# Optimal equilibrium for a reformulated Samuelson economical model

## TL;DR

This paper extends the Samuelson multiplier-accelerator model by incorporating memory effects, analyzing non-unique equilibria, and proposing a method to identify the optimal equilibrium in a delayed difference equation framework.

## Contribution

It introduces a reformulated model with memory, analyzes non-unique equilibria, and provides a novel method to determine the optimal equilibrium.

## Key findings

- Delayed difference equations of third order describe business cycles.
- Equilibrium is not always unique, requiring an optimal selection method.
- The model captures observed economic fluctuations with memory effects.

## 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.