Liquidity Provision. Tai-Wei Hu and Yiting Li. very, very preliminary, please do not circulate. Abstract

Transcription

1 Optial Banking Regulation with Endogenous Liquidity Provision Tai-Wei Hu and Yiting Li very, very preliinary, please do not circulate Abstract In a oney-search odel where deposits are used as eans-of-payents, banks have expertise to obtain higher returns fro assets with a cost and an econoy of scale but are subject to liited coitent and oral hazard. They can pledge a proportion of asset holdings to issue deposits. Optial regulation trades off efficiency in asset-anageent and liquidity service banks provide. An optial charter syste restricts banking licence to crate profits for banks to sustain a leverage ratio above the laissez-faire level to iprove liquidity. As oral hazard becoes ore serious, optial regulation allows banks to be larger and have higher profits to copensate for stricter capital requireent due to oral hazard. When banks are heterogenous, it is optial to allow higher leverage for larger banks. With uncertainty on bank returns, deposit insurance is optial as it akes bank liabilities inforation insensitive. Finally, with oral hazard under deposit insurance, we show that it is not always optial to exclude gabling behavior in equilibriu. Keywords: Capital requireent, banking, oral hazard, deposit insurance University of Bristol National Taiwan University 1

2 1 Introduction The serious interruption of the real econoy fro the Global Financial Crisis of 2008 has given rise to a renowned interest in understanding the role of financial interediaries and how to regulate the. Two particular issues have surfaced both in the ass edia and in the policy debate: first, bankers see to ake unjustified profits; 1 second, the banking sector see to be too concentrated in few big banks. These issues surfaced to the public doain partly because the banking sector has been under two governent protections: the deposit insurance that allows the to raise ore deposits, and governent bail-outs to any banking failures. These privileges see even ore unreasonable as it has been difficult to persecute any potential fraud in the sector. 2 While soe ay take for granted that these protections and regulations that lead to big banks and high profit are undesirable, to have a eaningful debate we need to first understand the role of financial interediaries in the working of the econoy and to understand why regulations ay be necessary. One particular aspect that ay result in externality which requires regulation is the liquidity role banks liabilities serve; 3 in ost advanced econoies, the ajority of oney supply consists of bank deposits. This role, which is ainly concerned with bank liabilities, otivates various regulations that proote stability, as bank failures would affect not only banks shareholders but also the welfare of the general public who rely on banks liquidity services. This liquidity service is provided by an asset transforation process: while banks issue deposits on the liability side, they also hold various assets to back those deposits. This process, as the financial crisis reveals, involves any credit ar- 1 In a coent about the Dodd-Frank refor, New Yorker article ( Banking s New Noral, 2016 issue) has argued that Bankers still ake absurd aounts of oney. 2 For a popular view on difficulty in such persecutions, see New Yorker article, Why Corrupt Bankers Avoid Jail, 2017 issue. 3 See, for exaple, the Controversies section in Econoic Journal, issue 106, May 1996, where all articles entioned that banking is special because they produce oney, or assets that can be used as eans-ofpayents. 2

3 ket frictions fro the banks, such as uncertainty to bank returns and liited coitent. The feature that banks supply liquidity and they are subject to frictions has iportant acroeconoic consequences, and thus, iplications on policy and regulations. In this paper, we propose a odel of financial interediaries with endogenous liquidity provision. We do this by introducing banks into a standard onetary odel à la Lagos and Wright (2005) to aintain tractability. On the asset side, banks are the only agents with the necessary expertise to anage/onitor long-ter loans (odeled as Lucas trees) to receive dividends. There is econoy of scale in the sector by way of a fixed cost of operation. On the liability side, banks ay issue deposits to finance their asset holdings, and, under the usual frictions that render eans-of-payents essential (lack of coitent and onitoring) fro the depositor side, this can generate a higher profits to banks by doing so. We consider two ain frictions in the banking sector. First, banks cannot fully coit to honor their future obligations; instead, they could only credibly pledge a fraction of their assets that can be seized by the court upon bankruptcy. This friction constraints the aount of liquidity banks can provide and ay prevent the first-best level of consuption for the depositors to be achieved. Second, the banks efforts in anaging the assets ay not be observable and this oral hazard issue ay hinder the liquidity role of the banks. We first consider the liited coitent of the banks but no oral hazard. When banks can only ake static contracts, the aount of deposits a bank can issue is constrained by liited pledgeability of assets through arket discipline; no one would deposit in a bank unless it can credibly repay. Under free entry of banks, bank sizes are deterined by a zero-profit condition that balances the variable cost of asset anageent and the fixed cost of entry, which coincide with the efficient level of asset holdings as far as the asset-anageent is concerned for the econoy of scale. However, unless the pledgeability constraint is slack, depositors cannot achieve first-best level of consuption due to lack of liquidity and asset pricing exhibits liquidity preiu. This pledgeability constraint also 3

4 iplies a capital requireent ipose by the arket: the bank will not repay any deposit beyond what is pledgeable because of liited coitent and hence the difference has to be financed by bank capital. Against this free arket arrangeent, we show that a charter syste with a banking regulator can iprove social welfare. Under the charter syste, the regulator can shut down a bank when it does not honor its obligation and hence allows for a dynaic incentive to relax the pledgeability constraint. For this dynaic incentive to be effective, however, it is necessary to liit the nuber of charters relative to the efficient nuber under free entry and to allow banks to earn econoic profits. This schee akes it incentive feasible for banks to issue unsecured deposits beyond the pledgable assets they own, and hence can increase the leverage ratio of banks. The optial policy then trades off two inefficiencies: on the one hand, a saller nuber of charters increases bank profits and hence helps increase liquidity, which iproves depositors welfare; on the other hand, a saller nuber of charters increases the overall cost of banking operations as each one gets inefficiently large. Our ain result deonstrate that, whenever liquidity is tight under static bank contracts, it is optial to liit the nuber of charters relative to the nuber under efficient asset anageent, and to relax the pledgeability constraint through a lower overall leverage ratio requireent that its laissez faire level. We then introduce the oral hazard issue with asset-anageent by banks, in which banks ay shirk on their asset anageent and obtain a lower return, although doing so would be socially suboptial. Precisely because of the two-sided nature of bank contracts, banks ay have incentives to shirk, as that ay increase their profits by lowering the cost while the depositors have to suffer the consequences. We show that, under static contract and free-arket arrangeent, arket discipline would ipose an additional proportional capital requireent to a bank s asset holding to ensure efforts. This, however, can be harful to liquidity provision as it lowers the level of deposits banks can offer. In particular, as 4

