Bank liquidity, interbank markets and monetary policy

Transcription

1 Bank liquidity, interbank markets and monetary policy Xavier Freixas Antoine Martin David Skeie January 2, 2009 PRELIMINARY DRAFT Abstract Interbank markets play a vital role or te lending o liquidity among banks wit idiosyncratic socks. Tis paper examines ow e ciently te interbank market distributes liquidity among banks ater socks, and weter tis a ects banks coice o liquidity provision to depositors. We sow tat tere are multiple ex-ante Pareto-ranked rational expectations equilibria. Tere exists a rst best equilibrium, in wic a low interbank lending rate provides e cient risk-saring among banks wen socks occur. A ig interbank rate is necessary in te state witout socks to induce banks to old optimal liquidity and provide optimal risk-saring or teir depositors liquidity needs. Te central bank can select te optimal interbank rate equilibrium, in wic rates vary according to te state o te nancial system, as an optimal monetary policy. Freixas is at Universitat Pompeu Fabra. Martin and Skeie are at te Federal Reserve Bank o New York. Corresponding autor david.skeie@ny.rb.org. We tank seminar participants at Te University o Paris X, Nanterre, Te Bundesbank, and te Conerence o Swiss Economist Abroad (Zuric 2008). Te views expressed in tis paper are tose o te autors and do not necessarily re ect te views o te Federal Reserve Bank o New York or te Federal Reserve System.

2 Introduction Te appropriate role o a central bank s interest rate policy response to nancial disruptions is te subject o continuing debate. A standard view is tat monetary policy only plays a role i a nancial disruption a ects in ation or te real economy. However, central banks appear to oten decrease interest rates during disruptions even wen output and in ation are not expected to all, wic as led to criticism. For example, in May 2008, Buiter (2008) asked Despite tese worrying in ation developments, and wit output not exactly alling o a cli (and probably not even weakening enoug to accommodate te necessary external rebalancing o te US economy) te Fed cut rates aggressively. Wat accounts or tis anomalous, and in my view misguided, monetary policy beaviour? Goodriend (2002), discussing earlier episodes, wrote Consider te act tat te Fed cut interest rates sarply in response to two o te most serious nancial crises in recent years: te October 987 stock market break and te turmoil ollowing te Russian deault in 998. Arguably, in retrospect, interest rate policy remained too easy or too long in bot cases. Te ramework we develop in tis paper suggests tat lower interbank market rates during nancial disruptions is part o an optimal policy by te central bank. A primary role or banks under incomplete markets is to provide greater risk-saring and liquidity tan markets can provide to depositors wo ace uninsurable idiosyncratic liquidity socks. During nancial disruptions, wic we tink o as states wen banks ace considerable uncertainty regarding teir idiosyncratic needs or liquid assets, banks temselves may ave large borrowing needs in te interbank market. We sow tat an interbank market can acieve te optimal allocation allowing banks to provide e cient risk-saring to teir depositors and insuring banks against teir idiosyncratic liquidity socks provided te interest rate on tis market is state-dependent and low in states o nancial disruption. Te need or a state dependent interest rate suggests a role or te central bank. In our model, te interest rate on te interbank market plays two roles: From an ex ante perspective, te expected rate in uences te banks portolio decision between liquid sort-term assets and illiquid long-term assets. Ex post, te rate determines te terms at 2

3 wic banks can trade teir assets in response to idiosyncratic socks. Tere is a trade-o between te two roles: I te ex ante expected rate is equal to te ex post realized rate, te e cient allocation cannot be acieved. I te rate is low, te redistribution o assets between banks subject to idiosyncratic socks will be e cient, but banks will coose a suboptimal portolio. At te rate tat induces banks to invest in te optimal portolio, te interbank market does not acieve an optimal redistribution. I te interbank rate is state dependent, owever, te ex ante expected rate need not be equal to te ex post rate in every state. A ig expected rate can induce banks to old te optimal portolio wile a low rate in states o nancial disruption allows te e cient redistribution o assets between banks. Tere are multiple rational expectations equilibria o our model, only one o wic is e cient. Te central bank can be tougt o as an equilibrium selection device. In particular, we sow ow te central bank can implement te e cient allocation by setting te interest rate in te interbank market. Despite te key role tey play or nancial stability, tere is relatively little work studying interbank markets. Tis may be related to te act tat until recently tere was no teory in wic interbank markets were part o an optimal arrangement. In teir seminal study, Battacarya and Gale (987) examine banks wit idiosyncratic liquidity socks rom a mecanism design perspective. Te optimal arrangement in teir paper is not an interbank market. More recent work by Freixas and Holtausen (2005), Freixas and Jorge (2008), Heider, Hoerova, and Holtausen (2008), assumes te existence o interbank markets despite te act tey are not part o an optimal arrangement. Bot Allen, Carletti and Gale (2008) and our paper develop rameworks in wic interbank markets are e cient. In Allen, Carletti and Gale (2008) te central bank can buy and sell assets, using its balance seet to acieve te e cient allocation. In contrast, te size o te central bank s balance seet does not cange in our model but te interbank market rate is state dependant. Bot o tese approaces seem to capture some aspects o actual central bank policy and it may be interesting, in uture researc, to explore ow tey may be combined. Our central bank intervention provides an alternative model to tat o Gutrie and Wrigt (2000) or te concept o open mout operations, by wic te central bank can 3

