Moral hazard, e ciency and bank crises

Transcription

1 Moral hazard, e ciency and bank crises S.Chatterji and S.Ghosal, Centro de Investigacion Economica, ITAM, and University of Warwick January 23, 2009 Abstract Under what conditions should bank runs be tolerated? We study a model with moral hazard in banking where using banking contracts with a zero crisis probability will fail to mobilize deposits and result in autarchy. Without any a priori restrictions on banking contracts, we show that there is positive crisis risk in banking scenarios where there is a tradeo between ex ante and interim e ciency: positive crisis risk is required achieve constrained ex ante e ciency although it is interim ine cient. We discuss the policy implications of our model. JEL classi cation numbers: G21, D82. Key words: moral hazard, e ciency, random contracts, bank runs. Contact details: shurojit@itam.mx, S.Ghosal@warwick.ac.uk. 1

2 1 Introduction Under what conditions should bank runs be tolerated? In the seminal paper by Diamond and Dybvig (1983) (see also Bryant (1980)), e cient risk-sharing between depositors with idiosyncratic and privately observed taste shocks creates a demand for liquidity. Banks invest in illiquid assets but take on liquid liabilities by issuing demand deposit contracts with a sequential service constraint. Although demand deposit contracts support e cient risk-sharing between depositors, the use of such contracts makes banks vulnerable to runs driven by depositor coordination failure. Prevention of bank runs driven by depositor coordination failure can be achieved if each individual depositor can be credibly assured that even if all other depositors withdraw, her deposit will continue to be safe. Indeed, as Diamond and Dybvig point out, when aggregate taste shocks are common knowledge, a demand deposit contract with an appropriately chosen threshold for suspension of convertibility eliminates bank runs while supporting e cient risk-sharing. In this paper we study a model of banking with ex ante moral hazard where there is a misalignment of incentives between banks and depositors. Under these conditions, as we argue in section 2 below, using banking contracts with a zero probability of a bank crisis will fail to mobilize deposits and will result in autarchy. We show that positive crisis risk is necessary to ensure constrained ex ante e ciency although a crisis is always interim ine cient. Without any a priori restrictions on banking contracts, we show that whenever there is a tradeo between ex ante and interim e ciency, the second-best contract involves a positive crisis risk. To make the point as simply as possible, we study banking in a closed region. Although the bank has no investment funds of its own, it has a comparative advantage in operating illiquid assets: no other agent in the economy has the human capital to operate an illiquid asset (i.e. the returns to the asset are premature. Consequently, the bank controls any investment made in illiquid asset. The bank chooses e ort (which can be interpreted as monitoring e ort). The probability distribution over asset returns depends on the bank e ort and a state of the world (which can be interpreted as a productivity shock). After depositors endowments have been mobilized, but before the realization of the shock, the bank chooses e ort. The asset generates a stream of "public" and "private" returns. We think of "public" returns as cash ows generated by the asset that the bank cannot access without depositors consent (for instance, such cash ows are generated by physical capital which can be monitored and seized by depositors). "Private" returns, then, are cash ows generated by the asset which can be accessed by the bank without depositors consent 1. Such "private" returns ensure that the bank s payo s are relatively immune to uctuations in 1 Following Hart and Moore (1998), we think of "public" returns as cash ows generated by the asset that the bank cannot steal because they are publicly veri able (for instance, they are embedded in physical capital which can be seized by depositors). "Private" returns, then, are cash ows generated by the asset which can be stolen by the bank. 2

