Liquidity hoarding

Gale, Douglas; Yorulmazer, Tanju Article Liquidity hoarding Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Gale, Douglas; Yorulmazer, Tanju (2013) : Liquidity hoarding, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 8, Iss. 2, pp. 291-324, https://doi.org/10.3982/TE1064 This Version is available at: https://hdl.handle.net/10419/150193 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Theoretical Economics 8 (2013), 291–324 1555-7561/20130291 Liquidity hoarding Douglas Gale Department of Economics, New York University Tanju Yorulmazer Federal Reserve Bank of New York Costly bankruptcy and incomplete markets cause inefficient liquidity hoarding. Banks are unable to trade contingent claims to liquidity, so they raise cash by selling illiquid assets on spot markets. Such trading increases asset-price volatility and creates the incentive to hoard liquidity. Hoarding creates a second inefficiency: the aggregate level of liquidity is inefficient too. A lender of last resort can implement the constrained-efficient allocation, but only if it intervenes so aggressively that it shuts down the private provision of liquidity altogether, becoming in effect the lender of first resort. Keywords. Interbank market, fire sale, market freeze, cash-in-the-market pricing, central bank. JEL classification. G12, G21, G24, G32, G33, D8. 1. Introduction One of the most interesting phenomena marking the recent financial crisis is the “freezing” of the interbank market. As early as the fall of 2007, following the collapse of the market for asset-backed commercial paper, European banks reported difficulty borrowing in the interbank market. At the same time, interbank borrowing rates reached record levels. Difficulty obtaining liquidity in interbank markets was subsequently experienced in many countries. As a result, central bank borrowing facilities became an essential source of liquidity for financial institutions. Douglas Gale: [email protected] Tanju Yorulmazer: [email protected] We thank Franklin Allen, Gadi Barlevy, Marco Bassetto, Markus Brunnermeier, Xavier Gabaix, David Martinez-Miera, Stephen Morris, Ned Prescott, Hyun Shin, Jeremy Stein, Anjan Thakor, Dimitri Vayanos, Vish Viswanathan, Wei Xiong, and seminar participants at the Board of Governors of the Federal Reserve System, the Boston University School of Management, the Centro de Estudios Monetarios y Financieros (CEMFI), the European Summer Symposium in Economic Theory (ESSET), the Federal Reserve Banks of New York, Philadelphia, and Chicago, the Fed Day Ahead Conference, Imperial College, the London School of Economics, Sabanci University, Seoul National University, Southern Methodist University, the Summer Workshop on Money, Banking, Payments, and Finance at the FRB of Chicago, the University of Chicago Booth School of Business, the University of Montreal, the University of Texas at Dallas, and the University of Western Ontario. Peter Hull provided excellent research assistance. The views expressed here are those of the authors and do not necessarily represent the views of the Federal Reserve Bank of New York or the Federal Reserve System. All errors remain our own. Copyright ©2013 Douglas Gale and Tanju Yorulmazer. Licensed under the Creative Commons Attribution- NonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE1064 292 Gale and Yorulmazer Theoretical Economics 8 (2013) One possible explanation for the market freeze was the fear of counterparty risk. Because of widespread exposure to subprime mortgage-backed securities, banks had good reason to be wary of lending to any bank that might be a credit risk. A second possible explanation was that banks were hoarding liquidity because they feared the loss of access to the interbank market in the future. There is substantial evidence that banks did in fact build up cash positions (Heider et al. 2009,Ashcraft et al. 2011,de Haan and van den End 2011,Acharya and Merrouche 2013).1The two explanations are not unrelated, of course. If access to the interbank market is expected to be difficult— because of fears of counterparty risk, for example—liquidity hoarding in the present is a rational response. In this paper, we use a simple model of liquidity management to analyze the possibility of liquidity hoarding and its impact on efficiency. We find that inefficient hoarding is a robust phenomenon in a laisser-faire equilibrium. Our model assumes a large number of “bankers,” who can hold two types of assets: a liquid asset and an illiquid asset. We refer to the liquid asset as “cash” and refer to the illiquid asset simply as “the asset.” Bankers are subject to stochastic liquidity shocks that require an expenditure of one unit of cash.2If a banker lacks the cash to make the required expenditure, he is forced to sell some of his holdings of the (illiquid) asset. If the demand for cash is high, the price of the asset will be correspondingly low. In extreme cases, the price of the asset may be so low (and the cost of liquidity so high) that the banker chooses to default. In that case, the banker is forced into a costly bankruptcy. In equilibrium, bankers weigh the cost of holding cash against the risk of having to sell assets at “fire sale” prices or experiencing a costly bankruptcy. We begin our analysis by solving the problem of a planner who determines how much cash to hold and when to distribute it. The solution to the planner’s problem takes a very simple form: after determining the efficient amount to hold at the first date, the planner supplies cash to every banker who needs it at a given date, until the supply runs out. Even though there may be a future need for cash, the planner never carries forward a positive balance as long as there is a banker who needs cash to meet a liquidity demand today. The simple form of the solution to the planner’s problem makes it easy to identify inefficient hoarding. Hoarding liquidity is inefficient if and only if it occurs when there are still bankers who need liquidity. We show that, in a laisser-faire market equilibrium, inefficient hoarding always occurs with positive probability. More precisely, when the demand for cash is sufficiently high, some illiquid bankers will be priced out of the market for cash and forced into bankruptcy at the same time as some liquid bankers are hoarding cash instead of supplying it to the market. A liquid banker has two reasons for hoarding cash. One is the precautionary motive. The banker may himself receive a liquidity shock in the future. If he uses his cash today, 1Afonso et al. (2011) document that while rates spiked and terms became more sensitive to borrower risk, borrowing amounts remained stable in the U.S. Fed Funds market during the Lehman episode. They argue that it is likely that the market did not expand to meet the additional demand, which is consistent with our result on rationing in the interbank market when demand for liquidity is high. 2For example, the liquidity shock could be interpreted as the random demand for immediate payment of a senior debt claim. Theoretical Economics 8 (2013) Liquidity hoarding 293 he can still obtain cash by selling the illiquid asset tomorrow, but the cost of obtaining cash may be very high. Another is the speculative motive. If the future demand for cash is very high, asset prices will be low. If he does not receive a liquidity shock, a hoarder may profit in those states by buying assets at fire sale prices. Clearly, these motives are two sides of the same coin: the same cash satisfies both motives. The precautionary and speculative motives are a function of the expected volatility of future asset prices. Asset-price volatility results from the use of the asset market as a source of liquidity. When liquid bankers first supply cash in exchange for assets, they create an imbalance in the system. They are increasing their holdings of illiquid assets and reducing their holding of liquid assets. If these large, illiquid bankers are subsequently hit by a liquidity shock, they have even more assets to dump on the market, producing a greater fire sale and reducing asset prices further. A laisser-faire equilibrium is inefficient because the incentives to hoard are simply too high. The buildup in volatility in one period is anticipated in previous periods and increases the precautionary and speculative motives for (inefficient) hoarding. To verify the role of fire sales in causing inefficient hoarding, we consider an alternative model in which the liquidity shock represents the demand for repayment of a nonrecourse loan. Because the loan is nonrecourse, only the assets that have been pledged as collateral for these loans are liquidated in the event of default. In the alternative economy, the large banks that buy up assets and later default do not create large fire sales. Only their initial holdings of assets are liquidated. We show that when loans are nonrecourse, both the aggregate amount of liquidity and its allocation are efficient. This result provides a rationale for the use of nonrecourse, securitized lending. It is a way to prevent or mitigate the fire sales that result from the liquidation of the entire financial institution. Another thought experiment shows an interesting link between hoarding and the level of aggregate liquidity in the economy. We consider an alternative model in which all liquidity shocks arrive in a single period. This means that when considering whether to supply liquidity, a banker knows that he will not receive a future liquidity shock (there is no precautionary motive) and that there will be no future fire sales (no speculative motive). In this case, it is no surprise that there is no (inefficient) hoarding and the market allocates the available liquidity efficiently. But, in addition, the aggregate level of liquidity in the economy is also efficient. By contrast, in the baseline model with hoarding, one inefficiency (hoarding) leads to a second inefficiency (the level of liquidity). Since laisser-faire is inefficient, there may be scope for intervention by the central bank (CB). The CB is subject to more constraints than a central planner, however. A central planner has exclusive control over the allocation of liquidity. The CB, by contrast, has to compete with markets in which cash and assets are exchanged. Generally speaking, when agents can trade in side markets, it is harder to improve welfare while satisfying incentive-compatibility constraints. In this case, however, the CB can successfully implement the planner’s solution. Because the central bank is a large player, it changes the prices at which markets clear. The optimal strategy is for the CB to accumulate and supply so much liquidity that private suppliers of liquidity are forced out of the market. More precisely, the CB makes liquidity cheap enough that none of the bankers wants to 294 Gale and Yorulmazer Theoretical Economics 8 (2013) supply liquidity in competition with the CB. In equilibrium, no one apart from the CB holds cash and everyone relies on the lender of last resort (LoLR) for liquidity. The fundamental reason for the inefficiency of the laisser-faire equilibrium is the incompleteness of markets. Illiquid bankers are forced to acquire cash ex post by selling the asset on the spot market, rather than entering into contingent contracts for the provision of cash ex ante. This might suggest that introducing markets for contingent claims to cash could restore the first best, but this is not correct. We show, to the contrary, that such markets cannot improve equilibrium welfare in the presence of asymmetric information. In an extension of our basic inefficiency result, we show that introducing a market for contingent liquidity cannot improve welfare in a laisser-faire equilibrium. More precisely, if bankers cannot be forced to deliver the liquid asset when they have received a liquidity shock or, conversely, cannot be forced to receive the liquid asset when they have not received a liquidity shock, the possibility of arbitrage in spot markets, together with private information about the liquidity shock, rules out any gains from trade.3This result provides a justification for the incompleteness of markets assumed in our baseline model. The rest of this paper is organized as follows. We begin our analysis in Section 2 by studying the constrained-efficient allocation chosen by a central planner who accumulates a stock of liquid assets and distributes them to the banks that report a need for liquidity. Then, in Section 3, we analyze a laisser-faire economy in which banks make their own decisions about liquidity accumulation and liquidity provision. In Section 4, we show that the central bank, in its role as lender of last resort, can achieve the same allocation as the planner, in spite of the competition from the asset market. We consider some variants of the model to shed more light on the sources of inefficiency in Section 5 and conclude in Section 6. All proofs not in the text are in the Appendix. 1.1 Related literature At a general level, our paper is reminiscent of Shleifer and Vishny (1992) and Allen and Gale (1994,1998). These papers show that when potential buyers of assets are themselves financially constrained, the asset prices may fall below their fundamental value and be determined by the available liquidity in the market, that is, we observe cash-in- the-market prices.4 Our paper is also related to the literature on portfolio choice and the liquidity of the financial system (e.g., Allen and Gale 2004a,2004b,Gorton and Huang 2004,Diamond and Rajan 2005,Acharya et al. 2011,andBernardo and Welch 2013). Recent work by Diamond and Rajan (2011) develops a model in which banks, in anticipation of future fire sales, have high expected returns from holding cash. Acharya and Skeie (2011) describe a model in which banks’ decision whether to provide term lending depends on 3This result has a family resemblance to an observation of Cone (1983) and Jacklin (1987). They show that in the Diamond–Dybvig (1983) model, banks cannot increase welfare if depositors have access to forward markets. Access to forward markets allows depositors to engage in arbitrage that undermines the bank’s ability to provide incentive-compatible liquidity insurance. 4Also see Allen and Gale (2005) for a review of the literature that explores the relation between asset-price volatility and financial fragility when markets and contracts are incomplete. Theoretical Economics 8 (2013) Liquidity hoarding 295 leverage and rollover risk over the term of the loan. Our paper differs from these papers in two respects. First, bankers hold liquidity to protect themselves against future liquidity shocks (the precautionary motive) as well as to take advantage of fire sales (the speculative motive). Second, bankers make an initial portfolio choice as well as a choice to lend to needy bankers or hoard liquidity. This adds to the richness of the model and allows us to analyze the interaction between the initial decision to hold liquidity and the later decision to hoard. A number of papers take different approaches to modeling disturbances in asset markets that affect liquidity. Some of them are based on informational problems. Caballero and Krishnamurthy (2008) show that liquidity hoarding can arise as a response to “unusual events or untested financial innovations” in a model with Knightian uncertainty. The increase in uncertainty affects investors’ preferences across asset classes, increasing the demand for liquid assets. Caballero and Simsek (2010) present a model where banks are uncertain about the network of cross-exposures. When conditions deteriorate, banks need to understand the financial network to assess counterparty risk. Knightian uncertainty amplifies the banks’ perceived counterparty risk, leading to a freeze in markets. Malherbe (2010) studies a model in which markets may be illiquid because of adverse selection. Anticipating a market “dry-up,” agents engage in liquidity hoarding that worsens the adverse selection problem and makes the market dry-up more severe.5Kurlat (2009) shows that adverse selection can lead to a market shutdown, and then considers how learning from past transactions can reduce adverse selection and improve market liquidity. Market downturns reduce learning and worsen the future lemons problem. Others adopt a search-theoretic model of over the counter markets and study the effect of preference shocks on market liquidity. Lagos and Rocheteau (2009) relax the indivisibility assumption found in many search models and investigate how this affects the market’s adjustment to a shock. Lagos et al. (2011) consider a model in which a shock reduces investors’ asset demands until some randomly determined date. If the shock is sufficiently severe, even well capitalized dealers are not willing to accumulate inventories and government intervention to increase demand for the asset may be welfare improving. Our paper focuses on the inefficiency of liquidity hoarding caused by incomplete markets rather than adverse selection, complexity, or extreme risk aversion. Goodfriend and King (1988) argue that, with efficient interbank markets, it is sufficient for the CB to provide an adequate level of aggregate liquidity and let the interbank markets determine the final allocation. In other words, the CB should not lend to individual banks, but simply provide liquidity via open market operations. Others, however, argue that interbank markets may fail to allocate liquidity efficiently due to frictions such as asymmetric information about banks’ assets (Flannery 1996,Freixas and Jorge 2008), banks’ free-riding on each other’s liquidity (Bhattacharya and Gale 1987), or on the central bank’s liquidity (Repullo 2005), market power and strategic behavior (Acharya et al. 2012), and regulatory solvency constraints and marking to market of the assets (Cifuentes et al. 2005). Our results provide support for both points of view. On the 5Also see Chapter 7 of Holmström and Tirole (2011) that uses the model described in Malherbe (2010). 296 Gale and Yorulmazer Theoretical Economics 8 (2013) one hand, inefficient liquidity hoarding does provide a rationale for intervention by the CB. On the other hand, the success of the CB’s policy could be seen as a vindication of the Goodfriend and King view. The fact that the LoLR has to take over the entire market to implement the planner’s solution should give us pause, however. 2. Constrained efficiency In this section, we characterize the constrained-efficient allocation as the solution to a planner’s problem in which the planner accumulates and distributes the liquid asset. The resulting allocation serves as the benchmark for our welfare analysis of laisser-faire equilibrium. 2.1 Primitives Time: Time is divided into four dates, indexed by t=0123. At the first date, bankers choose the amount of liquidity they hold as part of their portfolio. At the second and third dates, bankers receive liquidity shocks and trade assets so as to obtain the liquidity they need. At the final date, asset returns are realized. Assets: There are two assets: a liquid asset that we refer to as cash and an illiquid asset that we refer to simply as the asset. Cash can be stored from period to period and one unit of cash can be converted into one unit of consumption at any date. The asset can be stored from period to period. One unit of the asset has a return of R>1units of cash at date 3. Bankers: There is a continuum of identical, risk neutral agents, indexed by i∈[01], who we call bankers. To simplify the analysis, we focus on liquidity management and ignore other banking activities or treat them as exogenous in what follows. Each bank has an initial endowment consisting of one unit of the asset and one unit of cash at date 0, denoted by the vector (11), where the first and the second components represent the quantities of the asset and cash in the bank’s portfolio, respectively. The banker’s utility function is U(c0c3)=ρc0+c3 where c0denotes consumption at date 0,c3denotes consumption at date 3,andρ>1is a parameter. The interpretation of this utility function is the following: bankers prefer consumption at date 0to consumption at date 3, so holding cash after date 0(instead of converting it into consumption immediately) involves an opportunity cost ρ. In equilibrium, the banker has to weigh the cost of foregoing immediate consumption against the benefits of holding cash. These benefits include the capital gains realized when the future price of cash is high as well as the return to unused cash in the last period. Liquidity shocks:Wemodelaliquidity shock as a random demand for payment of one unit of cash. Each banker receives a liquidity shock at exactly one of the dates t= 123. The probability of receiving the shock at date 1is θ1,atdate2it is (1−θ1)θ2,and at date 3it is 1−θ1−(1−θ1)θ2=(1−θ1)(1−θ2). The aggregate liquidity shocks θ1 and θ2are assumed to be independent random variables with cumulative distribution Theoretical Economics 8 (2013) Liquidity hoarding 297 functions F1(θ1)and F2(θ2). We assume that θ1and θ2haveacommonsupport[01]. The “law of large numbers” convention dictates that the probability of receiving a shock at date tis equal to the fraction of bankers receiving the shock. Bankruptcy: A banker who is unable to make the required payment is considered to be bankrupt. If a banker becomes bankrupt, we assume that all his assets are immediately liquidated. For simplicity, we assume that the liquidation costs consume the entire value of the assets. This assumption can be relaxed, but it greatly simplifies the analysis and does not appear to affect the qualitative results too much. Note 1. In an earlier version of the paper, we modeled the liquidity shock as the demand for repayment of a callable bond. Each banker was assumed to issue a bond with face value equal to one unit of cash to a creditor with Diamond–Dybvig time preferences. That is, the creditor wanted to consume at date 1with probability θ1, at date 2with probability (1−θ1)θ2,andatdate3with probability 1−θ1−(1−θ1)θ2.Thus,thedemand for repayment arrives as a liquidity shock with the same probability as described above. For simplicity, we have eliminated the creditors from the model and instead treat the liquidity shock as a random cost of maintaining the banker’s portfolio. This approach is similar to the one used by Holmström and Tirole (1998). In the case of Holmstrom and Tirole, however, the amount of cash that must be paid is proportional to the fraction of the portfolio saved. In the present model, by contrast, the liquidity shock is a fixed cost: if it is not paid in full, all the assets disappear. Note 2. Ourmodelhasfour dates, rather than the three dates that are standard in much of the banking literature. More precisely, there are two dates at which bankers can receive liquidity shocks while they are still illiquid. (At date 3, the banker receives the cash returns from the illiquid asset and can use these to deal with the liquidity shock.) Although this seems to be a small extension, it is crucial for the analysis of liquidity hoarding. When a liquid banker is deciding at date 1whether to hoard cash or to supply it to the market, the possibility that he will be hit by a liquidity shock at the next date provides a precautionary motive for hoarding. Similarly, the possibility of an even greater fire sale at date 2provides a speculative motive for hoarding. We show in Section 5 that in a three-period model, i.e., a model without date 2, inefficient hoarding cannot occur in equilibrium. In fact, the equilibrium is constrained efficient. 2.2 The planner’s problem At date 0, all bankers are identical and risk neutral. Since it is possible to make transfers between bankers at date 3, we can redistribute the total surplus any way we like. So, maximizing ex ante welfare is essentially equivalent to maximizing total expected surplus. In what follows, we take this as the planner’s objective function. In addition 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 in this way, he would assign all assets at date 1to bankers who had already received a liquidity shock, thus rendering the liquidity shocks at date 2irrelevant. Alternatively, he could 298 Gale and Yorulmazer Theoretical Economics 8 (2013) transfer all the assets to a small set of bankers of measure ε>0,say,thusreducingthe need for liquidity to an arbitrarily small amount. To avoid this trivial outcome, we restrict the planner’s actions to accumulating cash at date 0, distributing cash at dates 1 and 2, and redistributing the consumption good at date 3. It is because of this constraint that we refer to the solution of the planner’s problem as a constrained-efficient allocation. The planner is assumed to face the same opportunity cost of holding cash, ρ,as the bankers. Suppose that the planner has m1units of cash at the beginning of date 2and the state is (θ1θ2).Thereare(1−θ1)θ2bankers who receive a liquidity shock in this period. The optimal strategy is to supply the lesser of (1−θ1)θ2and m1to the bankers in need of cash. Each unit of cash is worth one unit of consumption if it is held until date 3, but each unit distributed to a banker with a liquidity need saves an asset worth R>1at date 3. So it is optimal to save as many bankers as possible from default. Now suppose the planner has m0units of cash at the beginning of date 1and the state is θ1.Thereareθ1bankers who receive a liquidity shock in this period. Each unit of cash distributed to these bankers is worth Rif it saves an asset. On the other hand, the expected value of a marginal unit of cash held until date 2must be less than R.As we have seen, the value of cash is at most Randitwillbeonly1if the amount carried forward is greater than (1−θ1)θ2, which happens with positive probability if the amount carried forward is positive. So it is optimal to save as many bankers as possible from default at date 1, that is, the optimal strategy is to distribute the lesser of m0and θ1at date 1. At date 0, the choice of how much cash to hold is determined by equating the marginal cost of cash, ρ, to the marginal value of cash. As usual, a unit of cash held at the end of date 0is always worth at least one unit of consumption, but it may be worth Runits if it can be used to save an asset. The probability that the marginal unit of cash is used to save an asset is simply the probability that m0is less than θ1+(1−θ1)θ2.This probability is calculated to be Pr[θ1+(1−θ1)θ2>m 0]=1−m0 0 F2m0−θ1 1−θ1f1(θ1)dθ1 so the marginal value of cash carried forward at date 0is R1−m0 0 F2m0−θ1 1−θ1f1(θ1)dθ1+m0 0 F2m0−θ1 1−θ1f1(θ1)dθ1 =R−(R −1)m0 0 F2m0−θ1 1−θ1f1(θ1)dθ1 The solution to the planner’s problem is described by an array (m0m1(θ1) m2(θ1θ2)),wherem0≥0is the amount of cash carried forward from date 0,m1(θ1)is the amount of cash carried forward from date 1in state θ1,andm2(θ1θ2)is the amount of cash carried forward from date 2in state (θ1θ2). The previous argument leads to the following proposition. Theoretical Economics 8 (2013) Liquidity hoarding 305 3.1.3 Market clearing at date 0We can show that 0<α<1in equilibrium at date 0,so bankers must be indifferent between holding cash and spending it. The cost of holding liquidity is ρ. The benefit of holding liquidity equals the difference between the payoff of a liquid banker and the payoff of an illiquid banker.10 We have to consider three cases: (i) Shock occurs at date 1: In this case, a liquid banker uses his own cash to make the required expenditure and avoids default, whereas an illiquid banker needs to sell a fraction p1(θ1)of his assets. Hence, a liquid banker’s payoff is, in expectation, θ1p1(θ1)R more than an illiquid banker’s payoff. (ii) Shock occurs at date 2: In this case, a liquid banker can use his own cash to avoid default. However, a (small) illiquid banker needs to sell assets at date 2.For p2(θ1θ2)≤1, the (small) illiquid banker can get the needed liquidity by selling p2(θ1θ2)units of assets, but for p2(θ1θ2)>1he has to default. Hence, a liquid banker’s payoff, in expectation, is (1−θ1)θ2Rmin{1p2(θ1θ2)}more than an illiquid banker’s payoff. (iii) Shock occurs at date 3: In this case, a liquid banker can acquire p2(θ1θ2)units of theassetatdate2, which results in a liquid banker’s payoff, in expectation, to be (1−θ1)(1−θ2)p2(θ1θ2)R more than an illiquid banker’s payoff. When we combine these three cases and use the equilibrium condition p1(θ1)= E[p2(θ1θ2)|θ1], we get the following result. Proposition 6. In equilibrium, 0<α<1, which implies that bankers are indifferent at date 0between holding and not holding liquidity. Bankers are indifferent if and only if R1 0 p1(θ1)1−(1−θ1)(1−F2(θ∗∗ 2))E[θ2|θ2>θ ∗∗ 2]f1(θ1)dθ1=ρ (6) 3.2 Equilibrium An equilibrium is described by the endogenous variables α,λ(θ1),p1(θ1),andp2(θ1θ2) that satisfy (3), (4), (5)and(6). A comparison of the above equilibrium definition with the planner’s solution in Proposition 1 makes it clear that there are two major differences between the equilibrium allocation and the planner’s solution. First, the equilibrium value of αmust satisfy the first-order condition in (6), which differs from the first-order condition in (2). Second, it is clear from Proposition 5 that inefficient hoarding occurs in equilibrium, but not in the solution to the planner’s problem as described in (1). These differences result from the fact that illiquid bankers are forced to obtain liquidity by selling assets in the spot markets at dates 1and 2. This trade has a number of general-equilibrium effects. In the first place, it gives rise to large banks at date 1. This in turn causes greater assetprice volatility at date 2, when some of these large banks fail. The anticipation of this 10Note that in equilibrium large illiquid bankers and hoarders have the same payoff. Here, without loss of generality, we focus on the payoffs of the small illiquid bankers and the liquid bankers who choose to become hoarders. 306 Gale and Yorulmazer Theoretical Economics 8 (2013) asset-price volatility provides the incentive to hoard liquidity at date 1.Weconfirmthis explanation in Section 5, where we consider alternative specifications of the model and show that, absent these effects, the laisser-faire equilibrium is constrained efficient. Note 4. The constrained efficient allocation is defined by the planner’s problem, which leaves the original allocation of the asset unchanged, except in the case of default, where the asset is “consumed” by liquidation costs. On the other hand, in a laisser-faire equilibrium, the allocation is endogenous because assets are traded for cash at dates 1and 2. So the feasible set of allocations is in some sense larger than the set of feasible allocations for the planner’s problem. How can we be sure that the equilibrium allocation generates lower welfare than the planner’s problem? The key feature of the laisser-faire equilibrium is that assets are transferred from bankers who already received a liquidity shock to bankers who have not yet received a liquidity shock. This increases the demand for cash and hence increases the default costs, other things being equal. To put it another way, if the assets remained with their original owners and only cash changed hands, the outcome would be unambiguously better. Thus, the equilibrium allocation must be constrained inefficient. 4. The lender of last resort In this section, we introduce a central bank (CB) into the model. We describe an equilibrium in which the CB acts as the sole supplier of liquidity, all bankers choose to be illiquid (i.e., hold no cash at date 0), and the constrained-efficient policy characterized in Proposition 1 can be implemented as part of a competitive equilibrium. Our approach is constructive. We assume that α=1and that the CB chooses as its policy the solution to the planner’s problem (m0m1m2)given in Proposition 1.We define an equilibrium with the CB acting as a lender of last resort along the lines of the laisser-faire equilibrium. We continue to use the language of asset sales, but this is equivalent to supplying cash in the form of secured loans using the asset as collateral, as described in Section 3. At date 2, there are no large illiquid bankers, so the demand for liquidity comes from the (1−θ1)θ2small illiquid bankers who have received a liquidity shock at date 2. Since the supply of cash is max{m0−θ10}, the market-clearing price p2(θ1θ2)is defined by p2(θ1θ2)=R−1if (1−θ1)θ2<max{m0−θ10} 1if (1−θ1)θ2>max{m0−θ10}(7) Similarly, at date 1, the demand for liquidity comes from the θ1illiquid bankers who receive a liquidity shock at date 1and the supply is at most m0.Ifθ1>m 0, the marketclearing price must be p1(θ1)=1,butwhenθ1<m 0, the price may lie anywhere between E[p2(θ1θ2)|θ1]and 1. Since the CB can control the price, we assume that it sets p1(θ1)=E[p2(θ1θ2)|θ1]. Then the market-clearing price is p1(θ1)=E[p2(θ1θ2)|θ1]if θ1<m 0 1if θ1>m 0(8) Theoretical Economics 8 (2013) Liquidity hoarding 307 Market clearing at date 0requires that it is optimal for bankers to choose α=1.Wecan show that this is the case, which gives us the following proposition. Proposition 7. There exists an equilibrium in which the CB acts as the sole provider of liquidity; all bankers choose to be illiquid at date 0,thatis,α=1; market-clearing prices at dates 1and 2are given by (8)and(7), respectively, and the constrained-efficient policy (m0m1m2)given in Proposition 1 is implemented. Hence, the CB, by acting as the sole provider of liquidity, can implement the constrained-efficient allocation as an equilibrium. Note 5. We follow most of