Crashes and Recoveries in Illiquid Markets

Transcription

1 Crashes and Recoveres n Illqud Markets Rcardo Lagos, Gullaume Rocheteau and Perre-Olver Well September 11, 27 Abstract We study the dynamcs of lqudty provson by dealers durng an asset market crash, descrbed as a temporary negatve shock to nvestors aggregate asset demand. We consder a class of dynamc market settngs where dealers can trade contnuously wth each other, whle tradng between dealers and nvestors s subject to delays and nvolves barganng. We derve condtons on fundamentals, such as preferences, market structure and the characterstcs of the market crash (e.g., severty, persstence) under whch dealers provde lqudty to nvestors followng the crash. We also characterze the condtons under whch dealers ncentves to provde lqudty are consstent wth market effcency. Keywords: lqudty, asset nventores, executon delays, search, barganng J.E.L. Classfcaton: C78, D83, E44, G1 We thank Ruln Zhou for comments and Monca Crabtree-Reusser for edtoral assstance. We also thank semnar partcpants at Mannhem Unversty, the 26 Workshop on Money, Bankng and Payments at the Federal Reserve Bank of Cleveland, the 27 Mdwest Macro Conference, the 27 Conference on Mcrofoundatons of Markets wth Frctons n Montreal, and the 27 SED meetngs. The vews expressed heren are those of the authors and not necessarly those of the Federal Reserve Bank of Cleveland or the Federal Reserve System. Lagos thanks the C. V. Starr Center for Appled Economcs at NYU for fnancal support. Lagos: New York Unversty, rcardo.lagos@nyu.edu. Rocheteau: Federal Reserve Bank of Cleveland and Natonal Unversty of Sngapore, ecsrg@nus.edu.sg. Well: Unversty of Calforna, Los Angeles, powell@econ.ucla.edu.

2 1 Introducton Lqudty n fnancal markets s often provded by dealers who trade assets from ther own nventores. Even n markets where lqudty provson by dealers may be nconspcuous n normal tmes, t becomes crtcal durng tmes of large fnancal mbalances. Durng market crashes, for nstance, t can take a long tme for an nvestor to fnd a counterpart for trade, ether because of the technologcal lmtatons of order-handlng systems or, as s the case n over-the-counter markets, due to the decentralzed nature of the tradng process. 1 These stuatons appear to be very costly to nvestors, who concede strkng prce dscounts to unwnd ther postons (e.g., the 23% prce drop of the Dow Jones Industral Average on October 19, 1987). Some have argued that the socal cost could be even larger because of the rsk that the fnancal crss propagates to the macroeconomy (see, e.g., Boro (24)). It s commonly beleved that lqudty provson by dealers plays a crucal role n mtgatng these costs. In ths paper we study the equlbrum and the socally optmal nventory polces of dealers durng a market crash, whch we model as a temporary negatve shock to nvestors wllngness to hold the asset. We derve condtons under whch dealers wll fnd t n ther nterest to provde lqudty n the aftermath of a crash, as well as condtons under whch ther ncentves to provde lqudty are consstent wth market effcency. We also study how lqudty provson by dealers depends on the market structure, e.g., dealers degree of market power or the extent of the tradng frctons, and the characterstcs of the crash, e.g., severty and persstence of the shock to nvestors demands. Our work s related to a recent lterature that studes tradng frctons n asset markets. 2 In partcular, the market settng we consder s smlar to that of Duffe et al. (25) (DGP hereafter). Investors rebalance ther asset holdngs perodcally n response to random changes n ther utlty from holdng assets, and they must engage n a tme-consumng process to contact dealers and bargan over the terms of trade. Dealers get no drect utlty from holdng assets, and they can trade contnuously n a perfectly compettve nterdealer market. DGP focused on steady states, so ther analyss s slent about lqudty provson by dealers. 1 For a descrpton of the varous tradng problems that arose durng the market crashes of October 1987 and October 1997, see the report of The Presdental Task Force on Market Mechansms (1988) and the US Securtes and Exchange Commsson Staff Legal Bulletn No. 8 (September 9, 1998). 2 Examples nclude, Gârleanu (26), Longstaff (25), and Vayanos and Well (27). Conceptually, our analyss s also related to the nventory models of Stoll (1978) and Ho and Stoll (1983) (see Chapter 2 n O Hara (1997) for a revew of ths earler market-mcrostructure lterature). 2

3 Well (27) studes the tmng of lqudty provson by dealers n a dynamc verson of DGP. He asks under whch condtons dealers wll, and ought to, lean aganst the wnd n the mmedate aftermath of the crash. Well (27) and the lterature spurred by DGP, however, keep the framework tractable by mposng a stark restrcton on asset holdngs, namely, that nvestors can only hold ether or 1 unt of the asset. Lagos and Rocheteau (27) study a verson of DGP where nvestors can hold unrestrcted asset postons and fnd that, as result of ths restrcton on asset holdngs, exstng search-based theores of fnancal lqudty neglect a crtcal aspect of nvestor behavor n llqud markets,namely thefactthatmarketpartcpants can mtgate tradng frctons by adjustng ther asset postons so as to reduce ther tradng needs. Ths effect of tradng frctons on the demand for lqudty has been ponted out n a dfferent context by Constantndes (1986). In ths paper, we go beyond prevous studes by allowng both dealers and nvestors to hold unrestrcted asset postons. Ths turns out to generate new mplcatons for both the demand and the supply of lqudty. Absent extraneous upper bounds on asset holdngs, n tmes of crss, nvestors wth hgh utlty for the asset may absorb the sellng pressure comng from nvestors wth low utlty by holdng postons that are large relatve to what they would hold durng normal tmes. In other words, by removng the typcal restrctons on nvestors asset holdngs, we fnd that nvestors may provde lqudty to other nvestors n tmes of crss, much lke dealers do. These new effects on the supply and demand of lqudty mply that, n contrast to Well (27), dealers may sometmes not fnd t n ther nterest to provde lqudty durng a crash. Also, t may sometmes be effcent for them not to lean aganst the wnd. Whether or not dealers wll provde lqudty, and whether or not they ought to, depends on fundamentals, ncludng the detals of market structure and the characterstcs of the crash. Our stylzed descrpton of a market crash conssts of an aggregate negatve preference shock to nvestors asset demands, followed by a (possbly stochastc) recovery path. 3 We fnd that the amount of lqudty provded by dealers followng the crash vares nonmonotoncally wth the magntude of tradng frctons. When frctons are small, nvestors choose to take more 3 Ths scenaro could represent, for nstance, an nternatonal shock such as the 1997 Asan crss or the 1998 Russan soveregn default, domestc turbulence such as that trggered by the September 11 terrorst attack, or even some company-specfc shock, such as the collapse of Enron. Our crash follows the sprt of Grossman and Mller s (1988) crash dynamcs. In Grossman and Mller, dealers provde lqudty n order to share rsk wth outsde nvestors. In our model, dealers have no such utlty motve for holdng assets; nstead, they allow nvestors to trade faster. In related work, Bernardo and Welch (24) use the feature of nonsequental access of nvestorstomarketmakerstodescrbeamarketcrashasafnancal run. 3

4 extreme postons because they know that they can rebalance ther asset holdngs very quckly. Specfcally, nvestors wth hgher-than-average utlty for assets become more wllng to hold larger-than-average postons and absorb more of the sellng pressure comng from nvestors whose demands for the asset are lower than normal. In some cases, the former end up supplyng so much lqudty to other nvestors, that dealers don t fnd t proftable to step n. If, on the contrary, tradng frctons are large enough, dealers do not accumulate nventores ether, but for a dfferent reason: Tradng frctons reduce nvestors demand for lqudty. Indeed, n order to reduce ther exposure to the tradng frctons, nvestors choose to take less extreme asset postons. In fact, t s possble that they demand so lttle lqudty that dealers don t fnd t proftable to accumulate nventores followng a crash. Thus, f one consders a spectrum of asset markets rangng from those wth very small frctons, such as the New York Stock Exchange (NYSE), to those wth large tradng frctons, such as the corporate bond market, one would expect to see dealers accumulate more asset nventores durng a crash n markets whch are n the ntermedate range of the spectrum. We also fnd that, from the standpont of nvestors, an ncrease n dealers barganng strength s equvalent to an ncrease n tradng frctons. Hence, just as wth tradng frctons, dealers are less lkely to accumulate nventores f ther barganng strength s ether very small or very large. Ths fndng contrasts wth the commonly held vew that the market power of dealers (e.g., NYSE specalsts) s what gves them ncentves to provde lqudty. In our model, an ncrease n the dealers barganng strength may reduce the aggregate amount of nventory they accumulate, because nvestors endogenously take less extreme postons and demand less lqudty. Smlarly, a market reform that reduces dealers market power, as observed n equty markets n the 9 s, can rase dealers ncentves to provde lqudty durng a market crash. Our model can ratonalze why dealers ntervene n some crses and wthdraw n others. In lne wth Hendershott and Seasholes s (26) emprcal evdence on the nventory strateges of NYSE specalsts, n our model, dealers ncentves to provde lqudty are drven by antcpated captal gans. Therefore, dealers are more lkely to accumulate nventores when the crss s severe and expected to be short-lved: A large prce drop and the expectaton of a quck rebound make t more proftable for dealers to buy low early n the crash and sell hgh later, as demand for the asset recovers. From a normatve standpont, we fnd that the equlbrum asset allocaton across nvestors and the dealers nventory polces are socally effcent f and only f dealers barganng strength s equal to zero. Gven the nonmonotonc equlbrum relatonshp between 4