3 the "public" asset return. When the e ort decision of the bank is non-contractible, and beliefs are optimistic (i.e. a high probability is attached to a positive shock), we show that the bank will choose low e ort in absence of a positive crisis risk. Optimistic beliefs and bank e ort are thus substitutes so that a positive crisis risk is required for the bank to choose high e ort. Further, although depositors anticipate this possibility, they still deposit their wealth with banks as such second-best contracts dominate autarchy in expected payo s. Thus, banks improve on autarky, but also generate, endogenously, additional crisis risk. Policy is discussed in more detail in section 4 below. The rest of the paper is structured as follows. The remainder of the introduction relates the results obtained here with other papers on bank runs. Section 2 build the model of banking with moral hazard and section 3 is devoted to deriving the main results of the paper. Section 4 contains a brief discussion of policy. The nal section concludes. 1.1 Related literature A number of papers have studied the requirement of bank runs in banking contracts. Calomiris and Kahn (1991) study a model of embezzlement in banking where the bank s temptation to embezzle depends on the realization of an exogenous move of nature and depending on the prevailing state, either the bank will never be tempted to embezzle or will always be tempted to embezzle. Holmström and Tirole ((1997), (1998)), study a model where conditional on the realization of an exogenous liquidity shock, banks incentives have to be aligned with those of the depositors. Diamond and Rajan (2001) who show that the threat of bank runs o the equilibrium path of play impacts on the bank s ability and incentives to renegotiate loan contracts with borrowers. A common feature of Calomiris and Kahn (1991), Holmström and Tirole ((1997), (1998)), and Diamond and Rajan (2001), is their focus on issues of moral hazard that arise conditional on the realization of the liquidity shock. In contrast, here, we study moral hazard issues that arise ex ante before the realization of the liquidity shock. Another branch of the literature has focused on the relation between incomplete information about the future returns of the illiquid asset and bank runs (see, for instance, Gorton (1985), Gorton and Pennacchi (1990), Postlewaite and Vives (1987), Chari and Jaganathan (1988), Jacklin and Bhattacharya (1988), Allen and Gale (1998)). However, in these papers, the variation in the future returns of the illiquid asset is exogenous while here the variation in future returns is a function of the investment decision of the bank and is hence endogenous. Finally, in our paper, as in Aghion and Bolton (1992), bank runs can be interpreted as a way of allocating control of over banking assets to depositors. However, unlike Aghion and Bolton (1992), the reallocation of control rights isn t triggered by some exogenous event but endogenously via depositor s actions in the second-best banking contract. 3

4 2 The model In this section we study a model of banking in a closed region. There are three time periods, t = 0; 1; 2. In each period there is a single perishable good x t. There is a continuum of identical depositors in [0; 1], indexed by i, of mass one, each endowed with one unit of the perishable good at time period t = 0 and nothing at t = 1 and t = 2. Each depositor has access to a storage technology that allows him to convert one unit of the consumption good invested at t = 0 to 1 unit of the consumption good at t = 1 or to 1 unit of the consumption good at t = 2. Depositors preferences over consumption are identical ex-ante, i.e. as of period 0. For each agent, only total consumption (and not its period-wise decomposition) is publicly observable. Formally, at t = 1; each agent has a utility function which has the following form: U(x 1 ; x 2 ) = x 1 + x 2 There are two states of nature 2 f; g, 0 < < 1. At t = 0, it is assumed that the bank and each depositor have the same initial beliefs fq; 1 qg over f; g. The state of nature is realized at t = 1 2. In addition, there is a bank, denoted by b. Unlike depositors, the bank has no endowments of the consumption good at t = 0. However, the bank is endowed with a technology that converts inputs of the perishable good at t = 0 to outputs of the perishable consumption good at t = 1 or t = 2. The output of the perishable consumption good produced by the asset technology has two components: a private non-contractible component that only the bank can access and consume and a public component which depositors can access and consume. Both the public and the private component of both asset technologies are characterized by constant returns to scale and depend on costly e ort e 2 fe; eg, 0 < e < e 1 chosen by the bank at t = 0. For each unit of the consumption good invested in t = 0, the asset technology yields either 1 unit of the public component of the consumption good if the asset is liquidated at t = 1 or R units of the public component of the consumption good at t = 2 if the asset is liquidated at t = 2 where R = R L probability 1 e and R = R H with probability e. Further, we assume that e < e = e < e. In addition, for each unit of the consumption good invested in t = 0, the asset yields 1 unit of the private non-contractible component of the consumption good if the asset is liquidated at t = 1, or R b > 0 units of the private component of the consumption good at t = 2 if the asset is liquidated at t = Nothing essential in our analysis depends on this assumption: see below. At t = 0, it is assumed that the bank and each depositor have di erent initial beliefs over, f b ; 1 b g and f d ; 1 d g respectively, over f; g. The state of nature is determined at t = 1. An alternative formulation with the same formal implication would be that is determined at t = 0 before deposits are mobilized and the bank chooses e ort and (i) the bank and each depositor have a common prior f d ; 1 d g over f; g (ii) before choosing e ort, the bank privately observes a signal s which generates a posterior beliefs f b ; 1 b g over f; g. 3 The assumption that within the asset technology there is no choice as to how much of 4

