Lehner, Maria
Working Paper
Group versus Individual Lending in Microfinance
Munich Discussion Paper, No. 2008-24
Provided in Cooperation with:
University of Munich, Department of Economics
Suggested Citation: Lehner, Maria (2008) : Group versus Individual Lending in Microfinance, Munich
Discussion Paper, No. 2008-24, Ludwig-Maximilians-Universität München, Volkswirtschaftliche
Fakultät, München,
https://doi.org/10.5282/ubm/epub.7486
This Version is available at:
https://hdl.handle.net/10419/104267
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Group Lending versus Individual Lending in
Micro…nance
Maria Lehner
University of Munich
Septemb er 2008
Abstract
Micro…nance is typically associated with joint liability of group members. How-
ever, a large part of micro…nance institutions rather o¤ers individual instead of
group loans. We analyze the incentive mechanisms in both individual and group
contracts. Moreover, we show that micro…nance institutions o¤er group loans when
the loan size is rather large, re…nancing costs are high, and competition between
micro…nance institutions is low. Otherwise, individual loans are o¤ered. Inter-
estingly, our analysis predicts that individual lending in micro…nance will gain in
importance in the future if micro…nance institutions continue to get better access
to capital markets and if competition further rises.
JEL classi…cation: F37, G21, G34, L13, O16
Keywords: micro…nance, group lending, individual lending
University of Munich, Akademiestr. 1/III, 80799 Munich, Germany. Tel: +49 89 2180 2766. Email:
[email protected]uenchen.de
1 Introduction
In 2006, the Nobel Peace Prize was awarded to Mohammad Yunus. Since he founded the
Grameen Bank in Bangladesh in the late 1970s, micro…nance has experienced an impres-
sive growth. This is largely due to the many positive e¤ects attributed to micro…nance
programs. Micro…nance schemes have been found to reduce poverty and to positively
a¤ect nutrition, health and education as well as gender emp owerment (Little…eld et al.
(2003)). In 2006, micro…nance institutions reached around 130 million customers around
the world (Daley-Harris (2007)).
Typically, micro…nance is associated with joint liability lending. When borrowers
form groups and are held liable for each other, lending to the poor can be pro…table
even if borrowers do not possess any collateral and lack a credit history. Interestingly,
however, a large part of micro…nance institutions does not o¤er group but individual
loans. This gives rise to several questions: what are the incentive mechanisms that play
a role in individual and group lending schemes and how do they di¤er? Under which
circumstances do micro…nance institutions o¤er group or individual loan contracts? What
are the di¤erences between individual lending programs of micro…nance institutions and
the individual lending technology applied by commercial banks?
According to Giné and Karlan (2006), the di¤erent features of group and individual
lending schemes have not yet been studied in detail "despite being a question of …rst-order
imp ortance".
1
With the aim to contribute to a theoretical foundation of this topic, we
set up a model of spatial competition between micro…nance institutions. Micro…nance
institutions o¤er either group or individual loans and compete in the repayments they
charge their clients. Borrowers di¤er in their success probabilities and lack pledgeable
collateral. As borrowers have no documented credit history, they are unable to provide
hard information.
Consequently, in contrast to commercial banks, micro…nance institutions cannot screen
borrowers and secure loans with collateral. Screening borrowers is feasible only when a
relatively standardized evaluation procedure based on the analysis of hard information
such as balance sheet data is applicable. In addition, any lending strategy pursued by
micro…nance institutions must ensure monitoring of borrowers in order to prevent the
diversion of loans to urgent consumption needs.
When a micro…nance institution opts for the group lending technology, it transfers the
monitoring role to borrowers. Joint liability ensures strong incentives of group members
1
Giné and Karlan (2006, p. 3)
1
to monitor each other in order to make their peers succeed. Furthermore, self-selection of
borrowers into di¤erent credit contracts can be achieved.
In case it grants individual loans, the micro…nance institution specializes in closely
monitoring clients. Borrowers are o¤ered a pooling contract. However, borrowers are
exempt from negative e¤ects of group lending schemes such as bearing additional risk,
loss of privacy from disclosing their …nancial situation and investment projects to potential
peers, or time spent on group meetings.
Our …rst focus of interest lies on how the decision of a micro…nance institution to o¤er
either group or individual loans depends on the size of a loan. Controversial arguments are
brought forward in the so far rather descriptive literature on this topic. For instance, Kota
(2007) and Harper (2007) state that micro…nance institutions o¤er individual contracts if
clients are in need for larger loans. In contrast, Giné and Karlan (2006) advocate precisely
the reverse correlation. Our analysis aims to contribute to a theoretical foundation of this
discussion.
Another major focus of our study is to investigate how the choice of lending technology
depends on re…nancing conditions and competitive pressure in the micro…nance market.
According to Isern and Porteous (2005) as well as Reddy and Rhyne (2006), the world
of micro…nance currently changes substantially in both these respects. The emergence of
rating agencies specializing in the micro…nance business and the growing awareness re-
garding the potential of the micro…nance industry makes investors channel more and more
funds into this market. Enhanced access to capital markets, in turn, implies improved
re…nancing conditions for micro…nance institutions. In addition, competition among mi-
cro…nance banks steadily intensi…es. Our analysis provides a theoretical framework that
allows us to study in detail how changes in re…nancing conditions and competition a¤ect
a micro…nance institution’s lending strategy.
Interestingly, our results show that micro…nance institutions decide to o¤er individual
loans when the loan size is rather small. Moreover, micro…nance institutions favor individ-
ual over group contracts when re…nancing costs are low and when competition is intense.
Hence, our analysis allows for some interesting predictions concerning the future shape
of the micro…nance industry. Given that access to capital markets continues to improve
and competition between micro…nance institutions rises further, our results imply that
individual loan contracts in the micro…nance market will gain in importance over the next
years.
The remainder of this paper is organized as follows. The next section reviews the
literature. Section 3 describes the set-up of the model. In section 4, we study the choice
2
of lending technology of micro…nance institutions. We present our comparative statics
analysis in section 5. Empirical hypotheses are stated in section 6. Section 7 concludes.
2 Related Literature
Although individual loans account for a large portion of micro…nance loans, the literature
is heavily biased towards an analysis of group loan contracts. Individual lending schemes
have only very recently attracted the interest of researchers.
Numerous theoretical papers have addressed the positive e¤ects of group lending mech-
anisms. Ghatak and Guinnane (1999), Ghatak (2000) as well as Van Tassel (1999) show
that group lending achieves self-selection of borrowers and acts as a screening device.
Armendáriz de Aghion and Gollier (2000) …nd that even if borrowers do not know each
other’s type, group lending may b e feasible due to lower interest rates as a result of cross
subsidization of borrowers. Stiglitz (1990) outlines the role of peer-monitoring in group
lending schemes, which transfers the monitoring role from the bank to the borrowers and
acts as an incentive device. Armendáriz de Aghion (1999) demonstrates that the bene…ts
from peer monitoring are largest when risks are positively correlated among borrowers.
La¤ont and N’Guessan (2000) conclude that social connections facilitate the monitoring
and enforcement of joint liability loan contracts. This result has been con…rmed in an
empirical study by Karlan (2007). Furthermore, Armendáriz de Aghion and Morduch
(2000) point to a fall in transaction costs when - instead of individual visits of clients -
group meetings are held. In addition, the contact with banks to which poor borrowers
typically are not used to is facilitated.
However, certain drawbacks of group lending exist. Giné and Karlan (2006) state
that the demand for credit within a group may change over time, forcing clients with
small loans to be liable for larger loans of their peers. Furthermore, the growth of group
lending programs may slow down when new borrowers with looser social ties enter and,
consequently, the group lending technology loses some of its power. Besley and Coate
(1995) stress negative welfare e¤ects if the group as a whole defaults even if some members
had repaid under individual lending. In a case study, Montgomery (1996) outlines the
unnecessary social costs of repayment pressure. Stiglitz (1990) points to the higher risk
borrowers assume when they are not only liable for themselves but also for their group
partners.
The so far rather descriptive literature on individual lending schemes typically focuses
on the crucial role of closely monitoring borrowers. Navajas et al. (2003), Armendáriz
3
de Aghion and Morduch (2005) as well as Giné et al. (2006) describe the problem that
poor borrowers may divert a loan, at least partly, to urgent consumption needs. In order
to ensure the use of the loan for the agreed upon investment project, Champagne et al.
(2007) as well as Zeitinger (1996) stress the importance of regularly visiting clients. In a
theoretical analysis of individual lending schemes by Gangopadhyay and Lensink (2007),
the monitoring of borrowers by informal lenders plays a central role. Armendáriz de
Aghion and Morduch (2000) as well as Dellien et al. (2005) also point to the importance
of monitoring borrowers in individual lending schemes.
Only recently, researchers have been interested in comparing group lending programs
to individual lending schemes. Giné and Karlan (2006) conduct a …eld experiment in the
Philippines. They …nd that by o¤ering individual loans, a micro…nance institution can
attract relatively more new clients. Yet, both lending schemes do not di¤er in repayment
rates. In a recent empirical study, Ahlin and Townsend (2007) …nd a U-shaped rela-
tionship between joint liability contracts relative to individual contracts and a borrower’s
wealth. Furthermore, they conclude that higher correlation across projects makes group
lending contracts more likely relative to individual contracts. In her theoretical analysis,
Madajewicz (2008) shows that, in general, borrowers prefer individual loans the wealthier
they are. Nevertheless, she demonstrates that for very low levels of borrower wealth,
group loans are larger than individual loans. Moreover, she …nds that businesses funded
with individual loans grow more than those funded with group loans.
