Moving equilibrium theorem

From Wikipedia, the free encyclopedia

Consider a dynamical system

(1).......... $\dot{x}=f(x,y)$

(2).......... $\qquad \dot{y}=g(x,y)$

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 $\bar{x}(y)$ . Substituting this for $x$ in (2) yields

(3).......... $\qquad \dot{Y}=g(\bar{x}(Y),Y)=:G(Y)$ .

Here $y$ has been replaced by $Y$ to indicate that the solution $Y$ to (3) differs from the solution for $y$ obtainable from the system (1), (2).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. doi:10.1016/S0264-9993(96)01034-6.

Categories: Economics theorems

See also ebooksgratis.com: no banners, no cookies, totally FREE.

Moving equilibrium theorem

From Wikipedia, the free encyclopedia

[edit] References

Views

Navigation

Interaction

Search