Abstract
This paper studies the equilibrium of an extended case of the classical Samuelson’s multiplier–accelerator model for national economy. This case has incorporated some kind of memory into the system. We assume that total consumption and private investment depend upon the national income values. Then, delayed difference equations of third order are employed to describe the model, while the respective solutions of third-order polynomial correspond to the typical observed business cycles of real economy. We focus on the case that the equilibrium is not unique and provide a method to obtain the optimal equilibrium.
Highlights
1 Introduction Keynesian macroeconomics inspired the seminal work of Samuelson, who introduced the business cycle theory
The basic shortcoming of the original model is: the incapability to produce a stable path for the national income when realistic values for the different parameters are entered into the system of equations
For the case that we have infinite equilibriums, we provide an optimal equilibrium for the model
Summary
Keynesian macroeconomics inspired the seminal work of Samuelson, who introduced the business cycle theory. Lemma 2 The equilibrium(s) se of the reformulated Samuelson economical model (4) is given by the solution of the following algebraic system:. Proof From Lemma 1, the reformulated Samuelson economical model (4) is equivalent to (5). (a) If G is full rank, the solution Y ∗ of (5) is given by the following: Y ∗ = (I3 − F )−1V , and the unique equilibrium of the reformulated Samuelson economical model (4) is given by the following: se = (1 − c2 − c1)−1P. Further research is carried out for even higher order equations investigating qualitative results For this purpose, we may use an interesting tool applied to difference equations with many delays, the fractional nabla operator, see Atici and Eloe (2011), Dassios and Baleanu (2013, 2015), Dassios et al (2014b), Dassios (2017, 2015d), Klamka and Wyrwał (2008), Klamka (2010) and Podlubny (1999).
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