Competition and Microfinance
JEL Classifications: O12, O16, L31, L13
Keywords: Credit Markets, Competition, Asymmetric Information

Craig McIntosh*
University of California at San Diego

Bruce Wydick**
University of San Francisco

January 2005

Abstract: Competition between microfinance institutions (MFIs) in developing countries has
increased dramatically in the last decade. We model the behavior of non-profit lenders, andshow that their non-standard objectives cause them to cross-subsidize within their pool ofborrowers. As a result, when lenders with heterogeneous objectives compete, competitionis likely to yield an outcome that makes poor borrowers worse off. As competition
exacerbates asymmetric information problems over borrower indebtedness, the mostimpatient borrowers begin to obtain multiple loans, creating a negative externality that leads
to less favorable equilibrium loan contracts for all borrowers.

*Craig McIntosh: International Relations/Pacific Studies, U.C. San Diego,
9500 Gilman Drive, La Jolla, CA 92093-0519, e-mail: ctmcintosh@ucsd.edu**Bruce Wydick: Department of Economics, University of San Francisco, San Francisco, CA 94117,
e-mail: wydick@usfca.edu.

We wish to thank two very helpful anonymous referees, Chris Ahlin, Stephen Boucher, Xavier Gine,
John Kao, Michael Kevane, Man-Lui Lau, Edward Miguel, Jeff Nugent, Jeff Perloff, Betty Sadoulet,
Mark Schreiner, Dean Scrimgour, Alex Winter-Nelson, and seminar participants at Princeton
University, UC Berkeley, UC Davis, and the 2002 WEAI meetings in Seattle for helpful comments andencouragement. Any remaining errors are our own. Financial support from the Pew Charitable Trust
Evangelical Scholars Program, BASIS/USAID, and the Fulbright IIE Program is gratefully
acknowledged.

1. Introduction
Since Adam Smith, economists have nearly always favored policies that foster competition, as
competition typically results in lower equilibrium prices for consumers. It would be reasonable toexpect therefore that an increase in competition between microfinance institutions would unequivocallyresult in more favorable credit contracts for the entrepreneurial poor in developing countries. The
purpose of this paper, however, is to show that competition may actually prove detrimental to someor all of the borrowers in a microfinance market.
We develop a model in which a solitary client-maximizing microfinance institution (MFI)
competes with an existing informal moneylender to the benefit of each borrower captured in the
microfinance portfolio. Subsequently we show three potentially adverse effects of the entrance of new
MFIs into the same pool of borrowers. First, Bertrand competition between MFIs within the subset ofprofitable borrowers reduces the ability of a socially motivated lender to generate rents that support
lending to the poorest and potentially least profitable borrowers.1 This diminution of the capacity to
cross-subsidize means that the poorest borrowers in the client-maximizing portfolio are dropped ascompetition intensifies. Second, we show a number of instances in which failure to restrict grantfunding to the poorest potential borrowers can prevent the emergence of a competitive microfinancemarket altogether, as client-maximizing non-profit institutions undercut profit-maximizers to capture themost profitable borrowers in a given pool.
A third negative effect of MFI competition originates from the likelihood of increasing
asymmetric information between lenders. With a greater number of lenders in a market, we would
expect information sharing between lenders to become more difficult, all else equal. We show that thiscreates an incentive for some (impatient) borrowers to take multiple loans. Such instances of multiple
contracting both increase average debt levels among borrowers in the portfolio and decrease theexpected equilibrium repayment rate on all loan transactions, generating less-favorable Bertrand
1

See Shaffer (1996) for an example of cross-subsidization in U.S. lending markets.

1equilibrium credit contracts. This makes all patient borrowers worse off, and again results in thepoorest borrowers being dropped from the loan portfolio. In general, our results show that while
wealthier and impatient borrowers are likely to benefit from increasing competition among MFIs, very
plausible conditions exist under which an increase in the number of lenders in a market will lower thewelfare of the both the poor and the patient.
The widespread enthusiasm for microfinance has spawned a dramatic increase in the number of
MFIs in the developing world. Spurred by an accord reached at the Microfinance Summit in 1997 toreach 100 million of the world's poorest households with credit, there is arguably more widespread
support for microfinance today than any other single tool for fighting world poverty. The microfinancemovement has been both praised and supported by a broad range of academic scholars, major
development finance institutions such as the World Bank, and development practitioners themselves.
With the number of MFIs involved in this effort now 1,600 and growing, the overlap and competition
between these institutions is certain to increase.
The rapid early growth of the microfinance movement primarily consisted of non-profit, sociallymotivated lenders seeking to reach as many poor clients with credit as they were able, given theirlimited budgets.2 In the process they demonstrated that through the use of new lending technologies,
such as joint liability contracts and dynamic incentives, a substantial portion of this new market could infact be lent to profitably. This realization has drawn profit-motivated lending institutions into thesemarkets. The presence of competition from profit-driven lenders has forced MFIs in competitive
regions to rethink their strategies. Moreover, donors have questioned the need for continued subsidies,
resulting in the recent focus on “institutional sustainability” in the MFI sector.
Although some current research has begun to touch upon these issues, most of the burgeoning
economic literature on MFIs has been concerned with the impact on MFI clients, (e.g. Pitt andKhandker, 1998; Morduch, 1998; Wydick, 1999) or papers that examine the properties of the joint
2

Tuckman (1998) provides a good survey of competition among non-profit institutions.

2liability contracts often employed in microenterprise lending (e.g. Stiglitz, 1990; Besley and Coate,
1995; Ghatak, 1999). A limited amount of work, however, has undertaken a broader look at theindustrial organization of the microfinance movement.
Morduch’s (1999, 2000) work is one such example. In a detailed analysis of the Grameen Bank,
Morduch (1999) asserts that the failure to account for tradeoffs between sustainability and poverty
reduction has hamstrung discussion about the subsidies necessary for microfinance to move forward. In
a later paper (Morduch, 2000), he challenges the notion that microfinance provides a ‘win-win’ situation
for all players involved. Instead, he argues that subsidized lending to the poor as well as the creation ofsustainable for-profit institutions should be important, but separate, development goals.
Navajas, Conning, and Gonzales-Vega (2001) apply a variant of Conning’s (1999) model tolenders in Bolivia, showing that a high-screening, individually focused institution and a low-screening
group-lender with poorer clients can coexist in the same market. While their research discusses theclient sorting process that will take place under competition, its focus is different from our work in thatit does not explicitly model interactions between lenders in terms of strategic behavior and multiple
contracting by borrowers when more than one lender exists in the market. Rather, they model thesecond entrant to the market as a Stackelberg follower who focuses on collateral-based lending andhence captures the wealthiest and most productive segment of the market.
More similar to this paper in its approach to strategic behavior is Hoff and Stiglitz (1998), whoshow that subsidies to a lending market can generate the perverse consequences of market exit andincreasing equilibrium prices. In their model, these effects can arise either as a result of the loss of scaleeconomies, or due to a weakening of reputation effects as the number of lenders dilutes information in
the market. We also relate information asymmetries to poorer equilibrium loan contracts for borrowers
in Section 4 of this paper. However, in the Hoff and Stiglitz model, perverse effects are caused by the
weakening of dynamic repayment incentives, whereas in our model it is caused by asymmetric
information between lenders over borrower indebtedness.
3

Other papers have also examined the industrial organization of credit markets. Petersen andRajan (1995) show that multi-period, state-contingent contracts, or “relationship banking” is an efficient
contracting device for dealing with asymmetric information, and for this reason market power can leadto lower quality firms obtaining finance. Similarly, since the screening required in multi-lender markets
is wasteful, the authors demonstrate that borrower welfare can decrease as competition intensifies.
Broecker (1990), examining an equilibrium under Bertrand competition, shows that imperfect
information provides an additional reason for undercutting, as offering the prevailing interest rate causes
the lender to attract only those borrowers rejected by other lenders. Dell’Ariccia et al. (1999), illustrate
a “blockaded” market, where Bertrand competition leads to no more than two lenders being present in amarket in equilibrium. Marquez (2002), however, presents a model where entry is easier as turnoverincreases, weakening blockading. A more applied view of the escalating competition present inmicrofinance markets is provided by Rhyne and Christen (1999), who suggest that information sharing
in the form of credit bureaus is becoming increasingly necessary as the microfinance market matures.

2. Increasing Numbers and Competition among MFIs
We wish to motivate our theoretical model with evidence from three areas of the world in which MFI
activity has reached a relatively advanced stage, and where the effects of competition between MFIs
have become increasingly clear.
Bangladesh: The Grameen Bank, long the flagship of the microfinance movement, hasconsistently been upheld as a pinnacle of stability, self-sufficiency, and effectiveness in using
microfinance as a tool for lifting households from poverty. Yet the Grameen Bank's well-known
successes have encouraged imitators, which compete for borrowers’ attention along with two other verylarge microcredit providers, Bangladesh Rural Advancement Committee (BRAC) and Rural
Development Project 12 (RD-12), that have operated alongside the Grameen Bank for more than adecade. A front page Wall Street Journal article in November 2001 raised warnings about the financial

4health of the Grameen Bank, asserting “imitators have brought on more competition, making it harderfor Grameen to control their borrowers”. The article points in particular to the Grameen Bank’s lending
in the region of Tangail, in which competitive pressures have reduced interest rates for some borrowers,
but where 32.1 percent of the Grameen Bank’s loans have fallen more than two years overdue:
In Tangail, signboards for rival Micro lenders dot a landscape of gravel roads, jute fields and ponds withsimple fishing nets. Shopkeepers playing cards in the village of Bagil Bazar can cite from memory theterms being offered by seven competing microlenders--a typical repayment plan for a 1,000-taka ($17) loan
is 25 taka for 46 weeks. At an annualized rate, that works out to 30% in interest. Surveys have estimatedthat 23% to 43% of families borrowing from microlenders in Tangail borrow from more than one.
(WSJ: 11/27/2001)

