Asymmetric Information and Bank Runs

Transcription

1 Asymmetric Information and Bank uns Chao Gu Cornell University Draft, March, 2006 Abstract This paper extends Peck and Shell s (2003) bank run model to the environment in which the sunspot coordination device is imperfect. Consumers observe correlated but di erent sunspot information and make withdrawal decision simultaneously. We start with the assumption that consumers are endowed with the asymmetric information structure and show that if signals are closely correlated, there are banking contracts consistent with the coordination device. Due to the imperfectly correlated coordination device, both full bank run and partial bank run are the equilibrium phenomena if the probabilities of full run and partial run are small. The imperfect sunspot information reduces probabilities of full bank run and no run, and brings in partial runs as well. If the welfare gain from the reduced probability of full bank run is huge, an imperfect coordination device improves ex ante welfare. Thus employing an imperfect coordinating device can be socially optimal. We then relax the assumption to allow consumers to choose the information structure. The asymmetric information structure can be an equilibrium choice of a competitive economy. JEL Classi cation Numbers: D82, G2, P Keywords: Bank runs, coordination device, sunspot equilibrium, correlated equilibrium, imperfect information. I would like to thank David Easley, Todd Keister, Tapan Mitra and seminar participants at the Cornell/Pennsylvannia State University Macro Workshop for insightful comments. I am especially grateful to Karl Shell for numerous discussions and helpful guidance. All remaining errors are my own. Correspondence: Department of Economics, Cornell University, Ithaca, NY 4853, USA. cg223@cornell.edu.

2 Introduction In the classic Diamond and Dybvig (983) bank run model, bank runs occur due to panics. There are several ingredients in the model which account for the emergence of the bank as well as bring up the panic runs. First, consumers face preference shocks. However, the rigidity in the production technology does not allow people to diversify the investment in order to smooth the consumption ex ante. Banks arise as they can aggregate the preference shocks and provide a so-called "optimal" contract which admits the rst-best allocation. However, such a plan is subject to bank runs. Some people withdraw money from the bank not because they need to consume soon, but they are afraid if too many people withdraw money, the bank will be out of liquidity and has to shut down. Since the bank can not distinguish people who indeed need to consume immediately from those who do not, it pays to the consumers until it is out of resource. If panic run occurs, consumers welfare will be severely hurt as some consumers are left unpaid. Having that in mind, why do consumers put money in the bank in the rst place? This question requires us to model () the equilibrium selection ex post, and (2) the pre-deposit decision given the possibility that bank run might happen. The selection of equilibrium is traditionally based on some coordination device. In an economy, many types of information can serve as the coordination device. For example, a report on the performance of a bank in the news, the macro policy by the government, or the latest in ation rate (Boyd, et al, 200). The often used coordination device in the literature is a purely exogenous variable, that is, a sunspot variable. Peck and Shell (2003) are the rst who model bank runs formally as a sunspot phenomenon. This sunspot variable does not a ect any of the fundamentals such as endowment, preference, and technology. Consumers perfectly observe the realization of the sunspot, and make the withdrawal decision according to it given that everyone else makes the same decision. An individual consumer faces no uncertainty when he makes the withdrawal decision, as he knows all other consumers will behave in the same way since they observe the same thing. Thus, whether bank runs take place relies on the commonly acknowledged coordination device, and the probability of bank runs depends solely on the speci cation of the sunspot variable. 2

3 Knowing the probability of bank run, the bank o ers an optimal contract which takes the probability of run into account. Consumers then make the decision whether to deposit at the bank or to stay in autarky. This completes the pre-deposit decision. In this paper, we consider a more general sunspot coordination device in the sense that consumers receive imperfectly correlated sunspot signals. Consumers are grouped into networks according to their observation of the signals. People in the same network share the sunspot information but there is no communication between the networks. eceiving their own signals of the sunspot, consumers try to infer the signals others observe and the strategies they adopt. In this situation, a consumer faces uncertainty when he makes the withdrawal decision as other people might observe very di erent thing and make di erent decision. Soloman (2003, 2004) considers an imperfectly correlated sunspot coordination device in a twin crises model. Domestic and foreign investors observe correlated but not necessarily the same sunspot signals. Foreign investors are risk neutral and only paid in domestic currency (nominal asset) while domestic consumers are risk averse and paid in foreign currency (real asset). By these assumptions, foreign investors withdrawal decision actually does not a ect domestic consumers payo, but domestic consumers decision determines foreign investors returns. Thus, bank run occurs only when domestic consumers get a bad signal and can lead to currency crisis if reserves are depleted to serve domestic consumers. But foreign investors run on the bank does not cause the bank to collapse since no real asset is paid out during the bank run. If we focus on the banking aspect of the twin crises, the model is actually reduced to a traditional sunspot equilibrium model. Our model assumes ex ante identical agents and one asset. Consumers decisions always a ect others payo and in turn are a ected by others decisions. The special features of an imperfect coordination device and its welfare implications are addressed more seriously in this paper. Extrinsic uncertainty can be viewed as intrinsic uncertainty taken to the limit. In this regard, our model bears similarities to Goldstein and Pauzner s (2005). Goldstein and Pauzner construct a model in which fundamentals determine which equilibrium to occur. The fundamentals serve as a coordination device, and even with noisy signals of the fundamentals the model has a unique equilibrium. Consumers run if they observe a signal below the threshold, and do not run above. When the fundamentals are in the middle range, a proportion of con- 3

