Liquidity Hoarding 1

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1 Liquiity Hoaring Douglas Gale New York University Tanju Yorulmazer 3 Feeral Reserve Bank of New York 9 August, The views expresse here are those of the authors an o not necessarily represent the views of the Feeral Reserve Bank of New York or the Feeral Reserve System. We woul like to thank Franklin Allen, Gai Barlevy, Marco Bassetto, Markus Brunnermeier, Xavier Gabaix, Stephen Morris, Hyun Shin, Jeremy Stein, Anjan Thakor, Dimitri Vayanos, Vish Viswanathan, Wei Xiong an seminar participants at the New York Fe, Boar of Governors, Lonon School of Economics, Chicago Fe, European Summer Symposium in Economic Theory (ESSET) in Gerzensee, Summer Workshop on Money, Banking, Payments an Finance at the Chicago Fe, Sabanci University, Seoul National University, Southern Methoist University an the University of Texas at Dallas. Peter Hull provie excellent research assistance. All errors remain our own. Contact: Department of Economics, New York University, 9 West 4th Street, New York, NY, ouglas.gale@nyu.eu, Phone: , Fax: Contact: Feeral Reserve Bank of New York, Financial Intermeiation Function, 33 Liberty Street, New York, NY 45, tanju.yorulmazer@ny.frb.org, Phone: , Fax:

2 Abstract Banks hol liqui an illiqui assets. An illiqui bank that receives a liquiity shock sells assets to liqui banks in exchange for cash. We characterize the constraine e cient allocation as the solution to a planner s problem an show that the market equilibrium is constraine ine cient, with too little liquiity an ine cient hoaring. Our moel features a precautionary as well as a speculative motive for hoaring liquiity, but the ine ciency of liquiity provision can be trace to the incompleteness of markets (ue to private information) an the increase price volatility that results from traing assets for cash. J.E.L. Classi cation: G, G, G4, G3, G33, D8. Keywors: Interbank market, re sale, market freeze, cash-in-the-market pricing.

3 Introuction One of the most interesting phenomena marking the recent nancial crisis was the freezing of the interbank market. As early as the fall of 7, following the collapse of the market for asset backe commercial paper, European banks reporte i culty borrowing in the interbank market. At the same time, interbank borrowing rates reache recor levels. Di culty obtaining liquiity in interbank markets was subsequently experience in many countries. As a result, central bank borrowing facilities became an essential source of liquiity for nancial institutions. One possible explanation for this phenomenon is counter-party risk. Because of the wiesprea exposure to sub-prime asset-backe securities, banks ha goo reason to be wary of lening to any bank that might be a creit risk because of this or any other exposure. In this paper we explore a secon possible explanation, that banks were hoaring liquiity because of fears that their own future access to liquiity might be compromise. There is substantial evience that banks reuce their lening to other banks in orer to buil up cash positions (Acharya an Merrouche, 9; Heier, Hoerova an Holthausen, 8; Ashcraft, McAnrews an Skeie, 8). The two possible explanations are not unrelate. Even if the interbank market initially froze because of counter-party risk, liquiity hoaring woul still be a rational response to fears of future lack of access to liquiity. In this paper, we use a simple moel of liquiity management to analyze the e ciency of laisser faire equilibrium. Our moel assumes a large number of bankers, who can hol two types of assets, a liqui asset an an illiqui asset. We refer to the liqui asset as cash an refer to the illiqui asset simply as the asset. Bankers are subject to stochastic liquiity shocks that we interpret as unanticipate emans for repayment of a senior ebt. If a banker receives a liquiity shock an lacks the cash to meet the claim, he is force to sell some of his holings of the asset. If the eman for cash is high, the price of the asset may be corresponingly low. In equilibrium, bankers weigh the cost of holing cash against the See footnote 5 in Section 4. for a iscussion of the liquiity facilities introuce by the Feeral Reserve in the recent crisis.

4 cost of having to sell the asset at a re sale price. We begin our analysis by solving the problem of a planner who etermines how much cash to hol an when to istribute it. The solution to the planner s problem takes a very simple form: after etermining the e cient amount to hol at the rst ate, the planner supplies cash to every banker who nees it at a given ate until the supply runs out. Even though there may be a future nee for the cash, the planner never carries forwar a positive balance as long as there is a banker who nees cash to meet a liquiity eman toay. The simple form of the solution to the planner s problem makes it easy to ientify inef- cient hoaring. Hoaring liquiity is ine cient if an only if it occurs, at some ate, when there are still bankers who nee liquiity. Our secon result is to show that, in a laisser-faire market equilibrium, there is always (ine cient) hoaring. More precisely, when the eman for cash is su ciently high some bankers will be price out of the market for cash, while liqui bankers are hoaring cash rather than supplying it to the market. A liqui banker has two reasons for hoaring cash. One is a precautionary motive. The banker may himself receive a liquiity shock in the future. If he uses his cash toay an then receives a liquiity shock tomorrow, he can obtain cash by selling the illiqui asset, but the price may be very high. There is also a speculative motive. If the future eman for cash is very high, asset prices will be low. If he oes not receive a liquiity shock, a hoarer may pro t from buying assets at re sale prices. Clearly, these two motives cannot be separate: the same cash holings serve both motives. The incentive to hoar cash come from the expecte volatility of future asset prices an, in a laisser-faire equilibrium, the incentives to hoar are simply too high. Asset-price volatility results from the use of the asset market as a source of liquiity. When liqui bankers rst supply cash in exchange for assets, they create an imbalance in the system. If these bankers are subsequently hit by a liquiity shock, they have even more assets to ump on the market, proucing a greater re sale an reucing asset prices further. This buil up in volatility increases both the precautionary an speculative motives, which are responsible for ine cient hoaring.

5 As a thought experiment, we consier an alternative version of the moel in which the assets purchase in exchange for cash at one ate can be spun o into a special purpose vehicle (SPV) that is completely inepenent of the bank holing company. If the bank is threatene with efault in a future perio, it has no reason to ump the assets in the SPV, since these will survive the bank s efault. If all banks pursue this strategy, future volatility will be lower than in the baseline moel, the incentive to hoar is also lower, an there is no ine cient hoaring in equilibrium. Our thought experiment shows that the imbalance in bank portfolios only results in ine cient hoaring if banks become too big as well as too illiqui. Our thir result characterizes the optimal intervention by a central bank. A central bank is subject to more constrains than a central planner. A central planner has exclusive control of the allocation of liquiity. A central bank, by contrast, has to compete with markets in which cash an assets are exchange. Generally speaking, the existence of markets is a problem because it provies bankers with arbitrage opportunities that might unermine the central bank s e ort to improve welfare. In this case, however, the central bank can successfully implement the planner s solution. Because the central bank is a large player, it can in uence the prices at which markets clear. The optimal strategy is for the central bank to accumulate an supply so much liquiity that private bankers are force out of the market entirely. More precisely, the central bank makes liquiity cheap enough that none of the bankers wants to supply liquiity in competition with the central bank. In equilibrium no one, apart from the central bank, hols cash an every one relies for liquiity on the lener of last resort, who becomes in e ect the lener of rst resort. We also explore a number of smaller interventions in the market for liquiity. One of these allows the central bank to control the total quantity of the liqui asset in the system, but leaves it up to the market to etermine when an at what price this cash is use to purchase assets. We show that it is always optimal to increase the quantity of the liqui asset above the equilibrium level. A similar experiment allows the central bank to control the amount of liquiity hoare while allowing bankers to etermine freely the amount of liquiity in the 3

6 system. We show that the central bank can always improve welfare by reucing the amount of ine cient hoaring, while allowing markets to clear at other ates. These results con rm our intuition about the sources of ine ciency in laisser-faire equilibrium, speci cally, the inaequate incentive for banks to hol cash initially an the excessive incentive to hoar liquiity once liquiity shocks are realize. The funamental reason for the ine ciency of the laisser-faire equilibrium is the incompleteness of markets. Illiqui bankers are force to acquire the liqui asset ex post by selling the illiqui asset on a spot market rather than entering into contingent contracts for the provision of liquiity ex ante. We argue that contingent contracts cannot improve on equilibrium welfare in the presence of asymmetric information. More precisely, if bankers cannot be force to eliver the liqui asset when they have receive a liquiity shock or, conversely, cannot be force to receive the liqui asset when they have not receive a liquiity shock, the possibility of arbitrage in spot markets plus private information about the liquiity shock rule out any gains from trae. The rest of this paper is organize as follows. We begin our analysis in Section by stuying the constraine-e cient allocation chosen by a central planner who accumulates a stock of liqui assets an istributes them to the banks that report a nee for liquiity. Then, in Section 3, we analyze a laisser-faire economy in which banks make their own ecisions about liquiity accumulation an liquiity provision. In Section 4, we investigate the constraine (in)e ciency of the laisser-faire economy, an show that there are several simple interventions that can improve on the laisser-faire allocation. We conclue by iscussing some variants of the moel to she more light on various sources of ine ciency in Section 5. Constraine e ciency In this section, we characterize the constraine-e cient allocation as the solution to a planner s problem in which the planner accumulates an istributes the liqui asset. The resulting allocation serves as a benchmark in our welfare analysis. 4

7 . Primitives Time: Time is ivie into four ates, inexe by t = ; ; ; 3. At the rst ate, bankers choose the amount of liquiity they hol as part of their portfolio. At the secon an thir ates, bankers receive liquiity shocks an trae assets in orer to obtain the liquiity they nee. At the nal ate, asset returns are realize. Assets: There are two assets, a liqui asset that we refer to as cash, an an illiqui asset that we will refer to simply as the asset. Cash can be use to ischarge ebts an can be store from perio to perio. One unit of cash can be converte into one unit of consumption at any ate. The asset cannot be use to ischarge ebts (unless it is rst exchange for cash). The asset can be store from perio to perio. One unit of the asset has a return of R > units of consumption at ate 3. Bankers: There is a continuum of ientical, risk neutral agents, inexe by i [; ], whom we call bankers. Each bank has an initial enowment consisting of unit of the asset an one unit of cash at ate, enote by the vector (; ), where the rst an the secon components represent the quantity of the asset an cash in bank s portfolio, respectively. The banker s utility function is U (c ; c 3 ) = c + c 3 ; where c enotes consumption at ate an c 3 enotes consumption at ate 3 an > is a parameter. The interpretation of this utility function is the following: bankers prefer consumption at ate to consumption at ate 3, so holing cash after ate (instea of converting it into consumption immeiately) involves a cost >. Creitors: There is a continuum of ientical, risk neutral agents, inexe by j [; ], whom we call creitors. Each creitor j is owe a ebt by bank i = j that is payable on eman. The face value of the ebt is one unit of cash. Creitors are uncertain about their time preferences. More precisely, they want to consume at precisely one of the ates t = ; ; 3 but uncertain which ate they prefer. A typical creitor wants to consume at ate 5

8 with probability, at ate with probability ( ), an at ate 3 with probability ( ) ( ). The shocks an are ranom variables with ensity functions f ( ) an f ( ) an c..f. F ( ) an F ( ), respectively. We assume that an are ii with support [; ]. The creitor s expecte utility function is given by V (c ; c ; c 3 ) = E [ c + ( ) c + ( ) ( ) c 3 ] ; where c t enotes consumption at ate t = ; ; 3. Liquiity shocks: Bankers are sai to receive a liquiity shock if the banker s creitor emans repayment at ate or ate. If a banker is not hit by one of these shocks, he pays o his ebt at t = 3, after the return from the asset is realize. A banker who receives a shock must immeiately eliver one unit of cash to ischarge the existing ebt; otherwise he will be force to efault. If the banker becomes bankrupt, we assume that all his assets are immeiately liquiate an, for simplicity, we assume that the liquiation costs consume the entire value of the assets. This assumption can be relaxe, but it greatly simpli es the analysis an oes not appear to a ect the qualitative results too much. In orer to obtain cash, a banker can sell some or all of his holings of the asset. Bankers who receive a liquiity shock at ate will not receive a liquiity shock at ate.. The planner s problem There are two groups of economic agents, bankers an creitors, but each group consists of ex ante ientical agents at ate. Since it is possible to make transfers between the two groups at ate 3, we can reistribute the total surplus any way we like between the groups. So, in orer to maximize ex ante welfare, it is necessary an su cient to maximize total expecte surplus. In what follows, we take this as the planner s objective function. In aition to the usual feasibility constraints, the planner operates subject to the constraint that he cannot transfer assets between bankers. If the planner were able to transfer assets, he woul assign all assets at ate to bankers who ha alreay receive a liquiity shock, 6

9 thus renering the liquiity shocks at ate irrelevant. To avoi this trivial solution, we restrict the planner s actions to accumulating cash at ate, istributing cash at ates an, an reistributing the consumption goo at ate 3. Suppose that the planner has m units of cash at the beginning of ate an the state is ( ; ). There are ( optimal strategy is to supply the lesser of ( ) bankers who receive a liquiity shock in this perio. The ) an m to the bankers in nee of cash to ischarge their ebts. Each unit of cash is worth one unit at ate 3, whether it is hel by the planner or pai to a creitor an, in aition, each unit istribute to a banker with a liquiity nee saves an asset worth R at ate 3. So it is optimal to save as many assets as possible. Now suppose the planner has m units of cash at the beginning of ate an the state is. There are bankers who receive a liquiity shock in this perio. Each unit of cash istribute to these bankers is worth + R, because one unit of cash always prouces a return of one unit at ate 3 an it is worth an aitional R units if it saves an asset. On the other han, the expecte value of a marginal unit of cash hel until ate must be less than + R. We have seen before that the value of cash is at most + R an it will be only if the amount carrie forwar is greater than ( ), which happens with positive probability if the amount carrie forwar is positive. So it is optimal to save as many assets as possible at ate an the optimal strategy is to istribute the lesser of m an at ate. At ate, the choice of how much liquiity to hol is etermine by equating the marginal cost of cash,, to the marginal value of cash. As usual, a unit of cash hel at the en of ate is always worth one unit at ate 3 but it is worth an aitional R units if it can be use to save an asset. The probability that the marginal unit of cash is use to save an asset is simply the probability that m is less than + ( ). This probability is calculate to be Z m m Pr [ + ( ) > m ] = F f ( ) ; so the marginal value of cash carrie forwar at ate is Z m m R F f ( ) + : 7

10 The solution to the planner s problem is characterize by an array (m ; m ( ) ; m ( ; )), where m is the amount of cash carrie from ate, m ( ) is the amount of cash carrie forwar from ate in state an m ( ; ) is the amount of cash carrie forwar from ate in state ( ; ). The previous argument leas to the following proposition. Proposition The planner s optimal strategy is characterize by an array (m ; m ( ) ; m ( ; )) e ne by the following conitions: m ( ; ) = max fm ( ) ( ) ; g ; m ( ) = max fm ; g an R Z m m F f ( ) + = : Proof. See Appenix. We have assume so far that the planner has complete information about the banker s types. That is, he observes the realizations of an an knows which bankers have receive a liquiity shock at each ate. In the case where liquiity shocks are private information, the planner nees to use an incentive-compatible mechanism in orer to extract information from the bankers. A irect mechanism is e ne by an array ( ( ) ; p ( ) ; ( ; ) ; p ( ; )), where ( ) is the probability that an agent who reports a liquiity shock at ate in state receives one unit of cash an p ( ) is the price he pays for it an ( ; ) is the probability that an agent who reports a liquiity shock at ate in state ( ; ) receives a unit of cash an p ( ; ) is the price he pays for it. An agent who reports no liquiity shock is assume without loss of generality to receive no cash an make no payment. We can show that the constraine e cient allocation that solves the planner s problem can be implemente as a truth-telling equilibrium of a irect mechanism. We postpone this exercise until Section 4., where it appears as a corollary of another, stronger result. 8

11 3 A laisser-faire economy In this section, we provie an informal account of equilibrium in a laisser-faire economy. The formal e nition an analysis are containe in Appenix A. The time line illustrate in Figure shows the activities that occur in each of the four ates t = ; ; ; 3. We escribe these activities in more etail below. Figure about here Date Bankers are initially enowe with one unit of the asset an one unit of cash. At ate, bankers choose whether to consume their cash immeiately or retain one unit in their portfolios for future use. We call the bankers who retain the cash liqui an those who o not illiqui. Let enote the measure of illiqui bankers. The illiqui bankers en the perio with a portfolio consisting of one unit of the asset an no cash. The liqui bankers en the perio with a portfolio consisting of one unit of cash an one unit of the asset. Date At the beginning of ate, a fraction of bankers receive the liquiity shock. The liqui bankers who receive the shock can ischarge their ebt using their cash holings an en the perio with a portfolio consisting of one unit of the asset an no cash. The alternative is to efault an lose everything. The illiqui bankers who receive a liquiity shock sell part of their asset holings in exchange for cash to ischarge their ebt an en the perio with a portfolio consisting of p units of the asset an no cash, where p enotes the price of one unit of cash. If some of these bankers cannot obtain cash to ischarge their ebt, they must be ini erent between obtaining cash an efault. This will be the case if p =. An alternative to asset sales is secure borrowing, in which illiqui bankers who receive a shock put up p units of the asset as collateral against a loan of one unit of cash. The loan matures at ate 3, at which point the banker either repays p R units of cash or forfeits 9

12 the collateral. Notice that the interest rate on the loan is p R, which is high enough to make the banker ini erent whether he reclaims the collateral or not. Clearly, uner these assumptions, secure lening is equivalent to asset sales. The illiqui bankers who o not receive a shock o not trae an en the perio with their initial portfolio consisting of one unit of the asset an no cash. The liqui bankers who o not receive a liquiity shock have the option of acquiring p units of the asset using their one unit of cash. Liqui bankers who use their cash to purchase the asset are calle buyers; those who o not are calle hoarers. We enote by the fraction of these bankers that become buyers an en the perio with a portfolio of +p units of the asset an no cash. The complementary fraction become hoarers an en the perio with their initial portfolio consisting of one unit of the asset an one unit of cash. Date Some of the bankers at ate have nothing to trae an remain inactive. The bankers who receive a liquiity shock at ate have no cash an have no motive to trae the asset for cash since they cannot receive another liquiity shock. Similarly, the illiqui bankers who i not receive a liquiity shock at ate have no cash an have no motive to trae the asset for cash if they o not receive a liquiity shock. For the same reason, the buyers who o not receive a liquiity shock at ate have no cash an no motive to trae the asset for cash. Finally, the hoarers who receive a liquiity shock at ate will use their cash to ischarge their ebt an then have no gains from trae. This leaves three types of agents who can actively trae at ate, the hoarers who o not receive a liquiity shock an the buyers an illiqui bankers who receive a liquiity shock at ate. These agents trae cash for the asset at the market-clearing price p. The hoarers are willing to supply all of their cash at any price p R. The illiqui bankers, who hol one unit of the asset, are willing to supply the asset for one unit of cash at any price p (because the alternative is to efault). Similarly, the buyers, who hol + p units of the asset, are We will show that, in equilibrium, the price of cash at ate is equal to the expecte price of cash at ate. This is su cient to prove that an illiqui banker cannot improve his payo by purchasing cash at ate.

13 willing to supply the asset for one unit of cash at any price p + p. Again, an alternative of the moel is that banker s in nee of liquiity engage in secure lening at ate. In orer to obtain one unit of cash, the banker has to put up p units of collateral. At ate 3, he is oblige to pay p R units of cash to ischarge the ebt an reclaim the collateral. The allocation of assets in the rst two ates is illustrate in Figure an the allocation of assets at the en of ate is illustrate in Figure 3. Figure about here Figure 3 about here Date 3 At the last ate, bankers receive the payo s from the portfolios of cash an the asset carrie forwar from ate. Bankers who have not alreay ischarge their ebts must pay their creitors one unit of cash. The terminal payo s, which are easily calculate from the terminal allocation, are illustrate in Figure 4. Figure 4 about here Throughout, we assume that the liqui asset is inivisible. However, all our results go through when we allow the liqui asset to be ivisible. In particular, we can allow bankers to hol a fraction (; ) units of liquiity an consume the rest at t = ; an, if not hit by the liquiity shock at t =, use a fraction (; ) of his liquiity to purchase assets at t = while hoaring the rest, a fraction of his liquiity. We can show that such a strategy is not a pro table eviation from the equilibrium we construct below where we restrict f; g an f; g. 3 3 To keep the analysis short an simple we o not report these results that are available from the authors.

14 3. Market clearing In this section, we ientify the market clearing prices p an p, beginning at ate an working back to ate. The price at ate will be a function of the state at ate an the price at ate will be a function of the state ( ; ) at ate, but for the most part this notation will be suppresse as we take the state as given. 3.. Market clearing at ate Suppose that the state of the economy at ate is ( ; ). As we explaine above, the eman for cash comes from the buyers an illiqui bankers who receive a liquiity shock at ate. The supply of cash comes from the hoarers who o not receive a liquiity shock at ate. There are three regimes in the market for cash an assets at ate, e ne by two critical values of, enote by an an e ne by = ( ) ( ) an = : (i) Low eman for liquiity <. When the value of is low enough, the among of cash hel by the hoarers is more than enough to supply the buyers an illiqui bankers, so at the margin some hoarers have to be willing to hol cash. This means that they are ini erent between holing cash an the asset, which will only be true if the price of liquiity satis es p = R. (ii) Intermeiate eman for liquiity < <. When the value of is in an intermeiate range, the hoarers have enough cash to supply the buyers an some but not all illiqui bankers. Then the illiqui bankers must be ini erent between selling their assets for cash an efaulting. This will be true if an only if p =. (iii) High eman for liquiity >. Finally, when eman for cash is high, the hoarers have only enough cash to supply some but not all buyers, so the buyers must be ini erent between selling assets to obtain liquiity an efaulting. This occurs if p = + p.

15 We summarize the preceing iscussion in the following proposition, which is illustrate in Figure 5. Proposition The market-clearing price at ate is enote by p ( ; ) an e ne by 8 R >< for < ; p ( ; ) = for < < ; >: + p ( ) for < ; where = ( ) ( ( )) an = ( ): 3.. Market clearing at ate The analysis of market clearing at ate is a bit more complicate, because bankers ecisions epen on expectations about ate. The rst step is to show that, in equilibrium, there will always be some bankers who buy assets an some who hoar cash at ate. This requires that the bankers with spare cash are ini erent between buying an hoaring. We can show that it is optimal to hoar if an only if p E [p ] an, conversely, it is optimal to buy if an only if p E [p ]. Thus, ini erence is equivalent to p = E [p ]. Now consier what will happen if there are no buyers, that is, =. The excess eman for cash at ate implies that p =, but at ate the price p must be less than or equal to one (since there are no buyers) an will sometimes be less than one (when is su ciently small). Then E [p ] < = p contraicting the optimality of hoaring. Conversely, if =, the price at ate must satisfy p = +p because there will be excess eman for cash with probability one, at least from the buyers that get the liquiity shock at t =, but this violates the optimality conition for buying. Hence, we get the following proposition. Proposition 3 For every value of, < ( ) < in equilibrium at ate. Thus, bankers holing unneee cash at ate are ini erent between hoaring cash an buying the asset in equilibrium, which hols if an only if p ( ) = E [p ( ; ) j ] : 3

16 Proof. See Appenix. From Proposition 3, we know that p = E [p ] an from Proposition we know the istribution of p as a function of, which allows us to calculate the value of E [p ] as a function of. Let ~p () enote this value for each value of. There is a unique value of, call it (; ), such that ~p = an ~p () < if an only if <. If p <, then the market-clearing conition tells us that or ( ) ( ) = = ( ) ( ). On the other han, ~p () = implies that =. Putting these facts together, we can characterize the equilibrium values of p an in the following result. Proposition 4 The market clears at ate if an only if the equilibrium values of an p are given by an ( ) = min ( ) ( ) ; p ( ) = min ~p for every value of, where ( ) ( ) ; ; ~p () = F (( ) ( )) ( R ) F ( ) for every value of an is the unique value of (; ) satisfying ~p () =. Proof. See Appenix. Note that ( ) is weakly increasing in. Hence, there is a unique value of, enote by, where ( ) = for >. Thus, we observe rationing in the market for > ; where some banks cannot get liquiity an their assets nee to be liquiate prematurely. Figure 6a illustrates this point. Similarly, we can see that ~p ( ( )) is weakly increasing in since ~p () is increasing in. Hence, at =, price ~p () reaches, the market clearing 4

17 price p ( ) cannot increase any further an stays at for >. The market clearing price p ( ) is illustrate in Figure 6b. Figure 6 about here Example 5 Let be uniformly istribute over the unit interval [; ]: In this case, we can show that = ( ) ; where = ( ) (R ) R + ( ) (R ) : Furthermore, we obtain p ( ) = min ( ) R + ; : 3..3 Market clearing at ate Just as we showe that buyers an the hoarers have the same expecte return at ate, we can show that < < in equilibrium at ate an that bankers must therefore be ini erent between acquiring liquiity an not acquiring it. Note that the cost of holing liquiity is. We can erive the bene t from holing liquiity by comparing the expecte return of a liqui bank (that becomes a hoarer) with that of an illiqui bank. 4 We have three cases: (i) Shock at t = : In this case, a liqui bank uses her own liquiity to pay creitors an saves her project, whereas an illiqui bank nees to sell a fraction p of her assets. Hence, a liqui bank, in expectation, saves p R compare to an illiqui bank. (ii) Shock at t = : In this case, a liqui bank can use her own liquiity to pay creitors. However, an illiqui bank nees to sell assets at t =. For p 6, the illiqui bank 4 Note that in equilibrium buyers an hoarers have the same expecte return. For simplicity, we focus on a liqui bank that ecies to become a hoarer. 5

18 can get the neee liquiity by selling p units of her asset but for p > she has to efault. Hence, a liqui bank, in expectation, saves ( to an illiqui bank. ) R minf; p g compare (iii) No shock: In this case, a liqui bank can acquire p units of the asset at t =, which results in an expecte return of ( )( )p R for a liqui bank compare to that of an illiqui bank. When we combine these three cases an use the equilibrium conition p = E[p ], we get the following formal proposition. Proposition 6 In equilibrium, < <, which implies that bankers will be ini erent at ate between holing liquiity an not holing it. Agents are ini erent if an only if R Z Proof. See Appenix. p f ( )( F ( ))E [ j > ])g f ( ) = : The rst-orer conition in Proposition 6 i ers from the rst-orer conition for the planner s choice of m. The right-han sie, the cost of liquiity, is the same in both conitions but the left-han sies i er. First, the planner internalizes the bene t of the payment to the creitors whereas banks o not. Secon, liqui banks that receive a shock at ate use their own cash to avoi efault, even though there are illiqui banks that value the liquiity more highly. The planner, by contrast, minimizes welfare losses by supplying liquiity to the largest banks rst. These two e ects explain the i erence between the two rst-orer conitions. 3. Equilibrium An equilibrium is escribe by the enogenous variables, ( ), p ( ), an p ( ; ) satisfying the following conitions. De ne ~p () by putting ~p () = F (( ) ( )) ( R ) F ( ) 6

