A Search Approach to Joint Liability Lendingunder Imperfect Information ∗

Xinhua Gu †

December, 2002

Abstract

This paper utilizes a search model to examine how joint liability lendingenhances loan repayments compared with unsecured individual liability lending.Joint liability serves as a substitute for collateral in curbing default risk, wherethe effective cost of borrowing is positively related to group risk type, andwhere a safe group is more willing than its risky counterpart to trade jointliability commitments for lower interest rates. The efficiency gain under grouplending results from a risky borrower’s choice of less risky investment due tothe joint responsibility for default risk, and from more prudent actions taken bysafe borrowers since the effective interest reduction that they obtain outweighsthe effective ”collateral” imposed on them by joint liability. Moreover, jointliability has a favorable incentive effect if the loan rate is low and an adverseincentive effect otherwise; moral hazard is less severe for joint liability than forloan rates if the group size is not too large. All this makes it possible for jointliability to improve efficiency by lowering loan rates.

Keywords: group lending, joint liability, moral hazard, search framework.JEL classification: D83, G21, O16.

1 IntroductionJoint-liability lending programs under asymmetric information are designed for poorborrowers without collateral who are required to form groups by self-selection and

∗I am grateful to Varouj Aivazian for extensive discussions and continuous guidance, to JeffreyCallen and Miquel Faig for helpful suggestions, and to Kerstin Aivazian for much help. I wouldalso like to thank John Floyd, Angelo Melino, Shouyong Shi and other seminar participants at theUniversity of Toronto for valuable comments. The usual disclaimer applies.

†Corresponding address: Department of Economics, University of Toronto, 150 St. GeorgeStreet, Toronto, Ontario, M5S 3G7, Canada. Phone: (416) 978-4964. Fax: (416) 978-6713. email :xinhuagu@chass.utoronto.ca.

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take collective responsibility for group members’ repayments. Loans are issued toindividual members, but the entire group is held jointly liable for each member’sdebt. After repaying their own loans, each member has to make a joint-liabilitypayment to cover for defaulting partners, or else the whole group is denied access tofuture refinancing.Recent evidence on the remarkable successes 1 of joint liability lending has inspired

active researches in the micro-finance literature (see Ghatak and Guinnance (1999)for a detailed survey). Some authors (e.g., Stiglitz (1990), Varian (1990), Besley andCoate (1995), Conning (1996) and Aghion (1999)) focus on either the role of peermonitoring in utilizing local cohesion among borrowers to enhance their ability tomeet debt obligations, or the comparative advantage of peer pressure in employingsocial sanction to better enforce loan contracts. Some authors (such as Tassel (1999),and Ghatak (2000)) stress the importance of peer selection for group formation, andanalyze the screening mechanism of joint liability used by lenders to identify het-erogeneous borrower types and charge separate loan rates. Other authors (Laffonand N’Guessan (1999), Ghatak (1999) and Aghion and Gollier (2000)) emphasize theadverse selection aspects of asymmetric information, and examine how joint liabilitymitigates credit rationing in an environment where borrowers do not necessarily haveperfect knowledge of each other.However, most papers mentioned above deal with either one-period settings or

discrete-type borrowers. No study yet presents a general model with a continuum ofproject types and forward-looking borrowers to seriously examine why a reduction ininterest rates is made possible by joint liability. The empirical evidence 2 indicatesthat the interest reduction is an ultimate source of the improvement in repaymentsinduced by joint liability. Moreover, the existing treatment of continuous-type modelsis far from being rigorous or analytically satisfactory, and the role of joint liabilityin enhancing efficiency compared with individual liability lending remains to be for-mally analyzed. This paper develops a dynamic search model with continuous typesof projects. We will address moral hazard problems by treating project type as aborrower’s action type, and the selection of group partners is based on their chosenproject types. There are two features of the lending scheme in question: self-groupingand joint liability. Peer selection that leads to positive assortative matching is notour focus since one sees that the economic logic can naturally establish this kind ofgrouping if borrowers have complete information about each other 3.

1As cited by Ghatak (2000), the early default rate experienced by a leading group-lending in-stitution, the Grameen Bank of Bangladesh, was around 2% as opposed to 60-70% for comparableloans by conventional lending institutions (Hossain (1988)), and recent estimates of the default ratewere reported to be slightly over 7.8% (Morduch (1999)).

2As cited by Aghion and Gollier (2000), the Grameen Bank lends at a 20% interest rate perannum, in contrast with the yearly interest rates charged by usurious moneylenders in Bangladeshwhich often exceed 100% (Fuglesang and Chandler (1994)).

3By the principle of minimalism, one need not elaborate on an analytic proof of positive assor-tative matching (as done in the literature) since this is trivial especially in a one-period adverse

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Our dynamic analysis distinguishes this study from almost all others in the liter-ature by integrating the cross-sectional asymmetric information between lenders andborrowers with intertemporal uncertainty about investment prospects faced by bothparties. The static models implicitly assume that projects are always available toborrowers. Yet, it may be difficult to find acceptable projects or to choose from com-peting investment opportunities under imperfect information. Making wise choicesof projects is particularly important to the poor since credit is a very scarce resourceto them. The search approach provides a useful way of addressing such problemsunder uncertainty and of putting project selection into a tractable dynamic perspec-tive. We will keep the search framework as simple as possible because a simple searchmodel is enough to highlight its usefulness in addressing group lending and we needto concentrate our attention on the role of joint liability in alleviating asymmetricinformation.This paper extending Ghatak’s (1999) static model to a dynamic framework gener-

ates new findings. Greater investment availability leads to higher riskiness of acceptedprojects under individual liability so that the bank must raise the interest rate to off-set the decline in loan repayments. Dynamic borrowers are more sensitive to loanrates coupled with joint liability than without it, in that they are more prudent atlow rates and more averse to high rates. Dynamic borrowers become riskier the moreopportunities they have or the more forward-looking they are, regardless of liabili-ties. Borrowers tend to be choosier about projects in dynamic models than in staticones, which makes a difference to the incentive effect of joint liability. Although themoral hazard effect of interest rates is significant under imperfect information, jointliability may have a favorable incentive effect on borrowers’ investment behavior inthe dynamic setting, unlike the static case where the effect of joint liability is alwaysadverse.A problem with the extant literature is that the implication of moral hazard

associated with joint liability has not been well analyzed for the continuous-typestatic setting, let alone for a dynamic framework. Our answer to the question of whyjoint liability has a moral hazard problem is this. Since at some interest rate coupled

selection model if borrowers know each other perfectly. In this case, joint liability plays no role inscreening group heterogeneity by the bank except that borrowers in a group are now known to be ofthe same type. In our model, positive assortative grouping serves only as an institutional arrange-ment for ensuring joint-liability payments. In fact, as shown by Aghion and Gollier (2000), positiveassortative matching is not necessary for joint liability to improve efficiency. Joint liability is sopowerful that it is worthwhile to clarify its role more rigorously in a continuous-type model, whichis a main task of this paper. Models of assortative grouping are much less developed, and a sensiblejob in this regard has yet to be done. A dynamic framework may be interesting and more able thana static one to address the realistic situation of grouping, in which borrowers do not perfectly knoweach other ex ante, but over time receive more information about each other’s types that will stillbe unobservable to the bank. Using search models with learning would be a promising extension tothe present work, in that one can examine whether matching would be positively assortative (withcertain distortions before reaching a steady state) or groups could be stabilized as time goes on,given that partner search is costly.

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with joint liability, a borrower needs to select a risky project to obtain enough after-interest leftover income to cover for her defaulting partner(s), increasing the jointliability requirement of loans will imply financing riskier projects. We find in bothstatic and dynamic models that the moral hazard effect of joint liability is less severethan that of the interest rate, if the group size is not too large so that no ”free-riding”problem arises 4. In this case, the effect on borrowers’ actions of interest reductionspermitted by joint liability overwhelms its moral hazard effect, thereby making higherrepayment rates possible.We show that for borrowers in a dynamic search setting, joint liability has a

favorable incentive effect if interest rates are low, and an adverse effect otherwise. Tosee why this dichotomy happens, we look at borrowers’ greater sensitivity to loan rateswith joint liability than without it. Since such borrowers are more prudent at lowrates, joint liability along with a low rate could play a positive role in affecting theirinvestment decisions. In this case, borrowers take advantage of low borrowing costs byselecting safe projects to avoid joint-liability payments, and increasing joint liabilitycould make themmore prudent. If this favorable effect of joint liability is accompaniedby an interest rate reduction, the loan repayment rate will be increased further. Sincedynamic borrowers are more averse to high rates, joint liability together with a highrate could exert a negative impact on their project choice. In this case, a greaterreduction in loan rates under joint liability is needed to induce a positive impact onrepayments. Generally, we believe that any non-interest rate instrument such as jointliability, collateral or other default penalties may have dichotomous effects (while theinterest rate has only a moral hazard effect), and that this dichotomy can only beidentified in dynamic search models.There is as yet no formal proof in continuous-type models of how joint liability

triggers lower loan rates, though Aghion and Gollier (2000) provide a good analysisin a two-type model to suggest this fact. They correctly point out that the interestrate reduction is a direct consequence of the bank effectively transferring part ofdefault risks onto the peer group. However, they do not explicitly treat joint liabilityas a policy instrument for the bank, and exploring its optimal degree is thus notpossible within their model. Our analysis differs from their work in that we analyzethe welfare implication of both interest rates and joint liability. We show that aninterest rate reduction in equilibrium is easily established when joint liability hasa favorable effect; that this reduction is also possible if joint liability has a moralhazard effect on repayments and if this problem is not serious relative to the effectivejoint-liability payment (this is the case at the social optimum); and that the benefitfrom the interest reduction effectively dominates the burden of joint liability, boosting

4Aghion (1999) provides an insightful analysis of the group size issue by comparing the per-borrower expected payoffs between the two- and three-borrower cases. However, there has beenno meaningful study of the optimal group size in the continuous-type context. In the case of theGrameen Bank (reported by Hossain (1988)), it started with groups of 20 borrowers and graduallyreduced the group size to five which has remained ever since.

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loan repayments and enabling the bank to lower interest rates and to increase socialsurplus.This paper shows that joint liability serves as a substitute for collateral by making

safer (riskier) borrowers incur a lower (higher) effective cost of borrowing. Safer bor-rowers prefer joint liability to individual liability more than their risky counterparts,yet no borrower will take a riskier investment position under joint liability. Safeborrowers become more prudent since effective interest reductions allowed by joint li-ability dominate its implied effective ”collateral.” Borrowers with risky projects turnless risky due to the joint responsibility for their group’s high default risk. Given theoverall improvement in efficiency under joint liability, it may be that all borrowersare made better off; or, that safe borrowers are better off while risky borrowers areworse off, but the benefit to safe borrowers outweighs the loss to risky borrowers.Under certain fairly weak assumptions, the paper provides a rigorous proof for thesuperiority of joint liability over individual liability lending. We establish that a com-petitive equilibrium that maximizes intertemporal social surplus can be achieved asan interior solution if the bank’s breakeven curve does not intersect the lower boundof permissible combinations of the interest rate and joint liability.The rest of this paper is structured as follows. Sections 2 and 3 analyze a bor-

rower’s search model, the bank’s equilibrium condition and the social optimizationproblem for individual liability contracts. Sections 4 and 6 provide a similar dis-cussion for joint liability schemes, and Section 5 presents the relative advantage ofdynamic models in analyzing issues. Section 7 illustrates by example the implicationsof different liability arrangements for the individual welfare of borrowers. Section 8provides concluding remarks.

2 Search model under individual liabilityThis section develops a search model for borrowers to choose projects under individualliability, treating the interest rate and investment availability as search parameters.After deriving the optimality condition for the search model, we conduct a com-parative static analysis regarding the effect of parametric changes on the borrower’sinvestment choice. Although ex post informational asymmetry is incorporated inthe specification of the mean profit on a risky project, the search model is mainlyconcerned with ex ante uncertainty about project availability and return outcomes.Return on a project is assumed to equal sales revenue less the cost of non-capital

inputs. A project with the random return R each period yields R (p) > 0 with

probability p and zero with probability pc, where the notation xc4= 1− x for any x.

Here, p is the probability that a project succeeds while pc is its probability of failure.We assume that different projects have the same mean return:

E (R) = pR (p) = Ro for ∀p ∈ £p, 1¤ and p > 0. (1)

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This implies

σ2R (pc) = R2o

pc

p,

dσ2Rdpc

=

µRop

¶2> 0, R0 (pc) =

Rop2> 0.

Thus, projects differ only in their riskiness σ2R or pc. The more risky a project, the

higher its return R (p) conditional on success.The project success probability, denoted by p, is assumed to be identically and

independently distributed (i.i.d.). Its distribution Pr (p 6p) = G (p) and density g (p)are thus constant over time, with support

£p, 1¤, mean µ and variance σ2. The project

type p is used in this paper to characterize a borrower’s investment action. Whenapplying for loans to finance a project, the borrower observes its type p ex post, butbanks know only its distribution G (p); thus, information is asymmetric. For futureprojects, their types are unknown ex ante to either party, and only G (p) is commonknowledge; at this stage, information is symmetric, yet uncertainty about investmentprospects prevails.As assumed in the literature, borrowers and lenders are risk neutral. The borrower

has an amount of equity which is less than the total cost of the project, and needs toborrow one unit of a bank loan to launch the project. The supply of loanable fundsis unaffected by the interest rate the bank charges borrowers. We assume that eachborrower carries out one project each period, that the bank provides loan financingfor a project if it is accepted by the borrower, that sales revenue from a successfulproject is just enough to cover all costs (including debt repayment), and that thereare neither direct bankruptcy costs nor auditing costs (i.e., a borrower’s default isverifiable at no cost).Under limited liability, the random profit π each period from undertaking a type

p project is given by R (p) − r if it is successful (with probability p), and zero if itfails (with probability pc), where r > 1 denotes the gross interest rate (henceforth,the interest rate) paid to the bank by borrowers when successful. The mean profit,

πe = E (π) = Ro − pr = πe (p) , (2)

is downward sloping in p. The market value of a borrower’s equity is reduced to zeroif the project fails, and she must default on the loan that was used to finance it.Thus, we deal with a situation where the borrower is unable, rather than unwilling,to meet her debt obligation.We develop a search model V (p) = max Π (p) , U for a forward-looking borrower,

who has a type p project 5 in hand this period and makes an optimal choice of its

5One can also consider using a search model to deal with a more general case, in which there areseveral projects encountered each period, or in which a borrower may come across a random numberof projects in a period. The search model can be utilized by borrowers to choose from more thanone available project by trading off the benefit from investing in one project against its opportunitycosts (of having to give up other projects) under intertemporal uncertainties.

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acceptance or rejection while having the option to search for other opportunities lateron. V (p) denotes the search value, Π (p) the acceptance value, and U the rejection(or waiting) value. These values, fully specified below, are expected intertemporalpresent values.Suppose for simplicity that any project, whether a success or a failure, lasts only

one period 6. After one period, the borrower re-enters the market to look for a newproject. The acceptance value Π (p) goes to the borrower, who accepts the type pproject to obtain πe (p) this period, and re-enters the market to receive the waitingvalue U from then on. Thus, Π (p) is specified as

Π (p) = πe (p) + βU, (3)

where β is a discount factor. Since Π0 (p) = −r < 0 and Π (pc) increases with pc, theborrower has the incentive to choose a risky project.The rejection value U accrues to the borrower, who rejects the type p project this

period to wait in the market for possibly better prospects, and behaves optimallylater on. Clearly, U follows a recursive equation:

U = uo + β [αEV (p0) + αcU ] .

The borrower is assumed to have a reservation payoff uo (say, earning wage income)in the waiting period. In the next period, the borrower encounters no projects withprobability αc and continues receiving the waiting value U , or comes across a newtype p0 project with probability α and is in a position of deciding whether to acceptthis project and receives a new search value V (p0). One may set α = 1 to reflecta situation where projects are always available, or modify the model to deal withthe availability of multiple projects in a period from which a borrower chooses oneproject. Rearranging the above equation in U yields

U =uo + αβEV (p0)

1− βαc. (4)

It then follows that U > uoβc. Waiting in the credit market for investment is more

valuable than leaving it for employment.Substituting (3) and (4) into the search model yields the following Bellman func-

tional equation:

V (p) =maxacc,rej

Π (p) , U =½

Π (p) , if p 6 bpU, if p > bp . (5)

6This treatment fits in with the reality that poor borrowers in a rural area apply for only smallloans to undertake short-lived projects. The assumption can be relaxed without altering the mainresults in this paper. Additionally, agricultural projects are usually seasonal but industrial projectsare not. Thus, a borrower’s planning horizon varies with the nature of projects, and the underlyingdistributions G (p) may be different across project categories. This search model abstracts fromthese physical considerations to be concentrated on the economic aspect of loan financing problems.

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The borrower’s optimal search strategy is a reservation-type policy bp at which Π (bp)= U : accept any project of type p 6 bp and undertake it without delay; reject anyproject of type p > bp and continue to wait. Safer projects with type higher thanbp will not be carried out in the borrower’s individual equilibrium because of thedecreasing property of the acceptance value Π (p). The economy thus ends up withunder-investment and, as will be shown, the problem worsens with rising interestrates.We assume that the supply of loanable funds is perfectly elastic at a riskfree gross

deposit rate ρo > 1 (henceforth, deposit rate) paid by banks to depositors. The meanreturn on a project is more than enough to cover its opportunity cost in terms oflabor and capital: Ro > ρo + uo. We need to identify the range Ω = [r, r] of thesearch parameter r that makes the reservation type bp fall within the support £p, 1¤.In Appendix (A1), we show

Lemma 1 The upper and lower bounds of interest rates permissible to borrowers aregiven by

r =Ro − uop

, r =Ro − uo1 + αβµc

.

Thus, by construction, if r 6 r, then bp > p and p 6 p 6 bp; if r = r, then p = bp = p;if r > r, then p 6 bp < 1; if r 6 r, then bp = 1 and ∀p 6 1 = bp.Intuitively, if the interest rate r is no greater than its upper bound r, potential

borrowers are willing to borrow for investment. Once the interest rate hits the upperbound, the borrower will take on the riskiest project (of type p). When the interestrate rises above the upper bound, no projects are acceptable and nobody borrows forproduction. If the interest rate is equal to or less than its lower bound r, borrowerswill accept any type p project and we refer to this as a degenerate case.Define ep = Ep6bp (p) (< minbp, µ) as the average probability of success in the pool

of projects acceptable to borrowers. It follows from (5) that πe (bp) = βcU (> uo inΩ). Then, πe (p) (> πe (bp)) > uo for p 6 bp; that is, investing in an acceptable projectis more valuable than waiting in the market. If the bank earns zero mean profit, thenthe expected interest payment equals the deposit rate: rep = ρo. The difference invalue between investing and waiting, Ep6bpπe (p)−uo (= Ro−uo−ρo > 0), constitutesthe average opportunity cost of search per period if the borrower were to give up anacceptable project in the hope of finding a better one. Now that search is costly, theborrower should not postpone undertaking any project if its type is acceptable.Solving the search model yields its optimality condition below:

Ro − uo = rhbp+ αβ bG (bp− ep)i , (6)

This condition shows that the reservation type bp depends, among other things, onthe interest rate r and investment availability α in the form of bp = bp (r,α). There

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are two comparative static results associated with (6): ∂bp∂r< 0 and ∂bp

∂α< 0, which are

derived in Appendix (A2).[Figure 1 is here]The result, ∂bp

∂r< 0, captures the well-known adverse incentive (moral hazard)

effect of interest rates. Higher interest rates drive a borrower to take on a riskierproject by decreasing her reservation type bp. This effect, depicted in Figure 1 7 bythe curve corresponding to c = 0 (individual liability) within its domain Ω = [r, r],is presented in the following theorem:

Theorem 2 An increase in the interest rate r depresses the reservation type bp ofborrowers, and increases the riskiness of their accepted projects under asymmetricinformation.

The result, ∂bp∂α< 0, asserts that a rise in investment availability, α, increases

the riskiness of accepted projects; this is quite similar to the proposition in laboreconomics that a worker receiving more job offers has a higher reservation demandfor wages. This result, depicting the partial effect of α, will be used to derive itsgeneral counterpart, referred to as the total effect of α. In general, market playerswith more opportunities are choosier.

3 Individual liability equilibriumThe preceding section modelled how borrowers select acceptable projects for invest-ment at any given rate of interest under individual liability. This section will discusshow the interest rate for this liability scheme is set by a bank to maximize its expectedprofit, given asymmetric information about project types. It is assumed that banksare Bertrand competitors and that there may be multiple competitive equilibriuminterest rates. We then choose from these competitive lending rates a constrainedPareto optimal rate to close the model, and analyze the effect of investment avail-ability on the optimal lending rate under individual liability.Under asymmetric information, a bank cannot charge different rates of interest if

there is no way to find out the specific types p of projects to be financed. Faced withthe same interest rate r, borrowers who take on risky projects are cross-subsidized bythose who undertake safe projects. This problem cannot be resolved under individualliability without collateral or other instruments. From the bank’s perspective, ep= ep (bp (r)) is the average repayment rate of the loan pool corresponding to the interestrate r. Since

depdbp = bgbG (bp− ep) > 0 and

∂ep∂r=depdbp ∂bp∂r < 0, (7)

7In Figure 1, bp (r) = 1 (for r 6 r) is a line segment, and bp (r) = 0 (for r > r) coincides with thehorizontal axis.

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the average repayment rate ep increases with borrowers’ reservation type and decreaseswith the interest rate r. The bank may make a profit or loss on a particular loan thatis used to finance a project of type p (6 bp), i.e., rp− ρo R 0. However, what mattersto the bank is the average profit rate, R (r) = rep (bp (r)) − ρo, on all the loans. Thebank maximizes its expected profit R (r) by choosing the interest rate r.Suppose that there are many identical banks operating in the credit market. As

Bertrand price (r) setters, banks are free to compete for deposits and lending. Thebanking sector’s competitive equilibrium must then be characterized by a represen-tative bank’s breakeven condition R (r) = 0:

rep (bp (r,α)) = ρo, (8)

where the reservation type bp is parameterized by α only, and the other exogenousparameters of no interest are discarded from consideration. Since ep < 1, we knowthat r = ρo/ep > ρo; that is, the lending rate is higher than the deposit rate. Thesolution to this breakeven condition defines r = r (α). However, the competitiveequilibrium r (α) need not be unique. If there are multiple equilibria, those r (α) canbe Pareto ranked. One of them, denoted by r∗ = r∗ (α), must achieve a constrainedPareto optimum.Noting that pt is i.i.d. ∼ G (p) and Ω a compact (i.e., closed and bounded) set, we

specify the intertemporal aggregate surplusW accruing to all N borrowers, includingthose who invest in a project and those who wait in the market. Suppose that thereexists a central planner who chooses from r (α), and we assume that the bank acts asthe planner 8. Based on the partial equilibria of borrowers and lenders, the plannerselects an optimal r∗ to maximize W (r):

maxr∈Ω

βc

NW (r) =

bp(r)Zp

πe (p) dG (p) +

1Zbp(r)uodG (p) (9)

= (Ro − ρo − uo)G (bp (r)) + uo, s.t. R (r) = 0.

Here, (8) has been used in simplifyingW (r). Since the aggregate surplusW increaseswith reductions in the interest rate r as shown by dW

dr∝ ∂bp

∂r< 0, the social optimization

problem in (9) requires the planner to set as low an interest rate as possible. We areready to prove

8It is interesting to mention that our framework is similar to the Stackelberg model for a sequentialgame of price leadership in terms of strategic interactions among borrowers, banks and the planner.Borrowers are the price (i.e., r) followers, and banks or the planner the price leaders. The follower’ssearch policy bp depends on the leader’s predetermined choice of price r, as captured by the former’sreaction function bp (r). The bank should recognize that its choice of r influences the follower’sinvestment action with moral hazard. In the meanwhile, the leader must take into account both bp= bp (r) and R (r) = 0 in deriving social optimum r∗. At optimum, r∗ leads to bp∗ via bp (r∗), and thefollower’s project selection rule is thus determined eventually.

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Theorem 3 There exists a unique constrained Pareto optimum r∗ in the search equi-librium under the individual liability lending scheme if the opportunity cost of searchis not too large in the sense that 0 < Ro − ρo − uo < ρo

µc

µ(1 + αβ).

Proof : See Appendix (A3).A more general comparative static analysis is needed to derive the total effect

of parameter α. This total effect turns out to be consistent with the partial effectderived earlier. Denote by bp∗ and ep∗ the reservation type and the repayment rate atoptimum r∗, respectively. The comparative static analysis below is conducted afterthe optimal rate r∗ (α) has been parameterized by α. We have the following theorem:

Theorem 4 At the social optimum, an increase in investment availability α de-presses the reservation type bp∗ of borrowers, and increases the riskiness of their ac-cepted projects; this forces the bank to charge a higher interest rate r∗ to offset thedecline in the average repayment rate ep∗.Proof : See Appendix (A4).

4 Search model under joint liabilityThis section examines group formation among borrowers in an optimal sorting process,and investigates the effect of joint liability on effective borrowing costs to differentgroups. Joint liability serves as a substitute for collateral, and takes effect if theproject of a partner in the group fails. Thus, the investment action of partners be-comes a key element in peer selection for grouping, borrowing groups differ in theirriskiness of chosen projects, and one should explore the incentive effect of joint lia-bility on borrowers’ project selection. We develop a search model, with parametersdepicting investment opportunities and the terms of joint liability, to determine theacceptability of projects to borrowers under group lending.We assume that the enforcement of joint liability contracts is costless to banks,

that borrowers know about each other’s project types 9, and that returns on theprojects of group members are uncorrelated 10. As mentioned above, what ultimately

9Laffont and N’Guessan (1999) consider a case where borrowers do not have perfect informationabout each other. In this case, a learning process may be needed to reduce signalling errors in peerselection for grouping, and search models are a useful tool to incorporate sequential learning aboutpartners’ type. This paper will avoid such complications to focus on the role of joint liability.

10Under group lending, the investment decisions of members are independent of each other andtheir project returns are uncorrelated as well. Group members take on the same type projects; thesuccess probabilities of these projects are identical but the realizations of project outcomes may bedifferent. These independent risks enable successful members to cover for their defaulting partners,and so group members share financial responsibilities for loan repayments. The bank utilizes thegroup lending scheme to pool independent risks within a group, and will benefit from diversifyingaway default risk onto peer groups. Also, the effective risk the bank bears hinges on the group size.

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matters in peer selection is the project type of a partner. So, for expository conve-nience, we sometimes refer to those who accept a risky project as a risky borroweralthough project type and borrower type may not be the same 11.Suppose that a borrowing group without collateral has only two members: one

takes on a type p project and the other type p0. For a lending contract (c, r), theearnings of the type p borrower are equal to R (p) − r when both members succeed(with probability pp0), to R (p) − r − c when she succeeds and her type p0 partnerfails (with probability pp0

c), and to zero (under limited liability) when she fails (with

probability pc). Hence, the mean profit to this type p borrower with the type p0

partner isπe (p, p

0) = Ro −¡r + cp0

c¢p. (10)

In this group, the type p borrower, if successful in her own project and paying r, isheld liable for paying c if her partner fails. On the basis of the terms of this contract,we show the following:

Theorem 5 A borrower will reject all other borrowers who choose riskier projectsand will be rejected by all other borrowers who take on safer projects; this results ina group formed between borrowers undertaking identical type projects. The two-sidedpeer selection of this kind is socially desirable, with the mean profit to a member ofthe type p group given by

πe (p) = Ro − (r + cpc) p. (11)

Proof : It is straightforward to establish this theorem from the viewpoint of pri-vate optimization (as in Ghatak’s (1999)), given that borrowers have perfect knowl-edge of each other’s types. What we intend to do here is to explore the social desir-ability of private grouping behavior. It suffices to apply the supermodularity propertyto the borrower’s mean profit function in (10). One sees that ∂2πe(p,p0)

∂p∂p0 (= c) > 0 ifand only if

πe (pH , pH) + πe (pL, pL) > πe (pH , pL) + πe (pL, pH) , (12)

for a safe borrower of high type pH and a risky borrower of low type pL (< pH).Clearly, the project types of group members are complementary in their payoff func-tions. Thus, a combination of borrowing groups composed of identical types is pre-ferred to that of groups composed of mixed types from the viewpoint of social op-timization. Also, it is easy to show that private optimization in peer selection isconsistent with social optimization in group formation. With positive assortativematching, one can set p0 = p in (10) to obtain (11). ¥

11Strictly speaking, project type in this paper is characterized by p, which follows the distributionG (p | µ), while borrower type can be captured by say, µ, which follows a distribution denoted byF (µ | m). A risky borrower might accept a safe project while a safe borrower could choose a riskyproject, although the probabilities of these two events are not large. Ghatak (1999) treats p asborrower type.

12

To specify the bank’s expected profit function, consider two loans lent to a groupcomposed of two borrowers with projects of types p and p0 prior to the sorting process.The bank’s earnings are equal to 2r with probability pp0, to (r + c) with probabilitypp0

c, to (c+ r) with probability pcp0, and to zero with probability pcp0

cunder limited

liability. Since the bank has to pay its depositors 2ρo for loaning their funds to thisgroup, its expected profit per loan will be

R (p; c, r) = (r + c) (p + p0) /2− cpp0 − ρo.

After sorting with p = p0, the bank’s expected profit per loan, contingent on therealization of group type p, becomes

R (p; c, r) = rp+ (cp) pc − ρo. (13)

The bank takes the further expectation 12 of R (p; c, r) w.r.t. p over the loan poolsubject to the search policy bp (= bp (c, r) to be derived shortly) of borrowers underjoint liability. Thus, we have R (c, r) = Ep6bpR (p; c, r).To see why group lending improves efficiency, we work with (13) to show that joint

liability c serves as a substitute for collateral in curbing default risk. The findingsare presented in the following:

Theorem 6 The net effect of joint liability on effective borrowing costs is its inducedreduction in effective interest rates less its implied effective ”collateral.” 13 This effectis positive and large for groups with safe projects, and may be positive and small ornegative for groups with risky projects, thus increasing loan repayments.

Proof : Under secured individual liability with collateral co, the bank’s expectedprofit becomes

R1 (p; co, r) = rp+ copc − ρo, (14)

where pc is the borrower’s own default risk. Comparing (14) with (13), one sees thatunder group lending, pc is the partner’s default risk and cp is equivalent to collateralco. Effective joint liability (cp) pc in (13) plays a role similar to the effective collateralcop

c in (14) in compensating the bank for default risk.As mentioned before, borrowers with safe projects subsidize those with risky

projects under unsecured individual liability, and the bank makes profits on safeborrowers and losses on risky borrowers. Safe borrowers compensate the bank forthe default of risky borrowers since effective interest costs rp are inversely related to

12”Screening” by joint liability in this setting brings only about the result to the bank that loanapplicants from a group have the same type of projects, but the group type is still unobservable bythe bank.