4 sumers run since the noisy signals they get fall below the threshold. In our model, imperfect information on the sunspot results in the imperfect coordination in the sense that people act di erently when they receive opposite signals, and as a result, partial run occurs. However, when the imperfectness in signal disappears, people are perfectly coordinated by the sunspot, and the result will be the same as Peck and Shell s. Our model thus can be treated as a di erent approach from the global games to explain the coordination with imperfect coordination device. However, our model emphasizes more on the coordination device itself. By taking the extrinsic uncertainty seriously, we are able to explain the panic runs in the absence of fundamental weakness. In Goldstein and Pauzner s model, if the noise in the signals disappears, the fundamentals are fully revealed to the consumers. All patient consumers stay until the last period if the fundamentals are strong enough. The empirical studies show that most of the banking crises are extrinsic-driven panic runs (Boyd, et al, 200). That is, before most banking crises happen, there is no indicator of the economy which serves as a good predictor. In this regard, our model employs sunspots as the coordination device. Nevertheless, the coordination device in our paper can be understood in either way. It can be viewed as the intrinsic variable taken to the limit, or as the pure sunspots. For simplicity, there are only two networks in the model, and each network observes a sunspot signal which takes values of 0 and. Signals are correlated. This is the minimum structure which is required for the analysis of the imperfect coordination. Consumers are rst assumed to be "dropped" into one of the networks. We show that given the asymmetric sunspot information, a banking contract may or may not be consistent with the coordination device. The more correlated the information, the more likely that the coordination device will be consistent with a feasible contract, and will be employed in the equilibrium. We then calculate the optimal contract given the probabilities of run and partial runs determined by the information structure. Similar to Peck and Shell s work, if the probability of (partial) run is small enough, a run proof contract is not optimal. If an imperfectly correlated coordination device always performs worse than a perfectly correlated one, it can be conjectured that if there is a social planner, he would eliminate the asymmetry in the information structure and the problem we discuss here would not be 4

5 a problem at all. The second part of the paper shows that it is possible that an imperfectly correlated coordination device improves welfare. The reason is that since people receive di erent signals, probabilities of full bank runs and no bank runs are reduced. Instead, there occur partial runs. If the welfare gain from the reduced probability of full run is not completely o set by the welfare loss from the reduced probability of no run and the increased probabilities of partial runs, the ex ante welfare is improved. In the last section, the assumption is relaxed to allow people to choose the network. We answer the question that whether an imperfect coordination device can be an equilibrium choice of a competitive economy. In this case, there are two types of equilibria given a feasible demand deposit contract in the post deposit game. One is that all people choose to be in one network. In this equilibrium, consumers are perfectly coordinated, and the result will be the same as Peck and Shell s. In the other equilibrium, people choose to be in di erent networks. It occurs when consumers are indi erent between the expected utilities brought by the two networks. This equilibrium, ignored by most of the literature following Diamond and Dybvig (983), is picked up here by allowing for an asymmetric information structure. A simple demand deposit contract will be discussed. The economy has no aggregate uncertainty. Since our goal is not to eliminate bank runs, we assume the bank does not suspend convertibility even if it knows a bank run is taking place. The rest of the paper is organized as follows. Section 2 introduces the model set-up, discusses the equilibrium concept, and calculates the optimal contract. Section 3 shows how an imperfectly correlated coordination device can improve welfare. The model will be extended to allow for ex ante choice of networks in section 4. The last part gives the conclusions. 2 The Model 2. Set Up There are three periods (t = 0; ; 2: period 0, and 2 respectively) and a mass of consumers in the economy. Each consumer is endowed with unit of consumption good in period 0. 5

6 There are a measure of ( < ) impatient consumers, the rest are patient. Impatient consumers derive utility only from consumption in period. Their utility is described by u(c ); where c is the consumption received at t =. Patient consumers consume in the last period. If a patient consumer receives consumption at t = ; he can costlessly store it and consume it at t = 2: Thus, a patient consumer s utility is described by u(c + c 2 ); where c 2 is the consumption received at t = 2: The coe cient of relative risk aversion of the utility function, xu 00 (x)=u 0 (x); is greater than for x : The utility function is normalized to 0 at x = 0, i.e., u(0) = 0: Whether a consumer is patient or impatient is private information, and is revealed to the individual consumer at t =. The investment technology is as follows. One unit of endowment invested in period 0 yields unit of consumption good in period, and ( > ) units in period 2. The banking market is competitive. The representative bank o ers a demand deposit contract which describes the amount of consumption good paid to the consumers who withdraw money in period (c ) and 2 (c 2 ) respectively. The bank pays c to the consumers until it runs out of resource, and distributes the remaining resource plus the interest equally n o among the consumers who wait until the last period, therefore c 2 = max 0; nc ; where n n (0 < n < ) denotes the measure of consumers who withdraw the deposits in period. Consumers are isolated from each other. They do not know how many have already withdrawn the deposits before they come to the bank. There is a sunspot variable! which takes value of 0 and, that is = f0; g. Sunspots do not a ect production and preference.! is imperfectly observed by the consumers. Consumers are divided into two networks according to their ability to observe the sunspot. Each network gets a signal i, i = ; 2; which imperfectly re ects the true value of!. The set of values that i takes is denoted by i = f0; g. The joint distribution of and 2 is as the following. prob( = 0; 2 = 0) = p 00 ; prob( = ; 2 = 0) = p 0 ; prob( = 0; 2 = ) = p 0 ; prob( = ; 2 = ) = p : Networks do not communicate with each other. Consumers in the same network share the information of the sunspot they obtain, and conjecture the sunspot signal the other network 6

7 observes. The conditional distribution, derived from the joint distribution, is prob( 2 = 0 j = 0) = p 00 =(p 00 + p 0 ); prob( 2 = j = 0) = p 0 =(p 00 + p 0 ); prob( 2 = 0 j = ) = p 0 =(p + p 0 ); prob( 2 = j = ) = p =(p + p 0 ); prob( = 0 j 2 = 0) = p 00 = (p 00 + p 0 ) ; prob( = j 2 = 0) = p 0 =(p 00 + p 0 ); prob( = 0 j 2 = ) = p 0 =(p + p 0 ); prob( = j 2 = ) = p = (p + p 0 ) : Network i has a measure of n i consumers. By law of large number, the measure of impatient consumers in network i is n i ; i = ; 2. Impatient consumers withdraw deposits at t = regardless of what other people do as they have to consumer in period. Patient consumers make withdrawal decisions given all available information. Consumers know ex ante that there are two networks available, but they do not know which network they will be in until t =. The sequence of timing is as follows: 9 Bank announces the contract; = Consumers make investment decision. ; t = Consumption types are revealed; Information shock is realized; >= t = Sunspot variable is realized; >= Consumers make withdrawal decision. >; 9 Bank allocates the remaining resource = to the rest of the consumers. ; t = 2 >; post deposit game 9 >= >; predeposit game The post deposit game starts from t =, ends at t = 2. In the post deposit game, consumers determine whether to withdraw deposits or not given the banking contract and assuming they have already deposited at the bank. Knowing how consumers behave in the post deposit game, at t = 0, the bank decides which contract to o er and consumers decide whether or not to accept the o er. The whole game, played from t = 0 to t = 2, is de ned as the predeposit game. 7

8 2.2 Post Deposit Game A demand deposit banking contract in the post deposit game m = (c ; c 2 ) satis es c 2 = max 0; nc n ; where n ; c 0: () where n is the proportion of consumers who withdraw early. The participation incentive compatibility constraint is de ned as c u u(c ); (2) which means that the patient consumers should get at least as much as the impatient consumers get in period if all patient consumers wait. This is the minimum requirement for a banking contract such that the patient consumers are willing to wait. Given a demand deposit contract satisfying (2), if all other patient consumers are honest about their consumption type, an individual consumer should nd stay until t = 2 a better choice than withdraw immediately at t =. Let M denote the set which includes all banking contracts satisfying ()-(2). This is the set which includes all feasible banking contracts in the traditional bank run literature. A banking contract m = (c ; c 2 ) allows for a run equilibrium if Eu(Not unjall patient consumers run) < Eu(unjAll patient consumers run): (3) That is, given everyone else runs on the bank, the expected utility of a patient consumer if he chooses to stay is strictly lower than that if he also runs on the bank. A contract which allows for a run equilibrium is called a run admitting contract. In Diamond-Dybvig model, the rst-best allocation achieves the highest ex ante welfare assuming the consumption types are observable. c max u(c ) + ( )u c s:t:() (2): As the coe cient of relative risk aversion of the utility function s is greater than, the rst-best allocation (c ; c 2 ) satis es < c < c 2 <. The rst-best allocation is Nash 8

9 implementable even the types are not observable as there is a Nash equilibrium in the post deposit game in which all patient consumers choose to wait until the last period. However, since the bank can not observe consumers consumption type, if m = (c ; c 2 ) is o ered to the consumers, there is a run equilibrium in the post deposit game as c >. u(0) < c u(c ): In the event that all people run on the bank, the bank will be out of resource. If a patient consumer stays, he will be left unpaid for sure; if he also runs, his chance of getting paid of c is =c. unning on the bank is the dominant strategy for an individual consumer if everyone else is doing so. Hence, the so-called optimal contract m = (c ; c 2 ) is not necessarily optimal, and the discussion of the equilibrium selection in the post deposit game is not trivial. Also note that for any feasible contract satisfying c > ; the post deposit game has a run as well as a no run equilibrium. A banking mechanism m 2 M is run proof if it satis es (4). Eu(Not unjall patient consumers run) Eu(unjAll patient consumers run): (4) Let M P denote the set which contains all run proof contracts. It is a subset of M. The complement to M P ; MnM P, is the set of run admitting contracts. Given a run admitting contract, let consumers be coordinated by the sunspot variables if possible. De ne a function i :! i ; that is, when the value of the sunspot variable is realized, consumer j in network i; i 2 ; 2 only knows that a certain set in i has occurred. Given a banking mechanism, a correlated equilibrium (Aumann 987) is de ned as X f!j i (!)= i g (!j i )u j (s j (!); s j (!)) X f!j i (!)= i g (!j i )u j (s 0 j(!); s j (!)) 8s 0 j(!) 2 S j ; 8j in group i; i = ; 2: where (!j i ) denotes the probability of the realization of! conditional on observing the value of i ; and s j (!) denotes the strategy of individual j when! is realized. Observing his own realization of the sunspot signal, the consumer maximizes his expected payo given others strategies. Given m which admits a run equilibrium, from game theory s point of view, the game has equilibria in which consumers in the same group adopt di erent strategies 9

