Group Size Matters in Microlending under Adverse Selection or Moral Hazard

Transcription

1 Group Size Matters in Microlending under Adverse Selection or Moral Hazard Christian Ahlin September 2011 Abstract Department of Economics, Michigan State University; All errors are our own. 1

2 1 Introduction Ghatak (1999, 2000) showed how group lending could improve efficiency and equity in a lending environment where borrowers risk is common knowledge among borrowers but unknown to the lender. The efficiency gains come from the incentives for borrowers to sort homogeneously and the improved risk-pricing that results, both stemming from the joint liability contract. 1 Along with the vast majority of the sorting literature, Ghatak focused on two-person matches. In particular, group size is restricted to two in most of the analysis. This is helpful for two reasons. First, matching results can be significantly clearer with two-person groups. 2 Second, joint liability lending contracts can be summarized in just two parameters with two-person groups: a direct interest rate due on one s own loan, and an extra joint liability (bailout) payment due when one s partner fails. In short, two-person groups provide tractability. However, it would be helpful to know what optimcal contracts look like when group size is greater than two. Empirically speaking, many microlenders stipulate group sizes in the vicinity of five, and some allow for groups of more than ten; groups of size two are rare. Do joint liability contracts work the same way when groups are larger? Second, there are few formal results on optimal group size in micro-lending, though Ghatak (1999, 2000) provides insightful discussion (see also Baland and Somanathan, 2011). This paper aims to make some progress in understanding whether and when larger groups might be better. The goal of this paper is to study the optimal group contract and its properties when group size is n 2. We study a flexible contract that allows for nonlinearity in how a borrower s liability changes with the number of group failures. We also require contracts to satisfy the monotonicity constraint proposed by Innes (1990) and Ghatak et al. (2005), 1 Evidence for homogeneous sorting and improved risk-pricing can be found in Ahlin (2009). 2 See, e.g., Ahlin (2011). 2

3 namely that liability can be no greater than full. In other words, the lender cannot collect more money from the group when more people fail. However, due to potential complexities of sorting patterns when group size is greater than 2, we restrict attention to contracts that induce positive assortative matching as does the original literature. The closest to our work is the analysis of group size by Ghatak (1999, Appendix). There, the liability payment of a given borrower is assumed linear in number of group failures; under this assumption, group size is shown to be essentially irrelevant. However, this form of liability is not claimed to be optimal. This leaves open the question, is the optimal joint liability contract non-linear and does group size matter under the optimal contract? A preliminary finding of the paper is that not all joint liability contracts as we define them lead to homogeneous matching when n>2 (though they do when n =2). Nonhomogeneous matching had been shown possible in previous work by Sadoulet (1999) and Guttman (2008) in dynamic contracts with groups of 2. Our result is new in that it shows simple static group contracts need not lead to homogeneous matching when n>2. As a result, we restrict attention to contracts that induce homogeneous matching. We find that the optimal such contract imposes full liability, subject to the limited liability constraint. The intuition is as follows. In order to improve risk-pricing, which inefficiently and inequitably favors risky borrowers over safe, the lender would like to target more repayment burden toward risky borrowers. The way to do this is to load more repayment onto states of the world where there have been more failures. The monotonicity constraint limits the extent to which this can be done; thus the optimal contract causes this constraint to bind and imposes full liability whenever possible, i.e. when the limited liability constraint does not prevent it. A first set of results ignores the limited liability constraint upon success; that is, it is assumed a borrower can pay any amount when he succeeds. This condition can always be satisfied without affecting any key conditions if returns are high enough in both the borrowing-funded project and the outside option. In this scenario, the larger the group, the 3

4 better. As group size goes to infinity, perfectly equitable and efficient lending can always be attained, even with a continuum of types, simply by offering a standard contract with full liability to all comers. The intuition for the first set of results is that with full liability, the lender is fully repaid unless everyone fails. The probability of everyone failing goes to zero as group size gets large. Thus, all groups pay the same amount in the limit, that is, large groups eliminate the between-group cross-subsidy. The within-group cross-subsidy is also eliminated, by homogeneous matching. Thus, the cross-subsidy is asymptotically eliminated, allowing for perfect risk-pricing and efficient lending. The first set of results, though, requires successful borrowers to be able to bail out all their partners if needed. This is not necessary, as a second set of results makes clear. That is, imposing limited liability after success need not change the result that fully efficient lending is attainable as long as groups are big enough. The reason is that what matters as groups get large is whether a successful borrower can afford to bail out his average share of group defaulters which would be less than one defaulter s loan, for example, in a group where the success rate is above 50%. This is because, by the central limit theorem, probabilities are increasingly concentrated around mean default rates. If borrowers can afford this more typical bailout case, then in the limit the between-group cross-subsidy is again eliminated. In short, we find that larger groups allow for better risk-pricing and more efficient lending. This is not surprising. Given homogeneous matching in groups of size n, the lender essentially gets n observations from a single borrower s project distribution (since all his partners are identical to himself). Thus, the lender knows with increasing certainty a borrower s type as n gets large. This knowledge allows it to structure the contract so as to (asymptotically) eliminate the cross-subsidy. We show two extensions to these results. First, we switch to an analogous moral hazard framework (a simplified version of Stiglitz and Weiss, 1981, and Stiglitz, 1990) and show the same result holds. That is, large enough group size can eliminate the moral hazard problem. 4

