Macroprudential Regulation and Lending Standards

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Macroprudential Regulation and Lending Standards R. Matthew Darst, Ehraz Refayet, Alexandros Vardoulakis 2020-086 Please cite this paper as: Darst, R. Matthew, Ehraz Refayet, and Alexandros Vardoulakis (2025). “Macroprudential Regulation and Lending Standards,” Finance and Economics Discussion Series 2020-086r1. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.086r1. 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.

Macroprudential Regulation and Lending Standards∗ R. Matthew Darst† Ehraz Refayet‡ Alexandros P. Vardoulakis§ May, 2025 Abstract Weexaminehowmacroprudentialcapitalrequirementsinteractwithcompetitionbetween banks and non-banks to shape lending standards. Banks have private information and benefit from deposit insurance, while non-banks lack such advantages but are less regulated. We show that higher capital requirements raise banks’ incentives to screen, tightening lending standards despite a decline in lender protections at the contract level. Non-bank competition does not erode but rather strengthens aggregate standards by crowding out riskier bank lending. Optimal capital regulation is lower in the presence of non-banks. Our analysis helps rationalize dynamics in leveraged loan and private credit markets. JEL Classification: G01,G21,G28 Keywords: Lending standards, credit cycles, asymmetric information, non-banks, macroprudential regulation ∗WewouldliketothankMitchellBerlin,GiovanniDell’Ariccia,RobertMarquez,RaoulMinetti,Christine Parlour and seminar participants at the Federal Reserve Board, the IMF, the OCC, and the day ahead conference of the Federal Reserve System, the North American Econometric Society meetings, the Royal Economic Society, and IFABS. All errors are our own. The views expressed in this paper are those of the authors and do not necessarily represent those of the Federal Reserve Board of Governors or anyone in the Federal Reserve System. Refayet: The views expressed in this paper are my own and are based on independent research and do not necessarily reflect the views of the Office of the Comptroller of the Currency, the Department of the Treasury, or the United States government. This paper is the result of my independent research and has not been reviewed by the OCC. †Board of Governors of the Federal Reserve System: matt.darst@frb.gov. ‡Personal Affiliation: ehraz.refayet@gmail.com. §Board of Governors of the Federal Reserve System: alexandros.vardoulakis@frb.gov. 1

1 Introduction In the aftermath of the Global Financial Crisis (GFC), policy makers introduced stricter bank regulation to mitigate system-wide risks and complement the bank-specific risk assessments of microprudential regulators (Hanson, Kashyap, and Stein, 2011; Aikman, Bridges, Kashyap, and Siegert, 2019). At the same time, the stricter bank regulation has pushed activity to non-bank financial institutions, which operate outside the macroprudential regulatory perimeter (Buchak, Matvos, Piskorski, and Seru, 2018; Irani, Iyer, Meisenzahl, and Peydro, 2021). This migration of activity has raised concerns about higher systemic risk because non-banks have less oversight and may extend riskier loans. The left panel in Figure 1 shows that the rapid growth in nonfinancial business credit from non-banks coincides with the macroprudential capital regime that raised banks’ capital ratios. An often raised concern about credit migration to unregulated non-banks is that nonbank loans have fewer lender protections that could amplify losses during a crisis and depress economic activity. However, since the GFC and rise of non-bank lending, default rates have tended to remain low, often below even historic norms. The right panel in Figure 1 shows the relationship between the fraction of covenant-lite leveraged loans (blue bars), a proxy for lender protections, along with realized (red line) and expected (black line) default rates. Overall, the picture is clear: lender protections at the contract level are alarmingly low but realized default rates remain low and below expectations, suggesting that lending standards may not have deteriorated on aggregate. We present a model capturing the following dynamics: (i) the need for a macroprudential regulation, (ii) credit migration to non-banks, (iii) lower lender protection at the contract level, and (iv) lack of deterioration in aggregate credit quality and lending standards.1 The key to understanding how higher capital requirements can lower lender protection at the contract level without an associated deterioration in aggregate lending standards is through their differential impact on the intensive and extensive lending margins. First, consider an environment with only banks that benefit from subsidized deposit insurance. Tighter capital requirements are passed through to borrowers via higher interest rates and competition pushes each individual bank to lower collateral requirements and, 1More recently, these dynamics have also been observed in the rapidly expanding private credit market. See Cai and Haque (2024). 1

250 13 225 12 11 200 10 175 9 150 8 7 125 6 100 5 75 4 50 3 2 25 1 0 0 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 )2102 fo %( gnidnatstuO tnuomA % Risk Weighted Assets 100 12 95 90 11 85 10 80 75 9 70 65 8 60 7 55 50 6 45 40 5 35 4 30 25 3 20 15 2 10 1 5 0 0 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 Average Tier 1 Capital Bank loans Leveraged loans Outstanding nonfinancial business credit held by banks and nonbanks, normalized to 2012 (2012=100). % Risk Weighted Assets is the unweighted average of common equity Tier 1 capital as a percent of risk−weighted assets for G−SIBs, non G−SIBs with total assets above $100B,and other BHCs and IHCs with a substantial US commerical banking presence. Source: For % Risk−Weighted Assets, Federal Reserve Board, Form FR Y−9C, 'Consolidated Financial Statements for Holding Companies.' For bank loans, Statistical Release Z.1 'Financial Accounts of the U.S. For leveraged loan data: PitchBook Data, Inc. )%( snaol etil−tnanevoC Loan default rate (%) Covenant−lite loans Expected loan default rate Realized loan default rate EOM fraction of loans that are covenant−lite, expected default rate aggregate, realized default rate. Dec 2006 − July 2024. Source: PitchBook Data, Inc., Private Equity and Venture Capital Databases Research Platform,http://pitchbook.com/Products.html. For expected default rate: Moody's Analytics, Inc., EDF-X, https:// edfx.com/ Figure 1: Panel (a) - Tier 1 Capital Requirements and Leveraged Loans Commitments from Banks and Non-banks. Panel (b) - Fraction of Covenant-lite Loans and Loan Default Rates hence, lender protection at the contract level to avoid losing borrowers to other banks (the intensive lending effect). At the same time, higher capital requirements increase banks’ skin in the game, incentivizing them to screen out bad loans more frequently to avoid large losses in default (the external lending margin). Hence, the overall pool of loans consists of higher-quality borrowers despite each loan having less collateral pledged. Now, suppose that there is an alternative type of intermediary that, unlike banks, does not benefit from the deposit insurance subsidy akin to Donaldson, Piacentino and Thakor (2021). This puts them at a funding cost disadvantage relative to banks. Macroprudential regulation curtails the benefit accruing from the deposit insurance subsidy to banks and therefore allows for competition between banks and non-banks. Does macroprudential regulation allow non-banks to step in and offer loans to lower quality borrowers that banks screen out? The answer is no. This is because the deposit insurance subsidy is more valuable when making loans to bad borrowers. Hence, non-banks cannot offer a more competitive contract to screened-out bad borrowers. Instead, non-banks compete with banks to attract good borrowers: for a given lending rate, non-banks can profitably set a slightly lower collateral requirement than banks. Thus, macroprudential regulation results in a migration of good borrowers to non-banks with lower collateral requirements, but does not erode the average quality of loans extended in equilibrium. We build on Dell’Ariccia and Marquez (2006) along several dimensions to capture the 2

aforementioned dynamics in an environment with banks and non-banks, and optimally set capital regulation. There are two types of borrowers, good and bad. Good borrowers have projects with positive net present value (NPV), while bad borrowers have projects with negative NPV but with higher upside conditional on success. On average, a portfolio of both good and bad projects has positive NPV. Banks have private information about borrowers with whom they have existing relationships, but there is another set of borrowers whose projects’ quality is unknown. While the mass of known borrowers is fixed, the mass of unknown borrowers can vary and it represents the level of (new) demand for credit. Collateral requirements in loan contracts can be used to screen out bad borrowers but are costly because of inefficient liquidation of collateral in default. Alternatively, loan contracts withoutcollateralrequirementsfeatureonlyacompetitiveinterestrateandpoolallborrowers together. Thus, banks face an adverse selection problem because funding all projects mixes all unknown borrowers (good and bad) with competing banks’ known bad borrowers. There are two alternative lending regimes that determine the aggregate lending standards: (i) lending standards are tighter when collateral is required and bad borrowers are screened out, and (ii) lending standards are looser when all borrowers receive funding without collateral requirements. The level of credit demand determines the equilibrium lending regime—the external lending margin. But within the tighter regime, collateral requirements at the contract level may be higher or lower—the intensive lending margin. With respect to the external margin, higher credit demand by unknown borrowers mitigates adverse selection because their projects have, on average, a positive NPV . In particular, there exists a threshold for aggregate credit demand above which banks switch from a regime with tighter standards to one with looser standards. Capital regulation impacts this decision by affecting the relative cost of lending across the two regimes and intensifying competition from non-banks. We derive the following three results. First, a social planner sets a positive macroprudential capital requirement that trades off the cost of inefficient liquidation of collateral in the tighter-lending-standards regime versus the cost of funding some negative NPV projects in the looser-lending-standards regime. The level of the optimal macroprudential capital requirement determines the threshold for switching from tighter to looser lending standards. Absent non-banks, a higher macroprudential requirement—driven by the desire to decrease the cost of funding some negative NPV projects—improves the aggregate quality of funded 3

projects and reduces aggregate default rates. This is because higher capital requirements effectively erode the deposit insurance subsidy that is more valuable when making loans to riskier borrowers. Hence, banks are less likely to extend such loans and instead use collateral for screening. At the same time, to remain competitive, banks lower the amount of collateral required at the loan level to screen out bad borrowers. In sum, aggregate default rates go down but lender protections at the contract level weaken in response to stricter macroprudential regulation. Second, stricter macroprudential regulation increases competition from non-banks but, strikingly, aggregate lending standards tighten further. In other words, an economy with non-banks supports a tighter lending regime—whereby collateral is used to screen out bad borrowers—for higher levels of credit demand relative to an economy with only banks. The intuition behind this result is as follows. Non-banks can compete more easily with banks in the regime where collateral is used to screen out bad borrowers. The reason is twofold: first, as stated above, the deposit insurance subsidy to banks is more valuable when borrowers are not screened; second, collateral is an equally efficient screening mechanism for both banks and non-banks. This implies that non-banks determine lending costs in the tighter lending-standards-regime, while banks do so in the looser-lending-standards regime. Because, tighter macroprudential regulation does not impact non-banks, the lending cost in the tighter standards regime is unaffected. However, tighter regulation increases the lending cost in the looser standards regime, where credit continues to be intermediated by banks. In sum, compared to an economy with only banks, the cost of lending increases only in the looser standards regime, while it does not affect the cost of screening out bad borrowers in the tighter standards regime. Therefore, a higher level of credit demand is required to make lending profitable with looser standards. In equilibrium, expected default rates are lower. In conjunction with the first result above, this result rationalizes the four aforementioned dynamics observed in the data. Third, the optimal macroprudential capital requirement is lower in the presence of nonbank competition. Recall that the social planner trades off the inefficient collateral liquidation with the funding of some negative NPV projects when choosing macroprudential capital requirements. From the second result above, we know that aggregate lending standards are tighter under non-bank competition for the same level of capital requirement. Thus, the planner could soften capital requirements to economize on liquidation cost of collateral in 4

