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Federal Reserve Bank of Chicago Equilibrium Bank Runs Revisited Ed Nosal WP 2011-13 Equilibrium Bank Runs Revisited Ed Nosal Federal Reserve Bank of Chicago August 2011 This version: November 2011 Abstract Peck and Shell (2003) show that it is possible to get a bank run in a Diamond-Dybvig environment. The mechanism they use, however, is not an optimal one. When an optimal mechanism is used, the bank run equilibrium disappears. 1 Introduction Although Diamond and Dybvig’s (1983) seminal article is associated with bank runs, it’s actually di¢ cult to generate them. For example, when there is no aggregate risk, they demonstrate that a bank run equilibrium cannot exist when the deposit contract is appropriately designed. The optimal contract is a “standard” deposit contract augmented by a suspension of convertibility if too many people want to withdraw early. In the second part of their article, they assert, but do not demonstrate, that deposit contracts will be subject to bank runs when there is aggregate risk. It was not until Green and Lin (2003), GL, that an optimal deposit contract under aggregate risk was fully characterized. GL take a mechanism design approach and demonstrate that the optimal deposit contract does not have a bank run equilibrium. Subsequently, Peck and Shell (2003), PS, modify the GL environment, and produce a bank run equilibrium. In a departure from GL, PS assume that depositors do not know their positions in the service queue. This seems important. Among other things, it means that GL’s powerful backward induction argument— that appears to eliminate bank run equilibria— does not apply. One can interpret PS’s modi…cations as generalizing the GL environment. In particular, if depositors do not know their positions in the service I would like to thank Marco Bassetto, Todd Keister, Ali Shourideh, Nico Trachter, and Neil Wallace for helpful discussions, and for comments on earlier versions of the paper. 1 queue, as in GL, then, in principle, the mechanism (or planner) can choose to either inform or not inform depositors regarding their positions. (GL can be interpreted as restricting the mechanism to always inform depositors about their positions in the queue.) Independent of how one views the GL environment vis-à-vis the PS environment, the mechanism that GL adopt is optimal for their economic environment. Their mechanism is a direct revelation mechanism, where each depositor announces his private information or type to the planner. PS also use a direct revelation mechanism. But for their more general economic environment, the direct revelation mechanism may not be an optimal one. I pursue this idea by constructing an indirect mechanism and show that it uniquely implements the best allocation, or at least an allocation that is arbitrarily close to it. In other words, my indirect mechanism does not admit a bank run equilibrium. This result reinforces an earlier observation: When deposit contracts are appropriately designed, bank runs are hard to come by in the DiamondDybvig environment. A bank run equilibrium can arise in a GL environment when depositors’ types are correlated and allocations are implemented by a direct revelation mechanism, see Ennis and Keister (2009b).1 Cavalcanti and Monteiri (2011) examine indirect mechanisms in this environment and demonstrate that the best allocation can be uniquely implemented in dominant strategies. Their backward induction argument, however, will not work in the more general PS environment, where depositors do not know their positions in the queue.2 The indirect mechanism that I construct can uniquely implement the best allocation for either GL- and PS-type environments in Nash equilibrium strategies. The paper is organized as follows. The next section describes the economic environment. Section 3 characterizes the best implementable allocation. Sections 4 and 5 construct mechanisms that uniquely implements it. Some concluding comments are o¤ered in the …nal section. 