Financial Stability. Xingpeng Wei. University of Missouri. November 26, Abstract

Transcription

1 Capital Requirement, Bank Competition, and Financial Stability Xingpeng Wei University of Missouri November 26, 2017 Abstract In this paper, I examine the tradeoff between bank competition and financial stability resulting from the capital requirement. Built on the framework of iamond-ybvig, the model shows that a higher capital requirement tightens banks capacity for taking deposits, thus reducing the intensity of competition between banks and at the same time improving financial stability. However, the total effect of capital requirement on welfare is not monotone. JEL Classification: 82, E58, G21 Key words: Capital requirement, bank run, capacity constraint, subgame perfect Nash equilibrium, bank competition, financial stability I am especially grateful to Chao Gu for invaluable guidance and numerous discussions. Many thanks to Ron Harstad and Joseph Haslag for comments and suggestions. I also want to thank avid Mandy, Oksana Loginova, and Shawn Ni for helpful advice. All errors are my own. Correspondence: epartment of Economics, University of Missouri, Columbia, MO , USA. xwwmc@mail.missouri.edu. 1

2 1 Introduction What level of capital ratio should banks be required to keep? It has never been easy to answer this question. Although it is widely agreed that more capital will make individual banks as well as the banking system safer, 1 "the more the better" is not the answer to the question, given that tradeoffs between stability and liberalization may exist when higher capital is required. That is why a look back at the history shows banking policy has been repeatedly punctuated by the "regulation-liberalization" pattern. As banks become larger, this tradeoff between stability and liberalization becomes more important, since large banks keep lower capital to asset ratios than their medium and small counterparts, and at the same time their failure tends to be more disruptive. 2 The great recession caused by the failing financial system invoked stricter capital regulations through the Basel III Accord. New standards of complex capital requirement have been established, and capital requirement is much more stringent than before. 3 As the Basel III capital requirement phased in, 4 according to 1 It is widely agreed that higher bank capital can improve financial stability. Capital works on stability in two ways: Ex ante, it adjusts a bank s incentive to take risk, and Ex post, it works as cushions to protect depositors. With higher capital, the probability of banking failure is lowered, and what is more, post-crisis recovery may speed up. (See Holmstrom and Tirole 1997, Coval and Thakor 2005, and Mehran and Thakor 2011, among others). 2 The banking industry has become more concentrated with fewer and lager banks over the years, partly because large banks can achieve the economies of scale and can diversify their portfolio relatively easily, and partly because bank consolidation through mergers and acquisitions happening after financial crises reduced the number of banks and increased concentration in bank sector. Large banks tend to keep lower capital ratio than small and medium banks since they can diversify their risk easily, and they know they will be bailed out if they become insolvent since they are "Too big to fail". See Vives 2011, 2016 among others. Figure 9 in Appendix shows the historial concentration trend of bank sector in the whole world, as well as in several individual countries including the United States. 3 Compared with Basel II, required common equity increases from 2% to 4.5%, core Tier 1 ratio increases from 2% to 5%, and overall Tier 1 capital increases from 4% to 6%. It also requires banks to hold a 2.5% capital conservation buffer, as well as a countercyclical buffer ranging from 0% to 2.5%. 4 According to the newest report issued by the Basel Committee on Banking Supervision, a sample of 105 banks have Tier 1 capital more than 3 billion and are internationally active (categorized as group 1 banks) as well as 95 other banks (group 2 banks) are implementing Basel III following the arrangement, encompassing all economically important countries and regions. See "Basel III Monitoring Report, September 2017," by the Basel Committee on Banking Supervision. 2

3 the World bank data all the three indices used to capture bank competition intensity in current literature (i.e., the Lerner Index, the H-statistic, and the Boone Indicator) show that across the world, bank competition intensity decreased. 5 In this paper, I build a model based on the iamond-ybvig framework with bank capital to explain the tradeoff between bank stability and competition. Banks compete for depositors by offering the best deposit contract given their competitor s strategy. However, the total deposits a bank can take is capped by its own capital and the capital requirement. I show that as capital requirement increases, it reduces banks capacity of taking deposits and consequently, competition among the banks. epositors bid up the price and get a less favorable contract in normal times. On the other hand, the set of the contracts that admits bank runs shrinks. In the event of bank runs, depositors bear smaller losses. The effect of capital on this tradeoff can be decomposed into two parts. One is direct, since more capital provides depositors more insurance in crisis times. The other is indirect through its effect on the capacity of banks. With reduced capacity due to increased capital requirement, banks provide less favorable contracts, but these less favorable contracts provide better welfare in run time for depositors. When combined, the effect is not monotone on the welfare and there is a range of the capital to deposit ratio given which depositor s welfare increases with the ratio in crisis times at the cost of reducing welfare in normal times, but there is a threshold of capital ratio above which no improvement can be made. In the literature, there are two opposite views regarding the relationship between bank competition and financial stability. One is the "competition-fragility" view, holding that more bank competition will erode banks market power and decrease their profit margins, thus encouraging banks to take more risk on the asset side (see Keeley 1990, emsetz, Saidenberg, Hellmann, Murdock, and Stiglitz 2000, Repullo 5 See Appendix for more explanation on the indices. 3

4 2004, and Allen and Gale 2004, among others). The "competition-stability view, on the other hand, holds that with higher market power, higher interest rates are charged on the loan market; thus the moral hazard problem of borrowers is exacerbated and projects on the asset side of banks balance sheets become riskier. Thus, there is a positive relationship between competition and financial stability. (See, for example, Boyd and e Nicolo 2005, e Nicolo and Lucchetta 2009). More recent literature tries to accommodate both views considering correlation between loan defaults. (See Martinez-Miera and Repullo 2010 among others). To the best of my knowledge, all literature regarding bank competition and financial stability focuses on the asset side of a bank s balance sheet and addresses banks risk taking in the loan market. In contrast to the existing literature, I illustrate a competition-stability tradeoff in the deposit market. This is not to say that other frictions, such as asymmetric information and moral hazard, are not important. The point is even in the absence of these common frictions, capital requirement has a profound effect on competition and stability through the capacity constraint. To better expose this specific effect of capital requirement, this paper suppresses all other frictions except for the private information of depositor s consumption type, which is commonly used to generate panic runs in the iamond- ybvig framework. The conclusion of this paper is not to suggest that only panic run is to be considered when capital requirement is being set; rather, it is about the notion that the effect of capital requirement on capacity constraint of banks should not be ignored when considering optimal capital requirement policy. The rest of the paper is structured as follows. Section 2 introduces the environment of the economy. Section 3 describes the depositor s problem; section 4 describes the bank s problem. Section 5 solves the equilibrium. Welfare of normal and run time under different capital requirements are discussed in section 6, with a numerical example to visualize the results in section 7. Section 8 concludes. 4

