Sunspot Bank Runs and Fragility: The Role of Financial Sector Competition

Transcription

1 Sunspot Bank Runs and Fragility: The Role of Financial Sector Competition Jiahong Gao Robert R. Reed August 9, 2018 Abstract What are the trade-offs between financial sector competition and fragility when the potential for run behavior occurs with an exogenous, non-trivial probability? We study this question in a model of financial intermediation with limited commitment. When the potential for a crisis is relatively low, a monopolistic bank tends to be more stable than the competitive one since the monopolist is able to exploit its market power among investors and is less illiquid. As the potential for a financial crisis becomes more likely, the competitive bank becomes more prudent in altering its structure of liabilities while the monopolistic one takes on more risk to maximize expected profits. Therefore, either economy can be more susceptible to a financial crisis, but the monopolistic banking system tends to be less stable if the potential for run behavior is higher. Keywords: Bank Runs; Banking Concentration; Limited Commitment Department of Economics, Finance, and Legal Studies, University of Alabama, Tuscaloosa, AL 35487; jgao27@crimson.ua.edu; Phone: (205) Department of Economics, Finance, and Legal Studies, University of Alabama, Tuscaloosa, AL 35487; rreed@cba.ua.edu; Phone: (205)

2 1 Introduction In recent years, the degree of banking competition across countries has received a great deal of attention. For example, the banking sectors in some advanced economies such as Finland, Norway, and New Zealand are highly concentrated. The United States was once considered to have a relatively low concentration in comparison to many other nations. However, as documented by Janicki and Prescott (2006), during the past three decades, the banking system in the United States has become more concentrated. 1,2 Banking sectors in other developed economies have also experienced similar transitions. Moreover, available empirical evidence seems to suggest that concentration in the banking sector appears to be associated with less competitive outcomes. 3 Such variations in the competitive landscape of the banking sector have led to serious concerns about the relationship between banking competition and financial fragility. Notably, a large body of literature suggests that this relationship is complex. For example, one strand of empirical evidence favors the concentration-stability hypothesis. Notably, using data on 69 countries from 1980 to 1997, Beck et al. (2006) find that concentrated banking sectors tend to be more stable and argue that such intermediaries are protected by the monopoly rents that they obtain. Moreover, Bretschger et al. (2012) expand the number of countries to 160 for the period 1970 to 2009 and find similar results. In contrast, another strand of evidence supports the competition-stability hypothesis that challenges the stability effect of banking concentration. In particular, Boyd and De Nicolo (2005) demonstrate that the market power of large concentrated banks allows them to raise interest margins on loans. Such actions may induce adverse selection and moral hazard, undermining the stability of the banking sector. 4 Further, using data for 8,235 banks in 23 developed nations, Berger et al. (2009) obtain mixed results. Specifically, banks with a higher degree of market power have less overall risk exposure but market power can also increase loan portfolio risk. In light of these observations, the trade-offs between banking competition and fragility are potentially different under different market structures. For example, how does the possibility of a crisis affect a bank s optimal structure of liabilities? What are the implications of their ex- 1 According to Janicki and Prescott (2006), the largest banks, which make up less than 1% of active financial intermediaries in the United States, held over three-quarters of assets in the banking system. 2 See also Ennis (2001). 3 Numerous papers empirically study the implications of banking concentration for financial activity that may have resulted in less competitive consequences. For example, Hannan (1991) documents that banks in more concentrated sectors are more likely to be involved in some noncompetitive behavior. In particular, interest rates on loans depend on concentration ratios in markets. Corvoisier and Gropp (2002) conclude that less competitive pricing by banks may be the result of increasing concentration. In addition, Beck et al. (2004) find that the proxy of banking concentration is positively correlated with obstacles to obtain funding for firms across 74 countries. Demirguc-Kunt et al. (2004), in a similar manner, find evidence of positive association of banking concentration with higher net interest margins based on data on over 1,400 banks across 72 countries. 4 Similarly, De Nicolo and Lucchetta (2009) study a general equilibrium model and show that increased banking competition results in lower economy-wide risk. 2

3 ante actions for the buildup of banking vulnerabilities? How do different banking structures lead to different strategies in response to run behavior? The objective of this paper is to develop a framework that contributes to these important, yet rarely studied issues. Notably, we examine how the exogenous potential for a crisis affects a bank s optimal response and financial fragility in various banking systems. In particular, we compare a perfectly competitive banking economy to a fully concentrated one. To this end, our analysis is based on the classic paper of Diamond and Dybvig (1983) in which banks perform maturity transformation which leaves them illiquid. However, in contrast to their model, we consider that banks are potentially prone to a self-fulfilling run depending on the realization of a sunspot variable. Moreover, following Ennis and Keister (2009, 2010), we incorporate a limited commitment approach in which the bank will intervene optimally based on all information available. As a starting point, we begin the analysis by studying equilibrium outcomes among perfectly competitive banks. The objective of a competitive bank is to maximize the expected utility of a representative investor. In particular, the bank becomes cautious in allocating resources when the probability of a poor realization of a sunspot variable is relatively high. Such prudent behavior decreases an investor s incentive to run on a bank ex-ante, but would necessarily lead to underprovision of financial services. In addition, if investors can correctly anticipate that the bank is able to react quickly to poor sentiment, a self-fulfilling run might be eliminated in equilibrium. The intuition is that the bank can reschedule payments to the most appropriate level before a crisis deepens. In this manner, we fully characterize the conditions for the stability of a competitive bank. Further, in comparison to previous work such as Ennis and Keister (2010) and Keister (2016), we endogenize participation in the banking system. In particular, for given parameter values, there exists an upper bound on the probability of the panic state under which a competitive intermediary will emerge in equilibrium. Otherwise, the bank cannot deliver higher expected utility than autarky, and investors will choose not to deposit their endowments. To proceed, we turn to the analysis of a monopolistic banking economy. Unlike perfectly competitive banks, the risk-neutral monopolist earns positive expected profits. However, since the bank is assumed to earn zero profits once a crisis occurs, the banker focuses on maximizing monopoly rents from investors in normal times. In particular, when a run is unlikely, a bank s ability to exploit its market power to obtain rents is not significantly affected. As a result, the bank issues relatively smaller short-term obligations and realizes higher profits. Under this circumstance, the monopolist has more resources held in investment that can be used to increase payments in the long-term if the potential for run behavior emerges. By anticipating to receive a higher payoff, investors have less ex-ante incentives to withdraw early. In this sense, the stability of a monopolistic bank is augmented by its monopoly rents. In contrast, to respond to a higher probability of a crisis that would reduce its profits, the bank increases its short-term obligations. Though this aggressive 3

4 action increases its profits in good times, it unambiguously makes the bank become more prone to fragility. While the monopolist tends to implement more risky strategies at higher probabilities of a crisis, it is not necessarily more susceptible to fragility than its competitive counterpart. In particular, investors risk aversion seems to play an important role. Suppose, for example, that a run occurs at a certain probability and both banking sectors have the same speed of reaction to the crisis. As investors are more risk averse, they value risk-sharing more. As a result, under perfect competition, banks make larger payments in the short-term, which has a destabilizing effect on the banking system. By comparison, when facing sufficiently risk averse investors, the monopolist does not need to offer the same level of payments in the short-term to attract deposits. Consequently, its short-term obligations fall, and the bank obtains more monopoly rents which encourage stability compared to perfect competition. In addition to previously highlighted contributions that are most closely related to our work, other recent theoretical papers also study how banking structure matters for financial stability. For example, by constructing an overlapping generations model with random relocation, Boyd et al. (2004) demonstrate that when the rate of inflation is below some threshold, the probability of a banking crisis is higher under a monopolistic bank and vice versa. However, a banking crisis in their paper denotes a liquidity crisis in which banks have an inefficiently low level of money balances rather than a self-fulfilling run coordinated on the realization of a sunspot variable as in our framework. Papers closest to ours are Chang and Velasco (2001) and Matsuoka (2013). They study the existence of a bank run equilibrium based on the standard Diamond and Dybvig (1983) model, and both papers show that the condition for a run equilibrium to exist is less stringent for a competitive than for a monopolist bank. Though their question is similar to ours, their approach is not. In particular, they assume a crisis occurs unexpectedly and there are arbitrary restrictions on demand-deposit contracts. In contrast, a financial crisis occurs with a non-trivial probability and renegotiating obligations is allowed by the bank in our model. In this manner, we highlight the best responses of banks to a crisis and the potential for runs under payment schemes which adjust to available information at each point in time. Other papers study the relationship between banking structure and stability by incorporating costs of moving across banks in a version of the Diamond (1984) model. For example, Matutes and Vives (1996) construct a framework with portfolio risk where multiple equilibria emerge with different self-fulfilling beliefs among depositors regarding the possibility of bankruptcy. As they point out, their framework abstracts from the microeconomic foundations of financial intermediaries as in Diamond and Dybvig (1983) which make banks susceptible to fragility. Yet, along the lines of our work, they also show that even banks which act as local monopolies can be subject to 4

5 the possibility of bankruptcy. 5 The remainder of the paper is organized as follows. In the next section, we present the baseline model and discuss its key assumptions and the definition of financial fragility. In Section 3, we analyze the best-response allocation of a perfectly competitive bank and study equilibrium outcomes. In Section 4, we turn to the analysis of a monopolistic banking economy and show how the monopolistic intermediary can be vulnerable to a run. Finally, we compare these two banking systems in Section 5 and offer some concluding remarks in Section 6. Proofs of selected propositions are presented in Appendix. 2 The model The model follows Ennis and Keister (2010) which is based on a version of the Diamond and Dybvig (1983) model with limited commitment. We begin by describing the physical environment and the basic elements of the framework. 2.1 Environment There are three time periods indexed by t = 0, 1, 2. The economy consists of two types of agents: investors and bankers. In particular, the economy is populated by a 0, 1 continuum of ex-ante identical investors indexed by i. Investors have preferences characterized by: u (c 1, c 2 ; ω i ) = (c 1 + ω i c 2 ) 1, 1 where c t denotes private consumption in period t = 1, 2. For each investor, the parameter ω i which represents an investor s type and is privately observed is a random variable with support Ω {0, 1}. With probability π, an individual i is an impatient investor (i.e, ω i = 0) and values consumption in period 1; with probability (1 π), an individual i is patient (i.e, ω i = 1) and values consumption equally in both periods. In addition, due to the law of large numbers, the fraction of impatient or patient investors in the population is also π or (1 π). Following Diamond and Dybvig (1983), the coefficient of relative risk aversion is greater than unity. The population of bankers is given by N, which determines the competitive structure of the banking system. If N > 1, banks do not earn any profits since the banking system is perfectly competitive. Thus, in order to attract deposits, they must offer a contract that maximizes the expected utility of a representative investor. On the other hand, if N = 1, the banking system is 5 Allen and Gale (2004) also look at the connections between banking competition and fragility. However, in contrast to the previously mentioned papers, they do not formally model how banks attract deposits. Instead, they assume that banks face an upward-sloping supply curve of funds as a primitive in their model. 5

6 fully concentrated and the bank is a monopolist. In this case, the banker obtains positive expected profits. 6 Though the realization of expected profits depends on the competitive structure of the banking system, the preferences of all bankers are the same they are risk-neutral with preferences summarized by: v (Π; ω k ) = ω k Π = Π, where Π denotes a bank s consumption derived from expected profits. 7 For the bank k, the parameter ω k which represents a bank s type always takes the value of 1 since the bank s profit is realized from remaining revenues after banking contracts are made to all investors in period 2. As in other common approaches to equilibrium selection, we introduce an extrinsic sunspot signal on which investors can condition their actions, following Cooper and Ross (1998) and Peck and Shell (2003), among others. 8 There are two states of the economy, s S {α, β}, with probabilities {1 q, q}. The value of q is assigned strictly between 0 and 1, so the potential for a run occurs with a non-trivial probability. Though the sunspot variable is unrelated to fundamentals, it may affect investors withdraw strategies, which we will explain in more detail in the following section. The timing of events is summarized in Figure 1. In period 0, investors are each endowed with one unit of the all-purpose good. Next, they deposit all their endowments and banking contracts are set at this stage. In addition, the bank operates an investment technology in which one unit of the good invested yields R > 1 units of goods in period 2, but only one unit if liquidated early. At the beginning of period 1, investors choose to withdraw early or to wait until period 2 after learning her preference type and the aggregate state. As in Wallace (1988, 1990), the bank is subject to the sequential service constraint in which the bank pays investors who arrive one at a time in the order given by the index i which is assigned randomly at the beginning of period 0. In particular, a withdrawing investor i = 0 knows that she is the first one to withdraw in period 1 and investor i = 1 knows that she is the last one. Under the sequential service constraint, the bank can freely choose the amount to each investor, according to the objective of the bank. As in Ennis and Keister (2009, 2010), there is a lack of commitment, implying that the bank will choose contracts optimally given the situation at hand. 6 In the absence of switching costs or transport costs (as in Matutes and Vives (1996)) across different banks, Bertrand competition in the deposit market over deposit rates leads to the perfectly competitive outcome. This follows Salop (1976). While allowing for different levels of switching costs would produce intermediate levels of banking concentration, it is first useful to look at the different market structures with a relatively high degree of tractability by focusing on the two different cases of perfect competition and a monopolistic banking sector. 7 The assumption of risk neutrality of the banker or bank manager follows Cooper and Ross (2002), Boyd et al. (2004), and Ghossoub et al. (2012), among others. 8 An alternative approach to equilibrium selection is the global games technique which resolves the multiplicity of equilibrium. See also Goldstein and Pauzner (2005) and others for more information. 6

7 Figure 1: Timeline 2.2 Financial crises At the beginning of period 1, investors observe their own type ω i and the state of nature which banks may infer with a lag. Each investor makes her withdrawal decision to contact a bank in period 1 or to wait until period 2. Thus, for each combination of preference type and sunspot state, the function for a withdrawal strategy is: y i : {0, 1} {α, β} {0, 1}, where y i = 0 corresponds to withdrawing in period 1 and y i = 1 corresponds to waiting to withdraw in period 2. Let y denote a profile of strategies for all investors. Notably, the state s is thought to represent investor sentiment; it has no fundamental impact on the economy. As introduced by Ennis and Keister (2009, 2010), banks are able to react by rescheduling payments, and runs in this type of framework will be necessarily partial. However, it is assumed that the bank can infer that a run is underway from the flow of withdrawals. In particular, following Keister (2016) and Keister and Mitkov (2017), we study a setting in which the bank learns of the sunspot variable s with a delay θ (0, π. 9 Including the delay parameter is important as it allows us to investigate how equilibrium outcomes depend on the speed of the bank s reaction to a financial crisis. For example, if θ is arbitrarily close to zero, the bank learns of the sunspot state quickly right after investors do; if θ = π, the bank does not directly observe the sunspot state but infers it based on the fundamental withdrawal demand, which is equivalent to that in Diamond and Dybvig (1983) and Ennis and Keister (2009, 2010), among others. 9 When θ = 0, a run equilibrium does not exist. In this case, the bank learns of the sunspot state before any withdrawals taking place and will simply return each depositor their initial endowments. Please also see Keister (2016) for more information. 7

8 Without loss of generality, we assume that investors withdraw funds according to their preference types in which y i = ω i in state α, which is labeled the good state. Nevertheless, runs may occur in state β, which is called the panic state. We focus on actions taken by patient investors, since impatient investors only value consumption in period 1 and will choose to withdraw early certainly. In this manner, we consider the following partial-run strategy profile for investors: y i (ω i, α) = ω i for all i, and y i (ω i, β) = { 0 ω i } for { i θ i > θ }. (1) Under this specific profile, no one runs in state α but the patient investor with i θ chooses to withdraw in period 1 in state β. After θ withdrawals, runs will stop and all remaining patient investors wait to withdraw in period 2. We next summarize the formal definition of financial fragility below. Definition 1. The financial system is fragile if there exists an equilibrium strategy profile for a positive measure of investors in which y i (1, β) = 0. Fragility thus occurs if and only if there are some patient investors who withdraw early, and runs are triggered by self-fulfilling beliefs. 3 Perfectly competitive banks We begin our analysis with perfectly competitive banks. Since bankers are Nash competitors who do not earn any expected profits, their objective is to maximize aggregate welfare in order to attract deposits: W = 1 0 E u (c 1 (i), c 2 (i) ; ω i ) di. However, investors always have the option to invest on their own. Therefore, the competitive bank must offer sufficient incentives to attract these deposits. In other words, the intermediary faces a participation constraint, which guarantees investors to derive an expected utility higher than the autarky level. In addition, since there is a single asset, the portfolio choice for investing directly will be trivial. In other words, in the absence of financial intermediaries, individuals will place all initial endowments into the investment technology and liquidate their funds once subjected to the liquidity shock. Let u denote the reservation utility, which is characterized by: u πu (1) + (1 π) u (R). (2) 8

9 Investors will choose to deposit their endowments with banks if doing so yields a higher level of expected welfare: W u. (3) If the above inequality is reversed, investors are better off staying in autarky. Unlike previous work such as Ennis and Keister (2010) and Keister (2016), whether or not an investor will choose to participate in the banking system is an endogenous outcome of the model. Notably, in section 3.3, we demonstrate that under some restrictions on the probability q, a perfectly competitive bank will emerge in equilibrium. Thus, throughout the analysis below, we first assume that a bank is active in period 0. In the subsection that follows, we derive the best responses of banks to the strategy profile (1) by working backwards through two steps. In the first step, we investigate how intermediaries adjust their payments to depositors after the sunspot state is revealed. Subsequently, we look at what banks would choose to offer to the first fraction θ of investors. We then study fragility of the competitive banking system based on these best responses. 3.1 The best-response allocation As in Keister (2016), we first consider the allocation of remaining consumption after θ withdrawals have been taken by investors Ex-post efficient allocation of consumption The allocation for a bank j critically depends on how fast the bank is able to learn the state of nature. Before the state is revealed to the bank, it chooses to give the same level of consumption c j 1 to each withdrawing investor i θ where θ (0, π denotes the speed that the intermediary j becomes informed about the state. It is worth noting that in contrast to Diamond and Dybvig (1983), Ennis and Keister (2009, 2010), and Li (2017), the delay parameter θ implies the bank does not only rely on the fundamental withdrawal demand to infer the sunspot variable. In so doing, we aim to make banking arrangements fully contingent, allowing intermediaries to have increased flexibility to adjust payments. Once the bank observes the state s, it can calculate the fraction ˆπ s of these investors who are impatient and will withdraw in period 1 based on the strategy profile in (1). In particular, ˆπ s is either: ˆπ α π θ 1 θ or ˆπ β π. (4) Since there is no uncertainty at this point, the bank will be able to pin down the common amounts to remaining investors in both sunspot states in both periods. Specifically, the bank will 9

