Keeping all the settings unchanged, with one additional assumption: the bank provides a guarantee for a share \(\gamma\) of the loan that is sold to the market investors, against the realized losses in the loan's return-precisely, the bank covers \(L-x\) for investors, for any realized \(x

(a) Specify the bank's profit maximization problem and the bank's incentive compatibility constraint.

(b) Rewrite the bank's incentive compatibility constraint in terms of the loan's mean return. With positive loan sale \(s>0\), how is the bank's implied screening effort level, compared with the bank's screening effort level without the loan sale?

(c) Using the definition of compounded promised yield on the loan sale \(r_{l s}=\frac{1}{T} \ln \left(\frac{L s}{1-D}\right)\) in which \(D\) denotes the total deposits, assuming that the mean return of the loan \(\bar{x}(e)\) varies with the level of screening effort \(e\) as \(\bar{x}(e)=L[1-\alpha \exp (-\beta e)]\), solve for the bank's optimal volume on loan sale, \(s\).
i. How does the spread in the bank's funding sources, \(r_{d}-r_{s}\), affect the equilibrium \(s\) ?
ii. How does the premium in the loan sale, \(r_{l s}-r_{s}\), affect the equilibrium \(s\) ?
iii. How does the probability of the bank's insolvency affect the equilibrium \(s\) ?