$4.(18$ points$)$ Suppose there is a credit market with the fraction of $\backslash ($ a $\backslash )$ good borrowers, and the fraction of $\backslash (1-$a $\backslash )$ bad borrowers, with the total number of borrowers equal $\backslash ($ N$\_\{$b$\}\backslash ).$ Banks cannot differentiate between good and bad borrowers when making loans $($asymmetric information$)$ and loan out $\backslash ($ l $\backslash )$ units of goods to each borrower, good or bad. There are $\backslash ($ N$\_\{$d$\}\backslash )$ depositors$/$savers in the economy. Banks attract deposits in the amount of $\backslash ($ L $\backslash )$ from each of them and promise to pay a net real interest rate of $\backslash ($ r$\_\{1\}\backslash )$ to depositors. Banks charge net interest $\backslash ($ r$\_\{2\}\backslash )$ on loans. Good borrowers are identical and always repay their loans, while a debt collection agency makes bad borrowers pay a fraction $\backslash (0\backslash$leq f $\backslash$leq$\backslash$left$(1+$r$\_\{2\}\backslash$right$)\backslash )$ of their loans $($who$,$ in the absence of the agency, would pay nothing$).$ The banking sector is competitive, and the profit equals zero in equilibrium.

$($a$)(5$ points$)$ Using the bank balance sheet, find the relationship between $\backslash ($ N$\_\{$d$\},$ N$\_\{$b$\}\backslash ),\backslash ($ l $\backslash ),$ and $\backslash ($ L $\backslash ).$

$($b$)(10$ points$)$ Using the assumption of the competitive banking sector, find an expression for the interest rate on loans, $\backslash ($ r$\_\{2\}\backslash ),$ made by banks, as a function of $\backslash ($ a$,$ f $\backslash ),$ and $\backslash ($ r$\_\{1\}\backslash ).$

$($c$)(1$ points$)$ How will the interest rate change if the debt collection agency makes each borrower pay a higher fraction $\backslash ($ f $\backslash )$ of their loans taken?

$($d$)(2$ points$)$ What must $\backslash ($ f $\backslash )$ be for the interest rate on loans to equal $\backslash ($ r$\_\{1\}\backslash )?$