Actual Problems of
Economics and Law




DOI: 10.21202/1993-047X.10.2016.4.77-87

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Authors :
1. Valentina V. Tarasova, Master student, Higher School of Business
Moscow State University named after M.V. Lomonosov

2. Vasiliy E. Tarasov, Doctor of Physics and Mathematics, Leading Researcher, Scientific-Research Institute of Nuclear Physics named after D.V. Skobel’tsyn
Moscow State University named after M.V. Lomonosov

Deterministic factor analysis: methods of integro-differentiation of non-integral order

Objective: to summarize the methods of deterministic factor economic analysis, namely the differential calculus and the integral method.


Methods: mathematical methods for integro-differentiation of non-integral order, the theory of derivatives and integrals of fractional (non-integral)

Results: the basic concepts are formulated and the new methods are developed that take into account the memory and non-locality effects in the quantitative description of the influence of individual factors on the change in the effective economic indicator. Two methods are proposed for integro-differentiation of non-integral order for the deterministic factor analysis of economic processes with memory and non-locality. It is shown that the method of integro-differentiation of non-integral order can give more accurate results compared with standard methods (method of differentiation using the first order derivatives and the integral method using the integration of the first order) for a wide class of functions describing effective economic indicators.

Scientific novelty: the new methods of deterministic factor analysis are proposed: the method of differential calculus of non-integral order and the integral method of non-integral order.

Practical significance: the basic concepts and formulas of the article can be used in scientific and analytical activity for factor analysis of economic processes. The proposed method for integro-differentiation of non-integral order extends the capabilities of the determined factorial economic analysis. The new quantitative method of deterministic factor analysis may become the beginning of quantitative studies of economic agents behavior with memory, hereditarity and spatial non-locality. The proposed methods of deterministic factor analysis can be used in the study of economic processes which follow the exponential law, in which the indicators (endogenous variables) are power functions of the factors (exogenous variables), including the processes described by the Cobb – Douglas production function, since these methods allow to more accurately describe the total influence of the factors in comparison with the standard method. The proposed methods can be used in the study of economic processes described by equations with a power-law non-locality in factor space and in state space.

Keywords :

Economic and mathematical methods of Economics; Factor analysis; Method of differential calculus; Integral method; Processes with memory; Hereditarity; Derivative of non-integral order; Integration of non-integral order

Bibliography :

1. Samko, S. G., Kilbas, A. A., Marychev, O. I. Integrals and derivatives of non-integral order and some applications, Minsk: Nauka i Tekhnika, 1987, 688 p. (in Russ.).
2. Diethelm, K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Berlin, Heidelberg: Springer-Verlag, 2010, 247 p.
3. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006, 540 p.
4. Podlubny, I. Fractional Differential Equations, San Diego: Academic Press, 1998, 340 p.
5. Samko, S. G., Kilbas, A. A., Marichev, O. I. Fractional Integrals and Derivatives Theory and Applications, New York: Gordon and Breach, 1993, 1006 p.
6. Uchaikin, V. V. Technique of derivatives of non-integral order, Ulyanovsk: Artishok, 2008, 512 p. (in Russ.).
7. Tarasov, V. E. Models of theoretical physics with integro-differentiation of non-integral order, Moscow: Institut komp'yuternykh issledovanii, 2011, 568 p. (in Russ.).
8. Tarasov, V. E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, New York: Springer, 2011, 505 p.
9. Cartea, A., Del-Castillo-Negrete, D. Fractional diffusion models of option prices in markets with jumps, Physica A, 2007, vol. 374, No. 2, pp. 749–763.
10. Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M. Fractional calculus and continuous-time finance III: the diffusion limit. In: M. Kohlmann, S. Tang, (Eds.) Mathematical Finance. Trends in Mathematics, Basel: Birkhauser, 2001, pp. 171–180.
11. Kerss, A., Leonenko, N., Sikorskii, A. Fractional Skellam processes with applications to finance, Fractional Calculus and Applied Analysis, 2014, vol. 17, No. 2, pp. 532–551.

