Assume that there are \(n\) banks competing in the credit market. Instead of having each bank observe a private signal from each applicant, now assume that there is a credit rating agency providing credit scores of all potential borrowers for all banks; that is, when an applicant approaches multiple banks, these banks will see the same credit score of her. Precisely, a credit score is a noisy signal about the applicant's type, such that

- If the applicant is a truly good borrower, the probability that the signal indicates that she is good is \(\zeta(H \mid \mathcal{H})\);

- If the applicant is a truly bad borrower, the probability that the signal indicates that she is bad is \(\zeta(L \mid \mathcal{L})\).

(a) What is the average loan loss rate, if all banks rely on the common credit scores? Suppose \(p(H \mid \mathcal{H})=p(L \mid \mathcal{L})=0.9, n=20\), and \(\zeta(H \mid \mathcal{H})=\zeta(L \mid \mathcal{L})\), what is the minimum \(\zeta(H \mid \mathcal{H}) / \zeta(L \mid \mathcal{L})\) that achieves the same loan loss rate, compared with the case in Section 4.5 where each bank observes a private signal \(p(H \mid \mathcal{H}) /\) \(p(L \mid \mathcal{L})\) from each applicant?

(b) Suppose \(p(H \mid \mathcal{H})=p(L \mid \mathcal{L})=0.9, n=20, \zeta(H \mid \mathcal{H})=\zeta(L \mid \mathcal{L})\), \(\pi_{H}=1, \pi_{L}=0.5, a=0.8\), and \(r=0.1\). What is the minimum \(\zeta(H \mid \mathcal{H}) / \zeta(L \mid \mathcal{L})\) that achieves the same expected profit for each bank, compared with the case in Section 4.5 where each bank observes a private signal \(p(H \mid \mathcal{H}) / p(L \mid \mathcal{L})\) from each applicant?

(c) Continue with question (b), keep all parameters the same, except \(\pi_{L}=0.8\) and \(a=0.9\), compute the minimum \(\zeta(H \mid \mathcal{H})\) / \(\zeta(L \mid \mathcal{L})\). Comparing the results in questions (b) and (c), when are banks more likely to prefer using common credit scores instead of individual signals?