5 The intuition is similar larger group size increases information revelation and the lender s ability to cause the borrower to bear the full risk of his project. Second, we build in a cost to larger group size, namely deteriorating information quality. We show that in this context, if information quality approaches zero as groups get large, then the optimal group size is interior; that is, bigger is not always better. The point of this result is simply to formalize the well-known idea that a potential cost of increasing group size is worsened social cohesion, social information, and/or social cooperation (see, e.g., Ghatak, 2000). In sum, the theory uncovers a previously unexplored advantage to larger group size, which operates even when borrowers cannot afford large bailout payments. This advantage operates in both adverse selection and moral hazard contexts. This can help explain the group sizes often observed larger than two, and sometimes quite large. However, if information quality deteriorates sufficiently with group size, the optimal group size is bounded, and optimal group size can vary as it seems to do in the data depending on properties of the risk distribution and information attenuation. 2 Basic Model There is a unit measure continuum of risk-neutral agents, indexed by i (0, 1). is endowed with no capital, one unit of labor, a subsistence option, and a project. Each The subsistence option requires one unit of labor and gives expected output u. The project requires one unit each of capital and labor. the probability of success of the project, p i. Agents projects differ in risk, specifically, Agent i s project yields gross returns of R i ( succeeds ) with probability p i and yields 0 gross returns ( fails ) with probability 1 p i. The p i s are distributed over (p r,p s ) according to strictly positive and continuous density function g(p), where 0 <p r <p s < 1. Notationally, we let f(p) E[f(p)] = p s p r f(p)g(p)dp be the expected value of f(p) in the population. For example, p = E(p) andp 2 = E(p 2 ). 5

6 The Stiglitz-Weiss (1981) assumption is that projects have the same expected value but differ in risk, in the sense of second-order stochastic dominance: p i R i = R, i. (A1) The higher is p i, the lower is the agent s risk. Given their lack of capital, agents require outside funding to carry out their projects. We consider a non-profit lender that maximizes total borrower surplus subject to earning expected rate of return ρ>0 on capital. Define N R u. (1) ρ Since the numerator is the return to capital in the agents projects, and the denominator can be thought of as the return to capital elsewhere, N can be interpreted as the net excess return to capital embodied in these agents projects. It is assumed that R>ρ+ u N > 1. (A2) This implies that all projects are expected to return more than the cost of their inputs, capital and labor. Thus, social surplus is monotonically increasing in the number of projects funded. It is thus straightforward to show that if the lender exactly breaks even, its objective function is equivalent to lending to as many borrowers as possible, i.e. maximizing outreach. We assume limited liability, in particular that agents exposure in any financial contract is limited to project returns. It follows that an agent who fails owes nothing to an outside lender. Agents who succeed owe no more than their entire output. The importance of this constraint will often depend on the gross excess return to capital in this market, G where G R ρ. (2) 6

7 Note that the gross excess return, G, and the net excess return, N are related but can vary independently. For example, raising R and u in tandem raises G while leaving N fixed. Agents types are observable to other agents, but not to the lender. Further, following Ghatak (1999, 2000), we assume the lender can observe output only coarsely: it can distinguish between Y = 0 (fail) and Y > 0 (succeed), but not between different levels of Y > 0. This assumption along with limited liability makes debt contracts the only feasible financial contracts. 3 We also restrict attention to deterministic, symmetric contracts. 3 Generalized Joint Liability Lending In this section, we define generalized joint liability contracts and explore matching and optimality. We consider only pooling contracts; this is without loss of generality, at least in the 2-type case, because any optimal screening contract can be replicated with a pooling contract. 4 Consider contracts for borrowers in groups of size n 2, where n is an integer. Such a group contract can be summarized by n interest rates: (r 0,r 1,..., r n 1 ), where r k is the amount due from a borrower who succeeds and k of whose fellow group members fail, k {0, 1,..., n 1}. We define a joint liability contract as a group contract that satisfies: r 0 r 1 r 2... r n 1, and r 0 <r n 1. (A3) Thus, for a group contract to involve joint liability, the amount a borrower owes must a) vary and b) be non-decreasing in the number of failures in the group. In the baseline Ghatak model (1999, 2000), r 0 = r and r 1 = r + c; thus, by our definition 3 There are no enforcement issues by assumption: borrowers who can repay, do. 4 Ahlin and Waters (2011) prove this in a related context. This equivalence stems from the fact that the payoffs of the lender and any type of borrower are fixed-sum. The idea is that in an optimal menu of screening contracts, the risky borrower s incentive constraint must bind; thus, he must be indifferent between his own contract and the safe borrower s, and since the lender and risky-borrower payoffs are fixed-sum, so is the lender. Thus, offering only the safe borrower s contract achieves the same outcome. 7

