I'm reading an interesting book on mathematical economics written by a German professor. It contains many theory chapters and exercises, but not all of them have the solution at the end of the book. One of these exercises is the following and I would like to ask how to think in order to find law of motions and objective functions in these kind of problems.
"Imagine you are an economist only interested in maximizing the present value of your endowment. You own a renewable resource, for example a piece of forest. The amount of wood in your forest at a point in time $t$ is given by $x_t$. Trees grow at $b(x_t)$ and you harvest at $t$ the quantity $c_t$.
(a) What is the law of motion for $x_t$?
(b) What is your objective function if prices at t per unit of wood is given by $p_t$, your horizon is infinity and you have perfect information?
(c) How much should you harvest per period when the interest rate is constant? Does this change when the interest rate is time-variable?"
When I thought about the law of motion of $x_t$ I find difficult to link $b(x_t)$ and the unknown law of motion together, but I tried with $b(x_t)=x_t-c_t$. Next question is related to the objective function and I tried with $U_t=\sum_{\tau=t}^\infty=\beta^{(\tau-t)}\cdot p_\tau\cdot x_\tau$
I don't know how to go further, because I'm not really into this kind of stuff.
Thanks for your possible help.