NBER WORKING PAPER SERIES A GENERAL-EQUILIBRIUM ASSET-PRICING APPROACH TO THE MEASUREMENT OF NOMINAL AND REAL BANK OUTPUT

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1 NBER WORKING PAPER SERIES A GENERAL-EQUILIBRIUM ASSET-PRICING APPROACH TO THE MEASUREMENT OF NOMINAL AND REAL BANK OUTPUT J. Chrsna Wang Susano Basu John G. Fernald Workng Paper 466 hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 050 Massachuses Avenue Cambrdge, MA 0238 December 2008 Prepared for CRIW conference on Prce Index Conceps & Measuremen, Vancouver, June 28-29, We hank Erwn Dewer, Denns Fxler, Charles Hulen, Alce Nakamura, Em Nakamura, Marshall Rensdorf, Paul Schreyer, Jack Trple, and Km Zeschang for helpful dscussons, and Felx Momsen for daa asssance. The vews n hs paper are hose of he auhors, and should no be consrued as necessarly reflecng he vews of he Board of Governors, anyone else afflaed wh he Federal Reserve Sysem, or he Naonal Bureau of Economc Research. NBER workng papers are crculaed for dscusson and commen purposes. They have no been peerrevewed or been subjec o he revew by he NBER Board of Drecors ha accompanes offcal NBER publcaons by J. Chrsna Wang, Susano Basu, and John G. Fernald. All rghs reserved. Shor secons of ex, no o exceed wo paragraphs, may be quoed whou explc permsson provded ha full cred, ncludng noce, s gven o he source.

2 A General-Equlbrum Asse-Prcng Approach o he Measuremen of Nomnal and Real Bank Oupu J. Chrsna Wang, Susano Basu, and John G. Fernald NBER Workng Paper No. 466 December 2008 JEL No. E0,E44,G2,G32 ABSTRACT Ths paper addresses he proper measuremen of fnancal servce oupu ha s no prced explcly. I shows how o mpue nomnal servce oupu from fnancal nermedares neres ncome, and how o consruc prce ndces for hose fnancal servces. We model fnancal nermedares as provders of fnancal servces whch resolve asymmerc nformaon beween borrowers and lenders. We embed hese nermedares n a dynamc, sochasc, general-equlbrum model where asses are prced compevely accordng o her sysemac rsk, as n he sandard consumpon-based capal-asse-prcng model. In hs envronmen, we show ha s crcal o ake rsk no accoun n order o measure fnancal oupu accuraely. We also show ha even usng a rsk-adjused reference rae does no solve all he problems assocaed wh measurng nomnal fnancal servce oupu. Our model allows us o address mporan ousandng quesons n oupu and producvy measuremen for fnancal frms, such as: () Wha are he correc reference raes o use n calculang bank oupu? In parcular, should hey ake accoun of rsk? (2) If reference raes need o be rsk-adjused, should hey be ex ane or ex pos raes of reurn? (3) Wha s he rgh prce deflaor for he oupu of fnancal frms? Is jus he general prce ndex? (4) When f ever should we coun capal gans of fnancal frms as par of fnancal servce oupu? J. Chrsna Wang Federal Reserve Bank of Boson Chrsna.Wang@bos.frb.org Susano Basu Deparmen of Economcs Boson College 40 Commonwealh Avenue Chesnu Hll, MA and NBER susano.basu@bc.edu John G. Fernald Research Deparmen, Mal Sop 30 Federal Reserve Bank of San Francsco 0 Marke S San Francsco, CA 9405 john.fernald@sf.frb.org

3 In many servce ndusres, measurng real oupu s a challenge because s dffcul o measure qualy-adjused prces. In fnancal servces, however, here s no even an agreed-upon concepual bass for measurng nomnal le alone real oupu. Ths paper presens a dynamc, sochasc, general equlbrum (DSGE) model n whch nomnal and real values of bank oupu and, hence, he prce deflaor are clearly defned. We use he model o assess he (n)adequacy of exsng naonal accounng measures and derve a heorecally-preferred alernave. Our model s a general-equlbrum exenson of Wang s (2003a) paralequlbrum framework, and valdaes Wang s proposed bank-servce flow measure. The bgges challenge for measuremen s ha banks and oher fnancal servce provders ofen do no charge explc fees for servces. Insead, hey charge ndrecly by he spread beween he neres raes hey charge and pay. The Sysem of Naonal Accouns 993 (SNA93) hus recommends measurng hese fnancal nermedaon servces ndrecly measured (FISIM) usng ne neres: he oal propery ncome recevable by fnancal nermedares mnus her oal neres payable, excludng he value of any propery ncome recevable from he nvesmen of her own funds. 2 The so-called user-cos approach o bankng s aken o be he heorecal bass for measurng nomnal oupu va neres-rae margns and for nerpreng neres rae spreads as mplc prces for fnancal servces. 3 As a praccal maer, he SNA93 approach hus more or less equaes nomnal oupu from FISIM wh he ne neres ncome ha flows hrough banks. Ne neres ncome from lendng s concepually he dfference beween neres ncome receved and he opporuny cos of funds len. As currenly mplemened (for example, n SNA93), ne neres s mpued usng he dfference beween acual lendng raes and a rskless neres rae, such as a shor-erm Treasury rae, whch s mean o capure he opporuny cos of funds. However, Wang (2003a) shows ha ne neres conans no only nomnal compensaon for bank servces bu also he reurn due o he sysemac rsk of bank loans. Ths reurn s par of he opporuny cos of funds and, accordng o he essence of he user-cos framework, should be excluded from bank oupu. In modern fnance heores of asse prcng, he requred rae of reurn depends on rsk. Hence, he user cos of money needs o be adjused for rsk. Wang s (2003a) key conrbuon hus s o exend he user-cos approach o a world wh rsk. Ths For a recen sample, see chaper 7 n Trple and Bosworh (2004), he commen on ha chaper by Fxler (2003), and he auhors rejonder. 2 SNA 993, paragraph See, e.g., Fxler and Zeschang (992). Imporan conrbuors o he user-cos approach also nclude Dewer (974), Barne (978), and Hancock (985).

4 2 conrass wh he unrealsc rskless framework n he exsng leraure. The ne neres poron of Wang servce flow measure of nomnal bank oupu hus can also be characerzed as oal ncome ne of he opporuny cos of funds, provded hs opporuny cos s correcly adjused for rsk. (All agree ha explc fee ncome receved s also par of nomnal bank oupu.) The general-equlbrum model here verfes he paral-equlbrum concluson reached by Wang (2003a). As n Wang (2003a), we use a user-cos framework n whch banks opmal choce of neres raes mus cover he (rsk-adjused) opporuny cos of funds as well as he cos of mplcly-provded servces. The prmary conrbuon of our GE model s o endogenze he cos of funds, whch Wang (2003a) akes as exogenously gven by fnancal markes. Lke SNA93, he 2003 U.S. Naonal Income and Produc Accouns (NIPAs) benchmark revsons allocae he FISIM beween borrowers and deposors usng a reference rae. NIPAs mpue he nomnal value of servces o borrowers as he volume of neres-earnng asses mes he dfference beween he (average) lendng rae and ha reference rae he user cos of funds. Lkewse, mpues nomnal oupu of servces o deposors as he volume of deposs mes he dfference beween ha reference rae and he (average) depos rae (Fxler, Rensdorf, and Smh, 2003). The challenge s o deermne he approprae reference rae or raes. NIPAs use a bascally rsk-free rae for boh borrower and deposor servces. As n Wang (2003a), however, we show ha he rsk-free rae s no approprae for borrowers. Snce he cos of funds s rsk-dependen, he reference rae mus ncorporae he sysemac rsk of a bank s loan porfolo. Hence, he mpued value of banks mplc borrower servces excludes he rsk premum. The premum represens capal ncome o bank shareholders and unnsured deb-holders for bearng he sysemac rsk on funds used n producon ousde he bank. A smple example shows he nuon for excludng he rsk premum from bank oupu and llusraes he shorcomngs of he NIPAs approach. Consder wo oherwse-dencal borrowers who seek o oban addonal fnancng n a world wh no ransacons coss or nformaonal asymmeres; one borrows n he bond marke, he oher borrows from a bank. The bond-fnanced frm s expeced reurn equals he rsk-free rae plus a rsk premum. I s clear ha he enre reurn represens he value added of he borrower. Now consder he bank-fnanced frm. To keep hngs smple, suppose ha banks hre no labor or capal and produce no servces whasoever. Banks are merely an accounng devce ha records loans

5 3 (perhaps funded by bank shareholder equy) o borrowers. They are a perfec subsue for he bond marke, so hey charge he rsk-free neres rae plus he same rsk premum pad by he bond-fnanced frm: By consrucon, he rsk s he same and, n equlbrum, here s no arbrage. (Noe ha n equlbrum, bank shareholders are ndfferen beween buyng he bonds or holdng shares n he bank.) Bu NIPAs would arbue posve value added o a bank equal o he rsk premum mes he face value of he loan even hough, by assumpon, he bank does nohng! Concepually, he wo frms should be reaed symmercally: They are dencal apar from an arbrary and (o he frm) rrelevan fnancng choce. Bu under NIPA convenons, hey appear o have dfferen value added, npus of fnancal servces, and producves. In conras, he approach we recommend would rea he wo frms symmercally by excludng he rsk premum from he bank s nomnal fnancal oupu a premum he borrowng frm mus pay wheher s fnanced by bonds or bank loans. Thus, he naonal accounng measure leads o nconssency even n he very smples of possble models, where banks produce no servces ha use real resources. Our model as well as Wang s (2003a) shows ha he concepual nconssency exends o realsc cases where banks provde acual servces. The NIPAs measure corresponds o he emprcally rrelevan specal case when eher nvesors are rsk neural or bank loans have no sysemac rsk. Quanavely, he poenal msmeasuremen under he curren sysem s large. In 200, commercal banks n he Uned Saes had nomnal oupu of $87 bllon, 4 of whch half s fnal consumpon and half s nermedae servces o busnesses. Our heorecal resuls mply ha he measured fgures oversae rue oupu; Wang (2003b) suggess ha NIPAs bankng oupu measures are abou 20 percen oo hgh. Ths fgure reflecs boh an overesmae of lendng servces provded o consumers (hence, an oversaemen of GDP) and an overesmae of nermedae servces provded o frms (whch does no oversae GDP, bu dsors measures of ndusry oupu and producvy). Smlar consderaons apply o measurng he oupu of fnancal servces more generally, so he oal NIPAs msmeasuremen can be subsanal. Furhermore, he dsorons affec relave GDP measures across counres. For example, bankng servces accoun for 37 percen of Luxembourg s expors, whch n urn are 50 percen of GDP. Thus, our work suggess ha Luxembourg s GDP could be oversaed by abou 4 Fgures are from Fxler, Rensdorf, and Smh (2003), and reflec he December 2003 comprehensve revsons.

6 4 percen subsanal by any measure. In addon, me varaon n rsk prema dsors growh raes (Wang, 2003b). The dsoron s parcularly large durng ransons such as hose akng place now. For example, as banks securze a growng fracon of her loans, hey move he rsk premum off her books, even f hey connue o provde subsanally he same real servces, e.g., screenng and servcng loans. Several sudes fnd ha fnancal servces conrbued mporanly o he pos-995 U.S. producvy growh revval, so s mporan o measure he growh as well as he level of hese secors oupus correcly. The model provdes addonal nsghs. Frs, o measure real oupu (and, hence, he bankng prce deflaor), one wans o coun jus he producve acves of banks (such as hose relaed o screenng loans), no he (real) amouns of assocaed fnancal asses (.e., he loans). Imporanly, he wo are no generally n fxed proporon o one anoher, so one canno use he volume of loans as a proxy or ndcaor for he value of servces. The model provdes concepual gudance on how o wegh dfferen real servces n he absence of clearly arbuable nomnal shares n cos or revenue. Second, beng dynamc, our model hghlghs he poenal mng msmach beween when a servce s performed (e.g., screenng when a loan s orgnaed) and when ha servce s compensaed (wh hgher neres ncome over he lfe of he loan). Thrd, beng sochasc, he model pons ou ha he expeced nomnal oupu of monorng (servces ha are performed durng he lfeme of a loan, afer s orgnaed) can be measured from ex ane neres rae spreads, bu he acual monorng servces produced are dffcul o measure from ex pos revenue flows. Fnally, our servce-flow perspecve suggess major shorcomngs of he book-value oupu measures ha are unversally used n he emprcal mcroeconomc leraure on bank effcences. Our use of a DSGE model offers several advanages for sudyng measuremen ssues. Frs, naonal ncome accounng mposes a se of addng-up consrans ha mus hold n he aggregae. GE models mpose he same resrcons. By applyng acual naonal ncome accounng procedures o he varables generaed by he model, we can ask wheher and under wha condons he objecs measured n he naonal accouns correspond o he economc conceps we wan o measure. Second, and more specfc o our curren projec, he sudy of bankng nrnscally concerns boh goods- and asse-marke neracons among dfferen agens, whch endogenously deermne goods prces, quanes, and neres raes. Ths nexus of economc connecons s naurally suded n a GE seng, whch

7 5 ensures he comprehensve consderaon of all he key elemens of an economy. For example, one needs o specfy an envronmen n whch nermedaon s necessary: n he model, households canno or wll no lend drecly o frms for well-specfed nformaonal reasons. We also need o specfy how banks hen produce real nermedaon servces; and wha deermnes requred raes of reurn on bank asses. The DSGE model endogenzes he rsk premum on loans ha fund busness capal, as well as he requred rae of reurn on banks equy. Our model follows Bernanke and Gerler (989) and Bernanke, Gerler and Glchrs (999), among ohers, bu explcly models he screenng and monorng echnology of fnancal nermedares, whch hey use o resolve asymmerc nformaon problems n nvesmen. The model hghlghs he proper measuremen of bank servce oupu n boh nomnal and real erms. We absrac from some acves banks underake (manly ransacons servces o deposors) as well as from realsc complcaons (e.g., depos nsurance and axes). These absracons, whch could be ncorporaed, are unlkely o nerac n mporan ways wh he ssues we address here. For example, our approach exends naurally o valung acves by banks oher han makng loans and akng deposs, such as underwrng dervaves conracs and oher exoc fnancal nsrumens. We presen one such example. Thus, we begn he process of brngng measuremen no lne wh he new roles ha banks play n modern economes, as dscussed, for example, by Allen and Sanomero (998, 999). One mgh worry ha resuls from our bare-bones model do no apply o he far more complex real world. Bu our model provdes a conrolled seng where we know exacly wha neracons ake place and wha oucomes resul. Even n hs relavely smple seng, curren mehods of measurng nomnal and real bank oupu generae nconssen resuls ha can be economcally subsanal. I s mplausble ha hese mehods wll magcally succeed n he far more complex world. The paper has four man secons. Secons I and II presen he basc seup of he model wh mnmal echncaly, o buld nuon for he economc reasonng behnd our conclusons. (The rgorous soluon of he model s n he Appendx.) Secon I solves he model wh symmerc nformaon beween borrowers and lenders, and uses hs smple seup o show by example he flaws n exsng proposals for measurng bank oupu. Secon II nroduces asymmerc nformaon, and assumes ha banks and rang agences have a echnologcal advanage n resolvng such asymmeres. We derve he correc, model-based measure of bank oupu n hs seng, where fnancal nsuons provde real servces. Secon III

8 6 dscusses mplcaons of he model for measurng nomnal and real fnancal secor oupu. Secon IV dscusses exensons. Secon V concludes, and suggess prores for fuure research and daa collecon. I. The Model wh Symmerc Informaon A. Overvew Our model has hree groups of agens: households, who supply labor and who ulmaely own he economy s capal; enrepreneurs, who hre workers and buy capal o operae projecs; and compeve fnancal nsuons (banks and rang agences) ha resolve nformaon problems beween he owners and he fnal users of capal. I also has a bond marke, n whch enrepreneurs can ssue corporae deb. Frs, households are he only savers n hs economy, and hus he ulmae owners of all capal. Ther preferences deermne he rsk premum on all fnancal asses n he economy, and her accumulaed savng deermnes he amoun of capal avalable for enrepreneurs o ren n a gven perod. Second, enrepreneurs operae projecs whch produce he economy s fnal oupu. There s only one homogeneous fnal good, sold n a compeve marke, whch can be consumed or nvesed. Enrepreneurs projecs dffer from one anoher snce he enrepreneurs dffer n her ably levels (or, equvalenly, n he nrnsc producvy of her projecs). The echnology for producng fnal goods n any projec has consan reurns o scale. Thus, whou asymmerc nformaon, he socal opmum would be o gve all he capal o he mos effcen projec. Bu we assume ha, due o asymmerc nformaon problems, enrepreneurs face a supply curve for funds ha s convex n he amoun borrowed. 5 We also assume ha enrepreneurs are born whou wealh hey are he proverbal mpovershed genuses, whose heads are full of deas bu whose purses hold only ar so ha, one way or anoher, hey wll need o oban funds from households. The focus of hs paper s on how he enrepreneurs oban he funds for nvesmen from households, and he role of fnancal nermedares n he process. A large leraure on fnancal nermedaon explans (n paral equlbrum) fnancal nsuons role as beng o resolve nformaonal asymmeres beween he 5 Gven ha all enrepreneurs are borrowng whou collaeral, hs seems que realsc. Our specfc modelng assumpon s ha he cos of screenng s convex n he sze of he projec, bu oher assumpons such as leveragng each enrepreneur s ne worh wh deb would also lead o hs resul. See Bernanke, Gerler, and Glchrs (999).

9 7 ulmae supplers of funds (.e., he households n our model) and he users of funds (.e., he enrepreneurs who borrow o buy capal and produce). We ncorporae hs resul no our general-equlbrum model. 6 In hs paper, we consder boh ypes of nformaon asymmery hdden nformaon and hdden acons. Households face adverse selecon ex ane as hey ry o selec projecs o fnance: They know less abou he projecs (e.g., defaul probables under varous economc condons) han he enrepreneurs, who have an ncenve o undersae he rsk of her projecs. Moral hazard arses ex pos as savers canno perfecly observe borrowers acons (e.g., dverng projec revenue for her own consumpon). Thus, he hrd group of acors n our model are nsuons such as banks whch exs (n he model and, largely, n pracce) o mgae hese nformaonal problems. 7 We focus on wo specfc servces hey provde: screenng o lessen (n our model, o elmnae) enrepreneurs prvae nformaon abou he vably of her projecs, and monorng projec oucomes (e.g., audng afer a defaul) o dscover enrepreneurs hdden acons. 8 To conduc screenng and monorng, nermedares engage n a producon process ha uses real resources of labor, capal, and an underlyng echnology. The producon process s qualavely smlar o producng oher nformaon servces such as consulng and daa processng. 9 We call he fnancal nermedares banks manly for convenence, alhough he funcons hey perform have radonally been cenral o he acves of commercal banks. Bu he analyss s general, as we wll show ha loans subjec o defaul are equvalen o a rsk-free bond plus a pu opon. So our analyss also apples o mplc bank servces assocaed wh oher fnancal nsrumens, as well as o oher ypes of nermedares, such as rang agences and fnance companes. We assume ha banks and oher fnancal-servce provders are owned by households and are no subjec o nformaonal asymmeres wh respec o households. 0 6 Mos general-equlbrum models of growh or busness cycles absrac from hs ssue: mplcly, households own and operae he frms drecly, so here are no prncpal-agen problems. 7 Fnancal nsuons prevens marke breakdown (such as n Akerlof, 970), bu canno elmnae deadwegh loss. Anoher major funcon of banks s o provde servces o deposors, as dscussed n he nroducon. Bu we om hem from he formal model, snce her measuremen s less conroversal and has no bearng on our concluson abou how o rea rsk n measurng lendng servces. Ye, we noe praccal measuremen ssues abou hem n Secon III. 8 Many sudes, all paral-equlbrum, analyze he naure and operaon of fnancal nermedares. See, e.g., Leland and Pyle (977) model banks role as resolvng ex ane adverse selecon n lendng; Damond (984) sudes delegaed monorng hrough banks; Ramakrshnan and Thakor (984) look a non-deposory nsuons. 9 Only a handful of sudes analyze he effecs of fnancal nermedares on real acves n a general equlbrum framework. None of hem, however, consder explcly he ssue of fnancal nermedares oupu assocaed wh he process of screenng and monorng, nor he properes of he screenng and monorng echnology. 0 We could exend our model o allow for hs wo-er nformaon asymmery, a he cos of consderable added

10 8 As supplers of funds, households demand an expeced rae of reurn commensurae wh he sysemac rsk of her asses. Ths s, of course, rue n any reasonable model wh nvesor rsk averson, regardless of wheher here are nformaonal asymmeres. Banks hus mus ensure ha he neres rae charged compensaes her owners, he households, wh he rsk-adjused reurn n expecaon. Banks mus also ensure ha hey charge explc or mplc fees o cover he coss ncurred by screenng and monorng. The prmary focus of hs paper s how o correcly measure he nomnal and real servce oupu provded by hese banks when he servces are no charged for explcly bu mplcly n he form of hgher neres raes. Hence, we need o deal he naure of he conrac beween enrepreneurs and banks snce ha deermnes he neres raes banks charge. Indeed, mos of he complexy n he formal model n he Appendx comes from he dffculy of solvng for he neres rae charged under he opmal deb conrac, and from decomposng oal neres ncome no a compensaon for bank servces screenng and monorng and a rsk-adjused reurn for he capal ha households channel o frms hrough he bank. The payoff from hs complexy s ha he model provdes defne nsghs on key measuremen ssues. For he mos par, we ry o specfy he ncenves and preferences of he hree groups of agens n a smple way, n order o focus on he complex neracons among he agens. We now summarze he key elemens of he ncenves and preferences of each agen o gve he reader a workng knowledge of he economc envronmen. We hen derve he key frs-order condons for he opmal prcng of rsky asses, whch mus hold n any equlbrum, o draw mplcaons from he model ha are crucal for measuremen purposes. A he end of hs secon, he reader may proceed o he dealed dscusson of he model ha follows n he Appendx, or proceed o Secon III o sudy he mplcaons for measuremen. B. Households We assume households are nfnely lved and rsk averse. For mos of he paper, we assume ha households can nves her wealh only hrough a fnancal nermedary, because hey lack he ably o resolve nformaon asymmeres wh enrepreneurs drecly. In conras, households own and have no nformaonal problems wh respec o he nermedares. All households are dencal, and hey maxmze he expeced presen value of lfe-me uly here expressed n erms of a represenave household: complexy. We conjecure, however, ha our qualave resuls would be unaffeced by hs change.

11 9 H ( + s, + ) s E ρ V C N s= 0 s () subjec o he budge consran: C = W N + Π + R H X X +. (2) H H C s he household s consumpon, N s s labor supply, and ρ s he dscoun facor. E (.) s he expecaon gven he nformaon se a me. We assume ha he uly funcon V(.) s concave and ha V (0) =. W s he wage rae, X s he household s oal asses (equal o he capal sock n equlbrum), and Π s pure economc prof receved from ownershp of fnancal nermedares (equal o zero n equlbrum, snce we assume ha hs secor s compeve). s he ex pos gross reurn on he H R household s asse porfolo (real capal, len o varous agens o enable producon n he economy). Correspondng o he ex pos reurn s an expeced reurn he requred rae of reurn on rsky asses, whch H we denoe R. Ths s a key neres rae n he followng secons, so we dscuss furher. Defne he neremporal prcng kernel (also called he sochasc dscoun facor), m +, as, m + H (, +) H (, ) ρ VC C+ N V C N C (3) where V C s he paral dervave of uly wh respec o consumpon. In hs noaon, he Euler equaon for consumpon (whch s also a basc asse-prcng equaon n he Consumpon CAPM) s: H ( ) E m R. (4) + + = Now suppose a one-perod asse whose reurn s rsk-free because s known n advance. Clearly, f for hs asse, he rae of reurn f f R + sasfes E ( m R ) R E ( m ) = =. So, R f + =. (5) E ( m ) As s sandard n a CCAPM model, he Euler equaon (3) allows us o derve he rsk-free rae even f no such asse exss whch s he case n our economy, where he only asse s rsky capal. H From (4) and (5), he gross requred (expeced) rae of reurn on he rsky asse, R, s: + For more dscusson, see Cochrane (200, chap. 2).

