Bifurcation Analysis of the Keynesian Cross Model: Method and G is constant

Xinyu Li, University of Washington. 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 2. Method The Keynesian cross model builds upon two ordinary differential equations [6]: where C ≥ 0 is the rate of consumer spending, I ≥ 0 is the national income, and G ≥ 0 is the rate of government spending. The parameters α and β satisfy 1 < α < ∞, 1 ≤ β < ∞. Three relations between government spending and national income are discussed in the following subsections. 2.1. G is constant Consider a model consisting of equations (1) and (2) along with a constant government spending G. To determine the equilibrium state for this model, I find the point where = Ċ = 0. Rearranging terms, I obtain the following equilibrium: In order to calculate the stability of this fixed point, I compute the Jacobian matrix and eigenvalues: This paper is available on arxiv under CC 4.0 license. Xinyu Li, University of Washington. Xinyu Li, University of Washington. 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 and Introduction Abstract and Introduction 2. Method and 2.1. G is constant 2. Method and 2.1. G is constant 2.2. Linear Relation between G and I 2.2. Linear Relation between G and I 2.3. Nonlinear Quadratic Relation between G and I 2.3. Nonlinear Quadratic Relation between G and I 3. Results 3. Results 4. Conclusion and References 4. Conclusion and References 2. Method The Keynesian cross model builds upon two ordinary differential equations [6]: where C ≥ 0 is the rate of consumer spending, I ≥ 0 is the national income, and G ≥ 0 is the rate of government spending. The parameters α and β satisfy 1 < α < ∞, 1 ≤ β < ∞. Three relations between government spending and national income are discussed in the following subsections. 2.1. G is constant Consider a model consisting of equations (1) and (2) along with a constant government spending G. To determine the equilibrium state for this model, I find the point where = Ċ = 0. Rearranging terms, I obtain the following equilibrium: In order to calculate the stability of this fixed point, I compute the Jacobian matrix and eigenvalues: This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv