Frm Volaly and Cred: Macroeconomc nalyss Leo Kaas Ths paper examnes a racable real busness cycle model wh dosyncrac producvy shocks and bndng cred consrans on enrepreneurs. The model shows how frm volaly ncreases n combnaon wh cred marke developmen. I furher generaes he observed comovemen of cred and frm volaly wh oupu a busness cycle frequences n response o aggregae producvy shocks. (JEL E32, E44, O16) Federal Reserve ank of S Lous Revew, March/prl 2009, 91(2), pp. 95-106. Parallel o he declne n macroeconomc volaly over he pas decades (see lanchard and Smon, 2001, and Sock and Wason, 2002), here s some evdence ha volaly a he frm level has ncreased durng he same perod. For he Uned Saes, such evdence s avalable for dosyncrac sock reurns (Campbell e al., 2001), as well as for employmen, sales, and nvesmen. Comn and Mulan (2004) and Comn and Phlppon (2005) documen smlar resuls for oher counres. 1 There are dfferen explanaons for an ncrease n frm volaly. One s ha deregulaon and nensfed global compeon force frms o adjus prces and busness sraeges faser. noher s ha fnancal developmen leads o more rskakng by enrepreneurs or faclaes leverage, whch boh could poenally drve up frm volaly. Indeed, Comn and Phlppon (2005) fnd some suppor for boh hypoheses. They also show ha he ncrease n frm volaly s drven neher by he growng share of small frms n he sample nor 1 Davs e al. (2006) demonsrae, however, ha frm-level employmen volaly has ncreased only for publcly raded frms and no for prvaely held frms. changes n frm ownershp, ncludng merger and acquson acves. Ths paper pus he lnk beween fnancal developmen and frm volaly n a macroeconomc perspecve. To hs end, I develop a rac - able real busness cycle model wh dosyncrac producvy shocks and collaeral-based borrowng consrans. Producve enrepreneurs borrow up o he value of her collaeral. ecause her capal reurn exceeds he capal cos, he leveraged reurn on equy exceeds he equy reurn of less-producve enrepreneurs. n ncrease n cred marke developmen relaxes borrowng consran and ncreases leverage, and hereby also he spread beween nernal raes of reurn across frms. s a resul, frm growh raes become more volale. noher mplcaon of my model s ha boh cred marke developmen and frm volaly respond posvely o an aggregae producvy shock. Hgher producvy rases he value of pledgeable asses, hus sofenng cred consrans and leverage. Hence, boh he volume of frm cred and frm-level volaly are procyclcal. In he followng secon, I demonsrae ha such Leo Kaas s a professor of economcs a he Unversy of Konsanz and was a vsng scholar a he Federal Reserve ank of S. Lous when hs paper was wren. The auhor hanks Cosas zarads and Carlos Garrga for helpful commens and he German Research Foundaon for fnancal suppor (gran No. K 1519/3). 2009, The Federal Reserve ank of S. Lous. The vews expressed n hs arcle are hose of he auhor(s) and do no necessarly reflec he vews of he Federal Reserve Sysem, he oard of Governors, or he regonal Federal Reserve anks. rcles may be reprned, reproduced, publshed, dsrbued, dsplayed, and ransmed n her enrey f copyrgh noce, auhor name(s), and full caon are ncluded. bsracs, synopses, and oher dervave works may be made only wh pror wren permsson of he Federal Reserve ank of S. Lous. FEDERL RESERVE NK OF ST. LOUIS REVIEW MRCH/PRIL 2009 9 5
Fgure 1 usness Cred (Share of GDP) and Frm Volaly (nnual U.S. Daa 1955-2000) 0.8 0.7 0.6 0.5 0.25 0.20 0.15 0.4 0.3 0.10 0.2 0.1 usness Cred/GDP (lef axs) Frm Volaly (rgh axs) 0.05 0 0 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 procyclcaly s ndeed observable n poswar U.S. daa. In he quanave secon, I mach he model o he U.S. busness cycle and show ha replcaes reasonably well he comovemens among he hree key varables of oupu, cred, and frm volaly. However, he amplfcaon of frm volaly s wce as large as n he daa, and s cross-correlaons wh oupu and cred are somewha oo low. The model can also be used o nvesgae he effecs of a fnancal crss; n parcular, a severe crss where collaeral value drops emporarly by 20 percen feaures a declne of gross domesc produc (GDP) below rend by 3.5 percen. y adopng collaeral-based borrowng consrans n combnaon wh logarhmc uly and Cobb-Douglas producon echnologes, my model s essenally a varaon of he approach of Kyoak (1998) and Kyoak and Moore (2008), who also develop racable busness cycle models wh bndng cred consrans. Oher heorecal conrbuons on dosyncrac producon rsk and fnance consrans n dynamc equlbrum models are hose of Hopenhayn and Vereshchagna (2003) and Meh and Quadrn (2006). u whle hey examne rsk-akng n ncomplee-marke envronmens, he effec of borrowng consrans on frm leverage s he drvng force of hs paper. Furher, he model of hs paper has closed-form soluons, whch make s basc mechancs parcularly clear. THE EVIDENCE For he purpose of hs paper, he approprae measure of cred marke developmen s he share of busness cred n GDP, where busness cred ncludes all cred marke deb owed by nonfnancal frms, ncludng corporaons and noncorporaons. 2 Fgure 1 llusraes he subsanal fnancal deepenng durng he perod 1955 o 2000; as a share of GDP, busness cred roughly doubled. Real busness cred acually ncreased by a facor of 8.9. 3 2 See oard of Governors of he Federal Reserve (2008). 3 Real busness cred s defned as busness cred dvded by he GDP deflaor. Over he same horzon, by comparson, real household deb ncreased by a facor of 9.5, real governmen deb by a facor of 4.9, and real cred marke deb of he fnancal secor by a facor of 99.5. 9 6 MRCH/PRIL 2009 FEDERL RESERVE NK OF ST. LOUIS REVIEW
Fgure 2 Derended Real usness Cred and Frm Volaly (nnual U.S. Daa 1955-2000) 0.15 0.10 0.05 0 0.05 0.10 Real usness Cred Frm Volaly 0.15 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 NOTE: oh varables are repored n logs as devaons from a Hodrck-Presco rend wh smoohng parameer 100. Fgure 1 also shows he ncrease n frm volaly durng he same perod. I use he medan sales volaly repored n Comn and Phlppon (2005, able 1) as a volaly measure. Specfcally, Comn and Phlppon calculae a rollng sandard devaon (SD) of sales growh for nearly every frm n he Compusa daabase; he medan of he cross secon hen measures frm volaly a every pon n me. I becomes evden from he fgure ha frm volaly also doubled beween 1955 and 2000. Of noe, frm volaly s que dspersed n he cross secon. For example, n he 1990s, sales growh volaly was below 0.1 for 25 percen of frms and above 0.3 for anoher 25 percen of frms (see Fgure 2 n Comn and Phlppon, 2005). Fgure 1 furher suggess ha busness cred and frm volaly are posvely correlaed a he busness cycle frequency. To see hs more clearly, noe ha Fgure 2 shows he derended me seres of real busness cred and of frm volaly, where he rend s a Hodrck-Presco (H-P) fler wh λ = 100 and he cyclcal componens are repored as log devaons from rend. Parcularly snce Table 1 Descrpve Sascs (nnual U.S. Daa 1955-2000) Oupu Cred Volaly Sandard devaon 0.021 0.042 0.050 nnual auocorrelaon 0.546 0.826 0.675 Correlaon marx Oupu 1 0.460 0.257 Cred 1 0.567 Volaly 1 NOTE: ll varables are repored n logs as devaons from an H-P rend wh smoohng parameer 100. he 1970s, he wo cycles are closely synchronzed and he percenage devaons from rend are smlar n magnude. Table 1 summarzes he derended daa of oupu, real busness cred, and frm volaly. oh frm volaly and cred have hgher varance han oupu, and hey are posvely corre- FEDERL RESERVE NK OF ST. LOUIS REVIEW MRCH/PRIL 2009 9 7
laed wh conemporaneous oupu, where he correlaon beween cred and oupu s sronger han he one beween volaly and oupu. The correlaon coeffcen beween cred and volaly ncreases from 0.567 o 0.775 when he perod s resrced o he years 1975 o 2000. THE MODEL Consder a one-secor growh model wh nfnely lved enrepreneurs and workers n whch eher group s a connuum of mass one. ll agens derve logarhmc uly from consumpon and dscoun fuure uly wh he facor β < 1. ll workers supply one un of labor nelascally. Enrepreneurs do no supply labor; hey employ workers and capal o produce oupu wh a Cobb-Douglas echnology, whch s subjec o dosyncrac producvy shocks. Specfcally, enrepreneur n perod uses capal K and labor L o produce oupu Y = K L 1, where he enrepreneur s producvy aans he hgh level = (producve sae) wh probably π and he lower level = < (unproducve sae) oherwse. These producvy realzaons are ndependen across me and across enrepreneurs. Thus, a fracon π of enrepreneurs s producve n every perod. The assumpon ha producvy saes are ndependen of hsory smplfes he exposon bu can easly be generalzed o allow for auocorrelaed producvy saes. The only complcaon s ha he model mus hen be augmened by anoher sae varable, whch s he share of wealh n he hands of producve enrepreneurs. Facor producvy a he echnology froner s subjec o aggregae producvy shocks. In parcular, ln follows an R(1) process wh coeffcen ρ < 1, mean ln, and normally dsrbued shocks wh SD σ. The assumpon ha only he froner flucuaes whle he nferor echnology parameer s fxed agan smplfes he exposon and can be generalzed. Wha s crucal for he resuls, hough, s ha flucuaes less han proporonaely wh producvy a he froner. Oupu produced n perod becomes avalable for enrepreneurs nvesmen and consumpon purposes n he nex perod. To oban closedform soluons, capal fully deprecaes whn every perod. Equvalenly, Y can be nerpreed o nclude boh oupu and undeprecaed capal. In he calbraon exercse, I use hs nerpreaon and choose he capal share parameer accordngly. Each perod, all agens have access o a capal marke where hey can borrow and lend a gross neres rae R. orrowng can be agans collaeral only, however. ecause labor ncome canno be collaeralzed, workers are no permed o borrow. Furher, I show ha n any seady sae wh consraned enrepreneurs, R < 1/β holds, whch mples ha n any sochasc equlbrum near he seady sae, workers do no save; hence, workers smply consume her wage ncome n every perod. Enrepreneurs, n urn, can pledge a fracon λ < 1 of her oupu, where he collaeral share parameer λ plausbly depends boh on echnologcal feaures (e.g., wha par of capal s alenable) and on he nsuonal framework and marke envronmen (e.g., credors rghs and avalably of cred marke nsrumens). Every enrepreneur s prncpal and neres on deb D may no exceed he value of collaeral. Tha s, he cred consran akes he form R D λy. Cred repaymens occur a he begnnng of he nex perod before realzaon of he nex perod s producvy. For any realzaon of echnology shocks 0, a compeve equlbrum s a ls of consumpon plans, producon plans, and deb posons (C, K, L, D ) for every enrepreneur; consumpon plans and deb posons for workers (C w, D w ); and facor prces for labor and capal (w, R ) such ha n every perod 0 4 : () C, K, L, D maxmzes enrepreneur s expeced uly subjec o budge and deb consrans; ha s, solves 4 In he nal perod, = 0, all deb posons are assumed zero, and here s some gven dsrbuon of wealh across enrepreneurs. 9 8 MRCH/PRIL 2009 FEDERL RESERVE NK OF ST. LOUIS REVIEW
s β s s 1 s s s s ( Ks 1) ( Ls 1) w s 1Ls 1 Rs 1Ds 1, s, 1 s s λ s ( s ) ( s ),. max E lnc s.. C + K D = 1 () C w, D w maxmzes workers expeced uly subjec o budge and zero deb consrans; ha s; solves s β s w The appendx characerzes he soluons o he agens uly maxmzaon problems. Par - cu larly n he neghborhood of a seady-sae equlbrum wh bndng consrans, workers do no save; hence, D w = 0 for all 0. Furher, all enrepreneurs save a consan fracon β of her wealh. efore dscussng an equlbrum wh bndng deb consrans, s nsrucve o see how he economy acs when he collaeral value λ s large enough. In every perod, hen, all capal flows o producve enrepreneurs who also hre he oal workforce. ecause β s he enrepreneurs savngs rae and because oal enrepreneur wealh s share of oupu, he aggregae capal sock evolves accordng o K +1 = β K. The model s dynamcs hus resemble hose n he sandard real busness cycle model wh logarhmc uly, Cobb-Douglas producon, and full deprecaon. The followng secon characerzes equlbrum when producve enrepreneurs are cred consraned and unproducve enrepreneurs do no lend all her capal bu also produce. Hence, producon s neffcen and he seady-sae oumax E ln C s.. w Cs Ds = ws Rs 1Ds 1, s, w D 0, s. s R D K L s () Markes for labor and capal clear: 1 0 1 0 L d = 1, w s w w D d + D = 0. pu level s below he one n he frs-bes economy. parameer resrcon explaned below wll ensure ha such an equlbrum exss. EQUILIRIUM s s shown n he appendx, all enrepreneurs capal nvesmens are lnear n her equy. Hence, aggregaon over enrepreneurs wh dencal echnologes s sraghforward, and I wre o denoe aggregae capal nvesmen of producve and unproducve enrepreneurs. L and L are smlarly defned, and he absence of a super - ndex ndcaes an aggregae across all enrepreneurs. Le k s = K s /L s, s =,, denoe capal nenses for he wo ypes of enrepreneurs. ecause labor moves freely beween employers, he real wage s whch mples ha (1) ecause labor s perfecly moble and capal s no, unproducve enrepreneurs operae her echnology wh a hgher capal nensy han producve enrepreneurs. The labor and capal markes are n equlbrum f (2) (3) Le D denoe oal borrowng of producve enrepreneurs, whch equals oal lendng of unproducve enrepreneurs because workers do no parcpae n he cred marke. ecause producve enrepreneurs own πk uns of he capal sock, her capal npu s he sum of equy and deb: (4) : = : = K = K d and K = K d ( )( ) = ( )( ), w = 1 k 1 k k = ϕ k wh ϕ ( ) > 1. L + L = 1, L k + L k = K. L k = π K + D. 1 FEDERL RESERVE NK OF ST. LOUIS REVIEW MRCH/PRIL 2009 9 9
ecause he cred consran bnds on each producve enrepreneur, also holds wh equaly n he aggregae: (5) Unproducve enrepreneurs are ndfferen beween lendng capal a gross reurn R or producng hemselves, whch leads o he followng arbrage condon: (6) From equaons (1), (5), and (6), follows ha borrowng s proporonal o nvesmen: (7) and subsuon no equaon (4) yelds (8) To ensure ha unproducve enrepreneurs produce, her lendng may no exceed her capal holdngs; ha s, D mus be srcly smaller han 1 π K. Togeher wh equaon (8), hs necessaes λϕ < 1 π. ecause flucuaes around, ϕ flucuaes around ϕ / 1/. To guaranee a producon-neffcen equlbrum n he neghborhood of he seady sae, mus herefore be assumed ha (9) D R D λ k L. R = ( k ) = ( ) The explanaon of hs condon s as follows. If he collaeral share were oo large, producve agens would borrow all resources from her unproducve counerpars and producon would be effcen. The same would apply f eher ϕ or π were oo large: Wh a large producvy spread, producon becomes less aracve han lendng for unproducve agens, and a large share of borrowers rases cred above he funds suppled by lenders. Smlarly, a oo-low capal share would depress he neres rae, drvng up he demand for cred above lenders resources. 1. λ k k 1 L λϕ k L ; = ( ) ( ) = πλϕ D = K. λϕ λϕ < 1 π. ( ) Now equaons (1), (2), (3), (4), and (7) can be solved for he capal nensy of producve enrepreneurs as follows: where C < 1 follows from condon (9) when ϕ s close o s seady-sae value, so ha C s close o s seady-sae level, C < 1. Ths also mples ha k = ϕ C K > K. Employmen s allocaed accordng o L so aggregae oupu s wh oal facor producvy C <. ecause workers earn share 1 of oupu and do no save and because all enrepreneurs save share β of her wealh, he aggregae savng rae s β. Hence, he capal sock evolves accordng o In he absence of echnology shocks, he capal sock converges o s seady-sae level: Noe ha he seady-sae neres rae s R = 1/ ϕ C β < 1/β; hence, workers ndeed do no save when enrepreneurs are cred consraned. The seady-sae cred share n oupu s calculaed as (10) λϕ π + πϕ k = CK wh C < 1, ϕ λϕ = C π ( λϕ ) and L ( C π ) C λϕ =, C λϕ ( ) Y = ( k ) L + ( k ) L = C K K n unproducve enrepreneur s capal grows a rae βr, whereas a producve enrepreneur s capal grows a βr > βr, where ( ) + = 1 βc K. K = ( βc ) D Y βd = = K 1 ( 1 ). βπλϕ λϕ. ( ) R = ( k 1 λ ) λϕ, 1 0 0 MRCH/PRIL 2009 FEDERL RESERVE NK OF ST. LOUIS REVIEW
s he reurn on equy. 5 Therefore, he SD of a frm s growh rae n seady sae s (11) ( ) σ = π 1 π β ( ) = ϕ 1 R R. λϕ π + πϕ The closed-form expressons (10) and (11) capure he cenral message of hs paper. On he one hand, a rse n λ descrbes he effec of cred marke deepenng n hs model: When frms are able o pledge more of her asses as collaeral, he share of cred n oal oupu expands as shown n expresson (10). In andem wh he cred expanson comes a hgher frm volaly, as evdenced by equaon (11). Relaxed cred lms spur leverage, wdenng he gap beween frm growh raes, β R R. On he oher hand, a posve echnology shock rggers a rse n cred and n frm volaly. The ncrease of (relave o ) rases ϕ, whch unambguously ncreases D/Y and σ (whch s agan an mplcaon of nequaly (9)). Inuvely, a posve producvy shock booss he value of collaeral and hus he volume of cred. Noably, cred rses more han one-for-one wh oupu, so he share of cred n oupu also ncreases. ddonally, he posve echnology shock smulaes leverage, whch enlarges he spread beween frm growh raes, ncreasng frm volaly. QUNTITTIVE NLYSIS ( ) Ths secon explores he quanave properes of he qualave resuls obaned n he prevous secon: How well does hs model explan he observed dynamcs of oupu, busness cred, and frm volaly? To calbrae he seady sae, I frs choose he followng fve parameers: he capal share, ; he dscoun facor, β; he collaeral share, λ; he mean spread beween echnologes, ϕ ; and he share of producve enrepreneurs, π. The echnology level (and hus ) merely shfs he level of aggregae oupu and capal bu has no mpac on he capal-o-oupu rao 5 n enrepreneur wh equy E borrows D = λϕe/ λϕ and nvess K = E/ λϕ o earn prof π = k 1 K RD wk /k = k 1 K λ k 1 K = R E. or on any oher relevan economc varables. Therefore, I normalze = 1. ecause here s no deprecaon n hs model, I adjus he capal share o nclude he value of he undeprecaed capal sock. In he followng, he erm wealh refers o GDP plus undeprecaed capal, Y = GDP + 1 δ K. Wh an annual capal-o-gdp rao of K/GDP = 2.7 and a 5 percen deprecaon rae, he wealh-o-gdp rao s 3.57. Wh capal ncome n GDP a one-hrd, follows ha he capal share n wealh s = [0.33 + 0.95. 2.7 ]/3.57 0.81. Furher, n seady sae, β = K/Y = 2.7/3.57, whch yelds β = 0.938. I choose he collaeral share, λ, o mach a share of busness cred n GDP of around 0.55, he average over he perod 1955-2000. s equaon (10) gves he cred-o-wealh rao, he rghhand sde of hs equaon mus be equalzed o 0.55/3.57. Gven he above choces for and β, and for any choce of ϕ and π, λ s chosen o sasfy hs equaon. The remanng parameers π and ϕ are chosen o mach he followng wo arges: a 3 percen real neres rae (R = 1.03) and a value of frm volaly (measured by he SD of frm growh) of around 0.14, he average of medan frm volaly durng 1955-2000. Ths yelds ϕ = 1.13 and π = 0.08, whch n urn mples ha λ = 0.51. hese parameer values, assumpon (9) s sasfed by a wde enough margn. On he oher hand, f