5 dealers asset nventores and ther barganng strength, ths means that dealers may fal to buld up nventores n stuatons where t would be socally effcent to do so, and vce-versa. The rest of the paper s organzed as follows. Secton 2 lays down the envronment. Secton 3 characterzes nvestors and dealers behavor and Secton 4 defnes equlbrum. Secton 5 characterzes the socally optmal allocaton. Sectons 6 and 7 provde two alternatve descrptons of a market crash and determne the condtons under whch dealers act as provders of lqudty. Secton 8 concludes. Appendx A contans all proofs and Appendx B contans supplementary materal. 2 The envronment Tme s contnuous and the horzon nfnte. There are two types of nfntely-lved agents: a unt measure of nvestors and a unt measure of dealers. There s one asset and one pershable good, whch we use as a numérare. The asset s durable, perfectly dvsble and n fxed supply, A R +. The numérare good s produced and consumed by all agents. The nstantaneous utlty functon of an nvestor s u (a)+c, wherea R + represents the nvestor s asset holdngs, c R s the net consumpton of the numérare good (c < f the nvestor produces more than he consumes), and {1,...,I} ndexes a preference shock. The utlty functon u (a) s strctly ncreasng, concave, contnuously dfferentable and satsfes the Inada condton that u () =. We also assume that t s ether bounded below or above. Investors receve dosyncratc preference shocks that occur wth Posson arrval rate δ. Condtonal on the preference shock, the nvestor draws preference type wth probablty π,and P I =1 π =1. These preference shocks capture the noton that nvestors value the servces provded by the asset dfferently over tme, and wll generate a need for nvestors to perodcally change ther asset holdngs. 4 The nstantaneous utlty of a dealer s υ(a) +c, whereυ(a) s ncreasng, 4 Our specfcaton assocates a certan utlty to the nvestor as a functon of hs asset holdngs. Ths s a feature that we have borrowed from DGP. The utlty the nvestor gets from holdng a gven asset poston could be smply the value from enjoyng the asset tself, as would be the case for real assets such as cars or houses. Alternatvely, we can also thnk of the asset as beng physcal captal. Then, f each nvestor has lnear utlty over a sngle consumpton good (as s the case n most search models), we can nterpret u ( ) as a producton technology that allows the agent to use physcal captal to produce the consumpton good. The dosyncratc component can then be nterpreted as a productvty shock that nduces agents wth low productvty to sell ther captal to agents wth hgh productvty n an OTC market. As yet another possblty, one could adopt the preferred nterpretaton of DGP, namely that u (a) s n fact a reduced-form utlty functon that stands n for the varous reasons why nvestors may want to hold dfferent quanttes of the asset, such as dfferences n lqudty needs, fnancng or fnancal-dstress costs, correlaton of asset returns wth endowments (hedgng needs), or relatve tax dsadvantages (as n Mchaely and Vla (1996)). By now, several papers that buld on 5

6 concave and contnuously dfferentable. All agents dscount at the same rate r>. I N VE S T O D EA L E R S As s e t Marke t D EA L E R S I N VE S T O R S R S Fgure 1: Tradng arrangement There s a compettve market for the asset. Dealers can contnuously buy and sell n ths market at prce p (t), whle nvestors can only access the market perodcally and ndrectly, through a dealer. Specfcally, we assume that nvestors contact a randomly chosen dealer accordng to a Posson process wth arrval rate α. Once the nvestor and the dealer have made contact, they negotate the quantty of assets that the dealer wll acqure (or sell) n the market on behalf of the nvestor and the ntermedaton fee that the nvestor wll pay the dealer for hs servces. After completng the transacton, the dealer and the nvestor part ways. 5 The tradng arrangement s llustrated n Fgure 1. 3 Dealers, nvestors, and barganng In ths secton we descrbe the decson problems faced by nvestors and dealers, and the determnaton of the terms of trade n blateral meetngs between them. Investors readjust ther asset the work of DGP have formalzed the hedgng needs nterpretaton. Examples nclude Duffe, Gârleanu and Pedersen (26), Gârleanu (26) and Vayanos and Well (27). (See also Lo, Mamaysky and Lang (24).) Notce that nvestors n DGP, and therefore the nvestors n our paper, are akn to the lqudty traders whch arecommonplacenthelargebodyofthefnance mcrostructure lterature that uses asymmetrc nformaton nstead of search frctons to ratonalze bd-ask spreads, such as Glosten and Mlgrom (1985) and Easley and O Hara (1987). 5 In actual fnancal markets, there are poston traders who hold asset nventores n the hope of makng captal gans. There are also pure spread traders who don t hold nventores but nstead proft exclusvely from buyng low and sellng hgh. Stoll (1978), for example, calls the former dealers and the latter brokers. In our model, the agents that we refer to as dealers engage n both of these actvtes. The analyss would reman unchanged f we were to assume that these actvtes are carred out by two dfferent types of agents wth contnuous drect access to the asset market. 6

7 holdngs nfrequently, at the random tmes when they meet dealers. In between those tmes, an nvestor enjoys the utlty flow assocated wth hs current asset poston. A dealer s problem conssts of contnuously managng hs own asset poston by tradng n the asset market. At random tmes, the dealer contacts an nvestor who wshes to buy or sell some quantty of assets. At these tmes, the dealer executes the desred purchaseorsalentheassetmarketonbehalf of the nvestor and receves a fee for hs servces. 6 We begn wth the determnaton of the terms of trade n blateral trades between dealers and nvestors. Consder a meetng at tme t between a dealer who s holdng nventory a d and an nvestor of type who s holdng nventory a. Leta denote the nvestor s post-trade asset holdng and φ be the ntermedaton fee. 7 The par (a,φ) s taken to be the outcome correspondng to the Nash soluton to a barganng problem where the dealer has barganng power η [, 1]. Let V (a, t) denote the expected dscounted utlty of an nvestor wth preference type who s holdng a quantty of asset a at tme t. Then, the utlty of the nvestor s V (a,t) p(t)(a a) φ f an agreement (a,φ) s reached, and V (a, t) n case of dsagreement. Therefore, the nvestor s gan from trade s V (a,t) V (a, t) p (t)(a a) φ. Analogously, let W (a d,t) denote the maxmum attanable expected dscounted utlty of a dealer who s holdng nventory a d at tme t. Then, the utlty of the dealer s W (a d,t)+φ f an agreement (a,φ) s reached and W (a d,t) n case of dsagreement, so the dealer s gan from trade s equal to the fee, φ. 8 The outcome of the barganng s gven by [a (t),φ (a, t)] = arg max [V (a,t) V (a, t) p (t)(a a) φ] 1 η φ η. (a,φ) Hence, the nvestor s new asset holdng solves a (t) = arg max V (a,t) p(t)a, (1) a 6 In prncple, the dealer may fll the nvestor s order partally or n full by tradng out of, or for hs own nventory of the asset. For example, f at some tme t the dealer contacts an nvestor who wshes to buy some quantty a and the dealer s nventory s a d (t) >a, then n that nstant, the dealer may fll the buy order by gvng the nvestor a from hs nventory and chargng hm p (t) a plus the fee, and nstantaneously buyng back a d (t) a for hs own account n the asset market. Alternatvely, the dealer may nstead choose not to trade out of hs nventory and smply buy a n the market on behalf of the nvestor at cost p (t) a (and charge hm ths cost plus the ntermedaton fee). Clearly, the dealer s ndfferent between these modes of executon because he has contnuous access to the asset market and all the transactons he s nvolved n are nstantaneous. 7 In our formulaton we assume that the nvestor pays the dealer a fee. However, the barganng problem can be readly renterpreted as one n whch the dealer pays the nvestor a bd prce whch s lower than the market prce f the nvestor wants to sell, and charges an ask prce whch s hgher than the market prce f the nvestor wants to buy. See Lagos and Rocheteau (27) for detals. 8 The outcome of the blateral trade does not affect the dealer s contnuaton payoff because he has contnuous access to the asset market and hs trades are executed nstantaneously (see footnote 6). 7

