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 borrower 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. (a) determine the interest rate on loans made by banks. (b) how will the interest rate change if each borrower has more collateral? (c) explain your results, and discuss

Answer 1

(a) Determine the interest rate on loans made by banks whenW<L(1+r2). How will the interest ratechange if each borrower has more collateral?Suggested Answer.Suppose first thatW<(1+r2)L, which means that the bad borrowers will notrepay(1+r2)Lbut they will payW. Then, since the expected profit for the bank to making a loan iszero in equilibrium, we haveπ=NL(1+r2) + (1-N)W-L(1+r1) =0Solving forr2, we obtain:r2=1+r1-(1-N)WLN-1Therefore, the larger isW, the smaller isr2, the loan interest rate. The larger the quantity of collateralavailable, the larger the payoff the bank receives when a bad borrower defaults. This increases bankprofits, but in equilibrium expected profits are zero for the bank, borrowers benefit from having ahigher quantity of collateral with which to secure loans.

(b) Determine the interest rate on loans made by banks whenW≥L(1+r2). Discuss the results.Suggested Answer.IfWis sufficiently large, i.e.W≥L(1+r2), then the bank’s payoff on a loan to abad borrower is(1+r2)Lsince the bad borrowers agree to repay(1+r2)Lto the bank since otherwisethe bank will seizeW. Therefore, since the bank’s profit must be zero in equilibrium, we haveπ=L(1+r2)-L(1+r1) =0which impliesr2=r1. Thus, ifWis large enough then there is sufficient collateral to eliminate thecredit market friction.