4 determine sort term interest rates witout active trading intervention in equilibrium. Goodriend and King (987) argue tat wit e cient interbank markets, monetary policy sould respond to aggregate but not idiosyncratic liquidity socks. We nd a role or monetary policy to insure banks against socks to te distribution o liquidity. Results o our paper are similar to Diamond and Rajan (2008) in sowing a bene t to reducing interest rates during a crisis, but wic leads to moral azard or bank liquidity olding, and requires a symmetric interest rate policy wit ig rates in good times. In Diamond and Rajan (2008), providing liquidity to banks troug interest rate policy is ine ective and cannot lower interest rates witout taxing consumers outside o te banking system because o a Ricardian Equivalence argument. In our ramework, te central bank can lower interest rates ater distributional socks to bank liquidity or risk-saring reasons, because te inelasticity o banks sort term supply and demand or liquidity. Our paper also relates to Bolton et al. (2008) in examining te e ciency o nancial intermediaries coice o olding liquidity versus acquiring liquidity supplied by te market ater socks occur. E ciency depends on te timing o central bank intervention in Bolton et al. (2008), wereas te level o interest rate policy is te ocus o our paper. Ascrat, McAndrews and Skeie (2008) examine ex-post liquidity trading wit credit and participation rictions in te interbank market. Te model results explain teir empirical ndings o reserves oarding by banks and interbank rate volatility. 2 Model Te baseline real model adds distributional bank liquidity socks and an interbank market to te standard Diamond and Dybvig (983) ramework. Tere are tree dates, denoted by t = 0; ; 2. Tere is a large number o competitive banks, eac wit a unit continuum o consumers. Ex-ante identical consumers are endowed wit one unit o good at date 0 and learn teir private type at date. Wit a probability ; a consumer is early and needs to consume at date, and wit complementary probability a consumer is late and needs to consume at date 2: Tere are two possible tecnologies. Te sort-term liquid tecnology allows or storing goods at date 0 or date or a return o one te ollowing period. Te long-term investment tecnology allows or investing goods at date 0 or a return o r > at date 2: Investment 4

5 is illiquid and cannot be liquidated at date. 2 Banks are ex-ante identical at date 0. At date learn tey teir private type j = a wit prob 2 b wit prob 2, wit al o banks type a and al type b: Bank j as a raction o early consumers at date equal to j" = + " or j = a " or j = b; were 0 < b" " < ; and were " is a liquidity sock state o te world given by " = "0 > 0 wit prob " 00 = 0 wit prob : Bank j as a raction o late consumers at date 2 equal to j" : Consumer utility is U = u(c ) or early wit prob u(c 2 ) or late wit prob, were c t is consumption at date t = ; 2 and u is increasing and concave. At date 0, consumers deposit teir good in teir bank or a deposit contract tat pays a non-contingent amount or witdrawal at date o c 0, or pays an equal sare o te bank s remaining goods or witdrawal at date 2 o c j" 2 is E[U] = u(c ) + [ 2 ( 0. Consumer s expected utility a0 )u(c a0 2 ) + 2 ( b0 )u(c b0 2 )] + ( )( )u(c 00 2); were c 00 2 ca"00 2 = c b"00 2 ; 0 denotes " 0 and 00 denotes " 00 : Banks maximize teir depositors expected utility and make zero pro t because o competition or deposits at date 0. Banks invest 2 [0; ] in long-term assets and store in liquid goods. At date, consumers and banks learn teir private type. Bank j borrows j" 2 R on te interbank market and consumers witdraw. At date 2, bank j repays te amount j" l " or its loan and te bank s remaining consumers witdraw. We assume, or now, tat te interbank lending gross rate o return l ". We ceck later tat l " < cannot occur in equilibrium, since storage is available. 2 We extend te model to allow or liquidation at date in Section 5. 5

6 Te budget constraints or bank j in state " or dates and 2 are j" c = j" + j" or j 2 a; bg and " 2 " 0 ; " 00 g () ( j" )c j" 2 = r + j" j" l " or j 2 a; bg and " 2 " 0 ; " 00 g; (2) respectively, were j" 2 [0; ] is te amount o liquid goods tat bank j stores between dates and 2. We assume tat banks lend goods wen indi erent between lending and storing. We also assume tat banks cannot contract wit eac oter at date 0 and tat c is non-contingent. Late consumers bear all te risk o liquidity socks in c j" 2.3 Furter, we assume tat te coe cient o relative risk aversion or u(c) is greater tan one, wic implies tat banks provide risk-decreasing liquidity insurance. Trougout te paper, we disregard sunspot-triggered banks runs. For now, we consider parameters suc tat tere are no bank deaults in equilibrium. 4 olds: As suc, we assume tat incentive compatibility c j" 2 c or all j 2 a; bg and or " 2 " 0 ; " 00 g: Tis rules out bot standard multiple equilibria bank runs as well as bank runs based on very large " 0 socks. From te date budget constraint (), we can solve or j" = j" c j ( ) + j" : Substituting tis in te date 2 budget constraint (2) and rearranging gives 2 = r + j" [ j" c ( ) + j" ]l " ( j" : (3) ) c j" A bank s optimization to maximize its depositors expected utility is max 2[0;];c ; j" g j;" 0 E[U] (4) s.t. j" or j 2 a; bg and " 2 " 0 ; " 00 g (5) (3) or j 2 a; bg and " 2 " 0 ; " 00 g, (6) were te constraint gives te maximum amount o goods tat can be stored between dates and 2. 3 Allowing c to depend on " or j would complicate te solutions but would not cange te qualitative results o te model. Note, in particular, tat in te optimal allocation, c is constant. 4 Bank runs are considered in section?? 6

7 3 Results To nd te rst best allocation, consider a planner wo can observe consumer types: Te planner can consider te aggregate economy, or wic tere are no aggregate socks, wit no need to consider bank types a and b: Te planner s problem is to maximize a consumer s expected utility max 2[0;];c ;0 u(c ) + ( ) u(c 2 ) s.t. c + ( ) c 2 r + c ; : Te constraints are te pysical quantities o goods available or consumption at date and 2, and available storage between dates and 2, respectively. Te rst-order conditions and binding constraints give te well-known rst best allocations, wic are denoted wit asterisks, de ned by u 0 (c ) = ru 0 (c 2) (7) c = (8) ( ) c 2 = r: (9) = 0 (0) Equation (7) sows tat te ratio o marginal utilities between dates and 2 is equal to te marginal return on investment r: We next consider bank j 0 s optimization (4). Lemma. First order conditions wit respect to c and are, respectively, u 0 (c ) = E[ j" l" u 0 (c j" 2 )] () E[l " u 0 (c j" 2 )] = re[u0 (c j" 2 )]: (2) Proo. Te Lagrange multiplier or constraint (5) is j" : Te rst order condition wit respect to j" is 2 " u 0 (c j" 2 )( l" ) j" (= i j" > 0); 7

8 wic or l " > does not bind and implies j" = 0; and or l " = implies j" = 0 since banks are indi erent between storage and lending goods. Complementary slackness or constraint (5) implies j" = 0: First order conditions () and (2) ollow. Equation () is te Euler equation and determines te investment level given l " : Equation (2) states tat te expected marginal utility-weigted returns on storage and investment must be equal. Te return on investment between dates 0 and 2 is r: Te return on storage between dates 0 and 2 is te market rate l " : Banks can store goods at date 0, lend tem at date, and will receive l " at date 2. Te rates l 0 and l 00 are determined in equilibrium to make banks indi erent to olding goods and assets at date 0. Te interbank market clearing condition is a" = b" or " 2 " 0 ; " 00 g; wic wit te bank s budget constraints () and (2) determine c j and j" as unctions o : c () = j" () = ( )( j" ): Finding te market equilibrium is reduced to solving te two rst order conditions () and (2) in tree unknowns: ; l 0 and l 00 : 3. Single state: = 0; We start by nding solutions to te special cases o = 0; : We ten apply tese solutions to solve te general model 2 [0; ] below. Tere is certainty about te single state o te world " at date. First order conditions () and (2) can be written more explicitly as [ 2 u0 (c a0 2 ) + 2 u0 (c b0 2 )]l 0 + ( )u 0 (c 00 2)l 00 = [ 2 u0 (c a0 2 ) + 2 u0 (c b0 2 )]r + ( )u 0 (c 00 2)r (3) u 0 (c ) = [ a0 2 u0 (c a0 2 ) + b0 2 u0 (c b0 2 )]l 0 + ( )u 0 (c 00 2)l 00 (4) Equations (3) and (4) imply tat or = 0; te value o l 0 is indeterminate, and or = ; te value o l 00 is indeterminate. In eiter case, we will sow tat tere is an 8

9 equilibrium wit unique values or te allocation c ; c 2 and. Te indeterminate variable is o no consequence or te allocation. Te allocation is determined by te two rst order equations, in te two unknowns and l 00 (or = 0) or l 0 (or = ). Te rst order condition wit respect to ; equation (3), sows tat te interbank lending rate equals te return on assets: l 00 = r (or = 0) or l 0 = r (or = ): Wit a single state o te world, te interbank lending rate must equal te return on assets. In te case o no sock wit = 0; te banks budget constraints imply tat in equilibrium a00 = b00 = 0; no interbank lending occurs. Te interbank lending rate l 00 is te lending rate at wic banks net borrowing demand is zero. Te Euler equation or banks equation (4) is equivalent to tat or te planner equation (7). Banks coose te optimal and provide te rst best allocation c and c 2 ; wic are illustrated in Figure a. u(c t ) c * c a 2 ( *) c 2 * c b 2 ( *) c t Figure a Banks provide liquidity at date to early consumers by paying c > : Tis can only be accomplised by paying c 2 < r on witdrawals to late consumers at date 2. Te key or te bank being able to provide liquidity insurance to early consumers is tat te bank can only pay an implicit date to date 2 intertemporal return on deposits o c 2 c ; wic is less tan te return on assets r. Tis contract is optimal because te ratio o intertemporal marginal utility equals te marginal return on assets, u0 (c 2 ) u 0 (c ) = r: Proposition. For = 0; tere exists a rational expectations equilibrium caracterized by l 0 = r tat as a unique rst best allocation c ; c 2, : Proo. For = 0; equation (3) implies l 00 = r: Equation (4) simpli es to u 0 (c ) = u 0 (c 00 2 )r; and te bank s budget constraints bind and simpliy to c = 9 ; c 2 = r : Tese results

10 are equivalent to te planner s results in equations (7) troug (9), implying tere is a unique equilibrium, were c = c ; c 2 = c 2 and = : In te case o a certain sock wit = ; tere is interbank lending. Te banks budget constraints imply tat in equilibrium a0 = " 0 c and b0 = " 0 c. First, consider te outcome at date olding xed =. Wit l 0 = r; late consumers do not ave optimal consumption: c a0 2 ( ) < c 2 < cb0 2 ( ): Te deviation rom optimality is illustrated by te arrows in Figure a. A bank tat as to borrow at date at te rate l 0 = r aces a rate tat is iger tan te intertemporal return on deposits c 2 c consumers c 0 2 r : and cannot pay late Late consumers ace risk to teir consumption conditional on being a late type. Second, consider te determination o : In equilibrium, > : Compared to te rst best, banks store ewer liquid goods at date 0 and pay lower c at date in order to old more assets tat provide banks greater sel-insurance liquidity available at date 2 to pay to late consumers. Te di erence o equilibrium consumption written as a unction o te equilibrium compared to consumption or a xed = is demonstrated by te arrows in Figure b. Te result is c < c ; c0 2 > c 2 ; ca0 2 > ca0 2 ( ) and c b0 2 > cb0 ( ): For any " > 0 sock, banks do not provide te optimal allocation. u(c t ) c c * c a c a c 2 * c 2 c b 2 ( *) c b 2 ( *) 2 2 c t Figure b Proposition 2. For = ; tere exists a rational expectations equilibrium caracterized 0

11 by l 0 = r tat as a unique suboptimal allocation c < c c a0 2 < c 2 < c b0 2 > : Proo. For = ; equation (3) implies l 0 = r: By equation (3), c b0 budget constraints and market clearing, " 2( ) ca " 2( ) cb0 2 = r = c0 2; 2 > ca0 2 : From te bank s wic implies 2 ca cb0 2 < c0 2, since cb0 > c a0 2 : Because u () is concave, 2 u0 (c a0 2 )+ 2 u0 (c b0 2 ) > u 0 (c 0 a0 2 ): Furter, 2 u0 (c a0 2 ) + b0 2 u0 (c b0 2 ) > u0 (c 0 2 ) since a0 > b0, a0 2 + b0 2 Tus, u 0 (c ( )) = ru 0 (c 0 2( )) < r[ a0 2 u0 (c a0 2 ( )) + b0 2 u0 (c b0 2 ( ))]: = and ca0 2 < cb0 2 : Since u 0 (c ()) is increasing in and u 0 (c j0 2 ()) or j = a; b is decreasing in ; te Euler equation implies tat in equilibrium, > : Hence, c = and c a0 2 < c 2 : < c ; cb0 2 > c0 2 = r > c General sock: 2 [0; ] We now apply our results o te special cases o = 0; to examine te general case o 2 [0; ]: We will sow tat tere are multiple rational expectations equilibria wit di erent real allocations o c ; c 2 and. Tere are two possible states o te world at date : " 0 and " 00 : An equilibrium is determined by te two rst order condition equations (3) and (4) in tree unknowns ; l 0 and l 00. Te bank s budget constraints imply in te state o no sock wit " 00 = 0; no interbank lending occurs, ja = jb = 0, and c 00 2 = r ; as in te case o = 0: In te state o a positive sock wit " 0 > 0; tere is interbank lending wit a0 = " 0 c, b0 = " 0 c, c j" 2 = r (j" )c l " j" : (5)

12 In particular, tere exists a suboptimal rational expectations equilibrium wit l 0 = l 00 = r. Consider l 0 = r: Equation (3) implies l 00 = r: Equation (4) is a single equation wit a single unknown ; wic is determined. implicit unction o : Likewise, c 00 2 (); ca0 2 Equation (4) implies tat () is an () and cb0() are implicit unctions o. We can use te cases o = 0 and = to provide bounds or te general case o 2 [0; ]: Te equilibrium c () and c j" 2 () or j = a; b and " = "0 ; " 00 ; written as unctions o, are displayed in Figure 2a. Tis gure sows tat c () is decreasing in wile c j" 2 () in increasing in : In particular, c 2 = c 00 2(0) c 00 2() c 00 2() c j" 2 (0) cj" 2 () cj" 2 () c () c () c (0) = c ; or j = a; b; were c j0 2 ( = 0) = cj0 2 ( = ). Wit interbank rates equal to r in all states, tere is ine cient risk-saring among late consumers. To compensate, tere is ine cient liquidity provided to early consumers. 2 u(c t ) c a 2 ( ) c 2 ( ) c b 2 ( ) c ( ) c () c *(0) c a 2 (0) c a 2 () c 2 (0) c 2 () c b 2 (0) c c b 2 () t Figure 2a For < ; tere also exists a rst best rational expectations equilibrium wit l 0 = l 0 c00 2 c : (6) To sow tis, rst we substitute or l 0 into (5). and simpliy, wic gives c a0 2 = cb0 2 = c 0 2 = c00 2 = r : Wit l0 equal to te intertemporal return on deposits between dates and 2, tere is optimal ex-post risk-saring o te goods tat are available at date 2 troug 2

13 interbank lending at te low rate at date. Substituting or l 0 and c j0 2 into equation (3) and rearranging gives l 00 = r + (r c 2 c ) : (7) Substituting or c j0 2 ; l0 and l 00 into equation (4) and rearranging gives u 0 (c ) = r 0 u 0 (c 0 2 ): Tis is te planner s condition, and implies = ; c = c and c0 2 = c 2 ; a rst best allocation. To interpret, substituting tese equilibrium values into (7) and simpliying sows tat l 00 = l 00 r + (r c 2 c ) > r; (8) wereas l 0 = c 2 c < r: Wit l 00 greater tan r during te no-sock state, tere is no ex-post ine ciency because tere is no need or interbank lending. Wit l 0 less tan r or te sock state, tere is no ex-post ine ciency wit interbank lending because te rate is at te low optimal rate. Te ollowing result sows tat te expected interbank rate is equal to te return on assets. Tis result is based te rst order condition wit respect to ; wic requires banks to be willing to old bot storage and investment at date 0. Proposition 3. Te expected interbank rate is E[l " ] = r: Proo. E[l " ] = l 0 +( E[l " ] = r: )l 00 : Substituting or l 0 and l 00 rom (6) and (8) and simpliying, Since tere is no risk to late consumers, banks old optimal : Figure 2b illustrates te distinction o tis rst best equilibrium wit l 0 = c 2 c ; l 00 rom te equilibrium wit l 0 = l 00 = r: Arrows indicate tat in contrast wit te suboptimal l " = r equilibrium, in te l 0 = c 2 c equilibrium we nd te rst best outcome tat c j" 2 () = c 2 and c () = c or all j 2 ; bg, " 2 " 0 ; " 00 g and < : u(c t ) c a 2 ( ) c 2 '( ) c b 2 ( ) c ( ) c c c () c *(0) c a c a c 2 (0) 2 () c b 2 (0) c b 2 (0) 2 () 2 () t 3

14 Figure 2b For = ; l 0 = c00 2 c speci ed. Tereore we rule out l 0 = c00 2 c would imply l 00 is not nite and equations (3) and (4) are not well as an equilibrium value or = : As in te case o = above, tere are multiple equilibria since l 0 is indeterminate, but te allocation ; c, c j00 2 is unique and not rst best. Proposition 4. For 2 (0; ); tere exist multiple rational expectations equilibria wit di erent allocations. Tere exists a suboptimal rational expectations equilibrium wit l 0 = l 00 = r > c < c c 00 2 > c 2 c a0 2 < c 2 < c b0 2 ; and tere exists a rst best rational expectations equilibrium wit l 0 = c 2 c < r l 00 = l 00 > r = c = c c j" 2 = c 2 (or j 2 a; bg and " 2 " 0 ; " 00 g): Note tat all equilibria we consider are rational expectations equilibria. Gale (2004) sow tat tere exist sunspot equilibria in tis type o model. Allen and From te perspective o date only, an indeterminate continuum o l " is consistent wit ex-post individual rationality or banks lending in te interbank market. We sow tat multiple rational expectations equilibria exist rom te perspective o date 0 because tere are multiple " idiosyncratic liquidity states at date. Tere is a amily o l 0 ; l 00 at date, eac pair o wic can be anticipated and support a di erent rational expectations equilibrium. Witin a rational expectations equilibrium, l 0 and l 00 do not need to be equal. Te results rom tis section generalize in a straigtorward way to te case o N socks, as sown in appendix B. 4