10 pay c j 1s to remaining impatient investors in period 1 while the choice of c j 2s is a pro rata division of remaining resources in that state. These ex-post payments will be chosen to maximize: V ( 1 θc j 1; ˆπ s ) max {c j 1s,cj 2s} (1 θ) ˆπ s u ( ) c j 1s + (1 ˆπs ) u ( c2s) j (5) subject to the resource constraint: (1 θ) ˆπ s c j 1s + (1 ˆπ s ) cj 2s R = 1 θc j 1 (6) and appropriate non-negativity conditions. Note that V ( ) denotes the value function for the expected utility from distributing the bank s remaining resources in each sunspot state. Let µ j s denote the multiplier on the resource constraint in the respective sunspot state s, the first order conditions are given by: u ( c j 1s) = Ru ( c j 2s) = µ j s for s = α, β, (7) which says that the bank will equate the marginal value of consumption for the remaining impatient investors to the marginal value of consumption for patient investors, depending on the rate of return, R. In other words, since the bank j infers the state and can reschedule payments, the intermediary can implement the first-best continuation allocation. In this situation, withdrawing in period 2 becomes a strictly dominant strategy for the remaining patient investors, and runs, if any, will stop. Similar to Keister (2016), an equilibrium bank run is partial and can only involve investors who can withdraw before the state is observed by the bank (i.e, i θ) Early payments We now move to the next step in which the intermediary j needs to decide how much consumption to give to each of the fraction θ of investors who withdraw before the state is revealed. In addition, we define the sum of aggregate utilities for all investors based upon the realization of the sunspot state s in the following way: W s θu (c 1 ) + V s (1 θc 1 ) for s = α, β. (8) Hence, to maximize the expected utility of a representative investor, the initial payment c j 1 will be chosen to solve: W max {c j 1} (1 q) W ( ) ( ) 1 θc j 1; ˆπ α + qw 1 θc j 1; ˆπ β (9) 10

11 It is worth noting that the expected welfare consists of two terms, and each of them denotes the expected aggregate utilities for investors in the sunspot state defined in (8). The first-order condition for this problem is: u ( c1) j = (1 q) µ j α + qµ j β, (10) which implies that the intermediary equates the marginal value of resources before it observes the state to the expected future value of resources taking into account all possible outcomes in both sunspot states. 3.2 Equilibrium and banking fragility The above analysis describes the best responses of a perfectly competitive bank j to the strategy profile in (1). Since all intermediaries face the same decision problem (5) and thus problem (9), we simplify the notation by omitting the index j. Therefore, the best-response allocation can be summarized by a vector: c {c 1, {c 1s, c 2s} s=α,β }. We provide an explicit derivation of this allocation of resources in the Appendix. We now verify whether the strategy profile in (1) is part of an equilibrium and, hence, whether the banking system is fragile. Again, we only focus on actions taken by patient investors since impatient investors will always choose to withdraw early. Moreover, as it is straightforward to show that c 2α > c 1 holds for all q > 0, patient investors will find it optimal to wait until period 2 in the good state. In state β, the bank implements the first-best continuation allocation (i.e, c 2β > c 1β ) once the sunspot state is revealed. The remaining patient investors do not withdraw in period 1. However, a patient investor with early withdrawal opportunity i θ would receive c 1 if she runs and c 2β if she leaves her funds at the bank. Consequently, whether she chooses to withdraw early and thus the intermediary is said to be fragile depends on if c 1 c 2β holds. As a starting point, to understand intuitively the competing effects on banking stability, we write c 1/c 2β as: c 1 c 2β c 1 c 2α c 2α. c 2β Unlike Li (2017) that examines the relationship between interest rates and fragility, we focus on the effects generated by the probability q and the bank s response speed θ. The following proposition summarizes these results. Proposition 1 (Payments of perfectly competitive banks). Suppose a competitive intermediary exists in equilibrium. 11

12 (i) c 1 c 1α q < 0, c 1 c 2α q < 0, c 2α c 2β q = 0, and c 1 c 2β q < 0. (ii) c 1 c 1α θ < 0, c 1 c 2α θ < 0, c 2α c 2β θ > 0, and c 1 c 2β θ > 0. Proofs of all results are presented in the Appendix unless otherwise noted. Notably, increases in q and θ exhibit two different patterns depending on how the bank distributes consumption across the different groups of investors. On one hand, when the potential for a crisis becomes more likely, the bank reduces the earliest payment relative to the later payments in normal times. Thus, as q increases, banks set aside more income for investors to be paid upon the realization of the aggregate state. In turn, by holding back resources to the earliest investors, the shadow values of income in each aggregate state would adjust. Yet, payments c 2α and c 2β also depend in a significant way on the fraction of remaining patient investors in either state. Consequently, the spread between them depends on the true degree of liquidity risk rather than the ex-ante probability of the crisis state. Together, the combination of these two competing effects implies that the susceptibility to crises would be weaker. On the other hand, if the intermediary is slower to observe the sunspot state, this potentially increases the ex-ante incentives for patient investors to run. Notably, the bank provides less risksharing to the earliest investors in such settings since the bank knows that its reaction to a crisis is not as fast. This response leads to a lower payment ratio c 1/c 2α, which encourages investors to leave their deposits at the bank. However, the bank increases the spread between c 2α and c 2β in which an investor receives relatively less consumption in state β. Since some of the funds were distributed to patient investors who run in state β in period 1, there are relatively less resources held to maturity which earn productive returns. The later the response, the more this misalignment of payments is exaggerated among banks. Such an effect encourages patient investors to withdraw early we show that this effect dominates that through the ratio c 1/c 2α, which unambiguously increases the ratio c 1/c 2β. In what follows, we examine the explicit role of q and θ on fragility of a perfectly competitive bank The role of q and θ for fragility We begin by focusing on the role of q in determining banking fragility. To see precisely how susceptible the intermediary is to a run, we define a measure of financial fragility, taking into account the speed of the bank s reaction to a crisis. Definition 2. For a competitive banking economy, let q θ be the maximum value of q with which 12

13 a run can occur in equilibrium. In particular, q θ satisfies c 1 ( q θ ) = c 2β ( q θ). If c 1 < c 2β holds for all values of q, we then define q θ = 0. As shown in Proposition 1, banks respond to a higher probability of the panic state by lowering the ratio c 1/c 2β. In this manner, we demonstrate that when state β is highly likely to emerge, withdrawing early is no longer an equilibrium behavior for patient investors because intermediaries offer less incentives for investors to run. Moreover, banking fragility also significantly depends on the fraction of impatient investors in the economy and the speed of a bank s reaction to a run. We summarize our results in the next proposition. Proposition 2 (Maximum probability in which runs occur in equilibrium). Let R α be defined as R α ˆπ α R 1 + (1 ˆπ α ). Also, let R β ˆπ β R 1 + (1 ˆπ β ). Next, suppose a competitive intermediary is active in equilibrium. Further, let π (R 1) θ + 1 R 1 {R 1 (1 θ) (R 1 1 1)}, and θ ( R 1 1 ) (R 1) 1. (i) If θ < θ π < π, then q θ ( ) Rβ = R 1 Rα ( Rβ 1 Rα remains stable. If q q θ, the bank is fragile. ). Under this condition, if q > q θ, the bank (ii) Otherwise, q θ = 0. Under this condition, the competitive bank is always stable. The proposition pins down the maximum probability q θ > 0 in which runs can occur in equilibrium. In particular, suppose that the proportion of impatient investors is relatively high (i.e θ < π < π). As the intermediary anticipates a run will arise at a higher probability (i.e q > qθ ), it adjusts the payment scheme such that the payment ratio c 1/c 2β drops, as established in Proposition 1. In particular, the ratio falls below 1. Therefore, if q is sufficiently high, waiting to withdraw is the dominant strategy for all patient investors, and as a result, the banking system is not fragile. However, if a crisis is less likely to emerge (i.e q q θ ), the bank provides more risk-sharing in the short-term. Since the payoff from withdrawing early remains more attractive (i.e c 1/c 2β > 1), the intermediary becomes more susceptible to a run in the crisis state. On the other hand, if there are only a few impatient investors as in case (ii), the intermediary can observe the aggregate state quickly regardless of its response speed. As the bank still has sufficient resources left, the late payment c 2β is anticipated to be relatively high. Thus, investors 13

