# Exact Discretization of an Economic Accelerator and Multiplier with Memory

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## Abstract

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## 1. Introduction

## 2. An Accelerator and a Multiplier with Memory in the Continuous-Time Approach

## 3. Discrete-Time Approach to Dynamic Memory in Economics

## 4. Concept of Exact Discretization

- The Leibniz rule is a characteristic property of the derivatives of integer orders. Therefore, the exact discretization of these operators should satisfy this rule. The Leibniz rule should be the main characteristic property of the exact discrete analogs of the derivatives.
- The exact discretization should satisfy the semi-group property. For example, the second-order difference should be equal to the repeated action of the first-order differences.
- The exact differences of the power-law functions should give the same expression as an action of the derivatives. This allows us to consider the exact correspondence of the derivatives and differences on the space of entire functions.

## 5. Exact Discrete Analogs of the Standard Accelerator and the Standard Multiplier

## 6. Exact Discrete Analogs of the Accelerator and Multiplier with Memory

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Volterra, V. Theory of Functionals and of Integral and Integro-Differential Equations; Dover: New York, NY, USA, 2005; p. 288, Chapter VI, Section IV; ISBN 0486442845. [Google Scholar]
- Wang, C.C. The principle of fading memory. Arch. Ration. Mech. Anal.
**1965**, 18, 343–366. [Google Scholar] [CrossRef] - Coleman, B.D.; Mizel, V.J. On the general theory of fading memory. Arch. Ration. Mech. Anal.
**1968**, 29, 18–31. [Google Scholar] [CrossRef] - Saut, J.C.; Joseph, D.D. Fading memory. Arch. Ration. Mech. Anal.
**1983**, 81, 53–95. [Google Scholar] [CrossRef] - Day, W.A. The Thermodynamics of Simple Materials with Fading Memory; Springer: Berlin, Germany, 1972; p. 134. ISBN 978-3-642-65318-6. [Google Scholar]
- Rabotnov, Y.N. Elements of Hereditary Solid Mechanics; Mir Publishers: Moscow, Russia, 1980; p. 387. [Google Scholar]
- Evans, M.W.; Grigolini, P.; Parravicini, G.P. (Eds.) Memory Function Approaches to Stochastic Problems in Condensed Matter; Wiley: New York, NY, USA, 1985; p. 556. ISBN 978-0-470-14331-5. [Google Scholar]
- Alber, H.D. Materials with Memory; Springer: Berlin/Heidelberg, Germany, 1998; p. 171. ISBN 978-3-540-69689-6. [Google Scholar]
- Mainardi, F. Fractional Calculus and Waves Linear Viscoelasticity: An Introduction to Mathematical Models; Imperial College Press: London, UK, 2010; p. 368. ISBN 978-1-84816-329-4. [Google Scholar]
- Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media; Springer: New York, NY, USA, 2010; p. 505. [Google Scholar] [CrossRef]
- Amendola, G.; Fabrizio, M.; Golden, J.M. Thermodynamics of Materials with Memory: Theory and Applications; Springer: New York, NY, USA, 2011; p. 576. [Google Scholar] [CrossRef]
- Teyssiere, G.; Kirman, A.P. Long Memory in Economics; Springer: Berlin/Heidelberg, Germany, 2007; p. 390. ISBN 978-3-540-22694-9. [Google Scholar] [CrossRef]
- Beran, J.; Feng, Y.; Ghosh, S.; Kulik, R. Long-Memory Processes: Probabilistic: Properties and Statistical Methods; Springer: Berlin, Germany, 2013. [Google Scholar] [CrossRef]
- Robinson, P.M. (Ed.) Time Series with Long Memory: Advanced Texts in Econometrics; Oxford University Press: Oxford, UK, 2003; p. 382. ISBN 0199257302. [Google Scholar]
- Granger, C.W.J.; Joyeux, R. An introduction to long memory time series models and fractional differencing. J. Time Ser. Anal.
**1980**, 1, 15–39. [Google Scholar] [CrossRef] - Hosking, J.R.M. Fractional differencing. Biometrika
**1981**, 68, 165–176. [Google Scholar] [CrossRef] - Parke, W.R. What is fractional integration? Rev. Econ. Stat.
**1989**, 81, 632–638. [Google Scholar] [CrossRef] - Gil-Alana, L.A.; Hualde, J. Fractional integration and cointegration: An overview and an empirical application. In Palgrave Handbook of Econometrics; Volume 2: Applied Econometrics; Mills, T.C., Patterson, K., Eds.; Springer: Berlin, Germany, 2009; pp. 434–469. [Google Scholar]
- Ghysels, E.; Swanson, N.R.; Watson, M.W. Essays in Econometrics Collected Papers of Clive W.J. Granger. Volume 2: Causality, Integration and Cointegration, and Long Memory; Cambridge University Press: Cambridge, UK, 2001; p. 398. [Google Scholar]
- Tarasov, V.E.; Tarasova, V.V. Long and short memory in economics: Fractional-order difference and differentiation. IRA Int. J. Manag. Soc. Sci.
