Optimal Group Size in Joint Liability Contracts

Transcription

1 Optimal Group Size in Joint Liability Contracts Bahar Rezaei Anderson School of Management, University of California, Los Angeles, 110 Westwood Plaza, Los Angeles, CA 90095, USA; Sriram Dasu Marshall School of Business, University of Southern California, Los Angeles, CA 90089, USA; Reza Ahmadi Anderson School of Management, University of California, Los Angeles, 110 Westwood Plaza, Los Angeles, CA 90095, USA; Abstract We develop a model of repeated microcredit lending to study how group size affects optimal group lending contracts with joint liability. The story is that a benevolent lender provides microcredit to a group of borrowers to be invested in n projects. The outcome of each risky project is not observable by the lender; therefore if some of the borrowers default on their loan repayments, the lender cannot identify strategic default. The group will be entitled to a subsequent loan if all group members have repaid their loan obligations. We characterize the optimal contract and determine the optimal size of the group of borrowers endogenously. We discuss that, although joint liability contracts are feasible under a smaller set of parameter values than individual liability contracts, joint liability has positive effects on the repayment amount and borrowers welfare. Our analysis also suggests that group size should increase with project risk. Furthermore, we analyze the effect of less severe punishment, partial joint liability, and project correlation on the feasibility of joint liability contracts. Our results show that, first, less severe punishment doesn t affect the repayment amount or borrowers welfare, but the maximum loan that could be offered to borrowers should be lower when the punishment is less severe. This negative effect could be compensated by enlarging the group size. Second, although partial joint liability has a negative effect on the repayment amount and borrowers welfare, it can increase the 1

2 maximum feasible loan that can be offered to borrowers when collusion is not as likely between borrowers or when borrowers have high valuation for future loans. Third, when projects are correlated, joint liability contracts can increasingly offer larger loans to larger groups. [Keywords: Microcredit, Joint Liability, Strategic default] 1 Introduction Small-scale businesses are considered a major source of employment, particularly in developing countries where such businesses employ more than half of the economically active population (De Mel et al., One of the most efficient tools for microenterprise development is access to a source of finance (Berge et al., However, such enterprises are usually excluded from conventional financial services, either because they are unable to offer formal guarantees, or are located too far from the financial networks (Prior and Argandoña, In 2008 Peru, for example, although 3,080,000 microenterprises employed 76% of the country s labor force and accounted for 42% of the country s GDP, these small businesses were never regarded as relevant by the formal banking system (Chu, Microcredit may equip financial institutions with the right tools to carry out their specific social responsibilities of integrating people in the active population and combating the cause of social and financial exclusion (Lacalle-Calderón and Rico-Garrido, 2006 and reducing poverty in developing countries (Littlefield et al., Microcredit is generally defined as small loans with little or no collateral offered to microentrepreneurs who are usually excluded from conventional financial services. Although this type of lending had some success in lending to the poor and has been publicly seen as one of the recent improvements in financial institutions to support development, it has also been reported that such lending only reaches the moderately poor people and not the poorest of the poor (Scully, 2004; Marr, In the present paper, we study the following microcredit lending scenario. A benevolent lender (she wants to provides loans to a self-selected group of n microentrepreneur borrowers (he with joint liability. Members in a jointly liable group, are given individual loans but are held jointly liable for repayment. In addition, the entire group qualifies for a subsequent loan on the condition that all group members have repaid. The loans are then invested in n projects that have an equal 2

3 chance of success, 1 and can either be disjoint or correlated. We assume that the realized output of each project is known to the group members but unknown to the lender. 2 Thus, in this scenario both sides, the lender and the borrowers, have decisions to make. The lender decides on the optimal contract structure and the degree of joint liability, while the borrowers decide whether to repay the loan or default. Our study is primarily concerned with how large should the group size n be in order to maximize the borrowers benefit while leaving the lender to break even. On the one hand, a larger group size can have a positive effect on the repayment amount, as more people that are liable for repaying defaulted payments assures a higher rate of repayment. On the other hand, a larger group size can be a threat for members who repay their loans successfully, since they should repay all defaulting peers repayment, and everyone else in the group may default on repayment. We propose a method to find the optimal group size that enables the lender to provide larger loans to borrowers under the JL contract. We model the abovementioned lending situation as an infinitely repeated game. Inspired by Tedeschi (2006, we define two phases in our model, a lending phase, in which the game starts and continues until the group defaults on repayment, and a punishment phase, in which no new loans are extended to the borrowers. We stay aligned with Bhole and Ogden s (2010 simple group lending model designed for groups of two borrowers and extend their results to groups of n borrowers. In our model, borrowers receive contracts individually that determine the amount of loan L and repayment R. They invest their loans on n projects that are identical in terms of mean return and chance of success, however, the loans can be disjoint or correlated. Projects are either successful with a high return or unsuccessful with a low return. After the project outcomes are realized, each borrower decides whether to repay his loan. Only barrowers who completed successful projects are supposed to repay. We differentiate between strategic default and non-strategic default. Borrowers default strategically when they refuse to repay and have high outcomes from their projects. Borrowers default non-strategically when they do not repay due to little or no outcome from their projects caused by bad luck. Although the lender cannot identify strategic default, members of the group 1 In the literature, it is widely believed that when borrowers can self-select their group members, joint liability can induce borrowers to pool with borrowers of similar risk (see e.g., Stiglitz, 1990; Ghatak, Thus, it is not unrealistic to assume that the chance of project success of a self-selected group are equal. 2 This is a plausible assumption, as group members are self-selected and have better information about each other than the lender. 3

4 can. The lender deprives a defaulting group from future loans in order to decrease incentives for strategic default, however, non-strategic defaults that result from bad luck are unavoidable and repayment is not completely insured. We assume that group members play a grim-trigger strategy amongst themselves, meaning that, group members keep repaying their loans, as well as the loans of their non-strategically defaulting peers, as long as no one defaults strategically. If an individual defaults strategically, other members of the group will not repay his loan as well as their own loans, and no further loans are granted to the group. Our results show that, although a joint liability (JL contract is more feasible under a smaller set of parameter settings than an individual liability (IL contract, it has a higher performances than the IL contract in terms of borrowers welfare and the repayment amount. The JL contract can also outperform the IL contract in terms of the maximum loan that can be offered to borrowers if the group size is not too large. The intuition behind is that, because groups can offer higher repayment insurance than individuals, it is less risky to offer larger loans to groups. However, if the group size becomes too large, the risk of project failure and the number of defaulting members also goes up. This risk is even higher when borrowers invest in riskier projects. We calculate the optimal group size, given the maximum and minimum returns of projects, the discount factor of borrowers for future loans, and the chance of project success. Further, we consider relaxing some of our assumptions on the basic model. We seek to examine how much flexibility is possible in designing the JL contract in terms of the degree of joint liability and the penalty function employed against the defaulting borrowers. Should the successful members be held liable for full repayment of their unsuccessful group members or should they only be responsible for a fraction of the repayment? Should the defaulting borrowers be excluded from lending forever or be given another chance? We compare the JL contract, which deprives the strategically defaulting members from future loans forever, with a flexible joint liability (FJL contract, which deprives them for only T periods and lets them rejoin the group afterwards. We also compare the JL contract that holds successful group members responsible for their unsuccessful peers full repayment, with a partial joint liability (PJL contract, which asks the successful group members to repay only a fraction of the repayment. We show that the punishment phase does not have a significant effect on the borrowers welfare and repayment amount, and both stay the same under the JL and the FJL contracts. However, the 4

5 maximum loan that can be offered to borrowers under the FJL contract should be lower compared to the JL contract. More specifically, the maximum feasible loan under the FJL contract increases with the increase of the length of the punishment phase, and thus is highest when the length of the punishment phase goes to infinity. In other words, the maximum loan that can be offered to borrowers is highest under the JL contract, which is not surprising given what we know from studies on repeated games of cooperation. We also show that, in the FJL contract, when the punishment phase is not too overly long, larger groups should be formed. Intuitively, when the punishment is not as severe, members are more prone to default strategically, therefore, a larger group is needed to enhance repayment insurance. We also show that under partial joint liability, the repayment amount is higher and borrowers welfare is lower, because the lender incurs some auditing costs that should be compensated by the borrowers. However, partial joint liability can increase the maximum feasible loan that can be offered to borrowers when borrowers are likely to repay. More specifically, partial joint liability can increase the maximum feasible loan that can be offered to borrowers when the lender audits the states of projects randomly and so collusion is not as likely between borrowers or when borrowers have high valuation for future loans. As another extension on our basic model, we explore how project correlation can affect the lending outcome. It is natural to assume that the chance of project success is an increasing function of the group size, as jointly liable group members are likely to help each other to succeed. However, the marginal desirability of forming larger groups should decrease, as very large groups must confront higher tensions. We show that when any positive correlation exists between project returns, we could increase the feasibility of joint liability contracts by enlarging the borrower s group. 2 Related Literature Our study, in general, is in line with the literature that explores conditions under which group lending can help alleviate informational asymmetry, incentivize risk-sharing, and reduce enforcement problems (see e.g., Ghatak, 1999; 2000; Armendáriz de Aghion and Gollier, Within this literature, our study is related to studies concerned with the optimal design of joint liability contracts that should be offered to a group of microcredit borrowers who are subject to strategic 5

6 default, the degree of joint liability that should be imposed upon the group members, and the type of punishment that should be undertaken against defaulting group members (see e.g., Bhole and Ogden, 2010; Allen, 2016; Tedeschi, The literature generally concludes that when borrowers are unable to impose strong social sanctions on each other, a microcredit lender is better off offering each borrower an individual liability contract rather than a joint liability contract (see e.g., Besley and Coate, 1995; Armendáriz de Aghion, Contrary to the literature, we discuss that although joint liability contracts are feasible under a smaller set of parameters, such contracts can positively affect the borrowers welfare and repayment amount as well as the maximum loan that can be offered, even in the absence of social sanctions. The reason that our results differ is that we allow borrowers of one group to punish their strategically defaulting co-members by banishing them for at least T periods of the lending game. This strong punishment system compensates for the lack of social sanctions. The economic literature that has investigated different features of group lending with joint liability has paid little attention to group size as one of the potentially influential factors in the relative success of group lending. Theoretical studies mostly analyze lending models of groups of two borrowers, while experimental and empirical studies suggest the importance of group size (Abbink et al., 2006; Galak et al., We argue that group size is an important factor in increasing borrowers welfare and repayment amounts in microcredit lending, and we determine the optimal group size endogenously. However, existing literature that has considered group size as an important factor has concluded differently. On the one hand, although arguments in favor of larger group sizes have been made, such as Conning (2004 and Ahlin (2015, they also suggest that group size cannot grow too large. Conning (2004 specifies that it becomes increasingly costly to contain free-riding as group size increases. Ahlin (2015 shows that the presence of local borrower information is necessary in order for large groups to have any impact. Our assumption that borrowers play grim-trigger strategy against each other helps us to deal with free-riding inside a group of borrowers, and it also replaces the local borrower information effect. On the other hand, there are arguments in the literature that favor a smaller group size. For example, Bourjade and Schindele (2012 prove that if group members have social ties, a rational lender should choose a group of limited size. They explain that a trade-off exists between raising profits through increased group size and providing incentives for 6