Such conditions in Bangladesh, with perhaps the world’s most developed microfinance industry, have
put huge financial pressures on lenders, and have led to spiraling default rates. Where once Grameen
boasted repayment rates of 95% or higher, the Wall Street Journal reports 19% of the portfolio overdue
by a year or more. Alarming figures such as these have intensified efforts by the World Bank and
CGAP to help bring together a network of the largest 20 microfinance institutions in Bangladesh to
implement a centrally managed credit information system during 2004. It is hoped that more “centrally
managed” competition between lenders in Bangladesh will both help to foster healthy competition
between MFIs while bringing down arrears rates in MFI portfolios.
East Africa: While East Africa is at an earlier stage of competition, the major urban centers of
Uganda and Kenya are becoming saturated by competition among numerous MFIs (see Kaffu &Mutesasira, 2003). Markets for the more wealthy borrowers that were previously dominated by grantfunded, socially motivated lenders are now being contested by private institutions. For example,
CERUDEB and CMF, two private lenders with access to subsidized external lines of credit, have begun
competing with existing MFIs for larger microcredit borrowers. In response, there is increasing
competitive pressure on socially motivated MFIs, whose interest rates may be more than 1% per monthhigher than the new competition.
FAULU, one of the few major MFIs to operate in both Uganda and Kenya, is troubled by theincreasing presence of borrowers unknowingly receiving loans from multiple lenders. FAULU reports5that such behavior has become increasingly prevalent as the intensity of MFI activity increases. The
Kenya office is able to employ a risk management network based on the country’s national ID system todetect clients within their own portfolio with multiple loans. Uganda, however, has no such national ID
system, and so they are powerless to monitor the problem, even within their own institution.
Central America: The increase in both the size and number of MFIs operating in Central
America since the mid-1990s has been astounding. Much of the reason for this has been political: boththe United States and the European Union have desired to develop an entrepreneurial middle-class in theregion in order to try to bridge the societal divisions responsible for civil wars during the 1980s. As aresult, growth in MFI activity has been particularly heavy in Guatemala, El Salvador, and Nicaragua.
Table 1: Growth in Microenterprise lending, Nicaragua
Borrowers
Portfolio%Arrears Borrowers
Portfolio US$ %Arrears
1999
US$ 1999
1999
2001
2001
2001
10,568
1,711,801
8.00
24,0002,995,942
10.39
ASODENIC
6,087
6,230,864
0.21
16,402
6,948,907
1.63
FAMA
15,275
1,688,417
0.00
24,8882,680,631
9.00
FINCA
2,129
913,9454.40
5,940
1,609,631
5.20
FUNDENUSE
10,121
317,870
1.25
10,561
629,385
0.35
ProMujer
Five-MFI Total
44,18010,862,897
1.79
81,791
14,864,496
5.06
Source: USAID Microfinance Results Reporting. Averages for 5-MFI Total are portfolio weighted.

The case of Nicaragua is typical of the region. FAMA, an ACCION International affiliate,
enjoyed a virtual monopoly in microfinance lending in the Managua area for the few years after it
commenced operations in 1992. However, by 1996 approximately six other major MFIs entered themarket, though even by 1997 no MFIs in the region had a portfolio of more than 4,000 borrowers.
Moreover, according to ASOMIF, the association of Nicaraguan microfinance institutions, theportfolios of Nicaraguan MFIs grew at an annual rate of 47% between 1997 and 2001 (La Prensa,
10/02/2002). By 2001 the largest MFIs were carrying portfolios in the range of 15,000-25,000
borrowers, with considerable overlap in geographical operating regions. Table 1 gives USAID data
illustrating microlending growth during 1999-2001 of the five largest MFIs receiving USAID funding.
Note that as portfolios have increased dramatically, levels of arrears have also risen in all but one case.

6

Guatemala and El Salvador have experienced similarly dramatic growth in MFI activity.
FUNDAP in Quetzaltenango, Guatemala, like its sister ACCION institution in Nicaragua, experienced
very little competition from other MFIs since its inception in 1988 until the mid-1990s. New entrants
into its regional market, such as FUNDESPE and Fe y Alegria, have forced FUNDAP to cut interest
rates on its larger loans from 3% to 2.5% per month. To remain solvent under competition, it has pulledaway from its initial mission of offering smaller loans in the form of group-based credit, and instead
now lends to a wealthier, more lucrative segment of the market. While average initial loan size was
US$135 and average monthly sales were US$291 for borrowers receiving their first loans between 1988
and 1993, by 1999 these figures for new clients had grown to US$543 and US$672, respectively.3
Asymmetric information between lenders over borrower quality and indebtedness has been amounting issue in all three countries, but there have been great differences between the three countries
in the level of cooperation realized between MFIs to mitigate the problem. El Salvador, with its
internet-driven Info-Red borrower database4, represents the best example of a case where a network of
independent MFIs have built information-based institutions reminiscent of those in developed countries,
where nearly instantaneous credit checks are possible. In Guatemala, multiple contracting by MFIclients had become so damaging by the late 1990s that REDIMIF, an association of 19 MFIs embarked
on an effort to establish CREDIREF, a centralized microfinance credit bureau, which though now
functional, is still in its nascent stages. Cooperation between MFIs during the mid-1990s was fairly
strong in Nicaragua; institutions regularly shared information on poorly performing borrowers with one
another. However, as MFIs poured into the market in the late 1990s, cooperation has deteriorated tosuch an extent that, as one loan officer put it "our information-sharing consists of a trip to the local
cantina to ask neighbors if loan officers from other MFIs have been paying visits to a potential client.”

3
4

Figures are inflation-adjusted. Source: sample of 376 field surveys conducted in Guatemala in 1994-1999.
Info Red’s internet-based system can be accessed at http//www.infored.com.sv.

7

3. Basic Model: Full Information
3.1 Analytical framework
Consider an MFI operating in a large pool of potential borrowers who, for well-establishedreasons, are denied access to credit in the formal financial sector. New lending technologies such asgroup lending, community banking, and dynamic incentives (as well as the possibility of grant funding)have allowed an MFI access to these borrowers. Let this pool of potential borrowers be defined by theset B ≡ {1, 2 ,...n} where B is indexed in ascending order of an initial level of productive assets ki,
pertaining to each i ∈ B . The corresponding set of initial assets K ≡ {k1 , k 2 ...k n } is observable to the
MFI and uniformly distributed within B such that k i +1 − k i = ξ , where ξ =k −k, and for notationaln −1ease we denote the lowest level of endowment, k1, as k and the highest, kn, as k .
For each borrower i ∈ B , the MFI considers granting a loan Vi at an interest rate ri. The interest
cost of capital for the MFI is equal to c, and on each loan it incurs fixed administrative costs F. The
loan offered by the MFI is subject to a participation constraint, representing the best alternative sourceof financing by borrower i. We will represent this best alternative financing as offered by a local
moneylender competing over the same set of potential borrowers B , offering loans that return a profit()

Π iB Vi m , ri m to borrower i ∈ B . The moneylender has lower fixed costs per loan than the MFI, Fm < F,
but a higher cost of capital, cm > c, and thus operates at a cost disadvantage (advantage) for loan sizes
Vi > (< )F −Fm. We assume that prior to the entry of the MFI, the monopolistic moneylender operates
cm − cin a predatory manner as in Basu (1984), such that the moneylender is able to induce any borrower i toaccept a loan contract that leaves his client with only some very small level of profit, ε > 0.
If a borrower receives a loan, the resulting investment yields a low return per unit of borrowed
capital of β < 1 with probability pi (k i , Vi ) due to a lack of ability to post sufficient collateral, and a high8

return of β > 1 + ri with probability 1 − pi (k i ,Vi ) . We posit that the probability of the low return, in
which the borrower is forced to default on 1 - β of the principal is decreasing in ki and increasing in Vi .
This is equivalent to saying that a wealthier individual has a higher probability of repaying a loan of agiven size than a poorer individual, and that increasing the size of the loan to a given individual cannot
increase the probability of repayment.5 To simplify our model, we assume pv > 0 and pk < 0 to beconstant and that pi = p + p vVi + p k k i .6 We abstract from strategic default in the sense that any projectyield up to and including (1 + ri )Vi is forfeited to the lender. Profit to the borrower is thus

Π iB = (1 − pi )( β − (1 + ri ))Vi(1)

while profit to the MFI isΠ iMFI = (1 − pi )(1 + ri )Vi + pi βVi − (1 + c)Vi − F .

(2)

The profit function of the moneylender is identical to that of the lending institution, except for themoneylender’s lower fixed costs per loan, F m , and higher cost of capital, c m . The slope of theborrower’s isoprofit curves can be derived through total differentiation of (1) to obtain− Vi (1 − p )dVi=(β − (1 + ri ))( pvVi − (1 − p )) .
dri

(3)

The isoprofit curves for the borrower are strictly decreasing in ri and backward-bending with respect
to Vi because the borrower fears default as the loan becomes too large. Similarly we totallydifferentiate (2) to obtain the slope of the lender’s isoprofit curves:dVi
Vi (1 − pi )=.
( pvVi + pi ) 1 + ri − β − (ri − c )
dri

(

)

(4)

The lender’s isoprofit curves are increasing in ri and also backward-bending in Vi due to the riskof default in large loans. The lender’s isoprofit curves are positively (negatively) sloped for values of

5
6

[ ]

Boundary conditions on parameters given in the Appendix guarantee that pi ∈ (0,1) for all k i ∈ k , k .
For analytical tractability, we assume p to be independent of the interest rate.

9Vi > (<)⎤
1 ⎡ (ri − c )− p − p k k i ⎥ . Note that cm > c implies that the indifference curves of the⎢
2 p v ⎣⎢ (1 + ri − β )⎦⎥moneylender are steeper on the negatively sloped portion and flatter on the positively sloped portion;
the critical Vi at which the slope becomes vertical is also lower than for the MFI. The borrower’s
isoprofit curves at ri < β − 1 are positively (negatively) sloped for Vi < (>)(1 − p − p k k i ) / 2 p v . Thisimplies that the value of Vi at which the borrower’s isoprofit curve bends backward is higher than thatfor the lender if 1 + c and 1 + c m > β (see Assumption 3 below). This and other assumptions in ourmodel include the following:
Assumption 1: “Moneylender Feasibility”. In the absence of competition, the moneylender can offersome loan to every borrower in the pool that is weakly profitable for both. In other words for eachi ∈ B , ∃ some (Vi , ri ) for which Π ML (V (ki ), r(ki )) ≥ 0 and Π B (V (k i ), r (ki )) ≥ 0 . This implies k islarger than some minimum value whose terms, given in the parameters of the model, are provided in
the Appendix.
Assumption 2: “Microfinance Lending Environment”. When an MFI competes with a moneylender,
there exists more than one profitable borrower and at least one unprofitable borrower for the MFI in
the borrower pool. In other words, at its optimal loan contract (Vi* , ri* ) for each i ∈ B , we
have

∑i

> 1 andi∈Π MFI (V *,r *) > 0∑i≥ 1 . The parameters that satisfy this condition are alsoi∈Π MFI (V *,r *)< 0

detailed in the Appendix.
Assumption 3: “Potential Lender Loss”. We assume both the moneylender and the MFI will lose moneyin the bad state of nature for the borrower, or specifically that 1 + c m and 1 + c > β . This isconsistent with what we expect to be true in a developing-country context: a lender will lose money
under default and will require more interest to risk larger loans in equilibrium. It implies thatequilibrium contracts reside on the positive slope of both lender and borrower isoprofit curves.
These assumptions are necessary for equilibrium calibration of the model; we also believe they reflect
the lending environment within which most MFIs operate in the developing areas of the world.

10

3.2 The MFI as client-maximizer
The behavior of a profit-maximizing lender is well understood, but less clear is the objective of a nonprofit MFI, as are the great preponderance of NGOs involved in microlending. We argue here that theobjective of such an institution is typically to maximize its “outreach”, or the number of clients, n*,captured into its portfolio. The constraints are a net budget-balancing requirement, the participation
constraint, and non-negativity constraints. A non-profit, client-maximizing MFI which (potentially)receives grant funding, G, therefore solves the following problem:

Maximize n* subject to
Vi , ri

BC: ∑ Π iMFI (Vi , ri ) + G ≥ 0(Institutional Budget Constraint)

PC: Π iB (Vi , ri ) ≥ Π iB (Vi m , ri m )(Borrower Participation Constraint)

NC: Vi , ri ≥ 0(Non-negativity Constraints)

for all i captured in the MFI borrower portfolio.
The intuition of the MFI problem, given more formally in the proofs of LEMMAS 1 and 2, isstraightforward, and is as follows: The MFI maximizes its number of clients, n*, by first capturing theset of profitable borrowers, B P , from the moneylender. This is accomplished though offering a contract7(Vi , ri ) to i ∈ B P that leaves each i ∈ B P indifferent to the best contract the moneylender is able to offer()(at which Π iML Vi m , ri m = 0 and the PC binds). These equilibrium contracts are profitable for the MFI
but not to the moneylender, since on larger loans, the MFI’s lower cost of capital more than makes up
for its higher fixed lending costs. To maximize the number of borrowers in its portfolio, the surplus, A,~from lending to B P is used to subsidize loans to B NP , a subset of the non-profitable borrowers B NP .

7

Though little variation in interest rates is often observed in less competitive microfinance markets, much greater variation
in interest rates among different types of borrowers is observed as financial markets mature and become more competitive,
as seen for example in economies with highly developed financial systems. In the more mature microfinance markets inCentral America, we are already beginning to observe competitive pressure giving birth to multiple loan products at differinginterest rates that are targeted at a borrowers with varying levels of initial assets.

11~
B1NP thus consists of the set of m* unprofitable borrowers captured into the portfolio by means of cross{}subsidy A and grant funding G. Cross-subsidization begins with the borrowers ikˆ − 1, ikˆ − 2... onwhom losses are the smallest on the MFI contract at the point where the PC binds. The number of
unprofitable borrowers m* in the portfolio is then maximized by offering contracts to increasingly
poorer borrowers until the BC binds. Borrowers too poor to be reached by the MFI receive the()

exploitative monopoly moneylender contract for which Π iB Vi m , ri m = ε .8 The following lemmas arefoundational for the propositions that follow:
LEMMA 1: There exists a Pareto-efficient contract curve for every loan made between a lender and eachborrower i.
PROOF: See the Appendix.
By setting (3) equal to (4) we obtain the tangency points between the isoprofit curves of the MFI andborrower i. This condition reduces to( pvV + pi )(β − β ) − (β − (1 + c )) = 0(5a)which forms an Edgeworth-like contract curve for each borrower given by the dark horizontal curves in
Figures 1a and 1b. (The moneylender’s contract curve is shown by the lower dashed line.) By
substituting the default function into (5a) we obtain
Vi* (k i ) =⎤1 ⎡ β − (1 + c)
− p − pk k i ⎥⎢2 pv ⎣⎢ β − β
⎦⎥

(

)

(5b)

noting that the moneylender’s optimal contract is identical except that Vi m ∗ (k i ) < Vi* (k i ) since cm > c.
The contract curves are linear and horizontal, meaning that differences in the contractual interest rate
represent a direct transfer of profit between the borrower and lender. Figure 1a gives the example of aborrower i ∈ B P , for whom the MFI surplus is positive where the PC is binding. In Figure 1b, the8

If we relax Assumption 1, then the MFI may able to reach poorer borrowers than the moneylender. This complicates themodel, though the main results of the model are unchanged except that in order to reach the maximum number of borrowers
the true client-maximizing MFI will extract the entire surplus from loans to such borrowers.

12~
surplus is negative to the MFI for a borrower i ∈ B NP at the point where the PC binds. It is important tosee how borrower welfare increases by virtue of entry from the MFI, moving from points A to A* and B
to B*, respectively. Entry of the MFI makes borrower profits rise and moneylender profits fall, so thatthe equilibrium contract changes from a contract that lies at zero profit for the borrower to a contract~that lies at zero profit for the moneylender for all i ∈ B P ∪ B NP .

Vi

ΠiB > ΠiB∗

ΠiB∗ > εΠ iMFI= 0

ΠiB = εVi

ΠiB∗ > ΠiB

Π iMFI > 0

ΠiB > εΠiMFI < 0A*

ΠiB = εΠiMFI= 0B*A
ΠiML < 0 ΠiML = 0

BΠ iML = 0

ΠiML > 0ri
Figure 1a: i ∈ B

Π iML > 0

Π iML >> 0~
Figure 1b: i ∈ B NP

P

ri

LEMMA 2: The client-maximizing objective yields an equilibrium loan contract in which (i) V * (k i ) is
increasing in ki, (ii) equilibrium profit to the MFI is increasing in ki, (iii) lending to poorer
borrowers is subsidized by profits from wealthier borrowers, (iv) a unique number of clients receiveloans from the MFI for any microfinance subsidy G ≥ 0.
PROOF: See the Appendix.
In Figure 2 we can observe the two pivotal asset endowments: k̂ i ∈ K is the asset endowment of the~~
break-even borrower, while k i ∈ K is the asset endowment of the borrower i ∈ B NP who causes the BC~
to bind, and so is the poorest agent offered credit by the client-maximizing MFI. We denote k (0) as the~poorest agent served by the MFI in the absence of grant funding and k (G ) as the poorest served in thepresence of some G > 0.
We believe the equilibrium illustrated in LEMMA 2 is the typical case of microfinance markets.
In a borrower pool in which all borrowers have extremely low initial endowments and B P = {∅}, only
grant-funded agencies can survive. In cases in which all borrowers have very high endowments and13B NP = {∅}, we should observe all lending conducted by for-profit lenders rather than grant-supported
NGOs. The typical microfinance market is the intermediate case where cross-subsidization within a~given client pool is possible, and the losses realized on loans to borrowers ∈ B NP can be covered by a

combination of grants and profits made from lending to borrowers ∈ B P . One implication of this is thatwhen there is a single client-maximizing MFI that has access to the borrowing pool, there is no need totarget grant funding towards the poor in this intermediate case. All surplus is automatically directed tothe poorest clients. Figure 2 illustrates the surplus by borrower across the set K :

ΠiMFI

~
k (G > 0)

~
k (0)

A

k

k
G

B

k̂ i

Figure 2

3.3 Competition from a profit-maximizing market entrant
In many parts of the world, the new lending technology has begun to attract profit-seeking entrants
into the microfinance market.9 We now consider the effects of the entrance of a competing MFI, where
unlike with the incumbent moneylender, the incumbent MFI and entrant MFI possess the same lending
technology. Assume first that the incumbent is a client-maximizing MFI with G = 0. The profit
maximizer solves the following, more familiar problem:max∑ΠiMFI , subject to:
Vi ,rii =1

PC: Π iB (Vi , ri ) ≥ Π iB (Vi * , ri* ), Π iB (Vi m , ri m )9(Participation Constraint)

Two examples in Guatemala are BanRural and BanCafe, formal lenders that have entered the microfinance market, andnow are major providers of microcredit in the country; another example is the well-known BancoSol in Boliva.

