Shocks in Equilibrium

Use the same set$-$up as in problem $7.$

However, let's say the supply of capital $($maybe coming from the household$)$ is equal to $0\mathrm{.}5$ r$.$ Assume a competitive equilibrium.

Re$-$do part $($v$)$ from above, but in equilibrium:

How does the optimal choice of K depend on A$?$ Prove your answer.

$S=D$

$\frac{1}{2r}=(\frac{A}{r}{)}^{\frac{1}{1-}}$