Illiquid Banks, Financial Stability, and Interest Rate Policy

Transcription

1 Nov 008: Revied April 0 Illiquid Bank, Financial tability, and Interet Rate Policy Dougla W. Diamond Raghuram G. Rajan Univerity of Chicago and NBER Do low interet rate alleviate banking fragility? Bank finance illiquid aet with demandable depoit, which dicipline banker but expoe them to damaging run. Authoritie may chooe to bail out bank being run. Uncontrained bailout undermine the diciplinary role of depoit. Moreover, competition force bank to promie depoitor more, increaing intervention and making the ytem wore off. By contrat, contrained intervention to lower rate maintain private dicipline, while offetting contractual rigidity. It may till lead bank to make exceive liquidity promie. Anticipating thi, central bank can reduce financial fragility by raiing rate in normal time to offet their propenity to reduce rate in advere time. We thank the National cience Foundation and the Center for Reearch on ecuritie Price at Chicago Booth for reearch upport. Rajan alo thank the Initiative on Global Market at Chicago Booth for reearch upport. We benefited from comment from Guido Lorenzoni, Monika Piazzei, Tano anto, Joé cheinkman, the referee, and eminar participant at the Harvard Univerity, Princeton Univerity, Richmond Fed, tanford Univerity, the Univerity of Chicago, the American Economic Aociation meeting at an Francico in 009, and the February 009 Federal Reerve Bank New York conference on Central Bank Liquidity Tool. An earlier verion of thi paper wa titled Illiquidity and Interet Rate Policy.

2 hould central bank alter interet rate to deal with epiode of illiquidity and financial fragility? For intance, Greenpan (00) ha argued that while the Federal Reerve cannot recognize or prevent aet price boom, it can mitigate the fallout when it occur and, hopefully, eae the tranition to the next expanion. Other have reponded that by following an aymmetric interet rate policy colloquially known a the Greenpan put -- a central bank can engender the kind of behavior that make boom and but more likely. In thi paper, we ak why epiode of ytemic financial ector illiquidity arie, why private arrangement may be inufficient to alleviate them, how central bank intervention to alleviate financial tre by lending to market to reduce the hort term real interet rate may be better than other form of intervention, and how even uch intervention can create the potential for greater intability. Thi paper take a given a central bank that ha developed the credibility to maintain price tability, and therefore believe it ha the room to tabilize (and enhance) real activity. We how that a willingne to recapitalize bank directly at time of tre, will undermine the dicipline induced by rigid private capital tructure it i not jut that what i ex pot optimal i not ex ante optimal (e.g., Kydland and Precott (977)), but wore, intervention can undermine private commitment and make the ytem wore off. In thi ene, undirected interet rate intervention, where the central bank lend to any olvent bank that need fund may be better, becaue it preerve the commitment induced by private contract, even while retoring flexibility to the ytem by bringing down rate in a way that private contract cannot achieve. Interet rate intervention to relieve tre may be better than the alternative! However, the central bank willingne to intervene when liquidity need are high by puhing down interet rate ex pot doe not jut affect expectation of the real interet rate, but alo encourage bank to make commitment that increae the need for intervention becaue bank and depoitor do not internalize the cot of interet rate intervention. Expectation of low real interet rate (colloquially the belief the central bank will tay low for long ) can increae the future need for low rate. To mitigate thi, the central bank may have to commit to puh the interet rate above the natural equilibrium rate in

3 tate where liquidity need are low to offet the incentive created by it lowering them when need are high. Central bank may need credibility in a new direction. Let u elaborate. Our paper tudie the effect of policy intervention to alter interet rate in an economy where hort-term debt finance long term illiquid project. Bank themelve borrow from rik-avere houehold, who receive endowment every period. Houehold depoit their initial endowment in bank in return for demandable depoit claim (throughout the paper, we focu on demand depoit, though any form of overnight unecured debt could be a cloe ubtitute). Competition for fund induce bank to offer depoit contract that maximize ex ante houehold utility. There i no uncertainty about the average quality of a bank project in our model. However, there i uncertainty about future houehold endowment. Thi, coupled with the mimatch between the long getation period for the project and the demandable nature of depoit, i the ource of banking ector difficultie. Once houehold have depoited their initial endowment, and project have been tarted, ome houehold may have an unexpectedly high need to withdraw depoit. One poibility i that they uffer an advere hock to current endowment that caue them to want to run down financial aet to conume. But another i that they anticipate ignificantly higher income or endowment in the future and want to mooth conumption. Thu current adverity, a well a anticipated properity, can increae current houehold demand for conumption good ubtantially, a ource of financial ector tre when houehold claim on liquidity exceed available liquidity. We focu on anticipated properity a the ource of liquidity demand, though the reult apply for the mot part to tre from current adverity alo. A houehold withdraw depoit to atify conumption need, bank will have to call in loan to long getation project in order to generate the reource to pay them. The real interet rate will rie to equate the houehold demand for conumption good and the upply of thee good from terminated project. Thu greater conumption demand will lead to higher real rate and more project being terminated. It alo lead to lower bank net worth becaue the bank loan pay off only in the long run,

