Group lending, local information and peer selection 1

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1 Journal of Develoment Economics Ž. Vol Grou lending, local information and eer selection 1 Maitreesh Ghatak ) Deartment of Economics, niõersity of Chicago, Chicago, IL 60637, SA Abstract This aer analyzes how grou lending rograms use joint liability to utilize local information that borrowers have about each other s rojects through self-selection of grou members in the grou formation stage. These schemes are shown to lead to ositive assortative matching in grou formation. Faced with the same contract, this makes the effective cost of borrowing lower to safer borrowers: because they have safer artners, conditional on success their exected dues to the lender are lower than that of riskier borrowers. The resulting imrovement in the ool of borrowers is shown to increase reayment rates and welfare. q 1999 Elsevier Science B.V. All rights reserved. JEL classification: D8; L14; 01; 017 Keywords: Grou lending; Local information; Peer selection 1. Introduction Recent research on rural credit markets in develoing countries has focused on imerfect information and transaction costs in the lending rocess as the key to understand the reorted henomena of high interest rates, market segmentation and credit rationing. This has, led to a greater areciation of the fundamental disadvantages faced by formal lending institutions Že.g., the commercial banking ) [email protected] 1 This aer is based on the first chater of my PhD thesis entitled Essays on the Economics of Contracts submitted to Harvard niversity Ž June, that was circulated earlier as the working aer Grou Lending and the Peer Selection Effect Ž November, See Hoff and Stiglitz Ž for a review of the recent theoretical and emirical literature r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž. PII: S

2 8 ( ) M. GhatakrJournal of DeÕeloment Economics sector and government lending agencies. in this market owing to the costliness of screening loan alicants, monitoring borrowers, and writing and enforcing contracts due to imerfections in the judicial system, backward infrastructure Že.g., transort and communication., and low levels of literacy Ž Besley, It has also rekindled interest in the role of alternative institutional arrangements, such as grou-lending rograms, credit cooeratives, and rotating saving and credit associations, to overcome these roblems. In this aer we focus on grou-lending rograms under which borrowers who cannot offer any collateral are asked to form small grous. Grou members are held jointly liable for the debts of each other. 3 Formally seaking, what joint liability does is to make any single borrower s terms of reayment conditional on the reayment erformance of other borrowers in a re-secified and self-selected grou of borrowers. The remarkably successful exerience of some recent grou-lending rograms in terms of loan recovery rates, such as those in Bangladesh, Bolivia, Malawi, Thailand and Zimbabwe, has aroused a lot of interest in relicating them in other countries Ž Hui and Feder, A careful examination of the existing evidence on the relative erformance of these rograms comared to standard lending rograms across different countries yields a mixed icture ŽMorduch, 1998; Hui and Feder, Still, grou lending rograms where loans were made to homogenous self-selected grous of individuals belonging to the same village and with similar economic standing have tended to be more successful than others Ž Hui and Feder, We rovide a theory based on two contractual features of grou lending rograms to exlain why they can otentially achieve high reayment rates desite the fact that borrowers are not required to ut in any collateral: the existence of joint liability and the selection of grou members by borrowers themselves. As mentioned above, screening otential loan alicants is a costly activity for the lender. At the same time, borrowers from the same locality are exected to have some information about each other s rojects. Therefore, one way of looking at contracts based on self-formed grous is that they are a means of deliberately inducing borrowers to select their grou members in a way that exloits this local information. We use a simle adverse selection model to analyze this issue. In our model the borrowers know each others tyes, namely the robability of success of their 3 See Ghatak and Guinnane Ž for a discussion of how joint liability works in ractice. Credit cooeratives, which differ from grou lending rograms in that they borrow from outside sources as well as raise deosits from its own members, too often have some degree of joint liability. For examle, in German credit cooeratives, which first aeared in the middle of the last century and soon became very successful and influenced cooerative design everywhere else, all members of the cooerative were liable in whole or in art for any loan from an outside source on which a cooerative member defaulted. See Guinnane Ž

3 ( ) M. GhatakrJournal of DeÕeloment Economics rojects, but the outside bank does not. At the same time, collateral cannot be used because of the overty of the borrowers. This means loans have to be offered to all borrowers at the same nominal interest rate. Then, as in the lemons model of Akerlof Ž 1970., the resence of enough risky borrowers can ush the initial equilibrium interest rate high enough to drive the safe borrowers away from the market. We show that the joint liability asect of grou lending rograms induces borrowers of the same tye grou together in equilibrium. Given ositive assortative matching in grou formation, the effectiõe borrowing costs facing risky and safe borrowers are no longer the same. Conditional on success, a risky borrower faces a higher exected borrowing cost than a safe borrower because her artners are more likely to have failed. But this is recisely what a full-information credit contract would like to do borrowers with riskier rojects, because they succeed less often should ay more when they succeed. Facing a more favorable effective rate of interest, safer borrowers are shown to be attracted back into the market. This reduces the equilibrium interest rate, leads to an imrovement of the ool of borrowers, and increases the average reayment rate. Also, by attracting in safer and roductive rojects, which were not initially in the borrower ool as a result of the lemons roblem, joint liability imroves welfare from the oint of view of aggregate social surlus. Interestingly, even though riskier borrowers are burdened with higher exected joint-liability ayments because they have riskier artners, the overall decrease in the interest rate ermitted by the entry of safe borrowers maybe significant enough to imrove the welfare of all tyes of borrowers in the ool. Hence we show that by exloiting an intangible resource, namely local information, that is embodied in secific social networks the institution of joint liability based grou lending can alleviate credit market failures. Hence, it serves the objectives of both efficiency and equity by heling the oor escae from the tra of overty by financing small-scale roductive rojects. 4 We examine one ossible mechanism through which grou lending can imrove efficiency based on the self-selection of borrower grous and the effect on the ool of borrowers. The existing research on the toic, until very recently, has exlored other mechanisms focusing mainly on the effect of joint liability on the 5 behaõior of individual borrowers. Early work by Stiglitz Ž and Varian Ž exlore how joint liability may induce borrowers in a grou to monitor each other, thereby alleviating moral hazard roblems. Besley and Coate Ž address the question how joint-liability contracts affect the willingness to reay. They show how they may induce borrowers to ut eer ressure on delinquent 4 For an analysis of how an economy may get stuck in a overty tra due to credit market imerfections see Banerjee and Newman Ž See Ghatak and Guinnane Ž for a more detailed discussion.

4 30 ( ) M. GhatakrJournal of DeÕeloment Economics grou members, which may lead to an imrovement in reayment rates. However, none of these aers with the excetion of Varian Ž examine a crucial feature of these schemes, namely that grou members self-select each other. Varian Ž rooses a model where the bank directly screens loan alicants and joint liability takes the following form: if the member who is interviewed turns out to be a bad risk all grou members are denied loans. This induces safe borrowers to undertake the task of screening out bad risks on behalf of the bank. In contrast, we show that joint liability lending can imrove efficiency even if there is no direct screening so that risky borrowers too can form a grou and aly for a loan. Because borrowers are shown to end u with artners of the same tye, for the same joint liability contract offered to all borrowers, safer borrowers face lower exected borrowing costs conditional on success. Aart from the current aer, a number of recent aers have studied various roles grou lending can lay in alleviating adverse selection roblems in rural credit markets. Among them, Van Tassel Ž has analyzed grou lending in a similar informational environment and has obtained some similar results on its effect on the formation of grous and reayment rates. However, our aers differ in terms of the model in several resects. For examle, we allow for a general distribution of borrower tyes and arbitrary grou sizes, while Van Tassel allows for variable loan sizes. Armendariz de Aghion and Gollier Ž is another aer that looks at a similar environment and shows that joint liability can imrove the ool of borrowers if borrowers have erfect knowledge of their artners. However, their aer does not formalize the grou formation game. Also, it does not exlore the otimal degree of joint liability Ž by assuming full joint liability. or the welfare imlications of grou lending. On the other hand, their aer, and more recently that of Laffont and N Guessan Ž 1999., address a question we do not consider at all namely, whether grou lending can imrove efficiency in environments with adverse selection where borrowers do not necessarily have better information about each other.. The economic environment We take a simle version of the standard model of a credit market with adverse selection. 6 Everyone lives for one eriod. Borrowers are risk neutral and are endowed with a risky investment roject that needs one unit of caital and one unit of labor. There is no moral hazard and agents suly labor to the roject inelastically. But they have no initial wealth and need to borrow the required amount of caital to launch their roject. 6 Ž. See Stiglitz and Weiss 1981.

5 ( ) M. GhatakrJournal of DeÕeloment Economics A borrower is characterized by the robability of success of her roject, gw, 1x where )0. The return of a roject of a borrower of tye is a random variable y which takes two values, RŽ.)0 if successful, and 0 if it fails for all gw, 1 x. The outcome of a roject is binary random variable, xgs, F4 where S denotes success and F denotes failure. The outcomes of the rojects are assumed to be indeendently distributed for the same tyes as well as across different tyes. 7 Borrowers of all tyes have an exogenously given reservation ayoff u which can be thought of as the market wage rate. We assume that the tye of a borrower is rivate information so that lenders cannot distinguish between different tyes of borrowers. However, borrowers know each other s tyes. This informational environment is fundamental to our model and it may be helful to think of lenders as institutions external to the village Ž e.g., city-based. whereas borrowers are all residents of the same village. The outcome of a roject, i.e., whether it has succeeded or failed, is costlessly observable by the bank and is verifiable as well. But the return of a roject, i.e., how much it yields if successful, is not observed by the bank. Hence, lenders can use only outcome-contingent contracts such as debt contracts and not return-contingent contracts, such as equity. 8 Borrowers have no wealth they can offer as collateral and moreover non-monetary unishments are ruled out by a limited-liability constraint. We assume that enforcement costs are negligible once the bank receives the verifiable signal that a borrower s roject has been successful, the borrower cannot default. 9 We are going to focus on two tyes of credit contracts in this environment: indiõidual-liability contracts and joint-liability contracts. The former is a standard debt contract between a borrower and the bank with a fixed reayment r in the non-bankrutcy state Ž here x s S., and maximum recovery of debt in the bankrutcy state Ž xsf. which haens to be 0 in our model. The latter involves asking the borrowers to form grous of a certain size, and stiulating an individual liability comonent r, and a joint liability comonent, c. As in standard debt contracts, if the roject of a borrower fails then owing to the limited-liability constraint, she ays nothing to the bank. But if a borrower s roject is successful, then aart from reaying her own debt to the bank, r, she has to ay an additional 7 Elsewhere Ž Ghatak, we study the imlications of relaxing this assumtion. 8 In Ghatak and Guinnane Ž 1999., we show how this could be derived from an underlying costly state verification model Ž Townsend, The roblem of enforcement is undoubtedly of great ractical imortancein lending to the oor Ž because of limited sanctions against strategic default., and existing research Že.g., Besley and Coate, has shown how grou lending may alleviate this roblem. We make this assumtion to focus on the effect of joint liability on the selection of borrowers which is admittedly only one of several ossible channels through which grou lending can imrove reayment rates, but one which has been largely neglected in the literature so far.

6 3 ( ) M. GhatakrJournal of DeÕeloment Economics joint-liability ayment, c, er member of her grou whose rojects have failed. Thus, unlike standard debt contracts, reayment is not fixed in non-bankrutcy states: it is contingent on the roject outcomes of a re-secified set of other borrowers. 10 We model the lending side of the credit market as one where there is a single risk neutral bank that chooses the terms of the loans in order to maximize exected aggregate surlus subject to a zero rofit constraint and the relevant informational constraints. As a social welfare function, exected aggregate surlus can be interreted as the ex ante exected utility of the borrower before she knows her tye. 11 The bank can be thought of as a ublic lending institution or a non-governmental organization Ž NGO. which is most often the case for observed grou lending schemes. We assume that the bank faces a erfectly elastic suly of funds from deositors at the safe rate of interest, r. We model grou lending as the following sequential game: first, the bank offers a contract secifying the interest rate, r, and the amount of joint liability, c, to the borrowers; second, borrowers who wish to accet the contract select their artners; finally, rojects are carried out and outcome-contingent transfers as secified in the contract are met. Borrowers who choose not to borrow enjoy their reservation ayoff of u. 3. Equilibrium in the grou formation game In this section we study the grou formation game under grou lending. For simlicity of exosition we consider grous of size in the aer. In Aendix A, we show how all our results generalize to grous of any size ng. We require the equilibrium in the grou formation game to satisfy the otimal sorting roerty Ž Becker, 1993.: borrowers not in the same grou should not be able to form a grou without making at least one of them worse off. 1 Our main 10 While there are some similarities between standard debt contracts with a cosigner and joint-liability contracts used in grou-lending, there are two imortant differences: in the latter case the artner who can be viewed as a cosigner does not have to be an individual who is known to the bank andror owns some assets, and all members of the grou can be borrowers as well as co-signors on each other s loans at the same time. 11 Changes in social welfare can be measured by changes in aggregate surlus for any social welfare function when references are quasi-linear so long as the lanner can make lum sum transfers. Although borrowers are risk-neutral in our context and hence their references are quasi-linear, since there is rivate information, lum sum transfers across borrower tyes are not feasible. Hence, aggregate exected surlus is no longer a valid welfare measure for any social welfare function. 1 The size of a grou that qualifies for a loan under a grou lending rogram is fixed by institutional design. In game theoretic terms, an assignment satisfying the otimal sorting roerty is in the core given this restriction on the size of ossible coalitions.

7 ( ) M. GhatakrJournal of DeÕeloment Economics result is that for any given joint-liability credit contract Ž r,c. offered by the bank in the first stage, borrowers will choose artners of the same tye in the second stage. Consider a borrower with robability of success. The exected ayoff of this borrower under a given joint-liability contract Ž r,c. when her artner has robability of success X is: E X Ž r,c. s X RŽ. yr qž 1y X. RŽ. yryc., Ž. Ž. srž. yrqcž 1y X. 4. We establish the following imortant roerty of joint liability: Lemma 1: A borrower of any tye refers a safer artner, but the safer the borrower herself, the more she Õalues a safer artner. Proof: The difference in the exected ayoff of a borrower of tye from having a artner who has robability of success X instead of Y is E X r,c ye Y r,c sc X y Y, Ž., Ž. Ž.. Ž 1. Suose X ) Y. In choosing between two otential artners with different robabilities of success X and Y, any borrower will be willing to ay a strictly ositive amount to have the borrower whose robability of success is X. But the maximum amount a borrower of tye is willing to ay to have a artner of tye X Y X over a artner of tye, cž y Y., is increasing in her own robability of success. B The intuition is as follows: conditional on her own roject being successful, the maximum amount a borrower of any tye would be willing to ay to have a artner who is safer than her existing artner is the amount of joint liability times the difference in the resective robabilities of not defaulting. 13 But this exected gain from having a safer artner is realized only when the borrower herself is successful, and hence is higher the safer her tye. Let us assume that the oulation of borrowers is balanced with resect to grou size, i.e., there are NŽ. borrowers of each tye, where NŽ. is a ositive integer. This ensures that any borrower can always find another borrower 13 Since we assume borrowers have no wealth that can be used as collateral, when we talk about side ayments among borrowers, we mean that these transfers take forms that are not feasible with the bank. For examle, borrowers within a social network can make transfers to each other in ways that are not ossible with an outsider Ž namely, the bank., such as roviding free labor services, or writing contracts based on the outut Ž as oosed to outcome. of their rojects.

8 34 ( ) M. GhatakrJournal of DeÕeloment Economics of the same tye to form a grou. In this case, we rove the following result regarding the equilibrium in the grou formation stage: Proosition 1: If the oulation of borrowers is balanced with resect to grou size, the unique assignment satisfying the otimal sorting roerty under groulending schemes based on joint liability is one where all borrowers in a giõen grou haõe the same robability of success. Proof: Start with the assignment where all grous are erfectly homogeneous. Consider the ossibility that a risky borrower might try to induce a safe borrower to be her artner by offering a side ayment. By Lemma 1, the exected gain to a borrower of tye X from leaving a artner of the same tye and having a artner X of tye where -, namely, c X Ž y X., is less than the exected loss to a borrower of tye from leaving a artner of the same tye and having a artner X of tye,namely, cž y X.. Hence, a mutually rofitable transfer from a borrower of a riskier tye to a borrower of a safer tye to induce the latter to form a grou with the former does not exist and the initial assignment satisfies the otimal sorting roerty. Conversely, start with an assignment where all grous are not erfectly homogeneous and suose it satisfies the otimal sorting roerty. Within the set of all mixed grous, consider the subset of grous which have one borrower of the highest tye, namely, 1. Since the oulation of borrowers is balanced with resect to grou size, for every borrower of tye 1 in a mixed grou with a artner of tye -1, there will be another borrower of tye 1 in a mixed grou with a artner of tye X -1. By Lemma 1, if the two borrowers of tye 1 leave their existing artners and match together, their existing artners will not find it rofitable to induce them to remain by offering side ayments. Reeating this argument within the set of all remaining mixed grous iteratively, we comlete the roof that only erfectly homogeneous grous satisfy the otimal sorting roerty. B Intuitively, because a borrower with a high robability of success lace the highest value on having a artner with a high robability of success, they bid the most for these borrowers. As a result, borrowers of the same robability of success are matched together, just as artners of similar quality of are matched together in Becker s marriage model or to take a more recent examle, workers of the same skill are matched together in firms when they have Kremer s O-Ring roduction function. 14 The underlying force driving the ositive assortative matching result is also similar in these models: the tyes of agents are comlementary in the ayoff functions See Becker Ž 1993., Cha. 4, and Kremer Ž For examle, in our model ŽE E X Ž r,c.. ržee X. sc)0.,

9 ( ) M. GhatakrJournal of DeÕeloment Economics Notice that our roof uses only the fact that borrowers have different robabilities of success and does not deend on whether safe and risky borrowers have the same or different exected roject returns Ži.e., we make no assumtions about RŽ... In Aendix A we show that it also does not deend on whether borrowers have some wealth or not. However, if the oulation distribution of borrowers is not balanced with resect to grou size that requires some modifications to this result. In Aendix A we analyze this case. 4. Credit market equilibrium with adverse selection Let us assume now that there is a continuum of borrowers with robability of success gw,1x where ) 0 following the continuously differentiable density function of the robability of success gž.. The corresonding distribution function is denoted by GŽ.'H gž s. d s for gw,1 x. The size of the total oulation of borrowers in the village is normalized to unity. Following Stiglitz and Weiss Ž we assume that rojects have the same mean and differ only in terms of riskiness Žin the sense of second-order stochastic dominance.: 16 RŽ. sr for all g,1 Assumtion 1 We assume that the rojects of borrowers are socially roductive in terms of exected returns given the oortunity costs of labor and caital: R)rqu. Assumtion 4.1. Lending with indiõidual liability We model lending with individual liability as the following sequential game: the bank moves first and announces an interest rate, r. Borrowers who wish to borrow at the interest rate r do so, rojects are carried out and outcome-contingent transfers as secified in the contract are met. Borrowers who choose not to borrow enjoy their reservation ayoff of u. As a benchmark, consider the case where the bank has full information about a borrower s tye. It can then offer loan contracts under which a borrower whose robability of success is ays the full-information interest rate r s rr when her roject succeeds, and nothing otherwise Ž by limited liability.. Since safe borrowers reay their loan more often they are charged a lower interest rate than risky tyes. Given this contract, the bank earns zero exected rofit er loan, all 16 Ž. Elsewhere Ghatak, 1999 we study the imlications of relaxing this assumtion on the distribution of roject returns.

10 36 ( ) M. GhatakrJournal of DeÕeloment Economics tyes of borrowers borrow in the second stage and hence aggregate exected surlus is maximized. If the bank cannot identify a borrower s tye then charging searate interest rates to different tyes of borrowers would not work. A risky borrower would have an incentive to retend to be a safe borrower and ay a lower interest rate, but the bank would not be able to break even if all tyes of borrowers borrow at that rate. It, therefore, has to offer the same interest rate to all borrowers given the absence of collateral or any other screening instrument in our environment. The exected ayoff to borrower of tye when the interest rate is r is E Ž r. sryr, g w,1 x. Let denote the robability of success of the marginal borrower, i.e., one who is indifferent between borrowing and not. nder lending with individual liability, at the interest rate r the marginal borrower has a robability of success given by: Ryu s ˆ. Ž. r It follows that d Ryu sy -0. Ž 3. dr r Intuitively, borrowers who strictly refer to borrow at the interest rate r must have Ryr)u or, -. ˆ That is, they are riskier than the marginal borrower. This is a consequence of the fact under a debt contract borrowers of all tyes ay nothing if the roject fails and ay the same nominal interest rate r when their roject is successful. As a result, riskier borrowers face a lower exected interest rate and so an increase in r always reduces. ˆ Let denote the average robability of success in the ool of borrowers who choose to borrow at the interest rate r. In this case: H sg Ž s. d s s GŽ. Ž 4. Since all borrowers who strictly refer to borrow at the interest rate r are riskier than the marginal borrower, ). ˆ Ž 5. Moreover, differentiating Eq. Ž. 4 with resect to r and using Eqs. Ž. 3 and Ž. 5 we get: d g ˆ Ž. d s 1y -0 dr GŽ. dr ž /

11 ( ) M. GhatakrJournal of DeÕeloment Economics i.e., an increase in the interest rate reduces the aõerage robability of success in the ool of borrowers. The bank s objective is to choose an interest rate that maximizes exected aggregate surlus subject to the constraint that its exected rofit er loan is zero. Since the bank s rofits are negative for r-0, we will restrict our attention to non-negative values of r. If it offers an interest rate r in the first stage, it anticiates an average robability of reayment of er loan given the exected ool of borrowers in the second eriod. The equilibrium of the lending game with individual liability contracts is an interest rate r which is the outcome of the bank s otimization roblem: max rg0 H Ž Ryrs. g Ž s. d s s.t. ryrs0. Let VŽ r.'h Ž Ryrs. gž s. d ssž Ryr. GŽ. and Ž r.'ryr. Substitut- ing rsr from the zero rofit condition into the bank s objective function we get VŽ r.' Ž Ryr. GŽ ˆ.. As a result, the bank s otimization roblem can be reformulated as one where it chooses the minimum ossible value of r Žwhich ensures the maximum ossible level of and hence VŽ r.. subject to the zero rofit condition: r sminrg0: Ž r. s04 Therefore, by definition Ž r. s0. Ž 6. Let and denote the robability of success of the marginal borrower and the average robability of success of the borrower ool at rsr. Let m denote the unconditional mean of the robability of success in the borrower oulation: H 1 m' sg Ž s. d s. Now we are ready to rove: Proosition : An equilibrium interest rate r exists and is unique under lending with indiõidual liability for all arameter Õalues satisfying Assumtions 1 and. If in addition r) Ž R y u. m, then indiõidual liability achieões a lower leõel of exected reayment rate and aggregate surlus comared to full information. HoweÕer, for rfž Ryu. m indiõidual liability achieões the same exected reayment rate and aggregate surlus as under full information.

12 38 ( ) M. GhatakrJournal of DeÕeloment Economics X Proof: If the bank charges the interest rate r sž Ryu. r then only borrowers with the lowest robability of success will be willing to borrow. Now lim Ž r. sr X lim yr X r r X r r X H sg Ž s. d s sr lim ˆ GŽ. yr Alying L Hoital s rule we have lim ŽŽH sgž s. d s. ržgž... ˆ ˆ s. Hence, X X lim X Ž r. r r sr yrsryuyr)0 by Assumtion. Also, for r)r, no one borrows so that VŽ r. s0 and the zero rofit condition is satisfied trivially. On the other hand, lim Ž r. syr- 0. Since Ž r. is continuous in r Ž r 0 which follows from the fact that gž. and are continuous functions. there exists at least one value of r gž0, r X. and corresondingly gž,1. such that Ž r. s 0. By the continuity of Ž r., the set of values r satisfying Ž r. s0 is closed and bounded and so r s minr G 0: Ž r. s 04 exists and is unique. Since all borrowers of tye - borrow at this interest rate and so VŽr.)0. Hence, an equilibrium exists and is unique. Since Ž. 0-0, and by definition r is the lowest value of rgw0,r X x satisfying Ž r. s0, it must be the case the exected rofit of the bank should be ositively sloed with resect to r at r s r. That is, using Eqs. Ž. and Ž. 4, and the zero-rofit condition Žwhich yields s rrr.: X Ž r. s qr d dr rsr Ž. Ž. g Ryuyr s 1y )0 Ž 7. GŽ. r where X Ž r.'d Ž r. rdr. We are ruling out the degenerate case where Ž r. is tangent to the horizontal axis at rsr, i.e., the ossibility that X Žr. s0. If condition Ž. 7 is not satisfied, the bank can cut the interest rate which will attract in more safe borrowers Ž thereby raising aggregate exected surlus. and make ositive rofits, which is inconsistent with equilibrium. Together, the conditions Ž. 6 and Ž. 7 comletely characterize the equilibrium under lending with individual liability. Notice that if Ž r.)0 for rsryu, then s1. ˆ In that case, in equilibrium, r will go down such that Ž r. s0 and will continue to remain at 1. In this case, the equilibrium under individual lending will achieve the first-best. On the other hand if Ž Ryu. sž Ryu. myr-0 then -1 and -m and the exected reayment rate and aggregate surlus will be less than their first-best levels. B

