Question

Suppose a credit market with a good borrowers and 1 − a bad borrowers. The

good borrowers are all identical, and always repay their loans. Bad borrowers

never repay their loans. Banks issue deposits that pay a real interest rate r1 , and

make loans to borrowers. Banks cannot tell the difference between a good bor-

rower and a bad one. Each borrower has collateral, which is an asset that is worth

A units of future consumption goods in the future period. Determine the interest

rate on loans made by banks.

Answer 1

$A \lt(1 + r_{1})\times PV$

which means that bad borrowers will not pay back $(1 + r_{1})\times PV,$ but they will pay A. Since the bank's projected profit from lending is zero in equilibrium, we have $\pi= N \times PV\times (1 + r_{1}) + (1-N)\times A \times PV (1 + r) = 0$ . Hence,$r_{1} = 1 + r - (1-N)\times A\times PV \times N-1$

The more collateral, the more repayment the bank will receive if a bad borrower goes bankrupt. Borrowers benefit from having more collateral to secure loans.

If $A\ge PV\times(1 + r_{1})$ , then the payment to the bank on a loan to a bad borrower will be $(1 + r_{1})\times PV$ , since bad borrowers agree to give $(1 + r_{1})\times PV$ to the bank, otherwise the bank will arrest A. Since in equilibrium the bank's profit must be zero, then $\pi=PV\times(1 + r_{1}) - PV\times (1 + r) = 0$ , which means $r_{1} = r.$ If A is significant, then there is normal collateral for leveling problems in the credit market.