Strategic Interaction in Undeveloped Credit
Markets
Thomas Barnebeck Andersen
Institute of Economics, University of Copenhagen
Nikolaj Malchow-Møller
Centre for Economic and Business Research (CEBR); and
Department of Economics, University of Southern Denmark
April 10, 2005
Abstract
This paper studies the strategic interaction between informal and
formal lenders in undeveloped credit markets. In a model with adverse
selection, loan seniority, market power, and di¤erences in the cost
of lending, it is shown that under general conditions a co-funding
equilibrium will be a Nash outcome of the game. We demonstrate
that a collateral requirement in connection with formal loans always
generates a new co-funding equilibrium in which both lenders earn
higher pro…ts. A government subsidy to the formal lender may have
the opposite e¤ect. We relate our results to existing empirical evidence
and the emerging discussion of how best to ensure …nancial viability
and outreach of micro…nance institutions.
JEL Classi…cation: O1; D8; G2.
Keywords: Informal credit; Screening; Co-funding; Subsidies; Seniority.
Useful comments by Thomas Harr, Karl Ove Moene and two anonymous referees are
gratefully acknowledged.
1
1
Introduction
This paper explores the strategic interaction between formal and informal
lending institutions in less developed countries. There are several reasons as
to why an understanding of this interaction is important. First, despite the
recent trend of …nancial deregulation, government-owned agricultural banks
and, more generally, development banks continue to play a signi…cant role
in developing-country credit markets. Second, there is the increasing attention given to micro…nance institutions by policymakers and NGOs.1 Both
development banks and micro…nance institutions compete with informal …nancial institutions for borrowers, and many of the formal institutions are
not …nancially viable, but have to rely on signi…cant subsidies from donors
(Morduch, 1999b). Such interventions in rural credit markets obviously have
implications for the operation of informal lenders.
The literature on the interaction between formal and informal credit institutions can be divided into two groups according to the type of interplay
they model. First, in models of vertical interaction, borrowers obtain loans
exclusively from the informal sector. The informal sector, on the other hand,
borrows from the formal sector. Hence, the informal sector serves as middleman, and the e¤ects of changes in the formal sector will depend on the
structure of the informal sector, e.g., monopoly, monopolistic competition, or
perfect competition. Models of this type are analysed by Bose (1998), Floro
and Ray (1997), and Ho¤ and Stiglitz (1998).
Second, in models of horizontal interaction, borrowers can obtain loans in
both the formal and the informal sector. Such models have previously been
analysed by Bell (1990), Bell, Srinivasan, and Udry (1997), Kochar (1997),
and Jain (1999). This framework is also used in the present paper. However,
to our knowledge, it is assumed in all the existing models that at most one of
the lending institutions behave strategically. We argue that both the formal
and the informal lending agents have incentives to act strategically in the
market, and we explore a simple model that captures this.
Bell (1990) constructs a model in which the borrower seeks the cheaper
institutional credit …rst. Any recourse to the more expensive private moneylender stems from the borrower experiencing credit rationing at the institutional lender –often due to administered interest rate ceilings. If pro…table,
the moneylender then o¤ers a loan which is equal in size to the gap between
the rationed loan and the borrower’s notional demand for credit. Hence,
the existence of a private moneylender stems from the regulation-induced
1
In 1997, a high-pro…le group of policymakers, practitioners, and charitable funds initiated a drive to raise over $20 million for new micro…nance initiatives (Morduch, 1999a).
2
disequilibrium in the formal sector. As a comment to his model, Bell notes
that:
”[...] it must be ascertained whether the moneylender would
have any incentive in preventing the borrower from taking an
institutional loan. It seems probable that the moneylender will
have no objection. The institutional …nance will permit an expansion of the borrower’s activities, and if the moneylender is in
a position to exercise …rst claim on the returns produced by the
borrower’s activities, the institutional loan will, in general, improve the moneylender’s expected returns from his loan.” (Bell,
1990 p. 203).
Jain (1999) has in turn demonstrated that non-exclusive contracts may be
in the interest of the formal (institutional) lender as well. Thus, co-funding of
borrowers by formal and informal lenders need not per se be a disequilibrium
phenomenon. Jain constructs a model with two types of borrowers: safe and
risky. The formal lender is a monopolistic bank, which has a cost advantage
vis-à-vis the informal lender, but is uninformed about the borrowers’types.
The informal lenders are assumed to be fully informed and operate in a
perfectly competitive environment. As in Bell (1990), Jain assumes that
borrowers …rst obtain credit at the bank and then turn to the competitive
informal sector for residual …nancing.