5 oral hazard becoes ore serious, the liquidity service becoes poorer. In contrast, under the optial charter syste, although an additional proportional capital requireent is also necessary, the regulator would adjust the overall leverage ratio requireent as well as the nuber of licences to copensate for the liquidity loss. In particular, we show that as oral hazard becoes ore serious, the optial response is to allow for higher profits to sustain a higher unsecured deposit issuance, and to ake each banks larger. We extend our odel to address two policy debates. The first is the optial sizes of banks. In our odel, bank sizes are endogenously deterined by either free entry (in the absence of charter), or by the nuber of charters. We extend our odel by allowing for heterogeneous anageent costs. In the absence of regulation, ore efficient banks end up being larger in ters of asset holding, and it is efficient to do so. When liquidity is tight, we show that an optial charter syste would in fact ake large banks even larger by allowing ore generous unsecured borrowing. The intuition is siple: when the nuber of charters is liited, large banks ake higher profits and hence it is ore efficient to incentivise the to repay unsecured deposits. As a result, we obtain a positive correlation between bank size and leverage ratio under the optial policy arrangeent. Second, we consider dividend uncertainty where the return to each bank s assets are subject to an idiosyncratic shock that affects all assets the bank holds. Moreover, we assue that depositors receive noisy signals regarding the shocks and hence the value of the deposits ay be affected when used as eans-of-payents. We show that as the noisy signal becoes ore precise, the presence of shocks becoes a bigger hinderance to liquidity provision. A deposit insurance schee that charge a preiu on bank returns to bail out troubled banks, however, can ake deposit contracts inforation insensitive again, and hence iprove welfare. Nevertheless, as argued in the literature, deposit insurance ay further intensify the oral hazard issue. To understand this issue, we introduce oral hazard in the following way: banks ay secretly direct the asset to a ore risky projects that have higher return to 5

6 the bank that is not observable to the general public (and hence not subject to repayent to depositors) but have lower overall expected return. In the absence of a further capital requireent, there exists an equilibriu where all banks gable. However, introducing a harsh capital requireent faces a new trade-off that is not in the literature: on the one hand, it discourages gabling and hence increase overall return; on the other hand, it directly decreases liquidity provision. We show that how the trade-off resolves would depend on the fundaentals. Literature review This paper is not the first one to point out that future bank profits play an essential role in banking regulations. On the epirical side, Keeley (1990) provides soe evidence that charter value restricts banks risk-taking behavior. On the theory side, Hellann, Murdock, and Stiglitz (2000), in a odel where banks have arket powers and face oral hazard, show that it is optial to use a cobination of capital requireent and deposit-rate ceilings to create sufficient franchise value for banks to aeliorate the oral-hazard proble. Future bank profits are the ain incentive device for prudent behavior, using deposit-rate ceiling to aintain profits. In contrast, profits are aintained by restricting entry and deposit-rate ceilings would be sub-optial in our odel. Two ain odeling ingredients explain the difference: first, while deposit deand is exogenously given there, in our odel it is driven by endogenous liquidity needs; second, asset prices (and hence returns to the loans) are endogenously deterined in our odel. Our paper is also related to the literature on liquidity provision by banks. Using a eansof-payent-in-advance odel with currency and deposits, Chari and Phelan (2014) show that if there is insufficient deflation, fractionally backed banks which offer interest-bearing deposits ay be good, but such banks are subject to socially costly runs. Williason (2016), shows that, when banks face liited coitent, and when short-aturity governent debt has 6

7 a greater degree of pledgeability than long-aturity governent debt, quantitative easing can iprove liquidity. These papers, however, do not address optial financial regulations. Gorton and Winton (2017) also features a trade-off of raising capital requireent because bank debt is used for transactions purposes, while ore bank capital can reduce the chance of bank failure; however, they assue exogenously given banks charter value. Phelan (2016), in a odel where deposits serve the liquidity function, shows that leverage also increases asset price volatility and so liiting leverage decreases the likelihood that the financial sector is undercapitalized. However, the odel assues that deposits exogenously generate utility to depositors, and hence it is then not clear how regulations ay affect banks function in providing eans-of-payent, and through which, the econoic activity. Our rationale for deposit insurance differs fro the usual Diaond-Dybvig (1983) otive but is related to Gorton, Holstro, Ordonez (2017), who show that, to produce oneylike safe liquidity, banks keep detailed inforation about their loans secret, which provides a rationale for opaque banking exainations and capital requireents, and deposit insurance. With a siilar arguent, we show that deposit insurance akes deposit contract insensitive to inforation. This otive also iplies a different policy recoendation; in contrast to ost of the literature, we show that once we take endogenous liquidity provision into account, it is not necessary desirable to eliinate gabling behavior. 2 The Environent The environent is borrowed fro Rocheteau and Wright (2005). Tie is discrete and has an infinite horizon, t N 0. The econoy is populated by three sets of agents; each set has a continuu of infinitely-lived agents with easure one. The first set consists of buyers, denoted by B, and the second consists of sellers, denoted by S. The third set consists of potential banks. Each date has two stages: the first has pairwise eetings of buyers and 7

