In a calibrated representative agent model, Lucas (1987) shows a very insignificant welfare

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1 The Welfare Cos of Business Cycles wih Heerogeneous Trading Technologies YiLi Chien The auhor invesigaes he welfare cos of business cycles in an economy where households have heerogeneous rading echnologies. In an economy wih aggregae risk, he differen porfolio choices induced by heerogeneous rading echnologies lead o a larger consumpion inequaliy in equilibrium, while his source of inequaliy vanishes in an economy wihou business cycles. Pu simply, he heerogeneiy in rading echnologies amplifies he effec of aggregae oupu flucuaion on consumpion inequaliy. The welfare cos of business cycles is, herefore, larger in such an economy. In he benchmark economy wih a reasonably low risk aversion rae, he business cycle cos is 6.49 percen perperiod consumpion for an average household when he model is calibraed o mach he risk premium. (JEL C68, D61, D14, E32, G11, G12) Federal Reserve Bank of S. Louis Review, Firs Quarer 2015, 97(1), pp In a calibraed represenaive agen model, Lucas (1987) shows a very insignifican welfare gain from he eliminaion of business cycles. His work suggess ha he benefis of sabilizing he cyclical flucuaions in an economy are very limied. Hence, sudying he business cycle migh no be he op prioriy in macroeconomics. More recenly, Lucas (2003) has argued ha mos macroeconomics models sill fail o generae a sizable welfare cos associaed wih business cycles. In his aricle, I invesigae he welfare cos of business cycles in an economy where households have heerogeneous rading echnologies. In conras o mos research in he incomplee marke lieraure, he menu of asses available in his economy is quie rich. Moreover, households in his model have heerogeneous abiliies o access he menu of asses available on he marke. My aricle disinguishes beween passive raders, who hold fixed porfolios of socks and bonds, and acive raders, who frequenly adjus heir porfolios in response o changes in invesmen opporuniies. The welfare cos of business cycles is defined as he average welfare difference beween wo economies: one wih and one wihou aggregae oupu flucuaions. Given he heerogeneous agen economy, he average welfare is compued by aking he expecaion no only YiLi Chien is a senior economis a he Federal Reserve Bank of S. Louis. 2015, The Federal Reserve Bank of S. Louis. The views expressed in his aricle are hose of he auhor(s) and do no necessarily reflec he views of he Federal Reserve Sysem, he Board of Governors, or he regional Federal Reserve Banks. Aricles may be reprined, reproduced, published, disribued, displayed, and ransmied in heir enirey if copyrigh noice, auhor name(s), and full ciaion are included. Absracs, synopses, and oher derivaive works may be made only wih prior wrien permission of he Federal Reserve Bank of S. Louis. Federal Reserve Bank of S. Louis REVIEW Firs Quarer

2 over ime bu also across all idiosyncraic feaures of he populaion. In oher words, his welfare measure acs as if people were asked ex ane which economy hey would like o be born in. Hence, he measure of business cycles can be hough of as he amoun of consumpion compensaion newborns should receive such ha hey are indifferen in heir expeced uiliies beween he wo economies. In he equilibrium of he calibraed economy, heerogeneous rading echnologies resul in a clear difference beween acive raders and passive raders wih respec o heir porfolio choices. In response o he high risk premium, households wih more sophisicaed rading echnologies ake greaer aggregae risks by holding a large fracion of equiies in heir porfolios. They also opimally adjus heir porfolios in response o changes in invesmen opporuniies. On he oher hand, households wih less sophisicaed rading echnologies ake a more cauious approach. On average, hey bear less aggregae risk by holding a smaller fracion of equiies in heir porfolios and do no acively respond o changes in marke condiions. The acive raders ulimaely earn a high rae of reurn on heir porfolios, accumulaing more wealh and enjoying a high level of consumpion, while he passive invesors earn a low reurn on heir porfolios, hereby acquiring relaively low levels of wealh and consuming less. Hence, heerogeneous rading echnologies induce more consumpion inequaliy in his economy. Mos imporanly, he higher consumpion inequaliy across he populaion leads o lower welfare under my welfare measure. Clearly, he source of consumpion inequaliy depends heavily on he risk premium level and he variaion in he marke price of risk; boh are linked ighly o business cycle flucuaions. A reducion of aggregae oupu volailiy helps reduce no only he size bu also he ime variaion of he risk premium. This reducion downplays he role of porfolio choice and, hence, reduces consumpion inequaliy and improves welfare in he economy. Wihou aggregae risk, he inequaliy in consumpion caused by he heerogeneous rading echnologies disappears since he composiion and iming of porfolio choice no longer affec he rae of reurn. In shor, all asses are risk free and offer exacly he same rae of reurn. A more sophisicaed rading echnology does no offer any advanage in an environmen wihou aggregae risk. I conjecure ha heerogeneous rading echnologies may conribue o he welfare cos of business cycles. In an economy wih aggregae risk, he differen porfolio choices across households lead o a larger consumpion inequaliy in equilibrium, while his source of inequaliy vanishes in an economy wihou business cycles. In shor, he consumpion inequaliy caused by aggregae oupu flucuaion is amplified by he heerogeneous rading echnologies. The welfare cos of business cycles is, herefore, larger in such an economy. This aricle uses he modified macroeconomics model developed by Chien, Cole, and Lusig (2011) o evaluae he conjecure quaniaively. Their model incorporaes heerogeneous rading echnologies ino an oherwise sandard macroeconomics model. In my use of he model, he heerogeneiy in rading echnologies is calibraed o mach he high risk premiums seen in he hisorical U.S. daa. The welfare cos is measured by he average percenage of per-period consumpion compensaion received by a newly born household in an economy wihou business cycles, such ha his household is indifferen o an environmen wih aggregae flucuaions. I find ha he welfare gain from he eliminaion of business cycles is large, 68 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

