A credit market has two types of borrowers: $s($safe$)$ and $r($risky$)$; each

has proportion $\frac{1}{2}.$ Any borrower borrows $1$ unit of capital to invest in a project. A project can

result in either one of the two outcomes: good or bad.

Under bad outcome, the return is $0.$ Under good outcome, the return is $x_4=108$ for type $s$ and $x,$

$=111$ for type $r.$ The probability of good outcome is $p_s=\frac{2}{9}$ for type $s$ and $p_r=\frac{1}{6}$ for type $r.$

When the interest rate is $i,$ the borrower pays back $i$ to the lender if the outcome is good and pays

back nothing if the outcome is bad. The opportunity cost of a borrower is $B_0=12.$ The

opportunity cost of a lender is $L_0=7.$ Assume that the credit market is competitive, so any

lender makes zero net profit.

$($a$)$ Determine the maximum acceptable interest rates for type $s$ and type $r.$

$($b$)$ Consider the full information case where the lender knows types of individual borrowers. In

this case, determine:

$($i$)$ the interest rates offered to type $s$ and type $r$ borrowers,

$($ii$)$ which type borrows and which type does not borrow,

$($iii$)$ aggregate income.

$($c$)$ Consider the asymmetric information case where the lender does not know types of

individual borrowers. In this case, determine:

$($i$)$ the average repayment probability,

$($ii$)$ interest rate offered,

$($iii$)$ which type borrows and which type does not borrow,

$($iv$)$ aggregate income,

$($v$)$ if there is an under investment or an over investment problem.