A non-linear restatement of Kalecki’s business cycle model with non-constant capital depreciation

This paper deals with Kalecki's 1935 business cycle model, where a finite time lag in the investment dynamics is assumed. The time lag is the gestation period elapsing between orders for capital goods and deliveries of finished industrial equipment. Including the actual mainstream theory, the economic literature agrees on the consequences that time lag has on the economic activity. It is a cause of persistent economic fluctuations. Following some recent research lines on this model, here we restate the Kalecki approach, assuming sigmoidal functions in addition to Kalecki's linear treatment and further considering a non-constant capital depreciation. Never made until now, this last assumption is such that to yield, in place of a delayed differential equation, a Volterra delayed integro-differential equation. Taken the time delay and the rate of capital depreciation as critical parameters, a qualitative study of that equation is carried out. We proved that with a small-time lag stable equilibria arise. But, when the delay increases, equilibria are destabilized through Hopf bifurcations and stability switches occur. Consequently, a variety of cyclical behaviors appear.