Safe Assets As Collateral Multipliers

Safe Assets as Collateral Multipliers

Emre Ozdenoren Kathy Yuan Shengxing Zhang∗

London Business School London School of Economics London School of Economics CEPR FMG CFM CEPR CEPR March, 2020

Abstract Risky and information sensitive assets, such as mortgage and bank loans, are often pledged as collateral to overcome the limited commitment problem but they are costly to produce and their funding capacity is limited due to adverse selection. Safe assets, such as reserves and treasury bonds, are free of adverse selection and hence are information insensitive. Safe assets can also be used as collateral but are even costlier to hold than risky assets. We demonstrate a complementarity between the use of safe and risky assets as collaterals. Safe assets facilitate the production of risky collaterals, increase borrowers’ leverage and expand the balance sheet. Complementaries arise because safe assets lower adverse selection and coordinate beliefs in selecting the liquid equilibrium. We allow for joint design of borrowers’ asset and liability and show liquid debt/illiquid equity tranches emerge as optimal liability. In facilitating borrowing safe assets and liquid debts are substitutes, but in the optimal design of the borrower’s balance sheet they are complements. If safe assets become cheaper to hold then the borrower will also hold more liquid debt and vice versa. The theory has implications on the optimal balance sheet (asset-liability) compositions of banks and their role as liquidity producers, excess cash holdings for corporations, and also for monetary and ﬁnancial policies.

Keyword: Liquidity; Security Design; Financial Fragility; Collateral; Safe Assets, Information- Insensitive Assets; Equilibrium Coordination. JEL classiﬁcation: G10, G01 ∗Ozdenoren ([email protected]), Yuan ([email protected]), and Zhang ([email protected]). We thank Yue Wu for excellent research assistance.

1 1 Introduction

Safe assets in the form of reserves, treasury bills and bonds issued by the government have been viewed to take the following three roles: a medium of exchange, a unit of account and a store of value.1 In this paper, we argue that there is an additional role of safe assets: collateral multiplier. In an environment with limited commitment, risky but information-sensitive assets, such as mortgage and bank loans, are often used as collateral to obtain funding for productive opportunities. However, their funding capacity is limited due to information frictions. By comparison, safe assets are free from information frictions but costlier to produce. We show that by combining safe assets and risky collaterals, productive agents in the economy can improve the funding capacity of their liabilities backed by these assets. We start by studying a static benchmark where borrowers pledge the future cashflows from a portfolio of a safe asset and a risky collateral. By doing so they obtain private liquidity from potential lenders to use as inputs for a productive technology. The risky collateral is subject to adverse selection. It can be high or low quality and this information is privately known by the borrower. We measure the degree of adverse selection by the ratio of expected payoff between low to high quality collateral. When the portfolio consists of only the risky collateral, the equilibrium outcome depends on the degree of adverse selection. If the degree of adverse selection is low (high) relative to the productivity of the real technology then both the high and low quality risky (only the low quality) collaterals are used to generate liquidity which increases (lowers) the average quality of the collateral pool. As a result, uninformed investors are willing to pay a higher (lower) price for the risky collateral and more (less) funding is raised. In this case, we say that the risky collateral is liquid (illiquid). We next demonstrate the collateral multiplier role of the safe asset in this lemons environment. In reality, safe assets are costly to hold for investment (eg., due to value erosion from inflation). To starkly highlight the collateral multiplier role of the safe asset, we assume that its cost is so high relative to the productivity of the real technology that it is unprofitable to use it as collateral on its own. An immediate consequence of this assumption is that the safe asset is not desirable to hold if the risky collateral is liquid. If the risky collateral is illiquid, however, we demonstrate that the safe asset improves the funding environment. By combining the safe asset with the risky collateral in a portfolio, adverse selection is reduced, the economy switches from the illiquid to the liquid equilibrium, resulting in greater funding and real output. That is, by holding the safe asset on its balance sheet, the borrower influences the

1For example, Samuelson (1958) studies money as a store of value in an overlapping- generations model, and Kiyotaki and Wright (1989)study money as a medium of exchange in a model with search frictions.

2 lenders’ beliefs so that they rationally expect both borrower types to be present in the collateral market. Since, borrowers are willing to pay more to hold the safe asset and the risky collateral together in a portfolio, the assets would fetch higher prices jointly, resulting in a safety premium for the safe asset and a liquidity premium for the risky collateral. Further, there is a complementarity in holding both assets. When the cost of holding the safe asset is lower, more of the risky collateral will be held, expanding the borrowers’ balance sheet and increasing pledgeable cashflow in the economy. This finding is supported by the empirical evidence in Adrian and Shin (2014) where they document the expansion of banks’ balance sheet as monetary policy is loosened. So far we have been focusing on the asset side of borrowers’ balance sheet implicitly assuming that the borrower’s portfolio of assets are sold as a single security to raise funding. We next allow borrowers to optimally design the securities on the liability side of the balance sheet. Since these securities are backed by the borrower’s assets, security design opens the door to joint design of the asset and liability sides of the balance sheet. In our setting, optimal security design leads to tranching the liability into a liquid debt tranche that is issued by both the high and low quality borrowers (and hence it is information insensitive), and an illiquid equity tranche issued by only the low type. The debt tranche is a claim to the entire cashflow from the safe asset as well as the cash flow from the risky collateral up to a threshold. This threshold is chosen so that the high quality borrower is indifferent between selling the debt versus not. The liquid debt tranche is, hence, risky since it is subject to default. The interesting element of this design is the interaction between the two liquid instruments: the safe asset on the asset side and the liquid debt tranche on the liability side of the balance sheet. The safe asset can be considered as public liquidity (since safe assets are normally produced by the government) while the liquid debt tranche can be considered as private liquidity. In this way, our theory applies naturally to banking. The asset side of the banks’ balance sheet consist of safe assets such as reserves, and risky assets such as bank loans. The liability side consists of liquid short-term debt such as bank deposits and wholesale securitized products. Private liquidity production via the liquid debt tranche substitutes for the high cost of safe asset holding and lowers the funding cost in the economy. Without securitization, borrowers need to hold enough of the safe asset to make the entire asset portfolio liquid. With securitization, they are able to reduce the safe asset holding since only the debt tranche needs to be liquid. Private liquidity producing agents, such as banks, are hence able to economize on their safe asset holding using security design. They hold less of the safe asset per unit of the risky collateral, indicating a substitution effect. At the same time, there is a complementarity between the public and private liquidity. An increase

3 in the safe asset holding makes the cashflow of the debt tranche less information sensitive and reduces adverse selection on the debt tranche. When the bank holds more safe asset, it expands the liquid debt tranche and shrinks the illiquid equity tranche. Hence, by holding more of the safe asset, the bank reduces the implicit cost of holding illiquid liabilities. Conversely, by shrinking the liquid debt tranche and expanding the illiquid equity tranche, the bank can reduce its safe asset holding. Hence, by holding less liquid debt tranche as a liability, the bank reduces the explicit cost of its safe assets. The complementarity between the public and private liquidity leads to a multiplier effect at the aggregate level. If the cost of holding the safe asset goes down, then the bank not only holds more of the safe asset and the risky collateral, but also more safe asset per unit of the risky collateral. As a result, the liquid debt tranche increases, the balance sheet expands and total leverage is higher. This predicts pro-cyclical leverage which is also documented in Adrian and Shin (2014) for the financial intermediation sector of the economy (in particular shadow banks). Comparative statics of the model also has cross-sectional implications for monetary policy. For example, it predicts that when monetary policy tightens, that is, when the cost of safe assets increases, banks with loans that are more subject to adverse selections (eg., banks with a larger fraction of subprime mortgages) would shrink their balance sheet further and decrease their leverage more. This result is also consistent with the bank lending channel of monetary transmission in Kashyap, J. Stein, and Wilcox (1993). We next extend the static set up to a dynamic setting where both the safe asset and the risky collateral are long-lived and borrowers can issue security claims backed by the future price of the risky collateral (in addition to its current period dividend). The price for the risky collateral is determined in the dynamic equilibrium. This setting allows us to study the dynamics of the asset risk premium as well as that of funding liquidity and real output. Previous literature shows that such dynamic lemons markets exhibit multiple equilibria for a given set of economic fundamentals (See Ozdenoren, Yuan, and Zhang (2019)). This is because the liquidity of the funding market depends on the collateral asset price but the collateral asset price depends on the liquidity of the funding market. When lenders expect a high future collateral price, the current period adverse selection is ameliorated, leading to higher funding liquidity and output this period which in turn results in higher demand for collateral from borrowers and hence a self-fulfilling higher collateral price. Similarly when lenders expect a low future collateral price, the adverse selection this period is worsened, which makes the low collateral price self-fulfilling. In this dynamic environment, we find that the public safe asset has an additional role: removing

4 financial fragility by eliminating multiple equilibria. The intuition is the same as in the static benchmark of our paper: by holding the safe asset on their balance sheet, borrowers influence the lenders’ beliefs so that they rationally expect both borrower types to be present in the collateral market, achieving coordination on the liquid equilibrium. When borrowers are allowed to design both asset and liability jointly, we find that safe assets facilitates the dynamic feedback loop between asset prices and the liquid debt threshold as in Ozdenoren, Yuan, and Zhang (2019).2By holding the safe asset and a liquid debt tranche that is information insensitive, borrowers can lower adverse selection and obtain more funding liquidity. More funding liquidity increases the price of the risky collateral and makes lenders willing to accept a slightly larger liquid debt tranche, which increase the funding liquidity further, and so on. The safe asset hence facilitates the aforementioned dynamic loop to allow for a larger liquid debt tranche. There is a clear delineation of the role of safe assets as a collateral multiplier. In the presence of dynamic feedback loop, we find that monetary or financial policy (such as shocks to the cost or supply of safe assets) exert an amplification effect on asset risk premium and the amount of funding liquidity in the economy. Literature Review (Incomplete) The economics of safe assets has fascinated the economic profession due to its interesting features and pricing anomalies (G. B. Gorton (2016)). Krishnamurthy and Vissing-Jorgensen (2012), for example, find safe assets such as treasury securities have large convenience yields and hence are money-like. Recent work by Jiang et al. (2019) shows that the market value of outstanding government debt is significantly above the expected present discounted value of current and future primary surpluses. Our paper shows that the documented safety premium or convenient yield could come from the fact that borrowers are willing to hold these costly assets due to their collateral multiplier effect. Ordoñez (2013) show that public safe assets play an important stabilizing role in collateral production, irreplaceable by private assets such bank loans. Andolfatto and Spewak (2018) show that safe assets share has increased from 35% throughout early 2000 to 75% of GDP since 2007-2008 crisis, indicating that demand for safe assets is especially important during the crisis period where adverse selection is

2The feedback-loop between the asset price and the liquid debt threshold in Ozdenoren, Yuan, and Zhang (2019) is as follows. When initially a tiny slice of the liquid debt tranche is issued, the price of the risky collateral will rise since it helps to raise more funding liquidity and hence borrowers would demand more of it. A higher collateral price lowers the adverse selection which enables a slightly larger slice of the liquid debt trance. This feedback loop will go on until the debt threshold reaches a large enough level where high quality borrowers are indiﬀerent between issuing the debt tranche versus not. This feedback loop allows borrowers 1) to inﬂuence lenders’ beliefs and enable them to coordinate on the liquid equilibrium via information insensitive debt in the multiple equilibria region; and 2) to improve the funding capacity by tranching out the liquid debt in the separating illiquid equilibrium region.

5 severe. Our model formalizes the role of public safe assets played in lowering the adverse selection in the economy. Our theoretical finding of the complementarity between the supply of public safe assets and the production of private liquid debt adds an additional complexity to the finding that public safe assets crowds out private safe assets as shown in Greenwood, Hanson, and J. C. Stein (2015)and Carlson et al. (2016). We find that borrowers optimize on both public and private liquidity margins: when cost of public safe assets is relatively high, borrowers indeed substitute towards private liquidity per unit of the risky collateral; however this crowding out effect is subtle and only takes place when there is large enough supply of safe assets. When the cost of the safe asset is relatively low and the supply is limited, an additional holding of safe assets would lead to more production of private liquid short-term debts. Our finding on this complex interaction of public and private liquidity production potentially could shed lights on the impact of monetary policy on the role of banks in producing “money-like” short-term liquid securities. This role has been well recognized in both the macro and the banking literature (D. W. Diamond and Dybvig (1983);G. Gorton and Pennacchi (1990); J. Stein (2012); W. Diamond (2017); and Donaldson, Piacentino, and A. Thakor (2018)) while how the supply of public safe assets affects the private liquidity production and hence entrepreneurial activities/real output in this type of economic environment has been under studied (See Asriyan, Fornaro, et al. (2019) who study the role of monetary policy in a bubbly economy where bubble comes from private supply of liquidity). Our paper is also related to a long lineage of security design literature including Leland and Pyle (1977); Myers and Majluf (1984b); Nachman and Noe (1994);DeMarzo and Duffie (1995);DeMarzo and Duffie (1999) Biais et al. (2007); and Chemla and Hennessy (2014). By focusing on liquid debts, our paper is also related to endogenous asymmetric information in an optimal security design problem such as Yang (Forthcoming); Dang, G. Gorton, and Holmström (2013); and Farhi and Tirole (2015). The fact that information friction affects the moneyness of an asset has also been studied by Lester, Postlewaite, and Wright (2012) and Li, Rocheteau, and Weill (2012a). Our focus on the interaction between the public and private liquidity design is unique to this literature. By studying borrowers’ optimal asset-liability design, we are also related to the literature on reserve holdings in banks and cash holdings in non-financial firms. In particular, the theory of bank balance sheet in Bigio and Weill (2016) also points out the role of reserve and debt in creating liquidity. Our model on safe assets as collateral multiplier via coordination complements their finding. Our theory also finds empirical supports in Ennis and Wolman (2015). They have shown that cross-sectional distribution of reserves among banks are closely related to their liquidity position; holding excessive reserves allows

6 banks to expand their balance sheet; and their capital constraints are not restrictive. The structure of the paper is as follows. In Section 2, we lay out the basic setup and solve a static benchmark.. In Section 3, we extend the baseline case to a dynamic setup. Section 4 concludes and discusses potential applications.

2 Static Economy

In this section, we study a static benchmark, and in the next section, we present a dynamic extension of the model.

2.1 Model

In the static economy, there are two types of agents: a productive agent, which we call a firm, and a large number of investors which we call agent I s. There are two types of goods: an intermediate good and and a consumption good. All agents possess a basic technology that produces one unit of the consumption good from one unit of labor. An agent’s utility is given by Ut(x, l) = x − l where x is the amount of consumption good consumed and l is the amount of labor supplied by the agent. Note that production of the consumption good using the basic technology does not provide any surplus. In addition, investors, but not the firm, can produce one unit of the intermediate good from one unit of labor. Agents receive utility only from the consumption good; the intermediate good does not provide direct utility. This intermediate good can be interpreted as capital, or any other intermediate inputs that cannot be directly consumed. The firm possesses an inalienable productive technology that transforms one unit of intermediate good into z (where z > 1 ) units of consumption goods. This z- technology provides surplus but we assume that, due to limited commitment, its output is not pledgeable. That is, the firm cannot borrow the intermedate goods from investors against future cashflows from the z-technology project. In order to take advantage of this productive technology the firm must produce pledgeable cashflows and use them in exchange for the intermediate good from investors. In reality, storage technologies provided either by the government or private identities produce pledgeable cashflows. Examples in- clude information insensitive and relatively safe assets such as fiat money, reserve, treasury securites, or information-sensitive and riskier collateral assets such as mortgage loans. Accordingly, in our model, we assume that the firm has access to two storage technologies to generate pledgeable cashflows: an information insensitive and safe asset and an information sensitive and risky collateral asset. There are