13In a two-type adverse selection model, Aghion and Gollier (2000) refer to the effective reductionin interest rates caused by c as a ”collateral effect,” and to the effective ”collateral” imposed by cas a ”joint responsibility effect.” Their explanation of why the balance of these two effects improvesefficiency is not the same as ours.

13

risk type pc (i.e., rpH > rpL). This burden on safe types, however, is lessened undersecured individual liability since the effective collateral copc increases with risk typepc (i.e., copcH < cop

cL) so that risky borrowers have to become less risky. Although

safe types still subsidize risky ones via the interest rate (to a lesser extent since r ↓with co), the former are now subsidized by the latter via collateral.Note that ppc increases with p first, attains its maximum at p = 1

2and then

decreases. Thus, the effective ”collateral” (cp) pc under group lending is low for safeand risky borrowers, and high for borrowers of intermediate types (around p = 1

2). As

will be proved, the interest rate r in equilibrium is decreased (4r < 0) by introducingjoint liability c. Clearly, the effective reduction in interest costs is higher for a safetype pH than for a risky type pL (i.e., |4r| pH > |4r| pL). Denote by prc (= 1− |4r|

c)

the cut-off type such that the effective interest reduction is exactly balanced by theeffective ”collateral”. As shown in Appendix (A5), the effective interest reduction|4r| p outweighs the effective ”collateral” (cp) pc for safe borrowers of high type p> prc while the situation is reversed for risky borrowers of low type p < prc. In thiscase, safe borrowers are better off and will take on safer projects, type prc borrowersare exactly as well off as before and will keep their investment positions unchanged,and risky borrowers are worse off but have to become less risky due to higher effectiveborrowing costs. If the interest reduction is large enough relative to joint liability suchthat prc < p, then the net benefit |4r| p− cppc for all p > p is positive but decreasingwith group risk pc. In this case, any type p borrower will benefit from group lending,though riskier types benefit less 14; all borrowers can therefore be expected to takesafer investment actions. ¥To build a general search model under joint liability, consider a group of sizem+1

for m > 1 formed through optimal sorting. Given a lending contract (r, c), the payoffto a borrower from taking a type p project and havingm partners in her group dependson the outcomes of these partners’ projects as well as her own. Possible outcomesof the partners’ project failures can be represented by a binomial random variable

X = 0, 1, 2, · · · ,m with parameters (pc,m) and meanmXk=0

kP (X = k) = mpc. The

expected profit of this borrower is thus

πe (p) = Ro − (r +mcpc) p, (15)

which is downward sloping and convex since

π0e (p) = − (r +mc− 2mcp) < 0 for p 6 1 and r > mc,

and π00e (p) = 2mc > 0. Here, the restriction of r > mc, under which π0e (p) < 0, isreasonable since the interest rate r is the main cost of borrowing and joint liability

14The effect of joint liability on individual borrowers’ welfare has not been well proved for a generalcontinuous-type case in adverse selection or moral hazard models. Ghatak (1999) uses examples toillustrate this effect, but its establishment relies on the interest reductions caused by joint liabilitythat require a formal analytical treatment.

14

mc cannot be too large. Thus, permissible interest rates should lie above the line r= mc in the (c, r)-plane, as depicted in Figure 2.[Figure 2 is here]For the search model V (p) = max Π (p) , U based on joint liability, the accep-

tance value Π (p) is identical in functional form to that in (3), except that πe (p)’s in(2) and (15) are different. Clearly, Π (p) is decreasing and now convex. The searchvalue V (p) and the waiting value U are also the same as before in (5) and (4), re-spectively. The borrower’s optimal search strategy is still a reservation type policy bp,which now depends not only on the interest rate r but also on joint liability c. If c orm is equal to zero, the search model here reduces to that of individual liability. Fora given level of joint liability c, denote by Ωc = [r (c) , r (c)] the range of permissibleinterest rates such that bp ∈ £p, 1¤. In Appendix (A5), we showLemma 7 The lower and upper boundary curves of permissible interest rates for anyc are given by

r (c) =Ro − uo + αβmδc

1 + αβµc, r (c) =

Ro − uo −mcppcp

,

where δ = E (ppc). By construction, r 6 r (c) implies bp > p, r = r (c) causes p = bp= p, r > r (c) results in bp = 0, r > r (c) makes bp < 1, and r 6 r (c) leads to p 6 bp= 1.

Intuitively, the interest rate cannot be too large for a given level of joint liability.If the interest rate rises to its upper limit, borrowers will take the riskiest project; ifthe interest rate goes beyond its upper limit, then nobody borrows. Also, the interestrate cannot be too small for a given level of joint liability. If the interest rate is nogreater than its lower limit, then lending contracts (c, r) may not be profit-maximizingto the bank even though borrowers will now become less risky by taking any type ofproject (with safe projects included in the loan pool).The upper boundary curve r (c) has a higher vertical intercept than does the lower

boundary curve r (c). The latter is less steep than the line r = mc. These two curves,and the line r = mc, constitute the domain Λc for permissible (c, r) in the quadrantof (c, r) > 0 such that p 6 bp 6 1 and Π0 (p) < 0 for p 6 1. This permissibility domainis depicted as the shaded area in Figure 2.Solving the joint liability search model yields its optimality condition:

Ro − uo = (r +mcbpc) bp+ αβ

bpZp

G (p) [r +mc (1− 2p)] dp, (16)

which determines the decision rule bp = bp (c, r,α,m) in project search by borrowersunder joint liability. The incentive effects of lending contracts (c, r) and of projectavailability α based on (16) are presented in the following theorem:

15

Theorem 8 Under group lending, a rise in interest rates (r) or in investment op-portunities (α) increases the riskiness of a group’s accepted projects by depressing itsreservation type bp. A rise in joint liability (c) will increase the riskiness of acceptedprojects if the interest rate is high (r > ro for some ro), and reduce the riskiness ofaccepted projects if the interest rate is low (r < ro). For r > ro, moral hazard islower for joint liability (c) than for interest rates (r) if the group size (m) is not toolarge, and the size limit (m) ensuring this property will increase if borrowers’ searchis safer.

Proof : The comparative static results are provided in Appendix (A6) with thefollowing signs:

∂bp∂r

< 0,∂bp∂α

< 0,∂bp∂c< (or > ) 0 if r > (or < ) ro, (17)¯

∂bp∂c

¯<

¯∂bp∂r

¯for r > ro and m < m (bp) , where m (bp) > 1 and m0 (bp) > 0.

Notice from Appendix (A6) that the cut-off value bpo determines whether the effect ofjoint liability is favorable or not. It is easy to see that bpo depends on α, β and G, andcorresponds to some ro. Thus, bp < (or >) bpo means r > (or <) ro. Also, ∂bp

∂m= c

m∂bp∂c

< (or >) 0 for r > (or <) ro, where m is treated as a parameter rather than a choicevariable 15. ¥While higher interest rates always have a moral hazard effect on the investment

decision of borrowers, raising joint liability may exert a positive incentive effect ifthe interest rate is low, or an adverse incentive effect if the interest rate is high. Ifmoral hazard is kept lower for joint liability than for loan rates, the group size canbe made larger the safer the group’s selection of projects. These new findings havebeen generated in a dynamic search framework, and their implications are discussednext in comparison with static models.

5 Dynamic versus static approachesThis section examines the difference in borrowers’ investment selection between dy-namic and static approaches, explains why the dichotomous effects of joint liabilityarise in a search model, and interprets the implication of this dichotomy for equilib-rium lending contracts offered by competitive banks. In this section, we treat jointliability as a parameter of borrowers’ search policies to show the underlying reasonsfor its induced reduction in interest rates; one can see from the analysis the advantageof dynamic models.

15As pointed out by Aghion (1999), the group size m should be neither too small (due to ”cost-sharing” effects) nor too large (due to ”free-riding” effects). It may be that m can be used as achoice variable.

16

We refer to a borrower with a one-period horizon as a static borrower. Sucha borrower is assumed to have no difficulty in finding projects (but has no futureopportunity), and all she needs to do is to choose an acceptable project withoutconsidering the future. A borrower with a search perspective is called a dynamicborrower, and such a borrower is endowed with (uncertain) investment opportunitiesin the future. By comparing the dynamic and static choices of projects, we providethe result in the following theorem:

Theorem 9 For any given rate of interest, dynamic borrowers are riskier than staticborrowers in their investment actions, regardless of whether there are the provisionsof joint liability in lending contracts. Dynamic borrowers become choosier the moreopportunities they have or the more forward-looking they are.

Proof : See Appendix (A7).The reason why dynamic borrowers are choosier about projects for any given set

of lending terms is the following. First, a static borrower has only one period toinvest, and if she undertook a risky project, it would be more likely for her to gobankrupt while having an opportunity loss equal to uo. In contrast, if a dynamicborrower failed due to a risky investment action, she would be able to find otherprojects later on. Second, as shown in Appendix (A7), bps − bp = αβX for some X> 0, the difference between static and dynamic borrowers’ project choices increaseswith the time discount factor and investment availability. So, dynamic borrowers aremore selective the more opportunities they have or the more forward-looking they are.The implication of this phenomenon to the bank’s equilibrium is to charge dynamicborrowers higher interest rates for a given level of joint liability (if any).It is worth mentioning that ∂bps

∂c< 0 holds in a static model of group lending

as in Ghatak (1999) 16. The co-existence of efficiency improving and moral hazard(or adverse selection) associated with joint liability has not been well understoodin the literature. The intuition as to why joint liability has a moral hazard (oradverse selection) effect in the static setting is the following. Since at an interest ratecoupled with joint liability terms, a borrower needs to select a risky project so asto have enough after-interest leftover revenue to cover for her defaulting partner(s),increasing the joint liability requirement of loans will imply financing riskier projects(or causing marginal safer borrowers to drop out).

