Inefficient Liquidity Creation

Transcription

1 Inefficient Liquidity Creation Stephan Luck Paul Schempp February 13, 2018 We present a model in which banks can create liquidity by issuing safe debt claims. Safe debt claims can be generated either by holding assets to maturity and issuing risk-absorbing equity claims, or by fire-selling risky assets in downturns. We provide the novel insight that even though fire sales are always excessive, liquidity creation is not necessarily too high, but can also be inefficiently low. This result stems from the fact that fire sales increase liquidity creation only if the equilibrium fire-sale price is sufficiently high, but start to destroy liquidity once the fire sale price is too low. The finding implies that the constrained-efficient allocation can only be implemented by capital or liquidity regulation if liquidity creation is excessive, but not if it is too low. We further show that the model can be interpreted as an economy in which traditional banking and shadow banking coexist. The shadow banking sector necessarily grows too large, and optimal regulation may require a subsidy to bank equity issuance. Key works: banking; bank regulation; liquidity creation; fire sales; pecuniary externalities; shadow banking; regulatory arbitrage JEL codes: G21, G23, G28 Financial support by the Alexander von Humboldt Foundation and the Max Planck Society is gratefully acknowledged. The views presented are those of the authors and not of the Federal Reserve. stephan.luck@frb.gov, Federal Reserve Board. pschempp@uni-koeln.de, University of Cologne and Max Planck Institute for Research on Collective Goods, Bonn

2 1. Introduction After the financial crisis, policymakers as well as academics have held fierce debates on the role of banks in the economy. A particular focus has been on the role of bank debt, which is often argued to create liquidity as it fulfills a number of economic functions. 1 Importantly, any debate on the optimality of liquidity creation has automatically important implications for banking regulation: On the one hand, if one takes the view that bank liquidity creation is excessive as maturities of bank liabilities may tend to be too short, and leverage may be too high, 2 it is natural to be in favor of stricter capital and liquidity requirements to enhance financial stability. 3 On the other hand, if one emphasizes the virtues of short-term debt such as its value as a transaction medium, 4 or its ability to impose discipline on intermediaries 5, one tends to be skeptical about the effects of regulating banks financing decisions or even find it harmful. In this paper, we ask two questions: First, how and how much liquidity is optimally created by banks? And, second, how is liquidity creation optimally regulated? In our model, safe bank debt is associated with a liquidity benefit and generates utility beyond its guaranteed payoff. Banks can create such safe debt claims in two ways: The first option is to hold assets until maturity and issue a sufficient amount of risk-absorbing claims, such as equity (compare, e.g., Hellwig, 2015a). We refer to this type of banking as held-to-maturity banking or traditional banking. The second option is to sell off risky assets to outside investors upon the realization of adverse macroeconomic news (as, e.g., in Stein, 2012). We refer to this type of banking as market-based banking. Importantly, when purchasing banks assets in a fire sale, outside investors forgo other profitable 1 Bank debt claims may generate utility that does not stem from its expected payoff as it, for instance, allows insure agents against idiosyncratic liquidity risk, provides a safe asset to smooth consumption, and facilitates transactions and payments. 2 On maturities see, e.g., Brunnermeier and Oehmke (2013) and He and Milbradt (2016) and on leverage see, e.g., Admati et al. (2017). 3 See in particular Admati and Hellwig (2013). 4 See, e.g., Gorton and Pennacchi (1990), Dang et al. (2015a,b), and DeAngelo and Stulz (2015). 5 See, e.g., Calomiris and Kahn (1991), Diamond and Rajan (2000), as well as the Squam Lake Report (2010). 1

3 investment opportunities. Hence, fire sales are not merely redistributive (as, e.g., in Gale and Yorulmazer, 2017), but entail social costs. The key friction in our model results from banks inability to write state-contingent contracts with outside investors (e.g., as in Holmström and Tirole 1998, 2011). 6 This gives rise to a pecuniary externality, which induces a wedge between the constrainedefficient allocation and the equilibrium outcome. In particular, in line with the literature on collateral constraints and fire sales (Lorenzoni, 2008; Stein, 2012; Dávila and Korinek, 2017), we find that the degree to which extent banks rely on market-based banking and the resulting fire-sale effects are always excessive in equilibrium. In contrast to the existing literature, however, we argue that even though fire sales are always excessive, liquidity creation and leverage can also be too low. This results from a non-monotonic relationship between fire sales and aggregate safe debt creation. Banks can potentially sustain a higher level of safe debt ex-ante if they sell their risky assets in exchange for safe assets (e.g., cash) upon the arrival of pessimistic macroeconomic news ex-interim. However, the extent to which liquidity can be generated by fire sales crucially depends on the price at which risky claims are exchanged against safe claims. If only few assets are sold and the fire-sale price is sufficiently high, selling risky assets in exchange for a safe assets indeed allows banks to sustain a higher level of safe debt ex ante. However, if the extent of fire sales is relatively high, fire-sale prices may decrease to a point where further fire sales start to destroy aggregate liquidity and decrease the amount of safe debt that can be sustained ex ante. Importantly, both of these scenarios can prevail in equilibrium. In both cases, there is too much reliance on market-based banking compared to the constrained efficient. However, while in the former case there is excessive liquidity creation, there is too little liquidity creation in the latter. Given that too many assets are being sold in equilibrium, the straightforward regula- 6 The friction that agents cannot contract upon future financing also occurs in a banking context in Uhlig (2010) and Bolton et al. (2011). It also arises naturally in banking models in OLG environments, as in Qi (1994) and Martin et al. (2014a,b). 2

4 tory policy is to impose a tax on asset sales in order to implement the constrained-efficient allocation. However, there may be reasons to believe that such a policy, even though attractive in theory, may not be available in practice. Therefore, we investigate how the constrained-efficient allocation can be implemented by using standard tools of banking regulation such as minimum capital requirements or liquidity regulation. Here, our main finding has important implications for optimal bank regulation. We show that if liquidity creation is excessive in the competitive equilibrium, the constrainedefficient allocation can be implemented by a standard capital requirement or liquidity regulation. In contrast, if there is too little liquidity creation in equilibrium, there is no standard banking regulation that implements the constrained-efficient allocation. In this case, leverage is not excessive to begin with and the optimal minimum capital requirement or liquidity regulation will not be binding. Hence, capital requirements as well as liquidity regulation are no panacea when fire sales are excessive. While in our main analysis, we analyze banks that conduct both types of banking within one institution, the model has a natural interpretation as an economy in which two types of intermediaries coexist: On the one hand, there are traditional banks that rely on issuing equity to create safe debt claims, hold assets to maturity, and do not conduct any fire sales in downturns. On the other hand, there are shadow banks that rely on market-based liquidity creation, i.e., they generate safe debt by selling risky assets after the arrival of pessimistic news. Importantly, the two different types of intermediation coexist even in absence of any type of regulation. Hence, our model offers an explanation for the existence of shadow banking that is not based on regulatory arbitrage, as is often done in the theoretical literature on shadow banking (see, e.g., Luck and Schempp, 2014, Hanson et al., 2015, Plantin, 2015). When interpreting the model as a two-sectors model, the main finding can be rephrased as follows: If each single intermediary can decide between operating in one of the two sectors, the shadow banking sector grows to a size that it induces excessive fire sales in the 3

5 aggregate. Hence, the same notion of why a price-sensitive collateral constraint leads to excessive fire sales in the model with identical unregulated intermediaries carries through to the sectoral model: excessive fire sales at the bank-level translate into excessive shadow banking. Still, the aggregate liquidity creation can be both, too high or too low. We further show that if we also allow for regulatory arbitrage, 7 the constrained efficient can again only be attained if liquidity creation if excessive, but not if it is too low. Liquidity regulation or a capital requirement, however, not effective as intermediaries circumvent the regulatory requirements by migrating to the unregulated shadow banking sector. However, we argue that because the regulator cannot make the market-based liquidity creation of shadow banks more costly, a macroprudential regulatory authority should make low leverage cheaper by subsidizing the traditional, held-to-maturity banking model. In this case, the optimal policy is a Pigouvian subsidy to regulated held-to-maturity banking (or a direct subsidy for the issuance of bank equity), which reduces the incentives for excessive liquidity creation in the shadow banking sector. That way, the composition of the financial system and therewith the aggregate leverage can be controlled effectively. We proceed as follows: After briefly reviewing the related literature, Section 2 describes the setup of the baseline model. Section 3 describes the mechanics of the economy, derives the laissez-faire equilibrium and compares it to the constrained-efficient allocation. Section 4 shows under which conditions a social planner can or cannot implement the socially optimal allocation through macroprudential policy tools. Finally, Section 5 shows how the model can be interpreted as a model of traditional banks and shadow banks and discusses implications for regulation. 7 In our model, regulatory arbitrage is defined as the costless circumvention of a regulatory requirement. 4

6 1.1. Related Literature The banking literature has provided several arguments for why bank liabilities are different than liabilities from other corporations. In classic arguments, banks insure agents against idiosyncratic liquidity shocks by offering demand deposit contracts (Diamond and Dybvig, 1983) and credit lines (Holmström and Tirole, 1998), they provide claims that are immune to adverse selection and thereby facilitate trade among agents with liquidity needs (Gorton and Pennacchi, 1990; Dang et al., 2013, 2017), their debt claims may be informational undemanding and hence useful to minimize monitoring costs (Diamond, 1984), or they issue claims that are particularly valuable in the presence of search frictions (Gu et al., 2013; Hollifield and Zetlin-Jones, 2016). Like other recent papers (Van den Heuvel, 2008, 2016; Stein, 2012; Hanson et al., 2015; DeAngelo and Stulz, 2015; Moreira and Savov, 2016; Hellwig, 2015b), our analysis starts out by assuming that banks safe (short-term) debt unlike other bank liabilities is associated with a liquidity benefit. Our paper makes then two main contributions. First, we contribute to the literature that analysis how pecuniary externalities give rise to inefficiencies in models with collateral constraint. Our model features excessive fire sales, but liquidity creation and leverage is not necessarily too high, but can also be too low. This in turn has important implication for optimal banking regulation. Second, we contribute to the theoretical literature on the coexistence of traditional banking and shadow banking. In existing papers, the existence of shadow banks typically stems from regulatory arbitrage. In our model, the shadow banking sector arises endogenously and grows too large, even in absence of any regulatory interventions. Moreover, we show that the effects of regulatory arbitrage can be potentially mitigated by subsidizing the regulated banking system. In its basic structure, our model s setup is similar to Stein (2012): intermediaries can invest households funds in risky assets and generate liquidity benefits when issuing safe debt. In the model proposed by Stein, intermediaries can create safe debt only 5

