Shock Propagaton Through Cross-Learnng wth Costly Prce Acquston Jan Schneemeer Indana Unversty - Kelley School of Busness October 23, 2017 Abstract Ths paper shows that cross-learnng from other frms stock prces leads to the propagaton of unrelated shocks. Located n a crcular network, neghborng frms share a productvty shock. Stock prces are nformatve about future productvty and managers learn from them to mprove nvestment effcency. Wth costly prce acquston, each frm only learns from ts closest neghbors and s thus exposed to movements n ther prces reflectng ther own cross-learnng from more dstant frms. Ths non-local nose s then reflected n each frm s stock prce and transmtted further to other frms, partcularly n uncertan tmes and hghly correlated networks. Keywords: cross-learnng, feedback effects, nformaton acquston, manageral learnng. JEL Classfcaton: D83, D85, G14, G31. I thank Jean-Edouard Collard, Alex Edmans, Therry Foucault, Tarek Hassan, Juhan Lnnanmaa, Thomas Mertens, Johannes Tschler, Semh Uslu, Jesse Wang, and semnar/conference partcpants at the Federal Reserve Board of Governors, Frankfurt School of Fnance & Management, Indana Unversty (Kelley), Unversty of Vrgna (McIntre & Darden), FIRS, MFA, and SGF. Ths paper was prevously ttled "Shock Propagaton Through Cross-Learnng n Opaque Networks." Emal: jschnee@u.edu; Webste: www.jan-schneemeer.com
1 Introducton Informatve stock prces can gude real decsons because they aggregate the prvate nformaton of a large number of market partcpants. The dea that there exsts such a "feedback effect" from the fnancal market to frm decsons has receved emprcal and theoretcal support from the exstng lterature.1 In modern economes frms are hghly nterconnected. For example, they mght be lnked to each other through a producton network or operate n overlappng markets. As a result, frms should be able to mprove ther decsons by learnng from the prces of other frms as well. For nstance, consder a frm that has to decde whether to ncrease ts producton of a gven good and that observes an unusually hgh stock prce of a frm operatng n the same market. The frm does not know whether ths prce move s drven by fundamentals (lke hgher future demand) or other factors (lke postve market sentment). Overall, the frm should, however, nterpret ths ncrease as a postve sgnal about future demand and scale up nvestment. In ths paper, I analyze a settng n whch multple frms are located n a crcular network. Frms are parwse connected through ther exposure to a common productvty shock. Hence each frm s productvty shock s correlated wth that of ts left and rght neghbor but uncorrelated wth that of all other frms. Frm managers are mperfectly nformed about these shocks and have an ncentve to learn addtonal nformaton about them to nvest more effcently. Because stock prces reflect nformed traders prvate nformaton about local productvty shocks, each manager can mprove hs knowledge by relyng, n part, on the stock prces of the frm s neghbors when decdng on frm nvestment. As a frst result, I show that n ths setup frm managers can get the most precse sgnal about ther frm s productvty shock by combnng the prces of all frms n the network. 1See Bond et al. (2012) for a comprehensve survey of the feedback lterature. 1
Therefore, even though only the stock prce of each frm s drect neghbor reflects actual nformaton about ts future productvty shock, the prces of other, unconnected frms are useful sgnals as well. Intutvely, these prces are necessary to correctly nterpret movements n the stock prce of the drect neghbor and ther observaton allows the manager to gnore prce movements that are solely due to ths frm s learnng from ts neghbor. In ths benchmark equlbrum, frm nvestment and stock prces are correlated for neghborng frms that share a common productvty shock, but uncorrelated for all other frm pars. Thus, a gven locaton-specfc shock affects the two frms that are drectly exposed, but not other frms, such that there s no shock propagaton under costless prce acquston. In the man model, I ntroduce an nformatonal frcton by assumng that the collecton of prce sgnals s costly. Therefore, frm managers have to wegh the beneft of collectng more prce sgnals (more effcent nvestment) aganst ts cost (hgher nformaton acquston cost). In equlbrum, frm managers thus only observe a subset of prce sgnals and cannot perfectly flter out all non-local nose. One mght argue that n realty prce acquston costs should be neglgble because prce data s freely avalable to all market partcpants. However, ths cost should be nterpreted n a broader sense. It requres a sgnfcant amount of tme and resources by frms to properly analyze these prces or as Vves and Yang (2016) put t, "data can be vewed as nformaton only after t has been analyzed." That s to say, t s relatvely easy to observe the level of prces, but t requres a lot of n-depth analyss and background knowledge to map ths number nto an nformatve sgnal about a specfc frm s future fundamentals or payoffs. In the model, fnancal markets are populated by nformed nsders who trade clams to the local frm s termnal payoff. These traders receve a prvate sgnal about the local shock 2
and trade based on ths nformaton. The equlbrum stock prce therefore reflects the aggregated prvate nformaton about ths shock together wth a local nosy supply shock and serves as an endogenous publc sgnal. Benevolent frm managers can then combne these sgnals wth ther prvate nformaton to mprove the effcency of ther captal nvestment. Ths nvestment decson and two productvty shocks (the "fundamentals") determne the frm s termnal value and therefore fnal cash flow. Whle nformaton about the frst component (the "local" shock) s perfectly known by the local manager, he only observes the pror dstrbuton for the second component (the "non-local" shock). The manager can, however, learn addtonal nformaton about the non-local shock from stock prces of other frms. In equlbrum, each frm s stock prce reflects the nsders expectaton about both components of the fnal payoff, the composte productvty shock and future frm nvestment. Moreover, the prce s also affected by a frm-specfc random supply shock that adds nonfundamental nose. Each frm s nvestment decson depends on an endogenously chosen vector of stock prces that helps the manager to nvest more effcently. Importantly, I show that the equlbrum prces and nvestment decsons dffer from those n the benchmark equlbrum along several dmensons f each frm manager faces a prce acquston cost. Most mportantly, f frms can only learn from a subset of stock prces, ther nvestment decson and stock prce are no longer only exposed to local shocks, but depend on fundamental and fnancal shocks from multple remote locatons. Intutvely, each frm cannot flter out all non-local nose from the stock prce of ts drect neghbor such that ts manager s condtonal expectaton, and so the frm s nvestment decson, s affected by ths nose. Snce ths frm s stock prce, n turn, reflects the expected nvestment decson, the remote shock s further transmtted through the entre network of frms. Interestngly, 3