3 wih a reasonably low risk aversion coefficien. In my benchmark case where he risk aversion coefficien is 4, he business cycle coss each household in my economy 6.49 percen of perperiod consumpion. The welfare cos is significanly larger han ha calculaed by Lucas (1987). I also compue he case in which all households are acive raders and are endowed wih he same sophisicaed rading echnologies. Given he parameer values in my benchmark calibraion, he resuls show a low risk premium and a much smaller cos of business cycles. The imporance of his compuaional exercise is wofold. Firs, i shows how an inferior invesmen echnology among some of he invesors influences he paerns of reurn in asse markes. If all households make no invesmen misakes, he asse pricing resul is dampened compared wih ha in my benchmark economy. Second, i demonsraes a large welfare loss resuling from poor invesmen sraegies. This exercise shows ha he welfare cos of business cycles is much smaller if no household makes invesmen errors. The heerogeneous rading echnologies conribue significanly and mosly o he welfare cos number in my benchmark economy. The assumpion of heerogeneous rading echnologies is criical o my resuls. The quesion hus arises: How realisic is he assumpion of heerogeneous rading echnologies? The answer can be found in empirical sudies and daa ha have shown a high amoun of heerogeneiy in household porfolio choices. Differen households behave as if hey had access o differen menus of radable asses. In he Unied Saes, a majoriy of households do no inves direcly in equiy despie he sizable hisorical equiy premiums. Even for hose who paricipae in he equiy marke, mos do no frequenly adjus he composiion of heir porfolios, regardless of he large counercyclical variaion of Sharpe raios (SRs) in he equiy marke. Pu simply, hey miss he marke iming. However, a small fracion of households hold a large share of sock and consanly change heir equiy posiion in response o he high variable-risk premiums. Therefore, hese households end up richer bu have more exposure o aggregae risk. Parker and Vissing-Jorgensen (2009) show ha he consumpion of he riches 10 percen of U.S. households is five imes more exposed o aggregae risk han ha of average households. This aricle is closely relaed o he body of lieraure in which he disribuion effecs on consumpion inequaliy migh jusify a large welfare cos of business cycles. Krusell and Smih (1999) propose he idea ha business cycles migh worsen he consumpion inequaliy across he populaion while he impac on average households is insignifican. The higher cos of business cycles is due o he disribuional impac of consumpion among he rich and poor. Evidenly, he disribuional impac is missing in a represenaive agen economy. Krusell e al. (2009) use an incomplee marke model calibraed o he wealh disribuion in he Unied Saes o evaluae he welfare cos of business cycles. Using he same parameer for risk aversion as in Lucas (1987), hey find he welfare cos is approximaely 0.1 percen of household consumpion. Alhough he welfare cos is one magniude larger han ha calculaed by Lucas, i is sill negligible in an economic sense. Soresleen, Telmer, and Yaron (2001) consider he welfare cos of business cycles in an environmen wih counercyclical variaions in idiosyncraic shock. A more volaile idiosyncraic income risk during recessions can amplify he cos of aggregae risk in individual consumpion and leads o a higher disribuional impac. Alhough he welfare cos of business cycles is sill insignifican, he cos increases rapidly as Federal Reserve Bank of S. Louis REVIEW Firs Quarer

4 he risk aversion coefficien increases. Krebs (2007) exends he concep of idiosyncraic labor income shock by adding a permanen job displacemen risk. The risk of job displacemen is assumed o be closely associaed wih he business cycle. He finds a sizable cos of business cycles relaed o he imporance of displacemen risks. The cenral idea of his corpus of lieraure is o ranslae a low scale of aggregae risk ino a large consumpion inequaliy. I follow his concep, bu he large consumpion inequaliy in my model is caused by a novel feaure: heerogeneiy in rading echnologies. Mos aricles in his lieraure operae under incomplee marke models in which all households can rade only a very limied menu of radable asses. However, he acual menu of asses ha households can rade is quie rich. Insead of assuming a limied se of radable asses, I inroduce a heerogeneous abiliy o access he menu of asses, moivaed by he empirical evidence of heerogeneiy in porfolio choices. Wih heerogeneous rading echnologies, households oal incomes differ no only because of heir idiosyncraic risk in labor income bu also because of he variaions in heir invesmen reurns resuling from he heerogeneiy in rading echnologies. In addiion, heerogeneous rading echnologies affec he reurn of porfolio choices only in an economy wih aggregae risk. Wihou business cycles, he cos of consumpion inequaliy from differen rading echnologies disappears. Therefore, he heerogeneiy in rading echnologies only enlarges he consumpion inequaliy in an economy wih aggregae risk and, hence, amplifies he cos of business cycles. Alvarez and Jermann (2004) demonsrae a close link beween he cos of business cycles and risk premiums. Their work offers an alernaive and inuiive way o measure he cos of business cycles by using asse pricing daa. The cos of business cycles can be considered as he valuaion difference beween wo consumpion claims: One pays a consan sream of consumpion and he oher pays a sochasic sream of consumpion. In a represenaive agen framework, he former claim represens a consumpion sream in an economy wihou aggregae shocks and he laer represens a consumpion sream wih aggregae shocks. Hence, heir work illusraes ha, under a represenaive agen economy, he welfare cos of business cycles can be approximaed by he risk premium beween an aggregae sochasic consumpion claim and a risk-free asse, regardless of he assumpions abou uiliy funcion. Based on heir observaion, one can infer ha if a model generaes a high risk premium, hen i migh also imply a larger welfare cos of business cycles. My calibraed model produces a realisic resul for asse pricing; however, my model s mechanism differs from ha in Alvarez and Jermann (2004). The large welfare cos resuls mainly from he consumpion inequaliy induced by heerogeneous rading echnologies, no direcly from he variaions in aggregae consumpion over ime. This aricle also relaes o he fas-growing body of lieraure on household finance. Campbell (2006) poins ou ha some households migh make various misakes when facing complicaed financial decisions. This aricle evaluaes he welfare cos of some of hese misakes. In he model economy, passive raders make wo ypes of misakes. Households ha do no paricipae in he equiy marke forgo he large equiy premium. Also, hose equiy invesors who do no frequenly change heir porfolio choices miss he marke iming. By comparing he resuls of wo model economies, one wih heerogeneous rading echnologies 70 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

5 and he oher wihou heerogeneous rading echnologies, I demonsrae ha hese invesmen misakes no only affec he risk premium paerns bu also cause a large welfare cos. If all households consis of acive raders who do no make any invesmen misakes, hen he risk premium is low and sable in he calibraed economy. Moreover, he welfare cos of business cycles is almos negligible and similar o he resul found by Lucas (1987). This finding emphasizes he imporance of he sudy of household finance because prevening invesmen misakes can considerably improve welfare. The nex secion describes he environmen and rading echnologies, followed by a secion discussing he calibraion of he model. Then he resuls and sensiiviy analyses are presened. The final secion offers he conclusion. MODEL The model seup closely follows ha in Chien, Cole, and Lusig (2011). The novel feaure of heir model is he imposiion of resricions on he menu of asses ha households are able o rade, which defines he rading echnology a household owns. These resricions are imposed exogenously o capure he observed porfolio behavior of mos households. I refer o households as passive raders if hey ake heir porfolio composiion as given and simply choose how much o save or dissave in each period. Oher households consanly manage heir porfolios in response o changes in he invesmen opporuniy se. I refer o hese households as acive raders since hey opimally adjus he composiion of heir porfolios every period. Noe ha he passive raders are compleely raional excep in heir porfolio choice decisions. They fully acknowledge he rae of reurn on heir porfolios and adjus heir consumpion and saving decisions accordingly. Hence, he resuls are clearly driven by he only addiional novel assumpion heerogeneous rading echnologies in conras o mos research in he incomplee marke lieraure. Environmen This endowmen economy consiss of a coninuum of heerogeneous households subjec o boh idiosyncraic income shocks and aggregae endowmen shocks. The oal measure of households is normalized o 1. The heerogeneiy across households arises from wo assumpions. In he planning period = 0, households receive a one-ime permanen shock o heir rading echnologies, while all oher characerisics of he households are idenical. Saring a period 1, hese households also differ in erms of he realizaion of an idiosyncraic income shock a all subsequen periods. All households sar wih he same amoun of iniial wealh and face an idenical sochasic process of idiosyncraic income shocks. In he model, ime is discree, infinie, and indexed by = 0,1,2,. The firs period, = 0, is he planning period in which financial conracing akes place. I use z Z o denoe he aggregae shock in period and h N o denoe he idiosyncraic shock in period. The variable z denoes he hisory of aggregae shocks, and similarly h denoes he hisory of idiosyncraic shocks for a household. The idiosyncraic evens h are i.i.d. across households wih he Federal Reserve Bank of S. Louis REVIEW Firs Quarer

6 mean normalized o 1. I use p (z,h ) o denoe he uncondiional probabiliy of sae (z,h ) being realized. The evens are firs-order Markov, and I assume ha ( ) ( ) ( ) z π, η z, η = π z z π η η Noe ha he probabiliy of idiosyncraic evens does no depend on he realizaion of aggregae shocks. As I show laer, his aricle does no consider he counercyclical variaion of idiosyncraic risk. I inroduce some addiional noaion: z +1 z or h +1 h denoes ha he lef node is a successor node o he righ node. I use {z z } o denoe he se of successor aggregae hisories from z onward. There is a single nondurable good available for consumpion in each period, and is aggregae supply is denoed by Y (z ), which evolves according o 1 Y ( z ) = exp { z } Y( z ), wih Y(z 0 ) = 1. This endowmen good comes in wo forms. The firs par is nondiversifiable income subjec o idiosyncraic risk and is denoed by gy(z )h ; hence g is he share of income ha is nondiversifiable. Nondiversifiable income canno be raded in financial markes and may be considered labor income. The second par of he endowmen good is diversifiable income, which is no subjec o idiosyncraic shocks, and is denoed by (1 g)y (z ). All households are infiniely lived, and rank sochasic consumpion sreams according o he following uiliy funcion: (1) U( {}= c ) βπ z, η 1, ( z, η ) ( ) ( ) c z, η 1 α 1 α, where a > 0 denoes he coefficien of relaive risk aversion and c (z,h ) denoes he household s consumpion in sae (z,h ). Asses Traded Three asses are available in his economy: equiy, bond, and coningen claims on aggregae shocks. All of hese asses are claims on diversifiable income. The acual menu of asses ha a household can rade depends on is rading echnology. However, his is sill an incomplee marke economy because here is no sae-coningen claim on idiosyncraic shocks. Following Abel (1999), I simply consider equiy as a leveraged claim on aggregae diversifiable income ((1 g)y (z )). The leverage raio is assumed o be consan over ime and denoed by y. Le B (z ) denoe he supply of a one-period risk-free bond in period and R f, 1 (z 1 ) denoe he risk-free rae beween periods 1 and given he aggregae hisory z 1. Wih a consan leverage raio, he oal supply of B (z ) has o be adjused such ha B( z ) = ψ W ( z ) B( z ), 72 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

7 where W (z ) is he price of a claim on aggregae diversifiable income. Because he aggregae diversified income can be decomposed ino he ineres paymen o bond holders and he dividend paymen o shareholders, he dividend paymen, D (z ), is denoed by Traders who inves a fracion y/(1 + y) of heir wealh in bonds and he res in equiy hold he marke porfolio. I denoe he price of equiy (a claim on dividend paymen D (z )) by V (z ). The hird available asse is he aggregae sae-coningen claims. I denoe he price of a uni claim on he final good in aggregae sae z +1 acquired in aggregae sae z by Q (z +1,z ). I consider a household enering he period wih a ne financial wealh â (z,h ). This household buys securiies in financial markes (sae-coningen claims a (z +1,h +1 ), risk-free bonds b (z,h ), and equiy shares s D (z,h )) and wih consumpion c (z,h ) in he goods markes subjec o he following one-period budge consrain: z z, η η where â (z,h ), he agen s ne financial wealh in sae (z,h ), is given by he payoffs of his or her sae-coningen claim acquired las period, he payoffs from his or her equiy posiion, and he risk-free bond payoffs: Trading Technology f 1 1 D ( z ) = ( 1 γ ) Y ( z ) R ( z ) B ( z ) + B ( z )., Q z z a z s D z η π( η η) η V ( z ) + 1,, + 1 +, + b z, η + c z, η â z, η + γy ( z ) η, z, η, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D 1 1 f â z η = a z η + s z η D z + V z + 1 1, 1,, R, 1 z b 1 z. There are wo main classes of raders: acive raders and passive raders. Acive raders are able o rade sae-coningen claims on aggregae shocks. They change heir porfolio composiion of equiy and bonds opimally every period in response o he variaions of saeconingen prices. These acive raders make no misakes in heir invesmen choices. In conras, passive raders canno rade sae-coningen claims, and heir porfolio choice is limied by an exogenously assigned and fixed arge v for he equiy share. I refer o hese raders as passive precisely because of heir inelasic response o he changes in invesmen opporuniies. These passive raders poenially make wo kinds of invesmen misakes. Firs, hey miss he marke iming if he volailiy of he marke price of risk is no consan in he equilibrium. Second, for hose passive raders who hold small or zero fracions of equiy in heir porfolios, hey relinquish he risk premiums. The welfare cos of heir misakes may be large in he equilibrium, exhibiing a large risk premium and a volaile SR in equiy. In addiion, households face exogenous limis on heir ne asse posiions, or solvency consrains, 1 Federal Reserve Bank of S. Louis REVIEW Firs Quarer

8 (2) â ( z, η ) 0. Equaion (2) reflecs he fac ha raders canno borrow agains heir fuure nondiversifiable income. Measurabiliy Resricions The porfolio resricions implied by he differen rading echnologies can be ranslaed ino resricions on he evoluion of ne wealh. These resricions on ne wealh are called measurabiliy consrains. Measurabiliy consrains allow us o derive an aggregaion pricing kernel and o avoid searching for all he equilibrium prices ha clear all markes (see Chien, Cole, and Lusig, 2011, for a deailed discussion). Below, I lis he measurabiliy resricions for each ype of rader. Acive Trader. Since idiosyncraic shocks are no spanned for he acive raders, heir ne wealh needs o saisfy for all and h, h N. These consrains guaranee ha he ne asse posiions are he same across all realizaions of idiosyncraic shocks in each period since he acive raders are no allowed o rade sae-coningen claims on idiosyncraic shocks. Passive Trader. Passive raders who hold a fixed fracion v in levered equiy and 1 v in nonconingen bonds in heir porfolio earn a porfolio reurn: where R d, 1 (z ) denoes he equiy reurn beween periods and 1 given he realizaion of hisory sae z. Hence, heir ne financial wealh saisfies his measurabiliy resricion: for all, z, z Z, and h, h N. This resricion is sraighforward o undersand: The ne asse holding a he beginning of period, â, divided by he porfolio reurn beween periods and 1, R p, sae by sae, should all be equal o he ne wealh a he end of period 1. Two fixed porfolio choices are worh menioning here. Firs, if v = 1/(1 + y), hen his rader holds he marke in each period and earns he reurn on a claim on aggregae diversifiable income. Second, some passive raders do no paricipae in he equiy marke and hold only risk-free asses. I call hem nonparicipans, who can be hough of as having a zero equiy arge share, v = 0. Compeiive Equilibrium ( ) ( ) â z 1 η, â z, η = 1,, η, η p d f 1 R ( ϖ, z ) = ϖr ( z ) + ( 1 ϖ) R ( z ),, 1, 1 1 â ( z z â 1 η η 1 ) ( z z 1,,,,, η, η ) =, p 1 R z z R p ϖ 1,, ϖ, z, z ( ) ( ) The compeiive equilibrium for his economy is defined in a sandard manner. I consiss of a lis of bond, equiy, and sae-coningen claims holdings; a consumpion allocaion; and 74 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

9 a lis of bond, equiy, and sae-coningen prices such ha (i) given hese prices, a rader s asse and consumpion choices maximize his or her expeced uiliy subjec o he budge consrains, he solvency consrains, and he measurabiliy consrains, and (ii) all asse markes clear. CALIBRATION This secion discusses he calibraion of he parameers, he endowmen processes, and he composiion of rader pools. The nex secion uses a calibraed version of he model o evaluae he welfare effec of eliminaing business cycles. To compue he equilibrium of his economy, I follow he algorihm described by Chien, Cole, and Lusig (2011), who use runcaed aggregae hisories as sae variables. I rack he lagged aggregae hisories for up o seven periods. Preferences and Endowmens Lucas (2003) suggesed ha a reasonable risk aversion coefficien should lie beween 1 and 4. My benchmark calibraion se he coefficien of relaive risk aversion a o 4. To check he robusness of my resuls wih respec o he choice of risk aversion rae, I conduc a sensiiviy analysis as deailed in he Sensiiviy Analysis subsecion. The model is calibraed o annual daa. The ime discoun facor b is se o Following Chien, Cole, and Lusig (2012), he fracion of nondiversifiable oupu is se o 90 percen, which is also close o he value in Mendoza, Quadrini, and Ríos-Rull (2009): percen. The process of aggregae oupu is calibraed o mach he aggregae consumpion growh momens from Alvarez and Jermann (2001) and Mehra and Presco (1985). The average consumpion growh rae is 1.83 percen and he sandard deviaion (SD) is 3.15 percen. The auocorrelaion of consumpion growh is Expansions are more frequen han recessions: Seveny-hree percen of realizaions are saes of high aggregae consumpion growh. I calibrae he labor income process as in Soresleen, Telmer, and Yaron (2004, 2007), excep ha I eliminae he counercyclical variaion of labor income risk. The variance of labor income risk is consan in his model. An invarian labor income risk seup highlighs he role of he new feaure (heerogeneous rading echnologies) considered in his aricle. The main driving force of my resul is he heerogeneiy in rading echnology, no he counercyclical variaion of labor income risk. The Markov process for logh has an SD of 0.71, and he auocorrelaion is I use a four-sae discreizaion for boh aggregae and idiosyncraic risk. The elemens of he process for logh are 0.38 and 1.61 for low and high shocks, respecively. The equiy in my model is simply a leveraged claim on diversifiable income. In he Financial Accouns of he Unied Saes (formerly he Flow of Funds Accouns ables), he raio of corporae deb o ne worh is around 0.65, suggesing a leverage parameer y of 2. However, Cecchei, Lam, and Mark (1990) repor ha he SR of he growh rae of dividends is a leas 3.6 imes ha of aggregae consumpion, suggesing ha he appropriae leverage level is over 3. Following Abel (1999) and Bansal and Yaron (2004), I choose o se he leverage parameer y o 3. Federal Reserve Bank of S. Louis REVIEW Firs Quarer

