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Multiple borrowing and adverse selection in credit markets

Multiple borrowing and adverse selection in credit markets Abstract An entrepreneur planning a risky expansion of his small business project may prefer to fund the expansion by soliciting several loans from different lenders. While this is inefficient due to the duplication of screening and monitoring costs, it works to the entrepreneur’s advantage if he can lower his risk premium. When entrepreneurs are able to take out multiple loans in equilibrium, it takes place within a pooling contract, characterized by cross-subsidization. This kind of borrowing in the credit market leads to high interest rates and, in some cases, market failure due to adverse selection. 1. Introduction A number of empirical studies find that small firms often choose to finance their business investments by taking out several loans from different banks. For example, in a survey of over 5,000 small US firms who have at least one lender, Guiso and Minetti (2010) report that 49% of the firms rely on two or more lending institutions. Comparing small firms in the USA and Italy, Detragiache et al. (2000) find that on average a US firm has 2.3 banking relationships while an Italian firm has 12.3. There are similar findings for microfinance. In a study of microcredit markets in Nicaragua, Morocco, and Bosnia-Herzegovina, Chen et al. (2010) report incidences of multiple borrowing between 20% and 40% of active borrowers.1 What is surprising about this evidence is that small firms are generally viewed as opaque and thus costly to issue loans to. To mitigate information asymmetries, the lender must take a series of prudent and costly steps to ensure that the loan will be repaid. Part of these costs is independent of the actual size of the loan. The implication is that the cost of issuing a small loan necessitates a significant mark-up on the interest rate. Based on this logic, it should be cheaper for an entrepreneur to take out one large loan rather than several smaller loans. Microfinance is a good example. Looking at 346 institutions that issue microloans, Cull et al. (2009) find that the median bank, holding larger-sized loans on average, spends 12 cents on operating costs per dollar of loan, whereas the median non-governmental institution, holding smaller loans, spends 26 cents. Controlling for a number of different factors, the authors confirm that the institutions that issue the smallest loans on average are the same institutions that face the highest cost per unit lent and, furthermore, charge their customers the highest interest rates. To investigate this question, we develop a theoretical model of a credit market in which a subset of the entrepreneurs have an opportunity to scale up their projects. To fund the expansion, an entrepreneur can either take out one large loan or multiple smaller loans. In this setting, we find that if an entrepreneur’s expansion plan is viewed as high risk by the lenders, then the entrepreneur may find that it is cheaper to take out multiple loans. While multiple borrowing involves high transaction costs, this strategy allows him to conceal his true investment plans and thus avoid paying the additional risk premium that would get attached to the larger loan. In general, business expansion can involve more or less risk at the individual firm level. We embrace this idea by assuming that some firms face a higher risk of default under expansion, while other firms face a lower risk. The mechanism at work in our paper applies to the firms that face increased risk. One possible application is the developing economies, where entrepreneurs often run micro-sized businesses by relying heavily on their household resource endowments. As pointed out by Emran et al. (2011), these businesses can find that it is difficult to successfully scale up beyond the point where they start to require non-household resources, like wage labour and management or accounting skills. If banks view such plans as inherently risky, these type of micro-businesses may find that they can minimize their average cost of funds by relying on multiple lenders for small loans. Another example might be a credit market that contains an incumbent bank and an entrant bank, where the entrant is trying to gain market share. If there is heterogeneity in risk among the entrepreneurs who seek to expand, then the entrant may find that he can issue small loans to entrepreneurs who the incumbent views as having risky expansion plans. Under this scenario, the entrepreneur conceals his growth plans from the incumbent by continuing to take out a small but cheap loan, and then supplements these funds with a second, higher-priced loan from the entrant. The research in our paper is most pertinent to credit markets where information sharing between lenders is limited. Existing research, such as Pagano and Jappelli (1993) and Padilla and Pagano (2000), has clearly demonstrated how communication can be effective in resolving problems of asymmetric information between lender and borrower.2 However, in many emerging economies, information transparency in the credit markets is lacking. For example, in a survey of Latin America, Saaverdra (2012) documents widespread deficiencies in the operation of credit bureaus. Increasing transparency in these markets would likely go a long way towards alleviating the kinds of problems that we identify in our study. The model that we develop in our paper is related to a number of theoretical studies that examine competition and multiple borrowing in the credit market. In the microfinance literature, McIntosh and Wydick (2005) examine a dynamic setting with a population of clients that are heterogeneous in terms of a discount rate. In this setting, the authors argue that sufficiently myopic agents may take out multiple loans. As more lenders enter the credit market, increasing information asymmetries, contracts offered by lenders exhibit higher interest rates. Guha and Chowdhury (2013) look at a model where an agent can borrow two micro-loans from two different lenders, claiming that he will invest the funds, but in fact plans to invest one loan and consume the other. In this scenario, multiple borrowing leads to default. The authors argue that increased competition between lenders can increase incentives for borrowers to take two loans, leading to higher rates of default. In a related study, Guha and Chowdhury (2014) find that the possibility of multiple borrowing makes it easier to support equilibria where loans are issued to the poor. Casini (2015) studies competition between different microfinance lenders who have capacity constraints. In the model, an agent may choose to take out two loans and invest the second loan in an inefficient project. In equilibrium, multiple borrowing is most likely to occur when there is severe credit rationing. There are several models that examine multiple lending when lenders themselves are different. Papers such as Jain (1999), Andersen and Malchow-Moller (2006), and Gine (2011) explain how multiple borrowing can occur when two different kinds of lenders, namely a bank and moneylender, service the same population of borrowers. In the finance literature, papers like Sharpe (1990) and Rajan (1992) argue that borrowers may be able to protect themselves against rent extraction by relying on multiple lenders. Bennardo et al. (2015) and Bar-Issac and Cunat (2014) have models where multiple borrowing allows the entrepreneur to deceive the lender in specific ways, such as by consuming private benefits or concealing poor investment returns. We have organized the paper as follows. In Section 2, we introduce the model and explain the game. In Section 3.1, we examine a case where entrepreneurs have no incentive to take multiple loans. In Section 3.2, incentives change, and we document an equilibrium outcome where entrepreneurs take out several different loans in equilibrium. Next, in Section 4, we study adverse selection and look at a case of market failure. In Section 5, we explore an extension of the model where lenders can screen borrowers. Finally, in Section 6, we have the conclusion. 2. The credit market 2.1 The model Consider a one-period model with n risk-neutral entrepreneurs. Each entrepreneur faces a choice between investing in a production project or earning a fixed (wage) income of w ≥ 0. The project requires a $1 investment and generates revenue R with probability p and 0 otherwise. A fraction λ of the n entrepreneurs have a third alternative. These entrepreneurs can choose to scale up their production projects by investing $2 instead of $1. The outcome of the larger project depends on the entrepreneur’s type, and there are two types. A high ability type (h) generates revenue 2 R with probability ph, and 0 otherwise, while a low ability type (l) generates revenue 2 R with probability pl, and 0 otherwise. We assume that ph ≥ p > pl. Within the set of λn entrepreneurs who can scale up their projects, fraction α are type h and fraction 1 – α are type l, where 0 < α < 1. To invest in a project, the entrepreneur must obtain a loan from a lender. A loan contract specifies a loan size, of either $1 or $2, and an interest rate, r. There are m ≥ 3 lenders, who have access to an unlimited supply of funds at an interest rate of zero. A lender issues loans in order to maximize expected profit. Every loan issued by a lender costs c, regardless of loan size. Assumption A1 pR > 1 + c + w Assumption A2 pl2R > pR + 1 + c Assumption A1 implies that it is efficient for an entrepreneur to invest in the $1 project, as opposed to earning a wage. Assumption A2 implies that regardless of the entrepreneur’s type, it is efficient for an entrepreneur to choose his $2 project over his $1 project, even if the entrepreneur funds his $2 project using two separate $1 loans. Assumption A3  p(1+12c)>pl(1+c) Assumption A3 focuses our attention on a case where there is a significant increase in risk due to project expansion for low ability types. If this doesn’t hold, then the variation across risk types is too small to generate interesting results. 2.2 Information and the game Nature moves first, by distributing types across the population of entrepreneurs. The lenders do not observe which agents are awarded the opportunity to scale up their projects. However, lenders are familiar with the distributions, characterized by λ and α. Each lender simultaneously announces a set of loan contracts that are available to the entrepreneurs. A lender may offer as many different contracts as he wants. Next, lenders observe all offers, and then have the option to react by offering additional contracts. If one or more lenders do add contracts, then lenders again observe the offers and are allowed to add more contracts. This continues until all lenders choose not to offer any additional contracts. At no point may a lender withdraw a contract. We make two additional assumption regarding the offers. We assume that when a lender is indifferent between lending and not lending, he always lends. Second, we do not allow a lender to offer a contract where the lender expects to lose money on the contract. Once lenders have finished offering contracts, the entrepreneurs observe all offers and select loan contracts. Entrepreneurs who have access to the $2 project are allowed to either select one $2 loan, two $1 loans, or one $1 loan. When an entrepreneur demands two $1 loans, we say the entrepreneur is double borrowing. In this case, the entrepreneur uses the combined funds to invest in the $2 project. Entrepreneurs who do not have access to the $2 project may only select one $1 loan. After the entrepreneurs select their contracts, the lenders issue the loans and the entrepreneurs invest in their projects. Finally, project revenue is realized and the proceeds are used to repay the loans. At this point, the game ends. The primary motivation for allowing reactions is to avoid existence problems, as explained by Riley (1979). Given our intended application, we feel this assumption is not unreasonable. In practice, the contracts that lenders offer and the subsequent acceptance of these contracts tends to unfold continually over time. In light of this, it does not seem hard to believe that a lender might be able to offer a new contract in response to observing older contracts. The assumption that lenders are unable to withdraw contracts is perhaps harder to defend, but one could argue that once a contract is offered, if some agents accept it, then the lender is bound to honour the terms. Finally, our decision to disallow contracts where the lender knowingly expects to take a loss simplifies the analysis, by ruling out more complicated strategies where a lender might cross-subsidize within his own loan portfolio. 3. Competition in the credit market If the (1 – λ)n entrepreneurs who only have access to the $1 project select a $1 loan, and no other entrepreneurs do, then competition between the lenders leads to a competitive interest rate of r1=1p(1+c)−1. This leaves the λn entrepreneurs with access to both project sizes. Among this group there are αλn high ability and (1 – α)λn low ability entrepreneurs. If both types demand the same $2 loan, then a lender can afford a pooling rate r, where   [αph+(1−α)pl]2(1+r)−2−c=0, orr2=1αph+(1−α)pl(1+0.5c)−1. (1) The alternative is to take out two separate $1 loans. When choosing between these two methods of funding the project, the entrepreneur compares interest rates. That is, it is cheaper to fund the $2 project using one $2 loan as long as pj[2R−2(1+r2)]≥pj[2R−2(1+r1)], or r2 ≤ r1, where pj∈{ph,pl}. Observe that regardless of the entrepreneur’s type, the constraint is the same. Whether the $2 loan works out to be cheaper or not depends on how the transaction costs compare with the risk premium. The $2 loan has a lower transaction cost per dollar, but the $2 loan attracts low ability borrowers, who have a lower probability of success. In general, if average risk falls when projects grow, then both risk and transaction costs lead to a reduction in the rate. However, if average risks increase under expansion, then an additional risk premium is necessary for the $2 loan. To organize our analysis, we break the following discussion into two parts. In the first part we consider the case where r2 ≤ r1, and in the second part we consider r2 > r1. 3.1 Competition and efficiency The interest rate that lenders are willing to offer on the $1 loan depends on who the lenders expect to demand such loans. One possibility is that the lenders believe there will not be any double borrowing. In this case, competition between lenders generates a competitive interest rate of r1 on each $1 loan. To support this in equilibrium, it is necessary that entrepreneurs prefer not to double borrow. As we explained earlier, entrepreneurs will not want to double borrow as long as r2 ≤ r1, or   α≥(p−pl)(1+c)−p0.5c(ph−pl)(1+c).3 (2) Assumption A3 implies that this lower bound on α lies in the open interval 0, 1. Proposition 1 Let α≥(p−pl)(1+c)−p0.5c(ph−pl)(1+c). In equilibrium, the lenders offer $1 loans at r1 and $2 loans at r2. All entrepreneurs with access to the $2 project invest in the project using a $2 loan and all other entrepreneurs invest in the $1 project. The resulting allocation is efficient. Proof See appendix. This result describes equilibrium behaviour in the credit market when interest rates are decreasing in loan size. This occurs when a lower transaction cost per dollar dominates any added risk premium. The higher α is, the less significant is the presence of low ability entrepreneurs, which translates to a lower risk premium on the $2 loan. Since the interest rates decline with loan size, there is no reason for entrepreneurs to pursue double borrowing. This minimizes the transaction costs associated with funding the entrepreneurs’ investment projects. 3.2 Competition and inefficiency It makes sense for entrepreneurs to double borrow when r2 > r1. Note, this implies that αph+(1−α)pl must be less than p. Hence, while a larger loan saves on transaction cost per dollar lent, on average, project expansions are riskier. We now consider a scenario where all λn agents double borrow. The lenders anticipate this, and because of the added risk, the lenders increase the interest on the $1 loan accordingly. Consider a lender that offers a $1 loan. Denote this lender as lender A. Let all (1–λ)n entrepreneurs borrow from lender A. Say that the same offer also attracts all λn entrepreneurs, who double borrow. That is, these entrepreneurs take one loan from lender A and another $1 loan from a different lender. Given this composition of clients, lender A can afford   (1−λ)n[p(1+rx)−(1+c)]+αλn[ph(1+rx)−(1+c)] +(1−α)λn[pl(1+rx)−(1+c)]=0, orrx=1λ[αph+(1−α)pl]+(1−λ)p(1+c)−1. (3) One can verify that rx exceeds r1. This is because the contract pools types and some of the borrowers are low ability. In order for the λn entrepreneurs to double borrow, they need to take out a second $1 loan. However, the other lenders cannot match lender A on the interest rate rx. The reason is that lender A is lending $1 to all (1 – λ)n entrepreneurs, who only take one loan. If another lender tries to match lender A’s rate, then all (1 – λ)n entrepreneurs, who only take one loan, are indifferent between the two offers. Each lender then receives half of these entrepreneurs. But now, while these two lenders split the safe borrowers, they each attract all of the double borrowers. This alters the composition of agents who are borrowing on the lender’s contract. To be precise, if the two lenders both offer rx, then each lender earns   12(1−λ)n[p(1+rx)−(1+c)]+αλn[ph(1+rx)−(1+c)]+(1−α)λn[pl(1+rx)−(1+c)]<0. (4) This implies that when lender A exclusively offers rx, the other lenders who offer a $1 loan must charge a higher rate. Suppose that all lenders, except lender A, offer an identical $1 loan contract. Exactly m – 1 lenders offer this $1 contract. Each one of these lenders then attracts an equal share of borrowers, namely 1m−1, of the λn entrepreneurs. All of these entrepreneurs double borrow. On this offer, the lender can afford a rate where   [αph+(1−α)pl](1+rz)−1−c=0, orrz=1αph+(1−α)pl(1+c)−1.4 (5) The λn entrepreneurs who double borrow take out one loan at rx and the other loan at rz. Double borrowing is attractive to an entrepreneur only if it is cheaper than taking a $2 loan. That is, the entrepreneur prefers double borrowing when 12(rx+rz)<r2, or   λ<λ¯≡p−[αph+(1−α)pl](1+c)p−[αph+(1−α)pl]. (6) Proposition 2 Let α<(p−pl)(1+c)−p0.5c(ph−pl)(1+c). Also, assume that p[R−(1+r2)]≥w. I. If λ<λ¯, then in equilibrium one lender offers $1 loans at rx and all other lenders offer $1 loans at rz. All entrepreneurs with $2 projects invest in their projects by double borrowing and the remaining entrepreneurs invest in their $1 projects. The allocation is inefficient due to double borrowing, resulting in an efficiency loss of λnc. II. Say that λ≥λ¯. If rx ≥ r2, then in equilibrium all lenders offer both $1 loans and $2 loans at r2. If rx < r2, then all lenders offer $2 loans at r2 and one lender offers $1 loans at rx. All entrepreneurs with $2 projects invest in their projects using $2 loans and all other entrepreneurs invest in their $1 projects. The allocation is efficient. Proof See appendix. When r2 > r1, we identify two possible equilibrium outcomes. If λ<λ¯, then a relatively small fraction of the entrepreneurs have an opportunity to scale up their business projects. Rather than take out $2 loans, these entrepreneurs fund the expansion using multiple loans from different lenders. The incentive to double borrow comes from a cross-subsidy in one of the $1 loan contracts. Each entrepreneur who double borrows imposes a total transaction cost of 2 c on the economy. We interpret this as a case where lenders duplicate expensive screening and monitoring of the entrepreneurs, which is wasteful. It would be socially beneficial to bundle the two different $1 loans into a single $2 loan, all else equal. The reason this does not occur is that issuing a $2 loan would reveal that the borrower is a higher expected risk than the average entrepreneur taking the $1 contract at rx. Thus, issuing a $2 loan would require a higher risk premium. This risk premium makes the $2 loan unattractive from the point of view of the entrepreneur. Rather, the entrepreneur prefers to take out multiple loans and benefit from the subsidy inherent in interest rate rx.5 In contrast, when λ≥λ¯, there is no double borrowing and the equilibrium is efficient. This occurs when a relatively large portion of the credit market is trying to scale up their business projects. Since there are few entrepreneurs who only want one $1 loan, the potential for cross-subsidization does not generate much value. Thus, at interest rates where lenders can afford to tolerate double borrowing, the entrepreneurs are not interested. It is cheaper to borrow at r2. Interestingly, the lenders making these $1 loans now earn positive profit. Lenders will not undercut the offer in order to gain market share because this would initiate double borrowing and result in losses.6 4. Adverse selection We now investigate the possibility of market failure due to double borrowing. In particular, we are interested in whether competition between lenders can generate an outcome where the (1 – λ)n entrepreneurs choose to exit the credit market. In this section, we exclusively focus on a case where r2 > r1. Say that all (1 – λ)n entrepreneurs choose to earn wage income. Also, suppose that all lenders offer $2 loans at r2 and there are no $1 offers. The question is whether we can support this as an equilibrium or not. Say that a lender deviates and offers a $1 contract. To attract the entrepreneurs who plan to invest in the $1 project, the lender charges an interest rate where p[R−(1+r)]≥w, or r≤rw≡R−wp−1. Assumption A1 implies that rw > r1. Thus, as long as r∈[r1,rw], the deviation is interesting to both the entrepreneurs and the lender. To discourage this deviation, we require conditions such that there is an incentive for a second lender to react to the deviation by offering a new $1 contract. Furthermore, we then need to show that this reaction renders the initial deviation unprofitable. After the initial lender offers the $1 contract, suppose that another lender reacts to the deviation by also offering $1 loans. This opens the possibility for double borrowing. To make double borrowing unprofitable for the original lender, we focus on the case where at rx, the (1 – λ)n entrepreneurs elect to not borrow. That is, assume that p[R−(1+rx)]<w, which implies that rw < rx. This means that once the reacting lender offers his contract, and entrepreneurs begin to double borrow, the initial lender that is charging r∈[r1,rw] now expects a loss. The reacting lender does not attract any of the (1 – λ)n entrepreneurs. Rather, the lender only attracts double borrowers. This means that the reacting lender can afford to offer a rate as low as rz. To make the reaction profitable for the lender, entrepreneurs must prefer to double borrow. This is true for any r∈[r1,rw] as long as  12(rw+rz)<r2, or p[R−1αph+(1−α)pl]<w. This brings us to the following result. Proposition 3 Let α<(p−pl)(1+c)−p0.5c(ph−pl)(1+c) and p[R−1+cλ[αph+(1−α)pl]+(1−λ)p]<w. If p[R−1αph+(1−α)pl]<w, then in equilibrium all lenders offer $2 loans at r2 and no lenders offer $1 loans. All entrepreneurs with $2 projects invest in the projects and no entrepreneurs invest in the $1 project. The equilibrium is inefficient due to the entrepreneurs who do not invest, resulting in an efficiency loss of (1−λ)n[pR−1−c−w]. Proof See appendix. Entrepreneurs who only own the $1 project are unable to obtain the funds necessary to invest in their projects. If a lender tries to offer small loans to these entrepreneurs, then this offer instigates other lenders to also offer $1 loans, which in turn leads to double borrowing. Once entrepreneurs begin double borrowing, the lender that made the first offer finds that he is charging too low of a rate, and faces an expected loss. Anticipating all this, lenders choose to not issue any $1 loans. While entrepreneurs who have the opportunity to scale up their projects do so, using $2 loans, the remaining agents cannot access funding. This results in a version of adverse selection. While the risks of project expansion do not interfere with funding such a project, the presence of these risky expansions do interfere with the funding of small projects. 5. Screening In this section, we introduce the possibility that lenders can screen borrowers according to the risk in their expansion plans. Formally, we assume that the lender can screen entrepreneurs who select the $2 loan contract by identifying the individual type of a fraction φ of the αλn high ability entrepreneurs, where φ is a parameter and 0 ≤ φ ≤ 1. Consequently, if φ = 1, then lenders effectively have perfect information, but if φ = 0, then lenders are unable to verify the type of any entrepreneur, as in Section 3. Also, to simplify notation, we assume that ph = p. For entrepreneurs who are identified as high ability, lenders offer a conditional, competitive interest rate of rφ=2+c2p−1. Note that rφ < r1, so these entrepreneurs don’t want to double borrow. This leaves (1−φα)λn entrepreneurs with access to the $2 project, but who are not identified during screening. If all entrepreneurs in this group seek an unconditional $2 loan, the lender can afford to charge   [(1−φ)α(1−φα)p+(1−α)(1−φα)pl]2(1+r)−2−c=0, orr2(φ)=1−φα(1−φ)αp+(1−α)pl(1+12c)−1. (7) Note that if φ = 0, then r2(φ)=r2, as in Section 3. One possibility is that r2(φ)≤r1. When this holds, the entrepreneurs prefer the $2 loan over double borrowing. The more interesting case is when r2(φ)>r1. Consider strategies where lenders do not issue any unconditional $2 loans. Also, suppose that a single lender offers a $1 contract that attracts all (1 – λ)n entrepreneurs who take one loan, and all (1−φα)λn entrepreneurs who plan to double borrow. This lender can afford   rx(φ)=1−φαλλ[(1−φ)αp+(1−α)pl]+(1−λ)p(1+c)−1. (8) Furthermore, suppose all other lenders in the credit market offer $1 loans at a rate that only attracts the (1−φα)λn entrepreneurs who plan to double borrow. On this contract, the lenders can afford   rz(φ)=1−φα(1−φ)αp+(1−α)pl(1+c)−1. (9) Given the two different interest rates, rx(φ) and rz(φ), an entrepreneur prefers to double borrow if 12(rx(φ)+rz(φ))<r2(φ), or   λ<λ¯s≡p(1−φα)−[(1−φ)αp+(1−α)pl](1+c)p(1−φα)−[(1−φ)αp+(1−α)pl](1+φαc). (10) Proposition 4 Assume that p[R−1−αλλ(1−α)pl+(1−λ)p(1+c)]≥w. If λ<λ¯s then in equilibrium all lenders offer conditional $2 loans at rφ, one lender offers $1 loans at rx(φ), and all other lenders offer $1 loans at rz(φ). All entrepreneurs with $2 projects invest in the project: those who are screened as low risk take a $2 loan, and those who are not take two $1 loans. Entrepreneurs without the expansion take $1 loans. The allocation is inefficient due to the double borrowing, resulting in an efficiency loss of λn(1−φα)c. Proof See appendix. To understand how screening affects our results, we need to compare this result against Proposition 2. In Proposition 2, we are able to support double borrowing when λ<λ¯, whereas in Proposition 4 we now require λ<λ¯s. To illustrate the effect of screening, consider the case where φ = 1, such that screening is perfect. One can easily confirm that at φ = 1, λ¯s>λ¯, implying that double borrowing can now be supported for a wider range of λ values. The intuition is straightforward. The higher φ is, the more effective lenders are at identifying low risk entrepreneurs. The lenders issue cheap $2 loans to these entrepreneurs. This implies that the average quality of entrepreneur interested in a $2 loan at r2(φ) falls. This puts upward pressure on the interest rate r2(φ), making the loan less desirable. Consequently, the incentive to double borrow increases. However, there is an upside to having lenders screen entrepreneurs who plan project expansions. While screening worsens the average quality of the pool of entrepreneurs who go on to seek multiple loans, it also implies that less entrepreneurs do so. All borrowers who pass the screening test now get large loans. Comparing Propositions 2 and 4, one can indeed confirm that the efficiency loss is less when lenders screen, i.e., φ > 0. Thus, while screening may make double borrowing more likely, it also makes it less relevant. 6. Conclusions In our paper, we have argued that multiple borrowing can occur when entrepreneurs seek cheaper ways to fund relatively risky business expansions. By concealing their expansion plans from the lender, the entrepreneurs are able to achieve a lower average interest rate by taking out several small loans rather than one large loan. While this involves higher transaction costs, there is a benefit due to cross-subsidization in the small loan contract. Entrepreneurs who do not multiple borrow end up subsidizing those who do. While our study is theoretical and consequently limited in terms of its direct applicability to market settings, we can point to a few empirical implications. One is that our argument hinges on the presence of a cross subsidy, which is most valuable in a credit market where only a small fraction of the entrepreneurs try to expand. This suggests that if one was to look for evidence of multiple borrowing, it would be best to look at a market where growth and expansion is not too widespread, such as when lending standards are tight, or when the local business cycle is contracting. Second, one of our main predictions is that it is the riskier expansion plans that seek out multiple loans. Hence, one might test for this empirically by looking at whether all else equal, ex post default rates are higher for small businesses that relied on multiple loans to fund expansion. Third, when double borrowing takes place in equilibrium, a single borrower pays different interest rates on his small loans. This is how the entrepreneur lowers his average cost of funds. Along these lines, one might investigate empirically whether rates do in fact vary under multiple borrowing, and how significant the variation is. To a certain extent, our study resonates with a few of the empirical findings in microfinance markets. In a study of lending in India, Krishnaswamy (2007) looks at the number of days between when a borrower takes his first and second loan. He finds that a significant portion of the multiple borrowers take out consecutive loans in a short period time and argues that this suggests borrowers are trying to fund a single project by patching together different loans. Burki and Shah (2007) examine multiple borrowing in Pakistan and find that the overriding concern of borrowers is the small loan size cap enforced by lenders and that the high incidence of multiple borrowing is the result of borrowers trying to fund larger investment projects. Footnotes 1 For more examples, see McIntosh et al. (2005), Chua and Tiongson (2012), Frisancho (2012), and Khandker and Samad (2014). 2 On the empirical side, Doblas Madrid and Minetti (2013) find substantial evidence of the effectiveness of information sharing using US data. 3 Note that if expansion is a mean preserving spread, such that αph+(1−α)pl=p, then r2 < r1. 4 Note that this rate is similar in construction to r2. While it has the same risk premium as r2, rz has a higher transaction cost per dollar of loan. 5 The same argument applies to the question of whether lender A might offer two different $1 loan contracts. Say that lender A offers a second type of $1 loan, and that an agent accepts both types. Then it must be that the average of the two rates is less than r2, and this implies that lender A is expecting to lose money by funding this entrepreneur. 6 One may also notice that when rx < r2, a single lender offers $1 loans at rx, and earns positive profit. The reason these loans are not offered at a lower rate is that only one lender can offer $1 loans, and so, he offers the highest rate he can. 7 This is why we allow lenders to make reactions. Even though lender A is the only lender making the offer at rx, he will not deviate because other lenders will step in and replace his offer. 8 This is where it is important to assume a lender cannot offer a contract that generates a loss. For if the lender offers a rate r < r2, this improves the average quality of entrepreneurs taking the $1 contract. This is because on average, the double borrowing entrepreneur is riskier than the entrepreneur who only takes out one $1 loan. Hence, the lender can plan to take a loss on the $2 contract with the aim of earning a profit on the $1 contract. Acknowledgments I would like to thank the editor, Raoul Minetti, as well as two anonymous referees for their very helpful comments and suggestions. All remaining errors are mine alone. 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( 2010) Growth and vulnerabilities in microfinance, Focus Note, No. 61, Consultative Group to Assist the Poor, Washington, DC. Chua R.T., Tiongson E.R. ( 2012) Multiple borrowing in the Philippines: a pilot area study based on branch client data, Philippine Microfinance Discussion Paper, No. 2012‐02, Microfinance Council of the Philippines, Manila. Cull R., Demirguc-Kunt A., Morduch J. ( 2009) Microfinance meets the market, Journal of Economic Perspectives , 23, 167– 92. Google Scholar CrossRef Search ADS   Detragiache E., Garella P., Guiso L. ( 2000) Multiple versus single bank relationships: theory and evidence, Journal of Finance , 55, 1133– 61. Google Scholar CrossRef Search ADS   Doblas Madrid A., Minetti R. ( 2013) Sharing information in the credit market: contract-level evidence from U.S. firms, Journal of Financial Economics , 109, 198– 223. Google Scholar CrossRef Search ADS   Emran M., Morshed A., Stiglitz J. ( 2011) Microfinance and missing markets , Working Paper, Columbia University, New York. Frisancho V. ( 2012) Signaling creditworthiness in Peruvian microfinance markets: the role of information sharing, Working Paper Series No. IDB-WP-347, Inter-American Development Bank, Washington, DC. Gine X. ( 2011) Access to capital in rural Thailand: an estimated model of formal vs. informal credit, Journal of Development Economics , 96, 16– 29. Google Scholar CrossRef Search ADS   Guha B., Chowdhury P.R. ( 2013) Micro-finance competition: motivated micro-lenders, double-dipping and default, Journal of Development Economics , 105, 86– 102. Google Scholar CrossRef Search ADS   Guha B., Chowdhury P.R. ( 2014) Borrower targeting under microfinance competition with motivated microfinance institutions and strategic complementarities, Developing Economies , 52, 211– 40. Google Scholar CrossRef Search ADS   Guiso L., Minetti R. ( 2010) The structure of multiple credit relationships: evidence from U.S. firms, Journal of Money, Credit and Banking , 42, 1037– 71. Google Scholar CrossRef Search ADS   Jain S. ( 1999) Symbiosis vs. crowding out: the interactions of formal and informal credit markets in developing countries, Journal of Development Economics , 59, 419– 44. Google Scholar CrossRef Search ADS   Khandker S.R., Samad H.A. ( 2014) Dynamic effects of microcredit in Bangladesh, Policy Research Working Paper no. WPS6821, World Bank, Washington, DC. Krishnaswamy K. ( 2007) Competition and multiple borrowing in the Indian microfinance sector , Working Paper, Centre for Micro Finance, Institute for Financial Management and Research, Chennai. McIntosh C., de Janvry A., Sadoulet E. ( 2005) How rising competition among microfinance institutions affects incumbent lenders, Economic Journal , 115, 987– 1004. Google Scholar CrossRef Search ADS   McIntosh C., Wydick B. ( 2005) Competition and microfinance, Journal of Development Economics , 78, 271– 98. Google Scholar CrossRef Search ADS   Padilla A.J., Pagano M. ( 2000) Sharing default information as a borrower discipline device, European Economic Review , 44, 1951– 80. Google Scholar CrossRef Search ADS   Pagano M., Jappelli T. ( 1993) Information sharing in credit markets, Journal of Finance , 48, 1693– 718. Google Scholar CrossRef Search ADS   Rajan R. ( 1992) Insiders and outsiders: the choice between informed and arm’s-length debt, Journal of Finance , 47, 1367– 400. Google Scholar CrossRef Search ADS   Riley J.G. ( 1979) Informational equilibrium, Econometrica , 47, 331– 59. Google Scholar CrossRef Search ADS   Saavedra M.R. ( 2012) Public credit registries, credit bureaus and the microfinance sector in Latin America , Research Paper, Calmeadow, San Jose. Sharpe S. ( 1990) Asymmetric information, lender lending, and implicit contracts: a stylized model of customer relationships, Journal of Finance , 45, 1069– 87. Appendix Proof of Proposition 1 Let r2 ≤ r1. Working backwards, consider the entrepreneurs. When choosing between a $2 loan at r2 and one $1 loan at r1, the low ability entrepreneur prefers the $2 loan as long as pl[2R−2(1+r2)]>p[R−(1+r1)], or pl2R > pR + 1, which holds under Assumption A2. Given that low ability entrepreneurs prefer to scale up, high ability entrepreneurs do too. Since r2 ≤ r1, an entrepreneur clearly has no reason to take out two $1 loans at r1. Finally, given a choice between borrowing $1 at r1 or earning a wage, Assumption A1 implies that the entrepreneur prefers to borrow. Next, consider the lenders. With $2 loans, any lender deviation at r < r2 generates negative profit. If a lender deviates and offers a $1 loan at r < r1, then this attracts the (1 – λ)n entrepreneurs. In this case, there are two possibilities. If 12(r+r1)≥r2, then the deviation only attracts the (1 – λ)n entrepreneurs and the lender earns negative expected profit. On the other hand, if 12(r+r1)<r2, then the offer also attracts λn entrepreneurs who double borrow. However, the presence of low ability entrepreneurs among the double borrowers implies additional risk, which means that the lender offering r earns negative expected profit. □ Proof of Proposition 2 Under the conditions given, r2 > r1. Consider the following two cases. I. Let 12(rx+rz)<r2 (i.e., λ<λ¯). First consider the entrepreneurs. An entrepreneur prefers to borrow $1 over earning a wage if p[R−(1+rx)]≥w. This holds because for case (I.), rx < r2 and we have invoked the requirement p[R−(1+r2)]≥w for Proposition 2. A low ability entrepreneur prefers to double borrow rather than take one $1 loan as long as pl[2R−(2+rx+rz)]≥p[R−(1+rx)], or   pl2R+(p−pl)1+cλ[αph+(1−α)pl]+(1−λ)p ≥pR+pl1+cαph+(1−α)pl. (11) To show that this holds, it is sufficient to set λ = 0 and α = 0, in which case we have pl2R≥pR+plp(1+c). This holds under Assumption A2, and since the low ability entrepreneur prefers to scale up, so does the high ability entrepreneur. Second, consider the lenders. Denote the lender offering the rate rx as lender A. Prior to Proposition 2, in the main content of the paper we explained why no lender will deviate and offer a $1 loan at r ≤ rx. If the deviation is at r > rx, then the deviating lender only attracts the λn entrepreneurs, which results in a loss whenever r < rz. If lender A offers $1 loans at r > rx (instead of rx), then a different lender will react by offering the rate rx itself, and consequently, lender A will not attract any of the (1 – λ)n entrepreneurs.7  If a lender deviates and chooses to offer a $2 loan, when rx is available, then the $2 contract will only attract borrowers if r<12(rx+rz). But then r < r2 and the deviating lender expects negative profit on the contract.8 For the same reason, lender A does not have an incentive to offer a second type of $1 loan, to those who want such a loan, as this would reveal to the lender that the agent is double borrowing and hence earning the lender negative expected profit. If lender A deviates by withdrawing the contract rx and instead offering a $2 loan, a different lender reacts by offering rx, which then makes the $2 loan unattractive. II. Let 12(rx+rz)≥r2 (i.e., λ≥λ¯). All rates offered in equilibrium do not exceed r2. Thus, the entrepreneur prefers to borrow $1 rather than earn w. A low ability entrepreneur prefers to scale up rather than not, if pl[2R−2(1+r2)]≥p[R−(1+rx)], or   pl2R+p1+cλ[αph+(1−α)pl]+(1−λ)p ≥pR+pl2+cαph+(1−α)pl. (12) For sufficiency, we let λ = 0 and α = 0, in which case we have pl2R≥pR+1. This holds under Assumption A2. Since the low ability entrepreneur prefers to scale up, so does the high ability entrepreneur. Now consider the offers by the lenders. In general, either rx ≥ r2 or rx < r2. Case one let rx ≥ r2. All loans charge a rate r2. If a lender offers a $1 contract below r2, then it attracts all n entrepreneurs, and the λn entrepreneurs double borrow. They take one loan from the deviator and one loan at r2. However, since rx ≥ r2, the deviation is unprofitable. Case two let rx < r2. In this case, only one lender offers a $1 loan. Say that a different lender deviates and also offers a $1 contract. Any r < rx attracts all n entrepreneurs, but is unprofitable. At r = rx the offer attracts half of the (1 – λ)n entrepreneurs and, as we have discussed earlier, is unprofitable. Finally, at r > rx, the $1 offer attracts no (1 – λ)n entrepreneurs, in which case the lender must charge at least rz, and hence no entrepreneurs are interested, as double borrowing is more expensive than a $2 loan. Alternatively, still assuming rx < r2, say the lender offering the $1 loans deviates. An offer with r < rx lowers profit, and if r > rx, then a different lender will react and offer rx itself. □ Proof of Proposition 3 The conditions imply that r2 > r1. Entrepreneurs without access to a $2 project have no other choice but to earn wage income. Entrepreneurs with the $2 project prefer to borrow $2 at r2 rather than earn w if pl[2R−2(1+r2)]≥w, or pl2R≥plαph+(1−α)pl(2+c)+w. Assumptions A1 and A2 imply this holds. Given that low quality entrepreneurs prefer to borrow, it follows that high ability entrepreneurs do as well. Suppose a lender deviates and offers r ≤ rw on a $1 loan. Then other lenders react to the deviation and offer $1 loans at a rate rz. At these rates, entrepreneurs double borrow and the reacting lenders makes zero expected profit on their $1 loans. Due to the double borrowing, the lender offering r ≤ rw attracts all n entrepreneurs in the economy and earns negative expected profit because rw < rx. Thus, the deviation is not profitable. □ Proof of Proposition 4 This proof is very similar to part I of the proof of Proposition 2. First, the entrepreneurs. The Proposition applies only when at rx(φ), the entrepreneur prefers to participate and borrow $1 rather than earn w. A low ability entrepreneur prefers to scale up rather than not, as long as pl[2R−(2+rx(φ)+rz(φ))]≥p[R−(1+rx(φ))], or pl2R+(p−pl)(1+rx(φ))≥pR+pl(1+rz(φ)). Note that rx(φ) is increasing in λ. So, for sufficiency, we can use λ=0, which reduces the inequality to pl2R+(p−pl)1+cp≥pR+pl(1+rz(φ)). Furthermore, rz(φ) is increasing in φ. Hence, for sufficiency, we can use φ = 1, which reduces the inequality to pl2R+1+c≥pR+1pl(1+c)+plp(1+c). This holds due to Assumption A2. Since low ability entrepreneurs prefer to scale up, so do high ability entrepreneurs. Now the lenders. First, consider a deviation where a lender offers a $2 loan. If contract rx(φ) is available, then entrepreneurs will not want the $2 loan unless r<12(rx(φ)+rz(φ)). But, to be profitable, the loan requires r≥r2(φ), which means the deviation is not profitable. If instead, contract rx(φ) is not available (because the deviator is the lender that was offering rx(φ)), then a different lender will react and offer rx(φ), which again makes the $2 offer inviable. If the lender offering rx(φ) deviates and offers r > rx(φ), then a different lender will react and replace the offer rx(φ). If a lender other than the one offering rx(φ) deviates and offers a $1 loan at r < rx(φ), he will attract all n entrepreneurs in the economy, but due to the double borrowing will take a loss. At r = rx(φ) the lender only attracts half of the (1 – λ)n entrepreneurs, which implies a loss, and finally, at r > rx(φ) the lender does not attract any of the (1 – λ)n entrepreneurs. □ © Oxford University Press 2017 All rights reserved http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Oxford Economic Papers Oxford University Press

