A Model of Endogenous Loan Quality and the Collapse of the Shadow Banking System

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Model of Endogenous Loan Quality and the Collapse of the Shadow Banking System Francesco Ferrante 2015-021 Please cite this paper as: Francesco Ferrante (2015). “A Model of Endogenous Loan Quality and the Collapse of the Shadow Banking System,” Finance and Economics Discussion Series 2015-021. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.021. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

A Model of Endogenous Loan Quality and the Collapse of the Shadow Banking System∗ Francesco Ferrante † Federal Reserve Board First version August 2013 This version: March 2015 Abstract I develop a macroeconomic model with a financial sector, in which banks can finance risky projects (loans) and can affect their quality by exerting a costly screening effort. Informational frictions regarding the observability of loan characteristics limit the amount of external funds thatbankscanraise. InthisframeworkIconsidertwopossibletypesoffinancialintermediation, traditionalbanking(TB)andshadowbanking(SB),differinginthelevelofdiversificationacross projects. In particular, shadow banks, by pooling different loans, improve on the diversification of their idiosyncratic risk and increase the marketability of their assets. Due to their ability to pledgealargershareofthereturnontheirprojects,shadowbankswillhaveahigherendogenous leveragecomparedtotraditionalbanks,despitechoosingalowerscreeninglevel. Asaresult,on the one hand, the introduction of SB will imply a higher amount of capital intermediated. On the other hand it will make the economy more fragile via three channels. First, by being highly leveraged and more exposed to risky projects, shadow banks will amplify exogenous negative shocks. Second, during a recession, the quality of projects intermediated by shadow banks will endogenously deteriorate even further, causing a slower recovery of the financial sector. A final source of instability is that the SB-system will be vulnerable to runs. When a run occurs, shadow banks will have to sell their assets to traditional banks, and this fire sale, because of the limited leverage capacity of the TB-system, will depress asset prices, making the run selffulfillingandnegativelyaffectinginvestment. InthisframeworkIstudyhowcentralbankcredit intermediation helps reduce the impact of a crisis and the likelihood of a run. Keywords: Financial Frictions, Shadow Banking, Bank Runs, Unconventional Monetary Policy JEL classification: E44, E58, G23, G24 ∗The views expressed in this paper are those of the author and do not necessarily reflect the views of the Federal ReserveBoardortheFederalReserveSystem. IamindebtedtoMarkGertler,JaroslavBorovi˘ckaandDouglasGale for their guidance in the preparation of this paper. I would also like to thank Andrea Prestipino for the very useful discussions, and John Roberts and NYU seminar partecipants for their comments †Federal Reserve Board, francesco.ferrante@frb.gov

1 Introduction The years leading to the 2007-2009 (cid:133)nancial turmoil in the United States were characterized by the development of a new set of (cid:133)nancial institutions that formed the so-called "shadow banking" (SB) system. A precise and all-encompassing de(cid:133)nition of shadow banking is di¢ cult to obtain, but it can broadly be characterizedasanetworkof(cid:133)nancialsubjectsthatreplicatedthecreditintermediationprocessbydecomposing it in di⁄erent activities while heavily relying on securitization and sophisticated (cid:133)nancial products.1 These entities included, for example, broker-dealers, mortgage (cid:133)nance (cid:133)rms, asset-backed commercial paper (ABCP) conduits and money market mutual funds (MMMF), and provided an alternative chain of intermediation, parallel to the "traditional banking" (TB) o⁄ered by conventional banks. Simplifying considerably the complex structure of the shadow banking system, we can provide an example of the basic steps of the intermediation process as follows. Loans originated by non-bank lenders were pooled through securitization by a loan warehouse vehicle, for example a special purpose vehicle (SPV) supported by a broker-dealer.2 Thebroker-dealerfurthercombinedsuchloansintostructuredassetbackedsecurities(ABS) that were funded by issuing risk-free short term debt like commercial paper. MMMFs purchased the ABCP and (cid:133)nanced themselves with money-like securities with stable net-asset value. As a result, we can think of theaggregateshadowbankingsystemasengaginginthesamematurityandliquiditytransformationtypical of traditional banks, converting illiquid loans into demandable instruments. If, on the one hand, this (cid:133)nancial system increased the amount of credit available to borrowers, on the otherhanditprovedtobeinherentlymorefragilebecauseofaseriesofrisksa⁄ectingitsbusinessmodel. In this paper, I will focus on the following factors that increased the instability of the shadow banking system: high leverage, moral hazard in selecting the riskiness of loans, and exposure to bank runs. In addition, this paper will show how these channels can be ampli(cid:133)ed by the interaction with traditional banks, which are characterized by a lower leverage capacity. Shadow banking, became more and more important in the years leading to the crisis. By 2007 the SBsystem was intermediating a volume of credit comparable to that provided by traditional banks.3 Figure (1) provides an approximate measure of the SB-system, based on Pozsar et al (2010), considering all the liabilities linked to "non-traditional" intermediation (like ABS, commercial paper, repos, MMMF shares).4 Di⁄erent explanations have been given for this trend, including regulatory arbitrage or an increasing demand for riskless assets. Nonetheless, an important factor behind the fast growth of shadow banking can be clearly identi(cid:133)ed in (cid:133)nancial innovation. In particular, the securitization process, based on combining di⁄erent loans into diversi(cid:133)ed portfolios, increased the marketability of banks(cid:146)assets. As a result, by broadening the array of securities available for lenders, shadow banks were able to create a new stream of outside funding. Also for this reason, they had a higher leverage capacity than traditional banks. Even if the SB-system helped to expand credit and lower borrowing costs in the period preceding the (cid:133)nancial crisis, it also played a crucial role in making the whole banking sector more fragile and in causing 1A survey of the di⁄erent de(cid:133)nitions and measurements of the Shadow Banking system can be found, for example, in the IMF GlobalFinancialStability Report ofOctober2014 2Forexample mortgage companies or(cid:133)nance companies. 3Privatesectorestimatesofsizevaryfrom$10trillionto$30trillion(seeDeloitte,2012). A"ShadowBankIndex"developed by Deloitte put its size in the U.S.at $20 trillion in 2007. Assets intermediated by commercial banks in that period were approximately $10 trillion. 4The details on the data used can be found in Pozsar et al(2012),page8. Compared to Pozsar et al. I do not include GSE liabilities, since I want to focus on the part of the SB-system that did not have government sponsorship. In addition, these entitieswentundergovernmentconservatorshipin2008. ByalsoaddingFreddieMacandFannieMaethesizeoftheSB-system would be even larger. 1

Figure 1: Liabilities of the Shadow Banking System vs liabilities of Traditional Banking System, as constructed in Pozsar et al (2012). Data from Flow of Funds Accounts of the United States as of 2013:Q2 the "Great Recession". In fact, the rise in defaults among subprime lenders triggered runs in di⁄erent shadow banking markets, causing the collapse of most of these unregulated institutions and also a⁄ecting thetraditionalbankingsystem.5 Itseemsthereforecrucialtounderstandthelinkagesbetweentheserelatively new (cid:133)nancial structures and the real economy. In this paper, I will model how the introduction of an additional banking sector, with a higher level of "(cid:133)nancial sophistication", can make the economy more unstable, despite increasing credit availability; and how the interaction with the traditional banking system can create patterns similar to the ones observed in the (cid:133)nancial crisis. The novel contribution of this work is that it captures relevant features of the Great Recession by combining shadow banks, endogenous loan quality, and bank runs in a macroeconomic model. In particular, I consider (cid:133)nancial intermediaries that are able to originate risky projects and screen their expected quality by exerting costly e⁄ort. However, asymmetric information, on the characteristics of the loans that banks fund, limit the amount that they can raise from outside investors. This agency problem a⁄ects di⁄erently the two types of intermediaries that I model in this paper: traditional banks (TB) and shadow banks (SB). The former are able to invest only in the projects undertaken in a single geographical location, that I call "island", whose ex-post idiosyncratic performance is only observable by the originating bank. This implies a constraint on external funds linked to the worst possible idiosyncratic realization of the assets of TBs. On the other hand, intermediaries involved in shadow banking are assumed to have a higher level of (cid:133)nancial innovation, and are thus able to invest in a pool of loans originated from di⁄erent borrowers, improving on the ex-post observability of project outcomes. As a consequence, they can partially diversify 5See Covitz et al. (2013) for runs on ABCP programs orGorton and Metrick (2010) forruns on repos. 2

the idiosyncratic risk and pledge a larger share of the return on their projects to outside lenders, by writing contracts contingent on the realization of their pool of assets. In this way, shadow banks endogenously achieve a higher leverage than traditional banks, so that the presence of the SB-system helps to expand credit and to increase investments and output. The (cid:133)nancial sophistication of shadow banks, however, can be costly for the aggregate economy because of the higher fragility of the (cid:133)nancial sector. In this model such instability comes from three sources: higher leverage, lower quality of loans, and the possibility of bank runs. First of all, the higher aggregate leverage of the banking system will amplify negative exogenous shocks, through a mechanism similar to the (cid:133)nancial accelerator of Bernanke, Gertler and Gilchrist (1999) and Gertler and Karadi (2011). In addition, a novelty of this model is the interaction between asset quality and leverage, as a speci(cid:133)c feature of shadow banking: because the higher leverage is obtained by promising a higher payment to investors in case projects are successful, shadow banks have a lower incentive to screen projects, and will originate riskier loans. We can think of this as a stylized representation of the parallel boom of shadow banking and subprime lending. Themoralhazardproblemthatlinkso⁄-balance-sheet(cid:133)nanceandbankrisk-takingrepresentsoneimportant aspect of securitization, as shown in Pennacchi (1988) and Fender and Mitchell (2009). In addition to being theoretically signi(cid:133)cant, this characterization of shadow banking has received wide empirical support in recent years. For example, Keys et al. (2010) (cid:133)nd that securitized loans experienced higher default rates thansimilarmortgagesthatwereinsteadretainedbythebank. DruckerandPuri(2009)showthat(cid:133)nancial intermediaries usually sell riskier loans and provide covenants in order to reduce the problems arising from information asymmetry. Su(cid:133)(2007) shows that when borrowing (cid:133)rms require more intense due diligence, lenders retain a larger fraction of syndicated loans. In this paper, I present a novel mechanism that shows the implications of this agency problem for the cyclicality of bank asset quality. During recessions, as the value of shadow bank net worth declines, so does their"skininthegame". Asaresult,thequalityoftheprojectsthattheSB-systemcancrediblyintermediate will endogenously deteriorate even further, causing a slower recovery of this (cid:133)nancial sector. On the other hand, such a mechanism will be absent for traditional banks, since their funding capacity does not depend on their screening e⁄ort. The evolution of asset quality will also translate in endogenous volatility in the cross-sectional equity returns of (cid:133)nancial intermediaries. In fact, during a crisis the volatility in the returns of (cid:133)nancial intermediaries, and in particularof shadow banks, willrise considerably, a type of countercyclicality that has received great attention recently.6 Another important source of macroeconomic instability that a setup with two types of (cid:133)nancial intermediaries allows me to consider is the eventuality of a run on shadow banks. In particular, because of their highleverageandthetypeofsecuritiestheyissue,shadowbankswillbeexposedtobankruns. Ontheother hand,thelowleverageandtheincentiveconstraintontheirliabilitiesrulesoutthispossibilityfortraditional banks. As a result, when a run occurs, shadow banks will have to sell their assets to traditional banks in order to repay creditors, and this (cid:133)re sale, because of the limited leverage capacity of the TB-system, will depress asset prices and negatively a⁄ect investment. If prices drop enough, the run becomes self-ful(cid:133)lling and most of the shadow banks are liquidated, causing a prolonged recession and a slow recovery of the (cid:133)nancial system. 6See,forexample,Christiano,Motto,and Rostagno (2014);Ferreira (2014);Christiano and Ikeda (2014). 3

A series of run episodes were in fact at the heart of the collapse of the shadow banking system.For example,Covitz,LiangandSuarez(2013)showhowmostassetbackedcommercialpaper(ABCP)programs faced a run in 2007. As (cid:133)gure (2) shows, since then, the volume of outstanding ABCP, one of the main funding instruments of shadow banks, plummeted from $1.2 trillion in 2007 to about $250 billion in 2013. Figure 2: ABCP outstanding; data from the Federal Reserve Bank of Saint Louis Thereintermediationofassetsfromshadowbankstotraditionalbanks,akeyelementbehindthenegative ampli(cid:133)cation mechanisms in this model, was also an event that characterized the height of the (cid:133)nancial turmoil. As documented by He, Khang and Krishnamurthy (2012), during the crisis hedge funds and broker-dealers reduced their holdings of securitized assets by about $800 billion, whereas traditional banks and the government increased their holdings by $550 billion and $350 billion respectively. As a result, debt issued by the traditional banking sector increased dramatically and the related leverage increased. Such reintermediation is also visible in (cid:133)gure (1), where we see that traditional banks liabilities jumped in 2008 and increased steadily since then. Finally, I use this setup to analyze the implications of government policies similar to the ones that the Federal Reserve implemented in the aftermath of the crisis. With numerical experiments, I show how direct credit intermediation provided by an unconstrained government can help counter the negative feedback mechanisms described above. In particular, the presence of a central bank that helps to intermediate the assets of shadow banks in the case of a systemic run, can make the run less likely or even impossible. The notion of shadow banking encompasses a wide universe of (cid:133)nancial activities and economic forces, not all of which are captured in this paper. For example, I do not model any type of regulatory arbitrage motive behind shadow banking, or the sponsorship relationship between traditional banks and structured investment vehicles (SIVs) or MMMFs.7 Also, I do not introduce any type of deposit insurance or capital 7Forregulatoryarbitragesee,forexample,Ordonez(2013). Forspeci(cid:133)cmodelsonSIVsorMMMFssee,forexample,Segura 4

requirement for traditional banks, although the friction that I consider has the similar rationale of limiting the leverage of traditional banks in order to guarantee that they are always able to repay depositors. The modeling of all these phenomena is beyond the scope of this paper, whose focus is the interaction between (cid:133)nancial innovation, loan quality, shadow banking and macroeconomic instability. 1.1 Related Literature This paper draws from di⁄erent strands of literature related to agency problems in banking and their implications for the macroeconomy. Asregardsthemicrofoundationforthelimiton(cid:133)nancialintermediariesleverage,myframeworkcombines a "worst-case-scenario" constraint, similar to the one used in Carlstrom and Samolyk (1995) and Bernanke and Gertler (1987), with a moral hazard problem on monitored (cid:133)nance, like the one modeled by Pennacchi (1988) and Gorton and Pennacchi (1995). In particular, even if I do not model the details of securitization, the contract between shadow banks and outside investors that I use is similar to the loan sales contract described in these two papers. An informational friction similar to the one used in this paper for shadow banks is used by Christiano andIkeda(2014): alsointheirmodelbankscana⁄ecttheprobabilityofsuccessoftheirprojectsbyexerting costly unobservable e⁄ort. However, an important di⁄erence comes from the fact that in their framework the screening cost is not proportional to the amount of projects funded, so that no endogenous leverage constraint arises from their agency problem. In such a framework the focus of their paper is rather to study the possibility of improving welfare by introducing exogenous leverage restrictions. Another paper in which the recovery of the (cid:133)nancial sector is a⁄ected by an endogenous deterioration in assetqualityisBigio(2012). InBigio(2012),thisresultstemsfromanadverseselectionproblembetweenthe bank and the borrower, in which the latter provides lower quality collateral as the volume of intermediation shrinks. Inaddition, thismechanismisnottheresultof(cid:133)nancialinnovation, anda⁄ectsthebankingsystem as a whole, rather than being a speci(cid:133)c feature of the shadow banking system. My interpretation of shadow banking as a process to improve diversi(cid:133)cation is similar to the one used by Gennaioli et al. (2013). In their paper, banks can improve on their funding constraint by pooling di⁄erent assets in order to be able to pledge the worst aggregate realization on their loans, rather than the worst idiosyncraticone. Intheirframework,shadowbankingisdrivenbythedemandforrisklessassetsbyin(cid:133)nitely risk-averse depositors, and it becomes detrimental only when investors neglect tail risk. As mentioned in the introduction, other papers have modeled shadow banking as stemming from regulatory arbitrage, like Plantin (2012) or Ordonez (2013). Recently many macroeconomic models with a (cid:133)nancial sector have been developed (e.g. Gertler and Karadi (2011), Brunnermeier and Sannikov (2011), He and Krishnamurthy (2012) ), but there have been only a few attempts to include shadow banks and their exposure to runs in a general-equilibrium setting. For example, Meeks et al. (2013) introduce in the framework of Gertler and Karadi (2011) a SB sector that funds itself from traditional banks, and assume that traditional banks have a weaker friction when investing in shadow bank liabilities. However, in such a setup, there is no role for loan quality and a run on the SB-system, started byoutsideinvestors, is notpossible. Faia(2012)studiesthee⁄ectof asecondarymarket forloansinaDSGEmodelwithamoralhazardproblemsimilartotheoneIconsider,butwhereloanquality is determined exogenously and only one type of intermediary is present. (2014) orParlatore (2013). 5

As regards the modeling of bank runs in general equilibrium, my approach is similar to the one used by Gertler and Kiyotaki (2014). An important di⁄erence is that in their paper, when a run occurs assets are directly acquired by households that incur a real cost to manage capital. It is this cost that determines the liquidation price and that makes a run possible. On the other hand, in my setup a run occurs because of the (cid:133)re sale of assets from shadow banks to traditional banks with a lower leverage capacity. Other macroeconomic models of bank runs are Martin, Skeie, and Von Thadden (2012) and Angeloni and Faia (2013). The rest of the paper is organized as follows. Section 2 describes the asymmetric information problem in the (cid:133)nancial sector and the optimal contract for a (cid:133)nancial intermediary operating as a traditional bank or as a shadow bank. Section 3 presents the baseline model where both traditional banks and shadow banks are present. Section 4 explains how a run on the shadow banking system is possible in this model. Section 5 shows a (cid:133)rst set of numerical exercises with crisis experiments and run experiments. Section 6 introduces governmentinterventionandstudiesitsinteractionwith(cid:133)nancialcrisesandthepossibilityofarun. Finally, Section 7 concludes. 2 Risky Projects and Financial Intermediaries I begin by describing the agency problems a⁄ecting the two types of (cid:133)nancial intermediaries present in this framework, and by solving the related optimal contracts. I then proceed to embed the (cid:133)nancial system so characterized in a medium-scale macroeconomic model. One of the distinguishing features of banks in this model is that they are the only type of agent able to invest in risky projects, by (cid:133)nancing capital purchases of productive (cid:133)rms. In particular, I assume that therearetwo"regions",eachwithacontinuumof(cid:133)rmslocatedonacontinuumofislands.8 Oneveryisland, (cid:133)rms can invest in risky projects k , or "raw capital", that will be employed in a constant-return-to-scale t production technology at time t+1. Capital is risky because it can turn into (cid:18) k units of productive capital next period if the project H t succeeds or (cid:18) k if the project fails, with (cid:18) >(cid:18) . Projects on a speci(cid:133)c island will be perfectly correlated, L t H L so that they either all fail or all succeed. However, I assume that the probability of success p di⁄ers across thetworegions. Inparticularthetworegionswillbeperfectlynegativelycorrelated,sothateveryperiodone region will be "good", whereas the other one will be "bad". The di⁄erence between a region that turns out to be good and one that instead is bad is in the probability of success of loans pG and pB; where pG >pB. t t (cid:18) w.p. pG (cid:18) w.p. pB Good H Bad H t (cid:0)!( (cid:18) L w.p. (1 (cid:0) pG) (cid:0)!( (cid:18) L w.p. (1 (cid:0) pB t ) Therefore, the proportion of islands with successful projects will be pG in the good region and pB in the t bad one. In addition to assuming a higher probability of default in the bad region, I will also allow for iid disturbances to pB in order to capture, in a stylized way, a "subprime shock" that only a⁄ects the return on t lower quality loans. De(cid:133)ne the average realization of a project, conditional on the type of region as (cid:22)(cid:18) j for j =G;B, where (cid:22)(cid:18) G =pG(cid:18) +(1 pG)(cid:18) and (cid:22)(cid:18) B =pB(cid:18) +(1 pB)(cid:18) (1) H (cid:0) L t t H (cid:0) t L 8An alternative interpretation is that of two "sectors". What is going to be important in the characterization of the setup is just the presence ofa double layerofrandomness in the outcome structure ofprojects. 6

It is important to stress that (cid:133)nancial intermediaries are going to (cid:133)nance projects in an island in a given region at time t, without knowing whether that region will be good or bad at time t+1, and whether projects in a speci(cid:133)c island will be successful or not. However, banks can exert e⁄ort e in order to increase t the probability (cid:25) (e ) of selecting a loan in a region that will be good next period. For simplicity I assume t t thatthisprobabilityislinearine⁄ort,accordingto(cid:25) =e ,sothatwecanreferto(cid:25) alsoasscreeninglevel. t t t Imporantly, e⁄ort is costly, since it entails a non-percuniary convex cost c(e ) = c((cid:25) ), per unit of capital t t intermediated. In particular I assume c((cid:25) ) = (cid:28) (cid:25)2+(cid:19)(cid:25) and I allow for (cid:19) to be negative, meaning that t 2 t t there could be some bene(cid:133)ts from screening.9 However I consider calibrations where c ((cid:25) ) > 0 meaning (cid:0) (cid:1) 0 t that it is costly for (cid:133)nancial intermediaries to increase their screening e⁄ort. We can de(cid:133)ne the expected quality of a project with screening intensity (cid:25) as t E [(cid:2) ((cid:25) )]=E (cid:25) (cid:22)(cid:18) G +(1 (cid:25) )(cid:22)(cid:18) B (2) t t+1 t t t (cid:0) t t+1 h i Importantly, banks cannot perfectly diversify across all islands. This limit to diversi(cid:133)cation implies that the assets intermediated by each bank are risky and, as it will be clear below, it allows asymmetric information on bank portfolios to create a relevant agency problem. In addition, as I mentioned in the introduction, I consider two types of intermediaries, traditional banks and shadow banks, di⁄ering in the ability to diversify across islands. In particular I assume that TBs are onlyabletoinvestinprojectsinonesingleisland,thatnextperiodwilldeliver(cid:18) unitsofproductivecapital H in case of success and (cid:18) in case of failure. On the other hand, SBs are able to invest in a "pool" of loans L located in the same region. As a result, the outcome of the shadow bank(cid:146)s portfolio will be either (cid:22)(cid:18) G if the region is good or (cid:22)(cid:18) B if it is bad. t+1 This framework is equivalent to one in which shadow banks purchase loans originated by a set of traditional banks located in the same region.10 As long as these loans are purchased at their market value, implying zero pro(cid:133)ts for the TB on these projects, the structure of the model would be identical. What I am trying to model in this way, is the practice of pooling mortgages that was behind the rapid development of securitization and the shadow banking system. It needs to be noticed that even if SBs are more diversi(cid:133)ed than TBs, they will still be exposed to some idiosyncratic risk. As a supporting piece of evidence for this assumption, we can think of the fact that securitizedproductsmainlycomprisedloansbelongingtoasingleassetclass(credit-cards,mortagesstudent loans etc.), hence being far from perfect diversi(cid:133)cation. The di⁄erent level of diversi(cid:133)cation will play an important role in determining the funding constraints for the two types of (cid:133)nancial intermediaries because of two layers of information asymmetries: 1. Unobservable Outcome (UO): the default realization of loans ((cid:18) ;(cid:18) ), on a given island, is only observable by the originating L H (cid:15) bank the type-realization of a speci(cid:133)c region (good or bad) is public information. (cid:15) 2. Unobservable E⁄ort (UE): the screening level of the loans that a bank funds ((cid:25)j for j = tb;sb) is t private information 9The theoretical results of the paper hold for a generic quadratic cost function c((cid:25)t) = (cid:28) 2 (cid:25)2 t +(cid:19)(cid:25)t+" , but the speci(cid:133)c form used has the advantage ofproviding a closed form solution forthe optimal(cid:25)t chosen by each type ofintermediary. 10With this interpretation the (cid:25)t chosen by shadow banks would represent the probabilit (cid:0) y of purchas (cid:1) ing projects from traditionalbanks located in a good region. 7

Asaconsequenceofthe di⁄erentdiversi(cid:133)cationabilities of thetwointermediaries, the (cid:133)rstfriction(UO) will characterize the contracting problem between households and TBs whereas the second one (UE) will be at the core of the funding constraint for the SB system.11 Thecrucialdistinctionbetweenthetwotypesofcontractswilldependontheobservabilityoftheex-post realization of the loan portfolio held by each intermediary. In fact, because of the UO-friction, TBs can credibly commit only to a payment linked to the worst possible realization of their projects. On the other hand, since the outcome of their pool of loans is veri(cid:133)able, SBs will be able to write a contract contingent on the idiosyncratic realization of their assets. We can think of this framework as capturing the idea that, by combining several loans, shadow banks created securities that were easier to evaluate for a rating agency and hence easier to pledge to external investors.12 In particular I will show that traditional banks will have a smaller endogenous leverage than shadow banks. In addition, unlike the case for SB, the funding capacity of traditional banks will not depend on the expected quality of the loans they hold. The idea behind this setup is that of a SB system that, because of (cid:133)nancial products that exploit risk diversi(cid:133)cation, is able to increase the marketability of bank loans and to improve on the capital constraints a⁄ecting the traditional banking system, hence intermediating funds with a lower level of net worth. Given this characterization of the (cid:133)nancial system, I will (cid:133)rst derive the optimal contract for (cid:133)nancial intermediaries in a (cid:133)rst-best scenario when no asymmetric information is present and both the outcome and the riskiness of a project are observable. This will serve as a benchmark to identify the ine¢ ciencies arising from the agency problems of the two types of banks. I will then derive the optimal contract for a traditional bank and the one for a shadow bank. In the baselinemodel, Iwillconsideraneconomywherebothtypesofintermediariesarepresent, asitwasthecase for the U.S. economy in the years preceding the (cid:133)nancial crisis. The focus of this model is not to provide a speci(cid:133)c economic mechanism to explain the growth of the shadow banking system, but rather to take its existence, size and agency problems as given, in order to study its macroeconomic e⁄ects. 2.1 The Optimal Contract in the Frictionless Economy In this subsection I assume that there is no asymmetric information problem a⁄ecting the banking sector. I will refer to this scenario as the "Frictionless Economy" or "First-Best Economy". LetQ bethepriceofaunitofcapitalattime t, andRk thereturnperunitofe⁄ectivecapitalattime t t+1 t+1. As I will explain in more detail below, I assume that (cid:133)rms are competitive and that there is no agency problem between banks and entrepreneurs. Therefore, a bank will (cid:133)nance the total capital expenditures k Q facedbyeach(cid:133)rmandwillreceivetheriskyreturnperdollarinvested,(cid:18)jRk ,forj =H;L,depending t t t+1 on whether the speci(cid:133)c project is successful or not. At the beginning of time t, a bank enters the economy with an initial net worth n and has to decide the t amount of projects to (cid:133)nance k and the screening intensity (cid:25) . The required amount of external funding t t 11As an additional technical condition, I assume that at the moment of signing the contract with a bank (CB or SB), the individualhouseholddoesnotobservethedistributionof(cid:133)nancialintermediariesacrosstheislands. Equivalentlywecanthink that bank receives funds before having selected the speci(cid:133)c island(s) where to invest.This simply rules out that investors are able to perfectly foresee which sectoris good by inferring the monitoring levelselected by intermediaries. 12It has to be noted that here we are referring to ex-post observability. The ex-ante riskiness of loans, depending on (cid:25), will stillbe unobservable also forshadow banks and willbe behind the agency problem that a⁄ects shadow banking funding. 8

provided by households will hence have to be s =Q k n t t t t (cid:0) Inthisinstance,Iassumethatthebankhasaccesstothemoste¢ cientdiversi(cid:133)cationtechnologyavailable in the economy, that is the one used by shadow banks, which allows the (cid:133)nancial intermediary to invest in a pool of projects in a speci(cid:133)c region. As a result, the optimal contract will specify a pair of payments to outside lenders per unit of capital , bG;fb and bB;fb, contingent on whether the pool of loans is good or bad. t+1 t+1 In particular, these payments will have to satisfy the following participation constraint for the household Q k n E (cid:3) (cid:25)fbbG;fb+(1 (cid:25)fb)bB;fb Q k (3) t t (cid:0) t (cid:20) t t;t+1 t t+1 (cid:0) t t+1 t t h i where(cid:3) representsthehouseholdstochasticdiscountfactor, and(cid:25)fb isthescreeninglevelchoseninthe t;t+1 t frictionless case. In addition, I assume limited liability for the (cid:133)nancial intermediary, so that for every realization of projects outcome the payment to households cannot be larger than the assets available to the bank, that is bG;fb (cid:22)(cid:18) G Rk (4) t+1 (cid:20) t+1 bB;fb (cid:22)(cid:18) B Rk (5) t+1 (cid:20) t+1 t+1 Asmentionedintheintroduction,(cid:133)nancialintermediariesareabletoincreasetheprobabilityofselecting a good project, (cid:25) , by facing a non-pecuniary cost, c((cid:25) ), proportional to the value of the loans (cid:133)nanced. In t t particular, I assume c((cid:25) )= (cid:28)t (cid:25)2+(cid:19)(cid:25) , t 2 t To solve the optimal contract we have to maximize the following bank objective: (cid:0) (cid:1) max Q k E (cid:3) (cid:25)fb (cid:22)(cid:18) G Rk bG;fb +(1 (cid:25)fb) (cid:22)(cid:18) B Rk bB;fb c (cid:25)fb kt;(cid:25)t;bg t+1 ;bh t+1 t t n t t;t+1 h t (cid:16) t+1(cid:0) t+1 (cid:17) (cid:0) t (cid:16) t+1(cid:0) t+1 (cid:17)i (cid:0) (cid:16) t (cid:17)o subject to (3), (4) and (5). The objective function of the bank includes the expected return from the pool of projects, net of the payments to outside creditors and the screening costs. In particular, given that the bank is owned by the representative household, as will be explained in the next section, it discounts future pro(cid:133)ts with the same discount factor. In the appendix, it is shown that the (cid:133)rst order conditions of this problem imply the following c (cid:25)fb =E (cid:3) (cid:1)(cid:22) Rk (6) 0 t t t;t+1 t+1 t+1 (cid:16) (cid:17) E (cid:3) (cid:25)fb(cid:22)(cid:18) G + 1 (cid:25)fb (cid:22)(cid:18) B Rk R c (cid:25)fb =0 (7) t t;t+1 t (cid:0) t t+1 t+1(cid:0) t+1 (cid:0) t nh (cid:16) (cid:17) i o (cid:16) (cid:17) where (cid:1)(cid:22) = (cid:22)(cid:18) G (cid:22)(cid:18) B and I used E (cid:3) R =1. t+1 (cid:0) t+1 t t;t+1 t+1 Equation (6(cid:16)) determin(cid:17)es how the screening e⁄ort is optimally chosen in the frictionless scenario. It equates the marginal cost of screening to the social marginal bene(cid:133)t, which is given by the extra return generatedbygoodprojectswithrespecttobadones. Inaddition,equation(7)equalizestheexpectedreturn on capital, net of the screening cost, to the risk-free rate. This is a standard no-arbitrage condition for a model with perfect capital markets. 9

Combining these two equations we can determine the risk-adjusted return to capital E (cid:3) Rk , and t t;t+1 t+1 consequentlythelevelofcapitalintheeconomy. Inthisfrictionlessscenario,ifwefocuson(cid:133)rstordere⁄ects, this quantity will generally not move over time.13 On the other hand, as will become clear in the following sections,whenthereisabindingagencyproblemequation(7)willnotholdwithequality,implyingapositive premium on the adjusted return to capital, a distinguishing feature of macroeconomic models with (cid:133)nancial frictions. In addition, the movements in this premium will be important for the cyclicality of investments and , a unique feature of this model, asset quality. In the (cid:133)rst best contract, bG;fb and bB;fb are not uniquely determined; any pair of payments satisfying t+1 t+1 (3), (4) and (5) would be admissible. Finally, it is important to notice that in this case bank net worth does not play a role in determining aggregate demand for capital, and that the optimal contract does not constrain the (cid:133)nancial leverage (cid:30) =Q k =n . t t t t 2.2 The Optimal Contract for Traditional Banks Letusnowconsidertheoptimalcontractfora(cid:133)nancialintermediaryoperatingwiththetraditionalbanking technology. There is a continuum of traditional banks, each providing funds to non-(cid:133)nancial (cid:133)rms located in one single island. Each traditional bank (cid:133)nances the investment in its projects, Q ktb, by using its own t t net worth, ntb, and by issuing liabilities stb. t t The balance sheet of a traditional bank will then be Q ktb =ntb+stb (8) t t t t Becauseofthe(UO)frictiondescribedabove,traditionalbankswillbelimitedintheamounttheycanpledge torepaydepositors. Inparticular,similarlytoTownsend(1979),thepaymenttolenderscannotbecontingent on the idiosyncratic realization of the loans, since this is not observable, so that bG;tb = bB;tb = btb . In t+1 t+1 t+1 addition, for the amount that traditional banks commit to repay to be incentive-compatible, this will have to satisfy the following incentive constraint btb (cid:18) Rk (9) t+1 (cid:20) L t+1 This constraint comes from the fact that households cannot observe whether the loans held by the traditional bank have defaulted or not, hence, the only payment that can be enforced is linked to the worst possible idiosyncratic realization, since in this case the bank would not have incentives to misreport. Importantly, this "worst-case-scenario constraint" also guarantees that the traditional bank will always be able to repay its creditors, which is why we can also refer to (9) as a "solvency constraint". A similar type of funding constraint can be found in other papers such as Bernanke and Gertler (1987) and Carlstrom and Samolyk (1995). In addition, also Gennaioli et al. (2012) use a similar limit on bank deposits, but in their case it is motivated by extreme risk-aversion among depositors rather than by asymmetric information. Notice that in this setup, if (9) binds, the return on stb will be devoid of idiosyncratic risk but will be t exposed to aggregate risk. In this sense, we can think of stb as including both deposits and other types of t non-risk-free securities, like preferred equity.14 The important aspect is that since the payment is going to 13This is true unless there is a shock to (cid:22)(cid:18)B t+1 14In particular, it can be shown that the payment implied by the optimal contract can be implemented as a combination of risk-free debt,equity,and a bonus to bankers in case the project is successful. 10

