Financing Repeat Borrowers: Designing Credible Incentives for Today and Tomorrow

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1364 December 2022 Financing Repeat Borrowers: Designing Credible Incentives for Today and Tomorrow Anil K. Jain and Piruz Saboury Please cite this paper as: Jain, Anil K. and Piruz Saboury (2022). “Financing Repeat Borrowers: Designing Credible Incentives for Today and Tomorrow,” International Finance Discussion Papers 1364. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2022.1364. NOTE: International Finance Discussion Papers (IFDPs) 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 International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

FINANCING REPEAT BORROWERS: DESIGNING CREDIBLE INCENTIVES FOR TODAY AND TOMORROW ANIL JAIN∗ AND PIRUZ SABOURY† Abstract. Weanalyzerelationalcontractsbetweenalenderandborrowerwhen borrower cashflows arenotcontractibleandthe costsof intermediationvaryover time. Becauselendersproviderepaymentincentivestoborrowersthroughthecontinuation value of the lending relationship, borrowers will condition loan repayment on the likelihood of receiving loans in the future. Therefore, the borrower’s beliefs about the lender’s future liquidity and profitability become an important component of the borrower’s repayment decision. Consequently, the possibility of high lending costs in the future weakens repayment incentives and can cause the borrower to strategically default in some states and an inefficient under-provision of credit. We characterize the optimal relational contract and discuss the application of our model to the case of microfinance and trade credit. JEL Classification: G21, G33, O16 Keywords: Relationship Lending, Dynamic Incentives, Microfinance, Repeated Games, Trade Credit Date: November 2022. We would like to thank Bastian von Beschwitz, Emily Breza, Ricardo Correa, Logan Lewis and conference participants at AFBC conference (virtual), and WEAI conference. The findings and conclusions in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System, or the views of any other person associated with the Federal Reserve System. ∗Board of Governors of the Federal Reserve System. Email: anil.k.jain@frb.gov. †Department of Economics at the University of Houston. Email: psaboury2@uh.edu. 1

FINANCING REPEAT BORROWERS 2 1. Introduction In many forms of financial intermediation, such as microfinance and trade credit, a lender has little formal recourse or enforcement power to procure repayment. This friction is especially prevalent in developing countries with weaker institutions and fewer creditor protections. Therefore, to stop ex-post moral hazard and ensure repayment, a lender must design the contract to be incentive-compatible. This paper investigates the use of dynamic incentives with a lender whose funding cost varies over time. We propose a simple model of relational contracting—informal agreements sustained by the value of the future relationships (Baker et al. [2002])— where borrowers derive utility from the future value of lending relationships, which in turn, induces repayment incentives today. In other words, the long-run value from the bilateral relationship provides short-run repayment incentives. Specifically, we focus on how changes in a lender’s capacity to provide future loans affects the borrower’s incentive to repay today, and in turn, the equilibrium loan contract. We consider a world with two frictions: First, borrowers cannot commit to repay their loans and lenders cannot commit to providing loans in future periods. Second, lenders face shocks to the cost of intermediation that are uncorrelated with the borrower’s repayment ability. We believe the latter to be consistent with the case of microfinance, where global credit markets are virtually uncorrelated with the profits of micro-enterprises, and with the caseof trade credit, where costsfaced by suppliers may not be directly related to the demand for final goods faced by purchasers. We characterize how changes in the future value of a relationship affect today’s actionsandsubsequentlytheutilityofbothparticipants. Wedescribehowthelender structures incentives to maximize profits within the constraint of being unable to write binding long-term contracts. We show that the inability to commit to long-term actions, combined with changing intermediation costs, can lead to reduced lending and may even prohibit lending. Furthermore, we detail how the lack of enforceable contracts can change the distributionofeconomicrentssharedbetweenthetwoparties. Forinstance, environments with more variable lender costs (e.g. an economy characterized by deep financial crises and strong booms versus an economy with relative stable economy with shallow recessions and modest economic growth) cause the lending environment to be

FINANCING REPEAT BORROWERS 3 more likely to be characterized by only partial lending and some strategic borrower default. To gain intuition for our model, we model repeated lending where the lender’s cost of lending does not change over time. Due to moral hazard, no lending is possible without the prospect of future trading. We solve for the lender’s optimal stationary relational contract and characterize the equilibrium. We then consider the lending environment when the lender’s costs vary over time (according to a given Markov process). This added friction tempts the lender to not offer loans in some periods because of the higher realized costs in those periods. We investigate equilibria where lending is possible in all periods as well as equilibria where the lender may choose not to give loans in some periods. We show that in the latter case, partial default occurs along the equilibrium path solely due to the lender being unwilling to provide loans in some periods. After solving the equilibrium of the game, we detail how the equilibrium changes as we alter the game’s parameters. For instance, more persistent cost environments lead to a greater likelihood of the relationship breaking down (even when holding the ex-ante probability of each state constant). Therefore, this result would support the idea that infrequent long recessions may be worse for relational contracting than frequent short recessions. This paper is related to several strands of the economics and finance literature. First, it calls upon models of banking and intermediation under ex post moral hazard. Many related papers such as Townsend [1979] and Bernanke and Gertler [1989] show that under informational frictions, debt is the optimal type of financing by outside, uninformed investors. Bolton and Scharfstein [1996] consider the financing decision of the firm and the optimal debt structure when control can be transferred in different states of the world. In our setting, we also consider the characteristics of optimal financing, but we assume that borrowers have no net worth, that lenders cannot improve the contracting environment through monitoring, and that the borrowers cannot transfer control of their projects to creditors. Furthermore, a large financial literature (starting with Rajan [1992], Petersen and Rajan [1994], Berger and Udell [1995], Boot and Thakor [2000]) studies how strong borrower-lender relationships can help solve informational asymmetries. We contribute to this literature by modeling a lending relationship that is only possible