12 0 ( ) cov (, + ) H H f H R+ E R + = R+ m+ R, (6) where cov s he covarance condonal on he nformaon se a me. The rsk premum hen equals ( ) R R = R cov m, R. H f f H H Noe ha when R s he requred rae of reurn on deb (e.g., loans) subjec o he rsk of borrower H defaul, here s a suble bu mporan concepual dfference beween R and he neres rae (he so-called yeld for corporae bonds) ha s charged on loans he rae ha a borrower mus pay f no n defaul. To llusrae n a smple example, suppose here s probably p ha a borrower wll pay he neres rae charged (call R ), and probably ( p) oherwse, n whch case lenders ge nohng. Then R mus sasfy So ( ) p R p R R R p H H = + + = + /. H H R + exceeds he requred reurn R ; he margn R+ R+ s he so-called defaul premum. Thus, R dffers from he rsk-free rae for wo reasons. Frs, here s he defaul premum. The borrower repays less o nohng n bad saes of he world, so he mus pay more n good saes o ensure an adequae average reurn. Second, here s a rsk premum, as above. The rsk premum exss f he probably of defaul s correlaed wh consumpon (or more precsely, wh he margnal uly of consumpon). If defauls occur when consumpon s already low, hen hey are parcularly cosly n uly erms. Thus, he consumer requres an exra reurn, on average, o compensae for bearng hs sysemac, non-dversfable rsk. In addon o he neremporal Euler equaon, consumer opmzaon requres a sac radeoff beween consumpon and lesure whn a perod: ( H H, ) (, ) WV C N = V C N. (7) C N In equlbrum, households asses equal he oal capal sock of he economy: X = K. The capal sock evolves n he usual way: K ( δ ) + K I = +. Capal s used by nermedares o produce real fnancal servces, or s rened by frms for producon. 2 2 Snce we have assumed dencal households, we absrac from lendng among households (e.g., home morgages).

13 C. Enrepreneurs Each enrepreneur owns and manages a non-fnancal frm ha nvess n one projec, producng he sngle homogeneous fnal good and sellng n a perfecly compeve marke. So enrepreneur, frm, and projec are all equvalen and nerchangeable n hs model. Enrepreneurs are a se of agens dsnc from households n ha each lves for only wo perods, whch concdes wh he duraon of a projec. Thus, here are wo overlappng generaons of enrepreneurs n each perod. The number of enrepreneurs who are born and de each perod s consan, so he fracon of enrepreneurs s consan n he oal populaon of agens. The reason for havng shor-lved enrepreneurs n he economy s o creae a need for exernal fnancng and hus he screenng and monorng by fnancal nermedares. Long-lved enrepreneurs could accumulae enough asses o self-fnance all nvesmen, whou borrowng from households. In addon, by havng each borrower nerac wh lenders only once, we avod complex supergame Nash equlbra where enrepreneurs ry o develop a repuaon for beng good rsks n order o oban beer erms from lenders. We assume ha enrepreneurs, lke households, are rsk averse. 3 Bu we absrac from he ssue of rsk sharng and assume ha he sole ncome an enrepreneur receves s he resdual projec reurn, f any, ne of deb repaymen. 4 Tha also means enrepreneurs have no nal endowmen. 5 In choosng projec sze n he frs perod, enrepreneurs seek o maxmze her expeced uly from consumpon n he second perod, whch s he only perod when hey consume. Thus, he uly of enrepreneur born a me s E, ( ) + ( ) U C, where U > 0, U < 0, and U 0 = 0. (8) We denoe enrepreneurs aggregae consumpon by E C, whch s he sum over of C. E, Frms dffer only n her exogenous echnology parameers. Denoe he parameer A, for a frm creaed n perod, snce he owner produces n he second perod +. We assume ha A + = z A, where 3 If enrepreneurs were rsk-neural, hey would nsure he households agans all aggregae shocks, leadng o a degenerae and counerfacual oucome where lenders of funds would face no aggregae rsk. 4 In fac, hs model mplcly allows for he sharng of projec-specfc rsk (.e., z below) across enrepreneurs (e.g., hrough a muual nsurance conrac coverng all enrepreneurs), as all he resuls would reman qualavely he same. The model assumes ha here s no rsk sharng beween enrepreneurs and households, because he only conrac ha lenders offer borrowers s a sandard deb conrac. Gven our desre o sudy banks, hs assumpon s realsc. 5 The assumpon of zero endowmen s manly o smplfy he analyss. Inroducng paral nernal funds, e.g., wh enrepreneurs own labor ncome, affecs none of he model s conclusons. One poenal problem wh zero nernal funds s ha gves enrepreneurs ncenve o ake excessve rsk (.e., adopng projecs wh a hgh payoff when successful bu possbly a negave ne presen value), bu we rule ou such cases by assumpon. The usual prncpalagen problem beween shareholders and managers does no arse here because enrepreneurs are he owners-operaors.

14 2 A + s he sochasc aggregae echnology level n perod +, and z s s dosyncrac producvy level, drawn a me when he owner s born. z s assumed o be..d. across frms and me, wh bounded suppor, and ndependen of A +, wh E(z ) =. Condonal on z, he frm borrows o buy capal from he households a he end of perod. In keepng wh our desre o sudy bankng operaon n deal, we assume ha lenders offer borrowers a sandard deb conrac. (We dscuss he borrowng process, frs under symmerc nformaon and hen under asymmerc nformaon, n he nex several sub-secons.) The aggregae echnology level ( A ) s revealed a he sar of perod +, and deermnes A (= z A ). Bu snce A s unknown when he capal purchase decson s made, here s a rsk nvolved for boh he borrower and he lender. Condonal on A and he precommed level of capal npu, he frm hres he opmal amoun of labor a me (+) s gong wage, and produces he fnal good. Enrepreneurs frs pay her workers, hen pay he agreed-upon neres o households (as well as reurnng he loan prncpal, he value of he sock of capal rened for producon), and hen consume all he oupu lef over. If a bad realzaon of A leaves an enrepreneur unable o cover he gross neres on hs borrowed funds, he declares bankrupcy. The lenders (households) seze all of he asses and oupu of he frm lef over afer payng he workers, whch wll be shown o be less han wha he lenders are owed and expec o consume. Enrepreneurs are lef wh zero consumpon, less han wha hey expeced as well. The rsk o boh borrowers and lenders s drven by he aggregae uncerany of he sochasc echnology, A. D. Equlbrum wh Symmerc Informaon In order o make an mporan pon abou he SNA93 mehod for measurng nomnal bank oupu, we frs consder a case where households can coslessly observe all frms dosyncrac producvy, z. We assume ha he producon funcon of each poenal projec has consan reurns o scale (CRS): Y A z K N α α = ( ) ( ). (9) Gven CRS producon, households wll wan o lend all her capal only o he enrepreneur wh he hghes level of z or, o paraphrase n marke erms, he hghes-producvy enrepreneur wll be wllng and able o oubd all he ohers and hre all he capal n he economy. (We assume ha he wll ac compevely, akng prces as gven, raher han acng as a monopols or as a monopsons.)