λ would exceed 1 π /ϕ 0.66, assumpon (9) would be volaed, n whch case all capal would be used a he echnology froner. lhough producve enrepreneurs may sll be cred consraned, 6 he model behaves lke a sandard real busness cycle model and he value of λ has no effec on aggregae oupu. In he sochasc model of hs secon, I do no compue frm volaly defned over an nfne me horzon whch, n seady sae, gves rse o equaon (11) for every frm. Insead, I follow he procedure of Comn and Phlppon (2005) o calculae rollng SDs of frm growh raes over 10-year me wndows. Specfcally, a each pon 6 Precsely, when λ < 1 π 0.75, he seady-sae neres rae says below he margnal produc of capal of producve enrepreneurs. lhough he economy would be producon effcen, s consumpon neffcen because dosyncrac volaly sll maers. Frs-bes allocaons are aaned only when λ > 1 π. FEDERL RESERVE NK OF ST. LOUIS REVIEW MRCH/PRIL 2009 1 0 1
Fgure 3 Response o a 5 Percen Permanen Increase n he Collaeral Share λ n Perod = 5 Oupu 0.608 0.604 0.600 Cred 0.040 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.596 0.032 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Volaly 0.158 0.154 0.150 0.146 0.142 Capal Msallocaon 0.715 0.705 0.695 0.685 0.138 0 2 4 6 8 10 12 14 0.675 0 2 4 6 8 10 12 14 n me, I boosrap he dsrbuon of hese SDs from 10,000 frms drawng her growh raes from βr +τ,βr +τ 5 τ = 4. Then I use he medan of hs dsrbuon as he volaly measure. Fgure 3 shows he model s response o a permanen ncrease n he collaeral share λ by abou 5 percen. Oupu ncreases on mpac by 1.3 percen, convergng o he new seady sae, whch s more han 2 percen hgher. Cred ncreases by abou 23 percen, and volaly ncreases by 15 percen. The response of volaly s sluggsh; also, he response begns four perods before he shock because volaly s consruced usng rollng wndows ha are four perods backward lookng and fve perods forward lookng. The larges adjusmen of volaly occurs hree perods afer he shock. The lower-rgh graph shows he capal msallocaon, defned as he share of capal used by unproducve enrepreneurs. Noe ha only 8 percen of enrepreneurs have access o he echnology froner, bu hey sll use 28 percen of capal when λ =0.51. s λ ncreases o 0.535, 32 percen of capal s used a he echnology froner. Whereas hs expermen shows he effec of a permanen rse n collaeral value, also s llumnang o nvesgae he mpac of a emporary declne n collaeral value as a resul of a severe fnancal crss. To hs end, suppose ha he collaeral share drops by 20 percen for a perod of hree years before reurns o s orgnal value. I fnd ha he mpac of such a shock s a declne 1 0 2 MRCH/PRIL 2009 FEDERL RESERVE NK OF ST. LOUIS REVIEW
Fgure 4 Response o a 1 Percen Permanen Increase n Producvy n Perod = 5 Oupu 0.608 0.604 0.600 Cred 0.348 0.344 0.340 0.336 0.332 0.328 0.596 0.324 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Volaly 0.158 0.154 0.150 0.146 0.142 Capal Msallocaon 0.719 0.717 0.715 0.713 0.138 0 2 4 6 8 10 12 14 0.711 0 2 4 6 8 10 12 14 of GDP by 3 o 3.5 percen n he hree years of he crss, he larges occurrng n he hrd year. ecause he model has lle amplfcaon, oupu reurns o 0.5 percen below s seady-sae level n he frs year afer he crss. Durng he hree crss years, cred collapses dramacally: I falls by more han 40 percen, whch mples ha only 20 percen of capal s used a he froner echnology. Noe agan ha hese macroeconomc effecs of a declne of λ are due o he msallocaon of capal; hey would dsappear f λ exceeded he hreshold level mpled by assumpon (9). Publc polcy may aemp o preven he adverse effecs of he cred collapse o some exen, eher by resorng collaeral value or by he njecon of lqudy for example, by provdng unsecured cred lnes. Whou analyzng such polces formally, s clear ha hey mus be of a large scale, gven