8 the set of joint liability contracts is all (r, c) such that c>0, exactly as in his analysis. The Ghatak (1999) extension for n>2restricts attention to two-parameter contracts of the form r k = r + ck, fork {0, 1,..., n 1}; again, this is a joint liability contract by our definition iff c>0, exactly as in his paper. 3.1 Matching under generalized joint liability In both cases considered by Ghatak, homogeneous group formation occurs as the unique stable matching outcome iff c>0. This implies that the set of joint liability contracts coincides exactly with the set of homogenous-matching contracts in both of these settings. That is, joint liability implies unique homogeneous matching and vice versa. This is obviously a general result in the n = 2 case, but one might wonder if it also holds when n>2and attention is not restricted to contracts of the form r k = r + ck? The answer is negative: Lemma 1. Assume p s > 2/3 [MAKE MORE PRECISE.] For any n 3, joint liability contracts exist under which homogeneous group formation is not stable. Proof. See Appendix. An example of a joint liability contract that violates homogeneous group formation is (r, r,..., r, r + κ), for κ>0; that is, a fixed interest rate plus a joint liability payment κ due only from a borrower when all fellow group members fail. For any n 3, this gives rise to a group payoff function that is submodular over the domain (2/3, 1) n, ruling out homogeneous matching over this same domain. (The relevant domain is at least (1/2, 1) n for n 4.) The intuition is that since the contract penalizes only extreme failure, spreading out safe types across groups raises payoffs by lowering chances of paying the penalty. This example makes clear that joint liability contracts at least as we define them do not always give rise to homogeneous group formation. 5 Thus in the reality of micro-lending, where groups typically have more than two members, simply imposing some joint liability 5 A re-definition of A3 that made all inequalities strict would not help, since the example could include very small jumps in the interest rate before the jump to r n 1 without changing anything. 8

9 need not invoke homogeneous matching the specifics of joint liability matter. It is also easy to show that some non-joint liability contracts can induce homogeneous matching. Thus, there is no one-to-one relationship between homogeneous grouping and joint liability. From here we could proceed to seek the optimal joint liability contract, knowing that some will induce heterogeneous sorting patterns. However, this problem appears intractable, since submodularity of the group payoff function appears by itself not to go very far in pinning down matching patterns (see Ahlin, 2011), which can depend on distribution of types and specifics of the payoff function. Hence, our approach will be to restrict attention to contracts that induce homogeneous matching, and seek optimal contracts from among this set. 4 Homogeneous Matching, Monotonic Group Contracts We restrict attention to group contracts that induce homogeneous group formation, subject to the lender breaking even and borrower limited liability. Following Innes (1990) and Gangopadhyay et al. (2005), we also impose the following monotonicity constraint on the group contract: nr 0 (n 1)r 1 (n 2)r r n 2 r n 1, i.e. (n k)r k (n k +1)r k 1, k {1, 2,..., n 1}. (3) This guarantees that the total amount owed by the group is (weakly) increasing in the number of successes in the group: if there are (n k) successes, then (n k) borrowers owe r k, while if there are (n k + 1) successes, then (n k + 1) borrowers owe r k 1. The argument is that if it were cheaper to have more successes, group members could simply claim some of their fellow members succeeded in order to lower the total payment due from the group. 6 For example, if n = 2, this constraint is equivalent to requiring no more than full joint liability on the loan, c r. Whenn 3, it also requires no more than full liability. 6 This constraint can also be motivated as a reduced-form constraint from a costly state verification problem in which the lender only audits when a failure is reported. Since reports of success (with the required payments) go unverified, the constraint ensures there is no incentive to falsely report success. 9

10 An optimal contract maximizes total borrower surplus subject to borrower limited liability, monotonicity, and homogeneous matching. One can show that safer borrowers are the more likely to opt out of such contracts, 7 so our strategy is to maximize the safe-borrower payoff (corresponding to p i = p s ) subject to the above constraints. Limited liability after failure is incorporated into the contract in that nothing is due from a borrower after failure. Limited liability after success and homogeneous matching constraints are ignored, and verified ex post. Consider maximizing the safe-borrower expected payoff 8 n 1 R k=0 ( ) n 1 ps n k (1 p s ) k r k, k subject to monotonicity constraint 3 and the zero-profit constraint n 1 ( ) n 1 p n k (1 p) k r k ρ, (4) k k=0 where p n k (1 p) k times the probability of getting paid r k is the population average of p n k i (1 p i ) k, so the summed terms are r k from a randomly selected borrower. The zero-profit constraint assumes all borrowers borrow, which will be true if the safest choose to. Consider the isoprofit lines and safe-borrower indifference curves in (r k 1,r k ) space. The isoprofit line slope is and the slope of the indifference curve is ( dr k = pn k+1 (1 p) k 1 n 1 ) k 1 dr k 1 p n k (1 p) k( ) n 1 dr k = pn k+1 s (1 p s ) k 1( ) n 1 k 1 dr k 1 p n k s (1 p s ) k( ) n 1 = p s k k ( n 1 ) k 1 (1 p s ) ( n 1 k 7 Monotonicity guarantees that group payments are increasing in the number of successes. The distribution of number of successes of higher-probability groups stochastically dominates that of lower-probability groups. Thus, in expectation safer groups pay more and earn lower net payoffs. 8 This payoff reflects the fact that borrower i pays r k iff he succeeds probability p i andk of his n 1 fellow group members fail probability p n 1 k i (1 p i ) k( n 1 k ). ). 10