FINANCING REPEAT BORROWERS 4 through relationship lending—specifically, without the potential for a future lending relationship, the borrower has no incentive to repay the lender. In this paper, we consider a lender that faces liquidity risk in the capital markets. In some periods, financing costs for the intermediary are high, while in other states, these financing costs are low. The liquidity channel and the potential benefits of securitization have been studied by Carlstrom and Samolyk [1995] and Parlour and Plantin[2008]. DruckerandPuri[2009]considerthepotentialdiversificationbenefits ofloansalesbyfinancialintermediariesandshowthatborrowerswhoseloansaresold appear to receive more future credit from the same originator, potentially increasing the value of the lending relationship. Consistent with our model predictions, Berlin and Mester [1999] empirically show relationships between the structure of bank liabilities and relationship lending. Specifically, banks with greater deposit funding (relative to wholesale funding) provide greater loan smoothing than other banks. Ourpaperbuildsontheinsightsandtoolsdevelopedintherelationalcontractingliterature (see Levin [2003], Halac [2012], McAdams [2011], Macchiavello and Morjaria [2012], Li and Matouschek [2013], and Barron and Powell [2019] for example), which studies repeated contracting and reputation-building between parties. Our paper uses the relational lending literature tools to explain the relationship dynamics that we see in microfinance and trade finance. 1.1. Motivating examples. While our framework applies to any setting where contract enforcement between two parties today depends on the future value of the relationship between them, we highlight two applications—microfinance and trade credit. Microfinance. Microfinance has been a rapidly growing and well-publicized financial tool in developing countries.1 Microloans are intended for poor clients who generally do not have any pledgeable collateral and who live in countries where the formal legal institutions have little to no ability to enforce small credit contracts. To overcome these constraints, microlenders have developed two important contractual innovations to make lending feasible: social collateral and dynamic incentives. Microfinance is able to harness the power of social networks in a variety of ways 1According to mixmarket.org, microfinance institutions disbursed more than $150 billion in loans to over 30 million customers worldwide in September 2018.

FINANCING REPEAT BORROWERS 5 to provide repayment incentives.2 However, in this paper we focus on the use of repeated contracting to provide repayment incentives. The standard microloan is around$100andprimarilyaimedtofinancethepurchaseofassetsorworkingcapital for microenterprises. Borrowers make frequent installment payments for one year and upon completion, are disbursed a new loan.3 If borrowers default on their loans, then they are not offered future access to credit. This dynamic incentive is often credited as one of the main innovations of microfinance.4 In connection with our model, we highlight the potential fragility of microfinance institutions (MFIs) as a result of the reliance on future lending activities to provide repayment incentives today.5 Namely, if microlenders themselves become liquidityconstrainedinthefuture,doesthisconstrainthavefeedbackeffectsontherepayment ofexisting, outstandingloans? Inthistypeofscenario, commonfinancingstructures such as short-term debt or loan securitization may be suboptimal. This analysis is relevant to recent developments in microfinance in India, one of the largest microfinance markets, which has faced a similar crisis in the past.6 Throughout the crisis, microlenders faced difficulties accessing credit markets, despite the fact that the borrowers’ economic conditions remained unchanged. Some smaller MFIs failed to obtain financing and walked away from their loan portfolios. Larger MFIs were forced to delay new disbursements while still paying staff costs and assuring clients that their new loans were indeed coming. 2Manymicrofinancecontractshavejointliability,whereborrowersmustpayfortheirpeersifthose peersdecidetodefault(seeGineandKarlan[2006],GineandKarlan[2009],andGin´eetal.[2011] for empirical evidence on the effects of joint liability.) Research has also shown (see Feigenberg et al. [2010] and Breza [2012]) that even without contractual linkages between borrowers, there still may be social effects at play. 3In many settings the standard maturity for a microloan is 50 weeks, but some lenders offer variations on this product. 4See Morduch [1999]. Using an innovative empirical design, Karlan and Zinman [2009] find that borrowers dorespondto dynamicincentivesand aremore likelyto repay loansifthey know anew loan is forthcoming. 5In recent empirical work, Gertler et al. [2021] show how digital collateral can also be used to improve the long-term borrower incentives within microfinance contracts. 6In October 2010, the Indian state of Andhra Pradesh suffered from a debilitating wave of microfinance defaults sending repayment rates from close to 100 percent to 10 percent and below. The default crisis began when the state government passed an emergency ordinance to rein in microfinance institutions. The ordinance set out to curb usurious interest rates, cases of borrower harassment by credit officers, and over borrowing by the state’s poor. Stories of borrower suicides permeated the local and international press (see, for example, Indian Express [2010] and International Business Times [2010]). When banks refused to lend to MFIs, the localized crisis spread to all of India through a national liquidity crisis.

FINANCING REPEAT BORROWERS 6 To apply our model to microfinance, we make two financial friction assumptions. First, we assume that MFIs are monopolist lenders. We argue that this assumption is innocuous because some sort of monopoly power is required to maintain repeated lending relationships (Petersen and Rajan [1995]). Furthermore in the case of India, regulation gives MFIs some degree of market power, specifically, borrowers are allowed to take loans from a maximum of two lenders.7 Second, in our model, we assume that an MFI must have at least some debt financing. This assumption is consistentwiththemicrofinanceindustrywheretheaveragereportedcapitaltoasset ratio for Indian MFIs is just over 27 percent (World Bank [2019]). Trade credit and supplier relationships. Our model and explanation is focused on a traditional lender-borrower relationship; however, relabelling the agents in our model show that it can also be used to describe trade credit and supplier relationships. Specifically, we can analyze a buyer’s (borrower’s) incentive to renege on repaying the supplier (lender) as a function of the supplier’s capacity to continue to deliver future products (loans). Seminal early work on trade credit and supplier relationships, such as McMillan and Woodruff [1999] and Banerjee and Duflo [2000] analyze the value of reputation, repeated interactions, and relational contracts. We contribute to this literature by considering how a borrower’s incentive to repay trade credit depends on a supplier’s capacity to either finance or supply more goods—a channel that is missing from recent contributions to the literature such as Boissay and Gropp [2013], Jacobson and Von Schedvin [2015], McGuinness et al. [2018] as well as major prior contributions such as Burkart and Ellingsen [2004] and Cunat [2007]. This channel of potential default risk is particularly relevant to the developing world where contractual protections are weaker and expected loss due to default is significantly higher. In our model, we show how changes in the supplier’s financing—through changes in the value of the future relationship—directly affects the set of possible relationship’s today. In the next section, we describe our environment. Section (3) presents the baseline case where the lender’s cost does not change over time. Section (4) extends the analysis to our full model by allowing the cost of lending to vary over time. Section 7To protect borrowers against price gouging by monopolists, interest rates are also regulated at a maximum rate of 10 percent to 12 percent above the lender’s cost of capital.

FINANCING REPEAT BORROWERS 7 (5) analyzes how the optimal contract changes with different parameters and section (6) concludes. 2. Model Thissectiondescribesthemodel,assumptions,timing,andsolutionconcept. Section (3) characterizes the solution for a basic version of the model that has no change in the lender’s funding cost and section (4) extends the analysis to the full model by allowing the lender’s funding cost to follow a Markov process. 2.1. Set-up. There is an infinitely repeated game between a risk-neutral, profitmaximizing, monopolist lender and a risk-neutral capital-constrained agent. Each period, the agent has an investment project that requires a unit of capital and produces a deterministic output of v at the end of the period. The lender can provide a loan to the agent for the project. The lender’s cost of a unit of capital is c , which takes two possible values, c and c , where c is strictly less than c .8 Upon t l h l h lending, the lender requests a repayment of r to be made at the end of the period.9 t When the agent invests in their project, the output is observable and verifiable but, crucially, the lender does not have any enforcement power. Rather, the bank’s sole tool to extract repayment for the loan is through a relational contract and ensuring the value of a long-term banking relationship outweighs the short-term incentive to renege on payment. To highlight the importance of the future relationship between the lender and the agent, weassumethatthelender’scostofcapitalforthenextperiod, c , ispublicly t+1 observable before the agent repays the lender. Therefore, before the agent decides whether to repay, the agent can infer whether the lender will offer a future loan and, if so, at what interest rate. This assumption is motivated by two different phenomena: first, microfinance lenders have borrowers on different loan cycles, therefore, if a borrower’s friend is unable to renew their loan, this information will influence the 8c can be thought of as a reduced form formalization of the cost of lending in each period. In t particular,c ≡κc +(1−κ)c suchthatκisbetween0and1. c isthecostofequityfinancing, t e,t d,t e,t c is the cost of debt financing, and κ is the bank’s capital-to-assets ratio. d,t 9r includes both the loan’s principal and interest payment. t

FINANCING REPEAT BORROWERS 8 borrower’s expectation of whether they will get a loan tomorrow. Second, the information concerning microfinance institutions is well documented both in newspapers and by politicians.10 The evolution of lending cost (c ) follows a Markov process whose transition matrix t is common knowledge: (cid:34) (cid:35) p 1−p l l 1−p p h h where p = Prob(c = c |c = c ) is the probability of moving from a low-cost l t+1 l t l state to a low-cost state and p = Prob(c = c |c = c ) is the probability of h t+1 h t h moving from a high-cost state to a high-cost state.11 Thus, the long-term stationary (invariant) distribution of states is: (cid:34) (cid:35) (cid:34) (cid:35) θ 1−p h (1) Θ = l = 2−p l −p h θ 1−p l h 2−p−p l h where θ is the long-run probability of low cost and θ is the long-run probability of l h high cost. For ease of exposition, we make a number of simplifying assumptions. The agent’s and the lender’s outside options are both normalized to zero and the borrower is credit-constrained such that they cannot pay any amount ahead of time. The agent also has no storage technology to transfer wealth across time and we assume that the agent and the lender have the same common discount rate, δ ∈ (0,1). While the bank and the agent cannot enter into any form of binding contract, before the game begins, the lender offers a “relational contract” that lays out the conditions under which the lender will lend the required capital to the agent and the required repayment amount in any given time period.12 The relational contract also specifies the action undertaken by the lender or the borrower if the other party fails to undertake their specified action in the relational contract. Once the lender offers the relational contract, if the borrower accepts the contract, the repeated game begins. The lender’s goal is to choose a relational contract 10Forexample,duringtherecentIndianmicrofinancecrisis,journalistsandpoliticianswrotedamning articles about MFIs and bankrolled television commercials to further undermine the portfolios ofMFIsaftertheonsetofthecrisis. SeeBanerjeeandDuflo[2011]andBreza[2012]fordiscussions of the recent Indian microfinance crises. 11Theprobabilityofmovingfromalowstatetoahighstateistherefore(1−p )andtheprobability l of transitioning from a high state to a low state is (1−p ). h 12This lending structure can entail lending in every period or just in some periods.

FINANCING REPEAT BORROWERS 9 that maximizes their expected discounted stationary profits in the repeated game, conditional on being accepted by the borrower and being incentive-compatible at all times. We solve for the perfect public equilibrium of this game which requires that play following each history be a Nash equilibrium.13 Moreover, for simplicity and ease of exposition, wefocussolelyonstationaryrelationalcontractswheretheloanschedule and repayment amounts are independent of time. 2.2. Timing. The timing of the game is as follows: (1) The lender offers a relational contract to the potential borrower. (2) The borrower decides whether to accept the relational contract. If the borrower rejects the contract, the game ends and the lender and the agent both receive zero payoff. Otherwise, the game proceeds. (3) Both parties observe the lending cost in the first period (c ) and the repeated 1 game begins. (4) At the beginning of each period t (including t = 1): (a) The lender chooses whether to offer a loan. Their decision is denoted by l ∈ {0,1}, with l = 1 for lending. If the lender offers a loan, the t t lender also chooses a required repayment amount r . t (b) The agent either accepts or rejects the loan offer. Their decision is denoted by d ∈ {0,1} with d = 1 for accepting. t t (5) At the end of each period t (including t = 1): (a) The lending cost for the next period (c ) is observed by all parties. t+1 (b) The borrower then decides how much to repay, rˆ ∈ [0,r ]. t t (c) Payoffs are realized. (6) The game enters period t+1 and repeats from step (4). 13In this model, perfect public equilibrium imposes the same sequential rationality requirement that subgame perfection would impose in a complete information model [Fudenberg et al., 1994]. Hence, oncetherepeatedgamestarts, neitherthelendernortheborrowercancommittotherelational contract. Each agent will unilaterally renege on the agreement in any period if the benefits fromrenegingarelargerthantheexpectedbenefitsfromcontinuingwiththerelationalcontract. In other words, the bank’s choices are limited to relational contracts that are incentive-compatible— thatis,perfectpublicequilibria. Simplyput,thepre-gamestepworksasanequilibriumrefinement criterion for cases where the repeated game has more than one equilibrium. Such setting is consistent with the reality of microfinance where formal contracting is not possible but the bank can implement dynamic repayment incentives by putting preconditions on subsequent loans.

FINANCING REPEAT BORROWERS 10 We first analyze a benchmark case where lending cost is constant in section (3). Then we turn our attention to finding the equilibrium of the full model in section (4). 3. Equilibrium: Relational contracts with no change in cost over time (baseline) To gain intuition into the lender and borrower’s dynamic incentives, we begin with a simple environment that excludes any uncertainty. Specifically, we start with a constant cost of lending in all time periods, that is, c = c. t By abstracting from the time-varying cost structure in the full model, we consider a simple stationary version of our problem to demonstrate what inefficiencies in lending may arise because of the absence of binding contracts. In the simple baseline model, the lender chooses a feasible relational contract that maximizes their total discounted profits: ∞ (cid:88) (2) Π = δtl (r −c) t t t=0 Recall that the relational contract is not a binding contract and thus has to be sustainable in equilibrium—that is, it must be incentive-compatible for the lender and also satisfy the borrower’s repayment incentive compatibility constraint (ICC) at every point in time. In other words, the borrower must be willing to make the repayment the bank requires in every period, that is, rˆ = r . Using Abreu [1988] t t one-shot deviation principle, we know that the optimal lender punishment for noncompliance is breaking off trade for all future rounds. Therefore, the borrower’s long-term payoff for complying at the end of each period must be greater than or equal to the borrower’s outside option: ∞ (cid:88) (3) δtl (v −r )−r ≥ 0 t t 0 t=1 where (cid:80)∞ δtl (v − r ) is the discounted value of future borrower surplus from t=1 t t borrowing and r is the required repayment at time t = 0. Thus, the lender’s 0

FINANCING REPEAT BORROWERS 11 constrained maximization problem is as follows: max Π (as defined by equation [2]) {lt,rt}∞ t=0 (4) s.t. equation [3] holds. Furthermore, recall that we concentrate solely on stationary relational contracts where contract parameters are time-invariant—that is, l = l and r = r for all t. t t Therefore, there are only two possible types of relational contracts: no lending in any period (l = 0) or the following: i. The lender extends a loan and requires repayment r in the first period. ii. After the first period, at all histories where the lender has always received a repayment of at least r in the past, the lender will continue to extend a loan and require repayment r. At any other history, the lender will not offer any loans. iv. The borrower accepts a loan and repays r in the first period. v. After the first period, at all histories where a loan has been extended in all previous periods, the borrower accepts a loan and repays r. At any other history, the borrower accepts a loan and repays zero. The relational contract described above will be a perfect public equilibrium so long as both parties’ long-run discounted payoff is non-negative. However, because we assumethatthelendermakestheinitialoffer,thelenderoptsfortheequilibriumthat generates the largest profits, which is the equilibrium with the highest repayment (r∗) that is acceptable to the borrower. The resulting interest rate would be the solution to the lender’s maximization problem laid out in equation [4]. To pin down r∗, we begin by noting that in a stationary relational contract, the borrower’s ICC (equation [3]) simplifies to: δ (5) (v −r)−r ≥ 0 1−δ The left-hand side of equation [5] is the value of sustaining the relationship and the right-hand side is the borrower’s outside option. The latter is the result of the fact that, as discussed earlier, the optimal punishment for reneging is breaking the relational contract so that the game reverts to static Nash equilibrium actions, in which no loans are offered.

FINANCING REPEAT BORROWERS 12 Solving equation [5] for r reveals that the maximum repayment, r∗, that can be sustained in a stationary equilibrium is δv. Regarding the lender’s problem of choosing the most profitable stationary relational contract, equation [4] simplifies to maximizing r − c conditional on the expected profits in the sustained relationship being higher than the outside option of zero. Therefore, if δv is greater than or equal to c, the profit-maximizing monopolist lender will lend (that is, l∗ = 1) and require a repayment of δv in every period. Otherwise, nostationaryrelationalcontractwithpositivelendingispossiblebecause even with maximal punishment (no future loans), the borrower is unwilling to make a repayment that is high enough for the bank to cover its costs. Proposition 1. When intermediation costs are fixed over time at c, lending is possible if and only if δv is greater than or equal to c. Under this condition, the bank will lend (l∗ = 1) and require a repayment of r∗ = δv in every period. Otherwise, no lending occurs in equilibrium. Proposition 1 states that even though the lender is a monopolist, it cannot expropriate the full rent from the relationship by charging r = v because of the lack of formal contractual protection. In fact, the only obstacle that prevents the borrower from strategic defaulting on their loan is the “shadow value of the future,” equal to δv. The latter is only sufficient to constrain the borrower from reneging if the repayment does not exceed the shadow value of the future. Thus, the borrower retains a positive per-period payoff equal to (1−δ)v. Moreover, as a result of this limitation, lending does not occur at costs above δv even though it is socially efficient to lend if the cost of lending, c, is less than borrower’s value from a loan, v. 4. Equilibrium: Relational contracts with changing costs In this section we turn our attention to the full model where the cost parameter changes over time. In particular, we concentrate on the interesting case where the lender’s cost of lending c , take two possible values, c and c , such that c is less t l h l than δv and c is greater than δv (that is, c < δv < c ).14 Note that lending in the h l h 14Intheothertwopossiblecases,theequilibriumcharacterizationistrivial. Ifcostisalwaysbelow theborrower’sshadowvalueofthefuture(c <c <δv), thenlendingineveryperiodisprofitable l h for the bank and the lending equilibrium characterized in section (3) continues to hold. If cost is always more than the borrower’s shadow value of the future (δv <c <c ), then lending is never l h profitable and no lending occurs in equilibrium.

FINANCING REPEAT BORROWERS 13 high cost state is efficient if the borrower’s benefit, v, is greater than the lender’s cost of lending, c , but we do not place any parameter restrictions to ensure this h occurs. Recall the assumption that at the end of each time period t, the borrower observes tomorrow’s cost realization c before deciding whether to repay today’s loan. This t+1 assumption, combined with the variability of lending costs, captures the possibility that the borrower’s moral hazard—the borrower’s incentive to renege—changes over time. Asbefore,thelenderandtheborrowercannotenterintoabindingcontract. Nonetheless, they can agree on a relational contract (l ,r ) if it is incentive-compatible for t t both the lender and the borrower. The borrower’s ICC is that for all periods where the borrower received a loan, the borrower’s expected value of the future lending relationship is at least as large as their outside option at time t (not repaying the loan).15 Formally: ∞ (cid:88) (6) ∀t s.t. l = 1, δkE[l (v −r )]−r ≥ 0 t t+k t+k t k=1 Moreover, the lender should also find it profitable to abide by the relational contract at any point in the game, that is, for all periods the expected profits from lending are greater than or equal to zero. Formally: ∞ (cid:88) (7) ∀t Π = δkE[l (r −c )] ≥ 0 t t+k t+k t+k k=0 At the beginning of the game, the lender’s goal is to choose a relational contract that maximizes the lender’s long-term expected stationary profits, conditional on both ICCs (equations [6] and [7]) being satisfied. max Π = E (Π ) Θ 0 {lt,rt}∞ t=0 (8) s.t. equations [6] and [7] hold. As before, we limit our attention to stationary relational contracts, where lending and repayment are independent of time but can be state-dependent. 15In Abreu [1988], the optimal punishment for non-compliance is breaking off trade for all future periods.

FINANCING REPEAT BORROWERS 14 Definition 1. A relational contract (l ,r ) is stationary if, for all t, t t   l if c = c r if c = c h t h h t+1 h l = & r = . t t l if c = c r if c = c l t l l t+1 l Stated differently, the lender can offer a loan and promise to continue lending in both or one of the states as long as the borrower repays the agreed amount. Such an offer would be accepted by the borrower and respected by both parties only if it satisfies both the borrower’s and the lender’s ICCs (equations [6] and [7])— the relational contract’s feasibility conditions.16 Then, conditional on the relational contract being feasible, the lender offers the relational contract that generates the largest expected profit for the lender. There are three possible types of stationary relational contracts that the lender can choose from: (1) always-lending—offer to lend in all states. (2) partial-lending—offer to lend in some states, which is state-dependant. (3) no-lending—not offer to lend in any state. To determine the equilibrium of the game, we need to complete two steps. First, we need to solve for the parameter space where each relational contract is feasible. Second, from the set of feasible relational contracts, we need to solve for the contract that confers the largest expected profit for the lender. Since we assume the lender is able to choose their preferred relational contract (conditional on it being incentive compatible), the equilibrium of the game can be characterised using this contract. Inproposition(2), westatethelender’soptimalrelationalcontractandsubsequently explain the intuition for the lender’s preferred strategy and the game’s characterization. Then in sections (4.1) and (4.2), we explain in detail the lender’s and borrower’s strategies for each relational contract. The no-lending relational contract’s feasibility and equilibrium characterization are trivial. Proposition 2. At the beginning of the game, the lender’s optimal relational contractisoneofthreepossiblerelationalcontracts: “always-lending,” “partial-lending,” or “no-lending,” according to the following conditions. Always-lending relational contract The lender offers to lend in all states if, and only if, conditions (1) and (2) hold: 16Note by requiring the ICC to hold for all periods, including t equal to 0, we are implicitly subsuming the lender and borrower’s participation constraint into the ICC.

FINANCING REPEAT BORROWERS 15 (cid:16) (cid:17) (1) c −δv ≤ δ 1−p h (δv−c ), which guarantees always-lending to be feasible. h 1−δp l l (cid:16) (cid:17) (2) θ c ≤ δv − δvθ p +(1−p )δ(1−p h ) , which guarantees always-lending to h h l l l 1−δp h be more profitable than partial-lending. Partial-lending relational contract The lender offers to lend only in the low-cost state if, and only if, at least one of conditions (1) and (2) does not hold, and the following condition holds: (cid:104) (cid:105) (3) c ≤ δv p +(1−p )δ(1−p h ) , which guarantees partial-lending to be feasible. l l l 1−δp h No-lending relational contract Otherwise, if both conditions (1) and (3) do not hold, then lending is not feasible and the lender does not lend. The proof of proposition (2) is in the appendix. Conditions(1)and(3)inproposition(2)arethefeasibilityconditionsforthealwayslendingrelationalcontractandthepartial-lendingrelationalcontract, respectively.17 Condition (2) is the lender’s expected profit maximising condition, and if it holds, the always-lending relational contract is more profitable than partial-lending relational contract. Conversely, if condition (2) does not hold, partial-lending is more profitable than always-lending. Finally, if the always-lending and partial-lending relational contracts are both infeasible (conditions (1) and (3) do not hold), then the lender makes no loans (no-lending). Forconvenienceandclarity, table(4)summarizestheequilibriumstrategies, payoffs, and feasibility conditions. For completeness, sections (4.1) and (4.2) describe and characterize the feasibility of the always-lending and partial-lending relational contracts respectively. Section (5) analyses how the lender’s optimal relational contract varies in different economic environments. 17Recall that the feasibility conditions require that the relational contract is incentive compatible for both the lender and the borrower for any point in the game.

FINANCING REPEAT BORROWERS 16 Table 1. Summary of Relational Contract Characteristics Partial-lending Always-lending Lending l = 1 if c = c l = 1 for all t t t l t Strategy l = 0 if c = c t t h Gross δv if c = c r∗ = δv for all t t+1 l Repayment δvδ(1−p h ) if c = c (1−δp ) t+1 h h Expected θ [p r +(1−p )r −c ] r∗ −c¯ l l l l h l Stationary 1−δ 1−δ Profits Additional Feasibility None (1−δp )(r∗ −c )+δ(1−p )(r∗ −c ) ≥ 0 l h h l Condition Stationary θ [v −p r −(1−p )r ] l l l l h Borrower v 1−δ Payoff 4.1. Always-lending: The lender offers a loan in all states. In this subsection we start by walking through the conditions that determine the feasibility of the always-lending relational contract and the underlying intuition. Finally, we characterize the always-lending relational contract. Starting with the borrower’s incentive compatibility condition: If the borrower believes that the lender will offer a loan in every period, then the borrower’s participation decision problem will be identical to the case where the lender’s cost is deterministic, as discussed in section (3). Thus, the borrower’s maximum incentivecompatible repayment is δv for all time periods.

FINANCING REPEAT BORROWERS 17 Turning to the lender’s incentive compatibility conditions: Using the borrower’s maximumwillingnesstorepayandthelender’sexpectedprofit(asshowninequation [7]) reveals that by extracting a maximum repayment of r = δv in every period, t the lender’s expected discounted profit at time t = 0 simplifies to (cid:34) (cid:35) ∞ (cid:88) (9) Π = E δt(δv −c )|c 0 t 0 t=0 To ensure that offering loans in every period is a credible strategy, equation [9] must be greater than or equal to zero for both possible initial cost realizations, c = c 0 l and c = c . Otherwise, the lender could not credibly offer loans in all states. This 0 h equation can be written in recursive form by denoting the lender’s value function for each state as V , where s is the current state, and s(cid:48) is the state in the next period. s For the lender to avoid reneging on the relational contract, the following lender ICC must hold: V = (δv −c )+δE(V |s) ≥ 0 ∀s = {l,h} s s s(cid:48) This ICC requires the current value of the relationship to be sufficiently large such that the lender would prefer not to renege on the contract by stopping lending. Using this ICC we can state the following corollary of proposition (2) and the proof is in the appendix. Corollary 1. The lender will only be able to credibly offer loans in all states if the following condition is satisfied: (1−δp )(δv −c )+δ(1−p )(δv −c ) ≥ 0 l h h l Corollary1statestheintuitiveconditionforthealways-lendingrelationalcontractto be feasible: If the lender realizes a high cost state, the lender’s expected discounted losses from continuing to lend in this high state and future possible high states must be smaller than the lender’s expected discounted profits from future low states. This condition is the same as condition 1 in proposition (2), but rearranged in a more intuitive manner. To sum, the always-lending relational contract equilibrium is characterized by the following: • The lender lends in all states (l = 1 for all s ∈ {l,h}) and the borrower s always repays δv, i.e., r = δv for all s ∈ {l,h}. s