8 sellers in a decentralized arket (called the DM), and the second has centralized eetings (called the CM) where all agents eet. In each DM, the probability that a buyer has a successful eeting with a seller is σ. There is a single perishable good produced in each stage, with the CM good taken as the nuéraire. Agents labels as buyers and sellers depend on their roles in the DM where only sellers are able to produce and only buyers wish to consue. While all agents can produce and consue in the CM, potential banks do not consue nor produce in the DM. Buyers preferences are represented by the following utility function E β t [u(q t ) + x t h t ], t=0 where β (1 + r) 1 (0, 1) is the discount factor, q t is DM consuption, x t is CM consuption, and h t is the supply of hours in the CM. Sellers preferences are given by E β t [ c(q t ) + x t h t ], t=0 where c(q) is the seller s disutility of producing q in the DM. The first-stage utility functions, u(q) and c(q), are increasing and concave, with u(0) = υ(0) = 0. The surplus function, u(q) c(q), is strictly concave, with q = arg ax [u(q) c(q)]. Moreover, u (0) = c ( ) = and c (0) = u ( ) = 0. All agents have access to a linear technology to produce the CM output fro their own labor, x = h. There is only one type of real assets, Lucas trees, which are long-lived. Each unit of the trees pays off dividend τ units of the CM goods at the beginning of CM. The average supply (per buyer) of the trees is Ā. To receive dividends fro the Lucas trees, however, it requires a potential bank to perfor costly onitoring/anageent. This assuption is in line with the delegated onitoring odel of financial interediaries proposed by Williason 8

9 (1986) or Diaond (1984). 4 In contrast to those papers, however, our ain focus is on the role of banks in providing liquid deposits as eans-of-payents. Specifically, buyers ay use the bank s liability, or deposits, to finance their consuptions in the DM. Sellers have a technology that can access the records in the bank, and, upon buyers agreeent, ay transfer deposits to the sellers accounts in the bank. For instance, in a DM transaction the buyer gives to the seller a clai on deposits, which the seller ay present in the CM to the bank to acquire funds. There are two frictions associated with this financial interediation. The first friction is the cost associated with anaging/onitoring the asset. Only active banks can hold assets and issue deposits; to becoe active, a bank has to pay a fixed cost of γ each period. There is also a arginal cost of asset-anageent: for a banker to hold a units of assets, he needs to pay ψ(a) (as a labor cost) to onitor/anage the assets. We assue that ψ(0) = 0, ψ(a) is strictly increasing and strictly convex, and ψ(ā) =. Second, banks have liited liability and cannot coit to their future actions. However, we assue that if a bank files for bankruptcy, the court could seize ρ proportion of his assets. Thus, by holding a units of Lucas trees, a bank can credibly pledge ρ fraction of its value (plus dividends) but can take the rest away and sell the on the open arket. Banks axiize their life-tie profits. 3 Bank contracts The course of events. In the CM, the course of events is as follows: 1. first, banks settle deposit obligations with depositors; 4 Of course, in those odels one needs to introduce asyetric inforation between borrowers and lenders to give the financial interediaries a role, while here the return of the Lucas trees is certain. One can interpret the return of Lucas trees here as the diversified return in those odels where each bank represents a large nuber of depositors. 9

10 2. then, banks buys Lucas trees in copetitive arket at price φ (in ters of CM good); 3. finally, banks ay issue deposit contract, proising a gross return R (in exchange for CM good). We use d to denote the total aount of deposits that the bank proises to give out in the next CM (and hence it will receive d/r in the current CM). Note that there are two different arkets in the CM a spot arket for deposits, and a spot arket for assets. Because only banks can anage Lucas trees to receive dividends, with no loss of generality we assue that buyers and sellers do not participate in the asset arket. 5 We also assue that only buyers enter the deposit contracts in the CM but not sellers. In the DM, upon a successful eeting with a seller, the buyer akes a take-it-or-leaveit offer, (q, z), where q is the DM consuption and z is the aount of deposit (in ters of the coing CM goods) transfer. This is feasible because, as entioned earlier, there is record-keeping technology under which the accounts of the buyer can be transferred to the seller. 3.1 Static bank contracts Here we consider the case where the free entry of banks iplies a zero-profit condition, which in turn iplies that banks cannot credibly proise any aount beyond what could be seized by the court. As a benchark, we first begin with the situation where banks cannot issue deposits at all. In this case, the price of the Lucas trees can be easily pinned down by a no arbitrage condition (i.e., banks profit-axiizing condition) and the nuber of banks pinned down 5 We iplicitly assue that there is no friction within the two spot arkets in the sense that all agents (especially buyers and bankers) can ake proises to deliver the CM goods within the sae-date CM stage when aking the portfolio decisions, and hence, as usual in Lagos and Wright (2005) fraeworks, the tiing of the trades within CM does not atter and we can work with the net consuption in the CM for various agents. 10

11 by free entry. It is convenient to define Π(A) = ψ (A)A ψ(a). (1) We assue that [ ( )] τ Π(Ā) < γ < Π (ψ ) 1. (2) 1 + r Assuption (2) ensures that there is sufficient entry to the banking sector. Indeed, as will be clear below, Π(A) γ will be the profit for a bank with A units of trees. Free entry then requires banks to hold A = Π 1 (γ) and hence only a easure = Ā/Π 1 (γ) of banks will enter. (2) ensures that < 1 and hence a unit easure of banks is sufficient to provide free entry. Note that for a social planner who wants to iniize the cost of asset anageent will solve in γ + ψ(ā/). (3) 0 The easure also solves this proble. We ay define the fundaental value of the trees as φ = τ (1 + r)ψ ( Ā ) r, (4) which will be the price for the asset if the banks cannot issue any deposits. For expositional purposes, we define a variable, ι 1 + r R 1. Given R (and hence ι) and φ, and for a given asset holding, a, and a given deposit 11

12 issuance, d, (in ters of next CM proised value), a bank s profit is given by π(a, d; φ, R) = d φa γ ψ(a) + β{(φ + τ)a d} (5) R = β {ιd (rφ τ)a (1 + r)[ψ(a) + γ]}, and is subject to the pledgeability constraint, d ρ(φ + τ)a. (6) As entioned, under static contracts, banks can only pledge what could be seized by the court, naely, ρ fraction of the value of their assets; (6) captures this constraint. Let A(φ, ι) be the optial asset holding that axiizes (5) subject to (6). Note that whenever ι > 0, the constraint (6) is binding and A(φ, ι) is deterined by the following FOC: (r ιρ)φ + (1 + ιρ)τ = (1 + r)ψ (a). (7) When the pledgeability constraint is binding, the bank needs to own capital, φa d R, to finance soe of its asset holdings. Now we turn to depositors behavior. Given R, a depositor s proble is given by where ax d + β {σ[u(q(d)) c(q(d))] + d}, (8) d 0 R c(q(d)) = d if d < c(q ) and q(d) = q otherwise. Note that d is the proised value of the deposit in the coing CM. The FOC to (8) is ι = σ[u (q(d)) c (q(d))]. (9) c (q(d)) 12

