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Journal of Development Economics 105 (2013) 131–139
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Journal of Development Economics
j ourna l homepage: www.e lsev ie r .com/ locate /devec
Repayment incentives and the distribution of gains from group lending☆
Jean-Marie Baland a,⁎, Rohini Somanathan b, Zaki Wahhaj c,1
a University of Namur, Belgiumb Delhi School of Economics, Delhi University, Indiac University of Oxford, United Kingdom
☆ We thank Christopher Ahlin, Greg Fisher, Eliana la FerGhosh, Andrew Newman, E. Somanathan and two anoncomments. Jean-Marie Baland is grateful for the finanEuropean Research Council (AdG-230290-SSD).⁎ Corresponding author. Tel.: +32 81 724866; fax: +3
E-mail addresses: [email protected] (J.-M(R. Somanathan), [email protected] (Z. Wahhaj).
1 University of Kent, United Kingdom.
0304-3878/$ – see front matter © 2013 Elsevier B.V. All rhttp://dx.doi.org/10.1016/j.jdeveco.2013.07.008
a b s t r a c t
a r t i c l e i n f oArticle history:Received 12 May 2011Received in revised form 21 June 2013Accepted 12 July 2013
JEL classification:I38G21O12O16
Keywords:MicrocreditJoint-liabilityGroup lendingRepayment incentivesSocial sanctions
Group loans with joint liability are a distinguishing feature of many microfinance programs. While such lendingbenefits millions of borrowers, major lending institutions acknowledge its limited impact among the very poorand have shifted towards individual loans. This paper attempts to explain this trend by exploring the relationshipbetween borrower wealth and the benefits from group lending when access to credit is limited by strategic de-fault. In our model, individuals of heterogeneous wealth face a given investment opportunity so poor investorsdemand larger loans. We show that the largest loan offered as an individual contract cannot be supported as agroup loan. Joint liability cannot therefore extend credit outreach in the absence of additional social sanctionswithin groups. We also find that the benefits from group loans are increasing in borrower wealth and that opti-mal group size depends on project characteristics. By allowing formulti-person groups andwealth heterogeneityin the population, the paper extends the standard framework to analyze joint liability and contributes to an un-derstanding of the conditions under which microcredit can reduce poverty.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
The ideology and practice of poverty alleviation have been deeplyinfluenced by the idea that access to credit can empower the poor.Microfinance programs around the world cover millions of borrowersand are providedunder a variety of different institutional arrangements.The overall gains from such lending are widely acknowledged but thereis growing concern about their capacity to reach those at the bottom ofthe income distribution. This has resulted in a lively debate, but littleconsensus on how credit contracts can be better designed to improvecredit access and the welfare of poor borrowers. A recurring questionwithin this debate is whether group loans with joint liability can effec-tively achieve these objectives.
Group loans were first popularized by the Grameen Bank ofBangladesh in the 1970s. Itwas believed that joint liabilitywould gener-ate social pressure on borrowers to repay loans and create a financiallysustainable model of lending. This approach was questioned in the late
rara, Maitreesh Ghatak, Parikhitymous referees for very usefulcial support provided by the
2 81 724840.. Baland), [email protected]
ights reserved.
nineties when natural disasters triggered widespread default and bor-rowers protested against rigidities in the lending program. TheGrameen Bank responded by introducing Grameen II which made allmembers individually liable for their loans (Kalpana, 2006; Yunus,2004). Within a couple of years, membership doubled, suggesting thatindividual loans catered to a previously unmet demand for credit.2 Asimilar switch to individual contracts was made by Banco Sol ofBolivia, another pioneer in group lending.
This trend towards individual contracts is far from universal. A ma-jority of the 663 institutions reporting to the Microfinance InformationExchange (MIX) in 2009 relied on some form of joint liability andmany large microfinance institutions offer a combination of group andindividual loans.3 An interesting contrast to the Grameen case is provid-ed by the microfinance sector in India which is dominated by village-based groups that strictly adhere to joint liability.4
2 Wright et al. (2006) report that “Grameen took 27 years to reach 2.5millionmembersand then doubled that in the full establishment of Grameen II”.
3 De Quidt et al. (2012b) classify institutions reporting to MIX by the types of contractsused and Ghate (2008) does this for 129 recognized microfinance institutions in India.Many of these organizations use both individual and joint liability contracts. The Bankfor Agriculture and Agricultural Cooperatives (BAAC) in Thailand also allows its membersto choose between group and individual loans (Ahlin and Townsend, 2007).
4 There are currently an estimated 77 million microfinance clients in India and overtwo-thirds of these are in what are called Self-Help Groups or more popularly, SHGs.(Srinivasan (2012), p 3, Table 1.1).
6 According to their website (Grameen Bank, 2009):A destitute person does not have tobelong to a group…Bringing a destitute woman to a level where she can become a regularmember of a group will be considered as a great achievement of a group.
7 Dewanand Somanathan (2011) studymembership in newly formedSelf-Help Groupsin India while Coleman (2006) examines selection into village banks in Thailand.
8 Whilewe have inmind the self-employment projects financed bymanymicrofinanceorganizations, we do not explicitlymodel effort decisions. The returns to the project in ourmodel can be interpreted as net of effort costs.
9
132 J.-M. Baland et al. / Journal of Development Economics 105 (2013) 131–139
In this paper, we attempt to explain some of these patterns bymodeling the relationship between borrower wealth and the benefitsfrom group lending. We allow individuals to be in groups of arbitrarysize and characterize loan contracts available to them as a function oftheir initial wealth. To focus on the role of wealth heterogeneity, we as-sume that all borrowers face identical investment opportunities and thepoorest investors therefore need the largest loans. Ourfirst set of resultscompares joint liability contracts with individual loans assuming thatgroups have no additional social sanctions to discipline their membersinto repayment. We show that the largest loan available under agroup lending contract with joint liability is strictly smaller than thelargest individual loan. For borrowers with access to both types of con-tracts, the benefits from group loans are increasing in wealth.
