Investor Sophistication and Capital Income Inequality

Transcription

1 Investor Sophstcaton and Captal Income Inequalty Marcn Kacperczyk Imperal College London & CEPR Jaromr Nosal Boston College Lumnta Stevens Unversty of Maryland September 8, 018 Abstract Captal ncome nequalty s large and growng fast, accountng for a sgnfcant porton of total ncome nequalty. We study ts growth n a general equlbrum portfolo choce model wth endogenous nformaton acquston and heterogenety across household sophstcaton and asset rskness. The model mples captal ncome nequalty that grows wth aggregate nformaton technology. Investors d erentally adjust both the sze and composton of ther portfolos, as unsophstcated nvestors retrench from tradng rsky securtes and shft ther portfolos toward safer assets. Technologcal progress also reduces aggregate returns and ncreases the volume of transactons, features that are consstent wth recent U.S. data. We thank Boragan Aruoba, Bruno Bas, Laurent Calvet, John Campbell, Bruce Carln, John Donaldson, Therry Foucault, Xaver Gabax, Mke Golosov, Gta Gopnath, Jungsuk Han, Ron Kanel, Ka L, Matteo Maggor, Gustavo Manso, Alan Morera, Stjn van Neuwerburgh, Stavros Panageas, Alex Savov, John Shea, Laura Veldkamp, and Venky Venkateswaran for useful suggestons and Joonkyu Cho for research assstance. Kacperczyk acknowledges research support by a Mare Cure FP7 Integraton Grant wthn the 7th European Unon Framework Programme and by European Research Councl Consoldator Grant. Contact: m.kacperczyk@mperal.ac.uk, jarek.nosal@gmal.com, stevens@econ.umd.edu.

2 1 Introducton The rse n ncome and wealth nequalty has been among the most hotly dscussed topcs n academc and polcy crcles. 1 Among the possble explanatons, heterogenety n the returns on savngs due to d erences n rates of return or n the composton of the rsky portfolo has been hghlghted as an mportant drver. Ths factor has emerged n emprcal work on the wealth dstrbuton, such as Fagereng, Guso, Malacrno & Pstaferr (016a; 016b) and n research focused on the very top of the wealth dstrbuton (Benhabb, Bsn & Zhu, 011). However, as noted by De Nard & Fella (017), more work s needed to understand the determnants of such heterogenety. Ths paper studes captal ncome nequalty growth n a portfolo choce model wth nformaton constrants. When nvestors d er n ther capacty to process news about rsky asset payo s, both the sze and the composton of the rsky portfolos d er across nvestors. Not surprsngly, ths generates nequalty. More nterestngly, progress n the aggregate nformaton processng technology can exacerbate ths nequalty, and ths e ect can be economcally large, as less sophstcated nvestors get prced out of hgh-return assets. Ths pecunary externalty arses even n a settng wth a sngle rsky asset, but s amplfed n an economy wth heterogeneous assets. At the core of our model s each nvestor s decson of how much to nvest n assets wth d erent rsk characterstcs. Ths decson s shaped by the nvestors capacty to pro- 1 See Pketty & Saez (003); Atknson, Pketty & Saez (011). A comprehensve dscusson s also o ered n the 013 Summer ssue of the Journal Economc Perspectves and n Pketty (014). See also the revew by Benhabb & Bsn (017). Saez & Zucman (016) emphasze the role of dfferental savngs rates, rather than d erental rates of return, n generatng wealth nequalty. However, ther captalzaton method mposes homogenety across nvestors on the rates of return wthn asset classes, thereby rulng out one mechansm over the other. 1

3 cess nformaton about asset payo s, and by ther choce of how to allocate ths capacty across assets. 3 We model the learnng choce usng the theory of ratonal nattenton of Sms (003). Whle stylzed, the framework captures several appealng aspects of learnng. Frst, gettng nformaton about one s nvestments requres expendng resources. Second, learnng about more volatle assets consumes more resources. Lastly, nvestors can allocate ther nformaton capacty optmally across d erent types of assets, dependng on ther objectve and the characterstcs of the assets they nvest n. Our theoretcal framework generalzes exstng models Van Neuwerburgh & Veldkamp (010) n partcular by consderng heterogeneously nformed agents nvestng n multple heterogeneous assets. 4 We analytcally characterze three channels of how nvestor heterogenety generates captal ncome nequalty: Investors wth hgher nformaton capacty hold larger portfolos on average, tlt ther average holdngs towards rsker assets wthn the rsky portfolo, and adjust ther nvestments more aggressvely n response to changes n payo s. These patterns are consstent wth the emprcal lterature on portfolo composton d erences between wealthy and less wealthy nvestors, gong back to Greenwood (1983), and Mankw & Zeldes (1991), and dscussed more recently by Fagereng et al. (016b) andbach, Calvet & Sodn (015). Our central result s that growth n aggregate nformaton capacty, nterpreted as a general progress n nformaton-processng technologes, dsproportonately benefts the ntally more sklled nvestors, and leads to growng captal ncome nequalty. As the aggregate 3 In the model, we endow each nvestor wth a partcular level of nformaton processng capacty. However, ths capacty should be nterpreted more broadly, as a stand-n for the ndvdual s ablty to access hgh qualty nvestment advce, not lmted to hs or her own knowledge of or ablty to nvest n fnancal markets. 4 In fnance, ratonal nattenton models have been used successfully to address underdversfcaton puzzles, prce volatlty and comovement puzzles, overconfdence, and the home bas, among other applcatons. References nclude Peng (005), Peng & Xong (006), Van Neuwerburgh & Veldkamp (009; 010), Mondra (010). See also Maćkowak & Wederholt (009; 015), Matějka (015), and Stevens (018) for applcatons n macroeconomcs. Our applcaton to nequalty s new, to our knowledge.

4 capacty to process nformaton grows, all nvestors would lke to grow ther portfolos. However, n equlbrum, prces ncrease n response to the hgher demand, and only the sophstcated nvestors expand ther portfolos. The less sophstcated nvestors are prced out and retrench to lower-rsk, lower-return assets, whch amplfes captal ncome nequalty. Ths result holds regardless of the learnng technology assumed, and the specfc functonal form for nformaton acquston only a ects the magntude of the e ect. Our mechansm s amplfed n a settng wth heterogeneous assets, because the shfts n ownershp shares occur asymmetrcally across assets. Allowng nvestors to choose how to learn about d erent assets s crtcal here: Wth endogenous nformaton choce, the sophstcated ownershp share grows most for the most volatle assets, whch are precsely the assets that generate the largest captal ncome gans. As a result, the model wth multple rsky assets generates more nequalty growth compared wth a model wth one rsky asset. To provde some gudance regardng the magntudes of the e ects dentfed n our model, we conduct a set of numercal experments n a parameterzed economy. We show that a 5% annual growth n aggregate nformaton capacty 5 generates a rse n captal ncome nequalty of 38% over 4 years. In contrast, an economy wth a sngle rsky asset generates only 0% growth. Calbratng the nformaton capacty growth s challengng because the nformaton that nvestors have when they make ther nvestment decsons s not observable. However, for a range of plausble values of recent growth n nformaton capacty, nequalty growth ranges from 4% to 60%. The correspondng number n the Survey of Consumer Fnances (SCF) for the perod s 87%. 6 General progress n nformaton technology also generates 5 Ths annual growth rate s chosen to generate an average market return of 7% n the model. We dscuss the parameterzaton n detal n Secton 4. 6 We defne captal ncome nequalty as the rato between the average captal ncome of the top 10% of nvestors by wealth and that of the bottom 90% of nvestors by wealth, condtonng on partcpaton n 3

5 lower market returns, hgher market turnover, and larger and more volatle portfolos. These predctons are broadly consstent wth the data on turnover and ownershp from CRSP and Mornngstar on stocks and mutual funds over the last 5 years. Our fndngs connect to the dea that generatng the nequalty n outcomes observed n the data requres lnkng rates of return to wealth whch s our ndcator for access to better nformaton on nvestment strateges. 7 Ths dea has a long hstory, gong back to Ayagar (1994), who dscusses the wde dspartes n portfolo compostons across the wealth dstrbuton, emphaszng the fact that rch households are much more lkely to hold rsky assets. Subsequently, Krusell & Smth (1998) suggestthatthedatarequresthatwealthy agents have hgher propenstes to save, generate hgher returns on savngs, or both. Benhabb et al. (011) andgabax, Lasry, Lons & Moll (016) arerecenttheoretcaltreatments and Favluks (013), Cao & Luo (017), and Kasa & Le (018) are related quanttatve contrbutons. We complement ths lterature along two key dmensons. Frst, we study the wthn-perod portfolo problem wth multple rsky assets, rather than the dynamc savngs decson wth a sngle rsky asset. Second, we study nequalty n a general equlbrum context wth endogenous returns, rather than wth exogenous dosyncratc nvestment returns. Both asset heterogenety and the endogenous response of asset prces and hence returns are key sources of amplfcaton for nequalty. Our work contrbutes to a broader lterature on nequalty n captal ncome, ncludng the work on bequests (Cagett & De Nard (006)), lmted stock market partcpaton (Guvenen, 007; 009), fnancal lteracy (Lusard, Mchaud & Mtchell (017)), and entrepreneural talfnancal markets. The Appendx presents all varable defntons. 7 Ths connecton s motvated by evdence that has lnked tradng strategy sophstcaton to asset prces, wealth and ncome levels, such as Calvet, Campbell & Sodn (009), Chen, Cole & Lustg (011), and Vssng-Jorgensen (004). 4

6 ent (Quadrn (1999)). Our focus on d erences n access to nformaton bulds on the nsghts of Arrow (1987). The emphass on skll rather than rsk averson d erences s supported by the portfolo-level evdence of Fagereng et al. (016a). See Pástor & Verones (016) for aone-assetmodelwthheterogenetynrskaversonandexogenousentrepreneuralskll d erences. Also related s Peress (004) whoexamnestheroleofwealthanddecreasng absolute rsk averson n nformaton acquston and nvestment n a one-asset model. Secton presents the theory. Secton 3 derves analytc predctons, whch s quantfed n Secton 4. Secton5 presents addtonal corroboratng evdence, and Secton 6 concludes. Theoretcal Framework We set up a portfolo choce model wth nvestors constraned n ther capacty to process nformaton about asset payo s. Both asset characterstcs and nvestors are heterogeneous. Setup A contnuum of nvestors of mass one, ndexed by j, solve a sequence of portfolo choce problems, to maxmze mean-varance utlty over wealth W j n each perod, gven rsk averson coe cent >0. The fnancal market conssts of one rsk-free asset, wth prce normalzed to 1 and payo r, andn>1rskyassets,ndexedby, wthprcesp, and ndependent payo s z = z + ",wth" N(0, ). 8 The rsk-free asset has unlmted supply, and each rsky asset has fxed supply, x. For each rsky asset, non-optmzng nose traders trade for reasons orthogonal to prces and payo s (e.g., lqudty, hedgng, or lfecycle reasons), such that the net supply avalable to the (optmzng) nvestors s x = x +, 8 Under certan smplfyng assumptons about the nvestors learnng technology (namely the ndependence of sgnals across assets), assumng ndependent payo s s wthout loss of generalty. See Van Neuwerburgh & Veldkamp (010) for a dscusson of how to orthogonalze correlated assets under such assumptons. 5

7 wth N(0, x ), ndependent of payo s and across assets. 9 Followng Admat (1985), we conjecture that prces are p = a + b " c,forsomecoe centsa,b,c 0. Investors know the dstrbutons of the shocks, but not the realzatons (", ). Pror to makng ther portfolo decsons, nvestors can obtan nformaton about some or all of the rsky asset payo s, n the form of sgnals. The nformatveness of these sgnals s constraned by each nvestor s capacty to process nformaton. We consder two nvestor types: mass (0, 1) of nvestors, labeled sophstcated, havehghcapactytoprocessnformaton,k 1, and mass 1,labeledunsophstcated, havelowcapacty,k,wth0<k <K 1 < 1. Hgher capacty can be nterpreted as havng more resources to gather and process news about d erent assets, and t translates nto sgnals that track the realzed payo s wth hgher precson. A bound on ths capacty lmts nvestors ablty to reduce uncertanty about payo s. Gven ths constrant, they choose how to allocate attenton across d erent assets. We use the reducton n the entropy (Shannon (1948)) of the payo s condtonal on the sgnals as a measure of how much capacty the chosen sgnals consume. Startng wth Sms (003), entropy reducton has become a frequently used measure of nformaton n a varety of contexts n economcs and fnance. Entropy has a number of appealng propertes as a measure of uncertanty. For example, for normally dstrbuted random varables, t s lnear n varance. Moreover, the entropy of a vector ndependent random varables s the sum of the entropes of the ndvdual varables. Whle stylzed, ths learnng process captures the key trade-o s nvestors face when decdng how to allocate ther lmted capacty across multple nvestment decsons, as a functon of ther objectve and of the rsks they face. 9 For smplcty, we ntroduce heterogenety only n the volatlty of payo s, although the model can easly accommodate addtonal heterogenety n supply and n mean payo s. 6

