$1.$ 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 $=$alnSt

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$0.$ NotethatdA $=$d$($alnSt$)=$ 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$*.$

$(1\mathrm{.}4)$Leta$=615,$r$=0\mathrm{.}05,$$=1,000,($pt$$c$)=40,000.$WhatisthevalueofS$*?$

$(1\mathrm{.}5)$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.