I show that ths propagaton mechansm s stronger n tmes of hgh fundamental uncertanty and hgh parwse correlaton of fundamentals, when managers have a partcularly hgh ncentve to rely on prce sgnals. A novel mplcaton of the model s that the propagaton of shocks s non-monotonc. Thus, a gven shock can affect a certan group of nearby frms, then skp several locatons, and affect more dstant frms agan. Moreover, I show that a larger number of observed stock prces always allows frms to nvest more effcently because each manager s expectaton of hs frm s future productvty shock becomes more precse. However, as a byproduct, the senstvty of stock prces and nvestment decsons to unrelated, non-local shocks also ncreases. Next, I allow each frm to choose the number of observed prces compettvely. Interestngly, I show that each frm s prvate ncentve to acqure more prce nformaton s always hgher than the socal ncentve. Intutvely, each frm does not nternalze that t renders ts own prce more nosy for ts backward neghbor by collectng more prce sgnals from ts forward neghbors. As a result, the equlbrum number of observed prces s always neffcently hgh. In a numercal exercse, I explctly compute the number of observed prces and other key varables n equlbrum and compare t to the socal optmum that maxmzes all frms ex ante value collectvely. Overall, the man contrbuton of ths paper s to show the equlbrum mplcatons of frms cross-learnng when the collecton or nterpretaton of prce data s costly. Steppng away from the frctonless benchmark hghlghts several novel mechansms that can help to understand a varety of stylzed facts regardng the mpact and propagaton of dosyncratc macroeconomc and fnancal shocks. Frst, there s a substantal emprcal lterature concludng that reward for rsk reflects, to some extent, local factors that should 4
be dversfable.2 In ths paper, I show that local, frm-specfc shocks can be transmtted through the entre network of frms as long as each frm partally bases ts nvestment decson on the stock prce of another frm (but not on all prces). These shocks, thus, do not wash out quckly as the number of frms ncreases and cannot be dversfed away easly trough nvestment n an ndex fund for nstance. Moreover, I show that the rate of decay for dosyncratc shocks depends crucally on the nformaton envronment n the fnancal market. For example, hgher pror uncertanty or less nosy supply ncrease the propagaton ntensty and market-wde effect of frm-level shocks. Second, epsodes of hgh uncertanty (e.g. due to the arrval of novel technologes) are often assocated wth "exuberant" jont movements n asset prces and real economc actvty.3 In these epsodes, when frms have a hgh ncentve to learn from ther neghbors prces, the propagaton of shocks through the network of frms s strongest. As a result, a local fundamental (productvty) or non-fundamental (lqudty) shock can be transformed nto a (quas) systematc shock that affects drectly and ndrectly connected frms, such that a large postve shock to a specfc frm can lead to above-average ("exuberant") nvestment and a rally n stock prces for many frms. Taken together, ths paper helps to understand the real effects of fnancal markets wth many connected frms n general. In partcular, t shows that cross-learnng amplfes the degree of nterconnectedness n an economy because fundamentally ndependent frms appear correlated even though they do not drectly learn from each other. From a techncal perspectve, I provde a novel setup wth multple nterconnected frms and learnng from several prces. A specfc functonal form for the nosy supply of assets and the traders objectve functon, that have both been used n other contexts before, allow me to keep the model tractable. 2See e.g. Bekaert and Harvey (1995) for emprcal results or Garleanu et al. (2015) for a summary of ths lterature and a unfyng theory. 3See e.g. Angeletos et al. (2012) or Huang and Zeng (2015) for models along these lnes. 5
Ths paper bulds on the dea n Hayek (1945) that effcent markets aggregate prvate and publc nformaton nto prces. The extent to whch prevalng prces are nformatve about the future value of a frm s mportant for both traders and real decson makers, such as frm managers, central bankers, or poltcans. The more nformaton these agents can extract from stock prces, the more they can mprove on ther economc decsons, such as tradng, corporate nvestment, and polcy nterventons.4 Ths key nsght of nformatve prces led to the large lterature on nosy ratonal expectatons equlbra followng Grossman and Stgltz (1980), Hellwg (1980), Damond and Verreccha (1981), and Admat (1985). A recent lterature bulds further on ths nsght and models an nformatonal feedback effect from the fnancal market to frm decsons. For nstance, n Subrahmanyam and Ttman (2001) and Goldsten et al. (2013), nformatve sgnals orgnatng from the fnancal market nfluence a sngle frm s nvestment decson: frm managers are mperfectly nformed about future productvty and can learn some addtonal nformaton from stock prces.5 Ths paper s most closely related to Foucault and Fresard (2014), Huang and Zeng (2015), and Dessant et al. (2016). These three papers also consder setups n whch a frm can learn addtonal nformaton from the stock prces of other frms n the economy. There are two key dfferences wth respect to these papers. Frst, the crcular structure n my paper mples that there exst frms n the economy whch are not drectly lnked to each other, but only ndrectly because both have a common neghbor, for example. Consequently, ths novel framework allows me to study the mpact and mportance of dosyncratc shocks on economc aggregates and the equlbrum decsons of unrelated 4See Luo (2005), Chen et al. (2007), Bakke and Whted (2010), Edmans et al. (2012), and Edmans et al. (2017) for emprcal evdence of a feedback effect from the stock market to real decsons. 5See also Subrahmanyam and Ttman (2013), Goldsten and Yang (2014), Davd et al. (2016), Goldsten and Yang (2015a), and Hassan and Mertens (2017) for models wth a feedback effect from nformatve stock prces to a sngle frm s nvestment decson. 6