8 Indvdual optmzaton Optmzaton occurs n two stages. In the frst stage, nvestors solve ther nformaton acquston problem, and n the second stage, they choose portfolo holdngs. We frst solve the optmal portfolo choce n the second stage, for a gven sgnal choce. We then solve for the ex-ante optmal sgnal choce. Gven prces and posteror belefs, the nvestor chooses portfolo holdngs to solve U j = max E j (W j ) {q j } n =1 V j (W j ) (1)! nx nx s.t. W j = r W 0j q j p + q j z, () =1 where E j and V j denote the mean and varance condtonal on nvestor j s nformaton set, and W 0j s ntal wealth. Optmal portfolo holdngs depend on the mean bµ j and varance b j of nvestor j s posteror belefs about the payo z,andsgvenbyq j = bµ j Gven the optmal portfolo holdngs as a functon of belefs, the ex-ante optmal dstr- h Pn. The buton of sgnals maxmzes ex-ante expected utlty, E 0j [U j ]= 1 E 0j =1 =1 b j rp. (bµ j rp ) b j choce of the vector of sgnals s j =(s j1,...s jn ) about the vector of payo s z =(z 1,...,z n )s subject to the constrant I (z; s j ) apple K j,wherek j s the nvestor s capacty for processng news about the assets and I (z; s j )quantfesthereductonntheentropyofthepayo s, condtonal on the vector of sgnals (defned below). For analytcal tractablty, we assume that the sgnals s j are ndependent across assets and nvestors. Then, the total quantty of nformaton obtaned by an nvestor s the sum of the quanttes of nformaton obtaned for each asset, I (z ; s j ). We can thnk of the nformaton problem as a decomposton of each payo nto the sgnal component and a resdual component that represents the nformaton lost because of the nvestor s capacty constrant, z = s j + j. If the sgnal and the resdual are ndependent, then posteror belefs 7

9 are also normally dstrbuted random varables, wth mean bµ j = s j and varance b j = j. The nvestor chooses the precson of posteror belefs for each asset to solve 10 max {b j} n =1 nx G b =1 j G (1 s.t. 1 nx =1 rb ) + r c log x b j apple K j, (3) + (z ra ), (4) where G are the utlty gans from learnng about asset. These gans are a functon of equlbrum prces and asset characterstcs only; they are common across nvestor types, and taken as gven by each nvestor. Lemma 1. The soluton to the capacty allocaton problem (3)-(4) s a corner: Each nvestor allocates capacty to reducng posteror uncertanty for the asset wth the largest learnng gan G. If multple assets have equal gans, the nvestor randomzes among them. The lnear objectve and the convex constrant mply that each nvestor specalzes, montorng only one asset, regardless of her level of sophstcaton. For all other assets, portfolo holdngs are determned by pror belefs. If there are multple assets are ted for the hghest gan, the nvestor randomzes among them, wth probabltes that are determned n equlbrum. But she contnues to allocate all capacty to a sngle asset. Spreadng ndvdual capacty across multple assets even f they have equal gans from learnng would lower utlty. Ths result extends the specalzaton results of Van Neuwerburgh & Veldkamp (010) to the case of heterogeneous assets and nvestors. Equlbrum Gven the soluton to the ndvdual optmzaton problem, equlbrum prces are lnear combnatons of the shocks. 10 The nvestor s objectve omts terms from the expected utlty functon that do not a ect the optmzaton. See the Appendx for detaled dervatons. 8

10 Lemma. The prce of asset s gven by p = a + b " a = 1 r apple z x (1 + ), b = r (1 + ), c = c, wth r (1 + ), (5) m 1 e K m (1 ) e K 1, (6) where measures the nformaton capacty allocated to learnng about asset n equlbrum, and m 1,m [0, 1] are the fractons of sophstcated and unsophstcated nvestors who choose to learn about asset. Prces reflect payo and supply shocks, wth relatve mportance determned by amount of attenton allocated to each asset,. If there s no learnng, the prce only reflects the supply shock. As the attenton allocated to an asset ncreases, the prce co-moves more wth the payo. As K j!1,theprceapproachesthedscountedrealzedpayo,z /r. Gven prces, we can now determne the allocaton of attenton across assets. Let assets be ndexed so that > +1,andlet ( x + x )summarzethepropertesofasset. Lemma 3. Let k denote the endogenous number of assets that are learned about. The allocaton of nformaton capacty across assets, { } n =1, s unquely pnned down by the condtons G = max h{1,...,n} G h for all {1,...,k}, and G < max h{1,...,n} G h for all {k +1,...,n}, where n equlbrum the gan from learnng about each asset s G = 1+ (1+ ). The equlbrum gans from learnng are asset-specfc and depend only on the propertes of the asset,, and on the amount of attenton devoted to that asset, across all nvestors,. The model unquely pns down the number of assets that are learned about and the amount of attenton allocated to each asset. Aggregate capacty n the economy may be hgh enough that n equlbrum t s spread across multple assets. In ths case, each nvestor contnues 9