7 several ways that a firm can secure a safe asset. It can purchase a safe asset from a government agency; or produce one on its own by investing in a monitoring technology (eg., from a rating agency or a credible auditor) or a safe storage technology (eg., encryption). Similarly, there are several ways to generate a risky collateral such as producing mortgages or other loans. We assume that each unit of the safe asset generates one unit of pledgeable consumption good. We 0 00 specify the cost of producing M units of the safe asset as cm(M) where 0 = cm(0) < cm(M), cm(M), cm(M), for all M > 0. The safe asset production cost can be due to leakage that occurs during storage, payments made to monitoring/storage agencies or inflation. Each unit of the risky collateral generates s units of pledgeable consumption good in state s ∈ [sL, sH ] which can be interpreted as its dividend. Unlike the safe asset, the risky collateral is information sensitive. The realization of the state follows distribution FQ(s) where Q denotes the quality of the asset which can be low (L) or high (H). Expected R quality of the asset is given by EQs = sdFQ(s). The quality Q is L with probability λ and H with probability 1 − λ, and is privately observed by the firm. We refer to the firm that observes distribution

FQ as the type Q firm. We specify the cost of producing A units of the risky asset as ca(A) where 0 00 0 = ca(0) ≤ ca(A), ca(A), ca(A), for all A > 0. The risky asset production cost can be due to various expenses that are required to generate mortgages and other loans such as checking applicants, payments to loan officiers, etc. In the model, there are four dates. At date 0, the firm decides how to generate pledgeable cashflows by deciding on the amount of safe and/or risky collateral asset to be produced, designing securities backed by these assets to be issued and exchanged for intermediate goods later. After these decisions are made, the firm learns about the quality of the risky collateral assets. At date 1, the intermediate good is produced by agent I which is exchanged for the securities issued by the firm. At date 2, consumption good is produced via the z-technology using the intermediate good by the firm and/or the basic technology using labor by the firm or agent I. At date 3, the state is realised and consumption takes place. Any left over intermediate or consumption good perishes at the end of the period. In our leading interpretation of the model, the firm is a bank and the intermediate good is capital. The bank designs the (pledgeable) asset and liability composition of its balance sheet to increase its capacity to raise capital. The asset side of a bank’s balance sheet can be broadly categorized into two types: safe and risky. The safe assets are reserves, treasuries and AAA rated bonds. The risky collaterals are mortgages, car loans, and other collateral-backed debt that the bank issues. The inalienable productive technology could be other forms of loans that either are not backed by collateral such as loans to startups that have no immediate cashflows or loans that provide private benefits to the bank. The bank uses the

8 capital that it raises from investors to make these types of loans. In the rest of the paper, we refer the firm in the model as a bank but it can be any other types of firms that have collateral assets and are in need of financing projects that do not generate pledgeable cashflows. The liability side of a bank’s balance sheet consists of securities backed by the pledgeable assets. Formally, a security is a state-contingent promise at date 1 of a nonnegative amount of consumption good payment at date 3. Securities are traded in dedicated over-the-counter markets at date 1. We describe security trading in detail in the next section. Suppose that the bank holds J securities on the liability side of its balance sheet, and denote security j by yj. These securities must be backed by the PJ j bank’s assets. Hence, we impose the following collateral constraint: j=1 y (s) ≤ M + As.

2.1.1 Security Markets

For each available security, there is a dedicated over-the-counter market where the bank meets at least two randomly chosen investors, and trades the security in exchange for intermediate goods. We assume that the investors simultaneously make price offers per unit of the security. Hence, a price offer is the amount of intermediate goods that the investor is willing to pay for a unit of the security. The bank observes the price offers and decides whether to retain the security or sell a positive amount to the investor with the highest price offer. If several investors are tied for the highest offer, they receive equal amounts of the security. Consider the market for a security y. Suppose the bank sells A units of the security and denote the security price by q. Recall that the bank is type L with probability λ and H with probability 1 − λ. Due to Bertrand competition among investors, all investors make zero surplus in expectation. This means that in equilibrium security price must equal the expected value of a unit of the security given the investors’ expectations of the quantities that will be sold by the L and H type bank. Moreover, in equilibrium, these expectations must be correct. That is, if investors anticipate that a given type sells the security at per-unit price q, that type must indeed find it (weakly) profitable to sell the security at that price. The next proposition characterizes the equilibrium in security market.

Proposition 1. Let R = ELy/EH y. If R ≥ ζ ≡ 1 − (z − 1)/λz, the price of the security is q =

λELy + (1 − λ)EH y and both types sell A units of the security. If R < ζ, the price of the security is 3 q = ELy, type L type sells A units and type H sells zero units of the security.

3When R = ζ there are multiple equilibria. To simplify exposition, in this knife edge case, we select the pooling equilibrium.

9 In Proposition 1, R ∈ [0, 1] measures the degree of information insensitivity: higher values correspond to less adverse selection. When R is above the threshold ζ, the adverse selection problem is not too severe. In this case both types sell the security. The security price is higher since it is the pooling price. When R is below the threshold, the adverse selection problem is severe, only the low type sells the security. The security price is lower since it is the separating price. A security that is traded in a pooling equilibrium in the security market generates more liquidity for the bank than the one that is traded in a separating equilibrium because in a pooling equilibrium. Given the higher price, the bank obtains more intermediate goods in exchange for the security. Hence, we refer to a security that is traded in a pooling equilibrium as a liquid security, and one that is traded in a separating equilibrium as an illiquid security.

2.2 Balance Sheet with only the Safe Asset or the Risky Collateral

Suppose the bank holds only safe assets on its balance sheet. Let ym be a security that promises M units of the consumption good at date 3, backed by M units of the safe asset. The safe asset is always liquid and its price in the over-the-counter market is simply q = M. By holding M units of the safe asset, the bank generates M amount of pledgeable cash ﬂow for use in the z-technology. Hence the value of holding M units of the safe asset the bank is zM, and the bank chooses M to maximize zM − cm (M) .

Since cm is convex, the optimal holding of the safe asset when the bank is restricted to hold only the 0 0 0 safe asset is given by M = 0 if cm (0) ≥ z and cm (M) = z if cm (0) < z. Suppose the bank holds only the risky collateral on its balance sheet. The payoﬀ of A units of the risky collateral in state s is ya (s) = sA. Let ya be a security that promises sA units of the pledgeable consumption good at date 3, backed by A units of the risky collateral. By Proposition 1, the price of the P security in the over-the-counter market is given by qa = A [λELs + (1 − λ)EH s] if ELs/EH s ≥ ζ and S qa = AELs otherwise. The bank chooses A to maximize zA [λELs + (1 − λ)EH s]−ca (A) if ELs/EH s ≥

ζ and A [zλELs + (1 − λ)EH s]−ca (A) otherwise. Hence, the bank decides whether and how much risky collateral to produce depending on whether the risky collateral is liquid or illiquid. For the rest of the paper, to focus on the complementarity between the two assets in expanding the bank’s balance sheet, we restrict attention to the case where

0 cm(0) ≥ z, and (1)

0 z (λELs + (1 − λ) EH s) > ca(0) ≥ zλELs + (1 − λ) EH s. (2)

Inequality (1) implies that it is not proﬁtable for the bank to produce the safe asset on its own. Inequality

10 (2) implies that producing a positive amount of the risky asset is profitable if the risky asset is liquid but not if it is illiquid. We further assume that, ELs/EH s < ζ, i.e., there is enough adverse selection so that the risky collateral is illiquid on its own. Combined with (2), this implies that it is not profitable for the bank to produce the risky collateral on its own. As we show next, although the bank does not produce either asset on its own, it finds producing both assets and selling securities backed by a portfolio of both assets to be profitable. The above assumptions on the cost functions of producing the safe and risky assets are purely for expositional clarity. The insight that there is complementarity between these two assets in generating borrowing capacity/liquidity holds when these assumptions are relaxed.

2.3 Balance Sheet with a Portfolio of the Safe Asset and the Risky Collateral

Now consider a portfolio of M units of the safe asset and A units of the risky collateral. The cash ﬂow of the portfolio in state s is sA + M. Let y be a security backed by the cash ﬂow of this portfolio that pays y (s) = sA + M in state s. By Proposition 1, the price of the security in the over-the-counter market P S is given by q = M + [λELs + (1 − λ)EH s] A if (AELs + M)/(AEH s + M) ≥ ζ and q = M + AELs ∗ otherwise. Note that (AELs + M)/(AEH s + M) ≥ ζ if and only if M/A ≥ v where ζE s − E s v∗ ≡ H L . 1 − ζ The value of holding the portfolio (M,A) to the bank is  ∗ z ([λELs + (1 − λ)EH s] A + M) , if M/A ≥ v , V (M,A) = ∗ [λzELs + (1 − λ)EH s] A + (zλ + 1 − λ) M, if M/A < v .

For a ﬁxed A, V jumps upwards at M = v∗A. Everywhere else it is linear with respect to M and its 0 slope is less than or equal to z. Since cm (M) > z, v (M,A) − cm (M) is maximized with respect to M at either M = 0 or M = Av∗.

The bank chooses M ≥ 0 and A ≥ 0 to maximize V (M,A) − cm (M) − ca (A) . Since it is never optimal to produce either asset on its own, there are only two cases to consider. The optimal solution involves either A = M = 0 or A > 0 and M = Av∗. Hence, if the maximum is achieved at some A > 0, the optimal risky collateral holding is given by the maximizer of:

∗ ∗ ∗ ∗ V (Av ,A) − cm (Av ) − ca (A) = z ([λELs + (1 − λ)EH s] A + Av ) − cm (Av ) − ca (A)

Note that in that case the value of the optimization problem is strictly positive since the bank could obtain zero payoﬀ by choosing A = 0. Otherwise the bank chooses A = M = 0. Since the cost functions

11 are convex, the optimal A is strictly positive if and only if

0 0 ∗ ∗ ca(0) + cm(0)v < z ([λELs + (1 − λ)EH s] + v ) , (3) and if (3) holds then the optimal holding of risky collateral is given by A > 0 that satisﬁes

0 0 ∗ ∗ ∗ ca(A) + cm(Av )v = z ([λELs + (1 − λ)EH s] + v ) . (4)

We summarize the discussion so far in the following proposition.

Proposition 2. Suppose (1) and (2) hold, and ELs/EH s < ζ. Then the bank produces neither the safe asset nor the risky collateral on its own. If the bank is able to combine both assets in a portfolio and sell a security backed by the cash ﬂow from the portfolio, then the bank produces strictly positive amounts of both assets if and only if (3) holds. The amount of risky collateral produced by the bank is A and the amount of safe asset is v∗A, where A solves (4).

It is too costly to produce the safe asset on its own, if the marginal cost of the safe asset exceeds the value of the pledgeable funding that it generates for the z technology (condition (1)). It is also too costly to produce the risky collateral, if only the low type pledges its cash flow in the security market, but it is worth producing, if both types pledge the cash flow (condition (2)). In this situation, when the safe asset is combined with the risky collateral, it provides an additional service because it reduces adverse selection. To attract the high type risky collateral holder to the security market, each unit of risky collateral must be combined with at least v∗ units of the safe asset. Hence, the effective marginal cost of producing a unit of the risky collateral is increased by the marginal cost of producing v∗ units of the safe asset. This is the expression on the left hand side of (4). The right hand side is the marginal benefit of pledging the cash flow from a unit of risky collateral (given that both types participate in the security trading) and v∗ units of safe asset. The optimal amount of risky collateral equates the effective marginal cost and marginal benefit.

Next we construct a simple analytical example to illustrate the result. Suppose cm(M) = κM, and 2 ca(A) = (γ/2) A + χA, where z (λELs + (1 − λ) EH s) > χ ≥ zλELs + (1 − λ) EH s and κ > z. Hence,

0 0 ∗ ∗ ∗ ca(A) + cm(Av )v = γA + χ + κv .

0 0 ∗ ∗ ∗ For κ close enough to z, ca(0) + cm(0)v = χ + κv < z ([λELs + (1 − λ)EH s] + v ) and (3) holds. Optimal risky collateral holding is given by:

z (λE s + (1 − λ)E s) − χ − (κ − z) v∗ A = L H . γ

12 As the example illustrates, there is a complementarity in production of the two assets. As long as the marginal cost of producing the safe asset is not too high, the bank ﬁnds it proﬁtable to produce both assets and combine them in a portfolio. If this portfolio approach is not allowed, the bank does not produce either asset. Another way to see the complementarity is that, when the marginal cost of producing the safe asset falls, more risky collateral are created.

2.4 Balance Sheet with Security Design

In the previous sections, our focus has been on the asset side of the balance sheet. On the liability side, we considered a single security backed by the cash ﬂow of the bank’s assets. In this section, we consider the optimal joint design of the asset and liability. Intuitively, the bank can save the cost of producing the safe asset by instead creating a debt tranche carved from the risky collateral and impose a higher haircut on the residual equity tranche. On the liability side, we allow for the bank to choose the securities that it oﬀers to the investors.

Recall that a security, y :[s, s] → R+ is a state-contingent promise at date 1 of a nonnegative amount of consumption good payment at date 3. We assume that the bank offers monotone securities that are backed by its assets.4 Formally, the bank offers securities yj, j ∈ {1, ..., J} that are weakly increasing PJ j and j=1 y (s) ≤ M + As. We first analyze the bank’s optimal choice of securities given that on its balance sheet it has M units of safe and A units of risky asset. The following observation simplifies the analysis. Observe that if two securities yj and yk are both liquid (illiquid) then their sum, yj + yk, is also a feasible design and liquid (illiquid). Moreover, by offering the sum of the two securities, instead of the two securities separately, the bank raises the same amount of funding. Hence, w.l.o.g. we restrict attention to at most two securities, one liquid and the other illiquid. The following proposition characterizes the optimal securities for given M and A.

Proposition 3. Assume that fL(s)/fH (s) is decreasing in s. The optimal liquid security is a standard debt contract yD such that

yD(s) = M + A min(s, D), and the optimal illiquid security is the residual (equity) tranche yE (s) = A max (0, s − D). The prices

4Monotonicity can be easily relaxed, although the resulting optimal securities no longer look like standard debt and equity.

13 of the securities are given by

" Z D Z D # qD = M + A sL + λ FeL(s)ds + (1 − λ) FeH (s)ds , (5) sL sL and Z sH qE = A FeL(s)ds. (6) D ∗ Moreover, if M ≥ Av then the liquid debt threshold D is equal to sH and the illiquid tranche disappears. ∗ If M < Av then the debt threshold D ∈ (sL, sH ) is the unique solution to: " !# Z D Z D Z D h i (z − 1) M/A + sL + λ FeL(s)ds + (1 − λ) FeH (s)ds = λ FeH (s) − FeL(s) ds. (7) sL sL sL There are several notable features of the optimal securities. The debt tranche promises the cash ﬂow from the safe asset and each unit of the risky collateral up to the debt threshold D. That is, the face value of debt is M + AD. When the cash ﬂow from the risky collateral is below D, debt pays less than its face value, hence it is risky.

Recall that ELs/EH s < ζ. Under this assumption a security backed by the enture cash flow of the risky asset is not liquid. To create a liquid security, the firm needs to either combine the risky asset with the safe asset or tranche the risky asset or both. If M ≥ Av∗ then each unit of the risky asset is combined with enough safe asset to ensure that the participation constraint of the high type is not binding. In this case, there is no need to tranche the risky asset and the entire combined cash flow from the safe and risky assets back the liquid asset. When M < Av∗, the high type prefers not to trade a security that promises the entire combined cash flow. In this case, debt threshold must be lowered so that the high type retains the equity tranche which is the residual cash flow from the risky asset when the state exceeds D. The high type is then incentivized to trade the debt tranche. Debt threshold D is determined by the incentive constraint of the high type which is given by equation (7). By Proposition 1, the prices of the optimal liquid debt and illiquid equity securities in the over-the-counter market are given by (5) and (6).

2.4.1 Security Design vs the Safe Asset

To see the tradeoﬀs involved in the balance sheet design, we ﬁrst compare the following extreme cases:

Case 1. (Only Security Design) The borrower uses relies only on security design, and does not use any public liquidity to generate liquid funding.

Case 2. (Only Public Liquidity) The borrower uses only public liquidity to generate liquid funding.

14 In Case 1 the value of a unit of the risky collateral asset is: ! Z D Z sH z sL + λELs + (1 − λ) FeH (s) ds + (1 − λ) FeH (s) ds sL D where D is given by ! Z D Z D Z D h i (z − 1) sL + λ FeL(s)ds + (1 − λ) FeH (s)ds = λ FeH (s) − FeL(s) ds sL sL sL . In Case 1 the value of a unit of the risky collateral asset is:

∗ z (λELs + (1 − λ) EH s) − (κ − z) v .

Comparing we ﬁnd that the value of a unit of the risky collateral asset is higher in case 2 if and only if Z sH ∗ (z − 1) (1 − λ) FeH (s) ds ≥ (κ − z) v D where ζE s − E s λz v∗ ≡ H L = (E s − E s) − E s. 1 − ζ z − 1 H L H

2.4.2 Optimal Balance Sheet

Proposition 3 implies that the value to the bank of creating M units of safe assets and A units of risky collaterals is  z [M + A (λE s + (1 − λ)E s)] if M ≥ Av∗  L H  h R sH R D i V (M,A) = z M + A sL + λ FeL(s)ds + (1 − λ) FeH (s)ds  sL sL   R sH ∗ +A(1 − λ) D FeH (s)ds if M < Av where D solves (7). By deﬁning v = M/A, we can rewrite the value function as  Az [v + (λE s + (1 − λ)E s)] if v ≥ v∗  L H  n h R sH R D i AV (v, 1) = A z v + sL + λ FeL(s)ds + (1 − λ) FeH (s)ds  sL sL  o  R sH ∗ +(1 − λ) D FeH (s)ds if v < v where D solves: " !# Z D Z D Z D h i (z − 1) v + sL + λ FeL(s)ds + (1 − λ) FeH (s)ds = λ FeH (s) − FeL(s) ds. (8) sL sL sL

15 The bank’s optimization problem is then to choose v and A to maximize AV (v, 1) − cm (Av) − ca (A) where D is given by (8). First note that it is never optimal to choose v > v∗. Hence w.l.o.g. we focus on v ≤ v∗.

∗ Lemma 1. If v ≤ v then there exists a unique D ∈ (sL, sH ) that solves (8) which is strictly increasing in v.

Proof. Let Z x h i Z x Z x Υ (x) = λ FeH (s) − FeL(s) ds − (z − 1) v + sL + λ FeL(s)ds + (1 − λ) FeH (s)ds . sL sL sL ∗ It is easy to check that Υ (sL) ≤ 0,Υ (sH ) > 0 if v < v . Hence, Υ (y) = 0 at some y ∈ (sL, sH ) . Denote the smallest such y by D. At D, Υ intersects the horizontal axis from below so Υ 0 (D) > 0. Taking the derivative: 0 Υ (x) = λz FeH (x) − FeL(x) − (z − 1) FeH (x).

0 0 Note Υ (sL) < 0, Υ (sH ) = 0, and the second derivative is: 00 fL(x) Υ (x) = λzfH (x) − ζ . fH (x) 0 Since fL/fH is decreasing Υ is either decreasing for all s or it is increasing and then (possibly) decreasing. 0 0 Since Υ (sL) ≤ 0 and Υ (sL) < 0, if Υ is decreasing everywhere then Υ cannot cross zero. Hence, it must be that Υ 0 is initially increasing, crosses zero and becomes positive and then decreases so that 0 Υ (sH ) = 0. If Υ crosses zero a second time at De < sH ,it must remain below zero at all s > D,e contradicting that Υ (sH ) > 0. Finally, if v goes up, Υ shifts down and the point where it crosses zero shifts to the right, i.e., D increases.

∗ Let D (v) be the solution to (8) for v ∈ [0, v ]. Deﬁne g :[sL, sH ] → R as:

(z − 1)(1 − λ)F˜H (x) g (x) = h i . λz ˜ ˜ ˜ z−1 FH (x) − FL(x) − FH (x) As we see later, the function g captures the indirect beneﬁt of holding marginally more safe asset through its impact on the debt threshold. The following lemma characterizes the properties of this function.

Lemma 2. The function g is continuous and strictly decreasing everywhere, except at a single point

Db ∈ (sL, sH ) where it jumps from −∞ to ∞, and this jump always occurs strictly below D (0) . In addition, g (x) < 0 for all x < Db, g (x) > 0 for all x ∈ D,b sH ,and g (sH ) ≥ 0.

gcn l l l

l i 1 i i l I I I l s SL 1B DCO SH N

Cz 1 l X I 1

Figure 1: The function g.

Proof. The denominator of g is (z − 1)−1 Υ 0 (x) where Υ 0 is as deﬁned in the proof of 1. We know that 0 0 0 Υ (sL) < 0 and Υ (sH ) = 0. We know from the proof of 1 that Υ is initially increasing, crosses zero 0 0 and becomes positive and then decreases to Υ (sH ) = 0. Denote the point at which Υ is zero by D.b Taking the derivative of g we ﬁnd that

λz ˜ ˜ (z − 1)(1 − λ) FL(x)FH (x) f (x) f (x) g0 (x) = z−1 H − L h i 2 ˜ ˜ λz ˜ ˜ ˜ FH (x) FL(x) z−1 FH (x) − FL(x) − FH (x)

Recall that fL (x) /fH (x) is decreasing in x implying that FH dominates FL in hazard rate so the term in parentheses is negative for all x, and g is decreasing. Note also that the numerator is positive for all x so g (x) < 0 for all x < Db and g (x) > 0 for all x ∈ D,b sH .

We illustrate the function g in Figure 1. The marginal beneﬁt of holding v units of safe asset per unit of the risky collateral is

d ∂V ∂V ∂D V (v, 1) = + = z + g (D (v)) . dv ∂v ∂D ∂v

Holding marginally more safe asset per unit of the risky collateral has a direct benefit and an indirect benefit. The direct benefit is that the borrower can borrow more intermediate good and produce z units

17 of final good. The indirect benefit is that the debt threshold D is now higher. The value of the indirect benefit is g (D (v)). We see in Figure 1 that as the debt threshold increases the indirect benefit falls. The following proposition characterizes the optimal choice of safe asset per unit of risky collateral.

Proposition 4. Let ve be the optimal amount of money per unit of asset.

0 0 ∗ 0 1. If cm(0) < z + g (D (0)) and cm (v ) > z + g (sH ) then ve > 0 and satisﬁes cm(Ave) = z + g (D (ve)) , and the debt threshold is D (ve).

0 ∗ ∗ 2. If cm (v ) < z + g (sH ) then ve = v and the debt threshold is sH .

0 3. If cm(0) > z + g (D (0)) then ve = 0, and the debt threshold is D (0) .