5 The bank s preferences over consumption and e ort is represented by the utility function U b (x 0 ; x 1 ; x 2 ) = x 1 +x 2 c(e) 4 where c(e) = 0 and c(e) = c > 0. Further, we will assume that the size of the bank is large relative to the size of an individual depositor 5. We make some further assumptions on depositor s preferences and the asset technology: (A1) R H > 1 > R L > 0, (A2) 1 < R b c, (A3) e < e = e < e, (A4) qer H + (1 q) er L < 1 < qer H + (1 q)er L, (A5) er H + (1 e) R L < 1 while both er H + 1 e R L > 1 and er H + 1 e R L > 1. Assumption (A1) implies that production e ciency requires the bank not to shirk. Assumption (A2) implies that the bank prefers asset liquidation at t = 2 over asset liquidation at t = 1 whatever its e ort choice. Assumption (A3) ensures that no depositor can infer the e ort choice of the bank with certainty after observing the quantity e: to do so, would require the depositor also observes. Assumption (A4) implies that depositors wouldn t invest with the bank if the bank chooses low e ort. Taken together, assumptions (A2) and (A4) together imply that its is ex ante e cient for the bank to choose e = e. Assumption (A5) implies that it is interim e cient to liquidate the asset early at t = 1 if and only if the bank chooses e and the state of the world ; otherwise, interim e ciency requires that the asset should only be liquidated at t = 2. In order to operate the asset technology, the bank has to mobilize the endowments of the depositors. At t = 0, we assume that mobilizing depositors endowments requires a banking contract satisfying the following constraints: (a) the bank has sole control over the investment and operation of the asset technology as no other agent in the economy has the human capital to operate either of these two technologies; (b) no other agent can replace the bank to take over the operation of the asset from the bank at t = 1; (c) at t = 1 depositors observe the quantity e but do not directly observe either the e ort choice of the bank or the state of the world ; (d) the public return at t = 1 is observed by the depositors and/or an outside the investment goes into the public component and how much into the private component is a simpli cation and nothing essential in our results depends on this analysis. 4 The assumption that u b (:) is linear simpli es the computations and the notation considerably. All the results stated here extend, with appropiately modi ed computations, to the case where u b (:) is a strictly increasing and possibly concave in consumption. 5 As each individual depositor has a (Lebesgue) measure zero, if the bank has the same size as an individual depositor, transfers to the bank can be made without a ecting the overall resource constraint. In order to capture the trade-o between making transfers to the bank and e cient risk sharing between depositors, the bank has to be large relative to the depositors. Technically, the set of agents is modelled as a mixed measure space where each individual depositor has a Lebesgue measure zero (and therefore is part of an atomless continuum of depositors) while the bank is an atom with measure one. For details on how construct such a measure space see Codognato and Ghosal (2001). 5

6 agent (a court) only if the asset technology is terminated at t = 1 and the public return at t = 2 is observed by the depositors and/or the outside agent only at t = 2; (e) if the asset is liquidated early at t = 1, the proceeds of the asset are shared equally between all depositors. The consequence of making assumptions (a)-(d) is that the e ort choice of the bank at t = 0 is non-contractible. The combination of non-contractible actions together with the private non-contractible component of asset payo s and the privately borne cost of e ort by the bank is the source of moral hazard in banking. Assumption (e) ensures that there is equal treatment of depositors when the asset is liquidated early at t = 1. This leads us to de ne a bank crisis as a scenario where asset liquidation is interim ine cient i.e. when the there is a positive probability of early asset liquidation even when depositors know that the probability of a high return probability R H is strictly greater than e. Throughout the paper we assume that there is a benevolent social planner who maximizes the ex ante e ciency. An allocation is a vector (e; x; x b ) where e is the e ort choice of the bank chosen at t = 0, x = (x 1 ; x 2H ; x 1 2L ) is the consumption allocation of the depositors (x 1 is the consumption of a depositor at t = 1, and x 2H ; x 2L is a depositor in time period 2 in the two possible outcomes R H and R L ) and x b = x b 1; x2 b describes the consumption allocation to the bank. 3 Equilibrium crisis and e ciency Given that depositors observe the quantity e, in a banking contract, the probability of early asset liquidation can be made conditional on e. Clearly, when depositors observe e, early asset liquidation must occur with probability one. Alternatively, when depositors observe e, early asset liquidation must occur with probability zero. Let the probability of a bank crisis be denoted by, 0 1, when depositors observe e = e at t = 1. Then the expected payo to the bank from choosing e = e is q (1 ) R b + q + (1 q) while the expected payo from choosing e = e is [q + (1 q) (1 )]R b + (1 q) c. By computation, it follows that the bank will choose e = e if and only if the following inequality holds: [(2q 1) + (1 q)] c When = 0, the (1) reduces to the inequality (1 q) c while when = 1, (1) reduces to the inequality q c 6 (3): (2) (1)

