Group lending with endogenous social collateral

Transcription

1 frst verson: February 13, 2002 ths verson: September 24, 2002 Group lendng wth endogenous socal collateral BEATRIX PAAL STANFORD UNIVERSITY and UNIVERSITY OF TEXAS AT AUSTIN THOMAS WISEMAN UNIVERSITY OF TEXAS AT AUSTIN Department of Economcs, Landau Buldng Stanford Unversty, Stanford, CA phone: (650) fax: (650) Department of Economcs Unversty of Texas at Austn, Austn, TX phone: (512) fax: (512) Abstract Inspred by mcrocredt nsttutons that nteract wth groups of borrowers and rely on socal captal nstead of sezable physcal assets to collateralze ther loans, we construct a model of lendng to a communty of borrowers who are connected by rsk-sharng arrangements that are themselves subject to enforcement problems. We show that an outsde lender, f he condtons hs repeated nteractons wth each borrower on the hstory of hs nteractons wth all the group members (a jont lablty contract), can earn a hgher proft than he could through offerng ndvdual lablty contracts. The observaton drvng ths result s that wth ndvdual lablty loans, a jont-welfare maxmzng group may prefer to have one or more group members default on ther contracts, so that the group can consume a mx of outsde loans and the defaulters stochastc ncome. One contrbuton of our work s to gve economc content to the concept of socal captal as the surplus that an agent receves by adherng to the group rsk sharng arrangement nstead of retreatng to autarky. The group can deter the agent from defaultng on hs contract wth the outsde lender by threatenng to reduce ths surplus. We also derve predctons for how loan repayment performance changes as a functon of changes n the level of endogenous socal captal. We fnd a non-monotonc relatonshp, so that there s no presumpton that ncreased socal captal reduces loan delnquency. We thank Orazo Attanaso, Valere Bencvenga, Dean Corbae, Scott Freeman, Ken Hendrcks, Narayana Kocherlakota, Preston McAfee, Davd Sbley, and Maxwell Stnchcombe for helpful dscussons. We also thank semnar partcpants at Duke, Texas, and the 2002 SED meetngs for ther comments. Paal thanks the Donald D. Harrngton Fellows Program at the Unversty of Texas at Austn for fnancal support.

2 I. Introducton The concept of socal captal or socal collateral plays an mportant role n analyses of how group lendng contracts overcome enforcement problems n rural credt markets. In ths paper we propose a model that can be used to nterpret socal captal as an endogenous economc varable, a surplus that arses from repeated nteractons between members of a communty. We show how group lendng contracts can successfully use ths surplus as collateral aganst ndvdual default. We also derve predctons for how loan repayment performance changes as a functon of changes n the level of endogenous socal captal. We fnd a non-monotonc relatonshp, whch can help nterpret why emprcal nvestgatons of ths correlaton returned mxed results. Conventonal lendng to poor borrowers n developng countres s often unproftable because of weak repayment performance. Nevertheless, specalzed mcrocredt nsttutons have succeeded at mantanng reasonable ex post returns on ther loans to such borrowers. These nsttutons manage to overcome enforcement problems by nteractng wth groups of borrowers nstead of ndvduals. The jont lablty contracts they offer mply negatve consequences for each group member (typcally n the form of non-refnancng threats) should one group member default. Ths arrangement creates an ncentve for the group to apply ther own sanctons aganst a delnquent borrower. To the extent that such low cost socal sanctons are avalable to the communty when drect sanctons by the lender are mpossble or very costly, group contracts can domnate ndvdual contracts. The phenomenon of lenders nducng groups to enforce contracts for them by applyng socal penaltes or pressures s also referred to as collateralzng loans wth the socal captal of the borrower. In ths paper we explore how the value created by repeated economc nteractons between members of a group can be used as collateral towards the outsde. We construct an nfnte horzon model of a monopolstc lender offerng credt to a communty of agents who are connected by nformal socal nsurance arrangements. The members of the communty face random ncome fluctuatons and engage n rsk sharng wth each other n order to smooth ther consumpton. These rsk sharng contracts are themselves subject to lmted enforcement and are self-sustanng due to the repeated nature of communty nteractons. There s aggregate rsk, so that there s demand for outsde borrowng and lendng. However, the outsde lender also faces 1

3 the problem of lack of enforcement and must structure contracts wth borrowers n the vllage n such a way as to overcome these enforcement problems. In ths envronment nvestgate the advantages of lendng to a group rather than lendng to ndvduals separately. We show that a contract that condtons the lender s repeated nteracton wth any sngle member of the group on the hstory of hs nteractons wth all group members may be selfsustanng even f a contract based on ndvdual lablty s not self-sustanng. The ntuton behnd ths result s that the outsde lender, by threatenng to wthhold future funds from the group at large, can compel the group to pressure a defaultng member to repay. Ths socal pressure n our model takes the form of threatenng the defaultng borrower wth a reduced share of the surplus that communty rsk sharng generates over autarky. By contrast, gven our assumpton that group members may cooperate to maxmze jont surplus, a lender nteractng wth ndvduals separately can only punsh default by denyng further credt, wthout beng able to manpulate the terms of a defaulter s partcpaton n vllage rsk sharng. In fact, n some crcumstances the group s actually better off when one or more members default on ther contracts, so that the group can share a mxture of outsde lendng and the defaulters random ncome. Ths dfference between outsde optons drves the result. The analyss of our model suggests an operatonal defnton of socal captal as the surplus that an agent receves by partcpatng n the vllage rsk-sharng arrangement nstead of lvng n autarky. It s exactly ths surplus, the credt of the ndvdual wth the vllage, that an outsde lender can threaten to reduce n case of default. In ths sense, the surplus that we call socal captal of an ndvdual represents a form of collateral. (Of course, the outsde lender has no drect way of manpulatng ths surplus, but he can structure hs nteractons wth the group at large n such a way that the group has ncentves to apply the rght sanctons for hm.) Our model envronment s delberately parsmonous. We make assumptons on the preferences and the jont ncome process of our vllagers. We assume a sngle, rsk-neutral outsde lender. Both the vllagers and the outsde lender have access to perfect nformaton but no access to any enforcement technology. Our fnal assumpton s that the lender can commt to future actons but the vllagers cannot. Our am s to solate the role of economc nteractons wthn a communty n creatng socal captal, and we formulate a model that demonstrates ths possblty n a transparent manner. The stylzed nature of our model envronment also allows us to conjecture that the phenomenon we dentfy mght be relevant outsde of the mcrofnance context. For example, marred couples are typcally jontly lable for non-busness loans. Stll, t s useful to provde some motvatons for our modelng choces from the perspectve of the lterature on rural credt markets. Ths s what we turn to next. 2