7 borrowers with less social ties. The findings of Baland et al. (2013 are similar to ours, as they find that the optimal group size depends on project characteristics. We show that, although the repayment insurance provided by a larger group is higher, the weight of risk of being in charge of everybody else is also higher, specially when projects are risky. Therefore, riskier projects should be handled by relatively larger groups, however, the group size cannot become too large. It is blieved in the literature that group lending improves risk-sharing between group members (see e.g., Allen and Babus, Part of this literature emphasizes the role of social networks on the informal insurance that group members provide for each other and argues that group lending even without joint liability improves risk-sharing (Feinberg et al., Another part of this literature, however, gives credit to the joint liability feature of microcredit lending for improving risk-sharing. Among the well-known examples are Ghatak (2000, Fischer (2013, and Allen (2016. In accordance with this latter literature, we argue that joint liability positively influences risksharing. More specifically, we prove that when the degree of joint liability increases, borrowers can be charged a lower repayment amount as a result of the higher repayment insurance that they can provide. Furthermore, the literature that studies the effect of correlation between projects is also relevant to our study, such as Ghatak (2000 and Katzur and Lensink (2012. These authors argue that when projects are likely to succeed (or fail at the same time, the joint liability part of the contract is not applied as often. Based on this argument, Ghatak (2000 proves that joint liability contracts are less feasible when project outcomes are positively correlated. We argue, in contrast, that joint liability contracts maybe even more feasible when a positive correlation exists between project outcomes, because jointly liable groups are more likely to contribute to each other s success. In other words, joint liability can cause the formation of project externalities. Although Katzur and Lensink (2012 argue, similat to our argument, that positive correlation of project returns may improve the efficiency of group lending contracts, our model differs from theirs in various ways. For example, we model project correlation through the assumption that group members social interactions positively influence their chances of project success. 7

8 3 Model Consider an infinitely repeated lending game, with a benevolent lender and n borrowers playing a grim-trigger strategy against each other, that is, if at some period some members default strategically on their repayments, other members stop repaying themselves and stop repaying the defaulting players shares. The group will therefore not be eligible to obtain a loan in the next period. We consider a two-phase model in which the lender and borrowers begin in a lending phase. If one loan is successfully repaid by the group, another loan is then given in the next period. In any period, if borrowers default, the lender and borrowers engage in a punishment phase, in which no new loans are extended to borrowers. Each period of the game has three steps. s = 0 Each borrower receives an individual contract (L, R, specifying the amount of loan L and the repayment R > L (primary loan plus interest. s = 1 Each borrower invests L in his project that will either succed with a chance of α [0, 1] and yields a high return H > 0, or not succed with a chance of 1 α and yields a zero return. Project returns are public information to all group members. s = 2 Borrowers simultaneously decide whether to repay their repayment R. If i members default, the other n i members will be asked to pay additional amount ir n i to the lender for their defaulting peers. If the total repayment is equal to nr or more, the group receives future financing; otherwise, the lender will exclude the entire group from future loans. These three steps are repeatedly played until the lender realizes that the borrowers are not entitled to financing for the next period, and in each period of not receiving a loan, the borrowers utility will be zero. In the first step, we assume that projects do not differ in their riskiness (i.e., α is the same for all borrowers. We also assume that the lender does not know whether projects have high or low returns, and that each borrower always invests in the same project. Two types of defaults are possible: strategic default, in which the borrower does not repay although he had high outcome H, and nonstrategic default as a result of obtaining zero outcome resulting from bad luck. We assume that the lender is unable to observe whether a borrower s 8

9 default is strategic or nonstrategic, however, borrowers are able to observe strategic defaults of their peers without any costs. Borrowers play a repeated game among themselves. They start out cooperating and continue to repay if they can. If the other borrowers do repay or default non-strategically, they also repay the remaining share of their peers. In the first step, we assume that borrowers play the grim-trigger strategy against each other. In the second step, we relax the grim-trigger assumption by assuming that if some members default strategically at some point, other group members still repay their own loans as well as the defaulting members share, but they punish the strategically defaulting members by excluding them from the lending game for T periods. The benevolent lender strives to maximize the payoff of each borrower contingent on the following. First, each borrower must be willing to accept a loan (the repayment amount must be affordable; second, each borrower must have the incentive to repay for himself and for each defaulting peer, when he is able to pay (in the worst-case, that is, if all other members default, he must be still willing to repay for the entire group; and third, the lender must break even, meaning that she must maintain a sustainable lending operation over the entire loan portfolio by charging the appropriate repayment amount. 4 Joint Liability Contract (JL In this section, we formalize the model assuming that group members play a grim-trigger strategy between themselves. In other words, group members keep repaying their loans as well as the loans of their non-strategically defaulting peers, as far as no one defaults strategically. If someone defaults strategically, other members of the group will not repay that individual s loan, nor their own loans, and no further loans are granted to the group members. The expected repayment for a borrower who plays the repayment strategy in each period t can be stated as follows: n 1 ( n 1 i i=0 α n i (1 α i ( R + i n i R = [1 (1 α n ] R. 9

10 Thus, the expected utility of a borrower who plays a repayment strategy at any period in which he obtains financing, is determined as V R JL = αh [1 (1 α n ] R + δ [1 (1 α n ] V R JL, where 0 δ 1 is the borrower s discount factor that determines his valuation of the utility of future financing. The expected utility of a repaying borrower can be rewritten as V R JL = αh R [1 (1 αn ] 1 δ [1 (1 α n. (1 ] The benevolent lender wants to maximize the lifetime utility of a borrower who plays the repayment strategy. Therefore, considering the stationary nature of the model, the lender s optimization problem can be stated as max V JL R, subject to the following. L,R 1. The stipulated repayment amount for a successful borrower cannot exceed his output (i.e., the repayment amount must be affordable even in the worst-case, that is, when projects of everyone else has failed, nr H. (2 2. Each borrower is repaying when his project is successful is a subgame perfect Nash equilibrium (SPNE if, for each successful borrower, the payoff of strategically defaulting cannot be larger than the payoff of repaying and being refinanced, H H nr + δv R JL, (3 which also implies R < δvjl R, thus guaranteeing that a successful borrower pays the repayment R when all of his partners are successful. 3. The lender must be able to sustain the lending game over periods and at least break even. Thus, the expected repayment amount of each borrower has to be at least as large as L + ɛ, where ɛ is the interest rate, R [1 (1 α n ] L + ɛ. (4 10

11 If there are some (L, R that satisfy constraints (2, (3, and (4, then the JL contract is feasible, and these constraints define its feasibility region. Note that individual lending can be considered as a special type of group lending with n = 1. Proposition 1. There exist δ JL (n, α such that: a if δ δ JL (n, α, the JL contract is feasible if and only if L L JL (n, α, H. b if δ δ JL (n, α, the JL contract is feasible if and only if L ˆL JL (n, α, δ, H. Moreover, whenever the JL contract is feasible: for any α 0, the lender demands an optimal repayment R JL = will amount to V R JL = L+ɛ [1 (1 α n ] from each borrower, and the expected lifetime utility for each borrower αh (L+ɛ 1 δ[1 (1 α n ]. Proposition 1 is suggesting that the JL contract can be feasible if and only if L F JL (n, α, δ, H, where F JL (n, α, δ, H = min {ˆLJL, L } JL = ˆL JL if δ δ JL L JL if δ δ JL, (5 and will be called the feasibility function of the JL contract from now on. Note that the feasibility function defines an upper bound for feasible loans that can be offered to borrowers under the JL contract. Thus, wherever the feasibility function increases, the maximum feasible loan increases as well. The feasibility function, and consequently the maximum feasible loan, depend positively on the borrowers discount factor, as ˆL JL is strictly increasing and L JL is constant in δ. Intuitively, borrowers who highly value receiving future loans would have a higher incentive to repay their loans; the could also be trusted to repay larger loans, which is an intuitive result. From Proposition 1, it can also be inferred that the optimal repayment R JL decreases in group size n, which is also an intuitive result, as a lager group with joint liability can offer higher repayment insurance. Therefore, a larger group can be charged less, which in turn increases the borrower s welfare VJL R. Corollary 1 is a direct result from Proposition 1 and follows from substituting n = 1. Corollary 1. Individual lending is feasible if and only if L α 2 δh. For any α 0, the lender optimally demands repayment R IL = L+ɛ α αh (L+ɛ 1 αδ.. The borrower s expected lifetime utility will be V R We continue this section determining how the feasibility of the JL contract is affected by group size and the optimal group size that results in maximum feasibility. To describe the properties of the IL = 11

12 feasibility function with respect to changes of n, in Lemma 1 we take a closer look at the changes of δ JL, ˆL, L with respect to changes of n when other parameters (α, δ, H are given. Lemma 1. Assume δ JL, ˆL JL and L JL are functions defined in Proposition 1. 1 There exists ˆδ JL (n, α such that ˆL JL is strictly decreasing in n if 0 < δ < ˆδ JL and ˆL JL is strictly increasing in n if ˆδ JL < δ < δ JL. 2 L JL (n, α, H is strictly decreasing in n. 3 ˆδ JL is strictly increasing and δ JL is strictly decreasing in n and α. Lemma 1 shows that ˆL JL, and consequently the feasibility function, increase in n for any δ (ˆδJL, δ JL only if the interval (ˆδJL, δ JL is non-empty. As we see in this lemma, for any δ that does not belong to the interval (ˆδJL, δ JL, both ˆL JL and L JL strictly decrease in n. Lemma 1 shows also that the interval (ˆδJL, δ JL becomes tighter by the increase of n or α, and it becomes wider by the decrease of n or α. Therefore, to keep the interval (ˆδJL, δ JL non-empty, the larger the α is, the smaller the n must be chosen in order to offset the contraction of the interval resulting from a large α. In general, n = 2 always provides the widest interval for any given α. In Proposition 2, we prove that for very small α, regardless of the magnitude of n, we always have ˆδ JL < δ JL, and the interval (ˆδJL, δ JL might actually be non-empty for some n. Proposition 2. Assume δ JL (n, α and ˆδ JL (n, α are functions defined respectively in Proposition 1 and Lemma 1. 1 There exists ᾱ such that: a for any α < ᾱ, feasibility of the JL contract is increasing in n [2, N α,δ ], where { N α,δ = min ˆδ 1 JL (α, δ, δ 1 JL (α, δ } ; (6 b for any α > ᾱ, maximum feasibility of the JL contract happens at n = 2. 2 For very large n, feasibility of the JL contract will be decreasing. Proposition 2 tells us that under the JL contract, if projects have a higher chance of success, the maximum loan that can be given to a group of two members is higher than for any group with n > 2; while for projects with a lower chance of success, larger loans could be given to larger groups. Proposition 2 also suggests that in the latter case, the group size cannot grow too large. 12

13 Table 1: The Range of Feasible Group Sizes for Some Given α n = 2 n = 3 n = 4 n = 5 n = 10 n = 50 α < α < α < α < α < α < Intuitively, group size has two countervailing effects. On the one hand, a larger group can provide stronger repayment insurance, handle riskier projects, and repay successfully. On the other hand, a larger group can be a threat towards the feasibility of group lending. The threat comes from the fact that each successful member is in charge of all defaulting peers, that is, if one group member succeds and everybody else fails, that only successful member is in charge of the entire group loan. Obviously, this becomes very difficult if the group is too large. Proposition 2 also presents a characterization of the optimal group size, as a function of all other parameters, Eq. (6. Using the results of Proposition 2, it could simply be verified that ᾱ is approximately 0.5. For α < 0.718, the inequality ˆδ JL < δ JL holds at least for n = 2. We can see in Table 1 that the largest α for which the inequality ˆδ JL < δ JL holds for more than one n is α < Thus, feasibility can increase in n only if α < and if the given δ is such that ˆδ JL < δ < δ JL. Figures 1 and 2 can help to derive some intuition about Proposition 2. For simplicity, it is assumed that H = 1. In Figure 1, we depict δ JL and ˆδ JL with respect to n when the chance of project success is small (e.g., α = 0.3 and when it is large (e.g., α = 0.8, respectively. As shown in Figure 1a, for α = 0.3, there are some n for which the given δ = 0.85 belongs to the interval (ˆδJL, δ {ˆδ 1 } 1 JL, and the largest of such n lies at the min JL, δ JL, i.e. at ˆδ 1 JL = = 7. Note that if the given δ is very close to 1, then the largest n lies at δ 1 JL. However, in Figure 1b, when α is large, for any n > 1, the interval (ˆδJL, δ JL is empty. In Figure 2, we depict ˆL JL and L JL with respect to n when the chance of project success is small (e.g., α = 0.3 and when it is large (e.g., α = 0.8, respectively. As shown in Figure 2a, when α is small, ˆL JL defines the boundary for the maximum feasible loan, and it reaches its maximum at n = 7. Therefore n = 7 is the group size that maximizes the feasibility of the JL contract. However in Figure 2b, when α is large, L JL defines the boundary for the maximum feasible loan, 13

14 (a It is assumed that α = 0.3. (b It is assumed that α = 0.8. Figure 1: Larger group sizes are possible as long as the discount factor belongs to the interval (ˆδ, δ. which decreases in n. Therefore, n = 2 is the optimum group size that maximizes the feasibility of the JL contract. Up to this point, we have discussed that feasibility of the JL contract can increase in group size when the chance of project success is small. However, we do not yet know if it can perform more efficiently than the IL contract. Are there circumstances under which a JL contract outperforms an IL contract? Proposition 3 proves formally that the lender can charge borrowers less under the JL contract compared to the IL contract and can still break even. The reason for this may be that no repayment is something that happens less often under the JL contract than under the IL contract. In turn, a smaller repayment amount leads to a higher welfare level for the borrowers. Proposition 3. The following statements hold when both the IL and the JL contracts are feasible. 1 The borrower s repayment amount is lower and his welfare is higher under the JL contract than the IL contract. 2 There exists α, such that: a for any α < α, the JL contract is feasible for a larger loan than the IL contract only if at least for some n, αn [1 (1 α n ] α (n 1 [1 (1 α n ] < δ < [1 (1 αn ] nα 2 ; (7 b for any α > α, the IL contract is feasible for a larger loan than the JL contract. 3 For very large n, the IL contract can always offer a larger maximum loan than the JL contract. 14

15 (a It is assumed that δ = 0.85 and α = 0.3. (b It is assumed that δ = 0.85 and α = 0.8. Figure 2: The maximum loan increases in group size as long as the discount factor belongs to the interval (ˆδ, δ. Proposition 3 shows that the JL contract has a positive effect on borrowers welfare and the repayment amount compared with the IL contract. This proves that if the chance of project success is high, the maximum loan that can be given to a borrower under the IL contract is higher than under the JL contract. For projects with a lower chance of success, larger loans could be given only under the JL contract with the condition that the group size cannot grow too large. After seeing the results of Proposition 2, the results of Proposition 3 are not surprising, since the IL contract can be mathematically interpreted as a special case of the JL contract in which n = 1. As previously explained in Proposition 2, a larger group can create the certainty of enjoying a stronger repayment insurance as well as the threat of being in charge of the total repayment. Therefore when borrowers are investing in highly safe projects, strong repayment insurance is not necessary, and loans can be given to individuals. At the same time, when borrowers invest in highly risky projects, strong repayment insurance is urgent. For α < 0.764, the Eq. (7 can be satisfied for n = 2. Thus, for α < and any δ that satisfies Eq. (7, the JL contract does better than the IL contract at least for groups of n = 2. As we can see in Table 2, for any other smaller α, more than one group size exists for which the JL contract does better than the IL contract. Therefore, α = is the maximum α for which the JL contract outperforms the IL contract if the given δ is such that Eq. (7 holds. Thus, for any α > 0.764, the lender should offer an IL contract. 15

16 Table 2: The Range of Group Sizes for Some Given α for Which the JL Contract Outperforms the IL Contract. n = 2 n = 3 n = 4 n = 5 n = 10 n = 50 α < α < α < α < α < α < (a It is assumed that α = 0.4. (b It is assumed that α = 0.9. Figure 3: For smaller α, the JL contract offers larger loans than the IL contract, while for larger α, the IL contract offers larger loans than the JL contract. Figure 3 illustrates the situation discussed in Proposition 3. For the sake of simplicity, it is assumed that H = 1. As shown in Figure 3a, when the chance of project success is small enough so that Eq. (7 is satisfied (e.g., α = 0.4, for any 0 < δ < 1, ˆL defines the boundary for the maximum feasible loan under the JL contract, which is always equal to or larger than the maximum feasible loan under the IL contract. However, when the chance of project success is large (e.g., α = 0.9, the maximum feasible loan under the IL contract is strictly higher than the maximum feasible loan under the JL contract. 5 Partial Joint Liability Contract (PJL In our basic model, we assume that repaying group members must pay the total loan to qualify for a loan in the next period, however, this constraint may be too strong. It may be the case that 16

17 repaying members could pay a portion of the total loan but not the entire total loan, while the lender s NPV of the lender is still positive. In this section, we assume that a successful member has to pay γr for each defaulting member of his group, where 0 γ 1, that is, he has to pay only a fraction of his defaulting peers repayment. Note that γ = 0 and γ = 1 respectively resemble the case in which group members are not liable and the case in which group members are fully liable for each other s repayment. However, partial liability may give rise to a new type of strategic default: collusion. Because successful members may be better off if they pretend that only one of them is successful and together pay the minimum repayment acceptable by the lender (i.e., R + (n 1 γr. As collusion can particularlly be beneficial to borrowers when the degree of joint liability is low. Thus our partial joint liability mechanism also has to guard against collusion. To tackle this problem, we assume that if total group repayment is less than nr, the lender audits the states of projects at a cost c > 0, with probability q. In a case in which the lender verifies any collusion, she excludes them from future financing. Under partial joint liability, the expected repayment for a borrower, who plays the repayment strategy, can be calculated in each period as follows: n 1 ( ( n 1 αr + α n i (1 α i i i n i γr = αr + γ [(1 α (1 α n ] R. i=0 Thus, the expected utility of a borrower, who plays a repayment strategy at any period in which he gets financing, is determined as V R P JL = αh [α + γ (1 α γ (1 α n ] R + δ [1 (1 α n ] V R JL or VP R JL = αh [α + γ (1 α γ (1 αn ] R 1 δ [1 (1 α n. ] The benevolent lender s problem in the case of partial joint liability will be max V P R JL, subject to L,R the following conditions. 17

18 1. The worst-case repayment must be affordable for each successful borrower, that is, [1 + (n 1 γ] R H (8 or R H [1 + (n 1 γ]. 2. Each successful borrower must have the incentive to choose the SPNE, that is, to repay R for himself and γr for each unsuccessful peer when his project is successful. In the case of partial joint liability, there are two incentive constraints to satisfy. First, the lender must ensure that each successful borrower is better off repaying than defaulting, that is, H H [1 + (n 1 γ] R + δv R F JL (9 or R δv R F JL [1 + (n 1 γ]. Second, the lender must also ensure that each successful borrower is better off repaying than colluding, that is, H [ 1 + (n 1 γ R + (1 q δvf R JL H 1 + iγ ] R + δvf R n i n i JL, (10 where n i is the number of successful members. Note that collusion can exist if there are at least two successful member in the group (i.e., n i 2, Eq. (10 can be simplified to R (n i qδv R F JL (n 1 i (1 γ. 3. The lender must be able to sustain the lending game over periods and at least break even. Thus, the expected repayment amount of each member has to be at least as large as L+ɛ+cq, [α + γ (1 α γ (1 α n ] R L + ɛ + cq (11 18