10 or consumers adopt strictly mixed strategies. Here we focus on the case in which consumers in the same network adopt the same pure strategy as we want to show how the sunspot signals coordinate people s behavior. Consumer j s strategy set S j thus is described by S j ={run, not run}. Consumers adopt same strategy for the same value of i. De nition A run admitting banking contract m 2 MnM P has a coordinating equilibrium in the post deposit game if the following pure strategies construct a correlated equilibrium: (i) patient consumers in network run on the bank when observing =, and do not run when observing = 0; (ii) patient consumers in network 2 run on the bank when observing 2 =, and do not run when observing 2 = 0. This de nition is equivalent to the following four conditions. For a patient consumer in network : p 0 Eu(j, N) + p Eu(j, ) p 0 Eu(Nj, N) + p Eu(Nj, ) (5) p 0 Eu(NjN, ) + p 00 Eu(NjN, N) p 0 Eu(jN, ) + p 00 Eu(jN, N) (6) For a patient consumer in network 2: p Eu(j, ) + p 0 Eu(jN, ) p Eu(Nj, ) + p 0 Eu(NjN, ) (7) p 0 Eu(Nj, N) + p 00 Eu(NjN, N) p 0 Eu(j, N) + p 00 Eu(jN, N) (8) Eu(:) is the expected utility of an individual patient consumer. The rst argument in Eu(:) is the strategy of an individual consumer. stands for run, and N for not run. The second argument denotes the strategy of the consumers in network, and the third argument is the strategy of the consumers in network 2. The expected utility depends on individual consumer s own strategy, his network members strategies, and strategies of consumers in the other network. Other parameters such as c,, n, and n 2 are suppressed here. The expected utilities of a patient consumer given others strategies are speci ed in appendix. (5) (8) are also rewritten explicitly with the expected utilities in appendix. If a patient consumer in network observes =, by conditional probabilities, he knows that with probability p p +p 0 network 2 observes 2 =, and with probability p 0 p +p 0 network 2 gets 2 = 0. For a coordinating equilibrium, a patient consumer in network should 0

11 nd "run" the best response given the other network s and his network members strategies. Therefore, (5) must hold. If = 0, a patient consumer in network knows that network 2 runs on the bank with probability p 0 p 0 +p 00, and waits with probability p 00 p 0 +p 00. His group members will not run and he should nd "not run" the best strategy which maximizes his expected utility according to the de nition of the coordinating equilibrium. Thus, equation (6) should hold. Similarly, we have equation (7) and (8) for a patient consumer in network 2. Let M CE denote the set of run admitting banking contracts which satis es (5)-(8). (5) and (7) can be interpreted as the incentive compatibility constraints for run given the probability that some, but not all, patient consumers wait until the last period. (6) and (8) are the incentive compatibility constraints for wait given the probability that some, but not all, patient consumers run on the bank. Since a m in M CE has to satisfy four additional constraints besides that it is feasible and run admitting, M CE is a subset of MnM P. There are two noises in the coordination device, p 0 and p 0. If both of them are zero, the coordination device is perfect, and we are back to Peck-Shell sunspot approach. And for a perfect coordination device, M CE = MnM P. The following proposition shows that for any feasible run admitting contract, there are upper bounds of the noises, below which the contract satis es the de nition of a coordinating equilibrium. Proposition For any feasible demand deposit contract m = (c ; c 2 ) and p ; there exists " (p ; c ) 0 such that if p 0 " (p ; c ), there exists (p ; p 0 ; c ) 0: For p 0 " (p ; c ) and p 0 (p ; p 0 ; c ) the contract allows for a coordinating equilibrium in the post deposit game. " (p ; c ) = 0 and/ or (p ; p 0 ; c ) = 0 if and only if c = Proof. See the appendix. : + With the noises in the sunspot information, consumers face uncertainty when they make the withdrawal decisions. (5) (8) are the conditions for individual consumers to follow their signals given the probability that the other group run on the bank. When the participation incentive compatibility constraint is binding, only the minimum requirement for a patient consumer to stay is satis ed. Any increase in the measure of consumers running on the bank or any increase in the probability of more than measure of consumers running on the bank

12 breaks down the participation incentive compatibility constraint. Therefore, if c =, + unless p 0 = p 0 = 0, the contract does not allow for a coordinating equilibrium. On the other hand, if the participation incentive compatibility constraint is unbinding, there is room for the possible increase in the measure of consumers running on the bank, and to an individual consumers own interest, he still prefers to wait. Hence, all other feasible contracts bear a coordinating equilibrium if noises are small enough. Given p and p 0, denote the set of contracts which satis es (5) (8) by M CE (p ; p 0 ; p 0 ). The following proposition proves that the set M CE (p ; p 0 ; p 0 ) shrinks in p 0 and p 0. Proposition 2 If ( p 0 ; p 0 0; p 0 0) ( p ; p 0 ; p 0 ) ; (p 0 0; p 0 0) (p 0 ; p 0 ) and M CE (p 0 ; p 0 0; p 0 0) 6=?; then M CE (p ; p 0 ; p 0 ) M CE (p 0 ; p 0 0; p 0 0). Proof. See the appendix. The strategies in the post deposit game is complementary. If more people run on the bank or the probability of run is increased, an individual consumer s incentive to stay drops as the expected payo in the last period is lowered. Similarly, if more people stay in the system or the probability of no run is increased, a patient consumer s expected payo at t = 2 are raised, and he is more willing to stay. With a perfect coordination device, that is, p 0 = 0 and p 0 = 0, an individual consumer knows for sure that all others adopt the same strategies as he does. With the decrease in p and the increase in p 0 and p 0, the conditional probabilities prob( i = 0 j j = ) and prob( i = j j = 0) are increased. eceiving a signal, an individual consumer knows that the other group is more likely to use a di erent strategy. Given the contract, due to the strategic complementarity, it is better for an individual consumer to switch to the other group s strategy rather than follow his own signal as p 0 and p 0 are above the thresholds. Hence, there are fewer contracts consistent with the de nition of the coordinating equilibrium as p 0 and p 0 increase. In an extreme case, for example, there is no contract allowing for a coordinating equilibrium if p 0 = p and p is small enough. When a banking contract m falls in the subset of Mn(M CE [M P ), it neither allows for a coordinating equilibrium nor is run proof. In this situation, we move back to the original Diamond-Dybvig world in which a contract has a run as well as a no Soloman (2003, 2004) does not have this problem because of the assumptions that foreigners are paid in nominal asset and they are risk neutural. 2

13 run equilibrium. If such a contract is o ered, according to the traditional Diamond-Dybvig interpretation, at t = 0, consumers would either accept it believing no run equilibrium will take place or do not accept it believing the run equilibrium will occur. The following example is provided to show the partitions of feasible banking contracts. A Numerical Example: (c + b) b Let u(c) =, =, b = 0:5. = :5, = 0:4, n = n 2 = 0:5, p = p 0 = p 0 = 0:00. In this example, M, M P, M CE and Mn(M CE [ M P ) are as follows (summarized by c ): Banking Mechanism c M [0; :2500] M P [0; ] M CE (; :2495] Mn(M CE [ M P ) (:2495; :2500] In a coordinating equilibrium, there are three possible outcomes. If = 2 =, all consumers withdraw deposits. If = 0 and 2 =, or = and 2 = 0, only a fraction of patient consumers run on the bank. No patient consumer runs on the bank when = 2 = 0. We de ne full bank run, partial bank run and no bank run as follows. De nition 2 (Full Bank un) In the post deposit game, if all consumers withdraw deposits, full bank run occurs. De nition 3 (Partial Bank un) In the post deposit game, if some, not all patient consumers withdraw deposits, partial bank run occurs. De nition 4 (No Bank un) In the post deposit game, if all patient consumers do not withdraw deposits in period, there is no bank run in this post deposit game. By de nition, consumers in both networks interpret signal of value as the sign to run, and 0 the sign to stay in a coordinating equilibrium. Generally speaking, since the sunspots do not a ect the fundamentals, people can interpret the signals in the way they like. For example, one network views as the signal to run, and the other treats 0 as the signal to 3

14 run. So an imperfect information structure can allow for more than one type of coordinating equilibrium. This paper does not study how people select the coordinating equilibrium, so the initial interpretation are exogenously given. Instead of varying the interpretations, we can change the joint probability distribution of and 2 and achieve the same results. Given the imperfect coordination device, not every run admitting contract allows for a coordinating equilibrium. Before we start the welfare analysis, let us clarify the strategies of an individual consumer in the post-deposit game. Start with a banking contract m 2 M;. If m 2 M CE, that is, m allows for a coordinating equilibrium, patient consumers are coordinated by the sunspots. Patient consumers withdraw the deposits when i =, i = ; 2; is observed, and wait otherwise. 2. If m 2 M P, patient consumers do not run regardless of the realization of the sunspot variable. 3. If m 2 Mn(M CE [ M P ), the contract neither allows for a coordinating equilibrium nor is run proof. The coordination device fails. People either accept the o er ex ante and do not run ex post or reject the o er ex ante as they anticipate the run equilibrium always occurs. In the last two scenarios, sunspots do not matter in the post-deposit game, as the consumers strategies are independent of the realization of the sunspot signals. 2.3 Predeposit Game Knowing consumers strategies in the post deposit game, the bank chooses an optimal contract to o er at t = 0. As a result of competitive market, the bank o ers a contract which maximizes consumers expected utility. If the contract yields expected utility level higher than that under autarky, consumers will accept it so the post deposit game will be played. Otherwise, consumers prefer to stay in autarky. We use the same notation to denote the banking contracts in the predeposit game. The bank can choose from three types of contracts, corresponding to the partitions of M. 4

15 A run proof contract is a contract such that a patient consumer weakly prefers to wait even if everyone else withdraws the deposits. In the demand deposit contract framework, a banking contract is run proof if and only if c. It is equivalent to the autarky when c =. Since the coe cient of relative risk aversion is greater than, the banking contract which consumers are willing to accept ex ante should satisfy c. Thus the only ex ante acceptable run proof contract requires c =, which results in the same allocation as under autarky. We impose an assumption that if a contract yields expected payo equal to that under autarky, people still deposit in the bank. By this assumption, the bank can at least o er the run proof contract to the consumers. The highest level of expected utility under run proof contract is c ^W (m) = u(c ) + ( )u = u() + ( )u() (9) Let W P denote the value of (9). De nition 5 Given a feasible contract m 2 M, the predeposit game has a coordinating equilibrium if there is a subgame perfect Nash-Aumann equilibrium in which (i) consumers are willing to deposit, and (ii) the post deposit game has a coordinating equilibrium. We use partial run and partial run 2 to distinguish the partial runs conducted by di erent networks. If the contract admits a coordinating equilibrium, the probabilities of full bank run, partial bank runs ( and 2) and no run are determined by the information structure. They are p ; p 0 ; p 0 and p 00 respectively. We assume the social welfare is the aggregated utilities of individual consumers and all consumers are equally weighted. In the following context, welfare and ex ante expected utility are used interchangeably. When no run occurs, the welfare, denoted by W no run (m), is c W no run (m) = u(c ) + ( )u : When partial run occurs, the welfare, denoted by W p run (m), is 8 >< u(c ); if (n W p run c (m) = + n 2 )c > ; >: (n + n 2 )u(c (n + n 2 )c ) + ( )n 2 u ; otherwise. ( )n 2 5

16 Note that W p run (m); i = ; 2; is continuous in c. If partial run 2 occurs, the welfare, denoted by W p run 2 (m), is 8 >< u(c ); if (n W p run 2 c (m) = + n 2 )c > ; >: (n + n 2 )u(c (n + n 2 )c ) + ( )n u ; otherwise. ( )n Note that W p run 2 (m); i = ; 2; is continuous in c. When full run occurs, the welfare, denoted by W run (m), is W run (m) = c u(c ): Given the probabilities, the best contract which allows for a correlated equilibrium the bank o ers to the consumers is ^W (m) = max p W run (m) + p 0 W p run (m) + p 0 W p run 2 (m) + p 00 W no run (m) (PCE) c s:t: m 2 M CE : Lemma Given p, the value function of problem PCE is strictly decreasing in p 0 and p 0 for M CE (p ; p 0 ; p 0 ) 6=?: Proof. Let (p 0 0; p 0 0) (p 00 0; p 00 0) and (p 0 0; p 0 0) 6= (p 00 0; p 00 0). By proposition 2, M CE (p ; p 0 0; p 0 0) M CE (p ; p 00 0; p 00 0). Let m 0 = (c 0 ; c 20 ) solve PCE with p 0 = p 0 0 and p 0 = p 0 0. Denote the value function by ^W (m 0 ; p ; p 0 0; p 0 0). m 0 2 M CE (p ; p 0 0; p 0 0) and also in M CE (p ; p 00 0; p 00 0). Plug m 0 into PCE with p 0 = p 00 0 and p 0 = p Easy to see W (m 0 ; p ; p 00 0; p 00 0) > ^W (m 0 ; p ; p 0 0; p 0 0). The value of PCE with p 0 = p 00 0 and p 0 = p 00 0, denoted by ^W (m 00 ; p ; p 00 0; p 00 0) is at least as high as W (m 0 ; p ; p 00 0; p 00 0). Therefore, ^W (m 00 ; p ; p 00 0; p 00 0) > ^W (m 0 ; p ; p 0 0; p 0 0) for (p 0 0; p 0 0) (p 00 0; p 00 0) and (p 0 0; p 0 0) 6= (p 00 0; p 00 0). Proposition 3 There exists p > 0 such that for p p there is p 0 (p ) 0, p 0 (p ) = 0. For p p and p 0 p 0 (p ), there is p 0 (p ; p 0 ) 0, p 0 (p ; p 0 ) = 0. For p p ; p 0 p 0 (p ) and p 0 p 0 (p ; p 0 ), there is at least one feasible demand deposit contract m = (c ; c 2 ) allowing for a coordinating equilibrium and strictly better than the run proof contract. 6

17 Proof. The optimal contract allowing for a coordinating equilibrium solves problem PCE. Given p, if p 0 = p 0 = 0; the problem is the same as the traditional symmetric sunspot equilibrium problem. The conditions for a coordinating equilibrium are always satis ed. By Cooper and oss (998), there is a unique cuto level of p, above which a run proof contract is better, and below which the optimal contract is run admitting. Denote this cuto level by p. Given p and p 0 = 0, the value function of PCE is strictly decreasing in p 0 by lemma. If p = p, only p 0 = 0 can make an run admitting contract as good as a run proof contract. Hence, p 0 (p ) = p. If p < p and p 0 = 0, the value of PCE is strictly higher than that of a run proof contract if p 0 = 0. So p 0 can be increased a little bit and the value of PCE is still higher than that of a run proof contract. Note that the set of M CE is diminishing in p 0. Let p? 0 (p ) denote the cuto of value of p 0 below which the set of M CE is not empty, and above which it is empty. If the value of PCE at p? 0 (p ) is lower than W P, by the monotonicity of the value function, we can nd a cuto p 0 depending on p below which the contract allowing for coordinating equilibrium is better than the run proof contract and above which run proof contract is better. Denote such p 0 by p V 0 (p ). Let p 0 (p ) = min p? 0 (p ) ; p V 0 (p ). It is the cuto value of p 0 below which the contract allowing for a coordinating equilibrium is better than the run proof contract, above which the run proof contract is better or there is no contract which allows for a coordinating equilibrium. The value function of PCE is not necessarily continuous as the choice set can be nonconvex. We need to prove that p 0 (p ) is not equal to 0 for p < p. Let c denote the solution to problem PCE with p 0 = 0 and p < p. By Ennis and Keister (2004), c can not be. By proposition, given p +, c is a feasible contract which allows for a coordinating equilibrium for p 0 " (p ; c ) ; where " (p ; c ) > 0. The welfare at c on p 0 " (p ; c ) is continuous in p 0. So p 0 can be increased at least to minfp p ; " (p ; c )g and c still be better than the run proof contract. Thus when p < p, the cuto level of p 0 > 0. Let p p and p 0 p 0 (p ), we can repeat the same process to prove there exists p 0 (p ; p 0 ) 0: Therefore, if the probabilities of partial runs and full run are small, the 7

