Phillips model with exponentially distributed lag and power-law memory

  • Vasily E. TarasovEmail author
  • Valentina V. Tarasova


In this paper, we propose two generalizations of the Phillips model of multiplier–accelerator by taking into account memory with power-law fading. In the first generalization we consider the model, where we replace the exponential weighting function by the power-law memory function. In this model we consider two power-law fading memories, one on the side of the accelerator (induced investment responding to changes in output with memory-fading parameter α) and the other on the supply side (output responding to demand with memory-fading parameter β). To describe power-law memory we use the fractional derivatives in the accelerator equation and the fractional integral in multiplier equation. The solution of the model fractional differential equation is suggested. In the second generalization of the Phillips model of multiplier–accelerator we consider the power-law memory in addition to the continuously (exponentially) distributed lag. Equation, which describes generalized Phillips model of multiplier–accelerator with distributed lag and power-law memory, is solved using Laplace method.


Phillips model Multiplier–accelerator Distributed lag Delay Memory Fractional derivative Exponential distribution 

Mathematics Subject Classification

91B02 Fundamental topics (basic mathematics applicable to economics in general) 91B55 Economic dynamics 26A33 Fractional derivatives and integrals 

JEL Classification

C02 Mathematical Methods E00 Macroeconomics and Monetary Economics: General 



  1. Allen RGD (1960) Mathematical economics, 2nd edn. Macmillan, London. (ISBN 978-1-349-81547-0, first edition 1956) CrossRefGoogle Scholar
  2. Allen RGD (1968) Macro-economic theory. A mathematical treatment. Macmillan, London (ISBN 978-033304112) Google Scholar
  3. Allen RGD (2015) Mathematical economics. Andesite Press, New York (ISBN 978-1297569906) Google Scholar
  4. Bateman H (1954) Tables of integral transforms, vol I. McGraw-Hill Book Company, New York (ISBN 07-019549-8) Google Scholar
  5. Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 1(2):73–85. CrossRefGoogle Scholar
  6. Caputo M, Fabrizio M (2016) Applications of new time and spatial fractional derivatives with exponential kernels. Progr Fract Differ Appl 2(1):1–11. CrossRefGoogle Scholar
  7. Diethelm K (2010) The Analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Springer, Berlin. CrossRefzbMATHGoogle Scholar
  8. Erdélyi A (1950) Hypergeometric functions of two variables. Acta Math 83:131–164.
  9. Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG (1953) Higher transcendental functions (Bateman manuscript project), vol I. McGraw-Hill, New YorkzbMATHGoogle Scholar
  10. Fallahgoul HA, Focardi SM, Fabozzi FJ (2016) Fractional calculus and fractional processes with applications to financial economics, theory and application. Academic Press, London (ISBN 9780128042489) zbMATHGoogle Scholar
  11. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam (ISBN 9780444518323) zbMATHGoogle Scholar
  12. Kiryakova V (1994) Generalized fractional calculus and applications. Longman and J. Wiley, New York (ISBN 9780582219779) zbMATHGoogle Scholar
  13. Korbel J, Luchko Yu (2016) Modeling of financial processes with a space-time fractional diffusion equation of varying order. Fract Calc Appl Anal 19(6):1414–1433. MathSciNetCrossRefzbMATHGoogle Scholar
  14. Mainardi F, Raberto M, Gorenflo R, Scalas E (2000) Fractional calculus and continuous-time finance II: the waiting-time distribution. Phys A 287(3–4):468–481. CrossRefzbMATHGoogle Scholar
  15. Ortigueira MD, Tenreiro Machado J (2018) A critical analysis of the Caputo–Fabrizio operator. Commun Nonlinear Sci Numer Simul 59:608–611. MathSciNetCrossRefGoogle Scholar
  16. Paris RB (2010) Exponentially small expansions in the asymptotics of the Wright function. J Comput Appl Math 234(2):488–504. MathSciNetCrossRefzbMATHGoogle Scholar
  17. Paris RB (2014) Exponentially small expansions of the Wright function on the Stokes lines. Lith Math J 54(1):82–105. MathSciNetCrossRefzbMATHGoogle Scholar
  18. Paris RB (2017) Some remarks on the theorems of Wright and Braaksma on the Wright function pΨq(z). arXiv:1708.04824
  19. Paris RB, Kaminski D (2001) Asymptotics and Mellin–Barnes integrals. Cambridge University Press, Cambridge. CrossRefzbMATHGoogle Scholar
  20. Paris RB, Vinogradov V (2016) Asymptotic and structural properties of the Wright function arising in probability theory. Lith Math J 56(3):377–409. arXiv:1508.00863
  21. Phillips AW (1954) Stabilisation policy in a closed economy. Econ J 64(254):290–323. CrossRefGoogle Scholar