15 3 z = max z. 6 Then he economy s aggregae producon funcon wll be (ha of z s): Defne { } Y = A zk N. α The enrepreneur wh he z level of producvy wll hre capal a me o maxmze α ( α max + ) + ( ),0 + δ EU A zk N R K W N. (0) The expresson n (0) ndcaes ha he enrepreneur ges eher he resdual profs from hs projec, f he s no bankrup, or zero, f he has o declare bankrupcy. The labor choce wll be based on he realzaon of A + and he marke wage, and wll be α ( α ) za + N+ = K W +. () Producon, capal and labor paymens, and consumpon wll ake place as oulned n he prevous sub-secon. Noe ha producng a he hghes avalable level of z does no mean ha bankrupcy wll never ake place, or even ha wll necessarly be less lkely. Ceers parbus, a hgher expeced producvy of H capal rases he expeced reurn R, bu does no elmnae he possbly of bankrupcy condonal on ha hgher requred reurn. 7 Thus, deb wll connue o carry a rsk premum relave o he rsk-free rae. The naonal ncome accouns deny n hs economy s Y = C + C + I. H E E. The Bank ha Does Nohng There s no bank n he economy summarzed n he prevous sub-secon, nor any need for one. H Households lend drecly o frms, a a requred rae of reurn R. Suppose, however, a bank s formed, smply as an accounng devce. Households ransfer her capal sock o banks, and n reurn own bank equy. The bank rens he capal o he sngle mos producve frm a he compeve marke prce. Snce households see hrough he vel of he bank o he underlyng asses he bank holds rsky H deb ssued by he enrepreneur hey wll demand he same reurn (.e., R ) on bank equy as hey dd on 6 The maxmum s fne because we have assumed he z has a bounded suppor. 7 Le us assume, as n Secon II below, ha a connuum of enrepreneurs s born every perod, so ha we are guaraneed ha z s always he upper end of he suppor of z. Then all ha happens by choosng he mos producve frm every perod s ha he mean level of echnology s hgher han f we chose any oher frm, e.g., he average frm. Bu nohng n our dervaons urns on he mean of A; s smply a scalng facor for he overall sze of he economy, whch s rrelevan for consderng he probably of bankrupcy.

16 4 he deb n he economy whou a bank. Snce he bank acs compevely (and hus makes zero prof), H wll lend he funds a margnal cos (expeced reurn of R, hence a conracual neres rae of R ) o he frm, whch wll hus face he same cos of capal as before. However, applyng he SNA93 calculaon for FISIM o hs model economy, he value added of bank, effeced va book enres of he capal ransfer (he only sgn of he bank s exsence here), would be H f ( R R ). K s he value of bank asses as well as he economy-wde capal sock. Thus, by usng he rsk-free rae as he opporuny cos of funds nsead of he correc rsk-adjused neres rae, he curren procedure arbues posve value added o he bank ha n fac produces nohng. 8 A he same me, from he expendure sde he value of naonal ncome wll be unchanged sll equal o Y because he bank oupu (f any) s used as an nermedae npu of servce by frms producng K he fnal good. 9 Bu ndusry value addeds are msmeasured: for a gven aggregae oupu, he producve secor has o have lower value added, o offse he value added ncorrecly arbued o he bankng ndusry. Clearly, he producon secor s rue value added s all of Y H f ( ) Y R R K., bu wll be measured, ncorrecly, as: Thus, he general lesson from hs example s ha whenever banks make loans ha ncur aggregae rsk (.e., rsk ha canno be dversfed away), hen he curren naonal accounng approach arbues oo much of aggregae value added o he bankng ndusry, and oo lle o he frms ha borrow from banks. Ths basc nsgh carres over o he more realsc cases below, where banks do n fac produce real servces. We shall also argue laer ha our smplfyng assumpon of a fully equy-funded bank s compleely unessenal o he resul. The reason s ha n our seng, he heorem of Modglan and Mller (MM, 958) apples o banks. The MM heorem proves ha a frm s cos of capal s ndependen of s capal srucure. Thus, he bank ha does nohng can fnance self by ssung deb (akng deposs) as well as equy, whou changng he prevous resul n he slghes, eher qualavely or quanavely FISIM also mpues a second pece of bank oupu deposor servces. Bu snce bank deposs are zero n our model, FISIM would correcly calculae hs componen of oupu o be zero. 9 Msmeasurng bankng oupu would dsor GDP f banks oupu s used as a fnal good (e.g., lendng and deposory servces o consumers or, perhaps more mporanly, ne expors). 20 Assumng here s no depos nsurance. See Wang (2003a) for a full reamen of banks capal srucure wh rsk

17 5 Even n more realsc sengs, he lesson n hs sub-secon s drecly relevan for one ssue n he measuremen of bank oupu. Banks buy and passvely hold rsky marke asses, as n he example here. Even hough banks ypcally hold asses wh relavely low rsk, such asses (e.g., hgh-grade corporae bonds) sll offer raes of reurn ha exceed he rsk-free rae, somemes by a nonrval margn. Whenever a bank holds marke secures ha offer an average reurn hgher han he curren reference rae, creaes a cash flow he dfference beween he secures reurn and he safe reurn, mulpled by he marke value of he secures held ha he curren procedure mproperly classfes as bank oupu. II. Asymmerc Informaon and a Fnancal Secor ha Produces Real Servces A. Resolvng Asymmerc Informaon I: Non-Bank Fnancal Insuons Now we assume, more realscally, ha nformaon s n fac asymmerc. Enrepreneurs know her dosyncrac producvy and acual oupu, bu households canno observe hem drecly. In hs case, as we know from Akerlof (970), he fnancal marke wll become less effcen, and may break down alogeher. We nroduce wo new nsuons no our model. The frs s a rang agency. I screens poenal borrowers and monors hose who defaul, o allevae he asymmerc nformaon problems. The oher s a bond marke,.e., a porfolo of corporae deb. The wo combned fulfll he funcon of channelng funds from households o enrepreneurs so ha he laer can nves. Boh nsuons have real-world counerpars, whch wll be mporan when we urn o our model s mplcaons for oupu measuremen. The purpose of nroducng hese wo new nsuons wll become clear n he nex sub-secon when we compare hem wh banks. There we wll show ha a bank can be decomposed no a rang agency plus a porfolo of corporae deb, and he real oupu of banks nformaonal servces s equvalen o he oupu of he agency alone. Thus, makes sense o undersand he wo peces ndvdually, before sudyng he sum of he wo. Undersandng he deermnaon of bond marke neres raes s parcularly mporan when we dscuss measuremen, because we shall argue ha corporae deb wh he same rsk-reurn characerscs as bank loans provdes he approprae rsk-adjused reference rae for measurng bank oupu. We dscuss rang agences frs. These are nsuons wh specalzed echnology for assessng he qualy (.e., producvy) of prospecve projecs, and are also able o assess he value of asses f a frm goes and depos nsurance. Of course, n he real world axes and ransacons coss break he pure rrelevance resul of Modglan-Mller. Bu he basc lesson ha he reference rae mus ake rsk no accoun s unaffeced by hese realsc bu exraneous consderaons.