4 and thu fall in value a real interet rate rie, while the bank liabilitie, that i demandable depoit, do not fall in value. Eventually, if the rate rie enough, the bank may have negative net worth and experience run, which are detructive of value becaue all manner of project, including thoe viable at prevailing interet rate, are terminated. The fundamental problem i that demandable depoit are non-contingent promie of liquidity, which are cotly to ervice when the aggregate demand for liquidity i high. However, thee contractual promie may be contrained efficient given the friction in the environment. A key friction i the banker inability to commit hi human capital, an agency problem that explain why bank contract mut be demandable at all time o that the banker doe not renegotiate houehold down (ee Diamond and Rajan (00)). A econd friction that interact with the firt i that it take time to oberve, verify, and implement contractual contingencie flowing from the aggregate tate. If demandable contract are to be tate contingent, however, their term will need to be adjuted intantaneouly whenever the tate become known to ignificant egment of the depoitor, for a delay will caue informed depoitor to run whenever they anticipate a reduction in face value. Frequent change in tate and delay in contractual adjutment could explain why we do not ee tate-contingent demandable claim. Even if a ocial planner cannot overcome the friction which lead to the ue of depoit contract, there may be room for improvement if the planner can ue government power that are not available to the private ector to retore ueful tate contingency. The planner could, for example, expropriate houehold endowment at date through a uniform (acro houehold) tax after eeing the tate, and tranfer the proceed to the bank. Thi would make the bank reource tate-contingent, which would help it avoid ub-optimal bank run and liquidation. Relatively quick change in the face value of claim may be poible for financial intitution uch a mutual fund that hold only traded aet and offer quick but not immediate liquidity, but thee will be harder for intitution that hold illiquid non-traded loan and offer promie of immediate liquidity. 3

5 Uncontrained public intervention, however, undermine the dicipline induced by private contract. Banker rent extraction i uually limited by the banker fear that he will precipitate run if he take too much. When the authoritie are willing to intervene to prevent run, they exacerbate ex pot bank rent extraction. Ex ante, competitive bank will compenate by making even larger promie to houehold, making the ytem even more rigid, and ometime even exceeding the authoritie ability to ave the ytem. We how uncontrained intervention will make the ytem wore off. However, undirected lending, intead of grant, by the central bank (funded by poibly forced borrowing from houehold) to any olvent bank may help. uch liquidity intervention i till a bailout - - the private banking ector i inolvent at rate that would prevail abent intervention, and it i the central bank ability to lower overall market rate through it lending (at the expene of depoiting houehold) that allow it to bail out the banking ytem (a related point i treed in Goodfriend and King (988)). Alo, the central bank willingne to lend could leave it a hotage to market expectation expectation that it will intervene ubtantially and puh rate down very low could force it to intervene more. Neverthele, the prohibition on direct upport to inolvent bank prevent it from ubidizing bank even more, and undermining the diciplinary effect of depoit. Moreover, we how that mall non-pecuniary cot aociated with borrowing from the central bank will reult in unique interet rate equilibria that minimize ex pot central bank intervention. Thi form of interet rate intervention by the central bank dominate uncontrained bailout. A in Diamond-Dybvig (983) and Holmtrom and Tirole (998), the planner/government/central bank ability to partially acce houehold endowment allow it to do more than the private ector. But the authoritie coercive power i eential. Intervention, even if retricted to borrowing and lending at market rate, which i eemingly repectful of houehold property right, ha real effect only by forcibly changing houehold conumption opportunitie and thu influencing real liquidation and invetment 4

6 deciion. For intance, interet rate intervention effectively force ome houehold to lend at rate they would not chooe to voluntarily. If it prevent run, though, everyone may be better off ex pot. Even interet rate intervention i not without problem, however. A lower ex-pot interet rate impoe a lower penalty on bank for having high leverage (epecially becaue depoitor do not internalize the eventual cot to themelve a taxpayer of exceive bank promie) and lower reward for maintaining liquidity. If the central bank i expected to reduce interet rate at time of financial tre, bank will take on more hort-term leverage or make more illiquid loan, thu bringing about the very need for intervention. While the authoritie may want to counter thee incentive through ex ante regulation uch a minimum capital requirement, regulation are eaily evaded. It may be better then for the central bank to change bank incentive by altering it interet etting behavior. Knowing that it will be politically a well a economically undeirable to allow high rate in time of financial tre, the central bank may want to indicate it will raie interet rate in normal time above the market-determined level in order to preerve bank incentive to maintain low leverage and high liquidity. tability-focued central bank hould alo be reluctant to create expectation that real rate will be low for an extended period for fear that bank repone will make the ytem more fragile, and force the central bank to continue keeping rate low (which could work againt it goal of price tability). In um then, even while central bank have become more credible on fighting inflation by binding themelve with inflation target, they have created more room for themelve to vary real interet rate o a to affect real activity acro period (ee, for example, Borio and Lowe (00, 004)). Central bank may now have to build credibility in a new direction o a to enhance financial tability. The ret of the paper i a follow. In ection I, we lay out the baic model, in ection II, we olve it, in ection III we conider macroeconomic intervention on interet rate and it effect on ex ante choice of depoit level and aet liquidity. We then conclude. 5

