Question

1. Suppose that a newly planted forest has present value of W(T), where T is the age at which the forest is vested. Let (p-c)V(T) be the net value of a forest harvested at age T, with volume V(T) and with net price p-c. Let D equal the planting costs, which must be spent each cycle of the harvest. Suppose the interest rate is r so that the present value of a forest harvested at time T is e"" [p-c)V(T) - D. (a) Use the formula that W(T) = e-T (p-c)V(T) - D + (e-7) ( p c)V(T)-D] + (e-T) - c)V(T) - D +... to calculate an expression for W (T). (b) Derive the Faustmann formula the problem with planting costs D, with (p - c)V'(T) on the left-hand-side and r times the appropriate terms on the right-hand-side. Provide and economic interpretation of this equation. (c) Rewrite the Faustmann formula in terms of V'(T)/V(T) and graph this relative to the case where D = 0. Does the presence of planting costs increase or decrease T*? Explain.

Answer 1

1a. The above formula for W(T) can be expressed as -

W(T) = [(p - c)V(T) – DI- + +3+.....
  
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Here in the RHS of the equation the sum expression can be solved using formula for sum of geometric series i.e.

$\sum_{n=1}^{\infty} (\frac{1}{e})^{n}= \frac{\frac{1}{e}}{1- \frac{1}{e}} = \frac{1}{e-1}$

Now the W(T) expression can be written as,

W(T) = (p-c)V(T) - D et (e - 1)

b. Finding first-order total derivative w.r.t T of the formula given in part a.) we get the Faustmann formula.

So,

(p-c)V'(T) = r.e" (e-1W'(T)

where V'(T) = first order derivative of V(T)  w.r.t T , and

W'(T) is first order derivative of W(T)  w.r.t T.

The above expression means that the net present value of the forest harvested at age T is equal to the product of the ( rate of interest, present value of forest harvested at age T and rate of change of present value of forest with respect to age T)

c. The below image shows the expression for V'(T)/V(T)

v (T) = r. e (e-1) w' (T) (p-c) = W(T) ert (e-1) to .... (ü) (PC) v (T) Dividing V'(T) VCT) equation (i) by (4) we get; = wi(T) .vert (e- W (T)erle-1) ITD I WIT) e le-D VI(T) = r. WCT) Tit D V (T) - WCT) L ŞWCT) e Now o ; the expression asumine reduces D= ; to VI(T) VCT) = rotwi(T)) w (T)

det v lt be = be o and VITO wi(T) - W (T) be a w Therefore new enprension o = = r.o. r. w . (This is whe of the enpremno graph - below is Sv=r. w whe

Here D or planting cost is present as constant ( can be considered as sunk cost) in the above expression.Therefore any change in the expression due to change in other factors may cause change in T.

However in case of D, as derivative of constant(that is D) is zero therefore any change in planting cost will not impact the age of harvesting.