14 will always choose to leave their deposits in the banking system and, hence, the bank is not fragile if π is sufficiently small. The lemma below examines how the measure of fragility q θ is affected by parameters in the environment. Lemma 1 (Comparative statics for q θ ). q θ θ > 0, q θ π < 0, and q θ > 0. Banking stability can vary drastically depending on parameter values. Specifically, if the intermediary observes the aggregate state slowly, the bank has less resources left upon the realization of the sunspot state. As established in Proposition 1, the ratio c 1/c 2β increases as the misallocation of resources due to private information over the realization of depositors liquidity shocks is exaggerated. In this sense, the bank becomes more prone to a run, implying that q θ is strictly increasing in θ. By comparison, we find that increasing π may actually reduce the ex-ante incentives for patient investors to run. In particular, for a fixed delay parameter θ, if there are more investors with a real need to consume early, this would be equivalent to the fact that an intermediary reacts to a crisis relatively more quickly. In this manner, the ratio c 1/c 2β decreases, suggesting that the bank becomes less vulnerable to instability and, hence, q θ will be strictly decreasing in π. 10 Nevertheless, we find that investors incentives to withdraw early are stronger when they are more risk averse. That is, as investors value risk-sharing more, the intermediary makes a larger payment in the short-term, which has a destabilizing effect on the banking sector. We next look at the relationship between q θ and R. Since we are unable to proceed analytically, we present a numerical example that shows their relationship can be non-monotone in some cases. 11 First of all, suppose that the bank s response speed is relatively slow (i.e, θ = 0.3 and π = 0.4). As shown in Figure 2, q θ is strictly decreasing in R. As the intermediary expects to observe the aggregate state slowly, it offers a lower level of risk-sharing to investors. By waiting to withdraw, patient investors would obtain a higher payment if R is higher. Consequently, the range of q where the economy is fragile is smaller. 10 It is worth noting that this result is in stark contrast to that in Ennis and Keister (2010) and Li (2017). In their papers, the intermediary does not have the ability to observe the sunspot before π. When π is large, the bank realizes that runs are occuring relatively late. In this manner, the incentive for patient investors to run becomes stronger when π is larger. 11 Li (2017) provides a detailed discussion about the relationship between interest rates and fragility in an environment with multiple assets. However, in contrast, the bank has the ability to observe the sunspot at θ π in our model. Hence, the relationship between q θ and R is different from that in Li. 14

15 Figure 2: The relationship between q θ and R Yet, the effect of lower values of θ may appear to be counterintuitive. That is, when θ is relatively low (i.e, θ = 0.2), an increase in R may make the economy more vulnerable to a run. 12 Specifically, if the intermediary has the increased ability to react to a crisis fast, it issues relatively larger short-term liabilities. Thus, when R is relatively low, an increase in the return could lead to a higher ratio c 1/c 2β. Thereby, withdrawing early becomes more attractive for patient investors. By comparison, if the return R is high enough, an increase in R leads to a decreasing ratio c 1/c 2β, and the incentive to run falls. We proceed to look at the explicit impact generated by θ. We now define θ q to be the minimum level of the response timing to a run, which can also serve as a natural measure of financial stability. Definition 3. For a competitive banking economy, let θ q be the minimum value of θ such that θ q satisfies c 1 ( θq ) = c 2β ( θq ). If c 1 (θ) < c 2β (θ) holds for all values of θ, θ q does not exist. We derive a precise cutoff value of the bank s speed to react to a panic in which the bank is not fragile. That is, if θ < θ q, the banking system is immune to instability. Otherwise, the bank is susceptible to a self-fulfilling run because it responds slowly. Similar to the insights developed in Proposition 2, the cutoff value θ q is non-trivial if and only if the proportion of impatient investors is large enough. We summarize this result in the following Proposition. Proposition 3 (The role of the speed of the competitive bank s reaction to a crisis). Suppose a competitive intermediary exists in equilibrium. Let π 1 ) 2 ) ˆψ (ˆψ 4 (R 1 1 (ψ 1) 2 (R 1 1 1), 12 Similar results can be shown by fixing the delay parameter θ while increasing the fraction of impatient investors, π. In this manner, increasing π would also imply that the intermediary could react to a run quickly and hence makes a larger short-term obligation compared to that when π is low before considering the effect of the return R. 15

16 where ˆψ ) ) (ψr 1 1 (R 1 1 (ψ 1), ψ (R 1 q) 1 (1 q) 1, and q < R 1. (i) If π π 1 and q < R 1, then θ q competitive bank is fragile. (ψ 1) R β. Under this condition, if θ θ ψr 1 q, π, the R β (ii) Otherwise, θ q does not exist in which θ q stable regardless of θ. / (0, π. Under this condition, the bank is always It is worth noting that q < R 1 must hold otherwise θ q may not exist. That is, as implied in Proposition 2, the intermediary becomes sufficiently conservative when q is high enough. At the same time, if the reward for waiting, R, is sufficiently large, patient investors will never find it attractive to misrepresent their type and consume early. In general, the result in case (i) of the proposition demonstrates that θ plays a non-trivial role if and only if π is relatively large (π π 1 ). Otherwise, when π is low, regardless of the speed of its reaction, the intermediary can infer a run before a large number of patient investors withdraw early. In this sense, the bank is not vulnerable to a panic. First of all, suppose that θ q exists. If the intermediary is slow to observe the aggregate state (θ > θ q ) in which its ability to reschedule payments is compromised, the ratio c 1/c 2β is strictly increasing in θ as shown in Proposition 1. In this case, the bank has less resources available ex-post, thus increasing the susceptibility of the banking system to a run. By comparison, if the bank infers the sunspot state in an early stage of a crisis, the desirable effect associated with payment rescheduling is pronounced. That is, the bank can correct the misallocation of resources quickly. On the other hand, as in case (ii), θ q does not exist in equilibrium in which the intermediary is always stable regardless of speed of its reaction to a run. To develop more insights, please see Figure 3 which depicts c 1/c 2β as a function of θ at different values of q. Both curves are upward-sloping, as implied in Proposition 1. It is also worth noting that the bank is less vulnerable to a run if the probability of state β is more likely. In turn, the ratio c 1/c 2β is higher when the probability q is low. To see the existence of θ q, suppose that the bad realization of the sunspot state occurs at a low probability (i.e, q = 0.1). The ratio c 1/c 2β represented by the solid line is less than 1 when θ is small, and as a result, the banking system is stable. If a bank reacts to a run more slowly, there exists a cutoff value of θ above which the economy is fragile. In contrast, since the bank becomes sufficiently cautious in altering its obligations as the probability of the panic state is highly likely (i.e, q = 0.625) in which the ratio c 1/c 2β is denoted by the dashed line, the bank is immune to instability, and in this manner, θ q does not exist. 16

17 Figure 3: c 1/c 2β as a function of θ at different values of q We next look at how θ q changes according to parameter values in the environment. The results are presented in the following: Lemma 2 (Comparative statics for θ q ). θ q q > 0 and θ q π > 0. The intuition behind Lemma 2 goes as follows. Since the bank operates in a conservative manner as q increases, it delivers lower ex-ante utility to the first θ investors. As the ratio c 1/c 2β decreases, the scope for the bank to respond in order to maintain stability becomes larger. Therefore, θ q increases as the probability of state β is more likely. Further, if more investors have a real need to consume early, the range for the bank to react to a run becomes larger too. It is worth noting that in the presence of θ, an increase in π is equivalent to the fact that an intermediary could make a relatively faster reaction to the potential for financial distress, which promotes stability of the banking sector. By comparison, θ q is strictly decreasing in. The logic follows that of Lemma 1 that the intermediary becomes more susceptible to a crisis and, hence, if the bank does not react quickly, the banking system will be fragile. In addition, we also show numerically in Figure 4 that the relationship between the return R and the cutoff value for θ may be non-monotone. Again, this pattern occurs when there is an incentive for the intermediary to offer a relatively higher short-term payoff. To highlight such insights, we first look at the bank s behavior when the probability of state β is low (i.e, q = 0.01), in which values for θ q are represented by the solid line. In this case, the intermediary issues a higher shortterm payment, leading to an increase in the ratio c 1/c 2β as R increases. This generates a stronger incentive for patient investors to withdraw early since the late payment may not be attractive when R is low (i.e, R < 1.45). However, as long as the return R is sufficiently high (i.e, R 1.45), the ex-post payment to patient investors in the panic state becomes higher, and the ratio c 1/c 2β is strictly decreasing. In this manner, investors have sufficient incentives to wait to withdraw, thereby 17

18 implying that θ q is increasing in R. In contrast, whenever the bank prepares well for the emergence of a run (i.e, q = 0.2), the earliest payment c 1 would be relatively low. Hence, an increase in R always leads to a decrease in the ratio c 1/c 2β in which banks are less prone to a run. This is depicted by an increase in the cutoff value for θ as the return R increases from the dashed line. Figure 4: The relationship between θ q and R Based on the analytical results of the role of q and θ for fragility presented above, we also find it useful to study the fragile set of economies characterized by full parameters e (R, π,, θ, q). Let Φ denote the set of economies that are fragile in a perfectly competitive banking system. For example, Figure 5 below depicts the set Φ as the parameters q and θ are varied with three different values of. There are two general patterns from the figure. Notably, for a given value of q, the cutoff value of θ needs to be large enough for the bank to be fragile. When the speed of the bank s reaction to a crisis is slow, losses created by runs are larger, and the payment to a patient investor waiting until period 2 eventually drops to be lower than the payment from joining the run, c 1. In addition, the cutoff value for θ increases with q, as established in Lemma 2. If the probability of state β increases, banks tend to operate in a conservative manner. This cautious behavior makes the bank less susceptible to fragility, and the economy admits higher cutoff values of θ. Notably, when q is sufficiently large, all economies e are immune to fragility. Further, we show that the economy becomes more prone to fragility as investors are more risk averse. This result can be seen as we move from case = 4 to cases = 6 and = 8 in which the fragile set becomes larger. However, it is worth emphasizing that the scope for investors to participate in a financial system is also larger The maximum probabilities of state β such that a perfectly competitive bank exists in equilibrium are , , and for cases = 4, 6, and 8 when θ = 0.5, respectively. We will explain the upper bound for the probability q in more detail below. 18

