Cagan model of inflation with power-law memory effects
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Cagan model of inflation with power-law memory effects Vasily E. Tarasov1,2 Received: 20 December 2019 / Revised: 23 June 2020 / Accepted: 28 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper considers a generalization of the model that has been proposed by Phillip D. Cagan to describe the dynamics of the actual inflation. In this generalization, the memory effects and memory fading are taken into account. In the standard Cagan model, the indicator of nervousness of economic agents, which characterizes the speed of revising the expectations, is represented as a constant parameter. In general, the speed of revising the expectations of inflation can depend on the history of changes in the difference between the real inflation rate and the rate expected by economic agents. We assume that the nervousness of economic agents can be caused not only by the current state of the process, but also by the history of its changes. The use of the memory function instead of the indicator of nervousness allows us to take into account the memory effects in the Cagan model. We consider the fractional dynamics of the inflation that takes into account memory with power-law fading. The fractional differential equation, which describes the proposed economic model with memory, and the expression of its exact solution are suggested. Keywords Fractional differential equation · Growth model · Fading memory · Fractional derivative · Cagan model Mathematics Subject Classification 26A33 · 34A08
1 Introduction Inflation and seigniorage are important effects that should be described in economics (Romer 2006; Tumanova and Shagas 2004). Sometimes inflation reaches high levels, and goes into hyperinflations, which are defined as inflation that exceeds 50 percent per month. The basic cause of most cases of high inflation and hyperinflation is government’s need to obtain
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Vasily E. Tarasov [email protected]
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Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
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Faculty of Information Technologies and Applied Mathematics, Moscow Aviation Institute (National Research University), Moscow 125993, Russia 0123456789().: V,-vol
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seigniorage, i.e., revenue from printing money (Cagan 1956). However, even governments’ need for seigniorage cannot account for hyperinflations (Romer 2006, p. 543). If the public does not immediately adjust its money holdings or its expectations of inflation to changes in the economic environment, then in the short-run seigniorage is always increasing in money growth, and the government can obtain more seigniorage than the maximum sustainable amount, S ∗ . Thus, hyperinflations arise when the government’s seigniorage needs exceed S ∗ (Romer 2006; Cagan 1956). Gradual adjustment of money holdings and gradual adjustment of expected inflation have similar implications for the dynamics of inflation. The Cagan model focuses on the case of gradual
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