Multiple borrowing and adverse selection in credit markets

Van Tassel, Eric
Oxford Economic Papers , Volume 70 (1) – Jan 1, 2018
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0030-7653
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1464-3812
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10.1093/oep/gpx038
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Abstract

Abstract An entrepreneur planning a risky expansion of his small business project may prefer to fund the expansion by soliciting several loans from different lenders. While this is inefficient due to the duplication of screening and monitoring costs, it works to the entrepreneur’s advantage if he can lower his risk premium. When entrepreneurs are able to take out multiple loans in equilibrium, it takes place within a pooling contract, characterized by cross-subsidization. This kind of borrowing in the credit market leads to high interest rates and, in some cases, market failure due to adverse selection. 1. Introduction A number of empirical studies find that small firms often choose to finance their business investments by taking out several loans from different banks. For example, in a survey of over 5,000 small US firms who have at least one lender, Guiso and Minetti (2010) report that 49% of the firms rely on two or more lending institutions. Comparing small firms in the USA and Italy, Detragiache et al. (2000) find that on average a US firm has 2.3 banking relationships while an Italian firm has 12.3. There are similar findings for microfinance. In a study of microcredit markets in Nicaragua, Morocco, and Bosnia-Herzegovina, Chen et al. (2010) report incidences of multiple borrowing between 20% and 40% of active borrowers.1 What is surprising about this evidence is that small firms are generally viewed as opaque and thus costly to issue loans to. To mitigate information asymmetries, the lender must take a series of prudent and costly steps to ensure that the loan will be repaid. Part of these costs is independent of the actual size of the loan. The implication is that the cost of issuing a small loan necessitates a significant mark-up on the interest rate. Based on this logic, it should be cheaper for an entrepreneur to take out one large loan rather than several smaller loans. Microfinance is a good example. Looking at 346 institutions that issue microloans, Cull et al. (2009) find that the median bank, holding larger-sized loans on average, spends 12 cents on operating costs per dollar of loan, whereas the median non-governmental institution, holding smaller loans, spends 26 cents. Controlling for a number of different factors, the authors confirm that the institutions that issue the smallest loans on average are the same institutions that face the highest cost per unit lent and, furthermore, charge their customers the highest interest rates. To investigate this question, we develop a theoretical model of a credit market in which a subset of the entrepreneurs have an opportunity to scale up their projects. To fund the expansion, an entrepreneur can either take out one large loan or multiple smaller loans. In this setting, we find that if an entrepreneur’s expansion plan is viewed as high risk by the lenders, then the entrepreneur may find that it is cheaper to take out multiple loans. While multiple borrowing involves high transaction costs, this strategy allows him to conceal his true investment plans and thus avoid paying the additional risk premium that would get attached to the larger loan. In general, business expansion can involve more or less risk at the individual firm level. We embrace this idea by assuming that some firms face a higher risk of default under expansion, while other firms face a lower risk. The mechanism at work in our paper applies to the firms that face increased risk. One possible application is the developing economies, where entrepreneurs often run micro-sized businesses by relying heavily on their household resource endowments. As pointed out by Emran et al. (2011), these businesses can find that it is difficult to successfully scale up beyond the point where they start to require non-household resources, like wage labour and management or accounting skills. If banks view such plans as inherently risky, these type of micro-businesses may find that they can minimize their average cost of funds by relying on multiple lenders for small loans. Another example might be a credit market that contains an incumbent bank and an entrant bank, where the entrant is trying to gain market share. If there is heterogeneity in risk among the entrepreneurs who seek to expand, then the entrant may find that he can issue small loans to entrepreneurs who the incumbent views as having risky expansion plans. Under this scenario, the entrepreneur conceals his growth plans from the incumbent by continuing to take out a small but cheap loan, and then supplements these funds with a second, higher-priced loan from the entrant. The research in our paper is most pertinent to credit markets where information sharing between lenders is limited. Existing research, such as Pagano and Jappelli (1993) and Padilla and Pagano (2000), has clearly demonstrated how communication can be effective in resolving problems of asymmetric information between lender and borrower.2 However, in many emerging economies, information transparency in the credit markets is lacking. For example, in a survey of Latin America, Saaverdra (2012) documents widespread deficiencies in the operation of credit bureaus. Increasing transparency in these markets would likely go a long way towards alleviating the kinds of problems that we identify in our study. The model that we develop in our paper is related to a number of theoretical studies that examine competition and multiple borrowing in the credit market. In the microfinance literature, McIntosh and Wydick (2005) examine a dynamic setting with a population of clients that are heterogeneous in terms of a discount rate. In this setting, the authors argue that sufficiently myopic agents may take out multiple loans. As more lenders enter the credit market, increasing information asymmetries, contracts offered by lenders exhibit higher interest rates. Guha and Chowdhury (2013) look at a model where an agent can borrow two micro-loans from two different lenders, claiming that he will invest the funds, but in fact plans to invest one loan and consume the other. In this scenario, multiple borrowing leads to default. The authors argue that increased competition between lenders can increase incentives for borrowers to take two loans, leading to higher rates of default. In a related study, Guha and Chowdhury (2014) find that the possibility of multiple borrowing makes it easier to support equilibria where loans are issued to the poor. Casini (2015) studies competition between different microfinance lenders who have capacity constraints. In the model, an agent may choose to take out two loans and invest the second loan in an inefficient project. In equilibrium, multiple borrowing is most likely to occur when there is severe credit rationing. There are several models that examine multiple lending when lenders themselves are different. Papers such as Jain (1999), Andersen and Malchow-Moller (2006), and Gine (2011) explain how multiple borrowing can occur when two different kinds of lenders, namely a bank and moneylender, service the same population of borrowers. In the finance literature, papers like Sharpe (1990) and Rajan (1992) argue that borrowers may be able to protect themselves against rent extraction by relying on multiple lenders. Bennardo et al. (2015) and Bar-Issac and Cunat (2014) have models where multiple borrowing allows the entrepreneur to deceive the lender in specific ways, such as by consuming private benefits or concealing poor investment returns. We have organized the paper as follows. In Section 2, we introduce the model and explain the game. In Section 3.1, we examine a case where entrepreneurs have no incentive to take multiple loans. In Section 3.2, incentives change, and we document an equilibrium outcome where entrepreneurs take out several different loans in equilibrium. Next, in Section 4, we study adverse selection and look at a case of market failure. In Section 5, we explore an extension of the model where lenders can screen borrowers. Finally, in Section 6, we have the conclusion. 2. The credit market 2.1 The model Consider a one-period model with n risk-neutral entrepreneurs. Each entrepreneur faces a choice between investing in a production project or earning a fixed (wage) income of w ≥ 0. The project requires a $1 investment and generates revenue R with probability p and 0 otherwise. A fraction λ of the n entrepreneurs have a third alternative. These entrepreneurs can choose to scale up their production projects by investing $2 instead of $1. The outcome of the larger project depends on the entrepreneur’s type, and there are two types. A high ability type (h) generates revenue 2 R with probability ph, and 0 otherwise, while a low ability type (l) generates revenue 2 R with probability pl, and 0 otherwise. We assume that ph ≥ p > pl. Within the set of λn entrepreneurs who can scale up their projects, fraction α are type h and fraction 1 – α are type l, where 0 < α < 1. To invest in a project, the entrepreneur must obtain a loan from a lender. A loan contract specifies a loan size, of either $1 or $2, and an interest rate, r. There are m ≥ 3 lenders, who have access to an unlimited supply of funds at an interest rate of zero. A lender issues loans in order to maximize expected profit. Every loan issued by a lender costs c, regardless of loan size. Assumption A1 pR > 1 + c + w Assumption A2 pl2R > pR + 1 + c Assumption A1 implies that it is efficient for an entrepreneur to invest in the $1 project, as opposed to earning a wage. Assumption A2 implies that regardless of the entrepreneur’s type, it is efficient for an entrepreneur to choose his $2 project over his $1 project, even if the entrepreneur funds his $2 project using two separate $1 loans. Assumption A3  p(1+12c)>pl(1+c) Assumption A3 focuses our attention on a case where there is a significant increase in risk due to project expansion for low ability types. If this doesn’t hold, then the variation across risk types is too small to generate interesting results. 