the banking literature in treating “money” as a consumption good. In particular, when modeling the CB, we assume that it has the same opportunity cost of liquidity as private bankers do: to obtain a unit of cash, the CB has to give up ρ>1units of “consumption.” Our main interest is to identify the sources of market failure. For that purpose, it is appropriate to assume the regulator has access to the same technology as the market, so we ignore the possibility that the CB can supply liquidity more cheaply than the market. Allen et al. (2012) study a model in which the CB can create reserves “costlessly” and derive very different results from the standard “real” model. Their results are interesting and raise important policy issues that go beyond the scope of the present paper. Note 6. As a corollary of Proposition 7,weobtainProposition 2. The equilibrium allocation implemented by the CB defines a direct mechanism (μ1(θ1) t1(θ1) μ2(θ1θ2) t2(θ1θ2)) as μ(θ1)=min1m0 θ1 t1(θ1)=p1(θ1)R μ2(θ1θ2)=min1(1−μ1(θ1))m0 (1−θ1)θ2 t2(θ1θ2)=p2(θ1θ2)R The equilibrium conditions ensure that the mechanism is incentive compatible, so truth-telling is optimal for the bankers. 5. Sources of inefficiency So far, we have focused on the inefficiency of laisser-faire equilibrium and the appropriate intervention by the CB that restores efficiency. In this section, we try to identify the essential sources of inefficiency by analyzing variants of the model in which crucial distortions are removed. We also provide a justification for the incompleteness of markets, which is, in some sense, the fundamental cause of market failure. 308 Gale and Yorulmazer Theoretical Economics 8 (2013) 5.1 Hoarding We begin by considering a model in which there is no role for hoarding. Suppose there are only three dates, indexed by t=012. As before, bankers choose their portfolios (more precisely, the amount of liquidity in their portfolios) at date 0.Atdate1, they observe the liquidity shock θ1,andatdate2, the asset returns are realized. The specification of the rest of the model is the same as before, mutatis mutandis. We solve for equilibrium backwards, beginning with the second period. If a fraction 1−αof the bankers hold cash at date 0and the state is θ1at date 1, a fraction (1−α)θ1of the bankers can supply their own cash needs and a fraction (1−α)(1−θ1)of the bankers have spare cash that they can supply to the market. The measure of illiquid bankers who need cash is αθ1and it is clear that the market for cash clears at a price defined by p1(θ1)=R−1if θ1<1−α 1if θ1>1−α (9) The allocation of cash at date 1is efficient, since the number of bankers who can discharge their debts is min{θ11−α}, that is, every banker who receives a liquidity shock gets the cash she needs unless the number of bankers receiving a shock exceeds the supply of cash. To show that the equilibrium allocation is efficient, we have to show that the liquidity decision at date 0is also efficient. To see this, we need to compare the level of cash held in equilibrium with the level chosen by the planner. In equilibrium, bankers must be indifferent between being liquid and illiquid at date 0,thatis, 1 0 [R−θ1p1(θ1)R]f1(θ1)dθ1+ρ=1 0 [R+(1−θ1)p1(θ1)R]f1(θ1)dθ1 where the right-hand side and the left-hand side are the payoffs for a liquid and an illiquid banker, respectively. This, in turn, yields the equilibrium condition ρ=1 0 p1(θ1)Rf1(θ1)dθ1 Using (9)toevaluateE[p1(θ1)], we can rewrite the equilibrium condition E[p1(θ1)]= ρ/R as F1(1−α) =R−ρ R−1(10) In the planner’s problem, the marginal cost of cash is ρandthemarginalvalueofcash is 1if θ1<m 0and Rif θ1>m 0. So the planner’s first-order condition is R(1−F1(m0)) + F1(m0)=ρor F1(m0)=R−ρ R−1(11) Comparing (10)and(11), it is clear that m0=1−αand so the level of cash held in equilibrium is efficient. Theoretical Economics 8 (2013) Liquidity hoarding 309 Proposition 8. When the economy has only three dates, there is no (inefficient) hoarding in equilibrium. In fact, the equilibrium allocation is constrained efficient. The analysis of the simplified model demonstrates that the inefficiency of a laisserfaire equilibrium depends on inefficient hoarding, which can occur only when there are more than three periods. It is interesting to note that both types of inefficiency—the inefficient level of liquidity at date 0and inefficient hoarding at date 1—disappear when the third period is eliminated. In other words, one distortion leads to another. 5.2 Market liquidity and asset-price volatility When bankers supply cash at date 1, they acquire assets that make their portfolios larger and less liquid. When θ2is high, the default of large illiquid bankers at date 2creates a fire sale and increases asset-price volatility. The anticipation of this increased assetprice volatility in turn provides the incentive for inefficient hoarding at date 1.Weargued that this mechanism is the crucial distortion in the model of laisser-faire equilibrium. In this section, we show that asset-price volatility is responsible for inefficient hoarding. We do this by considering an alternative model in which default costs consume only the bankers’ original assets and not the assets acquired at date 1. Consider the model described in Section 3 with the following change. If a large illiquid banker receives a liquidity shock at date 2and is unable or unwilling to obtain one unit of cash, he defaults and has to liquidate his original unit of the asset, but not the assets he purchased at date 1. As before, liquidation costs consume the entire unit of the asset. One interpretation is that a liquidity shock takes the form of a demand for repayment of a nonrecourse loan for which the initial one unit of the asset was the collateral. That is, the creditor can seize the asset that serves as collateral, but cannot seize any other assets owned by the banker. Under the new assumption, a large illiquid banker who acquires p1(θ1)units of the asset in exchange for its one unit of cash at date 1is guaranteed to have a return of at least p1(θ1)R at date 3. Even if the large banker defaults on his loan, he loses only the unit of the asset originally pledged as security for the loan and retains the rest of his portfolio. Since only one unit of the asset is at risk, the large banker will only be willing to give up one unit of the asset in exchange for one unit of cash. Then the market-clearing price p2(θ1θ2)has the distribution p2(θ1θ2)=R−1with probability F2((1−α)(1−λ(θ1))) 1with probability 1−F2((1−α)(1−λ(θ1))) and the expected value of p2(θ1θ2)is E[p2(θ1θ2)|θ1]=F2(1−α)(1−λ(θ1))R−1+1−F2(1−α)(1−λ(θ1)) With probability θ2, large illiquid bankers receive a liquidity shock and have a payoff (1+p1(θ1)−p2(θ1θ2))R; with probability (1−θ2)they do not receive a shock and have 310 Gale and Yorulmazer Theoretical Economics 8 (2013) apayoff(1+p1(θ1))R. Thus, the large illiquid bankers’ expected payoff at date 1is 1 0(1+p1(θ1)−θ2p2(θ1θ2))Rf2(θ2)dθ2 Now consider the hoarders. With probability θ2, the hoarders receive a liquidity shock and have a payoff equal to R, and with probability (1−θ2), they do not receive a shock and have a payoff (1+p2(θ1θ2))R. Thus, the hoarders’ payoff at date 1is 1 0(1+(1−θ2)p2(θ1θ2))Rf2(θ2)dθ2 It is optimal to buy assets if and only if the large bankers’ payoff is at least as great as the hoarders’, that is, p1(θ1)≥E[p2(θ1θ2)|θ1]. Similarly, it will be optimal to hoard if and only if p1(θ1)≤E[p2(θ1θ2)|θ1]. Suppose that, in equilibrium, there is inefficient hoarding, that is, λ(θ1)< αθ1/((1−α)(1−θ1)). In that case, since illiquid bankers hit by the shock are willing to supply one unit of the asset for one unit of cash, the market clears at p1(θ1)=1.But, in equilibrium, we have E[p2(θ1θ2)|θ1]≥p1(θ1)=1, which requires that F2((1−α)(1−λ(θ1))) =0,thatis,α=1or λ(θ1)=1.Wecanruleoutα=1when ρ is not too high, and λ(θ1)=1means there is no hoarding, which is a contradiction. Hence, when shocks affect only assets pledged as collateral, rather than the entire bank, equilibrium is characterized by no (inefficient) hoarding. The intuition for this result is quite clear. Inefficient hoarding at date 1requires that p1(θ1)=1. However, the maximum number of assets that can be acquired by a hoarder at date 2(or saved when hit by the shock at date 2) is one and it will be less than one when θ2is small. Hence, liquid bankers prefer to buy the asset at date 1, rather than to hoard. We can also show that the amount of cash held in equilibrium is equal to m0.Then we have the following result. Proposition 9. In the economy with nonrecourse loans, there is no inefficient hoarding in equilibrium, that is, θ1>1−α⇒ λ(θ1)=1 and the constrained-efficient amount of cash is held at date 0, 1−α=m0 This result demonstrates the essential role of market liquidity in creating the distortions that lead to inefficient hoarding. 5.3 Incomplete markets In this section, we show that opening a forward market for contingent liquidity contracts at date 0cannot improve on the allocation provided by the laisser-faire equilibrium with Theoretical Economics 8 (2013) Liquidity hoarding 311 spot markets alone. In particular, we consider a market formed at date 0in which some bankers enter into a contract to acquire cash and supply it under certain conditions, and other bankers simultaneously enter into a contract to accept cash under certain conditions. The participants in this market are required to report their types at dates 1 and 2, that is, whether or not they received a liquidity shock in that period. In the event that suppliers do not report a shock, they may be required to supply one unit of cash, if they have not already done so, in exchange for a specified amount of the asset. The recipients of cash similarly report whether or not they have received a liquidity shock at date 1and date 2. If they report a shock, they may be supplied with one unit of cash, if they have not already received one, in exchange for a specified amount of the asset. We let ˆ p1(θ1)denote the price of cash at date 1in state θ1and let ˆ p2(θ1θ2)denote the price of cash at date 2in state (θ1θ2). (We continue to describe the provision of liquidity as an exchange of the asset for cash, but this is equivalent to secured lending at an appropriate interest rate.) Suppose that {α λ(θ1)p1(θ1)p2(θ1θ2)}is a laisser-faire equilibrium as described in Section 3 and consider the effect of opening a market for liquidity at date 0. Does any banker have an incentive to participate in the market at the equilibrium prices? The market must satisfy an incentive compatibility constraint to ensure that bankers report their types truthfully. At date 1in state θ1, one unit of cash can be traded for p1(θ1)units of cash on the spot market. If p1(θ1)> ˆ p1(θ1), a banker who has cash and has not received a liquidity shock is better off reporting a liquidity shock since he can always sell his unit of cash on the spot market for the higher price. Likewise, if p1(θ1)< ˆ p1(θ1), a banker who has no cash and has received a liquidity shock is better off reporting no liquidity shock since he can always buy cash at the lower price. Thus, incentive compatibility at date 1requires ˆ p1(θ1)=p1(θ1) for every value of θ1. A similar argument implies that ˆ p2(θ1θ2)=p2(θ1θ2) for every value of (θ1θ2). Since the prices are the same, it is clear that the market mechanism cannot improve on the allocation provided by the spot markets. 6. Conclusion In this paper we outlined a simple model of liquidity provision and characterized the constrained-efficient allocation as the solution to a planner’s problem. The salient feature of the constrained-efficient allocation is the absence of inefficient liquidity hoarding: the planner never carries cash balances forward if there are unsatisfied demands for liquidity. In a laisser-faire equilibrium, by contrast, inefficient hoarding always occurs with positive probability. The inefficiency of equilibrium results, among other things, from the incompleteness of markets. Although we take the market structure as exogenously given, we show 312 Gale and Yorulmazer Theoretical Economics 8 (2013) that, because of asymmetric information, the introduction of incentive-compatible contingent markets for liquidity cannot improve welfare as long as bankers can obtain liquidity by selling assets. This result suggests that an equilibrium in which markets are incomplete is robust to the opening of contingent forward markets, but it does not mean that no intervention can improve on the equilibrium allocation. In fact, a CB, operating as lender of last resort, can achieve the same allocation as the planner, in spite of facing competition from the market for liquidity. If the CB intervenes very aggressively, it can discourage bankers from holding liquidity. Thus, the CB becomes the sole supplier of liquidity. The crucial advantage of the CB is that, because it is a large player, it can change market prices. Bankers operating in a competitive market are, by contrast, price takers. We also explored the features of the model that account for the inefficiency of laisserfaire equilibrium. We showed, in particular, that it is necessary to have one more period than the usual model for inefficient hoarding to occur. We also showed that fire sales play an important role in providing incentives for inefficient hoarding and that a form of nonrecourse debt can avoid these fire sales and restore constrained efficiency. This benefit of nonrecourse debt