FINANCING REPEAT BORROWERS 18 • This relational contract is feasible only if the loss from lending in high cost states does not outweigh the expected profit from future low cost states: (1−δp )(δv −c )+δ(1−p )(δv −c ) ≥ 0 l h h l • The lender’s ex-ante expected profit is: δv −θ c −θ c l l h h Π = AL 1−δ • The borrower’s welfare (ex ante expected utility) is: U = v AL • This relational contract is socially efficient if the borrower project’s payoff is greater than the cost of capital in the high state (v > c ). h 4.2. Partial-lending: Lender offers a loan only in the low-cost state. In this subsection we analyze the partial-lending relational contract. We start by walking throughtheconditionsthat determine thefeasibility ofthepartial-lendingrelational contract and the underlying intuition. Finally, we characterize the partial-lending relational contract. Starting with the lender’s incentives: The cost of lending may be prohibitively high in the high cost state causing either the lender’s feasibility condition for lending in all periods to not be met (corollary (1)) or the profits from lending in only low cost states to be higher than always-lending. Regarding the borrower’s incentives, we assume that the borrower observes tomorrow’s state before deciding to repay; therefore, if the lender lends in only low cost states, the borrower’s willingness to repay a loan will always be relatively lower if tomorrow’s state realization is a high cost state rather than a low cost state. Taking the borrower and lender’s incentives together, the stationary partial-lending relational contract consists of lending only in the low-cost state and the required loan repayment being conditional on tomorrow’s state of the world.18 Specifically, the lender sets an interest rate (r ) for all loans, which the borrower has to pay in l full if tomorrow’s state is a low cost state (s = c ), or lose access to all future t+1 l 18To be precise, there is also a feasible stationary partial-lending relational contract that involves lending only in high cost states. However, that contract would always be weakly dominated by lendinginallstatesandpartial-lendinginlowcoststates;therefore,forconveniencethatrelational contract is ignored.

FINANCING REPEAT BORROWERS 19 loans. However, if tomorrow is a high cost state (s = c ), the borrower is allowed t+1 h to partially default and only repay r < r . If the borrower repays less than r then h l h the lender stops future lending.19 This partial-lending relational contract can be formally represented with (r ,r ) such that: l h (1) the lender’s obligation within the relational contract is described as i. offer a loan in the first period if, and only if, the cost is c , l ii. atallhistorieswhere(a)thelenderhasofferedaloanwhenthecostwas c , and (b), received at least r when tomorrow’s state was s, continue l s to offer a loan whenever the cost is c , l iii. at any other history, do not offer a loan, (2) the borrower’s obligation within the relational contract is described as: i. accept a loan and repay r in the first period if a loan is offered, where s(cid:48) s(cid:48) is the next period’s state, ii. at all histories where the lender has offered a loan whenever the cost was c , accept a loan and repay r , where s(cid:48) is the next period’s state, l s(cid:48) iii. at any other history, accept a loan and repay zero. For this relational contract to be feasible, we have to verify the lender’s and borrower’s ICCs are satisfied. The borrower’s incentive compatibility requires: (cid:34) (cid:35) ∞ (cid:88) (10) E δtl (v −r )|c ≥ r (c ) ∀c ∈ {c ,c } t t 1 0 1 1 l h t=1 where r (c ) = r . 0 s s Equation [10] can be rewritten in the following recursive forms where the subscripts denote today’s state and tomorrow’s state respectively: U = v −r +δ[p U +(1−p )U ] ≥ v LL l l LL l LH U = v −r +δ[p U +(1−p )U ] ≥ v LH h h HH h HL U = 0+δ[p U +(1−p )U ] ≥ 0 HH h HH h HL (11) U = 0+δ[p U +(1−p )U ] ≥ 0 HL l LL l LH BecauseU isaweightedaverageofU andU ,andU isafractionofU ,the HL LL LH HH HL last two inequalities are non-binding. Intuitively, the borrower has more incentive 19We do not rule out situations where r =0. h

FINANCING REPEAT BORROWERS 20 to renege when the current state is low and the borrower has already received a loan than when the borrower has not received a loan. Thus, we restrict our attention to the first two constraints in equation [11] that are potentially binding. They can be rewritten as: (12) vδ(1−δp ) ≥ [(1−δp )−δ2(1−p )(1−p )]r +δ(1−δp )(1−p )r h h l h l h l h (13) vδ2(1−p ) ≥ δ2p (1−p )r +(1−δp )(1−δp )r h l h l h l h Intuitively, inequalities (12) and (13) require that, conditional on getting a loan, the borrower is sufficiently incentivised to repay r when tomorrow’s state is low and to l repay r when tomorrow’s state is high, respectively. h The lender’s expected discounted profit at time t = 0 simplifies to: (cid:34) (cid:35) ∞ (cid:88) (14) Π = E δtl (r −c )|c ∀c ∈ {c ,c } 0 t t t 0 0 l h t=0 Because, in a partial-lending relational contract, loans are only disbursed in the low-cost state, the lender’s problem (that is, maximize Π as described in equation 0 [14)) can be simplified to maximizing the expected per-period profit in the low cost state: (15) max p r +(1−p )r −c l l l h l r,r l h s.t. inequalities [12] and [13] hold and r ≥ r ≥ 0 l h The solution to this optimization problem can be solved with a linear program. Using this solution, we can state the contract’s feasibility condition in lemma (1). Lemma 1. The lender will be able to credibly offer loans only in the low-cost state, if, and only if, the following condition is satisfied: (cid:20) (cid:21) δ(1−p )(1−p ) l h δv p + ≥ c l l 1−δp h The proof of lemma (1) is in the appendix. To sum, the partial-lending relational contract equilibrium is characterized by the following:

FINANCING REPEAT BORROWERS 21 • The lender only lends in the low cost state (l = 1 and l = 0). l h • The size of borrower’s repayment is conditional on the next period’s state: r∗ = δv l δ(1−p ) r∗ = δv h ≤ r∗ h (1−δp ) l h • This relational contract is feasible (and profitable) only if the expected repayment covers the cost of lending in the low cost state: (cid:20) (cid:21) δ(1−p )(1−p ) l h δv p + ≥ c l l 1−δp h • The lender’s ex-ante expected profit is: (cid:20)(cid:18) (cid:19) (cid:21) θ δ(1−p )(1−p ) l l h Π = p + δv −c PL l l 1−δ 1−δp h • The borrower’s welfare (ex ante expected utility) is: (cid:20) (cid:18) (cid:19) (cid:21) θ δ(1−p )(1−p ) l l h U = v − p + δv PL l 1−δ 1−δp h • This relational contract is socially efficient if the borrower project’s payoff is less than the cost of capital in the high state (v < c ). h 5. Comparative statics Relational contracts are often less robust to greater economic volatility than other contracts. To gain both more intuition for the relational contract characterization andtounderstandhowtheoptimalrelationalcontractschange, thissectionanalyzes how the feasibility and profitability of the always-lending and partial-lending relational contracts are affected by different economic parameters, such as persistence of economic recessions or depth of depressions. We introduce a number of comparative statics. 5.1. Deep financial crisis versus a shallow recessions. Economic depressions often make maintaining relational contracts less feasible. In this subsection, we analyze how volatility in lender costs affect always-lending and partial-lending. To do so, we examine the experiment where the lender’s expected cost does not change — only the dispersion in the cost varies. We show that with greater dispersion in costs, the lender’s optimal lending contract is more likely to be a partial-lending contract

FINANCING REPEAT BORROWERS 22 than an always-lending contract because, with greater cost dispersion, the lender’s expected profits from partial-lending rise and the feasibility of always-lending falls. Tostart, weintroducetheparameter, ∆, thatmeasuresthedispersioninthelender’s cost between the low and high cost states while holding the lender’s expected cost, c¯, and the stationary distribution of states (θ and θ ) fixed. Formally, ∆ is defined l h by the following two equations: (16) ∆ ≡ c −c such that c¯, θ , and θ , are constants h l l h where c¯≡ θ c +θ c l l h h Therefore, as we increase ∆ we increase the dispersion in the lender’s possible realization of costs. In effect, we consider large values of ∆ simulate deep financial crisesandlargeeconomicboomswhereassmallvaluesof∆simulateacalmeconomic period with only shallow recessions. Lemma 2. As the measure of cost dispersion, ∆, rises: (1) the expected profitability of the always-lending contract is unchanged but the expected profitability of the partial-lending contract increases, (2) the feasibility of the always-lending contract decreases whereas the feasibility of the partial-lending contract increases. To gain intuition for lemma (2), figure (1) plots the expected discounted profits for always-lending (blue line) and partial-lending (orange line), as well as the expected discounted profit from always-lending starting from a high state (dashed blue line) as we vary the cost dispersion parameter, ∆ (x-axis). Starting with the profitability and feasibility of the always-lending relational contract: The expected profitability of always-lending does not change because the stationary distribution of low and high cost states do not change (horizontal blue line). However, the feasibility of this contract decreases as dispersion rises because the expected future profits of lending when starting in a high cost state, linearly fall (the dashed blue line). The intuition for this result follows from corollary (1). Specifically, for the always-lending contract to be feasible, the lender must make non-negative expected profits in all periods; but as dispersion rises, if the lender is in a high state, the initial expected losses from lending rise.

FINANCING REPEAT BORROWERS 23 Figure 1. Lender’s expected profit as a function of cost dispersion Regarding the partial-lending relational contract, as dispersion rises, the partiallending contract is both more profitable and likely to be feasible (the orange line). The intuition for this result follows from that the lender only lends in the low cost state, therefore, as dispersion rises, the lender’s realized cost of lending (c ) linearly l falls and consequently the lender’s profitability rises. Moreover, since the feasibility of partial-lending solely requires that the lender makes positive profits in a low cost state(equation15), thefeasibilityofthepartial-lendingrelationalcontractalsorises. Finally, we see that as the cost dispersion rises sufficiently, the equilibrium relational contract switches from always-lending to partial-lending. In figure (1), the switch occurs because partial lending is more profitable (orange line is higher than blue line), however, it is also possible that the switch occurs because always lending becomes non-feasible (dashed blue line goes below zero). 5.2. Persistence of economic recessions. In addition to the depth of the economic crisis, the persistence or stickiness of economic crisis can effect the feasibility and profitability of relational contracts. This subsection analyzes how greater persistence (that is, tomorrow’s state is more likely to be the same as today’s state)

FINANCING REPEAT BORROWERS 24 affects the equilibrium. Moreover, to isolate persistence, we vary the probability of moving between states without changing the invariant distribution of states, that is, the ex-ante probability of either state. We show that greater state persistence is more likely to lead to a partial-lending relational contract than an always-lending relational contract. To start, we introduce the parameter, ρ, that measures the stationary probability of observing two consecutive periods with the same state. (17) ρ ≡ θ p +θ p l l h h Therefore, by varying ρ but ensuring that the stationary distribution of low and high cost states (θ and θ ) does not change, we can examine how the feasibility and l h profitability of the different relational contracts responds to changes in stability. Lemma 3. As the measure of state persistence, ρ, rises but the stationary distribution of each state (Θ) does not change: (1) the expected profitability of the always-lending contract is unchanged but the expected profitability of the partial-lending contract increases, (2) the feasibility of the always-lending contract decreases whereas the feasibility of the partial-lending contract increases. To gain intuition for lemma (3), figure 2 plots the expected discounted profits for always-lending (blue line) and partial-lending (orange line), as well as the expected discountedprofitfromalways-lendingstartingfromahighstate—thealwayslending feasibilitycondition(dashedblueline)aswevarythepersistenceparameter(x-axis). Starting with the profitability and feasibility of the always-lending relational contract. Similar to varying the cost dispersion parameter, the expected profitability of always-lending does not change because the stationary distribution of low and high cost states do not change. However, the feasibility of this contract decreases as persistence rises because the expected future profits of continuing to lend when starting in a high cost state falls (the dashed blue line). The intuition for this result follows from corollary (1). Specifically, for the always-lending contract to be feasible, the lender must make non-negative expected profits in all periods; but as

FINANCING REPEAT BORROWERS 25 Figure 2. Lender’s expected profit as a function of state persistence persistence rises, if the lender is in a high state, the expected losses from continuing to lend rise. Regarding the partial-lending relational contract, as persistence rises, the partiallending contract is both more profitable and likely to be feasible (orange line is increasing in persistence). Recall, the lender only lends in the low cost state, therefore, as persistence rises, the lender is more likely to receive the higher repayment associated with a low cost state (r ) than the partial defaulted payment (r ). In l h effect, this persistence causes the lender’s profitability to rise. In turn, since the feasibility of partial-lending solely requires that the lender makes positive profits in a low cost state (equation 15), the feasibility of the partial-lending relational contract also rises. To sum, as state persistence rises, the feasibility of an always-lending contract falls, whereas, both the feasibility and profitability of a partial-lending contract rises. Therefore, as persistence rises, the equilibrium lending strategy is more likely to be a partial-lending relational contract.