18 6 bankrup. Thus, hese nsuons are smlar o he rang agences found n he real world, such as Moody s and Sandard and Poor s, whch no only rae new ssues of corporae bonds bu also monor old ssues. The echnology of each rang agency for screenng (S) and monorng (M) s as follows: J J β JA J JA β JA Y = A ( K ) ( N ), J = M or S. (2) We use he superscrp A o denoe prces and oupu of he agency. JA K and JA N are he capal and labor, respecvely, used n he wo acves. S A and dffer when he pace of echnologcal progress M A dffer beween he wo acves. Dfference beween oupu elasces of capal M β and S β means ha neher knd of ask can be accomplshed by smply scalng he producon process of he oher ask. We assume here are many agences n a compeve marke, so he prce of her servces equals he margnal cos of producon. The represenave rang agency solves he value maxmzaon problem below: E 0 { )]}, (3) ( SV R ) [ SA SA MA MA A A 0 0 τ f Y + f Y WN I = τ = SA Y = S SA β SA A ( K ) ( N ) S S β, (4) MA Y = M MA β MA A ( K ) ( N ) M M β MA, and Y 0 = 0, (5) A SA MA N = N + N, and K = K + K, (6) A SA MA A A A K = K ( + δ ) + I. (7) In (3), SA Y and MA Y are he rang agency s respecve oupu of screenng and monorng servces. S f and f are he correspondng prces (mnemonc: fees) and, as assumed, equal o he respecve margnal M cos. W s he real wage rae, and N he agency s oal labor npu. (4) and (5) are he producon A funcons for screenng and monorng, respecvely, wh he npus defned as n (2). Toal labor and capal npus are gven n (6). (7) descrbes he law of moon for he agency s oal capal. problem,.e., The agency s fully equy-funded. Thus, he dscoun rae for he agency s value maxmzaon SV H R, analogous o SV R ( SV sandng for servces), s exacly s shareholders requred rae of reurn on equy. R + n equaon (6), hus s deermned by he sysemac rsk of he agency s enre cash flow. Accordng o he prcng equaon (6) (and equvalenly (4)), SV R equals,

19 7 The denomnaor ( A K R R m ( δ ) f Y + f Y W N + K SA SA MA MA A A SV f = + cov +, A K +. (8) ) n he covarance s he agency s capal used n producon a me +, funded by s shareholders a me. The numeraor s he ex pos reurn on ha capal, conssng of s operang profs (revenue mnus labor coss), plus he reurn of he deprecaed capal len by he sockholders a me. Even hough he agency s pad conemporaneously for s servces, he fac ha mus choose s capal sock a perod n advance creaes uncerany abou he cash flow accrung o he owners of s capal. Ths uncerany arses fundamenally because he demand for screenng and monorng s random, drven by he sochasc process for aggregae echnology, A +. Thus, he mplc renal rae of physcal capal n SV perod for hs agency s ( R + δ ), 22 where SV R wll generally dffer from he rsk-free rae. Snce a rang agency s of lle use unless one can borrow on he bass of a favourable rang, we assume ha a frm can ssue bonds of he approprae neres rae n he bond marke once s raed. Tha s, once an agency fnshes screenng a frm s projec, ssues a cerfcae ha reveals he projec s ype (.e., z ). Armed wh hs cerfcae, frms sell bonds o households n he marke, offerng conracual raes of neres R ha vary accordng o each frm s rsk rang. R depends on households requred rae of reurn on rsky deb, bu R s no he requred reurn per se. The wo dffer by he defaul premum, as dscussed n Secon I.B. (Deermnng he approprae neres rae o charge an enrepreneur of ype s a complex calculaon, n par because he probably of defaul s endogenous o he neres rae charged. We hus defer hs dervaon o he Appendx.) There s an addonal complcaon: Snce enrepreneurs are born whou wealh, hey are unable o pay her screenng fees up fron. Insead, hey mus borrow he fee from he bond marke, n addon o he capal hey plan o use for producon nex perod, and dash back o he rang agency whn he perod o pay he fee hey owe. In he second perod, hey mus pay he bondholders a gross reurn on he borrowed producve capal, plus a same reurn on he fee ha was borrowed o pay he agency. 2 2 The payoff o he shareholder depends, of course, on he margnal produc of capal. The assumpon of consanreurns, Cobb-Douglas producon funcons allows us o express he resul n erms of he more nuve average reurn o capal. Noe ha he capal reurn n equaon (8) s acually an average of he margnal revenue producs of capal n screenng and monorng, wh he weghs beng he share of capal devoed o each acvy. 22 Recall ha all R s are gross neres raes, so he ne neres rae r = R.

