FOREST MANAGEMENT $(12$ points total$)$

You are responsible for managing a stand of old$-$growth forest in Oregon. The old$-$ growth forest

yields an amenity flow of:

At $=$ a ln St

where a $>0$ and St is the stock of old$-$growth trees $($in hectares$)$ at time t$.$ If you cut down a

hectare of old$-$growth forest, the net benefits $($revenues minus costs$)$ are given by $($pt$$ c$)$qt$.$ The

land is then reforested with new$-$growth forests that are managed efficiently using the Faust$-$

mann rotation formula. Managed in this fashion, the stream of profits from a hectare of new$-$

growth forest equals $.$ The amount of new$-$growth forest that you initially have is given by Y$0.$

Note that dA

d$($a ln St $)$

a

$=$

$=$

dSt

dSt

St$.$

$(1\mathrm{.}1)$ Write down the optimization problem.

$(1\mathrm{.}2)$ Write down the first$-$order conditions for this problem.

$(1\mathrm{.}3)$ Derive an expression defining the economically efficient steady$-$state amount of old$-$

growth forest that will be maintained $($not cut$).$ Call this amount S$*.$

$(1\mathrm{.}4)$ Let a $=615,$ r $=0\mathrm{.}05,$ $=1,000,($pt$$ c$)=40,000.$ What is the value of S$*?$

$(1\mathrm{.}5)$ If the interest rate increases to $10\%($r $=0\mathrm{.}10),$ what happens to the optimal amount of old$-$

growth forest hectares preserved? Carefully explain the intuition behind this result