Costly Information Acquisition with Credit Scores

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1 Costly Information Acquisition with Credit Scores Sergio Vicente y Universidad Carlos III de Madrid November 25, 215 Abstract This paper analyzes the e ect of competition between large and small lenders on costly information acquisition about potential borrowers. I construct a model with two types of nanciers. Large lenders have an advantage at using credit scores, which allows them to make more accurate estimates of borrowers default probabilities at a negligible cost. Small lenders are more e cient at employing loan o cers to acquire costly soft information, which allows to improve upon credit scores predictions. I show that large lenders displace small lenders from part of the rm segment in which they are more e cient, lending to rms for which acquiring soft information before granting a loan would be more e cient. As a result, there is an excessive amount of defaults in the economy. The ine ciency created by the displacement of small lenders may more than o set the gains that credit scores induce in reducing the cost of evaluating borrowers risk. The paper shows that small lenders charge lower rates than large lenders to rms with similar quality. Moreover, while large lenders attract the highest quality rms, default rates on loans granted by large lenders are higher. These patterns are consistent with the received empirical evidence. Keywords: Credit scores, screening, hard and soft information, loan o cers. JEL Codes: G14, G21, G24, D82 1 Introduction Bank borrowing is the most important source of external nance for credit constrained small and medium enterprises around the world (Beck, Demirgüç-Kunt, and Maksimovic (28)). Small business lending has historically been a local activity, characterized by small community banks lending on the basis of "soft information" (Frame and White (24)). 1 However, over the past two decades, a growing number of large banks have incorporated credit score models in their small business lending operations (Akhavein, Frame, and White (25)). The evolution of credit markets I am thankful to David Martínez-Miera, Enrico Sette (discussant), Anatoli Segura and Enrico Perotti (discussant) and seminar audiences at the Barcelona GSE Summer Forum Financial Intermediation, Risk and Liquidity Workshop, the Oxford Financial Intermadiation Conference (OxFIT) and the CEPR European Summer Symposium in Financial Markets (Gerzensee, evening sessions). Financial support from Fundación Ramón Areces and the Ministerio de Educación y Cultura, under project ECO P is gratefully acknowledged. All remaining errors and shortcomings are mine. y sergio.vicente@uc3m.es. Url: Address: Calle Madrid, 126, Getafe (Madrid), 2893, Spain. 1 Soft information comprises aspects that are hard to reduce to a number, such as the ability of a manager or the way she reacts under pressure (Petersen (24)). It also includes the bank o cer s own assessments of "business prospects garnered from communications with [ rm] s suppliers, customers or neighboring businesses." (Berger and Udell (26)). 1

2 towards an increased use of credit scores by large banks has raised a number of concerns about the soundness of lending practices and the availability and price of credit to small businesses. 2 This paper provides a theoretical framework to address some of these concerns. I consider a competitive market with two types of nanciers: Large lenders and small lenders. The model builds on the insight, rst formalized by Stein (22), that small lenders have a comparative advantage in acquiring and using soft information, because there are organizational diseconomies of scale in the acquisition and use of soft information. 3 Indeed, Berger, Miller, Petersen, Rajan, and Stein (25) provide direct evidence consistent with small banks being more e cient at collecting and acting on soft information than large banks. 4 On the contrary, large banks have an advantage in using credit scores. While setting up and maintain a credit score model may require large amount of resources, in the form of dedicated researchers or access to databases, once the credit score model is in use, the marginal cost of appraising an additional loan is negligible. Therefore, credit scoring is subject to large economies of scale. More particularly, I assume that lenders have access to two types of information technologies. On the one hand, there is a credit scoring technology, which allows banks to appraise the quality of a rm at no cost. 5, 6 On the other hand, banks can acquire soft information about the rm s business prospects. Crucially, while his extra piece of information potentially allows to make better appraisals, it is costly to obtain. 7 I model the advantage of large banks at using credit scores by 2 Berger, Kashyap, Scalise, Gertler, and Friedman (1995) constitutes an early discussion of the potential e ects of the technological advances in the U.S. banking industry. Berger, Espinosa-Vega, Frame, and Miller (25) analyze the impact of credit scoring on the loan rates and availability of credit for small businesses, as well as on the risk of credit-score loans. Frame, Srinivasan, and Woosley (21) study the e ect of credit-scoring on the availability of credit to small businesses. DeYoung, Hunter, and Udell (24) assess the e ects of technological advances, among other factors, on community banks lending practices. DeYoung, Glennon, and Nigro (28) analyze the impact of credit-scores lending on loan performance rates. 3 In order to understand the intuition behid this result, consider a loan o cer deciding whether to grant a loan to a start-up company that has not yet established a credit reputation or to a rm without a su ciently realiable accounting information. The loan o cer may spend time interviewing with the rm s manager, appraising the particularities of the business plan for which funding is being sought or assessing the owner s character. However, given that this information is soft and cannot be veri ably conveyed, the incentives to spend time and e ort to produce a high quality report that would have to go through di erent hierarchical levels would be reduced in large organizations. Also, whenever soft information stands in contrast with the prediction of a credit score, it would be harder for a loan o cer within a hierarchical organization to justify his appraisal. 4 Other authors have provided direct evidence of the reluctance of large organizations to base their lending decisions on soft information. For instance Mian (26) shows that "foreign banks systematically shy away from lending to soft information rms" in Pakistan. Along these lines, Beck, Demirgüç-Kunt, and Pería (211) nd that in developing countries foreign banks are less likely to use soft information and to decentralize loan approval than domestic private banks. 5 One of the main bene ts of credit scoring is that it dramatically reduces the processing times and the labor inputs required (Frame, Srinivasan, and Woosley (21)). For instance, the average small business loan processing time at Barnett Bank decreased from three or four weeks to a few hours after credit scoring was implemented (Mester (1997)). Berger, Cowan, and Frame (211) write: "The primary motive for these banks is likely reduced underwriting costs. This method may exacerbate informational opacity problems, yield less accurate credit terms, and result in greater future credit losses, but may nevertheless be pro table because of the lower costs". 6 The assumption that the marginl cost of suing credit scores is zero is just made for simplicity. It su ces to assume that (automatized) hard information can be acquired and processed at a lower cost than (labor-intensive) soft information. 7 As an illustration of the interaction between a potential borrower and a loan o cer at a bank branch, Agarwal and Hauswald (21) write: "The application process typically takes four hours to a day to complete from the initial contact between rm and bank. During the branch visit, the manager owner or rm representative supplies all the relevant data, submits nancial and tax information, provides a list of assets, etc., which the local loan o cer transcribes [...] Concurrently, the loan o cer conducts an in-depth interview with the applicant and gathers soft information in the sense that it would be hard to verify by a third party. In about 8% of the cases, the branch o cer will invite the applicant back to follow up on open questions, review discrepancies in submitted information with 2

3 assuming that small banks credit scores are a noisy spread of large banks. On the other hand, I model the advantage of small banks at handling soft information by assuming that small banks face a lower cost of soft information acquisition than large banks. Before analyzing the competitive equilibrium of this economy, it is instructive to look into the e cient credit decisions. Consider a potential borrower with a given credit score. Would it be optimal to grant a loan, to go through a (costly) soft information acquisition to appraise the creditworthiness of the rm or to deny credit straightaway? The answer depends on the rm s credit score. It would be optimal to lend to rms with high scores, to acquire extra information on rms with intermediate scores and to deny credit to rms with low scores. The intuition is simple. When facing a rm of a given quality, we are confronted with a simple trade-o. Acquiring soft information improves upon credit scores appraisals and creates value through a reduction of the expected losses induced by a reduction of the number of nonperforming loans granted. But acquiring soft information entails a cost. Firms of high quality, as evidenced by their score, are unlikely to default. Therefore, the cost of acquiring information exceeds the expected gain of identifying nonperforming projects among these rms. Hence, e ciency prescribes that large lenders grant loans to rms with high credit scores without further scrutiny. On the opposite side of the quality spectrum, rms of low quality are highly likely to default, so that the cost of acquiring further information exceeds the bene ts from identifying creditworthy projects among them. Hence, the e cient action is to deprive these rms from borrowing. The bene ts of acquiring soft information only exceeds its cost for rms with intermediate scores. Hence, rms in the intermediate quality range, as evidenced by their scores, should be optimally matched with small banks, who are the most e cient institutions at handling soft information. In a competitive equilibrium, however, some rms with intermediate scores are ine ciently poached by large lenders. In order to better grasp the intuition behind this result, consider a simpli ed setting in which, on the one hand, large lenders cannot acquire any extra information beyond credit scores and, on the other hand, small lenders cannot observe any type of credit scores, but only acquire costly information through their loan o cers. From the perspective of small lenders, all borrowers are observationally equivalent ex- ante. Hence, any price discrimination amongst them would have to be conditional on soft information. Therefore, any competitive o er yielding non-negative pro ts would entail cross-subsidization of borrowers with di erent ex-ante risk. Large lenders, on the contrary, can use credit scores to cherry-pick the best borrowers within the pool of intermediate lenders. Although small lenders are more e cient in that range, they would be outcompeted by large lenders in the upper range because the best interest rate o er that small lenders can make has to target the average borrower within the pool, whereas large lenders can tailor their o ers to borrowers with di erent credit scores. Moreover, cherry-picking is reinforced by he fact that it leads to a reduction of the average quality within the pool of potential small-lender borrowers, which further reduces the value of acquiring soft information. Although credit scores generate a cost reduction in appraising the likelihood of nonperforming loans, there is a downside to the introduction of credit scores. With the aid of credit scores, large banks can displace small banks from the segment in which small institutions are more e cient. The overall e ect depends on the distribution of scores. To illustrate this point, consider the following two extreme examples. Suppose rst that all rms have either a high or a low score i.e. no rm has an intermediate score-, but that the average score lies on the range in which small lenders are more e cient. Then, small lenders would have to acquire soft information from each and every rm. The advent of credit scores would eliminate the need to ine ciently acquire all this information, as rms with high scores should be granted credit and rms with low scores should be deprived credit reports, discuss the prospects of the rm, etc...". 3

4 from borrowing straightaway. Hence, credit scores would induce a welfare gain. Consider now the situation in which all rms lay in the intermediate range. Then, credit scores would lead to a welfare reduction, as there would only be ine cient poaching by large lenders of rms in the upper range of the distribution. Matters are a little more subtle with general distributions, because the extent to which large lenders can poach rms is distribution-dependent, but the general lesson is that if the distribution of scores is very concentrated around the mean, introducing them will lead to welfare losses. There is an additional source of ine ciency in this economy, which depends on whether small lenders are totally ousted from the market for lending. 8 Consider rst the situation in which small lenders are driven away from the market. Then, large lenders will lend to all rms exceeding a certain credit score cuto and denying credit to the remaining. This credit score threshold will be the minimum such that large lenders would be able to make non-negative pro ts. Since small lenders are more e cient in the intermediate range, a small lender could make non-negative pro ts lending to a mass of (lower-medium scores) of rms with lower credit scores than the smallest that would get credit from large lenders. Hence, whenever small lenders are absent, there would be ine cient credit rationing of rms with lower-medium scores. Now, consider the case in which small lenders are present in the market. Then, since all rms are ex-ante observationally equivalent, they will acquire soft information from rms that should be deprived from credit without further scrutiny. Then, whenever small lenders operate, there will be excessive information acquisition. Crucially, whether small lenders are expelled from the market depends on which ine ciency is larger. Since small lenders internalize all the gains from eliminating the credit-rationing ine ciency and the losses from incurring in excessive information acquisition on the low end, they will be active in equilibrium if and only if credit rationing leads to larger losses than acquiring an excessive about of information. Hence, although competition will lead to a situation in which there will be an ine ciency of some sort, the competitive outcome is given by the market structure that minimizes the ine ciency. While the main results obtained in the simpli ed setting described so far generalize to the case in which small lenders have access to a noisy, yet informative, version of credit scores, some di erences are worth noting. The rst observation is that if noisy credit scores are very coarse, that is, if they carry very little information, we would be in a similar situation as portrayed above: Either small lenders would be ousted from the market or, if present, they would o er very similar loan terms to all rms obtaining credit from them. As the accuracy of noisy scores improve, small lenders would only acquire information for rms exceeding a certain (noisy score) threshold and deprive from credit those with lower scores. For rms exceeding the cuto, small lenders would o er lower interest rates the higher the (noisy) score. Not surprisingly, increasing the noisy score accuracy unambiguously increases welfare. When all borrowers are ex-ante observationally equivalent, small lenders may either acquire information from all of them or simply leave the market. However, credit scores allow small lenders to discriminate among borrowers and acquire information about those with su ciently high scores. Hence, although there would be some credit rationing (of those unlucky borrowers of lower-medium quality with a low noisy score) and ine cient information acquisition (of low quality borrowers with a high noisy score), the extent of both ine ciencies would be reduced. Moreover, noisy scores allow small lenders to tailor interest rate promises within a noisy score category. While there would be some cross-subsidization across borrowers, this would be con ned to a noisy score category, so that small lenders could o er more attractive o ers to ex-ante better debtors, reducing the margin for large lenders poaching. 8 We will return later to the conditions under which small lenders cannot survive in this market. 4

5 Letting large banks acquire soft information may however reduce welfare further. In order to see why, suppose that there is a large di erence between the cost of acquiring soft information by large and small banks, respectively. In this case, large lenders would not acquire soft information on any of their potential borrowers and there would be a rm, call it sorting rm, such that large banks would only grant credit to rms with at least as high a credit score as this sorting rm s. As long as the cost di erence is large enough, large lenders would not employ soft information in their appraisals. However, there is a critical level, call it hard-soft critical level, for this di erence such that large lenders become indi erent between lending to this sorting rm and acquiring soft information to appraise its creditworthiness. When the cost di erence falls below that hard-soft critical level, large lenders would strictly prefer to engage into soft information acquisition for a mass of rms around the sorting rm. There are two immediate e ects on welfare that go in opposite directions. On the one hand, the increase in e ciency at handling soft information would allow the bank to substitute out credit scores for loan o cers appraisals for some rms with higher scores than the sorting rm, which would lead to a welfare increase. On the other hand, large lenders would poach a mass of rms to the left of the sorting rm that would otherwise be matched with a small lender which, since small lenders are more e cient in soft information acquisition, would destroy social value. Whether the rst or the second e ect dominate depend on the extent of the cost di erence and the distribution of credit scores. However, when the cost di erence is not much larger than the hard-soft critical level described above, the poaching e ect dominates and welfare is thus reduced. The reason is that while the marginal e ciency gain from softening information around the sorting rm is close to zero, the marginal e ciency loss from poaching around the sorting rm is given by the cost di erence, which is bounded away from zero. As the cost di erence vanishes, information acquisition approaches the constrained e cient allocation. This paper explains some observed regularities on rms sorting by type of lender, loan pricing and relative performance of di erent types of loans. First, the model predicts that rms with the highest credit scores would borrow from large lenders. This pattern is explained by the fact that the value of soft information is lower the higher the credit scores. Hence, small lenders could not compete with large lenders in the high quality rm segment. The model also predicts that small lenders would charge lower interest rates than large lenders to rms of similar quality, which is a translation of their higher e ciency in that spectrum. Moreover, loan rates from small lenders would exhibit higher dispersion than those of large lenders for rms with the same quality. This pricing pattern is explained by the fact that large lenders credit decisions are exclusively based on rms scores. Notwithstanding the fact that small lenders charge lower rates to rms with similar quality, the (unconditional) average rate of loans granted by small rms may be higher than that of large lenders. High quality rms obtain credit from large lenders. Moreover, they obtain loans at lower rates than do rms with worse quality. Hence, if the proportion of high quality rms is high enough, the average price of loans granted by large nancial institutions would be lower. Finally, the paper predicts that rms borrowing from small lenders would experience less defaults, even though their credit scores are lower. This is due to the fact that soft information acquisition allows small banks to make better risk appraisals than those provided by credit scores, which large banks rely on. The remainder of the paper proceeds as follows. The model is laid out in Section 2. I analyze the e cient level of soft information acquisition and the optimal allocation of funds in Section 1. In Section 4 I study the competitive equilibrium of a simpli ed version of the model in which large lenders cannot acquire soft information and small lenders do not observe credit scores. This simpli ed setting allows me to highlight the main forces behind the full- edged equilibrium, which I study in Section 5. I discuss the predictions of the model in the light of the received empirical 5

6 evidence in Section 6 and then conclude. The appendix contains a small survey of the literature on the use of credit scores. All proofs are also displayed in the appendix. 2 The model I consider an economy populated by a continuum of rms and a nite number of lenders. 2.1 Firms and projects Each rm has an investment project that requires an outlay of 1 monetary unit. There are two types of projects in the economy: Good (G) and Bad (B). A good project pays M monetary units with certainty, whereas bad projects return. Firms are wealth constrained and therefore need to borrow in order to undertake a project. In order to make borrowing feasible, I assume that M > R, where R 1 is the gross risk-free rate. Project outcomes are veri able ex-post. Firms di er in their ability to select and implement projects. In particular, each rm is characterized by a quality index 2 [; 1], which represents the ex-ante probability that the rm carries out a good project. The distribution of rms qualities in the economy is described by a density function f with c.d.f. F. In order to have a well-behaved problem, I assume: Assumption 1.1: The density function f is continuous and positive everywhere in the interior of its support. Moreover, the function f is common knowledge to all agents in the economy. 2.2 Lenders and information There are two types of perfectly competitive lenders in this environment: Small Lenders (S-Lenders) and Large Lenders (L-Lenders). All lenders have access to funds at the (gross) risk-free rate R 1 and only di er in their information processing capabilities. The central assumptions of the model aim at capturing the following two features,which I formalize below. First, lenders use credit scores in order to appraise the probability that a rm defaults on a loan using "hard information" about the rm. Typically, constructing a credit-score model requires large amounts of data, as well as a dedicated research unit that identi es the key variables that better predict default. Hence, the creation of a credit-score model is subject to large economies of scale. Consequently, I assume that L-Lenders have access to better creditscore technologies than S-Lenders. 9 Moreover, while the xed costs of developing a credit-score technology are large, the marginal cost of assessing a loan default likelihood through a credit score is typically very small. I incorporate this characteristic in the model by assuming that acquiring and processing hard information about a particular rm is costless. Second, on top of using credit scores, lenders rely on information gathered and processed by loan o cers. In general, the evaluation process starts at a the branch level, where a bank o cer collects both hard and soft information about the rm through an interview with a rm manager. This information is then passed through various committees within the bank until a nal loan decision is made. The hard information component can be incorporated into the credit score, but evaluating and transmitting soft information through di erent hierarchical levels within the bank is costly in terms of the opportunity cost of o cers time. I draw on Stein s (22) insight that small organizations are better at processing "soft information" and assume that S-Lenders are more 9 Berger, Cowan and Frame (211) show that, while most large banks use Small Business Credit Scores (SBCS), community banks mostly rely on Consumer Credit Scores (CCS) acquired from public vendors, which provide less accurate estimates. See Section A in the Appendix for further details. 6

7 cost-e ective than L-Lenders at evaluating the probability that a particular project fails using soft information Credit scores: The Large Lenders advantage Formally, the relative advantage of L-Lenders over S-Lenders at assessing hard information is modeled as follows. I assume that L-Lenders observe the rm s quality at no cost. S-Lenders only observe a public noisy (informative) signal z, also at cost, which is the realization of a random variable Z with (conditional) densities fh (zj)g 2[;1]. 1 I make the following two assumptions about these conditional distributions: Assumption 2.1: For each 2 [; 1], the conditional density h (zj) is continuous everywhere in its support and common knowledge to all agents. Assumption 2.2: The family fh (zj)g 2[;1] satis es the Strict Monotone Likelihood Ratio Property (SMLRP), i.e.: For any z 1 < z 2, h (z 2j) h (z 1 j) is strictly increasing in. The SMRLP imposes that higher signal realizations are relatively more likely the higher the quality of the rm. The SMLRP implies that higher signals represent "good news" about the underlying parameter in Milgron (1981) s sense. As an illustrative example, suppose that, for a rm with quality, a (noisy) signal z is obtained as the realization of a normally distributed random variable Z N ; 2 with expectation and variance 2. The family of densities induced by this process clearly satis es the SMLRP. Higher (noisy) signals z are good news about the underlying quality of a rm in that they are relatively more likely to have been originated from higher quality rms Information acquisition by Loan O cers: The Small Lenders advantage On top of the information conveyed by credit scores, I assume that loan o cers can evaluate a potential borrower s project and gather additional information at a cost. In particular, I consider that lenders may obtain a signal x 2 fg; bg about the project s quality, which -remind from abovecan potentially be Good (G) or Bad (B). Signals are truthful with probability, independently of the underlying state, that is Pr (x = gjg) = Pr (x = bjb). For simplicity, I assume without further loss of generality that signals are unconditionally informative, that is, > 1 2. In order to capture the idea that S-Lenders are better at processing soft information than L-Lenders, I assume that the cost of gathering and processing the signal for each type of lender is k S and k L, respectively, with k S < k L. Notice that, without further restrictions, the cost structure assumed in the model may potentially encompass di erent type of L-Lenders. On the one hand, one may think of a highly decentralized decision-making L-Lender with local branch presence as facing similar assessment costs as an S-Lender, in which case k L and k S would be very similar. On the other hand, for lenders lacking the capacity to interact with borrowers and gather soft information, such as longdistance lenders, on-line lenders or credit card issuers, k L would be considerably larger than k S. The various implications of di erent cost structures are analyzed below. Finally, I assume that rms lack lenders expertise to evaluate their own quality and have to rely on the publicly available noisy signal z to assess it. 1 The assumption that the signal z is public is based on the observation that most US Community Banks use credit scores purchased from external vendors Berger, Cowan, and Frame (211). Section A in the Appendix contains a detailed description of the use of credit scores in the industry. 7

8 2.3 Timing There are three dates. t = 1 : Price competition for borrowers. Lenders compete for rms à la Bertrand making interest rate o ers to potential borrowers. 11 t = 2 : Loan contracts and information acquisition. Borrowers with loan o er rates at hand choose a lender to sign a loan contract with. Having signed a loan contract, lenders may (or may not) acquire soft information about the project to decide whether to grant credit or not. If the lender decides to nance the project, the borrower gets the loan at the interest rate previously o ered; that is, rate o ers are binding. 12 Credit may be denied on the basis of soft information. 13 If the loan is not granted, the rm can costlessly seek credit from another lender. For simplicity, I assume that Lenders can observe whether potential borrowers have previously sought credit elsewhere. 14 t = 3 : Project execution and payo realization n. Firms carry out projects and repay their loans if successful. 2.4 The lending game The lending game is composed of three types of players: a nite number n S 2 of Small Lenders (S-Lenders), a nite number n L 2 of Large Lenders (L-Lenders) and a continuous of Firms. A strategy for an S-Lender is a pair (a S (z) ; r S (z; x)). The strategy element a S () is an indicator function specifying whether to acquire costly information about a rm with (noisy) score z. The second strategy element r S () establishes which rate to o er to such rm; interest rate o ers are made contingent on the signal x 2 f?; b; gg that loan o cers may obtain about the project where, with a slight abuse of notation, I enlarge the signal space to include?, which stands for the case where a lender decides not to acquire soft information about the rm. 15 Similarly, a strategy for an L-Lender is a pair (a L (; z) ; r L (; z; x)), where actions depend on the score, the noisy score z, and the outcomenx of the loan o cer s information o acquisition process, if any. With a set of interest o ers r = rs 1 ; :::; rn S S ; rn S+1 L ; :::r n S+n L L at hand, a rm s strategy a F (r) speci es from which lender to borrow from. The payo for a Lender from granting a loan at an interest rate r is given by: (r) = 1 G r R, 11 I assume a perfectly competitive lending market in order to highlight that the distortion does not come from market power. Some small banks may act as a local monopolist, simply constrained by the presence of distant large banks that may attract some of their base costumers. 12 A common practice in small business lending consists of making conditional ex-ante o ers, specifying the loan terms if the loan is nally granted (Inderst and Mueller (27)). Some banks charge refundable application fees, which are returned to the applicant in case a loan is not granted in the terms speci ed in advance. 13 In order to avoid lenders "poisoning the well", I assume that Lenders cannot renege on a loan contract without acquiring soft information. In practice, lenders denying credit may have to provide (costly) evidence that a project has been evaluated before going back on a loan o er. 14 Lenders typically share information about borrowers applications either voluntarily, via "private credit bureaus", or forcefully, via "public credit registries". Pagano and Jappelli (1993) and Brown, Jappelli, and Pagano (29), and references therein, contain extensive information on information sharing among lenders. 15 Technically, the rate r S () should be made dependent also on the outcome of the costly information acquisition phase. However, in equilibrium, matters are very simple: if the lender does not acquire costly information about the rm, it grants the loan at the promised rate; if, on the contrary, the lender acquires costly information, it will issue the loan at the promised rate if it gets "good news" and it will not grant credit at all if it receives "bad news". 8

9 where 1 G is an indicator function taking on a value of 1 if the project is good and hence, it succeeds and otherwise. Signing a loan contract with a rm at an interest rate r yields a payo : (r) = 1 k (1 g (r) k) + (1 1 k ) (r). The function 1 k takes a value of 1 if the lender engages into information acquisition about the project and otherwise. Similarly, 1 g stands for an indicator function that takes the value 1 if a good signal arises. 16 On the other hand, payo s for rms are given by: w (r) = 1 G (M R). Beliefs (j) are a distribution function over rms types. L-Lenders observe, so that beliefs (j) take a non-degenerate distribution form only for S-Lenders. For expositional simplicity, we will refer to beliefs as those held by S-Lenders only. With a slight abuse of notation, I denote a lender s expected pro t from granting a loan to a rm with quality as: (; r) = Pr (Gj) r R. Similarly, I denote a type quality as: i lenders expected payo from signing a loan contract with a rm with i (; r) = max f k i + Pr (x = gj) (Pr (Gj; g) r R) ; (; r)g. (1) Info: Acq I solve for the Symmetric Perfect Bayesian Equilibrium (SPBE) of this lending game. Formally, an SPBE is a pro le of strategies and beliefs, which satis es the following characteristics: (i) (Sequential rationality) A player s strategy pro le maximizes its expected payo, given its beliefs and all other players strategies. (ii) (Belief consistency) Beliefs on the equilibrium path are derived from prior distributions and equilibrium strategies using Bayes rule. (iii) (Symmetry) Any two identical players with identical information pursue identical strategies E ciency In this section, I analyze the (constrained) e cient allocation of loans as the solution to the problem of a Social Planner that is only limited by the information constraints in the economy, which constitutes the benchmark for the welfare analysis. On the one hand, the Planner costlessly observes each rm s credit score, such as L-Lenders the most e cient agents in processing hard 16 Formally, a lender may decide to incur the cost of acquiring soft information and then ignore the signal, either by denying credit upon observation of a good signal, or by granting a loan following a bad signal. Since k i >, incurring the cost and then ignoring the signal is strictly dominated by not engaging into soft information acquisition. In order to simplify the notation, I implicitely assume that signals are not ignored.. 17 I impose symmetry solely for expositional simplicity. Without symmetry, some breaking-even lenders may be indi erent between either charging an interest rate that may lead to a lending transaction or asking for an outrageous rate that will be rejected by any rm. By imposing symmetry, identical lenders are required to pursue the same strategy, so that we get rid of a plethora of outcome-equivalent equilibria. 9

10 information do. In addition, the Planner has the ability to acquire a signal x 2 fb; gg about the project at the cheapest cost available in the economy, that is, at the cost k S faced by S-Lenders. The e ciency question reduces to two sequential decisions. Firstly, upon observation of a rm s quality, the Planner has to decide whether to acquire costly information about the rm s project. Secondly, the Planner must determine whether to grant credit to a rm, given all the available information. 3.1 Lending with credit scores only Consider rst the decision of granting a loan on the basis of the credit score only. A loan granted to a rm with a credit score succeeds and yields M with probability Pr (Gj), and returns otherwise. The opportunity cost of funding a project is R. Hence, the value of granting a loan with credit scores only to a rm with quality is given by: H () = Pr (Gj) M R. 3.2 The value of loan o cers information Valuable loan o cers information Before analyzing the optimality of acquiring costly information, consider the decision of granting credit to a rm with credit score for which a signal x has been obtained. Lending to that rm would yield: L (; x) = Pr (Gjx; ) M R, (2) where Pr (Gjx; ) stands for the probability that the project is good, given the signal x and the score. Suppose that L (; g) <, that is, the expected value of granting a loan to a rm for which a good signal has been obtained is negative. Then, it would be optimal to deny credit to the rm, regardless of the information gathered by a loan o cer. Similarly, if L (; b) >, a rm should obtain funds for its project with independence of the informational content of loan o cers assessments. In both cases, the information conveyed by soft information should be ignored. This observation motivates the concept of valuable soft information. De nition 1 The information obtained by loan o cers is valuable if the loan policy consists of lending upon observation of a good signal and denying credit when a bad signal arises. The information acquired by loan o cers is valuable when the following two conditions are satis ed: L (; b) < < L (; g). The following lemma describes the set of rms for which the information gathered by loan o cers is valuable. Lemma 1 De ne () and (), as the (unique) values satisfying L ( () ; g) = and L () ; b =, respectively. 18 For ; 1, loan o cers information is valuable only for rms with quality levels 2 () ; (). For = 1, the information gathered by loan o cers is valuable for every rm These equations are easy to solve, the respective solutions being () R R+(1 )(M R). 19 Notice that lim!1 () = and that lim!1 () = 1. (1 )R (1 )R+(M R) and () 1

11 The intuition for this result is straightforward. Consider rst the case ; 1, that is, when signals are noisy. At the low end of the quality space, good projects are very improbable. Hence, a good signal g is more likely to have arisen from a bad project than from a good one. 2 The opposite is true for high quality rms: A bad signal is more likely to be the outcome of noise than to originate from a bad project. Hence, information from either very high or very low quality rms should be ignored, as signals become too noisy towards the ends of the quality space. For = 1, signals perfectly reveal the underlying state. Hence, loan o cers information is always valuable The value of acquiring costly information In deciding whether to gather costly information about a project, the Planner must weigh its informational content against its cost. From the previous lemma, we know that the information gathered by loan o cers should be ignored for rms with either very high or very low credit scores. Clearly, it is suboptimal to engage into (costly) information acquisition when it is not valuable. For rms with scores 2 () ; (), all projects from which a good signal is obtained should be funded, whereas all rms with a bad loan o cer evaluation should be deprived from credit. Hence, the value of acquiring soft information about the project of a rm with quality 2 () ; (), and subsequently granting credit to the rm upon observation of a positive signal, is given by: ^S () = k S + Pr (x = gj) L (; g) where Pr (x = gj) = + (1 ) (1 ), and L (; x) is given by equation (2). Summarizing, the value of acquiring soft information is given by: 8 k S if () >< S () = ^S () if () < < () >: H () k S if (). 3.3 E cient matching of borrowers and lenders We now turn into the e ciency question. As described above, the Planner must choose whether to acquire costly information and whether to issue a loan based on all the available information. The rst decision is simply made by comparing the value of acquiring soft information with the alternative options, which are either granting a loan or depriving the rm from credit upon observation of the credit score alone. The second decision makes use of both the credit score and the information provided by loan o cers, if any, to assess whether issuing a loan to a rm has positive expected value. The following proposition characterizes the (constrained) e cient decision on costly information acquisition and credit granting. Proposition 1 De ne the thresholds l (1 )R+k S M (2 1)R, m R M, h M(1 )+R(2 1) and k m (2 1) (M R). Then, the (constrained) e cient decision on Loan O cers information acquisition and credit allocation is given by: 2 In particular, the relative likelihood of a good signal to have arisen from a good project versus a bad project is given by, which gets arbitrarily close to as we move towards the low quality end. (1 )(1 ) R k S 11

12 (a) For k S < k, loan o cers should acquire costly information for rms with scores 2 [ l ; h ]. Credit should be granted to rms with 2 [ l ; h ] if a good g signal is obtained, as well as to all rms with scores > h. Credit should be denied to the rest of rms. (b) For k S k, no costly information should be acquired. Credit should be granted to rms with scores m, whereas the rest of rms should be deprived from funds. This proposition establishes a role for Loan O cers when information acquisition is not too costly: They should acquire costly information for rms in the middle tranche [ l ; h ]. On the other hand, the proposition states that rms with high scores should be granted loans without further inquiry, whereas low score rms should be deprived from credit. When the cost of acquiring information exceeds the threshold k, there is no role for loan o cers and only rms with su ciently high scores should be granted credit. Figure 1: E cient Loan Assignment. Figure 1 illustrates the (constrained) e cient allocation of loans. The horizontal axis displays rms quality. The dashed line represents the value of lending with hard information H (), while the solid line depicts the value of acquiring soft information S (). H () is an a ne function that crosses the quality space at m. S () is a piecewise continuous function with a at portion for < (), a downward shift of H () for > (), and an a ne mapping in between. When k S < k, the function S () intersects with the quality space at l 2 ( () ; m ) and with the function H () 12

13 at h 2 m ; (). In the middle tranche ( l ; h ), we have that S () > H (), so that acquiring information is e cient. This inequality is reversed for rms with quality > h, so that optimality prescribes to grant credit without further inquiry. For < l, we have that H () < S () <, so that rms in the lower end should be excluded from borrowing. For k S k (not in the Figure), the function S () is shifted downward. In this case, whenever H () < S (), if at all, we have that S () <. Hence, acquiring information is always suboptimal when it becomes too costly. Granting credit is e cient only for > m. In order to highlight the role played by the cost of acquiring information, consider the case where soft information is perfectly informative, that is, = 1. Even when loan o cers assessments provide a perfectly accurate estimation of the quality of a project, it is e cient not to acquire costly information from rms with either high or low scores. By granting credit to a rm upon observation of a credit score alone, the issuer faces an expected surplus of (M R), which arises from funding good projects, and an expected loss of (1 ) R, due to wasting funds in defaulting loans. By acquiring soft information, the issuer avoids issuing non-performing loans. But eluding default entails a cost k S. For high scores, the cost k S of avoiding non-performing loans exceeds the expected loss (1 ) R induced by issuing loans that we know that will default for certain. Put di erently, it is preferable to bear the loss of some non-performing loans than to incur the cost to screen them out. On the other hand, for low scores, the cost k S that have to be incurred to single out good projects exceeds the expected surplus (M R) that would be generated from granting loans to those rms. 4 A decentralized economy with fully specialized lenders Before analyzing the equilibrium of the full- edged game, I analyze a simpli ed version with fully specialized lenders. This simpli cation allows us to highlight the minimal ingredients that are necessary for the ine ciency to arise, while the I refer to the full- edge game to make further predictions. In particular, in this section I solve for the SPBE of the game in which: (i) L-Lenders cannot acquire costly information about projects. Formally: k L k. (ii) S-Lenders (and rms) do not observe any (noisy) scores about rms quality. E [jz] = E [] for any z. Formally: When information acquisition is too costly for all lenders, that is, when k S k, there is no role for loan o cers information acquisition. Hence, for the remainder of the paper, I assume: Assumption 3: k S < k. In this setting, since L-Lenders cannot acquire soft information and there are not noisy scores, their strategy reduces to quoting an interest rate r L () for rms with quality. On the other hand, since S-Lenders do not observe noisy scores, their pricing strategy consists of a uniform rate r S (x), which only depends on the outcome of the soft information signal, if they decide to acquire information about a contracting rm s project. The following lemma states that S-Lenders do acquire information in equilibrium and condition their credit decisions to the signal outcome. Lemma 2 S-Lenders always acquire information about projects before issuing loans and grant credit at a rate r S = r S (g) if and only if the signal is good. 13

