Asset Retirement with Infinitely Repeated Alternative Replacements: Harvest Age and Species Choice in Forestry
At what
age should productive assets be retired? How should replacements be chosen when
they differ in their uncertain ability to generate future incomes? As a particular
version of that problem, we study the tree harvesting decision with two
possible replacement species whose values as timber are stochastic and whose
growth functions are deterministic. In the single-rotation (Wicksell) problem
starting with a bare piece of land (an empty shop), it is optimal to choose and
plant one species immediately if its current value is sufficiently high relative
to that of the other species (the alternative equipment). However, if the
species are insufficiently price-differentiated, it is preferable to leave the
land vacant (the shop empty) despite the opportunity cost of doing so. In the
repeated version of the problem, it is never optimal to leave the land bare
provided the cost of replacement is null. Furthermore, the optimal harvest
(tree retirement) age not only depends on the price and current productivity of
the trees in place but also on the price and productivity of the other species,
because it may replace the current one. The harvest age reaches a peak at some
critical threshold of the relative price that signals the necessity to switch
to the alternative species; indeed this is when the opportunity cost of
choosing one alternative replacement over the other is the highest. The land
value (and also the value of the firm) is similar to an American option with free
boundary, infinite expiry period, and endogenous payoff. The paper highlights
the opportunity cost of alternative replacement options, and the central role
of their volatility in both asset-retirement and replacement-choice decisions.
All results are derived analytically; a numerical treatment by the penalty
method completes the resolution.
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