Consider an economy with N savers and N borrowers. Among the borrowers a fraction a always pays its debts, while a fraction 1 a does not (it defaults on the debt). Savers are paid an interest rate rs and borrowers are charged an interest rate rb

The agents in this economy do not lend directly to each other, instead they use banks as financial intermediaries. There are many banks operating in perfect competition, so that savers will always save in the bank with that offers them the highest interest rate, and borrowers will always borrow from the bank that offers them the lowest interest rate.

All the borrowers look the same, so in principle a bank does not know if it is lending to a good or a bad borrower. Yet, it is possible to tell borrowers apart (good from bad) by conducting an interview and checking their past credit history. This process is called screening and it is costly. The larger the amount of the loan the costlier the screening. For a loan of size L the cost of screening is cL, where c < 1.

(a) Consider a bank that has L in deposits and does not screen any of its borrowers. What are the profits for this bank? Explain each term.

(b) Consider a bank that has L in deposits and screens all of its borrowers. What are the profits for this bank? Explain each term.

(c) When is it profitable for the bank to screen the borrowers?

Find an expression in terms of c, a and rb. Interpret it.

What happens when a increases? Is it more likely that the bank screens borrowers? Why?

What happens when rb increases? Is it more likely that the bank screens borrowers? Why?

(d) Assume that is is profitable to screen all the borrowers. In equilibrium banks must make 0 profits (if they made positive profits they would try to scale up their operation to infinity). Find an expression for the interest rate spread in terms of c.

(e) Define an equilibrium for this economy, and imagine that the screening technology improves and it is now less costly to screen borrowers. Explain the effects of a decrease in c on the equilibrium. Recall that rb is given by c and rs, and rs is obtained by market clearing (aggregate supply and aggregate demand).