Jain shows that it may sometimes be optimal for the bank to o¤er two
di¤erent contracts: A full funding contract at a high interest rate, and a
partial funding contract at a lower rate. The intuition for this result is
that the bank can use the informational advantage of the informal sector to
screen borrowers. Since risky borrowers face a higher competitive informal
interest rate, they will be more willing to accept a higher formal interest
rate in exchange for full formal …nancing. Thus, in Jain’s model, credit
rationing and co-funding is an optimal solution by a monopolistic bank to
an information problem.
By assuming the informal sector is perfectly competitive, Jain avoids the
complications associated with two-sided strategic interaction. The role of the
informal sector resembles that of a collateral requirement, and the problem
faced by the formal sector becomes a standard screening problem to which
the revelation principle is applicable.
In this paper, we build on the insights of Bell (1990) and Jain (1999). We
explore a model of strategic interaction based on the following observations:
i) the co-existence of formal and informal lenders, each enjoying considerable
market power; ii) borrowers are heterogenous with respect to the pro…tability
3
and risk of their investments; iii) the informal sector has an informational
advantage; iv) the formal sector has a lower deposit mobilisation cost; v) in
the absence of collateral requirements, the informal sector has seniority of
loans as a consequence of the informational advantage; and vi) both sectors
operate under imperfect competition and therefore behave strategically.
This results in a situation where the formal bank and the informal moneylender compete for borrowers and hence may try to crowd out each other. At
the same time, the bank has an interest in exploiting the informational advantage of the informal sector by screening the borrowers through co-funding
requirements. The moneylender, on the other hand, also has an interest in
co-funding solutions since he can use the cost advantage of the bank to increase his own pro…ts from the lending activity. The outcome is modelled as
a Nash equilibrium in both a simultaneous and a sequential game.
We demonstrate that the insights provided by Jain (1999) are robust to
the strategic consideration alluded to by Bell (1990). In addition, the model
leads to some novel insights. First, three di¤erent Nash equilibria can be
identi…ed. Apart from the co-funding and the pooling equilibria, which are
also found in Jain (1999), we show the existence of a segmented equilibrium2
in which the safe borrowers are fully …nanced by the moneylender, whereas
the risky types are fully …nanced by the formal lender. Second, a subsidy to
the formal sector may increase the fragility of the co-funding equilibrium by
increasing the bank’s incentives to deviate. As such, a subsidy can be counterproductive, since a breakdown of the co-funding equilibrium may imply
strictly lower pro…ts to both lenders. Third, both lenders have an interest in
eroding the informal lender’s loan seniority through a collateral requirement
in connection with formal loans. The reason is that the erosion of seniority
always generates a new co-funding equilibrium in which both lenders earn
higher pro…ts compared to the initial co-funding equilibrium.
Furthermore, the model’s predictions turn out to be consistent with several of the well-known facts about credit markets in less developed countries,
see Robinson (2001) and Banerjee (2001) for an overview. First, a common
characteristic of credit markets in developing countries is that borrowers are
often active in both the formal and the informal segments. Bell (1990) reports evidence from Punjab indicating that 45:1 per cent of all cultivators
and 32:4 per cent of all households borrowed from both informal and formal
sources. Jain (1999) reports evidence from a small sample of 17 powerloom
units from the Surat area, where all had access to informal …nance and 11
had access to formal …nance. Moreover, interest rate costs were lower, on
average, for the borrowers who were active in both segments of the market.
2
We are grateful to an anonymous referee for suggesting this term.
4
Second, informal moneylenders tend to charge higher interest rates than formal institutions. Aleem (1990) reports an interest di¤erential between formal
and informal lenders of more than 50 percentage points. Third, moneylenders tend to lend to relatively safe borrowers and experience low levels of
default. Timberg and Aiyar (1984) thus report default losses of as low as 0:5
per cent of working funds for the informal lenders in their study. In contrast,
they report default rates of up to 60 per cent for formal lenders (state-owned
commercial banks). Fourth, informal moneylenders tend to provide relatively
small loans, often just enough to cover working capital expenses (Robinson,
2001). We shall return to these issues in the …nal section of the paper.
The rest of the paper is structured as follows. Section 2 sets out the
basic model. In Section 3, the di¤erent Nash equilibria in the simultaneous
setting are derived and their intuition explained. In Section 4, a sequential
context is analysed and subgame-perfect Nash equilibria are found. Section
5 analyses comparative statics. In particular, we study the e¤ects of changes
in the cost of formal funding and changes in seniority. Section 6 concludes
with a discussion of the results and how they relate to the existing empirical
evidence. Proofs are relegated to the Appendix.
2
The Basic Set-Up
There are two types of potential borrowers, indexed by i = a; b. Each borrower is endowed with a project that requires a …xed investment of size K.
Borrowers have no wealth. In order to undertake the project, they have to
borrow the amount K. Project outcomes have a good state with a high payo¤ X i ; and a failure state with a low pay-o¤ X. Type a borrowers are assumed
to have a higher success probability than type b, pa > pb > 0. Moreover, it
is assumed that the expected outcome from type a borrowers undertaking a
project is higher than for type b: pa X a + (1 pa )X > pb X b + (1 pb )X. It is
also assumed that K > X > 0, which implies that failure is associated with
default as the borrower cannot meet his obligations at a nonnegative (real)
interest rate. The proportion of type a borrowers in the economy is given
by 2 (0; 1), and the reservation utility, U > 0; is assumed constant across
types.
Borrowers in need of funding for their projects have two sources of funds:
A monopolistic formal-sector bank and a monopolistic informal-sector moneylender. Only the moneylender is able to distinguish between the two types
of borrowers; the bank cannot observe the type. Thus, the moneylender is
assumed to have an informational advantage. In the default state, the information advantage also gives the moneylender …rst-move in the seizure of X.
5
That is, the moneylender has seniority. The cost of mobilising deposits in
the formal sector is c, whereas the cost of mobilising deposits in the informal
sector is m. It is assumed that the bank has a cost advantage in mobilising deposits vis-à-vis the moneylender, c < m.3 Finally, both lenders and
borrowers are assumed to be risk neutral.
Some comments on the assumptions of the model are perhaps required.
First, the neglect of physical collateral in the model is reasonable to the
extent that this is often not used in practice (Morduch, 1999b). In Section 5,
however, we analyse the consequences of introducing collateral opportunities
in connection with formal loans. Second, the neglect of enforcement and
moral hazard problems in the model can be justi…ed by the use of dynamic
contracts relying on progressive lending and the threat of cutting o¤ future
lending, etc. Third, the debt seniority of the moneylender might be justi…ed
by the fact that the moneylender, in contrast to the institutional lender, can
use the threat of violence to secure repayment. In addition, as pointed out
by Bell (1990), the moneylender often works as a trader buying the farmers’
crops, and therefore he can more readily exercise …rst-claim. Bell adds that
the moneylender-cum-trader often cooperates with other traders in securing
their dues.
Let ri be de…ned as the reservation interest rate of type i, which solves
pi X i (1 + ri )K = U . That is, ri ensures that the expected return to
a borrower of type i is equal to his reservation utility when o¤ered the full
funding contract: (L; r) = (K; ri ). Hence:
ri =
Xi
U =pi
K
1;
i = a; b
(1)
B
Moreover, de…ne ri0
as the interest rate that ensures the bank zero pro…t
B
B
from providing the loan (L; r) = K; ri0
to type i. That is, ri0
solves pi (1 +
M
B
ri0 )K + (1 pi )X (1 + c) K = 0. De…ning ri0 in the same way for the
moneylender results in:
(1 + c) K (1 pi )X
1; i = a; b
pi K
(1 + m) K (1 pi )X
=
1; i = a; b
pi K
B
ri0
=
(2)
M
ri0
(3)
Throughout the paper, it is assumed that the following inequalities are
satis…ed:
M
B
M
B
:
(4)
< ra0
< ra < rb0
< rb < rb0
ra0
3
The di¤erence, m c, can be interpreted as the moneylender’s cost of information
collection. Another interpretation would be that the di¤erence represents scale e¤ects.
6
B
M
B
M
Several things should be noted: First, ra0
< ra0
and rb0
< rb0
simply re‡ect
M
the cost advantage of the bank. Second, rb < rb0 implies that it is too
expensive for the moneylender to provide type b with full …nancing, whereas
B
B
<
< rb . Third, since ra0
the bank can earn a pro…t on type b since rb0
M
< ra ; both lenders can earn a pro…t on a full loan to type a. Fourth,
ra0
B
the assumption of ra < rb implies that X a < X b . Finally, ra < rb0
implies
that at the reservation rate of type a, bank pro…t from the risky type b is
negative.
As a …nal point, we establish some notation on loan contracts and pro…ts.
Because the moneylender is able to distinguish between types, he can o¤er
M
M
di¤erent contracts to types a and b. Hence, let M
i (Li ; ri ) denote the
expected pro…t to the moneylender from o¤ering a loan of size LM
i at an
M
M
interest rate ri to type i. While i also depends on the loan contracts
o¤ered by the bank, this dependence is suppressed in the notation to keep
it simple. The bank cannot distinguish between types, but may o¤er one or
two di¤erent contracts to all borrowers. Let B (LB ; rB ) be the pro…t to the
bank from o¤ering a single loan of size LB at the interest rate rB ; a pro…t
which also depends on the contracts o¤ered by the moneylender. If the bank
B
o¤ers two di¤erent loans to all borrowers, these are denoted by (LB
a ; ra ) and
B B
B B
B
B B
(Lb ; rb ) with expected pro…t:
(La ; ra ); (Lb ; rb ) .
