Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/collectiveriskmaOOcaba 1B31 Massachusetts Institute of Technology Department of Economics Working Paper Series Collective Risk Management in a Flight to Quality Episode Ricardo J. Caballero Arvind Krishnamurthy Working Paper 07-02 January 29, 2007 RoomE52-251 50 Memorial Drive Cambridge, 02142 MA This paper can be downloaded without charge from the Network Paper Collection at Social Science Research httsp://ssrn.com/abstract=960520 I MASSACHUSETTS TITUTE Kit- Management Collective Risk in a Flight to Quality Episode Ricardo J. Arvind Krishnamurthy* Caballero January 2007 29, Abstract We son present a model of optimal intervention in a flight to quality episode. for intervention model make their own risk stems from a management shocks, but are uncertain of how is They understand correlated their shocks are with systemwide shocks, treating the latter uncertainty as Knightian. uidity We show that when aggregate liq- low, an increase in uncertainty leads agents to a series of protective actions - decreasing risk exposures, hoarding liquidity, locking-up capital - that reflect a to quality. flight However, the conservative actions of agents leave the aggregate economy over-exposed to negative shocks. Each agent covers himself against his own scenario, but the scenario that the collective of agents are guarding against A rea- Agents in the collective bias in agents' expectations. decisions with incomplete knowledge. The lender of last resort, even if less is worst-case impossible. knowledgeable than private agents about individual shocks, does not suffer from this collective bias and finds that pledging intervention in extreme events JEL Codes: is valuable. The intervention unlocks private capital markets. E30, E44, E5, F34, Gl, G21, G22, G28. Keywords: Locked collateral, liquidity, flight to quality, Knightian uncertainty, collateral shocks, collective bias, lender of last resort, private sector multiplier. 'Respectively: MIT and NBER; Northwestern University. E-mails: caball@mit.edu, a-krishnamurthy@northwestern.edu. We are grateful to Marios Angeletos, Olivier Blanchard, Phil Bond, Jon Faust, Xavier Gabaix, Jordi Gali, Michael Golosov, William Hawkins, Burton Hollifield, Bengt Holmstrom, Dimitri Vayanos, Ivan Werning and seminar participants Imperial College, LBS, Northwestern, MIT, Wharton, Conference, Bank of England, Fed Liquidity Conference, and the Central Bank of Chile research assistance. Caballero thanks the replaces) "Flight to Quality NY NSF for their for financial support. and Collective Risk Management," NBER MIDM at Columbia, DePaul, Conference, UBC Summer comments. David Lucca and Alp Simsek provided excellent This paper covers the same substantive issues as (and hence NBER WP # 12136. Introduction 1 Policy practitioners operating under a risk-management paradigm may, at "... times, be led to undertake actions intended to provide insurance against especially When adverse outcomes uncertainty, human term commitments confronted with uncertainty, especially Knightian beings invariably attempt to disengage from in favor of safety and medium to long- The immediate response on liquidity... the part of the central bank to such financial implosions must be to inject large quantities of liquidity..." Alan Greenspan (2004). Flight to quality episodes are an important source of financial bility. Modern examples of these episodes in the US include the and macroeconomic Penn Central insta- default of 1970; the stock market crash of 1987; the events of the Fall of 1998 beginning with the Russian default and ending with the bailout of LTCM; Behind each of these episodes of 9/11. lies as well as the events that followed the attacks the specter of a meltdown that prolonged slowdown as in Japan during the 1990s, or even a catastrophe pression. 1 In this paper we much Great De- as needed to prevent a meltdown. present a model to study the benefits of central bank actions during flight to quality episodes. Our model has two key ingredients: capital/liquidity shortages Knightian uncertainty. The capital shortage ingredient and theoretical is like the lead to a In each of them, as hinted at by Greenspan, the Fed intervened early and stood ready to intervene as tainty may literature commonly less on financial crises a recurring theme in the empirical is and requires and little studied, but practitioners perceive it motivation. Knightian uncer- as a central ingredient to flight to quality episodes (see Greenspan's quote). Most flight to quality episodes are triggered by unanticipated or unexpected events. In 1970, the Penn-Central Railroad's default on prime rated commercial paper caught the market by surprise and forced investors to re-evaluate their models of credit risk. The ensuing dynam- temporarily shut out a large segment of commercial paper borrowers from a vital source ics of funds. In October 1987, the speed of the stock market decline took investors and market markers by surprise, causing them to question their models. Investors pulled back from the market while key market-makers widened bid-ask spreads. In the : See Table 1 (part A) during the 20th century. in Barro (2005) for a comprehensive list fall of 1998, the comovement of extreme events in developed economies of Russian government bond spreads, Brazilian spreads, and U.S. Treasury bond spreads was a surprise to even sophisticated market participants. These high correlations rendered stan- dard risk management models obsolete, leaving financial market participants searching new models. Agents responded by making decisions using "worst-case" scenarios testing" models. Finally, after 9/11, regulators were concerned that commercial and for "stress- banks would respond to the increased uncertainty over the status of other commercial banks by individually 2 hoarding liquidity and that such actions would lead to gridlock in the payments system. aspects of investor behavior across these episodes - re-evaluation of models, The common conservatism, and disengagement from risky activities - indicate that these episodes involved Knightian uncertainty and not merely an increase in risk. The emphasis on tail outcomes and worst-case scenarios in agents' decision rules suggests uncertainty aversion rather than simply risk aversion. It is also seldom repeats itself. noteworthy that when it comes to flight to quality episodes, history Similar magnitudes of commercial paper default (Mercury Finance in 1997) or stock market pullbacks (mini-crash of 1989) did not lead to similar investor responses. Today, as opposed to in 1998, market participants understand that correlations should be expected to rise in lending to during periods of reduced liquidity. Creditors understand the risk involved hedge funds. the recent default of the The While in 1998 hedge funds were still Amaranth hedge fund barely caused a a novel financial vehicle, ripple in financial markets. one-of-a-kind aspect of flight to quality episodes supports the view that uncertainty important ingredient is an in these episodes. Section 2 of the paper lays out a model of financial crises based on liquidity shortages and Knightian uncertainty. in We analyze the model's equilibrium and show that an increase Knightian uncertainty or decrease in aggregate liquidity can reproduce effects. When Knightian uncertainty leads agents to make decisions based on worst-case scenarios. the aggregate quantity of liquidity is limited, the Knightian agent grows concerned that he will be caught in a situation available to him. In this context, agents react of safe liquidity, but there by shedding is not enough liquidity risky financial claims in favor Investment banks and trading desks turn conservative They The main 2 where he needs and uncontingent claims. Financial intermediaries become self-protective and hoard liquidity. capital. flight to quality lock up capital and become unwilling to flexibly results of our paper are in Sections 3 and 4. in their allocation of risk move it across markets. As indicated by Greenspan's See Calomiris (1994) on the Penn-Central default, Melamed (1988) on the 1987 market crash, Scholes (2000) on the events of 1998, and Stewart (2002) or Ashcraft and Duffie (2006) on 9/11. comments, the Fed has We historically intervened during flight to quality episodes. analyze the macroeconomic properties of the equilibrium and study the effects of central bank actions environment. in our First, we show that Knightian uncertainty leads to a collective bias Each agent covers himself against in agents' actions: own his scenario that the collective of agents are guarding against so despite agents' uncertainty about the environment. is We worst-case scenario, but the known impossible, and to be show that agents' conservative actions such as liquidity hoarding and locking-up of capital are macroeconomically costly as they exacerbate the aggregate shortage of We to alleviate the over-conservatism. and