7 It follows that c (a) If > q, there is no banking contract which induces high e ort by the bank; (b) If (1 q) < c q, a strictly positive probability of a bank crisis is necessary to ensure that the bank chooses high e ort; (c) If (1 q) c, the bank chooses high e ort even with zero probability of a bank crisis. In what follows, we focus on the case (b) where (1 q) < c q. In this case, note that from the bank s (and depositors) forecasts that = exceeds a critical threshold value and given its forecast, in the absence of bank runs, chooses put in low e ort e = e: when agents believe that = with a high probability, q and e ort are substitutes for the bank. For case (b), we show that a strictly positive crisis risk is both necessary and su cient to implement e = e, a necessary condition for productive e ciency and we call an allocation with e = e constrained ex-ante e cient. Note, however, that in this case, necessarily interim e ciency is violated as at t = 1 there is early asset liquidation even when when depositors know that the probability of a high return probability R H e > e. Therefore, there is a trade-o between constrained ex-ante and interim e ciency. We summarize the above discussion as the following proposition: Proposition 1 When (1 q) < c q, a banking contract, all of whose equilibria at t = 1 involve a zero probability of early asset liquidation, will fail to implement an allocation where e = e. When (1 q) < c q, a zero probability of a bank run implies that the bank chooses e = e for sure. Therefore, if the banking contract has a zero probability of a bank run, each depositor will anticipate that the bank will behave opportunistically with probability one and prefer to invest in the storage technology thus resulting in autarchy. Therefore, even constrained ex ante e ciency requires that e = e: a positive probability of a bank crisis is necessary to mobilize deposits in the rst place. For the remainder of this section, we assume that (1 q) < c q and design a banking contract that implements the constrained ex ante e cient allocation where e = e. Consider the randomization scheme (S; ) where S = fs 1 ; :::; s M g, M 2, is some arbitrary but nite set of states of nature and = f 1 ; :::; M g, m 0, P m=m m=1 m = 1 is a probability distribution over S 6. The randomization scheme works as follows: at t = 0, no agent, including the bank, observes s m while at t = 1, before any choices are made, the realized value of s m is revealed to all depositors (but not the bank). A random allocation is a collection (~e; ~x; ~x b ) where ~e 2 fe; eg, ~x : S! < 3 + and ~x b 2 S! < 2 +. Let S = fs m 2 S : ~x 1 (s m ) 1g, M = m : s m 2 S and let = P m2m m. The interpretation is that whenever 6 Obviously, there are other ways of introducing randomness in the social planner s problem. We choose the randomization scheme presented here as a matter of convenience. 7