19 for every an let be the unique value of < < satisfying ~p () =. Then the equilibrium functions p ( ) an ( ) satisfy ( ) = min ( ) ( ) ; an for every value of. p ( ) = min ~p ( ) ( ) ; ; The equilibrium price function p ( ) must satisfy 8 R >< for < ( ) ; p ( ; ) = for ( ) < < ; >: + p ( ) for < ; where ( ) = ( ) ( ( )) an = ( ) : Finally, at ate, market-clearing requires ini erence between acquiring an not acquiring liquiity: Z p f ( )( F ( ))E [ j > ])g f ( ) = R : 4 Policy analysis In this section, we provie an analysis of various policies aime at improving liquiity an its allocation in markets. 4. Central bank as sole lener In this section, we introuce a Central Bank (CB) into the moel. We escribe an equilibrium in which the CB acts as the sole supplier of liquiity, all bankers choose to be illiqui, an the constraine e cient policy characterize in Proposition can be implemente. 7

20 Our approach is constructive. We assume that =, that is, all banks choose to be illiqui, an that the CB chooses as its policy the constraine e cient policy (m ; m ; m ) given in Proposition. We e ne an equilibrium with the CB acting as a LoLR along the lines of the laisser-faire equilibrium. At ate, there are no buyers, so the eman for liquiity comes from the ( ) bankers who have receive a liquiity shock at ate. Since the supply of money is max fm by ; g, the market clearing price p ( ; ) is e ne 8 < R if ( ) < max fm ; g ; p ( ; ) = : if ( ) > max fm ; g : Similarly, at ate, the eman for liquiity comes from the () bankers who receive a liquiity shock at ate an the supply is at most m. If > m the market clearing price must be p ( ) =, but when < m the price may lie anywhere between E [p ( ; ) j ] an. Since the CB can control the price we assume that it sets p ( ) = E [p ( ; ) j ], so that the bankers who i not receive a shock are ini erent between hoaring an buying. Then the market clearing price is 8 < E [p ( ; ) j ] if < m ; p ( ) = : if > m : Market clearing at ate requires that it is optimal for bankers to choose =. We can show that this is the case, which gives us the following proposition. Proposition 7 In an equilibrium where the CB acts as the sole provier of liquiity, all bankers choose to become illiqui, that is, = ; market-clearing prices at ate an are given in equations () an (), respectively; an the constraine e cient policy (m ; m ; m ) given in Proposition can be implemente. Proof. See Appenix. Hence, in equilibrium, the CB by acting as the sole provier of liquiity can implement the constraine e cient allocation from the planner s problem in Section. This results in the CB liquiity crowing out private liquiity. Next, we look at some simpler ex ante (ate ) an ex post (ate ) policies that can be use to improve welfare. 8 ()

21 4. Policy analysis with private liquiity In this section, we analyze two i erent policies that aim at maximizing the expecte total output that restrict (one at a time): (i) the portfolio choice (namely ) at ate ; an (ii) the level of lening (namely ) at ate. Other than the ate we impose the restriction, we assume that the markets will function as in Section 3. where we characterize the equilibrium. Since the planner in Section. is alreay restricte the resulting outcome from the planner s problem is constraine e cient, say secon best. The policies we analyze in this section constrain the policy maker more compare to the planner in Section.. Hence, the resulting outcomes qualify for a thir best an, for simplicity, we use the term socially optimal in this section. First, we try to n the socially optimal level of lening at t =, enote by soc, that maximizes the expecte output assuming that the market for asset sales at t = will function as in Section 3.. At t =, the liquiity shock is realize an we can n the expecte output for each realization of. Then we can n soc an compare it with the privately optimal level of lening given in Proposition (4). In calculating the expecte output at t =, we nee to consier three i erent regions for : (i) For <, there is enough liquiity for all agents that got hit by the liquiity shock at t =. Hence, no asset nees to be liquiate. (ii) For < <, there is enough liquiity for all buyers that got hit by the liquiity shock but not enough for all illiqui agents that got hit. Hence, some of the assets hel by illiqui agents that got hit nee to be liquiate prematurely. (iii) For > ; there is not enough liquiity even for all buyers that got hit by the liquiity shock. Hence, some of the assets hel by buyers that got hit an all the assets hel by illiqui agents that got hit nee to be liquiate prematurely. Using these we can calculate the total expecte output an n the level of lening soc that maximizes the expecte output. The following proposition characterizes the socially 9

22 optimal level of lening at t = an compares it with the equilibrium level of lening at t = characterize in Proposition (4). Proposition 8 We can characterize the socially optimal level of lening soc as follows: soc ( ) = min ( )( ) ; ~ ; where ~ is etermine implicitly by the conition Furthermore, we obtain ~ > : Proof. See Appenix. F (( ) ( ~ )) + F ( ~ ) = : The socially optimal level of lening has the same structure as the equilibrium level of lening. In particular, as in the equilibrium, the socially optimal level of lening requires that the liquiity nee of all illiqui agents that got hit by the shock at t = be satis e up to the threshol ~, which is higher than the threshol in equilibrium. Hence, in equilibrium there is ine ciently low level of lening, that is, equilibrium is characterize by an ine ciently high level of hoaring at t =. Thus, a policy that aims at facilitating lening at t = or lening irectly to banks can improve e ciency. One possibility is that the central banks can provie liquiity to markets in general through open market operations (OMO), which then is transferre to institutions in nee through the interbank market. Goofrien an King (988) argue that with e cient interbank markets central banks can provie su cient liquiity via OMOs an the interbank market will allocate the liquiity among banks so that the activities of central banks shoul be limite to monetary policy an they shoul not len to banks on an iniviual basis. The current crisis provies us evience that OMOs can have limite e ect in channeling liquiity to institutions that nee it in the presence of uncertainty about future liquiity shocks an hoaring incentives. For example, Governor of the Bank of Englan Mervyn King an the Chancellor of the Exchequer Alistair Darling, uring the hearings about the Northern Rock episoe in the Fall