We next ask whether social sanctions within groups can increaseborrower benefits from group loans and also whether they can extendcredit to unbanked households. We show that social sanctions alwaysimprove welfare for those who already have access to group loans.This is because, unlike bank sanctions, they are not imposed in equilib-rium. They do not necessarily extend credit outreach. For a reasonableset of parameter values, individual contracts support larger loans thanjoint liability contracts even when social sanctions are arbitrarily large.
Our analysis reveals that the two-person groups that have dominat-ed the theoretical literature on group lending are quite particular andmany of our results on the interaction of wealth and welfare areobtained by generalizing this framework. When groups have only twomembers, the fraction contributing to loan repayment is at least onehalf. This fraction can vary more in large groups and this variation gen-erates a positive relationship between the wealth of borrowers andtheir benefits from investment. Our last result explores optimal groupsize. We find that two-member groups are never optimal for smallloan sizes but that the benefits from group lending are not generallymonotonic in the number of members. We illustrate this non-monotonicity with a numerical example.
Our work contributes to a well-established theoretical literature onthe mechanisms through which joint liability can affect investment de-cisions and borrower welfare.5 It is most closely related to Besley andCoate (1995) who were the first to demonstrate its ambiguous effectson repayment rates. Several subsequent papers have shown howrelaxing the informational or contractual assumptions of the traditionaljoint liability model can be welfare improving. Rai and Sjostrom (2004)assume that borrowers with unsuccessful projects can report the pro-ject outcomes of other members to the bank. Bhole and Ogden (2010)allow co-payments for defaulting projects to be asymmetric. Bond andRai (2008) and De Quidt et al. (2012a) both leverage social sanctionswithin groups without joint liability. Our work is also related to modelsof income dynamics in which the poor have either no access to credit orface unfavorable interest rates (Banerjee and Newman, 1993; Galor andZeira, 1993; Matsuyama, 2000). These papers use individual credit con-tracts while we show that the relative disadvantage to poor borrowersmay be heightened when banks also offer group loans.
Takingmodels such as ours to the data on group lending programs isa challenge. Microfinance institutions operate in many different envi-ronments and their objectives vary from pure profit-maximization tobroader social missions. Besides, empirical work on lending contractstypically estimates average impact whereas we are concerned withthe relationship between benefits and loan size. Our theoretical resultsare however broadly consistent with patterns found in experimentaland observational data.
Several studies have highlighted targeting deficiencies in micro-finance programs. Morduch (1998) shows that eligibility criteria formembership in the Grameen Bank were frequently violated and thenon-poorwere admitted asmembers.With the introduction of GrameenII in 2002, the Grameen Bank explicitly acknowledged that the poor are
5 Useful surveys of this literature can be found in Armendariz de Aghion and Morduch(2005) and Ghatak and Guinnane (1999).
often best served outside groups.6 Dewan and Somanathan (2011) andColeman (2006)findparticipation rates rising in income in very differentregional contexts.7 Hermes and Lensink (2011) and Sebstad and Cohen(2001) survey programs in multiple countries and emphasize thetrade-off between the financial sustainability of microfinance institu-tions and their outreach among the poor.
In our model, larger bank sanctions allow groups to achieve higherrepayment rates. These sanctions could take the form of a rise in the fu-ture cost of credit. Karlan and Zinman (2009) collaborate with a South-African bank in an unusual field experiment and show that commit-ments of lower future interest rates improve repayment on currentloans. Abbink et al. (2006) and Cassar et al. (2007) provide experimen-tal evidence on the importance of social ties in repayment decisions.Giné and Karlan (2009) study a microfinance institution in thePhilippines that randomly assigned new areas to either joint or individ-ual liability. As in theGrameen case, the introduction of individual liabil-ity attracted many more clients. There was no significant difference inrepayment rates across the two types of contracts but it is hard to inter-pret this finding since repayment rates were close to 100% under bothregimes.
This empirical literature suggests that the benefits from micro-finance do vary with household wealth.We hope this paper encouragesmore specific tests of the mechanisms underlying this relationship. Therest of the paper is organized as follows. The next section describes ourmodel and derives optimal contracts for individual and group loans.Section 3 compares credit outreach and borrower welfare under thetwo types of loans in the absence of social sanctions. Section 4 extendsthe model to explore the effect of social sanctions on group contracts.Section 5 discusses optimal group size and Section 6 concludes.
2. The model
Our principal unit of analysis is a set of risk neutral households, eachof whom can choose to invest in a project. The project requires one unitof capital and no other inputs.8 It returns ρ with probability π and zerootherwise. Wealth, which we denote by w, varies continuously overthe (0, 1) interval and is used for investment whenever a loan of(1 − w) can be obtained. We are interested in characterizing the setof wealth levels over which households are eligible for individual andgroup loans and the benefits from these two types of contracts as a func-tion of initial wealth.
The banking system is competitive and offers depositors a gross re-turn r, equal to the opportunity cost of bank funds. Banks lend eitherto individuals or to groups of size n under joint liability. We assumethat groups are homogeneous in the wealth of their members. This al-lows us to abstract from questions of redistribution within groups andfocus on the risk-pooling function of joint liability contracts. In practice,many successful group lending programs have encouraged the creationof groups with similar members.9 Interest rates for each contract varywith repayment rates to equate the expected return from all lendingto r. Project returns are never observed by the bank. Under group lend-ing they are observed by members of the group.
The assumption of a single indivisible investment project requiressome justification. One might expect poorer borrowers to invest in
For example, when promoting micro credit through the commercial banking system,India's central bank explicitly recommended that savings and credit groups be formedwith households of “homogeneous background and interest” (National Bank for Agricul-ture and Rural Development, 1992).