improve outcomes. capital when show that a lender same incomplete knowledge that facing the agents' liquidity is It Central bank policy can be designed liquidity. of last resort (LLR), even one triggers agents' Knightian responses, can unlock does so by committing to intervene during extreme events depleted. Importantly, because these extreme events are unlikely, the expected cost of this intervention is minimal. If credible, the policy derives its power from a private sector multiplier: each pledged dollar of public intervention in the extreme event matched by a comparable private bailout was important not sector reaction to free for its direct effect, but because readiness to intervene should conditions worsen. The model value of the of bank environment is LLR facility is usually The LLR runs. up The capital. In this sense, the LTCM served as a signal of the Fed's it signal unlocked private capital markets. analyzed in terms of Diamond and Dybvig's rules out the "bad" run equilibrium in their model. a variant of Diamond and Dybvig's, it (1983) While our does not include the sequential service constraint that leads to a coordination failure and informs their discussion of policy. benefit of the LLR in in agent decisions is our model derives from a different mechanism, which is The the collective bias caused by an increase in Knightian uncertainty. Our model also provides a clear presciption to central banks on when to intervene in financial markets: The benefit of the LLR is highest when there is both insufficient aggregate liquidity and Knightian uncertainty. We also show that the LLR must be a last-resort policy: If liquidity injections take place too often, the policy exacerbates the private sector's mistakes and reduces the value of intervention. This occurs for reasons akin to the moral hazard problem identified with the LLR. Holmstrom and Tirole (1998) study liquidity provision (see also ment can issue how a shortage of aggregate collateral limits private Woodford, 1990). Their analysis suggests that a credible govern- government bonds which can then be used by the private sector provision. The key Woodford, is that difference between our paper and those we show how even aggregate collateral of Holmstrom and may be for liquidity Tirole, and inefficiently used, so that private sector liquidity provision is In our model, the limited. only adds to the private sector's collateral but also, and more government intervention not centrally, it improves the use of private collateral. Routledge and Zin (2004) and Easley and O'Hara (2005) are two related analyses of Knightian uncertainty in financial markets. 3 Routledge and Zin begin from the observation that financial institutions follow decision rules to protect against a worst case scenario. They de- velop a model of market liquidity in which an uncertainty averse market maker sets bids and asks to facilitate trade of an asset. quality: Their model captures an important aspect of flight to uncertainty aversion can lead to a sudden widening of the bid-ask spread, causing agents to halt trading and reducing market liquidity. Both our paper and Routledge and Zin share the emphasis on financial intermediation and uncertainty aversion as central ingredients But each paper captures in flight to quality episodes. different aspects of flight to quality. Easley and O'Hara (2005) study a model where ambiguity (uncertainty) averse traders focus on a worst case scenario when making an investment decision. Like us, Easley and O'Hara point out that government intervention in a worst-case scenario can have large effects. Easley and O'Hara study how uncertainty aversion while the focus of our study Finally, in our is affects investor participation in stock markets, on uncertainty aversion and financial crises. model agents are only Knightian with respect to systemic (2001) explores the home more uncertain about events. Epstein bias in international portfolios in a related setup, where agents are foreign than local markets. 4 As Epstein points highlights the difference between high risk aversion out, this modelling also and aversion to Knightian uncertainty. Moreover, our modelling shows that max-min preferences interact with macroeconomic conditions in We ways that are not present show that when aggregate differ, 3 liquidity However when there identically. in is models with an invariant amount of is plentiful, risk aversion. Knightian and standard agents behave an aggregate liquidity shortage, the actions of these agents leading to flight to quality in the Knightian model. There is a growing economics literature that aims to formalize Knightian uncertainty (a partial contributions includes, Gilboa and Schmeidler (1989), Dow and Werlang (1992), Epstein and Wang list of (1994), Epstein (2001), Hansen and Sargent (1995, 2003), Skiadas (2003), Epstein and Schneider (2004), and Hansen, et al. (2004)). As in much of this literature, treatment of Knightian uncertainty case 4 among a is we use a max-min device to describe agents expected utility. most similar to Gilboa and Schmeidler, in that agents class of priors. Epstein's model is closely related to our model in Caballero and Krishnamurthy (2005). Our choose a worst The Model 2 We study a model conditional on entering a turmoil period where liquidity risk and Knightian Our model uncertainty coexist. is silent on what triggers the episode. In practice, we think Penn Central that the occurrence of an unexpected event, such as the agents to re-evaluate their models and triggers robustness concerns. model to study the default or 9/11, causes Our goal to present a is role of a centralized liquidity provider such as the central bank. The environment 2.1 PREFERENCES AND SHOCKS The model has a continuum An may agent of competitive agents, specialist (e.g. a trading desk, first- wave 00,(1), and note With With wave The first randomly chosen group denote the probability of agent u> wave on average, across is probability 0(2| 1), We uj by a of one-half of the agents ^. all first wave (1) agents, the probability of an agent receiving a of being in this second — 0(1) > wave is Let 0(2) W (2), which = 0(1)0(2|1). satisfies, <w = *®. (2) l the economy experiences no liquidity shocks. note that the sequential shock structure means that, 0(1) > 0(2) > 0. This condition states that, in aggregate, a single-wave event event. hit a second wave of liquidity shocks hits the economy. In the second n 1 is ^p. probability for agent probability economy receiving a shock in the of liquidity shocks, the other half of the agents need liquidity. With We that, (1) states that shock in the [0, 1]. liquidity immediately. probability 0(1), the Ul)dw = Equation Q = € hedge fund, market maker). of liquidity shocks. In this wave, a We u> a sudden need for capital by a financial market for are correlated across agents. have liquidity needs. by some receive a liquidity shock in which he needs view these liquidity shocks as a parable The shocks which are indexed by We will refer to the (3) is more likely than the two-wave two-wave event as an extreme event, capturing an unlikely but 6 many severe liquidity crisis in which Relation agents are affected. (3), deriving from the sequential shock structure, plays an important role in our analysis. We Agent 1983). ui receives utility: U w (c With Q'i = 1, Q2 = second wave; and, a.\ shock date as "date u : 1Z+ — s- x , c 2 ,c T ) if the agent = 0, 1," if + a 2 u(c 2 + ot\u{c\) ) in the early wave; a\ is a2 = = the agent is = and the trader c 2 reflect the potentially ENDOWMENT AND of one, is = 1 if the agent is in the shock. We will refer to the first and the date as "date TV" final c\ and in the future, effectively risk neutral. c2 and linear over Cf. when market conditions The concave preferences over c\ and higher marginal value of liquidity during a time of market distress. SECURITIES endowed with is (4) . a2 0, Preferences are concave over oo. view the preference over ct as capturing a time, Each agent 2," Bc T twice continuously differentiable, increasing, strictly concave and satisfies 72. is are normalized = not hit by a the second shock date as "date the condition lim c _>o u'(c) We Diamond and Dybvig, liquidity shock as a shock to preferences (e.g., as in model the and then liquidated if Z units of goods. These goods can be stored at gross return an agent receives a liquidity shock. the model as financial traders, we may think of Z If we interpret the agents of as the capital or liquidity of a trader. Agents can also trade financial claims that are contingent on shock realizations. As we show, these claims allow agents who do not receive a shock to insure agents who do will receive a shock. We will assume all shocks are observable and contractible. There is no concern that an agent pretend to have a shock and collect on an insurance claim. Markets are complete. There are claims on all contingent claims histories of shock realizations. when we