Proof. The marginal beneﬁt of v is

d ∂V ∂V ∂D V (v, A) = + = A (z + g (D (v))) . dv ∂v ∂D ∂v

By Lemma 2, for D ≥ D (0), marginal benefit is continuous and decreasing in D. In addition, D (v) is continuous and increasing in v, so marginal benefit is continuous and decreasing in v. Marginal cost of v 0 0 is cm(Av)A which is continuous and increasing in v. Hence if cm(0) ≥ z +g (D (0)) then marginal cost of 0 0 ∗ v exceeds its marginal benefit for all v > 0 and ve = 0. If cm(0) < z + g (D (0)) and cm (v ) > z + g (sH ) ∗ 0 then there exists ve ∈ (0, v ) such that cm(Ave) = z + g (D (ve)). In this case, the optimal amount of safe asset per unit of risky asset is ve and the corresponding debt threshold is D (ve) ∈ (D (0) , sH ) . If 0 ∗ ∗ cm (v ) < z + g (sH ) then the optimal amount of safe asset per unit of risky asset is ve = v and the debt threshold is sH .

Figure 2 illustrates Proposition 4 and shows how to ﬁnd the amount of safe asset per unit of risky collateral, ve, and the corresponding debt threshold D (ve) in case 1. 0 Let D be such that cm(0) = z + g D . The threshold By the Lemma 2 there is a unique such D in

(sL, sH ] . When marginal cost of safe asset is constant, we obtain the following simpler characterization of the optimal safe asset holding and tranching.

0 Corollary 1. Suppose κ = cm(M) for all M. Let κ = z + g (D (0)) and κb = z + g (sH ). Let ve be the optimal amount of money per unit of asset and De be the debt threshold. 1. If κb < κ < κ then κ = z + g De , and De = D (ve) .

18 N r Ztgcr I

I I I I k I I i t I i 2 gait T l l l l l z l I l l l l l l l X I r l i I i i µ Dco 5 D F SH k

2 gcse n r I E t j L

I I k I I i I I r K T I I l l z l I l l l l l l l X I r l i I i i µ Dco 5 D F SH K

Figure 2: Construction of ve and D (ve).

∗ 2. If κ < κb then ve = v and De = sH .

3. If κ > κ then ve = 0, and De = D (0) .

If the marginal cost of creating safe asset is low, it is optimal to combine each unit of the risky collateral with v∗ units of safe asset. In this case, the pass-through security is traded in a pooling equilibrium and there is no need to create costly illiquid tranche. On the other hand, if the cost of creating safe asset is high, it is optimal to tranche the risky collateral into a debt tranche and an equity tranche and use no safe asset. If the cost of creating safe asset is intermediate, it is optimal to combine the risky asset with the safe asset and create a larger debt tranche. safe asset and risky asset are complements. Safe asset allows the borrower to reduce the illiquid equity tranche and tranching allows the borrower to use less safe asset which is costly to create. The next lemma shows that when tranching is allowed the value of safe asset is continuous and concave. This is in contrast with the case without tranching where value of money is decreasing except at the point Av∗ where equilibrium switches from separating to pooling.

Lemma 3. Suppose ve > 0 then V (v, A) − cm (v) is strictly concave in v.

Proof. The second derivative of V (v, A) − cm (Av) is ∂D A2 g0 (D (v)) − c00 (Av) < 0. ∂v m

19 Suppose there is a small negative shock to the supply of safe asset. Without tranching this shock has a first order effect. Since each unit of risky asset requires v∗ units of safe asset, the high type must retain some risky collaterals on its balance sheet. (Alternatively the bank can reduce the safe asset per unit of risky collateral below v∗, but this makes all the risky collaterals illiquid.) With tranching the effect of a small negative shock is second order. The shock can be spread over all the risky collaterals and reducing the amount of safe asset per unit of risky collateral. This in turn affects value function through the debt threshold D. The impact is still non-zero because, since the cost of money is sunk, V (v, A) is strictly increasing at the optimum. So moving away from the optimum reduces the value function. However we show that this impact is much smaller. (Alternatively: suppose the bank is required to produce slightly less safe asset than the optimal amount. Then the cost is also reduced so the impact is very low with tranching.)

3 Dynamic Coordination and Ampliﬁcation

In this section, we extend the analysis to a dynamic environment, where both the risky collaterals and the safe assets are long-lived.

The dividend payment of the risky collateral follows either distribution FL or FH , like in the static model. The quality of the collateral asset is i.i.d. over time. At the beginning of a period, an asset depreciates with probability δ. At the same time, Agent B is endowed with δA additional units of the collateral asset, so that the total supply of the risky collateral is fixed at A. The safe asset is modeled as a fiat asset. The supply of the fiat asset increases at rate µ where µ > βz. At the end of a period, agents can trade assets at a centralized market using a linear production technology that transforms h units of labor into h units of consumption goods. Denote the period-t a m resale price of the risky asset to be φt and the safe asset φt . m a For an agent B who chooses to hold At−1 units of safe assets and At−1 units of risky assets at the j centralized market at period t−1. The period-t securities issued by the agent, yt , respects the following feasibility constraint, X j m m a a yt (s) ≤ At−1φt + At−1(s + φt ), ∀s ∈ [sL, sH ]. j To determine the equilibrium value of money, we introduce another type of agents, agent C, who hold m m money as a medium of exchange, the liquidity value they get from φt at real balance of money is m m 0 00 0 + l(φt at ), where l (·) > 0 > l (·), with l (0 ) = ∞.

20 s m m m m Deﬁnition. A monetary equilibrium is φt ,φt > 0, aBt, aCt for all t such that (1) aBt solves the m optimization problem of agent B at the end of period t; aCt solves the optimization problem of agent C m m at the end of period t; (2) the market for money clears, aBt + aCt = Mt.

The optimization problem of agent C implies an Euler equation for money,

m m 0 m m φt = βφt+1l (φt+1act ), for all t. (9)

Because l0(0+) and l is concave, there always exists a monetary equilibrium, where the Euler equation m m holds. In a stationary monetary equilibrium where the money supply increases at rate µ, φt /φt+1 = µ. 0 m m Then ι = µ/β = l (φt+1act ), for all t.

3.1 Portfolio of Safe Assets and Risky Collaterals

m m In this section, we restrict the security space so that the security issued at period t is yt(s) = At−1φt + a a m a At−1(s + φt ) for a portfolio of At−1 units of safe assets and At−1 units of risky assets formed in the centralized market of the previous period. Similar to the static model, agents do not have incentive to hold safe assets separately when ι > z, because the cost of buying safe assets at the centralized market of period t − 1 exceeds the expected beneﬁt at period t.

Proposition 5. In the parameter space where there exists a unique separating equilibrium without money, there exists a unique pooling equilibrium where agent B holds a portfolio of money and risky collateral when ι < ι. In it,

βz [λE s + (1 − λ)E s + v∗] − µv∗ φa = L H 1 − ιβδ 1 EH s a when δ < β 1 − zλ , φ is decreasing in inﬂation rate, loose monetary policy increases the z−1 (EH s−ELs) collateral value of the asset. In the parameter space where there exists multiple equilibria without money, there exists a unique monetary equilibrium when ι < ι.

Proposition 5 implies that when money is valued, it can increase collateral liquidity and reduce ﬁnancial fragility. Ozdenoren, Yuan and Zhang (2019) has shown that in this dynamic lemons market, there exist multiple equilibria when agents trade only the long-lived risky assets. The above proposition shows that introducing safe assets to this dynamic environment, multiplicity in equilibrium is eliminated. The intuition is similar to that in the static case where safe assets allow coordination in belief that both high and low quality agents will issue this asset portfolio and the liquid pooling equilibrium will be

21 Figure 3: Asset price and dynamic cost of safe asset creation selected. Interestingly, this result draws parallel with that in Ozdenoren, Yuan and Zhang (2019) where they show that securitization also eliminates dynamic multiplicity. There they show that dynamic feedback between asset price and liquid debt threshold leads to the liquid pooling equilibrium. Here the safe asset takes place of the role of the liquid debt in facilitating the dynamic feedback between the µ asset price and the cost of safe assets, To see this, we denote the nominal interest rate to be ι = β . It represents the cost of safe asset creation. Like in the static model, when the cost increases, the risky asset price decreases. In the dynamic model, the decrease in the risky asset price feeds back into the total cost holding safe assets. Denote the real balance of safe assets held by agent B in the stationary ∗ equilibrium to be MB. Because the resale value of the portfolio per unit of the risky asset must be v , ∗ a a v = φ + MB. As φ decreases ι, MB must be increasing. So, the eﬀect of tightening monetary policy on the cost of safe asset creation is ampliﬁed by the response of risky asset price. Figure 3 illustrates the feedback.