11 The indifference curve is steeper than the isoprofit line, since ps p r p s p n k (1 p) k g(p)dp > ps ps p r p r (p s p)p n k (1 p) k 1 g(p)dp > 0. p n k+1 (1 p) k 1 (1 p s )g(p)dp This implies that r k will be at its upper bound defined by monotonicity constraint 3, for all k {1, 2,..., n 1}; if it were not, the safe-borrower payoff could be raised by increasing r k and lowering r k 1 along the zero-profit constraint while still satisfying all monotonicity constraints. Thus the safe-payoff maximizing contract causes all monotonicity constraints to bind, which means it is a joint liability contract that requires full liability: defining r r 0,the contract requires nr from the group regardless of number of failures (except of course if the whole group fails). The intuition is clear: tailoring the contract toward safe borrowers means requiring as much repayment as possible after outcomes with a higher number of failures, because these outcomes are rarer for safe borrowers. The zero-profit constraint becomes r[1 (1 p) n ]=ρ r = and the safe borrower chooses to borrow iff ρ 1 (1 p) n, R r[1 (1 p s ) n ] u N N Grp,n 1 (1 p s) n 1 (1 p) n, where the second inequality uses the expression for r and the definition of N. Clearly, N Grp,n > 1, and N Grp,n 1asn ; also, one can show that N Grp,n is declining in n. Thus, larger groups extend the parameter space over which efficient lending is achievable, and in the limit allow for fully efficient lending whenever lending is efficient. That is, large enough group size with full liability can potentially overcome the inefficiencies of hidden 11

12 information in the credit market. However, two constraints have been ignored. We first verify that the contract does indeed induce homogeneous matching. Consider a collection G of n borrowers. The group payoff is [ Π G = nr nr 1 i G(1 ] p i ), since the group pays nr unless all members fail. Choosing any x, y G, the cross-partial is 2 Π G p x p y = nr i G\{x,y} (1 p i ) > 0. Thus the group payoff function under this contract is strictly supermodular and homogeneous group formation is the unique equilibrium (see Ahlin, 2011). Next, the limited liability constraint dictates that the contract must be affordable from project proceeds. The largest payment a borrower may owe is r n 1 = nr, and the minimum payoff level is min i R i = R s R/p s. Thus, if group size is n we need R s nr = nρ G G n 1 (1 p) n np s 1 (1 p) n, where the second inequality multiplies the first by p s /ρ. One can show that G n is increasing in n, so the bigger the group, the more restrictive is the affordability constraint. Putting these facts together gives the following: Proposition 1. Consider monotonic, limited liability, homogeneous matching group contracts. Ignoring limited liability after success, efficient lending occurs over a larger parameter space the larger is group size, n. Further, whenever lending is efficient, i.e. for any N > 1, a large enough group size n and G G n guarantee efficient lending is achieved by a full liability group contract. Proof. Note that the best contract for the lowest-payoff (safe) borrowers, subject to all constraints except limited liability after success, is a full liability group contract, which we 12

13 have shown includes all borrowers iff N N Grp,n.Thus,ifN < N Grp,n, even the best contract for safe borrowers does not attract them, and some projects go unfunded, i.e. fully efficient lending is not achieved. Since N Grp,n declines in n, 9 the first claim follows. For the second statement, fix any N > 1. Since N Grp,n declines in n and N Grp,n 1 as n, there exists an n such that N N Grp,n for all n n. Thus efficient lending is achievable for n n as long as the omitted affordability constraint (limited liability after success) is satisfied by the full liability contract, i.e. if G G n. In sum, fully efficient lending is achieved by full liability group contracts, as long as group size n n and G G n. Thus, larger groups are better for achieving efficient lending, as long as borrowers can afford full liability (which means G needs to be high enough). A key reason for this is that larger groups give rise to greater information revelation. Essentially, the lender gets n draws from each borrower s distribution: each outcome in the group is equally informative about any one borrower, due to homogeneous matching. Larger groups also give more instruments for the lender to use, since there are n interest rates in the contract. 10 Both the greater number of instruments and greater information revelation allow the lender to price for risk better, in the limit eliminating the cross-subsidy from safe to risky borrowers that gives rise to potential inefficiency. The condition that G be high, for a fixed N, can be equivalently written that u/ρ be high enough. This suggests that for a given amount of value-added of external capital (net return N), a higher baseline level of payoffs makes efficiency more likely, in the sense that it weakens the affordability constraint associated with full liability. Thus, again holding net 9 Note that 1 (1 p s ) n 1 (1 p) > 1 (1 p s) n+1 p(1 p) n [1 (1 p s ) n ] >p s (1 p s ) n [1 (1 p) n ] n 1 (1 p) n+1 [ ( p(1 p) E [1 (1 p) n ][1 (1 p s ) n n ] 1 (1 p) n p s(1 p s ) n )] 1 (1 p s ) n > 0. p(1 p) n 1 (1 p) n One can show that is decreasing in p; thus the last inequality is true because each of the three terms in the product is positive for all p (p r,p s ). 10 Viewed as a contract with the group, however, only one instrument is needed: the total amount owed by the group whenever anyone succeeds. 13

14 returns to new capital fixed, the model predicts that joint liability lending has the potential for greater outreach in richer areas (areas with higher outside options). While the affordability requirement G G n may be reasonable for relatively low n, Proposition 1 requires G to grow without bound for efficiency to be attained as group size increases. This is because the contract requires that one successful borrower be able to cover the entire group obligation. This raises the question we turn to next: is efficiency attainable when full liability is not always affordable at the necessary group sizes? 4.1 Efficient Lending without Full Liability In this section, we analyze the case where a full liability contract at group sizes large enough for efficiency is not affordable. We proceed as in the previous section, maximizing the safeborrower payoff subject to all constraints except homogeneous matching, which we verify ex post. The limited liability after success constraint is imposed in the maximization: r k R s k {0, 1,..., n 1}. In the previous section, it was established that raising r k and lowering r k 1 along the zero-profit constraint raises the safe-borrower payoff. Thus, every interest rate r k will be at its upper bound, defined by either the monotonicity constraint 3 or the limited liability constraint. Again, define r r 0. One can show that binding monotonicity constraints imply r k = nr/(n k); imposing also limited liability gives the following interest rate schedule, { } nr r k =min n k,r s, k {0, 1,..., n 1}, (5) which maximizes the safe-borrower payoff subject to all constraints. Thus, the optimal contract in this setting imposes full liability until it cannot be afforded, at which point it asks for the maximum amount affordable from all successful borrowers. In other words, it is a debt contract with the group as a whole: a fixed payment, or if this is not affordable, 14

15 confiscation of all earnings of the group. This extracts maximal repayment from high-failure outcomes, at which safe borrowers are rarer. Note that according to interest rate schedule 5, r k increases in k until it hits the bound R s.fixr R s and define k (r) asthelowestk at which r k hits R s ; one can show that k (r) = ) n (1 rrs, (6) where the partial brackets denote the ceiling function and k (r) =n is understood to mean that r k <R s for all k {0, 1,..., n 1}. We also introduce the following notation. Let f n be the number of failures in independent draws from n different projects, and let P i (f n x) be the probability that f n x when the n projects are of risk-type i. Suppressing dependence of k on r, the safe payoff is R r P s (f n k 1) R s p s P s (f n 1 k ). (7) This is because when the number of failures in the group is k 1 or less, full liability is in effect, so the group pays nr and thus each borrower pays r (in expectation); and that iff the borrower succeeds and at least k of his n 1 group members fail, the borrower pays R s. Similarly, the zero-profit constraint can be written r P (f n k 1) + R s pp (f n 1 k ) ρ, (8) where again the bars denote population expectations. It is clear from equation 5 that borrower s payments are increasing in r, since a higher r shifts up the interest rate schedule; hence, lender receipts are also increasing in r. Thus the best contract for safe involves the r that satisfies the zero-profit constraint with equality. 11 It remains to verify that this contract induces homogeneous group formation. 11 Clearly this r>0, and one can show this r<r s iff G >p s /p. 15

16 Lemma 2. For any r (0,R s ), homogeneous group formation is the unique stable match under the contract defined by the interest rate schedule of equation 5. Proof. See Appendix. Thus, the contract outlined is the best for safe borrowers subject to all constraints. The final question: is it good enough to attract them? Once again, the result will require G to be high enough, but here this bound is independent of n and potentially quite unrestrictive: G > p s p r. (A4) Proposition 2. Whenever lending is efficient, i.e. for any N > 1, a large enough group size n and assumption A4 guarantee efficient lending is achieved by a full liability group contract. Proof. See Appendix. Interestingly, this result can hold even though a borrower may only be able to afford to bail out one fellow group member, or not even that. Hence, it is based on very different reasoning from Proposition 1, where it was important that one group member be able to bail out all other group members. Note that the difficulty in the current framework is that full liability is only possible if the number of failures is low enough; if the number of failures is high, the lender gets less than the full interest rate (in expectation) from each borrower. This creates a cross-subsidy to risky borrowers, who pay lower rates more often by more often having a high failure rate. In this context, what large groups do is concentrate failure rates around the mean; and if full liability is affordable at the mean failure rate, then the contract approximates full liability, which eliminates the cross-subsidy. Specifically, consider the limit case of infinitely large groups. Then if borrower i pays [ r 1+ 1 p ] i = r, p i p i 16

17 when he succeeds, he covers his own obligation plus his share of the group s defaulters obligations (since 1 p i need bailing out and p i are available to do so). If all borrowers can afford this rate, a) there is no default risk and r = ρ; and b) since the amount due increases in risk, it must be that the maximum interest rate R s is enough to cover the riskiest borrower s payment: r p r R s. Letting r = ρ in this inequality and rearranging gives assumption A4. The result is linked to the amount of risk in the joint liability group. The risk in mean group payoffs goes to zero as groups get large; hence, larger groups provide for greater insurance, in a sense. However, the reason that eliminating group risk is good here is none of the standard ones. It is not that the lender prefers a less risky return from a given group, or even that the borrowers prefer less risky payoffs all are risk-neutral, and in fact borrowerlevel payoff risk can increase with n for n not too high. The elimination of payoff risk at the group level is valuable here because it eliminates the cross-subsidy it ensures all groups can handle full liability and thus all pay the same, regardless of risk. However, the elimination of risk at the group level is not enough; homogeneous matching is also critical. Note that even with random matching, large enough groups eliminate group risk and full liability is possible with probability approaching 1 under assumption A4. However, while this eliminates the between-group cross subsidy from safe to risky borrowers, it does not eliminate the within-group cross subsidy. That is, the full liability within a group is borne disproportionately by the safer borrowers. For this reason, random matching reduces to individual lending regardless of group size. Thus, necessary for full efficiency in this context is homogeneous matching to eliminate the within-group cross subsidy and large groups to eliminate the between-group cross subsidy. In sum, this section has shown that full liability on large enough groups can always achieve fully efficient lending if gross payoffs are high enough. The two propositions highlight two different mechanisms. First, large group size sends the probability of the tail event n 17