23 of 7, pointe out the i culties with OMOs in channeling liquiity to neey banks as the primary reason for lening irectly to iniviual institutions. In particular, they pointe out that to channel the 4 billion that Northern Rock borrowe from the Bank of Englan to that institution woul have require many more billions of pouns to be injecte through the OMOs. In the same hearing, William Buiter suggeste: That woul take an enormous amount of money injections. We know for instance that espite all the money that the Fe an especially the ECB have put into these longer term markets, the actual spreas of three months LIBOR an the euro equivalent an the ollar equivalent over the expecte policy rate is no smaller in euro lan toay than it is here, so it really may take a large injection of liquiity to get an appreciable result if the market is really fearful. Early in the crisis of 7-9, the Feeral Reserve use OMOs to ease the strain in money markets. While OMOs ha some success in stabilizing the overnight rate, the rates on term loans continue to rise leaing to the introuction of several new liquiity facilities. These new facilities have extene maturities to inclue up to 9-ay loans, maturities at which money markets have rie up in the aftermath of sub-prime losses; extene eligible collateral to inclue investment-grae ebt securities (incluing high-rate but illiqui mortgage-backe securities); an extene these privileges not only to banks but also to securities ealers since they are also a ecte by funing problems cause by the rying up of liquiity extension from banks. 5 5 In particular, in aition to the traitional tools the Fe uses to implement monetary policy (e.g., Open Market Operations, Discount Winow, an Securities Lening program), new programs have been implemente since August 7: ) Term Discount Winow Program (announce August 7, 7) - extene the length of iscount winow loans available to institutions eligible for primary creit from overnight to a maximum of 9 ays; ) Term Auction Facility (TAF) (announce December, 7) - provies funs to primary creit eligible institutions through an auction for a term of 8 ays; 3) Single-Tranche OMO (Open Market Operations) Program (announce March 7, 8) - allows primary ealers to secure funs for a term of 8 ays. These operations are intene to augment the single ay repurchase agreements (repos) that are typically conucte; 4) Term Securities Lening Facility (TSLF) (announce March, 8) - allows primary ealers to plege a broaer range of collateral than is accepte with the Securities Lening program, an also to borrow for a longer term 8 ays versus overnight; an, 5) Primary Dealer Creit

24 Next, we show that the private choice of bankers to hol liquiity at t = oes not correspon to the level of liquiity that maximizes the expecte output. To show that we calculate the expecte output as we i in the analysis of the social optimum at t =. Then we show that at the equilibrium level, the expecte output is ecreasing in so that, at the equilibrium, by increasing the proportion of liqui agents, we can increase expecte output. This gives us the following formal proposition. Proposition 9 In equilibrium, expecte output increases as the fraction of illiqui bankers ecreases. Our results show that equilibrium is characterize by bankers choosing an ine ciently low level of liquiity in their portfolio. One policy measure to aress this issue can be liquiity requirements for banks. While some countries alreay have liquiity requirements, like the UK, others o not have such requirements an there is no international stanar on liquiity regulation like the Basel requirements for bank capital. The Basel III regulatory requirements that are being esigne propose two such measures for liquiity requirements, namely, the Liquiity Coverage Ratio (LCR) an the Net Stable Funing Ratio (NSFR). 6 Below, we provie simulation results that illustrate the wege between the equilibrium an the socially optimal levels of an. We use the parameter values R = 3, = an Facility (PDCF) (announce March 6, 8) - is an overnight loan facility that provies funs irectly to primary ealers in exchange for a range of eligible collateral; 6) Commercial Paper Funing Facility (CPFF) (announce November 7, 8) - is esigne to provie a liquiity backstop to U.S. issuers of commercial paper; 7) Money Market Investor Funing Facility (MMIFF) (announce November, 8) - is aime to support a private-sector initiative esigne to provie liquiity to U.S. money market investors; 8) Term Asset-Backe Securities Loan Facility (TALF) (announce November 5, 8) - is esigne to help market participants meet the creit nees of househols an small businesses by supporting the issuance of asset-backe securities (ABS) collateralize by auto loans, stuent loans, creit car loans etc. 6 LCR requires banks to hol a minimum level of liqui assets that can cover a net cash out ow uring a 3 ay stress perio, whereas NSFR establishes a minimum acceptable amount of stable funing base on the liquiity characteristics of an institution s assets an activities over a one year perio. For more etail see: BCBS () Basel III: International framework for liquiity risk measurement, stanars an monitoring.

25 assume that an are ii an U[; ]. We n that in equilibrium a fraction = :4 of agents choose to become illiqui at t =, whereas the socially optimal level of is :67. We also n that in the equilibrium = :364, whereas in the social optimum ~ = :46. We provie the simulation results for the equilibrium an socially optimal levels of as a function of (Figure 7a) an as a function of (Figure 7b). Figure 7 about here 4.. Comparative statics In this section, we provie comparative statics analysis for lening at t = : In particular, we analyze how the equilibrium an socially optimal levels of, an the wege between the two, are a ecte by the expectations of future liquiity shocks an increase uncertainty an volatility of such shocks. First, we focus on the case when higher liquiity shocks are more likely at t =. To capture the likelihoo of liquiity shocks at t =, we use two i erent probability istributions, f an g, for ; where g rst-orer stochastically ominates f. Hence, higher proportions of the liquiity shock at t = are more likely uner the probability istribution g. From the equilibrium conition we have F f + F ( ) f R = : Since g rst-orer stochastically ominates f, we obtain G f + G ( ) f R < : Note that the LHS of the above inequality is ecreasing in so that we obtain f > g : We can use a similar argument to show that ~ f > ~ g. This gives us the following formal proposition. Proposition Let f an g be two probability istributions over ; where g rst-orer stochastically ominates f. Let f ; ~ f an g ; ~ g be characterize as in Propositions (4) an (8) uner probability istributions f an g, respectively. We obtain f > g an ~ f > ~ g. 3

