An Economical Model with Allee effect

The Marx model for the profit rate r depending on the exploitation 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 hypothesis of Marx stating that the profit 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.