18 failures to zero and, if full liability is affordable at relevant group sizes, eliminates the benefit risky groups enjoy from all failing more often. Second, even if full liability is not affordable after even a handful of failures, larger group size concentrates the failure rate around its mean, and if full liability is always affordable at the mean, it eliminates the crosssubsidy. Both of these results are quite strong in that they hold for any number of types, even a continuum. 5 Further Results In this section, we explore two related directions. First, we show that the same logic works in a moral hazard context as in Stiglitz and Weiss (1981) and Stiglitz (1990). Second, we formalize the point that optimal group size may actually be interior under more realistic assumptions on the quality of local information. 5.1 Moral Hazard and Group Size Here we use a simplified version of the moral hazard problem studied by Stiglitz and Weiss (1981) and Stiglitz (1990). The issue in these models is the standard hidden action problem under which the lender cannot contract on project choice. Coupled with limited liability, this gives a borrower the incentive to take on excess risk, since part of the risk is borne by the lender. Specifically, borrowers choose between two types of projects to undertake with the borrowed capital, safe and risky, with 0 <p r <p s < 1, p s R s R s > R r p r R r, but R s <R r. The risky project represents an inefficient gamble, paying off more when it succeeds but returning less on average. We also assume that the safe project is more efficient than the outside option, i.e R s ρ>u. Hence, the efficient lending outcome is safe project 18

19 choice, which is also the outcome that would result from a set of self-financed entrepreneurs. However, given individual loan contract with interest rate r and limited liability, if the interest rate is high enough the borrower will prefer the risky project to the safe: R r rp r > R s rp s r> R s R r p s p r. When can efficiency be attained by individual lending? The break-even interest rate in that case would be r = ρ/p s. The safe project is preferred to the outside option at this rate but preferred to the risky project iff 12 r = ρ R s R r N R s R r p s p s p r ρ 1 p r p s. Previous assumptions (R s > R r ) only guarantee N is positive, so for N < 1 p r p s, the lending market gives rise to inefficiency, namely risky project choice or none at all. Now consider a size-n group contract (r 0,r 1,..., r n 1 ) as defined in section 3. The payoff of a borrower all of whose group chooses the safe project is n 1 ( ) n 1 R s p n k s (1 p s ) k r k. k If all groups choose the safe project, the zero-profit constraint is n 1 k=0 k=0 ( ) n 1 p n k s (1 p s ) k r k = ρ ; k this pins down the safe payoff at R s ρ, a level higher than the outside option. A key 12 N is redefined in this section to make clear the tight connection between the results here and those under adverse selection. 19

20 incentive compatibility constraint (IC) must be satisfied to ensure that groups prefer safe projects to risky: n 1 R s k=0 ( ) n 1 n 1 p n k s (1 p s ) k r k R r k k=0 ( ) n 1 p n k r (1 p r ) k r k. (9) k Note that this IC does not rule out switching a strict subset of members to the risky project; but we will show later the imposed constraint guarantees those deviations are also not profitable. Note also the critical assumption in this constraint that the group acts cooperatively, to maximize total group payoffs. Thus the constraint rules out a coordinated shift toward risky projects, but not a unilateral deviation. 13 This is the same assumption made and argued for by Stiglitz (1990), without which group lending in this context offers no improvement. Enforceable, cooperative behavior in project choice is the analog here to perfect local information of project type in the adverse selection setting. If the IC holds, safe project choice is guaranteed; otherwise, borrowers will choose risky projects or none at all. So, we proceed by minimizing the risky payoff, subject to the safe payoff equaling a fixed level defined by the zero-profit constraint, limited liability, and the monotonicity constraint 3. Reasoning very similar to section 4 s establishes that raising r k and lowering r k 1 along the safe payoff indifference curve leaves lender profit s unchanged but lowers the risky payoff. The intuition is familiar: charging more in states of the world with more failures maximally discourages risky project choice. Thus, r k will be at its upper bound defined by monotonicity or limited liability, and this gives rise to the full liability group contract defined by interest rate schedule 5. Again, this is a group debt contract: full liability when affordable, and otherwise confiscation of all returns. We now show that under this contract, the imposed incentive constraint guarantees all the other ones. This lemma is the analog to the homogeneous-matching proofs in the 13 Though written in terms of individual payoffs, the constraint is equivalent to a group-level constraint, that the sum of payoffs when all choose safe is no less than the sum of payoffs when all choose risky. We will show that imposing this constraint rules out any improvement to group payoffs from one member (or any number of members) switching to risky project choice. But, it will not rule out a unilateral deviation where the deviator enjoys higher payoffs than the rest of the group by essentially free-riding on their safe behavior. 20

21 adverse selection case, and it relies on similar properties of the payoff function, namely supermodularity and symmetry. Lemma 3. For any r (0,R s ) and the interest rate schedule of equation 5, incentive constraint 9 guarantees that a group cannot do better by undertaking any number of risky projects, the remainder being safe, than it can by choosing all safe projects. Proof. See Appendix. As before, we consider first the simpler case where full liability is always affordable. With full liability, the zero-profit constraint becomes r[1 (1 p s ) n ]=ρ r = ρ 1 (1 p s ) n, and the IC guaranteeing safe projects are preferred to risky becomes R s r[1 (1 p s ) n ] R r r[1 (1 p r ) n ] N N Grp,n (1 p r) n (1 p s ) n 1 (1 p s ) n, where the second inequality uses the expression for r and the (new) definition of N. Clearly, N Grp,n > 0, and N Grp,n 0asn ; also, one can show that N Grp,n is declining in n. So again, larger groups extend the parameter space over which efficient lending is achievable, and in the limit eliminate can eliminate any moral hazard problem. This is of course conditional on full liability being affordable, i.e. satisfying limited liability after success, which boils down to R s nr = nρ 1 (1 p s ) n R s ρ G G n np s 1 (1 p s ) n, wherewehaveredefinedg in the natural way. As before, G n is increasing in n, so the bigger the group, the more restrictive is the affordability constraint. Putting these facts together gives the following: 21