frms. Second, frms endogenously determne the precson and extent of ther composte prce sgnal. I show that ths nformaton acquston decson s closely related to the propagaton of shocks throughout the network. Furthermore, ths paper s related to the lterature on (nformaton-based) fnancal contagon such as Admat (1985), Kodres and Prtsker (2002), Pasquarello (2007), and Caballero and Smsek (2013).6 Pror work n ths lterature also emphaszes how local shocks can be transmtted across assets or frms and that ths transmsson can lead to aggregate fluctuatons. The man contrbuton relatve to ths lterature s to derve a novel contagon mechansm. In ths paper, contagon arses through a feedback effect between stock prces and nvestment. Therefore, local shocks are transmtted through a dfferent learnng channel: prces n one locaton affect nvestment n another locaton because they convey nformaton. Ths mpact on nvestment s then reflected n the local prce whch n turn affects nvestment at another locaton. Interestngly, ths contagon mechansm does not rely on the fact that one agent learns from unrelated prces and thus transmts dstant shocks drectly, as e.g. n Admat (1985) or Kodres and Prtsker (2002). In the man model, traders and managers only learn from a subset of prces but non-local shocks from other frms are stll transmtted throughout the network. The remander of ths paper s organzed as follows: Secton 2 sets up the model. Secton 3 solves for the equlbrum n a benchmark economy where all frms observe all prces. Secton 4 shows the equlbrum n the man model where each frm only observe a subset of prces. Secton 5 endogenzes the number of observed prces and Secton 6 concludes. 6See also Gabax (2011), Acemoglu et al. (2015), Barrot and Sauvagnat (2016), and Bgo and La O (2016). 7
2 The Model 2.1 Setup There s a large number of frms (or "locatons"), ndexed by N {1,..., N}. Each frm s run by a benevolent manager who decdes on ts captal nvestment. Together wth a random productvty shock, ths decson determnes the frm s termnal output. Each frm has access to a lnear producton technology Y e θ K, where K denotes captal nvestment and θ s a cross-correlated productvty shock descrbed n greater detal below. Clams to ths output are traded n a secondary fnancal market. Three tme perods exst. In t 0, frm managers acqure a set of n costly prce sgnals from the other frms stock prces to mprove ther nvestment decson. In t 1, the fnancal market s actve and stock prces are determned. In t 2, the frm managers make an nvestment decson, the termnal payoffs are realzed and traders get pad. Productvty Shocks and Informaton Sets The N frms are located n a crcular network as depcted n Fgure 1. I choose ths network structure prmarly because t allows for () the presence of connected and unconnected frms and () a tractable soluton. Neghborng frms are exposed to a common shock such that ther fundamentals (θ ) are correlated. For nstance, the fundamental shock for frm 1 s correlated wth that of ts forward neghbor ( 2) and that of ts backward neghbor ( N), but uncorrelated wth that of all other frms n the network. Defnton 1 Let x (j) x k, such that x (j) denotes the realzaton of the generc random varable x at locaton k, where k s j J frms clockwse away from frm. Let J {1,..., N 1} denote the set of neghbors for each frm. 8
1 e 1 e 2 N 2 e N e 3... Fgure 1: Crcular network wth N frms. I assume that θ, the productvty shock for frm, equals the sum of two components:7 θ e + ρe (1) wth ρ [ 1, 1]. (1) The two components are ndependent,.e. e d N ( 0, σ 2 e ) for all N. Intutvely, each frm s future productvty shock s manly determned by the "local" shock e, whch also affects the productvty of the frm s backward neghbor. Vce versa, θ also depends on the local shock of the frm s forward neghbor, e (1). The constant ρ determnes the strength and sgn of ths effect, and thus the degree of fundamental entanglement n the economy. Consequently, ths structure for θ mples that each frm shares a common shock wth ts forward and backward neghbor. Ths cross-exposure n productvty shocks s mportant to gve frm managers an ncentve to "cross-learn,".e. to learn from prces of other frms n the network. Each frm manager s perfectly nformed about the local component e, but unnformed about all other shocks, ncludng e (1) whch also affects hs frm s future productvty. 7Several papers n the fnance lterature assume that the fundamental value s affected by more than one shock. See Goldsten and Yang (2015b) or Kondor (2012) for recent examples. Note that the shocks of neghborng frms can be postvely or negatvely correlated. 9
Ths nformaton structure captures the fact that frm managers are most lkely precsely nformed about local factors that determne future productvty (e ). However, there mght be other, non-local factors, such as future product demand for a peer frm, that are also relevant for future proftablty. Because nformaton about these non-local factors s reflected n the stock prces of other frms, frm managers have an ncentve to partally base ther nvestment decsons on these prces. There are several possble reasons why two frms mght be exposed to a common fundamental shock. The man nterpretaton n ths paper s that each frm sells ts fnal product n the local market and the local market of ts forward neghbor. Because t sells ts output prmarly n the local market, total demand s mostly determned by local factors, captured by e. Frm managers are precsely nformed about ths pool of uncertanty and can adjust frm nvestment and thus output accordngly. However, frm managers are less precsely nformed about demand condtons (captured by e (1) ) n the local market of ts forward neghbor. The manager therefore tres to update hs belef based on the local stock prce n ths locaton. Alternatvely, two frms could rely on a common suppler or, more generally, be connected to each other n a producton network. To keep the model as smple as possble and to focus on the effects of cross-learnng, I treat the underlyng economc reason for the correlaton n fundamentals as exogenous. In addton to ther prvate sgnals, frm managers use the stock prces of other frms to update ther pror about the non-local productvty shock. A key feature of the model s the assumpton that acqurng these prce sgnals s costly. In partcular, at t 0 each frm manager has to pay a cost C(n ) for observng the stock prces of the next n frms n the network.8 8I show below that t s optmal for frms to focus on the prces of the next (n a clockwse drecton) frms,.e. they do not have an ncentve to "skp" frms n the nformaton acquston decson. 10