22 3.2 Portfolio and Tranching

In this section, we allow agent B to create any monotone securities.

Proposition 6. There exists a unique stationary monetary equilibrium when ι = g−1(D) for D ∈ a (sL, sH ), where equilibrium MB and φ are solved by the following equations λ Z D h i (1 − βδι)MB = (1 − βδz) FeH (s) − FeL(s) ds (10) z − 1 sL ! Z D Z D Z sH − sL + λ FeL(s)ds + (1 − λ) FeH (s)ds − βδ(1 − λ) FeH (s)ds sL sL D

n h R sH R D i R sH o β z sL + λ s FeL(s)ds + (1 − λ) s FeH (s)ds + (1 − λ) D FeH (s)ds − β(ρ − z)MB φa = L L (11) 1 − βδz As is illustrated in Figure 4, the qualitative implication of the theory is similar to that of the static model. As the cost of creating safe assets increases, the expected value of assets and the debt threshold decreases, a sign of complementarity between the cost of safe asset creation and endogenous asset liquidity. Different from the case where tranching is not allowed, the real balance of nominal safe assets also decreases in inflation. This is because when both safe assets and tranching are allowed, agent B economizes the cost of liquidity creation using both margins. As nominal interest rate ι increases, the cost of tranching and creating safe assets both go up. But because the asset price is lower, the cost of creating money still more sensitive to inflation. Because of the dynamic feedback from the asset price to the cost of safe assets, we would expect that the asset price and cost of creating safe assets are more sensitive to ι in the dynamic model than in the static model.

4 Conclusion

In this paper we find that safe assets have a role in coordinating equilibrium beliefs and help to achieve more funding liquidity in the economy. They have a multiplier effect on collateral production in the economy. We also find that there is complementary between safe asset holding and leverage of the borrowers. In a dynamic setting, safe assets facilitate dynamic coordination and eliminate financial fragility. The model can be used to understand the endogenous capital structure/balance sheet of the bank banks as the producer of pledgeable (information insensitive) securities: such as their optimal reserve holding. Studying the impact of various shocks to their balance sheet and hence risky asset risk premium and funding liquidity could offer a lens to understand the implication of various monetary and financial policies.

23 Figure 4: Asset price and dynamic cost of safe asset creation under pooling and tranching.

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A Proofs

A.1 Proof for Proposition 5

 z {[λE s + (1 − λ)E s] + v } , if v ≥ v∗ ∗  L H t+1 t+1 VB,t+1(qt+1; φt+1, Jt+1 (vt+1)) = ∗ [λzELs + (1 − λ)EH s] + (zλ + 1 − λ) vt+1 if vt+1 < v , where ζE s − E s v∗ ≡ H L 1 − ζ

 zφm , if v ≥ v∗, ∂ m ∗  t+1 t+1 m VB,t+1(qt+1; φt+1, Jt+1 (vt+1)) = ∂qt+1  m ∗ (zλ + 1 − λ) φt+1 if vt+1 < v ,

∗ ∗ The value function is linear when vt+1 is below or above v . It increases discontinuously at vt+1 = v ,

m ∗ ∗+ m ∗ ∗− VB,t+1(qt+1; φt+1, Jt+1 v ) − Vo,t+1(qt+1; φt+1, Jt+1 v )

∗ =(z − 1)(1 − λ)(v + EH s)

m m 1 Because βz < 1, and φt+1/φt+1 < βz , the objective function for investors portfolio choice at the end m ∗ a a m m m ∗ of period t, βVB,t+1(qt+1; φt+1, Jt+1 (vt+1)) − qt+1φt − qt+1φt , is decreasing in qt+1 when vt+1 < v or ∗ m a vt+1 > v . So, there are two local optimum of the objective function, one at qt+1 = 0, vt+1 = δφt+1, the

30 ∗ other at vt+1 = v . [Impose conditions for the adverse selection problem to be strong enough so that a ∗ φt < v ] At the local optimal where the agent does not hold cash,

a a −φt + β (zλ + 1 − λ) δφt+1 + λzELs + (1 − λ)EH s

At the second local optimal where the agent holds cash

m m a ∗ a φt m ∗ ∗ a ∗ a φt ∗ −φt − v − δφt+1 m + βVB,t+1(qt+1; φt+1, Jt+1 (v )) = −φt − v − δφt+1 m + βz [v + λELs + (1 − λ)EH s] φt+1 φt+1 The condition for an investor to have incentive to hold money is that

m a ∗ a φt ∗ a a −φt − v − δφt+1 m + βz [v + λELs + (1 − λ)EH s] ≥ −φt + β (zλ + 1 − λ) δφt+1 + λzELs + (1 − λ)EH s φt+1 ∗ a m zv − (zλ + 1 − λ) δφt+1 + (z − 1)(1 − λ)EH s φt ∗ a ≥ m v − δφt+1 βφt+1 For pooling to be an equilibrium, investors must find it optimal to bring sufficient amount of real balance per asset. In this case, the cost of holding the portfolio of money and assets equals to the benefit and the value of the portfolio is v∗.

m m ∗ a qt+1φt+1 = v − δφt+1 m a m m a ∗ a φt ∗ φ + qt+1φt = φt + v − δφt+1 m = βδz [v + λELs + (1 − λ)EH s] φt+1 m m In the stationary monetary equilibrium φt /φt+1 = µ where investors pool money with the asset with probability one,

βδz [λE s + (1 − λ)E s] − (µ − βδz)v∗ φa = L H 1 − µδ βδz [λE s + (1 − λ)E s + v∗] − µv∗ = L H 1 − µδ βz [λE s + (1 − λ)E s] − (µ − βz)v∗ qm φm = v∗ − δ L H t+1 t+1 1 − µδ v∗ − µδv∗ − δβz [λE s + (1 − λ)E s] + δ(µ − βz)v∗ = L H 1 − µδ −δβz [λE s + (1 − λ)E s] + (1 − δβz)v∗ = L H 1 − µδ v∗ − δβz [λE s + (1 − λ)E s + v∗] = L H 1 − µδ and

∗ a µ zv − (zλ + 1 − λ) δφt+1 + (z − 1)(1 − λ)EH s ≤ ∗ a β v − δφt+1

31 which can be simpliﬁed as

β {z + (z − 1)(1 − λ)(−1 + Q)} µ ≤ µ = 1 + βδ(z − 1)(1 − λ)Q where

∗ EH s + v Q = ∗ ∗ v − δβz [λELs + (1 − λ)EH s + v ] λz z−1 (EH s − ELs) = λz −EH s + (1 − δβ) z−1 (EH s − ELs) To check whether the inﬂation rate can be positive, we look at the limit of δ = 0. At the limit,

(z − 1) (1 − λ)(E s + v∗) µ = β z − (z − 1)(1 − λ) + H v∗ (z − 1) (1 − λ) ((1 − ζ)E s + ζE s − E s) = β z − (z − 1)(1 − λ) + H H L ζEH s − ELs (z − 1) (1 − λ)(E s − E s) = β z − (z − 1)(1 − λ) + H L ζEH s − ELs ( ) (z − 1) (1 − λ) = β z − (z − 1)(1 − λ) + 1 − z−1 EH s λz EH s−ELs