3
Simultaneous O¤ers
In the absence of collateral, a monopolistic bank can only discriminate between the two types of borrowers by raising the interest rate, in which case
it adversely selects the less pro…table b types. On the other hand, if the
bank wishes to keep the safe a borrowers by o¤ering a lower interest rate, it
cannot prevent the risky borrowers from taking the loan as well. Hence, the
bank has an incentive to …nd other means (than collateral) to screen borrowers. In the presence of fully informed informal lenders, the bank can extract
the information on types from the informal sector by requiring borrowers to
seek co-…nancing in this sector. More precisely, Jain (1999) argued that if
the risky borrowers …nd it more di¢ cult (expensive) to obtain co-funding in
the informal sector, the bank may seek to target cheaper loans at the safe
borrowers by o¤ering less than full …nancing.
In the present model, the above mechanism is complicated by the fact that
the informal sector is not simply a competitive sector where safe and risky
types face constant but di¤erent interest rates. Instead, the informal sector
is comprised of a monopolistic moneylender who also attempts to maximise
expected pro…ts from his lending activity. Hence, the formal and informal
7
sector engage in monopolistic competition for the borrowers. However, since
the moneylender is assumed to endure higher lending cost, m > c, the total
available rent from a given borrower’s project is increasing in the share that
the bank …nances. This provides the moneylender with an incentive to opt
for a co-funding solution.
In sum, we have a situation with monopolistic competition for borrowers, but where both lenders at the same time have an incentive to opt for
co-…nancing solutions. Technically, the strategic game is modelled as follows:
In stage one, the bank and the moneylender simultaneously decide on a menu
of contracts to o¤er. That is, the moneylender o¤ers an individual contract
for each type, whereas the bank o¤ers one or two contracts available for both
types of borrowers. In the second stage, the borrowers choose among the
contracts o¤ered and undertake their investment. For technical reasons, it is
assumed that when borrowers are indi¤erent between two (sets of) contracts,
they choose the contract(s) with the highest degree of formal …nancing. In
the third stage, loans are repaid by successful investors, and less successful investors have their income seized. The solution concept used is that of a Nash
equilibrium. More precisely, the contracts o¤ered must constitute a Nash
equilibrium in the simultaneous game between the bank and the moneylender. To simplify the analysis, we strengthen the Nash concept slightly by
requiring that no player plays a dominated strategy in equilibrium.4
The following Proposition characterises the possible Nash equilibria of
the simultaneous game:
4
This is a relatively standard assumption, see Tirole (1988), and in the present setting
it serves to rule out equilibria based on non-credible "threats" by one of the lenders. We
will return to this below.
8
Proposition 1 If a Nash equilibrium exists, both types of borrowers will
always be …nanced in equilibrium, and the bank always fully …nances type b.
Hence, a Nash equilibrium will always belong to one of the following three
types:
1. A pooling equilibrium where the bank fully …nances both types.
2. A segmented equilibrium where the bank fully …nances type b and the
moneylender fully …nances type a.
3. A co-funding equilibrium where the bank fully …nances type b and
where the bank and the moneylender co-…nance type a.
In addition, a pooling (type 1) equilibrium and a segmented (type 2) equilibrium cannot co-exist.
Proof. See Appendix.
Proposition 1 reveals that the model admits three types of Nash equilibria. Compared to Jain (1999), the segmented equilibrium in which the safe
borrowers are fully …nanced by the moneylender and the risky types by the
formal lender is novel.
Intuitively, both types will always be …nanced in equilibrium: The moneylender can always …nance type a, whereas the bank can target a pro…table
contract at type b. Furthermore, type b will always be fully …nanced by the
bank. To see this, note that if both types were co-funded, the bank could
always increase its pro…ts by o¤ering a full loan at an interest rate such that
it would only be accepted by one of the types, yielding exactly the same
rent to this type as the co-funding contract. Because of the lower aggregate
mobilisation costs of a full bank loan, the residual rent to the bank would be
larger in this case.
In addition, if type a is fully …nanced by the bank, it can never be optimal
for the bank to co-…nance type b. For type b to accept a co-funding contract,
it should yield a higher rent than the full funding contract which type a is
getting, since this contract is also available to b. Hence, the pro…ts to the
bank from a co-funding contract to b would necessarily be lower in this case
due to the higher mobilisation costs of such a contract.
Finally, a pooling equilibrium and a segmented equilibrium cannot coexist. The reason is that in a segmented equilibrium, the pooling equilibrium
is revealed "not preferred", as the bank could choose the pooling outcome
but does not …nd it optimal to do so, and vice versa. A co-funding equilibrium, on the other hand, may co-exist with either a pooling or a segmented
equilibrium.
9
In the following subsections, the three di¤erent Nash equilibria will be
characterised in more detail.