Tradng and Frm Investment Clams to each frm s fnal payoff Y are traded by a unt contnuum of dentcal "nsders" n each locaton, ndexed by j [0, 1]. These agents are rsk-neutral and trade compettvely based on the same nformaton set as the local manager, I M {e, P 1,..., Pn }. In partcular, nsder j n locaton chooses hs asset holdngs n the local stock to maxmze a quadratc objectve functon: max z j E [ ] z j (Y P ) I M 1 2 z2 j. (2) Intutvely, each nsder maxmzes hs expected tradng proft mnus a quadratc tradng cost. Ths specfc objectve functon ensures that each trader s demand remans fnte and has been used n the exstng lterature, lke Banerjee et al. (2017) and Vves (2011). To prevent the prce from fully revealng the nsders prvate nformaton, I assume that each asset s n nosy supply L (x, P ) wth x d N ( 0, σ 2 x). To get a closed-form soluton for the equlbrum stock prces and nvestment decsons, I assume a partcular form for ths nosy supply curve: L (x, P ) (e x 1) P, smlar to that used n Goldsten et al. (2013) or Huang and Zeng (2015).9 The market clearng condton for frm, then requres that aggregate demand equals the nosy supply: 1 0 z jdj L (x, P ), whch mples that the equlbrum prce for frm s gven by: P E [ ] Y I M e x. (3) Intutvely, each frm s stock prce s equal to the expected payoff (under the nsders nformaton set) dsturbed by an ndependent nosy supply shock. Frm managers are also rsk-neutral and choose captal nvestment (K ) to maxmze the 9These two papers use the functonal form L(x, P ) 1 2Φ ( x log P ) to obtan tractablty. 11
expected frm value, V.10 Followng the exstng lterature such as Goldsten et al. (2013), I assume that the frm s termnal value s equal to output net of a quadratc nvestment cost and the nformaton acquston cost:11 V Y 1 2 K2 C(n ). (4) Due to the curvature of each manager s objectve functon, they effectvely act rsk-aversely when choosng K and therefore have an ncentve to collect the most precse sgnal (subject to the nformaton acquston cost). 2.2 Optmal Tradng and Frm Investment For a gven set of observed stock prces, determned at t 0, each frm manager chooses K to maxmze the expected frm value condtonal on the nformaton set I M. As a result, the optmal log captal nvestment decson s gven by:12 k E [ ] θ I M 1 + 2 Var ( θ I M ). (5) Intutvely, each manager ncreases frm nvestment f the condtonal expectaton about hs frm s future productvty ncreases. Thus, k depends postvely on the manager s prvate nformaton about the local shock e and hs condtonal expectaton about the non-local component. Gven that ths expectaton depends on the set of observed stock prces, t represents the feedback channel n ths setup. From equatons (3) and (5), t follows that the equlbrum log prce for frm can be wrtten as: p 2E [ θ I M ] ( + Var θ I M ) + x. (6) 10Instead of explctly modelng the managers contracts, I take ths step as gven and assume that they act benevolently. 11Followng the exstng feedback lterature, I assume that the nvestment and acquston cost are prvate, such that the asset s only a clam to Y. 12Throughout, I denote log prces and nvestment decson by lower case letters: p log P and k log K. 12
Intutvely, each frm s stock prce depends on a constant (the condtonal varance term), the nsders expectaton of the frm s future fundamental, and a random supply shock. 2.3 Dscusson Before proceedng, I dscuss the two man assumptons of the baselne model. Frst, I assume that t s costly for frm managers to acqure prce sgnals. I model ths frcton through the nformaton acquston cost C(n ) that makes t costly for frms to observe the stock prces of ther peer frms. One mght argue that prce data s easly avalable to all market partcpants, such that frm managers should be able to collect as many prce sgnals as possble to get the most precse estmate about ther frms future shock. However, ths argument neglects the fact that t requres a sgnfcant amount of sophstcaton or attenton to nterpret these sgnals correctly. Therefore, the man dea s smlar to Vves and Yang (2016) who argue that prces can only be consdered nformatve after they have been analyzed (by traders) or to the lterature on optmal attenton allocaton (or nattenton), such as Abel et al. (2013) and Kacperczyk et al. (2016). Second, I assume that local nsders are the only nformed agents who trade frm s stock. Ths assumpton mples that p only reflects nformaton about the local shock e such that the frm s backward neghbor can use ths prce to mprove hs knowledge about hs frm s future productvty. If ths prce also reflected nformaton about the non-local shock e (1), each frm manager would have an ncentve to drectly learn from hs own prce as well whch would make the soluton more complcated. Ths assumpton s, however, not crucal for the underlyng economc mechansm whch only requres that each frm manager has an ncentve to nfer some nformaton from the prces of hs peer frms as well. 13
3 Full Informaton Benchmark Equlbrum In ths secton, I solve for the cross-learnng equlbrum wth freely observable prces,.e. C( ) 0, whch serves as a benchmark for the man results. Thus, all traders and managers observe the N equlbrum stock prces and use the entre prce vector p [p 1,..., p N ] (together wth prvate nformaton) n ther equlbrum decsons. Defnton 2 A symmetrc, benchmark equlbrum conssts of a log-prce functon for each frm p(e, x, p) : R N+2 R and a log-nvestment functon for each frm k(e, p) : R N+1 R such that: (a) nsders maxmze ther expected utlty, (b) each frm manager maxmzes the expected frm value by choosng K, and (c) the stock market clears for each frm. As a frst step, I can rewrte the equlbrum condton for frm s stock prce n equaton (6) as: p π 0 + 2e + 2ρE[e (1) p] + x (7) where π 0 s a constant that subsumes the condtonal varance term. Gven that each nformed trader receves an nformatve sgnal about the local fundamental e, the equlbrum log-prce p reflects ths nformaton to the frm s backward neghbor that tres to nfer nformaton about ths component. Furthermore, p s also affected by stock prces of other frms and the nosy supply shock, x. Lemma 1 In the benchmark economy, the vector of stock prces can be combned to get an unbased sgnal z ( p) e + 1 2 x about the fundamental shock of frm N. Proof: See Appendx A.2.1. 14
