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Equilibrium Value of the First Order Affine Difference Equation
When k grows very large (approaching infinity), we called it long run condition. Equilibrium value is the long term solution of the difference equation. Equation (3) shows the equilibrium value of the first order affine difference equation. In the long run condition _{ } produces _{ } . When _{ } , the equilibrium value is a constant, which is _{ } . When _{ } , however, _{ } has no solution (no equilibrium value) unless _{ } . If _{ } , there are infinite number of solution because any value of _{ } will do. Notice that the equilibrium value (unless _{ } ) does not dependent on the initial value, it only depends on the difference equation.
See:
Numerical Example
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Preferable reference for this tutorial is
Teknomo, Kardi (2015) Difference Equation Tutorial. https:\\people.revoledu.com\kardi\ tutorial\DifferenceEquation\