326
The Regulation of Banks
def
DðuÞ ¼ E½hju; C E½hju; S ; which is equal to ð þy ð þy DðuÞ ¼ h dHC ðhjuÞ h dHS ðhjuÞ; 0
0
or, after integrating by parts, ð þy DðuÞ ¼ fHS ðhjuÞ HC ðhjuÞg dh: 0
Continuing is optimal under complete information if and only if DðuÞ is nonnegative. To fix ideas, Dewatripont and Tirole assume that Dð Þ is increasing, so that the best rule can be described as follows: continue when u b u^; stop when u < u^. The threshold u^ is defined by Dð^ uÞ ¼ 0: From now on, assume that the e¤ort level of the manager, which can take only two values: e ¼ e (insu‰cient) or e ¼ e (correct), is not observable by others. However u and v are positively correlated with e. Higher realizations of u (or v) indicate a greater likelihood that e ¼ e. If f ðujeÞ and gðvjeÞ denote the conditional densities of u and v, this means that f ð jeÞ f ð jeÞ
and
gð jeÞ gð jeÞ
are both increasing functions. Let xðu; vÞ denote the probability of continuing when ðu; vÞ is observed. The second-best decision Ðrule Ð is obtained by maximizing the expected (incremental) profit from continuing xðu; vÞDðuÞ f ðujeÞgðvjeÞ du dv under the incentive compatibility constraint: ðð B xðu; vÞf f ðujeÞgðvjeÞ f ðujeÞgðvjeÞg du dv b c; which means that the expected loss from shirking is higher than the cost of e¤ort. The Lagrangian of this problem is ðð L¼ xðu; vÞfðDðuÞ þ mBÞ f ðujeÞgðvjeÞ mB f ðujeÞgðujeÞg du dv mc; where m is the multiplier associated with the incentive constraint. Pointwise maximization of L with respect to xðu; vÞ A ½0; 1 gives the second-best decision rule: