Withdrawal History, Private Information, and Bank Runs

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1 Withdrawal History, Private Information, and Bank Runs Carlos Garriga and Chao Gu This aer rovides a simle two-deositor, two-stage model to understand how a bank s withdrawal history affects an individual s decision about withdrawals, which could ossibly trigger bank runs. Individual deositors have rivate information about their ersonal consumtion tyes and receive noisy rivate signals about the quality of the bank s ortfolio. Deositors make ublicly observable withdrawal decisions in sequence. Comuted examles indicate that the otimal contract contingent on withdrawal histories can tolerate bank runs. These runs are triggered by unfavorable signals about a bank s ortfolio, and early liquidation of unsuccessful investments can avoid future losses. Because the signals are rivate, a deositor s action is the only way to artially reveal his rivate information. A runadmitting bank contract allows information to be revealed. However, if signals are too noisy, bank runs may occur too often when fundamentals are strong. In this case, a bank would offer a run-roof contract. Given the relevant role of information, a olicy that makes rivate information ublic would be useful to imrove welfare and eliminate bank runs. (JEL C73, D8, E59, G) Federal Reserve Bank of St. Louis Review, July/August 0, 94(4), The recent financial crises have drawn considerable attention on the regulation of financial intermediaries. One question that arises is whether bank runs should be revented in any circumstance. To answer this question, we first need to understand the underlying conditions that romt bank runs. One strand of the literature, following Diamond and Dybvig (983), argues that banks and bank runs are inherently intertwined because banks contracts rovide short-term liquidity, whereas banks ortfolios mature only in the long term. As a result, a anic-based bank run is self-fulfilling even in the absence of uncertainty about fundamentals and is not efficient. Some institutional arrangements for examle, deosit insurance or the romise from the central bank to serve as the lender of last resort can revent anic-based bank runs by roviding sufficient liquidity should a run occur. Another strand in the literature attributes the runs to fundamentals. The view on fundamental-based bank runs argues that bank runs occur when deositors receive negative information about their bank s ortfolio returns or about an aggregate liquidity shock. Unlike Carlos Garriga is a research officer and economist at the Federal Reserve Bank of St. Louis. Chao Gu is an assistant rofessor of economics at the University of Missouri Columbia; she thanks Karl Shell for guidance. 0, The Federal Reserve Bank of St. Louis. The views exressed in this article are those of the author(s) and do not necessarily reflect the views of the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Articles may be rerinted, reroduced, ublished, distributed, dislayed, and transmitted in their entirety if coyright notice, author name(s), and full citation are included. Abstracts, synoses, and other derivative works may be made only with rior written ermission of the Federal Reserve Bank of St. Louis. Federal Reserve Bank of St. Louis REVIEW July/August 0 305

2 anic-based runs, fundamental-based runs are not necessarily inefficient because liquidating unsuccessful investments early can mitigate future losses. In both strands of the literature, the arrival of information is the factor that determines whether a bank run occurs. For anic-based bank runs, the realization of an exogenous variable, called a sunsot, can trigger a bank anic. For fundamental-based runs, deositors lose confidence when there is unfavorable news about their bank s erformance. In either aroach it is generally assumed that uon receiving information deositors need to make a simultaneous withdrawal decision without observing the actions of others. In reality, at least some withdrawals are based on the information about revious withdrawals by others. 3 This sequential rocess of learning from the observed withdrawal history is imortant to understand not only bank runs, but also whether banks can use the rocess of revelation of information to design their deosit contracts. The objective of this aer is to understand how a bank s withdrawal history affects an individual s decision about withdrawals, which could ossibly trigger bank runs. A dynamic model is necessary to formalize the revelation of the withdrawal history. The model is a simle twostage game with two deositors and rivate information. In the game, bank runs are driven by signals on the fundamentals as oosed to sunsots. In the model, the deositors receive a rivate signal about their liquidity needs and a rivate noisy signal about the quality of the bank s ortfolio. Deositors make withdrawal decisions in sequence at a given stage and the withdrawal decisions are ublicly observable. The first deositor s action to withdraw or to wait can artially reveal his rivate signal about the bank s ortfolio, which affects the belief of the second deositor and thus his withdrawal decision. Under some arameterization, the otimal contract admits an equilibrium in which deositors strategies are contingent on their rivate signals and observed withdrawals. A dynamic model exlains some emirical results on bank runs that cannot be exlained by a static model. For examle, during the 00 run on Turkish secial finance houses, 4 deositors made sequential withdrawals influenced by the history of withdrawals by others, as noted by Starr and Yilmaz (007,. 4): Increased withdrawals by moderate-size account holders tended to boost withdrawals by [their] small counterarts, suggesting that the latter viewed the former as informative with resect to the SFH s [secial finance house s] financial condition. 5,6 Our model, although simle, sheds some light on whether bank runs should be comletely revented in an environment of rivate information. Comuted examles show that in some economies a contract that ermits bank runs is otimal, whereas in other economies a run-roof contract is otimal. This result is in line with the literature roosing that, if the robability of bank runs is low, a bank contract tolerates bank runs as deositors receive more consumtion insurance during normal times. Furthermore, in the environment considered here, a bank run is driven by the information about a bank s ortfolio return. In other words, it is driven by fundamentals. When fundamentals are weak, it is otimal for the bank to liquidate its ortfolio to avoid future losses. Since information is rivate, the only way that deositors can reveal their information is by their actions. A run-admitting contract allows deositors to do so, whereas a run-roof contract does not. However, (i) because a deositor s decision to withdraw carries noisy information about the signals he receives (the deositor might need to consume immediately or might receive an 306 July/August 0 Federal Reserve Bank of St. Louis REVIEW

3 unfavorable signal about the quality of the bank s ortfolio) 7 and (ii) because the information itself is imerfect, bank runs can occur when fundamentals are strong. In these cases, runs are misled. If the robability of such misled runs is high, a run-roof contract is better. The ayments to the deositors in our model have two functions. First, they rice fundamental risks. Second, deending on the quality of information, they give deositors an incentive to reveal or to hide their rivate information. Our results imly that ricing risk and incentive aroriately is the key to making financial markets efficient, 8 although ex ost inefficient runs can occur as a result of imerfect information. To show the imortance of information, it is useful to solve a numerical examle where signals on ortfolio returns are ublic and comare it with the one with rivate signals. With ublic signals welfare is higher and, most imortantly, there are no bank runs. 9 Hence, olicymakers may make more effort to ublicize the information of the fundamentals to imrove welfare. We focus on the numerical examles that yield a unique equilibrium. Hence, there is no sunsot-driven run (or anic-based run) in this aer. Although some bank runs occur when fundamentals are strong, since the runs are triggered by (imerfect) signals on fundamentals, these are still fundamental-based runs in our view. Runs on commercial banks have been rare in the United States since the introduction of deosit insurance. However, runs on the shadow banking system were the imortant events in the recent financial crisis (see Gorton, 00, and Anderson and Gascon, 009). Our model, which uses the customary terminology in the literature with regard to bank runs, alies to general financial intermediaries subject to systemic financial crises. The rest of the aer is organized as follows: The next section introduces the model setu and is followed by a discussion of the equilibrium given a banking contract. Next we calculate some examles of otimal contracts and then offer an examle of an otimal contract in an economy with ublic signals. The final section summarizes our findings and conclusion. THE MODEL Time. There are three eriods, indexed by t = 0,,. Period 0 is a lanning eriod called ex ante. Periods and are ex ost eriods. Period is divided into two stages. Deositors endowment and references. There are two deositors. Each deositor is endowed with one unit of consumtion good ex ante and nothing ex ost. Each deositor has robability α to become imatient in eriod and robability α to be atient. An imatient deositor values consumtion only at t =. His utility is described by u(c ), where c is the consumtion at t =. A atient deositor s utility is described by u(c + c ), where c denotes the consumtion at t =. The utility function is strictly increasing, strictly concave, and twice differentiable. The coefficient of relative risk aversion, xu (x)/u (x), is greater than when x. Whether a deositor is atient or imatient is revealed to the individual deositor at some stage in eriod. Technologies. The consumtion good can be stored at no cost. It can also be invested in a risky technology. The investment must be made ex ante and takes two eriods to mature. The return on the investment can be either R > or R < at t =. The ex ante robability of receiving R is 0. If the investment is liquidated at t =, the return is. Because the investment yields the Federal Reserve Bank of St. Louis REVIEW July/August 0 307

4 same return as storage at t =, all consumtion goods will be laced in the risky technology at t = 0 and will be artially or fully liquidated at t =, deending on the events occurring at t =. Withdrawal stages and information. Period is divided into two stages. At each stage, one deositor is informed of a air of signals. One signal tells him recisely his consumtion tye; the other imerfectly tells him the investment returns. The signal on investment return is accurate with robability q, where q 0.5. That is, ( i ) = i = = = Pr S = H R = R Pr S L R R q, where S i denotes deositor i s rivate signal of investment return. Deositors have an equal chance to receive signals at stage. The deositor who receives the signals at the first stage is called deositor ; the other is deositor. Each deositor can make withdrawals when he receives signals in eriod. If he does not withdraw in eriod, he receives ayment in eriod. For convenience, a deositor can withdraw in eriod only at the stage when he receives information. Deositors actions are ublicly observable. 0 Because there are only two deositors and two stages, allowing deositors to withdraw at any stage adds only two ossible simultaneous-move games to each stage and does not change the main results. The contract. A cometitive bank offers a contract to deositors ex ante. For convenience, the minimum deosit amount that the bank accets is one unit of a consumtion good. The bank allocates the funds between storage and investment and makes ayments to deositors uon withdrawals. The banking contract considered here ays deositors contingent on the withdrawal history., The contract secifies the ayments to withdrawals at t = deending on the number of withdrawals that have been made and the ayments to withdrawals at t = deending on the number of withdrawals at t = and the return on investment. Let x i {0,] denote deositor i s action in eriod, where 0 indicates wait and indicates withdraw. Let c (x ) be the ayment to deositor at stage, where c (0) = 0, and let c (x,x ) denote ayment to deositor at stage, where c (x,0) = 0. Similarly, let c (x,x,r) denote ayments at t =. All instances of c and c satisfy the following resource constraints: () () (3) + c x c x, x, x x c x, x, R c x c x, x max {,R}, x x c x, x, R c x c x, x min{,r}. Timing of the banking game. The timeline of the banking game can be summarized as follows: t = 0: The bank announces the contract. Deositors make deosit decisions. 308 July/August 0 Federal Reserve Bank of St. Louis REVIEW

5 t = : Stage : Deositor receives signals about his consumtion tye and roductivity. He decides whether to withdraw. Stage : Deositor receives signals about his consumtion tye and roductivity. He decides whether to withdraw. t = : The bank allocates the remaining resources to deositors who have not withdrawn in eriod. The ostdeosit game starts after deositors make deosits at the bank. An individual deositor decides when to withdraw. A bank run occurs if at least one atient deositor withdraws. Know - ing what deositors will do in the ostdeosit game, a reresentative bank offers a contract that maximizes the ex ante exected utility of the deositors. Deositors determine whether to deosit at the bank or stay in autarky. Starting at t = 0, the entire game is called the redeosit game. Solving the model backward, as in Peck and Shell (003), requires starting with the ostdeosit game and describing the equilibrium given a contract. Then the redeosit game is comleted by comaring the exected utilities in autarky with those in a banking economy. THE POSTDEPOSIT GAME The equilibrium concet is a erfect Bayesian equilibrium in which the strategies of the deositors are otimal given the deositors beliefs about investment returns and the beliefs are udated by Bayes rule whenever ossible. Let x ni and ni denote the strategy and osterior belief that the return is high, resectively, of deositor i at stage n. Given each deositor s references and the structure of the game, deositor s strategy at stage is and deositor s strategy at stage is x = if imatient and x = with robability θ S if atient, x = if imatient and x = with robability θ x,s if atient. Since deositors can make withdrawals only at their informed stage, x ni = 0 when n i. Bayesian Udates Suose a deositor has rior belief at the beginning of a stage. Let ρ() q + ( )( q) be the robability that an informed deositor will receive a favorable signal at that stage given the rior belief. When a deositor receives the signal, he udates his belief according to Bayes rule: Federal Reserve Bank of St. Louis REVIEW July/August 0 309

6 (4) As the signal is accurate with robability q 0.5, we have P H () P L (), where the equality holds if and only if q = 0.5. That is, if a favorable signal is received, a deositor is more confident in the ortfolio returns, whereas if an unfavorable signal is received, he is less confident. If a deositor is not informed at a stage, he still learns some information by observing the informed deositor s action. When deositor makes a decision at stage, his decision carries noisy information about the signals he has received. Given deositor s strategy, deositor s osterior belief at stage is (5) The denominator of P is the robability that deositor waits given deositor s strategies θ S. The numerator is the robability that the bank s ortfolio return is high and deositor waits. The same rule alies for P. L After deositor makes his decision, deositor udates his belief (although he has no chance to change his decision) in a similar way, as follows: (6) = q PH =, i ρ i = ( q) P =, L ρ H = P if S = H, i if S L. i = θl ( q)+ θh q P H = ( θl ) ρ +, θh ρ if deositor waits; P L x, H P x L α + ( α) θl ( q)+ θhq α + ( α ) θ ρ L θhρ if deositor withdraws. = { } + θx L q x H q, + θ, = ( ) +, θx ρ θ ρ, L x, H if deositor waits; α + ( α ) θx L q θx Hq, ( )+, α + ( α ) θ x L ρ, θx H ρ, if deositor withdraws., = { + },, 30 July/August 0 Federal Reserve Bank of St. Louis REVIEW

7 Again, the denominator of P ( ) is the robability that deositor waits given deositor s x, H strategies θ x,s. The numerator is the robability that the bank s ortfolio return is high and deositor waits. The same rule alies for P. Strategies The equilibrium strategies are a vector of θ = (θ S,θ x,s ), x = 0,, and S,S = L,H that solves the deositor s exected utility maximization roblem at each node. Working backward, deositor, if atient, chooses the withdrawal robability θ x,s to maximize his exected utility given his observation of deositor s action and his own rivate signal: (7) wˆ ( x, S ) = max θ x, S u( c ( x, ) ) + ( ) ( ) + θ x 0 ( ) ( 0 θ 0, S u c x,, R u( c x,, R) ), where if x = and otherwise. The first term on the righthand side is the ayoff if he withdraws given deositor s action. The second term in the closed PS P L 0 PS P H 0 bracket is the exected utility if he waits. For deositor, if he does not withdraw at stage, his exected utility at the end of stage is (8) x, S, = where if x = and otherwise. At stage, deositor P P 0, L S 0 P P 0, H S 0 chooses withdrawal robability θ S to maximize his exected utility given the robability that deositor will withdraw (i.e., the robability that reaches ŵ (0,,S )). This robability, in turn, is artially determined by deositor s action, as deositor udates his belief according to what he observes. Deositor solves = (9) wˆ ( S ) = max θs u( c )+ ( θ S 0, ( θs ) + x L, = + wˆ 0, x, S u c 0, x, R ( ) u( c ( 0, x, R) ), = ( = ( ( )) ( )+ α α θ, H ρ PS θ ρ, L PS 0 wˆ ( 0,, S )+ α ( θ, ) ( P ( 0 H ρ S 0 ))+( θ 0, L )( ( ρ P )) wˆ S,, S where the multiliers in front of ŵ (0,,S ) and ŵ (0,0,S ) are the robabilities that deositor withdraws/waits given that deositor receives S and withdraws. In equilibrium, deositor infers the investment status by watching deositor s action. His belief is udated by his rivate signal and deositor s action. When deositor makes a decision, he also knows his decision will affect deositor s belief and decision and, thus, his own ayoff., Federal Reserve Bank of St. Louis REVIEW July/August 0 3

8 Table Examle of a History-Deendent Contract Payments to deositors Variables Amount Deositor at t = c () Deositor at t = if withdraws c (,).000 Deositor at t = if waits c (0,).0000 Both deositors at t = if both wait and R = R c (0,0,R ).0000 Deositor at t = if deositor withdraws and R = R c (0,,R ).000 Deositor at t = if deositor withdraws and R = R c (,0,R ).000 Both deositors at t = if both wait and R = R c (0,0,R).0000 Deositor at t = if deositor withdraws and R = R c (0,,R) Deositor at t = if deositor withdraws and R = R c (,0,R).000 The solution to maximization roblems (7)-(9) given a contract is not necessarily unique. A simle way to illustrate the roerties of equilibria is to construct some numerical examles. In γ γ ( c + b) b all examles in the aer, the utility function is u( c) =, where b = 0.0 and γ =.5. γ The arameter b > 0 ensures that marginal utility is bounded low by a ositive number when c = 0. Examle shows a case in which a contract has more than one erfect Bayesian equilibrium. Examle : Multile equilibria in the ostdeosit game. The arameters in the economy are α = 0.5, R =.3, R = 0., 0 = 0.5, and q = 0.7. Table shows the history-deendent contract considered here. The contract in the examle satisfies the resource constraints ()-(3). That is, it is a feasible contract but it is not necessarily the best contract that a bank can offer. This contract has two ure strategy erfect Bayesian equilibria. They are (Equilibrium ) ( θl = 0, θh = 0, θ L =, θ0 L =, θ H = 0,,,, θ0, H = 0), and (Equilibrium ) ( θl =, θh = 0, θ L =, θ0 L = 0, θ H = 0,,,, θ0, H = 0). In the first equilibrium, deositor s signal of investment return does not affect his decision. He always waits if he is atient. Deositor cannot infer any information from deositor s action. Thus, deositor s decisions are based solely on his rivate signals, not the withdrawal history. In the second equilibrium, deositor reacts differently to different signals of investment return. His action artially reveals the signal he has received, which affects deositor s decision. Deositor s decision is deendent on the withdrawal history. A banking contract is run roof if θ = 0 is the unique solution. If a run-roof contract is rovided, deositors do not withdraw unless they are imatient. All other contracts are called run-admitting, as these contracts admit at least one equilibrium in which at least one atient deositor withdraws based on some realization of rivate signals and withdrawal history. 3 July/August 0 Federal Reserve Bank of St. Louis REVIEW

9 THE PREDEPOSIT GAME The ex ante exected utility of the deositors is determined by their strategies, which in turn are determined by the contract. Knowing the strategies of the deositors in the ostdeosit game given a contract, the reresentative bank offers a contract that maximizes the ex ante exected utility of the deositors. Given the contract, deositors decide whether to stay in autarky or to deosit at the bank at t = 0. If the ex ante exected utility in autarky is higher than that under the banking contract, the contract will be acceted and the ostdeosit game will be layed. Otherwise, deositors refer to stay in autarky. Autarky In autarky, deositors do not observe each other s actions. A deositor adjusts his investment ortfolio after he receives rivate signals at t =. If the deositor is revealed to be imatient, he immediately consumes all of his available assets and receives utility u(). A atient deositor s exected utility in eriod after receiving signal S is solved by where λ S denotes the roortion of assets liquidated after receiving the signal. The ex ante exected utility in autarky is the weighted average of the exected utility in eriod. That is, The Otimal Bank Contract The bank s otimal contract maximizes the deositor s ex ante exected utility. As each of the deositors has robability / of being the first to receive the signals and make a decision, a deositor s ex ante exected utility is the equally weighted exected utilities of deositors and at the beginning of eriod. Let ŵ (x ) be deositor s exected utility at the end of stage given deositor s action. Secifically, (0) () aut wˆ ( S) = max PS ( 0 ) u( λs + ( λs ) R)+ ( PS ( 0 )) u( λs + ( λ S ) R), λs 0, + w aut u w aut H w aut 0 = α + α ρ ˆ 0 ρ ˆ 0 L. { } wˆ ( 0) = αu( c ( 0, ) )+( α ) ρ( P wˆ, H P H ( 0 )) ( 0 )+ ρ( H ( 0 )) w 0 L ˆ (, ), { } () wˆ = αu( c (, ) )+( α) ρ( P wˆ, H P L ( 0 )) ( )+ ρ( L ( 0 )) w L ˆ (, ). The deositor s ex ante exected utility is given by { } w0 = αu( c )+ ( α ) ρ( 0 ) wˆ ( H )+ ( ρ( 0 )) wˆ ( L) ( ) ( )+( )( ) α ρ 0 θh ρ 0 θ ˆ L w ( 0)+ + + ( ) α α ρ + ( ( )) ˆ 0 θh ρ 0 θl w, Garriga and Gu Federal Reserve Bank of St. Louis REVIEW July/August 0 33

10 Table Examle of an Otimal Run-Admitting Contract ωˆ0 c () c (,) c (,0,R ) c (,0,R ) c (0,) c (0,,R ) c (0,,R) c (0,0,R) c (0,0,R ) where the multiliers in front of ŵ (0) and ŵ () are the ex ante robabilities that deositor will wait or withdraw. The reresentative bank offers a feasible contract that maximizes the ex ante exected utility of the deositors. That is, the bank seeks c = (c (),c (x,),c (x,x,r)) to solve ŵ = max w 0 0 c st ()-(3). aut The otimal contract will be acceted at t = 0 if and only if ŵ 0 w 0. The analytical solution to the otimal contract is comlicated to solve, so numerical examles are used to illustrate the roerties of a ure strategy equilibrium under an otimal banking contract. As demonstrated in the revious section, one of the challenges is the multilicity of equilibria in the ostdeosit game. Unfortunately, the conditions for the uniqueness of the equilibrium are too comlicated to derive. In the following examles, we check numerically that the otimal contract allows for a unique equilibrium in the ostdeosit game. 3,4 Examles and 3 illustrate two different cases of an otimal contract; one is run admitting and the other is run roof. Examle : The otimal banking contract is run admitting. Parameters in this examle are α = 0.6, R =., R = 0.8, 0 = 0.75, and q = 0.9. Table describes the ayment scheme that an otimal bank contract rovides. Given this contract, the equilibrium strategies of deositors in the ostdeosit game, if atient, are θ L =, θ H = 0, θ,l =, θ 0,L = 0, θ,h = 0, and θ 0,H = 0. Given the equilibrium strategies, we can calculate the robability of bank runs. Some bank runs are artial only one of the deositors withdraws but he does not need to consume immediately; some are full bank runs both deositors withdraw regardless of their consumtion tyes. The robability of having a artial run conducted by deositor in this examle is Pr( is atient)pr(s = L)Pr( is imatient or S = H) = The artial run conducted by deositor haens with robability Pr( is imatient and is atient)pr(s = L) = The robability of a full bank run is Pr( and are atient)pr(s = S = L) = We also reort the strategies of a deositor in autarky and comare the welfare in autarky with that under the otimal contract. In autarky, a deositor leaves all assets invested if a favorable signal is received and liquidates all assets when an unfavorable signal is received. The consumtion of a deositor contingent on the signals and the return is summarized as follows: {c =, c (H,R ) =., c (H,R) = 0.8, c (L,R ) =, and c (L,R) = }. The ex ante exected utility in aut autarky, w 0, is , which is lower than ŵ 0. Hence, the contract will be acceted, although bank runs will take lace with ositive robability. 34 July/August 0 Federal Reserve Bank of St. Louis REVIEW

11 Table 3 Examle of a Run-Proof Contract ωˆ0 c () c (,) c (,0,R ) c (,0,R) c (0,) c (,0,R ) c (,0,R) c (0,0,R ) c (0,0,R) The signals are 90 ercent accurate in this examle. In autarky, since the signal is highly accurate, a deositor will follow the signal. In the banking economy, a rivate signal still lays an imortant role deositor follows the signal as he would in autarky since he must make decisions before he learns information from deositor. However, deositor infers information from deositor s action and deositor does not rely solely on his rivate signals to make a withdrawal decision. Deositor s decision to wait reveals that a favorable signal has been received. If deositor receives an unfavorable signal, it will be offset by the favorable signal inferred and his osterior belief will become 0. As 0 still is fairly favorable, deositor will not withdraw. If deositor withdraws, however, the action sends noisy information that an unfavorable signal may be received. When deositor receives an unfavorable signal, his belief is lowered even more, such that he refers to liquidate the asset immediately to mitigate the loss in investment. But if deositor gets a favorable signal, his osterior belief becomes higher than 0. In this case, he still follows his rivate signal and waits. A run-admitting bank contract is otimal in some economies for the following reasons. First, the contract hels smooth the consumtion in an economy with aggregate consumtion shocks. Second, in an economy with roduction uncertainty, a bank run is not necessarily bad. In examle, if the true state of roductivity is low, then deositors receive ayments in the amount of either.0064 or But if both deositors wait, each will get A bank run is a means to terminate low-quality investments to mitigate future losses. In this sense, information about investment return is valuable and a run-admitting contract allows information to be artially revealed. In examle, the artial run conducted by deositor relies on the fact that deositor withdraws. Because of imerfect signals and the revelation of imerfect information by actions, a bank run can occur when roductivity is actually high. If the robability of a bank run in a high-return state is too high, a run-roof contract will be offered by the bank. Examle 3 illustrates this recise case. When the signal received by deositors contains too much noise, a runroof contract is otimal. Examle 3: The otimal banking contract is run roof. The arameters in this economy are the same as in examle excet that q = 0.5. In this case, a roductivity signal is not infor - mative. Table 3 describes the ayment scheme of the otimal banking contract. Given such a contract, there is a unique equilibrium in the ostdeosit game in which deositors withdraw if and only if they are imatient. In autarky, deositors leave all assets invested if they are atient. The rivate signal does not carry any information and if a deositor decides to invest ex ante, he will not change his decision if he is atient, as no useful information arrives ex ost. The ex ante exected utility in autarky Federal Reserve Bank of St. Louis REVIEW July/August 0 35

12 Figure Exected Utilities in Autarky and Under an Otimal Deosit Contract w aut, w Autarky Otimal Banking Contract is , which is equivalent to that in the banking economy. Deositors weakly refer to accet the contract and no bank run occurs ex ost. Because the signals carry too much noise in a banking economy, if a run-admitting contract were rovided, bank runs would haen too often when the fundamentals are strong. Therefore, a contract that does not allow for the disclosure of information is more desirable here. Comared with examle, the contract here rovides less consumtion to the first deositor who withdraws in t = (i.e., c () and c (0,)). The lower ayments in t = discourage deositors from withdrawing even when they receive unfavorable signals (although signals are not useful in redicting returns in this case). On the contrary, ayments of c () and c (0,) in examle are higher, so deositors are more encouraged to withdraw when the signals are unfavorable. As a result, deositors artially reveal their signals by their actions. Figure lots the exected utilities in autarky and under the otimal contract with different values of q given other arameters in examle. 5 The dashed line reresents the exected utility under the otimal banking contract, whereas the solid line reresents the exected utility in autarky. When q is small, the contract is run roof and the exected utility is the same as in autarky. As q increases, the otimal banking contract is run admitting and yields strictly higher exected utility than autarky. The difference in exected utilities between a banking economy and autarky is not monotone: As q aroaches, the welfare gain in the banking economy decreases (w aut = and ŵ 0 = if q = ). Why is that? In our model, deositors gain from articiating in banking through two functions of the bank. First, the bank rovides consumtion insurance for deositors as noted in the literature (see, for examle, Diamond and Dybvig, 983). Second, the bank rovides additional information on fundamentals to deositors since with- 36 July/August 0 Federal Reserve Bank of St. Louis REVIEW

13 Table 4 Examle of an Otimal Contract When Investment Return Signals Are Public c (,H) c (,L) c (,,H,S) c (,,L,S ) c (0,,H,H) c (0,,H,L) c (0,,L,H) c (0,,L,L) c (,0,H,H,R ) c (,0,H,H,R) c (,0,H,L,R ) c (,0,H,L,R) c (,0,L,H,R ) c (,0,L,H,R) c (,0,L,L,R ) c (,0,L,L,R) c (0,0,H,H,R ) c (0,0,H,H,R) c (0,0,H,L,R ) c (0,0,H,L,R) c (0,0,L,H,R ) c (0,0,L,H,R) c (0,0,L,L,R ) c (0,0,L,L,R) c (0,,H,H,R ) c (0,,H,H,R) c (0,,H,L,R ) c (0,,H,L,R) c (0,,L,H,R ) c (0,,L,H,R ) c (0,,L,L,R ) c (0,,L,L,R) drawals are ublicly observable. In autarky, deositors observe only their own signals. When q = (i.e., signals are erfect), observing the other deositors actions does not rovide additional information. Therefore, the gain from information aggregation disaears in this extreme case. PUBLIC SIGNALS To illustrate that the economy is inefficient because the information on investment is rivate, it is useful to solve a numerical examle with ublic signals on ortfolio returns and comare it with an examle with rivate signals. Here we continue with examle but with the conditions that the signals on investment return are now ublicly observable, although consumtion signals are rivate. Examle 4: Public signals on investment return. In this examle, the signals on investment return are ublicly observable. Deositors have rivate information about their consumtion tyes. The contract secifies the ayments contingent on a deositor s arrival time, the withdrawal history, and the ublic signals. The arameters are the same as in examle. Table 4 shows the otimal ayment scheme. The exected utility under the otimal ayment scheme is ŵ 0 * = , which is higher than that in the economy with rivate investment return signals. The strategies of deositors under the otimal contract are θ = 0. That is, no atient deositor withdraws. The otimal ayment scheme given ublic information encourages deositors to truthfully reort their consumtion tyes by their actions. Given any ublic history, the exected utility of a deositor in the last eriod is higher than the utility from immediate withdrawal. Although there are two sources of uncertainties in our model, the bank and deositors share the same information regarding investment return. In other words, the only information Federal Reserve Bank of St. Louis REVIEW July/August 0 37