Lemma 1 shows that each frm manager can combne the N stock prces to retreve the local traders prvate sgnal about e plus ( 1 2 tmes) the local supply shock. Interestngly, ths sgnal cannot be nferred from the local prce p alone because ths prce also contans non-local nformaton through the manager s cross-learnng. Consder for a moment that each manager only learned from ts forward neghbor s prce p (1). Then, the manager of frm s backward neghbor needs to observe p (1),.e. the prce of a frm two locatons away to wpe out ths source of uncertanty from p. However, f frm s manager also observes the prces of ts two forward neghbors, ts backward neghbor has to observe three prces, and so on. Contnung ths logc shows that the optmal prce sgnal z ( p) depends on all N stock prces. Therefore, f each frm s able to condton ts decson on all prces, t optmally uses all prces to get the most precse sgnal about the non-local component n ts composte productvty shock. Proposton 1 (Benchmark Equlbrum) There exsts a unque symmetrc log-lnear benchmark equlbrum n whch the log-nvestment decson and the log-prce for frm s gven by: ( ) k a 0 + e + a 1 ρ ( p 2 a 0 + e + a 1 ρ e (1) ( + 1 2 x(1) e (1) + 1 2 x(1) )) + x where N and the expressons for all coeffcents are provded n Appendx A.2.2. Proof: See Appendx A.2.2. Proposton 1 shows that n the benchmark economy, frm nvestment s affected by the local productvty shock and that of the frm s forward neghbor, gven that the frms fundamentals are correlated (ρ 0). recover the nformed trader s prvate sgnal about e (1) Because each frm manager s able to partally from the vector of prces, manageral expectatons and thus frm nvestment also depend on ths sgnal and on the local supply 15
shock x (1). Note, however, that other non-local shocks do not affect frm nvestment k snce managers are able to flter out these shocks perfectly when constructng ther prce sgnal z ( p). Smlarly, each frm s prce s affected by local fundamental and non-fundamental shocks, as well as both shocks orgnatng from ts forward neghbor s locaton. As a result, f all prces are observed, shocks from unrelated frms are not transmtted throughout the network. Thus, somewhat paradoxcally, even though each frm observes and uses all N prces (Lemma 1), only prce shocks from ts forward neghbor are reflected n ts nvestment decson (Proposton 1). Intutvely, precsely the fact that all prces are observed allows frm managers to flter out unrelated nose n prces and to recover the nformed trader s prvate sgnal about the non-local component of ther productvty shock e (1). 4 Cross-Learnng Equlbrum wth Costly Prce Acquston In ths secton, I solve for the equlbrum n the man model for a fxed choce of n n J {1,..., N 1} for all frms and endogenze ths number n Secton 5. Therefore, the manager and nsders n locaton only observe prce sgnals for the next n frms and can no longer use the optmal prce sgnal n Lemma 1. Defnton 3 A symmetrc, cross-learnng equlbrum wth costly prce acquston conssts of a log-prce functon for each frm p(e, x, p (1) for each frm k(e, p (1),..., p (n) ) : R n+1 R such that: (a) nsders maxmze ther expected utlty,,..., p (n) ) : R n+2 R, and a log-nvestment functon (b) each frm manager maxmzes the expected frm value by choosng K and n, and (c) the stock market clears for each frm. 16
As before, I can smplfy equaton (6) to get: p π 0 + 2e + 2ρE[e (1) p (1),..., p (n) ] + x (8) where the condtonal expectaton now only depends on the n observed prces and π 0 subsumes the condtonal varance term.13 As n the benchmark model, p reflects nformaton about the local productvty shock e through the local nsders perfect knowledge about ths shock. As before, ths prce also reflects local nosy supply, x. However, n sharp contrast to the benchmark model, equaton (8) cannot be easly nverted by the manager of frm s backward neghbor to back out ths unbased sgnal. Intutvely, the condtonal expectaton E[e (1) p (1),..., p (n) ] cannot be fltered out by the manager due to ts relance on p (n). Snce, ths prce s not observed by the frm s backward neghbor, t creates an addtonal source of uncertanty n the prce sgnal. Therefore, all shocks affectng ths prce are now transmtted from a frm s forward neghbor to ts backward neghbor (and further to other frms). Intutvely, there are now three sources of uncertanty n each stock prce: () local fundamental varaton (e ), () local non-fundamental varaton (x ), () non-local varaton nduced by p (n). Proposton 2 (Man Equlbrum for a fxed n) There exsts a unque symmetrc log-lnear cross-learnng equlbrum wth a fxed choce of n n J n whch the log-nvestment decson and the log-prce s gven by: k b 0 + e + b 1 ρ ( p 2 ( b 0 + e + b 1 ρ e (1) ( + 1 2 x(1) e (1) + 1 2 x(1) ) + b n ρp (n+1) + b n ρp (n+1) )) + x where N and the expressons for all coeffcents are provded n Appendx A.2.3. Proof: See Appendx A.2.3. 13As I show n Secton 5, there always exsts a symmetrc nformaton acquston equlbrum such that n n N. 17