16Based on his static formulation: Ro − (r +mcbpcs) bps = uo, the effects on borrowers’ borrowingchoices of interest rates and joint liability should be

∂bps∂r

= − bpsr +mc (1− 2bps) < 0, ∂bps

∂c= − mbpsbpcs

r +mc (1− 2bps) < 0,where r +mc (1− 2bps) > 0 since r > mc. Clearly, ∂bps

∂c =∂bps∂r mbpcs and ¯∂bps∂c

¯<¯∂bps∂r

¯if m < m ≡

(bpcs)−1, where m rises if bps is higher. These results, not presented in his paper, are provided herefor comparison. Denote by eps the average repayment rate in this case. If 4r < 0 is caused by 4c> 0, then using (7) shows that 4eps = ∂eps

∂r 4r + ∂eps∂c 4c > 0 leads to |4r| > epcsm4c.17

The implication of this for changes in efficiency depends on how large is the moralhazard (or adverse selection) effect of joint liability. If this effect is not severe relativeto the expected effective ”collateral”, then the introduction of joint liability will leadto a reduction in interest rates from the bank’s equilibrium perspective (to be shownin the next theorem). This interest reduction should be sufficient to cause repaymentsto rise. From borrowers’ search perspective, if the group size is small (< m) such thatno ”free-riding” problem arises, then the moral hazard (or adverse selection) problemof joint liability is less severe than that of interest rates, and the benefit from theinterest reduction dominates the burden of joint liability so that the net impact willbe a positive incentive (or selection) effect to increase the loan repayment rate.The dichotomous effects of joint liability have been identified in our dynamic

model, as depicted in Figure 1 in which the curve bp (c, r) rotates clockwise aroundpoint (ro, bpo) if joint liability c increases. When the interest rate is already high (r> ro), joint liability has a moral hazard effect. Borrowers’ search becomes risky inthat bp (c, r) < bp (r), and higher joint liability forces a risky group of type bp < bpo totake a riskier position in its project selection. On the contrary, if the interest rateis low (r < ro), joint liability has a favorable effect on the project search in thatbp (c, r) > bp (r), and higher joint liability induces a safe group of type bp > bpo to makea more prudent choice of investment. The dichotomy makes it easier to establish thefollowing theorem:

Theorem 10 The bank equilibrium interest rate is lower when coupled with jointliability than without it if the effect of joint liability is favorable; this is also true ifjoint liability has a moral hazard effect that reduces the expected return to the bankbut is not severe relative to the expected effective ”collateral” received by the bank.

Proof : To illustrate the fall in interest rates allowed by joint liability, setting thebank’s expected profit per loan R (c, r) equal to zero yields its breakeven conditionunder group lending:

rep+mcfppc = ρo, (18)

where fppc = Ep6bp (ppc) (< δ). Note that (18) reduces to (8) when c or m equals zero.Equations in (8) and (18), changed to ep = ρo

rand ep = ρo−mcgppc

r, are depicted together

in Figure 3 17. It is shown that in the bank equilibrium the group-lending rate r∗∗ islower than the individual-liability rate r∗ if the moral hazard effect of c is not severerelative to the expected effective ”collateral” cfppc (the dotted curve representing anopposite case, which cannot happen at the social optimum to be shown in the nexttheorem). It is straightforward to show that r∗∗ < r∗ if the effect of c is favorable, asdepicted in Figure 4. ¥[Figure 3 is here]

17This figure, equivalent to half of Figure 4 corresponding to r > ro, is drawn in this way to makethings clearer. Figure 3 can also be used to depict Ghatak’s (1999) case by modifying the labelingand adding subscript s (e.g., replacing epo = ep (bpo) and ro with µ and rs (c) = rs, respectively).

18

[Figure 4 is here]To see why the dichotomy of joint liability effects takes place under dynamic

search, notice the fact that r (c) > r and r (c) < r for any c > 0 in Figure 1.That implies that dynamic borrowers are more sensitive to interest rates under jointliability than without it, being more prudent at low rates and more averse to highrates. Borrowers’ current search decision is affected by forward-looking considerationsof many factors, not only the ability to pay a possible penalty for their defaultingpartners. At low interest rates (r < ro), borrowers try to take advantage of lowborrowing costs to make more profits by selecting safe projects to avoid joint-liabilitypayments. In this case, borrowers have enough leftover revenue to cover for partners’possible defaulting, and increasing joint liability can further boost their reservationtype. At high interest rates (r > ro), borrowers will have to choose risky projects inorder to offset high borrowing costs, earn themselves a profit and prepare for defaultpenalties that their partners may cause. In this case, joint liability has a moralhazard effect, and increasing joint liability will induce borrowers to become riskier.As mentioned earlier, this problem is not serious if the group size is not too large. Tochange borrowers’ risk-taking incentive, the net benefit from taking on a less riskyproject must be increased by lowering interest rates or lessening joint liability. Theresults obtained in this section hold for a given level of joint liability, and its optimumwill be determined next.

6 Joint liability equilibriumThis section provides a formal proof for the superiority of joint liability over indi-vidual liability by deciding upon the interest rate and joint liability simultaneouslyrather than separately. For this purpose, a social planner’s optimization problem isestablished, taking into account the borrower’s and lender’s decisions. We first an-alyze the bank’s breakeven condition, confirming that the interest rate is reduced ifthere is a positive level of joint liability in lending contracts. This interest rate reduc-tion must be balanced by joint liability in the bank’s equilibrium. We then examinethe properties of the planner’s indifference curve and demonstrate the existence ofa search equilibrium (c∗∗, r∗∗) that maximizes social surplus. We also show in analternative way the advantage of joint liability over individual liability in improvingloan repayments.Equation R (c, r) = 0 in (18) defines an iso-profit curve for the bank at the

breakeven level, r = r (c) |R≡0. Totally differentiating the identity R (c, r (c)) ≡ 0w.r.t. c yields r0 (c) |R≡0, and its sign is unclear, in general. We can find out thissign in a limiting case where c → 0+ for a simple example with standard uniformdistribution and full project availability (α = 1). The simplification produces

limc→0+

r0 (c) |R≡0= −µ1 +

2

3β

¶m < 0. (19)

19

This shows that if shifting individual liability lending (with r∗ and c = 0) to grouplending (with c > 0), the bank must lower the interest rate to a level of r∗∗ in order tokeep its breakeven status unchanged. Thus, r∗∗ < r∗. Yet, the bank’s breakeven curvemight not be downward sloping throughout the permissibility domain Λc. Figure 2depicts, for example, a case in which the constraint curve r (c) |R≡0 decreases firstand then increases, and so on.The central planner’s problem under group lending is to maximize the intertem-

poral aggregate surplus W , based on borrowing groups’ search policy bp and subjectto the bank’s breakeven condition R ≡ 0:

max(c,r)∈Λc

βc

NW (c, r) = (Ro − ρo − uo)G (bp (c, r)) + uo s.t. R (c, r) = 0. (20)

The gradient ofW = W (c, r) is ∇W ∝³∂bp∂c, ∂bp∂r

´. The arrows in Figure 2 indicate the

gradient direction. Since ∂bp∂c< (=, or >) 0 if r > (=, or <) ro, the surplus function

W (c, r) increases fastest in the lower leftward direction when interest rates are high,or in the lower rightward direction when interest rates are low. At the interest rater = ro, the aggregate surplus remains unchanged and its gradient points verticallydown regardless of joint liability c. This is because dW ∝ (∂bp

∂cdc + ∂bp

∂rdr) |bpo = ∂bp

∂rdr

r≡ro= 0.SettingW (c, r)≡Wo defines an indifference curve of the planner: r = r (c) |W≡Wo .

Its slope r0 (c) |W≡Wo = −∂bp∂c/∂bp∂r< (or>) 0 if r > (or<) ro. If r = ro, then r0 (c) |W≡Wo

= 0. The planner’s indifference curve is horizontal at the interest rate ro. Otherindifference curves are downward sloping when interest rates are high, and upwardsloping when interest rates are low. Particularly, in the limiting case where c → 0+for the example mentioned above, we find

limc→0+

r0 (c) |W≡W indi∗ = −m [6bpc + βbp (3− 4bp)]3 (2 + βbp) < (or > ) 0 if bp < (or > ) bpo,

(21)where bpo = 91% at β = 0.95, andW indi

∗ denotes the maximal aggregate surplus underindividual liability.Denote by ep∗∗ the optimal repayment rate under group lending. Based on the

above discussion, we can prove the existence of the solution to problem (20).

Theorem 11 Under certain plausible assumptions, there exists one optimal grouplending contract (c∗∗, r∗∗) which is welfare improving relative to the optimal individualliability lending r∗ such that ep∗∗ > ep∗, and the equilibrium interest rate is reduced dueto joint liability c∗∗ such that r∗∗ < r∗. Moreover, the optimal contract (c∗∗, r∗∗) willbe an interior solution if the bank’s breakeven curve does not intersect the lower boundof permissible combinations of the interest rate and joint liability.

Proof : We use Figure 2 to facilitate the proof. It is straightforward to verifyr0 (0+) |W≡W indi∗ > r0 (0+) |R≡0, for this leads to 0 < 2 + βbp. Hence, the curve W

20

≡ W indi∗ is flatter than the curve R ≡ 0 at the optimal point (c∗ = 0, r∗) of indi-

vidual liability. Clearly, this indifference curve of the planner lies above the bank’sbreakeven curve in an interval (0, c0) for some c0, and the planner can thus increasethe aggregate surplus W by deviating from individual liability to joint liability. Thiswelfare-improving move reduces r from r∗ and sets a positive c (< c0). When anindifference curve of the planner is tangent to the bank’s breakeven curve, the opti-mum under group lending is attained such that W join

∗∗ > W indi∗ if an interior solution

is assumed. This optimum is denoted by (c∗∗, r∗∗), where c∗∗ < c0 and r∗∗ < r∗.Obviously, the efficiency gain from group lending achieved at bp∗∗ (= bp (c∗∗, r∗∗)) isstrict, provided that individual liability r∗ fails to reach the situation of bp∗ = 1. Ifthe curve R ≡ 0 passes through the region below the line W ≡ W (bpo) and intersectsthe lower boundary curve r (c), then a corner solution is attained so that bp∗∗ = 1.Clearly, the uniqueness of solution (c∗∗, r∗∗) cannot be established without furtherrestriction.Alternatively, the existence of (c∗∗, r∗∗) can be shown in a general way without

involving the specifics used above. Since W ∝ bp (c, r), it suffices to maximize bp (c, r).Let us change the planner’s constrained optimization problem with two choice vari-ables (c, r) to an unconstrained one with one choice variable c in the following manner.Substituting the bank’s breakeven condition r = r (c) |R≡0 into the borrower’s opti-mal search strategy bp = bp (c, r) reduces the reservation policy to bp = bp (c). Then, theplanner’s problem becomes max

06c6cbp (c).

