Microfinance, subsidies and local externalities

Abstract

We analyse equilibrium borrowers’ effort and the cost of microcredit loans in the presence of moral hazard, project correlation and subsidies under group lending conditions. Our results show that under the assumption of endogenous effort, project correlation has significant effects on borrowers’ effort only when it is determined by asymmetric (positive or negative) shocks. These findings indicate that the well-known negative effect of within-group (symmetric) project correlation on group lending with joint liability disappears once endogenous effort is taken into account. We also analyse the effects of subsidised lending (and asymmetric correlation) on the relative convenience (in terms of borrowers’ effort) of the alternative (1) between group lending and individual lending with notional collateral and (2) among three different market structures of the microfinance industry.

1 Introduction

One of the main roles of the financial system is to match those who have productive ideas with those who have the money to finance them. This ambitious goal has been only partially achieved since the traditional banking system is generally unable to lend to uncollateralised borrowers. Microfinance was born to overcome such inability and has rapidly become very successful in providing small loans to poor uncollateralised borrowers.

The origin of modern microfinance can be traced back to the creation of the Grameen Bank in 1983 following the pioneering activity of Muhammad Yunus.¹ In 1976 Yunus began to experiment with the effects of lending small sums to poor borrowers without asking for collateral. Since then, the development of microfinance has been astounding. The Grameen Bank now has 6 million borrowers, and the Microcredit Summit Campaign at the end of 2006 documented the existence of 3,133 microfinance programs around the world reaching approximately 113 million borrowers and, among them, 82 millions in severe poverty conditions.

The most outstanding element of MFIs’ performance is their extremely low share of non-performing loans. According to the most systematic source of aggregate data on MFIs, the MicroBanking Bullettin (http://www.mixmbb.org/en), which has collated data from 200 MFIs throughout the world, the average loan loss for MFIs was 1% in 2005.

The success of microfinance represents a new “stylised fact” that is difficult to interpret using the instruments of traditional, small business financial literature and is therefore an interesting field of ongoing research. Grossly simplified, the standard literature debates whether some well-established empirical evidence, such as the correlation between cash flow and investment (Fazzari et al. 1988; Kaplan and Zingales 1997; Alti 2003), higher mortality of small and young firms and correlation between firm asset and survival (Cressy 2006), must be interpreted in terms of financing constraints and debt gaps under asymmetric information (Stiglitz and Weiss 1981; Cressy 1996) or rather in terms of self selection of businesses (where higher human capital entrepreneurs are more likely to accept bank offers). The latter is a natural phenomenon that must not be accommodated by public support in order to avoid overlending to small businesses (De Meza and Webb 1987; De Meza 2002) and support to unrealistic optimism (Cressy 2002).

Microfinance success represents something new within this framework. Though the phenomenon is mainly established in less developed economies, it nevertheless showcases new techniques (often backed by private capital rather than public subsidies) that enable lending to marginalised and poor, small entrepreneurs in both industrialised and non-industrialised countries without requiring collateral guarantees. This last characteristic is especially relevant since the crucial role of collateral in solving asymmetric information problems has been widely demonstrated from both theoretical (Bester 1994) and empirical (Cressy 1995) points of view. Based on these original characteristics (lending without collateral and subsidies from private donors) we may argue that, with microfinance, we are no more along the trade-off between credit constraints and overlending, but we are actually documenting something which seems to shift and slacken the trade-off itself.

In the attempt to solve the puzzle, the theoretical debate on microfinance has focussed on various facets of the borrower-lender relationship, such as adverse selection (e.g. Armendáriz de Aghion and Gollier 2000; Gangopadhyay et al. 2005; Ghatak 2000; Laffont and N’Guessan 2000), moral hazard (Chowdhury 2005; Conning 1999, 2005; Laffont and Rey 2003; Stiglitz 1990; Ghatak and Guinnane 1999), ex post hidden information and effects of project correlation under group lending (Armendariz de Aghion 1999).

Most of these papers try to provide explanations for the rapid development and outstanding performance of these financial institutions that challenge “the law of gravity” of lending activity. One of the most important keys of success is considered to be the group lending/joint liability mechanism: the bank provides small individual loans to a self-selected group of borrowers and enforces a contract in which an individual’s defaulting of repayment implies penalties for the other groupmates. In a framework of asymmetric information, this creates an incentive for virtuous group selection (assortative matching) among potential borrowers (noone wants to mix with unproductive groupmates in an attempt to minimise the probability of paying penalties) before together with peer monitoring after the loan has been provided.²

The role of subsidies and donors’ funds is one among the less explored, though important, aspects of this new literature. Even though many microfinance intermediaries pursue the ambitious goal of achieving moderate profitability, administrative costs on small loans are often prohibitive; hence, soft loans from donors and institutions still play a fundamental role in MFIs’ operating viability (Morduck 1999). As expected, the use of subsidies in microfinance has been widely discussed, and several arguments in favor or against it have been developed.

A second crucial issue is the puzzle of the substantial variability in MFI performance according to geographical area of operation. The MicroBanking Bulletin shows that the subsample of African MFIs has an average yearly return of −2% against the +9% of Asian MFIs. This suggests that local factors, such as positive externalities from the local infrastructure and/or productive environment, or, on the negative side, factors affecting systemic risk such as probabilities of weather or political shocks affecting projects’ correlation matter and their role need to be incorporated into theoretical models.