8 and that the ntermedaton fee s φ (a, t) =η {V [a (t),t] V (a, t) p(t)[a (t) a]}. (2) Accordng to (1), the nvestor s post-trade asset holdng s the one he would have chosen f he were tradng n the asset market hmself, rather than through a dealer. Accordng to (2), the ntermedaton fee s set so as to gve the dealer a share η of the gans assocated wth readjustng the nvestor s asset holdngs. 9 The value functon correspondng to a dealer who s holdng asset poston a t at tme t satsfes ½Z T ¾ W (a t,t)= sup E e r(s t) {υ[a d (s)] p(s)q(s)} ds + e r(t t) [ φ (T )+W(a d (T ),T)], q(s),a d (s) t subject to the law of moton ȧ d (s) =q (s), the short-sellng constrant a d (s), andthe ntal condton a d (t) =a t. Here, a d (s) represents the stock of assets that the dealer s holdng and q (s) s the quantty that he trades for hs own account at tme s. The expectatons operator, E, s taken wth respect to T, whch denotes the next random tme at whch the dealer meets an nvestor, where T t s exponentally dstrbuted wth a mean of 1/α. Snce the ntermedaton fee determned n a blateral meetng depends on the nvestor s preference type and asset holdngs, and gven that the nvestor s a random draw from the populaton of nvestors, at tme T the dealer expects to extract the average fee φ (T )= R φ j (a,t)dh T (j, a ), where H T denotes the dstrbuton of nvestors across preference types and asset holdngs at tme T. The dealer enjoys flow utlty υ[a d (s)] from carryng nventory a d (s), and gets utlty p (s) q (s) from changng ths nventory. Snce ntermedaton fees are ndependent of the dealer s asset holdngs, we can wrte ½Z ¾ W (a t,t)=max e r(s t) {υ[a d (s)] p(s)q(s)} ds + Φ (t), (3) q(s) t subject to ȧ d (s) = q (s), a d (s) and a d (t) = a t. The functon Φ (t) s the expected present dscounted value of future ntermedaton fees from tme t onward and satsfes Φ (t) = E{e r(t t) [ φ (T )+Φ(T)]}, where the expectaton s wth respect to T. Ths formulaton makes t clear that dealers trade assets n two ways: contnuously, n the compettve market, or at random tmes, n blateral negotatons wth nvestors. Snce dealers have quas-lnear preferences 9 Our choce of notaton for the barganng soluton n (1) and (2) emphaszes the fact that the terms of trade depend on the nvestor s preference type but are ndependent of the dealer s nventores. In addton, the nvestor s post-trade asset holdng s ndependent of hs pre-trade holdng, whle the ntermedaton fee s not. 8

9 and they can trade nstantaneously and contnuously n the compettve asset market, ther optmal choce of asset holdngs s ndependent from what happens n blateral negotatons wth nvestors. The followng lemma descrbes the soluton dealer s nventory accumulaton problem whchsnthefrst term on the rght-hand sde of (3). Lemma 1 Suppose that p(t) s a gven, pecewse contnuously dfferentable prce path. An nventory path, a d (t), solves the dealer s nventory accumulaton problem f and only f 1. for all t such that p(t) s dfferentable, a d (t) satsfes υ [a d (t)] + ṗ (t) rp (t) wth equalty f a d (t) > ; (4) 2. a d (t) =for any t for whch the prce has a negatve jump,.e., f 3. a d (t) satsfes the transversalty condton p(t + ) p(t ) <, then a d (t) =; (5) lm t e rt p(t)a d (t) =. (6) There s no bounded nventory path, a d (t), that solves the dealer s nventory accumulaton problem f p(t) has a postve jump, p(t + ) p(t ) >. The last part of Lemma 1, states that f the asset prce had a postve jump, a dealer could mprove hs utlty from any bounded nventory path by buyng assets just before the jump and re-sellng just after. 1 The opposte tradng strategy mples that the short-sellng constrant must be bndng whenever the prce jumps down. Accordng to (4), whenever the prce path s dfferentable and a dealer fnds t optmal to hold strctly postve nventory, the flow cost of buyng the asset, rp (t), must equal the drect utlty flow from holdng the asset, υ [a d (t)], plus the captal gan, ṗ (t). As t s well known from Mangasaran s results (see Theorem 13, Chapter 3 of Seerstad and Sydsaeter (1987)), together wth the other frst-order condtons, the transversalty condton (6) s suffcent for optmalty. Here, we show that t s necessary as well Note that, because there s a fnte measure of assets, and agents face short-sellng constrants, dealers asset holdngs wll have to be bounded n an equlbrum. Ths observaton, together wth the last part of the lemma wll mply that a prce paths wth upward jumps cannot be part of an equlbrum. 11 The necessty of such transversalty condtons for general formulatons of nfnte-horzon optmal control problems has been regarded as a delcate ssue snce Halkn s (1974) counterexample. See Benvenste and Schenkman (1982) for farly general results. 9

10 We now proceed wth an analyss of an nvestor s problem. The value functon correspondng to an nvestor wth preference type who s holdng a assets at tme t, V (a, t), satsfes Z T V (a, t) = E e r(s t) u k(s) (a)ds + t e r(t t) {V k(t ) [a k(t ) (T ),T] p(t )[a k(t ) (T ) a] φ k(t ) (a,t )}, (7) where T denotes the next tme the nvestor meets a dealer, and k(s) {1,..., I} denotes the nvestor s preference type at tme s. The expectatons operator, E, s taken wth respect to the random varables T and k(s), and s ndexed by to ndcate that the expectaton s condtonal on k(t) =. Over the nterval of tme [t, T ] the nvestor holds a assets and enjoys the dscounted sum of the utlty flows assocated wth ths holdng a (the frst term on the rght-hand sde of (7)). The length of ths nterval of tme, T t, s an exponentally dstrbuted random varable wth mean 1/α. Theflow utlty s ndexed by the preference type of the nvestor, k(s), whch follows a compound Posson process. At tme T the nvestor contacts a random dealer and readjusts hs holdngs from a to a k(t ) (T ). In ths event the dealer purchases a quantty a k(t ) (T ) a of the asset n the market (or sells f ths quantty s negatve) at prce p(t ) on behalf of the nvestor. At ths tme the nvestor pays the dealer an ntermedaton fee, φ k(t ) (a, T ). Boththefeeandtheassetprceareexpressedntermsofthenuméraregood. Substtutng the terms of trade (1) and (2) nto (7), we get Z T V (a, t) = E e r(s t) u k(s) (a)ds + (8) t e r(t t) {(1 η)max Vk(T ) (a,t) p(t )(a a) + ηv k(t ) (a, T )}. a From the last two terms on the rght-hand sde of (8), t s apparent that the nvestor s payoff s the one he would get n an economy n whch he meets dealers accordng to a Posson process wth arrval rate α, and nstead of barganng, he readjusts hs asset holdngs and extracts the whole surplus wth probablty 1 η; whereas wth probablty η he cannot readjust hs holdngs (and enjoys no gan from trade). Therefore, from the nvestor s standpont, the stochastc tradng process and the barganng soluton are payoff-equvalent to an alternatve tradng mechansm n whch the nvestor has all the barganng power n blateral negotatons wth dealers, but he only gets to meet dealers accordng to a Posson process wth arrval rate 1

11 κ = α(1 η). Consequently, we can rewrte (8) as Z T V (a, t) =E u k(s) (a) e r(s t) ds + e r( T t) {p( T )a +max[v a k( T ) (a, T ) p( T )a ]}, (9) t where the expectatons operator, E, s now taken wth respect to the random varables T and k(s), where T t s exponentally dstrbuted wth mean 1/κ. From (9), the problem of an nvestor wth preference shock, who gans access to the market at tme t, conssts of choosng a R + n order to maxmze Z T n E e r(s t) u k(s) (a ) ds p(t) E t e r( T t) p( T ) o a, or equvalently, t " Z T E e r(s t) # u k(s) (a ) [rp(s) ṗ(s)] a ª ds. (1) t If the nvestor had contnuous access to the asset market, he would choose hs asset holdngs so as to contnuously maxmze u (a) [rp(t) ṗ(t)] a, hsflow utlty net of the flow cost of holdng the asset. But snce the nvestor can only trade nfrequently, hs objectve s to maxmze (1) nstead. Intutvely, the nvestor chooses hs asset holdngs n order to maxmze the present value of hs utlty flow net of the present value of the cost of holdng the asset from tme t untl the next tme T when he can readjust hs holdngs. The followng lemma offers a smpler, equvalent formulaton of the nvestor s problem. Lemma 2 Let U (a) = (r + κ) u (a)+δ P I j=1 π ju j (a) r + δ + κ ξ(t) =(r + κ) p(t) κ Z (11) e (r+κ)s p(t + s)ds, (12) and assume that p(t)e rt s decreasng. Then a bounded process a(t) solves the nvestor s problem f and only f 1. a(t) =a (t), when the nvestor contacts the market wth current type, wth U [a (t)] = ξ(t) (13) 11

