$(6)$ Another variation. There is a large population of borrowers and lenders. Each active

borrower takes a loan $L$ from a lender at every date, produces $F(L)$ and repays $R.$ If

a borrower defaults on a loan, the lender tries to spread the word that the borrower has

defaulted. Just how successful he is in doing so is described as follows:

When a defaulting borrower appears to a new lender, the lender tries to remember if this

borrower has been complained about. But with probability $q$ between $0$ and $1$ he doesn't

know the old lender, so the new lender may not have heard about the default. In that case

the lender starts up a relationship with the borrower, and the past slate is wiped clean.

If the lender has heard about this borrower's past behavior, then he does not lend to the

borrower. The borrower waits one period, getting zero, and then meets a lender again the

day after, with exactly the same story repeating itself.

$[$a$]$ If every borrower borrows on the same terms from every lender, show that the lifetime

value to a borrower who has just defaulted $-$ but does not plan to default again $-$ is given

by

$V=\frac{q[F(L)-R]}{(1-)[1-(1-q)]}$

$[$Hint: it must be the case that for a defaulting borrower who does not plan to default again,

$V=q\frac{F(L)-R}{1-}+(1-q)V$

First explain very clearly what this equation is$,$ and then use it$.]$

$[$b$]$ Write down the no$-$default constraint using the solution from part $($a$),$ and show that

after some simplification, it is given by

$F(L)-R[1-(1-q)]F(L).$

$[$c$]$ Using this formula, prove that if $q$ is close to $1,$ then the market breaks down because no

lender will be unable to recover his loan.