133J.-M. Baland et al. / Journal of Development Economics 105 (2013) 131–139
projects requiring smaller loans. In the Grameen case, for example, thepoorest borrowers are often vegetable vendors or basket makers whileslightly richer households invest in looms or small shops. Our purposeis not to model investment choices and the distribution of wealth inan economy as a whole, but rather to illustrate that for a given project,access to and benefits from joint liability are systematically related toinitial wealth through differences in required loan sizes.
Banks have available non-pecuniary sanctions K which they can useto induce borrowers to repay their loans. These sanctions could representharassment by bank officials, unfavorable future contracts or a damagedreputation.We simply think ofK as the enforcement capacity of a bank.10
All loan contracts specify a loan amount, a gross interest rate and thevalue of the sanctionK∈ 0;K
� �that is levied if a borrower defaults. For in-
dividual contracts, the sanction is levied on a single borrower in case ofnon-repayment. For group contracts, all members are sanctioned K ifthe group fails to repay what it collectively owes. Since bank sanctionsprovide no direct benefits to the bank and are costly to the borrower,competition ensures that the value of K is all chosen to maximize bor-rower benefits for both types of contract and each loan size.
We refer to a contract as feasible if it induces repayment by bor-rowers whenever they are able to do so. For individual loans thismeans that all borrowers with successful projects repay their loans.For group contracts, the group reimburses the loan whenever enoughprojects within the group succeed. We will specify this more preciselybelow. An optimal contract for each loan type and loan size is the feasi-ble contract which maximizes welfare. Since banks always break even,this corresponds to the one with the highest benefits per borrower.We denote these benefits by Ui and Ug for individual and group con-tracts respectively.
Under individual contracts, those with unsuccessful projects neces-sarily default on their loans. If banks use sanctions to ensure all success-ful borrowers repay their loans, the repayment rate is π and thecompetitive gross interest rate charged by banks must be r
π for allloans. The smallest sanction that induces repayment for a loan of sizeL is thus
K ¼ rπL: ð1Þ
A loan contract is then given by L; rπ ;K� �
, where K depends on Laccording to Eq. (1). Expected benefits from investment depend on pro-ject returns, sanctions and the opportunity cost of the borrower's ownfunds and are given by
Ui ¼ π ρ− rπL
� �− 1−πð ÞK−rw:
On substituting for K and w, this expression simplifies to
Ui ¼ πρ−r− 1−πð Þπ
rL: ð2Þ
Those with smaller loans face lower bank sanctions and thereforeenjoy higher benefits. Since banks only lend if K≤K, the largest loan of-fered is
L ¼ πKr
: ð3Þ
L is a measure of credit outreach for individual loans. We assume that Lb1 so not all households have access to credit. We also assume that the
10 Timely repayment could produce future benefits in the form of lower interest rates,less monitoring and larger loans. The role of dynamic incentives is controversial. Karlanand Zinman (2009) find them to be quantitatively important in lending experimentswhile Bulow and Rogoff (1989) argue that the financial gain to the borrower from defaultcan often compensate for reputational losses. For our purpose, it is sufficient that bankshave some enforcement capacity in any form which makes lending possible.
poorest borrower with such access finds investment worthwhile. FromEqs. (2) and (3), we see that this requires the project return ρ to be atleast
ρ ¼ 1−ππ
K þ rπ: ð4Þ
We turn now to group contracts. A group of size n and homogeneouswealthw requires a loan of size n(1 − w) = nL. For each loan size L, wewould like to verifywhether there is a gross interest rate and bank sanc-tion that makes such a loan feasible and if so, compute borrower bene-fits to arrive at the optimal contract. While unsuccessful projects cannotcontribute to loan repayment, successful members honor the group'sobligation to the bank as long as the contribution required from eachmember is less than the bank sanction. The repayment rate for grouploans therefore depends on the probability of this event.
Let B(n,j,π) be the binomial probability of j or more successes in nindependent trials, with a success probability of π on each trial. Wedenote this by Bj when there is no ambiguity about the n and π inquestion. Suppose the loan is repaid whenever there are j or moresuccessful projects within the group, and not otherwise. Then the re-payment rate is Bj and banks break evenwhen charging a gross inter-est rate r
B j. The total loan obligation is nL r
B j. The contribution per
successful project depends on the number of successful projects.When exactly j projects succeed, each project contributes n
jrB jL. For
more favorable states, this contribution is smaller as shown below.A group lending contract with per-member loan size L is feasible ifthere exists an integer j ≤ n for which:
njrBj
L≤K: ð5Þ
The bank sanction which induces repayment by each of the j suc-cessful members is
K j ¼njrBj
L: ð6Þ
The largest loan size forwhichEq. (5) is satisfied gives us themargin-al householdwith access to group loans. This is determined by the valueof j for which the function jBj is maximized. The following lemma char-acterizes this function and is useful in deriving many of our results:
Lemma1. The function jBj is single-peaked. It starts at zero, takes the valuenπn at j = n and attains a maximum at j∗ ≤ ⌈nπ⌉, the smallest integergreater than or equal to nπ. If πb 1
n , it is decreasing for j ≥ 1 and, if πNn n−1ð Þ
1þn n−1ð Þ, it is increasing throughout.
Proof. See Appendix A
We can now derive the benefit from an optimal group contract
L; rB j;K j
� �. If there are fewer than j successes in the group, then each
member is sanctioned Kj and the successful members keep the returnfrom their projects. If j or more projects succeed, equal contributionsby all successful members are used to repay the loan. Using πk to denotethe probability of exactly k successes in n trials, the expected per-member benefit from the group loan is
Ugj ¼
Xj−1
k¼0
πkknρ−K j
� �þXnk¼ j
πkkn
ρ−nkrLBj
!" #−rw
¼ ρn
Xnk¼0
kπk−r Lþwð Þ− 1−Bj
� �K j
¼ πρ−r− 1−Bj
� �njrBj
L:
ð7Þ
134 J.-M. Baland et al. / Journal of Development Economics 105 (2013) 131–139
Our first result links this benefit to the repayment rate Bj:
Proposition 1. Consider the set of feasible group contracts L; rB j;K j
� �for a
per-member loan of size L. The optimal contract is the one with the highestrepayment rate.