2.2 Information and the game Nature moves first, by distributing types across the population of entrepreneurs. The lenders do not observe which agents are awarded the opportunity to scale up their projects. However, lenders are familiar with the distributions, characterized by λ and α. Each lender simultaneously announces a set of loan contracts that are available to the entrepreneurs. A lender may offer as many different contracts as he wants. Next, lenders observe all offers, and then have the option to react by offering additional contracts. If one or more lenders do add contracts, then lenders again observe the offers and are allowed to add more contracts. This continues until all lenders choose not to offer any additional contracts. At no point may a lender withdraw a contract. We make two additional assumption regarding the offers. We assume that when a lender is indifferent between lending and not lending, he always lends. Second, we do not allow a lender to offer a contract where the lender expects to lose money on the contract. Once lenders have finished offering contracts, the entrepreneurs observe all offers and select loan contracts. Entrepreneurs who have access to the $2 project are allowed to either select one $2 loan, two $1 loans, or one $1 loan. When an entrepreneur demands two $1 loans, we say the entrepreneur is double borrowing. In this case, the entrepreneur uses the combined funds to invest in the $2 project. Entrepreneurs who do not have access to the $2 project may only select one $1 loan. After the entrepreneurs select their contracts, the lenders issue the loans and the entrepreneurs invest in their projects. Finally, project revenue is realized and the proceeds are used to repay the loans. At this point, the game ends. The primary motivation for allowing reactions is to avoid existence problems, as explained by Riley (1979). Given our intended application, we feel this assumption is not unreasonable. In practice, the contracts that lenders offer and the subsequent acceptance of these contracts tends to unfold continually over time. In light of this, it does not seem hard to believe that a lender might be able to offer a new contract in response to observing older contracts. The assumption that lenders are unable to withdraw contracts is perhaps harder to defend, but one could argue that once a contract is offered, if some agents accept it, then the lender is bound to honour the terms. Finally, our decision to disallow contracts where the lender knowingly expects to take a loss simplifies the analysis, by ruling out more complicated strategies where a lender might cross-subsidize within his own loan portfolio. 3. Competition in the credit market If the (1 – λ)n entrepreneurs who only have access to the $1 project select a $1 loan, and no other entrepreneurs do, then competition between the lenders leads to a competitive interest rate of r1=1p(1+c)−1. This leaves the λn entrepreneurs with access to both project sizes. Among this group there are αλn high ability and (1 – α)λn low ability entrepreneurs. If both types demand the same $2 loan, then a lender can afford a pooling rate r, where   [αph+(1−α)pl]2(1+r)−2−c=0, orr2=1αph+(1−α)pl(1+0.5c)−1. (1) The alternative is to take out two separate $1 loans. When choosing between these two methods of funding the project, the entrepreneur compares interest rates. That is, it is cheaper to fund the $2 project using one $2 loan as long as pj[2R−2(1+r2)]≥pj[2R−2(1+r1)], or r2 ≤ r1, where pj∈{ph,pl}. Observe that regardless of the entrepreneur’s type, the constraint is the same. Whether the $2 loan works out to be cheaper or not depends on how the transaction costs compare with the risk premium. The $2 loan has a lower transaction cost per dollar, but the $2 loan attracts low ability borrowers, who have a lower probability of success. In general, if average risk falls when projects grow, then both risk and transaction costs lead to a reduction in the rate. However, if average risks increase under expansion, then an additional risk premium is necessary for the $2 loan. To organize our analysis, we break the following discussion into two parts. In the first part we consider the case where r2 ≤ r1, and in the second part we consider r2 > r1. 3.1 Competition and efficiency The interest rate that lenders are willing to offer on the $1 loan depends on who the lenders expect to demand such loans. One possibility is that the lenders believe there will not be any double borrowing. In this case, competition between lenders generates a competitive interest rate of r1 on each $1 loan. To support this in equilibrium, it is necessary that entrepreneurs prefer not to double borrow. As we explained earlier, entrepreneurs will not want to double borrow as long as r2 ≤ r1, or   α≥(p−pl)(1+c)−p0.5c(ph−pl)(1+c).3 (2) Assumption A3 implies that this lower bound on α lies in the open interval 0, 1. Proposition 1 Let α≥(p−pl)(1+c)−p0.5c(ph−pl)(1+c). In equilibrium, the lenders offer $1 loans at r1 and $2 loans at r2. All entrepreneurs with access to the $2 project invest in the project using a $2 loan and all other entrepreneurs invest in the $1 project. The resulting allocation is efficient. Proof See appendix. This result describes equilibrium behaviour in the credit market when interest rates are decreasing in loan size. This occurs when a lower transaction cost per dollar dominates any added risk premium. The higher α is, the less significant is the presence of low ability entrepreneurs, which translates to a lower risk premium on the $2 loan. Since the interest rates decline with loan size, there is no reason for entrepreneurs to pursue double borrowing. This minimizes the transaction costs associated with funding the entrepreneurs’ investment projects. 3.2 Competition and inefficiency It makes sense for entrepreneurs to double borrow when r2 > r1. Note, this implies that αph+(1−α)pl must be less than p. Hence, while a larger loan saves on transaction cost per dollar lent, on average, project expansions are riskier. We now consider a scenario where all λn agents double borrow. The lenders anticipate this, and because of the added risk, the lenders increase the interest on the $1 loan accordingly. Consider a lender that offers a $1 loan. Denote this lender as lender A. Let all (1–λ)n entrepreneurs borrow from lender A. Say that the same offer also attracts all λn entrepreneurs, who double borrow. That is, these entrepreneurs take one loan from lender A and another $1 loan from a different lender. Given this composition of clients, lender A can afford   (1−λ)n[p(1+rx)−(1+c)]+αλn[ph(1+rx)−(1+c)] +(1−α)λn[pl(1+rx)−(1+c)]=0, orrx=1λ[αph+(1−α)pl]+(1−λ)p(1+c)−1. (3) One can verify that rx exceeds r1. This is because the contract pools types and some of the borrowers are low ability. In order for the λn entrepreneurs to double borrow, they need to take out a second $1 loan. However, the other lenders cannot match lender A on the interest rate rx. The reason is that lender A is lending $1 to all (1 – λ)n entrepreneurs, who only take one loan. If another lender tries to match lender A’s rate, then all (1 – λ)n entrepreneurs, who only take one loan, are indifferent between the two offers. Each lender then receives half of these entrepreneurs. But now, while these two lenders split the safe borrowers, they each attract all of the double borrowers. This alters the composition of agents who are borrowing on the lender’s contract. To be precise, if the two lenders both offer rx, then each lender earns   12(1−λ)n[p(1+rx)−(1+c)]+αλn[ph(1+rx)−(1+c)]+(1−α)λn[pl(1+rx)−(1+c)]<0. (4) This implies that when lender A exclusively offers rx, the other lenders who offer a $1 loan must charge a higher rate. Suppose that all lenders, except lender A, offer an identical $1 loan contract. Exactly m – 1 lenders offer this $1 contract. Each one of these lenders then attracts an equal share of borrowers, namely 1m−1, of the λn entrepreneurs. All of these entrepreneurs double borrow. On this offer, the lender can afford a rate where   [αph+(1−α)pl](1+rz)−1−c=0, orrz=1αph+(1−α)pl(1+c)−1.4 (5) The λn entrepreneurs who double borrow take out one loan at rx and the other loan at rz. Double borrowing is attractive to an entrepreneur only if it is cheaper than taking a $2 loan. That is, the entrepreneur prefers double borrowing when 12(rx+rz)<r2, or   λ<λ¯≡p−[αph+(1−α)pl](1+c)p−[αph+(1−α)pl]. (6) Proposition 2 Let α<(p−pl)(1+c)−p0.5c(ph−pl)(1+c). Also, assume that p[R−(1+r2)]≥w. I. If λ<λ¯, then in equilibrium one lender offers $1 loans at rx and all other lenders offer $1 loans at rz. All entrepreneurs with $2 projects invest in their projects by double borrowing and the remaining entrepreneurs invest in their $1 projects. The allocation is inefficient due to double borrowing, resulting in an efficiency loss of λnc. II. Say that λ≥λ¯. If rx ≥ r2, then in equilibrium all lenders offer both $1 loans and $2 loans at r2. If rx < r2, then all lenders offer $2 loans at r2 and one lender offers $1 loans at rx. All entrepreneurs with $2 projects invest in their projects using $2 loans and all other entrepreneurs invest in their $1 projects. The allocation is efficient. Proof See appendix. When r2 > r1, we identify two possible equilibrium outcomes. If λ<λ¯, then a relatively small fraction of the entrepreneurs have an opportunity to scale up their business projects. Rather than take out $2 loans, these entrepreneurs fund the expansion using multiple loans from different lenders. The incentive to double borrow comes from a cross-subsidy in one of the $1 loan contracts. Each entrepreneur who double borrows imposes a total transaction cost of 2 c on the economy. We interpret this as a case where lenders duplicate expensive screening and monitoring of the entrepreneurs, which is wasteful. It would be socially beneficial to bundle the two different $1 loans into a