14NC: Π iMFI ,Vi , ri ≥ 0 ∀ i(Non-negativity Constraint)

Based on their respective objective functions, Bertrand competition by these identical MFIs will takeplace over i ∈ B P , which induces undercutting for all borrowers for whom the PC and the NC aresatisfied. Under competition solely with the moneylender, the PC of the cross-subsidizing MFI’s binds~
MLmm
for all borrowers i ∈ B P ∪ B NP at the level where Π i (Vi , ri ) = 0 . For the profit-maximizing MFI

under competition, it is the NC that binds for all i ∈ B P because the incumbent MFI, using identicaltechnology, then defines the PC. For the break-even borrower ikˆ ∈ B , both constraints are binding
under both market structures, and so the contract offers are the same. We denote the equilibrium MFI()

contract under moneylender competition as Vi* , ri* , and the Bertrand competitive equilibrium after()

entry of a second MFI by Vi** , ri** .
PROPOSITION 1: Bertrand competition between MFIs benefits wealthier borrowers, but makes a groupof poorer borrowers worse off.
PROOF: The proof follows in a straightforward manner from LEMMAS 1 and 2. Under Bertrand()

competition between MFIs, Π iMFI Vi ** , ri** → 0 for ∀ i ∈ B P . Since by LEMMA 1 all contracts in

(

)

(

)

equilibrium are Pareto efficient, this implies that Π iB Vi ** , ri** > Π iB Vi* , ri* for ∀ i ∈ B P . Because()

Π iMFI Vi** , ri** = 0 for ∀ i ∈ B P in equilibrium, this renders cross-subsidization impossible. Therefore,

(

)

(

)

(

)

~
borrower profits for ∀ i ∈ B NP fall from Π iB Vi* , ri* to Π iB Vi m , ri m where Vi m , ri m is the monopolistic

(

)

(

)

(

)

moneylender contract. We know that Π iB Vi * , ri* > Π iB Vi m , ri m because when two MFIs compete in()~
Bertrand equilibrium we have Π iML Vi m , ri m > 0 and Π iB Vi m , ri m = ε for ∀ i ∈ B NP . Conversely, with()~
only a single MFI competitor the PC binds for ∀ i ∈ B NP by LEMMA 2, implying that at Π iB Vi * , ri* we()

have Π iML Vi m , ri m = 0 . Since all contracts in equilibrium are Pareto efficient by LEMMA 1, an

(

)

(

)

equilibrium contract such that Π iML Vi m , ri m = 0 therefore implies that Π iB Vi* , ri* > ε.

15

Notice that faced with entry by a profit-maximizing MFI, the client-maximizing MFI respondsby conforming to the behavior of a profit-maximizing MFI. Profits to the Bertrand competitors go tozero in equilibrium; all surplus is captured in higher profits to the wealthiest borrowers in B .
3.4 Competitive equilibrium with non-targeted subsidies
We will refer to a subsidy that has been earmarked by the donor for underwriting the costs of loans
specifically to unprofitable (poorer) borrowers as a “targeted” subsidy, while we refer to subsidies thatare at the full discretion of the lender as “non-targeted”. Because a non-targeted subsidy can be used toundercut contracts to profitable clients, the presence of such grants will thwart the realization of the
Bertrand equilibrium when another MFI is unsubsidized.
PROPOSITION 2: The presence of a client-maximizing MFI with a non-targeted subsidy will prevent theentry, or force the exit of, any unsubsidized MFI.
PROOF: In order to maximize the number of borrowers in its portfolio, the client-maximizing MFI with~a non-targeted subsidy minimizes the loss from capturing each borrower into its portfolio, B P ∪ B NP .

Since it is the NC rather than the PC with the moneylender that is binding under Bertrand competition,
borrowers in B P require the smallest draw on the subsidy. Therefore, the subsidized MFI minimizes its(

loss on each borrower through a contract of Vi , ri − ζ****) to ∀ i ∈ B , where ζ is arbitrarily small.
P

Such a contract lies on the Pareto efficient contract curve by LEMMA 1, and hence is preferred by

(

)

(

borrower i since Π iB Vi ** , ri − ζ > Π iB Vi** , ri****

) . Since MFIs are assumed to be identical in lending()

technology, a contract on the contract curve that yields Π iB Vi** , ri − ζ implies that

(

**

)

Π iMFI Vi** , ri − ζ < 0 , therefore leaving no independently feasible contract for the unsubsidized MFI.
**PROPOSITION 3: Competition between two client-maximizing MFIs with non-targeted subsidies will leadto a Bertrand-Nash equilibrium in which the market share of each MFI will be proportional to its levelof grant funding.
PROOF: See the Appendix
16

The intuition to the proof is that since Bertrand competition eliminates profits on each profitableborrower i ∈ B P , all borrowers must be captured through competitive subsidy in the client
maximization process. One can think of each subsidized MFI as “purchasing” borrowers for its
portfolio, where in equilibrium, arbitrage behavior takes place such that the “market price” of capturing
each borrower is equilibrated across all those with access to MFI credit. The resulting Nash equilibrium
can be seen in Figure 3 (with the example of G1 > G2), where the specific borrower i served by each~
institution is undetermined. Note that and that all borrowers that remain in the portfolio B P ∪ B NP are(weakly) better off than under cross-subsidization.
Π i (Vi** (k ), ri (k ))
**

k

~
k

k̂ G1

G2

k

Figure 3

3.5 Competition from an MFI market entrant with a targeted subsidy
If one competitor is a client-maximizer with a targeted subsidy and the other a client-maximizer
with a non-targeted subsidy, then the competitive equilibrium is different from the previous case since
the targeting lender is prevented from expending grant resources on profitable clients. As a result, the(non-targeting lender must only offer Vi , ri − ζ****) in order to capture all i ∈ B . Since the nonPtargeting lender can capture i ∈ B P at negligible cost, it is likewise able to focus (nearly) all of its grant
resources on capturing as many i ∈ B NP as possible. The nature of the grant-based competition is thesame as the previous case, except only contracts to the poorer borrowers, i ∈ B NP , are contested.
~
PROPOSITION 3 now holds only over unprofitable borrowers; the share of B NP captured by each lender

now becomes proportional to grant funding. Thus, all i ∈ B P are better off if neither lender targets, but

17~
all i ∈ B NP are better off if even one lender targets. (Note that only M - 1 targeting lenders are required

to make all M client-maximizing lenders behave as if they were targeting.)
PROPOSITION 4: Market entry of a client-maximizing, subsidized MFI with funding specifically targeted
to poor borrowers may cause poor borrowers to lose access to MFI credit.
PROOF: Recall that A ≡ ∑ Π iMFI or the total profits realized on profitable borrowers by the incumbenti∈B P

client-maximizing MFI1 facing only moneylender competition (equal to the area A in Figure 2).
Case (a): The incumbent is unsubsidized. As the entrant (MFI2) and incumbent MFI1 engage in

Bertrand competition over i ∈ B P , A → 0 . In the Bertrand Nash equilibrium, the total subsidy forlending to the set of poorer borrowers B NP will be smaller unless G2 ≥ A. If and only if G2 > A, then
~
~
~
~
~
~
k (0) < k (G 2 ) , which implies ∃ i for which i ∈ B1NP , but i ∉ B2NP , where B1NP ,B2NP ⊂ B NP are the setsof non-profitable borrowers captured before and after entry of the competing MFI. Any i for which~~i ∈ B1NP but i ∉ B2NP is strictly worse off since Π iB Vi m , ri m < Π iB Vi* , ri* by PROPOSITION 1.

(

)

(

)

Case (b): The incumbent has a targeted subsidy. (This case is a hybrid between the unsubsidized

competitive equilibrium and the client-maximizer with an untargeted subsidy from section 3.5.)~
Without competition from the entrant MFI, the PC is binding for ∀ i ∈ B NP by LEMMA 2, and the total~
subsidy needed to capture each i ∈ B1NP into the portfolio is ∑ Π iMFI (Vi * , ri* ) = A + G 1 . Consider~i∈B NP

entry by a client-maximizing MFI with targeted subsidy G2. The ensuing competitive Nash equilibrium~
is similar to that in PROPOSITION 3 in that Π iMFI = Π MFI< 0 for i ∈ B2NP except that the targeted naturejof the subsidies results in Π iMFI = 0 for ∀ i ∈ B P . In contrast to case without MFI competition, the PC~binds in competitive equilibrium for i ∈ B NP only for the poorest borrower, ik~ (G +G ) , and the total MFI
1

(

)

2

~
subsidy allocated to B NP equals m1* + m2* Π iMFI = G1+G2 , i = ik~ (G +G ) . (Refer to the proof of
1

2

~
~
PROPOSITION 3). Thus for B2NP ⊇ B1NP , we must have G 2 ≥ m1* ⋅ Π iMFI
− G 1 , where i = ik~ , otherwise it′~~
must be that B2NP ⊂ B1NP . Observing that m1* ⋅ Π iMFI
− G 1 > A, note that the conditions for the poorest′
borrowers losing access to MFI credit are less restrictive in Case (b) than in Case (a).
If G2 indeed falls below the critical level shown in Case (b), then there are both winners and~losers within B NP under competition between MFIs with targeted subsidies. The set of borrowers B2NP
are better off since competition to attract them into the portfolios increases the subsidy level of their18~~equilibrium contract. In contrast, those in B1NP but left out of B2NP are worse off since upon being

(

)

(

)

dropped from the MFI portfolio, their profits fall from Π iB Vi * , ri* to Π iB Vi m , ri m .
An interesting implication of PROPOSITIONS 3 and 4 is that a maximum number of borrowers
could be reached under a collusive strategy between the two MFIs. In a collusive strategy, MFIs could
agree to split the market by jointly offering contracts at which the PC with the moneylender is binding.
By PROPOSITION 3, the collusive strategy is not a Nash equilibrium. However, collusion generates a total~
subsidy of A + G1 + G2 to capture borrowers in B NP , resulting in a larger set of poor borrowers, B2NP
C ,
~
~
~
with MFI credit access. Thus for any given G2 we have B2NP ⊂ B1NP ⊂ B2NP
C , noting, however, that~~
B2NP ⊂ B2NP
C would have less favorable contracts under collusion than competition. The difficulties ofsustaining collusion in this model are typical of any cartel, especially for loans to profitable borrowers.
Nevertheless, it is interesting that collusion between client-maximizing lenders leads to the maximum
redistribution towards the poor and the most evenly distributed benefits of microlending.

3.6 Summary of Conclusions from the Basic Competitive Model:•

Entry of an MFI is beneficial to all borrowers accessing MFI credit; it increases borrower profits
from the optimal contract for the moneylender to the reservation profit of the moneylender.

•

In competitive markets, profit-maximizers and unsubsidized client-maximizers always behave
the same way. Thus, under competition it is not the motivation, but rather the extent and thenature of the grant funding of a lender that matters.
•

Targeting of subsidies is unimportant in a market with a single client-maximizing MFI; thedistinction only becomes important under MFI competition.

•

Lenders with non-targeted subsidies can always drive any unsubsidized competitor out of themarket altogether, whereas targeted subsidies can never eliminate a competitor from the market.

•

Every competitive scenario involving a lender with targeted subsidies results in a market that isboth competitive and in which some of the poor receive loans.

•

Competition never makes any profitable borrower worse off.

•

The only way in which the poor can be reached without subsidies is if a client maximizer exists
as a solitary MFI in the market and competes only with a moneylender. However, in this casesubsidies are merely being generated from amongst the other, less poor borrowers.
19

•

If rents to the incumbent MFI from lending to the non-poor are higher than the grant funding ofan entrant MFI, then the entrant can actually decrease the funding available to underwrite loans
to the poor even if the grants are targeted specifically towards the poor.

•

Collusive behavior between client-maximizing MFIs can result in the maximum number ofborrowers reached by MFIs, and the most evenly distributed benefits among borrowers.

4. Extended Model: Asymmetric Information between Lenders

4.1 Asymmetric information and dynamic incentives
In the previous section, we developed a basic model in which information about borrower heterogeneity
was common knowledge between borrower and lender. An implicit assumption of the basic model isthat a lender is able to ascertain the optimal contract through observation of a borrower’s capitalendowment. We continue to assume that heterogeneity in K remains observable to the MFI. However,
we now assume that each borrower i ∈ B is characterized by a personal rate of time preference,

ρi ∈ [ ρ , ρ ] per lending period, information that is hidden to the MFI, whose per-period profit function
we assume is unchanged and homogeneous across institutions. Let g (ρ i ) represents the density
function of ρi and G (ρ i ) its associated distribution function. The distribution of time preference isassumed to be orthogonal to the distribution of k. Timing is more important in the extended model, so
we lay it out explicitly as follows:

Borrower endowedwith fully observable ki
and ρi observable only
to borrower.

Borrowers
apply for loans

Screening by lender
occurs. Borrowers
with other loans
denied new credit.

Capital provided toborrowerand invested

Returns realized oninvestment project;lender repaid fullyor partially

Figure 4

We now bring into our analysis the issue of dynamic incentives. Dynamic incentives provide
motivation for repayment when borrowers lack collateral to secure loans by implicitly promising20continued credit access as a reward for loan repayment. They are routinely used by MFIs (and other
lenders) in poor areas of developing countries to mitigate issues of moral hazard involved with credit
transactions. The present value to a borrower of the continued access to MFI credit in each period,

Γ(ki , ρi ) , is positively related to the advantage offered by MFI financing relative to the alternative(moneylender) contract, and a negatively related to a borrower’s rate of time preference.10
To focus our attention on issues related to problems of asymmetric information and competition,
we will concentrate on the case of two MFIs engaged in Bertrand competition over the pool ofborrowers with i ∈ B P (Competition over i ∈ B NP yields similar results.) Hoff and Stiglitz (1998)
show that dynamic incentives are weakened by new market entrants as this improves the reservationloan contract available to borrowers in the case of default. Their results imply borrowers must
somehow be punished for default by the financial system as a whole, through a system of negative
borrower information-sharing, i.e. each lender sharing its lista negra (as it is often referred to in Latin
American MFIs--the blacklist).
What we illustrate in this section is that even within a system that identifies defaulting
borrowers, other gaps involving asymmetric information between lenders must be bridged. We showthat it is also critical for lenders to share positive borrower information with one another, even regarding
well-performing loans, i.e. that a lista blanca (a list of positive information) is also necessary. This istrue even in a model without strategic default. In order to concentrate on this issue, we take as our
informational benchmark market with Bertrand competition in which all lenders fully share the lista

negra, where defaulters are denied future formal credit access, but no positive information is shared.
Calculating the present value of the relationship with the MFI, the objective function ofborrower i under dynamic incentives now becomes

[(

)

]

Max Π iB = (1 − pi ) β − (1 + ri ) Vi + Γ(k i , ρ i ) .
10(6)

To concentrate our analysis on borrower behavior we assume that lenders continue to maximize single-period profits;
the results derived are consistent with lender discounting provided that all lenders share the same discounting process.

21As ρi increases, the borrower becomes less concerned about lack of future credit access, and with any

k i ∈ K , prefers a larger loan size. We see this through total differentiation of (6) with respect to

Vi and ri which yields

dV i− V i (1 − p i )=(β − (1 + r )i )( p vV i − (1 − p i )) − p v Γ (k i , ρ i )
dri

(7)

Notice as ρi increases, the borrower’s indifference curve rotates clockwise, with the upwardsloping part of the curve becoming flatter and the downward sloping part becoming steeper as in
Figure 5. This implies that, for example, in a Bertrand equilibrium for ρ2 > ρ1
Vi** ρ 2 > Vi ** ρ 1 and ri** ρ 2 > ri** ρ 1(8)or that the equilibrium contract for a more impatient borrower is characterized by a higher loan size andinterest rate than the contract for a more patient borrower. Thus an MFI operating in the context of
information sharing between lenders about borrower indebtedness does not face a hidden information
problem; it can infer a borrower’s impatience from ki and the requested loan size. As ρi varies for any
given ki, it creates a unique tangency point on the isoprofit curve of the lender at a unique level of Vi forany given ki, i.e. Vi* (k i ) =⎤1 ⎡ β − (1 + c) − pv Γ(k i , ρ i )− p − pk k i ⎥ . As Figure 5 shows, with full⎢2 pv ⎣⎢
β −β
⎦⎥

(

)

information sharing, an MFI could make up for the increased likelihood of default on this larger loan
size by offering even the most impatient borrower a larger loan at a higher interest rate.
However, when there are multiple lenders and an absence of information sharing about borrower
indebtedness between them, there is an incentive for borrowers past a certain level ρi = ρ * to takemultiple loans, where we assume ρ* ∈ ( ρ , ρ ). Previous work typically adds the prevention of multiple
contracting as a constraint for the principal; instead we take the approach of seeing how multiple
contracting emerges without safeguards, and how the sharing of information can be used to prevent it.11

11

The problem of multiple contracting in markets with asymmetric information is not unique to microfinance; see Cawley
and Philipson (1999) for an application to life-insurance markets, and Kahn and Mookherjee (1998) for a more generalmodel of non-exclusive contracting in credit markets.

22

Let ρ * be the lowest value of ρi that satisfies the following inequality, in which expected profits frommultiple loans are higher for the borrower than on a single loan:(( ))[(β − (1 + ~r ))2V~ + Γ(k , ρ )] > (1 − p (V ))[(β − (1 + r ))V + Γ(k , ρ )] (9)~Π iB = α ⋅ ε ρ i + (1 − α ) 1 − pi 2Vi

i

i

i

i

i

i

i

i

i

i

where α represents an information parameter such that an MFI is able to identify a borrower’s true levelof indebtedness with probability α before a borrower receives a loan.12 Unless there are institutionalarrangements for information-sharing between lenders, field experience has shown that α typically falls
as the number of MFIs in a given market increases.
The difference in behavior between more patient and less patient borrowers is illustrated
graphically in Figure 5 where, for example, the most patient borrower with ρi = ρ receives an equilibrium contract for loan of size Vi (k i ) at interest rate ri (Vi (k i )) . Under full information sharing, theless patient borrower with ρi = ρ* receives an equilibrium contract for a loan 2Vi at the higher interest 
rate ri (2Vi ) . A single lender is thus willing to oblige such a borrower by providing a larger loan along
its zero-profit offer curve at a higher interest rate that accounts for the higher expected rate of default.
This is the contract that will exist in the absence of information asymmetries under exclusive
contracting between borrower and lender.

Vi~
2 V (k i ) > 2Vi (k i )
2Vi (k i )Π1MFI ( ki ) + Π 2MFI ( ki ) = 0

Π iB ( ρ *) ′′ΠiB (ρ*)′ ΠiB (ρ*)Π iMFI ( k i ) = 0~
Vi (k i )Vi (k i )12

We clarify three additional simplifying assumptions in the extended model: borrowers discovered with multiple loans areculled from the portfolio, a borrower who defaults on any loan defaults on all loans, and a borrower who defaults on an MFIloan is henceforth able to access credit only from the moneylender.

23

Π iB ( ρ i = ρ ) riri (Vi (k i )) ~ ri (2Vi (k i ))Figure 5riWith an absence of information-sharing between lenders, however, the borrower with ρi ≥ ρ*
can lower his overall cost of borrowing by obtaining two separate loans from two different lenders,
while creating the illusion for each lender that he is borrowing a fraction of his actual total—aubiquitous occurrence as reported by MFIs operating in competitive markets. In the example in
Figure 5, the less patient borrower with ρ* benefits by obtaining two loans from two different lenders


 
 
of size Vi (k i ) , and paying interest rate ri (Vi (k i )) < ri (2Vi (k i )) on each of the two different loans,increasing his utility to Π iB ( ρ *)′ . Moreover, the lower interest rate sensitivity on the joint offer curveΠ1MFI (ki ) + Π 2MFI (ki ) = 0 faced by the multiple-borrower induces the borrower to increase borrowing to~~~
2 Vi (k i ) > 2Vi (k i ) by taking two loans of size Vi (k i ) at interest rate ~ri (Vi (k i )) , further increasing theimpatient borrower’s utility from Π iB ( ρ *)′ to Π iB ( ρ *)′′ . Multiple contracting occurs in the absence offull information because the lender can no longer infer ρi from the requested loan size. The lendercannot distinguish between an impatient borrower taking two separate loans of the size a patientborrower with the same ki would demand. Note in our example that instances of multiple contracting
will occur whenever the full-information equilibrium Vi for the least patient borrower is two or moretimes greater than the equilibrium contract for the most patient borrower for any ki.13
This behavior creates an externality that raises the overall default rate for MFIs for two reasons:First, because the interest rate is lower under multiple borrowing than it would have been with a single
lender, total borrowing and indebtedness increases within the portfolio of borrowers. Second, and more

13

The result is generalizable to a borrower taking more than two separate loans, given sufficient borrower impatience andmarket competitors.

