A feedback policy for a modified Samuelson-Hicks model

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Under the assumption that the income Y t follows a Samuelson–Hicks type of model, we calculate a policy variable G t , in closed form, so that Y t satisfies an ideal law Y t ¼ FðU Ã t Þ, U Ã t being an influence variable. The approach uses some recently developed tools of non-linear feedback design; it is fully parameterized and allows the dynamic change of the influence variable U Ã t. From the control theory point of view, we solve a non-linear model matching problem and produce a policy instrument that shapes the income Y t within a desired family of behaviours. 1. Introduction Here, we consider a Samuelson–Hicks type of model which describes the dynamic behaviour of the income Y t , the investment I t and the consumption C t. The original purpose of such a system was to explain and quantify the existence of business cycles in the economy and this was the main theme of the articles written on this topic (Samuelson 1939, Hicks 1950). More recent research presented bifurcation studies and examination of the chaotic dynamics that appear in such models or modified ones (Nusse and Hommes 1990, Hommes 1995). The first such model was proposed by Samuelson (1939) and could only explain the existence of business cycles for specific values of the parameters (structural instability). This idea was further elaborated by Hicks (1950) producing a non-linear growth model with a unique unstable equilibrium growth path, a maximum growth path (the full employment ceiling) and a minimum growth path (the zero investment floor) all with the same constant growth rate. This model explained the business cycles as time paths oscillating between the floor and the ceiling around …