**2016**, 5, 327–334. [Google Scholar] [CrossRef] - Grunwald, A.K. About “limited” derivations their application. Z. Angew. Math. Phys.
**1867**, 12, 441–480. Available online: https://www.deutsche-digitale-bibliothek.de/item/7OWDE57YKUI6KP2N2ZI3SDEW2B2EZ6AP (accessed on 10 September 2017). (In German). - Letnikov, A.V. Theory of differentiation with arbitrary pointer. Matematicheskii Sbornik.
**1868**, 3, 1–68. Available online: http://mi.mathnet.ru/eng/msb8039 (accessed on 10 September 2017). (In Russian). - Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives Theory and Applications; Gordon and Breach: New York, NY, USA, 1993; p. 1006. ISBN 2881248640; 9782881248641. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1998; p. 340. ISBN 9780125588409. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006; p. 540. ISBN 9780444518323. [Google Scholar]
- Diethelm, K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type; Springer: Berlin, Germany, 2010; p. 247. ISBN 978-3-642-14573-5. [Google Scholar] [CrossRef]
- Tarasova, V.V.; Tarasov, V.E. Concept of dynamic memory in economics. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 55, 127–145. [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Marginal utility for economic processes with memory. Alm. Mod. Sci. Educ.
**2016**, 109, 108–113. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Marginal values of non-integer order in economic analysis. Azimuth Sci. Res.: Econ. Manag.
**2016**, 16, 197–201. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Economic indicator that generalizes average and marginal values. J. Econ. Entrep.
**2016**, 76, 817–823. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. A generalization of concepts of accelerator and multiplier to take into account memory effects in macroeconomics. J. Econ. Entrep.
**2016**, 75, 1121–1129. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Economic accelerator with memory: Discrete time approach. Probl. Mod. Sci. Educ.
**2016**, 78, 37–42. [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Logistic map with memory from economic model. Chaos Solitons Fractals
**2017**, 95, 84–91. [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Elasticity for economic processes with memory: Fractional differential calculus approach. Fract. Differ. Calc.
**2016**, 6, 219–232. [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Risk aversion for investors with memory: Hereditary generalizations of Arrow-Pratt measure. Financ. J.
**2017**, 36, 46–63. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Non-local measures of risk aversion in the economic process. Econ. Theory Pract.
**2016**, 44, 54–58. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Deterministic factor analysis: Methods of integro-differentiation of non-integral order. Actual Probl. Econ. Law
**2016**, 4, 77–87. (In Russian) [Google Scholar] [CrossRef] - Allen, R.G.D. Mathematical Economics, 2nd ed.; Macmillan: London, UK, 1960; p. 812. [Google Scholar] [CrossRef]
- Allen, R.G.D. Macro-Economic Theory. In A Mathematical Treatment; Macmillan: London, UK, 1967; p. 420. ISBN 978-1-349-81543-2. [Google Scholar] [CrossRef]
- Potts, R.B. Differential and difference equations. Am. Math. Mon.
**1982**, 89, 402–407. [Google Scholar] [CrossRef] - Potts, R.B. Ordinary and partial difference equations. J. Aust. Math. Soc. B
**1986**, 27, 488–501. [Google Scholar] [CrossRef] - Mickens, R.E. Difference equation models of differential equations. Math. Comput. Model.
**1988**, 11, 528–530. [Google Scholar] [CrossRef] - Mickens, R.E. Discretizations of nonlinear differential equations using explicit nonstandard methods. J. Comput. Appl. Math.
**1999**, 110, 181–185. [Google Scholar] [CrossRef] - Mickens, R.E. Nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl.
**2002**, 8, 823–847. [Google Scholar] [CrossRef] - Mickens, R.E. Nonstandard Finite Difference Models of Differential Equations; World Scientific: Singapore, 1993; p. 264. ISBN 978-981-02-1458-6. [Google Scholar]
- Mickens, R.E. (Ed.) Advances in the Applications of Nonstandard Finite Difference Schemes; World Scientific: Singapore, 2005; p. 664. ISBN 978-981-256-404-7. [Google Scholar]
- Mickens, R.E. (Ed.) Applications of Nonstandard Finite Difference Schemes; World Scientific: Singapore, 2000; p. 264. ISBN 978-981-02-41. [Google Scholar]
- Tarasov, V.E. Exact discrete analogs of derivatives of integer orders: Differences as infinite series. J. Math.
**2015**, 2015, 134842. [Google Scholar] [CrossRef] - Tarasov, V.E. Exact discretization by Fourier transforms. Commun. Nonlinear Sci. Numer. Simul.
**2016**, 37, 31–61. [Google Scholar] [CrossRef] - Tarasov, V.E. Lattice fractional calculus. Appl. Math. Comput.
**2015**, 257, 12–33. [Google Scholar] [CrossRef] - Tarasov, V.E. Toward lattice fractional vector calculus. J. Phys. A
**2014**, 47. [Google Scholar] [CrossRef] - Tarasov, V.E. United lattice fractional integro-differentiation. Fract. Calc. Appl. Anal.
**2016**, 19, 625–664. [Google Scholar] [CrossRef] - Tarasov, V.E. Exact Discretization of fractional Laplacian. Comput. Math. Appl.
**2017**, 73, 855–863. [Google Scholar] [CrossRef] - Scalas, E.; Gorenflo, R.; Mainardi, F. Fractional calculus and continuous-time finance. Phys. A Stat. Mech. Appl.