2 Environment There are three dates: 0, 1 and 2. The economy is endowed with Y > 0 units of date-1 goods. A constant returns to scale technology transforms y units of date-1 goods into yR > y units of date-2 goods. There are N ex ante identical agents. An agent is one of two types t 2 T = f1; 2g: patient, t = 1, or impatient, t = 2. The utility function for an impatient agent is u (c1 ) and the utility function for a patient agent is v (c1 + c2 ), where c1 is date-1 1 GL assume that depositor types are identically and independently distributed. Cavalcanti and Monteiri (2011) propose an alternative indirect mechanism when they examine a PS environment. In one example, they show that their indirect mechanism uniquely implements the best implementable allocation. However, in another example, their indirect mechanism has a bank run equilibrium. 2 2 consumption and c2 is date-2 consumption. u and v are increasing, strictly concave, and twice continuously di¤erentiable. Agents maximize expected utility. The number of patient agents in economy is drawn from the probability distribution = ( 0 ; : : : ; N ), where n > 0, n 2 f1; : : : N g N, is the probability that there are n patient agents. A queue is the vector tN = (t1 ; : : : ; tN ) 2 T N , where tk 2 T is the type of agent that occupies the k th position/coordinate in the queue. Let Pn = tN 2 T N j#2 2 tN = n and Qn = jjtj = 2 for tN 2 Pn , where ‘#2’is the number of patient agents. Pn is the set of queues with n patient agents and Qn is the queue positions of the n patient agents in tN 2 Pn . The probability that tN 2 Pn is n =#Pn = n = Nn , where #Pn is the number of queues tN 2 Pn . This speci…cation implies that all potential queues with n patient agents are equally likely. Agents are randomly assigned a position in the queue, where the (unconditional) probability that an agent is assigned to position k is 1=N . For convenience, call the agent assigned to position k agent k. The queue realization, tN , is observed by no one: not by any of the agents nor the planner. Each agent, however, privately observes his type t 2 T . The timing of events and actions is as follows. At date 0, the planner constructs a mechanism that determines how date-1 and date-2 consumption are allocated among the N agents, and queue tN is realized. A mechanism is a set of announcements, M and A, and a allocation rule, c = (c1 ; c2 ) where c1 = (c11 ; : : : c1N ) and c2 = (c21 ; : : : c2N ). At date 1, agents sequentially meet the planner, starting with agent 1. In a meeting with agent k, the planner announces ak 2 A and agent k responds with mk 2 M . Only agent k and the planner can directly observe ak and mk . (But the planner can reveal (ak ; mk ) to agent j k via announcement aj , if he wishes.) There is a sequential service constraint at date 1, which means the planner allocates date1 consumption to agent k 2 N based on the announcements of agents j k, i.e., 1 k 1 k 1 3 ck m ; mk , where m = (m1 ; : : : ; mk 1 ). Agents consume the date-1 good at their date-1 meetings with the planner. After all agents have met the planner, the planner simultaneously allocates the date-2 consumption good to each agent based on all of the date-1 announcements made by the agents, i.e., agent k receives c2k mN , where mN = (m1 ; : : : ; mN ) 2 M N . 3 Best Weakly Implementable Allocation An allocation is weakly implementable is if it is an outcome to some equilibrium of the mechanism; it is strongly (or uniquely) implementable if it is an outcome to every equilibrium of the mechanism. Among the set of weakly implementable allocations, the best weakly implementable allocation provides agents with the highest expected 3 1 ck is also a function of ak . The notation in the text anticipates the result that the best implementable allocation is consistent with ak = ;, i.e., the planner does not make an announcement (or does not reveal any information) to agents. As a result, the best implementable allocation is only a function of agents’announcements. 3 utility. To characterize the best weakly implementable allocation, it is without loss of generality to restrict the planner to use a direct revelation mechanism, where agents make truthful announcement, mk = tk 2 M D = f1; 2g. The economy-wide welfare— which is the expected utility of an agent before he learns his type— associated with allocation rule c when agents use strategies mk 2 M D is N X n=0 N X X n U N n tN 2Pn k=1 c1k mk 1 ; mk ; c2k mN 1 ; tk ; (1) where 1 k 1 U c1k mk 1 ; mk ; c2k mN ; mk 1 ; tk = u ck m if tk = 1 and 1 k 1 U c1k mk 1 ; mk ; c2k mN ; mk + c2k mN 1 ; tk = v ck m if tk = 2 The allocation rule c is feasible, i.e., there exists su¢ cient resources to pay for c for all mk 2 M D , k 2 N, if ! N N X X 1 k 1 c2k mN : (2) ck m ; mk R Y k=1 k=1 Allocation rule c must be incentive compatible in the sense that agent k has no reason to announce mk 6= tk . Since impatient agent k only values date-1 consumption, he always announces mk = 1.4 When A = ;,5 patient agent k has no incentive to depart from the strategy mk = 2, assuming that all other agents j announce mj = tj , if N X ^n n=1 N X n=1 ^n X 1 X v c1k tk 1 ; 2 + c2k tk 1 ; 2; tN k+1 n N k2Q t 2Pn n X 1 X v c1k tk 1 ; 1 + c2k tk 1 ; 1; tN k+1 n N k2Q t 2Pn (3) + ; n 4 This anticipates the result that the best weakly implementable allocation provides zero date-1 consumption to patient agents, which implies that the incentive compatibility constraint for impatient agents is always slack. 5 To characterize the best weakly implementable allocation, one wants to choose from the largest possible set of incentive compatible allocations. This occurs when A = ;, i.e., the planner makes no announcements. In particular, when A = ;, there is only one incentive compatibility constraint for all patient agents, (3). When A 6= ;, there will be distinct incentive constraints for agents k who receive information ak from the mechanism. For example, if ak = k, i.e., the planner announced to each agent his place in the queue, then there would be N incentive compatability constraints for patient agents; one for each queue position. Since an appropriately weighted average of these distinct incentive constraints reduces to the single incentive constraint (3), the set of incentive compatible allocations when A 6= ; is a subset of the set of incentive compatible allocations when A = ;. 4 where xji = (xi ; : : : ; xj ), 0 is a parameter, and ^ n = PN N n= n n=1 N n= n is the conditional probability that agent k is in a speci…c queue that has n patient agent. The 1=n terms that appear in (3) re‡ect that a patient agent has a 1=n chance of occupying each of the patient queue positions in Qn . Denote the solution to the problem max (1) subject to (2) and (3); c (4) where mk = tk for all k 2 N in (1) and (2), as c ( ) = (c1 ( ) ; c2 ( )). When agents use truth-telling strategies, c ( ) has the feature that impatient agents consume only at date 1 and patient agents consume only at date 2. The best-weakly implementable allocation is c (0); the allocation rule c (0) corresponds to the analysis contained in PS’s Appendix B. Both PS and Ennis and Keister (2009b) demonstrate, by example, that mechanism M D ; c (0) can have two equilibria: one where agents play truth-telling strategies, mk = tk for all k 2 N, and another where agents play bank run strategies, mk = 1 for all k 2 N.6 The bank run equilibria arise in these examples because the direct revelation mechanism M D ; c (0) is not an optimal mechanism. An optimal mechanism may be a direct mechanism with A 6= ; or an indirect mechanism, (or both). Direct Mechanisms with A 6= ; 4 When c (0) cannot be uniquely implemented by the direct mechanism M D ; c (0) , the optimal mechanism may be a direct mechanism with A 6= ;, i.e., A; M D ; c (0) . Consider …rst the example provided by Ennis and Keister (2009b), where, as in PS, agents do not know their place in the queue. Ennis and Keister (2009b) assume the preference speci…cation of GL, which implies that incentive constraint (3) does not bind for the allocation rule c (0). In addition, we know from GL that when A = N and ak = k, i.e., the planner announces the agent’s position in the queue, none of the N incentive compatibility constraints for patient agents bind for the allocation rule c (0). This means that mechanism A = N; M D ; c (0) can weakly implement the best allocation in c (0). And the main result of GL implies that mechanism A = N; M D ; c (0) can strongly implement c (0). Therefore, A = N; M D ; c (0) is an optimal mechanism; M D ; c (0) admits a bank run equilibrium only because it is a suboptimal mechanism. 