3.1
Pooling equilibrium
The following Proposition characterises the pooling equilibrium of the simultaneous game:
Proposition 2 A pooling (type 1) Nash equilibrium exists if and only if:
B
B
M
)
(K; ra0
(K; rb ):
(5)
M
), whereas
In equilibrium, the bank o¤ers a single contract, (LB ; rB ) = (K; ra0
M
M
M
the moneylender o¤ers the contract (La ; ra ) = (K; ra0 ) to type a, and any
M
M
M
contract (LM
LM
0 if accepted. In
b ; rb ) to type b such that
b
b ; rb
equilibrium, both types of borrowers choose the contract o¤ered by the bank
and earn a strictly positive rent.
Proof. See Appendix.
In the pooling equilibrium, the bank fully …nances both types of borrowers. The condition in (5) ensures that the bank prefers the pooling solution
to the case where it only lends to the risky types at the interest rate rb .
For the moneylender, any o¤er which is rejected by the borrowers is optimal, since it cannot possibly make a pro…t when the bank o¤ers full …nancM
: The requirement that the moneylender’s o¤er
ing at the interest rate ra0
M
to type b should give a non-negative pro…t if accepted, M
LM
0,
b
b ; rb
follows from the assumption that no player plays a dominated strategy in
M
equilibrium. By o¤ering a contract to type b such that M
LM
< 0
b
b ; rb
if accepted, the moneylender could push the bank to a lower pro…t in the
pooling equilibrium. Such behaviour is not in general ruled out by the Nash
requirement since in a pooling equilibrium type b never accepts the contract
o¤ered by the moneylender. However, the strategy is clearly dominated since
it entails a strictly negative pro…t if it was ever to be accepted by type b. We
M
therefore rule it out by assumption. For the same reason, ra0
is the lowest
rate the moneylender can o¤er on a full funding contract to type a.
M
Note that the interest rate charged by the bank in equilibrium, ra0
, is
lower than if the bank was the only actor in the market, in which case
it would charge ra . The reason is the presence of the moneylender which
M
requires the bank to charge the rate, ra0
, in a pooling solution. Otherwise,
the moneylender would have an incentive to undercut the bank and lend to
M
type a. However, for ra0
to be optimal for the bank, the moneylender has
10
to o¤er a competing loan to type a at the same interest rate. In a pooling
equilibrium, both types of borrowers therefore earn a positive rent by the
assumption in (4).
3.2
Segmented equilibrium
Proposition 3 characterises the segmented equilibrium:
Proposition 3 A segmented (type 2) Nash equilibrium exists if and only
if:
B
B
(K; rb )
(K; ra ):
(6)
In equilibrium, the bank o¤ers a single contract, (LB ; rB ) = (K; rb ), whereas
M
the moneylender o¤ers the contract (LM
a ; ra ) = (K; ra ) to type a, and any
M
M
M
contract (LM
LM
0 if accepted. In
b ; rb ) to type b such that
b
b ; rb
equilibrium, type a borrowers choose the contract o¤ered by the moneylender,
and type b borrowers choose the contract o¤ered by the bank. Both types are
pushed to their reservation utilities.
Proof. See Appendix.
In the segmented equilibrium, the bank fully …nances the risky types, and
the moneylender …nances the safe types. The condition in (6) says that for
this to be sustainable, the bank must prefer it to a pooling solution where
it fully …nances both types at the interest rate ra : The moneylender has no
M
attractive alternatives. By o¤ering (LM
a ; ra ) = (K; ra ) to type a, he pushes
type a borrowers to their reservation utilities, and he has no possibilities of
making a pro…t on type b. In this equilibrium, all rent is extracted from both
types.
Note the asymmetry caused by the simultaneous game. In a pooling
(type 1) equilibrium, the relevant alternative for the bank is the pro…t in a
segmented (type 2) equilibrium. In a segmented equilibrium, however, the
relevant alternative is not the pooling equilibrium, but a pooling contract at
the rate ra . At ra , the bank can attract both types of borrowers when the
M
moneylender o¤ers (LM
a ; ra ) = (K; ra ) to type a. This also reveals why a
Nash equilibrium might not exist in some situations.
3.3
Co-funding equilibrium
The most involved type of equilibrium in the model is a co-funding equilibrium in which contracts are designed to make borrowers self-select according
to their types. The following Proposition characterises the co-funding equilibrium:
11
Proposition 4 A co-funding (type 3) Nash equilibrium exists if and only
B
if the set
LB
given by:
a ; ra
B
LB
a ; ra =
(7)
B
2
LB
a ; ra 2 R+ :
M
a
B
M
M
K LB
a (K; ra )
a ; ra
B
B B
max
La ; ra ; (K; rb )
B
(K; rb );
B
(K; ra )
where:
raM
rbB
M
rb;low
=
U
pa
Xa
(
(1 + raB )LB
a
(K
= min rb ;
(
LB
a)
M
rb;low
K
(8)
1
B B
LB
a + r a La
K
(1 + m) K LB
(1
a
= max m;
(K LB
a ) pb
)
(9)
pb ) X
)
1 ,
(10)
B
B
LB
is non-empty. A given contract, (LB
a ; ra , then corresponds
a ; ra ) 2
B
to a speci…c type 3 equilibrium where the bank o¤ers the contracts (LB
a ; ra )
B
B
M
M
and K; rb with rb given by (9), and the moneylender o¤ers (La ; ra ) =
M
K LB
to type a, where raM is given by (8), and:
a ; ra
M
a. If rbB = rb (type 3a equilibrium), the moneylender o¤ers any LM
b ; rb
M
M
M
to type b such that b Lb ; rb
0 if accepted.