24

importantly to an individual MFI, the true probability of default is now a function of its own lending
and of some unknown quantity borrowed from elsewhere.
This makes the expected default rate of any given borrower i now equal toρ*ρ∫ρ p(V )g (ρ )dρ + (1 − α )ρ∫ p(2V )g (ρ )dρipˆ i =

i

~

i

i

i

*

ρ

(10)

ρ

∫ρ g (ρ )dρ − α ρ∫ g (ρ )dρ
i

i

i

i

*


pi (k i ) equal
Defining γ ≡ 1 − G (ρ *) as the probability of multiple contracting, and letting pi (k i ) and ~expected probabilities of default for borrowers (at a given level of ki) with single and multiple loans
respectively, we obtainpˆ i ≡ pi (k i , Vi ,α ) =(1 − γ ) p i + γ (1 − α ) ~pi
1 − γα

,

(11)

dρ *
> 0 and sodα

noting that γ = γ (α ) . Since an increase in α makes (9) less likely to be satisfied,γ α < 0. (Note since G ( ρ ) and α are both assumed to be independent of the distribution of k, we knowγ and γ α are independent as well.) As seen in (11), as α→1, pˆ i → pi , or the default rate approaches
the full information case. Using (2) and (11), expected profits to the lender for borrower i now become

Π

MFI
i

= (1 − pˆ i )(ri − c )Vi − pˆ iVi (1 + c − β ) − F =

(1 − γ )Π iMFI + γ (1 − α )Π~ iMFI
1 − γα

,

(12)

 


~
~
 
~
where ΠiMFI = (1− pi )(ri − c)Vi − pi 1+ c − β Vi − F and ΠiMFI = (1− ~pi )(ri − c)Vi − ~pi 1+ c − β Vi − F . Equation

(

)

(

)

(12) yields the information necessary for Proposition 5:
PROPOSITION 5: If asymmetric information between lenders increases as the number of competing
lenders increases, borrowers receive less favorable loan contracts after entry of new lenders.
PROOF: The details of the proof are given in the Appendix. However, first note the positive relationshipbetween profits to each MFI and α. The Appendix shows that partial differentiation of (12) yields

25

(

)


~
∂Π iMFIΠi − Πi= [γ (1 − γ ) − γ α (1 − α )]> 0.
∂α(1 − γα )2(13)Under Bertrand competition, each lender will adjust the contract to borrower i ∈ B P , so the resulting
equilibrium contract yields Π iMFI = 0 . If new entrants make information-sharing more difficult, total
differentiation of (12) reveals that each lender’s new zero-profit curve must lie strictly to the right and to
the interior of the old one. The lower-information equilibrium with more lenders in the market lies on alower isoprofit function for the borrower at which any size loan, offered at zero-profit to the lender, isoffered at a higher interest rate. The conclusion from the proof is illustrated in Figure 6, which shows
the change in the Bertrand competitive equilibrium as α falls.
Once the competing MFIs respond to multiple contracting by adjusting equilibrium contracts for all
clients, it is unclear whether or not the impatient have indeed benefited. The interest rate “discount”
received though multiple loan contracting may or may not compensate for the fact that every individualloan contract is marginally worse. What is unambiguous is that patient borrowers with ρi < ρ*, whofind it optimal to borrow only from a single lender, have been hurt by reduced informational flows
between lenders, and the ensuing instances of multiple contracting by other borrowers. By undertaking
action clearly observable to only one lender, the impatient create a classical externality whose costs are
spread across the whole population of borrowers.
Π iB

Vi
Π iB

α

< Π iBα <α ′αΠ iMFI = 0
Π

26

MFI
i

α

=0 ′

α <α

ri (k i )′ri (k i )Figure 6

ri

4.2 Effect of asymmetric information on the poorest borrowers
PROPOSITION 6: As asymmetric information between lenders increases, the poorest borrowers are
dropped from the lending portfolio.
PROOF: Consider the poorest borrower ikˆ ∈ B P who receives a loan with under Bertrand competition

(

( ))

( )

when Π iMFI (Vi** , ri** ) | α = 1 − pˆ i kˆi (ri − c )Vi − pˆ kˆi Vi (1 + c − β ) − F = 0 . Total differentiation yields
( p − ~p) > 0 by
∂pˆ
pˆ
dkˆ
= [γ (1 − γ ) − γ α (1 − α )]= − α , where pˆ k < 0 by assumption. We know that pˆ α ≡
∂α
dα
pˆ k
(1 − γα )2ˆdifferentiation of (11). Thus dk = − pˆ α < 0 , or as α declines to α′, the poorest borrower who receives an
dα

pˆ k

MFI loan must have initial assets kˆi + ψ > kˆi .
The intuition to the proof is straightforward. Increasing information asymmetries increase the cost oflending to the entire portfolio of borrowers. Consider the poorest borrower receiving MFI credit under
Bertrand competition at a given level α of information sharing, the borrower who has initial assets k̂ i .
As informational flows decline with an increasing number of lenders in a market to α′ < α, theprobability of default increases by p̂α such that marginally profitable loans to borrowers with level of
initial assets k̂ i (and slightly above k̂ i ) become unprofitable. As a consequence, the dilution of
information between lenders resulting from the entry of new MFIs in a given area, even non-profit
institutions, may cause the poorest borrowers to be dropped from MFI portfolios.
4.3 Conclusions from extended model with asymmetric information:•

With asymmetric information between competing MFIs, every loan contract yields a lower profit
to the borrower than under the full information benchmark. Patient borrowers are always worse
off with reduced information sharing. It is possible that the impatient borrowers are worse off aswell, if the negative externality of multiple contracting overwhelms the direct benefit.

27

•

If asymmetric information between lenders increases with the number of MFIs in the market,
competition has an unambiguously negative effect on both the most poor and the most patientborrowers in the portfolio.

•

Optimal information sharing between lenders must include not only data on defaulting
borrowers, the lista negra, but also continually updated information on current borrowers, eventhose who are not defaulting, or the lista blanca.

5. Policy Implications

It seems intuitive that the dramatic growth in the number of MFIs in developing countries, andthe ensuing competition between them, would have an unambiguously positive effect on low-income
entrepreneurs in developing countries. However, this research presents number of hypotheses thatprovide reason to question this notion.
We believe that a number of policy conclusions flow from this research. At the broadest level,
the results of the first part of our paper may extend to other instances in which altruistic motivationinduces a non-profit institution to cross-subsidize. Examples may include medical or health services,
socially motivated education programs, or provision of low-income housing, in which there exists somedegree of competition between for-profit and non-profit entities.
More specifically to microfinance, our research implies that the structure of funding is
unimportant in monopolistic markets, whereas the motivation of lenders is less important in competitivemarkets. Therefore, as competition increases, the onus for the inclusiveness of the market passes fromthe practitioners of microfinance to the donors. Yet the very existence of competitive markets hinges on
the idea that grant funding be used in a competitive market only to subsidize the cost of lending to thepoor. In light of this, our research supports the notion put forth by Morduch, that financially selfsufficient MFIs should co-exist with their subsidized counterparts, provided that these subsidies arecarefully restricted to the poorest borrowers.

28

Targeting of subsidies is difficult to achieve in practice, as such constraints are hard to monitor,
and MFIs themselves may not be able to easily identify the dividing line between profitable andunprofitable clients. In light of these issues, it may be most practical to reserve grants strictly for
geographical expansion of MFI activity into poor, unserved areas. In this way, donors can avoid
undermining competitive markets without placing expensive, ill-defined restrictions upon practitioners.
It also may be possible to restrict grant funding to methodologies that do not appeal to profitableborrowers, such as joint liability lending contracts where loan amounts are typically small andorganizational costs are high.
Another clear implication from our research is the need for credit bureaus, or internet-based
central risk-management systems, which identify outstanding debt in addition to cases of default. In
general the astounding growth in MFI lending in many areas has vastly outpaced the ability of MFIs tomonitor borrower quality and indebtedness. Among the Central American countries, for example, there
remains great heterogeneity in informational infrastructure. While some countries such as El Salvador
have established reasonably well-functioning centralized risk-management structures, others such as
Nicaragua lag far behind in this area, although the density of MFI activity is extremely high. At thisstage in the microfinance movement, the establishment of such centralized risk networks must become aleading priority to ensure the success and sustainability of the microfinance movement in LDCs.