5 2 Environment The economy has 3 periods: t = 0, t = 1, and t = 2. There are two large identical banks, bank I and bank II, as well as a continuum of depositors normalized to 2. Each depositor is endowed with 1 unit of consumption good; after normalization, they possess 2 measures of consumption goods in total. Each bank is endowed with K measures of consumption goods, which is its capital. 1 The consumption good can be stored costlessly across periods or invested in production by banks. epositors however, only have storage technology. Productive investment needs to be made in t = 0. One unit consumption good invested in t = 0 will grow to R in t = 2, R > 1. If it is liquidated in t = 1, the return is r = 1. Capital requirement θ is set by the government. In the real world, capital ratio is the fraction of assets that is supported by a bank s own capital, and capital requirement is the minimum capital ratio a bank needs to keep. In this paper for math simplicity, I use capital to debt ratio, which is equivalent to capital ratio. 2 For every unit of deposit accepted by a bank, the bank needs to have at least θ unit of capital to support the deposit. Thus K/θ is a bank s capacity for taking deposits. As in the iamond-ybvig framework, in t = 0, depositors are identical, and each depositor has probability α of becoming an impatient consumer in t = 1. An impatient consumer has to consume in t = 1 and derives no utility from consuming in t = 2. A patient consumer consumes in t = 2 and derives no utility in t = 1. epositor s type is private information. By the law of large numbers, α fraction of 1 This endowed capital assumption is made on the theoretical work as well as empirical fact that when a higher minimum capital ratio is required, banks are reluctant to issue new equity and new equity building takes time. Instead, they adjust their portfolio, or even reduce their asset size to meet the new requirement. According to the pecking order theory, raising equity is the last method of getting additional funds. And it is also true for bank capital. Research shows that banks were reluctant to build new equity when minimum capital becomes higher. See Jackson et al. 1999, Bernanke and Lown 1991, King 2001, Brana and Lahet 2006; PwC, 2015, "That Shrinking Feeling: Tracing the Changing Shape of the European Banking Industry," and irk Schoenmaker 2017, "What happened to global banking after the crisis?" 2 If capital to debt ratio is θ, capital ratio is θ/(1 + θ). 5

6 consumers are impatient, and 1 α fraction of consumers are patient. Let C 1, C 2 be the amount of consumption available to a consumer in t = 1 and t = 2, respectively. An impatient consumer s utility is described as u(c 1 ), and a patient consumer s utility is described as u(c 2 +C 1 ). The utility function u is strictly increasing, concave, and normalized so that u(0) = 0. Besides, relative risk aversion coeffi cient of the utility function satisfies cu (c)/u (c) > 1. In this environment, a bank can provide a contract that aggregates depositors consumption risk and provides a smoother consumption profile. Bank i s contract is a consumption plan c i = (c i 1, c i 2) that is contingent on when a depositor withdraws. If a depositor deposits her endowment to bank i in t = 0, bank i promises to provide her c i 1 if she withdraws in t = 1, and c i 2 if she withdraws in t = 2. In t = 2, banks realize total profit. A depositor makes a deposit decision in t = 0 to maximize her ex-ante expected utility in t = 0. Then she makes withdrawal decisions in the subsequent periods. Banks, as well as depositors, do not expect a panic run in t = 0. So in what follows, the strategies of depositors and banks are considered, assuming no run given an incentive-compatible contract. The timing is as follows: t = 0: Subperiod 1: Banks decide what contracts to offer. Subperiod 2: epositors make deposit decision. Subperiod 3: Banks make investment. t = 1: Nature draws a type for each depositor according to probability α. After depositors learn their types, impatient depositors withdraw and consume. At the same time, patient depositors choose to wait. t = 2: Patient depositors choose to withdraw and consume. Banks realize profits. The equilibrium concept I use is the subgame perfect Nash equilibrium. I focus on symmetric equilibria in which depositors of the same type adopt the same strategy. 6

7 Because banks and depositors do not expect bank runs to happen in t = 0, banks and depositors make their decisions based on the assumption that all patient depositors will wait. However, we know that as in the iamond-ybvig framework, in the post deposit game there is a run equilibrium under some contracts, in which all depositors withdraw in t = 1. If run happens, depositors welfare will be reduced severely. Also, banks will suffer loss because they need to take out some capital to serve depositors. Capital requirement helps improve run-time welfare of depositors; however, it may cause banks to suffer more and also may have negative effects on normal-time welfare. 3 epositor s Problem In t = 0, depositors make the decision regarding whether to deposit and to which bank to deposit to maximize the expected utility. Assuming no run happens given an incentive-compatible contract, a depositor withdraws in t = 1 if impatient, and waits until t = 2 to withdraw if patient. So I will focus on the strategies in t = 0. For an individual depositor, a strategy in subperiod 2 of t = 0 is a complete plan of distribution of probabilities (p I, p II, p A ) regarding depositing at bank I, bank II, or staying in autarky, given the contracts offered by banks I and II as well as other depositors strategies. Suppose each depositor can only choose from depositing her unit of good at one of the banks or not depositing so staying in autarky. If a bank cannot serve all the depositors who are willing to deposit, then each depositor will have some probability of not being served and ending up being in autarky. A depositor s payoff is the ex-ante expected utility that she realizes through the contract she chooses, given the strategy of banks and other depositors. enote the strategy of other depositors by (P I, P II, P A ). By the law of large numbers, if every depositor has probability of P i depositing to bank i, then the measure of population willing to deposit at bank i is 2P i. 7

8 It is noteworthy that when a depositor chooses at which bank to deposit, what she really cares about is the expected utility provided by the contract. Clearly, there are different contracts that provide same expected utility, and depositors strategies are no different under these contracts. Thus to simplify, I only discuss situations where the expected utility achieved by a contract is different, instead of where contracts are different. Let w A be depositors utility in autarky; that is, w A = αu(1) + (1 α)u(1) = u(1). The expected utility yielded by a contract of bank i is a function of (c i 1, c i 2): w i = αu(c i 1) + (1 α)u(c i 2). Let = K/θ be a bank s capacity. Given as well as expected utilities yielded by banks contracts w I, w II, and other depositors strategies (P I, P II, P A ), an individual depositor chooses (p I, p II, p A ) to maximize her payoff. Let W i denote the expected utility that a depositor can get if she chooses to deposit to bank i, then W i = 2P i wi + (1 2P i )wa w i if if 2P i < 1 2P i 1. If she decides to stay in autarky W A =w A. Given the utility under the contracts provided by the banks (w I, w II ), a depositor s problem is max p I W I + p II W II + p A W A (1) p I, p II s.t p I + p II + p A = 1. If W I > W II W A, a depositor chooses to deposit at bank I. If W II > W I 8

9 W A, the depositor chooses to deposit at bank II. If W I = W II W A, mixed strategy is adopted. 4 Bank s Problem Banks offer an incentive-compatible contract and expect no panic run. The contracts are decided in the first subperiod of t = 0. Each depositor can either deposit 1 or 0, and thus a bank s capacity is the amount of deposits, as well as the measure of depositors the bank can take. Let d i be the total amount of deposits that bank i attracts from the market, then d i depends on w I and w II, as well as depositors depositing strategy (P I, P II, P A ), which ultimately also depends on w I and w II. Thus d i is a function of w I and w II. If more than the capacity of depositors are willing to deposit at the bank, the bank has to turn away some depositors. Suppose banks value their reputation, and turning away depositors will harm their reputation very much and they never want to turn a depositor away. For simplicity, assume the cost of turning away a depositor is infinitely high for both banks so banks never want to attract more deposits than their capacity 6. Given θ and K, and thus, banks design contracts to make profits. Their payoff is the total profit realized in t = 2. A contract of a bank (c 1, c 2 ) yields an ex-ante expected utility level w for a depositor and a unit profit for the bank. The bank s expected profit on each unit of deposit accepted is (1 αc 1 )R (1 α)c 2, which is realized in t = 2 after patient depositors withdraw. For every level of w, there is only one contract by which unit profit is maximized. 6 The assumption that banks cannot turn away a depositor is made so that a pure strategy equilibrium exists when 1 < < 2. It will be shown later that within 1 < < 2, if a bank provides a better contract than its competitor, it will attract more deposits than its capacity. If banks can turn away depositors, then as long as the contract offered by the competitor does not yield the highest possible utility, a bank always has an incentive to offer a slightly better contract. However, if both banks provide contracts yielding the highest possible utility, both banks have an incentive to deviate to a slightly worse contract to make positive profit. There will be no pure strategy equilibrium. 9