3 The Model
Consider a continuum of borrowers with mass 2 that is uniformly distributed along a
straigt line of length 1. Each borrower can engage in one investment project that requires
an initial outlay of i; i > 0. Borrowers are not endowed with any initial wealth and
therefore need to apply for credit at a micro…nance institution, the only source of …nance
in our model. Borrowers have either safe or risky projects. It is common knowledge that
the fraction of borrowers with safe projects is and the fraction of borrowers with risky
projects is 1 ; 0 < < 1. We assume that borrowers with safe and risky projects
are distributed with density 2 and 2 (1 ) along the Hotelling line, respectively. As a
result, two borrowers of the same type are located at a certain point of the Hotelling street.
As will be explained in more detail later on, this assumption ensures costless formation of
groups. Individual b orrowers know about the type of their own and the other borrowers’
investment projects. In case a project is successful it generates a return of v > 0 and
zero otherwise. The success probability of safe and risky projects is given by p
S
and p
R
,
4
respectively, with 0 < p
R
< p
S
< 1. The returns of the projects are observable and
contractible. We assume that b orrowers must be monitored closely in order to prevent
the diversion of the loan to consumption needs.
2
The …nancial sector serving the borrowers consists of two representative micro…nance
institutions A and B that are located at the two ends of the Hotelling line. In our model,
both micro…nance institutions are pro…table and comp ete with each other. Note that
the pro…tability of micro…nance institutions has risen considerably over the last few years
(Christen and Cook (2001)). Some micro…nance institutions are now even listed at stock
exchanges, such as Compartamos in Mexico or Equity Bank in Kenya. Furthermore, due
to the immense growth of the micro…nance industry, in many countries, there is now …erce
competition between micro…nance institutions (Fernando (2007), McIntosh et al. (2005),
and Christen and Rhyne (1999)).
Micro…nance institutions A and B compete in the repayments they simultaneously
ask from borrowers. Micro…nance institutions incur re…nancing costs c > 0 per loan of
size i. We take it as given that each micro…nance institution disposes of enough funds to
…nance all borrowers applying for a loan. Micro…nance institutions do not know whether
borrowers have safe or risky projects.
A micro…nance institution may choose to o¤er either group or individual loans. We
abstract from the possibility that a bank o¤ers both group and individual contracts. In
fact, most micro…nance institutions o¤er either one or the other type of loan as is p ointed
out in Ahlin and Townsend (2007), Giné and Karlan (2006) as well as Madajewicz (2008).
If a micro…nance institution opts for the group lending technology, loans are o¤ered
to groups consisting of two borrowers each. Note that limiting the group size to two
borrowers is a standard assumption in the literature and greatly simpli…es our analysis
(see, for instance, Ghatak (2000) or La¤ont and N’Guessan (2000)). Group contracts
imply a transfer of the monitoring role to the group members. Due to joint liability,
group members have a strong incentive to monitor each other in order to ensure the
correct investment of the loan and to make their partners succeed. We assume that due
to close social ties between group members, borrowers monitor each other at zero cost.
3
The size of a loan a group receives is 2i so that each borrower receives an amount i of
the loan. In case both group members are successful, the loan is fully paid back with
interest. If both partners fail, no repayments are made. In the case that only one of
2
We could as well assume that borrowers divert a certain part of their loans for household needs if
not monitored. As a result, borrowers could only a¤ord low quality inputs for their investment projects
implying returns too low to pay b ack their loans.
3
One might argue that borrowers incur costs of monitoring. However, the monitoring costs should
clearly be lower for borrowers than for banks due to strong social ties between borrowers (Karlan (2007)).
By normalizing monitoring costs of borrowers to zero, our results remain qualitatively una¤ected.
5
the two group members is successful, the successful borrower pays back her part of the
loan plus interest and, in addition, the loan share of her peer with interest, weighed by
a joint liability parameter > 0. The joint liability parameter expresses the degree of
joint liability to which group members stand in for each other. Micro…nance institutions
compete both in interest rates and the joint liability parameters. Micro…nance institutions
can induce self-selection of borrowers by o¤ering two di¤erent contracts. A contract with
a low interest rate r
S
and a high degree of joint liability
S
will attract safe borrowers
whereas risky borrowers prefer a contract with a high interest rate r
R
and a low joint
liability factor
R
.
4
Borrowers incur some disutility d from group lending. The disutility
d captures drawbacks of group loans such as time spent on …nding a partner and group
meetings (Armendáriz de Aghion and Morduch (2000)), the higher risk b orrowers bear
due to joint liability (Stiglitz (1990)) or social costs of repayment pressure (Montgomery
(1996)). Borrowers may also su¤er from reduced privacy when disclosing details of their
investment project or their …nancial situation to their p eers (Harper (2007)).
In the case a micro…nance institution decides to o¤er individual loans, it has no mecha-
nism at hand to assess a borrower’s type. Note, …rst, that a collateralized contract cannot
be o¤ered since borrowers lack any pledgeable assets. Second, screening borrowers is not
an option as potential clients are unable to provide hard information. Hence, micro…-
nance institutions o¤er a pooling contract with repayment rate r
SR
per credit of size i.
In order to prevent the diversion of the loan for consumption needs once it is received,
micro…nance institutions need to closely monitor clients. This imposes a per borrower
cost of k on the micro…nance institution.
5
The crucial role of closely monitoring clients in
individual lending programs has been stressed, for instance, by Champagne et al. (2007)
as well as Zeitinger (1996). Armendáriz de Aghion and Morduch (2000) and Dellien et al.
(2005) con…rm the importance of regularly visiting clients in individual lending schemes.
Borrowers base their decision at which micro…nance institution to apply for credit on
the repayments r
j
, j = A; B, and the joint liability factors
j
asked by the micro…nance
institutions as well as on the transport costs they incur by travelling to a micro…nance
institution. We assume that transport costs tx are proportional to the distance x between
the borrower and the micro…nance institution. If b orrowers apply for a group contract,
transportation costs arise for both group members. Furthermore, we assume that the
return of a project v is high enough so that the market is covered at equilibrium prices.
Borrowers and micro…nance institutions are risk neutral and maximize pro…ts.
The time structure of the game is as follows. At stage 1, micro…nance institutions
4
See Ghatak (2000), Stiglitz (1990) or Van Tassel (1999) for a similar set-up.
5
Evidence for larger costs per loan in case of individual compared to group lending schemes is provided
by Giné and Karlan (2006). They …nd that credit o¢ cers spend more time per borrower when individual
contracts are o¤ered.
6
decide which lending technology to apply and simultaneously set interest rates and joint
liability parameters. At stage 2, borrowers decide at which institution to apply for credit
and form groups when applying for a group contract. At stage 3, returns realize and
borrowers make repayments if they have been successful. We solve the game by backward
induction.
4 Choice of Lending Technology
In this section, we derive the choice of lending technology of micro…nance institutions.
Both institutions can pursue the lending strategy "group loans" (G) or "individual loans"
(I). In order to solve the game we compare the pro…t of a micro…nance institution in
case it o¤ers group loans to the case it grants individual loans given that its competitor,
…rstly, o¤ers group loans and, secondly, concedes individual loans.
Both Micro…nance Institutions O¤er Group Loans
If both micro…nance institutions o¤er group contracts, borrowers form groups of two
borrowers each in order to apply for a loan. Before we turn to the pro…ts of micro…nance
institutions when they both o¤er group loans, we show that a borrower group always
consists of borrowers of the same type.
Consider, …rst, a safe borrower forming a group with another safe borrower. Any
contract with interest rate r and degree of joint liability gives the borrower a utility
U
S;S
= i + p
2
S
[v (1 + r) i] + p
S
(1 p
S
) [v (1 + r) i (1 + r) i] tx d: (1)
That is, the safe b orrower receives her part of the credit, i. With probability p
2
S
both
group members are successful so that the borrower receives return v and pays back her part
of the loan with interest (1 + r) i. With probability p
S
(1 p
S
) the borrower is successful
but her group partner is not. Then, the borrower pays back her part of the loan with
interest (1 + r) i and also stands in for her group partner with the amount (1 + r) i. If
the borrower is unsuccessful, her return is zero and she does not make any repayments to
the micro…nance institution. The borrower’s utility is reduced by the costs for travelling
to the micro…nance institution tx and the disutility related to group contracts d.
When the safe borrower has a risky partner, her utility from any contract with interest
rate r and degree of joint liability is given by
U
S;R
= i + p
S
p
R
[v (1 + r)i] + p
S
(1 p
R
)[v (1 + r)i (1 + r)i] tx d: (2)
7
Now, both group members are successful with probability p
S
p
R
. Then, the safe bor-
rower receives return v and pays back her part of the loan with interest (1 + r) i. With
probability p
S
(1 p
R
), the safe borrower is successful but her risky partner is not. In
that case, the safe borrower pays back her part of the loan with interest (1 + r) i and, in
addition, the amount (1 + r) i in lieu of her risky partner.
The di¤erence in a safe borrower’s utility stemming from the formation of a group
with a safe versus a risky borrower is given by U
S;S
U
S;R
= p
S
(p
S
p
R
) (1 + r) i. This
expression is clearly positive. Hence, a safe borrower always prefers to be part of a group
with a borrower of her own type.
Second, let us look at the preferences of a risky borrower concerning her partner. The
utility of a risky borrower when having a risky peer amounts to
U
R;R
= i + p
2
R
[v (1 + r) i] + p
R
(1 p
R
) [v (1 + r) i (1 + r) i] tx d: (3)
With probability p
2
R
both group members are successful. The borrower receives return
v and pays back her part of the loan with interest (1 + r) i. With probability p
R
(1 p
R
)
the borrower is successful but her partner is not. Then, the borrower pays back her part
of the loan with interest (1 + r) i and, in addition, she stands in for her peer with the
amount (1 + r) i.
When a risky borrower forms a group with a safe borrower, she attains the utility level
U
R;S
= i + p
R
p
S
[v (1 + r)i] + p
R
(1 p
S
)[v (1 + r)i (1 + r)i] tx d: (4)
Now, projects of both borrowers turn out to be successful with probability p
R
p
S
.
In that case, the risky borrower pays back her part of the loan with interest (1 + r) i.
With probability p
R
(1 p
S
), only the risky borrower is successful. Then, the risky b or-
rower pays back her part of the loan with interest (1 + r) i and, in addition, the amount
(1 + r) i for her partner.
The di¤erence in the utility of a risky borrower when being part of a group with a
risky versus a safe borrower amounts to U
R;R
U
R;S
= p
R
(p
S
p
R
) (1 + r) i. As this
expression is negative, a risky borrower clearly prefers to have a safe borrower as her part-
ner. However, as safe b orrowers prefer to form groups with safe borrowers, risky borrowers
will not …nd a safe borrower willing to form a group with them. As a consequence, risky
borrowers form groups with partners of their own type as well. Note that our assumption
concerning the density of the borrowers’distribution along the Hotelling line ensures that
two borrowers of the same type are located at a certain point of the Hotelling street.
Since we have shown that borrowers form groups with borrowers of their own type, we
8
can abstract from costs related to the formation of groups such as costs of searching for
a partner.
Let us now turn to the pro…ts micro…nance institutions achieve when they both o¤er
group contracts. Micro…nance institutions can induce self-selection of borrowers according
to their types into two di¤erent kinds of contracts. Safe borrowers will accept a contract
de…ned by a low interest rate r
S
and a high degree of joint liability
S
. In contrast, risky
borrowers prefer a loan contract based on a high interest rate r
R
and a low joint liability
factor
R
. Both interest rates and the joint liability factors are set endogenously.
When borrowers decide about where to apply for credit, they compare the utility they
obtain when borrowing from micro…nance institution A to the utility level they achieve
when accepting a loan from micro…nance institution B. The resulting marginal borrowers
in the segment of safe and risky borrowers x
S
(G; G) and x
R
(G; G), respectively, determine
the micro…nance institutions’pro…ts as given b elow. Note that throughout this pap er,
the …rst letter in brackets stands for the strategy pursued by micro…nance institution A
and the second letter for the strategy followed by micro…nance institution B.
A
(G; G) = x
S
(G; G)[2p
2
S
(1 + r
A
S
) + 2p
S
(1 p
S
)(1 +
A
S
)(1 + r
A
S
) 2(1 + c)]i+
(1 )x
R
(G; G)[2p
2
R
(1 + r
A
R
) + 2p
R
(1 p
R
)(1 +
A
R
)(1 + r
A
R
) 2(1 + c)]i (5)
B
(G; G) = [1 x
S
(G; G)][2p
2
S
(1 + r
B
S
) + 2p
S
(1 p
S
)(1 +
B
S
)(1 + r
B
S
) 2(1 + c)]i+
(1 )[1 x
R
(G; G)][2p
2
R
(1 + r
B
R
) + 2p
R
(1 p
R
)(1 +
B
R
)(1 + r
B
R
) 2(1 + c)]i. (6)
Due to our assumption concerning the distribution of borrowers, in the case of group
lending, a certain point on the Hotelling line represents a group consisting of two bor-
rowers. Hence, micro…nance institution A serves x
S
(G; G) safe and (1 ) x
R
(G; G)
risky clients. With probability p
2
S
, both members of a group of safe borrowers succeed
so that the micro…nance institution receives 2
1 + r
A
S
i. With probability 2p
S
(1 p
S
),
the project of only one group member turns out to be successful. Then, the successful
borrower stands in for her partner and repays the amount
1 +
A
S
1 + r
A
S
i. No re-
payments are made if both group members fail which happens with probability (1 p
S
)
2
.
Micro…nance institutions incur re…nancing costs 2 (1 + c) i per group of borrowers. Similar
considerations hold for the market share the micro…nance institution holds in the segment
of risky borrowers. The pro…t of micro…nance institution B is derived analogously.
Both micro…nance institutions maximize their pro…t with respect to the interest rates
and the degree of joint liability they demand from the two types of borrowers. The
resulting equilibrium pro…ts are stated in Lemma 1.
9
Lemma 1 If both micro…nance institutions o¤er group loan contracts, equilibrium pro…ts
of micro…nance institutions are given by
A
(G; G) =
B
(G; G) = t: (7)
Proof: see Appendix.
Both Micro…nance Institutions O¤er Individual Loans
We now analyze the situation in which both micro…nance institutions o¤er individual
loans. Borrowers compare the utility they achieve when borrowing from micro…nance
institution A to the utility they obtain when funded by micro…nance institution B. The
resulting marginal borrowers in the segment of safe and risky borrowers x
S
(I; I) and
x
R
(I; I), respectively, determine the pro…ts of micro…nance institutions given as follows:
A
(I; I) = 2x
S
(I; I) [p
S
1 + r
A
SR
i (1 + c) i k]+
2 (1 ) x
R
(I; I) [p
R
1 + r
A
SR
i (1 + c) i k] (8)
B
(I; I) = 2[1 x
S
(I; I)][p
S
1 + r
B
SR
i (1 + c) i k]+
2 (1 ) [1 x
R
(I; I)][p
R
1 + r
B
SR
i (1 + c) i k]. (9)
Due to our assumptions concerning the borrowers’ distribution, a certain point on
the Hotelling line represents two borrowers in the case of individual lending. Hence,
micro…nance institution A serves 2x
S
(I; I) safe and 2 (1 ) x
R
(I; I) risky borrowers.
Micro…nance institution A charges the pooled lending rate r
A
SR
to both safe and risky
clients. It receives the amount
1 + r
A
SR
i from safe borrowers with probability p
S
and
from risky borrowers with probability p
R
. Monitoring clients amounts to a per borrower
cost of k. Micro…nance institutions incur re…nancing costs of (1 + c) i per client. The
pro…t of micro…nance institution B is derived analogously.
Both micro…nance institutions maximize their pro…t with respect to the interest rates
they charge borrowers. Our results are stated in Lemma 2.
Lemma 2 If both micro…nance institutions o¤er individual loan contracts, equilibrium
pro…ts of micro…nance institutions are given by
A
(I; I) =
B
(I; I) =
t[p
S
+ p
R
(1 )]
2
(p
S
p
R
)
2
(1 )[k + (1 + c)i]
p
2
S
+ (1 )p
2
R
: (10)
10
Proof: see Appendix.
Micro…nance Institution A O¤ers Individual Loans and Micro…nance Institu-
tion B O¤ers Group Loans
Let us now turn to the situation in which micro…nance institution A o¤ers individual
loan contracts and micro…nance institution B o¤ers group loans. The marginal borrowers
in the safe and risky market segment x
S
(I; G) and x
R
(I; G), respectively, determine the
pro…ts of micro…nance institutions. Analogous to our reasoning above, pro…ts of both
micro…nance institutions are now given as follows:
A
(I; G) = 2x
S
(I; G)[p
S
(1 + r
A
SR
)i (1 + c)i k]+
2(1 )x
R
(I; G)[p
R
(1 + r
A
SR
)i (1 + c)i k] (11)
B
(I; G) = [1 x
S
(I; G)][2p
2
S
(1 + r
B
S
) + 2p
S
(1 p
S
)(1 +
B
S
)(1 + r
B
S
) 2(1 + c)]i+
(1 )[1 x
R
(I; G)][2p
2
R
(1 + r
B
R
) + 2p
R
(1 p
R
)(1 +
B
R
)(1 + r
B
R
) 2(1 + c)]i. (12)
Both micro…nance institutions set interest rates and micro…nance institution B, in
addition, the joint liability factors in order to maximize pro…t. The resulting equilibrium
pro…ts are stated in Lemma 3.
Lemma 3 If micro…nance institution A o¤ers individual loan contracts and micro…nance
institution B o¤ers group loans, equilibrium pro…ts of both micro…nance institutions are
given by
A
(I; G) =
2[p
S
+p
R
(1)]
2
(dk+3t)
2
9(p
S
p
R
)
2
(1)[k+(1+c)i][d+3t+(1+c)i]
18t[p
2
S
+(1)p
2
R
]
(13)
B
(I; G) =
4[p
S
+p
R
(1)]
2
(dk3t)
2
+9(p
S
p
R
)
2
(1)[dt+(1+c)i]
2
36t[p
2
S
+(1)p
2
R
]
. (14)
Proof: see Appendix.
Micro…nance Institution A O¤ers Group Loans and Micro…nance Institution
B O¤ers Individual Loans
Clearly, this case is symmetric to the situation described before. For the sake of com-
pleteness, the equilibrium pro…ts of both micro…nance institutions are stated in Lemma
4.
Lemma 4 If micro…nance institution A o¤ers group contracts and micro…nance insti-
tution B o¤ers individual loans, equilibrium pro…ts of both micro…nance institutions are
given by
11
A
(G; I) =
4[p
S
+p
R
(1)]
2
(dk3t)
2
+9(p
S
p
R
)
2
(1)[dt+(1+c)i]
2
36t[p
2
S
+(1)p
2
R
]
(15)
B
(G; I) =
2[p
S
+p
R
(1)]
2
(dk+3t)
2
9(p
S
p
R
)
2
(1)[k+(1+c)i][d+3t+(1+c)i]
18t[p
2
S
+(1)p
2
R
]
: (16)
Proof: analogous to proof of Lemma 3.
Nash Equilibrium
We now turn to the Nash equilibrium in this market. We determine micro…nance
institution A’s best response both given that micro…nance institution B o¤ers group
loans and individual contracts. With respect to micro…nance institution B, we proceed
analogously. Due to reasons of symmetry, we limit our exposition to the point of view of
micro…nance institution A. The matrix of the game is given in Figure 1.
A
B
group loans individual loans
group loans
A
(G; G);
B
(G; G)
A
(G; I);
B
(G; I)
individual loans
A
(I; G);
B
(I; G)
A
(I; I);
B
(I; I)
Figure 1: Matrix of the Game
Given that micro…nance institution B o¤ers group loans, micro…nance institution A
o¤ers individual contracts if
A
(I; G)
A
(G; G) > 0 holds. Our results are stated in
Proposition 1.
Proposition 1 Given that micro…nance institution B o¤ers group loans, micro…nance
institution A o¤ers individual loan contracts if
A
(I; G)
A
(G; G) > 0 holds. That is, if
9 (p
S
p
R
)
2
(1 ) [(k + i + ci) (d + i + ci) + (3k + 2t + 3i + 3ci) t]
2 (p
R
+ p
S
p
R
)
2
(d k) (d k + 6t) < 0. (17)
Proof: straight forward.
Given that micro…nance institution B o¤ers individual loans, micro…nance institution
A o¤ers individual contracts if
A
(I; I)
A
(G; I) > 0 holds. Our results are given in
Proposition 2.
12
Proposition 2 Given that micro…nance institution B o¤ers individual loans, micro…-
nance institution A o¤ers individual contracts if
A
(I; I)
A
(G; I) > 0 holds. That is,
if
9 (p
S
p
R
)
2
(1 ) [(d + i + ci)
2
+ (4k 2d + t + 2i + 2ci) t]
4 (p
R
+ p
S
p
R
)
2
(6t d + k) (d k) < 0. (18)
Proof: straight forward.
We cannot unambiguously determine whether the above two expressions are positive
or negative. That is why we now turn to a comparative statics analysis. In doing so,
we gain interesting insights in how the choice of lending technology is in‡uenced by the
size of credit, the re…nancing conditions of micro…nance institutions and the competitive
pressure of the market environment.
5 Comparative Statics Analysis
The …rst focus of our comparative statics analysis lies on the impact of the loan size for
a micro…nance institution’s decision to grant individual or group loans. Controversial
arguments are brought forward in the so far rather descriptive literature on this topic.
For instance, Kota (2007) and Harper (2007) state that micro…nance institutions o¤er
individual contracts if clients are in need for larger loans. In contrast, Giné and Karlan
(2006) advocate precisely the reverse correlation. Our analysis aims to contribute to a
theoretical foundation of this discussion.
We are further interested in how a micro…nance institution’s choice of lending tech-
nology depends on re…nancing conditions and competitive pressure in the micro…nance
market. According to Isern and Porteous (2005) as well as Reddy and Rhyne (2006),
the world of micro…nance currently changes substantially in both these respects. Micro…-
nance institutions get increasingly better access to capital markets which should transform
into improved re…nancing conditions. In addition, competition among micro…nance banks
steadily intensi…es, in large part due to the enormous growth of the industry. Our analysis
provides a theoretical framework that allows us to study in detail how changes in re…-
nancing conditions and competition a¤ect a micro…nance institution’s lending strategy.
13
5.1 Size of Credit
When we look at the role of the loan size, interestingly, we …nd that a micro…nance
institution prefers to o¤er individual contracts when the size of credit is rather small,
irrespective of whether its competitor grants individual or group loans. Conversely, when
a loan is relatively large, micro…nance institutions favor the group lending technology.
Our results are stated in Proposition 3.
Proposition 3 Micro…nance institutions o¤er individual contracts when a loan is rather
small. Group contracts are preferred by micro…nance institutions when a loan is rather
large. That is,
A
(I; G)
A
(G; G) > 0 if i < i
1
and
A
(I; G)
A
(G; G) < 0 if i > i
1
(19)
A
(I; I)
A
(G; I) > 0 if i < i
2
and
A
(I; I)
A
(G; I) < 0 if i > i
2
(20)
B
(G; I)
B
(G; G) > 0 if i < i
1
and
B
(G; I)
B
(G; G) < 0 if i > i
1
(21)
B
(I; I)
B
(I; G) > 0 if i < i
2
and
B
(I; I)
B
(I; G) < 0 if i > i
2
. (22)
Proof: see Appendix.
The intuition for this result is as follows. Consider, …rst, the situation of micro…nance
institution A given that micro…nance institution B o¤ers individual loans. If micro…nance
institution A o¤ers group contracts instead, it can charge lending rates according to
the borrowers’types implying a relatively larger interest burden for risky than for safe
borrowers. Since the interest repayments increase proportionally to the loan size, the
advantage of safe relative to risky borrowers under group contracts is the larger, the larger
the size of a loan is. Hence, the larger the loan, the more attractive group contracts of
micro…nance institution A become to safe borrowers and the less attractive such contracts
get for risky borrowers. Put di¤erently, an increasing loan size improves the quality of
the borrower pool of micro…nance institution A when it o¤ers group contracts relative
to micro…nance institution B. Therefore, the larger a loan is, the more micro…nance
institution A favors the group over the individual lending technology.
Consider now the situation of micro…nance institution A given that micro…nance insti-
tution B o¤ers group contracts. If micro…nance institution A o¤ers individual loans, its
pool of borrowers worsens with an increasing loan size as micro…nance institution B be-
comes relatively more attractive for safe and less attractive for risky borrowers analogous
to the reasoning above. Hence, the larger a loan is, the more micro…nance institution A
prefers to o¤er group contracts.
Consequently, irrespective of whether micro…nance institution B o¤ers group or in-
dividual loans, the group lending technology becomes more attractive for micro…nance
14
institution A with an increasing loan size. Analogous arguments hold for micro…nance in-
stitution B. Accordingly, a Nash equilibrium in which both micro…nance institutions o¤er
group contracts tends to emerge when the size of a credit is rather large. With a rather
small loan size, an equilibrium in which both micro…nance institutions o¤er individual
loans is more likely to result.
Our …ndings contradict the point of view of authors such as Kota (2007) and Harper
(2007). In a theoretical analysis, Madajewicz (2008) shows that individual loans tend to
be larger than group loans. However, her result only holds for borrowers that already
have accumulated a certain level of wealth. For low levels of individual wealth, she
demonstrates that group loans are larger than individual loans. In line with our results,
Giné and Karlan (2006) conclude from their empirical study that the loan size is smaller
for individual than for group contracts. However, their argument is somewhat di¤erent.
They state that when credit o¢ cers concede individual loans, they alone assume the
monitoring role and, thus, bear a higher responsibility. This is why they argue that credit
o¢ cers may be stricter on the size of individual loans.
5.2 Re…nancing Conditions
We now turn to the impact of re…nancing conditions on a micro…nance institution’s choice
of lending technology. We …nd that when re…nancing costs are relatively low, a micro-
…nance institution favors individual over group contracts, irrespective of the behavior of
its competitor. Conversely, group loans tend to be preferred when re…nancing costs are
rather high. Our results are stated in Proposition 4.
Proposition 4 Micro…nance institutions o¤er individual contracts when re…nancing costs
are rather low. Group loans are preferred in the presence of rather high re…nancing costs.
That is,
A
(I; G)
A
(G; G) > 0 if c < c
1
and
A
(I; G)
A
(G; G) < 0 if c > c
1
(23)
A
(I; I)
A
(G; I) > 0 if c < c
2
and
A
(I; I)
A
(G; I) < 0 if c > c
2
(24)
B
(G; I)
B
(G; G) > 0 if c < c
1
and
B
(G; I)
B
(G; G) < 0 if c > c
1
(25)
B
(I; I)
B
(I; G) > 0 if c < c
2
and
B
(I; I)
B
(I; G) < 0 if c > c
2
. (26)
Proof: see Appendix.
The intuition for this result is as follows. Consider, …rst, the situation of micro…nance
institution A given that micro…nance institution B o¤ers individual loans. If micro…nance
15
institution A o¤ers group contracts instead, it can adjust lending rates to the borrowers’
types. When re…nancing costs increase, lending rates of both micro…nance institutions go
up. However, rising re…nancing costs drive up the lending rate that micro…nance institu-
tion A charges risky borrowers more than the rate it o¤ers to safe borrowers. This makes
micro…nance institution A relatively less attractive for risky and relatively more attractive
for safe borrowers relative to the po oled rate charged by micro…nance institution B. It
follows that increasing re…nancing costs improve the quality of micro…nance institution
A’s po ol of borrowers when it o¤ers group contracts. Hence, the higher the re…nancing
costs are, the more micro…nance institution A prefers to o¤er group loans.
Consider now the situation of micro…nance institution A given that micro…nance insti-
tution B o¤ers group contracts. If micro…nance institution A o¤ers individual loans, its
pool of borrowers worsens with increasing re…nancing costs as micro…nance institution B
becomes relatively more attractive for safe and less attractive for risky borrowers. Hence,
the higher the re…nancing costs are, the more micro…nance institution A favors to o¤er
group loan contracts.
Consequently, irrespective of whether micro…nance institution B o¤ers group or in-
dividual loans, the group lending technology becomes more attractive for micro…nance
institution A when re…nancing costs increase. Clearly, the same arguments apply to
micro…nance institution B. Thus, a Nash equilibrium in which both micro…nance insti-
tutions o¤er group contracts tends to emerge in the presence of rather high re…nancing
costs. When re…nancing costs are rather low, an equilibrium in which both micro…nance
institutions o¤er individual loans is more likely to result.
The emergence of rating agencies specialized in the evaluation of micro…nance insti-
tutions and the growing awareness of the industry’s potential makes investors channel
more and more funds into this market. By now, some micro…nance institutions are listed
at stock exchanges, such as Compartamos in Mexico or Equity Bank in Kenya. Clearly,
enhanced access to capital markets implies reduced re…nancing costs. Given a continu-
ation of this trend, interestingly, our model predicts that individual lending schemes in
micro…nance will gain in importance in the future.
5.3 Competitive Pressure
Finally, we analyze the in‡uence of competition on a micro…nance institution’s choice of
lending technology. The competitive pressure of the market environment can be expressed
by the inverse of transportation cost,
1
t
. Note that the larger the transportation cost
parameter t and the more costly it b ecomes for borrowers to travel to a micro…nance
16
institution, the less intense price competition will be between micro…nance institutions.
Conversely, the lower t is, the stronger is competition.
We …nd that with increasing competitive pressure, a micro…nance institution prefers
to o¤er individual contracts, irrespective of whether its competitor grants individual or
group loans. Conversely, the less intense competition is, the more attractive group lending
becomes for micro…nance institutions. Our results are stated in Proposition 5.
Proposition 5 If the market environment is rather competitive, micro…nance institutions
prefer to grant individual loans. Group loans are o¤ered if competitive pressure is rather
low. That is,
A
(I; G)
A
(G; G) > 0 if t < t
1
and
A
(I; G)
A
(G; G) < 0 if t > t
1
(27)
A
(I; I)
A
(G; I) > 0 if t < t
2
and
A
(I; I)
A
(G; I) < 0 if t > t
2
(28)
B
(G; I)
B
(G; G) > 0 if t < t
1
and
B
(G; I)
B
(G; G) < 0 if t > t
1
(29)
B
(I; I)
B
(I; G) > 0 if t < t
2
and
B
(I; I)
B
(I; G) < 0 if t > t
2
. (30)
Proof: see Appendix.
The intuition for this result is as follows. Consider, …rst, the situation of micro…nance
institution A given that micro…nance institution B o¤ers group contracts. If micro…-
nance institution A o¤ers individual loans, the pooled interest rate it charges decreases
with increasing competition. The repayments asked by micro…nance institution B also
decline when competition becomes stronger. However, the fall in the interest rate is more
pronounced for risky than for safe borrowers. Hence, micro…nance institution B loses in
attractiveness concerning the segment of risky borrowers whereas micro…nance institution
A becomes relatively more attractive for safe borrowers. Consequently, the more com-
petitive the market environment is, the more micro…nance institution A prefers to o¤er
individual loans instead of group contracts.
The intuition is similar when we analyze the situation of micro…nance institution A
given that micro…nance institution B o¤ers individual loans. If micro…nance institution
A o¤ers group loans instead, it charges interest rates according to the borrowers’types.
Both repayment rates decline with increasing competition. However, interest rates fall
relatively more for risky than for safe borrowers. Now, with rising competition, micro-
…nance institution A becomes relatively more attractive for risky and less attractive for
safe borrowers if it o¤ers group loans and micro…nance institution B o¤ers individual
contracts. Hence, the more competitive the market is, the more micro…nance institution
A prefers the individual over the group lending technology.
17
Consequently, irrespective of whether micro…nance institution B o¤ers group or indi-
vidual loans, the individual lending technology becomes more attractive for micro…nance
institution A when competition toughens. Analogous considerations hold for micro…-
nance institution B. Hence, a Nash equilibrium in which both micro…nance institutions
o¤er individual contracts tends to emerge when competition is intense. In contrast, in
markets characterized by rather low competitive pressure, an equilibrium in which b oth
micro…nance institutions o¤er group loans is more likely to result.
According to Fernando (2007), McIntosh et al. (2005) and Christen and Rhyne (1999),
markets for micro…nance are often no more characterized by local monopolies of micro-
…nance banks. Instead, due to the immense growth of the micro…nance industry, there
is now …erce competition between micro…nance institutions in many countries. Given a
continuation of this trend, interestingly, our analysis again predicts that individual lend-
ing techniques will play a more important role in the future. This hypothesis is in line
with Dellien et al. (2005) who argue that due to rising competition, individual lending
schemes already gained in importance over the last few years.
Summing up our …ndings from the comparative statics analysis, we conclude that a
Nash equilibrium in which both micro…nance institutions apply the group lending technol-
ogy is the more likely to emerge when loans are rather large, re…nancing costs are relatively
high and competitive pressure is rather low. Otherwise, micro…nance institutions favor
individual loan contracts. Our results predict that when re…nancing conditions continue
to improve and competition rises further, individual lending schemes in micro…nance will
become more important in the future.
6 Empirical Hypotheses
Our model gives rise to several testable hypotheses concerning a micro…nance institution’s
choice of lending technology.
We found that the smaller the loan size, the more likely it is that micro…nance insti-
tutions o¤er individual loans. Hence, our …rst hypothesis is stated as follows.
Hypothesis 1 Micro…nance institutions are more likely to grant individual loans the
smaller the size of a loan. The larger the amount of credit is, the more likely it is that
micro…nance institutions o¤er group loans.
Next, we demonstrated that the lower re…nancing costs are, the more micro…nance
institutions prefer to o¤er individual loans. This gives rise to our second hypothesis.
18
Hypothesis 2 The higher re…nancing costs are, the more likely it is that micro…nance
institutions o¤er group contracts. The lower re…nancing costs are, the more likely micro-
…nance institutions are to grant individual loans.
Third, we showed that the more intense competition is, the more micro…nance in-
stitutions tend to o¤er individual instead of group contracts. Based on this result, we
formulate our third testable prediction.
Hypothesis 3 Micro…nance institutions are more likely to o¤er individual loans the
stronger competition is. The lower the competitive pressure, the more micro…nance insti-
tutions tend to o¤er group contracts.
Data best suited for testing our hypotheses concerning a micro…nance institution’s
lending strategy are cross country data. In that case, cultural e¤ects that may in‡uence
a micro…nance institution’s choice of lending technology could be controlled for. Fur-
thermore, panel data would render possible an analysis of how the relative importance of
group and individual loans altered following past changes in re…nancing conditions and
competitive pressure in the market for micro…nance.
7 Conclusions
In this paper, we have set up a model of competition between micro…nance institutions
in order to study a micro…nance bank’s choice of lending technology. We found that
micro…nance institutions tend to prefer individual loans over group loans when the size
of a loan is small, re…nancing costs are low, and competition is intense.
Currently, micro…nance institutions obtain increasingly better access to capital mar-
kets. Moreover, competition among micro…nance institutions increases steadily. Given a
continuation of these trends, our analysis predicts that individual lending schemes will
become more important in the micro…nance industry in the future.
Interestingly, when we interpret our results in the context of a recent trend in micro-
…nance, namely upscaling and downscaling, we can give further predictions about future
trends in the market for micro…nance. On the one hand, micro…nance institutions in-
creasingly start to invest in traditional banking technologies such as screening techniques,
a process called upscaling. On the other hand, commercial banks begin to downscale, i.e.
to invest in micro…nance technologies.
As mentioned earlier, micro…nance banks typically o¤er either group or individual
loans. Even more so, very often either group or individual loans dominate the market
19
for micro…nance in a given country or region (Madajewicz (2008)). Let us …rst consider
an environment in which micro…nance banks primarily o¤er group loans. Then, a micro-
…nance bank would only have an incentive to invest in screening if this technique were
better in terms of assessing a borrower’s type than the group lending technology. Anal-
ogously, commercial banks would have an incentive to invest in group lending only if
this technology would ensure a better evaluation of a borrower’s type. Hence, in such a
setting, upscaling and downscaling would constitute a form of substitutes.
Second, consider an environment characterized by micro…nance banks granting indi-
vidual loans. In such a situation, upscaling would allow a micro…nance institution to (more
or less e¤ectively) assess a borrower’s type through screening in addition to the realization
of high repayment rates by using the micro…nance monitoring technology. Similarly, a
commercial bank would gain from downscaling since in addition to assessing a borrower’s
type via screening, it can ensure higher repayment rates due to the micro…nance moni-
toring technology. Hence, in such a setting, upscaling and downscaling tend to work as a
form of complements. As a consequence, the gains from upscaling and downscaling should
be much higher in an environment where individual instead of group lending dominates
the market for micro…nance.
Coming back to our model, if due to rising competition and b etter access to capital
markets individual loan contracts in micro…nance will become more important in the
future, this development may at the same time boost upscaling of micro…nance institutions
and downscaling of commercial banks.
20
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Selection. European Economic Review 2000;44; 773-784
Little…eld E, Murdok J, Hashemi, S. Is Micro…nance an E¤ective Strategy to Reach the
Millennium Development Goals? CGAP FocusNote 2003;24
Madajewicz M. Joint Liability versus Individual Liability in Credit Contracts. Forthcom-
ing in Journal of Economic Behavior & Organization 2008
McIntosh C, de Janvry A, Sadoulet E. How Rising Competition among Micro…nance
Institutions A¤ects Incumbent Lenders. The Economic Journal 2005;115; 987-1004
Montgomery R. Disciplining or Protecting the Poor? Avoiding the Social Costs of Peer
Pressure in Micro-credit Schemes. Journal of International Development 1996;8; 289-305
Navajas S, Conning J, Gonzalez-Vega C. Lending Technologies, Competition and Consol-
idation in the Markets for Micro…nance in Bolivia. Journal of International Development
2003;15; 747-770
Reddy R, Rhyne E. How Will Buy Our Paper? Micro…nance Cracking the Capital Mar-
kets? Accion Insight 2006;18
Stiglitz J. Peer Monitoring and Credit Markets. The World Bank Economic Review
1990;4; 351-366
22
Van Tassel E. Group Lending under Asymmetric Information. Journal of Development
Economics 1999;60; 3-25
Zeitinger C-P 1996. Micro-lending in the Russian Federation. In: Levitsky J (Ed.), Small
Business in Transition Economies. ITDG Publishing: London;1996. p. 85-94
23
9 Appendix
Proof of Lemma 1:
The utility of a safe borrower if she receives a loan from micro…nance institution A is
given by
U
A
S
= i + p
2
S
[v
1 + r
A
S
i] + p
S
(1 p
S
) [v
1 + r
A
S
i
A
S
1 + r
A
S
i] tx d.
The utility of a safe borrower if she receives a loan from micro…nance institution B is
given by
U
B
S
= i + p
2
S
[v
1 + r
B
S
i] + p
S
(1 p
S
) [v
1 + r
B
S
i
B
S
1 + r
B
S
i] t (1 x) d.
Hence, the marginal borrower in the segment of safe borrowers is given by
x
S
(G; G) =
tip
S
[
A
S
(1p
S
)
(
1+r
A
S
)
+r
A
S
B
S
(1p
S
)
(
1+r
B
S
)
r
B
S
]
2t
.