7 via market-based banking, i.e., by selling assets in downturns. In contrast, our model also allows to create safe debt by holding assets until maturity and issuing equity. This key difference in our model stems from the assumption that the return of the banks technology in the lowest possible state is larger than zero. We argue that this assumption is almost always true for banks that are able to diversify idiosyncratic risks of its borrowers. Importantly, this assumption gives risk-absorbing claims next to assets sales the potential to create safe debt. Thus, the conclusion of our model deviates from the previous literature: While fire-sale effects and liquidation are unambiguously excessive, as e.g., in (Lorenzoni, 2008; Dávila and Korinek, 2017), liquidity creation and leverage are not necessarily excessive, but can also be too low. The fact that the amount of liquidity created can be too low in equilibrium has important implications for the efficacy of classical tool of banking regulation. In models that feature excessive liquidity creation (Stein, 2012) or excessive leverage (Lorenzoni, 2008; Dávila and Korinek, 2017), the constrained-efficient allocation can typically be implemented by imposing a capital requirement or by using liquidity regulation. However, we show that whenever liquidity creation and leverage are too low, such policies are not viable to implement the constrained-efficient allocation. Given that our model has a natural application as a model in which traditional banks and shadow bank coexist, our analysis related to number on papers that model the coexistence of traditional banking and shadow banking (e.g., Luck and Schempp (2014), Plantin (2015)). It is most closely related to Hanson et al. (2015), who use the setup proposed by Stein and contrast the comparative advantages of regulated banks and shadow banks with respect to their assets holdings and financing structures. There are two key differences in our approach. First, our model does not require any assumption on regulatory interventions to generate the coexistence of held-to-maturity banking and market-based banking. Second, our model contains a welfare analysis, and we show that 6

8 the shadow banking sector always grows too large from a social planner s point of view. Again, these difference have important implications for regulation. In particular, given that the shadow banking sector grows too large, we argue that the issuance of bank equity should be subsidized if liquidity creation is too high. This way, the social planner can control the composition of the financial system and maximize welfare. Given that our notion of shadow banking is one of market-based liquidity creation, we are also closely related to Bolton et al. (2011) and Gennaioli et al. (2013). In particular, Bolton et al. (2011) has a similar notion of liquidity creation by relying on asset sales. Our concept of market-based liquidity creation has similarities to their concept of outside liquidity. Their analysis, in contrast, focuses on asymmetric information: The fear of adverse selection can lead to excessive early trading and too little reliance on outside liquidity. They conclude that an optimal policy is to intervene in the asset market in order to stabilize prices, thereby overcoming the adverse selection issue. Finally, on a broader level, the spirit of our model is similar to recent papers in the macro-finance literature like Moreira and Savov (2016) and Begenau and Landvoigt (2016) which present quantitative models of shadow banking within a general-equilibrium context. Similar to our setup, the optimal regulatory regimes are determined by the fact that shadow banks are more exposed to runs and fire-sale dynamics in downturns compared to regulated banks. 2. Setup The basic setup is similar to the models by Lorenzoni (2008) and in particular Stein (2012) and Hanson et al. (2015). The economy goes through a sequence of three dates, t = 0, 1, 2. There is one good which can be used for investment and consumption. The economy is populated by three types of agents: households, intermediaries, and outside investors. We begin by describing the agents preferences, endowments and technologies. 7

9 Households are endowed with one unit of the good at t = 0. They maximize their expected lifetime consumption, given by U = c 0 + E[c 1 + c 2 ] + γd, where D denotes the amount of risk-free debt a household holds at t = 0, and γd denotes the liquidity benefits that households derive from holding safe claims. This implies that the required gross interest rate on risk-free debt claims is given by r = 1/(1+γ), whereas the required return on risky asset is given by 1. Next to households, there are intermediaries that have no goods endowment, but have access to a project (early production technology) that requires an investment of one unit. Households cannot invest in this technology directly, but can only hold claims issued by intermediaries. Intermediaries hence invest on behalf of the households. By choosing their capital structure (debt and equity), they are able to create safe claims that generate the households liquidity benefits. Intermediaries are risk neutral, maximize their expected final period consumption c I 2, and have full bargaining power vis-à-vis households. The technology of intermediaries is specified as follows: It has a fixed scale and requires the investment of one unit. There are two states of the world, high and low, s {H, L}. 8 In the high state, the intermediaries investment has a date-2 payoff of R H > 1; in the low state, it is given by R L < 1. The ex-ante probability of s = H is given by π + (1 π)θ, and s = L with (1 π)(1 θ). While the state realizes at date t = 2, news about the state of the world arrives in the form of a public signal date t = 1. With probability π, optimistic news arrive, and given this signal, the high state H will realize with certainty, i.e., the posterior of the high state is 1. With probability 1 π, pessimistic news arrive, and given this signal, the conditional probability of s = H is given by θ, and s = L with 1 θ. We define R 1 (θ) = R L + θ(r H R L ) as the expected payoff given the arrival of pessimistic news 8 The models of Stein (2012) and Hanson et al. (2015) assume that there exists a disaster state as a third state with a zero payoff. This implies that liquidity can only be created by selling assets Stein (2012), or by liabilities being insured by a deposit insurance in the disaster state. 8

