Consider the following $$portfolio choice$$ problem. The investor has initial wealth w and utility u$($x$)=$ln$($x$).$ There is a safe asset $($such as a US government bond$)$ that has net real return of zero. There is also a risky asset with a random net return that has only two possible returns, R$1$ with probability q and R$0$ with probability $1$ q$.$ Let Z be the amount invested in the risky asset, so that w $$ Z is invested in the safe asset. Note that w is her wealth.

a$).$ Find Z as a function of w$.$ Does the investor put more or less of his portfolio into the risky asset as her wealth increases?

b$).$ Another investor has the utility function u$($x$)=-$e$^(-$x$).$ How does her investment in the risky asset change with wealth?

c$).$ Find the coefficients of absolute risk aversion r$($x$)=-($u$"($x$)/$u$\text{'}($x$))$ for the two investors. How do they depend on wealth? How does this account for the qualitative difference in the answers you obtain in parts $($a$)$ and $($b$)?$