single $2 loan, all else equal. The reason this does not occur is that issuing a $2 loan would reveal that the borrower is a higher expected risk than the average entrepreneur taking the $1 contract at rx. Thus, issuing a $2 loan would require a higher risk premium. This risk premium makes the $2 loan unattractive from the point of view of the entrepreneur. Rather, the entrepreneur prefers to take out multiple loans and benefit from the subsidy inherent in interest rate rx.5 In contrast, when λ≥λ¯, there is no double borrowing and the equilibrium is efficient. This occurs when a relatively large portion of the credit market is trying to scale up their business projects. Since there are few entrepreneurs who only want one $1 loan, the potential for cross-subsidization does not generate much value. Thus, at interest rates where lenders can afford to tolerate double borrowing, the entrepreneurs are not interested. It is cheaper to borrow at r2. Interestingly, the lenders making these $1 loans now earn positive profit. Lenders will not undercut the offer in order to gain market share because this would initiate double borrowing and result in losses.6 4. Adverse selection We now investigate the possibility of market failure due to double borrowing. In particular, we are interested in whether competition between lenders can generate an outcome where the (1 – λ)n entrepreneurs choose to exit the credit market. In this section, we exclusively focus on a case where r2 > r1. Say that all (1 – λ)n entrepreneurs choose to earn wage income. Also, suppose that all lenders offer $2 loans at r2 and there are no $1 offers. The question is whether we can support this as an equilibrium or not. Say that a lender deviates and offers a $1 contract. To attract the entrepreneurs who plan to invest in the $1 project, the lender charges an interest rate where p[R−(1+r)]≥w, or r≤rw≡R−wp−1. Assumption A1 implies that rw > r1. Thus, as long as r∈[r1,rw], the deviation is interesting to both the entrepreneurs and the lender. To discourage this deviation, we require conditions such that there is an incentive for a second lender to react to the deviation by offering a new $1 contract. Furthermore, we then need to show that this reaction renders the initial deviation unprofitable. After the initial lender offers the $1 contract, suppose that another lender reacts to the deviation by also offering $1 loans. This opens the possibility for double borrowing. To make double borrowing unprofitable for the original lender, we focus on the case where at rx, the (1 – λ)n entrepreneurs elect to not borrow. That is, assume that p[R−(1+rx)]<w, which implies that rw < rx. This means that once the reacting lender offers his contract, and entrepreneurs begin to double borrow, the initial lender that is charging r∈[r1,rw] now expects a loss. The reacting lender does not attract any of the (1 – λ)n entrepreneurs. Rather, the lender only attracts double borrowers. This means that the reacting lender can afford to offer a rate as low as rz. To make the reaction profitable for the lender, entrepreneurs must prefer to double borrow. This is true for any r∈[r1,rw] as long as  12(rw+rz)<r2, or p[R−1αph+(1−α)pl]<w. This brings us to the following result. Proposition 3 Let α<(p−pl)(1+c)−p0.5c(ph−pl)(1+c) and p[R−1+cλ[αph+(1−α)pl]+(1−λ)p]<w. If p[R−1αph+(1−α)pl]<w, then in equilibrium all lenders offer $2 loans at r2 and no lenders offer $1 loans. All entrepreneurs with $2 projects invest in the projects and no entrepreneurs invest in the $1 project. The equilibrium is inefficient due to the entrepreneurs who do not invest, resulting in an efficiency loss of (1−λ)n[pR−1−c−w]. Proof See appendix. Entrepreneurs who only own the $1 project are unable to obtain the funds necessary to invest in their projects. If a lender tries to offer small loans to these entrepreneurs, then this offer instigates other lenders to also offer $1 loans, which in turn leads to double borrowing. Once entrepreneurs begin double borrowing, the lender that made the first offer finds that he is charging too low of a rate, and faces an expected loss. Anticipating all this, lenders choose to not issue any $1 loans. While entrepreneurs who have the opportunity to scale up their projects do so, using $2 loans, the remaining agents cannot access funding. This results in a version of adverse selection. While the risks of project expansion do not interfere with funding such a project, the presence of these risky expansions do interfere with the funding of small projects. 5. Screening In this section, we introduce the possibility that lenders can screen borrowers according to the risk in their expansion plans. Formally, we assume that the lender can screen entrepreneurs who select the $2 loan contract by identifying the individual type of a fraction φ of the αλn high ability entrepreneurs, where φ is a parameter and 0 ≤ φ ≤ 1. Consequently, if φ = 1, then lenders effectively have perfect information, but if φ = 0, then lenders are unable to verify the type of any entrepreneur, as in Section 3. Also, to simplify notation, we assume that ph = p. For entrepreneurs who are identified as high ability, lenders offer a conditional, competitive interest rate of rφ=2+c2p−1. Note that rφ < r1, so these entrepreneurs don’t want to double borrow. This leaves (1−φα)λn entrepreneurs with access to the $2 project, but who are not identified during screening. If all entrepreneurs in this group seek an unconditional $2 loan, the lender can afford to charge   [(1−φ)α(1−φα)p+(1−α)(1−φα)pl]2(1+r)−2−c=0, orr2(φ)=1−φα(1−φ)αp+(1−α)pl(1+12c)−1. (7) Note that if φ = 0, then r2(φ)=r2, as in Section 3. One possibility is that r2(φ)≤r1. When this holds, the entrepreneurs prefer the $2 loan over double borrowing. The more interesting case is when r2(φ)>r1. Consider strategies where lenders do not issue any unconditional $2 loans. Also, suppose that a single lender offers a $1 contract that attracts all (1 – λ)n entrepreneurs who take one loan, and all (1−φα)λn entrepreneurs who plan to double borrow. This lender can afford   rx(φ)=1−φαλλ[(1−φ)αp+(1−α)pl]+(1−λ)p(1+c)−1. (8) Furthermore, suppose all other lenders in the credit market offer $1 loans at a rate that only attracts the (1−φα)λn entrepreneurs who plan to double borrow. On this contract, the lenders can afford   rz(φ)=1−φα(1−φ)αp+(1−α)pl(1+c)−1. (9) Given the two different interest rates, rx(φ) and rz(φ), an entrepreneur prefers to double borrow if 12(rx(φ)+rz(φ))<r2(φ), or   λ<λ¯s≡p(1−φα)−[(1−φ)αp+(1−α)pl](1+c)p(1−φα)−[(1−φ)αp+(1−α)pl](1+φαc). (10) Proposition 4 Assume that p[R−1−αλλ(1−α)pl+(1−λ)p(1+c)]≥w. If λ<λ¯s then in equilibrium all lenders offer conditional $2 loans at rφ, one lender offers $1 loans at rx(φ), and all other lenders offer $1 loans at rz(φ). All entrepreneurs with $2 projects invest in the project: those who are screened as low risk take a $2 loan, and those who are not take two $1 loans. Entrepreneurs without the expansion take $1 loans. The allocation is inefficient due to the double borrowing, resulting in an efficiency loss of λn(1−φα)c. Proof See appendix. To understand how screening affects our results, we need to compare this result against Proposition 2. In Proposition 2, we are able to support double borrowing when λ<λ¯, whereas in Proposition 4 we now require λ<λ¯s. To illustrate the effect of screening, consider the case where φ = 1, such that screening is perfect. One can easily confirm that at φ = 1, λ¯s>λ¯, implying that double borrowing can now be supported for a wider range of λ values. The intuition is straightforward. The higher φ is, the more effective lenders are at identifying low risk entrepreneurs. The lenders issue cheap $2 loans to these entrepreneurs. This implies that the average quality of entrepreneur interested in a $2 loan at r2(φ) falls. This puts upward pressure on the interest rate r2(φ), making the loan less desirable. Consequently, the incentive to double borrow increases. However, there is an upside to having lenders screen entrepreneurs who plan project expansions. While screening worsens the average quality of the pool of entrepreneurs who go on to seek multiple loans, it also implies that less entrepreneurs do so. All borrowers who pass the screening test now get large loans. Comparing Propositions 2 and 4, one can indeed confirm that the efficiency loss is less when lenders screen, i.e., φ > 0. Thus, while screening may make double borrowing more likely, it also makes it less relevant. 6. Conclusions In our paper, we have argued that multiple borrowing can occur when entrepreneurs seek cheaper ways to fund relatively risky business expansions. By concealing their expansion plans from the lender, the entrepreneurs are able to achieve a lower average interest rate by taking out several small loans rather than one large loan. While this involves higher transaction costs, there is a benefit due to cross-subsidization in the small loan contract. Entrepreneurs who do not multiple borrow end up subsidizing those who do. While our study is theoretical and consequently limited in terms of its direct applicability to market settings, we can point to a few empirical implications. One is that our argument hinges on the presence of a cross subsidy, which is most valuable in a credit market where only a small fraction of the entrepreneurs try to expand. This suggests that if one was to look for evidence of multiple borrowing, it would be best to look at a market where growth and expansion is not too widespread, such as when lending standards are tight, or when the local business cycle is contracting. Second, one of our main predictions is that it is the riskier expansion plans that seek out multiple loans. Hence, one might test for this empirically by looking at whether all else equal, ex post default rates are higher for small businesses that relied on multiple loans to fund expansion. Third, when double borrowing takes place in equilibrium, a single borrower pays different interest rates on his small loans. This is how the entrepreneur lowers his average cost of funds. Along these lines, one might investigate empirically whether rates do in fact vary under multiple borrowing, and how significant the variation is. To a certain extent, our study resonates with a few of the empirical findings in microfinance markets. In a study of lending in India, Krishnaswamy (2007) looks at the number of days between when a borrower takes his first and second loan. He finds that a significant portion of the multiple borrowers take out consecutive loans in a short period time and argues that this suggests borrowers are trying to fund a single project by patching together different loans. Burki and Shah (2007) examine multiple borrowing in Pakistan and find that the overriding concern of borrowers is the small loan size cap enforced by lenders and that the high incidence of multiple borrowing is the result of borrowers trying to fund larger investment projects. Footnotes 1 For more examples, see McIntosh et al. (2005), Chua and Tiongson (2012), Frisancho (2012), and Khandker and Samad (2014). 