13 ( ) M. GhatakrJournal of DeÕeloment Economics If borrowing costs are low relative to roject returns less of labor costs, the roblem of adverse selection does not bind and the exected reayment rate and aggregate surlus under individual liability lending are at their first-best levels. However, safe borrowers cross-subsidize risky borrowers and hence are worse off comared to a full information environment. In such a situation, there will be no reason to exlore alternative mechanisms to imrove efficiency, such as grou lending. However, if borrowing costs are relatively high, then some socially rofitable rojects will not be carried out in equilibrium because of the informational roblem. Hence, reayment rates and social surlus will be lower than under full information. 17 This brings us to Section 4. where we consider whether grou lending schemes based on joint liability can achieve an imrovement in such a situation. 4.. Lending with joint liability Ž. The bank s objective is to choose a joint liability contract r,c that maximizes exected aggregate surlus subject to the constraint that its exected rofit er loan must be zero. There is an additional constraint that needs to be taken into account in this case: since borrowers do not have any wealth by assumtion, the sum of individual and joint liability ayments, r q c, cannot exceed the realized revenue from the roject when it succeeds for all tyes of borrowers that choose to borrow under that contract. 18 Formally, there is a limited liability constraint that requires: rqcfrž. for all F Ž. Ž. Since R sr by Assumtion 1, R is decreasing in. Hence this condition will be satisfied if rqcfrž. Ž 8. where is the robability of success of the marginal borrower. 17 This is the well-known under-investment result of Stiglitz and Weiss Ž in credit markets with adverse selection. Notice that we abstract from the ossibility of equilibrium credit rationing, another imortant result in the literature, by assuming a erfectly elastic suly of loanable funds at the given safe rate of interest r. The bank chooses a unique r corresonding to r and at that value of r the suly of loans is erfectly elastic and so the demand for loans at r equal to GŽŽr.. can be fully satisfied. 18 This constraint aears under lending with individual liability as well, since the bank cannot receive transfers from a borrower if her roject fails.

14 40 ( ) M. GhatakrJournal of DeÕeloment Economics For any joint liability contract Ž r,c. offered in the first stage by the bank, by Proosition 1, the ayoff of a borrower whose robability of success is can be written as E, Ž r,c. sryrycž 1y.. For the marginal borrower now we have: Ryrqc ˆ ˆŽ 1y. 4su Ž 9. Solving r from the indifference condition of the marginal borrower, we get: Ryu rs ycž 1y. Ž 10. The equilibrium of the grou lending game with joint liability is an interest rate r and an amount of joint liability c that solves the bank s otimization roblem: H max VŽ r,c. s RyrsycŽ 1ys. s gž s. d s rg0,cg0 Ž. subject to the new zero-rofit condition: H H sg Ž s. d s sž 1ys. gž s. d s r qc yrs0 Ž 11. GŽ. GŽ. and the limited liability constraint Ž. 8. Again, substituting the zero rofit condition into the bank s objective function, we get VŽ r,c.' Ž Ryr. GŽ Ž r,c... Hence the bank s otimization roblem can be reformulated as one where it chooses Ž r,c. so as to achieve the maximum ossible level of subject to the zero rofit condition and the limited liability constraint. Substituting Eq. Ž 10. in the zero-rofit condition and simlifying, we get ž / H H sg Ž s. d s sž ys ˆ. gž s. d s Ryu qc yrs0. Ž 1. GŽ. GŽ. Also, using Assumtion 1 and Eq. Ž 10. the limited liability constraint Ž 8. reduces to u cf. Ž 13. The bank s otimization roblem in this case is to choose a value of c such that is as high as ossible subject to Eqs. Ž 1. and Ž 13.. Given the values of c and, ˆ the value of r is determined by the condition Ž 10..

15 ( ) M. GhatakrJournal of DeÕeloment Economics Proosition 3: An equilibrium Žr, c. exists under lending with joint liability with c )0 for all arameter Õalues satisfying Assumtions 1 and. If r) Ž R y u. m the equilibrium with joint liability is unique and the aõerage reayment rate and aggregate exected surlus are both strictly higher relatiõe to the Ž equilibrium with indiõidual liability. If in addition Rmyu s qm. Gr) ŽRy u. m joint liability achieões the same reayment rate and aggregate exected surlus as under full information. Ž. Proof: First we consider the case where r) Ryu m so that individual liability lending does not achieve the first best. For cs0, the analysis of the revious section alies and s ˆ is the highest value of that satisfies Eq. Ž 1. by Proosition. Notice that for s 1, 1 Ž. Ž. Ž. Ž. Ž. Ž. H s ys ˆ g s d srg sh s 1ys g s d ssmy s qm )0 where s X is the Ž unconditional. variance of. Then consider the value of csc 'ryž Ry. Ž X u mrmy s qm.. It is readily checked that c and s1 ˆ satisfy Eq. Ž 1.. We X need to check if c satisfies Eq. Ž 13.. This requires rfrmyuž s qm.. 1 We are considering the case r)rmyum here. Also H sž 1ys. gž s. d ssm Ž. Ž y s qm )0. So Rmyu s qm.)rmyum and there exist arameter values that satisfy both r)rmyum and the above condition. For such arameter values, the unique equilibrium under lending with joint liability is c sc X, s1 and from Eq. Ž 10., r sryu. This achieves the same reayment rate and aggregate surlus as under full information. Ž However, if r) Rm y u s q m., then an equilibrium, if it exists, must involve 0Fc-c X. We show that cs0 cannot be an equilibrium. Totally differentiating the zero rofit condition Ž 1. with resect to c we get H y sž y ˆ s. gž s. d s E s Ec H sgž s. d s H s gž s. d s Ry u g ˆ Ž. c g ˆ Ž. g ˆ Ž. y 1 y y1 q 1 y q GŽ. ž / GŽ. ž GŽ. / GŽ. GŽ. As c aroaches 0 from above tends to ˆ, tends to and g ˆ Ž. rgž. Ž. Ž. Ž. tends to g rg ˆ. Also, lim H s g s d s s H s gž s. c 0q d s which exists and is ositive. Hence H ) ˆ Ž. Ž. y s ys g s d s E lim s )0. Ž 14. Ryu gž. Ž Ryuyr. y 1y Ž. GŽ. r c 0q Ec