11 to allocate her entre capacty to a sngle asset, but s now nd erent n terms of whch of these assets to learn about. The nvestor randomzes, wth the probablty of learnng about each asset beng determned by the equlbrum condtons n Lemma 3. Wth heterogeneous nvestor capacty, the model does not pn down how much attenton each nvestor class contrbutes: All that matters s the total capacty allocated to each asset. In the absence of emprcal evdence to gude us on how the two groups are dstrbuted, for our analytcal and numercal results we wll consder a symmetrc dstrbuton n whch nvestors of the two types contrbute capacty n proporton to ther sze n the populaton, so that m 1 = m. Ths assumpton s motvated by our result that the gans from learnng are the same for the two nvestor types, so that t s not obvous why they would choose d erent strateges. It also mples that capacty can be wrtten as = m,wth an exogenous measure of the economy s nformaton capacty, whch we wll vary to explore how the model responds to technologcal progress n nformaton Predctons In ths secton, we present analytc results mpled by our nformaton frcton. Heterogeneous nformaton mples d erences n portfolo szes, a d erent composton of the rsky portfolo across nvestors, and d erent responsveness to payo shocks. Moreover, technologcal progress amplfes these forces, resultng n further growth n nequalty. The E ects of Heterogenety on Inequalty How do d erences n capacty translate nto d erences n portfolo holdngs and captal ncome? Let q 1 and q denote the average 11 In Secton 4, we nvestgate the senstvty of our central results to ths assumpton. 10

12 per-capta holdngs of asset for sophstcated and unsophstcated nvestors, gven by q 1 = z rp + m 1 e K 1 1 z rp, (7) and q defned analogously. Equaton (7) showsthatper-captaholdngsarethequantty that would be held under the nvestors pror belefs plus a quantty that s ncreasng n the realzed excess return. The weght on the realzed excess return s asset and nvestor specfc. It s gven by the amount of nformaton capacty allocated to ths asset by ths nvestor group. Investors hold all assets, but nvest relatvely more n the asset they learn about. Hence, the model generates under-dversfcaton of portfolos, consstent wth the emprcal evdence (e.g., Vssng-Jorgensen (004) andreferencestheren). For actvely traded assets, heterogenety n capactes generates d erences n ownershp across nvestor types at the asset level. In a symmetrc equlbrum, the average per-capta ownershp d erence, as a share of the supply of each asset, s E [q 1 q ] x = e K 1 e K m 1+ m > 0. (8) Hence, the portfolo of the sophstcated nvestor s not smply a scaled up verson of the unsophstcated portfolo. Rather, the portfolo weghts wthn the class of rsky assets also d er across the two nvestor types. Proposton 1 (Ownershp). Let k>1 be the number of assets actvely traded n equlbrum. Then, for {1,...,k}, () E [q 1 q ] /x s ncreasng n and n E [z rp ]; () q 1 q s ncreasng n realzed excess returns z rp. Sophstcated nvestors hold a larger portfolo of rsky assets on average, tlt ther portfolo 11

13 towards more volatle assets wth hgher expected excess returns, and adjust ownershp, state by state, towards assets wth hgher realzed excess returns. To see the e ects of the portfolo scale and composton d erences on captal ncome, let captal ncome be j q j (z rp ). Average captal ncome dverges wth the gap n capactes, d erentally across assets : E [ 1 ]= 1 m G e K 1 e K > 0. (9) Proposton (Captal Income). Let k>1be the number of assets actvely traded n equlbrum. Then, for {1,...,k}, () E [ 1 ] s ncreasng n asset volatlty ; () 1 0, and s ncreasng n realzed excess returns z rp. The average sophstcated nvestor realzes larger profts n states wth postve excess returns, and ncurs smaller losses n states wth negatve excess returns. The bggest d erence n profts comes from nvestment n the more volatle, hgher expected excess return assets. It s these volatle assets that drve nequalty because they generate the bggest gan from learnng, and hence the bggest advantage from havng relatvely hgh capacty. To see the e ects of an ncrease n capacty dsperson, consder an experment n whch dsperson rses but wthout changng the aggregate capacty n the economy. Proposton 3 (Capacty Dsperson). Let k>1 be the number of assets actvely traded n equlbrum. Consder an ncrease n capacty dsperson, K1 0 = K >K 1, K 0 = K <K, wth 1 and such that the total nformaton capacty remans unchanged. Then, for {1,...,k}, () Asset prces and excess returns reman unchanged. 1

14 () The d erence n ownershp shares (q 1 q ) /x ncreases. () Captal ncome gets more polarzed as 1 / ncreases state by state. Increasng dsperson n capactes whle keepng aggregate capacty unchanged mples further polarzaton n holdngs and captal ncome. As dsperson reaches ts maxmum level, unsophstcated nvestors approach zero capacty and nvest based on ther pror belefs. However, dsperson n capacty has no e ect on asset prces. Both the number of assets learned about and the mass of nvestors learnng about each asset reman unchanged. Hence, the adjustment reflects a pure transfer of ncome from the relatvely unsophstcated nvestors to the more sophstcated nvestors wthout any general equlbrum e ects. The Consequences of Growth n Capacty Our central result consders the e ects of growth n aggregate capacty, nterpreted as general progress n nformaton-processng technologes. The e ect of capacty growth on asset prces and nequalty operate through ts e ects on the gans from learnng and on the mass of nvestors learnng about d erent assets. Fgure 1 shows the evoluton of masses and gans from learnng as aggregate capacty grows. At low capacty, all nvestors learn about the most volatle asset, but as capacty grows, the gans from learnng about ths asset declne, and strategc substtutablty n learnng pushes some nvestors to learn about less volatle assets. The threshold that endogenzes sngle-asset q 1+ learnng as an optmal outcome s gven by For capacty above 1, at least two assets are learned about and for su cently hgh nformaton capacty, all assets are learned about. 1 Nevertheless, not all assets are learned about wth the same ntensty: The mass of nvestors who learn about an asset s decreasng n ts volatlty. Ths allocaton 1 thus endogenzng the assumpton employed n models wth exogenous sgnals. 13

15 of attenton a ects the holdngs across assets, and hence the nvestors portfolo returns. Proposton 4 (Symmetrc Growth). Let k apple 1 be the number of assets actvely traded n equlbrum. Consder an ncrease n aggregate capacty generated by a symmetrc growth n capactes to K1 0 =(1+ ) K 1 and K 0 =(1+ ) K, (0, 1). Let k 0 k denote the new number of actvely traded assets. For {1,...,k 0 }, () Average asset prces ncrease and average excess returns decrease, approachng the rsk free rate n the lmt. () Average ownershp share of sophstcated nvestors E [q 1 ] /x ncreases and average ownershp share of unsophstcated nvestors E [q ] /x decreases, and the gap s ncreasng n asset volatlty. () As long as the return on the rsky portfolo exceeds the rsk-free rate, average captal ncome gets more unequal, as E [ 1 ] /E [ ] ncreases, wth nequalty beng hgher for the more volatle assets. Hgher capacty to process nformaton means that nvestors have more precse news about the realzed payo s, resultng n lower gans from learnng, lower average returns, and larger and more volatle postons. However, as asset prces ncrease and returns declne, nequalty keeps ncreasng. Sophstcated nvestors ncrease ther ownershp share at the expense of the less sophstcated nvestors, who retreat. Ths pecunary externalty arses regardless of the learnng technology, snce t s due to the fact that posteror varance s lower for the sophstcated nvestors, and hence on average they want to hold a larger quantty than the unsophstcated nvestors. Moreover, the ncrease n ownershp s larger for the more volatle assets that have hgher gans from learnng and generate hgher expected 14