4 Theoretcal work on why group lendng outperforms ndvdual lendng has focused on how group lendng overcomes varous frctons that hnder the operaton of rural credt markets. These postulated frctons nclude asymmetrc nformaton problems as well as enforcement dffcultes. Our purpose n ths paper s to further the understandng of how enforcement problems can be amelorated by group lendng practces. We are well aware that there are a number of other dmensons along whch the practces of mcrocredt nsttutons dffer from those of tradtonal lenders. For example, the group structure can be usefully exploted to better screen borrowers, to mprove the montorng of both the approprate use of funds and the outcomes of projects, to reduce the cost of transactng wth many small borrowers, or to provde educaton and techncal assstance. 1 Ultmately, however, any lender must collect repayment. We sngle out enforcement ssues because n envronments wth extremely weak legal systems and hardly any physcal collateral, every lender must confront the possblty of strategc default. To our knowledge the only systematc emprcal nvestgaton of the relatve mportance of enforcement problems versus varous nformatonal asymmetres s the work of Ahln and Townsend (2000), comparng four statc models of group lendng. Ther prelmnary results ndcate that data from Tha vllages (partcularly, the most rural areas) are best ft by the lmted enforcement model of Besley and Coate (1995). Zeller (1998) uses data from lendng programs n Madagascar and fnds that ncreasng the per capta land holdngs of members n a borrower group has an nsgnfcant (negatve) effect on the repayment rate of the group. He nterprets ths fndng to mean that the capacty to repay seems not to matter n actual repayment performance (p. 615). Regardng the nature of nteractons wthn a vllage communty, Lgon, Thomas, and Worrall (2002) show that the knd of dynamc lmted commtment envronment that we assume n our model s a good descrpton of the nformal socal nsurance arrangements operatng n three Indan vllages. Albarran and Attanaso (2001), usng data from a Mexcan welfare program, also fnd that a number of mplcatons of the lmted commtment envronment are generally supported. 2 1 Examples of theoretcal work on group lablty and varous asymmetrc nformaton problems nclude Varan (1990), Stgltz (1990), Banerjee, Besley, and Gunnane (1994), Connng (1996), Ghatak (1999), and Madajewcz (1999). A comparson of varous models s carred out by Ghatak and Gunnane (1999) and Ahln and Townsend (2000). Morduch (1999) provdes an extensve and nterestng revew of mcrofnance nsttutons. 2 Kocherlakota (1996) proposes a test that could dscrmnate between dynamc lmted commtment and dynamc asymmetrc nformaton models, but to our knowledge these alternatves have not been contrasted usng mcro data from rural communtes. 3

5 The most mportant modelng choce we make s to gve no nformatonal or enforcement advantage to the vllagers relatve to the outsde lender. Ths approach s qute dstnct from the standard approach n the lterature. For example, Ghatak and Gunnane (1999) emphasze that jont lablty lenders outperform conventonal lenders by gvng ncentves to members of a communty to ether use nformaton about one another or to apply non-fnancal sanctons aganst one another. Indeed, all theoretcal models of jont lablty lendng that we are aware of feature such an exogenous nformaton or enforcement advantage. By contrast, n our model, nformaton s freely avalable to everyone and all contracts must be self-enforcng. As a result, we can hghlght how socal captal s created through repeated communty nteractons, nstead of arsng from superor montorng or punshment technology that the communty possesses. In the lterature on group lendng and strategc default, t s commonly assumed that socety can mpose a socal penalty on an ndvdual who reneges on a group lendng contract. Ths penalty may be an exogenous constant, as n the paper of Armendarz de Aghon (1999), or t may be postulated as an exogenous functon of the contemporaneous loss suffered by the complyng partner and the repayment capacty of the defaulter, as n the work of Besley and Coate (1995). In contrast, we derve these penaltes endogenously, whch allows us to (1) study the ncentves of the communty to mpose such penaltes and (2) examne how the economc envronment facng the communty affects ther magntude and effectveness. The feature of our model that allows us to endogenze socal penaltes s ts explct dynamc nature. In ths regard, our work s closest to that of Sadoulet (2000) and of Che (2002). Sadoulet (2000) examnes how dynamc ncentves to repay dffer under ndvdual versus jont lablty contracts n an adverse selecton envronment. However, by assumpton, there s no nteracton between nsurance provson wthn a group and credt transactons wth the outsde. In our model, ths nteractons s an mportant element n makng jont lablty loans more proftable. Che (2002) studes a moral hazard envronment where agents repeatedly make effort choces. Jont lablty contracts turn effort choce nto an endogenous punshment devce much n the same sprt as the terms of partcpaton n vllage rsk sharng s an endogenous punshment devce n our model. We model vllage nteractons as rsk sharng by rsk averse agents who face exogenous stochastc ncome streams. The outcomes n ths envronment, n absence of an outsde lender, have been nvestgated by Kocherlakota (1996) and Lgon, Thomas, and Worrall (2002). A natural alternatve would have been to assume productve opportuntes and captal accumulaton. Lgon, Thomas, and Worrall (2002) show that extendng the model to nclude 4

6 stochastc storage leaves the mplcatons wth respect to nsurance provson qualtatvely unaffected. Emprcal evdence also suggests that provdng consumpton nsurance s an mportant component of rural credt transactons. The work of Townsend (1994, 1995) emphaszes the mportance of ncome rsk and the sharng of ths rsk wthn poor rural communtes. In hs classc study of credt arrangements n Northern Ngeran vllages, Udry (1994) also demonstrates that repayments on loans are state contngent, and depend on the realzatons of shocks to both the borrower s and the lender s ncomes. In another context, Calomrs and Rajaraman (1998) show that mplct nterest rates on funds placed n ROSCA (rotatng savngs and credt assocaton) accounts fluctuate sgnfcantly from month to month. They argue that ths observaton s more consstent wth nsurance provson aganst random shocks to tastes or ncomes than wth the presumpton that these accounts represent savngs n preparaton for antcpated large purchases (such as durables). Nevertheless, our man reason for assumng an endowment economy s to gan tractablty and mantan the transparency of our results. Fnally, the assumpton of a monopolstc outsde lender deserves comment. Frst, we note that t s consstent wth the observed operaton of mcrofnance nsttutons n developng countres. For example, van Bastelaer (1999) descrbes a typcal segmentaton between the roles of NGOs and local moneylenders n provdng credt to dfferent subsets of borrowers. He reports that the nterest rates on the loans from dfferent sources do not dsplay a strong tendency to converge. Second, the mcrofnance nsttutons n queston, although close to self sustanng, do not generate postve profts, so that ncentves for new entry are lmted. 3 We beleve that our defnton of the term socal captal captures the meanng that has been attrbuted to ths concept n the less formal lterature. We also beleve that t s useful to have a formal economc defnton, for example to motvate emprcal work. We llustrate ths by askng whether n our model hgher levels of socal captal are assocated wth hgher group loan repayment rates. Ths s an nterestng queston because the correspondng emprcal hypothess has been nvestgated wth qute mxed results. We show that our model smply does 3 From a theoretcal perspectve, the presence of multple outsde nsttutons would rase some nterestng ssues. The exstence of self-enforcng contracts would depend on how the nteracton between these outsde lenders s modeled. (See, for example, Bulow and Rogoff (1989), Krueger and Uhlg (2000), Kletzer and Wrght (2000), and Wrght (2002).) Whether group lendng would retan some advantage over ndvdual lendng s not mmedately clear. It would be worthwhle to pursue these ssues, partcularly to make predctons about the future of mcrocredt. However, n our current context, t would take us too far from the man pont. 5

7 not generate the hypothess: A plausble measure of loan repayment rates dsplays a nonmonotonc relatonshp to our measure of socal captal. We formally descrbe the model n Secton II. Dfferent lendng contracts are compared n Secton III, where we show that an outsde lender can obtan hgher profts usng group contracts nstead of usng ndvdual contracts. In Secton IV, we relate prevous uses of the term socal captal to our more formal defnton, and show that the emprcal hypothess that more socal captal should mprove the repayment performance of group loans does not follow from our model. In Secton V, we conclude by dscussng the robustness of our man message to varous plausble alteratons of the envronment. 6

8 II. Model A. Envronment The envronment (whch s smlar to the one that Kocherlakota (1996) studes) s as follows: A vllage comprses N nfntely-lved ndvdual agents. The ncome of ndvdual n perod t s denoted y t ; Y t s the vector of ndvdual ncomes n perod t. Jont ncome Y t s stochastcally determned accordng to the dstrbuton P, whch has fnte support and whch s..d. across tme. The dstrbuton P s symmetrc: If ncome vector Y s n the support of P, and vector Y s a permutaton of Y, then Y s also n the support of P, and P(Y ) = P(Y). For each ndvdual, then, the possble realzatons of y t are gven by a set {y(1),, y(s)}, where y(1) < y(2) < < y(s). We assume that ncome s non-negatve, so that y(1) 0, and that ndvduals face some uncertanty over ncome, so that S 2. Agents have dentcal preferences. An ndvdual who consumes x t n perod t receves utlty u(x t ), where the functon u s ncreasng, concave (ndvduals are rsk averse), and twce-contnuously dfferentable. The total utlty s the dscounted sum of per perod utlty, multpled by (1 δ) to put t on the same scale as utlty per perod: (1 δ) t= 1 δ t 1 u ( x ), δ (0, 1). t An ndvdual who consumes hs own ncome n each perod, for example, receves expected utlty t= 1 t 1 t E( 1 δ) δ u( y ) = Eu( y ) = P( y( s)) u( y( s)), S s= 1 where P(y), n an abuse of notaton, s the probablty that an ndvdual receves ncome y. (That s, P(Y) s the jont dstrbuton, and P(y) s the margnal dstrbuton.) B. Internal Rsk Sharng (No Outsde Lender) In the absence of an outsde lender, the rsk averse members of the vllage group can ncrease ther welfare by partcpatng n a rsk sharng (socal nsurance) arrangement, as long as 7

9 ndvdual ncome realzatons are not perfectly correlated. In each perod, agents wth hgh ncomes can make a transfer to agents wth low ncomes, n the expectaton of recevng smlar transfers n the future when ther own ncomes are low. Formally, n each perod the ncome vector Y t s realzed and s observed by all agents. Agents then have the opportunty to transfer ncome to some or all of the other agents; denote by R t the matrx of transfers made by the agents n perod t. The hstory h t at perod t s the sequence of past and present ncome realzatons and past transfers: h t = (Y 1, R 1,, Y t 1, R t 1, Y t ). A strategy for each ndvdual s a functon from the set of possble hstores to a vector of nonnegatve transfers. Indvdual s strategy specfes how much of hs ncome y t goes to each agent after every hstory, subject to the constrants that none of the transfers be negatve and that ther sum s no greater than y t. Indvdual s consumpton, after the transfers are made, s denoted x t. N Let y y N be the mean of the realzed ndvdual ncomes n perod t and let t = 1 t p be the dstrbuton of t x t depends only on y t, and not on hs own ncome arrangement, each agent consumes the average ncome y. In a full rsk sharng arrangement, each ndvdual s consumpton y t. In a symmetrc full rsk sharng y of all ndvduals n every perod. Because ndvduals are rsk averse, and the varance of y s less than the varance of y, Eu ( y) > Eu( y ), so the group members are better off when they share rsk. Full rsk sharng s sustanable as a subgame perfect equlbrum f the followng condton s satsfed: max Yˆ supp( P) (1 ) ( ˆ / ) ( ) (1 ) ( ˆ ) ( ) N j δ u + δ δ δ 0 1 j = y N Eu y u y Eu y (FRS) Gven an ncome realzaton, an agent s wllng to make hs specfed transfer f the expected utlty that he gets from dong so and remanng n the group exceeds the expected utlty that he would receve from consumng hs own ncome today, beng cast out of the rsk sharng group, and then consumng hs own ncome n every perod n the future. (An ndvdual s expected utlty n autarky, Eu ( y ), s hs mnmax payoff, so castng an agent nto autarky s the harshest punshment that can be enforced n a subgame perfect equlbrum.) If an agent prefers to reman n the rsk sharng arrangement after any realzaton of Y (f Condton FRS holds), then the arrangement s sustanable. We wll assume throughout that Condton FRS holds, so that full rsk sharng s subgame perfect. Increasng the dscount factor δ ncreases the sustanablty of full rsk t 8

