1.  Logistic function

Using Excel, for domain -3.5 < x < +3.5 in steps of 0.25, calculate the values of the standard logistic function y = 1/[1+exp(-x)], or equivalently, y = exp(x)/[1+exp(x)]. 

In an adjacent column use the NORMDIST function to obtain the corresponding values of the N(0,1) distribution function for the same x-values. 

Create an XY plot of the logistic and NORMDIST functions against x.

Determine the scale value k for x in the logistic function y = 1/[1+exp(-kx)] that best fits the standard logistic to the N(0,1) distribution.  (What value of k minimizes the sum of squared deviations between the logistic and normal?)

2.  Logistic growth

The general form of the logistic growth model y = K/(1+exp[-(β₀+β₁x)]) has 3 parameters: K is the asymptotic maximum value and β₀ and β₁ shift and rescale the x variable.  By algebraic manipulation this model can be transformed to ln[y/(K-y)] = β₀+β₁x.  Knowing K, you can calculate the LHS log ratio and then estimate the RHS parameters β₀ and β₁ via OLS.

The data for this project are timber stand ages t and per-acre marketable timber volumes V(t) for a sample of efficiently managed commercial stands of Douglas Fir.  Estimate the parameters β₀ and β₁ of the logistic growth function V(t) for this species if K=12,000 (the asymptotic maximum volume per acre).

Use your estimated function to model stand volume for ages 25 < t < 100.  "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 growth" (AIG) is defined as the extra stand volume obtained in each additional year of growth (dV/dt). Calculate this incremental 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 planting costs are $1,000; the price received at harvest is $1.00 per board foot; harvest cost is $0.30 per board foot; and the discount rate r = 0.02. The present value of the stand at future age t is thus:

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: the sooner you harvest, the sooner you can start the next rotation.  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

which collapses conveniently to

Calculate PV* at each harvest age in your spreadsheet. At what value of T (age at harvest) is PV* maximized?   Plot PV and PV* against stand age so you can clearly discern the peaks of these schedules.

The Federal government offers tax incentives and subsidies to induce forest landowners to replant after harvests.  Suppose the government pays $500 of the total $1,000 planting costs.  If the landowner's net replanting costs are only $500, calculate the new PV and PV* schedules (r = 0.02).  How does the replanting subsidy affect the optimal harvest age in the single rotation model?   How does it affect optimal harvest age in the multiple-rotation model?  Explain.