16 4 ( ) M. GhatakrJournal of DeÕeloment Economics w Ž. Ž.Ž..x Ž. Ž as lim 1 y g ˆ rg r ˆ y 1 s 1 y g rg.ž r y 1. c 0q s1y gž. rgž.ž Ryuyr. rr)0 from Eq. Ž 7.. At the same time, for arbitrarily small values of c, the limited liability constraint Ž 13. is always satisfied as is bounded below by ˆ. By the continuity of gž. and GŽ. the set of values of Ž c,. that satisfy Eq. Ž 1. is closed and bounded. Also, for gw ˆ,1 x, the set of values of c satisfying Eq. Ž 13. is closed and bounded as well. Hence, Ž even when the condition r F Rm y u s q m. is not satisfied so that the first-best is not attainable under joint liability, a unique equilibrium c gž0, c X. exists such that has the highest value and Žc,. satisfy Eqs. Ž 1. and Ž 13.. Moreover, given Eq. Ž 14., the robability of success of the marginal borrower, ˆ and hence the average quality of the borrower ool, is strictly higher comared to the equilibrium under lending with individual liability. Finally, we consider the case where rfž Ryu. m so that individual liability lending achieves the first best. nder individual liability lending r srrm. Suose we reduce this interest rate by )0 and introduce joint-liability by an amount such that the bank s zero-rofit condition is satisfied. The marginal borrower continues to remain s1. ˆ The reason is, this tye of a borrower never defaults, nor their artners. Hence, the amount of joint liability does not affect them directly, while a cut in the interest rate makes them strictly better off. Putting s 1 in the zero-rofit condition under joint liability, Eq. Ž 11., we get the Ž corresonding level of joint liability to be mrmy s q m.. We want to comare the exected ayoff of a borrower of tye under this new arrangement and that under the equilibrium with individual liability. A borrower of tye will Ž. Ž be worse off with this move if and only if rrm - rrm y qmrmy s q. Ž. Ž. 19 m 1y, or, - s qm rm. Also, such borrowers are worse off the greater is the amount of joint liability. Now so long as the articiation constraint of the most risky borrower Ž i.e., s. is satisfied, the new arrangement with joint liability is an equilibrium that achieves the first-best in terms of the reayment rate and aggregate surlus as well. Notice that since we have s1, ˆ the limited liability constraint Ž 13. imoses an uer bound on c equal to u. A borrower of tye will borrow if and only if R y u G Ž rrm. y q Ž1. Ž y mrmy s qm.. In the case under consideration, RyuGrrm by assumtion and so Ryu) rrm Ž since -1.. Hence for small values of the articiation constraint of the most risky borrower is satisfied under the new joint liability arrangement. Hence it is satisfied for all tyes of borrowers. Since the Ž ayoff borrowers of tye F s q m. rm is decreasing and continuous in the Ž Y amount of joint liability, mrmy s qm., there exists c gž0, ux such that for cfc Y borrowers of all tyes articiate under joint liability lending. Hence an equilibrium with joint liability exists and achieves the first-best even when rfž Ryu. m so that individual liability lending achieves the first best as well. 19 See Section 5 for an intuition for this.

17 ( ) M. GhatakrJournal of DeÕeloment Economics Ž The equilibrium with joint liability is not unique in this case however: any r, Y. Ž. Ž Ž c where c Fc and r s rrm y my s qm. rm. c is an equilibrium. B Intuitively, starting from a situation where the bank uses standard individual-liability credit contracts Ž i.e., c s 0., if it introduces joint-liability credit contracts by increasing c by a little bit, then that ermits a reduction in the equilibrium interest rate r. But now the effective interest rates facing risky and safe borrowers are no longer the same. Conditional on success, a risky borrower faces a higher effective interest rate than a safe borrower because her artner is more likely to have failed. But this is recisely what a full-information contract would like to do riskier rojects, because they succeed less often should ay more when it succeeds. This encourages entry of safe borrowers into the ool of borrowers as reflected in the increase in the robability of success of the marginal borrower,. ˆ But this raises the average robability of success of the ool, and hence reduces the equilibrium interest rate. Our analysis shows that joint liability will always imrove reayment rates and efficiency Ž in the sense of aggregate surlus. comared to lending with individual liability. These imrovements are strict when individual liability lending fails to achieve the first best. This is due to the fact that safer borrowers always refer joint liability to individual liability because they have safer artners. However, our result is based on various simlifying assumtions such as borrowers know each other s tyes erfectly, enforcement is costless, borrowers are risk neutral and roject return of grou members are uncorrelated. Relaxing some of these assumtions will reduce the relative effectiveness of grou lending and may hel to exlain the observed mixed reayment erformance of various grou-lending rograms in ractice Welfare analysis In Section 4 we showed that grou lending based on joint liability can raise exected aggregate surlus comared to lending based on individual liability. Indeed, it is ossible to achieve the same reayment rates and exected aggregate surlus as under full information under certain conditions. However, it is not guaranteed that every tye of borrower will be better off under grou lending. A fall in the interest rate, ermitted by banks receiving joint-liability ayments from borrowers with defaulting artners and the entry of safe borrowers leading to an imrovement in the borrower ool, hels all borrowers. But now each borrower 0 Ghatak Ž discusses the imlications of allowing borrower risk aversion and correlation of roject returns within this framework. Besley and Coate Ž show that it is ossible for grou lending to erform worse than individual lending under certain circumstances when strategic default is ossible.

