Collective Moral Hazard, Liquidity Evaporation and Time-Consistent Bailouts

Transcription

1 Collective Moral Hazard, Liquidity Evaporation and Time-Consistent Bailouts Ernesto Pasten August 2010 Abstract We study time-consistent bailouts when entrepreneurs (banks) correlate their aggregate risk exposure, events of liquidity evaporation may be triggered during nancial distress, and the authority lacks commitment but internalizes the future e ects of its policy actions. We nd that a nancial policy supporting smaller time-consistent bailouts generates large welfare gains. We propose a methodology based on six wedges to study the interaction of nancial policy and bailouts. In contrast to the related literature (Farhi and Tirole, 2009; Chari and Kehoe, 2009), exploiting these six wedges altogether may be more e ective and socially desirable than focusing on only one of them. JEL codes: E44, E52, E61, G38 Keywords: Sustainable Plans, Bailouts, Moral Hazard, Monetary Policy, Financial Policy I have greatly bene ted from conversations with Guillaume Plantin and Jean Tirole from early stages of this project. I also thank for their valuable comments Guido Friebel, Simon Gilchrist, Jerome Mathis and Randy Wright. The usual disclaimer for the Central Bank of Chile applies. All errors are my own. Contact info: epasten@cict.fr. Toulouse School of Economics and Central Bank of Chile. Manufacture des Tabacs, 21 Allée de Brienne, Toulouse, France.

2 1 Introduction Three critical ingredients shaped the sequence of events during the recent nancial crisis: Many nancial institutions (in short, banks) were engaged in similar investment strategies that correlated their aggregate risk exposure; the FED s concern of future moral hazard in its refusal to bailout Lehman Brothers; and the massive evaporation of liquidity in the whole nancial system after Lehman Brothers failed. We study in this paper the properties of the (socially) best time-consistent bailout plan when these three features are present. In addition, we propose a methodology to study the relation of time-consistent bailouts and other forms of nancial policy. By doing so, we seek to shed some light on the design of optimal regulatory and nancial policy. We also seek to discipline the discussion about the e ects of policy on the ex-ante incentives of banks to hoard liquidity and the ex-post incentives of the nancial authority to deviate from its commitments. There are two natural benchmarks for this paper. The rst is Farhi and Tirole (2009), who study Collective Moral Hazard. Banks hoard too little liquidity and correlate their aggregate risk exposure, the story goes, because an authority without commitment and with short-run horizon has higher incentives to implement larger bailouts when crises are deeper. The second is Chari and Kehoe (2009), who study time-consistent bailouts when the authority lacks commitment and has long-run policy horizon in an economy where bankruptcies are part of the contract designed by rms owners to induce managers high e ort. We depart from the Farhi and Tirole s benchmark in our target: We study time-consistent bailouts when the authority has a long-run horizon. We depart from the Chari and Kehoe s benchmark in the environment: We view the recent nancial crisis as a liquidity problem. We also depart from both benchmarks by studying the interaction between time-consistent bailouts, liquidity evaporation and other forms of nancial policy. Our model makes minimal modi cation to the Farhi and Tirole s environment to introduce a long-run policy maker. There is an in nite sequence of non-overlapping generations of entrepreneurs and investors who live for three periods. In each generation, entrepreneurs can invest in illiquid risky assets or liquid riskless assets (which have a lower return), while investors can only directly invest in riskless assets. Therefore, investors have incentives to lend to entrepreneurs when they are young. However, only a portion of entrepreneurs future income is pledgeable, so investors lending is rationed. When agents are mid-aged, a "distress" state may be realized, in which case risky assets need reinvestment to continue, otherwise the investment is lost. To face this state, entrepreneurs can either invest in the riskless asset when they were young (i.e., hoard liquidity), or get new loans from investors. 2

3 Critically, a benevolent authority may "bailout" entrepreneurs by decreasing the interest rate paid by these new loans at the cost of distorting investors saving decisions. If the authority could commit to a policy, it would choose no bailouts. Entrepreneurs would hoard enough liquidity to ensure full continuation of their risky investment, which maximizes social welfare. Under discretion, when the authority lacks commitment and has short-run horizon, entrepreneurs hoard less liquidity because they recognize the incentives of the authority to bailout ex-post. Even worse, if there is more than one distress state and entrepreneurs can choose which distress state they get exposed to, they will all choose the same distress state in equilibrium. This is because, by doing so, the authority will implement the largest possible bailout. Hence, as a result, social welfare is low because little liquidity is hoarded ex-ante and large distortionary bailouts must be implemented ex-post. This phenomenon is called Collective Moral Hazard. Then we focus on the case without commitment but with a long-run policy horizon. We look for Sustainable Plans (Chari and Kehoe, 1990), which could be seen as an equilibrium trigger strategy in which, if the authority deviates from a given committed extent of bailouts, then the social welfare under discretion will be realized for all future generations. 1 surprisingly, we nd that smaller sustainable bailout may be supported as the policy horizon is longer. Despite its triviality, this result has profound implications for our analysis. In comparison to the Farhi and Tirole (2009) benchmark, we nd that Collective Moral Hazard gets largely undermined as an ampli cation mechanism of the social costs of nancial crises. This is because the combination of three e ects: (i) the welfare cost of bailouts is convex in their range, so supporting smaller sustainable bailouts generates large welfare gains; (ii) the future cost of policy deviations increases as bailouts are smaller, which helps to support even smaller bailouts; and (iii) the future cost of policy deviations is convex in the authority s discount factor. Thus, the policy horizon has large e ect on the size of sustainable bailouts and may even support a no-bailouts policy. This result also motivates the introduction of liquidity evaporation into the analysis to recover the importance of Collective Moral Hazard. We do so because we seek in this paper to produce a sensible environment to study time-consistent bailouts. We see the correlation of aggregate risk exposure among banks implied by Collective Moral Hazard as essential for this task. We de ne "liquidity evaporation" as situation with a sharp increase in the liquidity premium that leads short-run securities markets to freeze, haircuts and collateral requirements to increase dramatically, etc. There exist a number of potential explanations 1 This approach has been applied in a variety of contexts, such as capital taxation (Chari and Kehoe, 1990), sovereign debt (Arellano, 2009), monetary policy (Chang, 1998) and new public nance (Farhi and Werning, 2009). Not 3