7 I. The Framework.. Agent, Endowment, Technology, Preference. Conider an economy with rik-neutral entrepreneur and banker, and rik-avere houehold, over three date, 0,, and. Each houehold i initially endowed with a unit of good at date 0. Houehold can invet their initial endowment in bank, which will lend the reource to entrepreneur. At date, houehold will get endowment e. They alo learn their date- endowment. ome houehold H L will get a high endowment, e at date while the ret will get a low endowment e. Let the tate at date be indexed by, with,,,...,. tate differ in the fraction of houehold that expect a high endowment in that tate, with. Higher tate thu reflect a higher degree of optimim about the economy. tate occur with probability p. The tate of nature and houehold type are not verifiable and in addition, houehold type are private information. Each entrepreneur ha a project, which require the invetment of a unit of good at date 0. The project produce an uncertain amount Y in good at date if it i not liquidated, with the realization becoming known at date. Y i ditributed uniformly over the range 0,Y. The project produce X at (or after) date if the project i liquidated, where X i greater than the return on torage. torage i a one period contant return to cale invetment available to entrepreneur and banker each period. Entrepreneur have no good to begin with, and the demand from entrepreneur looking for fund at date 0 i greater than houehold upply. Houehold maximize their expected utility of conumption, E(logC log C). Rik neutral entrepreneur and banker maximize EC ( C)... Financing Entrepreneur ince entrepreneur have no endowment, they need to borrow to invet. Each entrepreneur can borrow from a banker who ha, or can acquire during the coure of lending, knowledge about an 6

8 alternative, but le effective, way to run the project. The banker pecific knowledge allow him to generate Y from a project that ha jut matured, with. Once a banker ha lent, no one ele (including other banker) can learn thi alternative way to run the project. Becaue there are more entrepreneur than houehold, not all project are funded. Bank will lend only if entrepreneur promie to repay the maximum poible for a loan, Y, on demand. If the entrepreneur fail to pay on demand, he can make a counter-offer to the bank. If that offer i rejected, the bank take over the project and either harvet the date- cah flow Y or liquidate the project. 3 All financial contract are in real term we aume the central bank ha achieved a credible commitment to price level tability..3. Financing Bank ince banker have no reource initially, they have to raie them from houehold. But houehold have no collection kill, o how do bank commit to repaying houehold? By iuing depoit of face value D, repayable on demand. In our previou work (Diamond and Rajan (00)), we argued that the demandable nature of depoit contract introduce a collective action problem for depoitor that make them run to demand repayment whenever they anticipate the banker cannot, or will not, pay the promied amount. Becaue banker will loe all rent when their bank i run, they will repay the promied amount on depoit whenever they can. Later, we will detail why demandable depoit may be the optimal contract. Depoit financing introduce rigidity into the bank required repayment. Ex ante, thi enable the banker to commit to repay if he can (that i, avoid trategic default by paing through whatever he collect to depoitor), but it expoe the bank to detructive run if he truly cannot pay (it make inolvency more cotly): when depoitor demand repayment before project have matured and the bank 3 For a relaxation of thee aumption, ee Diamond and Rajan (00). In a more detailed model, we could think of the cah flow being generated over time, with the banker having a ufficient threat each period from hi ability to eize aet and redeploy them that the entrepreneur will pay out ome fraction of the cah flow that are generated. 7

9 doe not have the mean of payment, it will be forced to liquidate project to get X immediately intead of allowing them to mature and generate Y. We aume that each bank lend to enough entrepreneur that the ditribution of Y among the entrepreneur it finance matche the aggregate ditribution of entrepreneur and there i no aggregate uncertainty about project. Becaue the date-0 endowment i carce relative to project, bank will compete to offer the mot attractive promied depoit payment D to houehold per unit of endowment depoited (henceforth, all value will be per unit of endowment). ince thi i the face value repaid on a unit of good depoited, it i alo the date-0 gro promied depoit interet rate. Timeline Date 0 Bank offer depoit term and entrepreneur offer loan repayment term. Houehold get initial endowment and depoit in bank. Bank lend to entrepreneur. Date Uncertainty over tate, houehold date- endowment, and project outcome revealed. Houehold get date- endowment, decide withdrawal, and conumption. Bank decide what project to liquidate. Date Project mature, loan repaid, and depoit fully withdrawn from bank. All agent conume. II. olving the baic model In what follow, we will tart by olving the bank deciion vi a vi entrepreneur at date, then the houehold conumption and withdrawal deciion, and finally, the bank date 0 deciion on what level of depoit repayment D to offer to maximize houehold willingne to depoit... Bank deciion vi a vi entrepreneur Let u tart our analyi at date, once uncertainty i revealed. Let the interet rate houehold demand in tate for re-depoiting between date and be r. Through much of the paper, we will focu on the ituation where the date- interet rate exceed the return on torage, but torage i eaily handled. The bank can get X at date if the project i liquidated. The maximum it can collect from an 8