20 8 In he second perod of hs lfe, afer hs producvy s deermned by he realzaon of A +, an enrepreneur may approach hs bondholders and nform hem ha hs projec was unproducve, and he s unable o repay hs deb wh neres. The households canno assess he valdy of hs clam drecly. Insead, hey mus engage he servces of he rang agency o value he frm (s oupu plus resdual capal). The agency charges a fee equal o s margnal cos, as deermned by he maxmzaon problem n equaons (3) o (7). We assume ha he agency can assess he value of he frm perfecly. Whenever a rang agency s servces are engaged, he bondholders ge o keep he enre value of he projec, afer payng he agency s monorng fee. 23 The enrepreneur ges nohng. Under hese crcumsances, he enrepreneur always ells he ruh, and only clams o be bankrup when ha s, n fac, he case. Noe ha n hs asymmerc-nformaon envronmen, enrepreneurs requre addonal npus of real fnancal servces from he agences o oban capal. The producon funcon for gross oupu for a frm of ype s sll gven by (9). Bu now enrepreneurs have wo addonal coss. In he frs perod, when hey borrow capal, hey mus buy ceran uns of cerfcaon servces. The amoun of screenng vares wh he sze of he projec (see he Appendx for a dealed dscusson of he sze-dependence of hese nformaon S processng coss). A projec of sze needs ( K ) uns of screenng servces. Then, n he second K M M perod, a frm s requred o pay for Z υ uns of monorng servces, where Z M equals f he frm ) + ( K υ + defauls and 0 oherwse. Funcons υ S (.) and υ M (.) deermne how many uns of screenng and, possbly, monorng are needed for a projec of sze K. Eher υ S (.) or υ M (.) s srcly convex, and hs leads frms o effecvely have dmnshng reurns o scale. 24 Thus, s no longer opmal o pu all he capal a he mos producve frm, and he equlbrum nvolves producon by a srcly posve measure of frms. Gven hese wo addonal coss, frm producng n perod + maxmzes α ( α S M M max + ( ) ( ) ( ) ( ) ( ),0 ) δ υ υ EU A z K N R K W N K Z K. R + s he conracual neres rae approprae for a projec of ype, he analogue o he full-nformaon conracual rae for he hghes-producvy projec n equaon (0). As n he suaon wh perfec nformaon, eher he enrepreneur ges posve resdual profs, or he declares bankrupcy and ges nohng. 23 We assume ha a projec always has a gross reurn large enough o pay he fee. Ths assumpon seems reasonable even Enron s bankrupcy value was hgh enough o pay smlar coss (amounng o over a bllon dollars). 24 A convex cos of capal s needed o oban fne opmal projec scale; we dscuss hs ssue furher n Appendx.

21 9 K Defne R as he ex pos gross reurn on capal for he projec. I s he projec s oal oupu ne of labor cos and deprecaon, ( ) * K R + K+ = Y+ W+ N+ δ K, where N + * s he opmal quany of L labor. Thus, he ex ane requred rae of reurn on he bonds ssued by frm, R, s he requred reurn mpled by he asse-prcng equaon ( + υ ( ))( ) + υ ( ) K+ + f υ ( K+ ) ( ) S S M K M M M R K f K Z R K f K Z E m+ =. (9) S S L So, as usual, R depends on he condonal covarance beween he cash flow and he sochasc dscoun facor. The expresson n he numeraor of he fracon s he sae-conngen payoff o bondholders. If he realzaon of echnology (A + ) s suffcenly favourable, hen he projec wll no defaul (.e., Z M = 0), and ( ) S S he bondholders wll receve he conracual neres promsed by he bond R+ K+ + f υ ( K+ ). Oherwse, f he realzaon of echnology s bad enough, he frm wll have o declare bankrupcy, and bondholders wll receve he full value of he frm ne of he monorng cos R K M M ( K f υ K ) The conraced neres rae on he bond ssued by a projec ( R ) depends on s ex ane requred rae of L reurn R, whch n urn depends on he rsk characerscs of ha projec. For deals, see he Appendx. The denomnaor of (9) s he oal amoun of resources he frm borrows from households. K s S S he capal used for producon, whle f ( K ) υ + s he screenng fee. As dscussed above, enrepreneurs need o borrow o pay he screenng fees because hey have no endowmens n he frs perod of her lves. In general, households wll hold a porfolo of bonds, no jus one. For comparson n he nex subsecon wh he case of a bank, wll be useful o derve he requred reurn on hs porfolo. Snce each bond reurn mus sasfy (9), we can wre he reurn o he porfolo as a weghed average of he ndvdual reurns. Then, for a large porfolo of nfnesmal projecs, he requred rae of reurn s se by he equaon ( + υ ( ))( ) + υ ( ) K + + f υ ( K+ ) ( ) S S M K M M M R K f K Z R K f K Z K : 0 E + > m+ S S =, (20) K : + > 0

22 20 where he negral s aken over all frms whose bonds are n he nvesor s porfolo. 25 B. Resolvng Asymmerc Informaon II: Banks ha Produce Real Servces We are fnally ready o dscuss bank operaons. Now he bankng secor performs real servces, unlke he accounng devce n sub-secon I.E. We assume ha banks assess he cred rsk of prospecve borrowers, lend hem capal and, f a borrower clams o be unable o repay, banks nvesgae, lqudae he asses, and keep he proceeds. Tha s, n our model and n he world banks perform he funcons of rang agences and he bond marke under one roof. As mporanly, especally for measuremen purpose, noe ha banks, rang agences, and he bond marke all co-exs, boh n he model and n realy. Our banks are compleely equy-funded. 26 They ssue socks n exchange for households capal. Par of he capal s used o generae screenng and monorng servces, wh exacly he same echnology as n (2). The res of he capal s len o qualfed enrepreneurs. A me, a bank mus make an ex ane decson o spl s oal avalable capal no n-house capal (used by he bank for producng servces n perod +, denoed K B ) and loanable capal (len o enrepreneurs, used o produce he fnal good n perod +). Snce he bankng secor s compeve, banks prce her package of servces a margnal cos. The exac saemen of he bank s value maxmzaon problem s edous and yelds lle addonal nsgh, so oo s deferred o he Appendx. In summary, enrepreneurs are shown o be ndfferen beween approachng he bank for funds or gong o a rang agency and hen o he bond marke, 27 gven ha banks have he same screenng and monorng echnology as he agency (producon funcons (4) and (5)). 25 To llusrae he dervaon, consder an example of dscree projecs. Suppose a lender holds bonds from N frms. + ousde he Equaon (9) holds for every frm, and can be rearranged by pullng he denomnaor S S K ( ) + f υ K+ expecaons sgn, snce s known a me, Then mulply each frm s equaon (9) by he frm s share n he S S K+ + f υ ( K+ ) aggregae resources borrowed,.e., N S K f υ ( K ) = N S clearly sums up o, whle K+ + f υ ( K+ ) ( + υ ( ))( ) + ( υ ( = )) Consequenly, we fnd ha E m+ N S [ K+ + f υ ( K+ ) ], and add up he N resulng equaons. The rgh-hand sde [ ] [ ] becomes he common denomnaor for he lef-hand sde. = N S S M K M M M R K f K Z R K f K Z =. Tha = s, he weghed average of he N frms condons equals he sum of he numeraors over he sum of he denomnaors. 26 Agan, our assumpon ha he bank does no ssue deb s rrelevan for our resuls. See he dscusson of he Modglan-Mller (958) heorem a he end of Secon I.E. 27 We assume ha n equlbrum boh he bankng secor and agences/he bond marke ge he same qualy of applcans on average. In equlbrum, enrepreneurs wll be ndfferen abou whch roue hey should ake o oban