Hierarchical bank supervision

Transcription

1 SERIEs (2018) 9: ORIGINAL ARTICLE Hierarhia bank supervision Rafae Repuo 1 Reeived: 20 Otober 2017 / Aepted: 11 Deember 2017 / Pubished onine: 26 Deember 2017 The Author(s) This artie is an open aess pubiation Abstrat This paper presents a mode in whih a entra and a oa supervisor ontribute their efforts to obtain information on the soveny of a oa bank, whih is then used by the entra supervisor to deide on its eary iquidation. This hierarhia mode is ontrasted with the aternatives of deentraized and entraized supervision, where ony thoa or the entra supervisor oets information and deides on iquidation. Thoa supervisor has a higher bias against iquidation (supervisory apture) and a ower ost of getting oa information (proximity). Hierarhia supervision is the optima institutiona design when the bias of thoa supervisor is high but not too high and the osts of getting oa information from the enter arow but not too ow. With ow (high) bias and high (ow) ost it is better to onentrate a responsibiities in thoa (entra) supervisor. Presidentia Address deivered at the Symposium of the Spanish Eonomi Assoiation in Bibao in Deember I woud ike to thank the omments of Ósar Are, Manue Areano, Mahmoud Eamin, Luis Gariano, Charie Kahn, Lu Laeven, David Lua, Simone Manganei, David Martinez-Miera, David Pérez-Castrio, Enrique Sentana, André Sapir, Joe Sobe, Javier Suarez, Xavier Torres, Skander Van den Heuve, Wof Wagner, Ansgar Wather, an anonymous referee, as we as seminar audienes at CEMFI, the Bank of Portuga, the Bank of Spain, the European Centra Bank, the Federa Reserve Banks of Boston, Ceveand and New York, the University of Edinburgh, and the 2017 Congress of the European Eonomi Assoiation. I am aso gratefu to Wenbo Meng, Ávaro Remesa, and Yixin Yao for their exeent researh assistane. Finania support from the Spanish Ministry of Eonomy and Competitiveness, Grant No. ECO P, is gratefuy aknowedged. B Rafae Repuo repuo@emfi.es 1 CEMFI, Casado de Aisa 5, Madrid, Spain

2 2 SERIEs (2018) 9:1 26 Keywords Centraized versus deentraized supervision Strategi information aquisition Bank soveny Bank iquidation Supervisory apture Optima institutiona design JEL Cassifiation G21 G23 D02 1 Introdution Bank supervision, unike bank reguation, has not been unti reenty the subjet of muh aademi interest. As stated by Eisenbah et a. (2016), reguation invoves the estabishment of rues under whih banks operate, whie supervision invoves the assessment of safety and soundness of banks through monitoring, and the use of this information to request orretive ations. In ontrast to reguation, that is based on verifiabe information, supervision is about supervisory ations (party) based on nonverifiabe information. This paper ontributes to the theoretia iterature on bank supervision by onstruting a styized mode of a supervisor that oets nonverifiabe information on the soveny of a bank and, on the basis of this information, deides on its eary iquidation. The quaity of the information on the bank s soveny depends on the intensity of supervision (the nonverifiabe osty effort of the supervisor). I assume that the supervisor is not a soia wefare maximizer. In partiuar, its payoff funtion inorporates a iquidation ost, whih may be assoiated with either reputationa onerns or supervisory apture (e.g. revoving doors). The paper haraterizes the effort and thiquidation deisions of the supervisor, and shows that supervision wi be more intense thower the osts of supervisory effort and thower the supervisory bias against iquidation. The mode an be interpreted as a mode of deentraized supervision, in whih the bank is a oa bank and the supervisor is a oa supervisor, or as a mode of entraized supervision, in whih the bank is sti a oa bank but the supervisor is a entra (or supranationa) supervisor. It an aso be used as a buiding bok for a mode of hierarhia supervision, in whih the entra and thoa supervisors jointy supervise the bank in order to observe a nonverifiabe signa of the bank s soveny, and then the entra supervisor deides on thiquidation of the bank. Under hierarhia supervision, the entra and thoa supervisors simutaneousy hoose their efforts, so they wi be paying a game. The Nash equiibrium of this game desribes the outome of the hierarhia supervision mode. The main ontribution of the paper is to haraterize the onditions under whih one of the three institutiona arrangements, namey deentraized, hierarhia, and entraized supervision, dominates in wefare terms the other two. The anaysis is based on two key assumptions: (i) the ost of effort is higher for the entra than for thoa supervisor, and (ii) the ost of iquidating the bank is ower for the entra supervisor than for thoa supervisor. The first assumption may be justified by referene to distane between the entra supervisor and thoa bank. In the words of Torres (2015): The entra supervisor has informationa disadvantages reative to the nationa authorities, due to their better knowedge of banks, banking systems and