4 for such phenonomenon, each of them labelled by a di erent name. 2 Instead of focusing on a speci c explanation, we take a simple general approach: We assume that investors ask for a premium on loans to entrepreneurs in the distress state if entrepreneurs are forced to downsize. This happens when entrepreneurs hoard too little liquidity given a bailout policy, so sustainable bailouts are large again to avoid liquidity evaporation. With this model at hand, we turn to study the interaction of sustainable bailouts and other forms of policy. Our rst message is that this interaction is tricky. We study the main policy recommendation of the Chari and Kehoe (2009) benchmark: A "size cap" on banks (in our context, entrepreneurs risky investment) decreases the incentives of a policy deviation, alleviating the time-inconsistency of bailouts. We show that if capital from the restricted banks could go to open new banks or to increase the size of the smaller existing banks, then a size cap is largely ine ective. This is because what matter for the time-inconsistency of bailouts is the whole amount of resources in the economy that are exposed to the same aggregate risk. Because of Collective Moral Hazard, a size cap that reduces the exposition of one bank is o set by the increase in the exposition of other banks. 3 Regarding nancial policy, there is an explosion of literature that makes policy recommendations to prevent nancial crises. However, it is di cult to integrate these recommendations in a single environment. In addition, as pointed out above, the interaction of sustainable bailouts and nancial policy is tricky. To overcome these issues, we propose a general methodology based on "wedges." Wedges are de ned as deviations from the equilibrium level of some key variable in the time-consistency condition for sustainable bailouts. Wedges may be interrelated. In particular, we distinguish six wedges: a liquidity wedge on liquidity hoarding; a size wedge on the scale of risky investment; a pledgeability wedge on the degree of entrepreneurs credit rationing; an evaporation wedge on the liquidity premium during a crisis; a discounting wedge on the authority s time horizon; and a continuation wedge on the future costs of policy deviations. For instance, the call of Farhi and Tirole (2009) for macro prudential policy is captured by the liquidity wedge, and the call of Chari and Kehoe (2009) for a "too-big-to-fail" policy is captured by the size wedge. We also discuss other 2 These explanations are not fully independent. Some prominent examples are bank runs (Diamond and Dybdig, 1983), contagion (Allen and Gale, 2000), re sales (Acharya, Shin and Yorulmazer, 2009), rollover risk (Acharya, Gale and Yorulmaser, 2010), panic (Dasgupta, 2004), liquidity black holes (Morris and Shin, 2004), predatory trading (Brunnermeier and Pedersen, 2005), liquidity spirals (Brunnermeier and Pedersen, 2009), Knightnian uncertainty (Caballero and Krishnamurthy, 2008; Caballero and Simsek, 2010), and bubbly liquidity (Farhi and Tirole, 2010). 3 Along the same line, Farhi and Tirole (2009) show that liquidity requirements may induce banks to meet these requirements with "toxic assets", a cheaper form of liquidity in normal times that is useless in episodes of nancial distress. 4

5 policies that can manipulate these wedges. 4 Our main result is that policy manipulating any of these wedges may increase the ex-ante incentives of banks to hoard liquidity and reduce the ex-post incentives of the authority to deviate from its commitment. Therefore, di erent policies are complementary in the goal of supporting smaller time-consistent bailouts. This powerful result implies that, although nancial policy is typically di cult to enforce and policy always imposes social costs, large welfare gains may be obtained by small improvements in these wedges once managed in conjunction. In addition, in contrast with the recommendations of our two benchmarks, the optimal design of nancial policy should not concentrate only on one wedge, but in a battery of policy instruments that exploit all wedges together. The exact form of such optimal design depends on the policy toolkit, their social costs and their e ectiveness on manipulating wedges. We leave such task for future research. 5 The related literature is too massive to summarize here. Brunnermeier (2009) and Diamond and Rajan (2008) narrate the sequence of events during the nancial crisis. We borrow our baseline model from Farhi and Tirole (2009), but this model has been extensively used in Corporate Finance Theory (Tirole, 2006) and to study liquidity (Holmstrom and Tirole, 1998; Holmstrom and Tirole, 2008). Diamond and Rajan (2009) also develop a closely related idea to Collective Moral Hazard. We have designed our exercise to serve as input for studying nancial policy even in settings di erent than ours. For instance, most papers cited in footnote 2 that study mechanisms for liquidity evaporation also study their policy implications. Other examples in which our wedges approach could be valuable are Jeanne and Korinek (2010), who propose a tax on borrowing to smooth nancial cycles, and Reis (2010), who proposes a design for policy interventions during nancial crises. Section 2 displays the basic model with only one generation. Section 3 extends this model to an in nite sequence of generations and obtain sustainable bailouts. Section 4 compares results with our two benchmarks. Section 5 introduces the policy "wedges" and Section 6 concludes. All proofs are relegated to the Appendix. 4 Speci cally, the bailout design or a punishment policy on rescued banks to control the liquidity wedge; policy to increase pledgeability or mitigate liquidity evaporation during a crisis; institutional design to increase the time horizon of the nancial authority; and policy that imposes future costs on the authority if it deviates from its commitments. 5 This is because such task requires an empirically validated general equilibrium macro nance model, which lacks in the literature. The production of such model is above the ambition of this paper. 5