10 sharng, as does ncreasng the degree of rsk averson. Decreasng the correlaton between ndvdual ncome realzatons also ncreases sustanablty, by lowerng the varance of y and thus ncreasng the expected utlty that t provdes. Smlarly, a larger group sze decreases aggregate rsk (holdng the degree of ndvdual ncome correlaton fxed) and ncreases the gan from rsk sharng. On the other hand, a larger group sze may ncrease the sze of the maxmum transfer an ndvdual s called upon to make when one ndvdual receves the hghest ncome y(s) and the other N 1 receve the lowest ncome y(1) and thus make sustanng the rsk sharng agreement more dffcult. (Lgon, Thomas, and Worrall (2002) examne effcent partal rsk sharng when full rsk sharng s not sustanable, and Kocherlakota (1996) looks at rsk sharng between a par of agents.) We assume that the agents choose a symmetrc equlbrum, and that from the set of symmetrc subgame perfect equlbra they play one that s Pareto optmal. Because full rsk sharng s effcent, and Condton FRS holds, those assumptons mply that n the absence of an outsde lender the group wll engage n a symmetrc full rsk sharng arrangement. We defne socal captal, then, as the surplus that a group member receves each perod from partcpatng n nternal rsk sharng relatve to autarky: Eu( y) socal captal 1. Eu( y ) C. Monopolstc Outsde Lender Now consder the problem faced by a sngle outsde lender who approaches a group that engages n nternal rsk sharng. The lender maxmzes E 0 (1 δ) t= 0 t δ π t where π t s hs proft n perod t. We assume that the lender offers ndvduals contracts of the followng type: The lender guarantees a constant level of consumpton, c. At the begnnng of each perod, an agent s ncome y t s realzed and observed by all partes. If y t s less than c, then the lender transfers the dfference, c y t, to the agent. In perods when y t exceeds c, the agent turns over the surplus to the lender. In effect, the ndvdual trades hs entre stochastc 9

11 stream of future ncome to the lender n exchange for a guaranteed level of consumpton. 4 Because ncome s..d. across tme, then, the lender s problem s to maxmze a sequence of one-perod expected profts. The expected per perod proft, Eπ(c), can be wrtten as S Eπ ( c) = [ P( y( s)) y( s) c] = Ey c. s=1 The lender s able to commt hmself ex ante to such a contract. That s, the lender may specfy n the contract offered to agent condtons (on the hstory of the lender s nteractons wth all vllagers) under whch the lender wll permanently abandon the agreement. In any perod where those condtons are not met, the lender s constraned to provde c to agent. Contracts where the lender s contnued partcpaton depends only on the sequence of agent s transfers to the lender are ndvdual lablty contracts. Under jont lablty contracts, the lender condtons on the hstores of all vllagers. (Note that ndvdual lablty contracts are a specal case of jont lablty contracts.) The borrowers, on the other hand, cannot commt. They can walk away from the deal at any pont at whch they fnd t proftable to do so, because the lender lacks the ablty to enforce the terms of the contract. Thus, a strategy for the lender specfes only what contract (f any) wll be offered to each agent n perod 0. A vllager s strategy specfes whch contracts to accept n perod 0, and n all subsequent perods specfes a vector of non-negatve transfers to each agent and to the lender. As before, the sum of agent s transfers cannot exceed y t. Because of the one-sded commtment, the lender must offer self-enforcng contracts. A contract s self-enforcng f after every possble hstory no vllager fnds t n hs nterest to walk away from t. In the absence of a group rsk sharng arrangement, an agent who breaks the terms of the loan contract s dened future credt and must return to consumng only hs own ncome, so the condton for a contract to be self-enforcng s that u( c) (1 δ) u( y( S)) + δeu( y ). (SE) 4 The assumpton of a constant ncome stream offered by the lender s made for the sake of tractablty. However, because the lender s rsk-neutral and dscounts at the same rate as the vllagers, the assumed contract form s lkely to be only slghtly restrctve relatve to the unconstraned optmal contract. For example, f the lender nteracts wth only a sngle solated agent, the unconstraned optmal contract converges, wth probablty one, to a constant n a fnte number of perods. A smlar outcome wll occur when multple, suffcently patent, vllagers nteract wth each other n a closed vllage. (See Sargent and Ljungqvst (2000), pp ) 10

12 That s, even when an agent receves the hghest ncome level, y(s), he prefers consumng c n every perod to breakng the contract, consumng y(s) today, and returnng to autarky n all future perods. Note that f c CE (y), where CE (y) s the certanty equvalent of the stochastc average ncome y, then Condton FRS (guaranteeng that full rsk sharng s sustanable) mples that Condton SE s satsfed. If, on the other hand, the borrowers are also partcpatng n rsk sharng, then the outsde opton of an agent who walks away from a contract may be hgher. For example, f an agent s allowed to contnue sharng ncome wth the rest of the group even after beng dened access to credt from the lender, then hs expected payoff after breakng the contract s greater than the autarky payoff Eu(y ). In that case, self-enforcement requres a strcter condton than Condton SE: The lender must provde a hgher guaranteed consumpton level c to nduce borrowers not to abandon the contract. Furthermore, f the vllagers can collude to jontly devate from a contract, then an even hgher level of c s necessary. In the next secton, we explore that stuaton n detal and examne the condtons under whch the lender can do better by offerng jont lablty contracts rather than by dealng wth ndvdual borrowers separately. 11

13 III. Indvdual versus Jont Lablty The goal of the lender s to maxmze expected profts M Eπ(c) (where M N s the number of ndvduals offered contracts) subject to the constrant that the contract must be selfenforcng. The lender chooses the number M of vllagers to lend to, the level c of guaranteed ncome offered, and the structure (jont or ndvdual lablty) of the contract s renewal polcy. For smplcty, we restrct the lender to offerng the same c and the same renewal polcy to each borrower, although he may offer contracts to only a subset of the vllagers. We assume, as before, that the group of vllagers, gven the set of contracts offered, wll maxmze ther jont utlty subject to subgame perfecton. The soluton to the lender s problem, then, wll be a subgame perfect equlbrum that s coalton-proof (n the sense of Bernhem, Peleg, and Whnston (1987) and Moreno and Wooders (1996)) wth respect to the coalton of all vllagers. We provde general condtons under whch a lender may be able to ncrease profts by offerng jont rather than ndvdual lablty contracts to the members of a vllage, even when borrowers are arbtrarly patent. Frst, we derve an upper bound on the lender s profts from any contract. Second, we show n Proposton 1 that there s a jont lablty contract that can acheve that upper bound. Thrd, we demonstrate through an example that n some cases the optmal ndvdual lablty contract provdes a strctly lower proft, and n Proposton 2 we descrbe formally the crcumstances n whch ths wll occur. In nteractng wth a potental borrower who s part of a vllage rsk sharng arrangement, the lender must take that arrangement nto account, even f he offers only ndvdual lablty contracts. Consder agent n the vllage, who has stochastc ncome stream y. Suppose that the outsde lender approaches ths agent and offers to provde constant ncome c forever n exchange for that random ncome stream. If agent lved n solaton from the vllage and smply planned to consume c, then the proposed contract s self-enforcng f Condton SE s satsfed. In the lmt as δ 1, the lender s proft-maxmzng choce of c converges to CE(y ), the certanty equvalent of agent s random ncome. In fact, however, ndvdual s already nvolved n the vllage rsk sharng arrangement, so that hs effectve ncome s not y but y, the vllage s average ncome. Thus, maxmzng profts subject to Condton SE s not the rght problem for the lender to solve, because the borrower has a better outsde opton. Instead, the lender must provde a guaranteed consumpton level c at least as large as CE (y), 12

14 the certanty equvalent of the vllagers pooled ncome, or else the vllagers could do better through nternal rsk sharng. Furthermore, c must n fact be strctly greater than CE (y), so that the vllagers cannot gan by jontly abandonng the contract when they all receve hgh ncome realzatons. If c were equal to CE (y), then n any perod when y t > c, the vllagers would break the contract, consume y t today and receve Eu ( y) = u( CE( y)) n each future perod be revertng to nternal rsk-sharng. By honorng the contract, on the other hand, they would consume c < yt today and c = CE(y) n the future, whch gves them strctly lower utlty. To be sustanable, c must be at least CE ( y) + ε( δ), where ε(δ) s defned as the smallest value satsfyng {(1 δ ) u( y) δu( CE( y)) } u( CE( y)) + ε ( δ ) max + y supp( p) As the dscount factor δ approaches one, ε(δ) converges to zero. That lmt on c puts an upper bound, equal to the greater of N (Ey CE ( y) ε( δ) ) and zero, on the lender s expected proft per perod n a self-enforcng contract. The lender can acheve that maxmum proft wth the followng strategy, whch uses jont lablty contracts: If N (Ey CE ( y) ε( δ) ) > 0, the lender offers each vllager a contract guaranteeng ncome c J CE ( y) + ε( δ) n each perod, as long as no vllager has faled n any prevous perod to turn over hs realzed ncome y t to the lender (and as long as the lender hmself has not devated). Otherwse, the lender offers no contract. The vllagers strategy s to reject any contract wth c y(1) and revert to symmetrc nternal rsk-sharng; to accept any contract wth y(1) < c < c J and eventually devate from t; 5 and to accept any contract c c J and not devate n any perod before a devaton. In the thrd case, f vllager fals to turn over hs ncome, then all vllagers move to an nternal rsk sharng equlbrum where vllager s utlty s reduced to Eu(y ), hs autarkc utlty. If the lender devates, or f more than one borrower devates smultaneously, the vllage group reverts to symmetrc nternal rsk sharng. Those strateges consttute a subgame perfect equlbrum that s proof to self-enforcng jont devatons by the vllagers when δ s hgh enough, as shown n Proposton The optmal tme to devate s the frst perod n whch realzed mean ncome y satsfes the followng condton: ( 1 δ ) u( y ) + δeu( y) > (1 δ ) u( c) + δev ( c, y), where V(c, x) s gven recursvely by V ( c, x) max{(1 δ ) u( x) + δeu( y),(1 δ ) u( c) + δev ( c, y)}. 13