Recalling R0 (r∗) > 0 and m > 1, we find that bp (c) is increasing around c = 0+:limc→0+

dbp (c)dc

=

³1 + αβ bG´ bpEp6bp [p (bp− p)] + αβ (m− 1) ep bGfppc

r∗R0 (r∗)³1 + αβ bG´ > 0. (22)

This suggests that the bank’s deviation from individual liability r∗ by charging asmall c increases borrowers’ reservation type bp, reduces the riskiness of their acceptedprojects and improves the aggregate surplusW . Since bp (c) is a continuous function inthe compact set [0, c], there must exist at least one optimal level of joint liability c∗∗> 0 such that bp (c) is maximized to attain bp∗∗ = bp (c∗∗) > bp∗ 18. Then, r∗∗ = r (c∗∗)obtains. See Figure 5 for three possible shapes of curve bp (c). As shown by (22), allcurves bp (c) increase at point (0, bp∗) in this figure under fairly weak conditions. ¥

18The result obtained here under fairly weak conditions has important implications for the proofof the interest reduction caused by joint liability. Since bp∗∗ > bp∗ under c∗∗, the loan pool iswidened from

£p, bp∗¤ to £p, bp∗∗¤ including safer projects and the average repayment rate is increased

so that ep∗∗ > ep∗ as seen from (7). Then, one derives from (18) and (8) that (r∗ − r∗∗) ep∗∗ =c∗∗fppc∗∗ + r∗ (ep∗∗ − ep∗). Obviously, r∗ > r∗∗ in the bank equilibrium since ep∗∗ > ep∗; the literaturedid not provide such a rigorous proof. Also, this formula suggests that the effective reduction ininterest rates offered by the bank must exactly equal the expected joint-liability payment receivedby the bank plus the increase in income accruing to the bank and supplied by safer projects with p∈ (bp∗, bp∗∗].

21

[Figure 5 is here]The intuition behind this theorem is discussed below starting with unsecured

individual liability. A borrower chooses a project of any type p as long as it satisfiesthe search rule of p 6 bp. Under asymmetric information and individual liability, allborrowers are charged a non-discriminating interest rate r∗ (> r∗∗). A type p borroweris therefore forced to subsidize others with riskier projects of some types p0 (< p 6 bp),given that the bank’s breakeven condition is binding. The problem with this lendingscheme, however, is that risky borrowers have no incentive to correct their risk-takingbehavior without collateral. Then, the borrower who initially takes on a safe projectdecides to become riskier due to the high effective cost of borrowing caused by cross-subsidization. The average risk of the loan pool thus worsens, and a higher interestrate has to be charged by the bank to maintain its breakeven status. Borrowerschoose riskier projects the higher the interest rate, which in turn will be pushed upfurther as borrowers take even greater risk, and a vicious circle then follows. It is theintroduction of group lending which can possibly break this vicious circle and turn itinto a virtuous one, as shown below.Although each borrower under group lending faces one more possible burden of

joint liability c as opposed to unsecured individual liability, this burden generatesdifferent impacts on borrowing groups of different risks. The effective borrowing cost(r +mcpc) to a borrower, conditional on her own success, will be higher the riskierher group type pc. This forces the existing risky groups to change their risk-takingbehavior to selecting safer projects. On the other hand, safe groups are more willingthan risky ones to accept the imposition of joint liability c balanced by a cut in theinterest rate r. In equilibrium, the interest rate is decreased from r∗ to r∗∗ under jointliability c∗∗. Note that ppc is smaller the higher is p (for p > 1

2). Safe groups of high

types p have incentives to remain at least as prudent as before, because the effective”collateral” c∗∗ppc they face is lower than the effective interest reduction (r∗ − r∗∗) pthey enjoy under joint liability. The marginal groups will not take greater risk thanbefore because they can be compensated for the burden of joint liability by the lowerrate of interest. As borrowers become more prudent, or less risky, in their investmentactions to cash in on the low interest rate, and as the bank obtains joint liabilitypayments from borrowers with defaulting partners, the interest rate can be reducedfurther, which in turn induces borrowers to choose even safer projects, and a virtuouscircle then ensues. The favorable change resulting from group lending is an increasein the reservation probability of project success, bp∗ < bp∗∗. This raises the averagerepayment rate of the loan pool and the intertemporal social surplus of all marketparticipants.

7 Examples of welfare analysisThis section provides some examples to confirm our findings in the preceding sec-tions, and to analyze the effects of different lending contracts on individual borrowers’

22

welfare. In one example, we show that the bank’s expected profit under individualliability is increasing at the optimal interest rate, and this result is crucial to the proofof the superiority of joint liability over individual liability in equilibrium. We haveshown that compared with individual liability, group lending can be social-surplus-improving due to a reduction in interest rates allowed by joint liability. However,this does not necessarily result in making every type of borrower better off. We showby example in our dynamic setting that all borrowers can be made potentially betteroff, and that probably, with marginal borrowers just as well off, borrowers with safeprojects are better off while borrowers with risky projects may be worse off.We first discuss an example with individual liability. Suppose, for simplicity, that

the project distribution G is standard uniform. In this case, ep = bp − ep = 12bp, µ = 1

2

and δ = 16. The reservation project type bp is derived from (6) as follows,

bp = 1

αβ

"r1 +

2αβ

r(Ro − uo)− 1

#∝ 1

r. (23)

The borrower’s investment search policy bp is indeed inversely related to the interestrate r. The bank’s breakeven condition now becomes rbp = 2ρo. Combining this with(23) yields the socially optimal rate of interest r∗:

r∗ =2αβρ2o

Ro − uo − 2ρo , r0∗ (α) =r∗α> 0. (24)

The optimal rate of interest r∗ has to be raised with rising investment availability α.Substituting (24) to (23) gives the reservation policy bp∗ at the social optimum:

bp∗ = Ro − uo − 2ρoαβρo

, bp0∗ (α) = −bp∗α < 0. (25)

Clearly, the total effect of project availability α on the search policy bp is negative inthat the riskiness of accepted projects rises with investment opportunities. As shownin Appendix (A8), the bank’s expected profit under individual liability lending isincreasing at the optimal interest rate (i.e., R0 (r∗) > 0) if the project distribution isstandard uniform and the condition of 2ρo < Ro − uo < (2 + αβ) ρo holds.We look then at two corner solution examples to analyze the implications of

changing loan contracts for the individual welfare of different borrowers. Some re-sults obtained in our dynamic context resemble those in the static setting of Ghatak(1999) 19. Starting with an example of comparing group lending at bp∗∗ = 1 and itscorresponding individual liability lending, we show the following:

19The situation of deviating from bp∗ = 1 is used here as a special case while it was used by Ghatak(1999) in his proof of proposition 3 (as a general result) to show whether individual or joint liabilitylending achieves what he calls the first best.

23

• In the case where a joint liability contract with bp∗∗ = 1 corresponds to an individualliability scheme with bp∗ < 1, social surplus is increased under group lending,and all borrowers with various projects are made better off although the riskiesttype receives the smallest benefit.

Proof : See Appendix (A9).To show that risky borrowers may be worse off under group lending, we examine

an example of lending contracts slightly deviating from individual liability with bp∗= 1 to joint liability. It is found that

• By deviating from individual liability to joint liability in a case where bp∗∗ = bp∗= 1, safe borrowers of type p > µ2+σ2

µare made better off while risky borrowers

of type p < µ2+σ2

µare worse off, the greater the amount of joint liability c.

Although the social surplus W remains unchanged, the individual welfare ofborrowers is affected by the type of liabilities adopted in lending contracts.

Proof : See Appendix (A10).The first example involving group lending shows that all borrowers prefer joint

liability to individual liability. Obviously, safe borrowers do so because they have saferpartners. Their more prudent actions in investment under group lending improve themix of projects, raise the average repayment rate, and enable the bank to reduce theequilibrium interest rate. The fall in interest rates made possible by joint liability (asa compensation to the bank for default risk) may be significant enough to help allborrowers. This effect will be reinforced when the impact of joint liability becomesfavorable (∂bp

∂c> 0), if the interest rate r decreases below ro in the process. In this

case, falling interest rates and rising joint liability are complementary in boosting theprudent search behavior of all borrowers; they are thus made better off due to theresulting increase in social surplus.The second example suggests that joint liability is preferred to individual liability

by safe borrowers, but not by those with risky projects. Although each successfulborrower has to pay joint liability cpc for a partner’s default, this additional cost ofborrowing is higher for borrowers with riskier partners. For them, effective interestreductions (r∗ − r∗∗) p permitted by joint liability may be insufficient to compensatefor its implied effective ”collateral” c∗∗ppc, thus making them worse off. If a bankcannot reduce the interest rate that must be high to curb default risk, it may notbe able to rely on raising joint liability to boost repayments. This is because jointliability has moral hazard problems (∂bp

∂c< 0) when the interest rate r does not fall

below ro. In this case, the interest rate and joint liability are substitutes since aninterest rate reduction, coupled with a joint liability hike, creates opposing effects ona borrower’s investment search. Yet the net effect will be favorable to social surplusif the group is not large.It is now clear that group lending is social-welfare-improving, in that all borrowers

are better off, or that some of borrowers are better off and the benefit to borrowers

24

who are better off dominates the loss to borrowers who are worse off. This efficiencygain is due largely to the exploitation of effective risk pooling within a peer group andrisk diversifying from the bank to groups. The above examples rest upon a special setof assumptions, and their implications cannot be regarded as general. The conclu-sions reached in the main theorems are also based on some simplifying assumptions,such as the observability of partners’ project types, the costless enforcement of lend-ing contracts, the zero correlation among the project returns of group members, theabsence of deliberate fraud or corruption, and the profit-maximization objective ofcompetitive banks. As pointed out by Ghatak (1999), relaxing some of these under-lying assumptions may weaken the effectiveness of joint liability relative to individualliability lending, and will help address the observed mixed performance of variousgroup lending programs in the real world.