6 2 Baseline model This section displays the baseline model for our analysis, inspired in Farhi and Tirole (2009), that makes explicit distinction between expected policy and actual policy. 2.1 Setup Consider a static economy with three stages, s = 0; 1; 2. 6 There are two types of agents in this economy (each with total mass one): investors and entrepreneurs. Investors receive exogenous endowments e 0 and e 1 in s = 0; 1 and have utility function V = c 0 + u(c 1 ) + c 2 ; where c 0 ; c 1 ; c 2 denote consumption in stages s = 0; 1; 2 and u () is an increasing and concave function, u 0 > 0 and u 00 < 0. Entrepreneurs only receive an exogenous endowment A at s = 0 and have utility function U = c 0 + c 1 + c 2 : There are two type of assets in this economy: riskless assets and risky assets. Riskless assets are available for both investors and entrepreneurs and have net return equal to zero. In contrast, only entrepreneurs can invest in risky assets. These assets provide 1 > 1 of gross return in stage s = 2 if there is "no distress" in stage s = 1 (which has probability ). However, if there is "distress" in s = 1 (which has probability 1 risky assets yield zero gross return in s = 2 unless there is a reinvestment in s = 1. 7 ), In this case, these assets yield 1 gross return of the reinvestment scale in s = 2. For either state of nature, "no distress" or "distress", only a portion 0 < 1 of the total returns is pledgeable. 8 There is also a nancial authority in this economy (for instance, a Central Bank), which can tax the return of the riskless asset from s = 1 to s = 2, so its e ective return is R 1. This tax is interpreted as a "bailout", which is rebated to the taxed agents via lump-sum 6 Notation t is reserved for generations, which are introduced in Section 3. 7 The distress stage represents a scenario in which an aggregate liquidity shock hits the economy. 8 This limited pledgeability is assumed exogenous, but it may be justi ed as the terms of the optimal contract between investors and entrepreneurs when the latter have superior information than the former about the quality of the assets or their e ort. For instance, see Holmstrom and Tirole (1998). 6

7 transfers in s = 2. The authority maximizes social welfare, which is de ned as V + U where represents the relative weight of entrepreneurs welfare in social welfare. Timing. The timing of actions in this economy is as follows: In stage s = 0, investors receive e 0 and decide how much to invest in the riskless asset and to lend to entrepreneurs. Entrepreneurs receive A and decide how much to invest in risky assets, i, and in riskless assets, xi. All agents decide their consumption in this stage. In stage s = 1, investors receive e 1 and the aggregate state of nature is revealed: "no distress" (with probability ) and "distress" (with probability 1 ). In the "distress" state, investors decide how much to invest in riskless assets and to lend to entrepreneurs for reinvestment. Entrepreneurs receive no endowment and decide their reinvestment scale j in risky assets. The nancial authority decides taxes on the riskless asset. In the "no distress" state, risky assets do not need reinvestment, so investors only invest in riskless assets. All agents decide their consumption in this stage. In stage s = 2, no one receives endowment. Risky assets pay back, taxes are rebated, and all agents decide their consumption in this stage. 2.2 Equilibrium The main variables of interest are entrepreneurs investment in risky assets and riskless assets (also called liquidity hoarding) in stage s = 0, respectively i and xi, and their reinvestment scale j in risky assets in stage s = 1 if the "distress" state takes place. To nd out the equilibrium, we study the terms of the optimal contract between investors and entrepreneurs. Terms of the contract. entrepreneurs borrow from investors a total of For a given investment scale i and liquidity hoarding xi, i + xi A; i.e., the di erence between their total investment in s = 0 and their endowment. Investors endowment e 0 is assumed large enough such that it is not binding. This expression assumes 7

8 that entrepreneurs invest all their endowment A in the risky asset (i.e., they do not consume in s = 0), which is ensured by assuming 1 + (1 ) < 1 : (1) If there is no distress in s = 1, entrepreneurs pay ( 0 + x) i to investors in s = 2. This is the total amount that can be pledged by entrepreneurs (which includes the liquidity hoarding xi). Entrepreneurs pay zero net return to investors because the net return of the riskless asset is zero. However, if the distress state takes place in s = 1, entrepreneurs do not pay back their loans. Instead, entrepreneurs renegotiate with investors to obtain new loans. Optimal contract. Denote R as the policy in s = 1, i.e., the after-tax gross return at s = 2 of a riskless investment made in s = 1, and R e is the expected policy in s = 0. The break-even condition for investors in s = 0 is i + xi A = ( 0 + x) i; (2) such that the opportunity cost of the investors loan to entrepreneurs in s = 0 must be equal to the expected gross return of the loan. The opportunity costs is given by consumption (with marginal utility equal to one) or riskless investment (with gross return equal to one). The break-even condition implies that i (x) = A 1 + (1 ) x 0 : If the distress state takes place in s = 1, the reinvestment scale is 0 j + xi j (x; R; i) = min R ; i : because ( 0 j + xi) is the maximum amount pledgeable by entrepreneurs in s = 1, and return R must be paid to investors. Entrepreneurs have no incentives to continue their risky investment at a scale larger than i, which justi es the upper bound. This expression above 8

9 may be further simpli ed to x j (x; R; i) = min ; 1 i R 0 which depends on liquidity hoarding x as a proportion of the initial investment scale i, policy R, and the portion 0 of future returns that is pledgeable. Finally, we impose that entrepreneurs choose their liquidity hoarding such that their investment in the risky asset is expected to continue at the full scale, i.e. x (R e ) = R e 0 : (3) This result comes from the assumption in (1), which ensures that entrepreneurs prefer to concentrate their consumption in s = 2: In equilibrium, we get i (R e ) = A 1 + (1 ) R e 0 (4) R j (R; R e e ) = min 0 ; 1 i (R e ) (5) R 0 Note the distinction between e ective policy R, which a ects the re-investment scale j, and expected policy R e, which a ects liquidity hoarding x. Through liquidity hoarding, R e indirectly a ects the scale of investment and reinvestment in risky assets, i and j. 2.3 Optimal bailouts under commitment We derive now optimal bailouts assuming that the authority controls expected policy R e. To do so, we compute ex-ante welfare in stage s = 0 for any pair (R; R e ), and then look for the interest rate that maximizes ex-ante welfare after imposing R e = R. Our computations obtained here are used to study discretionary bailouts later in this Section and to study sustainable bailouts in Section 3. Ex-ante welfare. If there is no distress in s = 1, there is no need to re nancing risky investments, so policy is R = 1. Thus, investors welfare from s = 1 and on is V no distress (R = 1; R e ) = u (e 1 S) + S + [ 0 + x (R e )] i (R e ) 9

10 where u (e 1 S) is investors utility for consumption at s = 1 after investing S in the riskless asset. At s = 2 investors consume their riskless investment S (which has no net return), plus the return of their loans in stage s = 0 to entrepreneurs, [ 0 + x (R e )] i (R e ). The investment scale i (R e ) and liquidity hoarding x (R e ) are decisions taken by the entrepreneur at s = 0, so they depend on expected policy R e. This expression may be simpli ed after using equation (3) for x (R e ) and de ning ^V (R) = u (e 1 S) + S where u 0 () = R: (6) such that ^V (1) ^V (R) is the cost in welfare introduced by policy R after distorting investors savings decisions, so V no distress (R = 1; R e ) = ^V (1) + R e i (R e ) : In the distress state in s = 1, entrepreneurs do not pay back their loans taken at s = 0 and instead raise new funding for reinvestment. In this situation, investors welfare from s = 1 on is V distress (R; R e ) = u (e 1 S) + RS + (1 R) [S j (R; R e )] where u (e 1 S) is investors utility for consumption in s = 1. At s = 2, investors consume RS, the gross return of their investment in s = 1 either in the riskless asset or in new loans to entrepreneurs. In addition, investors also get in s = 2 the rebate for the tax levied to their riskless investment in s = 1, (1 R) [S j (R; R e )]. This expression may be simpli ed to V distress (R; R e ) = u (e 1 S) + S (1 R) j (R; R e ) (7) = ^V (R) (1 R) j (R; R e ) : Hence investors ex-ante welfare (at s = 0) is V ex ante (R; R e ) = [e 0 i (R e ) x (R e ) i (R e ) + A] (8) h i h i + ^V (1) + R e i (R e ) + (1 ) ^V (R) (1 R) j (R; R e ) where the rst term in brackets is investors consumption in s = 0, the second term is investor s welfare if there is no distress in s = 1 (with probability ), and the third term is investor s welfare if there is distress in s = 1 (with probability 1 ). 10

11 Using investors break-even condition in equation (2), V ex simpli ed to ante (R; R e ) may be further V ex ante (R; R e ) = e 0 + ^V h i (1) + (1 ) ^V (R) (1 R) j (R; R e ) : Entrepreneurs, on the other hand, receive in s = 2 a portion ( 1 0 ) of the surviving scale of their risky investment, which is i (R e ) if there is no distress and j (R; R e ) if there is distress, so U ex ante (R; R e ) = ( 1 0 ) [i (R e ) + (1 ) j (R; R e )] : (9) Combining (8) and (9), ex-ante social welfare is W ex ante (R; R e ) = e 0 + ^V h i (1) + (1 ) ^V (R) (1 R) j (R; R e ) + ( 1 0 ) [i (R e ) + (1 ) j (R; R e )] : (10) where is the weight of entrepreneurs in the nancial authority s objective. Optimal committed bailouts. when R e = R, ex-ante welfare (10) under commitment is Since risky investment always continues at full scale W ex ante (R; R) = e 0 + ^V h i (1) + (1 ) ^V (R) (1 R) i (R) + ( 1 0 ) i (R) : A bailout (R < 1) implies lower welfare for investors and higher welfare for entrepreneurs. For investors, lower R implies a larger distortion on their saving decision ( ^V (R) is smaller), and larger transfers to entrepreneurs in the distress state ((1 R) i (R) is larger). Entrepreneurs choose higher investment scale, (because i 0 (R) < 0), so lower R implies higher expected consumption (( 1 0 ) i (R) is larger). The overall e ect of decreasing R on ex-ante welfare is negative if the negative e ect on investors is stronger that the positive e ect on entrepreneurs, which is ensured by assuming ( 1 0 ) (1 ) + (1 0 ) : (11) so the optimal committed policy is R c = 1, i.e., no bailouts. 9 9 This condition is shown in the Appendix. 11

12 2.4 Optimal bailouts under discretion We now obtain the optimal discretionary policy Rd when the authority has no control on expected policy R e. This policy is time-consistent in the sense that if expected policy is R e < R d, then the authority has no incentives to deviate from R d. To nd out Rd, we rst compute the ex-post welfare in the distress state in s = 1 for any pair (R; R e ) and look for the set of time-consistent policies. Then we pin down the policy that maximizes ex-ante welfare in the set of time-consistent policies. Ex-post welfare. We make use of our computations in Section 2.3. Equation (7) computes the investors welfare in the distress state as V distress (R; R e ) = ^V (R) (1 R) j (R; R e ) : And equation (9) implicitly computes entrepreneurs welfare in the distress state as U distress (R; R e ) = ( 1 0 ) j (R; R e ) : The overall social welfare in the distress state is therefore as W ex post (R; R e ) = ^V (R) + [ ( 1 0 ) (1 R)] j (R; R e ) : (12) Optimal discretionary bailouts. = ^V (R) + [ ( 1 0 ) (1 R)] Re 0 R 0 i (R e ) : The set of time-consistent bailouts may be de ned < d = R j W ex post (R; R e ) W ex post (R e ; R e ) for any R e R : (13) This set is composed by policies such that if entrepreneurs expect a larger bailout (smaller R), the authority has no incentives to match these expectations. Following (13), a policy R is time-consistent if h ^V (R) ^V (R e )i [ ( 1 0 ) (1 0 )] R Re R 0 i (R e ) 0 8R e R: (14) We further restrict our attention to the parameters subspace that satis es ( 1 0 ) > (1 0 ) : (15) 12

13 Combining assumptions in (11) and (15), the weight of entrepreneurs welfare on the authority s objective must satisfy (1 0 ) < ( 1 0 ) (1 ) + (1 0 ) : The computation of the optimal discretionary policy Rd requires the speci cation of the investors utility function u (). However, we can show that Rd < 1 given the assumption in (15). For this, assume that the authority commits to R = 1. Since ^V 0 (1) = 0 from (6), the condition (14) is violated for any R e < 1, so R = 1 is not a time-consistent policy. Notice that the authority has no incentives to implement a bailout policy R < 0 because R = 0 ensures full reinvestment scale. Therefore, < d = 0 ; R d. Also note that ex-ante welfare is decreasing in R, so Rd = sup f< d g = R d the highest time-consistent policy is the optimal discretionary policy Rd. 2.5 Collective Moral Hazard We now sketch the central result from Farhi and Tirole (2009). Assume that there exist two states of distress at stage 1: distress 1 and distress 2 such that = : i.e., the total aggregate risk is identical to the case above with only one distress state. Also assume that entrepreneurs choose which of these distress states to get exposed to, i.e., in which of them their risky investment needs re nancing. Entrepreneurs problem is max ( 1 0 ) fk [ 1 i (R1) e + (1 1 ) j (R 1 ; R1)] e + (1 k) [ 2 i (R2) e + (1 2 ) j (R 2 ; R2)]g e k2f0;1g where R z ; R e z are respectively the actual and expected policy in state distress z, for z = 1; 2. The control is the binary variable k, such that k = 1 represents the exposure to distress 1 and k = 0 represents the exposure to distress 2. Denote optimal discretionary policy as R d;z in state distress z, for z = 1; 2. Since discretionary policy is time-consistent, then entrepreneurs hoard enough liquidity to ensure the continuation of their risky investment at full scale. Thus, given policy R d;1 ; R d;2, the 13

14 entrepreneur s problem is max ( 1 0 ) ki Rd;1 + (1 k)i R d;2 : k2f0;1g Since i Rd;z is decreasing in R d;z, entrepreneurs choose ( k Rd;1; Rd;2 0 if R = d;1 > R d;2 ; 1 if Rd;1 R d;2 : This is the best reaction function of entrepreneurs given policy. To nd the best reaction function of policy, i.e. Rd;1 ; R d;2, assume that a proportion of entrepreneurs choose to get exposed to state distress 1. Hence, a proportion of the total lending of investors to entrepreneurs in s = 0 is not paid back in s = 1, and a proportion of the total risky investment needs re nancing in s = 1. From (14), Rd;1 is the highest policy such that h ^V (R) ^V (R e )i [ ( 1 0 ) (1 0 )] R Re R 0 i (R e ) 0 8R e R: The key observation is that the second term in this condition is increasing in. Hence, the welfare cost of following a precommitted policy R is increasing in, i.e., the optimal discretionary policy Rd;1 is decreasing in. Since all entrepreneurs are identical, in equilibrium can be only 0 or 1. If = 0, then Rd;1 > R d;2, so all entrepreneurs choose to get exposed to state distress 2, so, = 0 is an equilibrium. Similarly, = 1 is also an equilibrium in which all entrepreneurs get exposed to state distress 1. Intuitively, entrepreneurs perfectly correlate their aggregate risk exposure in equilibrium because the nancial authority has incentives to implement larger bailouts (i.e., lower interest rate) when more risky investments need re nancing. This phenomenon is what Farhi and Tirole call Collective Moral Hazard. They focus on monetary bailouts, but the key feature is that bailouts are non-targeted. 10 In what follows, we also focus on monetary bailouts to keep Farhi and Tirole s analysis as a clean benchmark, but our results are also applicable to other forms of non-targeted., socially costly bailouts. 10 These authors show that monetary policy is an important component of the optimal design of bailouts. 14

15 3 In nite policy horizon We now study a non-overlapping generations version of the baseline model that keeps the three-stage structure of the economy but also allows the authority to internalize the future e ects of its policy actions. 3.1 Modi cation of the baseline model Consider an economy which in each period t = 0; 1; :::; 1 is populated by one generation of investors and entrepreneurs. Each generation lives for only one period. Agents in one generation have no possibility to interact with agents in the next generation, i.e., there are no intergenerational transfers. Each period is broken into stages s = 0; 1; 2, such that inside one period the setup is identical to the baseline model of Section 2. Both investors and entrepreneurs endowments remain exogenous. We also return to the assumption that there is only one distress state, which gets revealed in stage 1 every period. The equilibrium between investors and entrepreneurs inside one period is identical to Section 2. The only di erence is that now all relevant variables are indexed by t: i (R e t) = A 1 + (1 ) R e t 0 ; (16) x (Rt) e = Rt e 0 ; (17) R j (R t ; Rt) e e = min t 0 ; 1 i (R R t t) e ; (18) 0 where i (R e t) in (16) is the total risky investment scale of entrepreneurs at t, which depends on their endowment A in stage 0, the portion 0 of their future return that is pledgeable, and their expected policy Rt e if there is distress in stage 1. For simplicity, A and 0 are assumed constant across periods. The variable x (Rt) e in (17) is the proportion of i (Rt) e that entrepreneurs keep as liquidity hoarding, which depends on R e t and 0. Finally, j (R t ; R e t) in (18) is the entrepreneurs total reinvestment scale at t if there is distress in stage 1 which depends on the di erence between expected and actual policy. The main twist of this economy with respect to the baseline economy in Section 2 is that the nancial authority, unlike investors and entrepreneurs, is long-lived. Its objective is to maximize social welfare: ( 1 ) X E 0 t [V t + U t ] : t=0 15

16 which is the expected discounted sum across all generations of investors utility (denoted by V t ) and entrepreneurs utility (denoted by U t ). The parameter is the discount factor and is the weight of entrepreneurs in the authority s objective. Overlapping generations. The main feature that overlapping generations would add to our results is that bailouts would a ect both the re nancing capacity of one generation of entrepreneurs in their stage 1 and the nancing capacity of next generation of entrepreneurs in their stage 0. We abstract from the latter e ect because it is not central for our study of time-consistent bailouts and their interaction with other forms of nancial policy. 3.2 Optimal sustainable bailouts This Section applies Sustainable Plans (Chari and Kehoe, 1990). We proceed in a similar fashion as in the study of discretionary policy in Section 2.4. We rst look for the set of time-consistent policies and then pin down the policy that delivers the highest social welfare. The di erence between this policy and R = 1 is the optimal sustainable bailout. We rst compute the punishment the future welfare after the authority deviates from a given committed policy, in which case agents expect the optimal discretionary policy for all future periods. Then we compute the reward the future welfare after the authority sticks on its commitment, so agents expect the same policy for all future periods. 11 The future cost of policy deviations is the di erence between the punishment and the reward. Punishment. De ne W t as the generation t s ex-ante welfare: W t = V t + U t : The optimal discretionary policy Rd is time-consistent, so risky investment always continue at full scale, j t = i t. Thus, interpreting (10) as the generation t s ex-ante welfare, W ex ante t (R d; R d) = e 0 + ^V (1) + (1 ) h i ^V (R d) (1 Rd) i (Rd) + ( 1 0 ) i (Rd) ; the punishment at t is punishment t = W t ex ante (Rd ; R d ) 1 (19) 11 These assumptions to construct the punishment and the reward are con rmed in equilibrium. 16

17 because all generations are ex-ante identical. Note that the punishment does not depend on the policy R committed at t. Reward. Assuming that a given committed policy R is time-consistent, risky investment always continues at full scale, so the generation t s ex-ante social welfare for policy R is t ante (R; R) = e 0 + ^V h i (1) + (1 ) ^V (R) (1 R) i (R) + ( 1 0 ) i (R) : (20) W ex Hence, the rewards is reward t (R) = W t ex ante (R; R) : (21) 1 Note that, in contrast to the punishment, the reward depends on the level of the committed policy R. Recall from Section 2.3 that W ex ante (R; R) is increasing in R. Thus, the reward is higher than the punishment if R > Rd. This feature allows us to sustain time-consistent bailouts that are smaller than the discretionary bailout. Optimal sustainable bailouts. We now use our previous computation to nd the condition for a committed policy R to be time-consistent, i.e., that the authority has no incentives to deviate to a smaller expected policy Rt e once the distress state is realized in stage 1 at period t. The total value for the nancial authority of deviating is L deviation ex post t (R; Rt) e = Wt (Rt; e Rt) e + punishment t+1 : Similarly, the total value for the nancial authority of sticking on R is no deviation L ex post t (R; Rt) e = Wt (R; Rt) e + reward t+1 (R) : Thus, the set of sustainable policies may be de ned as < s = R j L no deviation t (R; R e t) L deviation t (R; R e t) for any R e t R : Compared to the set of discretionary policies < d in (13), the only twist of < s is that it involves the future e ects of sticking and deviating from policy plans. Both sets are identical 17

18 if = 0. In parallel with our analysis for < d, the condition for < s may be written as 8 < : h ^V (R) ^V (R e )i [ ( 1 0 ) (1 0 )] R Re R 0 i (R e ) 9 = ; ( punishment t+1 reward t+1 (R) ) 8R e R: (22) The left hand side of this condition is the same as for < d in (14). The rst term in brackets is the di erence in investors welfare after distorting their saving decisions by implementing R or Rt. e The second term in brackets is the di erence in welfare from the investment and reinvestment scales implied by R or Rt. e The di erence between < d and < s is on the right-hand side of (22). Instead of zero, (22) has the di erence between the punishment and the reward, which is negative for R > Rd. Proposition 1 The set of discretionary policies is contained in the set of sustainable policies, such that < d < s for > 0 and < d = < s for = 0. In addition, the assumptions to construct the punishment and the reward are con rmed in equilibrium, and the optimal sustainable policy is Rs = sup f< s g = R s ; which is increasing in the discount factor, so R s > R d for > 0. The proof of this Proposition is relegated to the Appendix. The main result of this Proposition is that the optimal sustainable policy Rs is the upper bound of the set of time-consistent policies < s. This is because ex-ante welfare is increasing in the level of the committed policy for R 1. In addition, Rs is increasing in and higher than the optimal discretionary policy Rs for > 0. Intuitively, the longer is the policy horizon, the higher importance the authority assigns to the stronger future moral hazard that arises after policy deviations, so the smaller is the expected bailout in the distress state. 4 Comparison to benchmarks This section focuses on the implications of Proposition 1 for the two benchmarks we have de ned for our exercise: Farhi and Tirole (2009) and Chari and Kehoe (2009). 18