22 Proposition 3. Consider monotonic, limited liability group contracts under moral hazard. Ignoring limited liability after success, efficient lending occurs over a larger parameter space the larger is group size, n. Further, whenever the safe project is more efficient, i.e. for any N > 0, a large enough group size n and G G n guarantee safe project choice is achieved by a full liability group contract. Proof. See Appendix. This result is essentially identical to the one obtained for adverse selection, Proposition 1. The intuition is the same. Larger groups move toward certainty the probability that a group will pay its entire obligation this allows for asymptotic elimination of the implicit subsidy afforded to borrowers by risky projects under limited liability. As before, though, an unattractive feature of Proposition 3 is that the affordability constraint gets more restrictive the larger the necessary group size n, since the contract presumes onesuccessful borrower can affordtheobligationsofalln group members. However, again this assumption is not necessary. We next show a result similar to Proposition 2 that only requires G to be boundedly high enough. Proposition 4. For any N > 0, a large enough group size n and assumption A4 guarantee efficient lending is achieved by a full liability group contract. Proof. See Appendix. Thus, large enough groups can overcome the moral hazard problem even if full liability is not always affordable. The reason, again, is that with large groups what matters is whether full liability is affordable at mean failure rates, since the probability is concentrated around these events. This translates into a potentially far weaker affordability condition than requiring one member to be able to bail out the entire group. And, with full liability typically resulting, the implicit subsidy to risk from limited liability lending is reduced, and eliminated asymptotically. We have shown in this section that the insights of this paper apply equally well to hidden action settings as they do to analogous hidden type settings. The reason is that larger 22

23 group size enables more accurate risk-pricing. In the hidden type setting, this eliminates the cross-subsidy and draws safe borrowers into the market; in the hidden action setting, this eliminates the implicit subsidy to risk embedded in limited liability lending. 5.2 Information Quality and Group Size The results of this paper show that full liability contracts achieve efficient lending under adverse selection or moral hazard, for large enough group size. Depending on parameters, this group size could be 2, 5, 20, 200, or larger. A remaining problematic aspect of this result is that past some point, larger group sizes weaken the plausibility of the informational/coordinational assumptions. More realistically, the quality of information and degree of coordination attenuates with group size. If so, optimal group size may not be as large as needed, but some interior value. While a full analysis is beyond the scope of this paper, we explore this issue briefly in the simple setting of adverse selection without the affordability constraint, i.e. where full liability is always affordable. 14 The goal is simply an understanding of how optimal group size is affected if information quality declines with group size. For simplicity, we formalize information quality somewhat mechanically. We imagine that a matching equilibrium occurs; then, before payoffs are realized, each borrower has an identical and independent probability of being removed from his group and being re-placed in another randomly selected borrowing group. The replacement process preserves group size. Every borrower s probability of not being replaced is π(n), with π(1) = 1, π(n) > 0, π (n) < 0, and lim n π(n) = 0; the probability of being replaced is 1 π(n). Thus, the expected fraction of type-p i borrowers that remain after replacement, in an initially homogeneous type-p i group, is π(n), declining in n and approaching zero. 15 Borrowers 14 Alternatively, this can be viewed as exploring the parameter space where G is high enough for full liability always to be affordable. 15 Unnecessary but allowable is to assume that bigger is still better in the sense that d[nπ(n)]/dn > 0, i.e. the expected number of borrowers of the same type in a group increases with n (even as the expected fraction declines). For example, π(n) =n a,fora (0, 1), satisfies all assumptions. 23

24 anticipate the replacement process in the group formation process and in calculating payoffs to decide whether to borrow. We again restrict attention to ex ante (i.e. pre-replacement) homogeneous group formation, and since our focus will be on when fully efficient lending is achievable, we write the conditions corresponding to all types borrowing. The ex post (i.e. post-replacement) expected type of a fellow group member of a type-p i borrower is p i π(n) p i +[1 π(n)]p, where p is the average risk-type and dependence of p i on n is suppressed. This expression holds since with probability 1 π(n) the initially identical borrower has been replaced with a borrower drawn at random from the pool of borrowers, i.e. from the entire distribution of types. Since replacement happens independently across all borrowers, one can essentially consider a borrower s ex post type as p i if his ex ante type was p i. Consider the ex post payoff of a borrower of type i. If he is not replaced, it is n 1 R p i k=0 ( ) n 1 p n k 1 i (1 p i ) k r k, k since he pays r k iff he succeeds probability p i andk of his n 1 fellow group members fail probability p n 1 k i (1 p i ) k( ) n 1 k. If he is replaced into a group of ex ante type j, his payoff is the same as the above with p j in place of p i ; since he is equally likely to be replaced into any group, his payoff if replaced involves the expectation over all types p j : n 1 R p i k=0 ( ) n 1 p n k 1 (1 p) k r k. k Since the replacement probability is 1 π, the expected payoff is n 1 R p i k=0 [ ] ( ) π p i n k 1 (1 p i ) k n 1 +(1 π) p n k 1 (1 p) k r k. k 24