The novelty of this paper is its attempt to integrate these two fundamental issues (the role of donors and the effect of local externalities on the observed performance variability) with a standard feature of the lending activity represented by moral hazard and endogenous borrowers’ effort.

The joint analysis of these features is essential if we want to provide answers to more articulated questions on the role of microfinance and its performance, such as: (1) the impact of positive and negative local externalities on borrowers’ effort within group lending schemes and (2) the effect of subsidised lending on the cost of credit, borrowers’ effort and utility in presence of moral hazard and local externalities; (3) the role of subsidies under different market structures in the microfinance industry and in the alternative between group lending and individual lending with notional collateral.

The paper is organised as follows. In Sect. 2, we provide a short survey on the pros and cons of subsidies in microfinance. In Sect. 3 we develop a model of endogenous effort with moral hazard, project correlation and subsidised group lending in which we examine the interaction among the three elements. In Sect. 4 we introduce the notion of asymmetric project correlation and local externalities and examine their effects on equilibrium borrowers’ effort and on the role of subsidies. In Sect. 5 we analyse the interaction among project correlation, subsidies and moral hazard under different market structures. In Sect. 6 we compare individual lending backed by notional collateral to group lending and evaluate the effect of subsidies under these two alternative options.

2 Microfinance and subsidies

The role of subsidies and donors in MFIs has been one of the most debated topics in the microfinance literature. The Microbanking Bulletin in (2003) shows that only 37% of microfinance lenders working with “low end” borrowers reach financial sustainability. Becchetti et al. (2005) show that subsidising institutions or donors assume part of the asymmetric information costs incurred by the microfinance institutions, thereby reducing the gap between the private and socially optimal volume of financed projects.

Townsend and Ueda (2000) use a dynamic general equilibrium model to evaluate the BAAC microfinance institution’s policy in Thailand and conclude that a nonzero subsidy is justified in such a framework. Armendariz de Aghion and Morduch (2005) argue that most of the discussion on the role of subsidies is based on static cost-benefit analyses, which miss the dynamic contribution that subsidies may provide to borrowers’ capacity-building and its relevant consequences on economic development. The two authors conclude that the evaluation of the welfare effects of microfinance subsidies depends on a few crucial model assumptions. More specifically, subsidies are generally beneficial when assuming a non-flat distribution of social weights, a credit demand that is elastic to interest rates, adverse selection effects and positive spillovers of microfinance credit to other lenders. Soliciting counter-arguments, we can also find reasons against subsidies, including the most traditional one based on the “infant industry” critique, or the weakening effect of subsidies on the efficiency of microfinance institutions.

However, none of these previous analyses adopts an integrated framework in which endogenous effort, project correlation and market structures are analysed jointly and which affect the impact of subsidies on borrowers effort and cost of debt.

A theoretical model that combines these features is developed in the following sections.

3 The framework of the model

The model incorporates the general characteristics of the Prescott exogenous effort model (1997) and the project correlation features of the Armendariz de Aghion (1999) theoretical framework into the Ghatak and Guinnane (1999) approach, which models cooperative endogenous effort among group borrowers under a microfinance loan contract that includes a joint liability element. To these elements we add and investigate the effect of subsidised funds from private donors that reduce the cost of financing for the microfinance institution.

In the model, two types of individuals (lenders and borrowers) exist. Lenders are risk neutral and endowed with 1/m, m > 1, units of the investment good. The investment good must be used to create consumption goods.

Risk neutral lenders have two alternatives: (1) a safe, low-return investment technology that takes x units of the investment good and turns it into Rx units of the good and (2) a risky investment yielding an output X in case of a positive outcome, with probability p, and an output f in the case of an unsuccessful investment, with probability 1 − p, where $p \in \left[ {0,1} \right]$ is therefore both the borrower’s effort and the probability of success.³ As a consequence, the expected value of the risky investment is $pX + (1 - p)f > R$. Risky investments are realised by risk neutral borrowers that are liquidity constrained. Their investment technology requires an input I of exactly one unit of the investment good (an investment of less than one produces an output of zero and any investment cost in excess of I is wasted).

The usual story of economies of scale in monitoring costs induces lenders not to lend directly and individually, but through a financial intermediary (Prescott 1997).

To model the impact of positive and negative environmental externalities in group lending (with groups being composed of two borrowers), we follow Armendariz de Aghion (1999) in formulating the following conditional probabilities.

We define $\alpha = pr\left( {Y_i = X|Y_j = X} \right)$ as the likelihood of obtaining the outcome X from project i given the outcome X obtained by project j and $\beta = pr\left( {Y_i = X|Y_j = f} \right)$. Consequently, $1 - \alpha = pr\left( {Y_i = f|Y_j = X} \right)$, and $1 - \beta = pr\left( {Y_i = f|Y = f_j } \right)$. From the inspection of these conditional probabilities it is evident that, with $\alpha = p = \beta$, the projects do not correlate, whilst with $\alpha > p > \beta$ ($\alpha < p < \beta$), the projects’ correlation is positive (negative). From these basic definitions we derive the following additional relationships, $p^2 = \alpha p$, $\left( {1 - p} \right)^2 = \left( {1 - p} \right)\left( {1 - \beta } \right)$, $p\left( {1 - p} \right) = \left( {1 - p} \right)\beta$ and $\alpha p + \left( {1 - p} \right)\beta = p$.