M
b. If rbB < rb (type 3b equilibrium), the moneylender o¤ers (LM
b ; rb ) =
M
M
K LB
a ; rb;low with rb;low given by (10) to type b.
In equilibrium, type b is …nanced by the bank, and type a is …nanced both
by the bank and the moneylender. In a type 3a equilibrium, both types are
pushed to their reservation utilities, whereas in a type 3b equilibrium, type b
earns a positive rent.
Proof. See Appendix.
In a co-funding equilibrium, the bank fully …nances type b, whereas the
bank and the moneylender co-…nance type a. Proposition 4 gives the set
B
of partial bank contracts to type a, (LB
a ; ra ), which are consistent with a
co-funding equilibrium, and it also speci…es the associated contracts o¤ered
by the bank and the moneylender in a given co-funding equilibrium. The
intuition underlying Proposition 4 is the following:
12
B
B
In a co-funding equilibrium, the bank o¤ers (LB
a ; ra ) with La < K to
M
B
type a, and the moneylender therefore provides La = K La to type a
while extracting all remaining rent from type a by setting raM according to
B
(8), i.e. as a function of LB
a and ra . However, since the bank is only o¤ering
a partial funding contract to type a, the moneylender can crowd out the bank
by o¤ering type a a full funding contract instead. Hence, for the co-funding
strategy to be optimal, it must be the case that the moneylender cannot
get a higher pro…t by o¤ering type a a full funding contract at the highest
possible interest rate that type a is willing to accept, i.e. ra . This is the …rst
M
M
K LB
constraint in (7), viz. M
a (K; ra ).
a ; ra
a
An alternative interpretation of the above condition is that the bank
should –through its cost advantage –increase the rent available to the moneylender. When LB
a is not too large, the bank does not get a share in X in the
failure state, and the condition simply reduces to 1 + raB pa 1 + m. That
is, the bank should charge a gross interest rate (corrected for risk) which is
lower than the moneylender’s gross mobilisation costs. As LB
a is increased
for a given raB , the bank starts receiving a share in X in the failure state.
This leaves less rent for the moneylender, and hence the maximum rate, raB ,
that can be charged by the bank in a co-funding equilibrium decreases.
Furthermore, in a co-funding equilibrium, the moneylender should not
have an incentive to o¤er a co-funding contract to type b which type b would
B
prefer to accept together with the partial bank contract, LB
a ; ra , targeted
B
M
at the safe type a.5 Given LB
a ; ra , the interest rate rb;low in (10) is therefore
de…ned as the lowest rate that the moneylender can possibly bear on a cofunding loan to type b of size K LB
a without incurring negative pro…ts.
B
For low values of K La , the seniority of the moneylender is su¢ cient to
M
guarantee his return in the failure state, and rb;low
therefore just equals m.
When the bank’s share, LB
;
is
decreased,
the
moneylender’
s cost cannot be
a
M
covered by his seniority in the failure state, and rb;low increases accordingly.
Now, to ensure that the moneylender does not have an incentive to o¤er this
contract, the bank should set the interest rate, rbB , such that type b will prefer
the full bank loan, (K; rbB ); even when the moneylender o¤ers the contract
M
B
(K LB
a ; rb;low ) to type b. The required rate, rb , is given by (9).
M
Note that when rb;low
and K LB
a are su¢ ciently high, the moneylender
poses no threat to the bank, and the bank extracts all rent from type b by
setting rbB = rb . This is the case in a type 3a equilibrium. On the other
M
hand, if rb;low
is su¢ ciently low, the bank may be forced to charge an rbB
5
Note that this is the only relevant deviation to consider for the moneylender in connection with type b, since the moneylender can never earn a pro…t on a full contract to
type b.
13
below rb , which is what we observe in a type 3b equilibrium. Thus, the
B
contract o¤ered to type a, LB
a ; ra , indirectly determines the interest rate,
rbB , which the bank can charge type b in a co-funding equilibrium, and hence
the type of co-funding equilibrium that can prevail.