12 2. a(t) satsfes the transversalty condton h lm E p(θ t )a(θ t )e rθ t =, (14) t where θ t denotes the nvestor s last contact tme wth a dealer before t. The assumpton that p(t)e rt s decreasng s wthout loss of generalty, because t wll be true n an equlbrum (ths follows from the dealer s frst-order condtons (4) and (5)). Intutvely, U (a) s the flow expected utlty the nvestor enjoys from holdng a assets untl hs next opportunty to rebalance hs holdngs, and ξ (t) sthecostofbuyngtheassetmnustheexpected dscounted resale value of the asset (expressed n flowterms). Notcethatwedonotneedto know the path for the prce of the asset, p (t), to solve for the nvestor s optmal asset holdngs. It s suffcent to know ξ(t). The followng lemma establshes the relatonshp between ξ(t) and p (t). Lemma 3 Condton (12) mples Lemma 3 allows us to rewrte (4) as rp (t) ṗ (t) =ξ (t) ξ (t) r + κ. (15) υ [a d (t)] + ξ (t) r + κ ξ (t) wth an equalty f a d(t) >. (16) Equatons (13) and (16) llustrate the man dfferences between dealers and nvestors n our setup. Relatve to nvestors, dealers get an extra return from holdng the asset, captured by ξ (t) / (r + κ). Thsreflects a dealer s ablty to make captal gans by explotng hs contnuous access to the asset market. Another dfference s the fact that the utlty functon for nvestors on the left-hand sde of (13) s a weghted-average of the margnal utlty flows that the nvestor enjoys untl the next tme he s able to readjust hs asset holdngs. 4 Equlbrum In ths secton, we study the determnaton of the asset prce, defne equlbrum, and show how to characterze t. Snce each nvestor faces the same probablty to access the market rrespectve of hs asset holdngs, and snce these probabltes are ndependent across nvestors, we appeal to the law of large numbers to assert that the flow supply of assets by nvestors 12

13 s α [A A d (t)], where A d (t) s the aggregate stock of assets held by dealers. (Note that A d (t) =a d (t), snce there s a unt measure of dentcal dealers facng the same strctly concave optmzaton problem). The measure of nvestors wth preference shock whoaretradngn themarketattmet s αn (t), wheren (t) s the measure of nvestors wth preference type at tme t. Therefore, the nvestors aggregate demand for the asset s α P I =1 n (t)a (t), and the net supply of assets by nvestors s α[a A d (t) P I =1 n (t)a (t)]. The net demand from dealers s Ȧd (t), the change n ther nventores. Therefore, market clearng requres " Ȧ d (t) =α A A d (t) The measure n (t) satsfes ṅ (t) =δπ δn (t) for all, and therefore, # IX n (t)a (t). (17) =1 n (t) =e δt n () + (1 e δt )π, for =1,..,I. (18) If we use (13) to substtute a (t) from (17), t becomes apparent that ths market-clearng condton determnes ξ(t). The ntermedaton fees along the equlbrum path are gven by (2). Usng (9), (11) and (12), (2) reduces to U [a (t)] U (a) ξ(t)[a (t) a] φ (a, t) =η. (19) r + κ Defnton 1 An equlbrum s a collecton of bounded asset holdngs [{a (t)} I =1,A d(t)], together wth pecewse contnuously dfferentable trajectores for prces and ntermedaton fees, [ξ(t),p(t),φ (a, t)], that satsfy, (4) (6), (12) (14), (17) and (19). We do not lst the dstrbuton of asset holdngs across nvestors n the precedng defnton becausetdoesnotaffect the dealer s problem, the nvestor s problem, nor any of the varables whch are relevant to our analyss. To characterze the equlbrum, we begn by establshng two mportant propertes of any equlbrum prce path. Lemma 4 In an equlbrum, lm t e rt p(t) =, and (2) Z " p(t) = e r(s t) ξ(s) ξ(s) # ds. (21) r + κ t 13

14 The no-bubble condton (2) follows from addng up the transversalty condtons (6) and (14) across all agents, whch after observng that agents holdngs must add up to A> mply lm t E p(θ t )e rθt A =. Ths, n turn, can be shown to mply (2). Wth (2), (21) follows from (15). If we combne (13), (16) and (17) and assume an nteror soluton for dealers nventores, the model can be reduced to a system of two frst-order dfferental equatons ( ) IX Ȧ d (t) =α A A d (t) n (t)u 1 [ξ (t)], (22) =1 ξ (t) =(r + κ) ξ (t) υ [A d (t)] ª, (23) wth n (t) gven by (18). Ths system s nonlnear and nonautonomous. The steady-state equlbrum s such that U (a )=υ (a d )=ξ = rp, whereξ s the unque soluton to υ 1 (ξ)+ IX =1 π U 1 (ξ) =A. (24) Consder the lmt as the tradng frctons vansh,.e., as α approaches. From (15), ξ(t) =rp(t) ṗ(t), so the nvestor s cost of nvestng n the asset s the flow cost rp(t) mnus the captal gan ṗ(t), the same as the dealer s. From (11), U (a) tends to u (a), so (11) mples that the nvestor s optmal choce of assets satsfes u (a )=rp(t) ṗ(t). Thsstheassetdemand of an nvestor n a frctonless Walrasan market. A very tractable specal case of (22) and (23) obtans when n = π for all,.e., when the dstrbuton of preference types across nvestors s tme-nvarant, snce the system s then homogenous. (Note that ths does not mply that the jont dstrbuton of assets and preference types across nvestors s constant, so the economy need not be n a steady state.) Lnearzng (22) and (23) n the neghborhood of the unque steady-state, (Ād, ξ), the steady state can be verfed to be a saddle-pont. For some ntal condton A d () n the neghborhood of the steady state there s a unque trajectory, the saddle-path, that brngs the economy to ts steady state. Ths trajectory also satsfes (6), so t s an equlbrum. Lemma 5 establshes that for a gven ntal condton, such a path s the unque equlbrum. Fgure 2 depcts the dynamcs of the system wth a phase dagram. 14

15 A d = A d A A d Fgure 2: Dealers and nventores: Phase dagram Lemma 5 Suppose that n () = π for all, and that the ntal condton a d () = A d () s close to the steady-state value Ād. Then, there s a unque dynamc equlbrum, and t converges to the steady state. As dealers margnal utlty for the asset decreases, the ξ soclne shfts downward. As υ (a d ) tends to, a case we wll focus on n the followng sectons, the ξ soclne approaches the horzontal axs for all A d > and the vertcal axs for A d =. The steady state s then at the ntersecton of the A d soclne and the vertcal axs, and there s a saddle-path that brngs the economy to the steady state. 5 Effcency In ths secton we characterze the effcent allocaton. We carry out an elementary varatonal experment to dentfy the socal gans assocated wth lqudty provson by dealers, and provde a more formal treatment of the socal planner s problem n Appendx A. Use m(τ,t) to denote the margnal utlty that an nvestor enjoys at tme t, from the asset 15

16 poston he acqured at tme τ t. Let M(τ,t) (r + α)e t Z T t e r(s t) m(τ,s)ds,.e., M(τ,t) s the flow expected present value of an nvestor s margnal utlty for the assets he acqured at tme τ, fromtmet τ untl hs next contact tme wth dealers, T. 12 Let represent the length of a small tme nterval, then M(τ,t) solves the recurson M(τ,t)=(r + α)m(τ,t) +(1 r α )E t [M(τ,t+ )]. (25) At each pont n tme t>, aquanttya d (t) of assets s held by dealers, and the remanng A A d (t) s held by nvestors. Because there s a contnuum of nvestors establshng contact wth dealers at Posson ntensty α, the law of large numbers mples that, durng any small tme nterval [t, t + ], there s a quantty A d (t)+α [A A d (t)] of assets that can be reallocated between those nvestors who are n contact wth dealers, and between nvestors and dealers. Holdng A d (t) fxed, an effcent allocaton of the remanng α [A A d (t)] assets must equalze the margnal value M(t, t) of all nvestors who are currently contactng dealers and holdng assets. Otherwse, one could mprove welfare by reallocatng assets from nvestors wth low margnal valuatons to nvestors wth hgh margnal valuatons. Ths means that, M(t, t) =λ(t), (26) for some λ(t), whch represents the shadow prce that the planner assgns to assets n the hands of dealers at tme t (assumng nvestors hold some assets). We now provde a necessary condton for dealers nventory holdngs, A d (t), tobepartof an effcent allocaton. Start from an allocaton such that (26) holds at each tme, and perturb t as follows: () keep the same allocaton durng [,t), () take a margnal asset from some postve measure of early nvestors at tme t and gve them to dealers untl tme t+. () If an early nvestor recontacts the market at tme t +, gve the asset back to hm. If he does not recontact the market at tme t +, gve the asset to some other late nvestor who contacted themarketattmet +. (v) Contnuewththentalassetallocatonaftert +. (Snce 12 We ntroduce a dfferent notaton here so that the present calculatons also apply to the envronment of Secton 7, whch features aggregate uncertanty. In the envronment of the prevous secton, for nstance, m(τ,s)=u k(s) ak(τ),andwthη =, M(τ,s)=U k(s) ak(τ),whereak(τ) s the asset poston chosen by the nvestor at tme τ, andk(s) s the nvestor s preference type at tme s. Also, n ths secton we use E t to denote the expectaton operator condtonal on the nformaton avalable at tme t. 16