8 ConclusionThis paper has explored why joint liability in group lending improves loan repaymentscompared with unsecured individual liability if borrowers are engaged in dynamicsearch for investment. We have shown that joint liability serves as a substitute forcollateral in curbing default risk. With the self-selected group of homogenous mem-bers, effective borrowing costs are positively related to group risk types, and this hasa favorable impact on borrowers’ investment selection. Borrowers who undertake safeprojects are more willing to do so since effective interest reductions permitted by jointliability are higher than its implied effective ”collateral.” Moreover, risky borrowersbecome less risky in their investment selection because of the joint responsibility forgroup default risks. All this enables the bank to lower the interest rate, and thus toincrease social surplus.We have established in this search framework that a forward-looking borrower is

choosier in investment than a static one regardless of liabilities adopted, that borrow-ers under individual liability tend to be riskier the more easily they can find projects,and that joint liability may have a favorable or unfavorable impact on investment.Thus, in order to overcome the risk-taking behavior of borrowers with dynamic searchand offset the decline in loan recovery rates under greater project availability, the bankshould charge dynamic borrowers a higher rate of interest under individual liabilityand choose a right combination of the interest rate and joint liability under grouplending.We have shown that joint liability has dichotomous effects in the dynamic setting,

and that the interest reduction is made possible and can be balanced by joint liability.If the effect of joint liability is favorable, the interest reduction can be firmly estab-lished and group lending surely improves efficiency. If joint liability has moral hazardand if this effect is not severe relative to the expected effective ”collateral” (this istrue at the social optimum), the bank equilibrium interest rate should be lower underjoint liability than without it. In this case, efficiency can also be enhanced if moral

25

hazard is lower for joint liability than for loan rates (or if groups are not large).We have proved under fairly weak assumptions that there exists an optimal com-

bination of the interest rate and joint liability in the search equilibrium, thoughthis optimum is not necessarily unique. An interior equilibrium combination thatmaximizes social surplus can be achieved under group lending as long as the bank’sbreakeven curve does not intersect the lower bound of the combination domain. Wehave also shown that borrowers with safe projects prefer joint liability to individualliability more than their risky counterparts. Given the overall improvement in effi-ciency arising from group lending, it may be that all borrowers are made better off, orthat some with safe projects are better off while others with risky projects are worseoff.

9 Appendix(A1) The upper bound r is determined by setting U = Π

¡p¢and V (p) = U for p

∈ £p, 1¤. It is easy to obtain U = uoβcand r = Ro−uo

p. The lower bound r can be found

by setting U = Π (1) and V (p) = Π (p) for p ∈ £p, 1¤. It is also straightforward toderive U = Ro−r

βcand r = Ro−uo

1+αβµc. Clearly, r < Ro − uo < r and bp ∈ £p, 1¤ if r ∈ Ω.

(A2) Applying the Leibniz rule and the implicit function theorem to (6) yields

∂bp∂r

= −bp+ αβ bG (bp− ep)r³1 + αβ bG´ < 0, (26)

∂bp∂α

= −β bG (bp− ep)1 + αβ bG < 0. (27)

(A3) By using L’hopital’s rule, the limiting values of R (r) are calculated in thefollowing:

limr→r

R (r) = r limbp→p1

G (bp)Z bpp

pdG (p)− ρo = rp− ρo = Ro − uo − ρo > 0,

limr→r

R (r) = r limbp→1

1

G (bp)Z bpp

pdG (p)− ρo =Ro − uo1 + αβµc

µ− ρo

∝ Ro − uo − ρo − ρoµc

µ(1 + αβ) < 0.

Since R (r) > 0, R (r) < 0 and the constraint function R (r) in (9) is continuous inr ∈ Ω, one sees that there is at least one point r in Λ such that R (r) = 0 and bp (r)∈ ¡p, 1¢. Since the objective function W (r) is continuous and W 0 (r) < 0, we findthat the optimal interior solution r∗ = min

©r ∈ Λ | R (r) = 0ª exists and is unique.

26

Also, we assume that the underlying distribution and exogenous parameters are suchthat R0 (r∗) 6 0 can be ruled out (see an argument for this in Ghatak (1999)). Thatimplies that R0 (r∗) > 0, and this will be illustrated using an example later.(A4) Substituting the social optimum r∗ = r∗ (α) in the equation R (r, bp (r,α))

= 0, totally differentiating the identity R (r∗ (α) , bp [r∗ (α) ,α]) ≡ 0 w.r.t. α, andusing the signs in (27) and (7), one sees

r0∗ (α) = −r∗

R0 (r∗)dep∗dbp ∂bp∗

∂α> 0. (28)

Noting that bp∗ = bp∗ (α) = bp [r∗ (α) ,α], and recalling the signs from (26), (27) and(28), we find bp0∗ (α) = ∂bp∗

∂rr0∗ (α) +

∂bp∗∂α

< 0. (29)

Also, since ep∗ = ep∗ (α) = ep (bp∗ (α)) and ep0∗ (α) = dep∗dbp bp0∗ (α), and from the signs in (7)

and (29), it follows that ep0∗ (α) < 0.(A5) Define the cut-off type prc by setting |4r| prc = cprcpcrc under group lending.

Then, prc = 1 − |4r|cis a borrower’s type whose benefit |4r| prc is completely offset

by the ”cost” cprcpcrc. And, prc > 0 implies |4r| < c. If pL < prc < pH , then

|4r| pH − cpHpcH = |4r| pH pH − prc1− prc > 0,

|4r| pL − cpLpcL = − |4r| pLprc − pL1− prc < 0.

So, safe borrowers obtain a net benefit from group lending while risky borrowersincur a net loss from group lending. If |4r| /c > pc such that prc < p, then Bn≡ |4r| p− cppc > 0 for all p > p and Bn = |4r| pp−prc1−prc ∝ p.r (c) is derived from V (p) = Π (p) for ∀p 6 1 and Π (1) = U while r (c) is obtained

by setting V (p) = U for ∀p > p and Π ¡p¢ = U . Denote E (ppc) by δ = µ−(µ2 + σ2),and 0< δ < min

©14, µ, µc

ª. For later use, denote Λc = (c, r) | r ∈ Ωc, r > mc, 0 6 c 6 c

for r < ∞, where c is the maximal possible point of c, i.e., a point at which r (c)intersects mc, as shown in Figure 2.(A6) The comparative static results for the joint-liability search model are pre-

sented below:

∂bp∂r

= −bp+ αβ

Z bpp

G (p) dp

m (r/m+ c− 2cbp)³1 + αβ bG´ < 0,∂bp∂c

= −bpbpc + αβ

Z bpp

G (p) (1− 2p) dp

(r/m+ c− 2cbp)³1 + αβ bG´ < 0, if bp < some bpo,27

∂bp∂α

= −β

Z bpp

G (p) (r/m+ c− 2cp) dp

(r/m+ c− 2cbp)³1 + αβ bG´ < 0.

Here, r + mc (1− 2bp) > 0 since r > mc. Use standard uniform distribution, setα = 1 and β = 0.95, and examine φ (bp) = 1 − ¡1− β

2

¢ bp − 23βbp2, which is concave

and positively related to the numerator of ∂bp∂c. Solving φ (bp) = 0 yields two roots.

Taking the positive root bpo = 91%, we know that φ (bp) > 0 for bp ∈ (0, bpo), and bpo isindependent of r or c. In general, bpo ∈ ¡1

2, 1¢.

For bp < bpo, ¯∂bp∂c

¯<¯∂bp∂r

¯if m < m (bp), where since m = 1 ⇒ a contradiction,

m (bp) =

bp+ αβ

Z bpp

G (p) dp

bpbpc + αβ

Z bpp

(1− 2p)Gdp> 1,

m0 (bp) =

³1 + αβ bG´"bp2 + 2αβ Z bp

p

G (p) (bp− p) dp#"bpbpc + αβ

Z bpp

G (p) (1− 2p) dp#2 > 0.

(A7) For comparison between static and dynamic models under no c, let bps bethe static reservation type such that πe (bps) = uo. Hence, bps = Ro−uo

r. It is easy to

show bps ∈ £p, 1¤ for r ∈ Ωs = [rs, r], where rs4= Ro − uo and, r = Ro−uo

p(the same

as the dynamic one). Transforming (6) leads to bp = bps − αβ bG (bp− ep). Since bp > ep,one sees bp (r) < bps (r) for r ∈ Ω\ (r, r) , Λ. So, the term of −αβ bG (bp− ep)(< 0) captures the effect of uncertainty about investment prospects on borrowers’investment choice.For comparison between static and dynamic models under c, look at Ghatak’s

condition of determining the static reservation type bps (c, r), which is Ro − uo =(r +mcbpc) bp, where rs (c) 4= Ro−uo and, r (c) = Ro−uo−mcppc

p(the same as the dynamic

one). In our dynamic model, bp (c, r) is derived from (16), which is rewritten asRo − uo = (r +mcbpc) bp+ Y , where

Y ≡ αβ

bpZp

G (p) [r +mc (1− 2p)] dp > 0

since r+mc (1− 2p) > 0 for p 6 1 and r > mc. Note that (r +mcbpc) bp is increasingin bp. In a graph with bp measured on the horizontal axis, drawing the horizontal line

28

Ro−uo and two upward-sloping curves with (r +mcbpc) bp+Y lying above (r +mcbpc) bp,one sees that bp < bps.Notice that bp = bps − αβX for X > 0, where X = bG (bp− ep) in the no-c case, and

X = 1r+mcbpc R bppG (p) [r +mc (1− 2p)] dp in the case with c.

(A8) We show that in this special case, one needs some restrictions on the exoge-nous parameters for the optimal rate to be meaningful. Clearly, r∗ > 0 if assumingRo−uo > 2ρo. This also gives rise to R (r) > 0. Since p = 0, we have r∗ < r =∞. Inorder to have an interior solution, we need r∗ > r =

2(Ro−uo)2+αβ

. This can be achievedby imposing a restriction: 2ρo < Ro−uo < (2 + αβ) ρo, which also leads to R (r) < 0as seen in Appendix (A3). It is easy to find that r∗ > ρo is implied by the restrictionfor r∗ > r. The derivation of this restriction is shown below. Setting r∗ > r leads toan inequality such as

η (x) = x2 − 2ρox− αβ (2 + αβ) ρ2o < 0,

where x4= Ro−uo (to be used below as well). Obviously, η (x) is convex. Discarding

the negative root of quadratic equation η (x) = 0 (since Ro > uo) and retaining itspositive root x = (2 + αβ) ρo, one sees that η (x) < 0 for x ∈ (0, x). Recalling othereconomic restrictions, we have r∗ > r > 0 if x ∈ (2ρo, x).To see the relation between r∗ and Ro − uo, setting r∗ in (24) to be greater than

Ro − uo yields an inequality:

ζ (x) = x2 − 2ρox− 2αβρ2o < 0.