14 S-Lenders must acquire information before issuing loans, as otherwise L-Lenders could use their superior information to o er better interest rates to the best rms, so that S-Lenders would be left out with the rms with negative net present value prospects. Proposition 2 fully characterizes the equilibrium of this lending game. Depending on parameters, di erent type of equilibria may arise. Proposition 3provide necessary and su cient conditions for each type of equilibrium to arise. Proposition 2 Let m and h as de ned in Proposition 1. Then, there exists a unique threshold, with m < h, such that the competitive equilibrium with fully specialized L-Lenders and S-Lenders is characterized by: (i) (Firms strategy) Firms sign loan contracts with L-Lenders if and only if r L r S +(1 ) (M r S ). (ii) (L-Lenders strategy) L-Lenders o er credit to rms with at rates: r L () = R. Firms with < are not o ered credit by L-Lenders. (iii) (S-Lenders strategy) Depending on parameters, the (unique) equilibrium belongs to one of these families: (a) If = m, S-Lenders are driven away from the market. (b) If > m, S-Lenders o er loan contracts at a rate of: rs = 1 R (1 ) M. (iv) (Beliefs) (a) If = m, any o -the-equilibrium-path belief density () such that R h () d l supports the equilibrium. (b) If > m, beliefs over the quality of rms accepting a credit o er from an S-Lender are given by: 8 f() < F ( ) if () =. : otherwise In equilibrium, rms sign loan contracts with the lender o ering the lowest rate, up to a premium (1 ) (M r S ) that rms are willing to pay in order to sign a loan contract with an L-Lender. The reason for this pricing wedge is that a rm with a good project at hand would earn M r i if the project got funded at an interest rate r i. However, signing a loan contract with an S-Lender entails a chance 1 that a good project be misperceived as a bad one, so that funds would not 14

15 Figure 2: Fully Specialized Competitive Equilibrium. Interest Rates and Firm Selection. be granted. 21 When soft information is perfectly accurate, that is, for = 1, the premium vanishes and rms simply opt for the lender quoting the lowest rate. Depending on parameters, we may have two types of equilibria. In the next section, we explore the conditions to have either one. Consider rst the situation such that S-Lenders are driven away from the market. With no competition from S-Lenders, L-Lenders target all rms for which it would be (ex-ante) pro table to lend to, namely rms with m. Recall from the analysis in Section 3 that granting credit without acquiring soft information yields positive surplus only for rms with quality > m. Hence, we have that = m. Now consider the scenario in which S-Lenders are active in equilibrium. This type of equilibrium is depicted in Figure 2. S-Lenders, who cannot observe rms quality, o er a single contract rs to any rm. The solid line represents this equilibrium price plus the pricing wedge (1 ) (M rs ) referred to above. The dashed line represents L-Lenders interest rate o ers rl (), which cater to rms quality. Firms with quality < only get loan o ers from S-Lenders and sign contracts with them. The rate rs is determined so that S-Lenders break even with the average rm in the pool that borrows from them, where expectations are taken according to the density function (). Hence, the lack of observability of rms quality by S-Lenders translate into cross-subsidization of low quality rms by rms above the average. Importantly, L-Lenders do not grant loans to rms with quality levels below the equilibrium threshold. If they did, they would have to charge an amount exceeding S-Lenders rates plus the premium. Hence, these rms could get better expected loan terms through an S-Lender. Hence, 21 Since rms enjoy limited liability, they are indi erent between L-Lenders and S-Lenders in the case that their projects are bad, in which case they get, regardless of the type of lender they borrow from. 15

16 if a rm were willing to accept a loan from an L-Lender, it would necessarily be the case that an S-Lender had rejected the rm on the basis of soft information. Hence, L-Lenders should refuse to lend to them. On the contrary rms with get better terms through L-Lenders and hence borrow from them. Therefore, we have perfect segmentation of the market by rm quality. The following corollary summarizes this nding: Corollary 1 There exists a unique threshold, with m < h, such that (i) Firms with quality get credit from L-Lenders. (iia) If S-Lenders are driven away from the market ( = m ), rms with quality < m do not obtain credit. (iib) If S-Lenders are active in the market, rms with quality < with good soft information signals get credit from S-Lenders. 4.1 When are Small Lenders driven away from the market? Equilibrium lending behavior is characterized by the threshold, which characterizes the segmentation of rms by type of lender. In this subsection, I lay out how to characterize this threshold, as well as necessary and su cient conditions for S-Lenders to be active in equilibrium. The expected S-Lenders pro t from signing a loan contract with a rm with quality at a rate r is given by S (; r), as given by equation (1). It follows that the expected S-Lenders pro t from signing a loan contract with a pool of rms with quality levels t 2 [; ], distributed according to a certain distribution (t), is given by R S (t; r) (t) dt. Letting the distribution of rms be (t) = f(t) F (), we can de ne the locus of break-even rates ^r S () for S-Lenders from signing contracts with rms with qualities below a given level as: Z ^r S () : S (t; ^r S ()) f (t) F () dt =. Since S (t; r) is strictly increasing in r, it follows that ^r S () is a function. Moreover, it is strictly decreasing, for a pool with a higher expected quality reduces the minimum interest rate that a lender would have to charge in order to break even. S-Lenders will be active in equilibrium if and only if they can outcompete L-Lenders, that is, if for some > m, the following condition is met: 22 ^r S () + (1 ) (M ^r S ()) r L (). Figure 3 depicts a situation in which a situation in which the curve ^r S ()+(1 ) (M ^r S ()) and rl () intersect once and above m. In this particular situation, S-Lenders are active in equilibrium. The dashed line represents the break even line rl () for L-Lenders, which intersects the largest interest rate M at m and the lowest interest price R at = 1. The thinnest solid line corresponds to the break even line rs () that S-Lenders would face if they were able to observe the quality of rms, plus the pricing premium. These two lines intersect at h, the latter laying below the former in the range [ l ; h ]. The thicker solid line represents the break even locus ^r S () of interest rates actually faced by S-Lenders, plus the pricing premium (1 ) (M ^r S ()) that rms consider when 22 It is straightforward to see that this condition is always trivially met for quality levels su ciently close to, so that the corresponding interest rates would exceed the maximum rate M that can be charged. 16

17 comparing interest rate quotations. In this gure, the curves r L () and ^r S ()+(1 ) (M ^r S ()) intersect once at, taking on a value below M. Suppose that L-Lenders o ered interest rates r L () to rms with. Then, if S-Lenders charged ^r S ( ) to any rm signing a loan contract with them, they would attract all rms in the range [; ] and would break even. Hence, the equilibrium in this situation would entail S-Lenders quoting r S = ^r S ( ) to any rm. This interest rate, together with the pricing premium, is depicted as the at solid line passing through the intersection of these break even loci. We now turn into the characterization of the conditions for S-Lenders to be active in equilibrium. A full characterization requires introducing the concept of a log-concave distribution function. A distribution with a density function f () is said to be log-concave if ln f () is a concave function. Intuitively, a density function is log-concave if "it is not too convex". Many standard distributions are log-concave. 23 The following provides this characterization. Proposition 3 If R m S () f () d >, then S-Lenders are active in equilibrium. Moreover, if the distribution f ()of rms types is log-concave, S-Lenders are active in equilibrium only if R m S () f () d >. Figure 3: Determination of Equilibrium Price and Firm Segmentation Threshold. Proposition 2 and Corollary 1 show that whenever S-Lenders are active in the lending market, they are left with the lowest quality tail of the distribution. Proposition 3 establishes that S-Lenders 23 Examples of log-concave distributions with support in [; 1] are the Uniform, Beta(; ) distributions with 1 and 1 or the Power Function distribution with 1. For a detailed account of log-concave functions, see (Bagnoli and Bergstrom, 25). 17

18 Figure 4: S-Lenders are driven away from the market. will be active in equilibrium if they create positive surplus by contracting with the pool of rms that L-Lenders would always neglect, that is, with rms in the range [; m ]. In this case, contracting with a larger pool of rms, with higher expected quality, would lead to a larger (positive) surplus. On the other hand, L-Lenders value from lending to rms with quality levels su ciently close to m gets arbitrarily close to zero. Hence, there must be a positive mass of rms to the right of m for which S-Lenders could underprice L-Lenders. This situation is depicted in Figure 3. We have that ^r S ( m ) + (1 ) (M ^r S ( m )) < M, which implies that S-Lenders would create positive surplus value by signing loan contracts with all rms in the range [; m ]. Figure 4 depicts the situation in which this equality is reversed and S-Lenders are driven away from the market. While this condition is su cient to have S-Lenders operating in the market, it is not necessary. Consider the situation in which we have R m S () f () d < = H ( m ), so that S-Lenders would not be able to create positive surplus by contracting with all rms to the left of m, but yet R ^ ^ S () f () d > H >, for some ^ > m. This would be the case, for instance, if there were a high concentration of rms around ^ that pushed up the average of the pool of borrowing rms when included in it. In this situation, we would have S-Lenders contracting in equilibrium. However, for these conditions to hold simultaneously, we would need that there be a su ciently high mass of rms to the left of l as well. Such distribution of rms would be U-shaped, at least in some range between and ^. If the distribution of rms quality is log-concave, this R ^ cannot happen. Under log-concavity, the function H ^ S () f () d is single peaked, so that R ^ ^ S () f () d and H satisfy a single crossing condition that ensures that if R S () f () d 18

19 Figure 5: The distribution of rms is not log-concave and single-crossing property is violated. = H ( ) for some > m, then R m S () f () d >. Figure 5 depicts a situation in which the loci ^r S () + (1 ) (M ^r S ()) and r L ( ) cross twice. This can only happen if the distribution of rms is not log-concave. While we have ^r S ( m ) + (1 ) (M ^r S ( m )) > M, S-Lenders are active in equilibrium and sign loan contracts with rms to the left of. This situation requires a su ciently high proportion of rms both to the left of l and somewhere in between m and. 4.2 Ine ciency of the competitive equilibrium Recall from Proposition 1 that optimality prescribes that S-Lenders should acquire soft information for rms in the range ( l ; h ). However, in equilibrium we have that < h, so that rms with quality 2 ( ; h ) are ine ciently cream-skimmed by L-Lenders. Even though S-Lenders are more e cient in this quality range, they are unable to attract these rms, because the maximum price that they could charge to attract all rms in the range ( ; h ): ^r = 1 R (1 ) M. h However, this is exactly the price that would make S-Lenders break even if only rms with quality h were to contract with them. Hence, since the average quality in the segment [; h ] is strictly smaller than h, S-Lenders would incur losses by charging this interest rate. Figure 6 illustrates this situation. The thick solid at line represents the maximum price (plus the price premium) that rms with quality h could be charged so as to prefer to contract with an S-Lender. This line lies below the thin solid decreasing line, which maps each quality with the break even rate (plus the price premium) faced by an S-Lender if she were to contract with a rm with that quality. 19

20 Hence, S-Lenders would incur losses with each and every rm that they would be contracting with if they charged a price ^r. Figure 6: Avoiding Cream-Skimming leads to S-Lenders Losses. The source for this ine ciency is the ex-ante informational advantage of L-Lenders. Since S- Lenders do not observe rms quality, they can only o er a at rate to all rms, targeting the average rm within the pool of potential borrowers. Therefore, the best rate that S-Lenders could o er is one that would make an S-Lender break even when signing a loan contract with all rms in the pool [; h ]. Clearly, the average quality of rms in this pool is lower than h. Hence, L-Lenders can poach a positive mass of rms to the left of h. Whether S-Lenders are active in equilibrium or driven away from the market, there is another source of ine ciency. Consider rst the case in which > m, so that S-Lenders sign contracts with all rms in the range [; ). From Proposition 1, we know that rms in the tranche [; l ) should be excluded from the lending market. However, in equilibrium, these rms get pooled with rms in the interval [ l ; ), because S-Lenders are unable to disentangle both groups. Hence, there is excessive information acquisition in this equilibrium. Consider now the case in which = m, so that S-Lenders are driven away from the market. It would be e cient that S-Lenders contracted with rms in the tranche [ l ; m ). However, since S-Lenders cannot observe their types, these rms are pooled with the rms in the lower tranche, making it not pro table to contract with them. Hence, this equilibrium entails credit rationing. The following corollary summarizes this nding. Corollary 2 The competitive equilibrium with perfectly specialized lenders is ine cient. (i) In any equilibrium, rms with quality 2 [ ; h ) are cream-skimmed by L-Lenders (instead of signing contracts with S-Lenders). 2