The natural growth of the forest stock provides part of the story in characterizing the evolution of the forest stock. The other part of the story depends on the harvesting behavior of the Easter Islanders. The key economic point is that under conditions of incomplete property rights, it is natural to expect that, for any given stock, the amount harvested at any given time period, H(t), is increasing with population size. In other words, for any given set of external conditions and any given forest stock, we expect more harvesting to occur if there is a larger population seeking to use the forest stock. Similarly, for a given population, harvesting is easier if the stock is larger, and therefore more harvesting would be likely to occur. In the case of pure open access (no enforceable property rights) and standard demand and production functions, the harvesting function can be shown to take on the form H = aSP, where a is a parameter reflecting demand conditions and the harvesting technology and P is the population. This functional form is used here for concreteness, although other related functional forms would have similar properties. The differential equation that governs the evolution of the forest stock can therefore be written as
dS/dt = G(t) – H(t) = rS(1 – S/K)- aSP, (1) where state variables S and P are functions to time t.