19 4.1 Implications for Farhi and Tirole (2009) The main result of Farhi and Tirole s analysis is that Collective Moral Hazard is a major ampli cation mechanism of the social cost of nancial crises: Entrepreneurs hoard little liquidity and get exposed to the same aggregate risk, which both increase the severity of crises. In addition, short-sighted policy makers are forced to implement massive bailouts, which involve large welfare costs for the society. Our analysis of Section 3 concludes that the equilibrium size of bailouts is smaller as the policy maker has longer policy horizon. We now study the implications of this conclusion on Farhi and Tirole s main result. Notice that, as long the optimal sustainable policy implies bailouts in distress states, Collective Moral Hazard remains an equilibrium phenomenon. This is because, no matter the size of bailouts, entrepreneurs coordinate their exposition to the distress state that delivers the largest expected bailout. Therefore, a long horizon policy maker does not per se solve the problem of Collective Moral Hazard. However, we show below that a long horizon policy maker mitigates Collective Moral Hazard as an ampli cation mechanism, and even may eliminate it when the authority is patient enough. To show this point, we need to study two separate issues: The gain in ex-ante welfare when credible bailouts are smaller, and how much smaller optimal sustainable bailouts are with respect to the optimal discretionary bailouts The welfare cost of bailouts De ne the social cost of a time-consistent bailout as W ex ante ex ante t (1; 1) Wt (R; R) ; which is the ex-ante welfare di erence delivered a no-bailouts policy (assuming that it is time-consistent) and a given time-consistent policy R. Ex-ante welfare Wt ex ante (R; R) is obtained in (20): t ante (R; R) = e 0 + ^V h i (1) + (1 ) ^V (R) (1 R) i (R) + ( 1 0 ) i (R) : W ex Given the assumption in (15), W ex t ante committed policy is R c < 1). In addition, W ex (R; R) is increasing in R (otherwise the optimal t ante (R; R) is concave because investors stage 1 utility function u (), which is implicit in ^V (R), is concave. Therefore, the ex-ante social cost of bailouts is convex and increasing in the size of bailouts 19

20 (smaller R). Intuitively, ex-ante welfare embeds two major channels by which the ampli cation mechanism of Collective Moral Hazard operates. Smaller time-consistent bailouts imply larger ex-ante liquidity hoarding, so the severity of crises is smaller. And, smaller bailouts also imply smaller ex-post transfers from investors to entrepreneurs, so the distortion on investors saving decisions is smaller. Hence, a small increase in the best sustainable policy generates a relatively large reduction on the social cost of a bailout Properties of the optimal sustainable policy Rs Proposition 1 shows that Rs > Rd for > 0 and that R s is increasing in. We now focus on how much higher is Rs with respect to Rd given > 0. The punishment of deviating from a committed policy is de ned in (19) as punishment t = W t ex ante (Rd ; R d ) : 1 Conversely, the reward of following a committed policy R is de ned in (21) as reward t (R) = W t ex ante (R; R) : 1 The di erence between the punishment and the reward (multiplied by ) is on the right hand side of condition (22). This condition must be satis ed by a time-consistent policy, including the optimal sustainable policy, which is the highest interest rate in this set. The punishment does not depend on the committed policy R. Conversely, the reward inherits the properties of W ex t ante (R; R), so it is increasing and concave in R. Thus, the right-hand side of (22) is negative and convex in R > Rd. Hence, for a given > 0, a policy R slightly above Rd has a relatively large e ect on relaxing the condition in (22). This force implies that, given, the optimal sustainable policy Rs is relatively high with respect Rd. How much higher is Rs with respect to Rd depends on. The right-hand side of (22) is increasing and convex in, approaching to 1 as goes to 1 while the left hand side of (22) is nite and independent of. Hence, there always exists a < 1 such that the optimal committed policy is time-consistent, so Collective Moral Hazard is ruled out in equilibrium. Proposition 2 Long policy horizon mitigates the welfare importance of Collective Moral Hazard. This result is obtained from three mechanisms: 20

21 (i) Welfare costs of bailouts are convex in the extent of bailouts, so reducing their extent has large e ect on reducing their welfare costs. (ii) The future cost of a current policy deviation is convex in the committed policy, so optimal sustainable policy Rs has the potential to be substantially larger than the optimal discretionary policy Rd (i.e., sustainable bailouts are smaller than discretionary bailouts). (iii) Rs is increasing and convex in, so Rs = 1 for 2 ; 1 (i.e., no bailout). This Proposition summarizes the results so far in this section, so it needs no proof The role of liquidity evaporation We now introduce liquidity evaporation to recover the importance of Collective Moral Hazard as an ampli cation mechanism of nancial crises when the authority is not short-sighted. Our motivation relies on the fact that little liquidity provisions and strong correlation of aggregate risk exposure among nancial institutions seem to be key ingredients that deepened the nancial crisis. In addition, the fact that Lehman Brothers was not rescued suggests that the U.S. nancial authority did not have a short-sighted policy horizon. To do so in a simple fashion, we slightly modify the model setup of Section 3. Assume that investors ask for a discount q > 1 to re nance risky investment when entrepreneurs must downsize. However, no discount is asked when entrepreneurs are able to continue at full scale. Intuitively, liquidity evaporation is triggered only when there is a massive downsizing in the nancial system. Critically, a bailout may prevent liquidity evaporation. Under this assumption, risky investment scale i (R e t) and liquidity hoarding x (R e t) remain identical to equations (16) and (17). The only di erence is that reinvestment scale is now j (R t ; R e t; q) = ( i (R e t ) if R t R e t; R e t 0 qr t 0 if R t > R e t: If the bailout is equal or larger than expected (R t R e t), risky investment continues at full scale. However, if the bailout is smaller than expected (R t > R e t), entrepreneurs must pay a premium q > 1 over the after-tax return R of the riskless asset to borrow from investors. 21