3.1 The base model with cooperative effort, joint liability and project correlation

After discussing project characteristics, we consider the following borrower’s utility in a (two borrowers) group lending scheme with joint liability and cooperative effort⁴ in presence of local externalities of which he is fully aware

$$ U_{{\text{GL}}} = \alpha p\left( {X - F} \right) + \left( {1 - p} \right)\beta \left( {X - F - c} \right) - \frac{1}{2}\gamma p^2$$

(1)

where F is the principal plus interest payment due to the microfinance intermediary, γ measures the disutility of the borrower’s effort, $c \in \left[ {0,X - F} \right]$ is the joint liability penalty for the solvent borrower if his groupmate fails. All other variables are defined as in the previous section.

In order to better understand the arguments of the utility function in (1), a short simplified summary of the taxonomy of the four different states of natures, the associated probabilities and the corresponding payoffs for the individual borrower are presented in the Table 1.

Table 1 Summary of the taxonomy of the four different states of natures, the associated probabilities and the corresponding payoffs for the individual borrower

It is easy to confirm the advantage of group lending in the two “mixed” states of nature (one borrower is successful, whilst the groupmate is not). In such cases, the bank recovers the value of only one loan under individual borrowing, whilst joint liability allows a full recovery under group lending to the bank that can therefore reduce its risk and also lower its total lending costs (from F_GL to F_IB) when it is not-for-profit (or in a perfectly competitive framework) and transfers this advantage onto borrowers. Note also that group lending introduces for the solvent individual borrower an additional (joint liability) cost (c) in the “mixed” state of nature where his groupmate is not solvent.

In the light of such a breakdown, what appears in the utility function (1) is merely the multiplication of values in the second and in the third column. As is easily confirmed, such multiplication produces non-zero results only in the first two states of nature.

To summarise the model’s output, project success depends on two dimensions. The first is the effort given by the borrowers in a given group. The second is the local environment, which is more or less conducive to positive and negative externalities among different projects.

To compare it with the case in which externalities are absent, we can evaluate (1) under $\alpha = \beta = p$. On the one side, the presence of (positive) project correlation has a positive effect on borrower’s utility as it increases the likelihood of joint success. On the other hand, it has a negative effect as it reduces the probability that he/she is successful while his/her groupmate is not, thereby reducing the expected value of this state-contingent outcome in the utility function.

To make the analysis and interpretation easier when deriving borrowers’ optimal effort and the other equilibrium values of the model, we define $\alpha = p + \theta$, with θ being project correlation, with $- 1 \le \theta \le 1$ and $0 \le \theta + p \le 1$. The new borrower’s utility is:

$$ U_{{\text{GL}}} = \left( {p + \theta } \right)p\left( {X - F} \right) + \left( {1 - p - \theta } \right)p\left( {X - F - c} \right) - \frac{1}{2}\gamma p^2 $$

(1')

By simplifying we obtain

$$ U = p\left( {X - F} \right) - p\left( {1 - p} \right)c + \theta pc - \frac{1}{2}\gamma p^2 $$

(1'')

The first order condition for an interior solution of (1″) yields the following borrower’s effort reaction function

$$ p = \frac{{X - F - c + \theta c}}{{\gamma - 2c}} $$

(2)

The zero profit condition of the microfinance institution is

$$ R = \alpha pF + \left( {1 - p} \right)\beta \left( {F + c} \right) $$

(3)

By substituting for $\alpha = p + \theta$, replacing (2) in (3) and solving for the highest effort solution we get⁵

$$ p_{{\text{GL}}(\theta )} = \frac{{X + \sqrt {X^2 - 4R\left( {\gamma - c} \right)} }}{{2\left( {\gamma - c} \right)}} $$

(4)

This solution shows that the optimal borrower’s effort is positively related to the good state output of the project and to the penalty paid when solvent in case of groupmate failure. Note also that the joint liability penalty (c) acts as a discipline device, increasing cooperative borrowers’ effort. The rationale is that the two borrowers know that by jointly choosing a higher level of effort, they will reduce the probability of paying the joint liability penalty.

Proposition 1

The joint liability cooperative effort solution dominates that of individual borrowing.

To demonstrate this proposition we compare (4) with equilibrium effort under individual borrowing. In such case the utility function of the borrower is

$$ U_{{\text{IB}}} = p\left( {X - F} \right) - \frac{1}{2}\gamma p^2 $$

(5)

By maximising this function with respect to p, and substituting into the zero profit condition of the bank, we obtain the following equilibrium level of effort

$$ p_{{\text{IB}}} = \frac{{X + \sqrt {X - 4\gamma R} }}{{2\gamma }} $$

(6)

By comparison with (4), it is easy to show that the denominator of the equilibrium effort under group lending with joint liability is smaller and the numerator is greater. Hence, the level of effort is higher under group lending than in the single borrower case.

The rationale is that, if borrowers act cooperatively, they understand that an increase of effort has more positive effects under group lending than under individual lending, as it increases not just the probability of the joint success state (p ²), but also the likelihood that, when one of the two lenders is successful, his groupmate is successful as well and therefore c is not paid.

Proposition 2

Symmetric project correlation does not change the borrower’s cooperative effort in (exogenous lump sum) joint liability lending contracts.

The comparison with the no correlation framework may be easily obtained by solving the problem for $\alpha = \beta = p$. In this case the solution becomes:

$$ p_{{\text{GL}}} = \frac{{X + \sqrt {X^2 - 4R\left( {\gamma - c} \right)} }}{{2\left( {\gamma - c} \right)}} $$

(6')

The comparison of (5) and (6′) clearly shows that, in equilibrium, the presence of project correlation and of an environment with significant externalities do not change borrowers’ effort.

This occurs since with (positive) project correlation, we are in the presence of six different compensating effects and their interactions. For the borrower, such correlation implies a higher probability of joint success (with positive effects on effort), lower probability of paying the joint liability penalty (with negative effects on effort) and higher probability of joint failure (which does not imply negative payoffs due to the limited liability assumption). For the bank, project correlation implies higher probability of full repayment in case of joint success, lower probability of partial repayment (only one successful borrower) and higher probability of no repayment (both borrowers fail).

Overall, these six effects compensate each other since, for the microfinance institution, the net impact of three effects on the expected repayment value is negative (see Eq. 3), but is offset by the higher effort of the borrower for whom the net impact of the three effects on effort is positive [as is clearly shown by the borrower’s reaction function in (2) which is increasing in θ]. The different impact of project correlation on the two sides depends on the differences between the borrower’s (residual claimant with limited liability) and the bank’s (whose main concern is the probability of borrower default) payoffs: correlation is a risk for the lender, but an opportunity for the residual claimant, the limited liability borrower. The importance of our results is that these two offsetting effects are such that the negative effect of correlation on cost of debt (Becchetti et al. 2005; Armendariz de Aghion 1999) no longer applies when effort is cooperative and endogenously determined.

In concluding our results and by way of comparison with the established moral hazard literature, we observe that, in the latter, an increase in project correlation reduces bank expected returns when the lender is risk neutral. This leads to an increase in the cost of debt and, in turn, reduces borrower effort (Chan and Thakor 1987). In our case, borrower’s effort is not lower in equilibrium since the negative effect of higher loan rates is compensated by the positive incentive on effort generated by the interaction of borrower limited liability with the peculiar contingent payoff structure induced by the joint liability mechanism.

When we finally calculate the indirect utility of the borrower by replacing the optimal level of effort and cost of debt in the utility function we obtain⁶

$$ {U_{{\text{GL}}(\theta )} = p_{{\text{GL}}(\theta )}^2 \left( {\frac{1}{2}\gamma - c} \right)}={U_{{\text{GL}}} = p_{{\text{GL}}}^2 \left( {\frac{1}{2}\gamma - c} \right)} $$

Hence, even in the borrower’s utility, the two effects of higher cost of debt and higher borrowers reaction function cancel each other out, and borrower satisfaction is unchanged by project correlation when effort is cooperative and endogenous.

To observe the effect of donors funds on effort, consider that:

$$ \frac{{{\text{d}}p_{{\text{GL}}\left( \theta \right)} }}{{{\text{d}}R}} = - \frac{1}{{\sqrt {X^2 - 4R\left( {\gamma - c} \right)} }} $$

(7)

It should be reminded that for real solutions the term under the square root is always positive (see footnote 3). This implies that the availability of donor funds (savings channeled to MFIs for which donors accept a remuneration R′ < R that is below the market rate) can raise borrowers’ equilibrium effort as it reduces their implicit “tax on success”⁷ (the effect is larger the higher the value of the project under the positive state of nature, the lower the non-pecuniary cost of effort and the higher the joint liability penalty).

The difference in effort translates into a higher borrower utility in the presence of subsidised microfinance funds:

$$ U_{GL(\theta ,R)} - U_{GL(\theta ,R')} = \left( {p_{GL(\theta ,R)}^2 - p_{GL(\theta ,R')}^2 } \right)\left( {\frac{1}{2}\gamma - c} \right) < 0 $$

4 Marshallian District (asymmetric project correlation)

It is interesting to analyse what happens when we remove the assumption of symmetric correlation. This assumption may be unrealistic in an environment characterised by (positive or negative) externalities, as in a Marshallian district, where the success of borrowers belonging to the same district can influence the success of the others, and positive externalities may be higher than negative ones. This is likely to happen when the positive result of one borrower can widen the local market, create positive externalities in the value chain, increase the demand for inputs made in the same district or enhance knowledge spillovers.

Proposition 3

Asymmetric project correlation increases borrowers’ cooperative effort in group lending schemes with moral hazard in presence of positive externalities.

From an analytical point of view, this means that the previous project correlation term ($\theta$) is decomposed into $\theta _\alpha$ (representing the contribution of project correlation to the joint probability of success) and $\theta _\beta$ ⁸ (representing the contribution of project correlation to the probability of one success and one failure within the lending group). In the presence of positive externalities we assume that $\theta _\alpha > \theta _\beta$. This new assumption leads us to modify our conditional probabilities.

$p + \theta _\alpha = pr\left( {Y_j = X|Y_i = X} \right)$ is now the likelihood of obtaining outcome X from project i given the outcome X obtained from project j and $p + \theta _\beta = pr\left( {Y_j = X|Y_i = f} \right)$ is the likelihood of obtaining outcome X from project i given the outcome f obtained from project j. Consequently, $1 - (p + \theta _\alpha ) = pr\left( {Y_j = f|Y_i = X} \right)$, and $1 - (p + \theta _\beta ) = pr\left( {Y_j = f|Y_i = f_j } \right)$.

Under this new framework the borrower’s utility is

$$ U_{{\text{GL}}(\theta _\alpha ,\theta _\beta )} = \left( {p + \theta _\alpha } \right)p\left( {X - F} \right) + p\left( {1 - p - \theta _\beta } \right)\left( {X - F - c} \right) - \frac{1}{2}\gamma p^2 $$

(8)

By simplifying we get

$$ U_{{\text{GL}}(\theta _\alpha ,\theta _\beta )} = \left( {1 + \theta _\alpha - \theta _\beta } \right)p\left( {X - F} \right) - p\left( {1 - p} \right)c + p\theta _\beta c - \frac{1}{2}\gamma p^2 $$

(8')

The first order condition of his utility maximisation problem is

$$ \left( {1 + \theta _\alpha - \theta _\beta } \right)\left( {X - F} \right) - c\left( {1 - 2p - \theta _\beta } \right) - \gamma p = 0 $$

(9)

Hence, the borrower’s effort reaction function becomes

$$ p = \frac{{\left( {1 + \theta _\alpha - \theta _\beta } \right)\left( {X - F} \right) - c\left( {1 - \theta _\beta } \right)}}{{\gamma - 2c}} $$

(10)

And the zero profit condition for the bank is

$$ R = p\left( {p + \theta _\alpha } \right)F + p\left( {1 - p - \theta _\beta } \right)\left( {F + c} \right) $$

(11)

By rearranging, substituting for the borrower’s reaction function and simplifying, we get the following equilibrium borrower’s effort⁹

$$ p_{{\text{GL}}(\theta _\alpha ,\theta _\beta )} = \frac{{\left( {1 + \theta _\alpha - \theta _\beta } \right)X + \sqrt {\left[ {\left( {1 + \theta _\alpha - \theta _\beta } \right)X} \right]^2 - 4R\left( {\gamma - c} \right)} }}{{2\left( {\gamma - c} \right)}} $$

(12)

This result shows that borrower effort is higher (lower) the greater is the positive (negative) difference between $\theta _\alpha - \theta _\beta$, while, with $\theta _\alpha = \theta _\beta$, we go back to (4). For MFIs it is therefore more attractive to work in a district with positive externalities because such externalities have a positive effect on project success via higher borrowers’ effort under cooperative group lending.

In addition, the presence of positive externalities increases borrower welfare since $U_{{\text{GL}}(\theta )} - U_{{\text{GL}}(\theta _\alpha ,\theta _\beta )} = \left( {p_{{\text{GL}}(\theta )}^2 - p_{{\text{GL}}\left( {\theta _\alpha ,\theta _\beta } \right)}^2 } \right)\left( {\frac{1}{2}\gamma - c} \right) < 0$ , which is always negative if $\theta _\alpha - \theta _\beta > 0$.

From closer inspection of (12) it is clear that ${{{\text{d}}p_{{\text{GL}}(\theta _\alpha ,\theta _\beta )} } \mathord{\left/ {\vphantom {{{\text{d}}p_{{\text{GL}}(\theta _\alpha ,\theta _\beta )} } {{\text{d}}R}}} \right. \kern-\nulldelimiterspace} {{\text{d}}R}}$ is lower in absolute value when the positive difference between $\theta _\alpha$ and $\theta _\beta$ grows. Hence, the positive effect of subsidies (reducing banks’ cost of financial resources) on borrower effort is reduced in presence of dominant positive externalities.

5 Moral hazard, symmetric correlation and subsidies under different market structures

In this section we want to investigate whether the analysis of market structure adds new relevant insights to the discussion of the effects of subsidised lending in presence of project correlation and endogenous effort. We therefore examine three possibilities: (i) a monopolistic, profit-maximising MFI; (ii) a zero-profit, monopolistic MFI and (iii) a perfectly competitive scenario in which a large number of MFIs compete, driving profits to zero.

A preliminary analysis of these three market structures leads us to formulate the following proposition:

Proposition 4

Borrowers’ cooperative effort is higher in the presence of a monopolistic, zero-profit MFI than either a monopolistic profit-maximising MFI or a zero-profit MFI in a perfectly competitive framework.

Consider the following case in which a profit-maximising, monopolistic MFI faces the following borrower’s utility function:

$$ U_{{\text{GL(PMM)}}} = \alpha p\left( {X - F} \right) + \left( {1 - p} \right)\beta \left( {X - F - C} \right) - \frac{1}{2}\gamma p^2 $$

(13)

In this case the borrower’s reaction function is $p = {{X - F - c + \theta c} \mathord{\left/ {\vphantom {{X - F - c + \theta c} {\gamma - 2c}}} \right. \kern-\nulldelimiterspace} {\gamma - 2c}}$ which, by rewriting F = 1 + r (where r is the lending rate), becomes

$$ p = \frac{{X - 1 - r - c + \theta c}}{{\gamma - 2c}} $$

(14)

We first consider the behaviour of a monopolistic MFI which maximises profits

$$ \mathop {\max }\limits_{\left\{ r \right\}} \left\{ {p\left( {p + \theta } \right)\left( {1 + r} \right) + p\left( {1 - p - \theta } \right)\left( {1 + r + c} \right) - R} \right\} $$