16 returns. Hence, asset heterogenety combned wth endogenous nformaton choce generates d erental ownershp growth that n turn amplfes the growth n nequalty. As capacty contnues to grow, the declne n returns eventually becomes a mtgatng factor n the growth of ncome nequalty. Intutvely, f market returns are close to the rsk free rate, then there s less scope n the economy for extractng nformatonal rents. Captal ncome nequalty peaks as rates of return reach the rsk free rate. It subsequently starts to declne, and eventually, t stablzes at a level mpled by the d erences n rsk-free return ncome earned on on prevously accumulated wealth. In the lmt, all nformaton s revealed and captal ncome nequalty becomes flat. Ths process s shown n Fgure. 4 Quanttatve Analyss So far, we have found that progress n nformaton technology can qualtatvely generate growng captal ncome nequalty, through changes n both portfolo sze and composton across nvestors. We now parameterze the model to provde some gudance for the magntudes mpled by ths mechansm. We use data on household captal ncome from the SCF and data on the fnancal market from CRSP. We parameterze the model based on data from the frst half of the SCF sample ( ), and then we consder an experment n whch aggregate nformaton capacty n the economy grows at a constant rate, to generate predctons for the second half of the sample ( ). 15

17 4.1 Technologcal Progress and Inequalty Growth Table 1 presents parameter values and targets for the baselne results. The parameters characterzng the fnancal market are the rsk free rate, r =.5%, whch matches the 3- month T-bll rate net of nflaton over the perod, the number of rsky assets, n, whchweset to 10 arbtrarly, and the means and volatltes of payo s and nose shocks. In the absence of detaled nformaton regardng holdngs of d erent types of securtes at the household level, we target volatlty moments from the U.S. equtes market. We set the dsperson n the volatltes of asset payo s to target a dsperson n dosyncratc return volatltes of 3.54, as measured by the the rato of the 90 th percentle to the medan of the cross-sectonal dosyncratc volatlty of stock returns. 13 We set the volatlty of shocks from nose traders to x =0.4 totargetanaveragemonthlyturnover(defnedasthetotalmonthlyvolumedvded by the number of shares outstandng), equal to 9.7%. We normalze the level of prces by normalzng the mean payo and the mean supply for each asset to z =10, x =5. 14 The nvestor-level parameters we need to pn down are the rsk averson coe cent, the nformaton capactes of the two nvestor types (K 1, K ), and the fracton of sophstcated nvestors ( ). We select those parameters to target the market return of 11.9% (correspondng to average); the fracton of assets that nvestors learn about, whch, n the absence of emprcal gudance, we set to 50%; the equty ownershp share of sophstcated nvestors of 69%; and the return spread between sophstcated and unsophstcated households of four percentage ponts. To compute the last two moments, we use data from the 13 We normalze the lowest volatlty to n = 1, and we set = n + (n )/n, whchmplesthe volatlty dstrbuton s lnear. The dsperson target generates a slope coe cent = Changng the number of assets n the parameterzaton does not have a major mpact on our results. 16

18 Survey of Consumer Fnances. Although not as comprehensve as tax records data, the SCF provdes detaled nformaton about the balance sheets of a representatve sample of U.S. households. 15 We restrct our sample to partcpants n fnancal markets, defned as households that report holdng stocks, bonds, mutual funds, recevng dvdends, or havng abrokerageaccount. Onaverage,34%ofhouseholdspartcpate. 16 We classfy as sophstcated nvestors the partcpants n the top decle of the wealth dstrbuton, and relatvely unsophstcated nvestors as the remanng 90% of partcpants. 17 Usng ths defnton, the equty ownershp share of sophstcated nvestors s 69%. 18 In order to quantfy the return heterogenety, for each household, we compute captal ncome dvded by holdngs of rsky securtes (stocks, bonds, and mutual funds), and then use these return measures to capture the heterogenety between the two groups of households. Specfcally, we compute the rato of the medan return of the unsophstcated households relatve to the medan return of the sophstcated households, whch s 69.% over the frst half of the sample. We use ths gap to obtan targets for the levels of returns of each household type, gven the market return. The weghts used n computng the aggregate return are the fracton of rsky securtes held by each type of household n the SCF (31% 15 We use the weghts provded n the publc use data sets of the SCF n order to make the results representatve of the populaton of U.S. households. These weghts account for both the oversamplng of wealthy households and for d erental patterns of nonresponse. For a dscusson of weghts and aggregate analyss of the qualty of SCF data, see Kennckell & Woodburn (1999) and Kennckell (000). See also Saez & Zucman (016) for a detaled comparson of the SCF to U.S. admnstratve tax data. In short, they fnd that the SCF s representatve of trends and levels of nequalty n the U.S., but understates nequalty nsde the top 1% of the wealth dstrbuton. 16 We also consder a broader measure of partcpaton that ncludes all households wth equty n a retrement account. Ths rases the partcpaton rates, but does not alter our man fndngs. 17 In the Appendx, we show that n the data people wth hgher ntal wealth use more sophstcated nvestment nstruments, hold larger portfolos, and nvest a lower proporton of ther assets n money-lke nstruments. Addtonal evdence that lnks wealth to nvestment sophstcaton ncludes Calvet et al. (009) and Vssng-Jorgensen (004). 18 To compute the number, we frst compute the dollar value of the rsky part of the fnancal holdngs of households (stocks, bonds, non-money market funds, and other fnancals) for each decle of the wealth dstrbuton. Then, we compute the share of these rsky assets held by the top decle. 17