analyze the We will be more precise in specifying these equilibrium. PROBABILITIES AND UNCERTAINTY Agents trade contingent claims to insure against their liquidity shocks. In making the insurance decisions, agents have a probability model of the liquidity shocks in mind. We assume that agents know the aggregate shock probabilities, (f)(1) and (f>(2). We may think that agents observe the past behavior of the economy and form precise estimates of these aggregate probabilities. However, and centrally to our model, the not reveal whether a given u> is more likely to treat the latter uncertainty as Knightian. be in the first wave same past data does or the second wave. Agents Formally, we use <fi u (l) to denote the true probability of agent and 0^(1) to denote agent-u/s perception and the </£j(2)). first We assume that each agent or second wave, </> + w (l) receiving the first shock, of the relevant true probability (similarly for to knows his probability of receiving a W (2) shock either in 5 W (2), and thus the perceived probabilities satisfy: C(l) + C(2) = We uj Ml) + MV = 0(1) (2) t^ (5) - define, % = C(2) - ^). That is, 9^ reflects how much agent (6) a/s probability assessment of being second is higher than the average agent in the economy's true probability of being second. This relation also implies that, -% = C(i) - ^y~- Agents consider a range of probability-models 9" (K < 4>{2)/2)), and design insurance in the set 0, with support portfolios that are robust to their follow Gilboa and Schmeidler's (1989) Maximin Expected [—K,+K] model uncertainty. We Utility representation of Knightian uncertainty aversion and write: max (ci,c 2> c T where mm£ )0£ee K captures the extent of agents' [[Hci,c2 ,cT )|^] (7) uncertainty In a flight to quality event, such as the Fall of 1998 or 9/11, agents are concerned about systemic risk and unsure of how this risk will impinge on their good understanding of their own markets, but are unsure other markets in the may affect of activities. how They may have a the behavior of agents in them. For example, during 9/11 market participants feared gridlock payments system. Each participant knew how much he owed to others but was unsure whether resources owed to him would arrive (see, e.g., Stewart, 2002, or Ashcraft and Duffle, 2006). In our modeling, agents are certain about the probability of receiving a shock, but are uncertain about the probability that their shocks will occur early relative to others, or late relative to others. 5 For further clarification of the structure of shocks and agents' uncertainty, see the event tree that detailed in the Appendix. is We view agents' max-min preferences The in (7) as descriptive of their decision rules. widespread use of worst-case scenario analysis in decision making by financial firms example an is of the robustness preferences of such agents. SYMMETRY To simplify our analysis agent's true Q w may be The symmetry we assume that the agents common applies in other dimensions as well: We common is drawn from the same 9. is <p u While each 0. , K, Z, and u(c) are the same knowledge. As noted above, cf>(l) for all and 0(2) are also knowledge. A 2.2 every agent different, the 9^ for u. Moreover, this information are symmetric at date benchmark begin by analyzing the problem insurance that is when economy valuable in our K= 0. as well. This case We a planning problem, where the planner allocates the will clarify the nature of cross- derive the equilibrium as a solution to Z across agents as a function of shock realizations. Figure 1 below describes the event tree of the economy. The economy two waves one, or two wave case, or of shocks. may An agent u> may be is denoted by C s affected in the first or second be affected or not affected in the one wave event. Agent w's allocation of waves, u/s shock) as the state for agent u. where in the event of agent may We receive zero, wave denote s in the = ( # as a function of the state being affected by a shock, the agent receives u> a consumption allocation upon incidence of the shock, as well as a consumption allocation For example, at date T. affected C by the = {C } is We note s if the economy is wave, we denote the state as first the consumption plan for agent that c\ is the same in ui An by two waves s = (2, 1) of shocks in which agent and agent w's allocation u> is as (c\, cf ). (equal to that for every agent, by symmetry). both state sequential shock structure in the economy. at that time, hit (2, 1) agent and state who (1,1). This is because of the receives a shock first needs resources and the amount of resources delivered cannot be made contingent on whether the one or two wave event transpires. Figure and K= 0w(2) wave = is 1 0, also gives the probabilities of each state s. Since agents are ex-ante identical each agent has the same probability of arriving at state 0(2)/2, which implies that the probability of one-half. Likewise, the probability of u u) s. Thus we know that being hit by a shock in the second being hit by a shock in the first wave is one-half. These computations lead to the probabilities given = (# s 2 waves, w's shock) 1. C s p s 1 (2.1) #2)/2 (d.4' (2.2) 0(2)/2 (c 2 Waves Prob ) 4>{2) Prob 1 - No Shocks 0(1)\ Figure The in Figure planner's problem is 1: (1,1) fo(l)-#2))/2 (l,no) (0(1) (0jno) x - _ 0(2))/2 ,4 ) 1 (d.4- (c^" o,n (c 0(1) 2 ) ) 0) Benchmark Case to solve, max^p s UJ t/ (C s ) subject to resource constraints that for every shock realization, the promised consumption amounts are not more than the total 1 - / \C\ endowment I (Ci + 1>1 CT " + I of Z: „0,no < z l,nO\ < * Z Cr ' J 2,2\ + C 2.1 T + C 2 + Cy . , ' J -. < rj ry Z, Zj as well as non- negativity constraints that for each 5, every consumption amount in C s is non-negative. It is for We obvious that consumption if shocks do not occur, then the planner will give at date T. rewrite the problem max 0(1) — 0(2) Thus.c T'"° =Z and we can drop Z to each of the agents this constant from the objective. as: , (u(ci) no + pcyn + /Jc^ J ! H subject to resource and non-negativity constraints. 10 — 0(2) v , (u(ci) + u(c 2 ) o 2,2\ + /3c2,1 r + /?c^ J , ' Observe that we of generality constraints. 1 c^' set c^ Without {Cl,C2,Cj enter as a c^'" = 1 sum + 4>{l)u{ Cl ) we both objective and constraints. Without in and Likewise, cj 0. loss of generality max \,no I subject and set Cj. T ' — 0. sum in both objective and The reduced problem 2 <K2)(u(c2) 2,2\ ,C enter as a cT + /3c /) + (0(1) - loss is: 0(2)) f3cY° ) to: c\ = oIZr/ 1,710 + , C\ Cj. + c2 + l,no ' Cl,C 2 ,C T = 2Z Cji 2,2 n v^ > ,C T (J. The Note that the resource constraints must bind. solution hinges on whether the non- negativity constraints on consumption bind or not. If the non-negativity constraints do not bind, then the first order condition for c\ and c2 yield: = c2 = 2(Z — c*), c1 The u'-\P) = c*. solution implies that, = Cy Thus the non-negativity lo Cr/ constraints do not bind sufficient aggregate liquidity. When Z plan in which marginal utility is is if = 2Z — Z > c*. We c*. refer to this case as large enough, agents are able to finance a equalized across all states. the marginal utility of early consumption with that of date one of consumption At the optimum, agents equate T consumption, which is (5 given the linear utility over ct- Now consider the case in which there to achieve full insurance. case (i.e. that c 2 = use cj no all first is — 2Z — Ci Z < towards shock and the problem 0(l)n(c 1 ) insufficient liquidity so that agents are not able the case where of the limited liquidity max with This is c* . It is states). obvious that c^ Thus, for 2 a given = c\ in this we have is, + 0(2)u(2Z-c order condition, 11 1 ) + (0(l)-0(2))/3(2Z-c 1 ) (8) Since u'(2Z — c\) > (3 (i.e. c2 f3 The who last inequality are affected by the < < c*) we can < u'(ci) u'(2Z on the right of (10) first- wave by the second-wave. There is is - the important result from the analysis. Agents more first of the limited liquidity to the more shocks. This because there Proposition • 1 is is scarce (small Z) On of each agent is are affected because the (or, a chance the economy it is optimal to allocate the other hand, when liquidity is not contingent on the order of the is enough liquidity to cover all shocks. results as follows: The equilibrium in The economy has is likely shock. Z) the liquidity allocation summarize these who than that of the first-wave wave having taken place there plentiful (large We liquidity than agents strictly smaller is spared of a second wave). Thus, when liquidity is (10) cross-insurance between agents. Intuitively, this equivalently, conditional on the more ci>Z. =» ci) of shocks receive probability of the second-wave occurring is order: the benchmark economy with insufficient aggregate liquidity if c* Agents are partially insured against > c\ > Z > Z< c2 c* . K= has two cases: In this case, . liquidity shocks. First wave liquidity shocks are more insured than second wave liquidity shocks. • The economy has sufficient aggregate liquidity if Cj and agents are = c2 c* . In this case, c* fully insured against liquidity shocks. Flight to quality effects, and a role for central (insufficient aggregate liquidity). 