Optimal Harvest of an Even-Age Timber Stand
In this exercise you will estimate an empirical logistic model
representing the growth in marketable
volume per acre of Douglas fir over time, and then use that
growth function to determine the economically optimal time to
harvest the stand.
The basic form of the logistic growth function is
V(t) = K /
[1+e-(β0+β1t)]
where V(t) is stand volume at time (age) t.
If you haven't already done so, you can download a
simple Excel demo to see how this function works
and how to express it in Excel.
The formula looks kind of ugly, but the lecture notes on
logistic growth functions show that it can be rearranged
in much simpler linear form:
ln[V/(K-V)] = β0 + β1t
So if you have data on the marketable volume V(t) of timber stands
at various ages t
and you know the maximum possible stand volume K,
than you can use the data to estimate the growth function parameters
β0 and β1.
You can then use these parameters to calculate the expected
marketable volume of a timber stand at any age.
The data for this project are timber
stand ages t and per-acre marketable timber volumes V(t) for
a sample of efficiently managed industrial stands of Douglas Fir.
-
Assuming K = 12,000, calculate the log ratios ln[V/(K-V)] and regress the
log ratios against stand ages. The regression coefficients
will be the parameters
β0
and β1 of the logistic growth function V(t)
for this species.
-
In another worksheet ply, enter stand ages between 25 and 120
years in the first column, and in the adjacent column
enter an Excel growth formula to calculate expected stand
volumes at these ages based on the parameters from your
regression.
-
Mean annual increment (MAI)
is defined as
stand volume divided by age (V/t). At what age does maximum MAI occur?
(The usual biological decision rule is to harvest the stand at the point where
MAI is maximized.)
- Annual incremental [aka marginal] growth (AIG) is
defined as
the extra stand volume obtained in each additional year of growth.
You can calculate this as
dV/dt = β1V[1-(V/K)].
(Alternately, you can divide each increment in volume by five,
the increment in time.)
Calculate the marginal growth at each age.
Create an XY plot of MAI and AIG;
this should remind you of something in basic production theory--explain!
-
Calculate the percent annual growth in stand volume at each
age (AIG/V). Viewing the forest as an investment, and assuming discount rate r
= 0.02, at what age does the rate of growth of the forest decline to the
discount rate? (If the net value of the stand is a simple multiple of stand
volume, you would harvest at that age.)
-
Now suppose initial per-acre planting costs are $500; the price received
at harvest
is $1.00 per cubic foot; harvest cost is $0.30 per cubic foot; and the
discount rate r = 0.025. The present value of the stand at future age
t
is thus:
PV = ($1.00 - $0.30)V/(1.025)t -
$500.
Use your spreadsheet to calculate PV at each age. At what age is
the
stand's PV maximized?
-
This single-rotation model fails to consider the opportunity cost
of deferring
subsequent rotations. If commercial forestry is profitable so that
replanting
is justified, and if prices and costs are assumed to remain constant
through
time, we would expect to see harvesting and replanting every T years,
providing
the landowner a perpetual stream of discounted returns with present
value
PV*, where
PV* = PV + PV/(1+r)T + + PV/(1+r)2T
+
PV/(1+r)3T + . . .
Explain. This formula collapses conveniently to
PV* = PV + PV/rT
Calculate PV* at each harvest age in your spreadsheet. At what value
of T (length of rotation) is
PV* maximized?
-
Plot PV and PV* versus harvest age. Manually set appropriate
lower and
upper bounds on the vertical scale so you can clearly discern the peaks
of these schedules. Print this graph.
-
Now change the discount rate to r = 0.035. Recalculate the PV and PV*
schedules and graph PV and PV* versus t.
What are the optimal
single rotation and multiple-rotation harvest times when r =
0.035? Print this graph.
-
The Federal government offers tax incentives and subsidies to induce
forest
landowners to replant after harvests. Suppose the government pays $400
of the total $500 planting costs. If the landowner's net replanting
costs
are only $100, calculate the new PV and PV* schedules (r = 0.025).
Plot these new PV and PV* schedules against stand age. Manually scale
your plot so you can clearly
discern the peaks of these schedules. Print this graph and your
complete spreadsheet. Compare this graph to the original PV and
PV* graph.
How do replanting subsidies affect the optimal harvest age in the
single
rotation model? Explain. How do replanting subsidies affect
optimal harvest
age in the multiple-rotation model? Explain.
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