10 ex-interim, and R 0 = πr H + (1 π)r 1 (θ) > 1 as the ex-ante expected payoff. Importantly, when pessimistic news arrive, intermediaries can sell their risky assets to outside investors, as specified below. Selling the risky assets makes sure that the cash flow obtained is larger than the lowest possible cash flow, thereby increasing the amount of safe debt that can be issued ex-ante. We will refer to generating liquidity benefits by selling assets after the arrival of pessimistic news as market-based banking or market-based liquidity creation. t = 0 t = 1 t = 2 π optimistic news R H > 1 good state θ 1 π pessimistic news 1 θ R L > 0 bad state Figure 1: Timing and payoffs of the early production technology. In general, intermediaries could raise funds from households by issuing a whole range of different types of claims. However, for the purpose of our model, the only important difference lies between safe debt claims and risky, residual claims. We will refer to the residual claims as equity. However, note that any other junior claim can be used to create safe claims, e.g., subordinated debt. Moreover, for the purpose of our model, it is not necessary to specify whether debt is short-term or long-term. 9 Because the gross interest rate on safe debt is given by r = 1/(1+γ), the intermediary can raise an amount d = D/r = (1 + γ)d through issuing safe debt with a face value D, while raising an amount of equity of e = 1 (1 + γ)d to absorb the losses in the low 9 In this model debt need not to be short-term to generate liquidity benefits. However, introducing further friction, e.g., an inability of intermediaries to commit to selling assets in the interim period, would induce the existence of short-term debt contracts. 9

11 state. In order to generate safe debt claims with a fixed repayment amount D, an intermediary must ensure that the cash flow in the low state amounts to at least D, i.e., intermediaries are subject to a collateral constraint. As long as D < R L, an intermediaries assets payoff in the low state exceeds the outstanding debt. Observe, however, that this can never be optimal. As long as the collateral constraint is non-binding, the assets payoff can be used to generate more safe debt, and hence intermediaries could create more valuable liquidity benefits at no cost. In contrast, an amount of safe debt D > R L is only feasible if the intermediary sells some fraction of its assets after the arrival of pessimistic news, i.e., conducts market-based banking. We denote this fraction of the asset portfolio that an intermediary sells by η [0, 1]. The remaining safe debt creation that does not come from assets sales is referred to as held-to-maturity banking. As indicated, assets can be bought by outside investors. A unit mass of such late investors is born at t = Note that the key friction of the model is that outside investors are not born before t = 1, and hence they cannot contract with intermediaries in t = 0, which is equivalent a limited commitment friction as in Holmström and Tirole (1998). This assumption can be summarized as: intermediaries cannot contract their future financing conditions in the initial period. In the absence of this friction, state-contingent contracts available at t = 0 would allow to implement the first-best allocation. However, we will show that in the presence of this friction, the laissez-faire equilibrium does not coincide with the constrained efficient allocation. Outside investors are risk neutral and maximize their date-2 payoff. Each investor is endowed with W units of the good and has access to a late production technology. The late production technology is specified as follows: an investment of k at t = 1 yields 10 Outside investor can either be interpreted as intermediaries that are not subject to the macroeconomic risk, or they can be interpreted as a different type of institutions that tend to buy bank assets in downturns such as insurance companies or pension funds. 10

12 an output of g(k) at t = 2 with certainty, where g( ) is a weakly concave function with g (W ) 1. Hence, investors need to decide how much of their endowment they use to operate their technology, and how much they use to purchase assets from intermediaries. 3. Equilibrium Analysis We start out with analyzing the asset market equilibrium at t = 0 and discuss the two different ways of creating liquidity creation in more depth before turning towards the analysis of the laissez-faire equilibrium and the constrained-efficient allocation Asset-Market Equilibrium at t = 1 Let M denote the amount of funds used by outside investors to buy assets when pessimistic news arrive. 11 The investment in the late technology in this case is thus given by k = W M. Let q denote the per-unit price of the intermediaries asset. Given some price q, spending one unit to buy assets gives an expected return of R 1 /q. The outside investor s profit for a given price q is thus given by Π outside (M) = g(w M) + M R 1 q. There are two extreme cases to consider. First, outside investors choose not to buy any assets. This happens only if the late technology s marginal productivity is high relative to returns that can be made in the asset market, i.e., M = 0 if g (W ) > R 1 /q. Second, at the opposite extreme, outside investors choose to use their entire endowment for asset purchases when asset returns are high and marginal productivity low. I..e, M = W if g (0) < R 1 /q. In case both of the above conditions are violated, outside investors prefer to purchase some assets, i.e., M (0, W ), and the price q must satisfy the first-order 11 Assets are never traded after the arrival of optimistic news and hence we can focus on the asset market after the arrival pessimistic news. 11

13 condition: q = R 1 g (W M). (1) This expression characterizes the optimal portfolio decision of outside investors, i.e., how much of her funds are used for asset purchases and how much for real investment at a given equilibrium fire-sale price. Recall that we denote the fraction of assets that are being sold by η. Market clearing for the asset market requires that supply equals demand, i.e., M = ηq. In the case of M < W, combining the outside investors FOC and the asset-market clearing condition gives rise to the following equation: η(m) = Mg (W M) R 1, (2) which characterizes a unique amount of funds used to purchase asset for a given amount of assets sold. In contrast, if M = W, the amount of assets sold is not uniquely determined. Using η(w ) as defined by (2), let η := η(w ) denote the amount of asset liquidation starting at which the entire budget of outside investors is being used for asset purchases. Then, any combination of M = W and η [ η, 1] constitutes an asset market equilibrium. Moreover, note that for η > η, assets are priced via cash-in-the-market pricing as in Allen and Gale (1994). Therefore, a more general characterization of asset prices as a 12