be contingent on the aggregate price Q , traditional banks will always be able to repay their creditors.15 In t addition, this result will also imply that they will not be exposed to bank runs. The implied objective for the traditional bank is therefore E Q ktb (cid:3) (cid:25)tb(cid:22)(cid:18) G + 1 (cid:25)tb (cid:22)(cid:18) B Rk btb c((cid:25)tb) t t t f t;t+1 t (cid:0) t t+1 t+1(cid:0) t+1 (cid:0) t g h(cid:16) (cid:0) (cid:1) (cid:17) i Finally, when solving for the optimal contract we also have to take into account the participation constraint (PC) that guarantees that creditors receive an appropriate return on their lending activity stb E (cid:3) btb Q ktb (10) t (cid:20) t t;t+1 t+1 t t This is going to be the same relationship implied by households (cid:133)rst order condition for the choice of traditional banks securities. Given these assumptions, the one period contract between the TB and households will have to solve max E Q ktb (cid:3) (cid:2) ((cid:25)tb)Rk btb c((cid:25)tb) ktb;(cid:25)tb;stb;btb t t t f t;t+1 t+1 t t+1(cid:0) t+1 (cid:0) t g t t t t+1 (cid:2) (cid:3) s.t btb (cid:18) Rk (IC) t+1 (cid:20) L t+1 Q ktb ntb E (cid:12)(cid:3) btb Q ktb (PC) t t (cid:0) t (cid:20) t t;t+1 t+1 t t It can be shown that when th (cid:0) e incentive c (cid:1) onstraint binds 16, then the following will be true E (cid:3) (cid:2) ((cid:25)tb)Rk R c((cid:25)tb)>0 t t;t+1 t+1 t t+1(cid:0) t+1 (cid:0) t (cid:2) (cid:3) This inequality shows the presence of a wedge between the discounted return on borrowers assets and the cost of funds (R ), two values that were equal in the (cid:133)rst best scenario. This is a classic result in models t+1 with (cid:133)nancial frictions, but in this framework it is enriched by the endogenous choice of asset quality. The incentive constraint will also limit the amount of assets that traditional banks can intermediate by implying a constraint on their leverage, (cid:30)tb = Qtk t tb , given by t ntb t 1 (cid:30)tb (11) t (cid:20) 1 (cid:18) E (cid:3) Rk (cid:0) L t t;t+1 t+1 (cid:2) (cid:3) Wecangiveanintuitiveinterpretationtothisrelationship. Firstofall,theleveragecapacityisincreasing in the expected aggregate return to capital E (cid:3) Rk , since it increases the amount that can be credibly t t;t+1 t+1 promised to external investors. For the same reason, leverage will be higher the higher (cid:18) , the recovery L rate on defaulted projects. It is also important to notice that the debt capacity of traditional banks is not directly linked to the riskiness of their loans, (cid:25)tb. This is a consequence of the fact that traditional banks t can only pledge the worst possible realization, independently from the outcome of their projects. Asaresult, the(cid:133)rstorderconditiononthescreeninglevelwilldetermine(cid:25)tb inasimilarfashiontowhat t 15It would be possible to slightly modify the assumptions of the agency problem in order to have the TB issuing risk-free debtaswell. Forexampleonecouldassumethatiftheprojectfailsitdeliversapredeterminedamountofgoods(cid:18)Lkt. However such featurewould notadd tothedynamicsofthemodeland itwould makethecharacterization ofthecontractlessintuitive. In addition,a framework in which also TB issue risk-free securities would amplify allthe mechanisms in this paper because of a higher(cid:133)nancialacceleratorin the TB sector. 16See appendix fora detailed solution ofthe contract. 11

occurred in the frictionless scenario, that is according to c((cid:25)tb)=E (cid:3) (cid:1)(cid:22) Rk (12) 0 t t t;t+1 t+1 t+1 The intuition for this result is the following: since the payment to households does not depend on whether theloanwillbeinagoodorbadregion, thetraditionalbankwillretainalltheexposuretotheidiosyncratic risk and hence it will equalize the marginal cost of monitoring to the expected social marginal bene(cid:133)t, given by the extra expected return that a good project delivers. In terms of the dynamics of traditional bank monitoring, this equation will imply countercyclical movements in the quality of their loans, since in recessions the marginal value of monitoring will be higher, due to a larger discounted expected return on capital. If we use the speci(cid:133)c functional form for the cost function , from (12) we can directly obtain the optimal level of (cid:25)tb set by the traditional bank as t E (cid:3) (cid:1)(cid:22) Rk (cid:28) (cid:25)tb = t t;t+1 t+1 t+1 t(cid:19) (13) t (cid:28) (cid:0) 2 t which also shows how (cid:25)tb is decreasing in the parameter a⁄ecting the marginal cost of screening, given t by (cid:28) . t At this point, we can de(cid:133)ne the return that households obtain after lending to traditional banks as (cid:30)tb R t tb =(cid:18) L R t k (cid:30)tb t (cid:0) 1 1 (14) t (cid:0) 1(cid:0) From the equations above, we notice how both the leverage ratio and the screening intensity of traditional banks only depend on aggregate quantities, allowing for an easy aggregation. In addition, equation (14) implies that Rtb only depends on aggregate variables. t 2.3 The Optimal Contract for Shadow Banks Shadow banks have access to the same screening technology of traditional banks. However, they can use a special diversi(cid:133)cation technology that allows them to "pool" projects within a single region. As described above, the type-realization of a region, that is whether it is good or bad, is publicly observable, enabling shadow banks to overcome the UO-friction that a⁄ects the relationship between households and traditional banks. However, since the diversi(cid:133)cation is not complete, shadow banks will still be exposed to some idiosyncraticrisk. Becauseofthis, itwillbetheunobservabilityofthemonitoringe⁄ortchosen, (cid:25)sb, coming t from the UE-friction, that will constrain the amount of funds that shadow banks can raise. The shadow bank will fund its capital, ksb; by using its net worth and by issuing securities, ssb. Its t t balance sheet will then be Q ksb =nsb+ssb (15) t t t t Unlike the case for traditional banks, the contract between shadow banks and outside investors will specify payments to the households, per dollar of loan, that are contingent on the realized type of the loan pool, that is bj for j =G;B. Again, because of limited liability, we require t+1 bj;sb (cid:22)(cid:18) j Rk for j =G;B (16) t+1 (cid:20) t+1 t+1 12

Note that this setup has some similarities to the pooling and tranching that was behind securitization, because we could interpret this contract as the shadow bank selling a contingent claim to the outcome of its pool of loans in return for an amount ssb, similarly to the notion of "loan sale" presented in Pennacchi t (1988). The expected return for the shadow bank, including the non-pecuniary monitoring costs, will be given by Q k E (cid:3) (cid:25)sb (cid:22)(cid:18) G Rk bG;sb +(1 (cid:25)sb) (cid:22)(cid:18) B Rk bB;sb c (cid:25)sb (17) t t t t;t+1 t t+1(cid:0) t+1 (cid:0) t t+1(cid:0) t+1 (cid:0) t n h (cid:16) (cid:17) (cid:16) (cid:17)i (cid:0) (cid:1)o Importantly, because of the UE-friction, the contract for SBs will be characterized by a moral hazard problem with hidden action. In particular, this is due to the fact that the payment to investors depends on the quality of the loans originated by the shadow bank, (cid:25)sb, which is unobservable by outsiders. t Therefore,anincentiveconstraint(IC),thatguaranteesthattheshadowbankwillselectagivenscreening level, will be required: (cid:25)sb =argmax E (cid:3) (cid:25)sb (cid:22)(cid:18) G Rk bG;sb +(1 (cid:25)sb) (cid:22)(cid:18) B Rk bB;sb c (cid:25)sb (18) t (cid:25)sb t t;t+1 t t+1(cid:0) t+1 (cid:0) t t+1(cid:0) t+1 (cid:0) t t n h (cid:16) (cid:17) (cid:16) (cid:17)i (cid:0) (cid:1)o In addition, because the simple way in which banks can a⁄ect the loans return distribution satis(cid:133)es the "convexity-of-distribution-function" condition described in Hart and Holmstrom (1986), we can write the (IC) in a more tractable way, by using the (cid:133)rst order conditions of (18), that is c (cid:25)sb E (cid:3) (cid:1)(cid:22) Rk bG;sb bB;sb (19) 0 t (cid:20) t t;t+1 t+1 t+1(cid:0) t+1 (cid:0) t+1 (cid:0) (cid:1) h (cid:16) (cid:17)i It has to be noted that such a constraint does not bind in the problem of TBs, since the payment that they promise to outsiders does not depend on the idiosyncratic realization of their projects. Finally,wehavetoconsidertheparticipationconstraintforlenders,whichguaranteesthatthehousehold obtainsanexpectedreturnequaltotheopportunitycostofitsfunds. AsinBernanke,Gertler,andGilchrist (1999), I assume that the shadow banker is willing to bear all the aggregate risk, guaranteeing a payment to the lender that is equal to the risk-free rate in expectation.17 As a result the participation constraint will imply restrictions on bG;sb;bB;sb contingent on the realization of the aggregate shock, according to t+1 t+1 R ssb (cid:25)sbbG;sb+(1 (cid:25)sb)bB;sb Q ksb (20) t+1 t (cid:20) t t+1 (cid:0) t t+1 t t h i If we focus on a parametrization that allows for a value of bG;sb;bB;sb satisfying (20) and (16) to exist t+1 t+1 for any aggregate state,18 then the household can diversify the residual idiosyncratic risk by investing in "mutualfunds"thatlendmoneytoseveralshadowbanks,andpromisearateofreturnequaltotherisk-free rate.19 Hence, in this framework we can think of the security ssb, as ABCP or shares of a MMMF. t Inthissetup,shadowbanksarehenceretainingalltheexposureto(cid:135)uctuationsinassetpricesanddefault rates, and they issue to the mutual fund a senior claim on the return from their loans. Such a con(cid:133)guration isinlinewiththeideathatevenwiththedevelopmentofthe"originate-to-distribute"model, whichmarked 17This assumption is mainly made to capture the fact that most of the liabilities issued by the shadow banks, like ABCP, were short-term non-contingent debt,that exposed the system to runs. 18AsIwillexplaininSection5,suchconditionwillnotholdinthecaseofarunonshadow banks. However,sincetherunis an unanticipated event it does not enterthe optimalcontract. 19Itisrelevanttonoticethat,becauseoftheUO-friction,diversi(cid:133)cationacrosscommercialbanksdoesnotalterthestructure of the contract, and its payments. This depends on the fact that households are only able to require the same individual payment of(cid:18)LR t k from allthe CB located on di⁄erent islands. 13

the growth of the shadow banking system, most of the risk remained inside the (cid:133)nancial sector, as noted, among others, by Acharya, Schnabl and Suarez. (2013). In addition, this will imply that when there is a low realization of Rk , bG;sb will have to rise, so that the banks with a good pool of loans will have to pay t+1 t+1 a higher amount to households, diminishing their net worth. Such mechanism will play an important role in all the quantitative experiments, including the run. Importantly, all these considerations are valid only in the "no-run equilibrium". In fact, as I will explain later in the paper, the economy will admit an alternative "run-equilibrium", in which the payment implied by the liquidation price and (20) would be such that bG > (cid:22)(cid:18) G Rk , thereby violating limited liability. t+1 t+1 t+1 However, ifweconsideronlyunanticipatedruns, thischaracterizationfortheoptimalcontractremainsvalid in the baseline economy. The problem of the shadow bank can therefore be written as max Q ksb E (cid:3) (cid:25)sb (cid:22)(cid:18) G Rk bG;sb +(1 (cid:25)sb) (cid:22)(cid:18) B Rk bB;sb c (cid:25)sb ksb;(cid:25)sb;bg ;bb t t t t;t+1 t t+1(cid:0) t+1 (cid:0) t t+1(cid:0) t+1 (cid:0) t t t t+1 t+1 n h (cid:16) (cid:17) (cid:16) (cid:17)i (cid:0) (cid:1)o s.t. R Q ksb nsb (cid:25)sbbG;sb+(1 (cid:25)sb)bB;sb Q ksb (PC) t+1 t t (cid:0) t (cid:20) t t+1 (cid:0) t t+1 t t (cid:0) (cid:1) h i c (cid:25)sb E (cid:3) (cid:1)(cid:22) Rk bG;sb bB;sb (IC) 0 t (cid:20) t t;t+1 t+1 t+1(cid:0) t+1 (cid:0) t+1 (cid:0) (cid:1) bG; h sb (cid:22)(cid:18) G Rk (cid:16) (LLG) (cid:17)i t+1 (cid:20) t+1 bB;sb (cid:22)(cid:18) B Rk (LLB) t+1 (cid:20) t+1 where the last two equations represent limited liability constraints for each idiosyncratic realization. The(cid:133)rstresultthatcanbeprovedisthatifthe(IC)bindsthenitwillbeoptimaltopaythebankonlyin case the pool belongs to a good region, so that bB;sb =(cid:22)(cid:18) B Rk .20 This result follows from the (cid:133)nding (see t+1 t+1 t+1 Hart and Holstrom (1986) ) that, in order to provide incentives to monitor, it is optimal to give the worst possible punishment to the agent when the bad realization occurs. Therefore, the contract for the shadow bank will imply that if the pool of loans reveals to be a bad one, the whole return will be given to creditors and the shadow bank will default, resembling a risky debt contract.21 As a result, we can rewrite the (IC) as c (cid:25)sb E (cid:3) (cid:22)(cid:18) G Rk bG;sb 0 t (cid:20) t t;t+1 t+1(cid:0) t+1 (cid:0) (cid:1) h i In addition, if the (IC) is binding then it can be shown that the following inequalities must be true E (cid:3) (cid:25)sb(cid:22)(cid:18) G + 1 (cid:25)sb (cid:22)(cid:18) B Rk R c (cid:25)sb >0 (21) t t;t+1 t (cid:0) t t+1 t+1(cid:0) t+1 (cid:0) t nh (cid:0) (cid:1) i o (cid:0) (cid:1) c((cid:25)sb)<E (cid:3) (cid:1)(cid:22) Rk (22) 0 t t t;t+1 t+1 t+1 The (cid:133)rst inequality is analogous to the one obtained in the problem for the traditional bank: also in this case the incentive constraint implies that the discounted return on bank assets is larger than the cost of funds. In addition, the second inequality implies that, given the same E (cid:3) (cid:1)(cid:22) Rk , the quality of loans t t;t+1 t+1 t+1 originated by shadow banks will be lower than the one of traditional banks.22 This result comes from the 20Details forthe solution ofthe optimalcontract can be found in the appendix. 21A similar result is derived in Pennacchi (1988), where the loan(cid:146)s return density is a continuous function with bounded support. 22This will be the case in the baseline model, where both TB and SB operate and Et(cid:3)t;t+1(cid:1)(cid:22) t+1R t k +1 only depends on 14

factthat,unlikethetraditionalbank,theshadowbankdoesnotretainalltheidiosyncraticriskcomingfrom the choice of (cid:25)sb. By being able to pledge a larger portion of the return on its loans, the shadow bank does t not internalize all the expected bene(cid:133)ts from monitoring, that is E (cid:3) (cid:1)(cid:22) Rk . As a result it will have t t;t+1 t+1 t+1 lower incentives to screen its projects. Furthermore, from the PC we obtain that 1 ((cid:30) 1) bG;sb = R t(cid:0) (1 (cid:25)sb)(cid:22)(cid:18) b Rk (23) t+1 (cid:25)sb t+1 (cid:30) (cid:0) (cid:0) t t+1 t (cid:20) t (cid:21) and by substituting the implied value of bG;sb in (19) we obtain that the (IC) imposes the following t+1 leverage constraint for shadow banks 1 (cid:30)sb (24) t (cid:20) (cid:25)sbc((cid:25)sb) E (cid:3) (cid:2) (cid:25)sb Rk 1 t 0 t (cid:0) t t;t+1 t+1 t t+1(cid:0) (cid:8) (cid:2) (cid:0) (cid:1) (cid:3)(cid:9) where Q ksb (cid:30)sb = t t t nsb t In this case (cid:30)sb is increasing in the total expected return on the pool of loans, whereas it is decreasing in t the expected payment due to the bank (cid:25)sb (cid:22)(cid:18) G Rk bG;sb = (cid:25)sbc (cid:25)sb . In particular, a very important t t+1(cid:0) t+1 t 0 t consequence of (24) is that it implies a neghative relationshiip between (cid:0) scr (cid:1) eening e⁄ort and leverage for the shadow banks. In fact, because of the moral hazard problem related to the unobservability of (cid:25)sb, in order t for the shadow bank to have incentives to exert a higher e⁄ort, it will need to have more "skin in the game" to internalize the bene(cid:133)ts of a larger (cid:25)sb. This is accomplished by requiring that the bank covers a larger t shareoftheinvestmentwithitsownnetworth,implyingalowerleverage. Infact,asequation(23)suggests, alowerleverageimpliesahigherpaymenttothebankincaseofsuccess,consequentlyincreasingitsincentive to screen projects, as shown in the (IC). Equation (24) represents an important di⁄erence from the model of Christiano and Ikeda (2014). Since in their framework the screening e⁄ort is not proportional to the amount of capital (cid:133)nanced by the (cid:133)nancial intermediary, the unobservability of bank e⁄ort does not imply any limit to the amount of debt that the bank can raise. This comes from the fact that the screening cost is not increasing with the amount of loans originated. As a result, in the unobservable e⁄ort scenario of Christiano and Ikeda (2014) equation (21) holds with equality and aggregate net worth does not directly a⁄ect investments. On the other hand, as it will be clear in subsequent sections, the interaction between loan quality, leverage constraints and net worth will play a crucial role in determining the aggregate dynamics of the shadow banking system and of the whole economy. In addition, it can be shown that (cid:30)sb >(cid:30)tb (25) t t This can be easily seen by using (22) when comparing (24) with (11). The intuition is that, when the IC binds, the pledgeable income per unit of capital of shadow banks, E (cid:3) (cid:2) (cid:25)sb Rk (cid:25)sbc (cid:25)sb t t;t+1 t+1 t t+1 (cid:0) t 0 t will be larger than the one of traditional banks, (cid:18)LE (cid:3) Rk . As a result, shadow banks need a lower t t;t+1 t+1 (cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1) net worth to fund the same quantity of loans. It has to be noted that, even if we assumed that also the type realization of a region was unobservable, aggregate quantities. 15

shadow banks would still have a higher leverage. This comes from the fact that, because of diversi(cid:133)cation, the worst possible outcome fora poolof projects (cid:133)nanced bya shadow bankwould be(cid:22)(cid:18) B , which is greater t+1 or equal than (cid:18)L. Therefore, even if they had to face the same type of contract used by traditional banks, shadow banks would still be able to promise a larger expected return to investors, thus obtaining a higher leverage. From this perspective, the relationship between shadow banking and diversi(cid:133)cation is similar to the one presented in Gennaioli et al. (2012), where by diversifying among themselves banks are able to o⁄er a payment linked to the aggregate "worst case scenario" rather than to the idiosyncratic one. In addition to this mechanism, my model also introduces endogenous screening performed by intermediaries and captures a link between shadow banking and laxer lending standards, which will play an important role in the crisis experiments shown in the next section. Theinverserelationshipbetweenleverageandscreeningwillbecrucialtodeterminethecyclicalityofthe asset quality of shadow banks. In particular, (cid:25)sb will be determined by the following equation t E(cid:3) (cid:1)(cid:22)Rk c (cid:25)sb (cid:25)sbc (cid:25)sb c (cid:25)sb = E (cid:3) (cid:2) (cid:25)sb Rk R c (cid:25)sb (cid:25)sbc (cid:25)sb t;t+1 t+1(cid:0) 0 t t 0 t (cid:0) t t t;t+1 t t+1(cid:0) t+1 (cid:0) t t 00 t (26) (cid:2) (cid:0) (cid:1)(cid:3)(cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) (cid:8) (cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1)(cid:9)(cid:2) (cid:0) (cid:1)(cid:3) implying @’ (cid:25)sb =’(E (cid:3) Rk ) where <0 (27) t t t;t+1 t+1 @E (cid:3) Rk t t;t+1 t+1 The quantity E (cid:3) Rk can be interpreted as the "external (cid:133)nance premium" de(cid:133)ned by Bernanke, t t;t+1 t+1 Gertler, and Gilchrist (1999). As equation (21) shows, an increase in the discounted return to capital is associated with a tightening of the incentive constraint, so that we can interpret equation (27) as a negative relationship between the quality of shadow banks loans and the severity of their agency problem. In fact, during a crisis the net worth of (cid:133)nancial intermediaries is eroded, causing a decrease in capital demand, a consequent drop in prices and an increase in leverage together with E (cid:3) Rk . As a result of the higher t t;t+1 t+1 leverage,theshadowbankwillhavealowerlevelof"skininthegame"intheprojectsthatitoriginates,and consequently it will be able to credibly commit to a lower level of (cid:25)sb. The relationship in (27) will play an t important role in the model dynamics. In fact, it will imply that when a negative shock hits the economy the quality of the loans intermediated by shadow banks will deteriorate, causing a slower recovery for the net worth of these intermediaries and their ability to invest. In addition, a lower aggregate quality will also imply a lower level of productive capital and output, making recessions more persistent. In particular, given the cost function c((cid:25) )= (cid:28)((cid:25)2+(cid:19)(cid:25)), I show in the appendix that we obtain t 2 t E (cid:3) R (cid:18)BRk (cid:25)sb =2 t t;t+1 t+1 (cid:0) t+1 (28) t E(cid:3) h (cid:1)(cid:22)Rk (cid:28)(cid:19) i t;t+1 t+1(cid:0) 2 (cid:2) (cid:3) As a result, also in this case it can be shown that both (cid:25)sb and (cid:30)sb only depend on aggregate quantities, t t facilitating aggregation in the shadow banking sector. At this point we can summarize the key di⁄erences between traditional banks and shadow banks in this model. First of all, shadow banks will have a higher leverage than traditional banks, achieved thanks to the possibility of pledging a larger share of the expected return on their loans. The larger amount of funds per unit of net worth that shadow banks can (cid:133)nance, will however be used towards lower quality projects, since (cid:25)sb <(cid:25)tb. Finally,theendogenousqualityofloans,dependingon(cid:25)i fori=tb;sb,willmovecountercyclically t t t for traditional banks but procyclically for shadow banks. As we will see in the quantitative exercises, all these features point to a shadow banking system much more sensitive to aggregate negative shocks. 16

2.4 Aggregation in the Financial System In the baseline model I assume that both types of (cid:133)nancial intermediaries are operating, each (cid:133)nancing a di⁄erent set of projects. Asexplainedindetailinthenextsection,IfollowGertlerandKaradi(2011)inassumingthateachbanker belongs to one of a continuum of households. In the baseline model each household will have three types of members: a worker, a traditional banker and a shadow banker. At the end of every period bankers (both traditional and shadow) exit the economy with probability (1 (cid:27)) and are replaced by an equal mass of (cid:0) workers that start their banking franchise with an initial endowment !j for j =tb;sb, according to whether they become traditional bankers or shadow bankers. As is standard in models with (cid:133)nancial frictions, the exogenous exit probability is used to prevent net worth from growing inde(cid:133)nitely because of the excess returns ensuing from the agency problem. Asshownin(11)and(24), wecanexploitthefactthatthemaximumleverageconstraintsareindependent of individual-speci(cid:133)c factors to aggregate across the two (cid:133)nancial sectors. In particular, if we de(cid:133)ne Nj for t j = tb;fb as aggregate net worth, then the demand for capital in the traditional banking sector and in the shadow banking sector will be determined by Q Ktb =(cid:30)tbNtb t t t t Q Ksb =(cid:30)sbNsb t t t t Therefore, the total capital intermediated by the (cid:133)nancial sector is given by Q K =(cid:30)tbNtb+(cid:30)sbNsb t t t t t t From the equation above, we notice that the overall asset demand by banks is going to be a⁄ected by variations in both Ntb and Nsb. In particular, given the higher leverage of shadow banks, aggregate capital t t is going to be a⁄ected more directly by (cid:135)uctuations in the net worth of non-traditional intermediaries. In addition, since (cid:25)sb and (cid:25)tb also depend only on aggregate variables, we can de(cid:133)ne the aggregate e⁄ective t t capital availalble for each type of (cid:133)nancial intermediary as K^j =(cid:2) ((cid:25)j )Kj for j =tb;sb t t t 1 t 1 (cid:0) (cid:0) If we aggregate across surviving and entering bankers we can obtain the following evolution of the aggregate net worth for the traditional banking sector and the shadow banking sector, which comprises the retained earnings of surviving bankers, Nj, and the initial net worth of new entrants Nj, that is st e Nj =Nj +Nj for j =tb;sb t st e Inparticular,foreachspeci(cid:133)csector,survivingbankers(cid:146)networthwillbegivenbythedi⁄erencebetween the earnings on the assets held and the cost of the liabilities issued in the previous period, multiplied by the share of surviving bankers (cid:27) Nj =(cid:27) Q K^jRk RjSj for j =tb;sb (29) st f t (cid:0) 1 t t (cid:0) t t (cid:0) 1g 17

Here we see how net worth depends on the average quality of the loans that are originated in a speci(cid:133)c (cid:133)nancial sector: (cid:2) ((cid:25)j ) = (cid:25)j (cid:22)(cid:18) G +(1 (cid:25)j )(cid:22)(cid:18) B for j = tb;sb. First of all, the lower (cid:25)i is the more exposed to "subpri t me t s (cid:0) h 1 ocks" t t (cid:0) o 1 (cid:22)(cid:18) B the n (cid:0) et w t o (cid:0) r 1 th t will be. In addition, a drop in the screenin t (cid:0) g 1 level at time t t; will negatively a⁄ect the earnings in the next period. Furthermore, because of the higher leverage and the risk-free return on liabilities, Nsb will drop much more in response to negative shocks, as we will see in the t next section. Ontheotherhand,theaggregatenet-worthofnewbankerswillbesimplygivenbytheirinitialendowment Nj =(1 (cid:27))!j for j =tb;sb e (cid:0) In particular, the ratio between !tb and !sb, together with leverage ratios and spreads, will be useful to determinetherelativesizeofeach(cid:133)nancialsectorinthesteadystateoftheeconomy. Fromthisperspective, this model will be agnostic about what forces determined the growth of the shadow banking system, and it will simply use a calibration where the relative size of this parallel (cid:133)nancial system is comparable to that of the traditional banking sector. 23 AssuggestedbyChristianoandIkeda(2014),thisframeworkwithendogenousprobabilityofbankdefault has also implications for the cross-sectional standard deviation of banks(cid:146)equity returns. Given the binomial structure of bank payo⁄s, the standard deviation for banks(cid:146)return per unit of net worth at time t is :5 (cid:27)~j = (cid:25)j 1 (cid:25)j (cid:30)j (cid:22)(cid:18) G Rk bj;G (cid:22)(cid:18) B Rk bj;B for j =tb;sb t+1 t (cid:0) t (cid:3) t t+1(cid:0) t+1 (cid:0) t+1(cid:0) t+1 h (cid:16) (cid:17)i h(cid:16) (cid:17) (cid:16) (cid:17)i For traditional banks this quantity is simply (cid:27)~tb = (cid:25)tb(1 (cid:25)tb) :5 (cid:30)tb(cid:1)(cid:22) Rk (30) t+1 t (cid:0) t t t+1 t+1 where the (cid:133)rst term is decreasing in (cid:25)tb as l (cid:2) ong as (cid:25)tb > (cid:3) :5, which will always be the case in the calibration t t of the model. On the other hand, for shadow banks, the fact that (cid:22)(cid:18) B Rk bB =0 implies t+1(cid:0) t+1 (cid:27)~sb = (cid:25)sb(1 (cid:25)sb) :5 (cid:30)sb (cid:22)(cid:18) G Rk bG;sb t+1 t (cid:0) t (cid:3) t t+1(cid:0) t+1 (cid:2) (cid:3) (cid:16) (cid:17) and by using (23) we can write (1 (cid:25) (e )) :5 (cid:27)~sb = (cid:0) t t (cid:30)sb (cid:2) (cid:25)sb Rk R +R (31) t+1 (cid:25) (e ) t t t+1(cid:0) t+1 t+1 (cid:20) t t (cid:21) h (cid:0) (cid:0) (cid:1) (cid:1) i where the (cid:133)rst term is decreasing in (cid:25)sb, while the second term is increasing in the spread between shadow t banksexpectedreturnoncapitalandtherisk-freerate. Importantly,duringacrisisbothtermswillincrease. In fact, as explained above, (cid:25)sb decreases when bank net worth deteriorates and spreads rise. As a result, t the agency problem of shadow banks will become more stringent, increasing the external (cid:133)nance premium. As can be seen from (31) these movements contribute to increase (cid:27)sb . t+1 Finally, the cross sectional standard deviation on the return on equity for the whole (cid:133)nancial sector will 23Onepossiblewaytoendogenizethisquantitymightbetoassumethepresenceofdi⁄erentcoststoaccesstheshadowbanking technologyorthetraditionalbankingone,inordertoobtainendogenousinitialin(cid:135)owsfromhouseholdsineach(cid:133)nancialsector. Such approach is beyond the scope ofthis paper,but could be used as a rationale to explain the growth ofshadow banking as resulting from a decrease in the cost to access alternative (cid:133)nancialproducts in the years leading to the (cid:133)nancialcrisis. 18

be given by (cid:27)~fin = N t TB 2 (cid:27)TB 2 + N t SB 2 (cid:27)SB 2 1=2 (32) t+1 N(cid:22) t+1 N(cid:22) t+1 "(cid:18) t (cid:19) (cid:18) t (cid:19) # (cid:0) (cid:1) (cid:0) (cid:1) where N(cid:22) =NTB +NSB. t t t 3 The Baseline Model Tocapturethemacroeconomice⁄ectsofshadowbanking,Iintroducethetwotypesof(cid:133)nancialintermediaries describedaboveinamedium-scalerealDSGEmodel. Inthemodelthereare(cid:133)vetypesofagents: households (HH), non-(cid:133)nancial goods producers, capital producers and two types of bankers: traditional bankers (TB) and shadow bankers (SB). Only the (cid:133)nancial intermediaries are able to invest in productive capital by (cid:133)nancing risky projects and they also own a unique technology allowing them to screen the quality of these assets.24 Households can only invest by lending funds to banks. We can think of the assumption of limited market participation for households as a result of bankers technological advantage in evaluating loans. Traditional banks and shadow banks have the same screening technology, but they di⁄er in their "diversi(cid:133)cation technology". In fact, as described in the previous section, I assume that intermediaries operating via shadow banking are able to (partially) diversify across a pool of projects, making the ex-post realization of their portfolio more easily observable and increasing the marketability of their assets. 3.1 Households As in Gertler and Karadi (2011), I assume that there is a representative household with a continuum of members of measure unity. Within each household there is a fraction fw of workers, a fraction ftb of (cid:147)traditional bankers" and a fraction fsb of (cid:147)shadow bankers", where fw +ftb +fsb = 1. In addition, I assume that the fractions of the two types of bankers are equal, so that ftb =fsb =fb. Workers provide labor and return wages to the household. Each type of banker manages a (cid:133)nancial intermediary, performing the screening decision, and transfers positive dividends back to the household. There is perfect consumption insurance across household members. Householdscannotdirectlyinvestincapital, andtheonlywayinwhichtheycansaveisbylendingfunds to the two types of (cid:133)nancial intermediaries. In particular, the relevant utility function for the worker is L1+(cid:17) logC (cid:31) t+i t (cid:0) 1+(cid:17) where C represents consumption and L labor. On the other hand, bankers utility is given by t t logC c (cid:25)j Q kj for j =tb;sb t (cid:0) t t t (cid:16) (cid:17) As a result, if we de(cid:133)ne (cid:31)(cid:22) =fw(cid:31), we can write the utility of the representative household as E 1 (cid:12)t logC (cid:31)(cid:22) L1 t +(cid:17) c (cid:25)tb Q Ktb c (cid:25)sb Q Ksb t " t (cid:0) 1+(cid:17) (cid:0) t t t (cid:0) t t t # t=0 X (cid:0) (cid:1) (cid:0) (cid:1) 24From now on Iwilluse the words "projects","capital" and "loans" interchangeably 19

Asdescribedintheprevioussection,the(cid:133)nancialsystemo⁄erstwotypesofsecuritiestooutsideinvestors. Shadow banks o⁄er securities , Ssb; that pay a risk-free return, Rsb =R in case a run on the SB-system t t+1 t+1 does not occur. Since I am modeling the variety of institutions that composed the SB-system as a single entity,wecanthinkofSsb asrepresentingthesetofinstrumentsthatallowedinvestorstochannelfundsinto t this parallel banking sector. For example we can refer to asset backed commercial paper (ABCP) or shares of money market mutual funds (MMMFS), that in normal times were perceived as basically risk-free assets. What is important is that lenders are paid according to a "sequential service constraint", so that if shadow banks do not have su¢ cient resources to repay all creditors, the latter have an incentive to withdraw their fundsassoonaspossible. Inthemodel, IassumethatarunontheSB-systemisacompletelyunanticipated event, so that I can characterize the household problem and the subsequent optimal contracts with the (cid:133)nancial intermediaries as if households do not expect a run event to occur next period. Traditional banks issue liabilities Stb, that pay a return Rtb that is exposed to aggregate (cid:135)uctuations. t t+1 Because of the structure of this security, traditional banks will always be able to repay the promised return on Stb. Even if I am not directly modelling a government-backed deposit insurance, the microfoundation for t the funding problem of traditional banks will endogenously imply that they are not exposed to a bank run. The budget constraint for households is given by Ch+Stb+Ssb =RtbStb +RsbSsb +(cid:5) +W L (33) t t t t t 1 t t 1 t t t (cid:0) (cid:0) where W represents real wage and (cid:5) are pro(cid:133)ts derived from the ownership of capital-producing (cid:133)rms. t t The (cid:133)rst order conditions for the choice of assets and and labor are given by E (cid:3) Rj =1 for j =tb;sb (34) t t;t+1 t+1 C 1W =(cid:31)(cid:22)L(cid:17) (35) t(cid:0) t t where (cid:3) =(cid:12) Ct . t;t+1 Ct+1 3.2 Physical Setup As wasshown above, all the traditional banks willchoose the same (cid:25)tb and allthe shadowbankswillchoose t the same (cid:25)sb so that if we de(cid:133)ne the aggregate capital (cid:133)nanced by one (cid:133)nancial sector at time t as Kj for t t j =tb;sb, then the e⁄ective capital available for production in each (cid:133)nancial system will be K^j =(cid:2) ((cid:25)j )Kj for j =tb;sb (36) t t t 1 t 1 (cid:0) (cid:0) and consequently, aggregate e⁄ective capital will be K^ =K^tb+K^sb t t t After the idiosyncratic default realization, projects become homogeneous raw capital again. Therefore, if we denote by I aggregate investment and by (cid:14) the rate of exogenous physical depreciation, then the t evolution of aggregate capital K =Ktb+Ksb, will be given by t t t K =(1 (cid:14))K^ +I (37) t t t (cid:0) 20

Inaddition,itwillalsobeusefultode(cid:133)nethefollowingmeasureforaggregatescreening(cid:25)^ ,whichweights t the monitoring level of each type of bank by the share of capital intermediated Ktb Ksb (cid:25)^ =(cid:25)tb t +(cid:25)sb t t t K t K t t Therefore, we can de(cid:133)ne "aggregate quality" as K^ (cid:2)^ ((cid:25)^ )= t t t (cid:0) 1 K t 1 (cid:0) At this point, comparing this setup with that of Gertler and Karadi (2011), we can think of (cid:2)^ ((cid:25)^ ) t t 1 (cid:0) as a way to endogenize the "capital quality shock" used in their paper, which a⁄ects the amount of capital available for production in every period. 3.3 Non-Financial Firms In the model there are two types of non-(cid:133)nancial (cid:133)rms: goods producers and capital producers. 3.3.1 Goods Producers GoodsproducersoperateaCobb-Douglasproductionfunctionwithe⁄ectivecapitalandlabor,underperfect competition. Since labor is perfectly mobile across islands we can write aggregate output Y as a function of t aggregate productive capital, K^ ; and aggregate labor L t t Y =A K^(cid:11)L1 (cid:11) (38) t t t t(cid:0) where (cid:11) (0;1) and A is aggregate productivity. t 2 Given the e⁄ective capital available for production, (cid:133)rms choose labor in order to satisfy Y W =(1 (cid:11)) t (39) t (cid:0) L t so that we can de(cid:133)ne gross pro(cid:133)ts per unit of e⁄ective capital as Y W L Y Z = t (cid:0) t t =(cid:11) t (40) t K^ K^ t t Firms(cid:133)nancethepurchaseofcapital/projectseveryperiodbyobtainingfundsfrom(cid:133)nancialintermediaries. As in Gertler and Karadi (2011), I assume that there are no frictions in the relationship between banks and goods producers. Banks can perfectly observe the realization ((cid:18) ;(cid:18) ) of projects purchased by a (cid:133)rm in L H a given island and can e¢ ciently enforce contractual obligations with these borrowers. As a result, goods producers can issue state contingent claims that are a claim to future returns from one unit of investments. Because of perfect competition, the price of these securities will be the same price of investment goods Q . t It is important to notice that, because of perfect labor mobility and constant returns to scale, we do not need to keep track of the distribution of default shocks, and consequently of e⁄ective capital, across islands. This allows us to consider a return per loan for an individual bank that is linear in expected quality, that is 21

E (cid:2) (cid:25)i Rk for i=tb;sb, where t t+1 t t+1 (cid:0) (cid:1) Z +(1 (cid:14))Q Rk = t+1 (cid:0) t+1 t+1 Q t 3.3.2 Capital Producers Capital producers create new capital by using the (cid:133)nal good as input and face convex adjustment costs in the gross rate of change in investment, f It I , where f(1) = f (1) = 0 and f () > 0. They sell new It 1 t 0 00 "raw" capital to (cid:133)rms in the di⁄erent islan(cid:16)ds(cid:0)at(cid:17)the price Q t . Given that households own capital producers, they choose I to maximize the following t max 1 (cid:12)(cid:28) t(cid:3) Q I I f( I (cid:28) )I I(cid:28) X (cid:28)=t (cid:0) t;(cid:28)+1 (cid:26) t (cid:28) (cid:0) (cid:28) (cid:0) I (cid:28) (cid:0) 1 (cid:28) (cid:27) so that the price of capital will be determined by I I I I 2 I Q =1+f t + t f t E (cid:12)(cid:3) t+1 f t+1 t I I 0 I (cid:0) t t;t+1 I 0 I (cid:18) t (cid:0) 1(cid:19) t (cid:0) 1 (cid:18) t (cid:0) 1(cid:19) (cid:18) t (cid:19) (cid:18) t (cid:19) Pro(cid:133)ts, arising out of the steady state, are redistributed lump sum to households. 3.4 Equilibrium in the Baseline Model without Runs Toclosethebaselinemodel(inwhichweabstractfromgovernmentinterventionandrunsontheSB-system) we need to specify the equilibrium in the labor market and the aggregate resource constraint. In particular, labor demand and labor supply will be equalized if the following holds Y C 1(1 (cid:11)) t =(cid:31)L(cid:17) t(cid:0) (cid:0) L t t Aggregate output is divided between household consumption C , and investment expenditures I t t I Y =C + 1+f t I (41) t t t I (cid:20) (cid:18) t (cid:0) 1(cid:19)(cid:21) The exogenous processes for productivity A , and monitoring cost (cid:28) , each follow an AR(1) process t t A =(1 (cid:26) )ASS +(cid:26) A +"A t (cid:0) A A t (cid:0) 1 t (cid:28) =(1 (cid:26) )(cid:28)SS +(cid:26) A +"(cid:28) t (cid:0) (cid:28) (cid:28) t (cid:0) 1 t whereasIassumethatthedefaultrateofbadloanspB followsani.i.d. processandisnotcorrelatedwith t "A;"(cid:28). t t 4 A Run on the Shadow Banking System Inthisframework,thepossibilityofhavingtwotypesof(cid:133)nancialintermediaries(cid:133)nancinginvestmentexpenditurescancausetheoccurrenceofanalternativeequilibriumcharacterizedbyarunontheshadowbanking 22

system. In particular, in a run scenario, households stop rolling over their debt with existing shadow banks. As a result, in order to repay their creditors, existing shadow banks have to sell their assets to traditional banks and entering shadow banks. In particular, the latter start operating with their small endowment !sb and no pre-existing debt, and hence are not exposed to runs in the period. If the (cid:133)re-sale value of these assets, arising from the re-intermediation towards banks with low leverage capacity (TB) or very low net worth(enteringSB),islowenough, arunequilibriumthatwipesoutexistingshadowbankswillbepossible. Aftertherunoccurs,newshadowbankswillaccumulatenetworthuntiltheeconomyslowlytransitionsback to the steady state. As in Gertler and Kiyotaki (2013), I assume that at time t 1 the run is a zero probability event for the (cid:0) agentsintheeconomy. Attimethouseholdswilldecidewhethertocontinuetoprovidefundstotheexisting shadow bankers or not. Therefore, the run equilibrium will exist together with the "normal" equilibrium in whichagentskeeplendingtoshadowbanks. Forthisreasonthepossibilityofarunisnottakenintoaccount when deriving the optimal contracts in section 2. A run equilibrium will be possible if investors believe that, if all households stop providing funds to shadow banks, then the SB-system might not be able to satisfy all its creditors. Therefore, if we assume some type of sequential service on the repayments from shadow banks liabilities, then all households have the incentive to withdraw their funds in order to avoid being left with nothing. As noted earlier, given the contract structure for traditional banks, a run on the TB-system will never be possible in this framework. By introducing some changes to the contractual framework we could also allow for this event, however the focus of this paper is on studying a run phenomenon similar to the one that occurred during the great recession, and that mainly a⁄ected the shadow banking sector.25 4.1 Conditions for a Run Asa(cid:133)rststeptocharacterizetherunequilibrium,wecandetermineathresholdvalueforthepriceofcapital Q(cid:22) , below which a run on shadow banks is feasible. In particular, the events of a run happen as follows. At t thebeginningoftimet,householdsdecidewhethertorollovertheirABCPornot. Iftheychoosetorun,the assets of all existing shadow banks are sold to the TB sector and to entering SBs, and the amount obtained is transferred to households. De(cid:133)ne Q as the liquidation price realized in this case. Then, a run will be (cid:3)t possible if the liquidation value of the assets of the SB system is smaller than the claim of creditors, that is (Z +(1 (cid:14))Q )(cid:2) ((cid:25)sb )Ksb <Ssb R (42) t (cid:0) (cid:3)t t t (cid:0) 1 t (cid:0) 1 t (cid:0) 1 t At the individual bank level, given the contract between shadow banks and households, condition (42) is equivalent to having a return on capital in case of a run, Rk ; so low that the payment to households, bG, t(cid:3) t implied by the participation constraint (20) would have to be higher than (cid:22)(cid:18) G Rk , violating limited liability. t(cid:3) That is, a run is possible if the price of assets is so low that even the shadow banks with a good realization for their pool of loans would not be able to repay creditors. When the condition above is satis(cid:133)ed, all investors have the incentive to run in order to avoid being left with zero after the run occurs. Notice that equation (42) can be equivalently stated as the condition such 25Forexamplewecouldassumethatpaymentspromisedbythecommercialbanksdependonthepriceoftheeconomywithout a run Qt,so that there could be the possibility forthe net worth ofcommercialbanks to turn negative as well. However, given the lower leverage of TBs and the high leverage capacity of SBs a TB-run scenario in which SBs continue operating seems quantitatively unfeasible, unless we also introduce some type of costly technology that allows households to directly invest in capitalas in Gertlerand Kiyotaki(2013). 23

thattheaggregatenetworthoftheshadowbankingsystem,notconsideringnewentrants,becomesnegative, that is (Z +(1 (cid:14))Q )(cid:2) ((cid:25)sb )Ksb Q Ksb R <NsbR t (cid:0) (cid:3)t t t (cid:0) 1 t (cid:0) 1(cid:0) t (cid:0) 1 t (cid:0) 1 t t t By using the balance sheet and the leverage equation for shadow banks, we can rewrite this condition as Q <Q(cid:22) (cid:3)t t 1 R 1 where Q(cid:22) = t 1 Q Z (43) t 1 (cid:0) (cid:14) " (cid:2) t ((cid:25)s t b 1 ) (cid:0) (cid:30)s t b 1! t (cid:0) 1 (cid:0) t # (cid:0) (cid:0) Fromthisrelationship,wecananalyzehowthepossibilityofarundependsondi⁄erentendogenouseconomic variables. Inparticular, allthevariablescontributingtoanincreaseinQ(cid:22) willincreasethemeasureofstates t in which a run is possible. Similarly to Gertler and Kiyotaki (2013), Q(cid:22) will be higher when the leverage of t shadow banks, (cid:30)sb , is higher. In addition, a feature unique to this setup is that Q(cid:22) will also be decreasing t 1 t (cid:0) in (cid:2) ((cid:25)sb ), the average quality of loans intermediated by the SB-system. This means that if the quality t t 1 (cid:0) of the assets intermediated by shadow banks starts deteriorating then the possibility of a run will become greater. Such mechanism might have played an important role in igniting the run on several components of the shadow banking system as the foreclosure rate on subprime securities began to surge. Alternatively, given Q(cid:22) and Q , we can characterize the possibility of a run by computing the recovery t (cid:3)t rate on SB assets, that is (cid:2) ((cid:25)sb )(Z +(1 (cid:14))Q )Ksb (cid:13)(cid:22) t = t t (cid:0) 1 t Ssb R (cid:0) (cid:3)t t (cid:0) 1 t 1 t (cid:0) Consequently, if (cid:13)(cid:22) > 1 an ex-post run will not be possible. This quantity can also be interpreted as the t fraction of "alert" withdrawing depositors needed for a run to be possible. We can now turn to the determination of the liquidation price, Q . As mentioned above, I assume that (cid:3)t at the time of a successful run, once the existing shadow banks are liquidated, only traditional banks and entering shadow banks will be able to intermediate capital. It is important to note that, unlike Gertler and Kiyotaki (2014), a run event does not entail any additional output cost, but will have real consequences becauseofthedropininvestmentscausedbythecollapseoftheshadowbankingsystem. Giventhefeatures oftherunequilibrium,itiseasytocharacterizetheeconomywhenarunoccurs. Infact,theonlymodi(cid:133)cation needed is to set the net worth of surviving shadow bankers to zero, so that Nsb =Nsb =(1 (cid:27))!sb t (cid:3) e (cid:0) As a result, at the time of the run, the shadow banking sector(cid:146)s ability to intermediate funds will be dramatically a⁄ected, since its net worth will fall almost to zero. Consequently, if the run occurs during a crisis, the traditional banking sector will have to absorb a substantially larger amount of capital compared to an equilibrium in which a run does not materialize. This will be crucial for the drop in Q that will make t a run possible. Moreover, a drop in the price of capital will also a⁄ect the net worth of traditional banks, making the liquidation price even lower and a run more likely. 24

5 Numerical Exercises Thequantitativeexercisesinthissectionaremeanttoillustratehowtheintroductionoftheshadowbanking system makes the economy more fragile, compared to a framework where only TB operate, the Traditional Banking Economy (TBE), or one where there are no informational asymmetries, the Frictionless Economy (FE). The impulse responses provided are the non-linear perfect foresight paths of the economy, in order to capture the non-linearities arising especially when the endogenous state variables move far from the steady state levels, as it occurs when a run takes place. 5.1 Calibration Table 1 reports the parameters used for the baseline model. Out of the (cid:133)fteen parameters of the model, the seven parameters pertaining to preferences and technology are fairly standard. I use a discount factor (cid:12) = :99, a utility weight on labor (cid:31)(cid:22) = 2 and a Frisch elasticity of 1. The capital share (cid:11) is set at 0.36, the exogenous depreciation rate is 2.5% and the elasticity of the price of capital to investments, given by f , is 00 set at 2.3. The remaining parameters are speci(cid:133)c to my model. I assume (cid:27) = 0:9, which implies that the average life of bankers is 10 quarters. In addition, I normalize pH = 1, and I calibrate the other seven parameters speci(cid:133)c to the banking sector, (cid:18) ;(cid:18) ;pb;(cid:28);(cid:19);!tb;!sb to hit the following targets: a shadow bank leverage of L H 12 compared to a leverage for traditional banks of 4; a level of (cid:25)sb equal to 0.95 and (cid:25)tb equal to 0.975; a steady-state quarterly spread of the return of capital over the risk free rate of 40 basis points, an aggregate quality (cid:2)((cid:25)^) equal to 1 (a normalization), and a share of aggregate capital intermediated by shadow banks equal to .5 . The values for (cid:30)tb and (cid:30)sb are meant to capture the di⁄erence in leverage between traditional banks (leverage around 10) and broker dealers (leverage above 30) in the period preceding the (cid:133)nancial crisis. The aggregatelevelsofleveragearebelowtheactualonesfor(cid:133)nancialintermediariesbecauseinthismodelbanks are directly investing in the equity of the goods producers, and non-(cid:133)nancial (cid:133)rms typically have a much lower leverage. The values for the screening levels are meant to capture a delinquency rate for the loans of traditional banks, 1 (cid:25)tb = 2:5%, similar to the delinquency rate of prime mortgages. On the other hand the implied (cid:0) delinquencyrateforloansoriginatedbyshadowbanksistwiceaslargeat5%,andisaconservativeestimate of the delinquency rate of subprimes.26 Finally, the spread on the return on capital is supposed to be a combination of the average spread on mortgage backed securities, around 100bp annually, and the Baa-treasury spread on corporate bonds, which was larger than 2% annually. Given these parameters, in table 2 I also report the implied steady state values for the "Frictionless Economy" (FE), where (cid:133)nancial frictions are absent, and the "Traditional Banking Economy" (TBE) where the only type of (cid:133)nancial intermediaries present are traditional banks.27 From table 2, we see how the introduction of shadow banks increases capital (by 20%), output (by 8%) and consumption (by 3%) with respect to the TBE, even if loan quality is lower. On the other hand the baseline economy implies lower values for these variables, when compared to the frictionless one. 26Delinquency rates on adjustable rate mortgages (ARM) were above 10% in 2005. 27Inparticular,inthiscaseIassumethatthetransfertotraditionalbanksisequaltothetotaltransfertothe(cid:133)nancialsector in the baseline economy,that is !tbe=!tb+!sb. 25