22 Thus, the condition that a time-consistent policy R must satis es becomes 8 >< >: h ^V (R) i ^V (R e t) [ ( 1 0 ) (1 0 )] qr Re t qr 0 i (R e t) (q 1) Re t 0 qr 0 Ri (R e t) 9 >= >; ( punishment t+1 reward t+1 (R) ) 8R e t R: (23) This condition may be compared with the time-consistency condition for a short-sighted policy maker in (14) and for an in nite horizon policy maker in (22). The rst observation is that the twist of (23) with respect to (14) and (22) is on the left hand side of the inequality, so liquidity evaporation a ects both optimal discretionary and optimal sustainable policies. The condition in (23) is equivalent to (14) if q = 1 and = 0 and to (22) if q = 1. Thus, we denote the optimal discretionary and optimal sustainable policies as R d (q) and R s (q), such that R d (1) and R s (1) coincide with R d and R s in previous sections. Proposition 3 Liquidity evaporation (q > 1) reduces the incentives of both the short-sighted and the in nite horizon policy maker to follow a given bailout policy. Hence, for <, both the optimal discretionary and the optimal sustainable bailouts are increasing in q. The proof of this Proposition is in the Appendix. This proof shows that, given policy R, liquidity evaporation a ects welfare in two opposite directions in (23). Entrepreneurs must pay a premium q > 1 over the interest rate R to obtain liquidity, so they are forced to do larger downsizing. But larger downsizing implies larger rebates to investors. This is because the authority rebates taxes levied from riskless investment. Thus, given policy R, the larger is entrepreneurs downsizing, the more investors invest in the riskless asset, so rebates are larger. The overall e ect on welfare is negative under the assumption in (15). Liquidity evaporation hence increases the welfare costs of following a given bailout policy. As result, the size of both optimal discretionary and sustainable bailouts (respectively 1 Rd (q) and 1 R s (q)) are increasing in q 1, the extent of the liquidity evaporation. We will go back to this result in Section 5, when we will study the interaction of sustainable bailouts and other forms of nancial policy. 4.2 Implications for Chari and Kehoe (2009) One of the main result of Chari and Kehoe s analysis is that a "too-big-to-fail" cap on the size of rms (in our context, entrepreneurs) via ex-ante regulation is an e ective tool to alleviate the time-inconsistency of a bailouts. To make this point, these authors build a model in 22

23 which bankruptcies are part of a mechanism to provide incentives to managers to exert high e ort. Bankruptcies are socially costly, so the policy maker has ex-post incentives to prevent them with bailouts, but such bailouts undermine the ex-ante incentives of managers to exert high e ort. A cap on rms size decreases the social costs of bankruptcies, so smaller bailouts may be promised and higher welfare is achieved. We show below that an important missing element in this story is Collective Moral Hazard. Once added, a too-big-to-fail policy may be largely ine ective. This is because a cap on rms size generates idle capital in the larger rms, which may go to open new rms or increase the size of the existing smaller rms. Collective Moral Hazard refers to the correlation of aggregate risk among rms in equilibrium, so the total amount of resources engaged in a single risky strategy remains about the same. This is the variable that matters for the ex-post incentives to implement larger-than-promised bailouts. As result, a too-big-to fail policy may have little e ect on supporting smaller sustainable bailouts. Unfortunately, to studying Collective Moral Hazard in the Chari and Kehoe s environment would force us to substantially modify our baseline model. However, for illustrative purposes, it su ces to rst show that a size cap increases the optimal sustainable policy when one entrepreneur can only own one rm, which recovers Chari and Kehoe s result. Then, we relax this assumption to show that the e ect of the cap on optimal sustainable policy vanishes Introducing a too-big-to-fail policy Assume that there is a cap i on the investment size. We focus on the case when this cap is binding, i.e., i (1) > i with i (1) de ned in (16). Assume also that an entrepreneur can own only one " rm". A rm is de ned as a decision unit that inherits from its owner the technology to invest in risky assets. In this environment, aggregate investment is simply i for any level of expected policy R e t 2 [ 0 ; 1]. Therefore, condition (22) for a sustainable bailouts becomes 8 < : h ^V (R) i ^V (R e t) [ ( 1 0 ) (1 0 )] R Re t i R 0 9 = ; ( punishment t+1 (i ) reward t+1 (R; i ) The cap i alleviates the time-inconsistency of bailouts by two channels: ) 8R e t R: (i) The momentary welfare cost of following a committed policy R when expected policy is R e t < R is decreasing in i (left-hand side). This is because the e ect of R on downsizing is proportional to the investment scale i. 23

24 (ii) A more subtle channel is that i increases the di erence between the punishment and the reward (right-hand side). The larger the expected bailout (the lower Rt), e the larger is the desired size of rms. Firms expect a larger bailout in the punishment scenario, so the cap i is more binding for this case. The result follows from noticing that, everything else constant, a higher investment scale increases ex-ante welfare. These two channels imply that a cap i relaxes this condition, so the optimal sustainable policy is increasing in the cap i Allowing entrepreneurs to own multiple rms We now allow entrepreneurs to use their idle capital. Given that entrepreneurs leverage is not restricted, we can use equation (16) to de ne a threshold A (R e t; i ) = [1 + (1 ) R e t 0 ] i : This is the level of "capital" (entrepreneur s endowment) such that the desired investment scale i (Rt) e equals the cap i. A (Rt; e i ) is increasing in Rt e because the larger Rt, e the larger is the rms liquidity hoarding, so the more capital is needed to attain scale i. The assumption that i (1) > i is equivalent to A (1; i ) < A, where A is the total amount of entrepreneurs endowment. Entrepreneurs idle capital is then A A (Rt; e i ). If entrepreneurs are allowed to use this capital to open another, potentially smaller rm, the total amount of risky investment in the economy is i + i (R e t; A A (R e t; i )) where, abusing of notation, i Rt; e ^A is the size of a rm with capital ^A < A (Rt; e i ): i R e ; ^A = ^A 1 + (1 ) R e 0 : By de nition i (R e t; A (R e t; i )) = i, so total investment is i (R e t; A) which is independent of the cap i and equal to the total desired investment. In principle, large and small rms may have di erent investment strategies, but Collective Moral Hazard implies that in equilibrium all rms choose to get exposed to the same distress state. The 24