FINANCING REPEAT BORROWERS 26 6. Discussion The model describes how the future affects the borrower’s decision today, and in turn affects the strategy undertaken by the lender. Furthermore, the potential to renege occurs for both parties: The borrower may renege on repaying the lender; the lender could renege on the implicit promise to offer loans. Thus, even if, ex-ante the lender’s profits were maximized by offering loans in all states, this outcome may not be credible due to the potential to renege when the cost of lending is high. The equilibrium of the game may stipulate states of the world where no loans are given and where full repayment is not expected. One implication is that the lender may have difficulties financing its operations and its loan origination activities. Because of the nature of the equilibria described above, equityfinancingonthepartofthelenderwouldlessentheseconcernsbecause the long-run equity investor would be able to reap the benefits of offering loans at a loss in some periods. In many real world settings, however, relationship lenders frequently access the short term debt markets for financing. In this case, our model suggests that short term debt, and therefore the securitization of individual loans, may be a problematic financing structure for loans that rely on dynamic repayment incentives.

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FINANCING REPEAT BORROWERS 30 Appendix Proof. of proposition (2) Thestationaryexpectedper-periodprofitswhenlendinginallstatesisδv−θ c −θ c l l h h and when lending only in the low-cost state is θ (p r∗+(1−p )r∗ −c ). Thus, after l l l l h l plugging in for r∗ and r∗, the lender’s expected profit is higher when lending in every l h period if, and only if,: (cid:20)(cid:18) (cid:19) (cid:21) δ(1−p )(1−p ) l h δv −θ c −θ c ≥ θ p + δv −c l l h h l l l 1−δp h which with some algebraic manipulation gives: (cid:20) (cid:18) (cid:19)(cid:21) δ(1−p )(1−p ) l h δv 1−θ p + ≥ (1−θ )c l l l h 1−δp h The remainder is based on lender credibility conditions (ICCs) given by corollary (1) and lemma (1). (cid:3) Proof. of corollary (1) Assuming the lender always offers loans we can write V as: V = (δv −c )+δ[p V +(1−p )V ] ∀s s s s s s −s (δv −c )+δ(1−p )V s s −s = 1−δp s (cid:20) (cid:21) (δv −c ) δ(1−p ) (δv −c )+δ(1−p )V s s −s −s s = + × ⇒ 1−δp 1−δp 1−δp s s −s (cid:20) (1−δp )(1−δp )−δ2(1−p )(1−p ) (cid:21) (cid:18) δv−c (cid:19) s −s s −s −s V = (δv−c )+δ(1−p ) ⇒ s s s 1−δp 1−δp −s −s (18) V (cid:2) (1−δp )(1−δp )−δ2(1−p )(1−p ) (cid:3) s s −s s −s = (1−δp )(δv−c )+δ(1−p )(δv−c ) −s s s −s Since δ ∈ (0,1) and p ∈ (0,1), the term that multiples V on left hand side of s s equation [18] is greater than zero. Thus, the right hand side of equation [18] tells us that the sign of V depends on the weighted values from the profits in each state. s Since we have assumed that c < c , then it can be shown that equation [18] implies l h that V > V .20 Therefore to ensure ICC for the lender to give loans in all states we l h 20We can write equation [18] as κV = (1−δp )u +δ(1−p )u , solving this equation and s −s s s −s noting that u >u we can conclude that V >V . l h l h

FINANCING REPEAT BORROWERS 31 only need to verify that: V ≥ 0 ⇒ h (19) (1−δp )(δv −c )+δ(1−p )(δv −c ) ≥ 0 l h h l If equation [19] is not satisfied, then the lender would renege on offering loans if a high cost state (c = c ) is observed. (cid:3) s h Proof. of lemma (1) The constraints in the lender’s problem in equation [15] form a non-empty compact set of feasible (r ,r ) pairs, and the linear objective function is continuous. Thus, l h by the Weierstrass extreme value theorem, an optimal (r∗,r∗) pair exists. Moreover, l h since the objective function is (weakly) convex, a local maximum is also the global maximum. Therefor, the problem in equation [15] has a unique solution that takes one of the three following forms: (1) r = 0, r > 0, only one of the constraints in equation [13] binds. l h (2) r > 0, r = 0, only one of the constraints in equation [13] binds. l h (3) r > 0, r > 0, both constraints in equation [13] bind. l h Direct payoff comparison for the lender between the above three possibilities reveals the optimal repayment rates. The first case imposes that r(1) = 0, hence the objecl tive function in equation [15] simplifies to (1−p )r −c and equation [13] simplifies l h l δ2v( 1−ph ) to r ≤ v and r ≤ 1−δph . Since the latter inequality implies the former, the h 1−p h 1−δp l l maximizer will be r(1) = δ2v( 1 1 − − δ p p h h ) . Thus, the maximum expected repayment will h 1−δp l be: (1−p )r(1) = δ2v(1−p l )(1−p h ) l h (1−δp)(1−δp ) l h The second case imposes that r(2) = 0, hence the objective function in equah tion [15] simplifies to p r − c and equation [13] simplifies to r ≤ v and r ≤ l l l l p l l δv(1−δp h ) . Since the latter inequality implies the former, the maximizer (1−δp )−δ2(1−p)(1−p ) h l h will be r(2) = δv(1−δp h ) . Thus, the maximum expected repayment will be: l (1−δp )−δ2(1−p)(1−p ) h l h p r(2) = δvp l (1−δp h ) l l (1−δp )−δ2(1−p)(1−p ) h l h In the third case, both constraints in equation [13] bind. Thus, the optimal repayment rates can be found by solving the system of equation when both inequalities in equation [13] bind. The solution is r(3) = δv and r(3) = δ2v( 1−p h ). The maximum l h 1−δp h expected repayment will be: p r(3)+(1−p )r(3) = δvp l (1−δp h )+δ2v(1−p l )(1−p h ), which is l l l h 1−δp h

FINANCING REPEAT BORROWERS 32 higher than the maximum expected repayment in the first and second cases. As a result, r∗ = r(3) and r∗ = r(3). l l h h Finally, in order for the lender’s commitment to lend when cost is low to be credible, the low cost should not exceed the expected repayment: δ(1−p )(1−p ) l h (20) δv[p + ] ≥ c l l 1−δp h (cid:3)