Essays in Honour of Karl-Göran Mäler
Edited by Bengt Kriström, Partha Dasgupta and Karl-Gustaf Löfgren
Chapter 16: What if Jevons Had Actually Liked Trees?
Robert M. Solow We usually credit W.S. Jevons with having provided a clear statement and analysis of the problem facing a producer with an intertemporal pointinput–point-output production technology. Suppose a tree is planted (costlessly, for simplicity) at time t. The real net value of its timber (after harvesting costs) is f (a) if the tree is cut down at time t+a, that is, at age a. The producer chooses the a that maximizes e–raf (a), where r is the appropriate discount rate, usually a market interest rate. The obvious necessary condition for an interior maximum at a* is that a* satisfy f ′(a)/f (a) = r. This defines a local maximum if f ″(a) rf (a), the natural growth of the tree is earning a better return than the interest rate on the proceeds from earlier harvesting, so it is better to wait. If there is an initial planting cost c, a* is still the best choice once the cost is sunk. Before that point, the producer would want e–ra*f (a*) > c, or else it would be better to abandon the project altogether. If the land had alternative uses, we would be dealing with a quite different problem. This solution is appropriate if tree planting is a one-time project, if nothing interesting happens after the tree is harvested. A more realistic presumption is that another tree is planted at t+a* on the land just cleared. Unless technology or the interest rate has changed, the second-generation tree...
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