The utility of a risky borrower if she receives a loan from micro…nance institution A is
given by
U
A
R
= i + p
2
R
[v
1 + r
A
R
i] + p
R
(1 p
R
) [v
1 + r
A
R
i
A
R
1 + r
A
R
i] tx d.
The utility of a risky borrower if she receives a loan from micro…nance institution B is
given by
U
B
R
= i + p
2
R
[v
1 + r
B
R
i] + p
R
(1 p
R
) [v
1 + r
B
R
i
B
R
1 + r
B
R
i] t (1 x) d.
Hence, the marginal borrower in the segment of risky borrowers is given by
x
R
(G; G) =
tip
R
[
A
R
(1p
R
)
(
1+r
A
R
)
+r
A
R
B
R
(1p
R
)
(
1+r
B
R
)
r
B
R
]
2t
.
Pro…ts of micro…nance institutions are given as follows:
A
(G; G) = x
S
(G; G)[2p
2
S
(1 + r
A
S
) + 2p
S
(1 p
S
)(1 +
A
S
)(1 + r
A
S
) 2(1 + c)]i+
(1 )x
R
(G; G)[2p
2
R
(1 + r
A
R
) + 2p
R
(1 p
R
)(1 +
A
R
)(1 + r
A
R
) 2(1 + c)]i
B
(G; G) = [1 x
S
(G; G)][2p
2
S
(1 + r
B
S
) + 2p
S
(1 p
S
)(1 +
B
S
)(1 + r
B
S
) 2(1 + c)]i+
(1 )[1 x
R
(G; G)][2p
2
R
(1 + r
B
R
) + 2p
R
(1 p
R
)(1 +
B
R
)(1 + r
B
R
) 2(1 + c)]i.
Micro…nance institutions maximize their pro…t with respect to interest rates and the joint
liability factors. Note that the following relationships hold:
d
(
j
(G;G)
)
dr
j
S
!
= 0 is equivalent to
d
(
j
(G;G)
)
d
j
S
!
= 0
24
d
(
j
(G;G)
)
dr
j
R
!
= 0 is equivalent to
d
(
j
(G;G)
)
d
j
R
!
= 0.
The …rst order conditions imply the following equilibrium interest rates dependent on the
joint liability parameters:
r
j
S
(G; G) =
t+(1+c)ip
S
[1+
j
S
(1p
S
)]i
p
S
[1+
j
S
(1p
S
)]i
r
j
R
(G; G) =
t+(1+c)ip
R
[1+
j
R
(1p
R
)]i
p
R
[1+
j
R
(1p
R
)]i
.
The resulting market shares and pro…ts are
x
j
S
(G; G) = x
j
R
(G; G) =
1
2
A
(G; G) =
B
(G; G) = t.
Proof of Self-Selection of Borrowers:
In order to induce self-selection of borrowers into di¤erent contracts o¤ered, the incentive
constraints for both safe and risky borrowers must be ful…lled:
If a group of risky borrowers truly reveals its type, the utility of a group member is given
by
U
R
(R) = i + p
2
R
[v
1 + r
j
R
(G; G)
i]+
p
R
(1 p
R
) [v
1 + r
j
R
(G; G)
i
j
R
1 + r
j
R
(G; G)
i] tx d.
If a group of risky borrowers pretends to be of the safe typ e, the utility of a group member
is given by
U
R
(S) = i + p
2
R
[v
1 + r
j
S
(G; G)
i]+
p
R
(1 p
R
) [v
1 + r
j
S
(G; G)
i
j
S
1 + r
j
S
(G; G)
i] tx d.
Note that U
R
(R) U
R
(S) > 0 is equivalent to
j
S
>
1
p
R
+p
S
1
.
6
If a group of safe borrowers truly reveals its type, the utility of a group member is given
by
U
S
(S) = i + p
2
S
[v
1 + r
j
S
(G; G)
i]+
p
S
(1 p
S
) [v
1 + r
j
S
(G; G)
i
j
S
1 + r
j
S
(G; G)
i] tx d.
If a group of safe borrowers pretends to be of the risky type, the utility of a group member
is given by
U
S
(R) = i + p
2
S
[v
1 + r
j
R
(G; G)
i]+
6
We will assume throughout our analysis that p
R
+ p
S
1 > 0 holds.
25
p
S
(1 p
S
) [v
1 + r
j
R
(G; G)
i
j
R
1 + r
j
R
(G; G)
i] tx d.
Note that U
S
(S) U
S
(R) > 0 is equivalent to
j
R
<
1
p
S
+p
R
1
.
As a consequence, if
j
R
<
1
p
S
+p
R
1
<
j
S
is ensured, self-selection of borrowers into the
di¤erent contracts can be achieved when interest rates are set accordingly.
We now show that r
j
S
(G; G) < r
j
R
(G; G) holds for contracts that achieve self-selection of
borrowers. This expression is equivalent to p
S
[1 +
j
S
(1 p
S
)] p
R
[1 +
j
R
(1 p
R
)] > 0.
We now de…ne
j
R
1
p
S
+p
R
1
with 0 < < 1. We can then rewrite the expression as
p
S
(1 p
S
) [
j
S
(p
S
+ p
R
1) 1] + p
R
(1 p
R
) (1 ) > 0. Note that
j
S
(p
S
+ p
R
1)
1 > 0 is equivalent to
j
S
>
1
p
S
+p
R
1
which holds when the incentive constraint of the safe
borrowers holds.
Hence, we have shown that if
j
R
<
1
p
S
+p
R
1
<
j
S
and r
j
S
(G; G) < r
j
R
(G; G) holds,
self-selection of borrowers can be achieved.
Proof of Lemma 2:
The utility of a safe borrower if she receives a loan from micro…nance institution A is
given by
U
A
S
= i + p
S
[v
1 + r
A
SR
i] tx.
The utility of a safe borrower if she receives a loan from micro…nance institution B is
given by
U
B
S
= i + p
S
[v
1 + r
B
SR
i] t (1 x).
Hence, the marginal borrower in the segment of safe borrowers is given by
x
S
(I; I) =
tp
S
(
r
A
RS
r
B
RS
)
i
2t
.
The utility of a risky borrower if she receives a loan from micro…nance institution A is
given by
U
A
R
= i + p
R
[v
1 + r
A
SR
i] tx.
The utility of a risky borrower if she receives a loan from micro…nance institution B is
given by
U
B
R
= i + p
R
[v
1 + r
B
SR
i] t (1 x).
Hence, the marginal borrower in the segment of risky borrowers is given by
26
x
R
(I; I) =
tp
R
(
r
A
SR
r
B
SR
)
i
2t
.
Pro…ts of micro…nance institutions are given as follows:
A
(I; I) = 2x
S
(I; I) [p
S
1 + r
A
SR
i (1 + c) i k]+
2 (1 ) x
R
(I; I) [p
R
1 + r
A
SR
i (1 + c) i k]
B
(I; I) = 2[1 x
S
(I; I)][p
S
1 + r
B
SR
i (1 + c) i k]+
2 (1 ) [1 x
R
(I; I)][p
R
1 + r
B
SR
i (1 + c) i k].
Micro…nance institutions maximize their pro…t with respect to interest rates which results
in the following equilibrium interest rates, market shares and pro…ts:
r
A
SR
(I; I) = r
B
SR
(I; I) r
SR
(I; I) =
[p
S
+p
R
(1)][k+t+(1+c)i][p
2
S
+(1)p
2
R
]i
[p
2
S
+(1)p
2
R
]i
x
S
(I; I) = x
R
(I; I) =
1
2
A
(I; I) =
B
(I; I) =
t[p
S
+p
R
(1)]
2
(p
S
p
R
)
2
(1)[k+(1+c)i]
p
2
S
+(1)p
2
R
.
Proof of Lemma 3:
The utility of a safe borrower if she receives a loan from micro…nance institution A is
given by
U
A
S
= i + p
S
[v
1 + r
A
SR
i] tx.
The utility of a safe borrower if she receives a loan from micro…nance institution B is
given by
U
B
S
= i + p
2
S
[v
1 + r
B
S
i] + p
S
(1 p
S
) [v
1 + r
B
S
i
B
S
1 + r
B
S
i] t (1 x) d.
Hence, the marginal borrower in the segment of safe borrowers is given by
x
S
(I; G) =
t+d+p
S
[r
B
S
r
A
RS
+
B
S
(1p
S
)
(
1+r
B
S
)
]i
2t
.
The utility of a risky borrower if she receives a loan from micro…nance institution A is
given by
U
A
R
= i + p
R
[v
1 + r
A
SR
i] tx.
The utility of a risky borrower if she receives a loan from micro…nance institution B is
given by
U
B
R
= i + p
2
R
[v
1 + r
B
R
i] + p
R
(1 p
R
) [v
1 + r
B
R
i
B
R
1 + r
B
R
i] t (1 x) d.
27
Hence, the marginal borrower in the segment of risky borrowers is given by
x
R
(I; G) =
t+d+p
R
[r
B
R
r
A
SR
+
B
R
(1p
R
)
(
1+r
B
R
)
]i
2t
.
Pro…ts of micro…nance institutions are given as follows:
A
(I; G) = 2x
S
(I; G) [p
S
1 + r
A
SR
i (1 + c) i k]+
2 (1 ) x
R
(I; G) [p
R
1 + r
A
SR
i (1 + c) i k]
B
(I; G) = [1 x
S
(I; G)][2p
2
S
(1 + r
B
S
) + 2p
S
(1 p
S
)(1 +
B
S
)(1 + r
B
S
) 2(1 + c)]i+
(1 )[1 x
R
(I; G)][2p
2
R
(1 + r
B
R
) + 2p
R
(1 p
R
)(1 +
B
R
)(1 + r
B
R
) 2(1 + c)]i.