10 Given that depositors only care about utility, this contract will be the contract that a bank provides if it determines to provide w. All other contracts providing the same utility, although equally attractive to depositors, are strictly dominated for the bank because they yield lower unit profits. enote the unit-profit-maximizing contract by (c 1(w), c 2(w)), and the maximized unit profit by π(w). The contract that reaches π at expected utility level w can be obtained from: max (1 αc c 1, c 1 )R (1 α)c 2 (2) 2 s.t w = αu(c 1 ) + (1 α)u(c 2 ). F OC yields u (c 1 ) = Ru (c 2 ). Let w be the highest possible utility for depositors, as in the iamond-ybvig framework w = max c 1, c 2 αu(c 1 ) + (1 α)u(c 2 ) s.t (1 αc 1)R 1 α = c 2. (i.e, π = 0). For any w [w A, w ], there is a unique contract (c 1(w), c 2(w)) that brings a bank the highest unit profit π(w). This uniqueness can be seen from the following proposition. Proposition 1 The maximized unit profit π is strictly decreasing in w, and c 1, c 2 are both strictly increasing in w. Proof. See Appendix. 10

11 Bank i s total profit is Π i = π i (w i )d i (w i, w i ) if d i if d i >. Bank i s problem is max Π i (3) w i s.t. w i w A (4) c i 1 c i 2 (5) Equation (4) is the participation constraint which says a contract that a bank provides should yield at least the same expected utility as autarky. Equation (5) is the incentive compatibility constraint that makes patient depositors willing to wait until t = 2. When w = w A, denote π A = π(w A ), it is the highest profit a bank can get per unit of deposit. Easily seen, π(w) [0, π A ], where w [w A, w ]. 5 Equilibrium of the Game Given, a symmetric equilibrium of the whole game is a profile of strategies (w I, w II, (p I, p II, p A )), where both banks and depositors have their payoff maximized. Under different capacity constraints, equilibrium will be different. I categorize into three cases. Case 1 is when 2. In this case, both banks are able to serve the whole market. Case 2 is when 1 < 2; that is, when banks can serve more than half, but not the whole market, and they together can serve the whole market. Case 3 is < 1; in this case the two banks together are unable to serve the whole market. Because banks are identical, w I w II is assumed without loss of generality. 11

12 Case 1 is the case where capacity constraint does not bind for either bank. This is the classic Bertrand competition model without constraint; the equilibria are (w I = w II = w ; p I [0, 1], p II = 1 p I ). Banks make zero profit, and depositors get the highest utility. Because both banks are able to take the whole market, a depositor can get served no matter which bank she chooses. Thus, she will just choose the bank whose contract provides higher utility, so banks will have incentive to offer a slightly better contract than the competitor unless they both offer the contract yielding the highest utility. When both banks contracts yield w, any combination of p I and p II such that p I + p II = 1 can be in equilibrium. Case 2 is where 1 < 2. When is in this range, there exists a continuum of equilibria in an interval that depends on, corresponding to different w and π. Perfect competition environment breaks down, and w is no longer achievable. In the following, Lemma 1 is used in showing that if both banks provide the contract yielding w, banks will have incentive to deviate. Proposition 2 shows that in equilibrium it must be true that both banks provide the same contracts and the utility provided is lower than w. Proposition 3 is about the existence of an interval of equilibria. Lemma 1 When 1 < 2, if 1, then 1. 2P I 2P II Proof. If 2P 1, then P I I 2, 2P = II 2(1 P I P A ) 2(1 ) = 2. 2 Since 1 < 2, 2 1. Thus if 2P 1, 1. I 2P II Lemma 1 says that when 1 < 2, if bank I is bounded by its capacity, bank II must be able to take all depositors willing to deposit at bank II. Proposition 2 When 1 < 2, in equilibrium it must be true that w I = w II < w and both banks can take all the depositors willing to deposit at its place, W I = w I, and W II = w II. Proof. When w I > w II, if 2P I, then depositors depositing at bank I will be served with certainty and get w I. The expected utility that a depositor can get 12

13 by depositing at bank II is at most w II. Given that w I > w II, for an individual depositor, depositing at bank I strictly dominates. Thus, every individual depositor will deposit into bank I, and P I = 1. Thus, 2P I is not the equilibrium result of the subgame played by depositors when w I > w II. When w I > w II, it must be true that 2P I >, which means bank I cannot serve all the depositors and needs to turn away some depositors at infinite high cost. It is something that bank I will not do. Thus, in equilibrium, a bank will not provide a contract that yields expected utility higher than its competitor. When banks contracts offer the same level of utility, depositors strategy must satisfy /2P I 1 and /2P II 1. If not, then W I W II, and some depositors will be better off if depositing at the other bank. Thus in equilibrium, W I = w I, and W II = w II. epositors getting w I = w II = w cannot happen in equilibrium because both banks have incentive to deviate to a worse contract that yields lower utility to make some positive profit. If bank II deviates to w II < w, then an individual depositor solves problem (1) given w I = w and w II < w. Because w I > w II, as previously shown it must be true that /2P I < 1. Then by Lemma 1, it must be true that /2P II 1. Thus W I = 2P I wi + (1 2P I )wa, W II = w II. If W I > W II, a depositor deposits at bank I. If W I = W II, a depositor is indifferent. If W I < W II, a depositor deposits at bank II. (The last case will not happen in equilibrium.) There is a low bar for w II, below which all depositors will deposit at bank I, thus P I = 1. It can be solved from 2 wi + (1 2 )wa = w II. As long as w II > 2 wi +(1 2 )wa, in equilibrium some fraction of depositors deposit 13

14 at bank II. Probability P I can be solved from 2P I wi + (1 2P I )wa = w II, which implies P II = 1 (wi w A ) 2(w II w A ), which is greater than 0 if w II > (/2)w I + (1 /2)w A. Thus, if w I = w, bank II can make a profit by deviating to a worse contract. As long as w II > (/2)w + (1 /2)w A, it makes a profit 2π II P II > 0. Thus in equilirium, w I = w II = w cannot happen. Given w I, there is a best deviation that bank II can make, by which bank II gets highest total profit. In equilibrium, it must be true that the total profit gained from this best deviation is less than or equal to the equilibrium total profit. Proposition 3 For any [1, 2), there exists π() > 0, such that for all π [π, π A ], (w I (π) = w II (π), p I = p II = 1 ) is equilibrium. Also, π is decreasing in, and 2 when = 1, π < π A. 7 Proof. See Appendix. For every given in [1, 2), there is an interval of π in which symmetric equilibria exist. When banks earn π, the lowest unit profit of all equilibria, they offer a contract that yields the highest expected utility of all equilibria, denote the utility by w. As becomes smaller, π increases and the interval shrinks, which means the highest utility decreases. See Figure 1. In Figure 1, 1 1 < 2 < 2. The interval where equilibria exist when = 1 is [ π( 1 ), π A ], while the interval where equilibria exist when = 2 is [ π( 2 ), π A ], π( 1 ) > π( 2 ), and correspondingly, w( 1 ) < w( 2 ). (irections of arrows show increasing w and π). 7 Actually, for any /2p I 1, /2p II 1 and p I + p II = 1, such an interval exists. See Appendix for proof. However, with p I = p II = 1/2, π is lowest. To pick the highest welfare level for the depositors, p I = p II = 1/2 is thus picked. 14

15 Figure 1. Change of the Interval of Equilibria Also noteworthy is that when = 1, π ( = 1) < π A, and w(1) > w A, which means higher-than-autarky utility is achievable. Intuitively, under = 1 suppose bank I and bank II provide the same contract that yields the same utility w, and they earn the same total profit π. To make neither bank want to deviate, π is not necessarily equal to π A. Suppose π < π A, if bank II is to deviate, it will not deviate to π A because no one will deposit there. As long as bank II s best deviation is nonprofitable, π I = π II = π is an equilibrium. Bank II will not want to deviate if no deviation can provide higher total profit than π. That is, if 2π II (w II )(1 (w wa ) 2(w II w A ) ) π, for all w II ((1/2)w + (1/2)w A, w), and accordingly, for all π II (π, π A ). The lower bar π ( = 1) is the lowest π where neither bank wants to deviate, and it should be that π > π A, because at π A, there is no room for either bank to deviate at all. More detail of the proof of π( = 1) < π A is shown in the Appendix. For a given [1, 2), a depositor s expected utility depends on which equilibrium 15