2.3 = Z> This is the case bank intervention, we analyze arise only in the first case in detail in the next sections. Implementation There are two natural implementations of the equilibrium: financial intermediation, and trading in shock-contingent claims. 12 In the intermediation implementation, each agent deposits and receives the right to withdraw > c\ Z, if Z in an intermediary initially he receives a shock in the first-wave. Since shocks are fully observable, the withdrawal can be conditioned on the agents' shocks. Agents who do — [Z not receive a shock in the < C\ The second group Ci). wave own claims first T and at date all redeem of agents either second wave of shocks, or at date T. Finally, if to the rest of the intermediary's assets their claims upon incidence no shocks occur, the intermediary is of the liquidated agents receive Z. In the contingent claims implementation, each agent purchases a claim that pays 2(ci Z) > the event that the agent receives a shock in the in — Z) identical claim to every other agent, paying 2(ci that this If is no shocks occur, agents consume their own Z. wave), to net first later agent has T as date wave. in case of the first Z— If — Z) = 2Z — (cj if there also is c\ units of liquidity to either finance a second shock, or is sufficient aggregate liquidity either the intermediation or Moreover, implementable by self-insurance. Each agent keeps his to finance a shock. The self-insurance implementation is Z in this case, and liquidates not possible when because the allocation requires each agent to receive more than his endowment of is 2.4 robustness case turn to the general problem when K > 0. Once again, by solving a planning problem where the planner allocates the shocks. Z < c*, Z if the hit first. K> We now first consumption. the allocation agent an wave shock. Note an agent receives a shock in the contingent claims implementation achieves the optimal allocation. < Z sells — Z) and pays out c\ — Z (since one-half of the agents are affected in c\ — Z. Added to his liquidity of Z, this gives total liquidity of c\. Any Finally, note that c* The agent a zero-cost strategy since both claims must have the same price. wave, he receives 2(c\ the first — When K > 0, agents This decision making process is make Z we derive the equilibrium to agents as a function of decisions based on a "worst-case" for the probabilities. encompassed planning problem by altering the planners in the objective to, raaxmmyp The only change the worst-case in the min problem relative to the rule. 13 s ' a, K [/(C = s (11) ) case is that probabilities are based on Figure 2 redraws the event tree the notation that 0^(2) is is unsure indicating agent's worst-case probabilities. when the economy is use going through a two-wave event in which other agents' shocks are going to occur before or after agent w's. if s 2 We agent u/s worst-case probability of being hit second. In our setup, this assessment only matters the agent now = (# waves, o/s shock) 6 b V (2^) Waves Prob 0(2) (2,2) Prob 1-^(1) simplify the problem following be equal to Z. Since the problem observe that c^ 1 (0(l)-0(2))/2 (l,no) (0(l)-0(2))/2 1-0(1) (0,no) Figure We (1,1) and c^"° enter some in the as Robustness Case 2: sum The reduced problem is V(CM = maxmin 0-(l)u( Cl )+(0(2) O.no of the steps of the previous derivation. c^"° one-wave node is must = both objective and constraint and choose c^ in we the same as in the previous case, 0. then, - 0^(2)) /?# +0^(2) (n(c 2 ) + ^ 2 ) ^ (1) + 0(2) l.no /3 (12) The is first two terms in this objective are the utility hit first (either in the consumption bundle when 6 the agent We is if one wave or two wave event). The third term the agent is derive the probabilities as follows, p -0£ (2) = #S(1)- Substituting forp 2 ' 1 The hit second. last term is is the agent if the utility from the the utility from the bundle not hit in a one- wave event. since the probabilities have to 0(2) from the consumption bundle -", sum up * (1) ~ 0(2) we which we can use to solve 2" 2-" = 2 <^(2) by definition. This implies that p to the probability of a using relation (5). The can rewrite this to find that? for p 1 ' 710 two wave event probability of w being hit 1 ' 1 '" ^ 14 = &ii±&2l. Finally, (0(2)). ' 1 '" We = 4>{2) = p = 0£(1) ' - <j%{2) rewrite p '"+p 1 no w first is 1 1 - ' 2 ' 1 <"J 2 < l +p < 1 w ' 1 = ' w . 0(l)-^(2), The resource constraints for this problem are, Cl c\ The optimization Proposition 2 + + 2,1 , c2 +4' no < 2Z Cni ' + . ^ < 2,2 c^ r,n> 2z. also subject to non-negativity constraints. is Let: R^ f(l) - 0(2) 4 /V(Z) V /? u 'i z ) K Then, the equilibrium in the robust economy depends on both • When there — For is Z as follows: insufficient aggregate liquidity, there are two cases: K < and < K, agents' decisions satisfy: C(lK(ci) = ^(2)u'(q > )-+/? * (1) ~ where, the worst-case probabilities are based on 6^ m ' . (13) = +K: In the solution, c2 < Z < c\ < c* We with C\{K) decreasing and C2(K) increasing. refer to this as the robust" case. — For K > K , agents' decisions are as if Ci We • When K=K = Z= C2 < , and C* refer to this as the "fully robust" case. there is sufficient aggregate liquidity (Z), agents' decisions satisfy, C\ = C2 15 = C* < Z. "partially The formal proof of the proposition to account for show controls, the Appendix, and complicated by the need is when possible consumption plans for every given &£ scenario all max-min problem. But, We in the is in the there is a simple intuition that explains the Appendix that c^ problem simplifies max _ min and c^ , solving the results. Dropping these are always equal to zero. to: 0£(l)u(ci) + ^(2)u(c 2 + ) m ~ ^~ 4>(1) - 3 -(3c , l,no C\,C2,Crp For the case of insufficient aggregate c2 Then the first liquidity, the resource constraints give: = 2Z-c 1} order condition for the 4' no max problem = 2Z-c a . for a given value of &£ is, >(1)"0(2) ^(1K( Ci = ^(1K(c2 )+/32 ) benchmark In the case, the uncertain probabilities are </>"(l) yields the solution calling for K> When 0, liquidity to decisions. Thus, they bias K is whoever is affected by the </£J(2) first = ^jp, shock [c\ which > C2). agents axe uncertain over whether their shocks are early or late relative to other Under the max-min decision agents. When more = ^- and agents use the worst case probability in making rule, up the probability small, agents' first order condition of being second relative to that of being first. is, (m- K M =(m +Ky, M+p t>m - «2) y As K becomes larger, c 2 increases toward This defines the threshold of K. In this c\. For K is set equal to c\. "fully robust" case, agents are insulated against their uncertainty over whether their shocks are likely to be 2.5 sufficiently large, c 2 first or second. Flight to quality A flight to quality K or Z. Let us fix episode can be understood in our model as a comparative static across a value of K and Z and suppose that contractual arrangement as dictated by Proposition (non-contracted) event that increases We may K (or 2. at a date At date 0, — 1, there agents enter into a is an unanticipated reduces Z) and leads agents to rewrite contracts. think of a flight to quality in terms of this rewriting of contracts. 16 In this subsection we discuss three concrete examples of flight to quality events in the Our context of our model. first two examples identify the model in terms of the financial intermediation implementation discussed earlier, while the last example identifies the model in terms of the contingent claims implementation. The the example first of 1998. fall We one of uncertainty-driven contagion and is interpret the agents of our Each trading desk concentrates bank. model drawn from the events is as the trading desks of in a different asset market. of an investment At date —1 the trading desks pool their capital with a top-level risk manager of the investment bank, retaining C2 of capital to cover They any needs that may arise in their particular also agree that the top-level risk manager shocks that hit whichever market needs capital agent in an unrelated market - An defaults. then before, so that likely </>£(l) + 0£(2) other trading desks will suffer shocks i.e. first an extra c\ ("trading capital"). — capital"). > C2 At date a market in which shocks are to cover 0, Russia now no more unchanged - suddenly becomes concerned that is first will provide market ("committed and therefore the agent's trading desk will not have as much capital available in the event of a shock. The agent responds by lobbying the top-level risk manager to increase every trading desk now has became up more committed capital up to a less capital in argues that during the 1998 desks) his crisis, the (likelier) level of ci = c\. As a result, event of a single