should be considered alongside recent criticisms of the “safe harbor” treatment of secured creditors (e.g., Bolton and Oehmke 2011). Goodfriend and King (1988) argue that it is sufficient to provide adequate liquidity to the system as a whole when interbank markets function efficiently. Our result, showing that the LoLR can implement a constrained-efficient allocation, provides some support for the Goodfriend and King position, but only if we accept a very large role for the CB. How seriously can we take the result? What are the limits on the role of the CB? In recent discussions, several concerns have been raised about the liquidity facilities rolled out by the Federal Reserve System during the financial crisis. One concern is the possibility that the expansion in the Fed’s balance sheet will result in inflation. Another is the possibility that the Fed will make losses from counterparty risk and lending against substandard collateral. Finally, there is the problem of unwinding its position as economic conditions change. Some writers doubt that the Fed will be able to shrink its balance sheet quickly or that the attempt to do so will destabilize the bond markets. These and other concerns should temper any enthusiasm for the possibility of achieving a constrained-efficient liquidity provision by having the Fed become the sole provider. Appendix:Proofs Proof of Proposition 1.Letm0≥0denote the quantity of cash held at the end of date 0,letm1(θ1)≥0denote the amount of cash held at the end of date 1in state θ1,and let m2(θ1θ2)≥0denote the amount of cash held at the end of date 2in state (θ1θ2). Feasibility requires m0≥m1(θ1)≥m2(θ1θ2)(12) for every value of (θ1θ2). The amount of cash distributed at date 1in state θ1is denoted by x1(θ1)and is defined by putting x1(θ1)=m0−m1(θ1)≥0 Theoretical Economics 8 (2013) Liquidity hoarding 313 for every value of θ1. The amount distributed at date 2in state (θ1θ2)is denoted by x2(θ1θ2)and is defined by putting x2(θ1θ1)=m1(θ1)−m2(θ1θ2)≥0 for every value of (θ1θ2). The expected output from the planner’s policy in state (θ1θ2)is R{x1(θ1)+x2(θ1θ2)+(1−θ1)(1−θ2)}+m2(θ1θ2) (13) The total amount of the asset at date 3is equal to the amount of cash distributed to bankers who receive a liquidity shock at dates 1and 2,thatis,x1(θ) +x2(θ1θ2),plus the number of bankers who do not receive a liquidity shock at either date, that is, (1−θ1)(1−θ2). The total amount of cash at date 3is equal to the amount held by the planner, m2(θ1θ2). Multiplying the amounts of cash and asset by their respective returns and summing them gives the expression in (13). The total surplus is equal to the expected output minus the cost of obtaining liquidity, that is, R{x1(θ1)+x2(θ1θ2)+(1−θ1)(1−θ2)}+m2(θ1θ2)−ρm0 =R{m0−m2(θ1θ2)}+R(1−θ1)(1−θ2)+m2(θ1θ2)−ρm0 (14) The planner chooses (x1x2)to maximize the expected value of (14) subject to the constraints in (12). We start the analysis at t=2and go backwards. Suppose that the planner has m1 units of cash at the beginning of date 2and the state is (θ1θ2).Thereare(1−θ1)θ2 bankers in need of cash and the optimal distribution strategy is to supply x2(θ1θ2)=min{(1−θ1)θ2m1} Thus, the value of m1units of cash in state (θ1θ2)is V2(m1θ1θ2)=Rmin{(1−θ1)θ2m1}+m1−min{(1−θ1)θ2m1} =(R −1)min{(1−θ1)θ2m1}+m1 For a fixed value of θ1, the value of m1unitsofcashattheendofdate1(before θ2has been realized) is V2(m1θ1)=E[V2(m1θ1θ2)|θ1] =m1/(1−θ1) 0 {(R −1)(1−θ1)θ2+m1}f2(θ2)dθ2+m1R1−F2m1 1−θ1 The derivative of V2with respect to m1is calculated to be V 2(m1θ1)=(R −1)(1−θ1)m1 1−θ1 +m1f2m1 1−θ11 1−θ1 +F2m1 1−θ1 −Rm1f2m1 1−θ11 1−θ1 +R1−F2m1 1−θ1 314 Gale and Yorulmazer Theoretical Economics 8 (2013) =Rm1f2m1 1−θ11 1−θ1 +F2m1 1−θ1 −Rm1f2m1 1−θ11 1−θ1 +R1−F2m1 1−θ1 =R1−F2m1 1−θ1+F2m1 1−θ1 The expression for V 2(m1θ1), the marginal value of cash carried forward to date 2,is quite intuitive. One unit of cash that has not been used can be converted into one unit of consumption, but in some cases it has a value of Rbecause it can be used to “save” one unit of the asset that would otherwise be lost in default. This happens if the total supply of cash at date 2,m1, is less than the demand (1−θ1)θ2and the probability of this happening is 1−F2(m1/(1−θ1)). So the value of an extra unit of cash is the probability that m1is less than (1−θ1)θ2times Rplus the probability that m1is greater than (1−θ1)θ2times 1. Now consider the planner’s problem at date 1. She has m0units of cash in state θ1 and must choose the amount x1to distribute to bankers. Feasibility requires 0≤x1≤m0 and, without loss of generality, we can assume x1≤θ1, since there is no point in giving cash to a banker who has not received a liquidity shock. Thus, the planner chooses x1to maximize Rx1+V2(m0−x1θ1) subject to 0≤x1≤min{m0θ1}(15) If the constraint (15) is nonbinding, the first-order condition R=V 2(m0−x1θ1) =R1−F2m1 1−θ1+F2m1 1−θ1 must be satisfied. This is possible only if F2(m1/(1−θ1)) =0or m1=m0−x1=0,a contradiction. Thus, the constraint (15) must bind and this implies that the optimal policy is x1=min{m0θ1}or m1(θ1)=max{m0−θ10} Substituting this decision rule into the objective above, we obtain the value function V1(m0θ1)=Rmin{θ1m0}+V2(max{m0−θ10}θ1) At the end of date 0, before θ1is realized, the value of m0units of cash is given by E[V1(m0θ1)]=1 0Rmin{θ1m0}+V2(max{m0−θ10}θ1)f1(θ1)dθ1 =m0 0 [Rθ1+V2(m0−θ1θ1)]f1(θ1)dθ1+Rm0(1−F1(m0)) Theoretical Economics 8 (2013) Liquidity hoarding 321 Since E[p2(θ1θ2)|θ1]=R−1F2m0−θ1 1−θ1+1−F2m0−θ1 1−θ1 =1−(1−R−1)F2m0−θ1 1−θ1 for θ1<m 0and 1otherwise, E[p2(θ1θ2)]=m0 01−(1−R−1)F2m0−θ1 1−θ1f1(θ1)dθ1+1−F1(m0) =1−(1−R−1)m0 0 F2m0−θ1 1−θ1f1(θ1)dθ1 Then E[p2(θ1θ2)R]=R−(R −1)m0 0 F2m0−θ1 1−θ1f1(θ1)dθ1=ρ as required.  Proof of Proposition 9. We can calculate the payoff of a liquid banker at date 0as in the proof of Proposition 6, assuming, without loss of generality, that the banker hoards cash at date 1, 1 01 0θ1R+(1−θ1)θ2R+(1−θ1)(1−θ2)(1+p2(θ1θ2))R −1 ×f1(θ1)f2(θ2)dθ1dθ2 =R+1 01 0 (1−θ1)(1−θ2)(p2(θ1θ2)R −1)f1(θ1)f2(θ2)dθ1dθ2 =R+1 01 0 p2(θ1θ2)Rf1(θ1)f2(θ2)dθ1dθ2 −1 01 0 (θ1+(1−θ1)θ2)p2(θ1θ2)Rf1(θ1)f2(θ2)dθ1dθ2 −1 01 0 (1−θ1)(1−θ2)f1(θ1)f2(θ2)dθ1dθ2 using the identity (1−θ1)(1−θ2)=1−(θ1+(1−θ1)θ2). Similarly, we calculate the payoff of an illiquid banker at date 0as in the proof of Proposition 6 to obtain 1 01 0θ1(1−p1(θ1))R +(1−θ1)θ2(1−p2(θ1θ2))R +(1−θ1)(1−θ2)(R −1)f1(θ1)f2(θ2)dθ1dθ2+ρ =R−1 01 0 {θ1p1(θ1)+(1−θ1)θ2p2(θ1θ2)R}f1(θ1)f2(θ2)dθ1dθ2 322 Gale and Yorulmazer Theoretical Economics 8 (2013) −1 01 0 (1−θ1)(1−θ2)f1(θ1)f2(θ2)dθ1dθ2+ρ =R−1 01 0(θ1+(1−θ1)θ2)p2(θ1θ2)Rf1(θ1)f2(θ2)dθ1dθ2 −1 01 0 (1−θ1)(1−θ2)f1(θ1)f2(θ2)dθ1dθ2+ρ using the facts that min{p2(θ1θ2) 1}=p2(θ1θ2)and p1(θ1)=E[p2(θ1θ2)]. 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