26 Hence, when expectations about high liquiity shocks in the future become stronger, both the equilibrium an socially optimal levels of lening are lower, resulting in higher levels of cash carrie into the future. Next, we analyze the wege between the equilibrium an the socially optimal levels of lening when liquiity shocks at t = become more likely an the volatility of liquiity shocks increases. First, we focus on the e ect of the likelihoo of liquiity shocks. Let be istribute uniformly accoring to the probability istribution f b = b a over the interval [a; b], with 6 a < b 6. Note that for a xe a, for b > b, f b rst-orer stochastically ominates f b. Using the characterization in Proposition 8, we obtain ~ = conition in Proposition 4, we obtain b+a : Using the equilibrium = br + a (R ) R + ( ) (R ) ; Furthermore, ( ~ ) b = ( ) (R + ( ) (R )) > : Hence, as higher shocks become more likely, in the rst-orer stochastic sense, the wege between the socially optimal level of lening soc an its equilibrium level increases. This suggests that uring perios where expectations of high liquiity shocks in the future become stronger, even though liquiity management requires hoaring from a social welfare point of view as well, hoaring becomes a more serious problem as the wege between the socially an privately optimal levels of lening wiens. Next, we look at how the wege between the equilibrium an the socially optimal levels of lening change with the volatility of shocks. Let be istribute uniformly accoring to the probability istribution f = b a over the interval [a; b] with a + b = so that the istribution is symmetric aroun. Note that for b > b, f b is a mean-preserving sprea of f b. From the equilibrium conition, we obtain = R + b R + ( ) (R ) : 4

27 Note that is ecreasing in b. Furthermore, in this case, we have ~ = so that the socially optimal level of lening is not a ecte by mean-preserving spreas. Hence, uring perios of heightene uncertainty about future liquiity shocks, moelle by a probability istribution that is a mean-preserving sprea, the wege between the socially optimal an the equilibrium levels of lening increases, an hoaring becomes a more serious problem. This result is relate to recent papers in the literature that explain breakown in markets using i erent frameworks. For example, Morris an Shin (8) show that even small amounts of averse selection in an asset market can lea to the total breakown of trae ue to the failure of market con ence, e ne as approximate common knowlege of an upper boun on expecte losses. Even though we use the expecte utility theory framework in our analysis, our result is consistent with the literature that uses the notion of Knightian uncertainty (see Knight, 9) an agents overcautious behavior towars such uncertainty to generate hoaring an market freezes. Routlege an Zin (4) an Easley an O Hara (9, ) use Knightian uncertainty an agents that use maxmin strategies to generate wiening bi-ask spreas an freeze in nancial markets. Caballero an Krishnamurthy (8) buil a moel to show that uring perios of increase Knightian uncertainty, agents refrain from making risky investments an hoar liquiity, leaing to ight to quality an freezes in markets for risky assets. 5 Robustness an iscussion In this paper, we have trie to investigate the welfare implications of liquiity hoaring when markets are incomplete. The attempt is complicate by the fact that hoaring is not the only source of ine ciency in the moel. In this section, we conclue by iscussing some variants of the moel to she more light on these sources of ine ciency. 5

28 5. A moel without hoaring We begin by consiering a benchmark moel in which there is no role for hoaring. Suppose there are only three ates, inexe by t = ; ;. As before, bankers choose their portfolios (more precisely, the amount of liquiity in their portfolios) at ate. At ate, they observe the liquiity shock an, at ate, the asset returns are realize. The speci cation of the rest of the moel is the same as before, mutatis mutanis. We solve for equilibrium backwars, beginning with the secon perio. If a fraction at ate an the state is at ate, a fraction ( of the bankers hol cash ) of the bankers can supply their own cash nees an a fraction ( ) ( ) of the bankers have spare cash that they can supply to the market. The measure of illiqui bankers who nee cash is an it is clear that the market for cash will clear at a price e ne by 8 < if > ; p ( ) = : R if < : The allocation of cash at ate is e cient, since the number of bankers who can ischarge their ebts is min f ; g, that is, every banker who receives a liquiity shock gets the cash she nees, unless the number of bankers receiving a shock excees the supply of cash. The equilibrium allocation is not e cient, however, because the liquiity ecision at ate is not constraine optimal. Bankers choose to hol too little cash at ate because they o not internalize the value of the cash provie to creitors. To see this, we nee to compare the level of cash hel in equilibrium with the level chosen by the planner. In equilibrium, bankers must be ini erent between being liqui an illiqui at ate, that is, Z [R p ( ) R ( )] f ( ) = Z (3) [R + ( ) (p ( ) R )] f ( ) ; where the RHS an the LHS are the payo s for a liqui an an illiqui banker, respectively. This, in turn, yiels the equilibrium conition E[p ] = =R, which gives us F ( ) = R R : (4) In the planner s problem, the marginal cost of cash is an the marginal value of cash is, if < m, an R+, if > m. So the planner s rst-orer conition is R( F (m ))+ = 6