Abstract
Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed.
Highlights
IntroductionMathematical economics is a theoretical and applied science, whose purpose is a mathematically formalized description of economic objects, processes, and phenomena
General Remarks about Mathematical EconomicsMathematical economics is a theoretical and applied science, whose purpose is a mathematically formalized description of economic objects, processes, and phenomena
The modern stage of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes
Summary
Mathematical economics is a theoretical and applied science, whose purpose is a mathematically formalized description of economic objects, processes, and phenomena. The marginal value of non-integer order [118,119,120,121,122,190] with memory and non-locality; The economic multiplier with memory [123,124]; The economic accelerator with memory [123,124]; The exact discretization of economic accelarators and multiplier [125,126,127,128] based on exact fractional differences [129]; The accelerator with memory and crisis periodic sharp bursts [130,131]; The duality of the multiplier with memory and the accelerator with memory [123,124]; The elasticity of fractional order [132,133,134,135] for processes with memory and non-locality; The measures of risk aversion with non-locality [136] and with memory [137]; The warranted (technological) rate of growth with memory [112,170,174,175,176,189]; The non-local methods of deterministic factor analysis for [138,139]; And some other The use of these notions and concepts makes it possible for us to generalize some classical economic models, including those proposed by the following well-known economists: Henry Roy F. We have entered the stage of forming a new direction in mathematical economics and economic theory, when concepts and methods are not borrowed from other sciences and areas, but their own are created
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