6 The Ennis and Keister (2009b) example that I refer to is their bank run example in section 4.2 of their paper, where agents do not know their position in the queue, as in PS, but where the utility functions of patient and impatient agents are the same, as in GL. 5 Consider now the example provided by PS in their Appendix B. Nosal and Wallace (2009) show that the best weakly implementable allocation, c (0), is not weakly implementable if the direct mechanism is characterized by A = N and M = f1; 2g. This implies that the mechanism used by PS, M D ; c (0) , is an optimal direct mechanism. But the optimal mechanism may not be a direct mechanism. 5 Indirect Mechanisms Suppose that mechanism M D ; c (0) weakly, but not strongly, implements the best allocation in c (0), and that mechanism A; M D ; c (0) , where A 6= ;, cannot weakly implement the best allocation in c (0). Since a direct mechanism cannot uniquely implement the best allocation in c (0), I construct an indirect mechanism that can uniquely implement an allocation that is arbitrarily close to c (0), i.e., allocation c ( ), where is arbitrarily close to zero. The indirect mechanism M I ; c has M I 2 f1; 2; gg. One can think of the payo¤ associated with the announcement g as providing the depositor with a (minimum) guaranteed payo¤ in date 2. This is in contrast to the date 2 payo¤ of agent k who announces mk = 2 when allocation rule c (0) is in place; his minimum guaranteed payo¤ is the lowest possible date-2 payo¤ associated with announcing mk = 2. Before I describe the payo¤s associated with announcements, the following notation is needed. The allocation rule for the indirect mechanism is c = (c1 ; c2 ), and the date-s payo¤ to agent j who announces mj is denoted as csj jmj . De…ne Z as the set of queue positions for agents who announce g, i.e., Z = fjjmj = gg and #Z as the number of agents in Z. De…ne m ^ k 1 as the message vector of length k 1 where for each j k 1, m ^ j = 1 if either mj = 1 or mj = g, and m ^ j = 2 if mj = 2. I now specify the allocation rule c for the indirect mechanism M I ; c . The basic construction of c uses c ( ), where > 0 is arbitrarily small. To reduce notational clutter I will suppress the ‘ ’when using allocations in c ( ) to describe c. If agent j announces mj = g, then c1j jg = 0 c2j jg = c1j (5) m ^ j 1 ; 1 (1 + "(#Z)) for all j; " > 0; where 0 < " (1) < " (2) < < " (#Z) < < " (N ), and R > 1 + " (N ). The date-2 payo¤s are feasible since R > 1 + " (N ). Note that the date-2 payo¤ from announcing mj = g is guaranteed to be at least c1j (mj 1 ; 1) (1 + " (1)). I will assume that " (N ) is arbitrarily small. If agent j announces mj = 1, then c1j j1 = c1j (m ^ j 1 ; 1) c1j (m ^ j 1 ; 1) + c2j j1 = 0: 6 if j < N ; if j = N (6) Allocation rule (5) has the feature that if some agents announce mk = g, then the planner accumulates “excess goods” since R > 1 + " (j) for all j 2 N. The total amount of this excess after all agents make their date-1 announcements, denoted as , is X (R 1 " (#Z)) c1 (m ^ z 1 ; 1) z : = R z2Z According to (6), agents who announce mk = 1 and occupy the …rst N 1 positions in the queue receive the consumption payo¤ that they would get under the direct revelation mechanism M D ; c ( ) , assuming that m ^ k 1 is used as the announcement vector. Agent N who announces mN = 1 receives an additional consumption payment of . Finally, if agent j announces mj = 2, then c1j j2 = 0; c2j j2 = (7) m ^N m ^N + c2j c2j if j < N : R if j = N The structure of the payments associated announcing mj = 2 resembles that of announcing mj = 1, except that in the former positive payments are made at date 2 and in the latter at date 1. Note that the allocation rule (5)-(7) has the planner sometimes throwing away goods. This happens when agent N announces mN = g. Proposition 1 The indirect mechanism