3 SERIEs (2018) 9: reguatory frameworks, as we as their geographia and utura proximity to them. The seond assumption may be justified by referene to thooser onnetions between the entra supervisor and the bank. In the words of Torres (2015): The existene of a supranationa supervisor aows to inrease the distane between supervisors and nationa obbies and poitiians, whih in prinipe shoud redue the risk of supervisors impementing exessivey ax poiies. 1 The resuts show that hierarhia supervision dominates deentraized supervision when the bias of thoa supervisor is high and the osts of getting oa information from the enter arow. But when these fores exeed ertain threshod, it is better to onentrate a responsibiities in the entra supervisor. The trade-off underying these resuts is ear: Deentraized supervision is better beause thoa supervisor finds it heaper to gather information, but it is worse beause its objetive funtion is biased against iquidation. 2 The resuts aso show that hierarhia supervision is more ikey to dominate when bank profitabiity is ow (e.g. as a resut of high ompetition) and when bank risk-taking is high (e.g. as a resut of soft reguation). Interestingy, the mode of hierarhia supervision is isomorphi to a mode in whih the entra supervisor gets a signa of the bank s soveny, thoa supervisor gets another signa whih truthfuy reports to the entra supervisor, who then deides on thiquidation of the bank. The origina mode orresponds to an institutiona arrangement in whih the supervisors work in teams, whie the aternative mode orresponds to an arrangement in whih the supervisors work independenty, but there is no probem of strategi information transmission à a Crawford and Sobe (1982). 3 The resuts provides a rationae for the design of the Singe Supervisory Mehanism (SSM), the new struture of bank supervision in Europe that omprises the European Centra Bank (ECB) and the nationa supervisory authorities of the partiipating ountries. Currenty, the ECB is responsibe for the supervision of signifiant (i.e. arge) banks that omprise about 80% of the banking assets of the euro area. For these banks, supervision is arried out in ooperation with the nationa supervisors via the so-aed Joint Supervisory Teams. Nationa supervisors are in harge of thess signifiant (i.e. sma) banks. To the extent that (i) the ost advantage of oa supervisors is smaer for arger, more ompex banks, and (ii) supervisory apture is more reevant for arger banks, the resuts of the mode are onsistent with the design of the SSM. 1 Lambert (2017) anayzes the reationship between bank obbying and supervisory deisions using bankeve information on the universe of ommeria and savings banks in the US. He onudes that these resuts appear rather inonsistent with an information based expanation of bank obbying, but onsistent with the theory of reguatory apture. 2 Bian et a. (2017) anayze baiouts in German savings banks, showing that banks that are baied out by oa poitiians (deentraized resoution) experieness restruturing and perform onsideraby worse than banks that are baied out by state assoiations (entraized resoution). In omparing the two aternatives they note the foowing trade-off: On the one hand, ose proximity has the potentia to improve the deision making proess, as it provides poitiians with good information about banks that get into distress. On the other hand, ose proximity oud impy that poitiians persona onsiderations distort the deision making proess, whih is eary undesirabe. Suh trade-off is simiar to the one onsidered in this paper. 3 SeeSobe(2013) for a reent survey. Hothausen and Rønde (2004) ontain an appiation of the resuts in Crawford and Sobe (1982) to the modeing of ooperation between host and home ountry supervisors of a mutinationa bank.

4 4 SERIEs (2018) 9:1 26 The mode an aso shed ight on issues reated to the organization of supervision in jurisditions in whih mutipe agenies are invoved in supervising banks. For exampe, state-hartered banks in the US are under a dua supervisory framework invoving both federa and oa supervisors. 4 Aso, the Federa Reserve is responsibe for the supervision of bank hoding ompanies, but subsidiaries may be supervised by other federa agenies. Two extensions of the mode are disussed. The first one examines what happens when the supervisory bias is inked to being responsibe for thiquidation deision. This means that moving from deentraized to hierarhia supervision impies a redution in thiquidation ost of thoa supervisor and an inrease in thiquidation ost of the entra supervisor. In this ase, the resuts show that there wi be an inrease in the (heaper) effort of thoa supervisor that wi ompensate the redution in the effort of the entra supervisor, so hierarhia supervision is morikey to dominate. A seond extension shows that imiting the size of the entra supervisor in the hierarhia supervision setup is wefare improving. The resut is osey reated to the we-known resut that a Stakeberg eader that optimizes over the reation funtion of the other agent does better than by paying its Nash equiibrium strategy. The intuition is that putting the entra supervisor in a situation of overoad fores thoa supervisor to inrease its (heaper) effort in a way that ompensates the redution in the effort of the entra supervisor. 5 Literature review Caretti et a. (2016) expore the working of a supervisory struture in whih a entraized ageny has ega power over deisions regarding banks, but has to rey on biased oa supervisors to oet the information neessary to at. They fous on how this institutiona design affets supervisors inentives to oet information and on how this, in turn, infuenes bank behavior, showing that when the ageny probem between the entra and thoa supervisor is severe, entraized supervision eads to ower information oetion and inreased risk-taking. 6 Coiard (2015) onsiders a mode in whih oa supervisors are morenient, so that banks aso have weaker inentives to hide information from them. These two fores an make a joint supervisory arhiteture optima. However, more entraized supervision enourages banks to integrate more ross-border. Due to this ompementarity, the eonomy an be trapped in an inferior equiibrium with both too itte entra supervision and too itte finania integration. Cazoari et a. (2016) study the impat of shared iabiity and deposit insurane arrangements on supervisors inentives to aquire information on the ativities of mutinationa banks, showing that entraized supervision an indue these banks to expand abroad through branhes rather than subsidiaries. 4 Agarwa et a. (2014) show that federa supervisors are systematiay tougher that state supervisors, downgrading supervisory ratings amost twie as frequenty as do state supervisors. However, they do not find support for supervisor sef-interest, whih inudes revoving doors, as a reason for enieny of state supervisors. 5 This is reminisent of resuts in thiterature on the theory of organizations. For exampe, Aghion and Tiroe (1997) write: It is aways optima for the firm to be in a situation of overoad so as to rediby ommit to rewarding initiative. 6 Gopaan et a. (2017) examine the effets of thosure of a US bank reguator s fied offies, onuding that distaning supervisors from banks to prevent apture an be osty as it inreases bank risk.

