I have derived the system of differential equations below, and am coming unstuck on the later parts of the question. This is in an econ course so I have changed some of the symbols so i dont offend anyone in here (we use for inflation rate!).
also note that and
the question is: examine the system above for stationary points and develop stability conditions for any such points.
The stationary point is when . The answer to this is apparently
Now I can kind of understand where this has come from if the fact that x and y are a function of a differential equation means that they are also 0 at a stationary point. Is this correct? Nevertheless I wouldn't mind a proof if possible as my answer is still missing its LHS
For the stability conditions, we use something called Hartman's Lemma which is pretty obscure but it basically involves finding the eigenvalues of the coefficient matrix, and using them to test for stability. What I cannot do, however, is find the eigenvalues.
Again I had scribbled down the characteristic equation as below, but I just can't see how to get to this from my coefficient matrix - and I'm sure its something to do with my poor technical math.
Any help would be much appreciated, it is no way near as long winded as it looks!
It's just multiplying out the matrix and equating both equations to 0. Nothing overly tricky!