(15)

By simplifying and substituting for p in the maxim, and we get

$$ \mathop {\max }\limits_{\left\{ r \right\}} \left\{ {\left( {\frac{{X - 1 - r - c + \theta c}}{{\gamma - 2c}}} \right)\left( {1 + r + c - \theta c} \right) - c\left( {\frac{{X - 1 - r - c + \theta c}}{{\gamma - 2c}}} \right)^2 } \right\} $$

(15')

The first order condition can be rearranged as

$$ r = \frac{{2c - 2\gamma - 2\gamma c + \gamma X + 2\theta \gamma c + 2c^2 - 2\theta c^2 }}{{2\left( {\gamma - c} \right)}} $$

(16)

By substituting (16) into the borrower’s effort reaction function we get

$$ p_{ ( {\text{PMM)}}} = \frac{X}{{2\left( {\gamma - c} \right)}} $$

(17)

This result shows that the optimal borrower’s effort in the presence of a profit-maximising, monopolistic MFI is lower than under both zero-profit monopolistic conditions and zero-profit, competitive frameworks. The zero-profit, monopolistic solution is the one shown in (4) and coincides with the zero-profit, competitive solution in which the market structure imposes a zero-profit solution to the MFIs.

Observe also that the optimal borrower’s effort under the profit-maximising, monopolist MFI is independent from Rhence donors funds, which have no positive effect on the expected value of the project in this case. The intuition is that in this specific example, the subsidies are rents that are appropriated by the profit-maximising monopolist, and the borrower will therefore not be induced to exert extra effort even in the presence of subsidised funds.

If these last two market structures (zero-profit monopoly and zero-profit competition) provide the same result in the uniperiodal case¹⁰, their outcome differs when we examine a two-period case.

In a two-period framework the generic borrower’s utility becomes

$$ \begin{gathered} U_{{\text{GL}}} = \left( {p + \theta } \right)p\left( {X - F} \right) + \left( {1 - p - \theta } \right)p\left( {X - F - c} \right) - \frac{1}{2}\gamma p^2 + \frac{v}{d}\left[ {\left( {p + \theta } \right)p\left( {aX - F} \right) + \left( {1 - p - \theta } \right)p\left( {aX - F - c} \right) - \frac{1}{2}\gamma p^2 } \right] \hfill \\ \end{gathered} $$

(18)

where ${1 \mathord{\left/ {\vphantom {1 d}} \right. \kern-\nulldelimiterspace} d}$, with $1/d \in \left[ {0,1} \right]$, is the borrower’s rate of time preference, $v \in \left[ {0,1} \right]$ is the probability of getting credit in the second period and (a > 1) is a multiplicative coefficient which makes the second loan higher than the first.

Solving for the borrower’s optimal effort we get

$$ \frac{{{\text{d}}U}}{{{\text{d}}p}} = X - F - pc + 2pc + \theta c - \gamma p + \frac{v}{d}\left( {aX - F - c + 2pc + \theta c - \gamma p} \right) = 0 $$

(19)

and

$$ F = \frac{{X\left( {d + av} \right) - c\left( {d + v} \right) + \theta c\left( {d + v} \right) - \gamma p\left( {d + v} \right) + 2pc\left( {d + v} \right)}}{{\left( {d + v} \right)}} $$

(20)

The zero-profit condition for the MFI in both the zero-profit competitive and monopolistic scenarios is

$$ R + \frac{{vR}}{d} = p\left( {p + \theta } \right)F + p\left( {1 - p - \theta } \right)\left( {F + c} \right) + \frac{v}{d}\left( {p\left( {p + \theta } \right)F + p\left( {1 - p - \theta } \right)\left( {F + c} \right)} \right) $$

(21)

By substituting for F we get

$$ R\left( {d + v} \right) = pX\left( {d + av} \right) - \gamma p^2 \left( {d + v} \right) + p^2 c\left( {d + v} \right) $$

(22)

Finally, the equilibrium effort is

$$ p_{(v)} = \frac{{X\left( {d + av} \right) + \sqrt {\left[ {X\left( {d + av} \right)} \right]^2 - 4R\left( {d + v} \right)^2 \left( {\gamma - c} \right)} }}{{2\left( {\gamma - c} \right)\left( {d + v} \right)}} $$

(23)

And may be rewritten as¹¹

$$ p_{(v)} = \frac{{X\left( {\frac{{d + av}}{{d + v}}} \right) + \sqrt {\left[ {X\left( {\frac{{d + av}}{{d + v}}} \right)} \right]^2 - 4R\left( {\gamma - c} \right)} }}{{2\left( {\gamma - c} \right)}} $$

(23')

This finding can be compared when a = 1, with the optimal effort in the profit-maximising monopolist market in which borrower’s optimal effort reduces to (17).

The two-period solution shows that borrower effort is increasing in v. Considering for simplicity, and without further specification of the progressive loan incentive ν = 1 in the perfectly competitive scenario, while ν <1 in both the monopolistic profit-maximising and zero profit frameworks, it becomes evident that borrower effort under the monopolistic, zero-profit scenario dominates effort in the other two cases. It is higher than under the zero-profit, maximising competitive scenario as a result of the different effect of the probability of getting credit in the second period. It is higher than the monopolistic, profit-maximising scenario for both the one period level and the effect of the probability of achieving credit in the second period. As a result, borrower effort in the monopolistic, profit-maximising MFI in the two-period model is lower than in either the zeroprofit, monopolistic or the perfectly competitive MFI in the same two-period scenario.

