Logistic equation with continuously distributed lag and application in economics

Tarasova, Valentina V.

doi:10.1007/s11071-019-05050-1

Cited by 15 publications

(9 citation statements)

References 32 publications

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“…Cushing [34] introduced and used distributed delays in mathematical biology, while Invernizzi and Medio [35] presented distributed delays into mathematical economics. Some examples in context of economic growth are provided in [36] and [37].…”

Section: Introductionmentioning

confidence: 99%

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

2020

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In this paper, we consider a model of economic growth with a distributed time-delay investment function, where the time-delay parameter is a mean time delay of the gamma distribution. Using the linear chain trick technique, we transform the delay differential equation system into an equivalent one of ordinary differential equations (ODEs). Since we are dealing with weak and strong kernels, our system will be reduced to a three-and four-dimensional ODE system, respectively. The occurrence of Hopf bifurcation is investigated with respect to the following two parameters: time-delay parameter and rate of growth parameter. Sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation are presented in case of time-delay parameter. Numerical studies with the Dana and Malgrange investment

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Section: Introductionmentioning

confidence: 99%

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

2020

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“…(b) We can use kernel D(t − τ) = e −δ (t−τ) (t − τ) α−1 /Γ(α), which describes the gamma distributed lag by Equation (17). The economic model with this type of distributed lag was considered in work [53]. Note that kernels of exponential amortization and power-law memory are special cases of this kernel for the case δ = 0 and α = 1, respectively.…”

Section: Discussionmentioning

confidence: 99%

Non-Linear Macroeconomic Models of Growth with Memory

Tarasov

2020

Mathematics

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In this article, two well-known standard models with continuous time, which are proposed by two Nobel laureates in economics, Robert M. Solow and Robert E. Lucas, are generalized. The continuous time standard models of economic growth do not account for memory effects. Mathematically, this is due to the fact that these models describe equations with derivatives of integer orders. These derivatives are determined by the properties of the function in an infinitely small neighborhood of the considered time. In this article, we proposed two non-linear models of economic growth with memory, for which equations are derived and solutions of these equations are obtained. In the differential equations of these models, instead of the derivative of integer order, fractional derivatives of non-integer order are used, which allow describing long memory with power-law fading. Exact solutions for these non-linear fractional differential equations are obtained. The purpose of this article is to study the influence of memory effects on the rate of economic growth using the proposed simple models with memory as examples. As the methods of this study, exact solutions of fractional differential equations of the proposed models are used. We prove that the effects of memory can significantly (several times) change the growth rate, when other parameters of the model are unchanged.

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“…• fading memory (forgetting) (for example, see [47][48][49][50] and references therein) and power-law frequency dispersion; • spatial non-locality (for example, see [63]) and power-law spatial dispersion (for example, see [64]); • lag (time delay) (for example, see [43,[53][54][55]65] and references therein); and • scaling (dilation) (for example, see section 9 in [43] and references therein).…”

Section: Seventh Example: Fading Memory Spetial Non-locality Time Dmentioning

confidence: 99%

“…x p x , p y , I . Note that the variables p x , p y enter into the ordinary demand function in (65) in two places. In 1915, Evgeny E. Slutsky proposed [85][86][87][88][89] an equation that allows us to calculate the compensated (Hicksian) demand function from observable functions, namely, the derivative of the Marshallian demand with respect to price and income.…”

Section: Fractional Generalization Of Slutsky Equationmentioning

confidence: 99%

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Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models

Tarasov

2019

Mathematics

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This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We formulate rules (principles) for constructing fractional generalizations of standard models, which were described by differential equations of integer order. Important requirements to building fractional generalization of dynamical models (the rules for “fractional-dynamic generalizers”) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics.

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Logistic equation with continuously distributed lag and application in economics

Cited by 15 publications

References 32 publications

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

Non-Linear Macroeconomic Models of Growth with Memory

Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models

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