biological systems, Contribution nr.106 of Centro Linceo interdisciplinare B. Segre, Atti Accad. Naz. Lincei, 113-128, 2002. Caputo M, The convergence of economic developments, Non Linear Dynamics & Econometrics, 16, 2 - 22 , 2012, DOI: 10.1515/1558-3708, 2012. Caputo M., The convergence of economic developments II, (with an econometric approach), presented at IV International Conference on New Trends in Fluid and Solid Models, Vietri, Italy, 4-6 April 2013, in press in Meccanica. Gandolfo G., Economic Dynamics, Study Edition, Macro-Economics, Springer Verlag, Berlin, 2009 (4th edition). Montesano A., Pantaleoni, Pareto e le loro scuole, lecture presented at the meeting: “Gli