An economic Economical Model with Allee effect
The Marx model for the profit rate r depending on the ex- ploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Supposing both e(t) and k(t) are continuous functions of time we derive a law for r(t) in the long term. Depending upon the hypothesis set on the growth of k(t) and e(t) in the long term, r(t) can fall to zero or remain constant. This last case contradicts the classical hy- pothesis of Marx stating that the pro t rate must decrease in the long term. Introducing a discrete dynamical system in the model and, supposing that both k and e depend on the profit rate of the previous cycle, we get a discrete dynamical system for r, rn+1 = fa(rn), which is a family of unimodal maps depending on the parameter a, the exploitation rate when the profit is zero. In this map we can have a fixed point when a is small and, when we increase a, we get a cascade of period doubling bifurcations leading to chaos. When a is very big, the system has again periodic stable orbits of period five and period three. But, when we incorporate the Allee effect in the profit rate the system turns completely previsible. Interesting is the fact that we can find in this model the behaviour of the Kondratieff waves in long term economical systems.