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The paper proposes an approach to the description of macroeconomic phenomena, which takes into account the effects of fading memory. The standard notions of the accelerator and the multiplier are very limited, since the memory of economic agents is neglected. We consider the methods to describe the economic processes with memory, which is characterized by the fading of a power-law type. Using the mathematical tools of derivatives and integrals of non-integer orders, we suggest a generalization of the concept of the accelerator and multiplier. We derive the equations of the accelerator with memory and the multiplier with memory, which take into account the changes of endogenous and exogenous variables on a finite time interval. We prove the duality of the concepts of the multiplier with memory and the accelerator with memory. The proposed generalization includes the standard concepts of the accelerator and the multiplier as special cases. In addition these generalizations provide a range of intermediate characteristics to take into account the memory effects in macroeconomic models.

<div><p><em>Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the </em><em>Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(p,d,q) models are the fractional-order difference equations with the Grunwald-Letnikov differences of order d. We prove that the long and short memory with power law should be described by the exact fractional-order differences, for which </em><em>the Fourier transform </em><em>demonstrates the power law exactly. The fractional differencing and the Grunwald-Letnikov fractional differences cannot give exact results for the long and short memory with power law, since </em><em>the Fourier transform</em><em> of these discrete operators satisfy the </em><em>power law in the neighborhood of zero only. We prove that the economic processes with t</em><em>he continuous time long and short memory, which is characterized by the power law, should be described by the fractional differential equations.</em></p></div>