5 SERIEs (2018) 9: In an earier ontribution, Boyer and Pone (2012) anayze whether banking supervision responsibiities shoud be onentrated in the hands of a singe supervisor, showing that spitting supervisory powers among different supervisors is a superior arrangement in terms of wefare when the apture of supervisors by bankers is a onern. The mode in the paper may be onsidered as a speia ase of modes in the iterature on the theory of organizations. 7 As noted in the semina paper by Jensen and Meking (1992), The assignment of deision rights infuenes inentives to aquire information. ( ) Determining the optima eve of deentraization requires baaning the osts of bad deisions owing to poor information and those owing to inonsistent objetives. It is aso osey reated to the reent iterature on strategi information aquisition, whih departs from thiterature on strategi information transmission by assuming that information is not exogenousy given, but is obtained through osty effort; see Argenziano et a. (2016). Struture of the paper Setion 2 presents the basi mode of bank supervision in whih a supervisor oets information on the soveny of a bank and, on the basis of this information, deides on its eary iquidation. This setup may be interpreted as a mode of deentraized supervision in whih supervisory responsibiities are aoated to a oa supervisor or a mode of entraized supervision in whih supervisory responsibiities are aoated to a entra supervisor. Buiding on this setup, Set. 3 presents the mode of hierarhia supervision in whih a entra and a oa supervisor jointy oet information and then the entra supervisor deides on iquidation. Setion 4 ompares in terms of wefare three possibe institutiona arrangements: deentraized, hierarhia, and entraized supervision. Setion 5 ontains the extensions, and Set. 6 the onuding remarks. The proofs of the anaytia resuts are in the Appendix. 2 Mode setup Consider an eonomy with three dates (t = 0, 1, 2) and two agents: a bank and a supervisor. The bank raises a unit amount of deposits at t = 0, and invests them in an asset that has a random fina return R at t = 2. The asset an biquidated at t = 1, in whih ase it yieds a random iquidation return L. Deposits are insured and the deposit rate is normaized to zero. It is assumed that L N R ( ) a R,σ 2 b, (1) 1 1 where R > 1, a < 1, b < 1, and > 0. Moreover, to ensure that the ovariane matrix is positive-definite it must be the ase that 2 < b,sob < 1 impies < 1. Thus, the expeted fina return E(R) = R is greater than the unit fae vaue of the deposits, and 7 See Jensen and Meking (1992), Aghion and Tiroe (1997), Dessein (2002), Aonso et a. (2008), and Dessein et a. (2010), among many others.

6 6 SERIEs (2018) 9:1 26 it is aso greater than the expeted iquidation return E(L) = a R. Moreover, the fina return has a higher variane than thiquidation return, and both returns are positivey orreated. Exept for the assumption of normaity, whih is made for tratabiity, the rest of the assumptions apture reaisti features of the distribution of banks returns. The supervisor hooses at t = 0theintensity of supervision e (nonverifiabe effort of the supervisor), whih eads to the observation at t = 1ofanonverifiabe signa s = R + ε (2) on the fina return of the bank s investment. The noise term ε is independent of L and R, and has a distribution N(0,σ 2 /e). 8 Thus, the effort e of the supervisor inreases the preision (inverse of the variane) of the noise term. From here it foows that L a b R N R 1,σ 2 s 1 By the properties of norma distributions we have e 1. (3) (s R) E(L s) = a R +, 1 + e 1 (4) E(R s) = R + s R. 1 + e 1 (5) Sine < 1 the sope of E(L s) is smaer than the sope of E(R s), whih impies E(L s) >E(R s) if and ony if s < s, where s = 1 1 a 1 (1 + e 1 ) R (6) is the effiient iquidation threshod. Note that higher supervisory effort e inreases the effiient iquidation threshod s. Aso, note that for e = 0wehaves =, so with a ompetey noisy signa the bank shoud never biquidated. Substituting s from (6)into(4) and (5) gives E(L s) = E(R s) = a 1 R. I wi assume that parameter vaues satisfy a R 1. (7) 1 8 This impies that L = (a )R +R+u, where u N(0,σ 2 (b 2 )) and Cov(u, R) = Cov(u,ε)= 0. Thus, signa s ontains information about L ony inasmuh as it ontains information about R.

7 SERIEs (2018) 9: This means that the effiient iquidation threshod s is suh that the orresponding expeted fina return is ess than or equa to the fae vaue of the deposits, so effiient iquidation ours ony if the bank is effetivey bankrupt. 9 The supervisor hooses its effort e at t = 0, observes the signa s at t = 1, and based on this observation deides on thiquidation of the bank at this date. Supervisory effort is osty. Speifiay, I assume that the ost funtion takes the simpe quadrati form (e) = γ 0 + γ 2 e2, (8) where γ 0 > 0 is a fixed ost of setting up the supervisory arrangements, and γ>0is the key ost of effort parameter. The supervisor iquidates the bank at t = 1if E(L s) δ>e(r s), (9) where δ>0 is aiquidation ost. This ost may be assoiated with either reputationa onerns or supervisory apture (e.g. revoving doors). Thus, the supervisor iquidates the bank when the soia benefits of iquidation, whih are E(L s) E(R s), are greater than thiquidation ost δ. Substituting (4) and (5) into(9) impies that the bank wi biquidated by the supervisor when s < ŝ, where ŝ = s δ 1 (1 + e 1 ) (10) is the supervisor s iquidation threshod. 10 Figure 1 shows the determination of the effiient iquidation threshod s by the intersetion of thines E(L s) and E(R s), and of the supervisor s iquidation threshod ŝ by the intersetion of thines E(L s) δ and E(R s). For signas in the range between ŝ and s the supervisor does not iquidate the bank when it woud be effiient to do so. The supervisor s effort deision at t = 0 is obtained by maximizing its expeted payoff at t = 1, given by Emax {E(L s) δ, E(R s)}, net of the ost of effort (e), that is ê = arg max v(e), e where ŝ v(e) = E(L s) δ df(s) + E(R s)df(s) (e), (11) ŝ F(s) denotes the df of the signa s, and thiquidation threshod ŝ is given by (10). 9 Interestingy, E(L s) = E(R s) does not depend on supervisory effort e. 10 Notie that, by (5) and(7), ŝ < s impies E(R ŝ) <E(R s) 1, so the supervisor iquidates the bank ony if it is effetivey bankrupt.