10 Composiion of Trader Pools and Equiy Targe Share I se he fracion of nonparicipans a 50 percen based on he fac ha 51.1 percen of households repored owning socks direcly or indirecly in a recen Survey of Consumer Finances. To mach he large equiy premium (7.53 percen) measured in poswar U.S. daa, a relaively small fracion of acive raders needs o bear he large amoun of residual aggregae risks creaed by nonparicipans. Hence, I se he share of acive raders a 10 percen and he share of passive equiy raders a 40 percen. Among hose households ha hold equiy, I am no able o disinguish beween acive raders and passive equiy raders in he daa. I is difficul o calibrae he arge equiy share of passive equiy raders, since I do no know who hey are. However, empirical sudies have shown ha rich households end o be more sophisicaed raders. Therefore, I consider he riches 10 percen of households o be acive raders and he poores 50 percen of households o be nonparicipans. The arge equiy share of passive equiy raders is herefore calibraed o mach he average fracion of equiy among hose households ha possess a percenile of wealh beween 50 percen and 90 percen (he middle wealhy). According o he daa from he Survey of Consumer Finances, he average equiy share among hese middle wealhy households is 24.2 percen. I, herefore, se he equiy arge share of he passive equiy raders a 24 percen. This calibraion also reflecs he observaion ha he rich end o hold a higher fracion of equiy han he poor. QUANTITATIVE RESULTS I consider wo cases in he quaniaive exercise. The firs case is he benchmark economy, where he parameers are calibraed as described earlier. The second case I consider is anoher economy wih no heerogeneiy in rading echnologies. All households are able o access all asses available on he marke wih no resricions. Table 1 repors momens of asse prices in boh of he economies considered. These resuls are generaed by simulaing daa from a model wih 12,000 agens for 10,000 periods. Panels A and B repor resuls for he benchmark economy and he economy wih no heerogeneiy in rading echnologies, respecively. Asse Prices The Asse pricing secion of Table 1 shows he maximum uncondiional SR, or marke ( ( )) D f ( + 1, R+ 1, ) ( ) σ ( ) Em ( ( E ) ) ( m ) σ ( m) m price of risk, he SD of he maximum SR Sd, he equiy risk premium E R σ ( + 1, + 1, ) D f, he SD of excess reurns R R, he SR on equiy, he mean risk- f f free rae E R + 1,, and he SD of he risk-free rae R + 1,. Benchmark Economy. In he benchmark economy, he maximum SR is 0.37 and he SD of he maximum SR is 4.04 percen. The equiy premium is 7.54 percen and he SR on equiy is The average risk-free rae is 1.91 percen and is volailiy is 2.27 percen. Clearly, he benchmark economy generaes several key feaures of asse pricing observed in he daa, σ ( ) 76 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

11 Table 1 Resuls of Benchmark and NHT Economy Panel A: Panel B: Benchmark economy NHT economy Acive raders (%) Passive equiy raders (%) 40 0 Nonparicipans (%) 50 0 Asse pricing σ ( m) Marke price of risk: Em ( ) σ ( m) SD of marke price of risk: Sd (%) E ( m) ( + 1, + 1,) Equiy risk premium: E R D f R (%) SD of equiy premium: R D f R (%) σ ( + 1, + 1,) SR ( ) f Risk-free rae: E R +1, (%) f SD of risk-free rae: (%) σ ( R +1, ) Approximaion R 2 > > Welfare cos Welfare cos of business cycle (%) NOTE: NHT refers o an economy wih no heerogeneiy in rading echnologies. Based on Soresleen, Telmer, and Yaron s (2007) calibraion of idiosyncraic shocks wihou counercyclical variaion risk and Alvarez and Jermann s (2001) calibraion of aggregae consumpion growh shocks. Parameers: a = 4, b = 0.95, and he collaeralized share of income is 10 percen. The resuls are generaed by simulaing an economy wih 12,000 agens and 10,000 periods. such as high equiy premiums, a low and sable risk-free ineres rae, and a relaively volaile SR. 1 The large fracion of nonparicipan raders is criical for he resuls wih high risk premiums. Those households ha hold only risk-free asses do no ake on any aggregae risk since heir porfolio reurn is independen of he realizaion of aggregae shocks. Addiionally, passive equiy raders ake on only a limied amoun of aggregae risk because of heir relaively low and consan arge equiy share. Therefore, a large amoun of aggregae risk has o be absorbed by a small fracion of acive raders. In equilibrium, a high risk premium is necessary so ha acive raders are willing o bear hese exra aggregae risks. The key mechanism is o concenrae he aggregae risk in a small fracion of he populaion. No Heerogeneous Trading Economy. In an economy where all households are acive raders, he asse pricing resuls are dampened. Compared wih he benchmark case, he maxi- Federal Reserve Bank of S. Louis REVIEW Firs Quarer