16 is picked. In the next section where welfare is analyzed, for every, I pick the equilibrium where highest utility w is achieved and where π = π(). Case 3 is the case where neither bank is able to serve half the market, so < 1. Any contract that yields w > w A will cause at least one bank to turn away some depositors. Under the assumption that banks turn away depositors at infinitely high cost, in equilibrium both banks provide autarky utility, and depositors choose between depositing at bank I, bank II, or staying in autarky randomly. Equilibrium outcome is (w I = w II = w A ; p I [0, /2], p II [0, /2]; p I + p II + p A = 1). Up until now, I have examined equilibrium for the whole game under different capacities of banks. When 2, the highest achievable utility is offered. When 1 < 2, there is a continuum of equilibria. When < 1, autarky utility is offered. If we pick the equilibrium (w, p I = p II = 1/2) when 1 < 2, then a depositor s utility depends only on. It is true that in some cases, there may be multiple equilibria where one level of w happens since multiple depositing strategies may constitute equilibrium; however, although depositors may deposit at different banks, they do get the same utility at every equilibrium. A bank s capacity is decided by its endowed capital K as well as the capital requirement θ. With any K, when θ increases, banks capacity reduces and eventually competition becomes less intense and depositors become worse off. With different K and θ, depositors welfare in normal and unexpected run time may be different. Banks welfare, described as the total profit earned, may also be different. In the next section, I examine welfare under different combinations of K and θ. 16

17 6 Analysis of Welfare 6.1 epositors Welfare Given a bank s capital being K, the magnitude of θ may put the bank under capacity constraint if it is too high. At the same time, higher θ provides better protection if unexpected run happens. In this section, welfare of depositors in normal time as well as under financial shock with different θ when K is different will be examined. It will be shown that the effect of changing θ on the welfare depends on the amount of capital. The welfare of depositors in normal time depends on banks capacity only, and thus is the same welfare as reached in equilibrium in the previous section. enote the normal-time welfare of depositors by W N. When = K/θ 2, w is achieved for depositors. enote the contract achieving w by (C1, C2). When 1 < 2, there is an interval of equilibria [w A, w], and w decreases as decreases. For every [1, 2), pick w, denote the unit-profit-maximizing contract that achieves w() by (C 1 (), C 2 ()). When < 1, only autarky welfare for depositors is achieved. enote the unit-profit-maximizing contract achieving autarky welfare for depositors by (C1 A, C2 A ). To sum up, in normal time depositors welfare changes with θ and takes the following form W N = αu(c 1) + (1 α)u(c 2) = w if K/θ 2 αu(c 1 (K/θ)) + (1 α)u(c 2 (K/θ) = w(k/θ), if 1 K/θ < 2 αu(c A 1 ) + (1 α)u(c A 2 ) = w A, if K/θ < 1. The run-time welfare of depositors is analyzed as follows. As shown in the standard iamond-ybvig model, without capital requirement, a contract that provides c 1 > 1 17

18 allows for a run equilibrium ex post. In the run equilibrium, a patient depositor prefers to withdraw at t = 1 if everyone else is doing so. With capital requirement, suppose a bank is obliged to pay c 1 to depositors showing up in t = 1, and for depositors who wait until t = 2 it is required to pay up to c 2 that it promised in the contract using the required capital. epositors get served with equal chance. If a bank served d measure of depositors in t = 0, total deposit is also d, and capital required on these deposits is thus dθ. Both required capital and deposits are put into productive investment in t = 0. If in t = 1 run happens, all of the depositors want to withdraw from the bank. If c 1 > 1, the bank needs to liquidate investment at rate r = 1. A patient depositor s expected utility of run is (1 + θ) c 1 u(c 1 ) if 1 + θ < c 1 u(c 1 ) if 1 + θ c 1. It is equal or smaller than u(c 1 ). If 1+θ c 1, a patient depositor has no incentive to run, because her second period consumption is protected, and u(c 2 ) u(c 1 ). Thus for any contract that provides (c 1, c 2 ), depositors welfare with financial shock, denoted by W R, is W R = 1 + θ c 1 u(c 1 ) if 1 + θ < c 1 W N if 1 + θ c 1. The capital requirement θ can provide some protection under a financial shock. However, as long as 1 + θ c 1, run is eliminated, and increasing θ no longer increases welfare. On the other hand, given a bank s endowed capital K, a bank s capacity decreases in θ. As long as θ K/2, = K/θ 2, and increasing θ imposes higher capacity constraint onto banks. It will be shown that the more suffi cient K is, the 18

19 less negative effect θ has when it is increasing. I categorize K into two cases. Case 1 is the case where K is suffi cient, and case 2 is the case where K is less suffi cient. In case 1, K can support w and eliminate run. That is, when increasing θ from zero, run elimination is achieved without causing the capacity constraint to bind. In this case, θ can be properly set so that run is eliminated without harming the competition environment. Case 2 is where run-proof contracts cannot provide w for depositors. That is, when θ increases to a level such that = K/θ = 2, 1 + θ is still smaller than C1, whereas when θ continues to increase and makes a run-proof contract happen in equilibrium for the first time, is less than 2. In the following, I examine how welfare in normal time as well as in crisis changes when θ increases from zero in each of the cases. Case 1. K 2(C1 1). This is the case where w can be achieved for depositors under run-proof contract with some θ. Run can be eliminated and w can be achieved if capital requirement is set properly. Proposition 4 When K 2(C 1 1), setting θ [(C 1 1), K/2] achieves w with run elimination. Proof. If setting θ < C 1 1, then = K/θ > 2, competition environment is not harmed. However, not enough protection is given for unexpected run given the K, because 1 + θ < C 1. So W N = w W R = 1 + θ u(c C 1). 1 Run-time welfare W R is increasing in θ. Increasing θ will improve run-time welfare without reducing normal-time welfare. 19

20 If setting C 1 1 θ K/2, then = K/θ 2, and 1 + θ C 1. Thus W N = W R = w. Capital requirement θ is optimally set, and changing θ in this range will not affect anything. It is noteworthy that as long as K = 2(C1 1), setting θ = K/2 achieves w by a run-proof contract. Bank capital does not need to be higher than 2(C1 1). If setting K/2 < θ K, then 1 < 2. Banks will become capacity constrained, and as shown in the previous section, w will no longer be achievable. Thus W N = W R = w(). Increasing θ in this range decreases banks capacity, and thus decreases depositors welfare w(). If setting θ > K, < 1, then W N = W R = w A. Increasing θ in this range no longer affects welfare. Thus, [(C1 1), K/2] is the interval of θ that achieves w with run elimination when K 2(C1 1). Case 2. K < 2(C1 1). When K < 2(C1 1), the range [(C1 1), K/2] no longer exists. Run-proof contract that yields w will no longer be achievable no matter what θ is set. If setting θ K/2, then = K/θ 2, and 1 + θ 1 + K/2 < C1. Thus, the 20