Proof. See Appendix A
This result highlights an important difference between individualand group contracts. For an individual loan, the borrower prefers thecontract with the lowest possible sanction since this is the penaltyincurred whenever the project fails. In the case of group loans,there is a possible trade-off. On the one hand, a higher sanction im-poses additional costs on all group members when the group de-faults. But raising the sanction also raises the repayment rate andthis lowers the interest rate because it increases the extent towhich successful members are willing to subsidize failed projects.With the Binomial distribution, which determines the number ofsuccesses in groups in our model, the increase in the repaymentrate Bj is large relative to the increase in the required sanction, Kj,and the net effect of a higher sanction is always positive.
The next section uses the optimal individual and group contracts todescribe the equilibrium relationship between borrowerwealth and thegains from investment.
3. Equilibrium credit contracts
Equilibrium in ourmodel consists of twowealth intervals correspond-ing tohouseholds that are offered individual and group loans respectively.A household, if offered both contracts, chooses the one with the higherbenefit. We start by showing that there is always a set of householdswho are offered individual loans but have no access to group loans. Inother words, joint liability per se cannot extend credit outreach.
Proposition 2. The largest loan available in a group lending contract isstrictly smaller than in an individual contract.
Proof. The largest individual loan L is not feasible as a group loan if, forall 0 b j ≤ n,
njrBj
LNK:
Substituting for L from Eq. (3), we can rewrite this condition as
nπN jB j:
The LHS is the expectation of a binomial distribution with parame-ters n and π. Writing this as a sum, we see that the above condition al-ways holds:
nπ ¼Xnk¼0
kπkNXnk¼ j
kπk≥ jXnk¼ j
πk ¼ jB j:
This result is driven by the fact that banks extract less from groupsthan from individual loans of size L . The maximum amount that thebank can recover from any borrower is K . Since only successful bor-rowers are able to repay their loan, expected repayments are boundedabove by nπK . With individual loans of size L, all successful borrowersdo repay the bank and so this bound is achieved. Under group lending,banks recover nothing from successful projects in defaulting groupsand cannot extract more than K from any project in a group that doesnot default. These leakages from the banking system cause the inequal-ity in Eq. (5) to fail at L.
Although joint liability does not extend outreach, it does result inhigher welfare for borrowers with small loans. Using Eqs. (2) and (7),
the difference between the benefit from a group loan and an individualloan of size L is given by:
Ug−Ui ¼ Lr1−ππ
−1−Bj
� �Bj
nj
24 35: ð8Þ
Thus, if a group contract L; rB j;K j
� �satisfies the condition
1−ππ
N1−Bj
� �Bj
nj; ð9Þ
then each member is better off than they would be under an individualcontract. The following result identifies one interesting case for whichthis condition always holds.
Proposition 3. If a per-member loan L is small enough to require only onesuccessful project for group repayment, borrower welfare is always higherunder group lending.
Proof. Rewriting Eq. (9) for j = 1, we get
1− 1−πð Þn−nπ 1−πð Þn−1N0:
The above expression is simply the binomial probability of morethan one success in n trials and is greater than zero for all n ≥ 2.
Group loans are always preferred to individual contracts in this casebecause groups default on their loan only when all projects fail. Butthese borrowers could not have repaid their loan with an individualcontract. For any other distribution of project outcomes, successful pro-jects in groups cover the unsuccessful ones, resulting in higher overallrepayment rates. The condition that one successful project in thegroup is enough to reimburse the entire group loan is quite restrictive.This is sufficient but not necessary. For medium and large groups, ben-efits from group loans are higher than from individual contracts underweaker conditions. With small groups however, this condition oftenbinds, as illustrated by the following example. If n = 3, π = .5, r =1.2 and K ¼ 1, the largest individual loan is .42 and the largest grouploan is .27. As long as the loan size is below .25, only one successful pro-ject is required for group repayment and welfare is higher under thegroup contract. For L = .24, the difference in benefits is maximizedwith Ug = .38 andUi = .21. For higher values of L, group repayment re-quires at least 2 successes and per member benefits fall below thosefrom individual contracts. As L increases marginally to .25, Ug fallsabruptly to .05 while Ui is only slightly lower at .20. If n = 4, with allother parameters at the above values, benefits from group loans arehigher than under individual loans for all loan sizes offered under bothtypes of contracts. We return to the effects of group size in Section 5.
4. Social sanctions
We have now obtained two main results. First, in the absence of so-cial sanctions, joint liability cannot increase credit outreach. Second,borrowers who need small loans prefer group lending. In this sectionwe explore how these results are modified if groups can augment theenforcement capacity of the bank with social sanctions. Our treatmentof these sanctions is standard and follows Besley and Coate (1995).Our contribution is to demonstrate how they interact with borrowerwealth to determine the benefits from group loans. Given the popularassociation of microcredit groups with these types of informal sanc-tions, we believe that this is a useful extension of our model.
We model social sanctions as a utility cost γ on members who areable to contribute their fair share of the group loan but choose not todo so. Since the size of this sanction does not depend on the numberof contributing members, it is best interpreted as a fall in a borrower'sstatus within the community or a partial or total exclusion from group
135J.-M. Baland et al. / Journal of Development Economics 105 (2013) 131–139
activities.11 Members with unsuccessful projects are not sanctionedsince they have zero returns and cannot contribute.12
We nowmodify the feasibility constraint in Eq. (5) for a group loan ofsize L:
njrBj
L≤min ρ;K þ γ� �
: ð10Þ
A group-lending contract with per-member loan size L is feasible ifthere exists some j for which members with successful projects areboth able and willing to pay their share of the loan. This requires thateach member's contribution be less than the return ρ and that it belower than the sum of bank and social sanctions. We can think ofthese two constraints as the liquidity and group incentive constraintsrespectively. We ignored the liquidity constraint in previous sectionsbecause ρ was larger than the total sanction K .13 This need not be thecase when total sanctions are K + γ.
We show that the effects of social sanctions on credit outreach areambiguous. They do improve repayment incentives but there are rea-sonable conditions under which even arbitrarily large sanctions donot support larger loans than those available as individual contracts.In contrast, higher social sanctions always improve borrower welfarefor those with group loans because, unlike bank sanctions, they arenever imposed in equilibrium. Since project outcomes are observedwithin groups but not outside them, banks sanction whenever defaultoccurs whereas groups use sanctions to induce repayment withoutpunishing those with failed projects.
Analogous to our previous analysis of group loans, the optimal contractcorresponds to the smallest value of j forwhich Eq. (10) is satisfied. If fewerthan this number of successes occur, the group defaults, the bank sanctionsall members and no social sanctions are used. In all other cases, the grouprepays the loan and each successfulmember contributes an equal share to-wards repayment. Comparing Eqs. (5) and (10), we see that social sanc-tions relax the incentive compatibility constraint and allow a larger shareof project returns to be extracted from each successful member. This in-creases possibilities for risk pooling and the group repays the bank loan instateswith fewer successful projects than in the absence of social sanctions.
Our next result outlines the conditions under which social sanctionsallow all individual loans to be supported as group loans, therebyextending credit outreach:
Proposition 4. Let social sanctionsγ≥ρ−K, whereρ is theminimumpro-ject return andK is themaximumbank sanction. ThenL, the largest individ-ual loan, is a feasible group loan for groups of size n = 2 or for a projectsuccess probability π≤ 1
4. In contrast, for large enough π, ρ ¼ ρ and n N 2,there always exist loan sizes which will be offered as individual loans butnot as group loans.
The proof is in the Appendix A and is based on the following idea.Whenγ≥ρ−K, repayment is constrained by liquidity and not by incen-tives. The minimum return ρ, is decreasing in π because risky projectsmust have higher returns to make them worthwhile. Social sanctionsallow these to be extracted under group lending contracts. If theexpected number of successful projects is an integer, the bound on πcan be considerably relaxed and L is a feasible group loanwhenever π≤12.14 The proposition once again illustrates that the two-member groups
11 There is no distinction between the total sanction and the sanction per contributingmember in the Besley–Coate model because with groups of two, each member can onlybe sanctioned once.12 One might ask whether it is plausible that group members are sanctioned if all mem-bers are successful and default collectively. We could allow for no sanctions in this case,but this would complicate the model and for moderately sized groups, this is a low prob-ability event.13 In Section 2, ourfloor on project returns given by Eq. (4) andLb1together implied this.14 When nπ is an integer, themean,median andmode of a binomial distribution coincideandwe are able to exploit this feature to relax the condition on π. Details are in the Appen-dix A.
that have been the focus of earlier studies on group lending are a specialcase for which high enough sanctions always extend outreach.
While social sanctions do not necessarily improve credit access, theyalways raise benefits from group loans by reducing the bank sanctionrequired for each loan size. To see this, consider a group contract
L; rB j;K j
� �that is feasible when γ = 0. For small γ, the same loan re-
mains feasible, but the bank sanction needed to enforce it is reducedby γ to:
K ′j ¼ K j−γ ¼ n
jrBj
L−γ: ð11Þ
For γ ≥ Kj, social sanctions substitute fully for bank sanctions andK′j ¼ 0.
Social sanctions can also improvewelfare through higher repaymentrates by relaxing the constraint in Eq. (10) and thereby reducing thenumber of successes in the optimal group contract. Utility from thegroup loan is now
Ugj ¼ πρ−r− 1−Bj
� �K ′
j: ð12Þ
This discussion is summarized in our next result. A complete proof isin the Appendix A.
Proposition 5. Social sanctions result in higher welfare for borrowersalready using group loans. If the largest per-member loan requires atleast two ormore successful projects within the group for repayment, socialsanctions also raise repayment rates.
5. Group size
The literature on group lending has focused on two-member groupsand fixed loan sizes. Our previous results have shown that group sizedoes influence both access to and the benefits from group loans. Inthis section, we explore the interaction between borrower wealth, pro-ject characteristics and group size. The following result shows that theeffects of group size on welfare depend both on the size of the loanand the project success probability.
Proposition 6. If π≥ n n−1ð Þ1þn n−1ð Þ, the largest loan available to a group of size n
is both feasible and generates higher per-member welfare in smaller groups.In contrast, for sufficiently small loans and no social sanctions within groups,a fall in group size lowerswelfare and two-member groups are never optimal.
The proof of the second part of the proposition is intuitive. Grouploans raise welfare whenever they lead to lower expected sanctions. Ifa loan is small enough to require only one success for repayment,large groups are favored because the probability of one or more suc-cesses, B(1,n,π), is increasing in n. The result will hold for zero or smallsocial sanctions. If social sanctions are large, they could substitute fullyfor bank sanctions and since no sanctions will then be implemented inequilibrium, larger groups would have no added benefits.
For the first part of the result, we know from Lemma 1 that when the
project success probability is greater than n n−1ð Þ1þn n−1ð Þ, the largest loan offered
to a group of size n requires allmembers to succeed. This ismore likely forsmaller groups. To complete the argument, we need to show that loans ofthis size are feasible for a smaller group. This is done in the Appendix A.