10 entrepreneur with project realization Y i Y at date, and thi determine how much the entrepreneur can commit to pay at date. o the bank liquidate project with X Y r, that i project with rx Y Y and continue project with Y Y in return for a promied payment of Y. Liquidated entrepreneur get nothing, while entrepreneur who are continued retain ( )Y. The preent value of the bank aet at date (before withdrawal) i Y Y Y XdY dy 0 Y Y Y r, which i eaily hown to fall in r... Houehold deciion Once uncertainty i revealed at date, houehold decide how much they want to withdraw and conume o a to maximize their expected utility of conumption. If they anticipate the bank will not be able to meet it obligation, they will collectively run on the bank, in which cae all project will be liquidated to pay houehold. We aume initially that houehold can coordinate on a Pareto preferred Nah equilibrium and thu rule out other, panic-baed, run (it will turn out that interet rate policy can help eliminate panic). We tart by conidering ituation where the bank will meet it obligation. When a houehold doe not withdraw all it depoit, it utility i maximized when the marginal rate of ubtitution between conumption at date and i equal to the prevailing depoit rate, r, that U( C) C i, when U( C ) C r. If houehold H (with high date- endowment) withdraw amount w H, at date then C e ( Dw ) r C e w H, H H, H, H,. A imilar expreion can be derived for houehold L with low date- endowment. For both houehold to have an equal marginal rate of ubtitution (and depoit at a common rate r ), it mut be that H houehold withdraw more o that w w. H, L, 9

11 Lemma : If r H H r H e, both houehold leave all their money in the bank at date. If e D e e, neither houehold withdraw fully from the banking ytem, but the H houehold e D e D withdraw more than the L houehold. If the L houehold maintain ome depoit. If.3. Equilibrium e e, the H houehold withdraw fully, while e D e D H L r L e e D r, both houehold withdraw fully. Aume firt that the bank can repay depoitor without failing. In equilibrium, market for good at date and date have to clear. At date, good are produced when bank liquidate project. Becaue all bank have the ame ditribution of project and will liquidate project with Y r X Y, date- liquidation proceed are Y r X XdY Y Y 0. Becaue thi hould equal the good conumed by withdrawing houehold, it mut be that where H, L, r X w ( ) w (.) Y i the fraction of H houehold in tate. An equilibrium at date in tate i an interet rate r and withdrawal by the H and L houehold, H, L, w, w uch that the date upply of good equal the date conumption, bank liquidate enough project to meet withdrawal, and houehold do not want to, or cannot, withdraw more. Theorem : (i) If it exit, the equilibrium i unique. (ii) If there i a et { r, w, w } that olve H, L, e e w D r H H,, e e w D r L L,, and (.), with r return on torage, w [0, D), w D, then that i the equilibrium, ele if there are H, L, 0

12 r, w H, Dand w L, that olve e e w D r L L,, and (.), with r return on torage and w L, [0, D) L, DY, then{ r, D, w } i the equilibrium, ele {,, } D D i the equilibrium. 4 X Proof: ee appendix. Corollary : (i) The interet rate r, total withdrawal, w ( ) w, and the fraction of project H, L, liquidated all (weakly) increae in the fraction value of depoit, D. of high endowment houehold and in the face (ii) The net worth of bank (date value of aet le depoit) decreae in the fraction of high endowment houehold and in the face value of depoit, D. Proof: ee appendix. ince H houehold have more date- endowment than L houehold, at any given interet rate they will conume (weakly) more at date, and hence will withdraw more. Thi mean total withdrawal will (weakly) increae in the fraction of high endowment houehold, which mean the interet rate and liquidation will have to increae to equilibrate the market. Becaue the preent value of the bank aet decreae with r while the value of it depoit liabilitie do not, it net worth decreae with. Note that all thee implication would alo hold if houehold differed, not in their date- endowment, but in their date- endowment, with H houehold receiving a lower date endowment (facing poor current condition) and, again, expecting higher growth in marginal utility. Much of pat analyi ha followed Diamond-Dybvig (983) and focued on advere liquidity hock that are tantamount to poor current condition, but it i ueful to remember that becaue conumption and interet rate depend on 4 Note that we have aumed the houehold marginal rate of ubtitution are not o low that they would be below the return on torage. Were that to be the cae, we would have to examine the torage return a a candidate, with any exce depoit being tored by the bank. The three equation would now olve for the withdrawal by high houehold, the withdrawal by low houehold, and the amount inveted by bank in the torage technology..

13 anticipated conumption growth, exuberant view of the future could be equally problematic from the perpective of demand for liquidity. We will focu on thi latter apect through the paper..4. Bank Fragility and the Ex Ante Choice of Depoit Rate Let u now examine each bank choice at date 0 of the promied payment D to offer for a depoit of a unit of good. ince the market i competitive and houehold endowment i carce relative to entrepreneur project, price-taking bank will have to offer a D that maximize houehold utility, given the properly anticipated future interet rate and action of houehold, bank and entrepreneur. o long a the bank i not run in any tate, a higher D make houehold wealthier and better off. But the bank net worth alo fall by Corollary, and for a given D, i lower in a tate with higher When D i high enough that the bank net worth i completely eroded in a tate, the bank i run in that tate which mean all project are liquidated to generate fund to pay depoitor, even though at the prevailing interet rate, ome deerve to be continued. Run are cotly preciely becaue depoitor withdraw money independent of conumption need or the prevailing rate. Becaue a depoitor place in X line i random, each running houehold get D with probability D and 0 with probability X. D Let D () be the promied payment above which a run will be precipitated in tate. From our dicuion above, D fall in. If depoit repayment could be tate contingent, then at date 0, the bank would offer to pay D () in tate at date. But if tate-contingent demandable depoit contract are not poible (we explain why later), promied payment will have to be fixed at date 0..

