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.

中文翻译：

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具有幂律记忆效应的通货膨胀的卡根模型

本文考虑了Phillip D. Cagan提出的模型的一般化，以描述实际通胀的动态。在这种概括中，考虑了记忆效应和记忆褪色。在标准的Cagan模型中，经济主体的紧张程度指标（代表刻画预期修订速度的特征）被表示为一个常数。通常，修订通货膨胀预期的速度可能取决于实际通货膨胀率与经济主体所期望的率之差的变化历史。我们认为，经济行为者的紧张感不仅可以由过程的当前状态引起，还可以由其变化的历史引起。使用记忆功能而不是紧张的指标使我们能够考虑Cagan模型中的记忆效应。我们考虑通货膨胀的小数动态，它考虑了幂律衰落引起的记忆。提出了分数微分方程，用记忆描述了所提出的经济模型，并给出了其精确解的表示。