15 Proposton 1: Let N, u, and P(Y) be gven. Then there exsts δ < 1 such that for all δ (δ, 1), there s a subgame perfect equlbrum that s proof to self-enforcng jont devatons by the vllagers that gves the lender expected profts of max{ N( Ey c ),0} per perod. J Proof: The strateges are as gven above. Frst, let δ be at least hgh enough that Condton FRS s satsfed. Then f the lender s patent enough, ts strategy s clearly optmal: Gven that the borrowers wll eventually reject any c less than c J, max{ N( Ey c J ),0} s the maxmum possble proft. The vllagers get hgher expected utlty from acceptng the contracts when c c J than from revertng to nternal rsk sharng, so t s a best response for them all to accept. Gven the punshment strateges, Condton SE guarantees that no borrower can ever gan from consumng hs ncome rather than turnng t over to the lender, even when he receves the hghest possble ncome y(s); Condton SE s satsfed because Condton FRS holds. After a devaton, the lender s strategy s not to nteract wth the vllagers, so not nteractng wth the lender s a best response for the vllagers. Therefore, the best the vllagers can do s an effcent (subject to the enforcement constrants) rsk sharng arrangement. As n Kocherlakota (1996), there s such an effcent self-enforcng arrangement that gves agent expected utlty equal to hs autarkc utlty Eu(y ) and gves the other agents utlty strctly hgher than Eu (y), the expected utlty from symmetrc rsk sharng. Therefore, the punshment path s subgame perfect and proof to selfenforcng jont devatons. Thus, the specfed strateges consttute a subgame perfect equlbrum that s coaltonproof wth respect to the coalton of the whole vllage and that gves the lender expected profts of max{ N( Ey c ),0} per perod. Q.E.D. J The maxmum proft that the lender can acheve by offerng ndvdual lablty contracts, on the other hand, s strctly lower than N (Ey CE ( y) ε( δ) ), as long as there s suffcent aggregate rsk n the vllagers pooled ncome y. That result s drven by the observaton that for c < Ey, a group of rsk averse agents that shares ncome fully may prefer to have one member receve random ncome y and the rest get c for sure n each perod rather than all of them gettng c. That s, t may be that ( N 1) c y Eu + N > u( c), 14

16 even though u(c) > Eu(y ). The ntuton follows from the defnton of a strctly concave functon: If dstnct alternatves A and B gve the same level of expected utlty, then a strct convex combnaton of the two alternatves s strctly preferred to ether. For example, f c = CE(y ) (the certanty equvalent of y ), and the alternatve s gettng y, then the group strctly prefers to have one or more members recevng ther random ncomes and sharng them wth the rest of the group, whch s gettng c: ( N 1) CE( y ) y Eu + N N 1 1 > E u( CE( y )) + u( y ) N N N 1 1 = u( CE( y )) + Eu( y ) N N = u( CE( y = u( c) However, the lender must offer at least c J, whch s greater than CE(y ). In that case, rsk averson does not mply that the group necessarly prefers to have one member (or more) recevng y. Nevertheless, gven a group sze N and preferences u, they stll do prefer t as long as ε(δ) s small enough and the certanty equvalent of pooled ncome y s not too much greater than the certanty equvalent of ndvdual ncome y that s, as long as there s suffcent aggregate rsk n the vllage. If the ncome dstrbuton P satsfes that condton, then, and f the lender has an ndvdual lablty contract wth every agent, t s certan that at least one agent wll break t. To avod that, the lender must offer a hgher c, lend to fewer agents (both of whch optons reduce expected profts), or offer only jont lablty contracts. If the lender offers credt only as long as all N agents accept and honor ther contracts, then the group no longer has the opton of mxng c wth y. As long as c gves greater utlty than nternal rsk sharng (that s, c s greater than the certanty equvalent of y ) and agents are patent enough, then the group does better to honor the lendng contracts than to break them and revert to rsk sharng. Thus, jont lablty contracts can provde hgher expected profts than ndvdual lendng. The followng numercal example demonstrates that possblty, and Proposton 2 formalzes the result. )) Example 1: Let the dscount factor δ be 0.95, and suppose that N = 5 and that utlty s gven by u(y) = 1/y, so agents have a constant coeffcent of relatve rsk averson equal to 2. The ncome vector s determned as follows: In each perod, there are two possble states, G and B, each of whch has probablty 0.5. Gven the state, each agent s ncome s chosen 15

17 ndependently. In state G, an agent s ncome s y H = 300 wth probablty a 0.5 and y L = 100 wth probablty 1 a. In state B, the probabltes are reversed. The correlaton between agents ncomes s ncreasng n the parameter a. Fgure 1 graphs the utlty from c J CE ( y) + ε( δ), the constant level of consumpton provded by the optmal jont lablty contract, aganst the expected utlty from 0.8 c J + 0.2y as the correlaton vares. For hgh enough levels of correlaton (correspondng to hgh aggregate rsk), the expected payoff from havng one vllager devate s greater than the payoff from all consumng c J. Now consder specfcally the case that a = 0, and let δ be close to 1. Snce a = 0, ncome realzatons are ndependent across ndvduals, and y H and y L occur wth equal probablty. The expected value of ncome s 200, and the certanty equvalent of the ncome y from group rsk sharng s approxmately Frst consder the optmal jont lablty contract, under whch the outsde lender guarantees a constant level of consumpton c J CE ( y) + ε( δ) to all agents n exchange for any excess ncome over that level. If any ndvdual breaks the terms of the contract, then there s no further nteracton between the lender and any of the agents. As the dscount factor δ approaches one, the group members wll accept any c greater than If they break the contract, they revert to nternal rsk sharng, whch gves them a lower expected payoff than u(c). In the lmt, then, the expected profts per perod for the lender approach roughly 5( ) = If the lender offers the same c J n ndvdual lablty contracts, so that only group members who have themselves broken ther contracts are dened credt, at least one agent wll certanly walk away from hs contract. The reason s that the group gets hgher expected utlty from sharng 4( ) + y (whch s sustanable for hgh enough δ) than from consumng each: > 2( ) 2( ) To avod that problem, the lender must ether offer contracts to fewer agents or provde a larger c. In ths case, the lender maxmzes expected profts by offerng c I to all fve agents, whch yelds expected profts per perod of Thus, the profts from ndvdual lendng are approxmately 4.9 percent lower than group lendng profts. 16

