At the Right Time:Ramsey-Cass-Koopmans Model with Declining Population

This paper investigates how population decline may affect the optimal path in two types of Ramsey-Cass- Koopmans (RCK) model with child rearing costs. An optimal path exists in both models under economically plausible conditions and a new type of optimal path exists in the model where the discount rate is only the time preference rate. Under population decline, the existence and properties of an optimal path depend on the range of the rates of population change, regardless of the child rearing costs. First, when population decline is mild, the optimal path is a saddle-point path converging to a finite steady state, as in the standard RCK model with increasing population. Second, when population decline is faster, the optimal path is a saddle-point path converging, by reversible investment, to a finite steady state (i.e., a balanced growth path (BGP)), at which per capita consumption is larger than per capita income. Third, when population decline is even faster, the optimal path can be an asymptotically BGP, along which both per capita consumption and income keep increasing permanently. We show empirical relevance of these optimal paths by Japanese data and World Population Prospects 2019.