14 function of the amount of assets sold is given by R 1 g q(η) = (W M(η)) if η η W η if η > η. (3) I.e., for η η, assets are priced by the marginal productivity of outside investors. Assuming g ( ) > 1 implies that outside investors investment is generally profitable. This in turn implies that q can be interpreted as a fire sale price, because assets are always traded at a price below their fundamental value, i.e., their expected future payoff. However, if sufficiently many assets are sold such that all endowment of outside investors is being used for asset purchases, η > η, the market clears at the cash-in-the-market price. 12 Note that, in our setup, the inefficiency that may arise from fire sales does not result from outside investors having a lower utility for intermediaries assets (e.g., as in Shleifer and Vishny, 1992), but from a reduced investment in their productive technology (see, e.g., Diamond and Rajan, 2011; Shleifer and Vishny, 2011, as well as Stein, 2012). Moreover, this automatically implies that in our model, fire sales are not entirely redistributive through cash-in-the-market pricing, as, e.g., in Gale and Yorulmazer (2017). Hence, M is a crucial variable in this model as it describes the amount of liquid funds that outside investors devote to purchasing assets from intermediaries in a fire sale instead of operating their productive technology. Increasing M has two opposing effects on welfare: On the one hand, it reduces the profitable late investment after the arrival of pessimistic news. On the other hand, it increases the intermediaries scope to generate safe debt and liquidity benefits by increasing the intermediaries cash-flow after pessimistic news. 12 A necessary condition for fire sale pricing to be possible is a violation of the Inada condition for the outside investors production technology. This assumption is not necessary for any of our results, but it does help to build intuition for the main findings. 13

15 3.2. Liquidity Creation As described above, the model features a direct link between the amount of safe debt issued D and the amount of assets sold η. In particular, η is a measure of the reliance on market-based liquidity creation relative to held-to-maturity banking. For the remaining analysis of the model, it is helpful to formalize this relationship. The degree of marketbased liquidity creation determines the amount of an intermediary s certain cash flows in t = 2, i.e., the amount of funds the intermediary has available in the low state, when selling assets after the arrival of pessimistic news. For a given bank-specific level of market reliance η i, the safe payoff bank i generates is given by d i (η i q) = (1 η i )R L }{{} + η i q }{{}, (4) held-to-maturity market-based i.e., the sum of the payoff from those assets that are being held until maturity and the proceeds from assets sales in the interim period. Observe that for a given price q = R 1 /g (W M) > R L, the safe payoff always lies in the interval [R L, q]. Given that issuing safe debt claims yields a liquidity benefit, safe debt is a cheaper source of funding than equity. Therefore, at a given liquidation policy, intermediaries find it optimal to issue the maximum amount of safe debt and the payoff the residual equity claims in the low state is zero. Given individual liquidation policies η i, the aggregate amount of assets sold is given by η = 1 0 η i. Accordingly, the aggregate level of safe debt and liquidity created can be written as follows: [ D(η) = (1 η)r L + min η R 1 g (W M(η)), W ]. (5) To understand the main mechanism of our model, it is helpful to contrast the private incentives to conduct market-based liquidity creation against the social trade-off. Notice 14

16 that from the perspective of an individual bank i that takes the price as given, the individual level of safe debt strictly increases in the liquidation policy η, the derivative being d i = q R L. In contrast, the derivative of the aggregate level of safe debt with respect to η, D (η), is strictly smaller and can even become negative when sufficiently many assets are being sold, i.e., for sufficiently large values of η. In particular, the elasticity of funds provided by outside investor, M, with respect to the fire sale price, q, can converge to zero. This mechanism is not excluded to, but particularly evident when there is cash-inthe-market pricing. For η > η, it holds that D (η) = R L < 0. Any further unit of assets sold will only further decrease the price. However, the budget used to purchase the assets and hence the liquidity generated from asset sales is constant. 13 Moreover, at the same time, the bank is giving up an asset with a safe payoff of R L. Hence, any additional unit sold decreases the overall liquidity generated by the banking sector. We refer to this as marginal liquidity destruction. To further strengthen the intuition for this effect, assume that the marginal productivity of the outside investors is sufficiently low such that asset pricing is governed cash-in-the-market pricing before all assets are sold. Formally, assume g (0) < and assume that η < 1. In this case, any additional unit sold decreases the aggregate amount of liquidity generated: Banks sell more assets, but since there assets are already priced via cash-in-the-market pricing, the amount of funds transferred, ηq = ηw/η = W is constant (see Appendix for an analysis when g is linear) 14 Notice that while the mechanism is particularly evident for the case of cash-in-themarket pricing, cash-in-the-market pricing is not a necessary condition. While D (0) > 0 13 This also implies that (5) is not necessarily invertible and a function η(d) that tells us which liquidation policy is necessary in order to support an amount of safe debt D does not necessarily exist. The reason is that there can exist some levels of safe debt D that can be supported two different liquidation amounts. 14 While in the case of g (0) = this literal cash-in-the-market pricing does not occur, the effect in this situation is very similar if g is large because in this situation, a change in η has a small effect on the revenue M, but a large effect on the price q. 15