14 asymmetry between the bank and the deositors comes from consumtion tyes. Now the bank s only requirement is to design an incentive-comatible contract to eliminate bank runs. Similar to Green and Lin s (000, 003) findings, a contract contingent on the withdrawal history can revent bank runs by eliminating the asymmetric information between the bank and deositors. Another lesson from this examle is that the direct revelation rule is, in general, not incentive comatible in the environment with rivate signals of investment return. Suose that deositors must announce their rivate signals (direct revelation) before they make decisions at t =. Deositors have incentives to lie (violate incentive comatibility) to the bank when they receive an unfavorable signal. A deositor can instead tell the bank that although he has received a favorable signal, he needs to consume immediately. If the bank believes him, he could receive a larger ayment than he should. Examle 4 illustrates this oint. Suose deositor is atient. He receives an unfavorable signal and he reorts truthfully. Deositor is also atient and he also receives an unfavorable signal. If he reveals the true signal and does not withdraw at t =, he will receive c (0,0,L,L,R ) = c (0,0,L,L,R) =.006 in the last eriod, whereas if he claims to be imatient but has received a favorable signal, he will receive c (0,,L,H) =.006. CONCLUSION This aer rovides a simle model to understand the dynamics during bank runs in an environment in which deositors have rivate information on bank fundamentals and the deosit contract can be made contingent on withdrawal history. Given such a contract, there is a erfect Bayesian equilibrium in which deositors beliefs and actions are affected by the actions of others. Under certain arameterizations, the comuted examles indicate that the otimal bank contract tolerates bank runs. Runs are tolerated because they are triggered by unfavorable signals on bank ortfolios and liquidating unsuccessful investments early can revent future losses. Because the signals are rivate, a deositor s action is the only way to artially reveal his rivate information. A run-admitting contract allows information to be revealed. Nevertheless, if signals are too noisy, bank runs may occur too often when fundamentals are strong. In this case, the bank would offer a run-roof contract. Given the relevant role of information, a olicy that can make rivate information ublic would be useful to imrove the welfare and eliminate bank runs. One of the model s main limitations is that the bank has no information on investment. A more sohisticated model in which the bank receives signals on investment would romt more interesting questions, such as how to eliminate a bank s moral hazard incentives related to the information asymmetry between the bank and its deositors and how the bank can reduce the robability of bank runs resulting from incorrect signals. NOTES See Allen and Gale (994), Goldstein and Pauzner (005), and Gu (0). The sunsot signals can be viewed as the uncertainty in the fundamentals taken to the limit (as in Manuelli and Peck, 99). 38 July/August 0 Federal Reserve Bank of St. Louis REVIEW

