Phillips model with exponentially distributed lag and power-law memory

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 memory-fading 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.