13 Let D(ι), the deposit deand per depositor, be the solution to (9). Note that for any ι > 0, D(ι) is uniquely deterined; when ι = 0, D(ι) is not pinned down but D(ι) c(q ). Without loss of generality we ay take D(0) as its iniu. Then, D(ι) is continuous and strictly decreasing in ι. Equilibriu then requires arket clearing conditions for deposits and assets: D(ι) = ρ(τ + φ)ā; (10) A(ι, φ) = Ā. (11) Finally, free-entry iplies that all active banks have to have zero profits. Lea 3.1. There is a unique equilibriu allocation, (, φ, ι, q, d), in which =, and (φ, ι, q, d) is characterized as follows. (a) Suppose that ρ 1 + r r [ ( )] Ā τ ψ Ā c(q ). (12) Then, φ = φ, q = q, and ι = 0. (b) Suppose that (12) does not hold. Then, φ = ( ) (ιρ + 1)τ ψ Ā (1 + r), (13) r ιρ with q = c 1 (D(ι)) < q and with ι (0, r ) as the unique solution to ρ ( τ ψ Ā D(ι) = ρ(1 + r) r ιρ ) Ā. (14) Condition (12) gives a precise condition for the first-best trades to occur in DM in equilibriu. It is obtained by plugging the fundaental value φ given by (4) into (6), which 13

14 iplies that the aount of deposits banks can credibly issue when assets are valued at their fundaental values is ore than enough to buy q for each buyer. In this case the net return to depositors is r. When (12) fails, Lea 3.1 (b) shows that φ will be higher than the fundaental price, and hence the asset price exhibits liquidity preiu, and R will be lower than 1+r. Finally, note that since always equals under free entry, there is distortion in asset-anageent; all the potential inefficiency coes fro insufficient liquidity provision. Here we show that, whenever the first-best is not ipleentable, higher pledgeability leads to higher welfare. Given the easure of banks,, and an allocation, (φ, ι, q, d), the associated welfare is given by W = σ[u(q) c(q)] ψ(ā/) γ. (15) Fro (8), if the first-best is not obtained, c(q(d)) = d. Using (14), d = D(ι), and given ( ) τ ψ Ā (1 + r), we have [ ] [ ( W r(1 + r)σ u (q(d)) 1 τ ψ Ā ρ = c (q(d)) )] Ā > 0. (r ιρ) 2 Indeed, welfare is increased by the pledgeability. For our discussion below, it is convenient to define a threshold of ρ, ρ, such that R 1, or, equivalently, ι r. If R < 1, then the governent could ake buyers better off by introducing fiat oney. Let y(d) = u (q(d))/c (q(d)). Then, (9) iplies that ( ) σ + ι D(ι) = y 1. σ 14

15 Thus, the solution to (14) satisfies ι r if and only if [ ( )] ( ) r + σ τ ψ Ā y 1 Ā ρ(1 + r), σ r rρ which is equivalent to ρ ρ y ( ) 1 r+σ σ [ ( )] (1+r) τ ψ Ā r Ā + y 1 ( r+σ σ ). (16) 3.2 The charter syste and dynaic bank contracts Here we introduce the charter syste with a banking authority or regulator. Under the charter syste, there are two policy paraeters for the regulator. The first paraeter is the nuber of banking licences, denoted by. The second is the aount of deposit issuance beyond what is allowed by the pledgeability constraint under static contract, denoted by κ. As entioned, since the pledgeability constraint, (6), effectively iplies that banks need to hold soe capital to finance their asset holdings, the paraeter κ ay also be interpreted as a requireent on the overall leverage ratio for a bank. Given κ, the pledgeability constraint (6) is thus odified to d ρ(φ + τ)a + κ. (17) The two policy paraeters, and κ, are intiately connected in the incentive provision for banks. As we shall deonstrate later, when <, chartered banks ake positive profits that will be lost if the charter is terinated. Hence, by terinating chartered banks who fail to repay their deposit obligations, the regulator can use future profits as a discipline device to ensure repayents beyond what can be seized by the court. Thus, to be incentive copatible, the aount κ has to be consistent with the equilibriu bank profits, which is deterined by. Each chartered bank still axiizes the profit given by (5), but subject to (17). Note 15

16 that the asset deand A(φ, ι) is independent of κ. A depositor s proble is still given by (8), and deposit deand, D(ι), reains the sae. For given κ and, the arket-clearing conditions are given by D(ι) = ρ(τ + φ)ā + κ; (18) A(φ, ι) = Ā. (19) Moreover, we only consider s that satisfy ( ) Ā τ ψ (1 + r). (20) By (2) and convexity of ψ, there exists a unique < such that (20) is satisfied for all [, ]. It can be verified that it is never optial to have <. Lea 3.2. Let [, ] and let κ 0 be given. There is a unique allocation (φ, ι, q, d) that satisfies the arket-clearing conditions that can be characterized as follows: φ = ( ) (ιρ + 1)τ ψ Ā (1 + r), (21) r ιρ and q = c 1 (D(ι)) with ι = ι(, κ) [0, r ) as the unique ι such that ι 0 and ρ D(ι) ρ(1 + r) ( ) τ ψ Ā Ā + κ, (22) r ιρ ( ) with equality whenever ι > 0. The profit for each bank is given by Π Ā γ + ι κ. 1+r It is straightforward to verify that ι(, κ) = 0 if and only if ρ 1 + r r [ ( )] Ā τ ψ Ā + κ c(q ). (23) 16