  22. Leeson R (ed) (2000) A. W. H. Phillips: collected works in contemporary perspective. Cambridge University Press, Cambridge. ISBN: 9780521571357Google Scholar
  23. Podlubny I (1998) Fractional differential equations. Academic Press, San Diego, p 340zbMATHGoogle Scholar
  24. Raberto M, Scalas E, Mainardi F (2002) Waiting-times and returns in high-frequency financial data: an empirical study. Phys A 314(1-4):749–755. CrossRefzbMATHGoogle Scholar
  25. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives theory and applications. Gordon and Breach, New York (ISBN 978-2881248641) zbMATHGoogle Scholar
  26. Scalas E, Gorenflo R, Mainardi F (2000) Fractional calculus and continuous-time finance. Phys A 284(1–4):376–384. MathSciNetCrossRefGoogle Scholar
  27. Skovranek T, Podlubny I, Petras I (2012) Modeling of the national economies in state-space: a fractional calculus approach. Econ Model 29(4):1322–1327. CrossRefGoogle Scholar
  28. Tarasov VE (2010) Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer, Berlin. CrossRefzbMATHGoogle Scholar
  29. Tarasov VE (2018a) No nonlocality. No fractional derivative. Commun Nonlinear Sci Numer Simul 62:157–163. arXiv:1803.00750
  30. Tarasov VE (2018b) Caputo–Fabrizio operator in terms of integer derivatives: memory or distributed lag? Comput Appl Math (submitted) Google Scholar
  31. Tarasov VE, Tarasova VV (2017a) Accelerator and multiplier for macroeconomic processes with memory. IRA Int J Manag Soc Sci 9(3):86–125. CrossRefGoogle Scholar
  32. Tarasov VE, Tarasova VV (2017b) Time-dependent fractional dynamics with memory in quantum and economic physics. Ann Phys 383:579–599. MathSciNetCrossRefzbMATHGoogle Scholar
  33. Tarasov VE, Tarasova VV (2018a) Criterion of existence of power-law memory for economic processes. Entropy 20(6):414. CrossRefGoogle Scholar
  34. Tarasov VE, Tarasova VV (2018b) Macroeconomic models with long dynamic memory: fractional calculus approach. Appl Math Comput 338:466–486. MathSciNetCrossRefGoogle Scholar
  35. Tarasova VV, Tarasov VE (2016a) Economic accelerator with memory: discrete time approach. Probl Mod Sci Educ 36(78):37–42. CrossRefGoogle Scholar
  36. Tarasova VV, Tarasov VE (2016b) Fractional dynamics of natural growth and memory effect in economics. Eur Res 12(23):30–37. CrossRefGoogle Scholar
  37. Tarasova VV, Tarasov VE (2017a) Accelerators in macroeconomics: comparison of discrete and continuous approaches. Am J Econ Bus Admin 9(3): 47–55. arXiv:1712.09605
  38. Tarasova VV, Tarasov VE (2017b) Exact discretization of economic accelerator and multiplier with memory. Fract Fract 1(1):6. CrossRefGoogle Scholar
  39. Tarasova VV, Tarasov VE (2017c) Economic growth model with constant pace and dynamic memory. Probl Mod Sci Educ 2(84):40–45. CrossRefGoogle Scholar
  40. Tarasova VV, Tarasov VE (2017d) Logistic map with memory from economic model. Chaos Solitons Fract 95:84–91. MathSciNetCrossRefzbMATHGoogle Scholar
  41. Tarasova VV, Tarasov VE (2018a) Concept of dynamic memory in economics. Commun Nonlinear Sci Numer Simul 55:127–145. arXiv:1712.09088
  42. Tarasova VV, Tarasov VE (2018b) Dynamic intersectoral models with power-law memory. Commun Nonlinear Sci Numer Simul 54:100–117. arXiv:1712.09087
  43. Tejado I, Valerio D, Valerio N (2015) Fractional calculus in economic growth modelling. The Spanish case. In: Moreira AP, Matos A, Veiga G (eds) CONTROLO’2014—Proceedings of the 11th Portuguese conference on automatic control. Volume 321 of the series lecture notes in electrical engineering. Springer International Publishing, pp 449–458.
  44. Tejado I, Valerio D, Perez E, Valerio N (2016) Fractional calculus in economic growth modelling: the economies of France and Italy. In: Spasic DT, Grahovac N, Zigic M, Rapaic M, Atanackovic TM (eds) Proceedings of international conference on fractional differentiation and its applications, Novi Sad, Serbia, July 18–20, pp 113–123Google Scholar
  45. Tejado I, Valerio D, Perez E, Valerio N (2017) Fractional calculus in economic growth modelling: the Spanish and Portuguese cases. Int J Dyn Control 5(1):208–222. MathSciNetCrossRefGoogle Scholar
  46. Tenreiro Machado JA, Mata ME (2015) Pseudo phase plane and fractional calculus modeling of western global economic downturn. Commun Nonlinear Sci Numer Simul 22(1–3):396–406. MathSciNetCrossRefGoogle Scholar
  47. Tenreiro Machado J, Duarte FB, Duarte GM (2012) Fractional dynamics in financial indices. Int J Bifurc Chaos 22(10):1250249. MathSciNetCrossRefGoogle Scholar
  48. Tenreiro Machado JA, Mata ME, Lopes AM (2015) Fractional state space analysis of economic systems. Entropy. 17(8):5402–5421. CrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Faculty of EconomicsLomonosov Moscow State UniversityMoscowRussia

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