Moving equilibrium theorem
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Consider a dynamical system
(1)..........
(2)..........
with the state variables x and y. Assume that x is fast and y is slow. Assume that the system (1) gives, for any fixed y, an asymptotically stable solution . Substitution this for x in (2) yields
(3)...........
Here y has been replaced by Y to indicate that the solution Y to (3) differs from the solution for y obtainable from the system (1), (2).
The Moving Equilibrium Theorem suggested by Lotka states that the solutions Y obtainable from (3) approximate the solutions y obtainable from (1), (2) provided the partial system (1) is asymptotically stable in x for any given y and heavily damped (fast).
The theorem has been proved for linear systems comprising real vectors x and y. It permits reducing high-dimensional dynamical problems to lower dimensions and underlies Alfred Marshall's temporary equilibrium method.
[edit] References
- Schlicht, E. (1985). Isolation and Aggregation in Economics. Springer Verlag. ISBN 0-387-15254-7.
- Schlicht, E. (1997). "The Moving Equilibrium Theorem again". Economic Modelling 14: 271-278.