15 3.3 Central bank monetary policy Te interest rate l " at wic banks lend in te interbank market is equivalent to te unsecured interest rate tat many central banks target or monetary policy, suc as wit te ed unds rate targeted by te Federal Reserve in te U.S. Te role o te central bank or monetary policy in te model is to select te optimal interbank rate equilibrium among multiple equilibria or 2 (0; ): Te central bank setting te interbank rate at a low rate l 0 = c 2 c ater te sock " 0 is equivalent to a transer rom bank b to bank a. An equilibrium wit l 0 = c 2 c as ex-post distributional e ects as bank b late consumers receive less and bank a late consumers receive more tan in te equilibrium wit l 0 = r, but te l 0 = c 2 c is ex-ante optimal because it reduces risk or banks late consumers. Banks ten do not need to sel-insure or te " 0 sock wit greater investment, and so will old greater liquidity or its early consumers o =. But extra ig rates o l 0 = l 00 > r are required ater te no-sock state " 00 ; suc tat expected rates equal te return on assets, E[l " ] = r; and banks are indi erent between olding goods and assets at date 0. Te model we ave used so ar is not ric enoug to adequately describe ow te central bank can implement te desired interbank interest rate. Te main bene t o tis simple model it to illustrate te key point o te paper witout te burden o too muc notation. However, te role o te central bank is central to our argument and we provide a generalized version o te model tat sows ow te central bank can actively select and enorce its coice o interbank rates in appendix A. In tat ricer model, banks can pay a at nominal rate rater tan a real rate on deposits, ollowing Skeie (2008). Te central bank can o er to borrow and lend unlimited amounts at its nominal policy rate contingent on te state " at date. Tis will orce banks to trade at tis rate in te interbank market, and te central bank does zero borrowing and lending in equilibrium. Te equilibrium o te nominal model is equivalent in real terms to te equilibrium o te real model in Section Bank runs We extend te model to consider bank runs, and we sow tat banks runs may occur i te CB does not ollow te optimal policy. In te state were " > 0, patient depositors 5

16 o banks wit many impatient agents will consume less tan patient depositors o oter banks i te CB does not set te interest rate equal to c 2 =c, te intertemporal return on deposits. I " is su ciently large it may be te case tat te consumption o patient depositors o banks wit many impatient agents would be lower tan te consumption o impatient depositors, wic would trigger a run. Tis argument can be presented in several ways. One can nd te equilibrium allocation assuming tat te CB does not ollow te optimal policy and sow tat, in equilibrium, banks runs would occur at banks tat ave many impatient depositors. Alteratively, one can consider an equilibrium assuming tat te CB ollows te optimal policy and sow tat, i te CB makes an unexpected mistake, a bank run occurs. We consider eac approac in turn. 4. Central bank makes unexpected mistake I te CB is assumed to adopt te optimal policy, te equilibrium allocation is optimal. Suppose tat, unexpectedly, te CB cooses interest rate l 0 = r > c 2 =c in te state were " = " 0 > 0. In tis case, te consumption, c a0 2, o patient depositor in banks wit many impatient agents is c a0 2 = r " 0 c r " 0 = r " 0 " 0 ; since c = ( )=. I we assume tat te utility unction is o te orm u(c) = c ; > ; ten we can rewrite te expression or c a0 2 as Recall tat " 0 min; c a0 2 = r " " 0 r " 0 # g. I is very small, ten " 0 must also be very small and te term in brackets will be close to. Tis implies tat c a0 2 will be close to c 2 and no bank run can occur since c 2 > c. In contrast, i =2, ten te term in brackets can be made arbitrarily close to zero, since r > so tat c a0 2 bank runs can occur. : will be close to zero. In suc cases, 6

17 Consider te ollowing example: = = =2, r = :5, and = 2. For suc parameters, we ave 0:4495, c :0, and c 2 :3485. Now assume tat "0 = 0:3, ten c a0 2 0:8939 < c, so tere would be a bank run. 4.2 Runs in equilibrium Consider te equilibrium allocation i banks anticipate tat te interbank market interest rate will be l 0 = l 00 = r. By continuity, tis allocation converges to te optimal allocation as! 0. We ave already seen tat at te optimal level, bank runs can occur i " is su ciently large and l 0 = r. Now since bank runs are anticipated, banks could coose a run preventing deposit contract, as suggested by Cooper and Ross (998). However, ollowing te argument in tat paper, banks will not coose a run-preventing deposit contract i te probability o a bank run is su ciently small. So or su ciently close to zero, bank runs will occur in equilibrium. 5 Liquidation o te long-term tecnology We extend te model to allow or liquidation o te investment at date. We sow tat tis restricts possible real interbank rates and may preclude te rst best equilibrium. At date, bank j liquidates j" o te investment or a salvage rate o return s at date and no urter return at date 2. Te bank budget constraints () and (2) are replaced by j" c = j" + j" s + j" or j 2 a; bg and " 2 " 0 ; " 00 g ( j" )c j" 2 = ( j" )r + j" j" l " or j 2 a; bg and " 2 " 0 ; " 00 g; and te bank optimization (4) is replaced max ;c ; j" ; j" g j;" E[U] s.t. j" or j 2 a; bg and " 2 " 0 ; " 00 g j" or j 2 a; bg and " 2 " 0 ; " 00 g: (9) Te rst order condition wit respect to j" is 2 " u 0 (c j" 2 )(l" s r) j" or j 2 a; bg and " 2 " 0 ; " 00 g (= i j" > 0); (20) 7

18 were j" is te Lagrange multiplier or constraint (9). Witout loss o generality, we assume tat no bank j liquidates all investment in state " unless all banks do. Because te interbank market is ex-post e cient, te equilibrium and allocation depends solely on te aggregate amount o liquidation, not te distribution o liquidation among banks. I tere is complete liquidation o investment, ten clearly te allocation is not rst best. Consider an equilibrium in wic tere is not complete liquidation o investment. Complementary slackness or constraint (9) implies j" as l " r s or all "; = 0: Condition (20) can be written wic gives a restriction on te equilibrium interest rate in state ": I tere is liquidation by any bank j in any state "; te equilibrium is not rst best. Alternatively, i l 00 > r s ; ten te equilibrium cannot be rst best. Te interest cannot be ig enoug in te " 00 state. At an interest rate o l 00 > r s ; all banks would liquidate investment and lend it on te interbank market, and no banks would borrow, wic cannot be an equilibrium. 6 Conclusion Tis paper examines te ex-ante coice o bank liquidity and te ex-post reallocation o bank liquidity troug te interbank market ater random idiosyncratic liquidity socks. An expected ig rate equal to te return on long-term assets is required or banks to old liquidity ex-ante. In te state wen te liquidity sock occurs, banks in need borrow rom banks wit surplus unds. A ig interbank rate, owever, is ine cient or lending in te interbank market. Te return on long-term assets is necessarily greater tan te implicit return tat banks pay on deposits to late consumers. Tis implicit rate is low to allow banks to provide teir key role o liquidity insurance to early consumers. Banks tat borrow on te interbank market at te ig rate pay teir late consumers a lower rate on deposits tan tat paid by banks tat lend on te interbank market. Te uncertainty o returns to late consumers implies tat banks old less liquidity as insurance to consumers against individual idiosyncratic liquidity socks in order to provide more liquidity as insurance or late consumers against bank idiosyncratic liquidity socks. However, tere are multiple rational expectations equilibria, wic are Pareto-ranked 8

19 on an ex-ante basis. An alternative equilibrium can reac te rst best allocation. An implicit interbank rate during te no-sock state tat is greater tan te return on assets allows or a lower interbank rate during te sock state. Te return on liquid goods in expectation, equal to te expected interbank rate, equals te return on assets and makes banks indi erent between olding liquid goods and assets. Te low interbank rate during te sock state allows or ex-post e cient interbank lending, suc tat late consumers receive equal rates rom banks tat ave positive or negative socks. Banks do not ave to provide extra liquidity at te late period to sel-insure against bank socks. Banks are able to old optimal liquid goods or liquidity insurance against consumer socks. Te interest rate policy o a central bank can select te interbank rate equilibrium and allocation to consumers. According to te model, a central bank sould lower interest rates ater an idiosyncratic liquidity sock. Tis as a distributional e ect tat bene ts banks wit negative liquidity socks and costs banks wit positive liquidity socks, but wic is ex-ante optimal or banks beore tey learn teir sock. In order to lower rates ater a sock and still ensure banks maintain incentives to old liquidity, a central bank must raise rates above te natural return on long-term assets during te no-sock state. Rates sould be raised in a symmetric manner to ow tey are lowered in te di erent states, adjusting or te probability o te sock occurring. Examining te impact o monetary policy on bank liquidity and te interbank market wen tere are aggregate liquidity socks and real socks to undamental asset values are steps or uture researc. 9

20 7 Appendix A: Monetary policy wit nominal rates We expand te real model to allow or nominal interbank lending rates. Wit nominal at interest rates, te central bank can explicitly enorce its target or te interbank rate, in order to actively select te rational expectations equilibrium. Te central bank o ers to borrow and lend to banks any amount o nominal, at money at te central bank s policy rate at date, wic ensures tat te interbank market rate equals te central bank s policy rate. Te equilibrium and allocation o te nominal rate model is equivalent to te real rate model. 7. Nominal rate model extension Te extension o te model to include nominal rates is based on Skeie (2008). A nominal unit o account, inside money and a goods market wit rms are added to te model o banks wit real deposits. To establis a at nominal unit o account, te central bank o ers at date 0 to buy or sell goods to te extent easible or at currency at a xed price P 0 =. Ater date 0, te central bank does not set te price o goods and does not o er to buy or sell goods. At date 0, eac bank makes a loan to a rm. Te rm buys te good rom te bank s unit continuum o consumers, and consumers deposit in te bank. All te transactions at date zero are paid or in te amount o one nominal unit o account. Tese nominal payments are called inside money. Te individual budget constraint or eac o te banks, rms and consumers requires tat te net inside money payments o eac party nets to zero at date 0. K " 2 Eac bank lends to its rm or loan repayments o nominal amounts ( ) K and payable in inside money at dates and 2, respectively. Uppercase variables denote nominal values and lowercase variables denote real values. Te rm buys te good rom consumers or price P 0 = : Te rm invests and stores o te good, were is cosen and can be enorced by te bank. 5 Consumers deposit in teir bank or a demand deposit contract repayment due in inside money o eiter D i witdrawn at date or D j" 2 i witdrawn at date 2. Altoug no currency circulates, te central bank s 5 I te bank could not enorce te rm s storage; te bank could alternatively buy and store goods, sell tem at date, and lend to te rm witout any storage requirements. Results o te model would be uncanged. 20