8 8 SERIEs (2018) 9:1 26 ERs ( ) _ 1 ELs ( ) ELs ( ) δ iquidate do not iquidate ŝ s ineffiient ontinuation s Fig. 1 Supervisor s iquidation threshod The foowing resut provides a osed form expression for v(e). For the resut it is onvenient to write s = E(s) + SD(s)x, where x N(0, 1), and define x to be suh that ŝ = E(s) + SD(s) x. Proposition 1 The supervisor s payoff funtion may be written as v(e) = R (1 a)r + δ ( x)) + φ( x) (e), (12) x where (x) is the norma df and φ(x) is the norma density, and x = (1 a)r + δ (1 )σ ( 1 + e 1) 1/2. (13) Figure 2 pots the payoff funtion of the supervisor v(e) for the parameters that wi be used in the numeria anaysis beow. 11 For e = 0wehave x =, so the bank is never iquidated, and v(0) = R γ 0 (the expeted fina return minus the fixed ost of supervision). For e = we have v(e) =, sine the expeted payoff at t = 1 is eary bounded whie the ost of supervisory effort goes to infinity. The funtion v(e) is initiay onvex, and then beomes onave, so there may be orner soutions with ê = 0 or interior soutions with ê > The foowing resut presents some omparative statis for the ase where the soution is interior. In partiuar, it shows that supervisory effort ê is dereasing in the ost of effort parameter γ and in thiquidation ost δ inurred by the supervisor. It aso shows that ê is dereasing in the expeted return R and inreasing in the standard deviation σ of the bank s investment return. 11 These vaues are R = 1.25, a = 0.9, = 0.5, σ = 0.25, δ= 0.05, and γ 0 = γ = Notie that these parameters satisfy assumption (7). 12 The nononavity of the funtion v(e) is a speia ase of the resut in Radner and Stigitz (1984).

9 SERIEs (2018) 9: ve () R γ 0 ê e Fig. 2 Supervisor s payoff funtion Proposition 2 Whenever the supervisor hooses a positiveve of effort ê we have ê γ ê < 0, δ < 0, ê ê < 0, and R σ > 0. Figure 3 iustrates these resuts. Panes A and B show that inreases in the ost of effort parameter γ and in thiquidation ost δ redue supervisory effort ê, whih jumps to zero for suffiienty high vaues of γ and δ. Pane C shows that inreases in the expeted return R redue supervisory effort ê, whih jumps to zero for suffiienty high vaues of R. Finay, Pane D shows supervisory effort ê is zero for suffiienty ow vaues of the standard deviation σ, jumping to positive and inreasing eves for vaues of σ beyond a ritia point. The omparative statis resuts are not surprising. Supervision wi be more intense when banks are easier to supervise (ower γ ), or ess profitabe (ower R), or riskier (higher σ ). And it wi bess intense when the supervisor is oser to obbies and pressure groups (higher δ) that aways prefer deaying intervention and gambing for resurretion. Summing up, I have set up a simpe mode of a bank and a supervisor in whih the supervisor exerts osty effort in order to observe a nonverifiabe signa of the bank s soveny, whih is used to deide the bank s eary iquidation. Importanty, the supervisor is not a soia wefare maximizer it has a bias against iquidation. I have haraterized the supervisor s effort deision, and derive some omparative statis resuts on its determinants. The mode may be interpreted as a mode of deentraized supervision, in whih the bank is a oa bank and the supervisor is a oa supervisor, or as a mode of entraized supervision, in whih the bank is sti a oa bank but the supervisor is a entra (or supranationa) supervisor. In this setup, it woud be reasonabe to assume that the ost of effort is higher for the entra supervisor than for thoa supervisor, an assumption that may be justified by referene to geographia as we as utura distane between the entra supervisor and thoa bank. It may aso be reasonabe to assume that the ost of iquidating the bank is ower for the entra supervisor than for thoa

10 10 SERIEs (2018) 9:1 26 e a e b ê ê γ d δ e e ê ê R Fig. 3 Comparative statis. a Effet of ost of effort. b Effet of supervisory iquidation osts. Effet of expeted asset return. d Effet of voatiity of asset return supervisor, an assumption that may be justified by referene to thooser onnetions between the entra supervisor and nationa obbies and pressure groups. The trade-off between the higher osts of supervision and thower inidene of supervisory apture wi be examined in Set. 4. But before doing this, the foowing setion presents a mode of hierarhia supervision in whih the entra and thoa supervisors jointy supervise the bank in order to obtain a nonverifiabe signa of the bank s soveny, and then the entra supervisor deides on the bank s eary iquidation. σ 3 Hierarhia supervision Consider an eonomy with a oa bank and two supervisors, a entra and a oa supervisor, denoted by subindies and. The supervisors independenty hoose at t = 0 nonverifiabe efforts e and, respetivey, whih eads to the observation at t = 1 of a singe nonverifiabe signa s = R + ε (14) on the fina return of the bank s investment. 13 The noise term ε is independent of L and R, and has a distribution N(0,σ 2 (e + ) 1 ). Thus, the higher the efforts e and the higher the preision of the noise term. 13 As noted above, this setup is intended to apture the working of the Joint Supervisory Teams of the Singe Supervisory Mehanism of the European Centra Bank.

11 SERIEs (2018) 9: From here it foows that L a b R N R 1,σ (15) s (e + ) 1 Compared to the ase of a singe supervisor presented in Set. 2, the ony differene is that now the preision of the signa s depends on the sum e + of the efforts of the two supervisors instead of on the effort e of the singe supervisor. 14 Supervisory effort is osty, and the ost of effort is assumed to be higher for the (distant) entra supervisor than for the (ose) oa supervisor. Speifiay, I assume that parameter γ in the ost funtion (8) takes the vaue γ for the entra supervisor and γ for thoa supervisor, where γ >γ > 0. The orresponding ost funtions wi be written (e ) and ( ), respetivey. The entra supervisor deides whether to iquidate the bank based on the observation of the signa s at t = 1. I assume that thiquidation ost δ of the entra supervisor is ower than that of thoa supervisor δ. Moreover, to simpify the presentation I wi assume that the entra supervisor does not have a bias against eary iquidation, so δ = Thus, the entra supervisor iquidates the bank at t = 1if E(L s) >E(R s). By the properties of norma distributions we have Hene, it foows that E(L s) = a R + E(R s) = R + (s R), (16) 1 + (e + ) 1 s R. (17) 1 + (e + ) 1 E(L s) >E(R s) if and ony if s < s, where s = 1 1 a (e + ) 1 R (18) is the effiient iquidation threshod. As before, I assume that parameter vaues satisfy assumption (7). This impies that the threshod s is suh that E(L s ) = E(R s ) 1, so iquidation takes pae when the bank is effetivey bankrupt. 14 A more genera setup woud be one in whih the preision of the signa is α e + α + α e, where α and α apture the produtivity of the effort of the two supervisors, and α possibe ompementarities in their efforts. For simpiity, I assume α = α = 1andα = Setion 5.1 anayzes a variation of the mode where supervisory responsibiities inrease thiquidation ost by an amount >0, in whih ase hierarhia supervision entais a positiviquidation ost for the entra supervisor.

