Screening by the Company You Keep: Joint Liability Lending and the Peer Selection Maitreesh Ghatak presented by Chi Wan 1. Introduction The paper looks at an economic environment where borrowers have some information about the nature of each other s projects that lenders do not, and analyses a contractual mechanism through which lenders can untilise information borrowers may have about each other, and thereby overcoming the problems of adverse selection in credit markets. This paper shows that by lending to self-selected groups of borrowers and making them jointly liable for each other s loan repayment, a lender can achieve high repayment rates even when these borrowers cannot offer any collateral. 2. Motivation This work is motivated by contractual methods successfully used by real world lending institutions, such as group-lending programmes and credit cooperatives. The practice of using joint liability to lend successfully to borrowers who cannot offer any conventional collateral actually goes well back in history. One of the dramatic success story of group lending is about the Grameen Bank.
3. Past Literature (brief and for the theoretically purposes of the paper) (a) The existing literature has largely treated group formation under joint liability lending as exogenous. (b) Van Tassel (1999) Tassel s paper shows that under imperfect information, lenders may be able to utilize joint liability contracts as a way to screen agent types by inducing endogenous group formation and self-selection among the borrowers. (c) Armendariz and Gollier (1998) Joint liability can improve the pool of borrowers if borrowers have perfect knowledge of their partners.
4. The model A one-period model of a credit market under adverse selection. Technology and Perferences A large number of borrowers who live in a same village. All borrowers are assumed to be risk-neutral and maximise expected returns. Every borrower is endowed with one unit of labor and a risky investment project To launch the project, it requires one unit of capital and one unit of labor. The outcome of the risk project is either a success (S) ora failure (F), denoted as a binary random variable x S, F Two types of borrowers: risky and safe borrowers. For risky borrowers, the probability of success of their project is p r, and for safe borrowers, that is p s. p r p s Risky and safe borrowers exist in proportions and 1 in the population. The return of the project is a random variable y i i s, r, y i R i,0. Borrowers of both types have a reservation payoff u. Risk neutral banks The opportunity cost of capital per loan is 1.
Information and Contracting The type of a borrower is unknown to the lenders (banks). Borrowers know each other s types. The information assumption of this model is that the outcome of a project of a borrower, x, is obervable by the bank at no cost and is verifiable. The realized returns of a project of a borrower, y i,isvery costly to observe for the bank. A limited liability constraint. In case their projects fail, borrowers are liable up to the amount of the wealth they posses, w. Wetakew 0 for simplicity. The only contractible variable is the vector of project outcomes of all borrowers, denoted as X. A lending contract can only specify a transfer from a borrower to the bank for every realization of X.
Two types of credit contracts: Individual Liability Contract That is a standard debt contract (SDC) between a borrower and the bank with a fixed repayment r in the case of non-bankruptcy state (x S ), and maximum recovery of debt in the case of bankruptcy state (x F ). We assume there is no recovery of debt in this model for simplicity. Joint Liability Contract Bank asks borrowers to form groups of a certain size. Every borrower has an individual liability component r and a joint liability component c. In the case of bankruptcy (x F), the borrower pays nothing back to the bank (zero recovery value of the debt). In the case of non-bankruptcy (x S), the borrower pays r back for his own debt and pays an additional joint liability payment c per member of his group whose projects have failed. Thus unlike SDC, repayment of joint liability contract is not fixed in the non-bankruptcy states.
5. Individual Liability Lending 5.1 The Full Information case If bank has full information about a borrower s type, then the optimal contracts are standard debt contracts (SDC) under which the borrower pays nothing when his project fails, and the interest rate when his project succeeds. The equilibrium interest rates: r i p i, i r, s. 5.2 The Asymmetric Information case Bank will offer a pooling individual liability contract with the interest rate r s r r r. Then the expected payoff to borrower of type i would be the difference between the expected payoff and the expected interest payment. U i r p i R i rp i, i r, s. The literature on the adverse selection problem in credit markets assumes that borrowers differ by a risk parameter which is not observed by the bank. (i) Assume that the risky and safe projects have the same mean return, that is p s R s p r R r R, but risky projects have a greater spread around the mean. (ii) Assume that risky projects have a lower mean than safe projects, but both types of projects have the same return when they succeed. That is to assume R s R r R, and p r p s.
The Underinvestment Problem Assume all projects have the same mean return, p s R s p r R r R, but risky projects have a greater variance around the mean. Assume that these projects are socially porductive, that means the expected return is larger than the opportunity costs of labor and capital: R u. In the full information case, the bank charges different interest rates for different types of borrowers, and the interest rates would be the opportunity cost of capital devided by the possibility of success r i /p i, i r, s. The expected payoff of each type of borrower will be equal to the difference between the expected payoff from the project and the expected interest payment: U i r i p i R i r i p i R. The average repayment rate will be equal to p, the average probability of success for the entire population. In the asymmetric information case, if the bank charges all borrowers the same interest rate r, and both types of borrowers borrow in equilibrium, from the zero profit constraint we get r / p. The participation constraints: For safe borrowers: U s r p s R s ps p R ps p u For risky borrowers: U r r p r R r pr p R pr p u R ps p u. If R ps p u, a pooling contract does not exist, and only the risky borrowers would borrow from bank. The repayment rate and welfare are strictly less than that under full-information. This is known as the lemons or the under-investment problem in credit markets with adverse selection.