17 dealers asset holdngs at t + are the same as n the ntal allocaton, the quantty of assets avalable n the market stays the same, and t s feasble to contnue wth the ntal allocaton after t +.) We can break up the net utlty of ths perturbaton as follows. Frst, durng [t, t + ] assets are held by dealers, wth a margnal utlty υ (t), nstead of the early nvestors, wth a margnal utlty of m(t, t). 13 Ths represents a net flow utlty of (r + α)[υ (t) m(t, t)]. Second, there s a fracton α of early nvestors who re-establsh contact wth dealers at t + and receve ther asset back, wth a net utlty of zero from t + onwards. For the fracton 1 α of early nvestors who do not re-establsh contact wth dealers, there s an expected dscounted cost of E t e r M(t, t + ) ' (1 r )E t [M(t, t + )]. (27) Ths represents the dscounted margnal value that s lost because early nvestors hold one unt less of assets untl ther next respectve contact tmes wth dealers. Lastly, snce the asset s transferred to some late nvestors at tme t +, there s an expected gan of (1 r )E t [M(t +,t+ )]. (28) As before, equaton (28) s the dscounted margnal value that s ganed because late nvestors hold one more unt of assets untl ther next respectve contact tme wth dealers. Ths dscusson shows that the net utlty of the perturbaton s (r + α) υ (t) m(t, t) +(1 α )(1 r )E t [M(t +,t+ ) M(t, t + )]. (29) The second term represents the gan from lqudty provson. The dscountng factor, (1 r ), appears because the gan occurs later n tme. The probablty factor, (1 α ), appears because the gan occurs only f the early nvestors do not manage to re-establsh contact wth dealers. The last factor, E t [M(t +,t+ ) M(t, t + )], s postve when the margnal utlty of the early nvestor, M(t, t + ), s, on average, smaller than the margnal utlty of the late nvestor, M(t +,t + ). Ths means that lqudty provson can rase welfare by mprovng ntertemporal matchng,.e., by creatng a mutually benefcal match between two nvestors who contact dealers at dfferent ponts n tme. 13 Note that, snce agents have quas-lnear preferences, one must gve equal weghts to all agents margnal utltes for the assets. 17

18 To a frst-order approxmaton, equaton (29) can be rearranged as follows 14 (r + α) υ (t) λ(t) +(1 r α ) {E t [λ(t + )] λ(t)}. (3) Dvde (3) through by and take to zero to fnd that ncreasng the amount of nventores held by dealers does not mprove welfare f υ (t)+ 1 r + α lm E t [λ(t + ) λ(t)] λ(t). (31) Consderng the opposte perturbaton of decreasng dealers nventores, we fnd that (31) holds wth equalty whenever A d (t) >. In the envronment of the prevous secton, wth no aggregate uncertanty and where υ () =, wecandervethesefrst-order condtons formally usng the Maxmum Prncple. Lemma 6 An effcent allocaton h {a (t)} I =1,a d(t) satsfes (r + α) u [a (t)] + δ P I j=1 π ju j [a (t)] = λ (t), r + α + δ (32) υ [a d (t)] + λ(t) = λ(t), r + α (33) the resource constrant (17), and the transversalty condton for some λ(t). In addton, f a d (t) satsfes then lm t e rt λ(t) =, (34) lm t e rt λ(t)a d (t) =, (35) h {a (t)} I =1,a d(t) s an optmal path. If we dentfy the equlbrum prce, ξ (t), wth the planner s shadow prce of assets, λ (t), and compare (4) and (11) wth (32) and (33), t becomes apparent that they would be dentcal f κ = α,.e., f η were equal to zero. The followng proposton formalzes ths observaton. Proposton 1 Equlbrum s effcent f and only f η =. 14 Use (25) to rewrte (29) as (r + α)v (t) M(t, t) +(1 r α ) E t [M(t +,t+ )]. The expresson (3) then follows from (26). 18

19 Whenever η >, anneffcency arses from a holdup problem due to ex-post barganng. Whenever they trade, nvestors antcpate the fact that they wll have to pay fees for rebalancng ther asset holdngs n the future. These ntermedaton fees ncrease wth the surplus that the trade generates. As a consequence, nvestors wll tend to avod postons that could lead to large rebalancng n the future. 6 Crash and determnstc recovery In ths secton, we descrbe the dynamc adjustment of the asset prce and the allocaton of assets between dealers and nvestors followng a market crash. We thnk of a market crash as a sudden rse n sellng pressure, and model t as a one-tme unexpected shock that modfes the dstrbuton of nvestors across preference types, {n (t)} I =1, n a way that causes the total demand for the asset to fall unexpectedly. 15 We suppose that the economy s n the steady state at the tme ths shock hts, whch we take to be t =. The total quantty of assets demanded by nvestors s lowest at t =, and then gradually recovers over tme as the ntal dstrbuton of preference types, {n ()} I =1, reverts back to the nvarant dstrbuton, {π } I =1. In order to hghlght the ntermedaton role of dealers, we assume that they start off wth no nventory, a d () =, and that they get no utlty from holdng the asset,.e., υ(a) =. In ths formulaton, dealers wll only buy assets for ther own account n an attempt to make captal gans over some holdng perod. Hence, A d =n the steady state, snce dealers cannot make captal gans f the asset prce s constant. For nvestors, we adopt u (a) =ε a 1 σ /(1 σ), whch mples U (a) = ε a 1 σ /(1 σ), wth ε = (r+κ)ε +δ ε r+κ+δ and ε = P I k=1 π kε k. The followng lemma summarzes the key propertes of the nvestor s and the dealer s optmzaton problems. Lemma 7 (a) An nvestor wth preference type who gans access to the market at tme t, demands 1/σ ε a (t) =. (36) ξ(t) (b) A dealer s asset holdngs satsfy [rp (t) ṗ (t)] a d (t) =. (37) 15 Ths s the same noton of market crash used by Well (26). We study a dfferent noton of market crash n the followng secton. 19

20 The second part of Lemma 7 formalzes the noton that f dealers do not enjoy any drect benefts from holdng the asset, then they wll only hold t to try to obtan captal gans. Dealers hold no nventores over perods when the prce s growng at a rate lower than the rate of tme preference. Conversely, they are wllng to take long postons n the asset only f ṗ (t) /p (t) =r. (Naturally,ṗ (t) /p (t) >rwould be nconsstent wth equlbrum.) We can use (15) to express the dealer s optmalty condton as " ξ (t) ξ # (t) A d (t) =, (38) r + κ wth ξ (t) /ξ (t) r + κ, wherea d (t) denotes dealers aggregate nventores. (Notce that ndvdual dealers need not hold the same nventores here.) Wth (18) and (36), the market-clearng condton (22) can be wrtten as n h Ȧ d (t) =α A A d (t) ξ(t) 1/σ Ē e δt o Ē E, (39) where Ē = P I =1 π ε 1/σ and E = P I =1 n () ε 1/σ. Intutvely, ξ(t) 1/σ Ē e δt Ē E s the total quantty of assets demanded by nvestors at tme t. Ths way of wrtng the nvestors aggregate demand reveals two sources of tme varaton. Frst, nvestors aggregate demand wll change n response to changes n the effectve cost of purchasng the asset, ξ(t). The second component, Ē e δt Ē E, captures changes n aggregate demand due to composton effects comng from varatons n the dstrbuton of nvestors over the varous preference types. The constant Ē s a measure of nvestors wllngness to hold the asset n the steady state,.e., when n (t) =π,whlee reflects the nvestors wllngness to hold the asset at tme, when the aggregate shock hts. Thus, E /Ē s a measure of the magntude of the composton shock to aggregate demand for the asset. In lne wth our market crash nterpretaton, we mantan E /Ē<1throughout the analyss,.e., lower preference types receve larger populaton weghts at tme relatve to the steady state. The dealers frst-order condton, (38), and the market-clearng condton, (39), are a par of dfferental equatons that can be solved for ξ (t) and A d (t). If A d (t) > for all t n some nterval [t 1,t 2 ], then (38) mples ξ(t) =e (r+κ)(t t2) ξ (t 2 ), and gven ths path for ξ (t), (39) s a frst-order dfferental equaton that can be readly solved for the path A d (t). Smlarly, f A d (t) =over some nterval, then (39) mmedately mples a path for ξ (t). In order to fully characterze the equlbrum path one needs to determne the tme ntervals over whch dealers 2