12 12 SERIEs (2018) 9:1 26 At t = 0 the entra and thoa supervisors independenty hoose their efforts e and, so they wi be paying a game. I wi haraterize the Nash equiibrium of this game, and show some omparative stati resuts. The payoff funtion of the entra supervisor is v (e, ) = s and the payoff funtion of thoa supervisor is v (e, ) = s E(L s)df(s) + E(R s)df(s) (e ), (19) s E(L s) δ df(s) + E(R s)df(s) ( ), (20) s where F(s) denotes the df of the signa s, and thiquidation threshod s is given by (18). It shoud be noted that in these expressions the entra (oa) supervisor does not take into aount the ost of effort of thoa (entra) supervisor. 16 More importanty, sine the threshod s depends on the efforts e and of the two supervisors, it must be the ase that eah supervisor observes theve of effort hosen by the other supervisor. 17 This may be rationaized by assuming that the efforts of the supervisors are reated to the quaity of the staff that they independenty hoose ex-ante, but whih an be observed ex-post. The reation funtions of the two supervisors are given by e ( ) = arg max v (e, ), e (21) (e ) = arg max v (e, ). (22) The intersetion of these funtions is a Nash equiibrium of the game payed by the supervisors. The foowing resut provides osed form expressions for v (e, ) and v (e, ). As before, it is onvenient to write s = E(s) + SD(s)x, where x N(0, 1), and define x to be suh that s = E(s) + SD(s)x. Proposition 3 The supervisors payoff funtions may be written as v (e, ) = R (1 a)r (x ) + φ(x ) x (e ), (23) v (e, ) = R (1 a)r (x ) + φ(x ) x δ (x ) ( ), (24) 16 This is without oss of generaity, given that subtrating ( ) and (e ) in the payoff funtions of the entra and thoa supervisor, respetivey, woud not hange their behavior. 17 What is key is that the entra supervisor observes, so in hoosing thiquidation threshod s it an take into aount the (out of equiibrium) deviations of thoa supervisor. In ontrast, thoa supervisor need not observe e, sine the (out of equiibrium) deviations of the entra supervisor do not affet its behavior.

13 SERIEs (2018) 9: e e Fig. 4 Nash equiibrium where x (1 a)r = (1 )σ 1 + (e + ) 1 1/2. (25) The anaysis in the previous setion shows that the supervisors payoff funtions are not everywhere onave. For this reason, the reation funtions (21) and (22) may have orner or interior soutions. In what foows, I wi restrit attention to parameter vaues for whih the soutions are interior. Figure 4 shows the Nash equiibrium of the game payed by the two supervisors, denoted (e, e ). Notie that for the hosen parameter vaues the reation funtions satisfy e ()<0, e (e )<0, and e ()e (e )<1. 18 That is, they are both downward soping (strategi substitutes), and the sope of the reation funtion of thoa supervisor is (in absoute vaue) ower than that of the entra supervisor, so the Nash equiibrium is stabe. A suffiient ondition for this to obtain is that the ost of effort parameters γ and γ are not too arge. 19 Figure 5 iustrates some omparative statis resuts of the game between the two supervisors. Pane A shows that inreases in the ost of effort of the entra supervisor γ shifts to theft its reation funtion, eading to a redution in the equiibrium effort e of the entra supervisor and an inrease in the equiibrium effort e of the oa supervisor. Pane B shows that an inrease in thiquidation ost δ of thoa supervisor shifts down its reation funtion, eading to a redution in the equiibrium effort e of thoa supervisor and an inrease in the equiibrium effort e of the entra supervisor. Pane C shows that an inrease in the expeted return R of the bank s investment shifts to theft the reation funtion of the entra supervisor and 18 In this figure the ost of effort parameters are γ = and γ = , and thiquidation ost parameters are δ = 0andδ = By the seond order onditions we have 2 v / 2 e 2 = h(e + )<0and 2 v / 2 e 2 = g(e + )<0. Sine e () = h(e + ) + γ /h(e + ) and e (e ) = g(e + ) + γ /g(e + ), a suffiient ondition for e ()<0isthatγ < h(e + ), and a suffiient ondition for e (e )<0isthat γ < g(e + ). Moreover, these onditions impy e () < 1and e (e ) < 1, so e ()e (e )<1.

14 14 SERIEs (2018) 9:1 26 a b e e e e d e e e e Fig. 5 Comparative statis of Nash equiibrium. a Inrease in ost of effort of entra supervisor. b Inrease in iquidation osts of oa supervisor. Inrease in expeted asset return. d Derease in voatiity of asset return shifts down the reation funtion of thoa supervisor, eading to a redution in the equiibrium effort of at east one of the two supervisors athough in the numeria resuts both e and e go down. Finay, Pane D shows that a derease in the standard deviation σ of the bank s investment return shifts to theft the reation funtion of the entra supervisor and shifts down the reation funtion of thoa supervisor, eading to a redution in the equiibrium effort of at east one of the two supervisors athough in the numeria resuts both e and e go down. These omparative stati resuts are in ine with the ones obtained in the mode with a singe supervisor, where supervisory effort is dereasing in the ost of effort γ, thiquidation ost δ, and the expeted return R, and is inreasing in the standard deviation σ. But in the game between the two supervisors the first two hanges ead to an inrease in the equiibrium effort of the supervisor not affeted by the parameter hanges. It is interesting to note that the mode of hierarhia supervision presented in this setion is ompetey isomorphi to a mode in whih the entra supervisor gets a signa s = R + ε, where ε N(0,σ 2 /e ), thoa supervisor gets a signa s = R + ε, where ε N(0,σ 2 / ), and then it truthfuy reports it to the entra supervisor, who deides on thiquidation of the bank. 20 By the properties of norma distributions we have 20 The noise terms ε and ε are assumed to be independent of L and R as we as from eah other.