12. Laskin, N. Fractional market dynamics, Physica A, 2000, vol. 287, No. 3, pp. 482–492.
13. Mainardi, F., Raberto, M., Gorenflo, R., Scalas, E. Fractional calculus and continuous-time finance II: The waiting-time distribution, Physica A, 2000, vol. 287, No. 3–4, pp. 468–481.
14. Scalas, E., Gorenflo, R., Mainardi, F. Fractional calculus and continuous-time finance, Physica A, 2000, vol. 284, No. 1–4, pp. 376–384.
15. Skovranek, T., Podlubny, I., Petras, I. Modeling of the national economies in state-space: A fractional calculus approach, Economic Modelling, 2012, vol. 29, No. 4, pp. 1322–1327.
16. Tenreiro Machado, J., Duarte, F. B., Duarte, G. M. Fractional dynamics in financial indeces, International Journal of Bifurcation and Chaos, 2012, vol. 22, No. 10, Article ID 1250249, 12 p.
17. Tenreiro Machado, J. A., Mata, M. E. Pseudo phase plane and fractional calculus modeling of western global economic downturn, Communications in Nonlinear Science and Numerical Simulation, 2015, vol. 22, No. 1–3, pp. 396–406.
18. Tenreiro Machado, J. A., Mata, M. E., Lopes, A. M. Fractional state space analysis of economic systems, Entropy, 2015, vol. 17, No. 8, pp. 5402–5421.
19. Tarasova, V. V., Tarasov, V. E. Criteria of hereditiarity of economic process and memory effect, Molodoi uchenyi, 2016, No. 14 (118), pp. 396–399 (in Russ.).
20. Tarasova, V. V., Tarasov, V. E. Marginal utility for economic processes with memory, Al'manakh sovremennoi nauki i obrazovaniya, 2016, No. 7 (109), pp. 108–113 (in Russ.).
21. Tarasova, V. V., Tarasov, V. E. Price elasticity of demand with memory, Ekonomika, cotsiologiya i pravo, 2016, No. 4–1, pp. 98–106 (in Russ.).
22. Tarasova, V. V., Tarasov, V. E. Marginal values of of non-integral order in economic analysis, Ekonomika i Upravlenie, 2016, vol. 5, No. 3 (16), pp. 197–201 (in Russ.).
23. Tarasova, V. V., Tarasov, V. E. Elasticity of over-the-counter cash turnover of the Russian currency market, Aktual'nye problemy gumanitarnykh i estestvennykh nauk, 2016, No. 07–1 (90), pp. 207–215 (in Russ.).
24. Tarasova, V. V., Tarasov, V. E. Elasticity for economic processes with memory: Fractional differential calculus approach, Fractional Differential Calculus, 2016, vol. 6, No. 2, pp. 219–232.
25. Cobb, C. W., Douglas, P. H. A theory of production, American Economic Review, 1928, vol. 18 (Supplement), pp. 139–165.
26. Gabaix, X. Power laws in economics and finance, Annual Review of Economics, 2009, vol. 1, No. 1, pp. 255–293.
27. Gabaix, X. Power laws in economics: An introduction, Journal of Economic Perspectives, 2016, vol. 30, No. 1, pp. 185–206.
28. Odibat, Z. M., Shawagfeh, N. T. Generalized Taylor's formula, Applied Mathematics and Computation, 2007, vol. 186, No. 1, pp. 286–293.
29. Tarasov, V. E. Fractional vector calculus and fractional Maxwell's equations, Annals of Physics, 2008, vol. 323, No. 11, pp. 2756–2778.
30. Grigoletto, E. C., De Oliveira, E. C. Fractional versions of the fundamental theorem of calculus, Applied Mathematics, 2013, vol. 4, pp. 23–33.
31. Allen, R. G. D. Mathematical Economics. 2nd edition, London: Macmillan, 1960, 812 p.
32. Tarasova, V. V., Tarasov, V. E. Economic indicator summarizing the average and the marginal values, Ekonomika i predprinimatel'stvo, 2016, No. 11–1 (76–1), pp. 817–823 (in Russ.).

Citation :

Tarasova V. V., Tarasov V. E. Deterministic factor analysis: methods of integro-differentiation of non-integral order, Actual Problems of Economics and Law, 2016, vol. 10, No. 4, pp. 77–87 (in Russ.). DOI: 10.21202/1993-047X.10.2016.4.77-87

Type of article : The scientific article

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