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Bifurcation Analysis of the Keynesian Cross Model: Method and G is constantby@keynesian

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

by Keynesian TechnologyJuly 19th, 2024
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This study investigates the Keynesian cross model of a national economy with a focus on the relationship between government spending and economic equilibrium.
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Xinyu Li, University of Washington.

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.