18 Proposton 2: Let N, u, and the margnal ncome dstrbuton P(y) be gven. Then there exsts an open and non-empty set Ψ of symmetrc jont ncome dstrbutons such that f P(Y) Ψ, then there s a δ < 1 such that for all δ (δ, 1), any subgame perfect equlbrum that s proof to self-enforcng jont devatons by the vllagers gves the lender expected profts per perod strctly less than N (Ey CE ( y) ε( δ) ) f Ey CE ( y) ε( δ) > 0. Proof: Let M be the number of elements n the support of the margnal dstrbuton P(y). Then a jont ncome dstrbuton P(Y) s a vector n the (M N 1)-dmensonal smplex. Let P(Y) j denote the jont dstrbuton where ndvdual ncomes are perfectly correlated, so that y t = y t for all t and for all, j {1,, N}. Gven P(Y), the dstrbuton of pooled ncome y s the same as the margnal dstrbuton P(y), so CE (y) = CE(y ). Because u s concave, therefore, ( N 1) CE( y) + y Eu N > Eu ( CE(y) ). Thus, when δ s hgh enough, the followng nequalty holds: ( N 1) CE( y) + y Eu N u. (CC) c J > ( ) When Condtons CC and FRS are satsfed, the lender cannot offer ndvdual lablty contracts promsng ncome c J CE ( y) + ε( δ) because, as n Example 1, at least one borrower s sure to break the contract and share hs future stream of random ncome wth the other borrowers, who are stll recevng c J. Instead, the lender must ether provde a hgher level of guaranteed ncome or lend to fewer borrowers. Because c J s contnuous as a functon of the jont dstrbuton P(Y), Condton CC holds n an open neghborhood (wth respect to Eucldean dstance) of P(Y). Around any pont n the nteror of that neghborhood, there s an open sub-neghborhood where for some δ < 1 Condton FRS also holds. That subneghborhood s the desred set Ψ. Q.E.D. 17

19 IV. Dscusson Above, we present a model that endogenzes socal captal and ts role as collateral n jont lablty contracts between an outsde lender and vllage members, who possess no nformatonal or enforcement advantage over the lender. Here we elaborate on the relatonshp between our operatonal measure of socal captal and the concept of socal captal dscussed n the mcrofnance lterature. We examne whether hgher levels of socal captal are assocated wth better repayment performance on jont lablty loans. A. The Concept of Socal Captal Grootaert and van Bastelaer (2001), n ther summary report on the results of the Socal Captal Intatve of the World Bank, provde an n-depth dscusson of the term socal captal. On a mcro level (whch s the relevant scope of the concept from our perspectve) they descrbe socal captal as those features of socal organzaton such as networks of ndvduals and households, and the assocated norms and values that create an externalty for the communty as a whole. They also emphasze that socal captal s an asset, n the sense that t generates a stream of benefts. Fnally, they note that one of the channels through whch socal captal produces payoffs s by ncreasng the benefts of complance wth expected behavor or by ncreasng the costs of non-complance. In the context of our model, partcpatng n nformal socal nsurance arrangements creates a surplus that arses from the repeated nteractons of a group of agents. Ths surplus can be used to repeatedly collateralze outsde loans even f loans to ndvduals are not sustanable. The group nteractons are not formally sanctoned; nstead they are mantaned n equlbrum by shared belefs about expected behavor and how devatons from t would be punshed. It s often asserted n the lterature that socal captal can be used to support credt contracts that would not be supportable n a more market based, anonymous context. Plunkett (1904) (quoted n Gunnane (1994)), wrtng about credt cooperatves n Ireland, cleverly expresses ths dea when he suggests these nsttutons perform the apparent mracle of gvng solvency to a communty of almost entrely nsolvent ndvduals. In our model, agents who may not be credtworthy ndvdually are credtworthy as a group. 18

20 B. Socal Captal and Repayment Performance The observaton that socal captal can perform the functon of collateral n jont lablty loans led to the hypothess n the emprcal lterature that ncreased socal captal lowers the default probablty on loans. Researchers have not found robust support for ths hypothess, however. Sharma and Zeller (1997) and Ahln and Townsend (2000) fnd that groups wth hgher levels of famly relatons are more lkely to default. On the other hand, Karlan (2002), usng Peruvan data, fnds a postve relatonshp between proxes for socal captal (such as geographc proxmty and cultural smlarty) and repayment performance. Wydck (1999) fnds a postve but statstcally nsgnfcant lnk between repayment and physcal dstance between borrowers. One clear reason for the dvergent results s the fact that the dfferent studes span a wde range n terms of the qualty of ther data, the defnton of default, and the varables that are used to proxy for socal captal. However, our model suggests that the hypothess s not justfed n the frst place. In order to relate repayment performance to the level of socal captal, we need to construct a measure for t wthn the context of the model. We want ths measure to be smlar to the delnquency rate measures reported by mcrofnance nsttutons and used n the emprcal lterature. The dffculty s that the type of lmted enforcement model that we examne has no explct default (n the sense of a devaton from the prescrbed contract) along the equlbrum path. We propose to use, as a proxy for the fracton of non-performng or delnquent loans, the probablty that n D 2 consecutve perods the realzed net transfer flows from the lender to the vllage. That s, the lender makes a loan n one perod and does not receve any repayment n the next D 1 perods. Consder the case where the outsde lender offers the optmal jont lablty contract, payng out c J to each agent and recevng N y t n every perod. In ths settng the probablty that a loan s delnquent can be expressed as = F ), delnquency rate [ ( ] D y c J where F y s the cumulatve dstrbuton functon of average vllage ncome. We contnue to assume that Condton FRS holds and therefore socal captal can be measured as Eu( y) socal captal = 1. Eu( y ) 19

