Phillips model with exponentially distributed lag and power-law memory
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Phillips model with exponentially distributed lag and power-law memory Vasily E. Tarasov1 · Valentina V. Tarasova2 Received: 3 August 2018 / Revised: 18 September 2018 / Accepted: 25 September 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract 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 memoryfading 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. Keywords 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
Communicated by José Tenreiro Machado.
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Vasily E. Tarasov [email protected] Valentina V. Tarasova [email protected]
1
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
2
Faculty of Economics, Lomonosov Moscow State University, Moscow 119991, Russia
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V. E. Tarasov, V. V. Tarasova
1 Introduction Macroeconomic models with continuous time are usually described by differential equations with derivatives of integer orders (Allen 1960, 1968, 2015). Examples of such models are well-known classical models, such as the natural growth model, the logistic model, the Harrod–Domar model, the Keynes model, and the Leontief model. Generalizations of such models are models, in which the time lag is taken into account. For example, a Harrod–Domar model with continuous time has been generalized by taking into account a continuously (exponentially) distributed lag by Phillips (1954, 2000, pp. 134–168) (see also Allen 1960, pp. 69–74, 1968, pp. 328–333). In the Phillips models, dynamic multiplier and accelerator are considered with a continuous (exponential) lag. Let us emphasize that in economic models with continuous time, the first time the continuously distributed lag was considered by Phillip
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