21 perfect competition environment does not break down, and run can not be eliminated, W N = w, W R = 1 + θ u(c C 1). 1 Run-time welfare W R increases with θ and gets its maximum at θ = K/2 in this range. If increasing θ above K/2, run-time welfare level will keep increasing, but competition environment will break down. If setting K/2 < θ K, then 1 < 2. Thus W N = w(), W R = 1 + θ C 1 () u(c 1()) if 1 + θ < C 1 () W N if 1 + θ C 1 (). Normal-time welfare W N is decreasing in θ, run-time welfare W R first increases in θ 8, and when the contract becomes run proof, the welfare is same as normal-time welfare and thus it decreases in θ. Notice that, in this range when θ increases, more run-admitting contracts are excluded from the interval of equilibrium. What is more, depositors will be better off because (1/(C 1 ())u(c 1 ()) is decreasing in θ. This tradeoff effect is achieved through the change of capacity only. With θ, even more run-admitting contracts are excluded, and depositors will be even more better off in run times under a run-admitting contract. If θ keeps increasing after run is eliminated, welfare decreases. Thus, setting θ > C 1 () 1 is never necessary. For all run-proof 8 This should be easily seen. ue to the character of the utility function, u(c 1 ())/C 1 () is decreasing in. When [1, 2), as θ increases, decreases, and thus u(c 1 ())/C 1 () increases. 1 + θ Therefore, C 1 () u(c 1()) increases. 21

22 contracts, welfare level achieved is above w A. If setting θ > K, then < 1, only autarky welfare is achievable. Run is eliminated. Proposition 5 For any θ, bank s unit-profit-maximizing contract that yields autarky level of utility is run-proof. Proof. Bank s unit-profit-maximizing contract (C A 1, C A 2 ) that yields autarky welfare should satisfy u (C A 1 ) = u (C A 2 )R, αu(c A 1 ) + (1 α)u(c A 2 ) = u(1). Thus, C A 1 < C A 2. It implies that C A 1 < 1 because if C A 1 1, then αu(c A 1 ) + (1 α)u(c A 2 ) > u(c A 1 ) u(1), which is contradiction. Because C A 1 < 1, for any θ, 1 + θ > C A 1, the contract is run-proof. Thus, as long as < 1, the following holds W N = W R = w A. 6.2 Banks Welfare Banks welfare is captured by their total profits combined. It is much more straightforward than depositors welfare. In normal time, when 2, a bank does not earn profit; thus, total profit of a bank Π i = 0. When 1 < 2, given the equilibrium (w I = w II = w, p I = p II = 1/2), a bank s total profit is π. When < 1, only autarky welfare is provided and depositors randomize. Assuming that both banks take deposits to their capacity, then both banks total profit is Π i = π A. 22

23 In run time, banks suffer loss if 1+θ < c 1. The total loss for a bank is θd, where d is the fraction of depositors a bank took from the market in t = 0, and d = min(1, ) 9. If 1 + θ c 1, then run is eliminated and the profit is the same as in normal time. enote banks total profit combined in normal time by Π N and their total profit combined in run time by Π R. Then Π N = 0 if 2 2π if 1 < 2 2π A d if < 1, Π N if 1 + θ c 1 Π R = 2θd, if 1 + θ < c 1. In the next section, I use a numerical example to visualize the effect of changing θ on the welfare of depositors as well as of banks. 7 Numerical Example The utility function is u(c) = (c + b)1 r b 1 r, 1 r where b = 0.5, r = 2. Let R = 1.5, α = 0.4. Three things are shown with the numerical example: the relationship between and π, as well as and w when 1 < 2, how depositors welfare in normal time and under financial crisis changes when θ changes in the two cases, and how banks total profit changes in the two cases. The relationship between and π, w when 1 < 2 is shown in Figure 2. It can be seen that π is decreasing in, and w is increasing in. As increases, the 9 However, as proposition 5 shows, when < 1, the contract banks provided offers autarky welfare and is run proof. Thus, d = 1. 23

24 interval of equilibria becomes larger and equilibrium with higher welfare of depositors becomes feasible. Also, when = 1, π is lower than π A, and accordingly w is higher than w A Figure 2. Relationship between and π, w. epositors welfare in the economies with different capital endowments is shown in Figure 3. It shows how welfare changes in both normal time and in crisis when θ changes, given different levels of K in the two cases discussed in the previous section. The black lines represent normal-time welfare, the red lines represent run-time welfare. When they overlap, it means that the contract is run proof. It can be seen that with K 2(C1 1), when θ keeps increasing from 0, there is a range of θ where w can be provided for sure. In this range of θ, 1+θ C1 and K/θ 2 are satisfied at the same time. In other words, when increasing θ from 0, run elimination is achieved before perfect competition environment breaks down. If K < 2(C1 1), then 1 + θ C1 cannot happen when K/θ 2; thus, as shown in the right graph of the panel, no θ can achieve w with run elimination. Thus the highest achievable welfare of depositors with run elimination depends on K. When K is suffi cient enough, w in normal time and run eliminating in financial shock can be both achieved if θ is properly set. When K is not suffi cient enough, there is a tradeoff between the two when [1, 2), 24

25 increasing θ increases run time welfare at the cost of decreasing normal time welfare, until run-proof contract happens. In both cases, there is an upper bar of θ, above which continuing to increase θ only decreases welfare. In case 1, it happens when θ = K/2. In case 2, it happens when θ = C 1 () Figure 3. epositors Welfare in the Two Cases Banks welfare (their total profit combined) also changes with θ. The total profit in normal time depends only on, the ratio of K/θ. As long as run is not eliminated, banks suffer loss in run time. Once run is eliminated, total profit is the same as in normal time. Figure 4 shows how banks total profit looks when changing θ in case 1 and case 2. When 1 + θ < c 1, the run-time total profit is negative, and the loss is increasing with θ. When 1 + θ c 1, the contract is run proof so run-time total profit is same as the normal-time total profit. When 2, normal-time total profit is 0. When 1 < 2, normal-time total profit is increasing, because π increases when θ increases, and banks together serve the whole market. When < 1, total profit decreases with θ, because decreases with θ, and unit profit is π A and it does not change with θ. 25

26 Figure 4. Banks Profit in the Two Cases 8 Conclusion In this paper, I examine how changes of capital requirement cause the competition intensity between banks to change on the deposit market, and what effect this change will have on financial stability. As capital requirement increases, banks capacity for taking deposits reduces and banks become less competitive, and thus depositors get less favorable contracts in normal times. On the other hand, with higher capital requirement, the set of contracts that admits bank runs shrinks and better insurance can be provided to depositors when a run happens. If it is high enough, it can even eliminate bank runs. There is a level of capital above which the best incentive feasible allocation for depositors can be strongly implemented if capital requirement is properly set. If capital is below this level, as capital requirement increases it increases run-time welfare of depositors at the cost of reducing normal-time welfare of depositors, and the highest possible welfare cannot be achieved under any runproof contract given any capital requirement. Bank s total profit is also analyzed. Bank s total profit earned also depends on endowed capital and capital requirement: 26

27 If capital requirement is high so that run is eliminated, banks will not suffer loss, and increasing capital requirement increases banks profit at the cost of reducing depositors welfare. If capital requirement is too high, it reduces banks profit without improving depositors welfare because it reduces the measure of deposits taken by banks. On the other hand, if capital requirement is very low and run is not eliminated, depositors run-time welfare increases at the cost of increasing banks loss in run time when capital requirement increases. In this paper, different levels of endowed capital are discussed. Further work will be done endogenizing the choice of capital. Also, run is unexpected in this paper; in the future, I will calculate the optimal contract given the probability of run and discuss the welfare cost of increasing capital requirement. 9 Appendix 9.1 Proof of Proposition 1 Proposition 1. The highest unit profit π given w is strictly decreasing in w, and c 1, c 2 realizing π are both strictly increasing in w. Proof. Since w is increasing in both c 1 and c 2, when w increases, we cannot have both c 1 and c 2 decrease, and at least one of them should increase. Let w > w, then u (c 1 (w)) =u (c 2 (w))r, (6) u (c 1 (w )) =u (c 2 (w ))R. (7) If c 1(w) c 1(w ), then it must be true that c 2(w) < c 2(w ), since if not w > w cannot happen. It follows that u (c 1(w)) u (c 1(w )) = u (c 2(w ))R < u (c 2(w))R, or u (c 1(w)) < u (c 2(w))R, it 27