15 SERIEs (2018) 9: E(L s, s ) = a R + (s R), (26) 1 + (e + ) 1 E(R s, s ) = R + s R, (27) 1 + (e + ) 1 where s is a weighted average of the two signas with weights proportiona to their preision, that is s = e s + s. (28) e + e + The random variabe s is normay distributed, with E(s ) = R and Var(s ) = σ (e + ) 1, so it has the same distribution as the random variabe s in (15). Moreover, it is aso the ase that Cov(s, L) = and Cov(s, R) = 1. Sine s has the same properties as s, it foows that a the resuts for the origina mode of hierarhia supervision, in whih the two supervisors put in effort to get a singe signa, extend to the aternative mode, in whih eah supervisor puts in effort to get a signa and then thoa supervisor truthfuy sends its signa to the entra supervisor. The origina mode orresponds to an institutiona design in whih the supervisors work in teams, whie the aternative mode orresponds to a design in whih the supervisors work independenty, but there is no probem of strategi information transmission, that is there are proedures in pae that prevent thoa supervisor from misrepresenting its signa. 4 Optima institutiona design This setion ompares in wefare terms three possibe institutiona arrangements: deentraized, hierarhia, and entraized supervision. Under deentraized supervision, ony thoa supervisor oets information and deides on thiquidation of the bank. Under hierarhia supervision, the entra and thoa supervisor jointy oet information and then the entra supervisor deides on thiquidation of the bank. Finay, under entraized supervision, ony the entra supervisor oets information and deides on thiquidation of the bank. The aim is to haraterize the onditions under whih one of the three institutiona arrangements dominates the other two. The omparison between these institutiona arrangements fousses on two key parameters of the mode: the ost of effort of the entra supervisor, aptured by parameter γ, whih is higher than parameter γ orresponding to thoa supervisor, and thiquidation ost of thoa supervisor, aptured by parameter δ, whih is higher than parameter δ = 0 orresponding to the entra supervisor. As noted above, the first assumption may be justified by referene to geographia as we as utura distane between the entra supervisor and thoa bank, whie the seond may be justified by referene to thooser onnetions between the entra supervisor and oa obbies and pressure groups. Soia wefare has two omponents. First, the expeted bank returns, given the effort and thiquidation deisions of the supervisors. Seond, with negative sign, the osts of supervisory efforts. Supervisory iquidation osts are not taken into aount,

16 16 SERIEs (2018) 9:1 26 sine they are assumed to binked to supervisory apture (e.g., possibe transfers from banks to supervisors that ane out in wefare terms). 21 By the resuts in Set. 2, soia wefare under deentraized supervision is given by w = ŝ E(L s)df(s) + = R (1 a)r + δ ( x ) + φ( x ) x where ê is the effort hosen by thoa supervisor and ŝ E(R s)df(s) (ê ) (1 a)r + δ ( x = ) 1/2 ê. (1 )σ Simiary, soia wefare under entraized supervision is given by w = ŝ E(L s)df(s) + = R (1 a)r (ê ), (29) ŝ E(R s)df(s) (ê ) where ê is the effort hosen by the entra supervisor and ( x ) + φ( x ) (ê ), (30) x (1 a)r ( x = ) 1/2 ê. (1 )σ Finay, by the resuts in Set. 3, soia wefare under hierarhia supervision is given by s w h = E(L s)df(s) + E(R s)df(s) (e s ) (e ) = R (1 a)r (x ) + φ(x ) x (e ) (e ), (31) where (e, e ) is the Nash equiibrium of the game payed by the supervisors and x (1 a)r = (1 )σ 1 + (e + e ) 1 1/2. 21 See Lua et a. (2014) for evidene on the revoving door between reguatory agenies and the banking industry. Their resuts appear to be inonsistent with a quid pro quo view aording to whih future empoyment opportunities in the private setor affets the stritness of ations of reguators. Instead, they find evidene in favor of a reguatory shooing view aording to shooing in ompex rues enhanes reguators future earnings, shoud they transition to the private setor.

17 SERIEs (2018) 9: δ C H D γ Fig. 6 Optima institutiona design I an now ompare in terms of wefare the three aternative institutiona arrangements by omputing w,w h, and w for different vaues of the ost of effort of the entra supervisor γ and thiquidation ost of thoa supervisor δ. The fores driving the omparison are easy to expain. The higher the ost of effort of the entra supervisor γ thower thikeihood that entraized supervision wi be optima, beause as shown in Fig. 3a the entra supervisor wi have an inentive to exert too itte effort. Simiary, the higher thiquidation ost thoa supervisor δ the ower thikeihood that deentraized supervision wi be optima, beause as shown in Fig. 3b thoa supervisor wi have an inentive to exert too itte effort. Reative to the ase of a singe supervisor, hierarhia supervision entais inurring twie the fixed ost γ 0, so by the onvexity of the ost funtion (8) it wi ony be optima when both supervisors have an inentive to exert signifiant effort. This requires that the bias of thoa supervisor is high but not too high and the osts of getting oa information from the enter arow but not too ow. Figure 6 shows the resuts. 22 Deentraized supervision dominates in Region D, where γ is arge and δ is sma, that is when the ost advantage of thoa supervisor is suffiienty arge to ompensate the bias in its iquidation deision. Hierarhia supervision dominates in Region H, where the ost advantage of thoa supervisor is not too arge or thiquidation ost of thoa supervisor is not too sma. Finay, entraized supervision dominates in Region C, where γ is sma and δ is arge, that is when the ost disadvantage of the entra supervisor is suffiienty sma to ompensate the bias of thoa supervisor. Next, I anayze the effet on the regions in Fig. 6 of hanges in the expeted return R and in the standard deviation σ of the bank s investment return. Figure 7 iustrates the resuts. Pane A shows that an inrease in R expands Regions D and C where deentraized and entraized supervision are optima at the expense of Region H where hierarhia supervision is optima. Simiary, Pane B shows that a derease in σ expands Regions D and C where deentraized and entraized supervision are 22 In this figure the origin orresponds to γ = γ = and δ = δ = 0.