21 We nvestgate the behavor of the probablty of the delnquency rate as the correlaton between an ndvdual s ncome and the average ncome of the rest of the vllage vares. 6 Socal captal s decreasng n ths correlaton, because rasng the correlaton ncreases the varance of the group s average realzed ncome y and thus lowers Eu (y). We show that the delnquency rate, on the other hand, s a non-monotonc functon of ths correlaton. Therefore, there s no presumpton that hgher socal captal must mprove loan performance. The ntuton s as follows: The optmal c J s less than the expected value of y, whch s also the expected value of y. (Otherwse, the lender could not make postve profts.) Decreasng the correlaton between ndvdual ncomes reduces the varance of each perod s sample average y. That decrease n varance has two, conflctng effects on the delnquency probablty. Frst, the realzaton of y s more lkely to be close to ts mean and thus greater than c J. On the other hand, lowerng the varance of y ncreases Eu (y), the value of the group s alternatve to borrowng from the outsde lender. That change means that the lender must offer a hgher c J, whch ncreases the probablty that the realzaton of y s less than c J. Whch effect domnates vares wth the level of correlaton. Furthermore, because the dstrbuton of ncomes has fnte support, the delnquency probablty s n fact not contnuous wth respect to the correlaton, as Example 2 demonstrates. The general result s stated n Proposton 3. Example 2: Suppose that N = 15 and δ = Suppose also, as n Example 1, that utlty s gven by u(y) = 1/y, and that ncome s determned as follows: In each perod, the state s ether G or B wth equal probablty. In state G, an agent s ncome s y H = 300 wth probablty a 0.5 and y L = 100 wth probablty 1 a, and n state B, the probabltes are reversed; condtonal on the state, agents ncomes are ndependent. The correlaton between agents ncomes s ncreasng n the parameter a. Therefore, the level of socal captal and the a optmal c J ( ) are both decreasng n a. The level of socal captal s hgh enough to sustan c J a full nternal rsk sharng when a < , and ( ) > Ey (so that lendng s proftable) for all a > Let n 15! P n (a) 0.5 [(1 a) (15 )!! = 0 15 a + (1 a) a 15 1 ] 6 Because of the symmetry of the jont ncome dstrbuton, that correlaton s the same for all agents. 20

22 denote the probablty that at most n agents receve ncome y H, so that mean ncome y s no 1 15 n more than y (n), defned as y( n) (300n + 100(15 )). When c J ( ) s at least y (n) and below y (n + 1), then Fy ( c J ( a)) = Pn ( a). For example, the probablty jumps, when c J ( a) = y(7) = 180 at a , from to As c J ( a) contnues to ncrease from ths level, P 7 (a) decreases. When c J ( ) reaches y (8) = 580/3 at a , the probablty jumps agan from to 0.5, at whch level t stays for all lower values of a. Fgure 1 shows the resultng relatonshp between socal captal and the three-perod delnquency probablty (D = 4). Ths probablty decreases n the level of socal captal except for three dscrete jumps. The fnal jump s to (0.5) 4 = a a Proposton 3: Let u and the margnal ncome dstrbuton P(y) be gven. Then there exsts an nteger N such that f the number of agents N s greater than N, the non-repayment probablty satsfes the followng condton: When the correlaton between an ndvdual s ncome and the mean ncome of the other agents s below the maxmum level at whch full rsk sharng s sustanable and above the mnmum level at whch Ey c J 0, then as the correlaton s decreased the probablty that y t < c J for D 2 perods n a row n the optmal jont lablty contract vares both non-monotoncally and dscontnuously. For smaller N, the probablty s ncreasng n the correlaton. Proof: Let the margnal dstrbuton P(y) of ndvdual ncome be fxed, and let ρ be the correlaton between own ncome and others mean ncome. Let F(ρ): R [0, 1] be the cumulatve dstrbuton functon of the average ncome y, gven ρ. Because y takes on values from a fnte set { y (1),, y (S ) }, F(ρ) s a step functon. Let c J (ρ) be the optmal level of guaranteed consumpton under a jont lablty contract, gven ρ; c J (ρ) s decreasng n ρ. For values of c that are not n the support of F(ρ) (that s, such that y (s ) < c < y ( s +1) for some s ), the dervatve df(ρ)(c)/dc equals 0. Ponts n the support are where the dstrbuton jumps. Because decreasng the correlaton ρ tghtens the dstrbuton of y, the dervatve df(ρ)(c)/dρ s non-negatve for c < E( y ). Thus, for values of ρ such that c J (ρ) s not n the support of F(ρ), the dervatve 21

23 df( ρ)( c J ( ρ)) dρ = df( ρ)( c) = df( ρ)( c) = 0 dρ + df( ρ)( c) dρ dc dc J ( ρ) dρ J At values of ρ such that c (ρ) les n the support of the dstrbuton F(ρ), however, a small ncrease n ρ wll decrease c J (ρ) enough that the probablty F(ρ)( c J (ρ)) jumps down to the next step. At such values, ncreasng ρ nfntesmally results n a dscrete drop n F(ρ)( c J (ρ)). Thus, as ρ decreases, the probablty F(ρ)( c J (ρ)) that y < c J (ρ) s contnuously ncreasng except at a fnte number of ponts, where t falls dscontnuously. The probablty of default, [F(ρ)( c J (ρ))] D, follows the same pattern. As N grows, holdng the margnal dstrbuton P(y) fxed, the number of ponts near Ey n the support of F(ρ) ncreases. If N s large enough, there are ponts n the support of F(ρ) that le n the range of values that ( c J (ρ)) takes on for ρ between the mnmum and maxmum. For such N, the probablty of non-repayment vares dscontnuously and non-monotoncally wth ρ. Otherwse, the probablty s ncreasng n ρ. Q.E.D. We have used the per perod surplus from nternal rsk sharng relatve to autarky, [ Eu ( y) / Eu( y )] 1, as our measure of socal captal. Alternatvely, we could defne socal captal as the present dscounted value of the stream of surpluses, 1 Eu( y) 1. 1 δ Eu( y ) By that defnton, the dervatve of the delnquency rate wth respect to socal captal s not welldefned. If socal captal ncreases due to a decrease n the dscount factor δ, then the delnquency rate weakly declnes, because c J falls whle F remans unchanged. On the other hand, f lower aggregate rsk s responsble for the ncrease n socal captal, then the effect on the delnquency rate s ambguous, as explaned n Proposton 2. Under ths alternatve defnton of socal captal, then, there s stll no smple lnk between repayment performance and the level of socal captal. 22