17 Lea 3.2 then generalizes Lea 3.1, the latter being a special case of the forer with κ = 0. When (23) holds, we also have q = q and φ = φ ; otherwise, we have q < q and φ > φ. The issuance of unsecured debt, κ, affects the equilibriu deposit holdings, as well as bank s profits. This iplies that κ has effects on the efficiency of both DM trades and asset anageents. Now we turn to the incentive copatibility of κ. Since the court can only seize ρ proportion of a bank s asset, the bank has teptation not to repay the κ coponent of his liability in (17). To deter this teptation, the regulator can reove the bank charter and stop the bank fro future business if the bank fails to honor his deposit obligations. Thus, if a bank defaults, he loses the pledged assets, ρ(φ + τ)ā/, as well as the charter to run the business, beginning fro the period when he fails to repay. As a result, a bank is willing to repay deposits if and only if κ ρ(φ + τ)ā/ + ( ) Ā β [Π t γ + ι κ ] ρ(φ r τ)ā/. t=0 This constraint can be siplified as [ ( ) ] Ā (r ι)κ + (1 + r) Π γ 0. (24) We have the following definition. Definition 3.1. A policy, (, κ) [, ] R +, is ipleentable if (24) holds for ι = ι(, κ). We have the following theore. Theore 3.1. Let [, ] be given. Then, there exists a greatest κ, denoted κ(), such that (, κ()) is ipleentable and whose allocation has the highest welfare aong all ipleentable policies for the given. 17

18 Theore 3.1 gives a full characterization of the best ipleentable equilibriu outcoe for a given nuber of charters,. The regulator s goal, however, is to design (, κ) such that the social welfare is axiized. Note that by Lea 3.2, for any given (, κ), there is a unique allocation (φ, ι, q, d) that satisfies the arket-clearing conditions (18)- (19). However, the regulator is constrained by banks repayent incentive constraint, (24). Theore 3.2. Assue (A0). There exists an optial policy (, κ) that axiizes (15) subject to ipleentability. (a) If (12) holds, then (, κ) = (, 0) is an optial policy. (b) Suppose that (12) does not hold and that ρ ( ρ, 1). Then, any optial policy has < and κ > 0. Theore 3.2 shows that, when designing an optial charter syste, the regulator has to balance efficiency in asset anageent and efficiency in liquidity provision. When there is abundant pledgable assets so that (12) holds, full efficiency can be achieved on both aspects, according to Theore 3.2 (a). Otherwise, according to Theore 3.2 (b), the constrained efficient arrangeent has to sacrifice full efficiency on both aspects. Restricting the nuber of charters reduces copetition and increases banks profits; this is suboptial regarding efficiency in asset-anageent. However, higher profits ake it easier for banks incentive constraint, (24), to hold and allow for a positive κ without having the banks defaulting on their debts. Thus, this financial stability in our fraework is possible because of positive profits banks enjoy, and it is useful to enhance social welfare because banks provides liquidity services as their liabilities are used as eans-of-payents. Since the optial κ regulates the aount of deposits a bank can issue through the pledgeability constraint (17), one can interpret the policy paraeter κ as an overall leverage ratio requireent. In contrast to the coon capital requireents that depend only on the asset characteristics a bank holds, optial κ also depends on other bank characteristics such 18

19 as ρ (proportion of asset that can be secured for repayent) and ψ (arginal cost of assetanageent). Our fraework then iplies a holistic approach to capital requireent that would take both idiosyncratic feature of a specific bank as well as the global environent into account when designing the optial capital regulations. 3.3 Moral hazard We have shown that when designing an optial policy in the presence of liited coitent of banks, the regulator needs to trade off efficiency and stability. Here we introduce another friction that is ore akin the conventional oral hazard issue discussed in the literature. Our ain focus is to what extent the copetitive arket can correct this issue and how this issue would interact with the optial overall leverage ratio requireents we obtained in the last section. Suppose that the dividends of Lucas trees that a bank holds are subject to oral hazard. By shirking the cost of anaging a units of assets is ψ(a) ea + γ, but the return is lower; it will be τ 0 < τ. The decisions to shirk are not observable, but the realized returns are. We assue that (1 + r)e < τ τ 0. (25) Condition (25) ensures that putting effort is socially beneficial. First we begin with the case where there is no regulation and hence only static contracts are feasible. We shall ipose free entry, but for now assue that the nuber of active banks is given by a fixed. Even in the absence of regulation, depositors can potentially discipline the bank to exert efforts by not depositing in a bank without sufficient capital in place. Of course, there could potentially be an equilibriu where all banks shirk and depositors, by rational expectation, understand this and deand a ore stringent capital requireent. Indeed, if a bank with a units of trees shirks, the return will then be τ 0 and hence would 19

20 only pay out the bank will only pay out d 0 = ρ(φ + τ 0 )a (26) to the depositors. As a result, in such an equilibriu, the pledgeability constraint would then be given by d ρ(φ + τ 0 )a. (27) The following lea, however, shows that such an equilibriu does not exist under assuption (25). Lea 3.3. Consider the static contract. There is no equilibriu with all banks shirking. Lea 3.3 is shown by a siple contrapositive arguent. If an equilibriu exists where all banks shirk, then depositors would deand the pledgeability constraint (27). However, under such a constraint, a bank receives all the additional return by exerting efforts, and hence (25) iplies that all banks are willing to exert efforts. This leads to a contradiction. By Lea 3.3, we can focus only on equilibria where all banks exert efforts. In this case, the constraint (6) ay no longer be appropriate as it ay not induce efforts. Instead, a ore general pledgeability constraint is needed: d ρ(φ + τ 0 )a + ω(τ τ 0 )a, (28) for soe ω [0, ρ]. Note that when ω = 0, (28) coincides with (27); when ω = ρ, (28) coincides with (6). The paraeter ω also has a siple interpretation: 1 ω stands for the share of the additional return that goes to the bank by exerting efforts. As argued earlier, when ω = 0 all banks are willing to exert efforts. Here we give a reark about what we ean by equilibriu under oral hazard. As entioned, here the pledgeability constraint is endogenous in the sense that a bank who reaches 20