14 At date 0, the bank will chooe the fixed promied payment to be amongt the et of D () where,...,. 5 If the promied payment i optimally et at * D ( ), it will be becaue the bank ee a net lo in houehold expected utility from having a run in tate * (in addition to the run in all tate * ) by etting the promied payment higher at D D * ( ), even though expected payment to houehold will be higher in all tate *. Thi then i the baic trade-off bank face i etting D higher commit to paying more in the le optimitic tate but have cotly run in more optimitic tate. The higher the probability of tate, and the more cotly the run, the le the deire to puh D higher from D () to D ( ). By contrat, the higher the probability of le optimitic tate, p, and the higher the increae in payout in thoe tate, D ( ) - D (), the greater the incentive to puh D higher. Let * D ( ) be the promied depoit payout that maximize ex ante houehold utility. We have Lemma : Conider an increae in the probability of tate the probability of tate * k accompanied by an equal decreae in * j k, where j and k are both poitive integer and j k that i, a rightward hift in probabilitie from a tate le optimitic than * to one more optimitic. Expected * houehold utility from lower promied depoit payment D ( )..., D ( ) increae relative to * expected utility from the old optimal D ( ), while expected utility from higher promied depoit * * payment, D ( k j)..., D ( ), decreae. If the probability of tate 5 uppoe not and the bank et the depoit rate at ˆD where ( ) ˆ D D D ( ) for two adjacent tate. By increaing ˆD to D (), the bank will not increae the probability of run, but will pay houehold more. Clearly, D () dominate ˆD. D ˆ D. D will alway be preferred to 3

15 * {,..., k j } i mall, the new bank-determined optimal promied payment will (weakly) decreae. imilarly, a leftward hift in probabilitie will lead promie to (weakly) increae. Proof: ee appendix. The lemma ugget higher probability weight on optimitic tate, where liquidity i expected to be tight and interet rate high, will tend to puh optimal promied payment lower. The implication i that in the abence of intervention by the authoritie, bank will be more cautiou about promiing to upply liquidity when optimim and the interet rate are likely to increae. Promied depoit payment are not pro-cyclical becaue competition force bank to internalize the cot of run to their depoitor..5. Example with Comparative tatic L Conider an economy with two future tate, low ( ) and high ( ). Let e 0.65, e 0.8, H e =.5, X =0.95, Y =0.0, Y =3.5, =0.9, 0.6, =0.3. Let D be et at the maximum depoit face value with no run in either tate, D ( ), which i.039. In the tate, the interet rate r i.5, H houehold withdraw 0.77, while L houehold withdraw In the tate, the interet rate r i.89, the H houehold withdraw 0.85, while L houehold withdraw 0.4. Now et D higher at D ( ) =.088. The bank i run in the tate, while in the tate, the interet rate i.94, the H houehold withdraw 0.86 while the L houehold withdraw probabilitie In Figure, we plot the optimal ex ante choice of D (dahed line) for different ex ante p. In thi no-intervention cae, the bank will et D high at.088 until the probability of the high tate exceed 0.4, at which point houehold are better off having afe bank with D et lower at.039. Note from Figure 3 that the expected utility of producer (the bank and entrepreneur) hift up ubtantially when the bank move to etting a lower afe face value of depoit: The banker get no rent when promied depoit payment are et at D ( ), and many project are liquidated at the high prevailing interet rate reducing the reidual value the entrepreneur expect. But banker cum entrepreneur 4

16 expected utility jump up when the promied payment i et lower at D ( ), both becaue the banker i not forced to pay everything he collect in the tate, and becaue interet rate are lower o fewer project are liquidated. Expected producer rent then decline a the probability of tate increae. In um then, in a competitive environment, the banking ytem can lever up to the point where it will fail with ome probability when a ignificant fraction of houehold become exuberant about the future (or peimitic about the preent). Exuberance create more preure for current conumption, which the economy may be too illiquid to provide. Conequently, real interet rate rie to retore equilibrium, and project are curtailed. The more levered the banking ytem i to begin with, the more project are liquidated, and higher the likelihood of ytemic bank failure. III. Optimal Contract Our analyi aim to provide a poitive theory of the effect of interet rate policy when demand depoit contract are taken a given. But we can alo point out the friction that make demand depoit part of an optimal mechanim. Demand depoit allow depoitor a choice of tate contingent amount to withdraw, but in a way that could exacerbate banking ytem tre depoit are firm promie of liquidity that are extremely cotly to deliver when the overall demand for liquidity i high. Could the bank (or any other intitution in the economy) offer contract with better propertie than depoit contract? The optimal full-information and full-commitment tate- and houehold-type contingent contract would equate the marginal utilitie of the houehold at each date, a well a their marginal rate of ubtitution to the marginal rate of tranformation between date- good and date- good. We would obtain a firt bet allocation (apart from the ditortion caue by the entrepreneur inability to fully pledge hi output) in a fairly tandard complete market production economy. However, a we how in the appendix, the friction decribed in thi ection enure there are no feaible contract with better outcome than the demandable depoit contract. The key friction are that banker need to commit to 5

