The delay Solow model with the Lambert W function

This paper shows how cyclical behavior can emerge in a simple Solow capital accumulation model with population not necessarily growing at a positive rate. For this purpose, it introduces a production delay (i.e., time-to-build) and the resulting capital accumulation equation is a delay differential equation having delay-dependent coefficients. With the use of the Lambert W function, it constructs a condition under which the interaction of the delays and the population growth rate can lead to the appearance of cyclical behavior in the economy and possibly to switch in stability near the steady state, and emergence of constant fluctuations and growth cycle, in consequence. The paper heavily draws from the properties of the Lambert W function, which has been so far used more often in exact sciences rather than social sciences. In addition, the paper demonstrates that (i) if population declines, the model is more likely to be unstable; (ii) introduction of the delay directly to the accumulation equation in an intensive form produces qualitatively similar and dissimilar dynamics.