21 o er to trade wit currency establises te nominal unit o account or transactions wit bank inside money. Tis is equivalent to Skeie (2008), were currency rater tan inside money transactions occur at date 0. At eac date t = ; 2, payments are made among banks wit eiter inside money or currency. Eac bank s net payments in a period must equal zero. At date, j" early consumers o bank j witdraw to buy goods rom a rm in te goods market. At date 2, j late consumers witdraw rom banks to buy goods. Te representative rm repays loans and banks borrow or lend i needed on te interbank market or rom te central bank. Te bank s budget constraints rom te real model () and (2) are replaced by budget constraints or nominal payments: s.t. j" D = ( )K + M j"d + M j"d ; 8 j 2 a; bg and " 2 " 0 ; " 00 (2) g; ( j" )D j" 2 = K 2 " M j"d D " M j"d D " ; 8 j 2 a; bg and " 2 "0 ; " 00 (22) g; respectively, were bank j s demand to borrow rom oter banks is M "jd and rom te central bank (in currency) is M "jd ; and were D " and D" are te returns on interbank loans and central bank loans, respectively. D " is te interbank market rate, wic is determined in equilibrium. At date, te central bank targets D " by coosing its policy rate D " at wic it o ers to borrow and lend to banks an unlimited amount. Speci cally, te central o ers to lend M j"s (D " ) 2 ( ; ) to bank j at rate D", were M S (D" ) is a correspondence. Te central does not ave a budget constraint to equate its borrowing and lending o central bank money. It can create and destroy money as needed. Te central bank s lending supply is perectly elastic at its cosen rate D " : Te way in wic we model te central bank o ering to borrow and lend at a single policy rate is similar to open market operations in practice. Many central banks in essence o er to borrow and lend an elastic amount o unds at a cosen rate to target te interbank rate at wic banks lend uncollateralized to eac oter. Open market operations lending is oten collateralized in practice, as in te orm o repos against government securities in te case o te Federal Reserve. We abstract rom collateralization since tere is no risk o loss or deault. Consumers buy goods rom rms at date t = ; 2 in a Walrasian market using inside 2

22 money as numeraire. Consumption or early and late consumers is were P " t c (P ) = D P (23) c j" 2 Dj" 2 (P ) = ; (24) P " 2 is te price o goods at date t = ; 2 and P (P ; P 2 " ) is a vector. We consider only P " t 2 (0; ); wic is or simplicity and does not e ect te results. Consumers aggregate demand is given by q D (P ) = q D 2 (P ) = 2 (a" + b" )D (25) P 2 [( a" ) + ( b" )]D j" 2 : (26) Te representative rm submits a supply scedule q t "S (P t " ) or te goods market. Te rm s optimization is to maximize pro ts: P " 2 max q "S;q"S r q "S q "S 2 (27a) s.t. q "S (27b) q "S 2 + r q "S (27c) q "S ( ) K P q "S (27d) 2 K" 2 : (27e) Te objective unction (27a) is te pro t in goods tat te rm consumes at date 2. Constraints (27b) and (27c) are te maximum amounts o goods tat can be sold at dates and 2, respectively. Constraints (27d) and (27e) are te rm s budget constraints to repay its loan at date and date 2; respectively. Te bank s demand or borrowing on te interbank market can be solved or rom P " 2 equation (2) as M "jd = "j D ( )K M j"d : (28) Substituting or M "jd rom equation (28) into equation (22) and rearranging, we nd tat bank j pays witdrawals to late consumers te amount D j" )M j"d 2 = K" 2 [ j" D ( )K ]D " + (D" D " j" : (29) 22

23 Te bank s optimization problem (4) is replaced by max 2[0;];D 0;M j"d g j;" 2R were c (P ) and c j" 2 (P ) are given by (23) and (24), respectively. E[U]; (30) s.t. (29) (3) An equilibrium is de ned as goods market prices and quantities (P; q ; q 2 ), deposit and loan returns and quantities D ; D " ; M j" g j;"; and investment () tat solve goods market clearing conditions and interbank market clearing condition qt D (P ) = 2 [qas t (P ) + qt bs (P )] or t = ; 2; M ad (M ad ) + M bd (M bd ) = 0; (33) were ; D ; M j" g j;" is a solution to bank j s optimization (30); q D t (P )g t=;2 is given by te consumers aggregate demand (25) and (26), and (q "S rm s optimization (27). (P ); q"s 2 (P )) is a solution to te 7.2 Nominal rate results Te results o te nominal model are equivalent to tose o te real model, wit te addition tat te central bank can coose its policy rate to target te interbank rate. Te rst order conditions or bank j 0 s optimization (30) wit respect to ; c and M "jd are E[ K" 2 P 2 " u 0 (c j" 2 )] = E[K P " 2 E[ P u 0 (c )] = E[ j" D " P " 2 D " u0 (c j" 2 )]; 8 " 2 "0 ; " 00 g (34) u 0 (c j" 2 )]; 8 " 2 "0 ; " 00 g (35) D " = D " ; 8 " 2 "0 ; " 00 g; (36) respectively. Loan returns are set according to a competitive loan market as suc tat te real returns K P = and K" 2 P " 2 K = P (37) K " 2 = rp " 2 ; (38) = r equal te marginal product o capital or teir respective terms and rms make zero pro ts in equilibrium. Substituting or K " t 23

24 rom equations (37) and (38), conditions (34) and (35) can be written as E[u 0 (c j" 2 )] = E[ D " P 2 " =P u 0 (c j" 2 )] (39) E[u 0 (c )] = E[ j" D " P " 2 =P Condition (36) states tat because o arbitrage, te interbank rate D " u 0 (c j" 2 )]: (40) equals te central bank s policy rate D " : Te real interbank rate equals te nominal rate divided by nominal goods price in ation between dates and 2: l " = D" P 2 " =P ; (4) wic implies tat te rst order conditions or te nominal model, equations (39) and (40), and or te real model, equations (3) and (4), are equivalent. Te central bank can target any real interbank lending rate l " at date by coosing D " = P " 2 P l " ; subject to satisying te date 0 rst order conditions or l 0 and l 00 : In particular, te central bank can implement te rst best allocation by coosing D " = D" P " 2 P l " : (42) Proposition 5. Te central bank can coose D " = D" ; and tere exists a unique equilibrium wit rst best allocation = ; c = c and cj" 2 = c 2 : Proo. Equilibrium prices and quantities satisy P = D q (43) P " 2 = ( )D 2 q 2 : (44) Te constraints in te rm s optimization (27) bind, wic gives q = (45) q 2 = r: (46) 24