B
with
Finally, for a co-funding equilibrium to exist at a given LB
a ; ra
B
associated rb from (9), it must be the case that it yields a higher expected
pro…t to the bank than o¤ering the pooling contract, (K; ra ), or the full
funding contract, (K; rb ), targeted at type b. That is, the bank must prefer
the co-funding equilibrium to both a (type 1) pooling contract with (K; ra )
and the (type 2) segmented equilibrium, as these are both outside options
to the bank. Formally, these are the constraints in the bottom line of (7),
which both result in lower bounds for raB given the loan size LB
a.
Intuitively, the bank will prefer the segmented solution at rb to a coB
funding solution with LB
when raB is too low, and vice versa. More
a ; ra
B B
precisely, for those pairs, La ; ra , where the associated rbB equals rb , i.e.
the potential co-funding equilibrium is of type 3a, the bank prefers this to
the segmented solution at rb as long as pro…ts from type a are not zero. Since
the moneylender has seniority, this requires pa 1 + raB to exceed the bank’s
mobilisation costs, 1 + c, as long as the bank does not get a share in X in
the failure state. On the other hand, if the associated rbB is less than rb , i.e.
the potential co-funding equilibrium is of type 3b, the bank earns less on
type b than in the segmented equilibrium, and earnings from type a should
therefore be higher. Hence, the required raB is higher.
Similarly, the bank prefers the pooling solution at ra to a co-funding
B
B
B
solution involving LB
a ; ra when ra is too low. However, a small La may also
cause pro…ts from type a to be small in a co-funding equilibrium compared
to a pooling equilibrium, in which case the required raB must increase as LB
a
becomes smaller.
B
To sum up, possible type 3a equilibria are given by pairs, LB
a ; ra , and
M
B
associated interest rates, rb and ra given by (9) and (8), respectively, which
satisfy the constraints in (7). As argued, these constraints result in upper
and lower bounds for raB for a given LB
a , and together they de…ne the set of
B
possible co-funding equilibria, (LB
;
r
a
a ). Type 3a (3b) equilibria are then
B
B
the subset of pairs, LB
a ; ra , for which the associated rb from (9) is equal
to (less than) rb .
Note that in a type 3a equilibrium, type b’s participation constraint is
binding, whereas in a type 3b equilibrium, type b is indi¤erent between taking
the full loan o¤ered by the bank and taking the partial bank loan intended
for type a in combination with co-funding from the moneylender, i.e. type
b’s incentive constraint is binding.
14
A distinguishing feature of the present model is the assumed seniority of
the moneylender in the failure state. It may thus be instructive to consider
the implications of this assumption. The main role of seniority in connection with co-funding equilibria is to increase the moneylender’s incentives to
B
provide co-funding contracts at a given partial bank contract LB
a ; ra . On
the one hand, this allows the bank to charge a higher raB and hence a higher
rbB in a co-funding situation, which increases the scope for co-funding solutions. On the other hand, the moneylender’s incentive to co-…nance type b
M
also increases. Speci…cally, the moneylender can o¤er a lower rate, rb;low
, to
type b, which may force the bank to lower rbB . This will decrease the scope
for co-funding solutions. We will return to this issue in Section 5.3.
A speci…c implication of the seniority is that in a co-funding equilibrium
M
(1 + m) (K LB
a ) must always be greater than or equal to X. Since ra
must always be strictly larger than m for the moneylender to support the cofunding solution, this implies that 1 + raM (K LB
a ) > X. Hence, the bank
will never receive anything from type a in the failure state. To see this, note
that if X > (1 + m) (K LB
a ), the moneylender would always be able to o¤er
type b a co-funding loan of size K LB
at an interest rate m + " raM ,
a
X.
where is " is a small positive number such that (K LB
a ) (1 + m + ")
This loan would be pro…table and risk free to the moneylender, and it would
preferred by type b, since it is "cheaper" than the contract o¤ered to type a.
Furthermore, the following two corollaries can be deduced directly from
Proposition 4:
Corollary 4.1 Whenever a co-funding (type 3) Nash equilibrium exists, it
has a (weakly) higher equilibrium payo¤ to both the bank and the moneylender
than any co-existing pooling (type 1) or segmented (type 2) Nash equilibrium.
Corollary 4.1 follows from the fact that both the bank and the moneylender have their pro…ts in type 1 and 2 equilibria as outside options in a type
3 equilibrium.
Corollary 4.2 Existence of a segmented (type 2) Nash equilibrium implies
existence of a co-funding (type 3) Nash equilibrium.
The intuition behind Corollary 4.2 is straightforward: Since c < m, we
can always increase total rent compared to the situation in a segmented (type
2) equilibrium by letting the bank co-…nance a small share, ", of type a. "
should just be small enough to deter the moneylender from providing type
M
b with a partial loan of size LM
". Since rb < rb0
; such an " can
b = K
always be found. By splitting the additional rent between the bank and the
moneylender, a co-funding equilibrium can be constructed.