21 the constraint cannot credibly issue ore deposits. Thus, in equilibriu, the pledgeability constraint (28) ust satisfy two conditions: first, it has to ensure that the banks are willing to exert efforts; second, it cannot be the case that any bank can credibly issue ore deposits than what the constraint requires. Equivalently, equilibriu requires the highest ω that is consistent with banks exerting effort under the constraint. To do this, we can odify our previous analysis and obtain arket clearing conditions. Recall that we assue a fixed nuber of active banks,. By exerting efforts, the bank profit is obtained by substituting (28) at equality into (5): π(a, d; φ, R) = d φa [ψ(a) + γ] + β{(φ + τ)a d} (29) R = β { (r ιρ)φa + [(τ + ιρτ 0 ) + ιω(τ τ 0 )]a (1 + r)[ψ(a) + γ]}. The FOC for (29) is thus (r ιρ)φ + (τ + ιρτ 0 ) + ιω(τ τ 0 ) = (1 + r)ψ (a). Thus, the equilibriu price for trees is pinned down by arket-clearing, a = Ā/: φ = ( ) (1 + ιω)τ + ι(ρ ω)τ 0 ψ Ā (1 + r) ; (30) r ιρ Note that, as before, the bank profit is then given by Π(Ā/) γ. Given φ, the equilibriu ι, denoted by ι (0, r ), is then the unique solution to (with equality whenever ι > 0) ρ D(ι) ( ) ρτ + rρτ 0 + ωr(τ τ 0 ) ρ(1 + r)ψ Ā Ā. (31) r ιρ Finally, we also need to consider the profit to a bank if he shirks, taking φ as given. Recall 21

22 that a shirking bank only pays d 0 = ρ(φ + τ 0 )a to depositors under return τ 0 ; hence, the bank profit is given by π s (a, d; φ, R) = d R φa [ψ(a) ea + γ] + β{(φ + τ 0)a d 0 } (32) (r ιρ)φa + [(τ 0 + ιρτ 0 ) + ιω(τ τ 0 ) + ω(τ τ 0 )]a = β (1 + r)[ψ(a) ea + γ]. The FOC iplies that the asset holding for a shirking bank is given by A s that solves (r ιρ)φ + (τ + ιρτ 0 ) + (ι + 1)ω(τ τ 0 ) = (1 + r)ψ (A s ) e. Hence, the bank profit under shirking is given by Π(A s ) γ. Thus, to ensure that banks have no incentive to shirk, we the following condition: Π(Ā/) Π(As ) 0. (33) To suarize, equilibriu conditions the consist (30), (31), and (33). We have the following lea. Lea 3.4. Consider the static contract and let be given. The highest ω under which all banks exert efforts in equilibriu is given by { ω 1 in 1 } (1 + r)e, ρ. (34) τ τ 0 Lea 3.4 shows that the arket can discipline banks to exert effort by deanding additional capital requireent paraeterized by ω 1. When ρ is relatively sall, i.e., when ρ 1 (1+r)e τ τ 0, ω 1 = ρ and the presence of oral hazard does not affect the equilibriu allocation. In contrast, when ρ is relatively sall and hence ω 1 < ρ, the presence of oral 22

23 hazard does liit the ability of the banks to provide liquidity. Under fee entry, we can use the sae arguents before and derive the equilibriu would be. Charter syste with oral hazard Now we turn to the charter syste with oral hazard. Relative to the literature, the novelty here is to study the two capital regulations together, one paraeterized by ω and the other by κ. Under the charter syste with oral hazard, the general pledgeability constraint is given by: d ρ(φ + τ 0 )a + ω(τ τ 0 )a + κ. (35) We reark here that Lea 3.3 can be generalized in this dynaic setting, and hence it is without loss of generality to consider only equilibria with all banks exerting efforts. Thus, the policy paraeter now becoes (, κ, ω). Now we ove to equilibriu analysis for a given policy paraeter. By exerting efforts, the bank profit is given by: π(a, d; φ, R) = d φa [ψ(a) + γ] + β{(φ + τ)a d} (36) R = β { (r ιρ)φa + [(τ + ιρτ 0 ) + ιω(τ τ 0 )]a + ικ (1 + r)[ψ(a) + γ]}. Note that the only difference between (29) and (36) is the ter βικ, they share the sae FOC s and hence the equilibriu φ is still given by (30). The profit to each bank in equilibriu is then Π(Ā/) γ + βικ. Given φ, the equilibriu ι, denoted by ι(, κ, ω) (0, r ), ρ is then the unique solution to (with equality whenever ι > 0) D(ι) ( ) ρτ + rρτ 0 + ωr(τ τ 0 ) ρ(1 + r)ψ Ā Ā + κ. (37) r ιρ 23

24 Again, note that the only difference between (31) and (37) is the ter κ. Now we turn the incentive copatibility of banks to exert efforts and to repay κ. We assue that banks with return τ 0 will have their charters terinated and it is easy to see that this is the optial punishent. Thus, a shirking bank only pays d 0 given by (26) to depositors under return τ 0. Thus, the profit to a shirking bank is given by π s (a, d; φ, R) = d R φa [ψ(a) ea + γ] + β{(φ + τ 0)a d 0 } (38) (r ιρ)φa + [(τ 0 + ιρτ 0 ) + ιω(τ τ 0 ) + ω(τ τ 0 )]a = β +(1 + ι)κ (1 + r)[ψ(a) ea + γ]. Again, note that the only difference between (32) and (38) is the ter β(1 + ι)κ and hence has no bearings on FOC; so the optial asset holding is still given by A s and the profit is Π(A s ) γ + β(1 + ι)κ. To ensure that banks follow equilibriu behavior, we have two incentive copatibility conditions, one for repaying κ, the other for exerting efforts. Since we assue that in equilibriu all banks exert effort, the first one is the sae as before, (24); note that, however, equilibriu ι is affected by ω through (37). The second condition is new and is given by [ Π(A s ) Π(Ā/) + βκ] + β [ ( ) Ā Π γ + ι κ ] 0. (39) 1 β 1 + r Theore 3.3. Let < be given. Suppose that ρ (0, 1). The optial capital requireent is such that ω = ω 1 given by (34), and κ is the highest κ that satisfies (24) with ι deterined by (37) and with ω = ω 1. Since Theore 3.3 holds for any given, it follows that we can extend Theore 3.2 to the case with oral hazard. In particular, Theore 3.3 states that the highest κ exists for which (24) holds with ι deterined by (37) and with ω = ω 1. One can then solve for the 24