18 44 ( ) M. GhatakrJournal of DeÕeloment Economics has an additional cost of borrowing: a joint-liability ayment to cover for the defaulting grou artner. This cost is relatively high for risky borrowers who end u with risky artners. Holding the borrower ool constant, a cut in the interest rate balanced by a rise in joint-liability ayments makes relatively risky borrowers worse off and the aõerage borrower in the giõen ool exactly as well off so long as the bank still earns zero-rofits after this change. However, the surlus generated from the rojects of safe borrowers who now find borrowing rofitable and enter the ool allows the bank to reduce the interest rate further. In this section we rovide an examle that shows if enough safe borrowers are attracted back into the market by a cut in the interest rate starting from an equilibrium with individual liability, all borrowers can be made otentially better-off, including the riskier borrowers in the re-existing ool. Examle: Assume that the robability of success of borrowers follow the uniform Ž distribution, and that the arameters of the model satisfy the condition Rmyu s qm. Gr) Ž Ryu. m in addition to Assumtions 1 and. Then all tyes of borrowers will be better off under joint liability lending comared to individual liability lending. Ž As we assume Rmyu s qm. Gr) Ž Ryu. m, we know from Proositions and 3 that individual liability lending does not achieve the first-best but joint liability lending does. If the robability of success of borrowers follow the uniform distribution, Ž. Ž. g s1r1y, G sy r1y, ms1q r and s s1y r1. Also sq ˆ r and Ž r. sž Ryuryr. qrr. sing Proosition, r sr ŽryŽ Ryu. r.. Since r) Ž Ryu. m by assumtion, and ms1q r)1r we have r )0. Also, r) Ž Ryu. m imlies Ryu-r ŽryŽ Ryu. r. s r,or sryurr -1. The exected ayoff of a borrower of tye in the equilibrium with individual liability is RŽ. yr yusž Ryu. yr ŽryŽŽ Ryu. r... The exected ayoff of a borrower of tye in the equilibrium with joint liability is RŽ. y r yc Ž 1y. yu. sing Proosition 3, we get r sryu and c sr Ž. Ž y Ryu mrmy s qm.. The equilibrium with joint liability rovides a higher ayoff to a borrower of tye comared to the equilibrium with individual liability if and only if r )r qc Ž 1y.. Since cancels out of both sides, it is sufficient to check if the borrower with lowest robability of success, i.e., s, is better off under joint liability lending. Substituting the values of r, r and c and simlifying, the required condition turns out to be r) Ž Ryu.ŽŽ 1q. r.. This shows that under our assumtions borrowers of every tye will be better off under joint-liability lending. This examle takes a very secific distribution and makes strong additional assumtions about the arameters. This is to get simle closed form solutions of

19 ( ) M. GhatakrJournal of DeÕeloment Economics the otimal loan contracts under individual and joint-liability lending that makes the welfare comarison straightforward. This examle should not be taken as imlying that joint liability will make borrowers of every tye better off in general. Rather it should be taken as showing the otential of the instrument of joint liability to imrove welfare. It also shows that the initial equilibrium with individual liability contracts is not necessarily constrained Pareto-efficient. In that equilibrium if safe borrowers were resent in the market the risky borrowers would benefit because even though the bank would make losses on them, that would be subsidized by the rofits made on safe borrowers. However, an individual risky borrower does not internalize the effect she has on the equilibrium interest rate. Therefore, she would enter the market even though that may ush u the interest rate just that much such that all safe borrowers would leave the market. This would cause the equilibrium interest rate to shoot u and make risky borrowers, the only kinds left in the market, worse-off. The reason why joint-liability contracts can imrove welfare Ževen in the Pareto sense as the examle shows. is because they use a valuable resource which individual liability contracts do not. It is the local information that borrowers have about each other through the rocess of endogenous eer selection. This allows grou-lending schemes to attract safe borrowers back into the market, which may make all borrowers, including risky ones better-off due to resulting increase in social surlus. 6. Conclusion In this aer we have rovided a theory to exlain how joint-liability credit contracts used by grou-lending schemes can achieve high reayment rates even when borrowers have no conventional collateral to offer. It is based on the fact that borrowers are asked to self-select grou members and this leads to differential exected costs of borrowing to borrowers deending on their tye because the tyes of their artners are different from the same joint-liability contract. An interesting imlication of the assortative matching roerty roved in the aer is that risky borrowers who will end u with risky artners will be less willing to accet an increase in the extent of joint liability than safe borrowers for the same reduction in the interest rate. This imlies that the degree of joint liability can be used as a screening instrument to induce borrowers to self select loans that differ in terms of individual and joint liability. Elsewhere Ž Ghatak, we have examined this screening roblem, and comared otimal joint liability contracts with other schemes suggested for similar informational environments by the mechanism design literature Ž e.g., cross reorting. in terms of efficiency and collusion-roofness. A related question is what is the nature of equilibrium in the credit market when lenders can comete by offering loans with varying amounts of individual and joint liability. For examle, starting with a cometitive equilibrium where

20 46 ( ) M. GhatakrJournal of DeÕeloment Economics lenders offer loans using only individual liability, can lenders offering joint-liability contracts can break the initial equilibrium? In this aer we have abstracted from these roblems by assuming that the bank is like a lanner who maximizes exected aggregate surlus subject to a break even constraint in order to focus on the effect of joint liability on reayment rates and welfare. In the aforementioned aer, we have addressed these questions as well. To conclude, the theoretical literature on grou lending to which this aer contributes, has roosed many different ways in which it may have informational and enforcement advantages over conventional forms of lending to borrowers who cannot offer collateral. However, emirical work testing the effect of the secific instrument of joint liability on reayment rates in grou lending rograms, and the relative imortance of eel monitoring, eer ressure and eer selection effects, has lagged behind theoretical work on the toic. 1 Aart from ure academic interest, such evidence could tell us whether joint-liability-based mechanisms can work only in very cohesive rural environments, or whether they can work in more urban environments where even though social enforcement mechanisms are weaker, eole still might have a lot of information about their neighbors or co-workers. Acknowledgements I am grateful to my thesis advisors Abhijit V. Banerjee, Eric Maskin and Jonathan Morduch for encouragement and guidance, and to Debraj Ray and two anonymous referees of this journal for helful suggestions. I would like to secially acknowledge the hel of one anonymous referee for critical comments on successive drafts of this aer from which it benefitted greatly. The usual disclaimer alies. Aendix A In this aendix we show that one of the main results of the aer concerning the comosition of borrowing grous under grou lending rograms that use joint liability, Proosition 1, continues to hold if we relax some of the simlifying assumtions. A.1. Grous of arbitrary size First we consider grous of size n and rove by induction that the ositive assortative matching result goes through. The unconditional exected ayoff of a 1 Ž. Ž. Wenner 1995 and Wydick 1999 are some recent aers that ursue this line of research.