8 s m 2 S, the asset needs to be liquidated at t = 1 and therefore, is the probability of a bank run. Therefore, at t = 1, only the depositors can condition any choices they make on s m but the payo s to depositors and the bank can be made contingent on s m. Ex-ante constrained e ciency requires that ~e = e. The contingent consumption allocation of the bank is de ned as follows: (4a) ~x b 1 (s m ) = 1; s m 2 S; (4 0 b) ~x b 1 (s m ) = 0; s m 2 SnS; (4 0 c) ~x b 2 (s m ) = 0; s m 2 S; (4 0 d) ~x b 2 (s m ) = RA b ; s m 2 SnS: The contingent allocation of the depositors are de ned as follows: (5a) ~x 1 (s m ) = 1; s m 2 S; (5b) ~x 2H (s m) = ~x 2L (s m) = 0; s m 2 S; (5c) ~x 1 (s m ) = 0; s m 2 SnS; (5d) ~x 2H (s m) = R H ; ~x 2L (s m) = R L ; s m 2 S: Let ~ be the lowest value of for which (1) is satis ed. If (1 q) < c q, it follows that for a xed pair (S; ), M 2, such that S is non-empty and ~, < 1, there is a unique random allocation satisfying (4) (5). For a xed pair (S; ), such that either S is empty or < ~, we have already established that there is no allocation that satis es (4) (5). Finally, for a xed pair (S; ), such that either SnS is empty or = 1, both ~x 1 (s m ) = 1, ~x b 1 (s m ) = 1 and ~x b 2 (s m ) = 0 for all s m 2 S. Note that at any optimal choice of (S; ), generating a unique random allocation satisfying (4) (5), both SnS and S will have to be non-empty. It follows that without loss of generality, we can restrict attention to S 0 such that M = 2. Finally, as the social planner wants to maximize ex-ante e ciency, she will choose the lowest value of 0 compatible with (1) being satis ed as a strict inequality i.e. choose 0 2 = ~ + " < 1, where " > 0 is small but strictly positive number so that (1) is satis ed as a strict inequality. Setting 0 2 = ~ will imply that (1) will be satis ed as an equality in which case the social planner will have to rely on the bank choosing a tie-breaking rule in favour of high e ort. It remains to specify a random banking contract that will implement the constrained ex ante e cient allocation satisfying (4) (5). A random banking contract 7 is described by the vector S 0 ; 0 ; ~e; ~r; ~ k where the pair (S 0 ; 0 ) are as in the preceding paragraph, ~e = e and ~r 1 (s 0 1) = 0, ~r 1 (s 0 2) = 1, ~r 2H (s 0 1) = R H, ~r 2L (s 0 1) = R L ; ~r 2 (s 0 2) = 0, ~r b 1 (s 0 1) = R b, ~r 2 (s 0 2) = 1; ~ k (s 0 1) = 0, ~ k (s 0 2) = 1. Let ~r m 2 (s 0 1) = er H + 1 e R L > 1 and ~r m 2 (s 0 2) = 0. The interpretation is that ~ k (s m ) is the suspension of convertibility threshold, each depositor who withdraws in period 1 obtains a random claim ~r 1 (s 0 m), s 0 m 2 S 0 per unit deposited at t = 0. If banking continues to t = 2, each agent who withdraws at t = 2, obtains a random claim ~r 2 (s 0 m), s 0 m 2 S 0 per unit deposited at t = 0. Given the sequential structure of the banking scenario 7 As before we assume that at t = 0, no agent, including the bank, observes s m while at t = 1, before any choices are made, the realized value of s m is revealed to only the depositors. Thererfore, at t = 1, only the depositers can condition any choices they make on s m. 8