25 The zero-profit constraint can be derived analogously: 16 n 1 k=0 [ ] ( ) n 1 π p p n k 1 (1 p) k +(1 π) p p n k 1 (1 p) k r k = ρ. k The information deterioration complicates the model, but it is straightforward to show the intuitive result that bigger groups are no longer necessarily better. Proposition 5. Consider monotonic, limited liability, homogeneous matching group contracts with information quality that declines in group size. Ignoring limited liability after success, if the probability of not being replaced is strictly positive but goes to zero as groups get large, then there exists an n 2 such that efficient lending occurs over the largest parameter space for n = n. Proof. See Appendix. This result is not surprising. It merely formalizes a potentially countervailing negative aspect of larger groups, the deterioration of match quality. If matching degenerates toward random matching as groups get large enough (which is equivalent in this model to lim n π(n) = 0), then large groups lose their value. In this case, they approach equivalence with individual lending. This guarantees that some interior group size with a somewhat effective matching outcome will outperform both individual lending and group lending with groups too large to be effective. A corollary of the above result would seem to be that efficient lending is no longer achievable for some N > 1. For, pick the n at which efficient lending occurs over the largest parameter space. At this n, even with a perfect matching process there is still a neighborhood of N above 1 where efficient lending cannot be achieved. Disrupting the matching process should only enlarge this neighborhood. So, not surprisingly in this context not every adverse 16 This is equivalent to an analog to equation 4: n 1 [ ] ( ) p n k (1 p) k n 1 r k = ρ. k k=0 25

26 selection problem can be overcome by large enough groups. The rate at which matching deteriorates more generally, the shape of π(n) will no doubt play a significant role in optimal group size. Thus, one might expect smaller optimal group sizes when the population within which matching occurs is smaller, and also when information is relatively localized or limited within a given matching population. 6 Conclusion We have extended the literature on group micro-lending to consider the role of group size. Interestingly, joint liability contracts do not necessarily lead to homogeneous group formation whenever groups of three or more are considered. But, restricting attention to group contracts that do lead to homogeneous matching, we find that larger groups extend the parameter space over which efficient lending can be achieved. They do this through full liability contracts, even when full liability is far from affordable for any one borrower. The reason is that they allow for more information revelation n draws from the borrower s distribution and for more meaningful contract instruments. These enable better risk-pricing, which eliminates (asymptotically) the cross-subsidy from safe borrowers to risky both the withingroup cross-subsidy, due to homogeneous matching, and the between-group cross-subsidy, due to high-probability full liability. These results hold in an adverse selection setting, where safe borrowers can be inefficiently priced out of the market. The same results hold in an analogous moral hazard context, where limited liability skews incentives toward risky project choice. The reason is the same: larger groups improve risk-pricing and eliminate (asymptotically) the implicit subsidy to risk-taking in limited liability lending. These results can be viewed as pointing out some previously unexplored advantages of larger groups in these settings. By maintaining similar assumptions to the previous literature as group size increases, they do not account for some of the potential disadvantages of larger 26

27 group size. We sketch an approach to doing so in section 5.2, assuming that match quality deteriorates with group size. Not surprisingly, in this case there is a tradeoff between the ability to observe more draws from the distribution and the amount of noise in each of the draws, and the tradeoff leads to an interior optimal group size. In short, the paper argues that larger groups can take greater advantage of local information in an adverse selection setting, and local coordination/enforcement in a moral hazard setting though there are likely to be limits. From a positive standpoint, this helps explain why observed group sizes are larger than 2 and sometimes quite large. It also points to potential factors behind optimal group size information quality, and the level of gross and net returns that can give rise to some of the variation observed in group size in contemporary micro-lending. 27

28 ProofofLemma1. We show by giving an example of a joint liability contract that does not induce homogeneous matching. Given n 3, consider the contract r k = r>0, k {0, 1,..., n 2}, and r n 1 = r + κ, for some κ > 0. We claim the group payoff function under this contract is strictly submodular over (2/3, 1) n. This will guarantee that homogeneous matching is not stable, at least for risk-types above p =2/3, since homogeneous matching violates the no-rankwise dominance requirement that anystable match must satisfy under payoff-function strict submodularity (see Ahlin, 2011). Let G be a collection of n agents with risk-types greater than 2/3. The payoff of a borrower i in group G is R r p i κ p i (1 p j ), j G\{i} since he pays r if he succeeds and an extra κ if he succeeds and the rest of the group fails. Summing, the group payoff function is Choose any x, y G. Then Π G p x Π G = nr r p i κ i G i G = r κ j G\{x} p i (1 p j ) i G\{x} j G\{i} p i (1 p j ). j G\{i,x} (1 p j ). Let G G \{x, y}. Consider the case where n 4, and choose m, n G. Then 2 Π G = κ 2 (1 p j )+ p i (1 p j ) p x p y j G i G j G\{i} = κ (2p i 1) (1 p j )+ (1 p j ) < 0, i {m,n} j G\{i} i G\{m,n} p i j G\{i} the inequality because all terms in the bracketed expression of the second line 17 are positive if risk-types are greater than 2/3. If n =3,letz = G. Then we can write 2 Π G p x p y = κ [ 2(1 p z )+p z ]= κ(3p z 2) < 0, again since risk-types are greater than 2/3. We have thus shown all cross-partials are strictly 17 If n = 4 the second term, i G\{m,n} p i j G\{i} (1 p j), is replaced by zero. A1