the tighter regime, while preserving the same aggregate lending standards in the presence of non-bank competition. Related Literature. Our paper contributes to the literature by linking macroprudential regulation and the rise of non-bank lending to aggregate lending standards and financial stability. To that extent, we relate to three broad strands of the literature. The first strand has studied how lending standards evolve along business and credit cycles. Dell’Ariccia and Marquez (2006) and Ruckes (2004) show how lending standards weaken during credit expansions, while Gormley (2014) studies how lender entry influences aggregatecreditextensionandoutputwhennewlenderscanpoachgoodborrowersfromother banks. Moreover, several recent papers study the dynamics of lending standards. Fishman, Parker and Straub (2024) study how screening intensity dynamically affects the quality of the borrower pool, while Farboodi and Kondor (2023) study how sentiment affects credit outcomes and how the quality of the borrower pool endogenously fluctuates with standards, generatingcreditcycles. GortonandOrdonez(2019)andAsriyan, LaevenandMartin(2022) study how the use of collateral in lending contracts affects information acquisition and the emergence of boom-bust cycles. We contribute to this literature by explicitly studying the effect of regulation on lending standards in a modern financial system that features both banks and non-banks. The second strand of the literature has focused more on the rise of non-bank financial intermediation and how they compete with banks. Buchak, Matvos, Piskorski and Seru (2018) and Irani, Iyer, Meisenzahl and Peydro (2021) study the rise of non-bank financial intermediation in mortgage and non-financial business lending markets and establish the role of bank regulation. Donaldson, Piacentino and Thakor (2021) develop a model where banks and non-banks co-exist in equilibrium and show that funding cost differences result in different lending strategies, but do not focus on lending standards. Parlour, Rajan and Zhu (2022)studytheimpactofFinTechcompetitioninpaymentservicesonbankprofitabilityand loan quality when information externalities accrue to banks. Vives and Ye (2025b) focus on the effect of informational technology (IT) on lenders’ competition and monitoring intensity. Vives and Ye (2025a) show how non-banks can exploit IT to price discriminate and poach borrowers from banks, with the relative funding costs between banks and non-banks driving the quality of non-bank loans. We contribute to this literature by explicitly examining how 5

capital regulation and non-bank competition affects lending standards. Moreover, these papers generally abstract from adverse selection in lending and, thus, cannot explain the worsening of lending protections at the contract level without a simultaneous increase in aggregate default rates, which our model delivers. The third strand of the literature examines more closely the effect of regulation on migration of lending activity from banks to non-banks. Begenau and Landvoigt (2021) show that tighter bank capital regulation leads to a shift of activity toward non-banks, though the overall financial system becomes safer due to reduced risk-taking incentives. Dempsey (2025) similarly shows that raising bank capital requirements reduces bank risk-taking and bank failures but prompts firms to shift toward non-bank lenders; yet banks adjust in the long-runandtheoverallquantityofaggregateinvestmentremainslargelyunchanged. Bengui and Bianchi (2022) and Davila and Walther (2022) show that the imperfect implementation and enforcement of regulation causes some activity to leak to unregulated institutions; optimal policy is still useful to reduce financial system’s vulnerability but it should account for these leakages. We contribute to this literature by deriving the optimal macroprudential regulation under adverse selection and non-bank competition. The rest of the paper is organized as follows. Section 2 presents the model with banks as the only financial intermediaries, establishes how capital requirements affect bank competition and lending standards, and set optimal macroprudential capital requirements. Section 3 introduces non-bank competition and the associated implications for macroprudential capital requirements and lending standards. Section 4 derives the optimal capital regulation with non-bank intermediation. Section 5 concludes. All proofs that are not immediately obvious from the text are included in the Appendix. 2 Model with Banks Thissectionpresentsthemodelandderivestheequilibriumwhenbanksaretheonlyfinancial intermediaries in the economy. 6

2.1 Time, Uncertainty, and Agents Consider an economy with two time periods, t = 0,1. Time 0—the most important time period—is broken into a three-stage game that is described below. Consider two types of agents: entrepreneurs/firms and banks, both of which have a discount factor of one. Suppose there is a continuum of firms with mass 1 + λ, each of which has an end of period wealth W that is sufficient to meet any collateral requirement. Each firm is endowed with a risky project that transforms $1 of input at t = 0 into a random output at t = 1. Firms differ in the quality of their projects: a firm has either a good or a bad project that produce y = G or y = B, respectively, when they succeed; for simplicity, both projects G B produce 0 when they fail. In addition, the probability that good (bad) firms produce G(B) is given by p (p ) where p > p . Let the average probability of success be defined by G B G B p = αp +(1−α)p . Moreover, we assume that the bad project has a higher payoff than µ G B the good project when successful, i.e., B > G, but bad projects have a negative net present value, i.e., p G > 1 > p B. Let the fraction of good and bad firms in the economy be given G B by α and (1−α), respectively. The mass of borrowers given by λ ∈ [0,∞) are unknown, i.e., none of the banks know the quality of their project, while the mass of borrowers equal to 1 are known, i.e., at least one bank knows the quality of their projects. There are N > 1 banks that compete for borrowers. Banks are symmetric and each bank knows the quality of a non-overlapping mass of 1/N different firms, i.e., each firm’s quality is known by only one bank. Private information exposes each bank to adverse selection from bad borrowers known only to other banks. Thus, each bank at t = 0 can either attempt to engage in screening and separate bad borrowers from the rest, or pool all borrowers together exposing itself to adverse selection. As described below, banks can use non-price terms, i.e., collateral, to separate good from bad borrowers. There are three stages in the game at time 0. In stage 1 banks offer a menu of contracts to unknown borrowers. The contracts are defined by the tuple (R ,C ), k = {S,P} where k k R is the face value of the debt in either separating (k = S) or pooling (k = P) contracts. k Cj is the corresponding required collateral, which banks can seize if projects fail. We use the k typical assumption that banks only recover a fraction of the value of the posted collateral upon project failure given by κC with κ < 1. Hence, collateral foreclosure is inefficient k and we will assume that the cost 1 − κ is sufficiently high that banks default if the bad 7

state realizes.2 In stage 2, banks observe the outcome of stage 1 and can offer competitive contracts to their known borrowers. Borrowers choose their preferred contract among those offered by all banks. In stage 3, banks may reject loan applicants.3 Debts are repaid or collateral is seized, and agents consume at t = 1. All firms are risk-neutral and maximize expected profits. Firms will consider the loan contract (R ,C ) if expected profits are positive, i.e., k k p (y −R )−(1−p )C ≥ 0. (1) j j k j k Our notion of collateral is quite general and can encompass the most common forms of collateral used in loan contracts as long as two conditions hold: 1) collateral is costly for the firm to pledge because it represents a wealth transfer to the lender, and 2) enforcing claims on the collateral in bankruptcy is costly to the lender. Bankruptcy costs can arise from the legal resources and time needed to settle the priority claims in bankruptcy resolution, from inefficient liquidation, or from the second-best use of assets. The collateral in the model can be physical collateral such as real estate, financial collateral, such as marketable securities, or going concern collateral, such as accounts receivable or blanket liens. Empirically, Caglio, DarstandKalemli-Ozcan(2021), usingloan-levelsupervisorydataaccompaniedwith private-firm balance sheets, show that private firms almost always post collateral in the form of one of the aforementioned types. 2.2 Banks, Risk-shifting, and Microprudential Capital Requirements Banks fund the loans to firms by raising equity capital and deposits in perfectly elastic markets. As in Allen, Carletti and Marquez (2015), we assume that there is a segmented investorbase, suchthattheownersofbanksarewillingtoinjectequityfunding, whileoutside investors are only willing to hold debt instruments. For simplicity, the outside option of the latter is a riskless technology with zero net yield, while the former demand a gross expected 2Aswewillshowindetail,bankswilldefaultinthebadstateformostparametersvaluesevenifκ→1for all admissible capitalization levels. For some other parameters, this requires that κ is below some threshold. Thisisareasonableassumptionbothtosimplifytheexpositionofthedifferentcases,andbasedonempirical observations. KermaniandMa(2023)findthattheU.S.industryaveragerecoveryrateforPPEisonly35%. 3If more than one bank offers the same contract to a group of borrowers, a sharing rule is invoked to guarantee the existence of equilibrium. In particular, all the borrowers that would choose a contract offered by more than one bank are randomly allocated to one of these banks. 8

return E > 1 to supply equity. Equity is long-term and receives payment only at t = 1 after deposits have been paid in full. On the contrary, deposits specify an uncontingent gross payment X ≥ 1 at withdrawal. Because deposits are fully insured, in equilibrium, X = 1, equal to the riskless gross return required by outside investors. Banks pay a premium for insurance, denoted by ι, implying a total cost of deposit funding equal to D = X +ι. The insurance premium cannot be conditioned on the loan type, which is private information to the banks. As such, banks have an incentive to risk shift and lend only to bad projects due to limited liability. The same regulator that insures deposits can eliminate the risk-taking behavior by setting a high enough equity capital requirement, denotedbyγ, suchthatbankshaveenoughskininthegame. Wecalltheserequirements"microprudential" capital requirements, as opposed to "macroprudential" capital requirements that target systemic-wide externalities and we study later on. We follow the literature that considers subsidized deposit insurance premium, which does not reflect the bank’s portfolio risk in equilibrium. As such, we take ι as given, i.e. independent of capital requirements, and for simplicity take ι → 0 in our proofs.4 We make the following two assumptions to ensure banks raise deposits and have an incentive to risk shift: Assumption 1 E > p G. G Assumption 2 Good and bad firms’ project payoffs satisfy G < (1+(p −p )D)/p and G B G B > (p /p )(G−D)+D. G B See Appendix for details how assumption 2 introduces risk-shifting incentives. 2.2.1 Microprudential capital requirements The bank regulator can set the capital requirement to prevent risk-shifting. The regulator knows that B is the maximum gross loan rate bad firms are willing to accept. Thus, it is sufficient to set γ high enough such that the profits from lending to bad firms are lower than the required return of equity, i.e., γE > p [B − (1 − γ)D]. Note that banks repay B 4See also Van den Heuvel (2008), Donaldson, Piacentino and Thakor (2021), and Jermann and Xiang (forthcoming) for the link between deposit insurance and bank risk-taking. 9

deposits only when the projects succeed because of limited liability. The microprudential capital requirement, γ, that prevents risk-shifting in equilibrium is given by p (B −D) B 1 > γ > γ ≡ . (2) E −p D B Under microprudential capital requirements, banks are willing to lend to known good borrowers if the expected profits are higher than the required return on equity. Hence, the minimum gross loan rate that makes banks break-even under microprudential capital requirements is given by E −p D G p [R(γ)−(1−γ)D] = γE ⇒ R(γ) = γ +D. (3) G p G The following assumption guarantees that it is rational for good types to borrow under microprudential capital requirements, i.e., G > R(γ). Assumption 3 pG(G−D) > pB(B−D). E−pGD E−pBD The following proposition summarizes the results of this subsection that relate optimal lending rates under microprudential capital requirements with risk-shifting incentives. Proposition 1 Let project returns satisfy assumptions 1, 2, and 3. The microprudential capital requirement that prevents risk-shifting is given by (2). 2.3 Equilibrium with Banks We solve the game at t = 0 by backward induction and focus on pure-strategy symmetric equilibria. Stage 3 is not interesting, but necessary to obtain a stable equilibrium. Recall that banks can reject borrowers in stage 3, so borrowers cannot coordinate on contracts that provide negative returns to banks. At stage 2, banks will offer their known good borrowers (cid:0) (cid:1) the contract RG,0 which makes them indifferent to their outside option; the contract that is offered to borrowers at stage 1. Note that the contract offered to known good borrowers does not entail collateral because it is costly for borrowers and banks already know their type. Known bad borrowers will not receive credit from their relationship banks in stage 2 because they have negative NPV projects for which banks make losses, in expectations, 10

under the microprudential capital requirements in (2). Therefore, known bad borrowers will only receive credit in stage 1 if in equilibrium all unknown borrowers are pooled into a common contract. We now proceed to solve the game at stage 1 and determine whether a separating or pooling equilibrium ensues. 2.3.1 Separating equilibrium with banks The first equilibrium concept we construct is the separating/screening equilibrium. In particular, banksusecollateralinloancontractstodistinguishbetweengoodandbadborrowers. Define this contract by (R ,C ). Banks want to attract only good borrowers while deterring s s bad borrowers. Good borrowers are willing to borrow if p (G−R )−(1−p )C ≥ 0, (4) G s G s while bad borrowers will not try to mimic the good borrowers if p (B −R )−(1−p )C ≤ 0. (5) B s B s In addition, perfect competition among banks drives expected profits to zero on the contract they offer to good borrowers, i.e., p (R −(1−γ)D)+(1−p )max{(κC −(1−γ)D),0} = γE, (6) G s G s The first term in the left-hand side of (6) are the expected profits when borrowers do not default, while the second term are the residual expected profits at default after repaying deposits, bounded below by zero due to limited liability. The right-hand side is the required return on equity capital. Then, the separating loan contract, offered to both G and B borrowers, is determined by the binding incentive compatibility constraint of bad borrowers (5) and the individual rationality constraint of banks (6). The IC constraint of good borrowers will be non-binding. We show in the proof of Proposition 2 that the second term in the left-hand side of (6) is zero in equilibrium, i.e., κC < (1−γ)D, under sufficiently low κ, which can be ashigh as one s for most parameterizations. Thus, banks’ expected profits are not affected by using costly collateral to screen out bad borrowers and the gross loan rate in the separating allocation is 11

given by R = R(γ) in (3). The following proposition summarizes the optimal contract in s the separating allocation offered by banks in stage 1 of the game at t = 0. Proposition 2 For γ given by (2), the loan contract that banks offer is characterized by 1 R = γE +(1−γ)D, (7) s p G p B C = (B −R ). (8) s s 1−p B We now proceed to derive the condition for a separating equilibrium. A separating allocation is an equilibrium when no bank would choose to offer a profitable pooling contract (Rothschild and Stiglitz, 1976). Define the pooling contract as (R ,0), which does not use p collateral because collateral is costly. A bank offering a pooling contract must, at minimum, break even, which puts a lower bound on R . Given that a pooling contract attracts all p unknown and known bad borrowers other banks reject, the break even condition requires: Expectedprofitsfrombadborrowers Requiredpayoff Expectedprofits rejectedbyotherbanks toequity fromunknownborrowers (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:20) (cid:125)(cid:124) (cid:21) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) N −1 N −1 λp [R −(1−γ)D]+(1−α) p [R −(1−γ)D] ≥ γE λ+(1−α) µ p B p N N (1−α) N−1 [γE +(1−γ)Dp ]+λ[γE +(1−γ)Dp ] ⇒R ≥ N B µ . (9) p λp +(1−α) N−1p µ N B Given R , a good firm would not deviate from the separating allocation if its profits are p higher compared to pooling, i.e., p (y −R ) ≤ p (y −R )−(1−p )C , which yields: G G p G g s G s 1−p G R ≥ R + C . (10) p s s p G Thisconditioncomparestheeffectiveborrowingcostunderpoolingandseparatingcontracts; the cost for the latter comprises both the loan rate and the loss of collateral. Substituting the lower bound for R from (9) in (10), we can derive the necessary and p 12

sufficient condition for the separating allocation to constitute an equilibrium: (1−α) N−1 [γE +(1−γ)Dp ]+λ[γE +(1−γ)Dp ] 1−p N B µ ≥ R + G C . (11) λp +(1−α) N−1p s p s µ N B G Condition (11) does not hold for all λ. Start with the two limiting cases λ → 0 and λ → ∞. Equilibrium is always separating when all borrowers in the economy are known, i.e., λ → 0.5 The intuition is that when borrowers are known, the pooling contract only attracts competitor banks’ known bad borrowers. For λ → ∞, the information asymmetry between competing banks becomes irrelevant because at the limit all borrowers are unknown to all banks. For this case a deviation from the separating contract requires: γE 1−p G +(1−γ)D < R + C . (12) s s p p µ G Condition (12) depends on the average borrower quality among types, α, through p . µ As α → 0, p → p and a deviation from a separating equilibrium is not profitable, which µ B we show in detail below in the proof of Proposition 3. Intuitively, banks always choose to separate borrowers when only bad types exist (α → 0) because they essentially extend credit exclusively to negative NPV projects under pooling. Alternatively, a deviation from a separating equilibrium is always profitable for α → 1, that is p → p , irrespective of µ G the value of λ. Intuitively, when only good types exist, separating borrowers is no longer the optimal strategy because collateral requirements are costly and unnecessary. We show there will be a threshold for α, denoted by α, such that (12) holds and a deviation from a separating equilibrium is profitable for λ → ∞. By continuity and the fact that the lefthand side of (11) is strictly decreasing in λ, we can conclude that there is a threshold for λ, denoted by λ, below which deviations from a separating equilibrium are not profitable. Proposition 3 There exists α > 0 such that condition (12) holds. The equilibrium purestrategies satisfy the following: 1. if α < α, banks offer unknown borrowers the unique separating contract (R ,C ); s s 2. for α > α, there exists 0 < λ < ∞ such that banks offer unknown borrowers the unique separating contract (R ,C ) when λ ≤ λ; s s 5The necessary and sufficient condition for (11) to hold as λ→0 is p >p . G B 13

3. there is no separating equilibrium if and only if α > α and λ > λ. The following Corollary is important for the subsequent analysis. (cid:0) (cid:1) Corollary 1 Known good borrowers receive the contract RG,0 , with RG = R + (1 − s p )/p C . G G s To show this Corollary observe that each bank can offer a contract to its known good borrowers that does not require any collateral. These borrowers will prefer this contract if the loan rate, RG, is less than or equal to the effective repayment under a separating contract offered by a competitive banks, given by R +(1−p )/p C . Hence, a bank can s G G s offer known good borrowers a contract that makes them indifferent and extract all surplus given by RG −R = (1−p )/p C . s G G s 2.3.2 Pooling Equilibrium with banks Pooling allocations are possible when profitable deviations from the separating equilibrium exist. In particular, in the pooling equilibrium, banks offer all firms the contract defined by the break-even condition (9) set to equality: (1−α) N−1 [γE +(1−γ)Dp ]+λ[γE +(1−γ)Dp ] R = N B µ . (13) p λp +(1−α) N−1p µ N B Proposition 4 If α > α and λ > λ, the unique equilibrium pure-strategy profile is the pooling allocation given by (R ,0). p Proposition 3 showed that there is no separating equilibrium for α > α and λ > λ. It then suffices to establish that the pooling strategy is a stable equilibrium. To see this, assume (11) does not hold so a pooling equilibrium is possible. Then, assume that some bank offers a “cream-skimming” deviation, (R(cid:48)(cid:48),C(cid:48)(cid:48)) that is preferred by good borrowers but not bad borrowers compared to the pooling contract (R ,0), i.e., R(cid:48)(cid:48)+[(1−p )/p ]C(cid:48)(cid:48) < R p G G p and R(cid:48)(cid:48) +[(1−p )/p ]C(cid:48)(cid:48) > R . Thus, all good borrowers will choose to borrow from that B B p bank, while all the other banks offering the pooling contract will attract only bad borrowers. The fact that all the contracts offered at stage 1 of the game are observable by all banks at stage 2 suffices to prevent such “cream-skimming” from being a profitable deviation in equilibrium. The reasoning behind this result can be described in the following steps: 14

(i) If the “cream-skimming” contract is offered by one bank, then all other banks will observe it and decline to offer the pooling contract at the final stage, because it would attract only bad borrowers. As a result, not only good but also bad borrowers may decide to borrow from the “cream-skimming” contract. (ii) To exclude bad borrowers, contract terms need to be such that C(cid:48)(cid:48) ≥ C + [p /(1 − s B p )](R −R(cid:48)(cid:48)) using (5). B s (iii) R(cid:48)(cid:48)+[(1−p )/p ]C(cid:48)(cid:48) < R , which is necessary for good borrowers to prefer the "cream- G G p skimming" over the pooling contract, also implies that R(cid:48)(cid:48) +[(1−p )/p ]C(cid:48)(cid:48) < R + G G s [(1−p )/p ]C or C(cid:48)(cid:48) < C +[p /(1−p )](R −R(cid:48)(cid:48)). G G s s G G s (iv) Satisfying the two restrictions on C(cid:48)(cid:48) requires (R −R(cid:48)(cid:48))(p −p )/[(1−p )(1−p )] > 0 s G B G B or R(cid:48)(cid:48) < R . As a result, the bank would make losses under these "cream-skimming" S contract terms, because R is the minimum lending rate under which the bank breaks s even in a separating equilibrium (see Proposition 2). (v) This means that a "cream-skimming" contract that can separate good from bad borrowers, while at the same time being more appealing to good types, cannot exist in equilibrium because banks would never offer it. (vi) Itfollowsthattheonly"cream-skimming"contractsthatcanbeofferedcannotseparate good from bad borrowers and, hence, will attract all borrowers should banks stop offering the pooling contract. (vii) Finally, an individual bank would never want to deviate by offering such a "creamskimming" contract because it would earn an R(cid:48)(cid:48) less that the minimum rate required to break even when all good and bad borrowers choose to borrow, which is equal to R in (9). p Taking all these out-of-equilibrium paths into consideration, all banks will offer the pooling contract (R ,0). Because of the sharing rule, one bank ends up lending to all unknown p borrowers and all but its known bad borrower through a pooling contract, while it offers its good known borrowers the contract (RG,0) with RG just below R in order to retain them. p p p Similarly, the other banks offer RG to their known good borrowers, but do not lend through p 15

a pooling contract in the final stage. The pooling equilibrium is unique and stable, and the pooling loan rate is given by (13). 2.4 Capital Requirements and Lending Standards How are lenders’ protection at the contract level and aggregate lending standards affected by changes in capital requirements? The degree of lenders’ protection is given by the level of collateral requirements, while aggregate lending standards are determined by whether the equilibrium lending regime features separating or pooling contracts. Hence, the effect of changing capital requirements, γ, on lenders’ protection is given by dC /dγ and on aggregate s lending standards by dλ/dγ. Recall that λ is the level of credit demand that corresponds to a switch from a separating to a pooling equilibrium. Also recall that collateral requirements are zero in the pooling regime so we will examine how they change with γ only in the separating one. Changes in capital requirements impact equilibrium loan terms for both separating and pooling contracts, (R ,C ) and (R ,0), respectively. Consider first how separating allocas s p tions are affected. Replacing γ with γ in Proposition 2, the separating contract terms as a function of a general γ can be written as R = γE/p + (1 − γ)D and C = [p /((1 − s G s B p )p )][p B −(γE +(1−γ)Dp )]. Taking the derivatives with respect to γ results in the B G G G following Proposition. Proposition 5 Considertheseparatingloancontract(R ,C ). Then, dR /dγ > 0, dC /dγ < s s s s 0, and d[R (γ)+[(1−p )/p ]C (γ)]/dγ > 0. s G G s Hence, borrowing costs in the separating region are increasing in capital requirements, butthecollateralrequirementisdecreasing. Theintuitionisthathighercapitalrequirements are passed on to borrowers through higher interest rates. Moreover, higher borrowing costs tighten bad borrowers’ incentive compatibility constraint (5), making it easier for banks to separate good from bad borrowers. As a result, banks can reduce the collateral required to separate borrower types. The net effect is that the effective borrowing cost, R + [(1 − s p )/p ]C , rises. G G s Similarly, by replacing γ with γ to get the pooling rate derived in Proposition 4 as a function of the general capital requirement, we can show that the borrowing cost in the 16

pooling regime also increases with capital requirements, i.e., dR /dγ > 0. Using these results p on the sensitivity of separating and pooling contract terms to γ, the following proposition establishes that lending standards tighten in equilibrium as γ increases. Proposition 6 Capital requirements tighten lending standards by increasing the domain over which the equilibrium strategy profile is the separating contract (R , C ) relative to the s s pooling contract (R ,0). Specifically, dα/dγ > 0 and dλ/dγ > 0. p Recall that α is the threshold for the portion of good borrowers above which pooling is possible, while λ is the threshold for the credit demand from unknown borrowers above which banks offer a pooling contract conditional on α > α. The proposition shows that the threshold, α, must be higher if banks are to fund all projects through weak lending standards. Intuitively, a higher capital requirement, γ, increases the amount of skin-in-thegame, therefore the average quality of the lending portfolio needs to be higher for banks to be willing to offer pooling contracts. Additionally, for every unit of increase in γ, it is more expensive to offer a pooling contract relative to a separating contract, because shareholders need to be compensated more for funding negative net present value projects. Thus, the threshold, λ, increases as well. 2.5 Optimal Macroprudential Capital Requirements The comparative statics in Proposition 6 considers γ ≥ γ to be exogenous. In this section we establish when a regulator would like to set γ > γ endogenously, i.e., macroprudential capital requirements are optimal. These requirements balance the deadweight loss from requiring collateral in the separating regime against funding negative NPV projects in the pooling regime. This distinguishes them for the microprudential capital requirement that address the possibility of risk-shifting at the level of an individual bank. We consider a planner that maximizes the overall surplus in the economy without consideration for how the surplus is distributed. The planner can implement any desired income redistribution through lump-sum transfers. Hence, the planner’s problem maximizes the exante net surplus from firm investment projects over the distribution of credit demand across the separating and pooling regimes. Consider, first, the credit demand in the separating regime, λ ≤ λ. The net surplus from firm projects is the expected net return to all good projects, α(1+λ)(p G−1), minus the G 17

expected loss from inefficient collateral liquidation, αλ(1−p )(1−κ)C (γ). The total mass G s of good projects funded is α(1 + λ), but only αλ are required to post collateral, because banks lend to their known good borrowers without requiring collateral. The net surplus from firm projects in the pooling regime is the expected net return to funding all good projects, α(1 + λ)(p G−1), plus the return to funding all but 1/N bad G projects, (1−α)((N −1)/N +λ)(p B −1).6 B The planner chooses γ to maximize expected surplus W: ˆ λ W = [α(1+λ)(p G−1)−αλ(1−p )(1−κ)C ]dλ G G s 0 ˆ ∞(cid:20) (cid:18) N −1 (cid:19) (cid:21) + α(1+λ)(p G−1)+(1−α) +λ (p B −1) dλ, (14) G B N λ subject to γ ≤ γ and γ ≤ γ , where γ is the maximum level of the capital requirement max max that makes lending profitable. The first line integrates over the level of demand in the separating regime i.e., λ < λ. The second line integrates over the level of demand in the pooling regime, i.e., λ ≥ λ. Denotingbyψ andψ theLagrangemultipliersforγ ≤ γ andγ ≤ γ ,respectively, min max max the optimality condition with respect to γ is   Expansionof Benefitfromfundingfewer separatingregion Costfrominefficientliquidation negativeNPVprojects (cid:122)(cid:125)(cid:124)(cid:123)  ofcollateral (cid:122) (cid:125)(cid:124) (cid:123) (cid:18) (cid:19) dλ (cid:122) (cid:125)(cid:124) (cid:123) N −1  −αλ(1−p )(1−κ)C +(1−α) +λ (1−p B) dγ  G s N B      2 dC λ s − α(1−p )(1−κ)+ψ −ψ = 0. (15) G min max dγ 2 (cid:124) (cid:123)(cid:122) (cid:125) Incrementaldecreasein requiredcollateral Equation (15) has the following intuitive interpretation. The terms in the first line capturetheeffectthatoperatesthroughaggregatelendingstandardsontheextensivemargin. 6Recall that each bank has 1/N known borrowers. Thus, the pooling bank will not fund its known bad borrower, but will fund all competitor banks’ bad borrowers. 18

Increasing the capital requirement expands the separating region, because dλ/dγ > 0 from Proposition 6. More separation in equilibrium comes with costs and benefits. On the one hand, more separation requires more aggregate collateral across projects, which imposes an inefficient liquidation cost–the first term multiplying dλ/dγ. On the other hand, more separation reduces the number of negative NPV projects undertaken by bad firms in the pooling region–the second term multiplying dλ/dγ. The third force comes from the first term on the second line, which captures the intensive margin of higher γ. Higher γ decreases the amount of collateral that each firm needs to pledge as shown in Proposition 5, which reduces the inefficiency from liquidating collateral. In an interior solution, these three forces balance each other. The optimal capital requirement is above the microprudential level if the positive effects from the extensive and intensive margins dominate the negative effect of possibly greater (inefficient) collateral liquidation. The following proposition establishes conditions under which macroprudential capital requirements are optimal. Proposition 7 Define the elasticity of λ w.r.t γ by η and similarly the elasticity of C λ,γ s w.r.t γ as η . The following statements hold: CS,γ 1. If η ≤ −0.5η , the optimal capital requirement, γ∗, is equal to the maximum λ,γ Cs,γ macroprudential level, γ . max 2. If η > −0.5η , and depending on parameters, the optimal capital requirement, λ,γ Cs,γ γ∗, is equal to either: (i) the microprudential requirement, γ; (ii) the maximum macroprudential level, γ ; or (iii) an interior macroprudential level between these two max extremes. Proposition 7 says that if the lending standards’ threshold, λ, is not sufficiently responsive to γ such that η ≤ −0.5η , then the optimal capital requirement is set at the λ,γ Cs,γ maximum level γ . The intuition is as follows. Increasing γ increases λ and, then, the remax gion of λ’s where collateral is inefficiently liquidated, but it also reduces the level of required collateral, C . The latter force mitigates the adverse effect from tightening standards and s always dominates the former force if the responsiveness of λ to γ is sufficiently lower than the responsiveness of C to γ. Given that tighter standards reduce the loss from funding s NPV projects, the optimal requirement is always γ . On the other hand, if λ is sufficiently max 19

responsive to γ such that η > −0.5η , the cost of the more frequent liquidation of collat- λ,γ Cs,γ eral dominates the benefit from the lower collateral requirements required. The net negative effect of the two needs, then, to be weighed with the positive effect of screening out negative NPV projects to determine the optimal capital requirement given the parameterization of the economy. Tosumup, lendingstandardsareincreasingincapitalrequirements, andmacroprudential capital requirements will be optimal in many cases. We now ask if the higher funding costs that macroprudential capital requirements impose on banks give space for other forms of intermediation, and if so, what is the equilibrium impact on lending standards? 3 Non-bank Competition, macroprudential Capital Requirements, and Lending Standards Given the concomitant rise of non-bank intermediation and macroprudential regulation post- GFC, it is natural to ask whether our lending standards and optimal regulation results continue to hold in the presence of non-bank competition. 3.1 Modeling Non-Banks Assume there are a large number of competitive non-banks that raise debt from the same outside investors as banks, demand a similar expected return, E, to supply equity, and have access to the same borrowers. However, non-banks do not have private information about any borrowers and treat the whole population of firms (1+λ) as unknown. This assumption captures the fact that banks are the incumbent institutions with existing relationships with some borrowers, while non-banks are the entrants without prior information.7 7Given our assumptions, banks and non-banks have no strict incentive to collaborate. There are two reasons for this. First, both banks and non-banks have access to the same underlying lending technology andcanscreenefficientlyusingcollateral. Second,bothbanksandnon-bankshaveaccesstoanelasticsupply of funds. Therefore, even though banks may enjoy cheaper funding due to deposit insurance, they would not lend to non-banks at a rate lower than the effective loan rate to firms, since they can extend as many loans as they want. Similarly, non-banks do not need to resort to bank funding as they have direct access to elasticfundingmarkets. Regulationmaychangetherelativeincentivesofbanktoextendloanstonon-banks if they get a preferential regulatory treatment compared to loans to firms. In the absence of this regulatory arbitrage, cooperation between banks and non-banks would not occur in our model. 20

Non-bank debt holders are not insured, which imposes some market discipline on nonbanks. However, the moral hazard problem from risk-shifting remains because the type of borrowers to whom non-banks lend is non-contractible. Therefore, non-banks need enough skin in the game to exclude risk-shifting. However, non-banks are unregulated and a market mechanism should ensure sufficient capital to resolve the moral hazard problem. We assume for simplicity that non-bank equity is contractible. Therefore, long-term debt withcovenantsdictatingthelevelofequitycanresolvethemoralhazardproblem(Holmström and Tirole, 1997).8 The contractibility of equity matters because non-banks can distribute dividends or repurchase shares after they have received the funds from debt-holders, so they may not have enough skin in the game to be discouraged from risk-shifting. If non-banks choose to operate with a level of equity less than what is required to discourage risk-shifting, the covenant would be violated and the debt-holder could seize the firm in the extreme, which would act as a deterrent. The minimum level of equity that long-term debt-holders would require non-banks to maintain is, then, given by p (B −DNB) γNB = B , (16) E −p DNB B where DNB > 1 captures the interest rate on non-bank debt, which will depend on the investment strategy of non-banks in equilibrium and will incorporate a default premium. In our model, DNB > D, because non-bank debt incorporates a default premium while bank deposits are insured, which also implies that γNB < γ.9 The funding cost difference between banks and non-banks has implications for the ability of non-banks to compete with banks. In fact, we show in Sections 3.2 and 3.3 that the combination of lower funding costs due to subsidized deposit insurance and information advantage that banks possess implies that non-banks cannot compete with banks in either the separating or pooling regions if banks are only subject to microprudential capital requirements.10 We, then, turn to the 8We show in Appendix B that an alternative assumption with non-contractible equity yields the same outcome presented here. In that case, non-banks have a fragile funding structure with demandable debt as in(DiamondandRajan,2000). Thecapitalrequirementsetbythemarketisidenticalinthisalternativeset up if the liquidation value of bank assets is sufficiently high. 9Note that we could derive a fair insurance premium, ιf, such that DNB = D. As long as the actual insurancepremiumιislowerthanιf,DNB >D andγNB <γ. Withoutlossofgeneralityandforsimplicity, we set ι→0. Our results generalize for ι∈(0,ιf), but the algebra is more cumbersome. 10Note that under fairly priced deposit insurance, banks and non-banks have the same overall cost of 21

effect of macroprudential capital requirements on non-bank entry and its implications for lending standards in Section 3.4. 3.2 Separating Contracts with non-banks We first ask whether non-banks can offer separating contracts to borrowers at least as attractive as the contract offered by banks. Recall that the terms of the separating contract banks offer, (R ,C ), are given in Proposition 2. For this contract, the interest rate that s s outside investors charge non-banks and break even is TotalPayoff Totalpayoff Outsideoption ingoodstate inbadstate onfundssupplied (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) p (1−γNB)DNB+(1−p )κC = 1−γNB G s G s 1 1−p κC ⇒DNB = − G s , (17) s p p 1−γNB G G since investors receive the contractual rate DNB on the 1 − γNB funds they supplied in s the good state where the non-bank does not default, while they receive the salvage value of collateral, κC , in the bad state where both the borrower and the non-bank default. s Then, the market-based non-bank capital requirement for a separating loan portfolio, γNB, s is determined by substituting (17) in (16). It is profitable for non-banks to participate and compete with banks and offer the sep- (cid:0) (cid:0) (cid:1) (cid:1) arating contract to screen borrowers if p R − 1−γNB DNB > γNBE. Substituting R G s s s s s from Proposition 2, we obtain the necessary condition for non-banks to compete with banks in separating allocations: (cid:2)(cid:0) (cid:1) (cid:3) E(γ −γNB) > p 1−γNB DNB −(1−γ)D . (18) s G s s The left hand side of (18) is the advantage that non-banks have from lower equity cost. The right hand side is the nonbank disadvantage from higher debt financing costs. Using (16) and (2), it can be shown that the inequality implies a contradiction as long as p > p , G B fundingforseparatingcontracts,butbankscanofferamorecompetitivepoolingrateduetotheirinformation advantage. Assuch,lendingstandardswillcontinuetobedeterminedasinPropositions2and4evenwithout underpriced deposit insurance. Hence, our assumption that deposit insurance is subsidized is without loss of generality. 22

which always holds by assumption. Intuitively, the benefit of the deposit insurance subsidy accrues to banks with probability p but the benefit of lower equity costs for non-banks G through risk-shifting occurs with probability p . Therefore, non-banks cannot compete with B banks under microprudential capital requirement via separating contracts. 3.3 Pooling Contracts with non-banks What are the terms of pooling contracts offered by non-banks? Recall that the pooling contract offered by banks is given by R in Proposition 4. Compared to banks, non-banks p do not have inside information about any borrowers. Therefore, they will attract the entire pool of bad borrowers, (1−α), when they offer pooling contracts rather than the fraction N−1 (1−α)thatbanksattract. Thedifferencebetweenthetwopoolsofborrowersthatbanks N and non-banks attract reflects banks’ information advantage from knowing their existing clientele. Outside investors, anticipating a more risky pool of borrowers, set the required repayment on nonbank debt, DNB, to break even with their outside option, i.e., p TotalPayofffrombad TotalPayofffromunknown Outsideoption borrowersingoodstate borrowersingoodstate onfundssupplied (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) p (1−α)DNB + p λDNB = 1−α+λ B p µ p 1−α+λ ⇒DNB = . (19) p (1−α)p +λp B µ Underthepoolingcontract,thenon-bankwouldfundallbadborrowers1−αandallunknown borrowers λ, i.e., investors supply 1−α+λ in total (recall that banks can and will keep their known good borrowers by offering a more competitive contract in stage 2 of the game). Since no collateral is posted, investors get zero in the bad state, while they receive the contractual rate DNB when borrowers do not default in the good state; with probability p the bad p B borrowers repay in total (1−α)DNB and with probability p the unknown borrowers repay p µ in total λDNB. p The market-based non-bank capital for a pooling loan portfolio, γNB, is given by substip tuting (19) in (16). Hence, the best pooling contract that non-banks can offer while breaking 23

even is (cid:2) (cid:3) (cid:2) (cid:3) (1−α) γNBE +(1−γNB)DNBp +λ γNBE +(1−γNB)DNBp RNB = p p p B p p p µ . (20) p λp +(1−α)p µ B Non-banks can compete with banks in the pooling region if they can offer borrowers a lower repayment amount, RNB ≤ R . Comparing (20) with (13) determines whether p p (cid:0) (cid:1) non-bank competition is feasible. It is straightforward to show that lim R −RNB < 0 p p λ→0 (cid:0) (cid:1) and lim R −RNB < 0. There are two reasons for these results. First, non-banks have p p λ→∞ an information disadvantage and charge a higher borrowing cost than banks because they attract bad borrowers, who are known and rejected by banks; non-banks must fund 1 − α unknown bad borrowers compared to N−1 (1−α) bad borrowers for banks. Second, similar N to the separating contracts, the underpriced deposit insurance gives banks an overall cost advantage even though γNB < γ¯. These two forces will be also important in the case of p macroprudential regulation, to which we later return. Finally, it is straightforward to show that both dRNB/dλ,dR /dλ < 0, which means, in conjunction with the two limits above, p p that the repayment amount non-banks require for pooling all borrowers is always higher than the repayment amount banks require. In sum, non-banks cannot compete with banks and affect lending standards with either separating or pooling contracts under microprudential regulation and underpriced deposit insurance. This result is summarized in the following proposition. Proposition 8 Under the microprudential capital requirements for banks in Proposition 1 and the market-based capital requirement for non-banks in (16), non-banks cannot compete with banks in any equilibrium allocation and lending standards are unaffected. 3.4 Equilibrium Lending Standards with Non-banks As established above, non-bank competition does not impact equilibrium lending standards when capital requirements are set at γ. By continuity, there exists macroprudential capital requirements, γ > γ, such that non-banks continue to be unable to compete with banks and lending standards are determined as in Proposition 6. However, capital requirements cannot increase without bound and keep non-banks at bay. At some point, increasing bank funding costs erodes banks’ advantage over non-banks. 24

We show that the crucial point that determines how non-banks impact lending standards is whether non-banks start competing with banks first in separating or pooling contracts as γ increases. We focus on two thresholds for macroprudential capital requirements. The first threshold, denoted by γˆ, indicates the level of bank capital at which non-banks can compete ˆ by offering separating contracts. The second threshold, denoted by γˆ, indicates the level of bank capital at which non-banks can compete by offering pooling contracts. We first derive γˆ, the level of bank capital that allows non-banks to compete through separatingcontracts. Non-bankswillparticipateintheloanmarketifandonlyifthereturnto lending, p R , is weakly greater than their funding costs. Using Proposition 2 to determine G s R and (16), non-banks will compete with banks through separating contracts for γ > γˆ s (cid:0) (cid:1) (cid:2)(cid:0) (cid:1) (cid:3) where γˆ is the solution to E γˆ −γNB = p 1−γNB DNB −(1−γˆ)D , or s G s s γNB(E −p DNB)+p (DNB −D) γˆ = s G s G s , (21) E −p D G Any capital requirement set above this level allows non-banks to enter the loan market and compete with banks through separating contracts. Now consider the possibility that non-banks can compete with banks through pooling ˆ contractsthatdonotrequirecollateral. Thisoccursformacroprudentialrequirementsγ ≥ γˆ. ˆ Using Proposition 4, the pooling contract that banks offer borrowers given γˆ must satisfy the following break-even condition: (cid:104) (cid:16) (cid:17) (cid:105) (cid:104) (cid:16) (cid:17) (cid:105) (1−α) N−1 γ ˆ ˆE + 1−γ ˆ ˆ Dp +λ γ ˆ ˆE + 1−γ ˆ ˆ Dp N B µ R(cid:98) (cid:98) = . (22) p (1−α) N−1p +λp N B µ Note once again that λ enters into the break-even pooling contract because the returns to pooling are a function of the mass of unknown borrowers, all of whom receive funding. Bankshavebothafundingadvantageduetosubsidizeddepositinsuranceandaninformation advantage due to known borrowers over non banks. Both of those advantages are eroded as λ → ∞. The reasons are, first, that banks do not have an information advantage when the mass of unknown borrows to both banks and non-banks dominates the mass of borrowers known to banks, and second, through equation (19), DNB is decreasing in λ and is at p its minimum for DNB| = D/p . Hence, non-banks most easily compete with banks p λ→∞ µ 25

ˆ in pooling contracts under macroprudential capital requirements for λ → ∞. Thus, γˆ is determined by equating R(cid:98) (cid:98) and RNB for λ → ∞: p p γNB(E −p D/p )+p (D/p −D) ˆ p µ µ µ µ γˆ = , (23) E −p D µ where γNB is given by (16) for DNB = D/p . The following proposition shows that γˆ < γ ˆ ˆ p µ and non-banks first compete with banks in separating contracts. Proposition 9 Non-banks compete first in separating allocations and then pooling alloca- ˆ tions as macroprudential capital requirements increase, i.e., γˆ < γˆ. Theintuitionisasfollows. First, bankspossessbothafundingandinformationadvantage over non-banks in pooling equilibria but only a funding advantage in separating equilibria; the information advantage in separating equilibria is negated due the collateral requirement screening out all bad borrowers. Second, the deposit insurance subsidy that banks enjoy is more valuable in pooling equilibria because bad borrowers, which are more likely to default, also receive funding. Hence, the necessary increase in regulatory capital requirement that allows non-banks to compete with banks is smaller in separating than pooling equilibria. Note that the information advantage banks possess does not play a role in Proposition 9 as it erodes for λ → ∞. By contrast, banks have an information advantage over non-banks ∀λ ∈ (0,∞) that makes it more difficult for non-banks to compete in pooling equilibria. Therefore, non-banks compete first in separating equilibria for any value of λ. However, ˆ macroprudential requirements in excess of γˆ raise bank funding costs to a level that begin to negate banks’ information advantage and allow non-banks to compete in pooling for values of λ bounded away from infinity. In fact, our next proposition shows how macroprudential regulation impacts the competition between banks and non-banks in separating and pooling contracts and that there is a ˆ threshold value γ˜ > γˆ above which banks are completely disintermediated. Proposition 10 There exist three thresholds for macroprudential capital requirements: γˆ < ˆ γˆ < γ˜ such that: 1. For γ < γˆ, non-banks cannot compete and banks fund all borrowers; 26

ˆ 2. For γˆ ≤ γ < γˆ, non-banks disintermediate banks in separating contracts; ˆ 3. For γˆ ≤ γ < γ˜, non-banks disintermediate banks both in separating contracts and in pooling contracts for high enough λ but banks continue to lend in pooling equilibria with lower λ; 4. For γ ≥ γ˜, banks are completely disintermediated and non-banks fund all demand for credit. Having established that different macroprudential capital requirements allow non-banks to compete with different types of contracts, separating vs. pooling, we now turn to their impact on both the level of lender protections at the contract level and aggregate lending standards. With respect to the former, collateral requirements in separating equilibria decrease as γ increases from its microprudential level γ up to γˆ when non-banks start offering more competitive separating contracts. This follows from Proposition 5. Thereafter, the lending rate and collateral requirement are given by the zero-profit condition for a non-bank that funds only good borrowers, and the individual rationality constraint of bad borrowers that are dissuaded from borrowing. Given that the latter implies the same inverse relationship between the rate and collateral requirement as in the economy with only banks, it follows that the collateral requirement that non-banks will offer, when they can effectively compete, will be associated with the interest rate a bank would charge for γ = γˆ. In other words, collateral requirements—capturing lenders’ protection at the contract level—do not improve once non-banks can compete with banks in separating contracts and are lower than before non-banks were able to be compete. Turningtotheaggregatelendingstandards, recallthattheyarecapturedbythethreshold λ where the economy switches from a separating equilibrium with collateralized lending to a pooling equilibrium with non-collateralized lending. Each threshold for the macroprudential capital requirement in Proposition 10 is associated with a different value of the λ threshold. For γ ∈ [γ,γˆ) the threshold λ is the level of credit demand that equalizes the effective rate that banks offer on separating and pooling contracts (see Proposition 6). For γ ∈ [γˆ,γ˜), the ˆ threshold λ equalizes the effective rate that non-banks offer on separating contracts with the ˜ rate that banks offer on pooling contracts. For γ ≥ γ˜, the threshold λ equalizes the effective rates that non-banks offer on separating and pooling. 27

The following proposition ranks these thresholds and summarizes how macroprudential regulation affects lending standards in the presence of competition by non-banks. ˆ Proposition 11 In the presence of non-bank competition, dλ/dγ > dλ/dγ > 0, λ(γˆ) = ˆ ˆ ˜ λ(γˆ), and λ(γ˜) = λ(γ˜). Recall from Proposition 6 that macroprudential capital requirements tighten aggregate lending standards by increasing the threshold value of aggregate credit demand for which banks screen borrowers. Proposition 11 establishes that non-bank competition amplifies this (positive) relationship. Higher capital requirements increase the cost to banks of offering pooling contracts but do not impact the cost of non-bank separating contracts. This means that the threshold value of λ that determines separating vs. pooling regions increases more for a given increase in γ when non-banks are present than when they are not. In other words, there is more screening and separating in equilibrium and less pooling. Finally, for sufficiently high macroprudential capital requirements, banks are completely disintermedi- ˜ ated and lending standards are kept at their higher level, λ(γ˜). In sum, Propositions 6 and 11 establish that neither non-bank competition or macroprudential regulation, even if coupled together, erode aggregate lending standards despite lenders’ protections being lower at the contract level. On the contrary, lending standards monotonically tighten with higher macroprudential requirements and the effect is stronger in the presence of non-bank competition. We should, however, note that it is possible for lending standards to deteriorate with higher macroprudential capital requirements if non-banks were first able to compete in poolingcontracts. Thereasoningisanalogoustowhytheyincreasewhennon-banksfirstcompete with separating. In particular, if non-banks initially compete using pooling contracts with weak standards while banks retain their advantage in screening, then higher macroprudential capital requirements that increase bank funding costs raise the relative costs of bank screening contracts compared to non-bank pooling contracts. This is not the case in the model we present and additional assumptions about how banks and non-banks differ would be needed to obtain such a result. 28

4 Optimal macroprudential regulation with non-banks We now address how the planner optimally sets capital requirements with non-banks competing with banks. Recall from Proposition 7 in Section 2.5 that the planner optimally sets γ∗ depending on the elasticity of λ with respect to γ and other parameters. The planner’s problem in the presence of non-bank competition is similar, but the planner needs to take into consideration the various threshold values of γ where the economy switches from the separating to pooling regime derived in Proposition 10. Define by Γ , k ∈ K, the different intervals for values of γ. For each interval the planner k chooses the γ that maximizes the social welfare function WNB(γ) (see below for detailed Γ k expressions). Denote this optimal γ for each interval by γ∗∗ ≡ argmax WNB(γ). Then, k γ Γ k the optimal γ∗∗ overall is the γ∗∗ that delivers the higher social welfare WNB(γ∗∗), i.e., k Γ k k γ∗∗ = argmax WNB(γ∗∗). γ k ∗∗ Γ k k Before presenting WNB(γ) for each interval, let us reiterate the different thresholds of γ Γ k that define these intervals: γ is microprudential capital requirement; γˆ is the threshold for macroprudential capital requirement where non-banks start competing in separating con- ˆ tracts; γˆ is the threshold where non-banks start competing in pooling contracts for very high credit demand; γ˜ is the threshold where banks are disintermediated in both separating and pooling contracts but can still lend to their known good borrowers; γ˜˜: threshold where banks cannot lend to their known good borrowers; and γ is the threshold where banks cannot max lend using separating contracts even in the absence of non-banks. For Γ = {γ ∈ [γ,γˆ)} the social welfare function is the same as in (14) given that 1 non-banks cannot compete for these levels of γ, i.e., ˆ λ WNB(γ) = [α(1+λ)(p G−1)−αλ(1−p )(1−κ)C ]dλ Γ1 G G s 0 ˆ ∞(cid:20) (cid:18) N −1 (cid:19) (cid:21) + α(1+λ)(p G−1)+(1−α) +λ (p B −1) dλ. (24) G B N λ Recall that the switch from the separating to the pooling regime happens for λ ≥ λ. ˆ For Γ = {γ ∈ [γˆ,γˆ)} non-banks lend through the separating regime and banks in the 2 29

pooling regime. The social welfare function is given by ˆ λˆ (cid:2) (cid:3) WNB(γ) = α(1+λ)(p G−1)−αλ(1−p )(1−κ)CNB dλ Γ2 G G s 0 ˆ ∞(cid:20) (cid:18) N −1 (cid:19) (cid:21) + α(1+λ)(p G−1)+(1−α) +λ (p B −1) dλ. (25) G B N λˆ The differences between (24) and (25) are that the collateral is given by C (γNB) for all s γ ≥ γˆ rather than C (γ), and that the switch from separating to pooling regime happens for s ˆ credit demand higher than λ rather than λ. ˆ For Γ = {γ ∈ [γˆ,γ˜)} non-banks can compete in the pooling region for credit demand 3 ˆ ˆ ˆ ˆ λ ≥ λ, where the threshold λ is given by the point where the pooling contract banks offer ˆ ˆ becomes as expensive as the one offered by non-banks, i.e., R (γ,λ ˆ ) = RNB(λ ˆ ). The social p p welfare function is given by ˆ λˆ (cid:2) (cid:3) WNB(γ) = α(1+λ)(p G−1)−αλ(1−p )(1−κ)CNB dλ Γ3 G G s 0 ˆ λ ˆˆ(cid:20) (cid:18) N −1 (cid:19) (cid:21) + α(1+λ)(p G−1)+(1−α) +λ (p B −1) dλ G B N λˆ ˆ ∞ + [α(1+λ)(p G−1)+(1−α)(1+λ)(p B −1)]dλ. (26) G B λ ˆˆ ˆ Note that the separating region still obtains for λ < λ given that banks continue to offer the ˆ ˆ ˆ pooling contract for credit demand λ ∈ [λ,λ]. Thus, the difference between (25) and (26) is ˆ ˆ that non-banks fund all bad projects conditional on high demand for loans, λ > λ. For Γ = {γ ∈ [γ˜,γ˜˜)} banks are disintermediated both in the separating and pooling 4 regimes,butcanstilllendtotheirknowngoodborrowers,hencetheonlycollateralliquidation cost in the separating regime comes from non-bank lending. The social welfare function is 30

given by ˆ λ˜ (cid:2) (cid:3) WNB(γ) = α(1+λ)(p G−1)−αλ(1−p )(1−κ)CNB dλ Γ4 G G s 0 ˆ ∞ + [α(1+λ)(p G−1)+(1−α)(1+λ)(p B −1)]dλ. (27) G B λ˜ Finally, for Γ = {γ ∈ [γ˜˜,γ ]} banks cannot even lend to their known good borrowers. 5 max The social welfare function is given by ˆ λ˜ (cid:2) (cid:3) WNB(γ) = α(1+λ)(p G−1)−(1+αλ)(1−p )(1−κ)CNB dλ Γ5 G G s 0 ˆ ∞ + [α(1+λ)(p G−1)+(1−α)(1+λ)(p B −1)]dλ. (28) G B λ˜ The difference between (27) and (28) is that collateral is required for all good borrowers in the latter, since non-banks do not have an informational advantage. Evaluating social welfare across intervals, the planner will never set γ ≥ γ˜˜. The reason is that she can do better setting γ ∈ [γ˜,γ˜˜) given that lending standards are unaffected because non-banks intermediate the whole market. Similarly, there is no scope to set γ > γ˜, because ˆ ˆ ˆ ˜ the planner can achieve the same level of welfare by setting γ → γ˜, since λ(γ˜),λ(γ˜) → λ and WNB(γ˜) → WNB(γ˜). Thus, the optimal solution when non-banks are present is between γ Γ3 Γ4 and γ˜. This result establishes that the optimal capital requirement is lower in the presence of non-banksforeconomiesparameterizedsuchthatγ∗ = γ withonlybanks(seeProposition max 7). The following Proposition generalizes this result for all parameterizations. Proposition 12 The optimal capital requirement with non-banks never exceeds the optimal requirement without non-banks, and is strictly lower if non-banks are active in the loan market. The intuition behind Proposition 12 is as follows. Recall that the planner trades off the cost from the inefficient collateral liquidation in the separating regime with the cost of funding negative NPV projects in the pooling regime. The collateral liquidation cost is relatively higher after a level of γ when non-banks disintermediate banks in separating 31

allocations. The reason is that the collateral requirement non-banks set, CNB, does not s vary with capital regulation contrary to the one set by banks that is decreasing in γ (recall that dC /dγ < 0 from Proposition 5). In addition, for high enough γ non-banks also start s ˆ ˆ disintermediating banks in pooling allocations for λ > λ. Hence, the social cost of funding negative NPV projects increases since non-banks fund all bad borrowers due to their informational disadvantage. Both forces work in the same direction and urge the planner to set a lower optimal macroprudential requirement in the presence of non-bank competition. 5 Conclusion This paper provides a theoretical framework for understanding how macroprudential capital regulation affects lending standards in an economy where banks and nonbanks compete. We show that while higher capital requirements reduce lender protections at the individual contract level—by lowering collateral requirements—they tighten aggregate lending standards and do not adversely affect aggregate default rates by incentivizing banks to screen out riskier borrowers. The entry of non-banks into the credit market, triggered by tighter bank capital regulation, does not lead to a deterioration in aggregate lending standards. Instead, non-banks compete with banks by offering loans to good borrowers under lower collateral requirements, without attracting the riskier borrowers that banks reject. Such dynamics have been observed in the leveraged loan markets after the Global Financial Crisis and in private credit markets more recently; dynamics that our model can rationalize. Weshowthatthepresenceofnon-banksamplifiesthepositiveimpactofcapitalregulation onaggregatelendingstandardsbyraisingthethresholdvalueofcreditdemandwherelending standardsstarttodeteriorate, therebypotentiallyincreasingtheresilienceofcreditqualityto financial expansions. At the same time, the optimal macroprudential capital requirement is lowerinthepresenceofnon-banks,mitigatingregulators’abilitytofurthertightenstandards. 32

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A Appendix - Proofs Derivation of assumption 2. We restrict attention to good and bad project’s payoffs, G and B, that generate the possibility of risk-shifting when there are no capital requirements, i.e., γ = 0. Consider that banks engage in risk-shifting offering a gross loan rate, R, that satisfies the following three conditions. First, it should be individually rational for bad firms to borrow, i.e., B ≥ R. Second, it should not be individually rational for good types to borrow, i.e., R > G; otherwise a pooling equilibrium would obtain, which does not constitute risk-shifting given that the average borrower has positive NPV. Third, risk-shifting should be individually rational for banks, i.e., profits should be higher than the maximum possible profit from lending to good borrowers or, R > (p /p )(G−D)+D. In addition, bad projects G B have negative NPV, i.e., p R ≤ p B < 1, which in combination with the previous condition B B yields p G − (p −p )D < 1. Combining these conditions, we derive assumption 2 as a G G B sufficient condition to obtain risk-shifting as the only equilibrium when capital requirements, γ, are zero. The proofs of Propositions 1, 4, and 5, are immediate from the text. Proof of Proposition 2. The equilibrium separating contract terms R and C are s s derived by solving jointly (5) and (6) given that the bank defaults in the bad state, i.e., κC −(1−γ)D < 0. We proceed to verify that this is true for sufficiently low κ. We, first, s examine the case that this condition is true for all κ ∈ [0,1). Take κ → 1 and assume that C = p /(1−p )(B−R ) ≥ 1−γ, which implies that R ≤ B−(1−p )/p (1−γ). Using the s B B s s B B equilibriumvalueofR andD = 1, thiscanonlybetrueforγ ≤ p (p B−1)/(p E−p ) < 0 s G B B G if E > p /p , or γ ≥ p (1−p B)/(p −p E) > 1 if E < p /p and E > p B. In other G B G B G B G B G words, for these set of parameters the bank defaults in the bad state not only for γ, but for any level of admissible capital requirement γ. For E < p /p and E < p B, there may G B G exist γ such that C > (1−γ)D. In such cases, we will impose that κ < κ ≡ (1−γ)D/C , s γ s such that the bank defaults if the bad state realizes. Proof of Proposition 3. We first establish the existence of the threshold α above which condition (12) is satisfied. The left-hand side of (12) is decreasing in α, while the right-hand 36