The Overinvestment Problem By solving the participation constraints: For safe borrowers: U s r p s R s ps p p sr ps p u For risky borrowers: U r r p r R r pr p p rr pr p u p r R p u. Assume p r R u, that means risky projects are unproductive. If p r R u, the risky borrower will find it profitable to borrow as p they are cross-subsidised by safe borrowers even through they make a negative contribution to social surplus. This is the overinvestment problem in the credit market with adverse selection.
6. Joint Liability Lending The joint liability lending can improve efficiency so long as borrowers have some private information about each other s projects. 6.1. Group Formation: The Assortative Matching Property The equilibrium in the group-formation game will satisfy the optimal sorting property: borrowers not in the same type could not form a group without making at least one of them worse off. Proposition 1. Joint liability constracts lead to positive assortative matching in the formation of groups. Comments & Intuition For any given joint liability contract r, c, borrowers will always choose partners of the same type. Borrowersareallowedtobeabletomakesidepaymentto each other. So in principle, a risky borrower can make a transfer to a safe borrower to have him as a partner. After compensating the safe borrower for the loss of having a risky partner, a risky borrower will never find it profitable to bribe a safe borrower to be in his group.
The expected payoff of type-i borrower under a joint liability contract r, c would be: U ii r, c p i p i R i r p i 1 p i R i r c p i R i p i r p i 1 p i c i r, s Assume that all projects have the same mean return, p s R s p r R r R. The indifference curve of a borrower of type i in the r, c plane is represented by this line rp i c 1 p i p i Const. The slope of an indifferenc curve of a type i borrower is dc dr U ii Const 1 1 p i. p s p r 1 1 p s 1 1 p r. Preferences of borrowers over joint liability contracts satisfy the single-crossing property.
6.2. Optimal Joint Liability Contracts: Joint Liability as a Screening Device The contracting problem is the following sequential game: stage 1: the bank offers a finite set of joint liability contracts r 1, c 1, r 2, c 2, ; In this two-type-borrower case, we restrcit the bank s choice of optimal contracts to a pair r r, c r and r s, c s designed for groups consisting of risky and safe borrowers respectively. stage 2: borrowers who wish to accept any one of these contracts select a partner and do so; stage 3: projects are carried out and outcome-contingent transfers as specified in the contract are met. Borrowers who choose not to borrow enjoy their reservation payoff of u. The bank s objective is to choose r r, c r and r s, c s to maximize a weighted average of the expected utilities of a representive borrower of each of the two possible types: V U rr r r, c r 1 U ss r s, c s, where 0, 1.
The bank is facing the following constraints: (1) The zero-profit constraint of the bank For the separating contracts r r, c r and r s, c s, it requires: r r p r c r 1 p r p r r s p s c s 1 p s p s. It ensures that the expected repayment from each loan is at least as large as the opportunity cost of capital. Let r, c denote the contract that satisfies the zero-profit constraints for both risky and safe borrowers with equality. rpr c 1 p r p r r rps c 1 p s p s pr p s 1 / p r p s c / pr p s Assume p r p s 1 to ensure that r 0.. For a pooling contract r, c, the zero-profit constraint is: r r p r c r 1 p r p r 1 r s p s c s 1 p s p s. (2) The participation constraint of each borrower. U ii r i, c i u, i r, s. (3) The limited liability constraint That requires the sum of individual and joint liability payments, r c, cannot exceed the realized revenue from the project when it succeeds: r i c i R i, i r, s. (4) The incentive-compatibility constraint U rr r r, c r U rr r s, c s U ss r s, c s U ss r r, c r That requires that it is in the self-interest of a borrower to choose a contract that is designed for his type.
By Proposition 1, for any given joint liability contract r, c offered by the bank in stage 1, assortative matching will result in stage 2 of this game. LEMMA 1. If r r, c r and r s, c s satisfy the incentive-compatibility constraints, then they will induce assortative matching in the group formation stage. LEMMA 1 ensures that even if the bank offers a menu of joint liability contracts in stage 1, assortative matching will still result in stage 2 of the game.
6.2.1. Joint liability lending and the underinvestment problem All projects have the same mean return, p s R s p r R r R. LEMMA 2. For any joint libaility contract r, c, ifr r and c c,then U ss r, c U rr r, c, andifr r and c c,thenu ss r, c U rr r, c. If we set U ii r, c equal u, then the indifference curve of each type will correspond the set of contracts that satisfy the zero-profit constraints for that type of borrowers. Then Point A is the contract r, c, that satisfies the zero-profit contraints for both risky and safe borrowers with equality. This lemma follows from the single-crossing property, and helps to identify the set of incentive-compatible contracts.