17 collect loan payment, that it i impoible to make the face value of demand depoit tate contingent, and that trade between depoitor or between depoitor and other bank cannot be retricted. pecifically, following Diamond and Rajan (00) and the logic in Hart and Moore (994), conider what happen if the banker borrow long term debt (or any other contract which can be renegotiated) rather than demand depoit. Following thee paper, aume the banker can make a credible threat to not collect the loan unle every lender to the bank make conceion. Any longer term debt contract will be negotiated down to the debt holder outide option in thi cae, the liquidation value of project loan X. Thi limit how much the banker can commit to pay, making it impoible for him to borrow more than liquidation (collateral) value of loan. However, when the banker ha iued demandable claim and then trie to renegotiate depoitor down, becaue depoit claim are paid firt-come-firt-erved, each depoitor ee their outide option a running to get the face value of their claim whenever other depoitor accept a reduced payment and do not run. Any attempt by the banker to renegotiate depoit will et off a run, lead to full liquidation of loan, and detroy the banker reidual claim. Demandable depoit thu allow the banker to pledge hi collection kill to unkilled depoitor, commiting him to pay out D, which i more than the underlying collateral value of the claim X. 6 Interetingly, demandability by itelf doe not contrain the et of feaible allocation all that i required for the firt bet allocation for each houehold to be renegotiation proof i that at date, the preent value of the optimal allocation hould be equal the tate-contingent demandable claim for that 6 We could alo think of an alternative model where the banker can undertake action that put aet or cah flow outide the reach of depoitor, a in Calomiri and Kahn (99) and Diamond (004). The need to top crime in progre then force depoit to be hort term. Demandability and the firt-come, firt-erved contraint give depoitor added incentive to monitor the bank. The key difference between Calomiri and Kahn (99) and Diamond and Rajan (00) i that the latter require demandability a a way to limit trategic default (alo ee Bolton and charftein ( 996)), which enhance the bank ability to borrow beyond the underlying collateral value of aet. 6

18 houehold type. The houehold can then achieve it optimal allocation by withdrawing what it need for conumption at date and the remainder at date (Hellwig (994) ha a related reult). Demandability, however, can prevent the value of depoit from being tate contingent. Depoit claim have to be demandable at every intant after contracting; otherwie houehold can be renegotiated down. Therefore, the initial promied payment on the depoit ha to be a tate-independent value, et ex ante at a level that make houehold indifferent between taking out the depoit immediately (and potentially re-depoiting elewhere) and waiting for the realization of the tate to redefine their face value. Thi mean the face value will ometime have to be reduced given the tate. 7 In uch tate, the bank and the authoritie enforcing claim would have to ee the tate and adjut the face value down before the houehold can withdraw, otherwie each houehold, anticipating a reduction with certainty, ha an incentive to withdraw before it happen and collect the higher initial face value. If it take a mall amount of time to alter D, and a ignificant number of depoitor can react fater than that, tate-contingent depoit will precipitate the very run they eek to avoid. We aume that the peed of face value adjutment i ufficiently low that any tate-contingency in the face value of depoit i undeirable Could houehold claim be directly contingent on houehold type? Again, not if the type i private information and unverifiable, a we aume. The bank can offer houehold a menu of date and date payment to try to get them to elf elect. However, becaue we aume that each houehold ha acce to the capital market (they can trade depoit with each other, a in Jacklin (987), or trade with other bank, and thi trade cannot be excluded by contract), all they care about when chooing a tream of payment i the preent value of the claim they are offered, dicounted at the market interet rate. Given 7 If the depoit face value i et at the lowet poible deired future tate-contingent preent value, all revaluation would be up. But in that cae, the banker would be able to negotiate depoitor down to a tate-contingent contract that promie a utility equivalent to the utility depoitor get from the lowet tate-contingent face value. To promie utility higher than thi, the banker would have to et the initial face value higher than the lowet future tate-contingent value (the tate-contingent contract i renegotiation proof only if the utility the depoitor obtain from withdrawing immediately equal the expected utility from the tate-contingent payment). The face value will therefore have to be reduced in ome ituation. 7

19 a choice, houehold will pick the highet preent value claim and tranform it to their preferred payment tream. Thi make it impoible to ue their (poibly) different marginal rate of ubtitution to get them to elf-elect tream with different preent value. In um then, the limited commitment by the banker, low adjutment to the tate, and the unverifiability of houehold type, all imply that the depoit contract i the contrained optimal contract in the abence of intervention by a ocial planner with power not poeed by the private ector. IV. Macroeconomic Intervention 4.. The Purpoe of Intervention We now examine how a government or planner that ha the ability to tax houehold endowment can help (or hurt). Aume initially that the planner (perhap elected by houehold) want to maximize expected houehold utility, ubject to meeting a minimum threhold utility for the producer (the bank and entrepreneur). If that threhold i low, the planner can be termed houehold friendly, which i the cae we will focu on in much of thi ection the detailed planner problem i laid out in the appendix. Later, we will examine what happen when the planner might become producer friendly when he maximize the expected utility of producer ubject to a low minimum threhold utility for houehold. In either cae, we will aume the planner want to avoid the indicriminate liquidation of entrepreneur -- the characteritic of a run -- and therefore alway want to prevent run ex pot. The bank work thu far becaue the rigidity of demand depoit commit the banker ex ante to pay out. However, if a ocial planner intervene in a pecific bank when depoitor run becaue the bank i inolvent, he will be unable to commit to not intervene when they run becaue the banker i being opportunitic. Indeed, the banker i diciplined by the collective action problem inherent in demand depoit (that i, no depoitor internalize the detruction in value, hence the threat of a run i credible), which will be undone becaue the ocial planner will internalize the externalitie that depoitor ignore. 8

