The profit of a firm harvesting a forest in a single$-$shot harvest is given by:

$=e^{-rT}(pQ(T)-c)$

a$)$ What is the firm's first order condition for profit maximization?

b$)$ What part of the firm's first order condition represent the marginal cost of waiting

to harvest? Explain.

Suppose a forest's volume is approximated by the function:

$Q(T)=aT+b{T}^2-T^3$

c$)$ When is MAI maximized $($maximum yield of forest$)$ as a function of $a,b,$ and $?$

d$)$ Suppose $a=10,b=2,$ and $=0\mathrm{.}02.$ Plot the forest volume over time. You can use

discrete time for the purpose of graphing, with t ranging from $0$ to $100.$

e$)$ When is MAI maximized $($using parameters from b$)?$

f$)$ Assuming continuous time discounting at rate of $r$ and constant cost of cutting,

$c,$ what is the firm's marginal cost of waiting to harvest $($in terms of $a,b,$ and

$,$ and $r)?$