Collective Moral Hazard, Maturity Mismatch, and Systemic Bailouts 2

A. Date-0 Interest Rate We assume that the government can tax the rate of return on the storage technology between dates 0 and 1. The proceeds are rebated lump sum to consumers. This is rigorously equivalent to assuming the government can subsidize investment in the banks at date 0, a subsidy financed by lump sum taxes on consumers. We show that it is optimal, both under commitment and under no commitment, to set the date-0 interest rate 0 to 1, even without assuming any distortion from lowering 0 .33 The borrowing capacity of banking entrepreneurs at date 0 is now given by (9) =  1−  0 − 0 0 + (1−)  0 where , as earlier, denotes the liquidity ratio. Banking entrepreneurs hoard liquidity  = 1 − 0  at date 0 if and only if it expects a date-1 interest rate  satisfying (10)  +  − 0 ≤ 0 µ ¶ 1 −1   and hoards no liquidity ( = 0) otherwise. This generalizes our analysis of Section II to arbitrary 0 ≤ 1. Denote by  ( 0 ) the corresponding investment scale chosen by banking entrepreneurs–obtained by replacing the optimal liquidity choice  of banking entrepreneurs in equation (9), a choice governed by equation (10). Similarly, let  ( 0 ) denote the corresponding reinvestment scale in case of a crisis. Ex-ante welfare under commitment is given by (11) 1−   ( 0 ) − (1 − )  ()  0 − [ ( 0 ) − ] (1 − 0 )   ex ante ( 0 ) ≡  ( 0 ) − (1 − ) The last term on the right-hand side is new and reflects the implicit subsidy to bank investment at date-0 (consumers invest a total amount  ( 0 ) − ). The following proposition shows that under the assumptions that we have maintained throughout the paper, the optimal interest rate policy under commitment is passive both at date 1 and at date 0. 33 These results would only be reinforced if we were to introduce a loss function 0 (0 ) as we did for the date-1 interest rate. 28 Proposition 6 The optimal interest rate policy under commitment features 0 =  = 1. The intuition is straightforward. Assumption 6 guarantees that redistributing resources from consumers to banking entrepreneurs is welfare reducing. Lowering the interest rate 0 or  below 1 induces such a redistribution of resources (with additional distortions in the case of ). Setting these interest rates equal to 1 is therefore optimal. Analyzing the no-commitment solution requires solving a dynamic game. We focus on subgame-perfect equilibria. We will need to impose some refinement. We know from our previous analysis that the set of equilibrium interest rates of the continuation game is expanding in the date-0 investment scale. We therefore find it natural to focus on subgame perfect equilibria that satisfy the following monotonicity requirement: Assumption 8 The continuation equilibrium of the no-commitment game is such that the date-1 interest rate is non-decreasing in the date-0 interest rate. This assumption would be automatically satisfied if for example, we always chose the worst possible continuation equilibrium (the one with the lowest ) for any value of 0 . Corollary 4 In every equilibrium of the no-commitment game, 0 = 1. This corollary follows directly from Proposition 6. Indeed, lowering the interest rate 0 below 1 only entails larger costs compared with the commitment solution, because it leads banking entrepreneurs to hoard less liquidity anticipating a lower interest rate  at date 1. B. Leverage decisions, going forward Consider a longer-horizon model, say the overlapping-generations version of this model, in which banking entrepreneurs live, like in this model, for three periods. A bailout at date  + 1 of those banks that borrowed at date  then also affects the financing decisions of the next generation of banks, which borrow at date  + 1. Interestingly, this operates through two channels: increased leverage and increased maturity mismatch.34 To preview the results, bailing out generation  induces leverage and maturity mismatch for generation +1 , sowing the seeds for a date- + 2 crisis and bailout. Indeed, an interest 34 Arguably, both channels seem to have operated during the long episode of very low interest rates in the wake of the 2000 Internet bubble crash. 