The intuition giving rise to this result is that effort is lowest under a profit-maximising monopoly where the MFI finds it optimal to raise interest rates on loans, even though this has the effect of increasing the tax on borrower success, thereby reducing optimal effort. The two other cases produce equal results in uni-periodal models, whilst, with two-period models, the incentive to repay is higher under a zero-profit monopoly where borrower insolvency prevents the possibility of receiving a loan in the second period. This does not occur under perfect competition where competitors are ready to offer credit to the insolvent borrower.

Proposition 6

Symmetric project correlation does not affect effort ranking among the three market structures (perfect competition, zero-profit and profit-maximising monopoly). Subsidised lending enhances the gap between borrower effort in monopolistic, profit-maximising MFIs and the two alternative scenarios. At the same time, the gap between these alternative scenarios, perfect competition and zero-profit monopoly, is reduced.

By inspecting the optimal effort solutions in the three cases, we can ascertain the impact of subsidised lending. More specifically, we observe that effort under monopolistic, profit-maximisation is unaltered (in 17 $\frac{{{\text{d}}p_{\left( {{\text{PMM}}} \right)} }}{{{\text{d}}R}} = 0$), while effort in the other two scenarios is increased. It increases more in the presence of perfect competition since the derivative of p with respect to R (marginal effect of subsidised financing on borrower effort) depends on the magnitude of v (see Eq. 23′).

As a consequence, SR savings enhance the (negative) gap between borrower effort in a monopolistic, profit-maximising MFI and the two alternative scenarios, whilst reducing the difference in outcomes in these latter scenarios.

6 Moral hazard, correlation and subsidies in the presence of notional collateral

In order to grant poor borrowers access to credit, MFIs need to find original solutions to replace standard individual collateral. Of these, an alternative to group lending has been found in the provision of so-called “notional collateral”. This concept hinges on the fact that even very poor borrowers may own assets that have high value for them despite their lack of market value and apparent worthlessness for the bank. In such cases, however, the “notional” collateral may still play a role of providing an incentive to repay.

We examine how microfinance with notional collateral works in the following example.

Imagine that the borrower owns an asset that has the value of w to him or herself and L for the bank, with w > L, and a second asset whose value is L for both him and the bank.

The borrower’s utility function when the second asset is used as collateral is

$$ U_L = p\left( {X - F} \right) - \left( {1 - p} \right)L - \frac{1}{2}\gamma p^2. $$

(24)

First order condition yields

$$ F = X + L - \gamma p $$

(25)

and the zero-profit condition of the MFI is

$$ R = pF + \left( {1 - p} \right)L. $$

(26)

Hence, the optimal borrower effort will be

$$ p_L = \frac{{X + \sqrt {X^2 - 4\gamma \left( {R - L} \right)} }}{{2\gamma }} $$

(27)

Consider now the borrower utility when the first asset with value w > L for the borrower is used as collateral

$$ U_W = p\left( {X - F} \right) - \left( {1 - p} \right)w - \frac{1}{2}\gamma p^2 $$

(28)

First order condition yields

$$ F = X + w - \gamma p $$

(29)

and the following zero-profit condition for the bank is

$$ R = pF + \left( {1 - p} \right)L $$

(30)

Hence, the optimal borrower’s effort is

$$ p_W = \frac{{\left( {X - L + w} \right) + \sqrt {\left( {X - L + w} \right)^2 - 4\gamma \left( {R - L} \right)} }}{{2\gamma }} $$

(31)

Inspecting (31) and (27) we find that $p_L < p_w$ since the borrower has a higher incentive to reduce the probability of the negative (default) event when the first asset is used as collateral.

The comparison of (31) with the optimal effort under group lending (6′) clearly shows that, for a sufficiently high w, the borrower’s effort is higher with individual lending and notional collateral than in a group lending scenario.

Again in this case a subsidy (in the form of lowered necessary returns on the MFI’s financing) reduces R and raises optimal borrower effort:

$$ \frac{{dp_W }}{{dR}} = - \frac{1}{{\sqrt {\left( {X + w - L} \right)^2 - 4\gamma \left( {R - L} \right)} }} $$

(32)

By comparing (32) with (7) we observe that, for L = 0, the denominator of the first expression is higher. As a consequence, the positive impact of subsidised loans on effort is lower in the case of notional collateral. The rationale is that the positive effect of subsidies on cooperative effort in group lending (an increased effort to reduce the probability of paying the joint liability penalty) does not apply in the case of notional collateral.

7 Conclusions

Theoretical research has so far analysed the impact of some important features of microfinance (project correlation, subsidised lending, borrowers’ moral hazard) when separately taken. In this paper we investigate what happens to borrowers’ effort, cost of debt and borrowers’ utility when, under a more realistic assumption, these features operate simultaneously and interact among each other.

