Bifurcation Analysis of the Keynesian Cross Model: Abstract and Introduction

Table of Links

• Abstract and Introduction
• 2. Method and 2.1. G is constant
• 2.2. Linear Relation between G and I
• 2.3. Nonlinear Quadratic Relation between G and I
• 3. Results
• 4. Conclusion and References

Abstract

This study rigorously investigates the Keynesian cross model of a national economy with a focus on the dynamic relationship between government spending and economic equilibrium. The model consists of two ordinary differential equations regarding the rate of change of national income and the rate of consumer spending. Three dynamic relationships between national income and government spending are studied. This study aims to classify the stabilities of equilibrium states for the economy by discussing different cases of government spending. Furthermore, the implication of government spending on the national economy is investigated based on phase portraits and bifurcation analysis of the dynamical system in each scenario

1. Introduction

The original Keynesian model was first proposed to explain the persistent unemployment after the Great depression [1]. Since then, it has evolved into many modified models such as the neoclassical Keynesian model [2], the post Keynesian model [3], and the IS-LM model [4]. This paper focuses on one modification of the original Keynesian model – the Keynesian cross model [5].

In this paper, a simple model of a national economy is studied based on the Keynesian cross model. The model in this paper aims to quantify the dynamic relationship between national income and consumer spending, with special attention to the impact of government spending on economic stabilization. This paper applies bifurcation analysis, a technique commonly used in dynamical systems in mathematics, to investigate the dynamics between national income and consumer spending. The rest of the paper is organized as follows: Section 2 describes the methodology, variables, and the mathematics model used in this study; Results of the economic equilibrium are discussed in Section 2. Finally, Section 4 concludes the paper.