19 versus 69%). That gves us the d erence between sophstcated and unsophstcated returns of four percentage ponts, whch together wth the target for market return above mples the target for sophstcated return of 13.1% and the unsophstcated return of 9.1%. 19 Table presents the model s response to aggregate capacty growth chosen to match the market return n the entre sample of 7%. It mples a 4.9% growth n capacty and addtonally generates an ncrease n tradng volume, as better nformed nvestors adjust ther holdngs more aggressvely. Quanttatvely, a capacty growth of 4.9% over 4 years generates adeclnenmarketreturnsto.6%, brngng the average return for the entre perod to 7%, as n the data, whle turnover ncreases from 9.7% n the frst half of the sample to 16.8% n the second half, versus 16.0% n the data. Ths technologcal progress leads to hgher captal ncome nequalty, whch grows by 38% over the perod. Ths fgure suggests that aggregate capacty growth s qute potent n generatng captal ncome nequalty growth. For reference, n the correspondng perod captal ncome nequalty growth n the SCF equals 87%. 0 Inequalty grows due to two man e ects: () larger relatve exposure of sophstcated nvestors to the asset market, marked by hgher ownershp shares across all assets, and () ashftofsophstcatednvestorstowardshghrsk,hghreturnassetsandthatofunsophstcated nvestors towards lower rsk and lower return assets. As capacty ncreases, less sophstcated nvestors are prced out of tradng the more rsky assets and shft ther portfolo weghts towards less rsky, lower-return assets. As a result, the ownershp share of sophstcated nvestors, relatve to ther populaton share, rses relatvely more for the assets that 19 We perform a detaled grd search over parameters untl all the smulated moments are wthn a 10% dstance from target. That gves sophstcated ownershp wthn 0.7%, sophstcated and unsophstcated returns wthn 7%, rato of volatltes wthn % and all other targets matched exactly. 0 We compute ths nequalty growth as follows. For each survey year, we sort the sample of partcpants by the level of total wealth, and we calculate nequalty as the rato of average captal ncome of the top 10% to that of the bottom 90% of partcpants. 18

20 are above the medan n terms of volatlty relatve to the assets that are below the medan n terms of volatlty. For both types of assets, sophstcated owners are over-represented relatve to ther sze n the populaton (both numbers are greater than 1), reflectng ther larger overall portfolos. But the d erence s larger for the more volatle assets: at the end of the smulaton perod, sophstcated nvestors hold 1% more of hgh-rsk assets relatve to ther populaton weght, compared to 14% more for low-rsk assets. Ths gap measures the retrenchment of unsophstcated nvestors from the most proftable assets. To solate the e ects due to portfolo composton and volatlty dsperson, we solve and parameterze our model wth just one rsky asset. In a one asset economy, the rates of return on rsky portfolos whch we use n the calbraton of the benchmark model are the same across the two types of nvestors, snce there s now only one rsky asset. The d erences n captal ncome come only from the d erences n holdngs of the rsky asset, both on average and state by state. Hence, we use ownershp and turnover to dscplne the one-asset numercal exercse. The resultng growth n captal ncome nequalty s almost half of the growth generated by the benchmark model: 0% versus 38%. Hence, the d erent exposure to assets wth d erent characterstcs, and the asymmetrc shftng of weghts across assets as capacty grows play a sgnfcant role n drvng captal ncome nequalty. 1 Our growth smulaton ncreases the relatve share of the sophstcated group n the 1 In terms of the parameterzaton, the model wth one rsky asset takes away three targets from the benchmark model: heterogenety n asset volatlty, fracton of actvely traded assets, and the return d erence between sophstcated and unsophstcated nvestors. We keep the value of the rsk averson coe cent the same as n the benchmark model and set the volatlty of the sngle asset payo equal to the medan payo volatlty of the benchmark model. That leaves three parameters: volatlty of the nose trader demand x, and the two capactes of sophstcated and unsophstcated nvestors. We choose these to match: the average market return, the average asset turnover, and the share of sophstcated ownershp. That gves (K 1,K, x) = (0.0544, , 0.37). In the dynamc smulaton, we pck the growth rate of aggregate capacty to match the declne n the market return (just as n the benchmark smulaton). That mples a 6.7% growth rate of technology. 19

21 economy s total nformaton capacty. To quantfy the relevance of ths force, we consder asmulatonnwhchwegrowcapactyd erentallysoastokeepthesharesofrelatve capacty of the nvestor types constant at the levels n the ntal perod. Ths change results n an nequalty growth of 3% versus the benchmark 38%. The relatvely lmted e ect reflects the fact that the sophstcated share n overall capacty s hgh to begn wth. Calbratng the nformaton capacty growth s challengng because the nformaton that nvestors have when they make ther nvestment decsons s not observable. Hence, our strategy s to set capacty growth so as to match the declne n market returns seen n the data, and to complement these results wth robustness checks on ths growth rate. We consder two alternatve annual growth rates: 4% and 8%, based on the annual growth rate of the number of stocks actvely analyzed by the fnancal ndustry, and the growth rate of the number of analysts per stock n the fnancal ndustry, respectvely. 3 These rates mply 4% and 60% nequalty growth. Although the results are senstve to the growth rate of nformaton capacty, the model generates a quanttatvely sgnfcant rse n captal ncome nequalty relatve to the data. 4. Robustness Two features of our specfcaton have mportant mplcatons for our results: the nformaton acquston technology and the equlbrum selecton mechansm. We now dscuss how changng our assumptons along these two dmensons a ects nequalty. In the Onlne Appendx, we also provde an exercse n whch the capacty grows n proporton to the rates of return of the portfolo, capturng explctly the dea that capacty s lnked to wealth. That exacerbates the growth n nequalty. Keepng the average capacty growth the same as n the benchmark economy, lnkng capacty growth to returns mples a 49% ncrease n captal ncome nequalty. 3 Our nformaton frcton mples that growth n nformaton capacty translates nto growth n actvely analyzed stocks, and also more nformaton capacty allocated per stock, consstent wth these growth trends 0