19 Figure 5: The fragile set Φ 3.3 Welfare analysis To study equilibrium welfare in a perfectly competitive banking system, it is helpful to substitute the best-response payments summarized by the vector c, which are functions of q, into equation (9). In this manner, equilibrium welfare can be rewritten as: W (q, θ) = (1 q) Wα (1 θc 1) + qwβ (1 θc 1), where Ws is defined by (8). We demonstrate some important properties of welfare in the next proposition. Proposition 4 (Properties of the expected welfare of a competitive bank). Expected welfare W of a perfectly competitive bank is strictly decreasing in q. The intuition behind Proposition 4 is straightforward. As some of the funds are received by patient investors, the intermediary needs to meet remaining real withdrawal demand. Hence, when the panic state occurs, the bank uses more resources in period 1 than does it in the good state. Consequently, the amount of assets held in investment that earns the higher rate of return R is lower in state β. By comparison, the intermediary owns a larger pool of resources that provides more ex-post utility in state α. In other words, runs in state β distort a bank s desirable risksharing. In this manner, upon the realization of the sunspot state, the total expected utilities from the remaining allocation, which is referred to as the first-best continuation allocation, is higher in state α than in state β: V α (1 θc 1) > V β (1 θc 1). (11) 19

20 Moreover, V β (1 θc 1) is lower if state β is more likely to occur. Consequently, the ex-ante expected utility is strictly decreasing in probability q. In a similar manner, a decrease in the speed that a bank observes the sunspot also negatively affects expected equilibrium welfare. The reason is intuitive: the earlier the intermediary is able to infer the state, the more the appropriate level of remaining payments adjusted by the bank, which unambiguously augments expected welfare. Further, as W (q) decreases in q, the following Proposition sheds some light on the existence of a competitive intermediary. Definition 4. A perfectly competitive banking economy is active if and only if the participation constraint (3) holds. In contrast to previous work such as Ennis and Keister, Keister, and Li, we endogenize participation in the banking system. Proposition 5 (The existence of a perfectly competitive bank). Let θ E ( {R 1 π + (1 π) R 1 } 1 R β ) (R 1 R β ) 1. If θ E θ π, then there exists a cutoff probability q (0, 1) such that a perfectly competitive bank is active in equilibrium if and only if 0 < q < q. Otherwise, q does not exist in the sense that the existence of the intermediary in equilibrium is independent of q. Notably, when the probability of state β is relatively small, a perfectly competitive bank will be active in period 0 in the sense that the intermediary can provide a higher expected utility than autarky. The intuition is straightforward. As established in (11), a representative investor expects to receive lower welfare in state β than state α. In this manner, when the probability q is higher, the aggregate expected welfare may fall below utility from autarky in which an intermediary is redundant. In addition to the expected welfare, the first-best allocation is achievable no matter how nature draws the state if the run equilibrium in state β is eliminated. Nevertheless, if withdrawing early is the unique equilibrium outcome in state β, different patterns for the realized welfare may arise. To see more insights, suppose c 1/c 2β 1 holds. The results are illustrated using numerical examples in Table We first study the role of q by looking at case (a). If q is higher, the bank would hold more resources prior to the realization of the aggregate state. Because less risk-sharing occurs for the first withdrawing investors in which the ratio c 1/c 2α declines as shown in Proposition 1, W α is decreasing in q. By comparison, in the panic state, though runs have occurred, the intermediary s earliest conservative payment scheme results in more remaining resources after it observes 14 Note that the autarky level of utility in these cases is and q = when θ =

21 the sunspot in the economy. In this manner, the ex-post aggregate utilities in state β would be increasing in q but all are evidently inferior to autarky. Furthermore, as illustrated in case (b), if the bank cannot observe the sunspot quickly, the best-response allocation makes the aggregate welfare in both states lower and vice versa. q c 1/c 2β Wα Wβ W θ c 1/c 2β Wα Wβ W (π, R,, θ) = (0.5, 1.2, 8, 0.5) (π, R,, q) = (0.5, 1.2, 8, 0.1) (a) (b) Table 1. The ex-post welfare of a competitive bank 4 A monopolistic bank In this section, we study equilibrium outcomes in a fully concentrated banking economy in which the population of bankers is equal to unity, N = 1. Unlike competitive banks whose objective is to maximize the expected utility of a representative investor, the monopolistic banker k maximizes its expected profits: Z = E v (Π (k) ; 1). Similarly, the bank faces a participation constraint, which guarantees investors to derive an expected utility higher than the autarky level in the sense that the sum of investors expected utilities is: Ŵ = 1 0 E u (ĉ 1 (i), ĉ 2 (i) ; ω i ) di = u. (12) In what follows, we derive the best responses of a monopolistic bank to the strategy profile in (1) by working backwards through two steps. We begin by characterizing the intermediary s optimal choice of payments once the state is revealed. We next examine how the monopolist decides the earliest payments to investors i θ in order to maximize its expected profits. We subsequently study the potential for fragility based on the intermediary s chosen obligation scheme. 21

22 4.1 The best-response allocation We first focus on the allocation of remaining consumption after θ withdrawals have been taken Ex-post profit maximization allocation Similar to competitive banks, the monopolistic bank is assumed to observe the sunspot state with a lag θ (0, π. To start, we look at how the intermediary decides the allocation of remaining consumption to maximize realized profits. Again, after the sunspot state has been observed, there is no uncertainty at this point, and runs, if any, have stopped. Hence, the monopolist can calculate the fraction ˆπ s of remaining impatient investors who will withdraw in period 1 as established in (4) based on the strategy profile (1). Note that since the monopolistic banker k is the only intermediary available to investors, we simplify the notation for payments by omitting the index k. To begin, let Π (1 θĉ 1 ; ˆπ α ) represent the amount of expected profits a bank would earn in the good state. In particular, bank profits depend on the return to productive investments, R, and the losses for payments to investors. We note that the total payments to investors in period 1 are θĉ 1 + (1 θ) ˆπ α ĉ 1α. However, if instead these payments could have been made in period 2, the intermediary would not have needed to liquidate its investment in the productive technology. Thus, the total cost of these payments is R θĉ 1 + (1 θ) ˆπ α ĉ 1α. In this manner, ĉ 1α and ĉ 2α are chosen to solve: Π (1 θĉ 1 ; ˆπ α ) max R }{{} {ĉ 1α,ĉ 2α } investment return θrĉ 1 + (1 θ)ˆπ α Rĉ 1α }{{} capital loss of early liquidation (1 θ)(1 ˆπ α )ĉ 2α }{{} total payments in period 2 (13) subject to the participation constraint: (1 q) Ŵ (1 θĉ 1; ˆπ α ) + qŵ (1 θĉ 1; ˆπ β ) u (14) and appropriate non-negativity conditions. Note that Ŵ s denotes the ex-post aggregate welfare based upon the realization of the sunspot states: Ŵ s (1 θĉ 1 ) θu (ĉ 1 ) + ˆV s (1 θĉ 1 ), where, in a similar manner as in perfect competition, ˆV ( ) represents the value function for the utilities the monopolistic intermediary delivers to the remaining (1 θ) investors in either sunspot state: ˆV s (1 θĉ 1 ) (1 θ) ˆπ s u (ĉ 1s ) + (1 ˆπ s ) u (ĉ 2s ) for s = α, β. (15) As in (2), the monopolistic institution offers a level of risk-sharing that has a lower bound. Fur- 22

23 ther, each investor is assigned individual index i randomly at the beginning of period 0. Therefore, with probability θ, investors are among the first θ individuals who can receive ĉ 1. With probability (1 θ), investor i withdraws after the intermediary has observed the sunspot. In particular, in the good state, the investor could obtain ĉ 1α if she chooses to withdraw early or ĉ 2α in the long-term. In contrast, in the panic state, the investor can consume ĉ 1β in the short-run or ĉ 2β if she waits until period 2. Let ˆµ α denote the multiplier on the participation constraint in state α, the first order condition is given by: u (ĉ 1α ) = Ru (ĉ 2α ) = R (1 q) ˆµ α. (16) It is straightforward to show that the values of both ĉ 1α and ĉ 2α are strictly increasing functions of (1 q). That is, if the probability of state α is more likely, the intermediary responds by allocating more remaining consumption in that state. In state β, as some of the funds are received by patient investors in period 1, the intermediary will need to choose to give ĉ 1β to each remaining impatient investor and ĉ 2β to each remaining patient one. Again, these payments will be chosen to maximize expected bank profits. Note also that the profit function is similar to that in (13) except the fraction ˆπ β of remaining investors who are impatient after the bank infers the sunspot state is different from that in state α. Hence, the intermediary s profit maximization problem becomes: Π (1 θĉ 1 ; ˆπ β ) max {ĉ 1β,ĉ 2β} R θrĉ 1 + (1 θ) ˆπ β Rĉ 1β (1 θ) (1 ˆπ β ) ĉ 2β subject to the participation constraint, (14), and also non-negativity conditions. Let ˆµ β denote the multiplier on the participation constraint in the panic state, the first order condition is given by: u (ĉ 1β ) = Ru (ĉ 2β ) = R qˆµ β. (17) Notably, the wedge between the payment to impatient investors and to patient ones is the same under monopoly versus competition in all sunspot states. Interestingly, the intermediary would choose to obtain zero profits in the crisis state: Π β (1 θĉ 1 ) = (18) It might initially appear counterintuitive that it can be optimal for the monopolistic intermediary 15 A monopolistic intermediary s objective is to maximize its expected profits. In this manner, when runs occur, the bank s best response would be to suspend convertibility in order to avoid liquidation to preserve its capital gains. However, throughout the paper, we do not allow the bank to carry out the suspension of withdrawals unilaterally as in Cooper and Kempf (2016). Instead, it would choose to allocate payments such that the intermediary earns zero profits. 23

