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If all the paths of equilibrium noncyclically converge to it, the equilibrium is a stable node. The conditions for the equilibrium to be a stable node are a function of the system being evaluated, in particular, whether the issue of concern involves discrete versus continuous change or is a linear versus nonlinear system. If the dynamic system is discrete and linear,
ŷ = ay + b
The equilibrium is stable if and only if a > 0.
For continuous linear dynamic systems, an equilibrium is defined as a stable node if the slope of the differential equation in the neighborhood of the equilibrium is negative. For nonlinear discrete or continuous dynamic systems, the Taylor expansion about xi*, a nonzero equilibrium, must satisfy the following for xi* to be stable: |f’ (xi*)| < 1. The concept of a stable node can be extended to simultaneous systems of discrete or continuous equations. The condition for a stable node is the same for discrete and continuous simultaneous systems of equation. If |λi| < 0, where λi are the eigenvalues that solve equation, then λi is stable. If the eigenvalues are complex, the real part must be negative for the eigenvalues to be stable.
The concept of a stable node is used to describe the dynamics of (1) an oligopoly—stability of a Cournot solution; 2) the IS-LM—stability after monetary and/or fiscal shocks; (3) the model of inflation and unemployment— stability of fiscal or employment policies; and (4) population models—stability of growth. The characterization of a node as stable is most useful in the qualitative analysis of differential equations using phase diagrams, which describes the paths of a system in and out of equilibrium.
- Hoy, Michael, John Livernois, Chris McKenna, and Ray Rees. 2001. Mathematics for Economics, 2nd ed. Cambridge, MA: MIT Press.
- Shone, Ronald. 2002. Economic Dynamics: Phase Diagrams and Their Economic Application, 2nd ed. Cambridge, U.K., and New York: Cambridge University Press.
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