21 ( ) M. GhatakrJournal of DeÕeloment Economics borrower with robability of success in a grou consisting of n members of tye from a given joint-liability contract Ž r, c. is: ny1 ny1 x ny1yx E, Ž r,c,n. srž. yryc Ý CxŽ 1y. x. xs1 Since the random variable x reresenting the number of grou members of tye who failed is binomially distributed with mean Ž 1y.Ž ny1. we can rewrite this exression as: E, Ž r,c,n. srž. yrycž 1y.Ž ny1.. Suose that ositive assortative matching holds for grous of size n. We want to show that, if the grou size is nq1 then this roerty still holds. The exected ayoff of a member of tye of a grou consisting of n members of tye from having an additional member of tye X is: RŽ. yrycž 1y.Ž ny1. q Ž 1y X. 4. Hence, the difference in the exected ayoff of a member of a grou with n members of tye from having an additional member of tye X over having an X X additional member of tye is cž y.. Suose ). Then the maximum total bribe that a grou consisting of n borrowers of tye will be able to offer to X X a borrower of tye to join their grou is ncž y.. The exected loss of a borrower of tye X to leave a grou of n borrowers of tye X and join a grou of n borrowers of tye is: X RŽ X. yr X yc X nž 1y X. 4 y X RŽ X. yr X yc X nž 1y. 4 snc X Ž X y.. X Ž. X X Since nc - -nc Ž -., Lemma 1 alies and hence there does not exist a mutually rofitable transfer from a grou consisting of risky borrowers to a safe borrower in order to ersuade the latter to leave a grou consisting of safe borrowers and join the former grou. It turns out that grou size and joint liability cannot be used as indeendent screening instruments. Rather, they are erfect substitutes as a candidate for a screening instrument that can be used with the interest rate. Starting with a given grou size n and amount of joint liability c, an increase in c must be accomanied by a decrease in n to kee the exected ayoff of a borrower of any tye constant. However, only if the rate at which n must be decreased for a given increase in c varies across tyes, it will be ossible to screen borrowers by varying grou size and joint liability. This is not the case as Ž dcrdn. sycrž ny1. Ž where we hold the exected ayoff of a borrower of tye constant. is indeendent of. In contrast Ž dcrdr. syž1ržž ny1.ž 1y... and Ž dnrdr. syž1rcž 1y.. Žagain, holding the exected ayoff of a borrower of tye constant. which suggests that either joint liability or grou size can be used as a screening instrument with the interest rate.

22 48 ( ) M. GhatakrJournal of DeÕeloment Economics A.. Borrowers haõe some wealth Next, suose a borrower of tye has some ositive level of wealth, w. So irresective of the outcome of her roject she receives a ayoff of w. The exected ayoff of this borrower when her artner is of tye X is E X Ž r,c;w. s X RŽ. qwyr, Ž. Ž. Ž. Ž. 4 qž 1y X. RŽ. qwyryc q Ž 1y. w sr qwy rqc 1y X. Y If her artner is of tye instead, her exected ayoff is E Y Ž r,c; w., s RŽ. qwyrqcž1y Y.4. The difference in her exected ayoff between X these two cases is cž y Y. which is the only determinant of her choice of artner. Since this does not deend on her wealth w Proosition 1 goes through even when borrowers have some wealth and there is heterogeneity in initial wealth levels. Of course if borrowers have enough initial wealth which can be offered as collateral then the adverse selection disaears. The bank can then offer contracts that differ in the amounts of interest charged and collateral demanded with safer borrowers referring loans with low interest and high collateral, and risky borrowers the oosite. The imortance of joint liability stems recisely from the fact that it allows a way of lending to the oor who cannot offer enough collateral. As mentioned above, elsewhere Ž Ghatak, we have examined the ossibility of joint liability as a screening device. In that context a question arises about the relative efficiency of joint liability and collateral as screening instruments. We show that if borrowers are risk neutral, it does not matter whether in any given state of the world Ž deending on roject outcome. the transfer to the bank takes lace in the form of individual liability, joint liability or collateral ayment. On the other hand, when borrowers are risk-averse, while both joint liability and conventional collateral involves less than full insurance in order to ensure incentive comatibility, joint liability involves a smaller dearture from efficient insurance. The reason is, collateral involves a transfer from the risk-averse borrower to a risk-neutral bank in a state of the world when her marginal utility of money is the highest, namely when her roject fails. But joint liability merely taxes her when her roject succeeds and that of her artner fails. A.3. Borrower oulation unbalanced with resect to grou size If we relax the assumtion that the suly of artners of each tye is balanced with resect to the required grou size, then some borrowers will not be able to borrow under joint-liability contracts. Since, in equilibrium, identical borrowers should receive the same ayoff under otimal sorting regardless of whom they get as a artner or whether they choose not to borrow given ositive assortative

23 ( ) M. GhatakrJournal of DeÕeloment Economics matching, it will be the borrower of the lowest tye who will fail to obtain a loan through grou lending. 3 Suose borrowers are arranged in terms of their tye by descending order: 1) )...) N. Suose there are two borrowers of each tye to begin with. Then from Lemma 1 we know that grou formation will dislay homogenous grous. Suose now one borrower of tye 1 is removed from the grou of borrowers. In that case by Lemma 1 the remaining borrower of tye 1 will always be able attract any borrower of tye Ž where i) 1. i away from the latter s current artner. Hence she will match with a borrower whose tye is closest to her own tye Ž from below., namely. But this means there will be a borrower of tye who will now fail to find a artner of her own tye. Reeating the above argument, she will be able to find a artner of tye 3.In order for a borrower of any tye to be indifferent between matching with a borrower of a safer tye and a riskier tye, there should be a corresonding set of transfers from a borrower of tye to a borrower of tye, t, which can i iy1 i,iy1 be solved from the following set of equations: Ei,iy1Ž r,c. yti,iy1 sei,iq1ž r,c. qt iq1,i, is,...,ny1 E Ž r,c. yt su. N, Ny1 N, Ny1 There is one otential comlication. Suose there are more than two borrowers of some tyes to start with. If i is such a tye then in equilibrium there may still be some homogenous grous consisting of borrowers of tye. In this case, a i borrower of tye must be indifferent in equilibrium between forming a grou i with three tyes of artners: a borrower of a safer tye, a borrower of a riskier tye and a borrower of the same tye. Suose there were Ni borrowers of tye to start with.then we need to determine a transfer t from each borrower of i tye i who gets to form a grou with a borrower of the same tye to the borrower of tye i who gets to form a grou with a borrower of tye iq1: E Ž r,c. yt se Ž r,c. qt qž N y1. t i,iy1 i,iy1 i,iq1 iq1,i i i,i se Ž r,c. yt. i,i Hence, for general distributions of the borrower oulation, there are two deartures from Proosition 1. First, now side ayments will take lace in equilibrium to ensure borrowers of the same tye receive the same exected ayoff. Second, while there will be some heterogeneous grous in equilibrium, the following weaker version of the assortative matching result resented in Proosition 1 will continue to hold the artner of a given tye of borrower will have a robability of success at least as large as the artner of a borrower with a lower robability of success. i,i i,i 3 This is similar to the case of sorting with unequal number of artners as in Becker s model of the marriage market.

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