Moving away from these limit cases we find non-monotonic effects ofgroup size on repayment rates and borrower welfare for given loan andproject characteristics. Fig. 1 illustrates this for a loan size L = .4 and aproject success probability, π = .5. Bank and social sanctions are 1 and.5 respectively and the risk free interest rate is 20%. For all group sizes be-tween 2 and 20, we show the repayment rate, the fraction of successfulprojects and the net benefit, (Ug–Ui), from the optimal group contract.Group sizes in microfinance programs are typically in this range.
repayment rate
fraction of successes
0.2
.4.6
.81
0 5 10 15 20group size
Fig. 1. Group size effects π ¼ :5; L ¼ :4;K ¼ 1;γ ¼ :5; r ¼ 1:2� �
.
136 J.-M. Baland et al. / Journal of Development Economics 105 (2013) 131–139
In this example, an increase in group size initially lowers welfare. Re-payment in a group of 2 requires one ormore projects to succeed, and thishappens with probability equal to 3
4. For a group of 3, repayment occurswhen at least 2 projects succeed and this probability is 1
2. As group size in-creases locally around 3, so do repayment rates and benefits from thegroup loan. When n = 6, repayment rates and net gains fall once more,since repayment now occurs when 3 or more projects succeed and thisprobability is roughly 2
3. The scissor-like patterns in the figure mirror thechanges in the value of j on which the optimal group contract is based.
It is useful to briefly summarize what we have learned about group-size effects. We know that when project success probabilities approach1, a decrease in group size raises welfare. We have also shown that forsufficiently small loans, two-member groups are never optimal. This isrelated to Proposition 3 in Section 3. Proposition 4 on the other hand,shows that two-member groups often maximize credit outreach. Thiscontrast highlights a central theme in this paper, namely that the char-acteristics of groups that increase gains from group loans are not alwaysthe ones that increase access to them. For other parameter values, thebenefit from group lending could either increase or decrease in thesize of the group. In small groups, additional members can dramaticallychange the fraction of successful projects needed for repayment.Changes in this fraction are mirrored in oscillating repayment ratesand expected benefits from group loans. General results on group sizeare elusive and need to be explored further.
6. Conclusion
This paper is motivated by the now common observation that grouplendingwith joint liability ismore successful formoderately poor house-holds than for the very poor.We highlight onemechanismwhich gener-ates this pattern of benefits by examining how credit contracts vary byloan size. We characterize conditions under which individual contractssupport larger loans than joint liability contracts. We also show thatfor those with access to group loans, wealthier households can poolrisk more effectively. This results in higher repayment rates, infrequentbank sanctions and greater benefits from group lending.
Our paper extends the standard approach in the group lending liter-ature in two important respects. First, we move away from two-persongroups to arbitrary group sizes. Second,we allow thenature of the creditcontract to vary by project and borrower characteristics.We believe thatthese extensions are significant, not least because both credit contractsand group sizes observed in the microfinance sector vary considerably.To focus on repayment incentives we ignored the effects of group lend-ing on borrower selection and on project and effort choices. There maybe interesting interactions between wealth and credit contracts along
these dimensions that remain unexplored, so generalizing existingmodels to address these aspects of group lending could be fruitful.
It is difficult to link ourmodel tightly to existing empirical studies ongroup lending because these rarely focus on the relationship betweenloan size and impact. Our analysis suggests that this is a useful directionto pursue in empirical research. In terms of policy, our results suggestthat effective strategies to expand rural credit require a mix of contrac-tual arrangements, and an exclusive focus on group lending may bemisplaced.
Appendix A
Proof of Lemma 1
Let ⌊nπ⌋ denote the largest integer below nπ and ⌈nπ⌉ the smallest in-teger above it. The function jBj takes the value zero at j = 0 and is pos-itive for all j N 0. We begin by showing that it is decreasing to the rightof ⌈nπ⌉. We then establish that it is single-peaked by showing that, if forsome j∗ b ⌈nπ⌉, jBj N (j + 1)Bj + 1, then this relationship holds for all j inthe range j* ≤ j ≤ ⌊nπ⌋. Finally, for the case where ⌈nπ⌉ = n, we derivethe lower bound on π given in the lemma for which jBj is increasingthroughout.
1. jBj is decreasing to the right of j = ⌈nπ⌉:Consider n N j ≥ ⌈nπ⌉. The func-tion jBj is strictly decreasing at j if jBj N (j + 1)Bj + 1. The LHS can bewritten as jBj + 1 + jπj and rearranging terms gives us the condition:
jπ jNBjþ1: ð13Þ
The binomial probabilities, πj and πj + 1 are respectively
π j ¼ π j 1−πð Þn− j n!j! n− jð Þ!
and
π jþ1 ¼ π jþ1 1−πð Þn− j−1 n!jþ 1ð Þ! n− j−1ð Þ! :
Writing πj in terms of πj + 1 and multiplying by j, we get
jπ j ¼j
n− j1−ππ
� �jþ 1ð Þπ jþ1:
But jn− j
1−ππ ≥1 if j ≥ nπ, which implies
jπ j≥π jþ1 þ jπ jþ1:
In the same way, we can express jπj + 1 as a function of πj + 2:
jπ jþ1 ¼ jn− j−1
1−ππ
� �jþ 2ð Þπ jþ2:
Since jn− j−1
1−ππ N1 for j ≥ nπ,
jπ jþ1Nπ jþ2 þ jπ jþ2:
Combining the expressions for jπj and jπj + 1 we get
jπ jNπ jþ1 þ π jþ2 þ jπ jþ2:
For all k ≤ n − 1, we can continue expressing jπk as a function ofπk + 1, and we obtain:
jπ jNXnk¼ jþ1
πk ¼ Bjþ1:
As a result, jBj is strictly decreasing to the right of ⌈nπ⌉.
