Commitment Game and the Optimal Rule

The design of the ECB veto right discretionary game drew on the formulation of the Treaty on the EU. In the following, we present an alternative approach to ambiguity that may be compatible with the prescription in the Treaty. In particular, the regulator is supposed to be able to commit to a rule on the identity of the LOLR, including the commitment to not specifying the identity. In other words, we ask whether and under which conditions it may be optimal not to explicitly appoint the LOLR responsibility to any of the regulators. The following lemma states the …rst necessary condition for constructive ambiguity.

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CONSTRUCTIVE AMBIGUITY 75

Lemma 10 For ambiguity of LOLR identity to be welfare improving, it must be that

E[pREG]2(minfpECB; pN CBg; maxfpECB; pN CBg):

Proof. See the Appendix.

The necessary condition in Lemma 10 appears to be quite restrictive. For example, merely randomising which threshold is binding will not produce a role for constructive ambiguity. The reason is that as the banks just adjust their expectations as to the probability of a threshold being binding, they always choose either one of the thresh- olds. In this case, however, it would always be welfare-improving to announce the threshold that is closer to the …rst best.15 _{Similarly, allowing the regulators to enter}

into negotiations on terms of rescue only lead the bank to expect rescue at the lowest regulatory threshold level.

Nevertheless, let us assume from now on that a mechanism that ful…ls the necessary condition exists. As an approximation of ambiguity, let us assume that the bank attributes a weight z on the ECB and (1 z)on the NCB, so that the bank forms its expectations such that the expected regulatory threshold becomesE(pREG) =zpECB+

(1 z)pN CB. The optimal policy is then the solution of the following maximisation problem: M ax USOC(p; z) s:t: p ₂ arg maxE( ) p pREG 2[pECB; pN CB] E[ (E(pREG))] E( (0)) USOC[E(pREG)] 0: (3.12)

Note that the individual rationality constraint on the line 4 of Equation (3:12) is always ful…lled inp₂(0; pREG)if the rationality constraint for the society is ful…lled.16 The solution is summarised in the following proposition.

15_{For illustration, the randomising game of incomplete information is presented in Section 3.A in}

the Appendix.

Proposition 7 a) As long as pECB; pN CB pF B, the optimal rule of the commitment game is a corner solution with pREG= maxfpECB; pN CBg;

b) If pN CB < pF B <minfpECB; C 1[pECBH s]g, or

pECB < pF B < min pN CB(S); C 1 pN CB(S)H

(s+1)(s+v_n)

v+s+1 , the optimal rule is

to choosez = _pECBpF B pN CB_{pN CB}, in which case E(pREG) = pF B;

c) In all the other cases, the optimal rule is to choose pREG= minfpECB; pN CBg.

Proof. See the Appendix.

Hence, the coexistence of the ECB and the NCB can indeed improve upon the NCB also in cases where the NCB was too lenient due to the time inconsistency problem in the …rst place. For those branch structure banks for which the NCB would be too lenient but the ECB is too strict, ambiguity is constructive. In addition, ambiguity allows improving upon the discretionary solution for those subsidiary structure banks, for which the NCB is too strict and the ECB too lenient.

Ambiguity in identity is constructive only in the case that either one, but only one, of the regulators is too strict and practises ine¢ cient closures of banks that would have a positive social value at the …rst best. This happens for banks that are relatively ine¢ cient. Therefore, if a mechanism exists to alter the expectations of the bank away from the regulatory thresholds, one too tight a regulator makes the …rst best solution achievable. In the case of both regulators being too strict, the bank will be kept open only if the rationality constraint of the social planner is ful…lled at the lower regulatory threshold. The optimal rule for those banks thus coincides with the discretionary solution.

Note that the commitment game always improves upon the discretionary game. This is not surprising because, if the discretionary solution would be optimal, the social planner could choose it in the commitment game; however, the possibility to commit enlarges the strategy space of the social planner.

Commitment to ambiguity in identity thus allows in theory to compensate some- what for the time inconsistency problem. However, the welfare improvement applies only for a limited class of banks for whom one of the regulators is too strict, as in Lemma 7. Next, we demonstrate how sensitive the result of constructive ambiguity is with the help of some robustness checks.

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