29 rate bailout of generation  makes generation +1 (a) more willing to take an illiquid position by loading up on short-term debt and (b) increase the size of its investment. These two effects–the increased maturity mismatch and the increased leverage channels respectively– distort generation +1 ’s incentives and generate a social cost that is a fixed cost (in the sense that it does not depend on generation  ’s size or illiquidity) when contemplating a rescue of generation  . The logic of the argument can be grasped by appending to the model of Section V.A a prior generation of entrepreneurs, generation −1 , living at dates −1 0 and 1.35 Generation −1 is in all respects similar to generation 0 born at date 0 and studied in this paper, except that its short-term (date-0) profit  0 is sufficiently large that not hoarding any liquidity is a dominant strategy (for any 0 ∈ [0  1]), which boils down to the condition that  0 +   1 +  (1 − 0 ). This assumption merely shortens the analysis by ensuring that generation −1 in case of a date-0 crisis is unable to withstand the shock unless the date-0 interest rate is brought down to 0 = 0 . The analysis of the generation 0 born at date 0 is exactly as in Section V.A. We make the following assumption, which is consistent with our previous assumptions.36 Assumption 9 (no liquidity hoarded when 0 = 0 ):   (1 − )0 . We rule out any deadweight loss from inefficient investments, so as to starkly illustrate that  can come from impaired incentives for the subsequent generation of entrepreneurs. We maintain this assumptions throughout Sections V.B and V.C. From our previous analysis and Assumption 8, we know that the optimal date-0 policy if one ignores the welfare of generation −1 consists in setting 0 = 1. Suppose that 0 = 1 is actually chosen. There are a number of possible continuation equilibria, corresponding to different expectations regarding the interest rate  Because we have assumed away any current distortion, all interest rates  ∈ [0  1] correspond to a possible continuation equilibrium, where banking entrepreneurs hoard liquidity  = 1 − 0 . Using the definition in 35 The model can be extended to infinite-horizon overlapping generations; we offer here the simplest illustration. 36 The analysis remains valid when this assumption is violated, but the cost of bailing out generation −1 in terms of generation 0 ’s incentives then comes solely from an increased investment scale, not from an increased maturity mismatch. 30 equation (11), this yields an ex-ante welfare for generation 0 £ 0 ¤   − (1 − ) (1 − ) ¡  0 ¢  ex ante ( 1) = 1 −  − 0 + (1 − ) 1 −  where  ex ante ( 1) increases with . In our analysis, we do not need to take a stand on which equilibrium is actually selected. Any 0 ∈ (0 ,1) is dominated: by Assumption 8, such interest rates reduce welfare from generation 0 and do nothing to help generation −1 . Therefore, we only have to analyze whether 0 = 0 is preferred to 0 = 1. When 0 = 0 , no liquidity is hoarded by generation 0 , and because there are no immediate distortions from interest rate policy, a full interest rate bailout with  = 0 follows. Using the definition in equation (11), welfare from the date-0 generation becomes " # )]  [ − (1 − ) (1 −  1 0 − (1 − 0 ) − 1   ex ante (0  0 ) = 1 −  − 0 1 −  −  0 Turning now to the decision of whether to rescue generation −1 at date 0, and remembering that choices outside {0 , 1} are dominated, we conclude that there exists a fixed cost of bailing out generation −1 by setting 0 = 0 . Depending on which continuation equilibrium is selected when 0 = 1, with corresponding welfare  ex ante ( 1), this fixed cost is given by (0 ) ≡  ex ante ( 1) −  ex ante (0  0 ). It is maximized when following 0 = 1, the continuation equilibrium with  = 1 is selected, and minimized when the continuation equilibrium with  = 0 is selected. It can be verified that even in the latter case, we have (0 )  0 as long as Assumption 6 holds. The selection of the continuation equilibrium with  = 0 isolates the increased leverage channel mentioned above. By contrast, the selection of the continuation equilibrium with  = 1 combines both the increased leverage and the increased maturity-mismatch channels. Proposition 7 When contemplating whether to rescue the generation −1 banks, the government faces a fixed cost equal to (0 ) =  ex ante ( 1) −  ex ante (0  0 )  0, where  ∈ [0  1] corresponds to the selected continuation equilibrium when 0 = 1. This cost is fixed in that it does not depend on generation −1 ’s investment to be rescued. 31 C. Central bank’s reputation Yet another deferred cost of bailouts is the loss of reputation by the central bank. This can be modelled by introducing a tough type and a soft type. A bailout then reveals the type of the central bank to be soft, raising the likelihood of future bailouts and pushing banks to take on more risk, hoard less liquidity and lever up, resulting in increased economywide maturity mismatch and in turn larger bailouts. Even a central bank of the soft type internalizes this reputation cost and is reluctant to engage in a bailout in the first place. To show how reputation concerns generate yet another fixed cost, we follow Section V.B but introduce uncertainty about the central banker’s preferences. With prior (date-0) probability 1 − , the central banker is “bailout-prone” as he puts weight   1 − 0 (so   0) on