Drop the negative root of ζ (x) = 0 and retain the positive one: xo =¡1 +√1 + 2αβ

¢ρo.

Then, under the economic restriction, ζ (x) < 0 for x ∈ (2ρo, xo) and ζ (x) > 0 for x∈ (xo, x). Thus, we establish that r∗ > Ro−uo if 2ρo < Ro−uo <

¡1 +√1 + 2αβ

¢ρo,

and r∗ < Ro − uo if¡1 +√1 + 2αβ

¢ρo < Ro − uo < (2 + αβ) ρo.

To show the sign of R0 (r∗) > 0, which has been used in (22), we calculate thevalue R0 (r) at r∗ for this example as follows:

R0 (r∗) = ep∗ ·1− Ro − uo2ρo (1 + αβbp∗)

¸.

Setting R0 (r∗) > 0 offers us an inequality:

ξ (x) = 3x2 − 8ρox+ 4ρ2 > 0.

Equation ξ (x) = 0 has two positive roots: x1 = 23ρo and x2 = 2ρo. Since ξ (x) is

convex, one sees that ξ (x) > 0 for x ∈ (0, x1)∪ (x2, x). Then, taking into account theeconomic restriction, we have R0 (r∗) > 0 if x ∈ (x2, x). The sign of R0 (r∗) is justifiedby the same condition as that for r∗ > r.

29

(A9) Suppose that an initial contract (c∗∗, r∗∗) is such that bp∗∗ = 1. Substitutingthis into (16) and (18) yields a set of two equations in two unknowns (c, r). Solvingthis set in the case of standard uniform distribution, setting m = 1 and α = 1 forsimplicity, we obtain:

r∗∗ =βρo +Ro − uo

1 + β, (30)

c∗∗ =3

1 + β[(2 + β) ρo − (Ro − uo)] .

Here, we need (2 + β) ρo > Ro−uo to make c∗∗ > 0, and Ro−uo > 3+β2ρo to have r∗∗

> c∗∗ (that satisfies the assumption of r > mc for m = 1). Since r∗∗ = r (c∗∗) holdsin this case, the lending contract (c∗∗, r∗∗) is indeed a corner solution.Substituting from (24) and (30) into r∗ > r∗∗ produces an inequality:

ψ (x) = x2 − (2− β) ρox− 2βρ2o (2 + β) < 0,

where ψ (x) is convex. Solving ψ (x) = 0 and taking its positive root x+ = (2 + β) ρo,one sees that ψ (x) < 0 for x ∈ (0, x+) where x+ = x |α=1. The economic restrictionrequires x ∈ (2ρo, x+). This suggests that the condition for r∗ > r∗∗ is identical tothat for c∗∗ > 0.Expressions (24) and (25) for α = 1 yield r∗ > 0 if Ro − uo > 2ρo (implying the

condition for r∗∗ > c∗∗), and bp∗ < 1 if Ro − uo < (2 + β) ρo (implying the conditionfor c∗∗ > 0). Thus, we know that W∗ < W∗∗ from bp∗ < 1 = bp∗∗. Because r0 (0+) |R≡0(= −δm

µ) < 0 is true for this example, there is some substitution between the interest

rate r and joint liability c to maintain the bank’s breakeven equilibrium.To see if joint-liability profit πjoine (p) is greater than individual-liability profit

πindie (p) for any type p, it suffices to check whether r∗ > r∗∗ + c∗∗ (> r∗∗ + c∗∗pc) orprc < p. This holds under the same condition as that for c∗∗ > 0, as shown below.Substituting from (24) and (30) into r∗ > r∗∗ + c∗∗ gives an inequality:

ϕ (x) = x2 − ρo (5 + 2β)x+ ρ2o¡6 + 5β + β2

¢> 0,

where ϕ (x) is convex. Solving ϕ (x) = 0 and dropping the larger root and taking thesmaller one x+ = (2 + β) ρo, one finds that ϕ (x) > 0 for x ∈ (0, x+). The economicrestriction requires x ∈ (2ρo, x+). For the riskiest type p = 0, since (r∗ >) r∗∗+ c∗∗pc> r∗∗ + c∗∗pc, it follows that all types are better off with group lending, with theriskiest type benefiting the least from it. All borrowers must prefer joint liability toindividual liability in this case, for the borrower who has the lowest success probabilitydoes so.(A10) Set m = 1 for simplicity. Suppose that an initial individual liability r∗ is

such that bp∗ = 1 = bp (r∗, c∗) |c∗=0(treating individual liability as a special case ofjoint liability). The bank’s breakeven curve passes through point (0, r) in Figure 2,where r∗ coincides with r as a corner solution. In this case, ep∗ = µ and r∗ = ρo/µ

30

in (8). Now we depress r∗ a bit by ε > 0, and change c∗ = 0 to τ > 0 such that thebank equilibrium is maintained and the point (r∗−ε, τ) still lies above line r = mc inFigure 2. Making bp∗∗ = 1 at this point and substituting (r∗− ε, τ , bp∗∗ = 1) into (18),one sees τ = ρo−(r∗−ε)µ

δ= εµ

δ(using r∗ = ρo/µ). This joint liability contract (r∗∗, c∗∗)

= (r∗ − ε, τ) generates the same social surplus as the individual liability contractsince W∗∗ = W∗. The move is in the iso-bp zone: bp (r, c) ≡ 1, and both contractssatisfy the search policy since p 6 1 = bp in this zone. Point (r∗ − ε, τ) is outside Λcsince r∗ − ε < r∗ = r < r (τ). At social optima (r∗∗, c∗∗) and r∗, we have πjoine (p)

= Ro − phρoµ− ε

¡1− µpc

δ

¢iand πindie (p) = Ro − pρoµ . If πindie (p) < πjoine (p), then µpc

δ

< 1; thereby p > µ2+σ2

µand d

dεπjoine > 0. If πindie (p) > πjoine (p), then p < µ2+σ2

µand

ddεπjoine < 0. Finally, we verify the following facts. Since r∗∗ = r∗ − ε, one sees that

r∗∗ < r∗. Since r∗ = ρo/µ and c∗∗ = τ = εµδ, we need a restriction ε < ρo

µ(1+µ/δ)to

make c∗∗ < r∗∗. Substituting the expressions of r∗ − r∗∗ and c∗∗ into prc = 1− r∗−r∗∗c∗∗

yields prc =µ2+σ2

µ.

References[1] Aghion, B. de Armendariz, 1999. ”On the Design of a Credit Agreement with

Peer Monitoring.” Journal of Development Economics. Vol. 60, 195-228.

[2] –––, Gollier, C., 2000. ”Peer Group Formation in an Adverse SelectionModel.” The Economic Journal. 110 (July), 632-643.

[3] Besley, T., Coate, S., 1995. ”Group Lending, Repayment Incentives and SocialCollateral.” Journal of Development Economics. 46 (1), 1-18.

[4] Conlin, M., 1999. ”Peer Group Micro-Lending Programs in Canada and theUnited States.” Journal of Development Economics. Vol. 60, 249-269.

[5] Conning, J., 1996. ”Group Lending, Moral Hazard and the Creation of SocialCollateral.” IRIS Working Paper #165. University of Maryland at College Park.

[6] Ghatak, M., 2000. ”Screening by the Company You Keep: Joint Liability Lend-ing and the Peer Selection Effect.” The Economic Journal. 110 (July), 601-631.

[7] Ghatak, M., 1999. ”Group Lending, Local Information, and Peer Selection.”Journal of Development Economics. Vol. 60 (1), 27-50.

[8] Ghatak, M., Guinnane, T.W., 1999. ”The Economics of Lending with JointLiability: Theory and Practice.” Journal of Development Economics, Vol. 60,195-228.

31

[9] Gu, X., 2002. ”A Search Equilibrium Model of Credit Markets with SuspensionPenalty against Default.” Working Paper. University of Toronto.

[10] Huppi, M., Feder, G., 1990. ”The Role of Groups and Credit Cooperatives inRural Lending.” The World Bank Research Observer. 5 (2), 187-204.

[11] Laffont, J.-J., N’Guessan, T., 1999. ”Group Lending with Adverse Selection.”Mimeo. IDEI, Toulouse.

[12] Morduch, J., 1998. ”Does Microfinance Really Help the Poor? New EvidenceFrom Flagship Programs in Bangladesh.” Working Paper, Princeton University.

[13] Stiglitz, J., 1990. ”Peer Monitoring and Credit Markets.” World Bank EconomicReview. 4 (3), 351-366.

[14] Variance, H., 1990. ”Monitoring Agents with Other Agents.” Journal of Institu-tional and Theoretical Economics. 146 (1), 153-174.

[15] Tassel, E.V., 1999. ”Group Lending under Asymmetric Information.” Journal ofDevelopment Economics. Vol. 60, 3-25.

32

A Search Approach to Joint Liability Lending under Imperfect Information

r ( )cr ( )cr r 0

op

1

p

( )rcp ,ˆ

( )rp

or

p

r

( )crr =

c 0

**c

**r

*r

W∇

( ) indiWrcW *, ≡

c′

( ) ( )opWrcW ˆ, ≡

or

r

r

Figure 1: The effect on search policies of interest rates and jo

Figure 2: The existence of socially optimal solutio

( ) joinWrcW **, ≡

r

mcr =

( )crr =

( ) 0, ≡ℜ rc(

int liability

n under group lendin

c

g

A Search Approach to Joint Liability Lending under Imperfect Information

( )r

ppcE cppo ˆ≤−ρ

F

µ

p

Fig

*~p

~p

op~ r

oρ

ig

( )rcp ,~

( )rp~

ur

*

**~~

pp

*

*

p

*r or

ure 3: The reduction in loan rates if moral ha

**r *r or

rcEpo −ρ

roρ

( )cr r

e 4: The interest reduction when c has a favor

r

**r

( )cr

zard is lower

( )ppcp≤

( )rp~

( )rcp ,~

able incentive

r

for c than for r

r

effect

A Search Approach to Joint Liability Lending under Imperfect Information

0 c **c

*p ( )cpp ˆˆ =

p

**p

Figure 5: The superiority of joint liability over individual

c

liability