25 optial and, as in Theore 3.2, we will have < and κ > 0 unless the first-best is ipleentable under = and κ = 0, as well as ω = ω 1. Copared against Lea 3.4, Theore 3.3 shows that under the charter syste the optial ω is the sae as that under arket-discipline, and that it is optial to use the dynaic incentive to increase κ and κ only. Note that, however, optial κ is indeed affected by oral hazard, since the choice of ω does affect the aount banks can provide through asset prices and returns on deposits. The following theore shows that when oral hazard affects welfare, it is in fact to allow bank profits to increase and hence to allow for higher unsecured lending to increase liquidity. Theore 3.4. Suppose that ω 1 < ρ and that the first-best is not ipleentable. Fix soe <. Then, when e increases, optial κ increases and banks ake higher profits under the optial arrangeent. As e increases and hence the oral hazard issue becoes ore serious, ω 1 increases as well by Theore 3.3. This directly decreases the aount of liquidity banks can provide. However, Theore 3.4 shows that the optial response to such change is to increase κ by allowing for higher profits to the banks. This iplies a nontrivial interaction between the conventional capital requireent designed to counter the oral hazard issue and the overall leverage ratio requireent in our charter syste that ais to balance stability and liquidity. Crucially, this result follows fro our explicit treatent of liquidity provision fro banks. 4 Heterogeneity, Leverage, and Deposit Insurance In this section we introduce heterogeneity across banks. We consider two types of heterogeneity. The first is concerned with efficiency of different banks. The ain focus is on the distribution of bank sizes and optial charter syste when banks differ in sizes. The second 25

26 is concerned with idiosyncratic shocks to individual bank returns. In this case, our ain focus is on how inforation leakage about bank returns affect liquidity provision fro banks. 4.1 Heterogenous bank sizes and profits Here we consider heterogenous banks in ters of their efficiency in asset anageent. Specifically, for each n {1,..., N}, the econoy has easure µ n of type-n banks with N n=1 µ n = 1, and the cost function for a bank of type-n is λ n ψ(a) + γ. The paraeter λ n is then a easureent of how efficient type-n banks are in ters of asset anageent. We assue that λ n [1, λ] is strictly increasing in n, and hence type-1 banks are the ost efficient ones while type-n are the least efficient ones. First we consider efficient asset anageent in this environent. Without deposit issuance, efficient asset anageent requires the easures of type-n active banks, denoted by n, to solve in n [0,µ n],a n 0,=1,...,N s.t. N n A n = Ā. n=1 N [ n γ + λ n ψ(a n )] (40) n=1 Parallel to (2), to ensure that there is sufficient entry we assue that N µ n Π 1 (γ/λ n ) > Ā. (41) n=1 To characterize the solution to (40), first for each = ( 1,..., N ) with 1 > 0, we define {A n ()} N n=1 as the solution to N n A n = Ā, λ 1ψ (A 1 ) = λ n ψ (A n ) if n > 0, A n = 0 otherwise. (42) n=1 26

27 We have the following clai. Clai 4.1. Assue (41). The solution to (40) is unique, denoted by, is characterized by n {1,..., N} and 0 < n µ n such that = (µ 1,.., µ n 1, n, 0..., 0), (43) λ n Π(A n ( )) γ, for all n = 1,..., n, (44) λ n Π(A n ( )) = γ if n < µ n, (45) λ n Π(A n ( )) < γ, for all n = n + 1,..., N. (46) Now we consider static contracts with free entry. Given R (and hence ι) and φ, and for a given asset holding, a, and deposits giving out, d, (in ters of next CM proised value), the profit of a type-n bank is given by π n (a, d; φ, R) = β {ιd (rφ τ)a (1 + r)[λ n ψ(a) + γ]}, and is subject to the pledgeability constraint, d ρ(φ + τ)a. (47) This gives rise to a well-defined asset deand A n (φ, ι) deterined by the following FOC: (r ιρ)φ + (1 + ιρ)τ = (1 + r)λ n ψ (a). (48) That is, A n (φ, ι) = (ψ ) 1 ( (r ιρ)φ + (1 + ιρ)τ (1 + r)λ n ). (49) 27

28 Let φ be the unique solution to N n=1 n A n (φ, 0) = Ā. As before, we ay call φ the fundaental value of the asset, the price for the trees if banks were not allowed to issue deposits; in that situation the easures of active banks would be given by. Let n be the easure of active type-n banks, n = 1,..., N. Then, equilibriu objects include asset price φ, returns to deposits ι, and the easure of active type-n banks, n for each n = 1,..., N ( n = 0 eans that no type-n bank is active). The arket-clearing conditions and free entry condition are given by (note that D(ι) is still given by (9)) D(ι) = ρ(τ + φ)ā; (50) N n A n (φ, ι) = Ā; (51) n=1 λ n Π[A n (φ, ι)] γ if n > 0, λ n Π[A n (φ, ι)] γ if n < µ n. (52) We have the following lea. Lea 4.1. Assue (41). There exists a unique equilibriu. The equilibriu easures of active banks are given by and equilibriu asset holding is given by A n ( ) according to (42) for type-n banks. If the equilibriu DM production is q, then equilibriu φ = φ ; otherwise, equilibriu φ > φ. Lea 4.1 shows that even with heterogenous banks, the result that without regulation efficiency of asset-anageent is achieved still holds. However, here we obtain an endogenous distribution of bank balance sheets. Specifically, (42) iplies that A n ( ) > A n+1 ( ) for all n = 1,..., n 1, and hence, under free entry, ore efficient banks are also larger in ters of asset holdings. Moreover, the FOC also iplies that the profit for bank of type-n is given by 28