22 Margnal Cost Predctons In our benchmark model, we endow each nvestor wth some level of capacty to process nformaton. What happens to nvestor choces and nequalty f we model a margnal cost of acqurng nformaton nstead? Let nvestors d er n ther margnal cost of nformaton, 0 <apple 1 <apple. Then the nvestor s objectve becomes P h max n {vj } n =1 =1 G apple j log bj,andthenformatonproblemsndependentacrossas- bj sets as nvestors decde how much nformaton to purchase for each asset separately. Hence, nstead of a corner soluton for learnng, each nvestor purchases nformaton about all assets whose gans exceed ther margnal cost, up to the pont at whch the gan from learnng reaches the margnal cost. In equlbrum, the gans from learnng declne endogenously the more nformaton nvestors purchase and the sophstcated, low margnal cost nvestors are the margnal buyers of nformaton, drvng the gans from learnng down to ther margnal cost for all assets. The unsophstcated nvestors, who have a hgher margnal cost, are now prced out of the nformaton market altogether. As n the benchmark case, there s apreferenceforvolatlty,wththequanttyofnformatonpurchaseddeclnngwthasset volatlty. The d erence s that now for each asset, ether the gans from learnng are too small relatve to the costs that nether nvestor learns about t, or only the sophstcated nvestors learn about t. For a gven amount of nformaton n the economy, the margnal cost specfcaton results n larger nequalty n both holdngs and captal ncome relatve to the endowed capacty case, n whch both types of nvestors learn. Moreover, technologcal progress n nformaton processng has no drect e ect on the unsophstcated nvestors: As long as ther margnal cost remans above that of the sophstcated nvestor, they purchase no nformaton and nvest n all assets accordng to ther pror belefs. 1

23 Asymmetrc Equlbrum Predctons In our benchmark model, we pn down the total amount of capacty devoted to each asset, but not the contrbuton of each nvestor group to that total. When dervng our analytc and numercal results, we mpose a symmetrc equlbrum, assumng that the fracton of nvestors that learn about each asset s the same for both nvestor types. We base ths assumpton on our result that the gans from learnng about d erent assets are the same for both sophstcated and unsophstcated nvestors. However, the same equlbrum allocaton of attenton (and hence asset prces) could be acheved wth a d erent dstrbuton of nvestors across assets. How senstve are our results to devatons from the symmetrc equlbrum? Frst, t s useful to note that all our results hold at the ndvdual level: If we compare two nvestors who both montor the same asset, one sophstcated and one unsophstcated, they wll d er n ther holdngs, captal ncome, and response to capacty growth as expected. But when we compare the average holdngs of the two groups, asset-level predctons depend on how many nvestors learn about the asset n each group. It s possble to conceve of an allocaton of nvestors across assets such that for some assets, the per capta ownershp of unsophstcated nvestors s larger than that of the sophstcated nvestors. But t remans the case that on average across all assets the per capta ownershp and hence captal ncome of the unsophstcated nvestors s strctly lower than that of the sophstcated nvestors. Moreover, growth n aggregate capacty contnues to ncrease captal ncome nequalty (as long as returns reman above the rsk free rate), even f we consder a reshu ng of masses most advantageous to the unsophstcated nvestors, namely one that assgns all the unsophstcated nvestors learnng about an asset to the hghest volatlty asset. Such a reshu ng yelds postve, albet lower, nequalty growth. Numercally, we fnd that n our parameterzed economy such a reshu ng has mnmal

24 e ects on nequalty growth (reducng t by less than one percentage pont), because the data favor a parameterzaton n whch the unsophstcated nvestors contrbute mnmally to the allocaton of attenton for each asset, so that how we reshu e them across assets has very lmted e ects on the dsperson of ownershp and captal ncome. 5 Emprcal Evdence We now provde auxlary evdence supportng our mechansm and ts mplcatons. Skll versus Rsk How much of the growth n nequalty comes from d erences n exposure to rsk versus d erences n skll? Our model s one n whch both rsk-takng d erences and pure compensaton for skll generate return heterogenety. Sophstcated nvestors are more exposed to rsk because they choose to hold a larger share of rsky assets (compensaton for rsk); and because they have an nformatonal advantage (compensaton for skll). Quanttatvely, n our model the compensaton for skll accounts for approxmately 75% of the return d erental between the two nvestor groups, wth the remanng 5% reflectng more rsk takng. 4 Emprcally, Fagereng et al. (016a) document that rsk takng s only partally responsble for the d erence n returns among Norwegan households, wth approxmately half of the return d erence beng attrbuted to unobservable heterogenety. Corroboratng ths fndng, we consder more aggregated data from the U.S. fnancal market. We compare returns from d erent types of mutual funds, usng data from Mornngstar, whch contans nformaton for two types of funds: those wth a mnmum nvestment of $100,000 (nsttutonal funds) 4 The Appendx presents the detals of the calculaton. 3

25 and those wthout such restrctons (retal funds). These two types of funds are suggestve of the knd of nvestment returns sophstcated versus unsophstcated nvestors can access. Snce the nsttutonal funds have a mnmum nvestment threshold, less sophstcated, less wealthy nvestors do not have access to the hgher returns earned by nsttutonal funds, even for plan vanlla assets lke equtes. 5 Our fund data span the perod 1989 through 01. We compare the returns of the two groups adjustng for d erences n exposure to common rsk factors, a methodology that s standard n asset prcng lterature. Our choce of common rsk factors follows Carhart (1997) andncludesmarketexcessreturns,return on the sze factor, return on the value factor, and return on the momentum factor. To compute quanttatve d erences between the two nvestor groups we calculate a hedge portfolo, defned as a d erence between monthly returns on the sophstcated portfolo and monthly returns on the unsophstcated portfolo. We then estmate the tme-seres regresson of the hedge returns on the four factors. Our coe cent of nterest s an ntercept, whch measures abnormal returns over and above prema for rsk. The hedge portfolo generates a statstcally sgnfcant postve return of 33 bass ponts per month, whch s almost 4% on an annual bass. Hence, we conclude that d erences n rsk exposures alone are unlkely to explan the d erences n returns between sophstcated and unsophstcated nvestors. Nevertheless, by shuttng down the rsk averson channel, we are lkely mnmzng the e ect that rsk has on nequalty outcomes. The overall growth n nequalty can be ncreased by assumng ether decreasng absolute rsk averson or d erences n rsk atttudes that, lke nformaton capacty, are correlated wth wealth. The less rsk averse nvestors would hold 5 In the Appendx, we present addtonal evdence that the there are both nsttutonal and nformatonal barrers that prevent unsophstcated households from ganng access and delegatng ther nvestment decsons to hgh qualty nvestment servces. 4