shock. Scholes (2000) the natural liquidity suppliers (hedge funds and trading liquidity demanders. In our model, uncertainty causes the trading desks to tie of the capital of the investment bank. shocks, suggesting reduced liquidity in all The average market has less capital to absorb markets. In this example, the Russian default leads to less liquidity in other unrelated asset markets. Gabaix, Krishnamurthy, and Vigneron (2006) present evidence that the mortgage-backed securities market, a and wider spreads effect if Z is market unrelated to the sovereign bond market, suffered lower in the 1998 crisis. Note also that in this example there large as the agents' trading desk will not be concerned capital to cover shocks when Z > c*. is no contagion about having the necessary Thus, any realized losses by investment banks during the Russian default strengthen the mechanism Our second example is liquidity we highlight. a variant of the classical bank-run, but on the credit side of a commercial bank. The agents of the model are corporates. The corporates deposit commercial bank at date receive a shock. wave — 1 and The corporates of shocks) the bank sign revolving credit lines that give are also aware that will raise lending if them the right to c\ Z in a if they banking conditions deteriorate (a second standards/loan rates so that the corporates will 17 effectively receive only c 2 suffering losses < The C\. flight to quality than when periods by the commercial bank will transpire. credit lines, effectively leading all firms to receive Gatev and Strahan (2006) present evidence C\. triggered is and corporates becoming concerned that the two-wave event They respond by preemptively drawing down less event of this sort of credit-line run during the spread between commercial paper and Treasury bills widens (as in the fall of 1998). The last example is agents of the model are one of the interbank market all commercial banks who have Each commercial bank knows that there from reserve account, which its that bank A is for liquidity can it is some by offset and the payment system. The Z Treasury bills at possibility that selling Treasury it will suffer To bills. advantageous terms (worth at worse terms of c 2 A is a bank outside (say, C\ of T-bills). If a second set of shocks the discount window). At date New York City which is 0, first, hits, its which and possible shock at 4pm. In first. this banks receive credit A is becomes concerned may arise before 4pm. Rather, it will hoard its Treasury Bills of Z to cover example, uncertainty causes banks to hoard resources, often the systemic concern in a is receives credit on interbank lending arrangements and be unwilling to provide will renegotiate its to any banks that receive shocks own suppose not directly affected by the events, but which that other commercial banks will need liquidity and that these needs c\ fix ideas, 9/11 occurs. Suppose that bank concerned about a possible reserve outflow at 4pm. However, now bank A a large outflow worried about this happening at 4pm. At date —1, the banks enter into an interbank lending arrangement so that a bank that suffers such a shock Then, bank the start of the day. payments gridlock (e.g., Stewart, 2002, and Ashcraft Duffie, 2006). The can be different interpretations mapped into the actors financial system. As is we have offered show that the model's agents and and actions during a apparent, our environment is flight to quality their actions episode in a a variant of the one that modern Diamond and Dybvig (1983) study. In that model, the sequential service constraint creates a coordination failure and the possibility of a In our model, the crisis realization of the is bad crisis equilibrium in which depositors run on the bank. an unanticipated rise in bad equilibrium. Our model of the rewriting of financial contracts triggered behavior of a bank's creditors. 7 There is Knightian uncertainty rather than the also offers interpretations of a crisis in terms by the uncertainty increase, rather than the 7 an underlying connection between the crisis in our model and that of Diamond and Dybvig (1983). As Diamond and Dybvig emphasize, a deposit contract implements an optimal shock-contingent allocation 18 of Collective Bias 3 In this section, we study the and the Value of Intervention benefits of central bank actions in the flight to quality episode of our model. Since the agents of the model are not standard expected utility maximizers (i.e. adhering to the Savage axioms), there are delicate issues that arise when defining the central bank's objective. On the one hand, the planner in (11), then since the central bank's objective if is the min-max we have previously derived the equilibrium a planner's problem, the central bank cannot improve on the allocation. throughout this section, we argue for objective of as the solution to On the other hand, a paternalistic central bank objective which does not We incorporate agents' worst-case probability assessments. paternalistic policies that arise in our setting. argue that there are "natural" 8 Collective bias 3.1 In the fully robust equilibrium of Proposition 2 agents insure equally against shocks. To conditions first and second arrive at the equal insurance solution, robust agents evaluate their first order (equation 13) at conservative probabilities: « (1) . ffi Suppose we compute the probability .iM(S] of (14) one and two aggregate shocks using agents' con- servative probabilities: 0(1) The two by each =2 f 0(2) <f>Z(l)dw, = 2 in front of these expressions reflects the fact that of the shocks. Integrating equation (14) liquidity. In the fully robust equilibrium of / <%(2)dw. only one-half of agents are affected and using the definitions above, we our model, agents choose an allocation that is find that non-contingent on each other's shocks instead of the contingent liquidity allocation of the benchmark economy. In this sense, robust agents' preference for shock-independent liquidity allocations is related to the behavior of panicked depositors in a bank run. 8 The appropriate notion the literature. choices The debate and welfare of welfare in models where agents are not rational is subject to centers on whether or not the planner should use the (see, e.g., 19 in same model to describe Gul and Pesendorfer (2005) and Bernheim and Rangel (2005) the argument). See also Sims (2001) in the context of a central bank's objective. some debate for two sides of agents' conservative probabilities are such that, m The - #2) = (0(1) - 0(2)) (£[§j) last inequality follows in the case of insufficient he receives a shock last Since each agent and there is little - 0(1) 0(2). aggregate liquidity (Z < c*). an agent's chances of being affected Implicitly, these conservative probabilities overweight second in the two-wave event. < concerned about the scenario in which is liquidity left, robustness considerations lead each agent to bias upwards the probability of receiving a shock later than the average agent. However, every agent cannot be probabilities violate the known later than the "average." Across all agents, the conservative and second wave events, implying that probabilities of the first agents' conservative probabilities are collectively biased. Note that each agent's conservative probabilities are individually range of uncertainty over 6^, of being second. it is u possible that agent plausible. Given the has a higher than average probability Only when viewed from the aggregate does it that the collective of conservative agents are guarding against become apparent the is scenario impossible. These observations motivate us to study how a central bank, which is interested in maxi- mizing the collective, can improve on outcomes. Central bank information and objective 3.2 The central bank knows the aggregate drawn from a common is common distribution for probabilities all u. We (f)(1), 0(2), and knows that the w 's are have previously noted that this information knowledge, so we are not endowing the central bank with any more information than agents. The central bank also understands that because of agents' ex-ante symmetry, agents choose the same contingent consumption plan C We denote pfj CB s . that the central bank assigns to the different events that central The may affect agent as the probabilities u>. bank does not know the true probabilities pw Additionally, pj CB may s s . central bank is concerned with the equally weighted ex-post all Like agents, the differ s from pJ^ utility that agents derive from their consumption plans: V CB = = [ J^pf s 5> c/(c*) 20 B U(C s ) (15) where the second line follows from exchanging the integral and summation, and using the fact that the aggregate probabilities are One can view common knowledge. this objective as descriptive of how As noted above, central banks behave: central banks are interested in the collective outcome, and thus natural that the objective it is adopts the average consumption utility of agents in the economy. The arise objective can also be seen as corresponding to agents' welfare. when making the consumption ignores any utility costs that agents "anxieties" about ex-post outcomes. economy, they may suffer Second, if we in this ignore the first because of date suffer we introduce some from the type of policies we discuss address the second concern below. In contrast, may potential issues based on ex-post First, since the objective is latter identification. utility, it Two rational agents into the and the next one, which is section. We what makes ours a paternalistic criterion. 