Proposton 2 shows frm s equlbrum nvestment and stock prce n the economy wth costly prce acquston. As before n the benchmark equlbrum, both equlbrum quanttes depend on the local shocks and those of the frm s forward neghbor. However, now shocks to the stock prce of the frm n + 1 locatons away also mpact k and p. Intutvely, frm would lke to use the prce of ts forward neghbor p (1) to learn about e (1). Importantly, frm would lke to flter out as much non-fundamental varaton from ths prce sgnal as possble. As a result, frm uses all n observed prces to do ths. However, as frm + 1 does the same wth ts forward neghbor (frm + 2), frm can only mperfectly subtract non-fundamental varaton from p (1). More specfcally, frm s unable to control for movements n ths prce that are due to changes n p (n+1), whch affects frm + 1 s nvestment decson, but s unobserved by frm. Therefore, all shocks that affect ths remote stock prce are reflected n frm s nvestment decson and stock prce. As a result, the fact that frms can no longer condton on the entre set of prces leads to the propagaton of non-local shocks. Importantly, the prce p (n+1) that affects k and p s tself exposed to shocks from three dfferent locatons. () ts local shocks e (n+1) x (n+1) ; () the local shocks of ts forward neghbor e (n+2) and and x (n+2) ; and () shocks to the stock prce n + 1 locatons apart that cannot be fltered out from ts forward neghbor s prce, p (2n+2). Contnung ths logc forward t follows that shocks from multple remote locatons affect nvestment decsons and stock prces. Paradoxcally, the fact that frms do not perfectly observe all prces leads to the propagaton of shocks. Thus, shocks from dstant locatons have an mpact on a gven frm s nvestment decson precsely when ths frm does not observe ts prce. Intutvely, f t could observe ts prce, t would be able to flter out ths source of unrelated nose. Ths 18
mechansm s therefore fundamentally dfferent from alternatve models of learnng-based fnancal contagon such as Admat (1985) or Kodres and Prtsker (2002), n whch all asset prces are observed and agents learnng from unrelated prces leads to non-local shock exposure. 4.1 Shock Propagaton Based on the precedng analyss, I now analyze the propagaton of shocks through frms cross-learnng n the man model, agan takng the chosen number of observed prces as gven. Proposton 2 shows that frm s nvestment decson depends on the manager s knowledge of the local shock as well as a feedback sgnal from the stock market. The latter sgnal helps the manager to mprove hs knowledge about the non-local shock that affects hs frm s composte productvty shock, but t also transmts non-local shocks as I formally show below. Corollary 1 If all frms have ndependent productvty shocks (ρ 0), only local shocks affect frm nvestment (k ) and stock prces (p ). for all N and j J. Proof: See Appendx A.2.4. k e (j) k x (j) p e (j) p x (j) 0 Corollary 1 shows that f all frms productvty shocks are ndependent, only local shocks affect frm nvestment and prces. Intutvely, n ths case frm managers do not have an ncentve to learn from ther neghbors stock prces and only use ther prvate sgnal about e when choosng frm nvestment. Smlarly, even f ρ 0 but managers gnore the nformatonal content of prces, non-local shocks cannot affect k and p. Therefore, frms 19
cross-learnng s a necessary condton for shock propagaton. Corollary 2 If neghborng frms have correlated productvty shocks (ρ 0), frm nvestment (k ) and stock prces (p ) are affected by non-local shocks. and k e (j) k x (j) 1 2 1 2 p e (j) p x (j) ( 1) D(j) (ρb 1 ) j j J n f 0 j J \ J n f 1 2 ( 1)D(j) (ρb 1 ) j j J n f 0 j J \ J n f for all N. The expresson for b 1 > 0 s gven n the proof of Proposton 2 and the set of nfectng frms s defned as: J n f {1, n + 1, n + 2,..., κ (n + 1), κ (n + 1) + 1} where κ s the largest postve nteger such that J n f J. The ndcator varable D(j) determnes the sgn of the senstvtes and s defned n Appendx A.2.5. Proof: See Appendx A.2.5. The results n Corollary 2 show that fundamental (e (j) ) and non-fundamental (x (j) ) shocks from dstant locatons affect each frm s stock prce and captal nvestment. Hence, ths result stands n stark contrast to the benchmark equlbrum n whch only local shocks affect each frm s equlbrum varables because all non-fundamental varaton can be fltered out of the forward neghbor s prce sgnal. It can be seen that the senstvty to these non-local shocks s proportonal to (ρb 1 ) j and thus depends on three factors: () the level of entanglement between neghborng frms (ρ), () each frm s weght on the optmal prce sgnal (b 1 - see Proposton 2), and () the dstance to the "nfectng" frm ( j). Intutvely, each frm manager bases hs expectaton about future productvty on the prce of ts forward neghbor and flters out as much unrelated varaton from ths 20
2.0 1.5 dp de (j) 1.0 0.5 0.0-0.5 0 1 2 3 4 5 6 7 8 9 10 j Fgure 2: Ths fgure plots the senstvty of each frm s stock prce to fundamental shocks, e (j), when each frm observes the stock prces of the next two frms (n 2). Other parameters: N 11, ρ 1, σ e 1, σ x 1. prce as possble. However, as shown n Proposton 2, frm cannot flter out varaton n p (1) comng from the frm n + 1 locatons away because that frm s prce s too costly to ( ) acqure. Thus, all shocks that affect ths prce are transmtted to frm s prce and nvestment. Consequently, fundamental and non-fundamental shocks n locaton + n + 1 are transmtted to frm. p n+1 Snce the dstant frm s also affected by ths transmsson mechansm, shocks to (the even more dstant prce) p 2(n+1) are also transmtted to frm. The mpact of each shock orgnatng n a nfectng locaton s proportonal to the weght that each frm assgns to the feedback sgnal from the stock market, ρb 1 (see Proposton 2), rased to the power of j because j 1 other frms have already attached a Bayesan weght to ths sgnal. Fgure 2 show the senstvty of frm s stock prce wth respect to e (j) for a specfc set of parameters, when each frm can only observe the stock prces of the next two frms n the network such that n 2. The fgure confrms the results n Corollary 2: () shocks from non-related frms are transmtted to frm s stock prce (and nvestment), () the absolute mpact of these shocks s stronger for locatons that are more closely located, () the sgn 21