21 accumulate nventores as well as the contnuty of the trajectory. The followng proposton provdes the salent features of the equlbrum path followng a market crash. Proposton 2 The unque equlbrum path, {ξ (t),a d (t)}, has the followng features: 1. It converges to the steady state, ξ, Ā d ª = { Ē/A σ, }. 2. There exsts a tme T [, ) such that A d (t) > for all t (,T) and A d (t) =for all t T. 3. Let p (t) denote the equlbrum asset prce that would obtan f dealers were constraned to hold no nventores. Dealers ntervene,.e., T>, f and only f, at the tme of the crss, t =, ṗ() p () >r, whch s equvalent to P I =1 n () [(r + κ) ε + δ ε] 1/σ P I =1 π [(r + κ) ε + δ ε] 1/σ < δσ r + κ + δσ. (4) Accordng to Proposton 2, the equlbrum path followng a market crash s characterzed by a swtchng tme T [, ) such that dealers hold the asset for all t (,T) and do not hold t for t T. It s possble that T =, n whch case dealers do not hold nventores at all. The last part of the proposton establshes that dealers wll ntervene f and only f ṗ ()/p () >r,.e., f and only f the rate of growth of the asset prce that would result at the tme of the crss f they dd not ntervene, exceeds the rate of tme preference. If dealers ntervene, the tme perod durng whch they hold the asset s an nterval, whch starts at the outset of the crss,.e., at t =. (See Fgure 4 for an llustraton.) Thus, dealers never fnd t benefcal to delay the acquston of the asset: If they wll buy at all, they start buyng from the very begnnng, when the nvestors sellng pressure s strongest. The economc reasonng behnd ths result s that snce dealers get no drect utlty from holdng the asset, they are only wllng to take long postons f the captal gans assocated wth those postons are large enough,.e., f the growth rate of the asset prce s greater than the dscount rate. It s possble to show that, n the absence of dealers nterventon, the prce of the asset, p (t), grows at a decreasng rate. Hence, f dealers don t have ncentves to hold nventores at t =, they never wll. In contrast, Well (26) fnds that dealers do not necessarly start accumulatng nventores rght after the crash, and that for some parameter values, delayng the nterventon 21

22 of dealers s socally optmal. 16 Notcethattheleftsdeof(4)equalsE /Ē. Thus, the last part of Proposton 2 states that the condton for dealers to partcpate,.e., ṗ ()/p () >r, can be expressed n terms of the exogenous severty of the crash, as measured by 1 E /Ē,.e., the magntude of the ntal drop n the nvestors wllngness to hold the asset. If ths drop s larger than the threshold 1 r+κ+δσ δσ, then dealers wll step n to take up the slack resultng from the reducton n nvestors demand. Conversely, dealers wll not ntervene f (4) s not satsfed. Condton (4) depends on all the fundamentals of the economy, e.g., preferences (σ), the extent of tradng frctons and the market-power of dealers (κ), the change n the dstrbuton of valuatons that trggers the crss ({n ()} I =1 ) and the frequency of the preference shocks (δ). Asshownnthenext corollary, there exst parametrzatons for whch condton (4) does not hold. Corollary 1 The set of parameter values under whch dealers do not accumulate nventores (.e., T =)snonempty. Corollary 1 contrasts wth Theorem 1 n Well (26), whch establshes that there s always a perod of tme durng whch dealers lean aganst the wnd before the nvestors sellng pressure subsdes. 17 Asuffcent condton for condton (4) to fal s that (r + κ) /δ be suffcently large. Suppose that preference shocks are very persstent (δ very small). In ths case the recovery s slow, the growth rate of the asset prce s small, and dealers fnd that the prospectve captal gans are smaller than the opportunty cost of holdng the asset. It s also nstructve to consder the lmtng case as α goes to nfnty and the economy approaches the frctonless Walrasan benchmark. In ths case, dealers no longer have the advantage of tradng contnuously vs-à-vs nvestors, and ther ablty to realze captal gans vanshes (recall our dscusson of (16)). Put 16 The key assumpton n Well (26) that les behnd ths result s that nvestors utlty functon s of the Leontef form, u(a) =mn{a, 1}, sotheyareeffectvely restrcted to hold zero or one unt of the asset. In contrast, here we allow nvestors to hold any nonnegatve poston. To reconcle our results wth Well s, we can nest Well s specfcaton wth ours by assumng an nvestor s utlty functon s σ +(1 σ)a θ 1 θ θ 1 θ u(a) =, 1 σ for some (σ, θ) R + (, 1). Our soelastc utlty functon s obtaned as θ 1, and Well s Leontef utlty functon s obtaned as θ +. Numercal calculatons (avalable upon request) suggest that, for θ close to zero, we would recover Well s result that dealers do not necessarly start accumulatng nventores at the tme of the crash. 17 Ths dfference n results s also due to the fact that nvestors asset holdngs are unrestrcted here but subject to a unt upper bound n Well (26) (see footnote 16). 22

23 dfferently, as frctons vansh, the market provdes dealers no ncentve to buy assets early n the crss, and they do not ntervene regardless of the severty of the crss. Below, we wll show that there are also parametrzatons for whch condton (4) s satsfed and dealers buy assets at the begnnng of the crss, hold them for a whle and sell them off as the nvestors sellng pressures subsde. In these cases, dealers choose postve asset postons (foregong nterest on ther stock of the numérare good) even though they get no utlty from holdng these assets. The reason why dealers may be wllng to carry assets s that they have contnuous access to the market whle nvestors do not: Ths tradng advantage allows dealers to tme the market contnuously n order to capture captal gans that nvestors cannot realze. Wthout dealers, or f dealers were unable to hold nventores, these captal gans would reman unexploted. In equlbrum, competton among dealers ends up equalzng these captal gans to the opportunty cost of holdng assets,.e., ṗ/p = r. Ths logc s consstent wth the frctonless lmt we dscussed above. Next,weusenumercalexamplestollustrateandexplanhowthekeyparametersnfluence the dealers ncentves to hold nventores. In what we wll consder to be the benchmark example, we set σ =1/2 and assume that the preference shock can ether be ε 1 =or ε 2 =1, wth equal probablty. Ths means that the nvarant dstrbuton has an equal measure of nvestors wth low and hgh valuatons. We also set r =.5 and α = δ =1, so that on average, nvestors get one preference shock and one chance to trade per perod. We also set η =so that the equlbrum allocaton of the benchmark parametrzaton corresponds to the soluton to the planner s problem. We consder an economy whch s at ts steady state, and at tme s subject to a shock that causes the fracton of nvestors wth the low preference shock to rse from π =1/2 to n 1 () =.95. The shaded (green) regons n Fgure 3 llustrate the combnatons of parameter values for whch condton (4) s satsfed so that dealers hold nventores after the crash. In each panel, we let the two parameters n the axes vary and keep the rest fxed at ther benchmark values. All panels have α our ndex of the degree of the tradng frctons on the horzontal axs. Markets wth large α are very lqud markets where trades get executed very fast. 23

24 Fgure 3: Parametrzatons for whch dealers lean aganst the wnd Fgure 3 allows us to address the followng normatve queston: Could t be socally effcent for dealers to accumulate nventores, even though they are pure speculators who don t derve any drect utlty from holdng assets? The answer s: yes. Recall (Proposton 1) that the equlbrum allocatons of an economy wth η =correspond to the Pareto-optmal allocatons. The thrd panel of Fgure 3 shows that there are parameterzatons nvolvng η =where dealers ndeed choose to ntervene. As explaned n Secton 5, the planner allocates assets to dealers n order to explot an ntertemporal trade-off between the margnal utlty of nvestors n the market at the current date and n the future. The average margnal valuaton of the asset across nvestors s low at the outset of the crss and hgher later on. The planner uses dealers nventores to smooth these margnal valuatons over tme. Specfcally, the planner may choose to put assets n the hands of dealers n the early stages of the crss (when the opportunty cost of not allocatng them to nvestors s relatvely low) to be able to transfer these assets wthout delays to nvestors n the later stages of the crss, when the margnal valuaton of the average 24

25 nvestor s hgh. Therefore, dependng on fundamentals, t can be optmal to have dealers act as a buffer stock. The opportunty cost of havng dealers carry an asset they don t value for a whle s the prce the planner pays to provde mmedacy to the future hgher-than-averagevaluaton cohorts of nvestors that wll gan access to the asset market at later dates. Let us now turn to the effects of fundamentals on dealers lkelhood to ntervene durng the crss. In turn, we wll consder the effects of the characterstcs of the crss, market structure and nvestors preferences. Characterstcs of the crss. The frst panel n Fgure 3 shows that for any gven α, dealers ntervene f n 1 () s large enough,.e., f the crash s suffcently severe. To explan ths result we resort to the connecton to the planner s problem (but there s an equvalent explanaton n terms of the dealers ncentves n the equlbrum). In the early stages of the crss, the cohorts of nvestors that contact the marketplace nvolve a very large fracton of low-valuaton nvestors who have relatvely low ndvdual demands for the asset. If the planner chooses not to use the dealers nventores, then n these early stages he wll be reallocatng more assets to the few hgh-margnal-valuaton nvestors. Such an allocaton wll mply a very low shadow prce of assets (denoted λ(t) n Secton 5) n the early stages of the crss. Conversely, the shadow prce of the asset wll be relatvely large at later dates, as the fracton of hgh-valuaton nvestors ncreases toward ts steady-state level, snce at that pont there wll be many more hgh-valuaton nvestors who are wllng to hold relatvely large quanttes of the asset. To larger values of n 1 () correspond larger dscrepances between the margnal utltes of earler and later cohorts of nvestors, among whch the planner can reallocate assets (ths dscrepancy s measured by the term λ(t + ) λ(t) of (3)). Dealers offer the planner a way to smooth these dfferences n ntertemporal margnal utltes across cohorts of nvestors, and are used as abuffer stock for large values of n 1 (),.e., whenever the crash s severe. The second panel n Fgure 3 shows that, gven α, dealersfnd t optmal to ntervene f the recovery s fast enough (.e., f δ s large enough), so that they would not have to hold the asset for very long. However, the fgure also shows that dealers won t ntervene f δ s too large. Ths s because δ not only measures the speed of the recovery but also the arrval ntensty of dosyncratc preference shocks. Wth a very large δ, the average type of an nvestor over hs holdng perod, e.g., ε, becomes very close to the mean, ε. In ths case, the economy becomes very smlar to an economy wthout dosyncratc preference shocks, so there s lttle need to 25