24 to forfeit all capital gains during a financial crisis. During normal times, only investors with a real need to consume early withdraw in period 1, and the intermediary could earn higher profits. However, in the crisis state, runs have occurred which causes the bank to earn lower profits than in state α. In this manner, the monopolist would choose to offer zero profits in the bad state doing so, by virtue of the participation constraint, would raise its profits in the good state. Since it is reasonable that the probability of a crisis is low, this strategy by the bank raises its expected capital gains. Further, ex-post payments in state β equal the remaining resources after θ withdrawals have been taken: (1 θ) ˆπ β ĉ 1β + (1 ˆπ β ) ĉ2β = 1 θĉ 1, R where the payments in state β are set according to (17). Notably, the above constraint is similar to (6) under perfect competition An incentive to react aggressively Moving backwards before the aggregate state is revealed, the monopolistic bank must choose how much consumption to pay to each of the fraction θ of withdrawing investors. In particular, the initial payment ĉ 1 will be chosen to maximize expected profits according to the optimal choice of remaining payments in (16) and (17): Z max (1 q) Π (1 θĉ 1; ˆπ α ) + q Π(1 θĉ 1 ; ˆπ β ). (19) {ĉ 1 } }{{} zero profits The first term corresponds to expected bank profits in state α in which the state occurs with probability (1 q). The second term represents the bank s capital gains in state β, however, the intermediary is expected to earn zero profits in this state. The first order condition for this problem requires: u (ĉ 1 ) = Rˆµ α. (20) Notably, compared to condition (10) in which all possible outcomes in both states are considered by competitive banks, condition (20) highlights the monopolistic bank s risk-taking strategy. That is, the monopolist ignores the value of resources in the event of a run. 16 The behavior is optimal for 16 It is worth noting that as in Keister (2016), when the government provides public assistance during a financial crisis, competitive banks have a similar incentive distortion created by the bailout policy. That is, intermediaries rationally ignore the value of resources in the run-state. However, as we show in the current paper in which the government does not bail out banks in an event of a run, the monopolistic bank can also find it optimal to ignore the consequences of its actions in state β. The reason lies in the fact that in the panic state, expected bank profits will be 24

25 the intermediary because it only earns positive expected profits during normal times. The larger the probability of a run, the greater the distortion will be on the earliest payments. It is also worth noting that the marginal value of early consumption is inversely related to the shadow value of utilities to investors after the state is revealed to be good. That is, if investors receive lower remaining welfare in the good state, which corresponds to a higher ˆµ α, the intermediary needs to allocate a larger payment to the earliest investors accordingly. In addition, using equations (16) and (20), we find the condition that satisfies ĉ 1 < ĉ 2α is: q < R 1 R. (21) Hence, we restrict our attention to this situation (21) in which a patient investor would prefer to wait until period 2 if all other patient investors choose to do so. In this manner, the investors strategy profile (1) is consistent with the equilibrium. Thereby, throughout the analysis of the monopolistic intermediary, we assume that the probability of the panic state is relatively low, and the partial-run strategy profiles in state α are ruled out. 4.2 Equilibrium and banking fragility The above analysis focuses on the best responses of a monopolistic bank to the strategy profile in (1), and this best-response allocation is summarized by a vector: ĉ {ĉ 1, {ĉ 1s, ĉ 2s} s=α,β }, in which we provide an explicit derivation of this allocation of resources in the Appendix. In particular, the components of this vector are jointly determined by equations (14) - (20). We now turn to our primary interests by verifying whether the strategy profile (1) is part of an equilibrium and, hence, whether the monopolistic intermediary is fragile. Similarly, we focus on actions taken by patient investors in state β. Again, a patient investor i θ receives ĉ 1 if she misrepresents her type to withdraw or ĉ 2β if she waits until period 2. Hence, patient investors will only find it attractive to withdraw early if ĉ 1 ĉ 2β. We begin by analyzing the underlying driving sources of banking fragility. We summarize the results in the following proposition. Proposition 6 (Payments of a monopolistic bank). (i) ĉ 1 ĉ 1α q > 0, ĉ 1 ĉ 2α q > 0, and ĉ 1 ĉ 2β q > 0. independent of its own choice of ĉ 1. Moreover, the level of risk sharing provided by the monopolistic intermediary is lower than in Keister since the monopolist only offers minimal returns to attract deposits. 25

26 (ii) ĉ 1 ĉ 1α θ = 0 and ĉ 1 ĉ 2α θ = 0. Apparently, an increase in q leads to a stronger incentive for investors to run on the bank. In particular, when the potential for a crisis is high, the intermediary issues a relatively larger shortterm obligation but will lower late payments to maximize expected profits, thereby raising the ratios ĉ 1/ĉ 1α and ĉ 1/ĉ 2α. Such effect builds up the bank s vulnerability to a run since withdrawing early becomes more attractive. At the same time, as the intermediary has paid out a larger payment in the short-run, the bank s remaining resources for the late payments in state β would drop, which increases the spread between ĉ 2α and ĉ 2β. In this manner, the susceptibility of the bank to a crisis would be stronger. While we cannot analytically show that the spread increases, please see panel (a) of Table 2 on the next page where we provide numerical examples in support of our claim. Combining both effects suggests that the monopolistic bank would be more prone to fragility. Notably, this is in stark contrast to the results presented in Proposition 1 in which a perfectly competitive bank is less vulnerable to a crisis as q increases. We next focus on the effects from the speed of the bank s reaction to an incipient crisis. If θ is higher, the intermediary makes payments to a larger fraction of investors before the realization of the sunspot state. However, expected bank profits in periods of financial distress are independent of the intermediary s own choice of ĉ 1. In this manner, both payments of ĉ 1 and ĉ 1α or ĉ 1 and ĉ 2α would depend significantly on the common marginal value of utility in state α. Hence, changes in the bank s response speed would not change the ratio between ĉ 1 and ĉ 1α or ĉ 1 and ĉ 2α. This behavior for the payment ratios is different from perfect competition. That is, as competitive intermediaries provide less risk-sharing services to the earliest investors, the ratio of ĉ 1/ĉ 2α or ĉ 1/ĉ 2α would decrease in θ. On the other hand, since the late payment in state β depends on all remaining investments and the monopolistic bank has less resources when responding late, the spread between ĉ 2α and ĉ 2β increases, which is presented in panel (b) of Table 2. Consequently, the composition of competing effects implies that the monopolist becomes more susceptible to a run as the bank infers the aggregate state slowly, with the ratio of ĉ 1/ĉ 2β strictly increasing. In this manner, such findings are similar to that under a perfectly competitive bank. Further, we next compare panel (b) to panel (c), which demonstrates the impact of θ on the bestresponse payments offered by competitive intermediaries. 26

27 q ĉ 1/ĉ 1α ĉ 1/ĉ 2α ĉ 2α/ĉ 2β ĉ 1/ĉ 2β θ ĉ 1/ĉ 1α ĉ 1/ĉ 2α ĉ 2α/ĉ 2β ĉ 1/ĉ 2β (π, R,, θ) = (0.5, 1.2, 8, 0.5) (π, R,, q) = (0.5, 1.2, 8, 0.1) (a) (b) θ c 1/c 1α c 1/c 2α c 2α/c 2β c 1/c 2β (π, R,, q) = (0.5, 1.2, 8, 0.1) (c) Table 2. The role of q and θ in risk-sharing under monopoly and perfect competition We find that when investors are sufficiently risk averse (i.e. = 8), the ratio c 1/c 2β is strictly greater than ĉ 1/ĉ 2β, implying perfect competition would be more vulnerable to a crisis than the one with market power as θ increases. Moreover, due to the differences in market power, the increase in the payment ratio in response to later timing is higher under perfect competition. In what follows, we examine in detail how q and θ play a role in affecting the bank s structure of liabilities and stability The role of q and θ for fragility We begin by looking at the explicit role of q for fragility. As implied by Proposition 6, unlike that for a competitive bank, an increase in q is associated with a monopolistic banking system that is more susceptible to fragility. We next demonstrate an important result about the monopolistic intermediary s vulnerability to a crisis. 27