investment as earlier. With probability , the central bank is “tough” as he puts no or little weight on investment and therefore always chooses interest rates equal to 1. The situation is otherwise the same as in Section V.B: Generation −1 optimally hoards no liquidity; when it faces a crisis, the rational choice for a bailout-prone central banker is again between 0 = 1 and 0 = 0 . A choice of 0 = 0 reveals that he is bailout-prone. Suppose first that the central banker sets 0 = 0 to bail out generation −1 . The equilibrium for generation 0 is then as in Section V.B. Welfare is  ex ante (0  0 ) for that generation. By choosing 0 = 1 by contrast, the central banker creates posterior beliefs b ≥ .37 If ̂ is large enough (̂ ≥), then hoarding liquidity is a dominant strategy for a generation-0 bank and welfare for generation 0 is  ex ante (1 1).38 Proposition 8 There exists   1 such that for  ≥, the bailout-prone central banker faces a fixed reputation-loss cost equal to (0 ) ≡  ex ante (1 1) −  (0  0 )  0 when rescuing the generation −1 banks. This cost is fixed in that it does not depend on generation −1 ’s investment to be rescued. 37 b =  in a pooling equilibrium, b ≥  in a partially revealing one, including a separating equilibrium for which b = 1. +[1−] 1 38 The threshold  is the solution of 1−− = 1−− +(1−)(1− . From Assumption 2,  1. To 0 0 0 ) understand this condition, note that the tough central banker (who has posterior probability b) will not bail out banks in case of a crisis, and so the probability of continuation is at most b + [1 − b] in the absence of liquidity hoarding. Conversely, hoarding liquidity (1 − 0 ) is needed in order to be able to continue in a crisis when the central banker is tough; but if the central banker turns out to be bailout-prone and lowers the interest rate at date 1 in case of crisis to  = 0 (this is the most optimistic hypothesis, which arises if other bankers hoard no liquidity), then this unneeded liquidity can be returned to investors. The cost of liquidity is therefore only (1 − )(1 − 0 )b  on average, which explains the term on the right-hand side. 32 The derivation of equilibrium behavior is then straightforward. For  ≥, the bailoutprone central banker pools with the tough one if and only if −1 −1 ≤ (0 ) is verified, where −1 =  −1 − (1 − 0 ) and  −1 and −1 are the weight on, and the investment of generation −1 . If −1 −1  (0 ), the equilibrium is separating. Finally, for  , the details of equilibrium behavior depend on equilibrium selection, but the overall pattern (a trade-off between the benefit of bailout and a cost of reputation loss) remains the same. VI. Optimal ex-post bailouts So far, we have restricted the set of policy instruments to the interest rate. We emphasized that this instrument was not targeted. Together with time inconsistency, this feature was a key ingredient in generating strategic complementarities in leverage choices. In this section, we relax the assumption of an exogenously specified policy instrument set. Instead, we follow a mechanism design approach and characterize the optimal ex-post bailout where policy tools are endogenous to the constraints of the economic environment. As we noted in the introduction, one may wonder whether interest rate bailouts, which involve both an implicit subsidy to banks and various distortions, are still desirable when other interventions are possible. For example, purchases of legacy assets, liquidity support and recapitalizations also involve direct transfers from consumers to banks–boosting their net worth and allowing them to refinance at a larger scale. However, these transfers do not reduce borrowing costs at the margin, and do not generate similar distortions. Interestingly, we show that interest rate policy still plays an important role within the optimal bailout scheme: It is always part of the optimal ex-post package, and over a range of parameters, bailouts boil down to pure interest rate policy. Furthermore, we recover a key insight from our previous analysis, the existence of strategic complementarities: to some extent, optimal bailouts are themselves untargeted. Setup. We allow the fraction  of banks that are distressed in a crisis to be less than 1 Denoting the probability of a crisis by 1 − ̂ we maintain the convention that  ≡ ̂ +(1 − ̂) (1 − ) represents the probability of being intact. In what follows the probability ̂ of a crisis is kept constant and the dependence of  with respect to  is left implicit. Moreover, we introduce an informational friction: the central bank can observe which banks are distressed, but the underlying auditing technology is imperfect. More precisely, we assume that the probability of generating a false positive when assessing if a bank is 33 distressed is equal to 39 As a