By adopting this framework the paper identifies a series of interesting findings. Firstly, the symmetric groupmates’ project correlation does not affect endogenous borrower effort in group lending. The explanation for this finding is that correlation is a risk for the lender, but an opportunity for the residual claimant, limited liability borrower: the first effect leads to an increase in the cost of debt, and a consequent decrease in the borrower’s effort, which is compensated by the second effect. This finding contradicts the commonly held opinion that positive project correlation reduces the advantage of joint liability in group lending by limiting the diversification gains achieved by the bank. Such a point is valid when effort is considered exogenous but not under the hypothesis of endogenous and cooperative effort.

Our results also extend and modify findings from the traditional moral hazard literature where an increase in project correlation reduces banks’ expected returns when the lender is risk neutral and leads to an increase in the cost of debt, in turn reducing borrower effort (Chan and Thakor 1987). In our models, however, this does not occur; borrower effort is not lower in equilibrium since the negative effect of higher loan rates is offset by the positive incentive on effort, generated by the interaction of borrower limited liability with the peculiar contingent payoff structure induced by the joint liability mechanism.

Secondly, in our framework subsidised lending significantly increases borrower effort as it reduces the implicit tax on success represented by the cost of the loan.

Thirdly, by investigating the relationship between endogenous effort, project correlation and subsidised lending, on the one side, and differing market structures for MFIs on the other, we find that borrowers effort is higher in a monopolistic, zero-profit scenario than in either a monopolistic, profit-maximising or a perfectly competitive, zero-profit MFI scenario. The intuition is that profit-maximising monopoly implies a higher loan cost and therefore an implicit increase in the borrower’s tax on success, while zero-profit competition reduces borrower discipline as it becomes easier for insolvent borrowers to obtain credit from a different institution in a successive time period or round.

Fourthly, we also observe that symmetric project correlation does not affect effort ranking among these different market structures, while subsidised lending enhances the gap between borrower effort in a monopolistic profit-maximising MFI and the two alternative scenarios, whilst closing the distance between these two alternatives.

Finally, we observe that donors funds are more effective in increasing borrower effort in scenarios of group lending than when individual lending is supported by notional collateral. The effect is a result of the positive effect of subsidies on cooperative effort in group lending and which does not apply in the alternative framework.

We believe that our results are relevant for policymakers for at least three reasons.

Firstly, most MFIs operate in areas with high systemic risk (the recurrent floods in Bangladesh throughout 1988, which led to the Grameen crisis and to the reform of its lending practices are one such example). In our model this translates into asymmetric correlation with higher probability of joint failure than of joint success).

What our paper tells us is not just that these local risk factors have an effect per se, but that they also depress borrowers’ expectations and effort. A guarantee fund covering lender losses in regions where systemic risk is higher could reduce part of borrower risks and their negative consequences on borrowers’ effort. A typical objection is that of the moral hazard effects that such funds might create. However, most of these systemic risk factors (weather, political and exchange rate shocks) are publicly observable and cannot be conditioned by affected borrowers’ behaviour.

Secondly, in the specific case of the microfinance model with endogenous effort, subsidies have a positive role on borrowers’ performance by reducing the lending rate and therefore the “tax on success”. If we consider that lending rates are far above those in standard lending practices (in our simple model this occurs simply because the bank needs to cover the higher expected cost of loan default due to the absence of collateral), we can recognise that private subsidisers decide to share part of the MFI’s lending risk via their investment decisions and therefore allow lenders to reduce their loan rates. Since equilibrium real lending rates are generally very high for the above-mentioned reasons, the additional projects that become profitable with lower cost of debt are highly likely to have returns above the opportunity cost of funds.

Furthermore, the interesting feature of most MFIs is that they do not ask for public subsidies, but rely on private donors or even (socially responsible) depositors accepting lower rates of return.

As of 31 December 2004, one of the leading financial institutions channeling socially responsible investment to microfinance, Oikocredit, had supported over 398 project partners with an outstanding capital of 114 million euros. Its total share capital was 203.5 million euros invested by 24,000 members and 37 Support Associations. In Italy, Banca Etica issues “microfinance bonds” remunerated below the market level and channels collected funds to finance microfinance institutions around the world.

The mechanism may therefore be win–win in nature. When socially responsible investors voluntarily decide to sacrifice some of their gains in exchange for the satisfaction they receive from financing poor borrowers, there is no “arbitrary” use of public resources or diversion of taxpayers’ contributions and profitable projects are financed.

A third relevant policy consideration from our model concerns MFIs market structures.

Profit maximising monopolist MFIs have higher lending rates in equilibrium. Hence, they set higher “taxes on success” and reduce borrowers’ effort and productivity in the areas in which they operate. The comparison of loan rates of two of the biggest institutions—Grameen (a not-for-profit monopolist with 11% real loan rate in 2004 in Bangladesh, in line with average lending rates in the country) and Bancosol (a profit-oriented monopolist with 25% real loan rate in 2004 in Bolivia, 18% above the country’s average lending rate)—confirms such a gap.

As in any industrial sector, competitive states may offer improvements, since they may improve efficiency of MFIs and lead to a reduction of lending rates in equilibrium, though setting up an important drawback. Our two-period version of the model shows how this may enhance borrower moral hazard as it reduces or even eliminates the borrower’s penalty from failure in the first period if, in the second period, it is easy to receive a loan from an alternative MFI that has no knowledge of the borrower’s previous track record.

Policymakers should therefore investigate under which conditions MFIs might be compelled or induced to share information on local borrowers or to create credit bureaux in order to avoid borrowers’ moral hazard.