Optimal Timing

Timber Cutting Problem

Mathematical Model

Suppose the value of timber:

(1)
\begin{align} V={2}^{\sqrt{t}} \end{align}

Present value:

(2)
\begin{align} A(t)={2}^{\sqrt{t}}{e}^{-rt} \end{align}
(3)
\begin{align} \ln A={t}^{\frac{1}{2}}\ln2-rt \end{align}

To maximize lnA
FOC

(4)
\begin{align} \frac{d\ln A}{dt}=\frac{\ln 2}{2\sqrt{t}}-r=0 \end{align}
(5)
\begin{align} \frac{\ln 2}{2\sqrt t}=r \end{align}
(6)
\begin{align} {t}^{*}={(\frac{\ln 2}{2r})}^{2} \end{align}

Economics Interpretation

The key of this model is the interest element. It constrains that our present value has an optimal value.

Optimal condition:
Growth rate of the tree equals interest rate

This is shown in equation (5),

(7)
\begin{align} \frac{V'}{V}=\frac{\ln 2}{2\sqrt t}=r \end{align}

When we consider interest rate here, we consider opportunity cost as well. So when the return of continuing growing trees is less than opportunity cost (saving), we cut them and put money into bank.

Further Study

If we include planting cost S here,
Optimal condition:

(8)
\begin{equation} rV(t)+s=V'(t) \end{equation}

when s=0, the same as equation (7).

Other Applications

Wine Storage Problem
The same as Timber Cutting Problem. May be more romantic than the tree problem for some people.

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