26 reallocate assets across nvestors (see our dscusson of Corollary 1). Market structure. We dentfy the structure of the market wth two parameters: α, the extent of the tradng frctons, and η, dealers barganng strength. The frst panel n Fgure 3 shows that, for a gven sze of the aggregate shock, dealers provde lqudty f tradng frctons are nether too severe nor too small. For large α, nvestors face short delays to rebalance ther asset holdngs, 1/α on average. Ths ncreases ther wllngness to take more extreme postons. In partcular, nvestors wth hgher-than-average utlty become more wllng to hold largerthan-average postons and absorb more of the sellng pressure. In some cases, when α s large enough, they end up supplyng so much lqudty to other nvestors that dealers don t fnd t proftable to step n. Conversely, f α s very small, then ε becomes close to ε, and all nvestors choose very smlar asset holdngs regardless of ther preference type. In ths case, the economy becomes smlar to an economy wthout dosyncratc preference shocks, and dealers are not needed to reallocate assets across nvestors. The thrd panel n Fgure 3 reveals that for any gven α, dealersaremorelkelytohold nventores f ther barganng power s nether too large nor too small. Snce α and (1 η) enter the equlbrum condtons as a product, an economy wth large η s, from an nvestor s standpont, payoff equvalent to an economy where nvestors access the market very nfrequently,.e., an economy wth small α. Recall that f η =, the economy s constraned-effcent. Therefore, the thrd panel shows that there are parametrzatons for whch dealers ntervene n equlbrum although the planner would not have them ntervene, as well as parametrzatons for whch the opposte s true. Preferences. The fourth panel of Fgure 3 llustrates the role that σ, the curvature of the nvestor s utlty functon, plays n the dealer s decson to hold the asset. Frst, σ<1s a necessary condton for dealers to ntervene. In the case of the most severe crss possble,.e., n 1 () = 1 (no nvestor values the asset at t =), one can show that dealers ntervene f and only f σ<1 regardless of the value of α. Ifσ =1, the trajectory of the prce s p(t) = ε ra + e δt (r + δ) A IX n ()(ε ε), =1 26

27 whch s ndependent of α. In fact, n ths case p(t) concdes wth the prce that would preval n a frctonless Walrasan market. 18 But as we argued earler, n a Walrasan market dealers wouldholdnoassetssncearbtragebynvestors would prevent the asset prce from growng faster than the dscount rate. Therefore, dealers never hold nventores for σ =1. Forlower values of n 1 (), the dealers ncentves to hold the asset are nonmonotonc n σ. In partcular, for a range of values of α they only hold t f σ s n some ntermedate range, but not f t s too bg or too small. Suppose that σ s very bg, so that margnal utlty s very steep. One could thnk that there s more room for dealers to smooth dfferences n ntertemporal margnal utltes across cohorts. However, f σ s very large, then the ndvdual asset demand of an nvestor wth hgh valuaton tends to be very close to the asset demand of an nvestor wth low valuaton, and ths reduces the beneft from transferrng assets between them. In the extreme case σ, a (t) =A for all and all t. But of course ths means that shockng the nvarant dstrbuton from {π } I =1 to {n ()} I =1 has no effect on asset holdngs. So effectvely, there s no shock and thus no gan from lqudty provson, even f {π } I =1 frst-order stochastcally domnates {n ()} I =1. Alternatvely, one can nterpret 1/σ as the elastcty of asset demand, a, wth respect to the preference shock, ε.asσ, asset demand becomes nelastc to the preference shock. In ths case, the planner s shadow prce (λ (t) n the notaton of Proposton 6) s constant over tme, so there s no need nor scope for hm to reallocate assets over tme. It s also nstructve to look at the opposte extreme of very low σ. For example, consder what happens as σ so that nvestors preferences become lnear. Suppose that ε 1 <ε 2 <... < ε I. 19 From (11) t follows that a for {1,...,I 1},.e., only nvestors wth the hghest margnal utlty, ε I, hold the asset. Furthermore, ξ(t) ε I for all t and, from (21), p(t) p = ε I /r for all t. Thus, the prce of the asset s constant and equal to ts steady-state level. There s clearly no ncentve for dealers to buy the asset, regardless of the ntal shock to the populaton weght of nvestors wth hgh valuaton. In ths extreme case, nvestors desred holdngs change dramatcally n response to preference shocks, but margnal utlty s constant at all tmes among those who demand the asset, so a planner would have no need to use dealers to store the assets n order to smooth the margnal utltes of cohorts of nvestors at varous 18 Wth log preferences an nvestor s demand s lnear n ε, so the aggregate demand for the asset only depends on ε,.e., t s ndependent of α, see Lagos and Rocheteau (27). For a related result under a CARA utlty functon, see Gârleanu (26). Also, note that for σ =1, condton (4) reduces to I =1 n ()ε <, ndcatng that dealers never ntervene. 19 Ths utlty specfcaton s the same one used n Duffe et al. (25) and Well (26), except that they assume a unt upper bound on nvestors holdngs. 27

28 ponts n tme. We can summarze the dscusson above as follows. Dealers provde lqudty by accumulatng asset nventores f: () the market crash s abrupt and the recovery s fast; () tradng frctons are nether too severe nor too small; () dealers market power s not too large; (v) dosyncratc preference shocks are not too persstent and nvestors asset demand s not too nelastc wth respect to preference shocks. Fgure 4: Dealers asset holdngs Whle Fgure 3 llustrates the condtons under whch dealers accumulate nventores, t s not nformatve about the extent of dealers nterventon, e.g., what quantty of the asset do dealers accumulate, and how long s the holdng perod? To answer these questons, Fgure 4 plots the trajectory for dealers nventores for the parameter values of our benchmark example. 2 In both panels one can clearly dentfy T, namely the swtchng tme at whch A d (t) becomes zero after a perod over whch dealers have held assets. The frst panel llustrates the relatonshp between market structure (κ) and dealers nventory polcy. Tradng frctons have a nonmonotonc effect on T : the length of the holdng perod s ncreasng n κ for low values of κ (because nvestors take more extreme postons, whch ncreases the dscrepancy between ther margnal utlty at dfferent dates), and decreasng for large values of κ (because nvestors need less lqudty from dealers when tradng frctons are mld). The second panel of Fgure 4 descrbes dealers nventory behavor as a functon of the severty of the crash. As n 1 () decreases, the holdng perod shrnks and the quantty of assets held by dealersatanypontntmebecomessmaller. So n a more severe crash, dealers provde more lqudty and for a longer perod of tme. 2 Together wth Proposton 2, Lemma 13, whch s stated and proved n Appendx A, provdes a full characterzaton of the equlbrum path followng a market crash, ncludng closed-form expressons for the paths of ξ (t) and A d (t). 28

29 The followng proposton compares the trajectory for ξ(t), the equlbrum effectve cost of holdng the asset, to the trajectory of ξ (t), theeffectve cost of holdng the asset that would result f dealers were constraned to hold no nventores. ^ T A d ( t) > ( t) = A d Fgure 5: Paths for ξ (t) and ξ(t) Proposton 3 If condton (4) holds, then there exsts t such that ξ(t) >ξ (t) for all t [,t) and ξ(t) <ξ (t) for all t (t,t). Accordng to Proposton 3, the presence of dealers mtgates the effect of the market crash on the effectve cost of holdng the asset. By accumulatng nventores rght after the crash, dealers prevent ξ from fallng too much: ξ() s hgher than t would have been had dealers not stepped n to buy assets. Ths s llustrated n Fgure 5. 7 Crash and stochastc recovery In the prevous secton, our operatonal defnton of a market crash was a shock to the dstrbuton of nvestors across valuatons whch caused the nvestors total demand for the asset to fall. The recovery path corresponded to the transtonal dynamcs leadng to the steady state, so t was determnstc and t started mmedately after the shock. It may be 29