137J.-M. Baland et al. / Journal of Development Economics 105 (2013) 131–139
2. jBj is single-peaked to the left of ⌈nπ⌉:Weuse the property of a Binomialdistribution that the modeM of the distribution is either ⌊nπ⌋ or ⌈nπ⌉(Kaas and Buhrman, 1980).Let us first consider any value j, 1 ≤ j ≤ ⌊nπ⌋ for which jBj is increas-ing at j. From Eq. (13), this implies that jπj ≤ Bj + 1. But since πj ismaximized at the mode, which is at least as large as j, the LHS jπj isincreasing for all j b j. The upper tail probability, Bj + 1 is strictly de-creasing in j throughout. It follows that ∀ j b j, jBj must be increasingat j.Now suppose that j′′ ≤ ⌊nπ⌋ is the smallest value of j atwhich Bj is de-creasing, or equivalently, jπj ≥ Bj + 1. Since j″ is less than the mode,this inequality must hold for all j between j″ and the mode by thesame reasoning given above, i.e. jπj is increasing in j until the modeand Bj + 1 is strictly decreasing in j. If no such value exists, jBj is in-creasing throughout.
3. For jBj to be increasing throughout,we require themode to be greaterthan (n − 1) and in addition, using Eq. (13), (n − 1)πn − 1 b πn. Wecan rewrite this inequality as
n n−1ð Þπn−1 1−πð Þbπn
or
πNn n−1ð Þ
1þ n n−1ð Þ :
However, at the values of π satisfying the above condition, nπ N n − 1so thatM ≥ n − 1.
Proof of Proposition 1
Recall that Ujg is the benefit to the group from a contract in which
repayment occurs whenever there are j or more successes. The
contract is given by L; rB j;K j
� �, where r
B jis the interest rate charged
and K j ¼ njrB jL , is the sanction applied to each member when the
group defaults. We will show that Ujg N Uj + 1
g whenever both con-tracts are feasible.
We restrict our consideration to contracts with j + 1 ≤ ⌈nπ⌉ (thesmallest integer greater than or equal to nπ) since we know byLemma 1 that contracts with higher values of j are never optimal. Wecan write the difference in benefits from Uj
g and Uj + 1g as
Ugj−Ug
jþ1 ¼ Δ j−1 þ Δ j þ Δ jþ1:
Δj − 1 is the difference in benefits from the two contracts for states withfewer than j successes, Δj is this difference when exactly j successesoccur andΔj + 1 is the difference in benefits whenmore than j successesoccur. In the first case, the group is sanctioned under both contracts, inthe third case, it is not sanctioned under either contract, and in the sec-ond case, when exactly j successes occur, it is sanctioned under
L; rB jþ1
;K jþ1
� �but not under L; r
B j;K j
� �. These three differences are
given by
Δ j−1 ¼Xj−1
l¼0
πl lρ−nK j
� �− lρ−nK jþ1
� �h i¼ n 1−Bj
� �K jþ1−K j
� �Δ j ¼ π j jρ−nLr
Bj
!− jρ−nK jþ1
� �" #¼ π jnK jþ1−π j
nLrB j
Δ jþ1 ¼Xnl¼ jþ1
πlnLrB jþ1
−nLrBj
" #¼ nLr
Bj−Bjþ1
BjBjþ1Bjþ1 ¼ π j
nLrB j
:
Summing the three terms we obtain
Ugj−Ug
jþ1 ¼ n 1−Bj
� �K jþ1−K j
� �þ π jnK jþ1
and substituting for Kj and Kj + 1, we see that Ujg − Uj + 1
g N 0 as longas
n2Lrπ j
jþ 1ð ÞBjþ1þ 1−Bj
� � jB j− jþ 1ð ÞBjþ1
jB j jþ 1ð ÞBjþ1
" #N0
or
π j þ 1−Bj
� � jB j− jþ 1ð ÞBjþ1
jB jN0:
Using Bj + 1 = Bj − πj we need
jþ 1ð Þπ jNBj 1−Bj þ π j
h ior
jþ 1ð Þπ jNBj π0 þ π1 þ…þ π j
h i:
The sumof the (j + 1) terms on the RHS of the above inequality is al-ways smaller than the LHS if πj is increasing in j. Since any contract underconsiderationmust have (j + 1) b ⌈nπ⌉, we always have j ≤ ⌊nπ⌋ andweknow that πj is increasing in this range. So Uj
g N Uj + 1g .
Proof of Proposition 4
We start with the first part of the proposition and show that underthe stated conditions on n, π and γ, the largest individual loan L can beoffered as a group contract even when the project return takes its min-imum value of ρ.
Let γ≥ρ−K and ρ ¼ ρ. We know from Eq. (10) that repayment isconstrained by the project return ρ. L is therefore feasible if there existsj such that
njrBj
L≤ρ:
Substituting for L from Eq. (3) and for ρ from Eq. (4), this conditioncan be rewritten as
njπBj
K≤ 1−ππ
K þ rπ:
Now, since L ¼ πKr b1; rπ NK . It is therefore enough to show that
njπBj
K≤ 1−ππ
K þ K;
or equivalently
njπBj
≤ 1π: ð14Þ
If n = 2, set j = 2and sinceB2 = π2, Eq. (14) holdswith equality, soa loan of size L is always feasible for a two-person group.
The rest of the proof uses a well-known result on the relationshipbetween the mean (nπ) and the median (m) of a binomial distribution(Kaas and Buhrman, 1980). This states that the median is either ⌈nπ⌉,the smallest integer above the mean or ⌊nπ⌋, the largest integer belowthe mean. For integer values of the mean,m = nπ.
We will first prove the result for integer values of nπ and then con-sider non-integer values.