12 mum SR is only 0.15 and he SD of he maximum SR decreases o 1.01 percen. The equiy premium decreases o 3.01 percen and he SR on equiy is only The average risk-free rae increases o 3.09 percen and is volailiy remains roughly he same, 2.25 percen. The heerogeneiy in rading echnologies considerably affecs he paerns of asse pricing resuls. The reason for he low equiy premium is clear: The aggregae risk is equally borne by all households, and here is no concenraion of risk in a small fracion of households as in he benchmark economy. Approximaion. In general, he prices of sae-coningen claims depend on he enire aggregae hisory, which is inracable in compuaion. Following Chien, Cole, and Lusig (2011), I use runcaed aggregae hisories as sae variables o forecas sae-coningen prices. To show he accuracy of my approximaion, I repor he implied R-squared value from a linear regression of he acual realizaion of sae-coningen prices on he prediced sae-coningen prices, which are based on he runcaed aggregae hisories. This measure of precision is close o ha of Krusell and Smih (1998). As shown in Table 1, he R-squared value for his regression is higher han in he benchmark case and higher han in he case wihou heerogeneous rading echnologies. This resul shows ha he approximaion is accurae and comparable o ohers repored in he lieraure for models wih heerogeneous agens and incomplee markes. Welfare Coss of Business Cycles The welfare cos of eliminaing business cycles is defined as he average welfare difference beween wo economies: one wih aggregae shocks and he oher wihou aggregae shocks. Given he fac ha households are heerogeneous in erms of heir wealh, income shocks, and rading echnologies in he long-run equilibrium, he average welfare of one economy is compued by aking he expecaion across all idiosyncraic feaures of he populaion. In addiion, I measure he average welfare gap beween he wo economies by calculaing he percenage of per-period consumpion. Therefore, he welfare cos is defined as he expeced percenage of consumpion compensaion for a household in an economy wihou business cycles, so ha his household is indifferen o joining he benchmark economy. The welfare cos is repored a he boom of Table 1. Benchmark Economy. In he benchmark economy, he welfare cos of business cycles is 6.49 percen. This number means ha he average household in he benchmark economy is willing o relinquish up o 6.49 percen of is per-period consumpion o be in he oher economy wihou aggregae uncerainy, all else being equal. The welfare cos is much larger han mos of he findings in he body of lieraure. This resul demonsraes ha heerogeneous rading echnologies play an imporan role no only in he paerns of asse pricing bu also in he disribuional effecs of consumpion. In he benchmark economy, he households wih beer rading echnologies earn a higher reurn on heir wealh, while he households wih less sophisicaed rading echnologies earn a lower reurn. This phenomenon generaes a disribuional impac on consumpion and evenually widens he welfare gap across households. However, he welfare inequaliy caused by heerogeneous rading echnologies vanishes in an economy wihou business cycles. The reason for his is quie simple: Since all asses are risk 78 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

13 free, he porfolio choice beween equiy and risk-free bonds does no affec he reurn on he porfolios. There is no invesmen advanage for a household ha has an advanced rading echnology. The reurns on wealh beween acive raders and passive raders are idenical in an environmen wihou aggregae risk. As discussed earlier, he large welfare cos is mainly driven by he consumpion inequaliy caused by heerogeneous rading echnologies. A reasonable quesion is To wha exen does he consumpion inequaliy in our model compare wih ha in he daa? This is especially imporan since he model is no calibraed o mach he consumpion dispersion observed in he daa. Krueger and Perri (2006) repored a narrow variaion of he Gini index in U.S. consumpion inequaliy ranging from 0.23 o 0.26 beween 1980 and The simulaed daa of he benchmark model generae a Gini index of 0.248, which is in he range of he repored daa even hough he consumpion inequaliy is no argeed in calibraion. This finding enhances he confidence of he welfare calculaion. No Heerogeneous Trading Economy. In my second exercise, where all households are acive raders, he welfare cos is only 1.45 percen. 2 This low welfare cos is consisen wih he findings in he lieraure on he cos of business cycles. This resul suggess ha he welfare cos of business cycles is less significan in an environmen where all agens have sophisicaed rading echnologies and make no invesmen misakes. This oucome can easily be undersood: Because all households have he same rading echnologies, here is no heerogeneiy in porfolio choice. The income and consumpion inequaliy are grealy reduced in his case. The aggregae risk no longer amplifies he disribuional impac on consumpion, so he welfare cos of business decreases considerably. The amoun of reducion in he welfare cos of business cycles can be seen as he average welfare gain from prevening he invesmen misakes made by passive raders in my model. Clearly, he resuls show ha he average welfare loss resuling from hese invesmen errors is large: 5.04 percen of per-period consumpion (he welfare cos difference beween he benchmark economy and he economy wih no heerogeneiy in rading echnologies). This number implies ha he welfare cos of inferior rading echnologies is sizable. My findings also shed ligh on he imporance of undersanding he invesmen misakes made by passive raders, since avoiding hem can improve he average welfare of he sociey. Sensiiviy Analysis Risk Aversion Coefficien. The benchmark calibraion ses he risk aversion coefficien o 4. Alhough my choice of risk aversion is in he range considered in many macroeconomics models, i is differen from he choice made by Lucas (1987), who uses a log uiliy. More imporanly, he welfare cos of business cycles migh be sensiive o he risk aversion rae. Here, I invesigae he sensiiviy of he resuls o changes in he risk aversion coefficien by conducing wo sensiiviy analyses wih respec o changes in he risk aversion rae. In each analysis, I vary he risk aversion coefficien from 3 o 1. The firs analysis considers only changes in he risk aversion rae while keeping all oher parameers unchanged. Table 2 repors he resuls of my firs analysis. The decrease in he risk aversion rae lowers he risk premium as well as he welfare cos. The risk premium drops Federal Reserve Bank of S. Louis REVIEW Firs Quarer