( z−1 EH s ) = β z + (z − 1) (1 − λ) λz EH s−ELs 1 − z−1 EH s λz EH s−ELs For inﬂation rate to be positive,

z − 1 E s 1 − βz z − 1 E s (z − 1) (1 − λ) H > 1 − H λz EH s − ELs β λz EH s − ELs z − 1 E s 1 − βz H > λz EH s − ELs 1 − βz + β(z − 1) (1 − λ) Notice that λz v∗ = −E s + (E s − E s) H z − 1 H L For v∗ to be positive, z − 1 E s 1 > H . λz EH s − ELs This shows that µ¯ could be positive. At µ = µ+,

βz [λE s + (1 − λ)E s] − (µ − βz)v∗ φa = L H 1 − µδ β [(1 − λ)E s + λzE s] = H L 1 − βδ(λz + 1 − λ)

32 In a stationary equilibrium where money is not used in equilibrium and the asset is traded only if it is of low quality, β [λzE s + (1 − λ)E s] φa = L H , S 1 − βδ(λz + 1 − λ) and m a ∗ a φt ∗ a a −φt − v − δφt+1 m +βz [v + λELs + (1 − λ)EH s] < −φt +β (zλ + 1 − λ) δφt+1 + λzELs + (1 − λ)EH s φt+1 This means that ∗ a µ zv − (zλ + 1 − λ) δφS + (z − 1)(1 − λ)EH s ≥ µ = ∗ a = µ. β v − δφS The RHS of the inequality is increasing in φa. The condition reﬂects the fact that money is not held in equilibrium.

A.2 Proof of Proposition

Suppose the designer anticipates that the liquid security will be liquid as long as ELy/EH y ≥ ζ. In this case, the prices of equity and liquid debt are

Z sH a qE = qt FeL(s)ds, D " Z D Z D # a qD = v + qt sL + λ FeL(s)ds + (1 − λ) FeH (s)ds sL sL where D satisﬁes the participation constraint of the high quality seller in the debt market: " Z D Z D !# Z D a a h i (z − 1) v + qt sL + λ FeL(s)ds + (1 − λ) FeH (s)ds = λqt FeH (s) − FeL(s) ds. sL sL sL So,

∂D 1 = λz h i ∂v qa=1 ˜ ˜ ˜ t z−1 FH (D) − FL(D) − FH (D)

Z sH a ∗ a VB,t+1(qt+1, Jt+1 (v)) = z(qD + λqE) + (1 − λ)qt+1 FeH (s)ds D " !# Z sH Z D Z sH a a = z v + qt+1 sL + λ FeL(s)ds + (1 − λ) FeH (s)ds + qt+1(1 − λ) FeH (s)ds sL sL D

∂VB,t+1 = z ∂vt+1 a qt+1=1

∂VB,t+1 ˜ = (1 − λ)(z − 1)FH (D) ∂D a qt+1=1

∂ a ∗ m VB,t+1(qt+1, Jt+1 (vt+1)) ∂qt+1 a qt+1=1 " # ∂vt+1 ∂VB,t+1 ∂VB,t+1 ∂D = m + ∂q ∂vt+1 a ∂D a ∂vt+1 t+1 qt+1=1 qt+1=1   (1−λ)(z−1) ˜ m z FH (D) = φt+1z 1 + h i  λz ˜ ˜ ˜ z−1 FH (D) − FL(D) − FH (D)   ˜ m (1 − λ)λ(1 − ζ)FH (D) = φt+1z 1 + h i  λz ˜ ˜ ˜ z−1 FH (D) − FL(D) − FH (D)

2 ∂ a ∗ VB,t+1(qt+1, Jt+1 (vt+1)) m 2 ∂ qt+1 a qt+1=1 n h i o λz ˜ ˜ ˜ −(1 − λ)λ(1 − ζ)fH (D) z−1 FH (D) − FL(D) − FH (D) n o ˜ λz +(1 − λ)λ(1 − ζ)FH (D) z−1 [fH (D) − fL(D)] − fH (D) ∂D = φm z t+1 n h i o2 m λz ˜ ˜ ˜ ∂qt+1 z−1 FH (D) − FL(D) − FH (D) n h i o λz ˜ ˜ ˜ −fH (D) z−1 FH (D) − FL(D) − FH (D) n o ˜ λz +FH (D) z−1 [fH (D) − fL(D)] − fH (D) ∂D = φm z(1 − λ)λ(1 − ζ) t+1 n h i o2 m λz ˜ ˜ ˜ ∂qt+1 z−1 FH (D) − FL(D) − FH (D) n o λz ˜ ˜ λz λz z−1 FL(D)fH (D) + FH (D) z−1 [fH (D) − fL(D)] − z−1 fH (D) ∂D = φm z(1 − λ)λ(1 − ζ) t+1 n h i o2 m λz ˜ ˜ ˜ ∂qt+1 z−1 FH (D) − FL(D) − FH (D)

fH (D) fL(D) ˜ − ˜ ∂D = φm z(1 − λ)λF˜ (D)F˜ (D) FH (D) FL(D) t+1 L H n h i o2 m λz ˜ ˜ ˜ ∂qt+1 z−1 FH (D) − FL(D) − FH (D)

< 0

a a m m where vt+1 = qt+1φt+1 + qt+1φt+1. The value of the whole portfolio ( " !# ) Z sH Z D Z sH m m a a a φt qt+1+φt = βιM+φ = β z Z + δφ + sL + λ FeL(s)ds + (1 − λ) FeH (s)ds + (1 − λ) FeH (s)ds sL sL D

34 n h R sH R D i R sH o β z sL + λ s FeL(s)ds + (1 − λ) s FeH (s)ds + (1 − λ) D FeH (s)ds − β(ρ − z)M φa = L L 1 − βδz

n R sH R D R sH o βδ z sL + λ s FeL(s)ds + (1 − λ) s FeH (s)ds + (1 − λ) D FeH (s)ds + [1 − βδz + βδ(z − ρ)] M δφa + M = L L 1 − βδz

n R sH R D R sH o βδ z sL + λ s FeL(s)ds + (1 − λ) s FeH (s)ds + (1 − λ) D FeH (s)ds + (1 − βδρ)M = L L 1 − βδz (12)

The Euler equation for money is

m ˜ φt (1 − λ)(z − 1)FH (D) ι = m = z + h i βφt+1 λz ˜ ˜ ˜ z−1 FH (D) − FL(D) − FH (D)

= z + g(D)

ι − z = g(D) (13)

From the participation constraint of high quality sellers,

" Z D Z D !# Z D a h i (z − 1) M + δφ + sL + λ FeL(s)ds + (1 − λ) FeH (s)ds = λ FeH (s) − FeL(s) ds. sL sL sL

Z D Z D Z D ! a λ h i δφ + M = FeH (s) − FeL(s) ds − sL + λ FeL(s)ds + (1 − λ) FeH (s)ds z − 1 sL sL sL

n R sH R D R sH o βδ z sL + λ s FeL(s)ds + (1 − λ) s FeH (s)ds + (1 − λ) D FeH (s)ds + (1 − βδρ)M δφa + M = L L 1 − βδz ! λ Z D h i Z D Z D = FeH (s) − FeL(s) ds − sL + λ FeL(s)ds + (1 − λ) FeH (s)ds (14) z − 1 sL sL sL

Combining the information in (12) and (14), we have ! λ Z D h i Z D Z D (1 − βδι)M = (1 − βδz) FeH (s) − FeL(s) ds − (1 − βδz) sL + λ FeL(s)ds + (1 − λ) FeH (s)ds z − 1 sL sL sL ! Z sH Z D Z sH − βδz sL + λ FeL(s)ds + (1 − λ) FeH (s)ds − βδ(1 − λ) FeH (s)ds sL sL D ! λ Z D h i Z D Z D = (1 − βδz) FeH (s) − FeL(s) ds − sL + λ FeL(s)ds + (1 − λ) FeH (s)ds z − 1 sL sL sL Z sH − βδ(1 − λ) FeH (s)ds D

35 and

n h R sH R D i R sH o β z sL + λ s FeL(s)ds + (1 − λ) s FeH (s)ds + (1 − λ) D FeH (s)ds − β(ρ − z)M φa = L L . 1 − βδz