Time Lags in Capital Accumulation

We formulate an optimal control capital accumulation model with an exogenously given time lag between investment and the accumulation of the capital stock to analyze the qualitative and quantitative influence of time lags to the system dynamics. Optimal investment paths for a finite time lag are shown to be cyclic as opposed to the monotone paths for instantaneous capital accumulation. Furthermore, we show how to reformulate the retarded differential equation in a suitable way so that sophisticated numerical methods for optimal control can be applied to analyze the impact of the length of the time lag. It turns out that both the frequency and the amplitude of the cycles depend on the length of the investment period.