17 holds by assumption (R 1 /g (W ) > R L ), implying that optimal liquidity creation is associated with some level of asset sales, we can very well get D (η) < 0 for some η < η. Finally, observe that a necessary condition for the existence of liquidity destruction is to assume that R L > 0. In contrast, in the case in which the banks assets generate no return in the low state, i.e. R L = 0, it always holds that D (η) 0. This is the crucial difference to the model proposed by Stein. Assuming R L = 0 implies automatically that assets that remain on the balance sheet after bad news has arrived do not generate any liquidity benefits. Therefore, increasing asset sales can never destroy any liquidity. 15 We argue, however, that it is unlikely for a diversified bank to ever find itself in a state in which its assets are totally worthless. As will become clear below, this has important implications Laissez-Faire Equilibrium In order to derive the laissez-faire equilibrium of the economy, we start out with characterizing the maximization problem of intermediaries. A single intermediary s problem can be formalized as follows: Taking as given the asset purchase budget M and thus the asset price q, an intermediary maximizes her profit by choosing the amount of safe debt d i and the degree of market reliance η. The profit is thus given by [ Π(η i, d i q) = πr H + (1 π) (1 η i )R 1 + η ] i 1 + γd i. (6) q The first term describes the payoff after the arrival of optimistic news. The second term describes the payoff after the arrival of pessimistic news, when the intermediary sells η i units of her assets at the fire sale price q. Finally, the third term is the liquidity benefit that the intermediary generates by issuing safe debt, and that she can fully appropriate. While it is more intuitive to think of a bank s problem as choosing the amount of safe 15 This also means that in Stein, 2012, there is no difference between D and M the amount of liquidity creation ( money ) is exactly equal to the amount of funds that are used to purchase assets. 16

18 debt d i while taking into account the implied required assets sales, η i (d i q), it simplifies the formal analysis to consider the reverse mechanic, i.e., to have a bank choose it s liquidation η i taking into account the amount of safe debt it can thereby create at the given market price of assets. Plugging in d i (η q) = (1 η)r L +ηq and q = R 1 /g (W M), the profit as a function of the market-based liquidity creation η is given by [ R πr H +(1 π) (1 η i )R 1 + η 1 i [ ] R +γ (1 η i )R L + η 1 i g Π(η i η, M) = (W M) [ ] W πr H +(1 π) (1 η i )R 1 + η i η 1 [ ] W +γ (1 η i )R L + η i η g (W M) ] 1 if if η η η > η (7) An equilibrium is characterized by a pair (η, M) such that η i = η maximizes the intermediaries profit (7), and that also satisfy the asset market clearing condition (2). Let us define ξ := (1 π + γ)r 1 (1 π)r 1 + γr L. Lemma 1. The laissez-faire equilibrium is characterized by the one of the following three cases: (i) if g (W ) ξ, there is no market-based liquidity creation, η = 0 and M = 0, (ii) if g (W ) < ξ g (0), there is market-based liquidity creation and marginal productivity pricing, η = M g (W M ) R 1 (0, η), where g (W M ) = ξ, i.e., M (0, W ) and (iii) if g (0) < ξ, there is market-based liquidity creation and cash-in-the-market pricing, η = (1 π+γ)w (1 π)r 1 +γr L > η and M = W. The lemma can be derived as follows: As long as η < η, the first derivative of the 17

19 profit is given by [ Π(η i ) = (1 π) η i R 1 g (W M) R 1 ] [ + γ R 1 g (W M) R L ]. For an interior optimum as described in (ii), the condition characterizing M is obtained by setting (3.3) equal to zero: g (W M ) = (1 π + γ)r 1 (1 π)r 1 + γr L. (8) However, the model generally allows for corner solutions for both, η and M. For instance, intermediaries may find it optimal not to rely on market-based liquidity at all, as described in (i). In this case, the issuance of debt is limited by the asset payoff in the low state, D = R L, and no asset sales occur after the arrival of pessimistic news, η = 0, and hence outside investors do not use any funds to buy assets, M = 0. This is only optimal if the derivative of the profit is negative for any value of M, and this corner solution occurs if the low payoff, R L, is sufficiently close to the expected payoff after pessimistic news, R 1 = R L +θ(r H R L ). This in turn is the case if the probability of recovery θ is very small, i.e., the chance of being in the high state after receiving pessimistic news. 16 The reason is that intermediaries only desire to engage in selling assets at a fire sale discount after pessimistic news, if this allows them to capture sufficiently high liquidity benefits from issuing additional safe debt. However, a sufficient amount of additional liquidity can only be created through asset sales if the fundamental value of assets and their sales price is sufficiently larger than R L. For instance, in the extreme case of θ = 0, no additional liquidity can be created from selling assets, and given that assets can only be sold at discount, asset sales cannot be optimal. Finally, at the other extreme, as described in case (iii) intermediaries may desire to 16 For the special case of g (W ) = 1, this corner solution never occurs unless θ = 0. 18

20 rely on market liquidity to an extent that exhausts the endowment of outside investors, M = W, and hence assets priced are governed by cash-in-the-market pricing. This is the case if the marginal productivity of outside investors must be relatively low, even if not a single unit is invested in the late technology Constrained-efficient allocation We now turn towards the socially optimal allocation. The constrained-efficient allocation is defined as the allocation that maximizes welfare by a choice of η, D, and M, subject to the asset market clearing condition (2). Using (5) to substitute for the amount of safe debt issued D and ignoring constants, 18 the welfare is given by W(η, M) = γ[(1 η)r L + M] + (1 π)[g(w M) + M]. (9) The first term denotes the liquidity benefit, and the second term denotes the net return of late investment after the arrival of pessimistic news. This illustrates the trade-off the social planners faces: creating liquidity benefits ex ante at the cost of reducing efficient late investment ex-interim after the arrival of pessimistic news. The constrained-efficient allocation is characterized by a par (η, M) that maximize welfare (9) subject to the asset market clearing condition (2). First, observe that it is never optimal for social planner to allows for cash-in-themarket pricing. Lemma 2. The constrained-efficient liquidation level is limited by η η. As described above, in case of cash-in-the-market pricing, any additional unit sold will destroy liquidity rather than creating it. This is driven by the fact that any additional 17 As indicated above, note that for the existence of this case, it is crucial to assume that the marginal productivity at a zero investment is finite (we do not assume Inada conditions). 18 The payoff following optimistic news is dropped because it is constant, and so is the net return of the intermediaries early investment. The extensive welfare is given by W = πr H + (1 π)r γ[(1 η)r L + M] + (1 π)[g(w M) + M] + πg(w ). 19