18 18 SERIEs (2018) 9:1 26 a b δ C δ C H H D D γ γ Fig. 7 Comparative statis of optima institutiona design. a Inrease in expeted asset return. b Derease in voatiity of asset return optima at the expense of the Region H where hierarhia supervision is optima. 23 The intuition is that when the bank is more profitabe (higher R) or safer (ower σ ) the supervisors wi exert ess effort (see Fig. 5, d), Thus, given the onvexity of the ost funtion (8), the reative advantage of having two supervisors wi derease. Summing up, I have shown that hierarhia supervision dominates deentraized supervision when the possibe apture of thoa supervisor is a signifiant onern and the osts of getting oa knowedge are not too arge. But when these two fores go beyond ertain threshod it is better to eiminate thoa supervisor and onentrate a responsibiities in the entra supervisor. Moreover, hierarhia supervision is morikey to dominate when the risk of bank faiure (due to ow profitabiity or high risk-taking) is high. Thesatter resuts iustrate the way in whih banking supervision shoud be adjusted in response to hanges in the environment, and hene may interpreted as maroprudentia poiies. 5 Extensions 5.1 Linking iquidation osts to supervisory responsibiities This extension examines what happens when part of thiquidation ost inurred by the supervisors is inked to whether they are responsibe for thiquidation deision. To formaize this idea, suppose that δ = 0 and δ > 0 are thiquidation osts for the entra and thoa supervisor, respetivey, in the absene of supervisory responsibiities, and that suh responsibiities add to these osts. Thus, thiquidation ost of thoa supervisor wi be δ + under deentraized supervision and δ under hierarhia supervision, whereas thiquidation ost of the entra supervisor wi be under both entraized and hierarhia supervision. 23 Interestingy, both panes iustrate that for reativey sma vaues of γ and δ the optima institutiona design may swith from Region D to RegionC without passing through Region H. Notie aso that thine that separates the two regions goes through the origin, sine when γ = γ and δ = δ both supervisors have the same payoff funtion.

19 SERIEs (2018) 9: By our previous resuts, under deentraized supervision the effort deision of the oa supervisor is given by where { ê = arg max R (1 a)r + δ + (x ) + φ(x } ) ( ), x x = x ( ) = (1 a)r + δ + (1 )σ ( 1 + e 1) 1/2. Soia wefare under deentraized supervision is w = R (1 a)r + δ + ( x ) + φ( x ) + (δ + ) ( x ) (ê ), x where x = x (ê ). Simiary, under entraized supervision the effort deision of the entra supervisor is given by where { ê = arg max R (1 a)r + (x ) + φ(x } ) (e ), e x x = x (e ) = (1 a)r + (1 )σ ( ) 1 + e 1 1/2. Soia wefare under entraized supervision is w = R (1 a)r + ( x ) + φ( x ) + ( x ) (ê ), x where x = x (ê ). Finay, sine thiquidation ost of the entra supervisor is now >0, the payoff funtions of the supervisors beome v (e, ) = R (1 a)r + (x) + φ(x) x v (e, ) = R (1 a)r + (x) + φ(x) x (e ), (δ ) (x) ( ), where (1 a)r + x = x(e + ) = 1 + (e + ) 1 1/2. (1 )σ Let (e, e ) denote the Nash equiibrium of this game. Soia wefare under hierarhia supervision is then w h = R (1 a)r + (x ) + φ(x ) x + (x ) (e ) (e ),

20 20 SERIEs (2018) 9:1 26 δ C H D γ Fig. 8 Linking iquidation osts to supervisory responsabiities where x = x(e + e ). Figure 8 shows the resuts. 24 The assumption that part of thiquidation ost inurred by the supervisors is inked to whether they are responsibe for thiquidation deision expands Region H where hierarhia supervision is optima at the expense of Regions D and C where deentraized and entraized supervision are optima. The intuition is as foows. Consider the effet of introduing the additiona ost on a point in Fig. 8 on the origina boundary between Regions D and H. By our omparative statis resuts in Set. 3, this wi ead to a shift to theft of the reation funtion of the entra supervisor (due to the inrease in its iquidation ost by ) and a shift upwards of the reation funtion of thoa supervisor (due to the equivaent redution in its iquidation ost by ), eading to a redution in the equiibrium effort e of the entra supervisor and an inrease in the (heaper) equiibrium effort e of thoa supervisor, whih expains that the origina boundary point is now inside Region H. Simiary, onsider the effet of introduing the additiona ost on a point in Fig. 8 on the origina boundary between Regions H and C. By our omparative statis resuts in Set. 3, this wi ead to a shift to theft of the reation funtion of the entra supervisor (due to the inrease in its iquidation ost by ), eading to a redution in the equiibrium effort e of the entra supervisor and an inrease in the (heaper) equiibrium effort e of thoa supervisor. This inrease, whih does not obtain under entraized supervision, expains that the origina boundary point is now inside Region H. 5.2 Limiting the size of the entra supervisor This extension onsiders whether it woud be desirabe from a wefare perspetive to imit (in some statutory manner) the size of the entra supervisor. In terms of the 24 In this figure =

21 SERIEs (2018) 9: Centra supervisor indifferene urves e e Fig. 9 The entra supervisor as a Stakeberg eader mode, a sizimit impies an upper bound to the effort of the entra supervisor. How oud this be wefare improving? To answer this question it is onvenient to start onsidering the ase where the entra supervisor were a Stakeberg eader. By standard resuts for games with strategi substitutes, in a Stakeberg equiibrium the entra supervisor woud redue its effort e and thoa supervisor woud inrease its effort, reative to the Nash equiibrium (e, e ). Figure 9 iustrates the resut. By the definition of Nash equiibrium, the indifferene urve of the entra supervisor at the point (e, e ) is tangent to the horizonta ine = e. But sine the reation funtion of thoa supervisor is downward soping, reduing the effort of the entra supervisor inreases its payoff. The question is whether this argument aso appies when we repae the indifferene urve of the entra supervisor by the soia indifferene urves. To hek that this is indeed the ase, notie that using the definition (23) of the payoff funtion of the entra supervisor v (e, ) and the definition (31) of the soia wefare funtion w(e, ), the soia indifferene urve going through the Nash equiibrium point (e, e ) may be written as w(e, ) = v (e, ) ( ) = wh, where wh = w(e, e ). This impies w(e, ) e = v(e, ) (e,e ) e = 0, (e,e ) where the seond equaity foows from the definition of Nash equiibrium. Thus, the soia indifferene urve at the point (e, e ) is aso tangent to the horizonta ine = e, so moving up the reation funtion of thoa supervisor inreases soia wefare. Figure 10 iustrates the resut. Hene, the ommitment to redue the effort of the entra supervisor in the hierarhia supervision setup is wefare improving. The intuition is that a redution in the effort of the entra supervisor fores thoa supervisor to inrease its (heaper)