3.3 management and wasted Collective risk Starting from the robust equilibrium of Proposition agents' decisions by increasing the same amount. The First, note that if by an infinitesimal there is m amount, and decreasing sufficient aggregate liquidity, c\ Aci) c-i and c^ Turning next to the = C2 = c* = u'~ l ((3). no by is: For this .m AC2) .msm,^ (16) implies that there there robustness equilibrium consider a central bank that alters value of the reallocation based on the central bank objective case, and equation c\ 2, liquidity is no gain to the central bank from a reallocation. insufficient liquidity case, the first order condition for agents in the satisfies, C(iK(ci) - C(2K(c2 ) - ^ffl-M = o or (m- K y Xci) -(m +K y, Rearranging this equation we have that, f»'W - ^<*> - {C2) J« 21 _ 4>(l)-0(2) - AV(c) + . Ac,)). Substituting this relation into (16), bank + u'(c 2 )) K{u'(c\) is Summarizing these which is it follows that the value of the reallocation to the central positive for all K > 0. results: K> Proposition 3 For any if the 0, economy has insufficient aggregate liquidity [Z < c*), agent decisions are collectively biased. Agents choose too much insurance against receiving shocks second relative to receiving shocks first. The central bank that economy can improve outcomes by post) utility of agents in the toward the A maximizes the expected reallocating agents ities. reallocation central valuable to the central bank because from is bank reaches this conclusion requiring only As we have remarked, the about individual O^s. central its perspective agents are In fact, going beyond of the central bank objective is 2, knowledge of aggregate probabil- bank may be the least informed agent of the that the central bank does not suffer from collective bias. 9 local perturbations, the linearity of the planner in Section risks. bank may be more confused than individual agents In this sense, the central economy. The important point 3.4 insurance first shock. wasting aggregate liquidity by self-insuring excessively rather than cross-insuring The ' with respect to individual probabilities in (15) implies that the central regardless of how uninformed it bank may solves the K= problem be about individual 9w s. Welfare and paternalism To what extent does the arguments central bank's objective correspond to welfare? We offer in favor of our paternalistic welfare criterion in this subsection. First, two further suppose that a fraction of the agents in the economy are rational and had probabilities such that 6U these agents i.e. Suppose the agents all is know that the worst-case probabilities are truly their rest of the agents are zero. it If the central bank is utility cost to the rational agents is uncertain about the values of we may imagine a <p(l) situation in which the central objective function overweights liquidity crises first shocks hurts the (i.e. and bank is second order, while, since the 0(2), then we could overturn the result. In uncertain about these probabilities, and its the incidence of both shocks occurring). In this case, the central bank will also be subject to the "overinsurance" bias of agents. However, this "bias" nature than the one probabilities. helps the average Knightian agent. However, note that the envelope theorem implies that the particular, own = +K; Knightian agents with 6Js such that the average #w across In this case, reallocating insurance from second to rational agents while 9 (ex- we emphasize as it would not be is of a different collectively inconsistent with conditional probabilities. 22 envelope theorem does not apply to the average Knightian agent, the policy results in a Thus, although the central bank's policy order utility gain to the Knightian agents. Pareto improving, involves asymmetric gains to the Knightian agents. it the asymmetric paternalism criterion that Camerer, policy when some agents Another justification the liquidity episode agent u> al., is policy satisfies (2003) propose for evaluating through the following thought experiment: Suppose that we repeat we have described infinitely many times. At the beginning i.i.d. of each episode, across episodes, and the agent be zero. In each episode, since agent his 8^ will not are behavioral. draws a 8^ G 0. These draws are on average et. Our is first episode, the agent's worst-case decision rule will have u does not know knows that the 8U for that him using #w = +K. The central bank's ex-post utility criterion corresponds to the expected consumption utility of an agent across where the expectation of these episodes, all taken using agents' known probabilities. 10 is Risk aversion versus uncertainty aversion 3.5 Given the centrality of collective bias in Proposition 3, increased risk aversion may should be apparent that while it generate positive implications (flight to quality) to those of Knightian uncertainty, its normative implications are not. and regardless of the agent's degree or to reallocate liquidity toward the first change wave in risk aversion, of shocks ti'(ci) = = j—. Then the first Cl y 7 we think C\ falls and of a flight to quality event in c 2 rises as 7 rises. sector's choices. order condition in the is, <P{2) is, W 0(2) si) If beyond the private order condition ' collective bias, 0(1)' u'(ca) Suppose that u(c) Without our central bank sees no reason We can make this point precise by returning to the agents' first = case of Proposition 1. Equation (9) simplified by setting (3 = K that are similar terms of increased risk aversion then However, since K= 0, there is no it is clear that role for the central bank in this case. 10 9U . In living through repeated liquidity events, an agent will learn over time about the true distribution of However, it is still the case that along this learning path, and hence the qualitative features of our argument will 23 K go through remains for strictly positive (while shrinking) a small enough agent discount rate. We conclude that there a role for the central bank only in situations of Knightian is Of course not uncertainty and insufficient aggregate liquidity. these ingredients. and the The are many recessionary episodes exhibit scenarios where they are present, such as October 1987 Fall of 1998. An 4 But there all Application: Lender of Last Resort abstract reallocation experiment considered in Proposition 3 flight to quality episodes the central bank will find against second shocks and of this result more against and consider a lender first it clear that during desirable to induce agents to insure less shocks. In this section of last resort makes (LLR) policy we discuss an application in light of the gain identified in Proposition 3. As in Woodford (1990) and Holmstrom and Tirole to collateral that private agents do not (or at least, (1998), we assume the LLR has has access at a lower cost). Woodford it this collateral. Our that, because of the reallocation benefit of Proposition 3, the value of the LLR and Holmstrom and Tirole focus on the direct value of intervening using novel result is access exceeds the direct value of the intervention. Thus our model sheds light on a new benefit of the LLR. The model also stipulates when the benefit will be highest. As we have remarked previously, the reallocation benefit only arises in situations where directly to our analysis of the show that the LLR must LLR > benefits are highest be a last-resort policy. In often, the reallocation effect 4.1 LLR: the K and when Z < K> c*. and This carries over Z< fact, if liquidity injections works against the policy and reduces c*. We also take place too its value. policy Formally, the central bank credibly expands the resources of agents in the two-shock event by an amount Z G That . is, agents who are affected second in the two- wave event (s have their consumption increased from c 2 to c 2 agents are affected by the second shock) problem) . The + 2Z° (twice ZG = (2, 2)), will because one-half measure of resource constraints for agents (for the reduced are: Cl + ci 4'"° + c2 < 2Z < 2Z + 24 (17) 2Z G . (18) may In practice, the central bank's promise be supported by a credible commitment to costly ex-post inflation or taxation and carried out by guaranteeing, against default, the liabilities who have of financial intermediaries sold financial claims against extreme events. we study an are interested in computing the