21 unit sold does reduces the equilibrium price and hence the amount of funds transferred into the banking sector at t = 1, but requires banks to give up an additional unit that generates a cash flow of R L at t = 2. Hence, in the constrained efficient allocation, we necessarily have that M W, and the social planner never chooses to destroy liquidity. Recall that for M W, the asset market clearing condition Equation (2) is invertible and describes how much of the outside investors funds are allocated to purchasing assets at a given liquidation policy η. Its derivative is given by M/ η = R 1 [g (W M) Mg (W M)] 1. Using this relationship, the total derivative of the welfare is given by dw dη = γr R 1 [ L + γ + (1 π)(1 g g (W M) Mg (W M)) ]. (10) (W M) Let us now consider whether it is socially optimal to choose a corner solution for M. Observe that it can only be optimal to have M = 0 if the derivative at this point is negative. This is the case if and only if condition for case (i) of Lemma 1 is satisfied. Hence, in the laissez-faire equilibrium, intermediaries rely on market-based liquidity creation (i.e., η > 0) if and only if intermediation in the constrained efficient does so (i.e., if η > 0). Or, said differently, there is no wedge between the social optimum and the equilibrium outcome if condition (i) is fulfilled. Let us now consider the case of an interior solution. In this case, M is characterized by and g (W M ) = (1 π + γ)r 1 (1 π)r 1 + γr L + γr LM g (W M ) (1 π)r 1 + γr L, (11) η = M g (W M ) R 1. (12) Comparing these to the interior solutions in the laissez-faire equilibrium, (8) and (11), 20

22 we get g (W M ) = g (W M ) + γr LM g (W M ). (13) (1 π)r 1 + γr L }{{} pecuniary externality Observe that it holds that g (W M ) < g (W M ) as we assumed that g < 0. Hence, it must hold that M > M and η > η. Translating these into economic terms: Assuming the social planner finds some marketbased liquidity creation optimal, we find that there is a wedge between social and private interests. In particular, there is necessarily too much reliance on market-based liquidity creation in the laissez-faire equilibrium and too many assets are sold in a fire sale. 19 Proposition 1 (Excessive fire sales). If η (0, 1), the laissez-faire equilibrium is not constrained efficient. It features strictly too much reliance on market-based liquidity creation (i.e., too much fire sales), η > η. Figure 2 illustrates the wedge between constrained efficient and the laissez-faire equilibrium by plotting the amount of asset sales for different values of R 1 (the fundamental asset value after pessimistic news) for a chosen parametrization of the model. It also displays η, i.e., the liquidation level above which assets pricing is governed by cash-inthe-market pricing. The horizontal axis is given by R 1, which increases in θ. For low levels of θ, it is not socially desirable to engage in asset sales, and liquidity creation entirely relies on equity issuance. Only if θ is sufficiently high, it becomes socially optimal to rely on asset sales as well. However, in the example shown, asset sales are always desirable from a private perspective, and the amount of assets sold in the equilibrium exceed the social optimal level necessarily. However, observe that even though market-based liquidity creation and hence the 19 Note that there are parameter constellations in which intermediaries fully rely on market-based liquidity creation, i.e., sell all their assets after pessimistic news, η = 1, whereas more equity and less fire sales would be constrained efficient. However, whenever η = 1 it must hold that also η = 1. 21

23 η R 1 Figure 2: Aggregate amount of asset sale (liquidation): constrained efficient η (red) and laissez-faire equilibrium η (blue) as a function of R 1, for π = 0.99, γ = 0.5, R L = 0.8, W = 0.6, and g(k) = 1.8 ln(k + 1). The dotted green line denotes η, the liquidation level above which assets prices are governed by cash-in-the-market pricing. amount of assets sold in a fire sale are always excessive, the amount of liquidity created is not necessarily. By Lemma 2 we know that η > η cannot be socially optimal because in this region, the marginal welfare effect of asset sales is clearly negative: They reduce valuable late investment and reduce aggregate liquidity creation. Nonetheless, it is possible that for some parameters, banks behavior implies that marginal welfare of asset sales in the laissez-faire equilibrium is negative, implying that liquidity is being destroyed in equilibrium. Proposition 2 (Inefficient liquidity creation). The amount of safe debt in the laissezfaire equilibrium, D(η ), can be higher or lower than in the constrained efficient, D(η ). Observe that marginal liquidity creation at the constrained efficient must be positive, i.e., D (η ) > 0. Note that the welfare function can be rewritten as the sum of the 22