18 contract which allows for a coordinating equilibrium is better than the run proof contract. Since the best run proof contract is equivalent to autarky, we have the following corollary. Corollary For p p ; p 0 p 0 (p ) and p 0 p 0 (p ; p 0 ), there exists at least one feasible banking contract such that the predeposit game has a coordinating equilibrium. If m 2 Mn(M CE [ M P ), the best outcome one can hope for is that all consumers anticipate the no run equilibrium in the post deposit game, so consumers deposit at the bank ex ante and that they do not run at t =. The welfare of the best outcome is given by c ^W (m) = max u(c ) + ( )u c (PDD) s:t: m 2 Mn(M CE [ M P ): A su cient condition for allowing for a coordinating equilibrium is that the welfare of the optimal m 2 M CE is higher than that of the optimal run proof contract and that of the best outcome of the best m in Mn(M CE [ M P ). Similarly, a run proof contract will be the best if the welfare is higher than that of the optimal m in M CE and than the best outcome of the optimal contract in Mn(M CE [ M P ). The objective function of PDD is not a ected by p, p 0 or p 0, but the choice set increases in p 0 and p 0 given p. So the value function of PDD is increasing in p 0 and p 0. However, since the choice set is not necessarily convex, the value function can jump at p 0 = 0 and/or p 0 = 0. Thus, it is not necessarily true that when p, p 0 and p 0 are small, an optimal contract always allows for a coordinating equilibrium. However, we have some examples which show that in some economies, the optimal contract allows for a coordinating equilibrium, and in other economies, the run proof contract is optimal. Proposition 4 In some economies, the optimal demand deposit banking contract allows for a coordinating equilibrium. Proof. Prove by example. All parameters are the same as in the previous example. The welfare of the optimal contracts in M CE ; M P and the best outcome in Mn(M CE [ M P ) 8

19 are as follows. m in c ^W (m) M P :4333 M CE :0707 :434 Mn(M CE [ M P )! :2495! :4286 The optimal contract in this example is c = :0707; which yields a welfare level of :434. It is better than the best run proof contract (and autarky). If m in Mn(M CE [M P ), in the best situation, that is, consumers accept the contract and do not run ex post, the highest welfare level is :4286, which is still lower than that under M CE and also lower than that under autarky. Thus consumers will not accept the contract in the rst place. Hence, the optimal m is in M CE in this example. Corollary 2 In some economies, the optimal demand deposit banking contract is run proof. Proof. Prove by example. We still use the example in the previous section, but vary p, p 0 and p 0. Let p = p 0 = p 0 = 0:005. The welfare of the optimal contracts in M CE ; M P and the best outcome in Mn(M CE [ M P ) are as follows. m in c ^W (m) M P :4333 M CE :060 :4330 Mn(M CE [ M P )! :2476! :4287 The run proof contract is the best one. The best outcome in Mn(M CE [M P ) can achieve higher welfare than the best run proof contract and the best contract which allows for a coordinating equilibrium. Let us continue the example, but let p = 0:, p 0 = 0:2 and p 0 = 0:4. The welfare is as follows. m in c ^W (m) M P :4333 M CE!! :3933 Mn(M CE [ M P )! :407! :4335 9

20 In this economy, it is hard to tell which contract is optimal. A run proof contract is better than a contract which allows for a coordinating equilibrium. But if this coordination device is not used, we might achieve higher welfare level. In appendix 3, we plot the welfare levels achieved by run proof contracts and contracts allowing for a coordinating equilibrium for any combinations of p and p 0 while holding p 0 = 0. To make the results more distinguishable, is exaggerated to be Discussions If a contract neither allows for a coordinating equilibrium nor is run proof, the description of the equilibrium needs more discretion. From the game theory point of view, at least one Nash equilibrium exists. If we accept Diamond-Dybvig s logic, such a contract can be o ered on a trial base. If consumers accept the o er and make deposits at the bank, no-run equilibrium is the only pure strategy Nash equilibrium in the post deposit game. If they do not accept the o er, consumers anticipate a run equilibrium ex post, so they would prefer to stay in autarky. When this situation occurs, the bank should change the o er. The bank either o ers a run proof contract or a contract allowing for the coordinating equilibrium, depending on the welfare levels that the two contracts would achieve. However, this argument sends us back to the question why we actually observe panic runs since the empirical studies (Boyd, et al, 200) show that the most banking crises cannot be explained by the weakness in the fundamentals. The second plausible approach is to introduce mixed strategies to the banking game. There are two reasons that we do not follow this approach. First, it requires a large amount of calculations for a N-player game. Second, if mixed strategies are allowed, we implicitly add a coordination device to the economy, such as a "coin", which is a potential con ict with the sunspot coordination device introduced in the rst place. Another possible way to tackle the problem is to simply rule out market participation if m is in Mn(M CE [ M P ). This can be done with the assumption of ambiguity aversion. Consumers simply stay away from the bank if it provides a contract which admits a run equilibrium but the consumers do not know for sure the probability of bank runs. ecent work on ambiguity aversion (Easley and O Hara, 2005) provides an explanation for the 20

21 statistically signi cant market e ect after the "too big to fail" doctrine was established in 985. This can be the future research direction. 3 Welfare Implication of an Imperfect Coordination Device In the rst part of the paper we have proved that an imperfect coordination device also results in equilibrium. How is this equilibrium compared with the equilibrium resulted from a perfect coordination device? If adding a sunspot always results in a lower level of welfare, suppose there is a social planner, he would prefer to put all consumers into one network, and the problem we discuss in section 2 would not be a problem at all. In this section, we compare the welfare levels achieved by the perfect and imperfect coordination devices. We will see that in some circumstances, an imperfect coordination device is better. In general, since the sunspot signal does not a ect the fundamentals, people can interpret the signal freely. In the best scenario, consumers interpret both and 0 as the signals to stay. So no run equilibrium always occurs. In this case, an imperfect coordination device works just as well as the perfect one. So is the worst scenario. In this section, we still impose the assumption that consumers interpret as the signal to run and 0 the signal to stay. So whichever coordination device is employed, the probability of bank runs is positive. We will compare the welfare level resulting from using a single sunspot signal with that from using an additional sunspot signal. The welfare, represented by the ex ante expected utility, from an imperfect coordination device is described by problem PCE. If there is only one of the sunspot signals is observed, for example, all consumers observe ; the problem is reduced to a traditional sunspot problem with only full run and no run outcomes. If only is provided, the probability of bank run is p + p 0, and that of no run is p 00 + p 0. The ex ante welfare of a run admitting contract is ^W (m) = max (p + p 0 ) c c u c + (p 00 + p 0 ) u c + ( s:t: m 2 MnM P : c ) u (PP) 2

