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First Order ODE Stability

  1. Sep 9, 2011 #1
    1. The problem statement, all variables and given/known data
    In the market for a certain good, the price p(t) adjusts continuously in the presence of excess supply or demand:
    [itex]\frac{dp}{dt}[/itex] = F(D(p)-S(p)) where F(0) = 0, F'>0.
    Obtain the condition for stability of the equilibrium price p* in terms of the slopes D'(p*) and S'(p*), and illustrate stable and unstable equilibria graphically for straight-line supply and demand schedules.


    2. Relevant equations
    General I suppose, p' = F(p)


    3. The attempt at a solution
    This is what I've come up with so far, but I'm not sure if it's correct or delves deeply enough:
    p'=F(D(p)-S(p))
    We know there would be an equilibrium point where p'=0 and therefore where F(D(p)-S(p))=0.
    Since F(0) = 0, we know that at least one equilibrium exists at p=0, so p*=0. From that, we can see that p'=0=F(D(0)-S(0)) and therefore at p*, D(0) = S(0).

    To test stability we would look at the sign of p'' at p*.
    p'' = F'(D(p*)-S(p*))(D'(p*)-S'(p*))
    Since we're given F'>0, we know that the D'(p*)-S'(p*) dictates stability.
    So if D'(p*) > S'(p*) then p'' >0 and p* is unstable.
    Else, if D'(p*) < S'(p*) then p'' <0 and p* is stable.

    That's what I've come up with. It seems a bit simple which leads me to believe I may be missing something. Any help or insight would be greatly appreciated.

    Thanks!
     
  2. jcsd
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