28 Proposition 7. There does not exist a maximum probability above which the bank with market power remains stable in equilibrium. The reason is straightforward. As the panic state becomes highly likely, a monopolistic institution ends up issuing a large payment before the sunspot state is revealed. Thus, when the state of nature turns out to be good, the bank realizes the maximum expected profits given the situation at hand. Nevertheless, such aggressive action makes itself more prone to a run in state β. Further, given that the bank takes on more risk, there exists a threshold probability below which a monopolistic intermediary remains stable. We will show this property through illustrations of the different fragile sets below in detail. We next ask the following question: how does the speed of the bank s reaction to a run impact the stability of the financial system? Due to the highly non-linear system of equations from the bank s best responses, we are unable to proceed analytically to characterize the general case for the cutoff value for the speed of the bank s reaction to an incipient crisis. Therefore, in the following limited case, we show that there exists a critical value ˆθ 0 when the probability q is arbitrarily small. Under this condition, if the bank reacts to a crisis slowly (i.e, θ ˆθ 0 ), the monopolistic bank will be fragile. We summarize these results in the following Proposition. Proposition 8 (The role of the speed of the monopolistic bank s reaction when the probability of a run is negligible). Suppose that the probability q is arbitrarily close to zero and a monopolistic bank exists in equilibrium. Let χ (i) If χ R π (R R ) β condition, if θ { π + (1 π) R 1 π + (1 π) R } R 1, β then ˆθ0 ˆθ0, π, the monopolistic bank is fragile. (R χ R β ) (R R β ) χ 1. Under this (ii) Otherwise, ˆθ 0 does not exist in which ˆθ 0 regardless of θ. / (0, π. Under this condition, the bank is stable Insights for the measure ˆθ 0 are similar to those for its analogue demonstrated under perfect competition. That is, if the bank can correct the misallocation of resources due to private information over the realization of the liquidity shock among depositors in a timely fashion, the intermediary is not fragile. Further, if ˆθ 0 does not exist in equilibrium, the intermediary would remain stable, and the speed of its reaction to a run does not affect banking fragility. To see more precisely, we illustrate the fragile set ˆΦ in a monopolistic bank as the parameters q and θ are varied with three different values of in Figure 7. Notably, there are two general patterns 28

29 from the cases illustrated in the figure. Again, for a given value of q, the cutoff value of θ needs to be large enough for the banking system to be fragile. Another pattern is that the cutoff value of θ decreases with q. That is, as q approaches the bound in equation (21), the monopolistic bank increases its short-term obligations so that it can maximize expected bank profits in state α. However, this higher return, ĉ 1, makes withdrawing early more attractive to a patient investor, leading to a more fragile banking system. Thus, the economy e admits lower cutoff values of θ. In addition, when probability q is sufficiently small, the cutoff value of θ for all three cases does not exist. Alternatively, when the protential for a run is more likely, the monopolist s risk-taking strategy makes the bank become susceptible to a run. This contrasts sharply to the situation under perfect competition. That is, there exists a cutoff probability q θ above which the competitive banking system is immune to fragility, as discussed in Proposition 2. Interestingly, when investors are more risk averse, the economy becomes less prone to fragility in which the set ˆΦ shrinks as we move from case = 4 to cases = 6 and 8. The reason is, when facing sufficiently risk averse investors, the monopolistic intermediary does not need to offer the same level of payments in the short-term to attract deposits. The bank sets lower obligations to the earliest investors, which increases banking stability. Figure 7: The fragile set ˆΦ 4.3 Welfare analysis We begin to conduct welfare analysis for a fully concentrated bank. As a starting point, we look at the expected utility for an investor. 29

30 4.3.1 Welfare for investors Notably, the expected utility for an investor in a monopolistic intermediary is exactly the same as the reservation utility u, which is independent of q and θ. This is also true for the ex-post welfare to investors, which equals the autarky utility, if the run equilibrium is eliminated. Nevertheless, suppose that partial runs are an equilibrium outcome in state β. Though the intermediary uses more resources to augment payments in the panic state, interestingly, the aggregate ex-post welfare will always be higher in state α. The results are summarized in the following proposition. Proposition 9 (Properties of ex-post welfare with a monopolistic bank). (i) For any e where ˆΦ =, Ŵ α = Ŵβ = u. (ii) For any e where ˆΦ with q > 0, Ŵ α (1 θĉ 1 ) > u > Ŵβ (1 θĉ 1 ). To see more precisely, we focus on the latter situation when the run equilibrium occurs in the panic state, and total welfare Ŵs will be based on the realization of the sunspot signal. Numerical examples are provided in Table It is worth noting that, when q is higher, but the aggregate state is revealed to be good, total welfare will be higher than would have occured in the bad state as the bank will have more resources available to pay the remaining (1 θ) depositors. Interestingly, similar results are obtained if the delay parameter θ is large. Thus, together, we see that Ŵ α is an increasing function of q and θ. Notably, such results for the utilities to investors offered by a monopolistic bank differ sharply from those by a competitive bank. By comparison, these results in Table 1 show how Wα is strictly decreasing in q and θ. In addition, as panels (a) and (b) of Table 3 show, bank profits drop in both cases. On the other hand, given these banking arrangements, if nature chooses state β, investors welfare is negatively affected by both q and θ. Specifically, the intermediary s early aggressive response to an increase in q leaves less resources available and, hence, makes the ex-post utilities worse. The same is true if the bank reacts to a run slowly in which the crisis has deepened. 17 Note that the autarky level of utility in these cases is , which is also the expected welfare for investors. 30

31 q ĉ 1/ĉ 2β Ŵα Ŵβ Π α θ ĉ 1/ĉ 2β Ŵα Ŵβ Π α (π, R,, θ) = (0.5, 1.2, 8, 0.5) (π, R,, q) = (0.5, 1.2, 8, 0.1) (a) (b) Table 3. The ex-post welfare in a monopolistic bank In what follows, we turn to examine the monopolistic banker s utility Welfare for a monopolistic banker We next turn to the analysis of utility for a monopolist. Notably, given the profit-maximizing choice of the monopolistic bank, the intermediary may not exist under some situations. In particular, the residual revenues are negative if the payments in both periods are set to be sufficiently high. Note that all banking contracts contingent on the sunspot states can be considered as functions of the earliest payment ĉ 1. Moreover, the bank earns positive expected profits if and only if: (1 q) Π (1 θĉ 1 ; ˆπ α ) > 0, which reduces to: ĉ 1 < R Rθ + (1 θ) R α R (1 q) c1, (22) where c 1 represents the threshold or the cap of the earliest payoff. That is, if the earliest payment to the first θ withdrawals does not exceed this threshold, the remaining payments can meet the participation condition. Otherwise, the bank does not exist in equilibrium, and investors are better off investing directly on their own. Definition 5. Given the bank s best-response scheme, a monopolistic bank exists in equilibrium if it earns positive expected profits, and the participation condition by investors is not violated. We then present an important property of the bank s expected profits and the existence of the intermediary. 31

32 Proposition 10 (The behavior of expected bank profits and the existence of the bank). Expected bank profits are strictly decreasing in the probability q. In this manner, there could exist a maximum value of the probability ˆq such that a monopolistic intermediary exists in equilibrium. In particular, ˆq is defined by condition Π α (1 θĉ 1 (ˆq)) = 0. That is, if q > min { R 1, ˆq}, the bank R will not emerge. As previously stated, the monopolist earns zero profits in the panic state. In response to an increase in q, the bank ignores state β and best responds by increasing payments to the earliest investors i θ. At the same time, the monopolist has to meet the investors participation constraint. Since the bank only earns profits from investment assets that are held to maturity, a larger short-term liability ĉ 1 leads to less resources that are kept in investment long enough to obtain the higher rate of return, R. In this manner, when the potential for a crisis becomes more likely, the resource allocation will become more distorted from the bank s perspective, thereby lowering the intermediary s expected profits. As q becomes large enough (i.e, q approaches to ˆq if ˆq < R 1 R ), anticipated bank profits drop to zero, and the bank will not operate in period 0. Furthermore, to understand the effects of q and θ on the monopolistic banker s utility, we illustrate a set of examples in Figure 6. In panel (a), the bank s capital gains in the good state are strictly decreasing in q, as established in Proposition 6. Moreover, when investors are less risk averse (i.e, = 3), the bank may not exist if the probability q is relatively high in which profits are non-positive and thus the participation constraint is violated. In this manner, the figure also shows how ˆq is determined as the point where Π α crosses 0. In contrast, when investors are more risk averse, a monopolistic intermediary exists in equilibrium for all q < R 1 and earns higher levels R of profits compared to when is low. In panel (b), we show the speed of the bank s reaction can also play an important role. When the institution has the ability to react quickly, a bank s earnings may still be positive. Otherwise, the late response of rescheduling payments may result in negative profits. In this manner, there could exist a maximum value of the bank s response speed ˆθ, above which the intermediary is not active in equilibrium, and the figure also shows the existence of ˆθ. 32