22 22 SERIEs (2018) 9:1 26 Soia wefare indifferene urves e e Fig. 10 Limiting the size of the entra supervisor effort. However, the mode assumes that the efforts of the supervisors are not verifiabe, so there is an issue about how this oud be impemented. One obvious way to move in this diretion woud be to imit the size of the entra supervisor. 6 Conuding remarks This paper onstruts a mode of hierarhia supervision in whih a entra and a oa supervisor independenty hoose supervisory efforts whih determine the preision of a signa of the soveny of a oa bank. The entra supervisor uses this information to deide on the bank s eary iquidation. Importanty, thoa supervisor is assumed to be haraterized by a ower ost of effort (due to proximity) and a higher ost of iquidating the bank (due to apture). The mode of hierarhia supervision is ompared in terms of wefare with two aternative arrangements, namey deentraized supervision, where thoa supervisor is fuy in harge, and entraized supervision, where the entra supervisor is fuy in harge. The resuts show that moving supervision from oa to hierarhia to entra is morikey to be optima when the ost of getting oa knowedge is suffiienty ow and/or the possibe apture of thoa supervisor is a signifiant onern. To the extent that supervisory apture may be more reevant for arge banks, the resuts provide support for the design of Singe Supervisory Mehanism of the European Centra Bank (ECB), in whih signifiant banks are supervised by the ECB in oaboration with the Nationa Competent Authorities (NCAs), and ess signifiant banks are supervised by the NCAs. I woud ike to onude with a few remarks. First, the mode assumes that the iquidation ost is fuy driven by apture. However, it may be the ase that (at east part of) thiquidation ost is aso a soia ost, that is internaized by thoa supervisor but not by the entra supervisor. In this ase, having a more biased (oa) supervisor oud be better than having a ess biased (entra) supervisor, so the region in the parameter spae where deentraized supervision dominates beomes arger.

23 SERIEs (2018) 9: Seond, the mode assumes that bank is ompetey passive. As in Caretti et a. (2016), it woud be interesting to endogenize bank s hoie of risk under different supervisory arrangements. Moreover, it woud aso be interesting to introdue bank apita reguation, and to anayze the possibe trade-offs between reguation and supervision. In partiuar, one may onjeture that the tougher the reguation, and hene thower the bank s risk-taking, thess vauabe supervision wi be, whih as noted above redues the region of the parameter spae where hierarhia supervision dominates. Third, the mode is stati, but one oud easiy think of some dynami impiations. For exampe, it may be the ase that in good times supervisors graduay redue their apabiities, so they may not be abe to inrease them quiky when bad times arrive, in whih ase we oud end up having (invountary) supervisory forbearane. Finay, the mode fouses on supervision of a domesti bank. It woud be interesting to expore the ase of an internationa bank that operates aross severa oa jurisditions. In fat, one oud argue that the rationae for the reation of the Singe Supervisory Mehanism of the ECB was the fat that in a muti-ountry system, ike the euro area, the supervisory deisions of a biased nationa authority may entai negative externaities on other ountries. But these potentia externaities aso appear in the simpe setup of this paper. In partiuar, sine deposits are assumed to be insured, a bank faiure may impat on the soveny of the ountry in whih the bank is oated, and through a fisa hanne affet other ountries in the area. Open Aess This artie is distributed under the terms of the Creative Commons Attribution 4.0 Internationa Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided you give appropriate redit to the origina author(s) and the soure, provide a ink to the Creative Commons iense, and indiate if hanges were made. Appendix Proof of Proposition 1 By (3) one an write s = R + σ ( 1 + e 1) 1/2 x, where x N(0, 1). Substituting this expression into (4) and (5)gives E(L s) = a R + σ ( 1 + e 1) 1/2 x, E(R s) = R + σ ( 1 + e 1) 1/2 x. Hene, one an write v(e) = x + a R + σ ( 1 + e 1) 1/2 x δ φ(x)dx x R + σ ( 1 + e 1) 1/2 x φ(x)dx (e),

24 24 SERIEs (2018) 9:1 26 where x is impiity defined by the equation R + σ ( 1 + e 1) 1/2 x = ŝ. Soving for x and using the definition (10) ofŝ gives (13). Then, using the fat that xφ(x) = φ (x) and the definition (13) of x, one onudes v(e) = ( a R δ ) ( x) σ ( 1 + e 1) 1/2 φ( x) + R (1 ( x)) + σ ( 1 + e 1) 1/2 φ( x) (e) = R (1 a)r + δ ( x) + (1 )σ ( 1 + e 1) 1/2 φ( x) (e) = R (1 a)r + δ ( x) + φ( x) (e). x Proof of Proposition 2 Given that d ( x) + φ( x) = φ( x) d x x x 2 and x e = x 2e (1 + e), the first-order ondition that haraterizes an interior soution ê > 0is v (1 a)r + δφ( x) (e) = γ e = 0, 2e(1 + e) x and the seond-order ondition is v (e) <0. Hene, by the impiit funtion theorem one has ê γ = 1 v (e) v = e (e) γ v (e) < 0. Next, v (e) <0 and x < 0impy ê δ = 1 v (e) v (e) δ = 1 1 v (e) 2e(1 + e) φ( x) + (1 a)r + δ ( d φ( x) x d x x = 1 1 φ( x) v (1 + x2 )φ( x) (e) 2e(1 + e) x x = 1 1 v <0, (e) 2e(1 + e) xφ( x) v (e) ê R = 1 v (e) R ) x δ