he subsanal declne of he cred marke. Fgure 4 shows he mpulse response o a permanen ncrease of producvy a he echnology froner by 1 percen. s he value of pledgeable asses rses, cred expands by 6 percen, and he hgher leverage leads o an ncrease of frm volaly by abou 15 percen. Oupu ncreases by 2 percen, whch comes abou hrough wo effecs: The frs s hgher producvy a he echnology froner; he second s ha capal s now more effcenly allocaed as capal msallocaon falls from 0.72 o 0.711. To explore he sochasc model wh auocorrelaed shocks, FEDERL RESERVE NK OF ST. LOUIS REVIEW MRCH/PRIL 2009 1 0 3
Table 2 Smulaon Resuls wh Sochasc Producvy for 10,000 Model Perods Oupu Cred Volaly Sandard devaon 0.022 0.061 0.130 nnual auocorrelaon 0.534 0.542 0.643 Correlaon marx Oupu 1 0.980 0.177 Cred 1 0.183 Volaly 1 NOTE: ll varables are repored n logs as devaons from an H-P rend wh smoohng parameer 100. I choose he wo parameers ρ and σ o arge he SD and auocorrelaon of he cyclcal componen of he model-generaed GDP me seres. s explaned prevously, GDP n perod s he dfference beween Y and he undeprecaed capal sock, whch s 0.95 K = 0.95 βy 1. ll me seres agan are derended wh an H-P fler wh smoohng parameer 100, and he cyclcal componens are he log devaons from rend. The resul of hs exercse s ha ρ = 0.95 and σ = 0.014 mach he frs wo momens n Table 1 reasonably well. These and all oher model-generaed momens from a smulaon wh 10 5 perods are lsed n Table 2. Relave o he daa, he amplfcaon of cred s mached reasonably well, whereas he SD of frm volaly s more han wce as large as n he daa. lhough all conemporaneous correlaons have he rgh sgn, he one beween oupu and cred s larger han n he daa, whereas he cross-correlaons wh volaly are oo low. Volaly s srongly amplfed because producvy of he nferor echnology says consan. If were o flucuae wh accordng o / = / γ, he SD of frm volaly would halve n value for γ = 0.5, and would (counerfacually) become smaller han he SD of oupu for γ = 1. Proper calbraon of he sochasc dynamcs of boh and would requre machng he meln ρln 1 σε, ε 0, 1, = + ( ) seres properes of he mean and he SD of he cross-seconal producvy dsrbuon, whch s beyond he scope of hs paper. CONCLUSION Ths paper has developed a racable real busness cycle model wh dosyncrac producvy shocks and collaeral-based cred consrans. Imporan feaures of he model are ha oupu s below he effcen level because no all capal s used a he echnologcal froner and ha frm growh raes are volale. The model allows dervaon of closed-form expressons for he dynamcs of oupu and capal, for he volume of cred, and he SD of frm growh raes. I accouns qualavely for he observed smulaneous long-erm ncrease of he cred-o-oupu rao and of frm volaly. Quanavely, he model s able o generae he correc comovemen among oupu, cred, and frm volaly, alhough frm volaly s oo srongly amplfed. REFERENCES lanchard, Olver J. and Smon, John. The Long and Large Declne n U.S. Oupu Volaly. rookngs Papers on Economc cvy, March 2001, 2001(1), pp. 135-64. Campbell, John Y.; Leau, Marn; Malkel, uron G. and Xu, Yexao. Have Indvdual Socks ecome More Volale? n Emprcal Exploraon of Idosyncrac Rsk. Journal of Fnance, February 2001, 56(1), pp. 1-43. Comn, Dego and Mulan, Sunl. Dvergng Trends n Macro and Mcro Volaly: Facs. NER Workng Paper 10922, Naonal ureau of Economc Research, November 2004; www.nber.org/papers/ w10922. Comn, Dego and Phlppon, Thomas. The Rse n Frm-Level Volaly: Causes and Consequences. NER Workng Paper 11388. Naonal ureau of Economc Research, May 2005; www.nber.org/ papers/w11388. 1 0 4 MRCH/PRIL 2009 FEDERL RESERVE NK OF ST. LOUIS REVIEW