M I ; c uniquely implements in Nash equilibrium an allocation that is arbitrarily close to the best weakly implementable allocation in c (0). Proof. First, there does not exist an equilibrium where all patient agents j randomize between announcing mj = 1 and mj = 2 or where all patient agents j announce mj = 1 with probability one. Suppose that such an equilibrium exists. Then, suppose that patient agent k defects from proposed play and announces mk = g with probability one. The payo¤ associated with this announcement, given by (5), is c1k mk 1 ; 1 (1 + " (1)), which strictly exceeds the proposed equilibrium payment associated with announcing mk = 1, c1k mk 1 ; 1 ; a contradiction. Second, there does not exist an equilibrium where all patient agents j announce mj = g with probability one. Suppose such an equilibrium exists. Then, the equilibrium expected utility to patient agent k is N X n=1 ^n X 1 X f v c1k n k2Q N t 2Pn n 7 m ^ k 1 ; 1 (1 + " (n)) g: (8) Suppose that patient agent k defects from the proposed equilibrium and announces mk = 1. Using (6), his expected utility is N X ^n n=1 X 1 X f v[c1k n N k2Q t 2Pn m ^ k 1; 1 + X (R 1 " (n 1)) c1j m ^ j 1 ; 1 ]; (9) j2Qn j6=k;N n where = 1 if k = N; : 0 otherwise Note that as " (N ) ! 0, the di¤erence between (9) and (8) is N X n=2 ^n X 1 X f v[c1k n N k2Q t 2Pn n m ^ k 1; 1 + X j2Qn j6=k;N (R 1) c1j m ^ j 1 ; 1 ] v c1k m ^ k 1 ; 1 g > 0: Hence, for any given N , , and R > 1, the mechanism can choose " (N ) > 0 su¢ ciently small so that the value of (9) strictly exceeds that of (8), a contradiction. Third, there does not exist an equilibrium where patient agents j randomize over announcing mj = g and other announcements. Suppose that the proposed equilibrium has patient agents announcing mj = g with probability g , where 0 < g < 1. Since patient agent k randomizes he must be indi¤erent between announcing mk = g and announcing mk = 1 and/or mk = 2, (depending on the speci…cation of the proposed equilibrium). However, given (5), if agent k announces mk = g with probability one, he can increase his expected payo¤, compared to the proposed equilibrium payo¤, since the expected number of agents who announce mj = g increases compared to the proposed equilibrium. Therefore, there cannot be an equilibrium where patient people announce mj = g with probability g , where where 0 < g < 1. Finally, consider an equilibrium where agents of type tj announce mj = tj with probability one. Since > 0 in contract c ( ), incentive constraint (3) implies that all patient agents j strictly prefer to announce mj = 2 to mj = 1. Note that patient agent k strictly prefers to announce mk = g to mk = 1 when all other agents j 2 Nn fkg announce mj = tj . But for any > 0, there exists an " (N ) > 0 su¢ ciently small so that patient agent k strictly prefers announcing mk = 2 to mk = g, (since > 0 implies that agent k strictly prefers announcing mk = 2 to mk = 1). Therefore, for > 0 arbitrarily small, c ( ) c (0), and the unique equilibrium for mechanism I M ; c is characterized by mj = tj for all j 2 N. Agents do not know their positions in the queue for the indirect mechanism M I ; c . Suppose that the economic environment is modi…ed so agents not only learn their type, but they also (somehow) learn their position in the queue. Proposition 1 and the basic proof remains valid for the modi…ed economic environment, where agents know their positions in the queue.7 7 Of course, the allocation rule c ( ) for the modi…ed environment may be di¤erent than the 8 6 Final Comments In a way, the message that underlies this paper is a rather negative one: A well designed deposit contract can prevent bank run equilibria in the classic DiamondDybvig environment. The message is negative because the Diamond-Dybvig model is supposed to be a model of banking instability. Green and Lin (2000, 2003) conjectured that the overlapping generations nature of depositors in the real world and/or moral hazard associated with the people who operate banks may prevent agents from using e¢ cient mechanisms, which has implications for banking instability. These conjectures, unfortunately, do