Question: Cut and replace

Consider a tree that is planted at time t=0, and let V(t) denote its value at time t. The rate of interst is 100r per cent per year and is continously compounded.

a) Derive an expression for the present value of the tree.

b) The optimal time to cut down the tree is the value of t that maximise the present value of the tree. Derive the condition this optimal time must satisfy.

c) Explain what the condition means in plain language.

After a tree is cut down another tree is planted to takes its place. Taking this into account might change the optimal time to cut trees. If each tree is cut after t periods a new tree would be planted at time t, 2t, 3t etc.

d) Derive an expression for the present value of all trees.

e) Find the sum of the terms in this expression.

f) Show that the optimal time to cut tree satifies the condition     V'(t) / V(t)  =  r / 1 -e

g) By comparing the condition (f) with the previous condition (b), explain how taking into account the possibility of planting new tree changes the optimal cutting time.

Pls help to solve this question....