Micro…nance bank A chooses repayment rates and micro…nance B both interest rates and
joint liability factors to maximize pro…t. This results in the following equilibrium interest
rates, market shares and pro…ts:
r
A
SR
(I; G) =
[p
S
+p
R
(1)](d+2k+3t+3(1+c)i)3[p
2
S
+(1)p
2
R
]i
3[p
2
S
+(1)p
2
R
]i
r
B
S
(I; G) =
p
S
[p
S
+p
R
(1)](d+2k+3t+3(1+c)i)+[p
2
S
+(1)p
2
R
]f3t3d+3(1+c)i6p
S
[1+
B
S
(1p
S
)]ig
6p
S
[p
2
S
+(1)p
2
R
][1+
B
S
(1p
S
)]i
r
B
R
(I; G) =
p
R
[p
S
+p
R
(1)][d+2k+3t+3(1+c)i]+[p
2
S
+(1)p
2
R
]f3t3d+3(1+c)i6p
R
[1+
B
R
(1p
R
)]ig
6p
R
[p
2
S
+(1)p
2
R
][1+
B
R
(1p
R
)]i
x
S
(I; G) =
2[p
2
R
+
(
p
2
S
p
2
R
)
](dk+3t)(1)p
R
(p
S
p
R
)[3(1+c)i+d+2k+3t]
12t[p
2
R
+
(
p
2
S
p
2
R
)
]
x
R
(I; G) =
2[p
2
R
+
(
p
2
S
p
2
R
)
](dk+3t)+p
S
(p
S
p
R
)[3(1+c)i+d+2k+3t]
12t[p
2
R
+
(
p
2
S
p
2
R
)
]
A
(I; G) =
2[p
S
+p
R
(1)]
2
(dk+3t)
2
9(p
S
p
R
)
2
(1)[k+(1+c)i][d+3t+(1+c)i]
18t[p
2
S
+(1)p
2
R
]
B
(I; G) =
4[p
S
+p
R
(1)]
2
(dk3t)
2
+9(p
S
p
R
)
2
(1)[dt+(1+c)i]
2
36t[p
2
S
+(1)p
2
R
]
.
Proof of Proposition 3:
Note that
A
(I; G)
A
(G; G) =
1
18t[p
2
R
+
(
p
2
S
p
2
R
)
]
f2[p
R
+ (p
S
p
R
)]
2
(d k) (d k + 6t)
9 (1 ) (p
S
p
R
)
2
[(1 + c)
2
i
2
+ (d + k + 3t) (1 + c) i+
dk + 3kt + 2t
2
]g.
Solving
A
(I; G)
A
(G; G) = 0 for i, we arrive at
i =
d+k+3t
2(1+c)
1
2(c+1)
q
d
2
+ k
2
+ t
2
2 (dk 3dt + 3kt) +
8[p
R
+(p
S
p
R
)]
2
(dk)(dk+6t)
9(p
S
p
R
)
2
(1)
.
As we only look at positive values of i, only the larger one of both thresholds is relevant
for our analysis. We de…ne
i
1
d+k+3t
2(1+c)
+
1
2(c+1)
q
d
2
+ k
2
+ t
2
2 (dk 3dt + 3kt) +
8[p
R
+(p
S
p
R
)]
2
(dk)(dk+6t)
9(p
S
p
R
)
2
(1)
.
28
Furthermore,
d
(
A
(I;G)
A
(G;G)
)
di
=
(1)(p
S
p
R
)
2
(1+c)(d+k+3t+2i+2ci)
2t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0 and
d
2
(
A
(I;G)
A
(G;G)
)
di
2
=
(1)(p
S
p
R
)
2
(1+c)
2
t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0.
Hence,
A
(I; G)
A
(G; G) describes a parabola with its maximum at i =
d+k+3t
2(1+c)
.
It follows from the analysis ab ove that
A
(I; G)
A
(G; G) > 0 if i < i
1
and
A
(I; G)
A
(G; G) < 0 if i > i
1
.
Note that
A
(I; I)
A
(G; I) =
1
36t[p
2
R
+
(
p
2
S
p
2
R
)
]
f4[p
R
+ (p
S
p
R
)]
2
(k d) (d k 6t)
9 (1 ) (p
S
p
R
)
2
[(c + 1)
2
i
2
+ 2 (d + t) (1 + c) i + d
2
+
t
2
+ 4kt 2dt]g.
Solving
A
(I; I)
A
(G; I) = 0 for i, we arrive at
i =
d+t
1+c
1
(1+c)
q
4t (d k)
4[p
R
+(p
S
p
R
)]
2
(dk6t)(dk)
9(p
S
p
R
)
2
(1)
.
As we only look at positive values of i, only the larger one of both thresholds is relevant
for our analysis. We de…ne
i
2
d+t
1+c
+
1
(1+c)
q
4t (d k)
4[p
R
+(p
S
p
R
)]
2
(dk6t)(dk)
9(p
S
p
R
)
2
(1)
.
Furthermore,
d
(
A
(I;G)
A
(G;G)
)
di
=
(1)(p
S
p
R
)
2
(1+c)[d+t+i(1+c)]
2t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0 and
d
2
(
A
(I;G)
A
(G;G)
)
di
2
=
(1)(p
S
p
R
)
2
(1+c)
2
2t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0.
Hence,
A
(I; I)
A
(G; I) describes a parabola with its maximum at i =
d+t
1+c
.
It follows from the analysis ab ove that
A
(I; I)
A
(G; I) > 0 if i < i
2
and
A
(I; I)
A
(G; I) < 0 if i > i
2
.
Proof of Proposition 4:
Solving
A
(I; G)
A
(G; G) = 0 for c, we arrive at
c =
d+k+3t+2i
2i
1
2i
q
d
2
+ k
2
+ t
2
2 (dk 3dt + 3kt) +
8[p
R
+(p
S
p
R
)]
2
(dk)(dk+6t)
9(1)(p
S
p
R
)
2
.
29
As we only look at positive values of i, only the larger one of both thresholds is relevant
for our analysis. We de…ne
c
1
d+k+3t+2i
2i
+
1
2i
q
d
2
+ k
2
+ t
2
2 (dk 3dt + 3kt) +
8[p
R
+(p
S
p
R
)]
2
(dk)(dk+6t)
9(1)(p
S
p
R
)
2
.
Furthermore,
d
(
A
(I;G)
A
(G;G)
)
dc
=
(1)(p
S
p
R
)
2
[d+k+3t+2i(1+c)]i
2t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0 and
d
2
(
A
(I;G)
A
(G;G)
)
dc
2
=
(1)(p
S
p
R
)
2
i
2
t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0.
Hence,
A
(I; G)
A
(G; G) describes a parabola with its maximum at c =
d+k+3t+2i
2i
.
It follows from the analysis ab ove that
A
(I; G)
A
(G; G) > 0 if c < c
1
and
A
(I; G)
A
(G; G) < 0 if c > c
1
.
Solving
A
(I; I)
A
(G; I) = 0 for c, we arrive at
c =
d+t+i
i
2
i
q
(d k) t
[p
R
+(p
S
p
R
)]
2
(dk6t)(dk)
9(1)(p
S
p
R
)
2
.
As we only look at positive values of i, only the larger one of both thresholds is relevant
for our analysis. We de…ne
c
2
d+t+i
i
+
2
i
q
(d k) t
[p
R
+(p
S
p
R
)]
2
(dk6t)(dk)
9(1)(p
S
p
R
)
2
.
Furthermore,
d
(
A
(I;G)
A
(G;G)
)
dc
=
(1)(p
S
p
R
)
2
[d+t+(1+c)i]i
2t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0 and
d
2
(
A
(I;G)
A
(G;G)
)
dc
2
=
(1)(p
S
p
R
)
2
i
2
2t[p
2
R
+
(
p
2
S
p
2
R
)
]
< 0.
Hence,
A
(I; I)
A
(G; I) describes a parabola with its maximum at c =
d+t+i
i
.
It follows from the analysis ab ove that
A
(I; I)
A
(G; I) > 0 if c < c
2
and
A
(I; I)
A
(G; I) < 0 if c > c
2
.
Proof of Proposition 5:
(1) Shape of
A
(G; G)
A
(G; G) = t
30
d
(
A
(G;G)
)
dt
= 1
d
2
(
A
(G;G)
)
dt
2
= 0.
(2) Shape of
A
(I; G)
A
(I; G) =
2[p
S
+p
R
(1)]
2
(dk+3t)
2
9(p
S
p
R
)
2
(1)[k+(1+c)i][d+3t+(1+c)i]
18t[p
2
S
+(1)p
2
R
]
Note, …rst, that
A
(I; G) is not de…ned at t = 0.
Note, second, that
d
(
A
(I;G)
)
dt
=
[p
R
+(p
S
p
R
)]
2
p
2
R
+
(
p
2
S
p
2
R
)
[p
R
+(p
S
p
R
)]
2
(dk)
2
9[p
2
R
+
(
p
2
S
p
2
R
)
]t
2
+
(1)(p
S
p
R
)
2
[d+(1+c)i][k+(1+c)i]
2[p
2
R
+
(
p
2
S
p
2
R
)
]t
2
.
If we solve
d
(
A
(I;G)
)
dt
= 0 for t, we arrive at
t =
p
[p
R
+(p
S
p
R
)]
2
(dk)
2
9
2
(1)(p
S
p
R
)
2
[d+(1+c)i][k+(1+c)i]
3[p
R
+(p
S
p
R
)]
.
In order for the above expression to be de…ned, we assume [p
R
+ (p
S
p
R
)]
2
(d k)
2
9
2
(1 ) (p
S
p
R
)
2
[d + (1 + c) i][k + (1 + c) i] > 0. In what follows, we will refer to
this assumption as Condition (1).
Note, third, that
d
2
(
A
(I;G)
)
dt
2
=
2
9t
3
[p
2
R
+
(
p
2
S
p
2
R
)
]
f[p
R
+ (p
S
p
R
)]
2
(d k)
2
9
2
(1 ) (p
S
p
R
)
2
[d + (1 + c) i][k + (1 + c) i]g > 0 due to Condition (1).