15 3 Brunnermeier (00,. 4) says that Although withdrawals by deosit holders occur sequentially in reality, the literature tyically models bank runs as a simultaneous move game. 4 Secial finance houses are like commercial banks, but their deosits are not insured. 5 Also see Schumacher (000) for details on the Argentine banking crisis. 6 Bank runs here have the features of the herd effect (Banerjee, 99, and Bikhchandani, Hirshleifer, and Welch, 99). 7 Unlike Green and Lin ( 000, 003), the asymmetric information between the bank and deositors cannot be fully eliminated by deositors simle zero-one (i.e., withdrawal-or-wait) decisions. 8 See Anderson (009) for ricing risk. 9 This result agrees with the findings of Green and Lin (000, 003) and Andolfatto, Nosal, and Wallace (007) that the ayment schedules contingent on withdrawal history can eliminate bank runs in an economy with i.i.d. consumtion shocks. 0 Besides the additional dimension of uncertainty, there are two other distinctions between Green and Lin s (000, 003) setu and ours. First, Green and Lin use a direct revelation mechanism in which deositors reort their rivate information about consumtion tyes to the bank. The direct revelation mechanism is not feasible in an economy with two dimensions of uncertainty the bank makes a bigger ayment if the future return is higher. So a deositor would always reort that he receives a favorable signal on the bank s ortfolio when he decides to withdraw. Second, deositors do not observe the decisions of others in Green and Lin s economy but they do in ours. Whether a decision is observable is not crucial to Green and Lin s model (see Andolfatto, Nosal, and Wallace, 007). However, it is crucial in our model because the observed withdrawals rovide information on fundamentals. In a similar model setu, Gu (0) studies the herding effect on bank runs given a simle demand deosit contract. The consideration of a ayment scheme contingent on history has been widely discussed in the banking literature. See Diamond and Dybvig (983) for details on full susension of convertibility in an economy with no aggregate uncertainty and Wallace (990) for artial susension of convertibility. Green and Lin (000, 003) show that in a finite economy with i.i.d. consumtion shocks, the otimal banking contract that ays deositors deending on their arrival time and the withdrawal history can comletely eliminate anic-based bank runs. 3 Only ure strategy equilibria are considered. The exected utility under a true otimal contract (that is, if we consider a mixed strategy equilibrium) can be higher. However, considering mixed strategies significantly comlicates comutation. Note that the strategies under a run-roof contract are ure strategies. Hence, the bottom line is that the otimal contract is not run roof in some economies (examle ), while it is otimal in others (examle 3). 4 We check the uniqueness in the following way: There are 7 ossible ure strategy rofiles. We comute the bank s otimal contract given each strategy rofile and then check whether the given strategy rofile is the equilibrium strategy under the comuted otimal contract. And if so, whether the ex ante exected utility is higher than that under autarky. In all the numeric examles in the aer, only one of the 7 cases is the equilibrium strategy and yields ex ante exected utility higher than the autarky. Hence, it is sufficient to conclude the otimal contract allows for a unique equilibrium. 5 Again, only ure strategy equilibria are considered. The bottom line is that when q increases, the otimal contract is no longer run roof. REFERENCES Allen, Franklin, and Gale, Douglas. Limited Market Particiation and Volatility of Asset Prices. American Economic Review, Setember 994, 84(4), Anderson, Richard G. Bagehot on the Financial Crises of 85...and 008. Federal Reserve Bank of St. Louis Monetary Trends, February 009; htt://research.stlouisfed.org/ublications/mt/00900/cover.df. Anderson, Richard G. and Gascon, Charles G. The Commercial Paer Market, the Fed, and the Financial Crisis. Federal Reserve Bank of St. Louis Review, November/December 009, 9(6), ; htt://research.stlouisfed.org/ublications/review/article/7857. Andolfatto, David; Nosal, Ed and Wallace, Neil. The Role of Indeendence in the Diamond-Dybvig Green-Lin Model. Journal of Economic Theory, November 007, 37(), Federal Reserve Bank of St. Louis REVIEW July/August 0 39

16 Banerjee, Abhijit V. A Simle Model of Herd Behavior. Quarterly Journal of Economics, August 99, 07(3), Bikhchandani, Sushil; Hirshleifer, David and Welch, Ivo. A Theory of Fads, Fashion, Custom, and Cultural Change as Informational Cascades. Journal of Political Economy, October 99, 00(5), Brunnermeier, Markus. Asset Pricing under Asymmetric Information-Bubbles, Crashes, Technical Analysis, and Herding. Oxford, UK: Oxford University Press, 00. Diamond, Douglas and Dybvig, Phili. Bank Runs, Deosit Insurance, and Liquidity. Journal of Political Economy, June 983, 9(3), Goldstein, Itay and Pauzner, Ady. Demand-Deosit Contracts and the Probability of Bank Runs. Journal of Finance, June 005, 60(3), Gorton, Gary. Slaed by the Invisible Hand: The Panic of 007. New York: Oxford University Press, 00. Green, Edward J. and Lin, Ping. Diamond and Dybvig s Classic Theory of Financial Intermediation: What s Missing? Federal Reserve Bank of Minneaolis Quarterly Review, Winter 000, 4(),. 3-3; Green, Edward J. and Lin, Ping. Imlementing Efficient Allocations in a Model of Financial Intermediation. Journal of Economic Theory, March 003, 09(),. -3. Gu, Chao. Herding and Bank Runs. Journal of Economic Theory, January 0, 46(), Manuelli, Rodolfo and Peck, James. Sunsot-like Effects of Random Endowments. Journal of Economic Dynamics and Control, Aril 99, 6(), Peck, James and Shell, Karl. Equilibrium Bank Runs. Journal of Political Economy, February 003, (), Schumacher, Liliana. Bank Runs and Currency Run in a System without a Safety Net: Argentina and the Tequila Shock. Journal of Monetary Economics, August 000, 46(), Starr, Martha and Yilmaz, Rasim. Bank Runs in Emerging-Market Economies: Evidence from Turkey s Secial Finance Houses. Southern Economic Journal, Aril 007, 73(4),. -3. Wallace, Neil. A Banking Model in Which Partial Susension Is Best. Federal Reserve Bank of Minneaolis Quarterly Review, Fall 990, 4(4),. 3-6; 30 July/August 0 Federal Reserve Bank of St. Louis REVIEW

17 Close Research Focus Carlos Garriga Research officer and economist, Federal Reserve Bank of St. Louis htt://research.stlouisfed.org/econ/garriga/ Carlos Garriga s recent research has focused on house rice movements, mortgage finance, and the macro - economic imlications of housing. Past research has analyzed the issues of otimal fiscal olicy, the design of social security reforms, and the financing of education. Chao Gu Assistant rofessor of econonics, University of Missouri Columbia Research Focus Chao Gu s research focuses on money, banking, and credit markets. Her research analyzes bank runs, credit market cycles, ricing in the ayment system, and unconventional monetary olicy.