result, in a crisis, a fraction  (1 − ) of banks are mistaken by the authorities as banks that need liquidity. These banks are aware that they belong to the false positive group. We assume that banking entrepreneurs and their investors form perfect coalitions, and that banks have full bargaining power in these coalitions. Instruments. We assume that the government cannot directly hold bank securities. The only available instruments are the interest rate (the borrowing cost of banks) and direct transfers to banks perceived as being distressed. We have already commented on the possible interpretations of interest rate policies as policies that lower the borrowing cost of banks. Direct transfers capture policies used in practice to boost the net worth of banks, such as the purchase of legacy assets at inflated prices. The assumption that the government cannot directly hold bank securities deserves some comments, as it rules out some forms of government intervention that are used in practice. For example, a recapitalization involves a transfer from the government in exchange of shares or warrants. However in most practical cases, the government usually sells its stake relatively quickly. What remains is a transfer from the government to the bank. At a theoretical level, this assumption limits the ability of the government to screen between intact and distressed banks. In Farhi-Tirole (2009) we analyzed the case where the government could hold stakes in banks and use this as a screening device. We showed that our insights were robust in this environment. However the analysis was quite involved. The assumption that the government cannot directly hold a stake in banks allows us to considerably streamline the analysis. If direct transfers could be perfectly targeted to distressed banks, they would dominate interest rates as a policy instrument. However, the government cannot perfectly recognize if a given bank is distressed and some banks might engage in rent seeking by successfully portraying themselves as distressed. Moreover, low interest rates benefit distressed banks comparatively more than intact ones. As a result, there is a non-trivial policy tradeoff between interest rate policy and direct transfers to institutions perceived as being distressed.40 Although we will consider only symmetric equilibria, we analyze the general case where the authorities face an arbitrary distribution  ( ) of banks with scale and liquidity ( ). 39 Note that for simplicity, we assume that there are no false negatives. The possibility of false positives is crucial for the following reason. If  = 0 or  = 1 then the authorities do not face any information extraction problem. As long as Assumption 7 holds, then banks in distress are always rescued through a direct transfer and are allowed to continue at full scale even if banks are completely illiquid. In equilibrium, banks would then choose to be completely illiquid, interest policy would not be used. 40 34 A bailout specifies an interest rate  and for every scale and liquidity ( ) a transfer  ( )  ≥ 0 for banks perceived as being distressed. This implies the following reinvestment scale for distressed banks ¾ ½  +  ( )  1   ( ) = min 1 − 0 Timing. The timing within period 1 is as follows: (1) the government announces a rescue scheme {  ( )}; (2) each banking entrepreneur accepts the plan if and only it makes him better off given that investors must be as well off as they would in the absence of participation in the scheme. To simplify the expressions, we assume that the government places no Pareto weight on banking entrepreneurs   = 0. Transfers to banks that are perceived to be distressed but are in fact intact therefore represent a pure welfare loss. Planning problem. In order to solve for the optimal ex-post bailout, we use the variables {  ( )} rather than {  ( )}  The transfers  ( ) can be inferred as follows:  ( ) = (1 − 0 )  ( )  −  ≥ 0 since without loss of generality  ≤ 1 − 0 . Up to a constant, the optimal ex-post bailout maximizes41 (12) Z ³ 0 ´ − () +   − (1 − )  ( )  ( )  Z h³ i  ´ − [ + (1 − ) ] 1 − 0  ( ) −   ( )  s.t.  ≥  ( ) ≥   1 − 0 To obtain equation (12), note that for distressed banks, the total subsidy is given by  ( ) + (1 − ) 0  ( ) . For intact banks that are perceived as distressed, the subsidy is  ( ) as those banks do not need to borrow more funds. The first two terms in equation (12) are exactly as in Section III. The third term corresponds to the additional implicit subsidy associated with the direct transfer  ( ). This is a transfer to distressed firms and intact firms that are perceived as distressed. Consumers get a rate of return equal to 0 (and not  as when they reinvest in the banks) on this transfer. 