Inaddition,ifwemeasurethesedi⁄erencesintermsofthechangeinconsumptionthatwouldequalizethe steady state utilities across the di⁄erent economies, we have that the introduction of shadow banks provides a consumption equivalent gain of 3.6% but it falls short of the frictionless economy by about .02%.28 5.2 Crisis experiments InthissetofexamplesIcomparetheresponseofthebaselineeconomy(solidline),intheno-runequilibrium, withtheresponseofthetraditionalbankingeconomy(reddashedline)andofthefrictionlesseconomy(green dotted line) to the same shock. As a (cid:133)rst experiment, I consider a 1% drop in productivity with a persistence of 0.95. Figure (3) reports theresultsforthedi⁄erenteconomies. Inthebaselinemodel,thedropinA negativelya⁄ectsthenetworth t of both banks. As a result, because of the tightening in their balance sheet, banks will have to sell their assets, depressing asset prices and further a⁄ecting the net worth of (cid:133)nancial intermediaries. This sequence of events is in line with the (cid:133)nancial accelerator mechanism described in Gertler and Karadi (2011) and other macroeconomic papers with (cid:133)nancial frictions. However, there are several aspects of a crisis that are unique to this setup. First,wenoticethatduringthedownturn,traditionalbanks(cid:146)assetholdingsincreasebyabout8%,whereas projects funded by shadow banks decrease by about 12%. This is similar to the reintermediation of credit that I have described in the introduction. In fact, because of their higher initial leverage, the decline in net worthwillmakethe(cid:133)nancialconstraintofshadowbanksmorebinding,sothattheywillhavetoo›oadassets to the other (cid:133)nancial sector. However, because of their lower leverage capacity, for traditional banks to be abletoabsorbthecapitalheldbyshadowbanks,pricesneedtoadjustdownward. Thisampli(cid:133)esthe(cid:133)nancial accelerator channel, resulting in an initial drop in Nsb of about 25%, and a prolonged reintermediation as t we can see from the evolution of Ktb and Ksb in the graph. Consequently, aggregate investments and prices t t decrease, causing a slower recovery of the capital stock in the economy. An additional variable that is going to determine the recovery of the economy is the endogenous quality of loans. As we can see in the bottom-right part of (cid:133)gure (3), (cid:25)sb decreases by more than 7% on impact, t whereas(cid:25)tb increasesbyabout0.5%. Asexplainedinsection2,theseoppositemovementsareaconsequence t of the di⁄erent contract structures. Given the drop in prices and the higher expected return on capital, traditional banks (cid:133)nd it optimal to increase their screening e⁄ort. On the other hand, the considerable deterioration in shadow banks net worth decreases the "skin in the game" that they can credibly promise to investors, implying a lower level of screening. The lower quality of loans intermediated by shadow banks during a recession makes the crisis more persistent for two reasons. First, it causes a slower recovery for Nsb, because of the implied lower average t returnintheshadowbankingsector. Inaddition, sincethedropin (cid:25)sb islargerthantheincreasein(cid:25)tb, the t t "aggregate quality" of capital, (cid:2)^, deteriorates by about .3%. As a result, e⁄ective capital falls, contributing to a slower recovery of output. Importantly, this e⁄ect is basically absent when we consider the other two types of economy. If we compare the path of the baseline economy with the TB economy and the Frictionless economy, we see that the productivity shock has larger consequences when shadow banks are present. One reason for this is the lower aggregate leverage in the economy with only traditional banks. This implies a weaker (cid:133)nancial 28It has to be noted that a proper welfare evaluation would require to solve the model globally accounting also for the frequencyofcrisesarisingfrom thebindingincentiveconstraints. Howeverthisisbeyondthescopeofthispaperandisleftfor furtherresearch. 26

accelerator, as evidenced by the fact that the drop in prices is about 80% smaller than in the baseline economy. In addition, in the traditional banking economy there is no deterioration in asset quality during a recession. As a consequence, the net worth of traditional banks su⁄ers a moderate drop and recovers relatively quickly, and the decrease in aggregate capital and labor (not reported) is about one fourth of that experiencedinthebaselineeconomy. TheTB-economybehavessimilarlytothefrictionlesseconomybecause of the low leverage and the absence of interaction between loan quality and funding capacity. However it has to be remembered that, as shown in Table 2, the steady state values of consumption and output are substantially lower in the traditional banking economy. Next,Iconsidera"subprimeshock"thata⁄ectsonlytheaveragequalityofbadloans,(cid:22)(cid:18) B . InparticularI t consideranincreaseinthedefaultrateinthebadregion(adecreaseinpB), withnopersistence, thatcauses t a 10% drop in (cid:22)(cid:18) B . We can think of this experiment as the initial rise in subprime defaults that ignited the t (cid:133)nancial crisis. The results of this experiment are reported in (cid:133)gure (4). Given that bad loans represent only a small portion of total projects, the drop in the aggregate quality of e⁄ective capital at time t, (cid:2)^ is only .3%, t so that this can be considered as a relatively small "capital quality shock" as the one used by Gertler and Karadi (2011) or Gertler and Kiyotaki (2010). In this case, the initial shock hits the net worth of shadow banks more directly because of the higher exposure to bad projects. Once the (cid:133)nancing conditions of banks become tighter, we will have all the same mechanisms described in the previous experiment: the (cid:133)nancial accelerator,thereintermediation,andthedeteriorationofthequalityofprojects(cid:133)nancedbytheSB-system. Asaconsequence,aggregatecapitaldeclinesupto1%sothatoutputandconsumptionexperienceaprolonged decline, despite the absence of persistence in the shock. Ifweconsidertheresponsesinthetraditionalbankingeconomyandthefrictionlesseconomy,weseethat the di⁄erence from the baseline economy is even larger than in the previous experiment. First of all, the initial impact on existing e⁄ective capital is about 50% larger in the shadow banking economy, as can be seen from the di⁄erent drops in (cid:2)^ , because of the larger proportion of bad loans (cid:133)nanced. t In addition, since the shock is i.i.d, this experiment captures in a stark way the slower recovery caused by shadow intermediaries. In fact, in the economy with only traditional banks, the (cid:133)nancial sector is more stable both because of the lower leverage and because bank liabilities are contingent on the aggregate state. Consequently, after the initial shock to Ntb, the economy is able to quickly restore capital by increasing t investments, so that prices are almost unchanged. As a result, at the trough, the drop in output and consumption is less than one-tenth of the one occurring in the baseline economy. Finally, (cid:133)gure (5) presents the evolution of the cross-sectional volatily of equity returns in the (cid:133)nancial sector, for each shock. The (cid:133)rst thing to notice is the spike in (cid:27)~sb , which increases by 65% with the tfp t+1 shock and 40% with the subprime shock. Looking at equation (31), we can see that this is due to three e⁄ects,allresultingfromadeteriorationinNsb: thedropin(cid:25)sb,theincreaseinleverage,(cid:30)sb;andthehigher t t t expected return from capital (due to the tightening of the incentive constraint). On the other hand, (cid:27)~tb is characterized by a much smaller movement in the opposite direction, due to t+1 the opposing movements in (cid:25)sb, (cid:30)tb and E (cid:3) Rk , as can be seen from equation (30). As a result, the t t t t;t+1 t+1 standard deviation in the whole (cid:133)nancial sector increases in both experiments, as it occurred at the peak of the (cid:133)nancial crisis. 27

5.3 An Increase in the Screening Cost After analyzing how the di⁄erent banking systems react to real disturbances, we can now focus on shocks thatdirectlyinteractwiththenatureofthe(cid:133)nancialfrictions. Figure(6)showsthee⁄ectofanunanticipated 5% increase in the variable (cid:28) , which implies a higher marginal cost of screening projects for both banks, t with a persistence of .5. As can be seen from (13) and (28), the immediate e⁄ect of such a shock is a drop in both (cid:25)tb and (cid:25)sb, t t due to the fact that monitoring is now more costly for banks. In the TB-system this is the only e⁄ect that takes place. However, in the shadow banking system, the higher cost of monitoring will also make the UE-friction moresevere,becausenowthecosttoprovideincentivestobankerswillbehigher. Thiswillimplyanegative pressure on the maximum leverage constraint for shadow banks, as described in (24), so that shadow banks willhavetostartsellingtheirassets. Asaresult,spreadswillincreaseandcurrentpriceswilldropgenerating all the ampli(cid:133)cation e⁄ects described in the previous experiments. The interaction of (cid:28) with shadow banks t leverage implies that (cid:25)sb drops about 1% more than (cid:25)tb. In addition, the recovery in aggregate quality is t t slower in the baseline model, because it is also driven by a slower recovery of prices. Interestingly, even if this shock does not have any real consequences on impact, it causes output and consumption to drop up to 0.3% and consumption up to .4% . In the traditional banking economy, the only e⁄ect comes from the initial drop in (cid:25), which however quickly returns to the steady state level, since prices and investments stay almost unchanged. As a result, output experiences a contraction 50% smaller than in the baseline model. 5.4 A Wealth Transfer Within the Financial System In order to illustrate the relevance of the di⁄erent leverage capacities of the two (cid:133)nancial sectors and the consequences of reintermediation, I consider a simple experiment in the baseline model that consists in a transfer from the net worth of shadow banks to the one of traditional banks. In particular, (cid:133)gure (7) shows the e⁄ects of a transfer T equal to 1% of the steady state level of Nsb. t Given the lower leverage of traditional banks, the amount of funds that can be intermediated with T unitsofinternalfundsislowerinthetraditionalbankingsectorthanintheshadowbankingone. Asaresult, aggregateinvestmentsdeclineignitingadropinpricesandtheconsequentampli(cid:133)cationmechanismspresent in the baseline model, implying (cid:133)re sales and a decline in aggregate loan quality. Given the drop in Q , the t actual initial drop in Nsb is actually larger than 1%. t The real e⁄ects of reintermediation will play an important role in the run experiment that I consider in the next subsection. 5.5 A Run Experiment Duringthe(cid:133)nancialcrisis,investorsstoppedconsideringsomeofthesecuritiesissuedbytheshadowbanking system as substitutes for risk-free assets. This caused a run (cid:133)rst on the ABCP market and then on the MMMFs. To model the consequences of these events, I now allow for the possibility of a run on the shadow banking system. Given the liquidation price, Q , and the threshold, Q(cid:22) , determined in (43), a run will be possible in the (cid:3)t t baseline model at time t if and only if Run =Q(cid:22) Q >0 t t (cid:0) (cid:3)t 28

The model calibration implies that a run equilibrium is not possible in steady state. However, during a recession Q(cid:22) will increase and Q will decrease, so that the quantity Run will indicate whether a run is t (cid:3)t t feasible after a shock hits the economy. First, I consider a run occurring after a tfp shock. In addition, in line with the narrative of the recent (cid:133)nancial crisis, I also consider the eventuality of a run happening after an increase in the default rate of riskier loans. In particular, I consider the same shock magnitudes as in (cid:133)gures (3) and (4). Figure (10) shows how, after the initial shock is realized, a run becomes possible for four periods in both cases. In (cid:133)gure (8) and (9), I compare the path of the economy when the run happens in the third period (the dashed red line) to the same response of the baseline economy (solid blue line) when a run does not occur. Both lines are deviations from the steady state of the baseline economy. When the run occurs, traditional banks absorb most of the capital of defaulting shadow banks. As a result, the capital holding of traditional banks increases much more dramatically in this case. Due to the lowleveragecapacityoftraditionalbanks,inorderforthemtobeabletointermediatesuchahigheramount of capital, prices need to drop by almost 10%, instead of the much smaller drop that would have occurred without a run. It is this drop in Q that makes the run possible. At the time of the run, the net worth t of shadow banks drops by more than 95%, causing a much slower recovery for this (cid:133)nancial sector. The considerably larger decrease in asset prices and Nsb implies a noticeable ampli(cid:133)cation of all the feedback t mechanisms that I have described in the previous set of experiments. In fact, in both (cid:133)gure 7 and (cid:133)gure 8, after a run occurs investment drops by more than 10% and asset quality by more than 1.5%, implying a decrease of output and consumption at the trough of more than 2%. In addition, the decrease in prices negatively a⁄ects the net worth of the surviving traditional banks, impairing their ability to invest and further reducing prices. Interestingly, as (cid:133)gure 9 shows, the possibility of a bank run can create a deep and prolonged recession even after a relatively small shock like the subprime one. In both cases, the initial increase in consumption occurring at the time of the run is due to the fact that households are not able to invest in the TB-system all the funds that they have withdrawn from the SB-system. This path for consumption is consistent with that observed during the (cid:133)nancial crisis. 6 Government Intervention In this section, I consider the possibility that the government is willing to facilitate lending by directly purchasing securities in the asset market, similarly to some of the unconventional policies that the Fed put in place during the (cid:133)nancialcrisis. De(cid:133)ne the total amountof assets that are privately intermediated bythe (cid:133)nancial sector as Kp = Ktb+Ksb. Therefore, if the government funds an amount Kg; the total value of t t t t assets will be Q K =Q Kp+Q Kg t t t t t t The government can fund itself frictionlessly by issuing risk-free bonds, Dg, but it does not have the monit toring technology owned by the banking sector. In particular, I assume that the government(cid:146)s projects will havea(cid:133)xedquality(cid:25)(cid:22). Itisalsoimportanttoassumethatthegovermentwillbeabletodiversifyalsoacross regions. On the other hand, as in Gertler and Karadi (2011), I assume that government intermediation causes e¢ ciency costs proportional to the amount funded, (cid:24)Q Kg. t t If we de(cid:133)ne T as government transfers, the government(cid:146)s budget constraint will be t (cid:24)Q Kg+Q Kg+R Dg =Dg+(cid:2) ((cid:25)(cid:22))RkQ Kg +T t t t t t t 1 t t t t 1 t 1 t (cid:0) (cid:0) (cid:0) 29

and if we assume that at every period the government is going to fund its asset purchases through risk-free bonds then Dg =Q Kg, so that we can rewrite the budget constraint as t t t (cid:24)Q Kg =T + (cid:2) ((cid:25)(cid:22))Rk R Q Kg t t t t t (cid:0) t t (cid:0) 1 t (cid:0) 1 (cid:2) (cid:3) To characterize government policy, I assume that the central bank intermediates a fraction of total t assets, Kg = K t t t To model , I assume that the government intervenes when spreads E Rk R rise. In fact, this t t t+1(cid:0) t+1 will occur during a crisis, when prices drop and (cid:133)nancial frictions are tighter. In particular, I consider the (cid:0) (cid:1) following simple rule for government intervention E (cid:2) ((cid:25)(cid:22))Rk R if E (cid:2) ((cid:25)(cid:22))Rk R >0 = 1 t t+1 t+1(cid:0) t+1 t t+1 t+1(cid:0) t+1 (44) t ( (cid:8) (cid:2) 0 (cid:3)(cid:9) (cid:8) (cid:2) otherwise (cid:3)(cid:9) whereIparameterize(cid:25)(cid:22) insuchawaythatE (cid:2) ((cid:25)(cid:22))Rk R iszerointhesteadystateofthebaseline t t+1 t+1(cid:0) t+1 model. This implies that (cid:25)(cid:22) < (cid:25)sb < (cid:25)tb. Therefore, the government will start intermediating assets only (cid:2) (cid:3) when it makes a positive excess expected returnn. The parameter will determine the intensity of the 1 government reaction. 6.1 Crisis Experiment with Government Intervention Figure 11 and 12 show the response of the baseline economy to the same shocks considered in (cid:133)gure (3) and (4), when the credit policy is in place. The parameter is set to 5 (or to 20), so that the government 1 intermediates 2% (or 3%) of assets when the tfp shock hits and 1.5% (2%) when the subprime shock occurs; the ine¢ ciency parameter (cid:24) is set for now at .0025. By intervening in the asset market, the central bank initially prevents the reintermediation from shadow bankstotraditionalbanks. Inthisway,governmentinterventioncontainsthedropinprices,causingafaster recoveryinthenetworthofshadowbanksandreducingtheinitialdeteriorationinassetquality. Theimpact ofcentralbankcreditintermediationontheendogenousqualityofassetsisanovelmechanismofthispaper, highlighting an additional positive e⁄ect of government intervention during a crisis. Under this policy, the drop in output and consumption is up to 50% smaller at the trough than in the baseline model. 6.2 Run Experiment with Government Intervention In this framework, it is interesting to investigate how government intervention interacts with the possibility of a run. Even if the run is unanticipated, I can look at the fraction of "alert" investors, (cid:13)(cid:22) , necessary for a t run to be possible, as an indicator of the exposure of the economy to a run on the SB-system. In particular, I assume that the government policy described above, is in place also in the run equilibrium. The idea is that now, at the time of the run, investors know that the government will act in order to increase the leveragecapacityofthe(cid:133)nancialsystem,whetherthisiscomposedonlyoftraditionalbanksorofbothtypes of (cid:133)nancial intermediaries. In (cid:133)gure (13) I consider the same run experiments analyzed before, for di⁄erent levels of government interventions , and plot the implied (cid:13)(cid:22) . This exercise is performed for three di⁄erent levels of e¢ ciency 1 t 30

costs, (cid:24): 0, 0.0025 and 0.005. As increases, the fraction of investors needed for a run to be possible increases as well, and if (cid:24) =0, a 1 run is not possible already with around 5. The intuition for this result is that if a central bank absorbs 1 part of the assets of SBs, then the liquidation price will be higher in the event of a run. On the other hand, as (cid:24) increases, the government is less e⁄ective in containing the drop in Q , and fewer alert agents are (cid:3)t necessary to have a run. In particular, the parameter (cid:24) can be thought of as either the cost that the central bank has to face to (imperfectly) replicate the unique monitoring technology that banks have, in order to achieve a (cid:25)(cid:22) > 0, or as the cost for diversifying across islands. The key feature of this experiment is that it captures how, the fact that the Fed tried to replace the private demand for di⁄erent types of asset-backed securities probably also helped to prevent additional run episodes on "shadow intermediaries". 7 Conclusion There is no univocal de(cid:133)nition for the term "shadow banking", since it encompasses a variety of (cid:133)nancial markets and institutions that are interconnected in very complex ways. In this paper, I try to capture some of the salient features of this alternative banking system that played a signi(cid:133)cant role in the recent crisis. In particular, I show how (cid:133)nancial innovation and higher diversi(cid:133)cation, which were at the heart of the "originate-to-distribute" model, can make the real economy more fragile in di⁄erent ways. First, the SB-systemincreasestheaggregateleverageofthe(cid:133)nancialsector,amplifyingexogenousshocks. Inaddition, it reduces banks(cid:146)incentives to invest in high quality loans, increasing their exposure to "subprime shocks" and causing procyclical asset quality. Furthermore, I also perform an experiment involving a run on the shadow banking system, aimed at replicating the events that caused turmoil in the markets for ABCP and MMMF shares. As the exercise shows, if a run occurs, the shutdown of the markets for securities on which the modern (cid:133)nancial system heavily relies can have long-lasting consequences for real investment, output, and consumption. Finally, I show that there is scope for government intervention. In fact, by directly purchasing assets that were previously intermediated by the SB-system, the central bank can intervene on the reintermediation process, from shadow banks to traditional banks, that is responsible for the drop in asset prices during a crisis. As a result, such policies can dampen the e⁄ects of a recession and even prevent a run on the SB-system. This framework could be used to study other important policies aimed at improving (cid:133)nancial stability. For example, a leverage restriction similar to the one considered by Christiano and Ikeda (2014) could have thetwofoldbene(cid:133)tofprovidingincentivesforbankscreeningofprojectsanddirectlyreducingthelikelihood of a run, as suggested by Gertler and Kiyotaki (2014). 31

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8 Appendix In this appendix I derive the optimal contracts for three di⁄erent scenarios regarding the observability of projectscharacteristics. Africtionlessscenarioinwhichboththeex-postoutcomeandtheex-antequalityof projects are observable. Then I derive the optimal contract for traditional banks, when the outcome of the project is not observable. And (cid:133)nally I solve the optimal contract for the shadow bank, when the outcome of the project is observable but the ex-ante quality is not. In particular these results hold for a generic increasing linear function (cid:25)(e ) = (cid:20) e +(cid:20) and a convex t 1 t 2 quadratic cost function c(e )=(cid:28) e2+(cid:28) e +(cid:28) . In the paper I assume (cid:20) =1;(cid:20) =0 and (cid:28) =0: t 1 t 2 t 3 1 2 3 8.1 Optimal Contract in the First Best Scenario As explained in the main text, the optimal contract solves maxQ k E (cid:3) (cid:25) efb (cid:22)(cid:18) G Rk bG;fb +(1 (cid:25) (e )) (cid:22)(cid:18) b Rk bB;fb c efb t t t t;t+1 t t t+1(cid:0) t+1 (cid:0) t t t+1(cid:0) t+1 (cid:0) t n h (cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)i (cid:16) (cid:17)o Q k n E (cid:3) (cid:25) efb bg +(1 (cid:25) efb )bb Q k ((cid:22) ) t t (cid:0) t (cid:20) t t;t+1 t t t+1 (cid:0) t t t+1 t t t h (cid:16) (cid:17) (cid:16) (cid:17) i bG;fb (cid:22)(cid:18) G Rk (cid:31)g t+1 (cid:20) t+1 t+1 bB;fb (cid:22)(cid:18) B Rk (cid:0) (cid:31)b (cid:1) t+1 (cid:20) t+1 t+1 The FOCs for efb;k ;bg ;bb are (cid:0) (cid:1) t t t+1 t+1 c (e )=(cid:25) (e )E (cid:3) (cid:1)(cid:22)Rk bg bb +(cid:25) (e )(cid:22) E (cid:3) bg bb 0 t 0 t t t;t+1 t+1(cid:0) t+1(cid:0) t+1 0 t t t t;t+1 t+1(cid:0) t+1 (cid:2) (cid:0) (cid:1)(cid:3) (cid:0) (cid:1) E (cid:3) (cid:25) (e ) (cid:22)(cid:18) g Rk bg +(1 (cid:25) (e )) (cid:22)(cid:18) b Rk bb c(e )=(cid:22) 1 E (cid:3) (cid:25) (e )bg +(1 (cid:25) (e ))bb t t;t+1 t t t+1(cid:0) t+1 (cid:0) t t t+1(cid:0) t+1 (cid:0) t t (cid:0) t t;t+1 t t t+1 (cid:0) t t t+1 n (cid:0) (cid:1) (cid:3) (cid:25) (cid:16) (e )((cid:22) 1)=(cid:31) (cid:17) g o (cid:8) (cid:2) (cid:3)(cid:9) t;t+1 t t t(cid:0) t+1 (cid:3) (1 (cid:25) (e ))((cid:22) 1)=(cid:31)b t;t+1 (cid:0) t t t(cid:0) t+1 From the last two equations we see that either (cid:31)G and (cid:31)B are both positive or they are both zero. t+1 t+1 However if they were both positive the bank would not be obtaining any payo⁄from funding the projects. As a result (cid:31)g =0 and (cid:31)b =0, so that the main equations become t+1 t+1 c efb =(cid:25) efb E (cid:3) (cid:1)(cid:22)Rk 0 t 0 t t t;t+1 t+1 (cid:16) (cid:17) (cid:16) (cid:17) E (cid:3) (cid:25) efb (cid:22)(cid:18) g Rk +(1 (cid:25) efb )(cid:22)(cid:18) b Rk c efb =1 t t;t+1 t t+1 (cid:0) t t+1 (cid:0) t n (cid:16) (cid:17) (cid:16) (cid:17) o (cid:16) (cid:17) (cid:22) =1 t whereIusedE (cid:3) R =1. The(cid:133)rsttwoequationsarethesameas(6)and(7)inthemaintext,whereas t t;t+1 t+1 thelastequationsimplystatesthatinthe(cid:133)rstbestthemarginalvalueofaunitofnetworthisequaltoone. 34

8.2 Optimal Contract for the Traditional Bank As explained in the paper, the one period contract for the traditional bank will be given by the solution of the following max Q ktb E (cid:3) (cid:25) etb (cid:22)(cid:18) G Rk +(1 (cid:25) etb )(cid:22)(cid:18) B Rk btb c etb ktb;etb;btb t t t t;t+1 t t+1 (cid:0) t t+1(cid:0) t+1 (cid:0) t t t t+1 n h (cid:0) (cid:1) (cid:0) (cid:1) i (cid:0) (cid:1)o b (cid:18) Rk (IC) (! ) t+1 (cid:20) L t+1 t+1 Q ktb ntb E (cid:3) b Q ktb (PC) (cid:21)tb t t (cid:0) t (cid:20) t t;t+1 t+1 t t t The (cid:133)rst order conditions w (cid:0) ith respect (cid:1) to ktb;btb ;etb are (cid:16) (cid:17) t t+1 t E (cid:3) (cid:2) (etb)Rk btb c((cid:25)tb) (cid:21)tb 1 E (cid:3) btb =0 (45) t t;t+1 t+1 t t+1(cid:0) t+1 (cid:0) t (cid:0) t (cid:0) t t+1 t+1 (cid:2) Q ktb (cid:3) (cid:3) (cid:21)tb 1 =! (cid:2) (cid:3) (46) t t t+1 t (cid:0) t+1 h i c(etb)=(cid:25) (etb)E (cid:3) (cid:1)(cid:22) Rk (47) 0 t 0 t t t+1 t+1 t+1 where! and(cid:21)tb aretheLagrangemultipliersontheincentiveconstraintandtheparticipationconstraint. t+1 t The last equation directly determines the screening level for traditional banks, as reported above. In addition, from (46), we see that the SC will bind if (cid:21)tb 1 > 0 , a condition that we assume to hold in a t (cid:0) neighborhood of the steady state. Then, substituting the incentive constraint into (45) this condition can be rewritten as E (cid:3) (cid:2) (etb)Rk (cid:18) Rk c(etb) (cid:21)tb = t t;t+1 t+1 t t+1(cid:0) L t+1 (cid:0) t >1 t 1 (cid:18) E (cid:3) Rk (cid:2) (cid:0) L t t+1 t+1 (cid:3) which implies E (cid:3) (cid:2) (etb)Rk R c(etb) t t;t+1 t+1 t t+1(cid:0) t+1 (cid:0) t >0 1 (cid:18) E (cid:3) Rk (cid:2) (cid:0) L t t+1 t+1 (cid:3) that indicates how the incentive constraint for the traditional bank implies a wedge between the expected return on capital and the risk-free rate. Finally,wecancombinethe(IC)the(PC)inordertoobtainanexpressionfortheleverageratioreported in (11) 1 Q ktb = ntb =(cid:30)tbntb t t 1 (cid:18) E (cid:3) Rk t t t (cid:0) L t t+1 t+1 In addition from the PC we can think(cid:2)of the face value of(cid:3)the debt raised by the TB as being given by btb Q ktb (cid:30) RTB = t+1 t t =(cid:18)LRk t t+1 Q ktb ntb t+1(cid:30) 1 t t (cid:0) t t(cid:0) (cid:0) (cid:1) 8.3 Optimal Contract for the Shadow Bank In this section I report the complet solution to the problem to the optimal contract of the shadow bank. As explained in the paper, the problem to be solved is the following: 35

max Q k E (cid:3) (cid:25) esb (cid:22)(cid:18) G Rk bG;sb +(1 (cid:25) esb ) (cid:22)(cid:18) B Rk bB;sb c esb ksb;esb;bg;sb;bb;sb t t t t;t+1 t t+1(cid:0) t+1 (cid:0) t t+1(cid:0) t+1 (cid:0) t t t t+1 t+1 n h (cid:0) (cid:1)(cid:16) (cid:17) (cid:0) (cid:1) (cid:16) (cid:17)i (cid:0) (cid:1)o R (Q k n ) (cid:25) esb bG;sb+(1 (cid:25) esb )bB;sb Q k (cid:22) (PC) t+1 t t (cid:0) t (cid:20) t t+1 (cid:0) t t+1 t t t+1 h (cid:0) (cid:1) (cid:0) (cid:1) i (cid:0) (cid:1) c esb (cid:25) esb E (cid:3) (cid:1)(cid:22)Rk bG;sb bB;sb ((cid:26) ) (IC) 0 t (cid:20) 0 t t t;t+1 t+1(cid:0) t+1 (cid:0) t+1 t (cid:0) (cid:1) (cid:0) (cid:1) bG;sb (cid:22)(cid:18) h G Rk (cid:31) (cid:16) g (LL) (cid:17)i t+1 (cid:20) t+1 t+1 bB;sb (cid:22)(cid:18) B Rk (cid:0) (cid:31)b (cid:1) (LL) t+1 (cid:20) t+1 t+1 where (cid:22) ;(cid:26) ;(cid:31)g and (cid:31)b are the multipliers asso (cid:0) ciated (cid:1) with each constraint. The implied FOCs are t+1 t t+1 t+1 esb :c esb =(cid:25) esb E (cid:3) (cid:1)(cid:22)Rk bG;sb bB;sb +(cid:25) esb E (cid:22) bG;sb bB;sb (cid:26) c esb t 0 t 0 t t t;t+1 t+1(cid:0) t+1 (cid:0) t+1 0 t t t+1 t+1 (cid:0) t+1 (cid:0) t 00 t (cid:0) (cid:1) (cid:0) (cid:1) h (cid:16) (cid:17)i (cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1) k : E (cid:3) (cid:25) esb (cid:22)(cid:18) G Rk bg +(1 (cid:25) esb ) (cid:22)(cid:18) B Rk bb c esb = t t t;t+1 t t t+1(cid:0) t+1 (cid:0) t t t+1(cid:0) t+1 (cid:0) t E (cid:22) R n (cid:0) (cid:25)(cid:1)(cid:16) esb bg +(1 (cid:17) (cid:25) esb )bb(cid:0) (cid:1) (cid:16) (cid:17)o (cid:0) (cid:1) t t+1 t+1 (cid:0) t t t+1 (cid:0) t t t+1 (cid:8) (cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)(cid:9) bG;sb :(cid:26) (cid:25) esb) (cid:3) =(cid:25) esb) (cid:22) (cid:3) (cid:31)g t+1 t 0 t t;t+1 t t t+1(cid:0) t;t+1 (cid:0) t+1 bB;sb : (cid:26) (cid:25) es(cid:0)b) (cid:3) (cid:1) =(cid:31)b (cid:0) (1 (cid:1)(cid:0) (cid:25) esb) ) (cid:22) (cid:1) (cid:3) t+1 t 0 t t;t+1 t+1(cid:0) (cid:0) t t t+1(cid:0) t;t+1 (cid:22) :R (cid:0) (cid:30)sb(cid:1) 1 = (cid:25) esb bG;sb+( (cid:0) 1 (cid:25) (cid:1) (cid:0) esb )bB;sb (cid:30)sb(cid:1) t+1 t+1 t (cid:0) t t t+1 (cid:0) t t t+1 t (cid:16) (cid:17) h (cid:0) (cid:1) (cid:0) (cid:1) i (cid:26) : c esb) =(cid:25) esb) E (cid:3) (cid:1)(cid:22)Rk bG;sb bB;sb t 0 t 0 t t t;t+1 t+1(cid:0) t+1 (cid:0) t+1 (cid:0) (cid:1) (cid:0) (cid:1) h (cid:16) (cid:17)i where in the (cid:133)rst equation I have used that (cid:25) (e )=0. 00 t First notice that if (cid:26) > 0 then it can(cid:146)t be that (cid:31)g = 0 and (cid:31)b = 0 otherwise this would imply t t+1 t+1 (cid:22) (cid:3) = 0 and then (cid:26) = 0, a contradiction, therefore at least one of the two payment has to be t+1(cid:0) t;t+1 t at the maximum. In addition, setting both payments to the maximum would not be optimal since it would (cid:0) (cid:1) imply that the bank does not receive any payo⁄, so that only one limited liability constraint can be binding. In particular, by combining the (cid:133)rst order conditions for bG;sb and bB;sb it can be seen that the only case t+1 t+1 compatible with (cid:26) >0 is (cid:31)g =0 and (cid:31)b >0 = bB;sb (cid:22)(cid:18) b Rk , the intuition being that setting bB;sb t t+1 t+1 ) t+1 (cid:20) t+1 t+1 to its maximum improves on the incentive constraint on monitoring. As a result, the FOCs for bG;sb implies t+1 (cid:25) esb (cid:22) =(cid:3) (cid:26) 0 t +1 (48) t+1 t;t+1 t (cid:25) esb " (cid:0) t (cid:1) # (cid:0) (cid:1) and if we substitute this relationship in the FOC for ksb we obtain t (cid:25) esb 1 E (cid:3) (cid:2) esb Rk R c esb =(cid:26) 0 t (49) t t;t+1 t+1 t t+1(cid:0) t+1 (cid:0) t t (cid:25) esb (cid:30)sb (cid:0) t (cid:1) t (cid:8) (cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1)(cid:9) (cid:0) (cid:1) where (cid:30)sb =Q ksb=nsb. t t t t 36