9 studied here, our notion of implementation requires that agents use dominant actions in every subgame of the banking contract. With such a contract, given s 0 m 2 S 0, the payo to per unit of deposit withdrawn at t = 1, which depends on the fraction of depositors who have chosen to withdraw their deposits, f, is given by the expression ~v 1 (f; ~r 1 (s 0 m) ; k ~ (s 0 m) ; s 0 ~r1 (s 0 m) = m) ; if f k ~ (s 0 m) 0; f > k ~ (s 0 m) while the period 2 payo per unit deposit withdrawn at t = 2, which depends on total fraction of deposits withdrawn in period 1, k (s 0 m), is given by the expression ~r ~v 2 (k (s 0 m) ; ~r 1 (s 0 m) ; s 0 m m) = 2 (s 0 m) ; if 1 > k (s 0 m) ~r 1 (s 0 m) 0; otherwise At t = 1, for each value of s 0 m 2 S 0, the above contract induces a noncooperative game between depositors where each depositor chooses what fraction of their deposits to withdraw. Fix s 0 m 2 S 0. Suppose depositor j withdraws a fraction j (s 0 m) Then, a depositor obtains a payo of j (s 0 m) ~v 1 (f; ~r 1 (s 0 m) ; k ~ (s 0 m) ; s 0 m)+ 1 j (s 0 m) ~v 2 (k (s 0 m) ; ~r 1 (s 0 m) ; s 0 m). For s 0 1, as k ~ (s 0 1) = 0 and ~r 2 m (s 0 1) > 1, it follows that ~v 2 (f; ~r 1 (s 0 1) ; s 0 1) > ~v 1 (k (s 0 m) ; ~r 1 (s 0 1) ; k ~ (s 0 1) ; s 0 1) and for all depositors, j (s 0 1) = 0 strictly dominates all other actions. For s 0 2, as ~r 1 (s 0 2) = 1 while ~r 2 m (s 0 2) = 0, it follows that for all depositors, j (s 0 2) = 1 strictly dominates all other actions. Therefore, (i) for s 0 1, the unique Nash equilibrium in strictly dominant actions is j (s 0 1) = 1 if j is a type 1 depositor while j (s 0 1) = 0 if j is a type 2 depositor and (ii) s 0 2, the unique Nash equilibrium in strictly dominant actions is j (s 0 1) = 1 for all j. At t = 0, as (1 q) < c q and the probability of early asset liquidation is 0, (1) will hold as a strict inequality and the bank will choose e = e. Therefore, the above random banking contract implements the allocation satisfying (4) (5). We summarize the above discussion with the following proposition: Proposition 2 When (1 q) < c q, the constrained ex-ante e cient allocation determined by (4) (5) is implemented by the random banking contract S 0 ; 0 ; ~; ~r; k ~. The above result makes clear that whenever the moral hazard constraint binds, bank runs are an endemic feature of the banking contract and limit e cient risk-sharing (equivalently, e cient liquidity provision) by banks. Remark 3 A corollary of proposition 1 is that when (1 q) < c q, although the second-best banking contract has a positive probability of a bank run, it improves on autarchy. Therefore, although depositors anticipate that a bank run will occur with positive probability, they still deposit their wealth with banks as such a second-best banking contract still dominates autarchy in expected payo s. Thus, banks improve on autarky, but also generate, endogenously, additional crisis risk. 9

10 4 Policy implications The additional randomness introduced by banking contracts studied here adds noise to the randomness generated by fundamentals and is driven purely by incentives. In our model, bailouts are equivalent to building in a suspension of convertibility clause in the banking contract. In this sense, the random bank contracts studied here provides a rationale for the banking regulator not to make any ex ante commitment to a speci c bailout policy. In the absence of monitoring and the ability to con scate bank payo s, our result provides a rationale for the doctrine of "creative ambiguity", wherein the banking regulator makes no ex-ante commitment to a particular bailout policy but instead leaves the banking sector in doubt about its intentions (Goodhart (1999)). Although intervention by central banks or government agents takes place typically after the onset of a crisis (see, for instance, OECD (2002)), plans for a contingent intervention regime can be put in place ex ante. Such an intervention regime would entail an ex ante commitment by a public authority (such as a central bank or a nancial services regulator) to (a) monitor bank actions, (b) followed by a threat of early termination of bank assets as a function of the information revealed. In principle, instead of a threat of early termination, the regulatory authority could set in place a system of positive transfers to the bank conditional on the information revealed by monitoring. However, incentive compatible transfers to the bank will lower consumption for all depositors. A consequence of our analysis is that it is still possible to implement e cient risk sharing between depositors, without sacri cing consumption, by using a contract which embodies the threat of bank runs o the equilibrium path of play. A key assumption of our formal analysis is that once the investment decision by a bank has been made, the bank cannot be prevented from accessing its "private" bene ts. When this assumption is relaxed, other intervention options become available such as directly con scating "private" bene ts of the bank. In essence, these are like negative transfers ( nes). Con scation of bank payo s contingent on the information revealed by monitoring will deter ex-ante opportunistic behavior. 5 Conclusion We interpret the signi cance of our results in two distinct ways. First, our results show that with moral hazard, crisis are necessary elements in banking scenarios where there is a trade-o between ex ante and interim e ciency. Second, under conditions of borrower moral hazard, we have shown that appropriately designed random demandable debt contracts Pareto improve on autarky. Extending the model to examine contagion, risk averse depositors and episodes of twinned bank runs and currency crises is a topic for future research. 10

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