22 Without loss of generality, let p 0 = p 0. So providing only or 2 results in the same level of expected utility. It is generally hard to describe the conditions for problem PCE achieves higher value than problem PP since problem PCE is de ned on a non-convex set and we do not have a more detailed description of how the set varies with the probabilities of partial runs. However, we have examples to show that in some economy, providing an imperfect coordination device is a better choice. Proposition 5 In some economies, providing an imperfect coordination device which results in a coordinating equilibrium is better than a perfect coordination device. Proof. Prove by example. Let us use the example in the proof of proposition 4. That is, b u(c) = (c+b), =, b = 0:5. = :5, = 0:4, n = n 2 = 0:5, p = p 0 = p 0 = 0:00. In this example, a contract consistent allowing for a coordinating equilibrium is better than the run proof contract and also better than the best outcome if the coordination device fails. The welfare level achieved by the optimal contract allowing for a coordinating equilibrium is :434. If only is observable to all consumers, solving problem PP, we get the welfare of an optimal run admitting contract as follows: m c ^W (m) MnM P :0689 :4340 The welfare level it achieves is lower than what the imperfect coordination device can do. The intuition behind proposition 5 is that, by providing an imperfect coordination device, the probability of full run is reduced. Although the probability of no run is also decreased and partial runs are introduced to the problem by adding the noises to the coordination device, if the welfare loss from the full run is reduced dramatically and is not o set completely, the overall ex ante welfare is improved. Hence, an imperfect coordination device performs better in this economy. 22

23 4 Choice of Networks In previous sections, networks are endowed. So the imperfect coordination device is by assumption has to be used by the economy. The question is if people have choice of the coordination device, is an imperfect coordination device the equilibrium choice? In this section, networks are chosen instead of endowed. Networks are di erentiated by the sunspots the network members observe. People in the same network obtain the same sunspot signal and adopt the same strategy in the post deposit game. For simplicity, we assume two available networks here. Two networks observe imperfectly correlated sunspot variables. If all consumers choose to be in the same network, the perfect coordination device is used. If in the equilibrium, some choose to be in network, and the others in network 2, the imperfect coordination device is employed. Consumers know the available networks ex ante, and choose either of them. They observe perfectly the number of consumers in each network, and can move freely between networks. We will show that given some run admitting contract, there exists a coordinating equilibrium in the post deposit game. For some special probability distribution, any run admitting contract allows for a coordinating equilibrium. The sequence of timing at t = 0 is as follows. Bank announces the contract; Consumers choose network; Consumers make investment decision. 9 >= >; t = 0 The sequence of timing after t = 0 is the same as in previous sections. If the contract is run proof, the observation of sunspots does not matter at all. However, if the contract is run admitting, ex ante welfare may be di erent for members in di erent networks. Consumers calculate the ex ante welfare each network brings observing the measures of consumers in both networks, and choose the one which brings higher expected utility. Consumer j s strategy set S net j is described by S net j = {network, network 2}. The expected payo of an individual consumer depends on which network he is in, the measure of people in each network, and each network s withdrawal decisions. The expected payo then can be written as W = W (network i j S post j ; S post j ; m; n ; n 2 ; p ; p 0 ; p 0 ). The analysis in this section takes two steps. First, assuming people deposit at the bank, 23

24 we prove that given a run admitting contract, there are two types of equilibria. The rst type is symmetric in the sense that all people choose to be in the same network, and are perfectly coordinated by the sunspots at t =. The other is asymmetric in the sense that a fraction of the consumers choose network, and the rest choose network 2. Consumers are coordinated by the imperfect coordination device at t = in this equilibrium. Knowing the strategies taken by the consumers, in the second step, we compare the run proof and run admitting contract, and provide the better one of the two so that consumers are willing to deposit at the bank ex ante. 4. Symmetric Equilibrium Proposition 6 Assume people deposit at the bank. Given a run admitting contract m = (c ; c 2 ) 2 MnM P, there is a subgame perfect Nash equilibrium in the predeposit game such that (i) all consumers choose to be in the same network i, (ii) all consumers run on the bank if i = is observed, and do not run on the bank if i = 0 is observed in the post deposit game. Proof. Let us check the equilibrium by one step deviation. Assume (ii) is true. Without loss of generality, let consumer j observe that all other consumers choose network, and according to the strategies in the post deposit game, all other consumers run on the bank when i = ; and do not run when i = 0. If consumer j also chooses network, his expected utility will h be W = (p + p 0 ) u(c ) + (p c 0 + p 00 ) u(c c ) + ( )u i. If he chooses network 2, given network 2 s signal, consumer j runs if 2 = ; and waits if 2 = 0. The expected payo if he is doing so is p u(c ) + p c 0 u(0) + p 0 u (c ) + p 00 hu(c c ) + ( )u i, which is strictly lower than W. In the same way, we can prove that when all consumers choose network 2; it is to an individual consumer s best interest to choose network 2. Assume (i) is true, by Peck and Shell (2003), the post deposit game have a sunspot equilibrium since the contract is run admitting. This sunspot equilibrium is a special case of the coordinating equilibrium given a perfect coordination device. This completes our proof. 24