Therefore, when the (IC) binds there will be a positive spread between the expected return on capital and the risk-free rate, as indicated by equation (21). In addition, if the (IC) binds then we can rewrite the FOC for e and the incentive constraint as t (cid:26) c esb =(cid:25) esb E (cid:22) bG;sb (cid:22)(cid:18) B Rk t 00 t 0 t t t+1 t+1 (cid:0) t+1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:16) (cid:17) c esb =(cid:25) esb E (cid:3) (cid:22)(cid:18) G Rk bG;sb 0 t 0 t t t;t+1 t+1(cid:0) t+1 (cid:0) (cid:1) (cid:0) (cid:1) h i From the (cid:133)rst equation we see that if (cid:26) >0 then E (cid:22) bG;sb (cid:22)(cid:18) b Rk >0, and because of (48) this t t t+1 t+1 (cid:0) t+1 also implies E (cid:3) bG;sb (cid:22)(cid:18) B Rk >0: As a result, if we(cid:16)rewrite the sec(cid:17)ond equation as t t;t+1 t+1 (cid:0) t+1 (cid:16) (cid:17) c (e )=E (cid:3) (cid:1)(cid:22) Rk E (cid:3) bG;sb (cid:22)(cid:18) B Rk 0 t t t;t+1 t+1 t+1(cid:0) t t;t+1 t+1 (cid:0) t+1 (cid:16) (cid:17) you obtain equation (22) c esb <E (cid:3) (cid:1)(cid:22) Rk 0 t t t;t+1 t+1 t+1 (cid:0) (cid:1) This is an important relationship since it implies that the screening e⁄ort of shadow banks is lower than the one of traditional banks, so that (cid:25)(esb)<(cid:25)(etb) (50) t t Next, from the (PC) we can obtain the payment to the bank in the good state (cid:30)sb 1 bG;sb = 1 R t (cid:0) (1 (cid:25) esb )(cid:22)(cid:18) B Rk (51) t+1 (cid:25)(esb)2 t+1(cid:16) (cid:30)sb (cid:17) (cid:0) (cid:0) t t t+13 t t (cid:0) (cid:1) 4 5 and by substituting this in the (IC) we obtain the leverage constraint reported in the main text (cid:25) esb) (cid:30)sb 0 t (52) t (cid:20) (cid:25) esb c esb (cid:25) esb E (cid:3) (cid:2) esb Rk 1 t t 0 t (cid:0) 0 t (cid:0) t t(cid:1);t+1 t+1 t t+1(cid:0) (cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:2) (cid:0) (cid:1) (cid:3)(cid:9) Finally, substituting (48) and (51) in the FOC for esb one obtains t (cid:25) esb (cid:26) c (esb)=(cid:25) esb) (cid:26) 0 t +1 E (cid:3) bG (cid:22)(cid:18) B Rk t 00 t 0 t " t (cid:25) (cid:0) es t b (cid:1) # t t;t+1 t+1(cid:0) t+1 (cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1) and by using the (IC) at equality (cid:25) esb (cid:26) c esb =(cid:25) esb (cid:26) 0 t +1 E (cid:3) (cid:1)(cid:22) Rk c esb) t 00 t 0 t " t (cid:25) (cid:0) es t b (cid:1) # t t;t+1 t+1 t+1(cid:0) 0 t (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3) (cid:0) (cid:1) Finally if we substitute for (cid:26)sb from (49) we obtain the equation determining esb t t (cid:25) esb E(cid:3) (cid:1)(cid:22)Rk c esb (cid:25) esb c esb (cid:25) esb c esb = (53) 0 t t;t+1 t+1(cid:0) 0 t t t 0 t (cid:0) 0t t t (cid:2) (cid:0) E (cid:1) (cid:3) (cid:2) esb Rk (cid:0) (cid:1) R (cid:3)(cid:8) (cid:0) c (cid:1) esb(cid:0) (cid:1) (cid:25) esb(cid:0) c (cid:1) (es(cid:0)b) (cid:1)(cid:9) t t;t+1 t t+1(cid:0) t+1 (cid:0) t t t 00 t (cid:8) (cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1)(cid:9)(cid:2) (cid:0) (cid:1) (cid:3) 37

At this point, we can use such equation to study the determinants of esb. t Let(cid:146)s de(cid:133)ne g(esb;E (cid:3) Rk )= E (cid:3) (cid:2) esb) Rk 1 c esb (cid:25) esb c esb t t t;t+1 t+1 t t;t+1 t+1 t t+1(cid:0) (cid:0) t t t 00 t (cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9)(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) (cid:25) (e )E(cid:3) (cid:1)(cid:22)Rk c esb (cid:25) esb) c esb (cid:25) esb) c esb =0 (cid:0) 0 t t;t+1 t+1(cid:0) 0 t t t 0 t (cid:0) 0t t t (cid:2) (cid:0) (cid:1)(cid:3)(cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) Therefore we can obtain dg = (cid:25) esb c esb) +(cid:25) esb c esb E (cid:3) (cid:2) esb Rk 1 c esb desb 0 t 00 t t t 000 t f t t;t+1 t t+1(cid:0) (cid:0) t g t + (cid:2) c (cid:0) es(cid:1)b) (cid:0) (cid:25) e (cid:1)sb c (cid:0) esb (cid:1) (cid:25) (cid:0) es(cid:1)b(cid:3) c esb >0 (cid:0) (cid:1) (cid:0) (cid:1) 00 t t t 0 t (cid:0) 0t t t (cid:0) (cid:1)(cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) Given (49), and the fact that (cid:25) esb c esb) (cid:25) esb c esb 0 since this quantity is proportional to the t t 0 t (cid:0) 0t t t (cid:21) objective of the banker when the constraint binds, then, as long as c esb 0 (as it is implied by the cost (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 000 t (cid:21) function we use) we have that dg >0. desb (cid:0) (cid:1) t In addition, dg = (cid:25) esb) c esb) E (cid:2) esb) (cid:25) esb) E (cid:1)(cid:22) (cid:25) esb) c esb) (cid:25) esb) c esb) dE (cid:3) Rk t t 00 t t t+1 t (cid:0) 0 t t t+1 t t 0 t (cid:0) 0t t t t t;t+1 t+1 = (cid:25) (cid:0) esb (cid:1) c (cid:0) esb E (cid:1) (cid:22)(cid:18) B +E (cid:0) (cid:1)(cid:22) (cid:1) c es (cid:0) b + (cid:1) (cid:25) esb E (cid:8) (cid:1)(cid:22) (cid:0) (cid:25) (cid:1) es (cid:0) b c (cid:1) esb (cid:0) c (cid:1) esb (cid:0) > (cid:1) 0 (cid:9) t t 00 t t t+1 t t+1 t t t t+1 t t 00 t (cid:0) 0 t (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) where the term in the square brackets is positive for the class of cost functions that we consider. At this point, if we employ the implicit function theorem we will have desb dg=dE (cid:3) (cid:1)(cid:22)Rk t = t t;t+1 t+1 <0 dE (cid:3) (cid:1)(cid:22)Rk (cid:0) dg=desb t t;t+1 t+1 t so that the monitoring intensity of shadow banks will be decreasing in the expected return on capital. Finally, if we use the cost function c(e)= (cid:28) e2+(cid:19)e we obtain 2 (cid:0) (cid:1) E (cid:3) R (cid:18)BRk (cid:25)sb =esb =2 t t;t+1 t+1 (cid:0) t+1 t t E(cid:3) h(cid:1)(cid:22) Rk (cid:28)(cid:19)i t;t+1 t+1 t+1(cid:0) 2 (cid:2) (cid:3) 38

9 Tables and Figures Table 1: Parameters Parameter Value Description Conventional (cid:12) 0.99 Discount rate (cid:17) 1 1 Frisch Elasticity (cid:0) (cid:31) 2 Labor Utility weight (cid:11) .36 Capital Share in Production (cid:14) .025 Depreciation Rate f 2.3 Elasticity of Price to Investments 00 Bank-speci(cid:133)c Target (cid:18) 1.036 High Idiosyncratic Realization (cid:30)sb =12 H (cid:18) 0.747 Low Idiosyncratic Realization (cid:30)tb =4 L pG=pB 1.6 Ratio of Success Rates Rk R=:004 (cid:0) !tb .18 Transfer to Traditional Banks Ktb=K =:5 !sb .05 Transfer to Shadow Banks Ksb=K =:5 (cid:27) .9 Bankers survival probability 10qtr horizon (cid:28) .19 Monitoring Cost Parameter (cid:25)sb =:95 (cid:19) -.94 Monitoring Cost Parameter (cid:25)tb =:975 39

Table 2: Steady State Values Steady State Values Baseline Model Traditional Banking Economy Frictionless Economy Q 1 1 1 Y 2.2390 2.0706 2.3234 Ch 1.7264 1.6659 1.7822 K 20.5062 17.1137 22.5962 (cid:2)^ 1 1.0014 1.0011 Ktb=K .5 1 Ksb=K .5 (cid:30)tb 4 4.0502 (cid:30)sb 12 (cid:25)tb .975 0.9771 (cid:25)sb .97 (cid:25)^ .9625 0.9771 0.9738 Rk R .004 .008 .002 (cid:0) 40

Figure 3: TFP shock, 1% drop in A , persistence .95 t A 0 -0.2 -0.4 -0.6 -0.8 -1 0 20 40 ss morf ∆% Y C 0 0 -0.5 -0.5 -1 -1 -1.5 -2 -1.5 0 20 40 0 20 40 K 0 -0.5 -1 -1.5 -2 0 20 40 Baseline TBE FE ss morf ∆% Q Θˆ 1 0.1 0 0 -0.1 -1 -0.2 -2 -0.3 -3 -0.4 0 20 40 0 20 40 Ksb 0 -2 -4 -6 -8 -10 -12 0 20 40 ss morf ∆% Nsb πsb 0 2 -5 0 -10 -2 -15 -4 -20 -25 -6 -30 -8 0 20 40 0 20 40 Ktb 8 6 4 2 0 -2 0 20 40 ss morf ∆% Ntb πtb 8 0.4 6 0.3 4 0.2 2 0.1 0 -2 0 -4 -0.1 0 20 40 0 20 40 41

Figure 4: Subprime shock, 10% drop in θ¯B t θB 0 -2 -4 -6 -8 -10 0 20 40 ss morf ∆% Y C 0 0.2 -0.1 0 -0.2 -0.2 -0.3 -0.4 -0.4 -0.5 -0.6 0 20 40 0 20 40 K 0 -0.5 -1 -1.5 0 20 40 Baseline TBE FE ss morf ∆% Q Θˆ 0.5 0.1 0 0 -0.1 -0.5 -0.2 -1 -0.3 -1.5 -0.4 0 20 40 0 20 40 Ksb 0 -1 -2 -3 -4 -5 -6 0 20 40 ss morf ∆% Nsb πsb 0 1 0 -5 -1 -10 -2 -3 -15 -4 -20 -5 0 20 40 0 20 40 Ktb 4 3 2 1 0 -1 0 20 40 ss morf ∆% Ntb πtb 4 0.3 3 0.2 2 1 0.1 0 0 -1 -2 -0.1 0 20 40 0 20 40 42

Figure 5: Cross-sectional standard deviation of equity returns 70 60 50 40 30 20 10 0 -10 0 10 20 30 40 ss morf ∆% Cross-sectional stdev Tfp-shock 40 35 30 25 20 15 10 5 0 -5 0 10 20 30 40 ss morf ∆% Cross-sectional stdev Subprime-shock std-dev SB std-dev TB std-dev Financial Sector 43

Figure 6: Screening cost shock, 5% increase in τ , peristence .5 t τ 6 5 4 3 2 1 0 0 20 40 ss morf ∆% Y C 0.1 0.2 0 0 -0.1 -0.2 -0.2 -0.4 -0.3 -0.6 0 20 40 0 20 40 K 0.5 0 -0.5 -1 0 20 40 Baseline TBE FE ss morf ∆% Q Θˆ 0.4 0.1 0.2 0 0 -0.1 -0.2 -0.2 -0.4 -0.3 0 20 40 0 20 40 Ksb 0 -0.5 -1 -1.5 -2 -2.5 -3 0 20 40 ss morf ∆% Nsb πsb 0 1 0 -2 -1 -4 -2 -6 -3 -8 -4 0 20 40 0 20 40 Ktb 1.5 1 0.5 0 -0.5 0 20 40 ss morf ∆% Ntb πtb 1.5 1 1 0 0.5 0 -1 -0.5 -2 -1 -1.5 -3 0 20 40 0 20 40 44

Figure 7: One time transfer from SB to TB equal to 1% of SB net worth Y 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 0 20 40 ss morf ∆% C I 0.04 0.1 0.02 0 0 -0.02 -0.1 -0.04 -0.2 -0.06 -0.08 -0.3 0 20 40 0 20 40 K 0 -0.05 -0.1 -0.15 -0.2 0 20 40 ss morf ∆% Q Θˆ 0.05 0.01 0 0 -0.05 -0.01 -0.1 -0.02 -0.15 -0.2 -0.03 0 20 40 0 20 40 Ksb 0 -0.5 -1 -1.5 0 20 40 ss morf ∆% Nsb πsb 0 0.2 -0.5 0 -1 -0.2 -1.5 -0.4 -2 -2.5 -0.6 -3 -0.8 0 20 40 0 20 40 Ktb 1 0.8 0.6 0.4 0.2 0 0 20 40 ss morf ∆% Ntb πtb 1 0.04 0.8 0.03 0.6 0.02 0.4 0.01 0.2 0 0 -0.01 0 20 40 0 20 40 45

Figure 8: Run on the SB-system after a TFP shock Y 0 -1 -2 -3 -4 0 20 40 ss morf ∆% C I 0.5 5 0 0 -0.5 -1 -5 -1.5 -10 -2 -2.5 -15 0 20 40 0 20 40 K 0 -1 -2 -3 -4 -5 -6 0 20 40 Baseline No-Run Run at t=3 ss morf ∆% Q Θˆ 2 0.5 0 0 -2 -4 -0.5 -6 -1 -8 -10 -1.5 0 20 40 0 20 40 Ksb 0 -10 -20 -30 -40 -50 -60 0 20 40 ss morf ∆% Nsb πsb 0 10 -20 0 -10 -40 -20 -60 -30 -80 -40 -100 -50 0 20 40 0 20 40 Ktb 50 40 30 20 10 0 0 20 40 ss morf ∆% Ntb πtb 50 2 40 1.5 30 1 20 0.5 10 0 0 -10 -0.5 0 20 40 0 20 40 46

Figure 9: Run on the SB-system after a subprime shock Y 0 -0.5 -1 -1.5 -2 -2.5 0 20 40 ss morf ∆% C I 2 5 1 0 0 -5 -1 -10 -2 -3 -15 0 20 40 0 20 40 K 0 -1 -2 -3 -4 -5 0 20 40 Baseline No-Run Run at t=3 ss morf ∆% Q Θˆ 2 0.5 0 0 -2 -4 -0.5 -6 -1 -8 -10 -1.5 0 20 40 0 20 40 Ksb 0 -10 -20 -30 -40 -50 -60 0 20 40 ss morf ∆% Nsb πsb 0 10 -20 0 -10 -40 -20 -60 -30 -80 -40 -100 -50 0 20 40 0 20 40 Ktb 50 40 30 20 10 0 0 20 40 ss morf ∆% Ntb πtb 50 2 40 1.5 30 1 20 0.5 10 0 0 -10 -0.5 0 20 40 0 20 40 47

Figure 10: Run feasibility x 10-3 Run Variable Tfp-shock x 10-3 Run Variable Subprime-shock 6 4 3 4 2 2 1 0 0 -2 -1 -4 -2 -6 -3 0 10 20 30 40 0 10 20 30 40 48

Figure 11: TFP shock and credit policy Y 0 -0.5 -1 -1.5 -2 0 20 40 ss morf ∆% C I 0 1 0 -0.5 -1 -2 -1 -3 -4 -1.5 -5 0 20 40 0 20 40 K 0 -0.5 -1 -1.5 -2 0 20 40 Baseline Ψ =5 Ψ =20 1 1 ss morf ∆% Q Θˆ 1 0.1 0 0 -0.1 -1 -0.2 -2 -0.3 -3 -0.4 0 20 40 0 20 40 Ksb 0 -2 -4 -6 -8 -10 -12 0 20 40 ss morf ∆% Nsb πsb 0 2 -5 0 -10 -2 -15 -4 -20 -25 -6 -30 -8 0 20 40 0 20 40 Ktb 8 6 4 2 0 0 20 40 ss morf ∆% Ntb Ψ 8 4 6 3 4 2 2 1 0 -2 0 -4 -1 0 20 40 0 20 40 49

Figure 12: Subprime shock and credit policy Y 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 20 40 ss morf ∆% C I 0.2 1 0.5 0 0 -0.2 -0.5 -0.4 -1 -0.6 -1.5 0 20 40 0 20 40 K 0 -0.5 -1 -1.5 0 20 40 Baseline Ψ =5 Ψ =20 1 1 ss morf ∆% Q Θˆ 0.5 0.1 0 0 -0.1 -0.5 -0.2 -1 -0.3 -1.5 -0.4 0 20 40 0 20 40 Ksb 0 -1 -2 -3 -4 -5 -6 0 20 40 ss morf ∆% Nsb πsb 0 1 0 -5 -1 -10 -2 -3 -15 -4 -20 -5 0 20 40 0 20 40 Ktb 4 3 2 1 0 -1 0 20 40 ss morf ∆% Ntb Ψ 4 2.5 3 2 2 1.5 1 1 0 0.5 -1 0 -2 -0.5 0 20 40 0 20 40 50

Figure 13: Fraction of ”alert” investors necessary for a run (γ¯) as government intervention Ψ 1 increases (TFP-shock on the left and subprime shock on the right) 1.035 1.03 1.025 1.02 1.015 1.01 1.005 1 0.995 0 5 10 15 20 ¯γ 1.035 1.03 1.025 1.02 1.015 1.01 1.005 1 0.995 0 5 10 15 20 Ψ 1 ¯γ Efficiency costs=0 Efficiency costs=.005 Efficiency costs=.0025 Ψ 1 51