28 is contradiction to equation (6). So it must be that c 1(w) < c 1(w ). If c 2(w) c 2(w ), similarly it follows that u (c 2(w))R u (c 2(w ))R = u (c 1(w )) < u (c 1(w)), contradiction to equation (7). So c 1(w) < c 1(w ), c 2(w) < c 2(w ). Since π = (1 αc 1)R (1 α)c 2 is strictly decreasing in c 1 and c 2, it follows that π is strictly decreasing in w. Q.E Proof of Proposition 3 Proposition 3. For any [1, 2), there exists π that depends on, for all π [π, π A ], (w I (π) = w II (π), p I = p II = 1 ) is equilibrium. Also, π is decreasing in, and 2 when = 1, π < π A. Proof. To prove that π I = π II = π (thus w I = w II = w) is an equilibrium, it only needs to be shown that no bank wants to deviate from π. By proposition 2, when 1 < 2, if bank provides higher utility than its competitor, the amount of depositors deciding to deposit at its place will exceed its capacity. Given that cost of turning away depositors is infinitely high, neither bank has incentive to deviate to π i < π (thus w i > w). A bank may have incentive to deviate to π i > π (w i < w). Without loss of generality, examine the behavior of bank II. Given π I, if bank II chooses to earn unit profit π II = π I, then w II = w I, bank II gets half the market and earns total profit π II = π I given depositors strategy p I = p II = 1. If bank II deviates, there 2 will be a best deviation it can make, which gives it the highest possible total profit. To make bank II not want to deviate, best deviation must be nonprofitable. Let Π II be the highest profit bank II can earn if it deviates to a higher unit profit, then π I Π II must be satisfied in equilibrium. 28

29 Part 1. The existence of π when 1 < 2. Bank II cannot make the contract too worse and there is an lower bar of welfare level, below which no one will decide to deposit at its place. Let w II be the low bar, as is shown in proposition 2, w II = (/2)(w I w A ) + w A. When w II = w II, P II = 0. enote the corresponding level of unit profit at w II as π II, π II is decreasing in w I, thus increasing in π I. If w II w II < w I, (i.e., π I < π II π II ), P I = (wi w A ) 2(w II w A ), P II = 1 (w I w A ) 2(w II w A ). Thus Π II= max π II (2 (wi w A ) (w II w A ) ) s.t π I < π II π II. Let Π II be the maximized value solved from max Π II = π II (2 (wi w A ) w II w A ) (8) s.t. 0 π II π II. That is, the same maximization problem without the constraint that π II > π I. First order condition of equation (8): Π II = 2 (wi w A ) π II w II w A Second order condition: 2 ΠII π 2 II = +π II (w I w A ) wii π II (w II w A ) 2 + (w I w A ) wii π II (w II w A ) 2 = 0. (w I w A ) wii π II (w II w A ) 2 + (w I w A 2 w II ) (π π II ) 2 (wii w A ) 2 (w I w A ) wii 2(w II w A ) wii π II π II II, (w II w A ) 4 29

30 2 ΠII π 2 II is negative if w II is concave in π II. It can be shown that w is concave in π. Equation (2) is equivalent to w = max αu(c 1 ) + (1 α)u(c 2 ) (9) s.t(1 αc 1 )R (1 α)c 2 = π. Substitute c 2 : w = max αu(c 1 ) + (1 α)u( (1 αc 1)R π ). 1 α First order condition needs to hold for maximization: u (c 1 ) u (c 2 )R = 0. using Implicit Theorem, take derivative with respect to π on both sides: u (c 1 ) c αr c 1 1 π u (c 2 ) π 1 R = 0. 1 α c 1 π 1 Since c αr 1 π < 0, it must be true that < 0 when maximum is reached. 1 α Using Envelope Theorem, take first derivative of maximized w with respect to π : w π = u (c 2 )( 1). Second derivative: 2 w αr c 1 π = 2 ( 1)u (c 2 ) π 1. 1 α αr c 1 Since u (c 2 ) < 0, π 1 1 α 2 w π < 0. 2 Thus w is concave in π. < 0, Since w is concave in π, for equation (8) SOC < 0. At π II = 0, Π II = 0. As π II increases, Π II becomes positive, when π II = π II, P II = 0, Π II = 0. Thus, there must be a unique maximum Π II realized in the interval 30

31 (0, π II ), where F OC satisfies. enote the unit profit that realizes this maximum as π II, clearly π II (0, π II ). For a given π I, there exists π II (0, π II (π I )) that maximizes (9). It can be shown that when π I increases, both π II and Π II increase. What s more, Π II is convex in π I. From F OC condition of 9, by Implicit Theorem π II can be calculated: π I wi (w II w A ) (w I w A ) wii π π I π II II π I + (w I w A ) wii π II π II + (w II w A ) 2 π I (w II w A ) 2 π II[ [ wi π I w II π II + (w I w A ) 2 w II π II π 2 ](w II w A ) 2 π II I (w II w A ) 4 (w I w A ) wii π 2(w II w A ) wii π II π II II π I ] (w II w A ) 4 = 0, thus (w I w A ) wii π II π w I w II II π I π II So [2(w I w A ) wii π II π II π I wi π I (w II w A )+ π II + π II(w I w A ) 2 w II π 2 II + π II(w I w A ) 2 w II w I (w II w A ) π w I w II II π I π I π II π I = 2(w I w A ) wii π II π II Thus, π II is increasing in π I. Also, by Envelope Theorem,, (w I w A ) wii + π I π II π I π II π 2 II π II π II 2(w I w A )( wii π ) 2 π II II π I (w II w A ) 2(w I w A )( wii (w II w A ) w I (w II w A ) π w I w II II π I π I π II + π II(w I w A ) 2 w II π 2 II π II π ) 2 II ] π II π I = 2(w I w A )( wii (w II w A ) = 0. π ) 2 II > 0. 31

32 Π II = Π II π π I π II = π II I wi π I > 0. wa w II Thus, Π II is increasing in π I. What s more, when π I increases, Π II π I increases, so Π II is convex in π I. wi = π π I II wii wa Thus when π I increases, π II and Π II both increase, and Π II is convex in π I : π II > 0, Π II π I π I of π can be shown. > 0, and 2 Π II π 2 I > 0. Using features of π II and Π II, existence I will show that for all 1 < 2, there is a π, when π I [0, π), Π II = Π II > π I, bank II will deviate, so π I = π II [0, π) is not equilibrium. When π I [π, π A ], Π II π I, so bank II will not deviate, and π I = π II [π, π A ] is equilibrium. (1). It can be shown that when π I = 0, 0 < Π II < π II. At π I = 0, bank II benefits if it deviates. Thus, π II > π I = 0. Π II = π II(2 (wi w A ) w II ( π II) w ), since A π II > π I, w II ( π II) < w I, 2 (wi w A ) w II ( π II) w ) < A 1, thus 0 < Π II < π II. (2). When π I approaches π A, π II < π I, and Π II > π I. Split (0, π II ) into (0, π I ) and [π I, π II ). when π I approaches π A, w I approaches w A, w II = 2 (wi w A ) + w I approaches w A, so π II approaches π A. Thus, if choosing π II [π I, π II ), when π I approaches π A, π II approaches π A, Π II approaches π A (2 ). If choosing π II = π I δ, where δ being small, then when π I approaches π A, 2 (wi w A ) w II ( π II ) w A approaches 2, Π II approaches 2(π A δ), which is greater than π A (2 ). Thus, When π I approaches π A, π II < π I, and Π II > π I. (3). Examine the relationship between Π II and Π II. Since the difference between Π II and Π II is that Π II is solved under constraint 0 π II π II while Π II is solved under constraint π I < π II π II, it s easily seen that when π II > π I, Π II = Π II, when π II π I, since π II is restricted to π II > π I, Π II > Π II. Some other relationships of Π II, π II, and π I are needed before final proof 32