41 Because of the linearity of the objective and the constraints in  ( )  we need to take a stand on government actions when the government is indifferent. We assume that it sets  ( ) equal to its minimal possible value in case of indifference. 35 The government could achieve a given pattern of continuation scales  ( ) entirely through direct transfers by setting  ( ) =  ( ) − . This would economize on interest rate distortions. However, it increases undesirable transfers to intact banks perceived to be distressed. Lowering the interest rate  increases the collateral value 0  ( )  of distressed firms, allowing them to continue at a larger scale, while at the same time reducing the funds transferred to intact firms perceived to be distressed. By affecting the terms at which banks can borrow on the market, the government is able to better target firms that are actually distressed. However, it pays a cost in terms of distortions  (). We need to ensure that the latter is convex enough. Assumption 10 (enough convexity). The function −2 0 () is decreasing in  ∈ [0  1]. Optimal ex post bailout. The optimal bailout policy is summarized in Figure 1. Due to the linearity of the program, there can be two equilibrium configurations. We discuss them informally here and refer the reader to the appendix for the full details. (i) No liquidity hoarding ( = 0 for all banks). In this equilibrium configuration, banks load up on short-term debt as they expect to be rescued in case of a crisis. Given the absence of liquidity in the banking sector, the least-cost rescue policy for the government is a mix of transfer and interest rate policies. To see this, note that low interest rates reduce the wasteful transfer  to intact banks given that distressed ones require a unit refinancing from both sources: 1 =  + 0 . Basically, distressed banks can lever up the direct transfer to refinance at a greater scale. However low interest rates also increase the deadweight loss (); the marginal distortion is 0 at  = 1, and increases as the interest rate is reduced. When the severity () of the crisis increases, undue transfers to intact banks become less of a concern, and the optimal interest rate ̄ () increases to reach 1 when there are no false positives ( = 1).42 It must also be the case that the government wants to bail out the banks, that is that there is not too much adverse selection. Thus if the crisis is not very severe (   in the figure), the no-liquidity-hoarding equilibrium disappears. (ii) Liquidity hoarding. In this equilibrium configuration, banks hoard just enough liquidity (issue just as little short-term debt) as to be able to continue at full scale in case of a crisis. This requires two conditions. 42 ¡ ¢ 2 The function ̄ () is defined by the equation −̄ () 0 ̄ () = 36 (1−)0 1−−0 . First, given the expected date-1 interest rate, an individual bank must not expect that the government will make up for insufficient liquidity through a targeted transfer. Otherwise it would hoard no liquidity at all. Put differently, liquidity hoarding relies on the expectation of a pure interest rate policy bailout. Whether the government is willing to bail out a bank with insufficient liquidity depends on the trade-off, discussed in (i), between wasteful transfers to intact banks and rescues of distressed ones but in reverse: when  increases, the needed transfer  = 1 − 0  increases and so bailouts become less attractive. This implies that  has to be greater than or equal to (), where () is an increasing function of  as the government is more tempted to rescue banks as the crisis becomes more severe. It reaches 1 when  = ̄.43 Figure 1: pure and mixed bailouts Second, because the bailout is a pure interest rate intervention, the equilibrium must be an equilibrium when no transfers are feasible; that is, it must satisfy equation (3). So the equilibrium set (the shaded area in Figure 1) is the set of interest rates in Proposition 2 that satisfy  ≥ (). 43 A simple inspection of equation (12) shows that this threshold () is defined as the solution of the  1− 0  following equation in : +1−0 = +(1−) . The threshold () is increasing in  with (0) = 0 and  (̄) = 1 where ̄ ∈ [0 1] is the solution of the following equation in : 37  +(1−) = 1−0 +1−0  Once again banks’ leverage and liquidity choices are strategic complements, and there can be multiple equilibria. More generally, in this region, the insights gleaned from our analysis of pure interest rate bailouts in Sections III and IV carry over to optimal bailouts. There is one difference in the form of an additional lower bound () ≥ 0 on the interest rate. For   ̄ these equilibria disappear. The function ̄ () is increasing in . For  close to ̄, ̄ () is below () and the opposite is true for  close to 0 provided that ̄ (0)  0 . The following assumption ensures that these two functions cross