29 λ n Π[A n (φ, ι)] γ, and hence Clai 4.1 iplies that even under the efficient arrangeent for asset anageent, soe banks ay ake positive profits. Strict convexity also iplies that λ n Π(A n ( )) > λ n+1 Π(A n+1 ( )) for all n = 1,..., n 1, and hence, ore efficient banks also ake higher profits. In what follows, we assue that the solution satisfies n < µ n. Heterogenous bank leverages Here we consider the charter syste. We assue that bank efficiency, λ n, is observable. Given this assuption, the policy paraeters are now a easure of banks for each type, = ( 1,..., N ), and a unsecured deposit liit κ n for each type n with n > 0. Note that the deand for assets fro banks of type-n is still given by (49) (since κ n does not affect the FOC). For given and {κ n }, the arket-clearing conditions are given by (note that D(ι) is still given by (9)) N n=1 N D(ι) = ρ(τ + φ)ā + n κ n ; (53) n=1 n A n (φ, ι) = Ā. (54) We have the following lea. Lea 4.2. Let with 1 > 0 and {κ n } be given. There is a unique allocation (φ, ι, q, d) that satisfies the arket-clearing conditions, and can be characterized as follows: A n = A n (), φ = (1 + ιρ)τ (1 + r)λ 1ψ (A 1 ), r ιρ (55) D(ι) ρ(1 + r) τ λ 1ψ (A 1 ) N Ā + n κ n, with equality if ι > 0. r ιρ (56) Let the unique ι that satisfies (56) be denoted by ι(, {κ n }). Moreover, the profit for bank 29 n=1

30 of type n is given by λ n Π(A n ()) γ + ι(, {κ n})κ n. (57) (1 + r) The assuption that 1 > 0 is with no loss of generality; if, instead, 1 = 0 but n > 0 for soe other n, then we can siply replace 1 by n in (55) and (56). Note also that since we are only concerned with arket clearing and not entry, banks ay ake negative profits (because of the fixed cost γ). However, a full equilibriu analysis also requires incentive copatibility for repayent of κ, which would require nonnegative profits. As before, banks fail to repay depositors will be closed and hence lose their future profits. Thus, given a policy, and {κ n }, a bank of type-n is willing to repay deposits if and only if κ n ρ(φ + τ)a n () + β t [λ n Π(A n ()) γ + ι(, {κ n })κ n /(1 + r)] ρ(φ + τ)a n (). t=0 This constraint can be siplified as rκ n + (1 + r)[λ n Π(A n ()) γ + ι(, {κ n })κ n /(1 + r)] 0. (58) The regulator then chooses policy paraeters to axiize the social welfare. For a given policy and {κ n } and the DM trade q, the regulator axiizes the welfare given by N σ[u(q) c(q)] n [λ n ψ(a n ()) γ], (59) n=1 subject to equilibriu ipleentation c(q) = D(ι(, {κ n })) and incentive copatibility condition (58). The following lea characterize optial {κ n } for a given. Lea 4.3. Let be given such that λ n Π(A n ()) γ for all n with n > 0. 30

31 (a) Let ˆκ n () = 1+r[λ r nπ(a n ()) γ] for each n = 1,..., N. If c(q ) ρ(1 + r) τ λ 1ψ (A 1 ()) Ā + r N nˆκ n (), (60) then {ˆκ n ()} is optial under. In this case, ι = 0 and q = q in equilibriu. (b) Suppose that (60) does not hold. Then, there exists an optial {κ n } under, denoted by { κ n ()}, such that the constraint (58) is binding for all n with n > 0. n=1 Now we are ready to characterize optial policy. Theore 4.1. There exists an optial policy and {κ n }; in any optial policy, we have that, and that n = µ n or n = 0 except for at ost one n. (a) Suppose that (60) holds for =, then (, {ˆκ n ( )}) is an optial policy. (b) Suppose that (60) does not hold for =. (b.1) Any optial policy (, { κ n ()}) have n < n. (b.2) Suppose that ψ(a) = A x for soe x > 1. Then, for any optial policy (, { κ n ()}), L n = ρ(φ + τ)a n() + κ n () A n () is strictly decreasing in n. Theore 4.1 (b.1) shows that unless the first-best is ipleentable, restriction in banking licence is optial. This generalizes Theore 3.2. Moreover, (b.2) shows that under the optial arrangeent, not only the regulator would allow higher unsecured deposit issuance for larger banks, the ration between total debt and total asset also increases with the bank size. If we assue that all banks issue less debts than their assets (for exaple, by having ρ not too large), this also iplies that it is optial to allow for a higher leverage ratio requireent for larger banks. 31

32 4.2 Dividend uncertainty and deposit insurance Here we introduce another diension of heterogeneity, naely, fluctuation over each bank s return on their assets. Specifically, the dividends of Lucas trees that a bank holds are subject to bank-specific shocks: by holding a units of the trees, the return, denoted by τ s, is deterined by the state of the bank, s, which can be either h or l and τ h > τ l. State s fully realizes in the CM and occurs with probability p s. We assue that the state is observable to all in the CM. Moreover, this fluctuation in dividends can cause disturbances to transactions, by way of noisy signals transitted to agents in the DM before it realizes in the following CM. Specifically, when the state for a bank will be s in the coing CM, a buyer who has ade deposits with the bank and the seller who et the buyer in the DM receive the sae signal, which can be either g or b, and the conditional probability is given by p(g h) = p(b l) = ν > 1/2. The paraeter ν then easures the inforativeness of the signal. Because uncertainty in asset returns ay affect the value of deposits, the DM consuption ay depend on the noisy signals and hence ν. Because of uncertainty in the realization of dividends, the fundaental price, φ, of Lucas trees changes as well. Suppose there is a easure of banks. Let E(τ) p h τ h + p l τ l. Then φ = ( ) E(τ) (1 + r)ψ Ā, (61) r which would be the price for the asset if banks cannot issue any deposits. We focus only on the charter syste here. For now we only consider policy paraeters and κ. The pledgeability also requires soe odification. When the realized state is s, the regulator can seize ρ(φ + τ s )a fro each bank and require κ fro each bank (both in ters of CM goods). This in turn affects the bank contract with the depositors. We focus 32