Davs, Seven, J.; Halwanger, John; Jarmn, Ron and Mranda, Javer. Volaly and Dsperson n usness Growh Raes: Publcly Traded versus Prvaely Held Frms. NER Workng Paper 12354, Naonal ureau of Economc Research; June 2006; www.nber.org/papers/w12354. oard of Governors of he Federal Reserve Sysem. Federal Reserve Sascal Release, Z.1. Flow of Funds ccouns of he Uned Saes: Flows and Ousandngs Thrd Quarer 2008. Cred Marke Deb Ousandng (December 11, 2008). Washngon, DC: oard of Governors, 2008; www.census.gov/compenda/saab/ables/ 09s1130.xls. Hopenhayn, Hugo. and Vereshchagna, Galna. Rsk Takng by Enrepreneurs. Rocheser Cener for Economc Research Workng Paper 500, Unversy of Rocheser, prl 2003; hp://rcer.econ.rocheser.edu/rcerppers/ rcer_500.pdf. Kyoak, Nobuhro. Cred and usness Cycles. Japanese Economc Revew, March 1998, 49(1), pp. 18 35. Kyoak, Nobuhro and Moore, John. Lqudy, usness Cycles, and Moneary Polcy. Workng Paper, Prnceon Unversy, prl 2008; www.prnceon.edu/~kyoak/papers/chkm6-1.pdf. Meh, Cesare. and Quadrn, Vncenzo. Endogenous Marke Incompleeness wh Invesmen Rsks. Journal of Economc Dynamcs and Conrol, November 2006, 30(11), pp. 2143-65. Sock, James H. and Wason, Mark W. Has he usness Cycle Changed and Why? NER Workng Paper 9127, Naonal ureau of Economc Research, ugus 2002; www.nber.org/papers/w9127. PPENDIX Ths appendx characerzes he soluons o he workers and enrepreneurs uly maxmzaon problems. Consder enrepreneurs frs, and suppose ha producve enrepreneurs are deb consraned, whch requres ha (1) 1 ( 1 ) R < ( 1 ) w for all 0. ll enrepreneurs hre labor o equalze margnal produc o he wage; hence, ( ) L = K 1 w Consder frs producve enrepreneurs =. They borrow up o her deb lm because he margnal produc of capal exceeds he neres rae because of equaon (1); hence, λ K ( 1 ) D = R ( 1 ) w. Le S = K D denoe equy (savngs) of he enrepreneur. Wealh a he end of perod s proporonal o he borrower s capal nvesmen and also proporonal o equy: 1. FEDERL RESERVE NK OF ST. LOUIS REVIEW MRCH/PRIL 2009 1 0 5
1 ( ) ( ) K L w L R D ( 1 ) = ( λ) ( 1 ) w K R ( λ) = S 1 ( 1 ) R w ( 1 ) λ = R S. Here R s he reurn on equy for an enrepreneur who s producve n perod. Consequenly, he budge consran n perod +1 reads as C +1 + S +1 = R S. Consder nex an unproducve enrepreneur n perod ; ha s, =. There are wo possbles. Frs, eher he neres rae exceeds he margnal produc of capal of hese enrepreneurs n whch case hey do no produce and her reurn on savngs s smply R or, as s assumed n he man ex, he neres rae equals her margnal produc of capal, 7 whch requres ha gan, S = K D s savngs and D > 0 are fnancal asses of unproducve enrepreneur. Wealh a he end of perod s agan proporonal o savngs: Therefore, any enrepreneur s budge consran n perod +1 reads as C +1 + S +1 = R S, where R = R f he enrepreneur s unproducve n perod and R = R f he enrepreneur s producve n. The Euler equaon for enrepreneur s uly maxmzaon problem s hen Clearly, consan consumpon/savng shares C +1 = 1 β R S and S +1 = β R S are he only soluon o hs equaon ha sasfy he ransversaly condon. For workers, he Euler equaons for her problem specfed n () of he equlbrum defnon s wh complemenary slackness. In he man ex, s shown ha R β < 1 holds n he neghborhood of he seady sae. Furher, n he neghborhood of he seady sae (.e., small-enough producvy shocks), w w and E 1/w +1 1/w hold, where w s he seady-sae wage level. Therefore, follows from he complemenary slackness condon ha D w = 0 and C w = w for all 0. 1 ( 1 ) R = ( 1 ) w. 1 1 ( ) ( ) = ( ) ( ) K L w L R D K L ( ) = = R K D R S. 1 1 = R E C β. C + 1 1 1 w β R E D 0 w, C C w, + 1 R D 7 The hrd case ha he neres rae s below he margnal produc of capal of all enrepreneurs s ncompable wh equlbrum; hen all enrepreneurs would be lenders, whereas workers do no save (as shown below). Hence, he capal marke canno be n equlbrum. 1 0 6 MRCH/PRIL 2009 FEDERL RESERVE NK OF ST. LOUIS REVIEW