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.

  1. 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.

  2. 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.

  3. 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.)

  4. 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!

  5. 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.)

  6. 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?

  7. 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?

  8. 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.

  9. 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.

  10. 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.