33 can be done: (4). For 1 < < 2, if Π II = π II, Π II = π II < π I ; if π II = π I, Π II < π I. By definition, Π II = π II(2 (wi w A ) w II ( π II) w A ). If Π II = π II, (2 (wi w A ) w II ( π II) w A ) = 1 (wi w A ) w II ( π II) w A = 1 wii > w I π II < π I. If π II = π I, Π II = π II(2 (wi w A ) w II ( π II) w A ) = π I(2 ) < π I. Thus, For 1 < < 2, if Π II = π II, Π II = π II < π I,if π II = π I, Π II < π I. For = 1, easily seen if Π II = π II, Π II = π II = π I, if π II = π I, Π II = π I. (5). Since π II is increasing in π I, and when π I = 0, π II > 0. When π I approaches π A, π II < π I. Then it must be true that π II intersects π I for odd times before π A. According to (4), for 1 < < 2, at every intersection, π II = π I > Π II ; for = 1, at every intersection, π II = π I = Π II. It may be that Π II intersects π II several times, but at every intersection, according to (4), if 1 < < 2, Π II = π II < π I, if = 1, Π II = π II = π I. Next I examine 1 < < 2 and = 1 separately. For 1 < < 2, Π II must intersect π I twice between (0, π A ). Since if not, when π II = π I, π I Π II. This cannot happen. (See Figure 4. Figure 4 shows what will happen if Π II does not intersect π I. Case where Π II only touches π I once before π A can also be easily imagined. Neither of these should happen). 33

34 Figure 5. Π II Not Touching π I When Π II intersect π I for the first time, denote this interception point as π. When Π II intersects π I for a second time, denote this second interception as π 2. Since when Π II = π II, π I > Π II = π II according to (4), Π II and π II only intersect within the interval (π, π 2 ). When π I [0, π), Π II > π I, π II does not intersect with Π II or π I. So π I < Π II < π II, and π II = π II, Π II = Π II > π I. Bank II will deviate. When π I [π, π 2 ], Π II Π II π I. So this part is equilibrium. For π I (π 2, π A ), Π II > π I, π II < π I, thus Π II approaches π I(2 ) < π I, so bank II will not deviate. Thus (π 2, π A ) is equilibrium. Thus for all π I [π, π A ), π II = π I is equilibrium. Thus, π exists when 1 < < 2. Figure 5 visualizes what happens. When π I < π, Π II > π I, π II > π I, bank II will deviate. When π I [π, π 2 ], Π II Π II π I, bank II will not deviate. When π I (π 2, π A ), Π II > π I, π II < π I, Π II approaches 34

35 π I (2 ) < π I, bank II will not deviate. Figure 6. Equilibrium When 1 < < 2 For = 1, it may be that Π II touches π I once or intersect π I twice before π A. If twice, proof will be similar as the case 1 < < 2. If once, the point can also be easily proven to be π, and π < π A. It s already shown that: π II must intersect π I for odd times, if π II intersects Π II (or π I), it must also intersect π I (or Π II ) when = 1. Also, π π A. Since if π = π A, then when π II = Π II, π II = Π II > π I. This couldn t happen. So if Π II touches π I once, it must be before π A. Since π II must intersect π I at least once, and when they intersect, they both intersect Π II, π is the point where all three curves intersect, and the only point where π II = π I, because if π II intersects π I at other points, Π II > π I = π II. This cannot happen. When π I [0, π), Π II > π I, π II > π I, π II = π II, Π II = Π II > π I, bank II will deviate. When π I = π, Π II < Π II = π I, bank II will not deviate from π I. When π I (π, π A ), π II < π I, Π II < π I, bank II will not deviate. So [π, π A ) are equilibria. 35

36 See Figure 6. Figure 7. = 1, Π II Touching π I Once Above, I proved that for 1 < 2, there exists [π, π A ), when π [π, π A ), π I = π II = π is equilibrium. Next, it can be easily proven that π I = π II = π A is equilibrium for any 1 < 2. At π I = π A, autarky welfare is provided by bank I. Bank II will not offer a higher welfare since it will need to turn away depositors. It will not offer a lower welfare either. It will offer autarky level of welfare as bank I. Its profit is determined by depositor s depositing strategy, and can be any number between [0, π A ]. So π I = π II = π A is also equilibrium. So far, I proved the existence of π when 1 < 2. Also, when = 1, π < π A. Part 2. When decreases, π increases. Proof. I showed that π is π such that if π I < π, Π II > π I, if π I π, Π II < π I, where Π II = max π II(2 (wi w A ) w II w A ) s.t 2 (wi w A ) + w A < w II < w I. Look at Π II, π II : Π II = π II (2 (wi w A ) w II w A ), where π II can be chosen from [0, π II ]. 36

37 Π II = max π II(2 (wi w A ) w II w A ) s.t 0 π II π II. Given w I, by Envelope Theorem: Π II = π II (or π I ), when decreases, Π II increases. Given 1, 2, where 1 > 2, there are Π II1 (π1 ) =π 1 Π II2 (π2 ) =π 2. Since when decreases, Π II increases, Given π 1 : Π II2 (π1 ) > Π II1 (π1 ) = π 1. Given π 2 : π 2 = Π II2 (π2 ) > Π II1 (π2 ). so π 1 is before the intersection of Π II2 and π A, which is π 2. (w I w A ) w II w A < 0. Thus, Given wi, So π 2 > π 1 when 2 < 1. When decreases, π increases. See Figure 7. Figure 8. 1 > 2 Q.E.. It can also be easily shown that for any for any /2p I 1, /2p II 1 and p I + p II = 1, such an interval exists. Since although when π I = π II, there may be 37

38 different symmetric equilibria of depositors game, once π I π II, depositors best response is unique. Thus, for π = π I = π II to be equilibrium for the game, if /2p I 1, /2p II 1 and p I + p II = 1, p I p II are depositors strategy, so that bank I makes total profit 2πp I and bank II makes total profit 2πp II, it should be that the bank which makes fewer total profit has no incentive to deviate to a worse contract. Given, for π that works for p I = p II = 1/2 no longer works, since the bank with smaller total profit will deviate. Thus, π needs to be higher. So if p I p II, the interval of equilibria is smaller, and the highest achievable welfare at is lower than when p I = p II. 9.3 Equilibrium of subgames played by depositors In this part, I examine all the symmetric equilibria of subgames played by depositors under different capacity constraints of banks and at different value of (w I, w II ). For each individual, p = P. Case Banks are not constrained technically. Since 2, /2P 1 because P 1. No matter which bank depositor deposits at, the probability of being served is 1. Thus, depositor will just choose the bank that offers higher expected utility. If banks provide same level of welfare, the depositors will be indifferent and will take mixed strategies. Symmetric equilibria of the subgames at each possible pair of (w I, w II ) will be (p I = 1, p II = p A = 0) if w I > w II w A (p I [0, 1], p II = 1 p I, p A = 0) if w I = w II > w A (p I [0, 1], p II [0, 1], p A [0, 1]; p I + p II + p A = 1) if w I = w II = w A Case 2. 1 < 2. In this case, both banks are capacity constrained, each of them can take more 38