24 liquidity creation and the return on late investment, i.e., W (η) = γd(η)+(1 π)[g(w M(η)) + M(η)]. In the constrained-efficient allocation, it must hold that γd (η) + (1 π)m (η)[1 g (W M(η))] = 0 The second term displays the marginal costs of reducing late investment and is hence necessarily negative. The first term displays the the marginal liquidity benefit and, given that the second term is negative, must hence be positive in the socially optimal allocation. However, in the equilibrium, the sign of D (η ) can also be negative, i.e., the equilibrium may feature liquidity destruction. If there is liquidity destruction at the equilibrium, the liquidity creation in equilibrium can be higher compared to the constrained efficient, D(η ) < D(η ), but it can also be too low, D(η ) > D(η ). Hence, the excessive reliance on market-based banking does not necessarily translate into an excessive creation of safe debt. In fact, the excessive reliance on market-based banking may imply that liquidity is destroyed in equilibrium. Figure 3 shows an example in which the amount of liquidity created in the constrained efficient is larger than in the competitive equilibrium. It also illustrates how the transition from asset pricing according to real outside opportunities to cash-in-the-market pricing affects the liquidity creation. When η lies right of the threshold η, asset prices are governed by cash-in-the-market pricing as opposed to the marginal productivity of outside investors, and leads to a change in the marginal amount of liquidity destroyed. It is worth noticing that the slope of liquidity creation becomes negative to the left η, emphasizing that cash-in-the-market pricing is a sufficient but not a necessary condition for marginal liquidity destruction in the laissez-faire equilibrium. This result contrasts to the existing literature on collateral constraints and pecuniary externalities. In such models, the equilibrium outcome typically features not only exces- 23

25 sive fire sales, but also excessive leverage (Lorenzoni, 2008; Dávila and Korinek, 2017) or excessive liquidity creation (Stein, 2012). In contrast, we find that even though firesales effects are always excessive in equilibrium, the leverage 20 and amount of liquidity created are not necessarily too high, but can also be too low. The result is driven by the assumption that the lowest possible return a bank can attain is always strictly positive. This in turn implies that fire sales, even if always excessive, can destroy liquidity and constrain the amount of safe debt created. As we will argue next, this has important implications for the optimal regulation of intermediaries. 4. Optimal Regulation Given the wedge between the laissez-faire equilibrium and the constrained-efficient allocation, there is scope for regulatory interventions to improve the allocation attained in the laissez-faire equilibrium. In particular, a social planner would desire to limit the amount of asset sales to the constrained efficient level η, and thereby also implement the optimal amount of safe debt to D. While by Proposition 1, we know that asset sales are unambiguously excessive, Proposition 2 shows that the wedge for safe debt can go in both directions. Taking our model seriously, the straight-forward method of a macroprudential regulation is to target η directly, either trough quantity or price regulation. After shortly analyzing this approach, we will discuss how policy tools that are more similar to tools that are being used in practice, e.g., equity or liquidity regulation Transaction tax on asset sales There are two ways the constrained efficient allocation can be implemented: either the aggregate amount of assets sold to η can be restricted, a quantity regulation, or a transaction tax on assets sales is imposed, a price regulation. 20 Defining leverage as the ratio of safe debt to all other claims. 24

26 q marginal productivity pricing cash-in-the-market pricing D η liquidity creation liquidity destruction η Figure 3: Fire-sale price (top) and liquidity creation (bottom) as a function of asset sales. In the top panel, the orange dotted line displays the hypothetical cash-in-the-market for the case that the outside investors outside option is just storage, i.e., g(k) = k. The first dashed vertical line is the constrained-efficient level of asset sales η, the second is the η (start of cash-in-the-market pricing), and the third is the equilibrium level η. The solid vertical line denotes the liquidation that maximizes liquidity creation, i.e., the point were marginal liquidity creation turns into marginal liquidity destruction. This numerical result was obtained by choosing π = 0.99, γ = 0.5, R H = 2, R L = 0.8, W = 0.6, θ = 0.9 and g(k) = 1.8 ln(k + 1). 25

27 Using quantity regulation, the constrained-efficient allocation can be implemented by setting η as the uniform limit on asset sales for each intermediary. If there was unobservable heterogeneity across intermediaries, one can also use a cap-and-trade approach: Intermediaries initially receive permits specifying individual amounts of assets that may be sold, with the sum of these permits being η. Banks can then freely trade these permits. The corresponding price regulation approach would be given by a positive transaction tax on asset sales ( Pigouvian tax ), the proceeds of which would have to be redistributed to banks in a lump-sum fashion. The most simple tax schedule is a linear transaction tax τ that is levied on the proceeds of asset sales. Given this tax, for an aggregate liquidation level 21 η < η, the profit function (7) changes to [ ] R 1 Π(η i τ) = πr H + (1 π) (1 η i )R 1 + η i (1 τ) g 1 (W M) [ ] R 1 +γ (1 η i )R L + η i (1 τ) g, (14) (W M) implying that funds used for asset purchases M τ satisfies the FOC g (W M τ ) = (1 τ) (1 π + γ)r 1 (1 π)r 1 + γr L. (15) The optimal tax that implements M τ = M > 0 is given by 22 τ = 1 g (W M τ ) (1 π)r 1 + γr L (1 π + γ)r 1. (16) In practice, however, we do not see any policies that target or tax asset sales in this fashion. One reason for why such a policy might not be desirable or feasible in practice is that banks also engage in asset trading for reasons that are independent of 21 Notice that we only have to consider this interval because the constrained-efficient lies in this interval (η < η), and the marginal profit is strictly decreasing. 22 In case that M = 0, the exact tax level does not matter as long as it is prohibitively high. 26