19 or R L + ɛ + cq [α + γ (1 α γ (1 α n ]. Clearly, if γ 1, Eqs. (8 and (9 will be equal to individual rationality and incentive constraints of the JL contract i.e., Eqs. (2 and (3. In order to solve the lender s optimization problem, two main cases should be considered. First, when the probability of auditing is high and thus collusion is unlikely. In such a case, the incentive constraint (10 is slack, and the lender needs only to satisfy the incentive constraint (9: δv R F JL [1 + (n 1 γ] (n i qδv R F JL (n 1 i (1 γ or q (n 1 i (1 γ q (n, γ, i. (12 (n i [1 + (n 1 γ] and the situation will be similar to the JL contract. Second, when the probability of auditing is small and thus borrowers are more prone to collude. For any probability of audit q < q (n, γ, i, the incentive constraint (10 is the one to be satisfied and the incentive constraint (9 is slack. Note that q (n, γ, i is decreasing in γ, that is, a higher degree of joint liability decreases the need for frequent auditing. Borrowers find it increasingly less beneficial to collude when their degree of joint liability goes higher. Proposition 4. There exist q (n, γ, i, δ P JL (n, α, γ, and δ P JL (n, α, γ, q, i such that: if q q (n, γ, i, a for any δ δ P JL (n, α, γ, the PJL contract is feasible if and only if L ˆL P JL (n, α, δ, γ, c, q, H. b and for any δ δ P JL (n, α, γ, the PJL contract is feasible if and only if L L P JL (n, α, γ, c, q, H. if q < q (n, γ, i, a for any δ δ P JL (n, α, γ, q, i, the PJL contract is feasible if and only if L L P JL (n, α, δ, γ, c, q, H, i. b and for any δ δ P JL (n, α, γ, q, i, the PJL contract is feasible if and only if L L P JL (n, α, γ, c, q, H. Moreover, whenever the PJL contract is feasible for any α 0, the lender demands optimal repayment R P JL = L+ɛ+cq [α+γ(1 α γ(1 α n ] borrower will amount to V R P JL = αh (L+ɛ+cq 1 δ[1 (1 α n ]. from each borrower, and the expected lifetime utility for each 19

20 A comparison between Proposition 4 and Proposition 1 shows that the PJL contract compared to the JL contract has disadvantages in terms of the repayment amount and the borrower s lifetime benefit. Under the PJL contract, the lender asks for a higher repayment amount to compensate for the expected cost of auditing, and this higher repayment amount results in a lower borrowers lifetime benefit. Apart from the expected cost of auditing, the degree of joint liability also influences the repayment amount. As the degree of joint liability lowers, the higher the borrower s repayment amount should be under the PJL contract. However, under the PJL contract, similar to the JL contract, optimal repayment R P JL decreases and borrowers welfare V R P JL increases in group size n, thus, enlarging the group can, at least to some extent, reverse the negative effect of the partial joint liability on the borrower s welfare and repayment amount. Proposition 4 suggests another version of the feasibility function F P JL (n, α, δ, γ, q, i = min {ˆLP JL, L P JL, L } P JL = ˆL P JL L P JL L P JL L P JL if q q and δ δ P JL if q q and δ δ P JL. (13 if q < q and δ δ P JL if q < q and δ δ P JL The feasibility function of the PJL contract, F P JL, defines an upper bound for feasible loans that can be offered to borrowers under the PJL contract. The feasibility function of the PJL contract, similar to that of the JL contract, depends positively on the borrowers discount factor, as ˆL JL and L P JL both strictly increase and L JL is constant in δ. Intuitively, borrowers who highly value receiving future loans have a higher incentive to repay their loans; they could also be trusted to repay larger loans whether offered a JL or a PJL contract. What is the effect of the degree of joint liability on the feasibility of a PJL contract? How should the group size be adjusted when partial joint liability replaces full joint liability? The answer to these questions depend on whether the probability of auditing is sufficiently high. First, assume the probability of auditing is high. This is the case where q q, and collusion is unlikely. In Lemma 2 and Lemma 3, we look at the changes in the components of the feasibility function of the PJL contract when q q, i.e. LP JL, ˆL P JL, and δ P JL, with respect to changes of group size n and degree of joint liability γ, respectively. Lemma 2. Assume q q, and assume also L P JL, ˆLP JL, and δ P JL are functions defined in 20