marginal benefit of intervention, intervention of ZG Since we infinitesimal . the central bank offers more insurance against the two-shock event, this insurance has If a direct benefit in terms of reducing the disutility of an adverse outcome. The direct benefit of the LLR is, yCB,direct = f 2 ^u' = (c 2 ^) dlO 0(2) W '(c 2 ). Jn The anticipation of the central bank's second-shock insurance leads agents to reoptimize their insurance decisions. Agents reduce their private insurance against the publicly insured The second-shock and increase their first-shock insurance. total benefit of the intervention includes both the direct benefit as well as any benefit from portfolio reoptimization: trCB, total v g n , s ,/ ,dci tU first M - ^w + simplify the expression for substituting for u'{c\) from the we Last, v 0(1) ^' dZ G ^M - no I, 4>{2) dcj dZ G 2 du. order condition for agent decisions, from (13), in the robust equilibrium gives, *®* We dc 2 ,u , rww dZ G z The , y G ®> first total + KMci) + Aci)) . by integrating through 0^(1) and ^(2) and then order condition. These operations yield, differentiate the resource constraints, (17) _ dci dc 2 dZ G dZ G and dci dZ G ' (18) with respect to dc^ no ZG to yield, _ dZ G Then, CB, V^ r total tatM = i/ n \ // \ 0(2)u'(c 2 ) rst ti + ^(u'( Cl + = V^GB direct + ' The additional benefit we identify is \ . . ) K(u'( Cl ) + ii dc\ w u'( C2 ))- dZ G dc\ u'(c 2 ))- dZ G ' due to portfolio reoptimization: Agents cut back on the publicly insured second shock and increase 25 first shock insurance, thereby moving their what the central bank would choose decisions closer to for them. policy can help to implement the policy suggested in Proposition We also note that without Knightian uncertainty direct benefit) from the Moreover, policy. it (K = 0), In this sense, the 3. there is no gain (beyond the straightforward to see that is LLR if Z > c* then agents will not use the additional insurance to cover their liquidity shocks, but will reoptimize in a way as to use the insurance at date T. In this case also there public insurance (since ^fc- Proposition 4 For exceeds its direct K > = ). and We summarize these Z < , total important to note that under the As we only rarely. failure in our t rCB Z° V LLR , direct policy the central bank does not promise many Diamond and Dybvig's differences in the > bank associate the second-shock event with an extreme in expectation the central similar to results as follows: the total value of the lender of last resort policy c* , rC B V Z° is no gain to offering the value: < It is (1983) analysis of a mechanism through which the resources. injects resources and unlikely event, This aspect of policy LLR. However there are a few important policies work. As there is no coordination model, the policy does not work by ruling out a "bad" equilibrium. Rather, the policy works by reducing the agents' "anxiety" that they will receive a shock last the is economy has depleted its liquidity a high 0^(2) in their decision rules. ingredient in the policy is resources. From anxiety that leads agents to use It is this this standpoint, when it is also clear that an important that agents have to believe that the central bank will have the necessary resources in the two-event shock in order to reduce their anxiety. Credibility and commitment LLR 11 policy. Moral hazard and early interventions 4.2 The are central to the working of our policy we have suggested interventions. cuts against the usual moral hazard critique of central bank The moral hazard of public insurance critique by cutting back on is their predicated on agents responding to the provision own insurance activities. In our model, in keeping with the moral hazard critique, agents reallocate insurance away from the publicly insured n In this sense, the policy relates to the government bond policy of Woodford (1990) and Holmstrom and Tirole (1998) who argue that government promises are unique because they have greater than private sector promises. 26 collateral backing shock. However, when outcomes on average. the concern, the reallocation improves (ex-post) flight to quality is 12 Public and private provision of insurance are complements in our model. This logic suggests that interventions against shocks first may be subject to the moral hazard critique as agents' portfolio reoptimization would lead them toward more insurance against the second shock. bank credibly from c\ to c\ To consider the consumption of agents who are affected offers to increase the + 2Z° The ' . in the first shock resource constraints for agents (for the reduced problem) are: Ci + 4'"° < 2Z + 2Z G ci The "early intervention" case, suppose that the central + < 2Z + 2Z G c2 direct benefit of intervention in the first yCBJirectJirst = is: j^^fa = 0(1)^). f g shock . JQ. We compute the total benefit as previously except that we substitute agents' first order con- dition using, Also, using the fact that, we dc\ dc2 dZ G dZ G dZ G ' no dZ G find that, trCB, total VZG The expected = -trCB, Vz a direct, first - A t,*/ bank to the two-shock event, which We is resort policy is if . "'-l 1/ is much . t larger than the second shock the direct effect of intervention time inconsistent. This result is the private sector reaction, not from the policy cost of insurance exactly the opposite of what the central bank in intermediate events are subject to the conclude that the lender of private financial markets, has to be a last Note that \\ .direct, first + u (C2))^c < V-CB za Agents reallocate the expected resources from the wants to achieve. In this sense, interventions hazard critique. \ bank rather than the private sector bears the against the (likely) single-shock event. central 1/ (u (ci) cost of the early intervention policy intervention, since the central 12 dch dc\ last resort facility, to and not an intermediate is moral be effective and improve resort. insufficient to justify intervention, then the lender of last not surprising as the benefit of the policy comes precisely from itself. 27 Multiple shocks 4.3 It is LLR clear that the should not intervene during early shocks and instead should only pledge resources for late shocks; but realistic we move away from our two-shock model context with multiple potential waves of aggregate shocks, To answer the if we extend this question economy may experience n probability of the = for all 1...N waves of shocks, each affecting economy experiencing n waves is policy takes the following form: The denoted focusing on the fully robust case where c„ Z < the intervention The c* (i.e. is N). injected to a measure effects > M n = is r with late? We all <fi(n) N _^- +1 < — </>(n 1). The Also, = ^~. units of liquidity also simplify our analysis n and by on c^ assume of the agents. ^{njduj injects We -^ more no . setting (3 = c n rises to c n 0, by thereby + N 3 +i in of agents). jj direct value of the intervention as a function of j vzcGB bank the same for and allowing us to disregard N 2-+\ < (j 4>(n), satisfies J^ en central shocks after (and including) the jth wave assuring that late the model to consider multiple shocks. each w's probability of being affected in the nth wave The LLR how to a is, N j^j^JJZUny^)^ J n=j Agents reduce insurance against the publicly insured shocks and increase their private insurance for the rest of the shocks. direct benefit as well as The total benefit of the intervention includes both the any benefit from portfolio reoptimization: N -,/CB, total VZ° From the resource constraint = f V^ ^^\ n U J Q, „ we have / i \ 1/ ) \. dCn,LO \ j C n^)-T^Ga.UJ. 1 that N dcni dZ G In the fully robust case, c n ^ and dc n ^§- are the same ^ V™*** = «'(ci)^£>(n) = rv dZ G N n=l 28 Then, for all n. ' i)4E^) «'( c l v ' n=l ( 19 ) Note that this expression yCB> direct that decreasing with respect to j since the is Thus, the independent of the intervention rule is V-CB, direct it is apparent are monotonically decreasing. </>(n)'s strictly greater than one Of course, the above shock. Instead, it for all j > pledge resources to the it 1 \— i/„\ ^ nn ±i is \ ) increasing with respect to j. of resources available for intervention, Nth shock and where the shadow cost of the continue to do so until N— 1st shock, it and so LLR LLR become increasingly less likely. resources for the central bank is If completely on. also clarifies another benefit of late intervention. events that are being insured by the falls amount should then move on to the The multiple shock model policy Y^ w result does not suggest that intervention should occur only in the iVth replaces private insurance, LLR and suggests that for any given LLR should first cost of the 1 1 J^j+1 En=j ZO the In contrast, ratio: trCB, total is j. As j rises, we take the case constant, the expected as j rises, while the expected benefit remains constant. In other words, as j rises, it is the private sector that increasingly improves the allocation of scarce private resources to early and more