Consider the following optimal control problem:

max

In$($x$)$e dt

$0$

subject to

$9=$y$-2$ V$>=0$

with $3(0)=$ yo $>0$ given.

$(2\mathrm{.}1)$ Write down the Hamilton$-$Jacob$-$Bellman equation, its corresponding first order condition and express a in term of the Bellman Value function.

$(2\mathrm{.}2)$ Using maximum principle write down the first order condition.

$(2\mathrm{.}3)$ From the first order condition obtained in $(2\mathrm{.}2),$ obtain the long$-$run steady state of the dynamic system of $($x$*,$ y$*).$