29

Appendix

EXPLICIT FORMULATION OF ASSUMPTIONS AND BOUNDARY CONDITIONS:
Assumption 1: “Moneylender Feasibility”. Since the maximum interest rate acceptable to a borrowerlies very close to β − 1 , we can approximate the condition for the least favorable loan to a borrower that()leaves zero profit to the moneylender as Π iML = (1 − pi ) β Vi + pi βVi − (1 + c m )Vi = F m . Substituting
the expression for Vi* (ki ) in (5b) into the moneylender zero-profit condition, and solving for k shows()

2 pv F m β − 1 + c m ⎤−1 ⎡that for moneylender feasibility k must be ≥
−
⎥.
⎢p +
β −β
p k ⎣⎢
β − β ⎦⎥

Assumption 2: “Microfinance Lending Environment”. Lemma 2 shows that MFI profits in competitive
equilibrium can be expressed as Π *i =
∆=

[(

)

]

1
(∆ − pi ) 2 − (∆m − pim ) 2 ( β − β ) + F m − F , wherepv

β − (1 + c)
β − (1 + c m )and ∆m =. Setting this expression equal to zero and substituting in
β −β
β −β

pi = p + p k k i + pvVi shows that the equilibrium break-even level of initial borrower assets for the MFI
is kˆ =

2 p v ( F − F m ) ∆ + ∆m ⎤
−1 ⎡
+
p
−
⎢
⎥ . By setting the above boundary condition in Assumption 1 for2 ⎦pk ⎣cm − ck less than k̂ and solving for parameter conditions, we find that an unprofitable borrower will exist

provided that2 pv ( F − F m ) >cm − cβ −β, which holds given sufficiently high β . Provided thatk > kˆ + ξ , more than one profitable borrower will exist, which satisfies the 2nd part of Assumption 2.
Boundary Conditions on Default Rate: Here we delineate parameters for the model to ensure that[ ]0 < p(Vi , k i ) < 1 for ∀ k i ∈ k , k . Substituting Vi* (k i ) =yields pi (k i ) =()1[∆ − p − pk ki ] into the default function
2 pv

1( p + ∆ + p k k i ) . Recalling that pk < 0, our model parameters must ensure p(k ) < 1 and
2p k > 0 . The conditions are thus that k >1( p + ∆ − 2) and k < 1 ( p + ∆ ) . The former always
− pk
− pk

holds since p + ∆ − 2 is always < 0. The latter condition that k <301( p + ∆ ) is consistent with the− pk

boundary condition in Assumption 2 provided
or 3∆ + ∆m + 2 p k ξ >

m
m
⎡
⎤
1
( p + ∆ ) > − 1 ⎢ p + 2 pv ( mF − F ) − ∆ + ∆ − ξp k ⎥ ,
− pk
2
pk ⎣
c −c
⎦

4 p v (F − F m ), which holds given sufficiently high β relative to pv.
cm − c

Proof of LEMMA 1: We show in turn that (i) the constraint set impose by any PC (given as a reservationlevel of borrower profit) is compact and convex; (ii) the contract that maximizes lender profit is uniquefor every PC; (iii) that (i) and (ii) yield a set of points that form a Pareto-efficient contract curve foreach borrower and lender by the assumptions of the model.
(i) Assumption 1, which says that every borrower in the pool is offered some contract by amonopolistic moneylender, ensures that a PC exists and is defined by some level of borrower profit Π iB
for each borrower i. Re-arranging the borrower’s profit function in (1) and solving for ri at any(non-negative) constant level of profit of Π iB , yields ri = β −Differentiation gives

Π iB
−1.
Vi (1 − p − p k k i − p vVi )dri Π iB (1 − p − p k k i − 2 p vVi ), meaning that borrower isoprofit curves are
=
2 2
dVi
Vi (1 − p − p k k ) − p vVi[]

positively (negatively) sloped for Vi < (>)(1 − p − p k k i ) / 2 p v . Furthermore, we have[d 2 ri− 2Π iBVi (1 − p − p k k i − p vVi ) p vVi (1 − p − p k k i − p vV ) + (1 − p − p k k i − 2 p vV ) 2

dVi

Vi (1 − p − p k k i − p vV )

=
2

4

4

] < 0 . Thus, theborrower’s isoprofit curves are concave and decreasing in ri, which implies that the upper contour set
above any isoprofit contour is a convex set. Setting ri = 0 for the borrower’s profit function in (1) andsolving for Vi, note that the each isoprofit curve of the borrower, for any Π B ≥ ε , intersects the Vi axis
H

at the positive values V and VL⎛⎛ Π B ⎞ ⎞⎟1 ⎜2
⎟ . That the=(1− pi − pk ki ) ± (1− pi − pk ki ) − 4( pV )⎜⎜ β −1⎟ ⎟⎟2 pV ⎜⎜⎝⎠⎠⎝

constraint set is closed follows from the observation that (a) Π B (Vi , ri ) is a continuous function; and(b) the constraint set is the intersection of {(Vi , ri ) VL ≤ Vi ≤ VH and ri ≥ 0} and the inverse image, under{}Π B , of the set Π iB Π iB ≥ Π iB (Vi , ri ) . Boundedness follows from the fact that the constraint set lies atand below the isoprofit curve defined by (1) within and including the positive values V H and V L ,together with non-negativity constraint that ri ≥ 0 . Since all inequalities are weak, the constraint set isbounded. Therefore, the constraint set is compact and convex, and a maximum exists.

31

(ii) The properties of the lender’s isoprofit contours can be similarly derived re-arranging (2)Vi (c + p i (1 − β )) + F L + Π iL,
while holding lender profits constant and solving for ri, yielding ri =Vi (1 − p i )p vVi (1 + c − β ) − ( Π L + F L )(1 − pi − p vVi )dri
L = {MFI, ML}, for which
, and
=
2
dVi
Vi (1 − pi ) 22

d 2 ri[2 p v2V i (1 − p i )(1 + c − β ) + 2V i ( Π L + F L )(1 − p i ) (1 − p i − p vV i )(1 − p i ) + p v2V i
4=

V i (1 − p i )drdrSince for the borrower and lender isoprofit curves i < 0 and i > 0 respectively for
dVi
dVi
dV i

2

4

4

].

Vi > (1 − p − p k k i ) / 2 p v (i.e. for (1− pi − pvV ) < 0 ), a sufficient condition to establish a lender maximum
at the tangency point to Π iB is that ri (Vi ) is convex for VL ≤ Vi ≤ (1− p − pk ki ) / 2 pv . When (1− pi − pvV ) > 0,
which is true at V * (k i ) by Assumption 3, it is easily shown that for sufficiently large p k , Π iL + F L ≥ 0 .
Thus the lender’s isoprofit curve is increasing and convex over VL ≤ Vi ≤ (1− p − pk ki ) / 2 pv and hence at()

V * (k i ) . That the tangency point Vi* , ri* defined by V * (k i ) at each level of Π iB and i ∈ B is unique isshown by equation (5b) and the fact that ri (Vi ) is one-to-one over the domain [VL ,V = (1− p − pk ki ) / 2 pv ] .
We know that the optimal V * (k i ) that defines each horizontal contract curve lies within [V L , V H ] (andthus satisfies the non-negativity constraints) for ∀ k i ∈ [k , k ] , since re-arranging (5b) yields the⎤
2 pv F m−1 ⎡1( p − ∆ ) , which satisfies the boundary condition that k ≥ ⎢ p +− ∆ ⎥ incondition k >p k ⎣⎢
− pk
β −β
⎦⎥

Assumption 1. Furthermore, note that Vi* (k i ) can never exceed V H since V * (k i ) < (1 − p − p k k i ) / 2 p v .
(iii) Assumption 1 guarantees that the PC (and thus a contract curve) is defined for all i ∈ B with eachlender. That the locus of points on the contract curve are Pareto-efficient can be proven bycontradiction: If a point on the contract curve were not Pareto-efficient, another contract would exist for
which Π iL (Vi , ri ) could be increased holding Π iB (Vi , ri ) constant. But then this point could not havebeen a maximum of the objective function, and hence could not be on the contract curve.

Proof of LEMMA 2: We prove each part of the LEMMA sequentially:(i). Partial differentiation of equation (5b) shows that32

∂Vi∗ ∂Vi m ∗ − pk
=
=
> 0.
∂ki
∂ki
2 pv

(ii). To show that MFI profits in competition with a moneylender are higher for higher k i , we solve
explicitly for the equilibrium MFI loan as a function of parameters. First, we utilize the fact that thezero-profit constraint of the moneylender will bind for all loans under competition. Hence we can solve
for the equilibrium rate of interest charged by the moneylender, ri m ∗ for ∀ i in the MFI portfolio. Letting
pim be the borrower’s probability of default to the moneylender and where again ∆m =β − (1 + c m ),β −βpv F m− pim β + (1 + c m )mmmm∆ − pi∆ − pi, so the zero-profit interest rate will be ri m∗ =then Vi m =− 1.
pv
1 − pimIn the continuous equilibrium, the PC will bind such that Π iB (Vi m∗ , ri m∗ ) = Π iB (Vi ∗ , ri∗ ) for all
borrowers. (To see this, suppose it did not; by increasing ri to the point where the PC binds, the MFI~
could retain that borrower in the portfolio B P ∪ B NP and relax the BC. Therefore, the original contract

could not have been an equilibrium.) We substitute ri m ∗ into the binding PC
pv F m− pim β + (1 + c m )mm∆ − pi(1 − pi )(∆ − pi )( β − (1 + ri* )) = (1 − pim )(∆m − pim )( β −) and solve for the
1 − pimcompetitive equilibrium MFI rate of interest ri* as a function of the parameters of the model, obtaining
ri = β −*(∆m − p im ) 2 ( β − β ) − p v F m(1 − pi )(∆ − pi )− 1 . Substituting for ri* and Vi* , we can now solve explicitly forthe equilibrium level of MFI profit with the PC and the moneylender zero-profit constraints binding,
Π

(

)

2
⎡
∆m − p m (β − β ) − pv F m ⎤[(∆ − pi ) 2 − (∆m − pim ) 2 ](β − β ) m⎡ ∆ − pi ⎤⎢⎥+F − F .
= [(1− pi ) β −+ pi β − (1+ c)]⎢
⎥−F =
p
p
(1− pi )(∆ − pi )
⎢
⎥
v
⎣ v ⎦
⎣
⎦

MFI *
i

Taking the derivative of this equilibrium MFI profit level with respect to k, we getmm∂Π iMFI * − 2 p k ( β − β )[(∆ − pi ) − (∆ − pi )], which > 0 if c m − c > ( pi − pim )( β − β ) . Substituting
=
∂k
pv

33

in the default function, pi = p + p k k i + pvVi , into this expression, we can rewrite this inequality asc m − c > p v (Vi − Vi m )( β − β ) . Solving explicitly for the optimal loan size as a function of k i ,Vi =∆m − p − p k k i∆ − p − pk ki∆ − ∆m
and Vi m =, so Vi − Vi m =. Hence the required condition for
2 pv
2 pv
2 pv

⎛ β − (1 + c) − β + (1 + c m ) ⎞∂Π iMFI *cm − cmm⎜⎟.
> 0 reduces to c − c > p v ( β − β )or c − c >⎜⎟2 pv ( β − β )∂k2⎝⎠
By Assumption 2, Π * (k ) > 0 , and Π * (k ) < 0 . Π * (k i ) is a continuous, increasing function of k
because p is a continuous function of k and p v is a positive constant, so by the Intermediate Value
Theorem there exists some k̂ , with k < kˆ < k , for which Π * (kˆ) = 0 . We denote as the “break-evenborrower” the individual iˆ whose capital endowment k̂ i is the closest to k̂ while being ≥ k̂ ; thus forthe break even-borrower it will be the case that Π * (k iˆ ) ≥ 0 but Π * (k iˆ −1 ) < 0 .
(iii).