thereby reducing the likely aggregate shocks, extent of the flight-to-quality phenomenon. In contrast, the central bank plays a decreasingly small role in terms of the expected value of resources actually disbursed, as j increases. Thus, while a well designed unlikely events, the policy is LLR policy not irrelevant may indeed have a direct effect only in highly for likely unlocking private markets to insure more likely and less Its main benefits come from extreme events. Final remarks 5 Flight to quality and outcomes. is a pervasive financial accelerators. phenomenon that exacerbates the impact The starting point of this paper is a model of this phenomenon We based on Knightian uncertainty aversion among financial specialists. aggregate liquidity when there is is plentiful, agents' uncertainty of recessionary shocks show that when does not affect the equilibrium. However, both Knightian uncertainty and agents think that aggregate liquidity they take a set of protective actions to guarantee themselves safety, is scarce, but which leave the aggregate economy overexposed to negative shocks. In this context, a Our model Lender of Last Resort policy is valuable prescribes that the benefit of the LLR 29 is when used highest when to support rare events. there is both insufficient Many aggregate liquidity and Knightian uncertainty. this paper (1987 market crash, are times when examples we have discussed of the Fall of 1998, 9/11) satisfy these criteria. call for central by a hedge fund - even one that is bank action. Likewise, we large - should not elicit central prescribe that a default bank reaction unless the default triggers considerable uncertainty in other market participants financially weak. The model also suggests that it is understanding of whether a given market dynamic is just as important the "dog did not bark." For example, a recession in which the ingredients are not both central, does not There But in important is currently considerable uncertainty over affect the credit derivatives for and hedge funds are the central bank to have an driven by risk or uncertainty. how the downgrade of a top name will market. Our model suggests that ex-ante actions to reduce the extent of uncertainty during a flight to quality episode are valuable. For example, recent moves to increase transparency and risk assessment in this market as well as streamline back-office settlement procedures can be viewed in this light (Geithner, 2006). The implications of the framework extend beyond the particular interpretation given to agents and policymakers. For example, in the international context one of our agents as countries prescribes that the IMF and the policymaker as the not support the to intervene once contagion takes place. first The IMF we have may or other IFI's, Then, our think model country hit by a sudden stop, but to commit benefit of this policy comes primarily from the additional availability of private resources to limit the impact of the initial pullback of capital flows. 30 References [1] Ashcraft, Adam and Darrell Duffie, 2006, "Systemic Dynamics in the Federal Funds Market," Mimeo, Stanford University. [2] Barro, R.J., 2005, "Rare Events and the Equity Premium," Mimeo, Harvard University. [3] Bernheim, B.D. and A. Rangel, 2005, "Behavioral Public Economics: Welfare and Policy Analysis with Non-Standard Decision-Makers," Stanford Mimeo, June. [4] Caballero, R.J. and A. Krishnamurthy, 2005, "Financial System Risk and Flight to Quality," [5] NBERWP # Calomiris, Charles 11996. 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[29] Woodford, M., "Public Debt as Private Liquidity," American Economic Review, Papers and Proceedings 80, May 1990, pp. 382-88. 33 A Event Tree and Probabilities Prob 2 Waves 1 Wave 2nd Wave (f)(2) 1st Wfcve No More Shocks Prob The event (f)(2); of tree is 1 pictured above. - No Shocks 0(1 The probability of two waves affecting the the probability of one wave affecting the economy no waves economy affecting the is 1 — (f)(1). We is (f)(1) — economy is 0(2); and, the probability used the dashed box around "1st wave" to signify that agents ate unsure whether they are in the upper branch (one more wave will occur) or the middle branch (no further shocks). Consider an agent u who may be when being affected by a shock affected in these waves. the event is the middle branch ("1 wave") that his probability of being of being affected by a branch ("2 waves") is if) is shock when the event first uncertain about first t/> is agent's probability of being affected by a first = shock is, 1 -(f)(2)) -. is, 0(2)(l-&,). that, 1 &,(!) + <M2) =0(2) + (^(1) 34 the upper is 1 , agent's probability of being affected by a second shock &,(2) Suppose — ip^. w which we interpret as the agent or second, in the case of a two ^(l) = ^(2)^ + ^(1) Note one-half. , uncertain about his likelihood of being The is his probability of u while his probability of being affected by a second shock Moreover, suppose that the agent The Suppose that -0(2)) _ (f)(1) + (f)(2) wave event. is and, <Mi) —y = — = &/(2) These expressions show that the event tree ^ 2 W" — + -<f>{2)il> u 2~ — consistent with agents being certain about their is probability of receiving a shock, but being uncertain about their relative probabilities of being or second. In the text, first terms of in B il) u we describe the uncertainty terms of (f) UJ {2) — ^~ rather than . Proof or Proposition We in 2 focus on the case of insufficient aggregate liquidity (Z same logic as the (12). We K — We case. < The other c*). case follows the are looking for a solution to the problem in equation can describe this problem in the game-theoretic language often used in max-min problems. The agent chooses C to maximize V(C; &£) anticipating that "nature" will choose 9% to minimize V(C; &£) given the agent's choice of C. The solution (&£,C) has to satisfy a pair of optimization problems. First, 6% That is, That is, We Let us cT ' 5 = the agent chooses C optimally given nature's choice compute: dV — = \ - I \ m(cx) + , this value of 9% let us consider the agent's 0. With To 2,2 - problem utility gain of we ' < ) 0. If so, first > 0. Then we can reduce c^' by when 9% > 5(<^(2) - 0(2) + </£(2)) > + PC?2,2 < u{c\) => J^- C\ < > C before nature chooses 9" does not only gives the agent an advantage if set 5 = +K. First note that and increase c^ by 0. as, c2 affect our problem. To see this, note the agent can induce nature to choose a 6% different / C to increase V('). Clearly this choice reduces V below V(0£,C). 6% = 9" and make the agent strictly worse off than at the choice C = C. than 6%. Suppose the agent chooses C Thus, nature can always choose to then clearly 0" in equation (12). 1 rewrite the condition that fact that the agent chooses 13 2,1\ cT ' this knowledge, that choosing at p{cf of #£. see this, suppose that c r u{c 2 ) The I u{c2 ) ask whether there exists a solution in which J^r and produce a 13 9^ nature chooses &£ optimally given the agent's choice of C. Second, C € argmaxcV(C; 0£). first Taking £ argmineu.V(C; 35 If C\ > and c2 Z< c* it follows from the resource constraint that 2 from the agent's problem, we must have that T if = c^' (i.e. c*. But if c2 < do not save any resources c* then for date first order condition for the agent at &£ +A = in (12) for values of c-i,C2,c^ We note that for K= 0, As K This > 0. < the unique solution to the agent's problem small values of A", a solution exists in which the agent chooses = +A. n0 is, (m_ Ky. iCl)= (m +Ky, {C2)+p &£ < these resources could be used earlier). Thus, we only need to consider the agent's problem The c2 > c\ > is c 2 c2 c2 . Thus, for and nature chooses the partially robust solution given in the Proposition. is becomes larger, Ci/c 2 falls and at some point C\ = c2 = Z. This occurs when K solves, (m_ Ry, {z)= (m +Ry, {z)+0 m^m. which gives the expression Note that if K > K, the solution 6% The nature's optimization problems. nature is K defined in the Proposition. = +K and = c solves both for the value of c\ agent's choice is uniquely optimal at 6^ G [—A, +A]. This indifferent over values of &£ the agent's and 2 still is = +K, while the fully robust solution given in the Proposition. We have thus Proposition is far shown that considering the case where -^ < the only solution to the problem in (12). are no other solutions to the problem. which exists a solution in Suppose there does ^> To do this, We 0, the solution given in the conclude by showing that there we only need to consider whether there 0. exist such a solution. If J^ > 0, then 6% = -A. We 09 through arguments similar to those previously offered to show that c^ and zero in this case. Then the condition that Jjj£ < c\ The first order condition for the agent > is solution to this problem exist a solution in which is -^ > that c\ can go back i cT ' must both be to, c2 is, (m +K M ±(m- K y The equivalent 9 > c2 , Ae )+0 which 0. 36 i . ) is a contradiction. Thus there does not 521