# Phase Diagram - growth model

The dynamics of Ramsey-Cass-Koopman growth model is usually summarized in phase diagrams with the 2 equations (conventional symbols apply): \begin{align*} \frac{\dot c}{c}=&\frac{r-\rho}{\theta}\\ \\ \dot k =& f(k) - (n+g+\delta)k \end{align*}

From these equations you work out the equilibrium and whether it is a saddle path.

However this analysis is static; some of those parameters are time varying (e.g., 1950 population growth isn't the same as in 2017). How can you represent these changes in this essentially static graphical framework?

• I've just analyzed the stability of the system at different points in time corresponding to periods when there were large changes in the deep parameters. Seminar audiences have vaguely complained though that there's something wrong with such an approach. Apr 23, 2017 at 16:11
• Unfortunately that problem does seem vague. Asking a critic for advice would be the academic thing to do. Apr 23, 2017 at 16:54

In control systems engineering, there was the "Aizerman Conjecture" (the transliteration of Aizerman varies) that argued that a linear time-varying system was stable if the parameters (state matrices) at all times corresponded to stable systems. However, counter-examples were found to this conjecture.

About the only way to show that a time-varying system is stable (for example) is to find a Lyapunov function that holds at all times. This is generally going to be difficult.

Something like graphical analysis would be technically invalid, as you are violating assumptions that are in place. You are pretty much going to have to derive any results from first principles.

What time varying-parameters do to a model like this is to make the long-term equilibrium to stop being a specific point in the phase diagram. The zero-change loci shift and move around, and so does their intersection.

A simple thing that you could do, at least from a pedagogical point of view, is to reproduce the effect of "structural breaks" and not smooth time-variation. So for a certain time period, a parameter of the model was fixed at a certain level, but then it jumped to a different level (say depreciation changed because in the past it was more buildings and machinery, now it is more IT, software and intangibles).

Such an abrupt change essentially discretizes a smooth time-variation, and it is an acceptable approximation to it.

This means that one draws two pairs of zero-change loci in the phase diagram, one representing the situation before and one after the structural break. And one shows how the economy behaves by jumping from the one saddle-path to the other.

See this blog post of mine where I implemented this approach to reflect in simple descriptive terms the current depression of the Greek Economy.