**2000**, 284, 376–384. [Google Scholar] [CrossRef] - Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E. Fractional calculus and continuous-time finance II: The waiting-time distribution. Phys. A Stat. Mech. Appl.
**2000**, 287, 468–481. [Google Scholar] [CrossRef] - Laskin, N. Fractional market dynamics. Phys. A Stat. Mech. Appl.
**2000**, 287, 482–492. [Google Scholar] [CrossRef] - Raberto, M.; Scalas, E.; Mainardi, F. Waiting-times and returns in high-frequency financial data: An empirical study. Phys. A Stat. Mech. Appl.
**2002**, 314, 749–755. [Google Scholar] [CrossRef] - West, B.J.; Picozzi, S. Fractional Langevin model of memory in financial time series. Phys. Rev. E
**2002**, 65, 037106. [Google Scholar] [CrossRef] [PubMed] - Picozzi, S.; West, B.J. Fractional Langevin model of memory in financial markets. Phys. Rev. E
**2002**, 66, 046118. [Google Scholar] [CrossRef] [PubMed] - Scalas, E. The application of continuous-time random walks in finance and economics. Phys. A Stat. Mech. Appl.
**2006**, 362, 225–239. [Google Scholar] [CrossRef] - Meerschaert, M.M.; Scalas, E. Coupled continuous time random walks in finance. Phys. A Stat. Mech. Appl.
**2006**, 370, 114–118. [Google Scholar] [CrossRef] - Cartea, A.; Del-Castillo-Negrete, D. Fractional diffusion models of option prices in markets with jumps. Phys. A Stat. Mech. Appl.
**2007**, 374, 749–763. [Google Scholar] [CrossRef] [Green Version] - Mendes, R.V. A fractional calculus interpretation of the fractional volatility model. Nonlinear Dyn.
**2009**, 55, 395–399. [Google Scholar] [CrossRef] - Blackledge, J. Application of the fractional diffusion equation for predicting market behavior. Int. J. Appl. Math.
**2010**, 40, 130–158. [Google Scholar] [CrossRef] - Tenreiro Machado, J.; Duarte, F.B.; Duarte, G.M. Fractional dynamics in financial indices. Int. J. Bifurc. Chaos
**2012**, 22. [Google Scholar] [CrossRef] - Korbel, J.; Luchko, Y. Modeling of financial processes with a space-time fractional diffusion equation of varying order. Fract. Calc. Appl. Anal.
**2016**, 19, 1414–1433. [Google Scholar] [CrossRef] - Kerss, A.; Leonenko, N.; Sikorskii, A. Fractional Skellam processes with applications to finance. Fract. Calc. Appl. Anal.
**2014**, 17, 532–551. [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Influence of memory effects on world economics and business. Azimuth Sci. Res.: Econ. Manag.
**2016**, 17, 369–372. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Economic growth model with constant pace and dynamic memory. Probl. Mod. Sci. Educ.
**2017**, 84, 38–43. [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Fractional dynamics of natural growth and memory effect in economics. Eur. Res.
**2016**, 23, 30–37. [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Hereditary generalization of Harrod-Domar model and memory effects. J. Econ. Entrep.
**2016**, 75, 72–78. [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Memory effects in hereditary Harrod-Domar model. Probl. Mod. Sci. Educ.
**2016**, 74, 38–44. (In Russian) [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Keynesian model of economic growth with memory. Econ. Manag. Probl. Solut.
**2016**, 58, 21–29. (In Russian) [Google Scholar] - Tarasova, V.V.; Tarasov, V.E. Memory effects in hereditary Keynes model. Probl. Mod. Sci. Educ.
**2016**, 80, 56–61. (In Russian) [Google Scholar] [CrossRef] - Tarasova, V.V.; Tarasov, V.E. Dynamic intersectoral models with power-law memory. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 54, 100–117. [Google Scholar] [CrossRef] - Tarasov, V.E.; Tarasova, V.V. Time-dependent fractional dynamics with memory in quantum and economic physics. Ann. Phys.
**2017**, 383, 579–599. [Google Scholar] [CrossRef] - Tejado, I.; Valerio, D.; Valerio, N. Fractional Calculus in Economic Growth Modelling. The Spanish Case. In CONTROLO’2014—Proceedings of the 11th Portuguese Conference on Automatic Control; Moreira, A.P., Matos, A., Veiga, G., Eds.; Springer: New York, NY, USA, 2015; pp. 449–458. [Google Scholar]
- Tejado, I.; Valerio, D.; Valerio, N. Fractional Calculus in Economic Growth Modeling. In Proceedings of the 2014 International Conference on Fractional Differentiation and Its Applications, Catania, Italy, 23–25 June 2014; ISBN 978-1-4799-2591-9. [Google Scholar]
- Tejado, I.; Valerio, D.; Perez, E.; Valerio, N. Fractional calculus in economic growth modelling: The Spanish and Portuguese cases. Int. J. Dyn. Control
**2017**, 5, 208–222. [Google Scholar] [CrossRef] - Tejado, I.; Valerio, D.; Perez, E.; Valerio, N. Fractional calculus in economic growth modelling: The economies of France and Italy. In Proceedings of the International Conference on Fractional Differentiation and Its Applications, Novi Sad, Serbia, 18–20 July 2016; pp. 113–123. [Google Scholar]