side is independent of α. Using the participation constraint for good types in a separating equilibrium, p (G−R ) − (1−p )C ⇒ G > R + 1−pGC . Re-writing (12) and taking G s G s s pG s α → 0 as γE p (B −D) p (G−D) B B +(1−γ)D > G ⇒ γ = > p E −p D E −p D B B B which always holds because B > G. Hence, there is no pooling equilibrium even for sufficiently high λ for α → 0. Letting α → 1, condition (12) becomes p (R −(1−γ)D) + G s (1−p )C > γE, whichalwaysholdsbecauseγE = p (R −(1−γ)D). Hence, ∃α ∈ (0,1) G s G s such that condition (12) holds, and the separating allocation cannot be an equilibrium for sufficiently high λ and α. Putting this together with the fact that there is no pooling equilibrium for λ → 0, part (i) follows immediately. To establish the threshold λ in part (ii), note that the left-hand side of (11) is continuous and decreasing in λ and approaches γE/p +(1−γ)D. Thus, if (12) holds, then there must be a λ > 0 such that equilibrium B is separating if λ ≤ λ. Moreover, the zero-profit condition from which the contract (R ,C ) s s is derived ensures that no bank can profitably offer a different contract. From Rothschild- Stiglitz argument, no separating strategy exists when condition (11) is violate. Therefore, part (iii) shows the conditions for violating condition (11) while preserving condition (12) and eliminating all separating equilibria. There is no pooling equilibrium under the conditions established in parts (i)-(iii) because a necessary condition for pooling to be an equilibrium is that condition (9) holds. But, for λ < λ, condition (11) implies that a bank could offer a deviating contract (R +(cid:15),C ) for s s (cid:15) > 0 sufficiently small that attracts only good borrower and make a profit. Thus, there is no pooling equilibrium for λ < λ. Lastly, for α < α, a bank could offer a deviating contract (R +(cid:15),C ) for (cid:15) > 0 sufficiently small that attracts only good borrowers and make a profit s s while still preserving the relationship γE +(1−γ)D > R +(cid:15)+ 1−pGC . pµ s pG s Proof of Proposition 6. The equilibrium value of λ is implicitly defined by indifference ¯ ¯ condition of the pooling and separating contracts: R = R , where R = R +(1−p )/p C . s p s s G G s Totally differentiating, dR¯ s = ∂Rp +∂Rp∂λ ⇒ ∂Rp∂λ = dR¯ s −∂Rp. Re-writing the equilibrium dγ ∂γ ∂λ ∂γ ∂λ ∂γ dγ ∂γ price of the pooling contract, (cid:0) (cid:1) (cid:0) (cid:1) γE λ+(1−α) N−1 +(1−γ)D λp +(1−α) N−1p R = N µ N B (A.1) p λp +(1−α) N−1p µ N B 37

It is straightforward to see that ∂Rp = (1−α)N N −1(γE(pB−pµ)) < 0 ⇒ p < p . Therefore, ∂λ (·)2 B µ (cid:16) (cid:17) (cid:16) (cid:17) sign ∂λ = −sign dR¯ s − ∂Rp . PluggingintheseparatingcontracttermsR = γE−pGD+D ∂γ dγ ∂γ s pG and C = pB(B−R(γ)) from Proposition 2 we obtain s 1−pB (cid:20) (cid:18) (cid:19) (cid:21) E p −p (1−p ) p ¯ G B G B R = γ −D +D + B. (A.2) s p p (1−p ) p (1−p ) G G B G B (cid:16) (cid:17) NotingthatγdR¯ s = γ E −D pG−pB ,onecanexpressγdR¯ s = R ¯ − pG−pB D−(1−pG)pBB. dγ pG pG(1−pB) dγ s pG(1−pB) pG(1−pB) Lastly, we need to find ∂Rp. Note that R = γ E(λ+(1−α)N N −1) −γD+D ⇒ R = γ∂Rp +D. ∂γ p (cid:16) λpµ+(1−α)N N(cid:17) −1pB p ∂γ Hence, one can write γ∂Rp = R −D. Thus, γ dR¯ s − ∂Rp 1 = R ¯ − pG−pB D−(1−pG)pBB− ∂γ p dγ ∂γ γ s pG(1−pB) pG(1−pB) ¯ R +D. Using the equilibrium relationship that R = R , p s p (cid:18) ¯ (cid:19) dR ∂R (1−p )p (1−p )p s p G B G B − = − B + D < 0, (A.3) dγ ∂γ p (1−p ) p (1−p ) G B G B because B > D. To conclude, ∂λ > 0. ∂γ Proof of Proposition 7. First, consider the case that κ = 1. Then, ψ = 0 and min ψ > 0, because dλ/dγ > 0 from Proposition 6. In other words, the planner imposes max the maximum capital requirement because there is no cost from liquidating collateral. Now consider κ < 1, and rewrite (15) as: (cid:104) (cid:105) −dCsλ 2 α(1−p )(1−κ) η λ,γ + 1 + dλ(1−α) (cid:0) N−1 +λ (cid:1) (1−p B)+ψ −ψ = 0, dγ G ηCs,γ 2 dγ N B min max where η dλ/dγ C (γ) λ,γ s = < 0, η dC /dγ λ(γ) Cs,γ s i.e. the ratio of the elasticities of λ(γ) and C (γ) with respect to γ is negative from Proposs itions 5 and 6. Dividing through by dλ, we have dγ −1 dλ F + [ψ −ψ ] = 0, min max dγ 38

(cid:104) (cid:105) where F ≡ −λ·C ·α(1−p )(1−κ) 1+ 1ηCs,γ +(1−α) (cid:0) N−1 +λ (cid:1) (1−p B). s G 2 η N B λ,γ We already know that F > 0 for κ = 1. Therefore, if dF/dκ < 0 ⇒ ψ > 0,∀κ ∈ [0,1]. max It is straightforward to see that ηCs,γ ≤ −2 ⇒ dF/dκ < 0. Hence ηCs,γ ≤ −2 or η ≤ η η λ,γ λ,γ λ,γ −0.5η is sufficient for γ∗ = γ ∀κ. Cs,γ max Now consider 0 > ηCs,γ > −2 or η > −0.5η , implying dF/dκ > 0. Taking the limit η λ,γ λ,γ Cs,γ of F as κ → 0 (cid:20) (cid:21) (cid:18) (cid:19) 1η N −1 limF = −λC α(1−p ) 1+ Cs,γ +(1−α) +λ (1−p B). s G B κ→0 2 η λ,γ N If lim F > 0, then again γ∗ = γ ∀κ. If lim F < 0, γ∗ ∈ [γ,γ ). κ→0 max κ→0 max Proof of Proposition 8. Equation(18)showsthatforp > p ,thebenefitoflowerequity G B costfornon-banksneveroutweighstheirhigherfinancingcostsundermicroprudentialcapital requirements. Thus non-banks cannot compete with banks in separating contracts. Now consider pooling contracts. For λ → 0, substituting the equilibrium values of γ, (cid:0) (cid:1) γNB, and DNB into equations (13) and (20) yields R = RNB. Thus, lim R −RNB = 0. p p p p p λ→0 Note that for λ → 0, the separating contract always dominates the pooling contract for banks: all good borrowers are known to at least one bank and thus receive funding, while all remaining (unfunded) borrowers offer negative NPV projects and thus banks cannot do better by offering a pooling contract when λ → 0. Additionally, we have shown that non-banks cannot compete with banks through separating contracts under microprudential capital requirements. Thus for λ → 0, non-banks cannot compete with banks. For λ → ∞, substituting the equilibrium values of γ, γNB, and DNB into equations (13) p (cid:0) (cid:1) and (20) yields R < RNB for p < 1, which holds by definition. Thus, lim R −RNB < 0 p p µ p p λ→∞ and non-banks cannot compete with banks through pooling contracts. For intermediate values of λ, using equations (13) and (20) we obtain (cid:0) (cid:1) ∂R (1−α) N γ¯E(p −p ) p = N−1 B µ < 0, (A.4) ∂λ (cid:2) λp +(1−α) (cid:0) N (cid:1) p (cid:3)2 µ N−1 B and ∂RNB (1−α)γNBE(p −p ) p B µ = < 0, (A.5) ∂λ [λp +(1−α)p ]2 µ B 39

(cid:0) (cid:1) (cid:0) (cid:1) since p < p . In combination with lim R −RNB = 0 and lim R −RNB < 0, (A.4) B µ p p p p λ→0 λ→∞ and (A.5) imply that non-banks cannot compete through pooling contracts for any value of λ under microprudential capital requirements. Therefore, under microprudential capital requirements, non-banks are unable to compete with non-banks either through separating or pooling contracts. Proof of Proposition 9. γˆ is the capital requirement that equates bank and non-bank ˆ participation constraints when offering a separating contract given by equation (21). γˆ| λ→∞ is the capital requirement that equates bank and non-bank participation constraints when D−(1−pG)κ offering a pooling contract given by equation (23). Note that DNB = 1−γs NB < D ⇒ s pG pG DNBp < D. Using this inequality, we can re-write equation (21) as s G γNBE −p D D(1−γNB) γNB(E −D) D(1−p ) γˆ < s G + s = s + G . E −p D E −p D E −p D E −p D G G G G Using lim DNB = D, γ ˆ ˆ can be expressed as λ→∞ p pµ γNB(E −D)+D(1−p ) ˆ p µ γˆ = E −p D µ ˆ We derive a sufficient condition such that γˆ > γˆ and show that is always holds when pooling is possible in equilibrium. Using the above relationships we need γNB(E −D)+D(1−p ) γNB(E −D)+D(1−p ) p µ > s G E −p D E −p D µ G ⇒γNB(E −p D)−γNB(E −p D)+D(p −p ) > 0. p G s µ G µ Using (16) for the respective separating and pooling non-bank equity requirements, the required condition becomes p (B −DNB) p (B −DNB) B p (E −p D)− B s (E −p D)+D(p −p ) > 0. E −p DNB G E −p DNB µ G µ B p B s Since DNB > DNB, substituting DNB into the denominator of the first term on the left p s s decreases the l.h.s of inequality. Then the following becomes sufficient: p (B −DNB)(E − B p 40

p D)−p (B−DNB)(E−p D)+D(p −p )(E−p DNB) > 0. Re-groupingandre-arranging G B s µ G µ B s we have D(p −p )(E−p B)−p (DNB−DNB)(E−p D) > 0. Note that we can re-group G µ B B P S G the above condition irrespective of whether p D ≷ p B and maintain sufficiency. Hence, G B substitute p D for p B and re-group to obtain (E −p D)[D(p −p )−p (DNB−DNB)] > G B G G µ B P S 0. Using D lim DNB = λ→∞ p p µ and D− (1−pG)κ D DNB = 1−γ s NB < , s p p G G the sufficient condition can be written as (E −p D)[D(p − p ) − pB (p − p )D] > 0. G G µ pGpµ G µ Hence, if this holds, the original inequality holds. Once again re-grouping and cancelling terms, the sufficient condition simplifies to p p −p > 0. Plugging in p = αp +(1−α)p , µ G B µ G B we obtain p (αp +(1−α)p ) > p ⇒ 1 > α > pB(1−pG) . For this to be met, it is necessary G G B B pG(pG−pB) that pB(1−pG) < 1 ⇒ p2 > p . pG(pG−pB) G B This condition is sufficient condition for non-banks to first compete in separating allocations as macroprudential regulation gets tighter. The interpretation is that good types must be sufficiently more likely to produce good outcomes than bad, formally given by p2 > p , G B which is stronger than requiring p > p . This stronger condition is implied by requiring G B that α ≥ α derived in Proposition 2, which gives the minimum level of α such that there can exist a pooling equilibrium for high enough λ. The intuition is simple. Note that if α < α only the separating equilibrium is possible and, thus, non-banks necessarily can only compete in separating contracts. ˆ Proof of Proposition 10. The relationship between thresholds γˆ and γˆ accrue from Proposition 9. Here in we show that γ˜, i.e., the macroprudential requirement that allows non-banks to compete in pooling contracts for any λ, satisfies the additional relationships stated in Proposition 10. 41