14 Table 2 Resuls of Sensiiviy Analysis 1 Panel A Panel B Panel C Risk aversion rae (a) Acive raders (%) Passive equiy raders (%) Nonparicipans (%) Asse pricing σ ( m) Marke price of risk: Em ( ) σ ( m) SD of marke price of risk: Sd (%) E ( m) ( + 1, + 1,) Equiy premium: E R D f R (%) SD of equiy premium: R D f R (%) σ ( + 1, + 1,) SR ( ) f Risk-free rae: E R +1, (%) f SD of risk-free rae: (%) σ ( R +1, ) Approximaion R Welfare cos Welfare cos of business cycle (%) NOTE: Based on Soresleen, Telmer, and Yaron s (2007) calibraion of idiosyncraic shocks wihou counercyclical variaion risk and Alvarez and Jermann s (2001) calibraion of aggregae consumpion growh shocks. Parameers: b = 0.95 and he collaeralized share of income is 10 percen. The resuls are generaed by simulaing an economy wih 12,000 agens and 10,000 periods. subsanially, from 5.18 percen wih a risk aversion coefficien of 3 o 1.28 percen in he log uiliy case. In addiion, he welfare cos of eliminaing business cycles decreases in a nonlinear paern. Wih a risk aversion rae of 3 or 2, he welfare coss are sill very significan: 5.27 percen and 4.22 percen, respecively. However, he coss are sharply reduced o 0.6 percen when I consider he case of log uiliy. This analysis demonsraes a close relaionship beween he risk premium and he welfare cos of business cycles. This is no surprising, because he welfare cos of business cycles in my aricle depends criically on he magniude of consumpion dispersion, which is based on he reurn difference beween equiy and risk-free bonds. As he risk premium decreases, he heerogeneiy in wealh reurns is reduced along wih he welfare cos. The firs analysis indicaes ha when households become less risk-averse, he model misses he calibraion arge, he equiy premium, by a wide margin. Therefore, I conduc a second 80 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

15 Table 3 Resuls of Sensiiviy Analysis 2 Panel A Panel B Panel C Risk aversion rae (a) Acive raders (%) Passive equiy raders (%) Nonparicipans (%) Asse pricing σ ( m) Marke price of risk: Em ( ) σ ( m) SD of marke price of risk: Sd (%) E ( m) ( + 1, + 1,) Equiy premium: E R D f R (%) SD of equiy premium: R D f R (%) σ ( + 1, + 1,) SR ( ) f Risk-free rae: E R +1, (%) f SD of risk-free rae: (%) σ ( R +1, ) Approximaion R Welfare cos Welfare cos of business cycle (%) NOTE: Based on Soresleen, Telmer, and Yaron s (2007) calibraion of idiosyncraic shocks wihou counercyclical variaion risk and Alvarez and Jermann s (2001) calibraion of aggregae consumpion growh shocks. Parameers: b = 0.95 and he collaeralized share of income is 10 percen. The resuls are generaed by simulaing an economy wih 12,000 agens and 10,000 periods. sensiiviy analysis. For each risk aversion rae considered earlier, I adjus he composiion beween acive raders and passive equiy raders o mach he hisorical risk premium as much as possible, while keeping all oher parameers fixed. The resuls of he second analysis are shown in Table 3. Panel A of Table 3 repors he resuls when he risk aversion coefficien is 3. To mach he high hisorical risk premium, he fracions of acive raders and passive equiy raders are adjused o be 3 percen and 47 percen, respecively. The asse pricing resuls are similar o hose in my benchmark economy. The risk premium is high (7.38 percen) and volaile (SD of percen), while he risk-free rae is low (2.25 percen) and sable (SD of 1.66 percen). Mos imporanly, he welfare cos of business cycles increases o 9.37 percen. The higher welfare cos resul can be undersood as follows: Firs, acive raders are hose who respond o he change in sae-coningen prices and bear exra aggregae risk. Pu simply, hey are mar- Federal Reserve Bank of S. Louis REVIEW Firs Quarer

16 ginal raders who price he risk premium. Second, if hese acive raders sill bear he same amoun of aggregae risk as in he benchmark case, hen he risk premium will drop since heir risk aversion rae is lower now. To mainain he same high risk premium while having a lower risk aversion rae, a larger amoun of aggregae risk has o be concenraed and borne by a smaller fracion of acive raders. As he fracion of acive raders is adjused from 10 percen o 3 percen, each acive rader bears more aggregae risk bu is able o enjoy an even higher level of consumpion in erms of compensaion. The reducion in he fracion of acive raders worsens he consumpion inequaliy and, consequenly, increases he welfare cos of business cycles. Panels B and C of Table 3 repor he resuls of he case wih a = 2 and 1, respecively. In boh cases, I am unable o mach he high risk premium shown in he daa even when he fracion of acive raders is se o be only 1 percen of he oal populaion. The risk premiums of boh cases are significanly smaller: 5.79 percen wih a = 2 and only 3.04 percen wih log uiliy. Neverheless, he welfare cos of business cycles is even higher, 9.56 percen, when he risk aversion coefficien is 2. The reason for his is simply ha here is a higher inequaliy in consumpion. Alhough he lower risk premium reduces he inequaliy of consumpion by decreasing he heerogeneiy of wealh reurns across he populaion, he smaller fracion of acive raders amplifies he consumpion inequaliy even more. The second effec on consumpion inequaliy caused by he diminishing size of acive raders dominaes he firs effec resuling from he lower risk premium. Consequenly, he welfare cos increases slighly. The las panel repors he resuls for he log uiliy case. The welfare cos drops subsanially from 9.56 percen o 3.81 percen when he risk aversion coefficien changes from 2 o 1. This resul is no surprising given ha he composiion of raders is he same in boh Panels B and C. The second sensiiviy analysis demonsraes ha he welfare cos of business cycles is even larger wih a lower risk aversion coefficien whenever he hisorical, high risk premium can be mached in my calibraion economy. Addiionally, for he log uiliy case, he welfare cos of business cycles is sill significan even if my calibraion fails o mach he risk premium. The welfare cos is 3.81 percen when acive raders comprise 1 percen of he oal populaion. CONCLUSION This aricle demonsraes ha heerogeneous rading echnologies can play an imporan role no only in he paerns of asse pricing bu also in he welfare cos of business cycles. In my calibraed model, a large amoun of aggregae risk is borne by a small fracion of households, while a large fracion of households bear lile or even no aggregae risk. The concenraion of risk in a limied se of households drives he large risk premium in my model. As a resul, sophisicaed invesors who hold a large fracion of equiy in heir porfolios are compensaed wih a much higher reurn on wealh, while less sophisicaed invesors earn a lower reurn on heir wealh. A larger wealh reurn difference worsens he income and consumpion inequaliy. In addiion, he new feaure of my model heerogeneous rading echnologies has no disribuional effec on consumpion in an economy wihou aggregae shocks because he reurn difference beween socks and bonds vanishes. Cessaion of aggregae shocks can 82 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW

17 grealy improve he consumpion inequaliy caused by he heerogeneiy in invesmen behavior. Therefore, he welfare cos of business cycles is more pronounced in an economy wih heerogeneous rading echnologies. For economies wih homogeneous rading echnologies, he resuls show an insignifican welfare cos of business cycles. This resul implies a large welfare difference beween economies wih and hose wihou heerogeneous rading echnologies, which can be hough of as he welfare cos of invesmen misakes made by passive raders. These misakes include relinquishing high risk premiums and missing he marke iming. The significan welfare cos of invesmen errors highlighs he imporance of he sudy of household finance. If a way can be found o avoid hese invesmen misakes, he average welfare of he sociey can be improved considerably. Addiionally, he resuls indicae ha he welfare improvemen from avoiding hese invesmen errors is comparable o ha of eliminaing business cycles. Therefore, if he eliminaion of aggregae oupu volailiy is infeasible or exremely expensive, hen concenraing more resources on prevening household invesmen misakes may be a reasonable alernaive. Federal Reserve Bank of S. Louis REVIEW Firs Quarer

18 NOTES 1 The SR esimaed from he daa is enormous and highly counercyclical. My model sill falls shor of maching he daa quaniaively. However, Chien, Cole, and Lusig (2012) exend a similar version of his model by inroducing ineria invesmen behavior among some of he households. Their work shows ha he ineria invesmen behavior helps significanly o explain he large counercyclical variaion in he SR. 2 This welfare cos number is significanly larger han hose in he sandard complee marke lieraure for wo reasons. Firs, he endowmen growh shock is assumed o be permanen and hence has an infinie variance, which gives he larges uncerainy for fuure aggregae consumpion. Second, he risk aversion parameer is higher han hose in he sandard lieraure. REFERENCES Abel, Andrew B. Risk Premia and Term Premia in General Equilibrium. Journal of Moneary Economics, February 1999, 43(1), pp Alvarez, Fernando and Jermann, Urban J. Quaniaive Asse Pricing Implicaions of Endogenous Solvency Consrains. Review of Financial Sudies, Winer 2001, 14(4), pp Alvarez, Fernando and Jermann, Urban J. Using Asse Prices o Measure he Cos of Business Cycles. Journal of Poliical Economy, December 2004, 112(6), pp Bansal, Ravi and Yaron, Amir. Risks for he Long Run: A Poenial Resoluion of Asse Pricing Puzzles. Journal of Finance, Augus 2004, 59(4), pp Campbell, John Y. Household Finance. Journal of Finance, Augus 2006, 61(4), pp Cecchei, Sephen G.; Lam, Pok-sang and Mark, Nelson C. Mean Reversion in Equilibrium Asse Prices. American Economic Review, June 1990, 80(3), pp Chien, YiLi; Cole, Harold and Lusig, Hanno. A Muliplier Approach o Undersanding he Macro Implicaions of Household Finance. Review of Economic Sudies, 2011, 78(1), pp Chien, YiLi; Cole, Harold and Lusig, Hanno. Is he Volailiy of he Marke Price of Risk Due o Inermien Porfolio Rebalancing? American Economic Review, Ocober 2012, 102(6), pp Krebs, Tom. Job Displacemen Risk and he Cos of Business Cycles. American Economic Review, June 2007, 97(3), pp Krueger, Dirk and Perri, Fabrizio. Does Income Inequaliy Lead o Consumpion Inequaliy? Evidence and Theory. Review of Economic Sudies, 2006, 73(1), pp Krusell, Per; Mukoyama, Toshihiko; Şahin, Ayşegül and Smih, Anhony A. Jr. Revisiing he Welfare Effecs of Eliminaing Business Cycles. Review of Economic Dynamics, July 2009, 12(3), pp Krusell, Per and Smih, Anhony A. Jr. Income and Wealh Heerogeneiy in he Macroeconomy. Journal of Poliical Economy, Ocober 1998, 106(5), pp Krusell, Per and Smih, Anhony A. Jr. On he Welfare Effecs of Eliminaing Business Cycles. Review of Economic Dynamics, January 1999, 2(1), pp Lucas, Rober. Models of Business Cycles. New York: Blackwell, Lucas, Rober. Macroeconomic Prioriies. American Economic Review, March 2003, 93(1), pp Mehra, Rajnish and Presco, Edward C. The Equiy Premium: A Puzzle. Journal of Moneary Economics, March 1985, 15(2), pp Mendoza, Enrique G.; Quadrini, Vincenzo and Ríos-Rull, José-Vícor. Financial Inegraion, Financial Developmen, and Global Imbalances. Journal of Poliical Economy, June 2009, 117(3), pp Parker, Jonahan A. and Vissing-Jorgensen, Annee. Who Bears Aggregae Flucuaions and How? NBER Working Paper No , Naional Bureau of Economic Research, January 2009; hp:// 84 Firs Quarer 2015 Federal Reserve Bank of S. Louis REVIEW