~
That MFI client-maximization implies cross subsidization from B P to B NP flows directly fromthe preceding arguments; the MFI maximizes its number of clients by maximizing profits for B P and~minimizing losses in capturing B NP , both subject to the PC, and hence uses profits from the former tocross-subsidize the latter. To show that it is able to do so, first note that∂ 2 Π MFI * − 2 p k ( β − β )( p k − p k )== 0 , so the profits of the MFI in equilibrium are an increasing andpv∂k 2

linear function of k . Since we have assumed that k̂ refers to the borrower who weakly satisfies a zeroprofit condition for the MFI under competition, Π MFI * (kˆi ) ≥ 0 . The linearity of the profit function
combined with the uniform distribution of k implies that Π * (k iˆ ) + Π * (k iˆ +1 ) ≥ −(Π * (k iˆ −1 )) , and since
Assumption 2 tells us that there exist one unprofitable borrower and more than one profitable borrower,
1 < iˆ < n , so kiˆ+1 and k iˆ−1 exist and thus cross-subsidization of at least k iˆ −1 is possible.

34

n(iv). Let S + ≡ ∑ Π *i (k i ) + G be the surplus profits from lending to profitable borrowers plus the
i =iˆ

m

subsidy received by the MFI. In addition, let S m = S + + ∑ Π *i (k iˆ − j ) for m = 1, . ., iˆ − 1 be the finite,
j =1

decreasing sequence of total profits from lending to successive borrowers below k̂ . The maximum
number of borrowers n * , and hence the equilibrium, will be reached when S m is as close to zero as
possible while remaining non-negative. Consider three cases: (a) If S m = 0 for some m * , then the BC
and the PCs all bind in equilibrium, and n * = n − iˆ + m * + 1 . For simplicity of exposition we haveassumed case (a) to be operative in our analysis. The alternative cases do not affect the fundamental
insights of our model, but are nevertheless interesting. (b) If S m > 0 for m * but S m < 0 for m * +1, thensome small residual profit exists due to the discrete distribution of k i . In this case an infinite number of
equilibrium contracts exist in which this residual profit may be distributed within the portfolio, andeither the BC or some borrowers’ PCs do not bind, but m * and hence n* remain unchanged. (c) If
S m > 0 when n * = n , the MFI has surplus funds even after reaching the entire group of potential
borrowers. Note, however, even in this case n * = n remains unique regardless of the rule used to[]dispose of excess profits. Thus for any given MFI subsidy G ≥ 0, ∀i ∈ 1, iˆ − m * − 1 receive themonopolistic moneylender contract (Vi , ri ) = (Vi m , ri m ) , and no loan contract from the MFI, and[]∀i ∈ iˆ − m * , n receive the MFI loan contract (Vi , ri ) = (Vi* , ri* ) and no loan from the moneylender.
PROOF OF PROPOSITION 3: Consider competition between two client-maximizing MFIs, h ={1, 2}
with non-targeted subsidies G1 and G2, respectively. A best response by MFIh to any an existing subsidy
pattern involves a re-allocation of subsidy over B so as to minimize the subsidy cost of capturing eachborrower, subject to the PC and BC. Therefore, a Nash equilibrium must satisfy the following~for ∀i, j ∈ B P ∪ B NP and h ={1, 2}, or that the loss fromnecessary conditions: (i) Π iMFI = Π MFI
j35

capturing all n1* + n2* borrowers in the total portfolios of 1, 2 must be equal. (Suppose not: If
Π iMFI > Π MFI
and i and j belong to the portfolio of different MFIs, this implies that MFIh with i canjrelease i and capture j from MFI~h while relaxing its BC; if i and j both belong to MFIh, then this implies
that Π iMFI or Π MFI
must be > or < Π MFI, where borrower x is some borrower in the portfolio ofjx
MFI~h. If Π iMFI or Π MFI> Π MFI, then borrower x can be captured from MFI~h while relaxing the BCjxof MFIh. If Π iMFI or Π MFI< Π MFI, then either i or j can be captured from MFIh by MFI~h, thusjx~
relaxing the BC of MFI~h.) (ii) The PC must be binding for the poorest borrower i ∈ B P ∪ B NP , wherei = ik~ . (Suppose not: If the PC is not satisfied, then this is a contradiction because ik~ cannot be in the~
total MFI portfolio, B P ∪ B NP . If the PC is satisfied, but non-binding for ik~ , then by LEMMA 2,borrower ik~ -1 can be captured for a lower subsidy cost than ik~ by either MFIh or MFI~h (assuming
sufficiently small ξ), thus relaxing the BC of either MFI.) (iii) The BC must bind for both 1, 2.
(Suppose not: In this case, an additional borrower can be captured by MFIh if its BC is not binding, andn1* and n2* are not maximized.) The unique Nash equilibrium that satisfies all three necessary conditions~
is the subsidy pattern in which n1* + n2* borrowers in B P ∪ B NP receive the subsidy equal to Π iMFI ,i = ik~ , so that G 1 + G 2 = (n1* + n2* ) ⋅ Π iMFI . The subsidy allocation is a Nash equilibrium since thesubsidy patterns by 1, 2 constitute a mutual best response; it is a unique Nash equilibrium because it is
the only subsidy pattern that satisfies the necessary conditions, (i),(ii), and (iii), for a Nash equilibrium.
~
G + G2Thus each borrower i ∈ B P ∪ B NP will receive a subsidy equal to *1
and MFIh will lend to
n1 + n 2*()

G h n1* + n2*
Ghborrowers, and the share of borrowers for MFIh will be
as seen in Figure 3,
G1 + G2
G1 + G2

where the specific types of borrower i served by each institution are undetermined.

36

PROOF OF PROPOSITION 5: (For simplicity, and without loss of generality, consider the case in which
G = 0 for all competing MFIs.) Let the set of possible contracts {Vi , ri }α , for some borrower i ∈ B P

satisfy Π iMFI (Vi , ri ) = 0 α , or zero MFI profits at information-sharing level α. Note that the Bertrand
equilibrium MFI contract {Vi** , ri** }α ∈ {V i , ri }α , and that by LEMMA 1 the Bertrand equilibrium contract
is Pareto efficient. First we show that as α declines to α′, we have Π iMFI (Vi , ri ) < 0 for all {V i , ri }α .
Partially differentiating the right-hand-side of (12) we obtain

[

[

]

[

]]




∂Π iMFI
1
~
~
~
~
~
which
= (1 − γα ) γ α Π − αγ α Π − γ α Π − γΠ + (αγ α + γ ) Π − γΠ + γΠ − αγΠ
∂α(1 − γα )2

[(

)

(




 ~
~ 
~
~
~
~
~
= γ α Π − Π − αΠ − γαΠ + γα 2 Π + γαΠ + αΠ − γαΠ + γαΠ − γα 2 Π + γ (1 − γ ) Π − Π)]

1(1 − γα )2and through algebraic manipulation and cancellation of terms, reduces to

(

)

 ~
∂Π iMFIΠ−Π= [γ (1 − γ ) − γ α (1 − α )]> 0 . Thus as α falls to α′, we have Π iMFI (Vi , ri ) < 0 for all {V i , ri }α ,2∂α(1 − γα )and therefore Π iB (Vi** , ri** ) α cannot be a Bertrand equilibrium contract for borrower i at α′, i.e.

{V , r }α ∉ {Vi , ri }α ' . Second, we show that the Bertrand equilibrium contract Π (V , r ) = 0 α ' isless favorable to the borrower than Π (V , r ) α . Totally differentiating the left-hand-side of (12)
**

i

**

B
i

i

B
i

(

**

i

**

i

**

i

**

i

)



pˆ 1 + ri − β( p − ~p ) < 0 from∂pˆ
dryields i = α< 0, noting that pˆ α ≡= [γ (1 − γ ) − γ α (1 − α )]∂α(1 − pˆ i )dα(1 − γα )2differentiation of (11). Thus we know that Π iMFI (Vi** , ri** ) = 0 α ' is characterized by a higher ri for any
given level of Vi relative to the {V i , ri }α pair that satisfies Π iMFI (Vi** , ri** ) = 0 α . Noting that borrower
profits are monotonically decreasing in ri, i.e. that(

Π iB Vi ** , ri**

∂Π iB
= −Vi (1 − p i ) < 0 , it must be true that∂ri

) α ′ is less than Π (V , r ) α since under α′, borrower i ∈ B P pays a strictly higher
B
i

**

i

**

i

interest rate ri for any level of borrowed capital Vi at α′ than at information-sharing level α > α′.

37

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39