20 The bank will no longer be able to promie more than the collateral value of it aet, and the banker will collect rent ex pot. In addition, ex ante, the banker will be preured by competition to offet the effect of the anticipated ex pot intervention. He will increae the bank commitment by promiing to pay depoitor even more, making everyone wore off. Thi i why it i helpful to contrain the extent to which the planner can intervene ex pot, for example by limiting him to making loan to any olvent bank. Indeed, to limit intervention to the minimum needed, thi contraint will need to be accompanied by mall (non-pecuniary) penaltie for additional borrowing from the planner. We now elaborate, leaving a poible remedy to the next ection. 4.. Bailout by the ocial planner without commitment. We aume the following: (i) The planner know the date- tate of the world (perhap after the paage of time) and can alter it action accordingly. The planner cannot bind himelf to a tate contingent plan, however, and thu ha no advantage in thi repect over private contract. (ii) The planner ha no ability to collect cah flow from the entrepreneur or ditinguih between high Y and low Y entrepreneur. he cannot oberve houehold type and cannot tax them differentially baed on their realized endowment. (iii) The planner ha no reource other than what can be obtained from houehold or bank. (iv) Houehold and bank anticipate the action of the planner and act accordingly. 8 8 The planner taxation authority play a imilar role in our model a the depoit inurance in Diamond and Dybvig (983 ) and the government in Holmtrom and Tirole (998) by making available for contracting houehold endowment that are inacceible to the banking ector. However, our focu i very different from thee model where the government provide new ource of collateral or explicit tranfer of value. While the planner can alter houehold conumption deciion and influence interet rate in our model, it can alo undo ex pot private commitment, and thereby alter ex ante deciion. 9

21 Knowing only what i public information, the planner can levy a tax t on houehold endowment at date L t in tate, where eand e, and tranfer thee reource to the bank. uch a direct bailout would bolter the bank net worth and could prevent a run. Unfortunately, though, the ocial planner cannot commit to intervene only in the tate where the bank i actually inolvent given prevailing interet rate. Anticipating the willingne of the ocial planner to prevent run, the banker could alo threaten to not collect on loan even when the bank i olvent (a trategic default) unle the planner infue fund into the bank. Demand depoit no longer erve to dicipline the banker, for knowing that the ocial planner will do anything to prevent run, depoitor will not run the planner can be relied on to internalize the externality depoitor ignored. Thi mean, however, that houehold will be taxed to pay for what the banker extract. How much will the banker extract? Firt conider only date- taxe. The ocial planner will internalize not jut the utility lo to depoitor from a run but the entire lo in value of project that hould be continued at the prevailing interet rate. If the value of aving them i high, a we have aumed, the banker can extract everything the ocial planner can obtain in taxe from houehold. The banker will demand and obtain e at date. 9 In tate where the bank i inolvent without intervention, ome of that e will go to depoitor. In tate where the bank i olvent, all the e will go to the banker. Knowing that he will get a direct additional rent in the future, each banker will promie it away in the ex ante competition for depoit. If D i the optimal level of depoit when there i no expectation of I intervention, D De i the optimal level of depoit once the ocial planner i expected to intervene the banker pay for the rent he expect to get by increaing the face value of promie to depoitor. A a reult, outcome with lump um taxe are exactly the ame with intervention a without indeed, now 9 One can conider cae where the banker threat i le potent, and he extract le than the taxation capacity of the ocial planner. Thi will require additional aumption on relative bargaining power and outide option, and we do not purue it further in thi paper. 0

22 ex ante promied payment are o high that even after the maximum intervention, the ocial planner cannot ave the bank. The bank i run in the ame tate a it wa without intervention; the ocial planner taxe houehold and tranfer to bank but the houehold withdraw an equivalent additional amount from their higher depoit holding, o net withdrawal from the bank in each tate do not change, and nor doe houehold conumption. 0 If the banker can extract more from the planner in high tate than in low tate, then hi pot-intervention net worth i more poitively correlated with optimitic tate than hi net worth in the abence of intervention. Houehold are better off becaue the banker will promie them more ex ante (net of taxe) and/or fail in fewer tate. The revere i true if the banker can extract le in high tate than in low tate. Thi ugget what happen when the planner can tax houehold endowment at both date and date. he can give the bank e at date and guarantee the bank e L at date (a guaranteed capital injection). The date- guaranteed inflow, however, ha a preent value that fall in high tate where the market clearing r i higher. The cum-tranfer net worth of the bank relative to the no-intervention cae now fall with. o houehold are wore off becaue they are either promied le (net of taxe) than in the no-intervention cae and/or the bank default more. Theorem : If the ocial planner can tax houehold and tranfer to bank, limited only by the extent of the L houehold endowment, houehold will be wore off than without intervention. 0 If the ocial planner cannot ave a bank becaue it liabilitie are too great given the planner ability to tax and tranfer, he will not have an ex-pot incentive to intervene. Intervention will not change the expected net flow that go to houehold. However, the ditribution of run proceed to houehold will be affected. If the authoritie do intervene in a run (by providing all of e to bank) but cannot prevent it, houehold who do not ucceed in withdrawing during the run will be left with zero conumption at date. Thi i a terrible outcome when houehold have log utility (or any preference where the marginal utility of zero conumption i infinite).