Since we only consider t > 0,
A
(I; G) must be a parabola with a minimum at
t =
p
[p
R
+(p
S
p
R
)]
2
(dk)
2
9
2
(1)(p
S
p
R
)
2
[d+(1+c)i][k+(1+c)i]
3[p
R
+(p
S
p
R
)]
.
Furthermore, it holds that lim
t!1
d
(
A
(I;G)
)
dt
=
[p
R
+(p
S
p
R
)]
2
p
2
R
+
(
p
2
S
p
2
R
)
.
Finally, note that
[p
R
+(p
S
p
R
)]
2
p
2
R
+
(
p
2
S
p
2
R
)
< 1 since this expression is equivalent to (p
S
p
R
)
2
(1 ) <
0. Hence, in the limit, the …rst order condition of
A
(I; G) approaches
[p
R
+(p
S
p
R
)]
2
p
2
R
+
(
p
2
S
p
2
R
)
, a
value that is smaller than the constant …rst order condition of
A
(G; G) which is equal
to 1. As a consequence, it must be true that there is exactly one intersection of
A
(I; G)
and
A
(G; G). We now calculate the exact intersection point.
Calculation of the Intersection Point
Note that
A
(I; G)
A
(G; G) = 0 is equivalent to
31
t
2
+
9(1)(p
S
p
R
)
2
[k+(1+c)i]4[p
R
+(p
S
p
R
)]
2
(dk)
6(1)(p
S
p
R
)
2
t+
9(1)(p
S
p
R
)
2
[(1+c)
2
i
2
+dk+(d+k)(1+c)i]2[p
R
+(p
S
p
R
)]
2
(dk)
2
18(1)(p
S
p
R
)
2
= 0
We de…ne
B
1
9(1)(p
S
p
R
)
2
[k+(1+c)i]4[p
R
+(p
S
p
R
)]
2
(dk)
6(1)(p
S
p
R
)
2
C
1
9(1)(p
S
p
R
)
2
[(1+c)
2
i
2
+dk+(d+k)(1+c)i]2[p
R
+(p
S
p
R
)]
2
(dk)
2
18(1)(p
S
p
R
)
2
Solving the ab ove expression for t, we arrive at
t =
1
2
B
1
1
2
p
B
2
1
4C
1
.
Due to the above analysis, the larger one of both thresholds is the one that is relevant for
our analysis. We de…ne
t
1
1
2
B
1
+
1
2
p
B
2
1
4C
1
.
It follows from the analysis above that
A
(I; G) >
A
(G; G) for t < t
1
and that
A
(I; G) <
A
(G; G) for t > t
1
.
(3) Shape of
A
(I; I)
A
(I; I) =
t[p
S
+p
R
(1)]
2
(p
S
p
R
)
2
(1)[k+(1+c)i]
p
2
S
+(1)p
2
R
d
(
A
(I;I)
)
dt
=
[p
S
+p
R
(1)]
2
p
2
S
+(1)p
2
R
d
2
(
A
(I;I)
)
dt
2
= 0.
(4) Shape of
A
(G; I)
A
(G; I) =
4[p
S
+p
R
(1)]
2
(dk3t)
2
+9(p
S
p
R
)
2
(1)[dt+(1+c)i]
2
36t[p
2
S
+(1)p
2
R
]
Note, that this is equivalent to
A
(G; I) =
36
f2[p
2
R
+
(
p
2
S
p
2
R
)
](dk3t)p
R
(p
S
p
R
)(1)(d+2k+3i+3ci+3t)g
2
t[p
2
R
+
(
p
2
S
p
2
R
)
]
2
+
1
36
f2[p
2
R
+
(
p
2
S
p
2
R
)
](dk3t)+p
S
(p
S
p
R
)(d+2k+3t+3i+3ci)g
2
t[p
2
R
+
(
p
2
S
p
2
R
)
]
2
Note, …rst, that
A
(G; I) is not de…ned for t = 0.
Second, note that
d
(
A
(G;I)
)
dt
=
36t
2
[p
2
R
+
(
p
2
S
p
2
R
)
]
2
f2[p
2
R
+ (p
2
S
p
2
R
)] (d k + 3t) p
R
(p
S
p
R
) (1 )
32
(d + 2k + 3i + 3ci 3t)gf2[p
2
R
+ (p
2
S
p
2
R
)] (d k 3t) p
R
(p
S
p
R
)
(1 ) (d + 2k + 3i + 3ci + 3t)g
(1)
36t
2
[p
2
R
+
(
p
2
S
p
2
R
)
]
2
f2[p
2
R
+ (p
2
S
p
2
R
)] (d k 3t) + p
S
(p
S
p
R
)
(d + 2k + 3i + 3ci + 3t)gf3p
S
[(1 + c) i + d + t] (p
S
p
R
) + 2p
R
[p
R
+ (p
S
p
R
)] (d k + 3t)g.
We now show that
d
(
A
(G;I)
)
dt
> 0 holds. Therefore, we use the following four conditions:
Condition (2) follows from x
S
(I; G) > 0 and is given by
2[p
2
R
+ (p
2
S
p
2
R
)] (d k + 3t) p
R
(p
S
p
R
) (1 ) (d + 2k + 3t + 3i + 3ci) > 0.
From Condition (2) we get Condition (3) which is given by
d k + 3t > 0.
Condition (4) follows from x
S
(I; G) < 1 and is given by
2[p
2
R
+ (p
2
S
p
2
R
)] (d k 3t) p
R
(p
S
p
R
) (1 ) (d + 2k + 3i + 3ci + 3t) < 0.
Condition (5) follows from x
R
(I; G) < 1 and is given by
2[p
2
R
+ (p
2
S
p
2
R
)] (d k 3t) + p
S
(p
S
p
R
) (d + 2k + 3i + 3ci + 3t) < 0.
It can be easily seen that these four conditions ensure that
d
(
A
(G;I)
)
dt
> 0.
Third, note that
d
2
(
A
(G;I)
)
dt
2
=
4[p
R
+(p
S
p
R
)]
2
(dk)
2
+9(1)(p
S
p
R
)
2
(d+i+ci)
2
18t
3
[p
2
R
+
(
p
2
S
p
2
R
)
]
> 0.
Hence, it follows from the above analysis, that
A
(G; I) is a parabola with a minimum
where only the increasing part of the parabola is of interest for us.
Note, further, that
d
(
A
(G;I)
)
dt
can be written as
d
(
A
(G;I)
)
dt
=
3[p
R
+(p
S
p
R
)]
2
+[p
2
R
+
(
p
2
S
p
2
R
)
]
4[p
2
R
+
(
p
2
S
p
2
R
)
]
[p
R
+(p
S
p
R
)]
2
(dk)
2
9[p
2
R
+
(
p
2
S
p
2
R
)
]t
2
(p
S
p
R
)
2
(d+i+ci)
2
(1)
4[p
2
R
+
(
p
2
S
p
2
R
)
]t
2
.
Hence, it holds that lim
t!1
d
(
A
(G;I)
)
dt
=
3[p
R
+(p
S
p
R
)]
2
+[p
2
R
+
(
p
2
S
p
2
R
)
]
4[p
2
R
+
(
p
2
S
p
2
R
)
]
.
Note, further, that
3[p
R
+(p
S
p
R
)]
2
+[p
2
R
+
(
p
2
S
p
2
R
)
]
4[p
2
R
+
(
p
2
S
p
2
R
)
]
>
[p
S
+p
R
(1)]
2
p
2
S
+(1)p
2
R
is equivalent to 2 (p
S
p
R
)
2
(1 ) [p
2
R
+
(p
2
S
p
2
R
)] > 0. Hence, in the limit, the …rst order condition of
A
(G; I) approaches
3[p
R
+(p
S
p
R
)]
2
+[p
2
R
+
(
p
2
S
p
2
R
)
]
4[p
2
R
+
(
p
2
S
p
2
R
)
]
, a value that is larger than the constant …rst order condition
of
A
(I; I) which is equal to
[p
S
+p
R
(1)]
2
p
2
S
+(1)p
2
R
. As a consequence, there must be exactly one
33
intersection point of
A
(G; I) and
A
(I; I) that is interesting for our analysis. To the
left of this intersection, it must hold that
A
(I; I) >
A
(G; I) and to the right of this
threshold, it must hold that
A
(I; I) <
A
(G; I). We now calculate the exact intersection
point.
Calculation of the Intersection Point
A
(I; I)
A
(G; I) = 0 is equivalent to
t
2
+ 2
4[p
R
+(p
S
p
R
)]
2
(kd)+3(1)(p
S
p
R
)
2
[(1+c)i+2kd]
3(1)(p
S
p
R
)
2
t+
4[p
R
+(p
S
p
R
)]
2
(dk)
2
+9(1)(p
S
p
R
)
2
[d
2
+(1+c)
2
i
2
+2d(1+c)i]
9(1)(p
S
p
R
)
2
= 0.
We de…ne
B
2
2
4[p
R
+(p
S
p
R
)]
2
(kd)+3(1)(p
S
p
R
)
2
[(1+c)i+2kd]
3(1)(p
S
p
R
)
2
C
2
4[p
R
+(p
S
p
R
)]
2
(dk)
2
+9(1)(p
S
p
R
)
2
[d
2
+(1+c)
2
i
2
+2d(1+c)i]
9(1)(p
S
p
R
)
2
Solving the ab ove expression for t, we arrive at
t =
1
2
B
2
1
2
p
B
2
2
4C
2
.
Due to the above analysis, the larger one of both thresholds is the one that is relevant for
our analysis. We de…ne
t
2
1
2
B
2
+
1
2
p
B
2
2
4C
2
.
It follows from the analysis above that
A
(I; I) >
A
(G; I) for t < t
2
and that
A
(I; I) <
A
(G; I) for t > t
2
.
34