39 than half of the market, and together they can take the whole market. 1. If w I > w II > w A, then staying in autarky is not in the support since depositing at either bank gives a depositor some probability to improve above autarky. Further split the case w I > w II > w A into three subcases. First, if in equilibria it results that W I = W II, then 2P I wi +(1 2P I )wa =w II, which implies P I = (wi w A ) 2(w II w A ). Next, if W I > W II, then P I = 1 implies 2 wi +(1 2 )wa > w II, or (wi w A ) 2(w II w A ) > 1.. Lastly, if W I < W II, then P II = 1, W II = w II. However, this cannot happen in equilibrium, since individual depositor has the incentive to deviate and deposit at bank I and get w I, which is higher than w II. Thus the case w I > w II > w A can be split into two specific cases. Symmetric equilibrium in each case is as following: p I = (wi w A ) 2(w II w A )., pii = 1 p I, p A = 0) if (wi w A ) 2(w II w A ) 1 (p I = 1, p II = p A = 0) if (wi w A ) 2(w II w A ) > If w I > w II = w A : then all depositors will deposit at A. Equilibrium will be: (p I = 1, p II = p A = 0). 39

40 3. If w I = w II > w A : again staying in autarky will not be in the support. If W I > W II, everyone will deposit at bank I, p I = 1, /2P I = /2 < 1, W I = (/2)w I + (1 /2)w A < w I = w II. This is a contradiction. Thus in equilibrium, it must be true that W I = W II. More specifically, W I = W II means 2P I wi + (1 2P I )wa = w II, if 2P I < 1, 2P II 1 2P II wii + (1 2P II )wa = w I, if < 1, 2P II 2P I 1 2P I wi + (1 2P I )wa = 2P II wii + (1 2P II )wa, if 2P I < 1, 2P II < 1 w I = w II, if 2P I 1, 2P II 1 It can be easily shown that only the last case can happen. So equilibria in this case are (p I /2, p II /2; p I + p II = 1). 4. If w I = w II = w A, then depositors are indifferent between depositing at either bank or staying in autarky. Thus, equilibria will be (p I [0, 1], p II [0, 1], p A [0, 1]; p I + p II + p I = 1). To sum up, in the case where 1 < 2, according to the values of, w I, and 40

41 w II, symmetric equilibria will be (p I = (wi w A ) 2(w II w A ), pii = 1 p I, p A = 0) if (wi w A ) 2(w II w A ) 1 (p I = 1, p II = p A = 0) if (wi w A ) 2(w II w A ) > 1 (p I = 1, p II = p A = 0) if w I > w II = w A (p I 2, pii 2 ; pi +p II = 1) if w I = w II > w A (p I [0, 1], p II [0, 1], p A [0, 1]; p I +p II +p I = 1) if w I = w II = w A Case 3. < 1. In this case both banks can only take less than half of the market, and together they cannot take the whole market. 1. If w I > w II > w A, it must be true that /2P I < 1 (meaning that not all depositors depositing at bank I will be served). Since if /2P I 1, then all depositors will deposit at I, P I = 1 implies /2P I < 1, this is a contradiction. Next, look at /2P II. If depositors at bank II are served with probability 1, it requires that /2P II 1.Then W II = w II, W I = (/2P I )w I + (1 /2P I )w A. If W I > W II, depositors will all deposit at I. If W I = W II, depositors mix their strategy. If W I < W II, depositors all deposit at II, but this cannot happen in equilibria since given everyone depositing at bank II, an individual depositor will want to deviate to deposit at bank I. If depositors at bank II are not served with probability 1, then it requires that /2P II < 1. 41

42 If /2P II < 1, then W I = W II, since if not individual depositor expecting lower welfare will have incentive to deviate to the other bank. Thus, 2P I wi + (1 2P I )wa = 2P II wii + (1 2P II )wa. It leads to 2P I (wi w A ) = 2P II (wii w A ), which implies Thus the following are equilibria: P II P I = wii w A w I w A. (p I = (wi w A ) 2(w II w A ), pii = 1 p I ), if 1 2 (wi w A ) 2(w II w A ) 1, (p I w I w A = (w I w A ) + (w II w A ), w II w A pii = (w I w A ) + (w II w A ) ), if (wi w A ) 2(w II w A ) < 1 2, (p I = 1, p II = 0), if (wi w A ) 2(w II w A ) > If w I > w II = w A, (p I = 1, p II = p A = 0) will be the equilibria, since as long as depositing at bank I, depositor s welfare will be higher than autarky. 3. If w I = w II > w A, then it must be true that W I = W II. Since if W I > W II, then all depositors deposit at bank I, then for an individual depositor, W I = 2 wi + (1 2 )wa, W II = w II = w I > W I. An individual depositor will deviate to deposit at bank II. 42

43 So in this case, 2P I wi + (1 2P I )wa = 2P II wii + (1 2P II )wa, implies p I =p II = If w I = w II = w A, then clearly (p I [0, 1], p II [0, 1], p A [0, 1]; p I +p II +p A = 1) are equilibria, since depositors are indifferent between depositing at either bank or staying in autarky and they just randomize. To sum up, when < 1, according to, w I and w II, equilibria will be (p I = (wi w A ) 2(w II w A ), pii = 1 p I ) if 1 2 (wi w A ) 2(w II w A ) 1 (p I = w I w A (w I w A ) + (w II w A ), pii = w II w A (w I w A ) + (w II w A ) ), if (wi w A ) 2(w II w A ) < 1 2 (p I = 1, p II = 0), if (wi w A ) 2(w II w A ) > 1 (p I = 1, p II = p A = 0), if w I > w II = w A (p I =p II = 1 2 ), if wi = w II > w A (p I [0, 1], p II [0, 1], p A [0, 1]; p I +p II +p A = 1), if w I = w II = w A. 9.4 Math proof: w α < 0, c 1 α < 0. First order condition of equation (9) is 43

44 u (c 1 ) = u (c 2 )R. By Envelope Theorem, w α = u(c 1) (u(c 2) + (1 α)u (c 2) c 1R 1 α ) < 0, since c 1 < c 2; c 1 can be solved by applying Implicit Theorem onto first order condition, α u (c 1 ) c 1 α = Ru (c R(c 2 ) 1 + α c 1 α )(1 α) [(1 αc 1)R π]( 1), (1 α) 2 Which implies c 1 α = Rc 1(1 α) + [(1 αc 1)R π] (1 α) 2 Ru (c 2) u (c 1) + R2 α 1 α u (c 2) Rc 1 + R π u (c (1 α) 2)R = 2 < 0. Q.E.. u (c 1) + R2 α 1 α u (c 2) Thus, both w and c 1 are decreasing in α. 9.5 Tables and Figures Figure 9. Historical 5-Bank asset concentration, Graph from World Bank data. China is an exception, due to series of bank reform acts aiming at continually enhancing bank competition. 44

45 Table 1 Basel III Phase-in Arrangements Figure 10-12: indices of comeptition showing decrease in bank comeptition intensity as capital requirement of Base III phased in. Figure 10. Lerner Index of bank sector, world,

46 Figure 11. H-statistic of bank sector, world, Figure 12. Boone Indicator of bank sector, world, Measure of bank competition in current literature follows the new empirical industrial organization, which directly assess the competitive conduct of firms. There are three types of indices that are popularly used: the traditional Lerner Index, the Panzar and Rosse H-statistic (see Panzar and Rosse 1982, 1987), and the Booner Index (see Hay and Liu 1997; Boone 2001; Boone, Griffi th, and Harrison 2005). The Lerner Index is based on markups and defined as the difference between product prices and marginal costs, and a higher Lerner Index indicates that the industry is less competitive. The H-statistic shows elasticity of bank interest revenues to input prices, thus lower H-statistic indicates less competition. The Boone indicator calculates the elasticity of profits to marginal costs. It assumes more-effi cient banks achieve higher profits. Thus a less negative Boone indicator means a lower level of competition. All these indices show that competition in banking industry worldwide reduced after Basel III when new higher capital requirement started to phase in. All graphs are from Federal Reserve Economic ata, originally from the World Bank and based on the most updated data available. 46