of the senstvtes alternates and (v) some locatons can be skpped (e.g. j {2, 5, 7} n Fgure 2). Corollary 3 The absolute senstvty of frm s stock prce and frm nvestment wth respect to non-local shocks from nfectng locatons J n f (defned n Corollary 2) s hgher f: a. σ e ncreases b. σ x decreases c. ρ ncreases. Proof: See Appendx A.2.6. Corollary 3 shows how the senstvty of nvestment and prces to non-local shocks depend on the three key parameters of the economy: (a) the volatlty of fundamentals σ e, (b) the volatlty of the supply shock σ x, and (c) the nterdependence of productvty shocks ρ. A hgher ex ante varance of the fundamentals mples that managers have a more dffuse pror about the non-local component of ther productvty shock. Therefore, they have a hgher ncentve to learn nformaton about ths shock from other frms prces such that they place a hgher weght on these sgnals. As a result, shocks to these non-local prces are reflected to a larger extent n the frm s equlbrum varables such that the propagaton of non-local shocks s stronger n more uncertan tmes featurng hgher shock volatlty. Smlarly, lower supply rsk renders the neghbor s stock prces more nformatve about the non-local component such that frm managers place a larger Bayesan weght on stock prces whch agan leads to stronger shock propagaton. Lastly, f two neghborng frms are more strongly entangled (hgh ρ ), the non-local shock plays a more mportant role for the frm s overall nvestment decson such that frm managers have a hgher ncentve to place a larger weght on other frms prces. Fgure 3 plots ρb 1, 22
ρb1 0.7 0.6 0.5 0.4 0.0 0.5 1.0 1.5 2.0 τ e ρb1 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 2.0 τ x Fgure 3: These two fgures plot ρb 1 (a measure of each frm s absolute senstvty to non-local shocks), aganst τ e (left plot) and τ x (rght plot). Other parameters: n 2, τ x 1 (left), and τ e 1 (rght). The sold lne corresponds to ρ 2 1, the dashed lne to ρ 1. a measure of each frm s absolute senstvty to non-local shocks, aganst τ e σ 2 e τ x σ 2 x. The plots confrm the analytcal results n Corollary 3 that shock propagaton ncreases (decreases) n τ x (τ e ) and that t s stronger for a hgher degree of entanglement between neghborng frms. and 4.2 The mpact of n In ths secton, I dscuss the mpact of n, the number of observed stock prces, on the key results n the man model. Whle ths secton analyzes exogenous changes n ths number, the next secton endogenzes n. Increasng the number of observed prces allows each frm to collect more prce sgnals about the non-local component n ts productvty shock. Thus, each frm manager becomes better nformed about e (1) and can nvest more effcently. Moreover, a hgher value of n also ncentvzes the managers to rely more heavly on ths prce sgnal,.e. to choose a hgher value of b 1 n Proposton 2. As shown n Corollary 2, ths ncrease n b 1 ncreases the mpact of non-local shocks orgnatng from nfectng frms. At the same tme, the proporton of nfectng frms n the economy decreases wth more observed prces, as 23
0.8 0.45 0.7 0.40 ρb1 0.6 0.5 0.4 Var(e (1) ) 0.35 0.30 0.25 2 4 6 8 10 n 0.20 2 4 6 8 10 n Fgure 4: Left plot: ρb 1 (a measure of each frm s absolute senstvty to non-local shocks), aganst n; parameters: τ e τ x 1; ρ 1 2 (sold lne) and ρ 1 (dashed lne). Rght plot: condtonal varance of e (1) aganst n; parameters: τ e τ x ρ 1. frms are able to flter out more non-local varaton n ther neghbor s stock prce. Corollary 4 Increasng the number of observed stock prces n always ncreases a. the absolute senstvty of stock prces and frm nvestment wth respect to non-local shocks, ρb 1 b. nvestment effcency, E[Y 1 2 K2 ]. Proof: See Appendx A.2.7. Corollary 4 formalzes these results and Fgure 4 confrms the results for a specfc set of parameters. It can be seen that ρb 1, whch determnes the frm s exposure to non-local shocks, ncreases n n. Moreover, ths ncrease s hgher f ρ s hgher. The left panel, plots each manager s condtonal varance of the non-local shock and shows that the frm manager becomes more nformed about the non-local shock as he observes more prces. It follows from the expresson for V n equaton (4) that ths decrease n the condtonal varance leads to a more effcent nvestment decson. 24
4.3 Emprcal Implcatons The framework presented above yelds several emprcal mplcatons. Frst, t mples that purely fnancal shocks, lke e.g. lqudty shocks or market sentment, can affect real decsons such as frm nvestment. If frm managers just rely on ther prvate nformaton these shocks do not play any role for real decsons and the stock market s just a "sdeshow." However, wth a feedback effect from the fnancal market to frm decsons, purely fnancal shocks affect managers expectatons and nvestment decsons. Ths exposure then gets mplemented nto stock prces. Importantly, f frm managers learn from ther neghbors prces, these shocks are transmtted through the entre network such that they not only affect local frms. Thus, even f a certan fnancal shock s fundamentally frm-specfc, mperfect cross-learnng mples that t affects many other frms n the network as well. Surprsngly, as shown n Corollary 2 and Fgure 2, the mpact can "skp" several frms,.e. a certan shock can affect a number of frms, spare ts drect neghbors, then affect other, more dstant frms agan, and so on. In addton to the propagaton of non-fundamental shocks, the framework also mples that fundamental shocks for a gven frm affect real decsons for a completely unrelated frm. As a result, fundamentally unrelated frms have correlated nvestment decsons and generally appear more hghly correlated than they are (based on ther fundamentals). In partcular, ths endogenous comovement s stronger n tmes of hgher uncertanty, when frms face a hgher ncentve to cross-learn. The model thus provdes a learnng-based explanaton for the emprcal fndngs, such as n Barrot and Sauvagnat (2016), that frmlevel dosyncratc shocks propagate n networks. In the cross-learnng equlbrum wth mperfect learnng, frm-specfc shocks effectvely become systematc shocks that affect multple frms n the economy. 25
Moreover, the model emphaszes that the fact that frm nvestment s affected by msprcng n a gven frm s stock prce, does not mply that the frm learns from ths prce. On the contrary, f two frms are fundamentally unrelated, msprcng n a gven stock only splls over f the frm does not observe ts prce and cannot learn from t. Intutvely, the msprcng could be fltered out f ths prce was observed.14 5 Endogenous Prce Acquston In ths secton I allow each frm to choose the number of observed prces, n. In partcular, at t 0 frm chooses n to maxmze the expected future frm value, V : [ ] max E n Y 1 2 K2 C(n ). (9) In ths expresson, C(n ) captures the prce acquston cost for each frm whch s assumed to be strctly ncreasng n n. Moreover, I rule out corner solutons by assumng: C(1) 0 and C(N). Therefore, frms wll always learn from at least one prce, but never observe the entre vector of prces. The next secton focusses on the equlbrum outcome when each frm chooses n compettvely. Subsequently, I contrast ths outcome wth the socal optmum n whch a benevolent socal planner assgns n to all frms wth the objectve to maxmze each frm s ex ante value. Both sectons wll focus on the frms beneft to acqure prce nformaton wthout assumng a partcular functonal form for C( ). Secton 5.3 numercally solves for the equlbrum values assumng specfc functonal forms for the prce acquston cost. 14Recent work by Dessant et al. (2016) also emphaszes ths pont. 26