If nπ is an integer, set j = nπ = m. The inequality in Eq. (14) isnow
π≤Bm:
138 J.-M. Baland et al. / Journal of Development Economics 105 (2013) 131–139
But sincem is themedian,Bm≥ 12 so withπ≤ 1
2 this is always true. Forinteger values of n, we therefore have an upper bound for π of 12, which islarger than stated in the proposition.
Now consider non-integer values of the mean. The median in thiscase must either be ⌈nπ⌉ or ⌊nπ⌋. If the median m = ⌈nπ⌉ then set j =⌈nπ⌉ in Eq. (14). Since nπ b ⌈nπ⌉, the LHS of Eq. (14) is smaller thanwhen nπ is an integer and the result goes through a fortiori.
If instead m = ⌊nπ⌋, consider first j = ⌈nπ⌉ = 1. In this case nπ isstrictly less than 1 and Eq. (14) holds if π b B1. But B1 = 1 − (1 − π)n
so this is always true. If ⌈nπ⌉ N 1, set j = ⌊nπ⌋ = m. We can now rewriteEq. (14) as
nm
πBm
≤ 1π:
But since nπ b ⌈nπ⌉, it is enough to show that
π≤ mmþ 1
Bm:
The ratio mmþ1 is increasing in m and takes its minimum value of 1
2
when j = 2. The minimum value taken by Bm is also 12, so the above in-
equality holds whenever π≤ 14.
We now turn to the second part of the proposition and show that forlarge enough π and n N 2, there always exist loan sizes that will be of-fered as individual but not group loans.
By Lemma1,whenπN n n−1ð Þ1þn n−1ð Þ, the function jBj ismaximized at j⁎ = n.
The largest feasible group loan therefore has a repayment rate Bn = πn. Ifthe group loan is equal to L, each successful member in the group is re-quired to contribute r
πn L, while the minimal return to the project ρ is
ρ ¼ 1−πð Þπ
K þ rπ¼ 1−πð Þ
πrLπþ rπ:
Required payments are therefore higher than ρ whenever
LNπn
π−πn−1 þ πn :
If n N 2, the RHS of the above expression is smaller than one andthere exist loan sizes L that are not feasible group loans even for arbi-trarily high social sanctions.
Proof of Proposition 5
Let L; rB j;K j
� �be the optimal group contract for a loan of size Lwhen
γ = 0. This contract results in repaymentwhen there are j ormore suc-cessful projects in a group so the repayment rate is Bj.
Denote by ej , the number of successes in the optimal contract withγ N 0. Without loss of generality, we can restrict ourselves to valuesof γ less than ρ−K . Beyond this value, contributions are constrained byproject returns ρ and higher social sanctions have no effect on incentives.
Since the increase in γ relaxes (Eq. (10)), we know thatej≤ j. If, for allfeasible loans andγ = 0, the parameters ρ andK are large enough to en-able repaymentwith only one success in the group,ej ¼ j ¼ 1 and repay-ment rates cannot change since B1 is themaximum possible repaymentrate. We see from Eq. (12) that the benefit from a group loan increases.
If, on the other hand, some group loans require more than one suc-cess for repayment, repayment rates also increase, or equivalently, ejb jfor some loan sizes. Denote by L1
g the largest loan per member that canbe repaid with only one success in the group when social sanctionsare equal to γ. From Eq. (10), this is given by
Lg1 ¼ K þ γ� �B1
nr:
Moving from zero to positive social sanctions, this increases by γ B1nr.
All thosewith loans in the interval K B1nr ; K þ γ� � B1
nr
� ihave higher repay-
ment rates than in the absence of social sanctions, since they now re-quire only one success to be reimbursed while they previously
required at least two successes. For these loansej ¼ 1b j. More generally,
groups with loan sizes in the interval K jB j
nr ; Lgj γð Þ
� irequired at least
(j + 1) success for repayment in the absence of social sanctions andnow require atmost j successes. Sincewealth has a continuous distribu-tion on (0, 1), all these intervals are non-empty. Repayment rates there-fore go up for some loan sizes, causing an increase in the overallrepayment rate.
Proof of Proposition 6
For a group of size n, we know from Lemma 1 that if π≥ n n−1ð Þ1þn n−1ð Þ ¼ π
nð Þ, the largest loan requires all nmembers to succeed. Since π(n) is in-creasing in n, smaller groups will also require all members to succeed ifπ≥π nð Þ. For these values of π, we see from Eq. (5) that the largest loanavailable to groups with n members is
Lgn ¼ Kπn
r
which is decreasing in n. The largest loan available to a group of size n istherefore also available to a group of size n − k.
For a loan of sizeLgnð Þ, the difference in borrower benefits for groups
with (n − k) members and n members is at least
rLg nð Þ 1−πn
πn −1−πn−k
πn−k
" #N0:
This is a lower bound on the relative benefits of the smaller groupsince, for a loan of size L
gnð Þ , a group with (n − k) members may
need less than (n − k) successes for repayment and benefits increasingin the repayment rate.
We now show that borrower benefits are increasing in group size forvery small loans. Consider a loan size Lwhich is repaid by a group of sizen and a group of sizen − 1with only one success. The difference in ben-efits to a borrower in an n member group relative to an (n − 1) mem-ber group is given by
Ugn−Ug
n−1 ¼ 1−B 1;n−1;πð Þð Þ n−1ð ÞLrB 1;n−1; πð Þ− 1−B 1;n;πð Þð Þ nLr
B 1;n; πð Þ¼ Lr
1−πð Þn−1
1− 1−πð Þn−1 n−1ð Þ− 1−πð Þn1− 1−πð Þn n
" #
¼ Lr1−πð Þn−1
1− 1−πð Þn−1� �
1− 1−πð Þnð Þ nπ−1þ 1−πð Þn �:
The term in parenthesis can be written as
nπ− 1− 1−πð Þn� � ¼Xnl¼0
lπl−Xnl¼1
πl
which is strictly positive for all n ≥ 2.
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