γ˜ is the solution to R (γ˜,λ) = RNB(λ). From (13) and (20), this can be written as p p (1−α) N−1 [γ˜E +(1−γ˜)Dp ]+λ ˜ [γ˜E +(1−γ˜)Dp ] N B µ (1−α) N−1p +λ ˜ p N B µ (cid:104) (cid:105) (cid:104) (cid:105) (1−α) γNB(λ ˜ )E +(1−γNB(λ ˜ ))DNB(λ ˜ )p +λ ˜ γNB(λ ˜ )E +(1−γNB(λ ˜ ))DNB(λ ˜ )p p p p B p p p µ = , ˜ (1−α)p +λp B µ (A.6) where γNB(λ ˜ ) and DNB(λ ˜ ) are given by (16) and (19). λ = λ ˜ is the value of λ for the p p exogenous credit demand for which separating and pooling equilibria are equivalent for nonbanks and banks are completely disintermediated. Because of Proposition 9, non-banks ˜ compete first in separating allocations as macroprudential requirements increase, thus λ is thecreditdemandthatmakesnon-banksswitchfromofferingseparatingtopoolingcontracts (absent competition from banks), i.e., 1−p RNB + G CNB = RNB(λ ˜ ), (A.7) s p s p G where RNB = γNBE/p + (1 − γNB)DNB and CNB = [p /[(1 − p )p ]][p B − (γNBE + s s G s s s B B G G s (1−γNB)DNBp )]—the separating contract terms for non-banks are derived using the same s s G steps as for the separating contract terms for banks in Proposition 2. From(A.6)and(A.7)wealsogetthatRNB+(1−p )/p CNB = R (γ˜,λ ˜ ),i.e.,theeffective s G G s p rate on the non-banks’ separating contracts is equal to the pooling rate banks offer. Now, consider a γ(cid:48) < γ˜. Then, RNB+(1−p )/p CNB > R (γ(cid:48),λ ˜ ), and hence the threshold λ(cid:48) that s G G s p equate the two is strictly less than λ ˜ . From (A.7), this implies that RNB(λ(cid:48)) > R (γ(cid:48),λ(cid:48)), p p i.e., non-banks can compete in pooling contract for λ ∈ [λ(cid:48),λ ˜ ]. This confirms that γ˜ is the minimum threshold for the macroprudential requirement such that non-banks can compete ˆ ˜ ˆ ˜ in pooling contracts for all λ, i.e., λ > λ, and that for λ > λ banks do not fund any unknown borrowers. 42

Proof of Proposition 11. From Proposition 6, we know that dλ/dγ = (∂R /∂γ)(dR /dγ −dR /dγ) > 0. p s p Following the following steps for the determination of λ ˆ from RNB = R (λ ˆ (γˆ)) we get that s p ˆ dλ(γ)/dγ = −(∂R /∂γ)(dR /dγ) > 0, because (∂R /∂γ) < 0. Note that this also implies p p p that dλ ˆ /dγ > dλ/dγ. Finally, because R (γˆ) = RNB and by continuity, we have that s s ˆ ˆ ˆ ˜ R (λ(γˆ)) = R (λ(γˆ)), and thus λ(γˆ) = λ(γˆ). Similarly, λ(γ˜) = λ(γ˜) as a direct consequence p p of (A.6). Proof of Proposition 12. Recall that the optimal capital requirements in the absence and in the presence of non-banks are denoted by γ∗ and γ∗∗, respectively. For the parameterization in Proposition 7 such that γ∗ ∈ [γ˜,γ ], we already established in the body of max the paper that γ∗∗ < γ˜ and, hence, the optimal capital requirement is strictly lower in the presence of non-banks. We now turn to the other possible cases starting with γ∗ ∈ Γ , i.e., γ ∈ [γˆ,γ ˆ ˆ). Using (15) 2 we get that (cid:18) (cid:19) N −1 −αλ(γ∗)(1−p )(1−κ)C (γ∗)+(1−α) +λ(γ∗) (1−p B) G s B N (cid:18) dλ (cid:19)−1(cid:20) dC (γ∗)(λ(γ∗))2 (cid:21) s = α(1−p )(1−κ)−ψ < 0, (A.8) G min dγ dγ 2 which also implies that −α(1−p )(1−κ)C (γ∗)+(1−α)(1−p B) < 0 G s B & −α(1−p )(1−κ)CNB +(1−α)(1−p B) < 0, (A.9) G s B because CNB = C (γˆ) > C (γ∗). s s s 43

Evaluating the first-order optimality condition for (25) at γ = γ∗ yields dλ ˆ (γ∗) (cid:20) (cid:18) N −1 (cid:19) (cid:21) −αλ ˆ (γ∗)(1−p )(1−κ)CNB +(1−α) +λ ˆ (γ∗) (1−p B) dγ G s N B dλ ˆ (γ∗) (cid:20) (cid:18) N −1 (cid:19) (cid:21) < −αλ(γ∗)(1−p )(1−κ)C (γ∗)+(1−α) +λ(γ∗) (1−p B) , G s B dγ N (A.10) because CNB = C (γˆ) > C (γ∗) and λ ˆ (γ∗) > λ(γ∗) from Proposition 11. Moreover, the last s s s term in (A.10) is negative due to (A.8). Hence, γ∗ cannot be a optimal solution to (25). Given that γ∗ ∈ (γˆ,γ ˆ ˆ), the expression in (A.8) is positive evaluated at γ = γˆ because the Lagrange multiplier drops out and capital requirements do not affect the non-bank collateral requirement, so the derivative is equal to zero. This implies that the l.h.s of (A.10) evaluated at γˆ can be either positive or negative because the r.h.s is positive. If the l.h.s is positive, then there exists γ∗∗ ∈ (γˆ,γ∗) because λ ˆ (γ∗) is increasing in γ, which implies that the capital requirement could be raised to the level equating it to the r.h.s. If the l.h.s is negative, we get a corner solution and γ∗∗ = γˆ. To conclude the proof for this case, we need to show that ˆ social welfare is not higher in interval Γ = {γ ∈ [γˆ,γ˜)} (if welfare is higher for γ ∈ Γ , then 3 1 trivially γ∗∗ < γ∗). The optimality condition for (26) is given by ˆ (cid:20) (cid:18) (cid:19) (cid:21) dλ(γ) N −1 −αλ ˆ (γ)(1−p )(1−κ)CNB +(1−α) +λ ˆ (γ) (1−p B) dγ G s N B ˆ ˆ dλ(γ)1−p B B + , (A.11) dγ N which is negative from (A.10) because dλˆ(γ) > 0, λ ˆ (γ) > λ ˆ (γ∗), and dλ ˆˆ(γ) < 0. The latter dγ dγ accrues from the fact that non-banks start competing with banks in pooling contracts first for high λ and then for lower ones, as γ increases further. Combining all the results above, we can conclude that γ∗∗ < γ∗ if γ∗ ∈ Γ . 2 Using similar logic we can also show that γ∗∗ < γ∗ is true for γ∗ ∈ Γ . Hence, the optimal 3 capital requirement is strictlylowerinthe presence of non-banks forparameterizations where non-banks are active in loan markets (either in separating or both separating and pooling 44

contracts). Finally, for γ∗ ∈ Γ , i.e., under parameterization where non-banks are not active 1 in loan markets, we can similarly show that the optimality conditions for (25) and (26) are negative. Since WNB = W, this implies that γ∗∗ = γ∗ concluding the proof. Γ1 B Appendix - Nonbank Contracting If the equity choice is not contractible, then long-term debt is not a viable solution. In this case a fragile funding structure consisting of runnable debt can restore incentives and non-banks would voluntarily maintain a level of equity that suffices to signal that they have enough skin in the game to deter them from risk-shifting. For simplicity, assume that debtholders are promised a gross interest rate greater or equal to one if they withdraw early, and an interest rate greater than one if they withdraw late, thus compensating them for credit risk. This is essentially a Diamond and Dybvig (1983) contract accounting for the possibility of non-bank default. Given that equity capital is observable, a drop below the required level would immediately induce debt-holders to withdraw early and a run would ensue. Because equity is worthless in a run, non-banks would voluntarily maintain the required level of capital. The type of run described above is driven by bad fundamentals due to risk-shifting (see, for example, Jacklin and Bhattacharya, 1988, and Allen and Gale, 1998). As expected, there can be other type of runs driven by the type of coordination failure described in Diamond- Dybvig. In order to simplify the analysis, we make a technical assumption that eliminates thepossibilityofsuchpanic-basedruns.11 Inparticular, weassumethattheliquidationvalue ξ is high enough to cover early withdrawals by all debt-holders. Because their debt would be riskfree in the short run and because debt-holders are risk-neutral, the gross interest rate for early withdrawals can be set equal to their outside option, i.e., equal to one. Then, the level of equity, γNB(cid:48), that non-banks need to hold would need to satisfy the following two 11Otherwise, multiple equilibria would exist as is typically the case in coordination failure games. The multiplicity could be resolved by assuming that the withdrawal decision is driven by sunspots (Cooper and Ross, 1998) or, preferably, by modeling an incomplete information game (Goldstein and Pauzner, 2005; Kashyap, Tsomocos, and Vardoulakis, 2024). 45

conditions: γNB(cid:48)E ≥ p [B −(1−γNB(cid:48))DNB(cid:48)] (B.12) B ξ ≥ 1−γNB(cid:48), (B.13) where DNB(cid:48) is the gross interest rate for late withdrawals. Condition (B.12) guarantees that there will be no risk-shifting in equilibrium, while condition (B.13) guarantees that there will be no panic-based runs. Combining the two and realizing that γNB(cid:48) takes its lowest value for DNB(cid:48) = D(cid:48)/p , leads to the following assumption for the liquidation value. B Assumption 4 The liquidation value ξ is higher than ξ ¯ = E−pBB < 1. E−1 Hence, the theory of the non-bank capital structure we describe in this paper could combineelementsofexistingtheorieswithfrictionsthatcharacterizeallfinancialinstitutions, namely risk-shifting due to the unobservability/noncontractability of the lending choice and instability due to run risk. If equity is contractible, similar to Holmström and Tirole (1997), then long-term debt is the solution. Otherwise, a fragile funding structure can induce the discipline needed to deter moral hazard and run risk in equilibrium (given assumption 4) similar to Calomiris and Kahn (1991) and Diamond and Rajan (2000). Hence, our theory can be applied to the various diverse non-bank financial institutions that have either stable or runnable liabilities. Moreover, the market-based non-bank equity capital will be the same for both types of institutions given the absence of panic-based runs for the latter, i.e., γNB = γNB(cid:48) and DNB = DNB(cid:48). 46