23 If the ocial planner i anticipated to intervene maximally ex pot, ex ante promie by the banker do not affect the degree of intervention. A cap on D would then only preerve rent for banker and make houehold wore off. Even if the ocial planner were to et D, anticipating her own intervention ex pot he would favor the higher D Arm Length Central Bank Lending to Bank Could contraint on the ocial planner ability to intervene ex pot improve matter? One alternative i to allow the ocial planner to only make loan with tax proceed, not grant, and only at a market rate (the equilibrium depoit rate) to bank that will be olvent at thoe rate. Thi could be accomplihed by requiring that lending be contingent on imultaneou voluntary lending by houehold. We will ee that mall direct loan may be ineffective, but that larger loan can lower economy-wide interet rate and effectively bring inolvent (at the higher no-intervention rate) bank back to olvency. However, while arm length lending can eliminate direct hold up by bank, they can till obtain rent if the planner i anticipated to intervene heavily, o we may need to contrain the planner even more. pecifically, uppoe now the planner, henceforth called the central bank, borrow b (poibly forcibly) from houehold endowment at date and lend the proceed directly to the bank. It coerce repayment from the banker at date and give it to the houehold (or equivalently, ditribute demand claim on bank of face value rb back to houehold at date ). It i eaily hown that a mall intervention may have no effect on interet rate or bank olvency when neither houehold ha fully withdrawn it depoit at date. Intuitively, becaue houehold wealth remain unchanged, houehold have no reaon to change their conumption if they can withdraw an additional amount equivalent to the amount they are forced to lend out of endowment today. With unchanged conumption, the interet rate doe not change, nor doe the amount the bank ha to liquidate ince the additional loan it receive from the central bank i completely exhauted by additional withdrawal. In hort, becaue houehold have

24 acce to the capital market (through their depoit), houehold choice perfectly offet central bank action, a form of Ricardian equivalence (ee Barro (974)). Central bank lending need not be neutral. uppoe that H houehold withdraw their entire depoit at date even if their endowment i not taxed, or uppoe the central bank taxe a larger amount o that even if H houehold withdraw their depoit fully, they cannot compenate for the lot date- endowment. In either cae, b D w. Thi mean H houehold date conumption will fall by H, b Dw relative to the no-intervention cae, and their marginal rate of ubtitution will go up. H, ( ) But thee houehold are on a corner: they no longer chooe to hold depoit in the banking ytem and cannot borrow againt their future endowment, o their marginal rate of ubtitution ha no effect on the interet rate. The L houehold do hold depoit and affect the interet rate. To clear the market, their date- conumption will have to go up to aborb ome of the conumption given up by the H houehold. In the new equilibrium, their marginal rate of ubtitution fall, the interet rate fall, and L houehold do not make up entirely for the fall in conumption of H houehold. Overall date- conumption fall, and the required liquidation to meet conumption need fall, and bank net worth rie. Lemma 3: (i) If the prevailing no-intervention equilibrium ha w H, D, then a loan to the bank, raied from houehold, of b Dw ha no effect on the interet rate, on conumption, or bank net H, worth. (ii) A loan to the bank of b D w, raied by taxing houehold, will reduce the interet H, rate the bank face, increaing bank net worth and reducing project liquidation. The central bank in our model affect equilibrium rate, and hence bank olvency, becaue ome of the participant who are taxed and forced to lend would not chooe to lend in the capital market at market interet rate. Indeed, H houehold would like to withdraw more to offet the tax at prevailing There are limit to how much effect planner lending can have. Firt, the planner cannot forcibly borrow more than the endowment, e. econd, once it puhe gro interet rate down to the torage return, it cannot puh them down any further ince the bank will imply tore any fund they obtain, and not pa on lower rate to borrower. 3

25 interet rate, but they cannot once they have withdrawn their entire depoit. Compenation at market interet rate at date- doe not fully make up their lo (becaue their marginal rate of ubtitution i higher). L houehold that till participate in the market find an altered invetment opportunity et lower conumption by H houehold implie project with lower rate of return can be continued. A lower putative interet rate (than the earlier no-intervention equilibrium) give L houehold the incentive to conume more, reduce their marginal rate of ubtitution, and allow the new equilibrium to etablih at a lower interet rate than before. In a richer model where there are a continuum of type, the effect of intervention will typically be moother, with a et of type on a corner and not lending voluntarily at market rate, and policie to affect real rate alway having traction. Intervention to lower rate make all houehold wore off, unle the intervention i neceary to top a run; H houehold (who have withdrawn all depoit in the range where the central bank can affect interet rate) are forced to conume le at date. Even though they are compenated for thi at date, it i at an interet rate that i even lower than their already high marginal rate of ubtitution, o they are wore off on net. L houehold ue the financial ytem to ave. For them, a reduction in interet rate reduce their invetment opportunity et, making them wore off. Indeed it i eay to how Lemma 4: Houehold are better off in tate if the depoit level i et at D () and the central bank doe not intervene than if it i et above D () and the central bank reduce rate to retore olvency. Proof: ee appendix Interet Rate Equilibria and Interet Rate Intervention The above dicuion ugget the houehold friendly central bank will want to intervene to reduce rate only to the extent needed to prevent run. The problem, however, i that even under arm length lending, the houehold friendly central bank may be coerced by market expectation to lower the interet rate below the level that make the bank jut olvent, leaving rent for the banker at the expene 4