In an economy there is a large number of risk-neutral entrepreneurs who are protected by limited liability, and have no initial wealth. The entrepreneurs undertake projects at a cost \(I=1\). The financing of each project is done by a monopolistic and risk-neutral bank. Projects can be of two types: good and safe (undertaken by good entrepreneurs), or bad and risky (undertaken by bad entrepreneurs). The type of an entrepreneur is known only to the entrepreneur himself. The fraction of good entrepreneurs in the economy is \(a\) with \(0

\(1>p>q>0, B>G\) and \(p G>q B>1\). All agents know the probability distributions. To focus on the adverse selection problem, we further assume that the bank does not have deposit insurance so that her behavior is not distorted by the limited liability.

(a) First, assume complete information. Show that the bank can extract all profits by extending loans at terms that depend on the type of the entrepreneur.

(b) Suppose next that only the entrepreneurs know their own type. Illustrate how the mixture of loan applicants will change as the gross rate of return demanded by the bank changes. Explain how the bank's expected return will vary with the gross rate of return demanded. Is there a gross rate of return for which only good entrepreneurs demand a loan?

(c) Which gross rate of return will a profit-maximizing monopolistic bank choose? What is critical to your answer?

(d) Assume now that all entrepreneurs have wealth \(W\) such that \(0