26 agreatershareofrskyassets,andhencetheywouldhavehgherexpectedcaptalncome. 6 In a CRRA framework, the model soluton under no capacty d erences predcts the same portfolo shares for rsky assets, ndependent of wealth. Intutvely, f agents have common nformaton, then wealth d erences a ect the composton of ther allocatons between the rsk-free asset and the rsky portfolo, but not the composton of the rsky portfolo, whch s determned optmally by the (common) belef structure. As a result, d erences n capacty are a necessary component for the model to generate any rsky return d erences across agents. Smlarly, wthn our mean-varance specfcaton, a growng d erence n rsk averson produces growng aggregate ownershp n rsky assets of less rsk averse nvestors, and a unform, proportonal retrenchment from all rsky assets of more rsk averse nvestors. However, heterogenety n rsk averson alone cannot generate the emprcal nvestor-specfc rates of return on equty, d erences n portfolo weghts wthn a class of rsky assets or dfferental growth n ownershp by asset volatlty. Hence, the nformaton asymmetry remans central to matchng several recent trends n U.S. fnancal markets. 7 The Extensve Margn of Lmted Partcpaton Lmted partcpaton n U.S. fnancal markets has long been a source of nequalty n total ncome and wealth (e.g., Mankw &Zeldes(1991)). How mportant s the lmted partcpaton margn for generatng captal ncome nequalty? Usng data from the SCF, we fnd that much of the recent growth n fnancal wealth nequalty has occurred among household who partcpate n fnancal markets, and that trends n captal ncome growth mrror trends n total fnancal wealth 6 Such settng would also encompass stuatons n whch nvestors are exposed to d erent levels of volatlty n areas outsde captal markets, lke labor ncome. 7 Addtonally, Gomez (016) shows that when macro asset prcng models wth heterogenous rsk averson are parameterzed to match the volatlty of asset prces, they requre a degree of heterogenety n preferences that leads to counterfactual predctons about wealth nequalty. 5

27 nequalty. Our evdence on captal ncome nequalty renforces exstng results usng more detaled U.S. and European data, e.g. Saez & Zucman (016), Fagereng et al. (016b) and Bach et al. (015). Frst, partcpaton s hump-shaped over tme. Moreover, nequalty n total fnancal wealth has grown wthn the group of households who partcpate n fnancal markets, but t has remaned essentally unchanged along the extensve margn (defned as the rato of average fnancal wealth of the bottom 10% of partcpatng households to that of the nonpartcpatng households). Thus the dynamcs of fnancal wealth nequalty do not appear to be drven by the partcpaton margn. These trends are shown n Fgure 3 and Fgure 4. 8 Second, among partcpants, the ncrease n nequalty n fnancal wealth tracks the accumulaton of captal ncome from the rsky assets (namely, ncome from dvdends, nterest ncome, and realzed captal gans). To see ths, we consder the counterfactual fnancal wealth obtaned from accrung captal ncome only. 9 Fgure 5 suggests that past captal ncome realzatons may be su cent to explan the evoluton of fnancal wealth nequalty, wthout resortng to mechansms that nvolve savngs rates from other ncome sources. 30 Thrd, among partcpatng households, captal ncome nequalty s large and growng fast. Panel (a) of Fgure 7 shows that n the cross-secton, captal ncome s an order of magntude more unequal than ether labor or total ncome. For example, n 1989, the average captal ncome of the top 10% of partcpants was 1 tmes larger than that of the 8 Fnancal wealth n the SCF contans holdngs of rsky assets (stocks, bonds, mutual funds), passve assets (lfe nsurance, retrement accounts, royaltes, annutes, trusts), and lqud assets (cash, checkng and savngs accounts, money market accounts). 9 For example, the counterfactual fnancal wealth level n 1995 s equal to the actual fnancal wealth n 1989 plus 3 tmes the captal ncome reported n the pror survey years (n ths case, 1989 and 199). 30 By constructon, the two wealth levels are dentcal n 1989, so the fgure also mples that the counterfactual levels of fnancal wealth for each group are very close to those n the data. Stll, we treat ths evdence as suggestve, snce our exercse mposes a panel nterpretaton on a repeated cross-secton. 6

28 bottom 90% of partcpants. Ths rato ncreased to nearly 40 n 013. By comparson, the correspondng rato for wage ncome was.4 n 1989 and 3.9 n 013. To compare the dynamcs of nequalty across ncome sources, we normalze the nequalty of each ncome measure to 1 n 1989, and plot growth rates for captal, labor, and total ncome nequalty n panel (b) of the fgure. As s well known, labor ncome nequalty has grown sgnfcantly durng ths perod, and so has captal ncome nequalty, whch nearly doubled. We complement ths evdence wth addtonal data on flows nto and out of mutual funds from Mornngstar by sophstcated (nsttutonal) and unsophstcated (retal) nvestors. As shown n Fgure 6, the cumulatve flows from sophstcated nvestors nto equty and nonequty funds ncrease steadly over the entre sample perod. In contrast, snce 000, unsophstcated nvestors have been shftng ther funds out of equty mutual funds and nto less rsky non-equty funds. To the extent that drect equty holdngs are more rsky than dversfed equty portfolos, such as mutual funds, ths mples that unsophstcated nvestors have been systematcally reallocatng ther wealth from rsker to safer asset classes. Ths trend s consstent wth our model, whch predcts that as aggregate capacty grows, sophstcated nvestors expand ther ownershp of rsky assets by order of volatlty: startng from the hghest volatlty assets and then movng down. 6 Concludng Remarks What contrbutes to the growng captal ncome nequalty across households? We propose a theoretcal nformaton-based framework that lnks captal ncome to nvestor sophstcaton. Our model mples ncome nequalty that rses wth the total nformaton n the 7

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32 φ(1) φ(4) φ(7) φ(1) φ(4) φ(7) 0.6 φ(10) 50 φ(10) (a) Masses (b) Gans Fgure 1: The evoluton of masses and gans from learnng as aggregate capacty s ncreased. (k) ndcatesthelevelofaggregatecapactyforwhchk assets are learned about n equlbrum. Gans are hgher for hgher volatlty assets. As capacty ncreases, gans fall. Gans are equated for all assets that are learned about n equlbrum. On the x-axs, assets are ordered from most (1) to least (10) volatle Years Fgure : Model: captal ncome nequalty n the long run. 31