5.1 Equlbrum Prce Acquston Frst, note that from equaton (9) and the equlbrum expressons for Y and K, t follows that the expected frm value can be wrtten as: V 1 (2(1 2 exp + ρ 2 )τe 1 ρ 2 (τ e + τ z, ) 1) C(n ) (10) ( ) where τ z, Var 1 e 1 p1,..., p(n ) denotes the precson of the prce sgnal, frm can compose usng the stock prces of the next n frms. As shown before, a more precse prce sgnal allows each manager to make a more nformed nvestment decson. Importantly, when each frm manager decdes on the optmal number of prce sgnals, he performs the followng cost-beneft analyss. On the one hand, an ncrease n n s assocated wth hgher prce acquston cost C(n ), on the other hand t also leads to a more precse prce sgnal and hgher nvestment effcency (captured by τ z, ). Lemma 2 In a symmetrc equlbrum wth n n for all N, a sngle frm j can remove all non-local nose from the composte prce sgnal by observng n j n + 1 prces. The precson of ths sgnal equals τ z,j 4τ x. Proof: See Appendx A.2.8. Lemma 2 formalzes the beneft for each ndvdual frm to collect an addtonal prce sgnal. In partcular, assume all frms observe the prces of the next n frms, but frm j unlaterally observes n j n + 1 prces. Then, ths frm s able to remove all non-local varaton n ts forward neghbor s stock prce. As a result, t s able to recover the optmal sgnal e (1) j Lemma 1). + 1 2 x(1) j from the benchmark equlbrum wthout prce acquston cost (see Of course, ths sgnal s always more precse than the sgnal wth only n observed prces such that all frms margnal beneft of ncreasng n s always postve.15 15The lemma also shows why frms always want to observe the prces of followng frms n the network. Only the prce of frm n + 1 reduces the non-local nose n ther exstng prce sgnal. 27
3.0 1.5 2.5 2.0 1.0 Δτz,j 1.5 Δτz,j 1.0 0.5 0.5 0.0 5 10 15 20 0.0 5 10 15 20 n n Fgure 5: Both plots show the ncrease n τ z for frm j f t observes n + 1 prces, when all other frms observe n prces; ρ 1 for both plots. Left plot: τ x 1 and τ e 1 (sold), τ e 1 2 (dashed). Rght plot: τ e 1 and τ x 1 (sold), τ x 1 2 (dashed). Fgure 5 plots the ncrease n τ z for frm j f t chooses to observe an addtonal prce sgnal whle all other frms observe n prces. It can be seen that the ncrease n prce nformatveness for frm j decreases n n, the number of observed prces by ts peers. Thus, the margnal beneft of collectng addtonal prce nformaton s decreasng. Furthermore, the left panel shows that t s more benefcal for an ndvdual frm to ncrease n n tmes of hgh volatlty (dashed lne). Intutvely, hgher values of σ e mply low pror knowledge about the realzaton of e (1) and thus render prce nformaton more valuable. Smlarly, the rght plot shows that the margnal ncrease n prce nformatveness s hgher when σ x s lower,.e. when the nosy supply shock s less volatle and the prce sgnal s more nformatve n general. Proposton 3 Assume C(n ) s strctly ncreasng, C(1) 0 and C(N). There exsts a unque, symmetrc nformaton acquston equlbrum n whch all frms choose to observe n J stock prces. Proof: See Appendx A.2.9. Proposton 3 shows that the nformaton acquston equlbrum s unque. Intutvely, unqueness follows from the decreasng margnal beneft of nformaton acquston (Fg- 28
0.5 0.5 0.4 0.4 Δτz,j 0.3 0.2 Δτz,j 0.3 0.2 0.1 0.1 0.0 5 10 15 20 0.0 5 10 15 20 n n Fgure 6: Both plots show the ncrease n τ z for each frm f t observes n + 1 prces, when all other frms also observe n + 1 prces; ρ 1 for both plots. Left plot: τ x 1 and τ e 1 (sold), τ e 1 2 (dashed). Rght plot: τ e 1 and τ x 1 (sold), τ x 1 2 (dashed). ure 5) and the strctly ncreasng cost. Secton 5.3 explctly computes the equlbrum choces of n gven a specfc cost functon C(n ) and dscusses the mplcatons for prce effcency, shock propagaton, and welfare. 5.2 Socally Optmal Prce Acquston In ths secton, I compare the nformaton acquston equlbrum to the socally optmal allocaton. In partcular, I consder a socal planner who assgns n to all frms wth the objectve to maxmze all frms ex ante expected value V. The man dfference to the equlbrum choce s that a socal planner nternalzes the mpact of frm s nformaton acquston decson on other frms. In partcular, he nternalzes that an ncrease n n poses a negatve externalty for the frm s backward neghbor. Intutvely, the prce of frm contans more non-local varaton f t reles on more prces tself such that frm 1 can extract less precse nformaton from ths sgnal. Fgure 6 plots the ncrease n prce nformatveness (τ z ) for each frm f all frms observe an addtonal prce sgnal. There are two man dfferences to the equlbrum change n prce nformatveness depcted n Fgure 5. Frst, t can be seen that the ncrease n prce 29
nformatveness s an order of magntude smaller f all frms smultaneously ncrease n. Second, the mpact of σ e can now be negatve,.e. for a gven n the socal beneft of acqurng addtonal prce nformaton can be lower n tmes of hgher uncertanty (and small values of n). Corollary 5 The equlbrum number of observed prces s neffcently hgh. Proof: See Appendx A.2.10. Corollary 5 formalzes ths nformatonal externalty. From each frm s ndvdual perspectve, ncreasng the number of observed frms promses a very nformatve prce sgnal that s not contamnated wth non-local nose. Ths reasonng does, however, not survve n equlbrum such that the prvate beneft of nformaton acquston does not equal the socal beneft und the equlbrum prce acquston decson s neffcently hgh. 5.3 Numercal Results In ths secton, I provde numercal results regardng the equlbrum choce of observed prces and ts mplcatons for the man equlbrum. I also compare ths soluton to the socally optmal choce of observed prces and evaluate the loss n frm values resultng from the nformatonal neffcency dscussed before. For all of the numercal results n ths secton, I use a specfc (ntentonally smple) functonal form for the prce acquston functon: 0 for n 1 C(n ) cn for n 2,..., N 1 (11) for n N. Ths prce acquston cost s therefore strctly ncreasng n the number of observed 30