not appear to supported by subsequent research.8 An important assumption in the Diamond-Dybvig environment is that the planner can ex ante commit to implement contract allocations. Relaxing this assumption may result in bank run equilibria; see, for example, Ennis and Keister (2009a). Perhaps assuming that agents cannot not fully commit is a fruitful avenue for future work. References [1] Andolfatto, David, and Nosal, Ed. “Bank Incentives, Contract Design and Bank Runs,”J. Econ.Theory 142 (2008): 28-47. [2] Cavalcanti, Ricardo, de O., and Monteiri, Paulo, K. “Enriching Information to Prevent Bank Runs.”manuscript, February 2011. [3] Diamond, Douglas, and Dybvig, Phillip. “Bank Runs, Deposit Insurance, and Liquidity.”J.P.E. 91 (June 1983): 401-19. [4] Ennis, Huberto, M., and Keister, Todd. “Bank Runs and Institutions: The Perils of Intervention,”A.E.R. 99 (2009a): 1588-1607. [5] Ennis, Huberto, M., and Keister, Todd. “Run Equilibria in the Green-Lin Model of Financial Intermediation.”J. Econ. Theory 144 (2009b): 1996-2020. [6] Green, Edward J., and Lin, Ping. “Diamond and Dybvig’s Classic Theory of Financial Intermediation: What’s Missing?” Fed Res. Bank of Minn. Quarterly Review 24 Winter (2000): 3-13. solution to (4). The best weakly implementable allocation for the new environment is given by maxc (1) subject to (2) and N incentive compatibility constraints, one for each patient agent in position j 2 N in the queue. 8 The Peck and Shell (2003) can be interpreted as a way of modeling overlapping generations in the sense that there is no “last” depositor in an overlapping generations model, and in Peck and Shell (2003) depositors do not know if they are the last depositor. Andolfatto and Nosal (2008) assume that one of the agents operates the bank in a Diamond-Dybvig environment, and …nd that there do no exist bank run equilibria. 9 [7] Green, Edward J., and Lin, Ping. “Implementing Allocations in a Model of Financial Intermediation,”J. Econ. Theory 109 (2003): 1-23. [8] Nosal, Ed, and Wallace, Neil. “Information Revelation in the Diamond-Dybvig Banking Model.” Policy Discussion Paper no. 7, Chicago, IL: Fed. Res. Bank of Chicago, December 2009. 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Ron Alquist and Benjamin Chabot WP-10-13 Introduction to the Macroeconomic Dynamics: Special issues on money, credit, and liquidity Ed Nosal, Christopher Waller, and Randall Wright WP-10-14 Summer Workshop on Money, Banking, Payments and Finance: An Overview Ed Nosal and Randall Wright WP-10-15 Cognitive Abilities and Household Financial Decision Making Sumit Agarwal and Bhashkar Mazumder WP-10-16 Complex Mortgages Gene Amromin, Jennifer Huang, Clemens Sialm, and Edward Zhong WP-10-17 4 Working Paper Series (continued) The Role of Housing in Labor Reallocation Morris Davis, Jonas Fisher, and Marcelo Veracierto WP-10-18 Why Do Banks Reward their Customers to Use their Credit Cards? Sumit Agarwal, Sujit Chakravorti, and Anna Lunn WP-10-19 The impact of the originate-to-distribute model on banks before and during the financial crisis Richard J. Rosen WP-10-20 Simple Markov-Perfect Industry Dynamics Jaap H. Abbring, Jeffrey R. 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Campbell and Zvi Hercowitz WP-11-05 Survival and long-run dynamics with heterogeneous beliefs under recursive preferences Jaroslav Borovička WP-11-06 A Leverage-based Model of Speculative Bubbles (Revised) Gadi Barlevy WP-11-07 Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen WP-11-08 Fertility Transitions Along the Extensive and Intensive Margins Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder WP-11-09 Black-White Differences in Intergenerational Economic Mobility in the US Bhashkar Mazumder WP-11-10 Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options? Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein WP-11-11 5 Working Paper Series (continued) Business Networks, Production Chains, and Productivity: A Theory of Input-Output Architecture Ezra Oberfield Equilibrium Bank Runs Revisited Ed Nosal WP-11-12 WP-11-13 6
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