Q1. The cost of an information producer (i.p.) to produce information is 10, which is non-monetary. The i.p. is risk averse with a utility function, √x, defined over monetary wealth, x. The i.p. demands a minimum expected utility level of 20; otherwise, he/she will work elsewhere. A risk-neutral firm that wishes to attract capital has to hire the i.p. to release information to the capital market. The firm can monitor the i.p.'s effort. This monitoring produces a noisy signal. If the 1.p. produced information, the signal says that he/she did only with probability 0.8 and is erroneous with probability 0.2. If the 1.p. did not produce Information, the signal says that he/she did only with probability 0.2 and that he/she did not with probability 0.8.
(a) What would be the best way for the firm to compensate the i.p. so as to induce information production?
(b) Now, there are three identical i.p.s, each of them deals with a separate firm. The three i.p.s pool their payoffs that are shared equally among themselves. The three i.p.s are cooperating in that they either all produce information or all do nothing. Signals between the three firms are mutually independent. Does the solution in part (a) remain incentive compatible to ensure the i.p.s to produce information? Is it possible that the i.p.s in this case are made better off as compared to the case in part (a)?
Q2. Consider a bank that is faced with two types of borrowers whom it cannot distinguish, G and B. The type G borrower wishes to borrow $100 to invest in a single-period project that yields $135 with probability 0.9 and 810 with probability 0.1 at the end of the period. The type B borrower wishes to borrow the same amount to invest in a single-period project that yields 8150 with probability 0.7 and $10 with probability 0.3 at the end of the period. If the borrower comes to the bank for a loan in the second period, he/she will use the loan to invest in exactly the same kind of project as in the first period. There are no taxes and no bankruptcy costs. Everybody Is risk neutral and the riskless rate of Interest per period is 5%. The bank's prior belief is that any given borrower is of type G with probability 0.7 and of type B with probability 0.3. The bank is competitive in that it earns zero expected profits in each period.
(a) What are the interest rates charged by the bank on the first-period loans for the type G and type B borrowers, respectively?
(b) What are the interest rates charged by the bank on the second-period loans for the type G and type B borrowers, respectively, conditioned on their first-period loans being performing?
(c) What are the interest rates charged by the bank on the second-period loans for the type G and type B borrowers, respectively, conditioned on their first-period loans being non-performing?
(d) Since a borrower's first-period repayment behavior may not be freely observable by other banks, this gives some market power to the incumbent bank that made the first-period loan. The incumbent bank as such is no longer competitive in the second period. Comment on the possible interest rates that would be charged by the incumbent bank on the second-period loan under this scenario.