30 argued that durng actual market crashes, the nvestors behavor and the dealers decsons about whether to ntervene and when to ntervene, may be affected by uncertanty about the duraton of the crss. For ths reason, n ths secton we study the dealers ncentves to provde lqudty n the aftermath of a crss wth an uncertan recovery. We consder the followng scenaro. At tme zero all nvestors receve an unantcpated multplcatve shock that temporarly scales down ther margnal utlty from holdng the asset. Ths consttutes the crash. Subsequently, the economy awats a recovery shock that follows a Posson process wth arrval rate ρ whch causes all nvestors to smultaneously revert back to ther pre-crss wllngness to hold the asset. Formally, we let T ρ be an exponentally dstrbuted random varable wth mean 1/ρ, wheret ρ denotes the tme at whch the economy reverts to normal. An nvestor wth preference type gets utlty u (a) from holdng a for all t< and all t T ρ. For t [,T ρ ), the nvestor gets utlty Ru (a), wthr<1. Thus, a small R ndcates that the crash s severe, and a small ρ that t s expected to be long-lved. 21 We assume that the stochastc process that descrbes the recovery s ndependent of the one that governs an nvestor s transtons between preference types. Furthermore, n ths secton we assume {n ()} I =1 = {π } I =1,.e., that the ntal dstrbuton of preference types s the nvarant dstrbuton. We dscuss the equlbrum dynamcs usng Fgure 6. (Appendx B provdes an analytcal soluton of the model.) We let A d (t) be the dealers nventores at tme t, condtonal on t<t ρ,.e., gven that the recovery has not occurred untl tme t. We denote ξ (t) the effectve cost of holdng the asset before the recovery takes place. Smlarly, we use the superscrpt h to denote varables after the recovery has occurred. The soclne Ȧ d =s located to the rght of the soclne Ȧd =mpled by (22). Ths s because, for any gven ξ, dealers need to hold more of the asset n order to clear the market. The soclne ξ =s downward-slopng and located underneath the saddle-path leadng to the long-run steady state, ξ,. The equlbrum unfolds as follows. The economy starts at A d () =, and at the tme of the crash, ξ jumps down to the saddle-path leadng to ( ξ, Ā d ). (Ths saddle-path s represented by a dotted lne n the fgure.) The economy then evolves along ths saddle-path untl the random recovery shock occurs. In the meantme, along ths path, dealers nventores ncrease and ξ (t) decreases. At the random tme when the recovery occurs, say t ρ, the system jumps to the saddle-path leadng to ξ,. 21 One vrtue of ths formulaton s that t dsentangles the speed of the recovery, ρ, and the frequency of the dosyncratc preference shocks, δ. In the prevous secton, both were captured by δ. 3

31 Ths saddle-path, denoted ξ = ψ(a d ), s represented by a dashed lne n the fgure. At the tme the recovery shock occurs, the cost of holdng the asset jumps from ξ to ξ h, and dealers begn sellng ther nventores gradually untl they are completely depleted. A d = l A d = l ξ >>>> ψ l ξ A d = A d ( t ρ ) l A d A d Fgure 6: Stochastc recovery: Phase dagram The followng proposton provdes a condton under whch A d (t) > for all t>before the recovery occurs,.e., a condton for dealers to lean aganst the wnd durng a crss of random duraton. It s convenent to defne Ṽ (a) as the expected sum of dscounted utlty flows from holdng asset poston a for an nvestor of preference type untl the next tme he contacts a dealer, and U (a) =(r + κ)ṽ (a).22 Proposton 4 Let p be the asset prce durng the crss, and ph be the prce after the stochastc recovery, that would obtan f dealers were constraned to hold no nventores. Dealers hold nventores durng the crss f and only f ρ(ph p ) >r, whch s equvalent to p IX =1 µ π U 1 ρ ξ r + κ + ρ <A. (41) Proposton 4 provdes a condton on fundamentals such that dealers fnd t benefcal to buy assets durng the crss. Analogously to what we found for the case of a determnstc recovery, 22 We report the expresson for U (a) n the proof of Lemma 15 n Appendx B. 31

32 dealers ntervene f and only f ρ(p h p )/p >r,.e., f and only f the expected captal gan that they would obtan by buyng the asset durng the crash and re-sellng t once the economy recovers, n an economy where dealers do not ntervene, exceeds the rate of tme preference. Condton (41) need not always hold, as the followng two lmtng cases show. Consder frst the frctonless lmt α. Then, U (a) Ru (a), andtheleftsdeof(41)approaches. If nvestors can access the market as frequently as dealers, there s no role for dealers to provde lqudty by buyng assets. Next, consder the case where ρ,.e., the crss becomes permanent. Agan, the left sde of (41) can be shown to approach. If the shock s permanent, dealers cannot expect to make captal gans, and therefore they do not nvest n the asset. We summarze these fndngs as follows. Corollary 2 The set of parameter values under whch dealers do not accumulate nventores s nonempty. Corollary 2 s the analogue of Corollary 1 for a crss of random duraton. Next, we show that there exst parameterzatons for whch condton (41) s satsfed. To ths end, let u (a) = a 1 σ /(1 σ) and u (a) =ε u (a). Then, durng the crss, an nvestor s flow expected utlty U (a) =ˆε u(a), whereˆε s gven n the followng corollary. Corollary 3 Let u (a) = ε a 1 σ 1 σ. Dealers hold nventores durng a crash f and only f where ˆε = r+κ+ρ r+κ P I =1 π ˆε 1/σ P I =1 π ε 1/σ µ < (r+κ+ρ)ε +δ I j=1 π jε j r+κ+δ+ρ R + ρ r + κ + ρ ρ r+κ+ρ 1 σ, (42) (r+κ+ρ) ε +δ I j=1 π j ε j r+κ+δ+ρ. Condton (42) s a condton on fundamentals, ncludng the degree of tradng frctons, preferences and the propertes of the crash. The shaded (green) regons n Fgure 7 llustrate the combnatons of parameter values for whch condton (42) s satsfed so that dealers hold nventores n tmes of crss. The benchmark parametrzaton s: σ =.5, r =.5, π 1 = π 2 =.5, α = δ =1, ρ =1, R =.2 and η =. In each panel, we let the two parameters n the axes vary and keep the rest fxed at ther benchmark values. All panels have α our ndex of the degree of the tradng frctons on the horzontal axs. 32

33 Characterstcs of the crss. The frst panel confrms one of our fndngs from Secton 6: dealers are more lkely to accumulate asset nventores when the market crash s severe (R low). If the crash s severe, dealers expect a larger captal gan when the economy recovers and hence they have an ncentve to buy assets durng the crash. Accordng to the second panel of Fgure 7, for dealers to buy the asset, the crash must be antcpated to be short-lved (ρ must be suffcently large). From the planner s standpont, f ρ s small, the opportunty cost of havng dealers hold assets (.e., the utlty foregone by nvestors) s hgh. Thus, for ρ low enough, the planner would not use dealers nventores to reallocate the asset across nvestors over tme. Market structure. As before, we dentfy the market structure wth the parameters α and η. The thrd panel of Fgure 7 shows that dealers accumulate the asset f ther barganng power s nether too large nor too small. If η s close to 1, nvestors only enjoy a small gan from rebalancng ther asset holdngs. As a consequence, when n contact wth a dealer they put more weght on ther average preferences n order to reduce ther need to readjust ther asset holdngs n the future. As dscussed above, f dosyncratc preference shocks become less relevant, there s less scope for dealers to help reallocate the asset over tme. To understand why dealers have lower ncentves to provde lqudty when η s small, recall that a reducton n η s smlar from the pont of vew of nvestors payoffs toanncreasenα: If tradng frctons are reduced, there s less need for the bufferstockofassetsprovdedbydealers. Preferences. The fourth panel shows that the curvature of nvestors utlty functon must be suffcently small for dealers to accumulate asset nventores. As before, f σ s hgh, nvestors demand for the asset s relatvely nelastc wth respect to the dosyncratc preference shock, whch reduces the usefulness of dealers. To conclude, we study how the characterstcs of the crash and the structure of the market affect the amount of lqudty provded by dealers. For our baselne parameter values, n Fgure 8 we plot the maxmum quantty of assets that dealers are wllng to accumulate durng the crash, namely, Ā d =lm t A d (t).23 The frst panel confrms the nonmonotonc relatonshp between dealers provson of lqudty and the degree of frctons that preval n the market. The second panel shows that dealers wllngness to provde lqudty ncreases wth the severty of the crss. Accordng to the thrd panel, the relatonshp between the maxmum amount of 23 In Appendx B we report a closed-form expresson for Ā d. 33

34 lqudty that dealers are wllng to provde and the expected duraton of the crss (1/ρ) s nonmonotonc. If the crash s very persstent (ρ small) dealers are not wllng to accumulate large postons snce the expected dscounted captal gan of these nventores s small. If the crash s antcpated to be short-lved (ρ large), dealers wll not accumulate too much nventores because the crash reduces nvestors asset demand only by a small amount. The fourth panel shows that dealers nventores decrease as nvestors ntertemporal elastcty of substtuton (1/σ) gets smaller. In partcular, as nvestors utlty functon becomes lnear (σ ) dealers are wllng to accumulate the entre stock of assets n the economy (Ā d A). Fgure 7: Parametrzatons for whch dealers lean aganst the wnd 34