only once. We denote by  the crossing point. Assumption 11 The function ̄ () defined by the solution of the equation −2 0 () =  (1 − ) 0  [1 −  − 0 ] crosses the function () once from above on the interval [0 ̄]. Proposition 9 The symmetric equilibria of the no-commitment economy are as follows: (i) if    then there is an equilibrium with  = ̄ (); it features  = 1(1 −  − 0 )  = 0 and  = ; (ii) if  ≤ ̄ then there is an equilibrium associated with each fixed point of the equation  ∈ R() ∩ [() 1]; it features  = 1[1 −  − 0 + (1 − )(1 − 0 )]  = 1 − 0 , and  =  − (1 − 0 ). To sum up, interest rate policy is always used in equilibrium; indeed direct transfers are not even used in some regions of the parameter space; in the remaining regions, interest rate policy and direct transfers are used in conjunction. Interest rate policy is a market-driven solution, in that it benefits primarily those institutions with actual borrowing needs. Direct transfers better focus on strategic actors, but they entail a greater waste of resources by supporting entities that have no need for, or should not engage in refinancing. When direct transfers are not used (for equilibria of type (ii)), the insights from Sections III and IV carry over to optimal bailouts: multiple equilibria, macroprudential regulation, and endogenous macroeconomic uncertainty.44 44 Marvin Goodfriend and Robert G. King (1988) argue that the central bank should just provide liquidity anonymoulsy through open market operations and leave it to the market to allocate this liquidity efficiently. They argue against providing liquidity to specific institutions. Our interpretation of interest rate policy is broader than open market operations, complicating the comparison with their view. Nevertheless, in our framework, optimal policy under commitment is entirely passive and can therefore be seen as vindicating their view. Even under limited commitment, the central bank finds it optimal ex post to use only interest rate policy for equilibria of type (ii). However for equilibria of type (i), a combination of interest rate policy and direct transfers to (perceived to be) distressed institutions is used. Direct transfers are used only when 38 VII. Conclusion We have built a simple framework to jointly analyze leverage and maturity mismatch in the banking sector, bailouts, and optimal regulation. In order to derive a simple yet general model, we have made a number of simplifying assumptions. Refining our analysis would require a richer model. For example, we have argued that one interpretation of interest rate policy in our framework was as a reduced form for conventional monetary policy. We could introduce an explicit nominal structure with sticky prices and imperfect competition to model conventional monetary policy as in New-Keynesian models. The framework could be further enriched to study the consequences of bailouts in a setup where interactions between the central bank’s balance sheet, inflation, and the government budget are non trivial. We could also introduce the possibility of sovereign default. We have also maintained the assumption of risk neutrality. Introducing risk aversion would allow studying how different policy interventions can impact banks net worth and borrowing costs by affecting risk premia. Finally, in our model, liquidity is costless in the sense that there are no liquidity premia. If liquidity were costly instead, interest rate policy under commitment would not necessarily be passive: it could be optimal for the government to provide liquidity in bad times. However, our insights would carry over: authorities would provide too much liquidity in the time-consistent outcome.45 We leave these questions for future research. the losses from mistaken transfers to intact institutions are smaller than the distortions arising from interest rate policy. Note that these policies are not Pareto improving ex post since they transfer resources away from consumers, but they do lead to an increase overall ex post welfare. These policies reduce ex ante welfare. 45 This could be formalized along the lines of Holmström-Tirole (1998). Imagine that consumers cannot commit at  = 0 to provide funds to banks at  = 1. Banks then have to hoard liquidity by purchasing assets at  = 0 and selling them at  = 1 in the even of a crisis. Banking entrepreneurs are willing to pay a liquidity premium (in the form of a lower return) on these assets, which can happen if there is a scarcity of stores of value. Then, the lack of commitment of consumers is a form of market incompleteness which can be alleviated through government intervention (even under the commitment solution). For example, through a commitment to a loose interest rate policy in crisis. See Farhi-Tirole (2009) for a discussion. 39