Proposition 2. Suppose that assumptions A1, A2, A3 and A4 hold. Then optimal separating joint liability contracts r r, c r and r s, c s exist which have the property r s r r and c s c r. The average repayment rate and welfare under these contracts are equal to their full-information levels and strictly higher than those under individual liability contracts. A1: R u A2: R ps p u A3: p r p s 1 A4: R 1 ps p r ZPC r : the set of contracts that satisfy the zero-profit constraints for risky borrowers with equality. ZPC s : the set of contracts that satisfy the zero-profit constraints for safe borrowers with equality. AC: The set of incentive-compatible contracts for risky borrowers. DA: The set of incentive-compatible contracts for safe borrowers. LLC: the set of contracts that satisfy the limited liability constraint with equality. AC: the optimal contracts for risky borrowers. BA: the optimal contracts for safe borrowers.
Proposition 3 show that the optimal pooling contract achieves higher repayment rates and welfare than individual liability contracts under more general conditions than separating contracts. Proposition 3. If assumptions A1, A2, A3 and A4 hold then a unique optimal pooling joint liability contract exists and is equal to r, c. Even if A4 is not satisfied so that optimal separating joint liability contracts do not exist, so long as A5 is satisfied together with A1, A2, and A3, optimal pooling joint liability contracts exists and achieve higher repayment rates and welfare than individual liability contracts. Assumption A5: R ps p u, where p r 2 1 p s 2 /p s p. A5 ensures that the limited liability constraint is satisfied. In the r, c plane the line r p r 1 p s c p r 1 p r 1 p s 1 p s Const represents an indifference curve of an average borrower as well as an iso-profit curve of the bank when offering a pooling contract.
ZPC r,s : the set of contracts that satisfy the pooled zero-profit line. ZPC r : the set of contracts that satisfy the zero-profit constraints for risky borrowers with equality. ZPC s : the set of contracts that satisfy the zero-profit constraints for safe borrowers with equality. LLC: the set of contracts that satisfy the limited liability constraint with equality. PC s : represents the assumption A2 R ps p u, and ensure the existence of a pooling contract. In the figure, point A is above the line LLC, soa4isviolated. A set of optimal pooling contracts nevertheless exists as illustrated by the line segment BC, and that is guaranteed by Proposition 3.
6.2.2 Joint liability lending and the overinvestment problem In this part, the paper shows that joint liability contracts can discourage unproductive risky borrowers from borrowing, and solve the overinvestment problem under individual liability lending. Since the proof of the assortative matching property in group-formation, and consequently, the single-crossing property of indifference curves do not depend on the distribution of the revenues of the projects, Proposition 1 still applies in this case. As the incentive-compatibility and zero-profit constraints are the same as before, the set of incentive-compatible contracts that satisfy the zero-profit constraints for risky and safe borrowers are the same as well. Proposition 4. Suppose that the assumptions A1, A2, A3 and A4 hold. If projects have different mean returns and risky projects are unproductive in terms of expected returns, joint liability contracts will discourage risky borrowers from borrowing and thereby achieve a strictly higher average repayment rate and expected social surplus compared to individual liability contracts. A1 : p r R u. A2 : p r R p u. A3: p r p s 1. A4 : R 1 p s 1 p r.
8. Some Extersions 8.1. Risk Averse Borrowers When the borrowers are risk averse, the main results go through! However, there s only a unique pair of optimal separating contracts in this case. 8.2. Multiple States of the World and Costly State Verification It s showed that if verifying returns is costly, the optimal contract is a standard debt contract. If the borrower doesn t announce bankruptcy, pays his dues to the bank, and bank doesn t undertake costly output verification. And borrower announces bankruptcy only when he realized that return is less than his debt to the bank, and bank collects all output net of verification costs. 8.3. Correlated Project Returns If project returns are positively correlated, then there is less heterogeneity and less room for the effects of joint liability to work. The joint liability works particularly well if project returns are negatively correlated. 8.4. Optimal Size of the Group Joint liability works better than other financial contracts because group members have superior information on one another. This advantage is likely to be diluted in larger groups. On the other hand, if the project returns are uncorrelated, an increase in group size improves the effectiveness of joint liability because it increase the number of states of the world in which the group as a whole can repay its members loans.
9. Conclusion This paper proposed a theory to explain how joint liability contracts can achieve high repayment rates even when borrowers have no conventional collateral to offer. It is based on the fact that borrowers are asked to self select group members, which is shown to economize on information costs by exploiting local information. 10. Criticisms & Extersions (1) The whole argument is built on the assumption that the preferences of borrowers over joint liability contracts satisfy the single-crossing property. (2) The group size is assumed to be the same over the whole population. (3) The peer-pressure may also play a role in joint liability lending programme.