2 On the empirical side, Doblas Madrid and Minetti (2013) find substantial evidence of the effectiveness of information sharing using US data. 3 Note that if expansion is a mean preserving spread, such that αph+(1−α)pl=p, then r2 < r1. 4 Note that this rate is similar in construction to r2. While it has the same risk premium as r2, rz has a higher transaction cost per dollar of loan. 5 The same argument applies to the question of whether lender A might offer two different $1 loan contracts. Say that lender A offers a second type of $1 loan, and that an agent accepts both types. Then it must be that the average of the two rates is less than r2, and this implies that lender A is expecting to lose money by funding this entrepreneur. 6 One may also notice that when rx < r2, a single lender offers $1 loans at rx, and earns positive profit. The reason these loans are not offered at a lower rate is that only one lender can offer $1 loans, and so, he offers the highest rate he can. 7 This is why we allow lenders to make reactions. Even though lender A is the only lender making the offer at rx, he will not deviate because other lenders will step in and replace his offer. 8 This is where it is important to assume a lender cannot offer a contract that generates a loss. For if the lender offers a rate r < r2, this improves the average quality of entrepreneurs taking the $1 contract. This is because on average, the double borrowing entrepreneur is riskier than the entrepreneur who only takes out one $1 loan. Hence, the lender can plan to take a loss on the $2 contract with the aim of earning a profit on the $1 contract. Acknowledgments I would like to thank the editor, Raoul Minetti, as well as two anonymous referees for their very helpful comments and suggestions. All remaining errors are mine alone. References Andersen T.B., Malchow-Moller N. ( 2006) Strategic interaction in undeveloped credit markets, Journal of Development Economics , 80, 275– 98. Google Scholar CrossRef Search ADS   Bar-Issac H., Cunat V. ( 2014) Long-term debt and hidden borrowing, Review of Corporate Financial Studies , 3, 87– 122. Google Scholar CrossRef Search ADS   Bennardo A., Pagano M., Piccolo S. ( 2015) Multiple bank lending, credit rights, and information sharing, Review of Finance , 19, 519– 70. Google Scholar CrossRef Search ADS   Burki H.B., Shah M. ( 2007) The dynamics of microfinance expansion in Lahore , Working Paper, Pakistan Microfinance Network, Islamabad. Casini P. 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( 2012) Public credit registries, credit bureaus and the microfinance sector in Latin America , Research Paper, Calmeadow, San Jose. Sharpe S. ( 1990) Asymmetric information, lender lending, and implicit contracts: a stylized model of customer relationships, Journal of Finance , 45, 1069– 87. Appendix Proof of Proposition 1 Let r2 ≤ r1. Working backwards, consider the entrepreneurs. When choosing between a $2 loan at r2 and one $1 loan at r1, the low ability entrepreneur prefers the $2 loan as long as pl[2R−2(1+r2)]>p[R−(1+r1)], or pl2R > pR + 1, which holds under Assumption A2. Given that low ability entrepreneurs prefer to scale up, high ability entrepreneurs do too. Since r2 ≤ r1, an entrepreneur clearly has no reason to take out two $1 loans at r1. Finally, given a choice between borrowing $1 at r1 or earning a wage, Assumption A1 implies that the entrepreneur prefers to borrow. Next, consider the lenders. With $2 loans, any lender deviation at r < r2 generates negative profit. If a lender deviates and offers a $1 loan at r < r1, then this attracts the (1 – λ)n entrepreneurs. In this case, there are two possibilities. If 12(r+r1)≥r2, then the deviation only attracts the (1 – λ)n entrepreneurs and the lender earns negative expected profit. On the other hand, if 12(r+r1)<r2, then the offer also attracts λn entrepreneurs who double borrow. However, the presence of low ability entrepreneurs among the double borrowers implies additional risk, which means that the lender offering r earns negative expected profit. □ Proof of Proposition 2 Under the conditions given, r2 > r1. Consider the following two cases. I. Let 12(rx+rz)<r2 (i.e., λ<λ¯). First consider the entrepreneurs. An entrepreneur prefers to borrow $1 over earning a wage if p[R−(1+rx)]≥w. This holds because for case (I.), rx < r2 and we have invoked the requirement p[R−(1+r2)]≥w for Proposition 2. A low ability entrepreneur prefers to double borrow rather than take one $1 loan as long as pl[2R−(2+rx+rz)]≥p[R−(1+rx)], or   pl2R+(p−pl)1+cλ[αph+(1−α)pl]+(1−λ)p ≥pR+pl1+cαph+(1−α)pl. (11) To show that this holds, it is sufficient to set λ = 0 and α = 0, in which case we have pl2R≥pR+plp(1+c). This holds under Assumption A2, and since the low ability entrepreneur prefers to scale up, so does the high ability entrepreneur. Second, consider the lenders. Denote the lender offering the rate rx as lender A. Prior to Proposition 2, in the main content of the paper we explained why no lender will deviate and offer a $1 loan at r ≤ rx. If the deviation is at r > rx, then the deviating lender only attracts the λn entrepreneurs, which results in a loss whenever r < rz. If lender A offers $1 loans at r > rx (instead of rx), then a different lender will react by offering the rate rx itself, and consequently, lender A will not attract any of the (1 – λ)n entrepreneurs.7  If a lender deviates and chooses to offer a $2 loan, when rx is available, then the $2 contract will only attract borrowers if r<12(rx+rz). But then r < r2 and the deviating lender expects negative profit on the contract.8 For the same reason, lender A does not have an incentive to offer a second type of $1 loan, to those who want such a loan, as this would reveal to the lender that the agent is double borrowing and hence earning the lender negative expected profit. If lender A deviates by withdrawing the contract rx and instead offering a $2 loan, a different lender reacts by offering rx, which then makes the $2 loan unattractive. II. Let 12(rx+rz)≥r2 (i.e., λ≥λ¯). All rates offered in equilibrium do not exceed r2. Thus, the entrepreneur prefers to borrow $1 rather than earn w. A low ability entrepreneur prefers to scale up rather than not, if pl[2R−2(1+r2)]≥p[R−(1+rx)], or   pl2R+p1+cλ[αph+(1−α)pl]+(1−λ)p ≥pR+pl2+cαph+(1−α)pl. (12) For sufficiency, we let λ = 0 and α = 0, in which case we have pl2R≥pR+1. This holds under Assumption A2. Since the low ability entrepreneur prefers to scale up, so does the high ability entrepreneur. Now consider the offers by the lenders. In general, either rx ≥ r2 or rx < r2. Case one let rx ≥ r2. All loans charge a rate r2. If a lender offers a $1 contract below r2, then it attracts all n entrepreneurs, and the λn entrepreneurs double borrow. They take one loan from the deviator and one loan at r2. However, since rx ≥ r2, the deviation is unprofitable. Case two let rx < r2. In this case, only one lender offers a $1 loan. Say that a different lender deviates and also offers a $1 contract. Any r < rx attracts all n entrepreneurs, but is unprofitable. At r = rx the offer attracts half of the (1 – λ)n entrepreneurs and, as we have discussed earlier, is unprofitable. Finally, at r > rx, the $1 offer attracts no (1 – λ)n entrepreneurs, in which case the lender must charge at least rz, and hence no entrepreneurs are interested, as double borrowing is more expensive than a $2 loan. Alternatively, still assuming rx < r2, say the lender offering the $1 loans deviates. An offer with r < rx lowers profit, and if r > rx, then a different lender will react and offer rx itself. □ Proof of Proposition 3 The conditions imply that r2 > r1. Entrepreneurs without access to a $2 project have no other choice but to earn wage income. Entrepreneurs with the $2 project prefer to borrow $2 at r2 rather than earn w if pl[2R−2(1+r2)]≥w, or pl2R≥plαph+(1−α)pl(2+c)+w. Assumptions A1 and A2 imply this holds. Given that low quality entrepreneurs prefer to borrow, it follows that high ability entrepreneurs do as well. Suppose a lender deviates and offers r ≤ rw on a $1 loan. Then other lenders react to the deviation and offer $1 loans at a rate rz. At these rates, entrepreneurs double borrow and the reacting lenders makes zero expected profit on their $1 loans. Due to the double borrowing, the lender offering r ≤ rw attracts all n entrepreneurs in the economy and earns negative expected profit because rw < rx. Thus, the deviation is not profitable. □ Proof of Proposition 4 This proof is very similar to part I of the proof of Proposition 2. First, the entrepreneurs. The Proposition applies only when at rx(φ), the entrepreneur prefers to participate and borrow $1 rather than earn w. A low ability entrepreneur prefers to scale up rather than not, as long as pl[2R−(2+rx(φ)+rz(φ))]≥p[R−(1+rx(φ))], or pl2R+(p−pl)(1+rx(φ))≥pR+pl(1+rz(φ)). Note that rx(φ) is increasing in λ. So, for sufficiency, we can use λ=0, which reduces the inequality to pl2R+(p−pl)1+cp≥pR+pl(1+rz(φ)). Furthermore, rz(φ) is increasing in φ. Hence, for sufficiency, we can use φ = 1, which reduces the inequality to pl2R+1+c≥pR+1pl(1+c)+plp(1+c). This holds due to Assumption A2. Since low ability entrepreneurs prefer to scale up, so do high ability entrepreneurs. Now the lenders. First, consider a deviation where a lender offers a $2 loan. If contract rx(φ) is available, then entrepreneurs will not want the $2 loan unless r<12(rx(φ)+rz(φ)). But, to be profitable, the loan requires r≥r2(φ), which means the deviation is not profitable. If instead, contract rx(φ) is not available (because the deviator is the lender that was offering rx(φ)), then a different lender will react and offer rx(φ), which again makes the $2 offer inviable. If the lender offering rx(φ) deviates and offers r > rx(φ), then a different lender will react and replace the offer rx(φ). If a lender other than the one offering rx(φ) deviates and offers a $1 loan at r < rx(φ), he will attract all n entrepreneurs in the economy, but due to the double borrowing will take a loss. At r = rx(φ) the lender only attracts half of the (1 – λ)n entrepreneurs, which implies a loss, and finally, at r > rx(φ) the lender does not attract any of the (1 – λ)n entrepreneurs. □ © Oxford University Press 2017 All rights reserved

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Oxford Economic Papers – Oxford University Press

Published: Jan 1, 2018

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