33 Figure 6: The existence of a monopolistic bank 5 The impact of market structure on fragility and welfare The analysis in the previous two sections has illustrated the best responses of financial institutions under distinct competitive structures during normal times and financial crises. We next turn to the comparison of fragility of both banking sectors, providing some analytical results followed by examples. We consider two major issues. The first is the following: How does the probability of the panic state lead to different behavior across banking systems? As we show, the evolving patterns of a bank s short-term liabilities to investors are distinct across the competitive landscapes. We then turn to study our primary interest: which financial sector is more vulnerable to a crisis? In the case of θ = π, we prove that the fragile set of a monopolistic bank is strictly contained in that of a competitive bank if the probability of state β is arbitrarily small. Simply put, the monopolistic intermediary is able to exploit its market power among investors and, hence, is less illiquid. As q increases, however, there exist some economies that the competitive bank is more stable than the one with market power which can become more illiquid. In this manner, there may exist a cutoff probability under which the monopolist is less prone to a run. To make these points, we also provide a set of illustrative numerical exercises. In addition to the major results demonstrated above, we find that investors risk aversion is likely to play an important role. If investors are highly risk averse, in particular, we illustrate that the fragile set of perfectly competitive banks is strictly larger than the set of the monopolistic one. Hence, our results show the non-trivial trade-offs between banking competition and instability. Further, either economy can deliver higher aggregate welfare, depending on the realization of the sunspot signal and model parameters. 33

34 5.1 Comparison of banking illiquidity and fragility We now compare the critical measure that affects financial fragility across the two banking structures. That is, we start by analyzing the ex-ante incentives for runs by investors in both economies. To this end, following Keister (2016), we define a natural measure of illiquidity, denoted by ρ and ˆρ for competitive banks and the monopolistic bank respectively. Specifically, the degrees of illiquidity are: ρ θc 1 and ˆρ θĉ 1. It is worth noting that the measure represents the short-term liabilities of the banking system in which a fraction θ of investors will withdraw c 1 or ĉ 1 before their banks react to an incipient crisis. Alternatively, the total payout to the investors before the sunspot state is revealed is interpreted as the total short-term liability. To compare illiquidity across economies, we first show that two completely different patterns arise. The following proposition summarizes how the probability of state β affects bank s illiquidity. Proposition 11 (Properties of banking illiquidity across systems). Suppose that banks are active in equilibrium in both sectors. ρ is strictly decreasing in q for all q < q while ˆρ is strictly increasing in q for all q < R 1 R.18 As shown in the proposition, one notable feature of the short-term liabilities of a competitive bank is that, as the probability q increases, the bank reduces its short-term payments to the earliest investors. For ˆρ, on the other hand, an opposing pattern arises. Unlike competitive banks, a monopolist increases its short-term liabilities and thus becomes more illiquid as the probability q increases. Notably, the monopolistic bank can be more illiquid than the perfectly competitive one if investors are less risk averse. Moreover, the speed of the bank s reaction is also likely to play an important role. These results are illustrated in Figure 8, which depicts the difference in illiquidity across the two financial sectors as a function of q and θ, respectively. Starting from panel (a), when the probability q is small, we see the competitive bank would be more illiquid compared to the monopolistic one, while the short-term liabilities can be larger under a monopolistic bank too. As a crisis becomes more likely, the monopolistic bank becomes more illiquid, and eventually ˆρ > ρ as the lines representing the difference in illiquidity exceed zero. Further, as can be observed in the figure, this only occurs when investors are less risk averse; the illiquidity of the competitive banks dominates that of the fully concentrated one. Panel (b) plots the illiquidity differential between the competitive structures as a function of the speed of the bank s reaction to a crisis. For example, suppose that a run occurs at a certain probability. When θ is arbitrarily small, the measure for illiquidity is negligible. As θ increases, 18 Please note that q was introduced earlier in Proposition 5. 34

35 the competitive bank is more illiquid as its earliest payments are larger. However, both banking sectors decrease their short-term obligations, and c 1 falls by a larger amount than ĉ 1. For higher values of θ, however, the competitive intermediary eventually becomes more cautious while the monopolist only decreases the short-term liabilities slightly in order to realized larger expected profits in state α. The pattern of difference in illiquidity begins to reverse; a monopolistic bank becomes more illiquid as θ increases. Figure 8: Comparing the degree of illiquidity We now ask: given the different behavior for illiquidity, which banking sector is more vulnerable to a financial crisis? We demonstrate that the implications for fragility critically depend on the probability q. In particular, the monopolistic intermediary tends to be less susceptible to a financial crisis when the potential for a panic is unlikely, but this result can be overturned as q increases. Proposition 12 (Comparison of banking fragility across systems). Let θ = π. If q is arbitrarily small, Φ ˆΦ. As the Proposition states, when the probability q is small enough, the competitive bank holds less resources in reserve as a provision against run behavior this is because its payments to the earliest depositors who have the ability to withdraw funds before the bank realizes that the bad state has occurred would be higher. That is, although a crisis is rare, the bank s ability to adjust payments to remaining patient investors will be compromised once runs are underway. This leaves less income available for the patient investors, increasing the vulnerability of the banking system to a run. On the other hand, since the monopolistic bank issues less short-term liabilities, ĉ 1, the payment to a patient investor from waiting to withdraw, ĉ 2β, remains at a higher level even if some investors run on the bank. The economy is then not prone to a run and, hence, the fragile set ˆΦ is strictly contained in Φ. By comparison, once q becomes relatively large, the scope for fragility shrinks 35

36 in a competitive bank, since the bank allocates its liabilities in a sufficiently conservative way. The monopolistic bank, on the other hand, increases short-term payments in response to a larger probability of state β in order to maximize expected profits in state α, leading to a widening set of fragility. In this sense, there exist economies e such that the fragile set Φ is strictly contained in ˆΦ. Such result conveys a message that a monopolistic bank is more stable when the probability of the panic state is relatively small. Nevertheless, the net effect of the different incentives associated with the best responses under the two different financial sectors for banking fragility is ambiguous. When q is relatively small, the advantage of banking stability is apparent in a monopolistic banking economy, while the competitive banks tend to be more stable once q is large. Yet, the latter may not be the case if investors are sufficiently risk averse. To consider how model parameters impact the intermediary s vulnerability to a crisis when comparing two different structures, we construct a number of numerical examples. Interestingly, we find that, if investors are not highly risk averse (i.e, (1, 2), both the competitive and monopolistic banks are immune to banking fragility if q is low enough. Table 4 illustrates the results. 19 q c 1/c 2β ĉ 1/ĉ 2β Table 4: The banks susceptibility to crises when is small We also compare the fragile sets of the competitive banks and the monopolistic one when q < min { R 1, ˆq}, as can be observed in Figure 9. We first focus on panel (a). When q is R small, the cutoff value of θ for fragility to arise is much higher under a monopolistic bank than a competitive one. Economies in the lightly shaded region on the left side of the graph are not fragile under a monopolist but the competitive banks are unstable. In this sense, when a crisis is unlikely, a competitive bank is more susceptible to a self-fulfilling run. As the probability of state β increases, the incentive to obtain higher profits in state α causes the cutoff value for θ to decrease under a monopolistic bank as the bank increases its short-term potential obligations. On the other hand, the competitive bank reacts in an opposite way by reducing the degree of illiquidity to account for both sunspot states. When the probability of the panic state is large enough, for economies in the medium-shaded region in the lower right-hand corner of the panel, a monopolistic bank is more fragile than a perfectly competitive one. 19 The other parameter values are given by (π,, R, θ) = (0.5, 2, 1.3, 0.5). Under this set of parameters, ˆq = at which the monopolistic intermediary earns zero expected profits. Moreover, note that we consider the worst case scenario in which θ = π. That is, suppose that the bank infers the sunspot state based on the fundamental withdrawal demand. Under this condition, if the intermediary is not prone to fragility, the bank will also remain stable if it has the ability to reveal the sunspot signal before π. 36

37 When the coefficient of relative risk aversion is sufficiently large, as in panel (b), we see that ˆΦ is strictly contained in Φ, implying that the competitive bank is more fragile than the monopolistic banking economy. The competitive bank provides a relatively large payment to the earliest investor i θ. The bank thus is more illiquid and has less resources which shrinks the consumption level (ĉ ) 1β, ĉ 2β when some investors misrepresent their type and withdraw early. With fewer assets left, sufficiently risk averse investors have a stronger incentive to join the run. However, the payoff to the earliest investors is smaller in a monopolistic bank. When a crisis is unlikely (i.e, q is less than approximately 7.5%), the best-response scheme of a monopolist leads to an economy which is immune to fragility. Since only a small proportion of resources are paid to investors i θ, a substantial high level of resources would be left. As the bank uses all resources to provide risksharing services, investors then have a small incentive to withdraw early. 20 Figure 9: Comparing the sets Φ and ˆΦ 5.2 Welfare comparison A perfectly competitive intermediary maximizes the expected utility of investors, taking into account the exogenous probability of state β while investors are always kept indifferent to an autarky level in a monopolistic bank. In this manner, a competitive bank would deliver higher expected welfare than a bank with market power. Nevertheless, we find that the ex-post utilities for investors can be higher under either banking sector. We next summarize some of the results in the following proposition. Proposition 13 (Implications for expected and ex-post welfare across systems). 20 As in Keister (2016), the set-theoretic approach to measuring fragility in terms of changes in the probability of a run is helpful in this type of comparison of fragility across structures. The ex-ante probability assigned to a run by this process can be strictly lower under either financial sector, implying the tradeoff between banking concentration and fragility. 37