21 Proposition 4. 1 There exists ˆδ P JL (n, α, γ such that ˆL P JL is strictly decreasing in n if 0 < δ < ˆδ P JL, and ˆL P JL is strictly increasing in n if ˆδ P JL < δ < δ P JL. 2 L P JL and δ P JL are strictly decreasing in n. From Lemma 2 it can be inferred that when the probability of auditing is high and therefore collusion is unlikely, the maximum feasible loan that the PJL contract can handle can increase in group size n only if δ (ˆδP JL, δ P JL, which in turn requires (ˆδP JL, δ P JL to be non-empty for some n. For any δ / (ˆδP JL, δ P JL, the maximum feasible loan under the PJL contract will decrease in the group size. Lemma 3. Assume q q, and assume also δ JL, L JL, and ˆL JL are functions defined in Proposition 1, and δ P JL, L P JL, and ˆL P JL are functions defined in Proposition 4, and finally that ˆδ P JL is the function defined in Lemma 2. 1 L P JL and ˆL P JL are strictly decreasing in γ. Moreover, LP JL L JL cq and ˆL P JL ˆL JL cq. 2 ˆδ P JL is strictly increasing and δ P JL strictly decreasing in γ. Moreover, ˆδ P JL ˆδ JL and δ P JL δ JL. From Lemma 3, it can be understood that when collusion is unlikely, the maximum feasible loan that can be offered to each borrower under the PJL contract can grow larger with a lower degree of joint liability γ. Moreover, the interval (ˆδP JL, δ P JL becomes looser by the decrease of the degree of joint liability γ, that is, with a lower degree of liability, the circumstances for offering the maximum feasible loan to borrowers become easier. In addition, larger groups can be formed. To gain an intuition, consider a situation in which only one member of the group is successful and all the others have failed in their projects. If the loan size is large, it may happen that this only successful member cannot afford to repay a large portion of the total repayment of his group on top of his own repayment, however, he can afford to repay a smaller portion of the total repayment of his group. When the degree of joint liability is too high, such a borrower may refuse repaying even for himself; with a lower degree of liability, the successful member repays his own loan obligation plus a small portion of the entire group loan. Therefore, repaying larger loans and being a member of a larger group is less risky for moderately liable borrowers compared to highly liable borrowers. 21

22 In short, when the lender audits states of projects with high probability, and thus collusion is less likely, the PJL contract has advantages compared to the JL contract. The advantages of the PJL contract are in terms of the larger maximum loan that can be offered to borrowers and the more relaxed circumstances necessary for offering such a loan. The PJL contract is feasible for a larger set of parameters compared to the JL contract, as larger groups are possible under the PJL contract. Now assume that the probability of auditing is not high. This is the case in which q < q and collusion is likely. In Lemma 4 and Lemma 5, we look at the changes in the components of the feasibility function of the PJL contract (i.e., L P JL, L P JL, and δ P JL with respect to changes of group size n and degree of joint liability γ, respectively. Lemma 4. Assume q < q and assume also L P JL, L P JL, and δ P JL are functions defined in Proposition 4. 1 There exists ˆδ P JL (n, α, γ, i such that L P JL is strictly decreasing in n if 0 < δ < ˆδ P JL, and ˆδ P JL is strictly increasing in n if ˆδ P JL < δ < δ P JL. 2 L P JL and δ P JL are strictly decreasing in n. A comparison between Lemmas 4 and 2 shows that the general behavior of the feasibility function of the PJL contract in response to changes of n is the same whether the probability of auditing is high or low. Lemma 2 shows that when the probability of auditing is not high enough, and therefore borrowers are more prone to collude, the maximum feasible loan that the PJL contract can handle can increase in group size n only if δ (ˆδ P JL, δ P JL, which requires (ˆδ P JL, δ P JL to be non-empty for some n. For any δ / (ˆδ P JL, δ P JL, the maximum feasible loan under the PJL contract will decrease in the group size. Lemma 5. Assume q < q and that δ JL, L JL, and ˆL JL are functions defined in Proposition 1, and δ P JL, L P JL and L P JL are functions defined in Proposition 4 and finally that ˆδ P JL is the function defined in Lemma 4. 1 L P JL is strictly decreasing and L P JL is strictly increasing in γ. Moreover, L P JL L JL cq and L P JL ˆL JL cq. 2 ˆδ P JL and δ P JL are both strictly decreasing in γ. Moreover ˆδ P JL > ˆδ JL and δ P JL > δ JL. 22

23 Comparing Lemmas 5 and 2, we can see that when the probability of auditing is low but the borrowers discount factor is large, the maximum feasible loan under the PJL contract can still increase with a decrease of the degree of joint liability. However, when the probability of auditing is low and the borrowers discount factor is also low, the maximum feasible loan under the PJL contract should decrease with a decrease of the degree of joint liability. The reason for this is that the larger borrowers discount factor can compensate for the low probability of auditing and make the strategic default less interesting to the borrowers. Lemma 5 also shows that when the probability of auditing is low, the interval (ˆδ P JL, δ P JL shifts increasingly more to the right as the degree of joint liability γ decreases, that is, when the probability of auditing is low, borrowers with a lower degree of joint liability should have a higher discount factor in order to qualify for the maximum feasible loan. Intuitively, when a lack of sufficient auditing exists, borrowers with high valuation for future loans can be convinced not to default strategically by means of dynamic incentives, i.e., deprivations from future loans. In summary, when the lender audits the states of projects with a low probability and thus a higher chance of collusion, the PJL contract can still have an advantage compared to the JL contract, if borrowers have a higher valuation for future loans. In this case, the advantage of the PJL contract is in terms of the larger maximum loan that can be offered to borrowers. However, the circumstances necessary for offering such a loan are more difficult to satisfy, as borrowers must have a higher discount factor to qualify. 6 Flexible Joint Liability (FJL Contract The grim-trigger strategy that group members played against each other under the JL contract may be too harsh. Consider a situation in which some borrowers default strategically, but it is still beneficial for the other group members to take care of the entire repayment and thus entitle to a new loan in the next round. Moreover, strategically defaulting members may have good explanations, and the other group members may want to punish them less severely. In this section, we examine joint liability under a strategy that is less severe than the grim-trigger. Players start in the lending phase and cooperate until someone defaults strategically. They then go to the punishment phase and exclude the defaulter for T periods of receiving loans. Later, when punishment is served, 23