**Table 1.**The action of derivatives and standard and exact finite differences on some elementary functions.

$\mathbf{X}(\mathbf{t})$ | $\mathbf{d}\mathbf{X}(t)/dt$ | ${\u2206}_{b}^{1}\mathbf{X}(t)$ | ${\u2206}_{T}^{1}X(t)$ |
---|---|---|---|

$\mathrm{exp}(\mathsf{\lambda}\xb7\mathrm{t})$ | $\mathsf{\lambda}\xb7\mathrm{exp}(\mathsf{\lambda}\xb7\mathrm{t})$ | $\frac{\mathrm{exp}(\mathsf{\lambda})-1}{\mathrm{exp}(\mathsf{\lambda})}\xb7\mathrm{exp}(\mathsf{\lambda}\xb7\mathrm{t})$ | $\mathsf{\lambda}\xb7\mathrm{exp}(\mathsf{\lambda}\xb7\mathrm{t})$ |

$\mathrm{sin}(\mathsf{\lambda}\xb7\mathrm{t})$ | $\mathsf{\lambda}\xb7\mathrm{cos}(\mathsf{\lambda}\xb7\mathrm{t})$ | $2\xb7\mathrm{sin}\left(\mathsf{\lambda}\xb7\mathrm{t}-\frac{\mathsf{\lambda}}{2}\right)\xb7\mathrm{cos}\left(\frac{\mathsf{\lambda}}{2}\right)$ | $\mathsf{\lambda}\xb7\mathrm{cos}(\mathsf{\lambda}\xb7\mathrm{t})$ |

$\mathrm{cos}\left(\mathsf{\lambda}\xb7\mathrm{t}\right)$ | $-\mathsf{\lambda}\xb7\mathrm{sin}(\mathsf{\lambda}\xb7\mathrm{t})$ | $-2\xb7\mathrm{sin}\left(\mathsf{\lambda}\xb7\mathrm{t}-\frac{\mathsf{\lambda}}{2}\right)\xb7\mathrm{sin}\left(\frac{\mathsf{\lambda}}{2}\right)$ | $-\mathsf{\lambda}\xb7\mathrm{sin}(\mathsf{\lambda}\xb7\mathrm{t})$ |

${\mathrm{t}}^{2}$ | $2\xb7\mathrm{t}$ | $2\xb7\mathrm{t}-1$ | $2\xb7\mathrm{t}$ |

${\mathrm{t}}^{3}$ | $3\xb7{\mathrm{t}}^{2}$ | $3\xb7{\mathrm{t}}^{2}-3\xb7\mathrm{t}+1$ | $3\xb7{\mathrm{t}}^{2}$ |

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**MDPI and ACS Style**

Tarasova, V.V.; Tarasov, V.E.
Exact Discretization of an Economic Accelerator and Multiplier with Memory. *Fractal Fract.* **2017**, *1*, 6.
https://doi.org/10.3390/fractalfract1010006

**AMA Style**

Tarasova VV, Tarasov VE.
Exact Discretization of an Economic Accelerator and Multiplier with Memory. *Fractal and Fractional*. 2017; 1(1):6.
https://doi.org/10.3390/fractalfract1010006

**Chicago/Turabian Style**

Tarasova, Valentina V., and Vasily E. Tarasov.
2017. "Exact Discretization of an Economic Accelerator and Multiplier with Memory" *Fractal and Fractional* 1, no. 1: 6.
https://doi.org/10.3390/fractalfract1010006