25 Substitution or quantities and prices rom (43) - (46) into (23) and (24), To nd D ; substituting or M j"d simpliying gives c = (47) r c 2 = : (48) rom (28) into te market clearing condition (33) and D = ( ) K + M a"d + M b"d 2 : (49) Substituting rom (49) or D into (28) and simpliying gives te demand or interbank borrowing by bank j as M j"d = ( j" Rearranging, aggregate bank borrowing is ) ( ) K + j" a"d (M + M b"d ) M j"d : M j"d Using (50), we can sow tat + M j"d = ( ) K ( j" ) + j" a"d (M + M b"d ); (50) (M a"d + M b"d ) + (M a"d + M b"d ) = 2(M a"d + M b"d ): (5) By market clearing equation (33), aggregate net interbank borrowing is zero, M a"d + M b"d (M a"d = 0; wic by equation (50) implies (M a"d + M b"d ) = 2(M a"d + M b"d ). Hence, +M b"d ) = 0: Aggregate net borrowing rom te central bank is zero in equilibrium. Te central bank lends zero net supply o liquidity to te market. Wile bank j aggregate net borrowing rom te interbank market and te central bank is determined by equation (50) as M j"d + M j"d = ( ) K ( j" ); te individual components M j"d and M j"d are not determined. Te central bank does not need to lend to any banks in equilibrium. Lending by te central bank is equivalent and a substitute or interbank lending. Substition into (29) or K t rom (37) and (38), or D " rom (36), or D" rom (42), or = c rom (47), and or D = P q = P ( ) gives c j" 2 = Dj" 2 P 2 " = r (j" )c l " j" ; rom (43) and (45), and rearranging wic is identical to c j" 2 in te real model given by equation (5). Te bank as an optimization identical to tat in te real model and cooses = : Hence, te equilibrium is identical to tat o te real model and te allocation is c = c 25 and cj" 2 = c 2 :

26 8 Appendix B: Generalization to N socks Consider te case o te baseline real model (witout te central bank, nominal rates, runs or liquidation o assets) were " can take N values, " ; :::; " N 0. We maintain te assumption tat " = 0. Te probability o " i is i, P N i= i =. A bank s problem is tus max 2[0;];c 0 u(c ) + NX i [ 2 ( i= s.t. j" c + j" + j" ( j" )c j" 2 r j" j" l " or j 2 a; bg and " 2 " ; :::; " N g: a" i )u(c a" i 2 ) + 2 ( b" i )u(c b" i 2 )] Te rst order conditions wit respect to and c are, respectively, NX i= i [ 2 u0 (c a" i 2 ) + 2 u0 (c b" i 2 )]l" i = NX i= i [ 2 u0 (c a0 2 ) + 2 u0 (c b0 2 )]r (52) u 0 (c ) = NX i= i [ a" i 2 u0 (c a" i 2 ) + b" i 2 u0 (c b" i 2 )]l" i (53) By te same logic as in te case wit two socks, te interest rate in te interbank market sould be equal to c 2 =c wenever " i > 0 in order to acilitate risk saring between banks. Witout loss o generality, assume tat " i or all i 2. Ten we ave l " i = c 2 =c and c a" i 2 = c b" i 2 = r or all i 2. Let P N i=2 " i =, ten we can write interest rate l " as wic is te same as in te two sock case. 6 l " = r + (r c 2 c ) ; (54) 6 We can sow tat i tere is no state wit a zero-size sock, ten a rst best equilibrium does not exist because an equilibrium requires an interest rate o l " > c 2 c or at least one state "; wic is ten always distortionary. I te baseline real model is modi ed suc tat " 0 > " 00 > 0; we can sow tat tere is a constrained e cient equilbrium wit l 0 < l 0 < r < l 00 < l 00 ; wic is cosen by te central bank. 26

27 Reerences [] Allen, Franklin, Elena Carletti and Douglas Gale (2008). Interbank Market Liquidity and Central Bank Intervention, working paper. [2] Allen, Franklin and Douglas Gale (2004). Financial Fragility, Liquidity, and Asset Prices, Journal o te European Economic Association 2. [3] Ascrat, Adam, James McAndrews and David Skeie (2008). Precautionary Reserves in te Interbank Market, working paper. [4] Battacarya, Sudipto and Douglas Gale (987). Preerence Socks, Liquidity and Central Bank Policy, in W. Barnett and K. Singleton (eds.), New Approaces to Monetary Economics, Cambridge University Press, [5] Bolton, Patrick, Tano Santos and Jose Sceinkman (2008). Inside and Outside Liquidity, working paper. [6] Buiter, Willem, (2008). Lessons rom te Nort Atlantic nancial crisis, Manuscript (ttp:// wbuiter/nacrisis.pd). [7] Cooper, R. and T. W. Ross (998). Bank runs: Liquidity costs and investments distortions, Journal o Monetary Economics 4, [8] Diamond, Douglas W. and Pillip H. Dybvig (983). Bank Runs, Deposit Insurance, and Liquidity, Journal o Political Economy 9. [9] Diamond, Douglas W. and Raguram G. Rajan (2008). Illiquidity and Interest Rate Policy, working paper. [0] Freixas, Xavier and Cornelia Holtausen (2005). Interbank Market Integration under Asymmetric Inormation, Review o Financial Studies 8, [] Freixas, Xavier and Jose Jorge (2007). Te Role o Interbank Markets in Monetary Policy: A Model wit Rationing, working paper. [2] Goodriend, Marvin and Robert G. King (988). Financial deregulation, monetary policy, and central banking, Economic Review, Federal Reserve Bank o Ricmond, issue May/Jun, pages

28 [3] Gutrie, Graeme and Julian Wrigt (2000). Open Mout Operation, Journal o Monetary Economics 46. [4] Heider, Florian, Marie Hoerova, and Cornelia Holtausen (2008) Inormation Asymmetries in te Interbank Market: Teory and Policy Responses, working paper. [5] Skeie, David R. (2008). Banking wit Nominal Deposits and Inside Money, Journal o Financial Intermediation 7. 28