15
Corollary 4.1 states that if a co-funding equilibrium exists, both lenders
will prefer this equilibrium. This makes co-funding equilibria natural focal
points (equilibria) when they exist, see Schelling (1980) for a discussion of
this. Speci…cally, since the existence of a segmented equilibrium always implies the existence of a co-funding equilibrium, cf. Corollary 4.2, and since
the latter always Pareto dominates the former (both lenders and borrowers
are better o¤), one may argue that we should never expect to observe the segmented equilibrium outcome. The lenders should naturally focus/coordinate
on a co-funding solution.
However, it is di¢ cult to predict which equilibrium will actually be selected. The experimental evidence seems quite convincing in terms of dispelling the view that coordination problems will not occur in simple strategic
settings, see Cooper (1999). The problem in the present case is that the segmented equilibrium, although inferior in terms of payo¤s, is more "secure"
than the co-funding equilibrium. To be more precise, if there is a probability that the bank plays the "wrong" equilibrium strategy, such a mistake
does not in‡ict a cost upon the moneylender when he plays the segmented
equilibrium strategy. Conversely, when the moneylender plays a co-funding
equilibrium strategy, it will be quite costly for him should the bank happen
to play the segmented equilibrium strategy –or just another co-funding equilibrium strategy. In sum, since there is a risk-return trade-o¤, theory o¤ers
little guidance in terms of choosing among the existing Nash equilibria (see
also Camerer, 2003).
4
Sequential O¤ers
In other models of horizontal formal-informal sector interaction, it has been
assumed that the borrowers seek the institutional …nance …rst, see Jain (1999)
and Bell (1990). In the present setup, this would imply that the moneylender observes the contracts o¤ered by the bank. Hence, consider the following
sequential structure: In the …rst stage, the bank o¤ers contracts which the
moneylender observes. In the second stage, the moneylender makes an o¤er
in response to the …rst-stage o¤er; and in the third stage, borrowers decide
among the contracts o¤ered and undertake their investments.6 In the …nal stage, loans are repaid by successful investors and defaulters have their
income seized.7
6
As previously, it is assumed that when borrowers are indi¤erent between two (sets of)
contracts, they choose the contract(s) with the highest degree of formal …nancing.
7
An analysis of the sequential game where the moneylender moves …rst is omitted
for two reasons. First, it seems unreasonable to assume that the moneylender cannot
16
In this setting, the bank can always o¤er full …nancing at either the
M
interest rate ra0
, in which case it gets the same pro…t as in a pooling (type 1)
equilibrium in the simultaneous game, or the rate rb , in which case it gets a
pro…t as in the segmented (type 2) simultaneous equilibrium. At any interest
M
; ra ], the moneylender will choose to undercut the bank with
rate rB 2 (ra0
respect to type a borrowers, and the bank can therefore earn a higher pro…t
on type b by charging rb .
However, the bank might prefer to opt for a co-funding solution. The set
of potential co-funding solutions in this sequential game is de…ned as those
contracts where the moneylender has no incentive to deviate, and which the
bank prefers to its two outside options above. This set is equivalent to the
set from Proposition 4 with the exception that the second argument in the
M
max operator in the last line of (7) becomes B (K; ra0
) instead of B (K; ra ).
In the simultaneous game, a pooling contract at the interest rate, ra , was the
relevant outside option; in the sequential game, the relevant outside option
M
is a pooling contract at ra0
< ra , as the moneylender will otherwise crowd
out the bank subsequently. Whenever this extended set is non-empty, the
bank will choose a co-funding contract in the sequential game. This follows
from Corollary 4.1. Since the set
has been extended compared to the
simultaneous game, this means that a co-funding solution will exist for a
larger set of parameter values in the sequential game. Using Corollary 4.2,
this leaves only two potential equilibria in the sequential game, summarised
in the following proposition, which is stated without a formal proof:
Proposition 5 In the sequential game where the bank makes the …rst offer, the subgame-perfect Nash equilibrium is either: i) a co-funding (type 3)
equilibrium; or ii) a pooling (type 1) equilibrium. The former will be chosen
whenever it exists.
5
Comparative Statics
In this Section, we provide comparative statics results from the simultaneous
model. Speci…cally, we analyse the e¤ects of lowering the bank’s mobilisation
respond to the contract(s) o¤ered by the bank given his proximity to borrowers. Second,
if the moneylender moved …rst, he would only have to contribute marginally to the cofunding contract to reveal the borrower type. Since overall rents are decreasing in the
moneylender’s share, it would be optimal for him to do so (charging a very high interest
rate). But then anybody could for all practical purposes become moneylenders (even
the borrower himself), and we would then have to assume that either the bank knows the
identity of the moneylender in advance or that there is some exogenous minimum loan size
which the moneylender must provide. Neither of these assumptions are very attractive.
17
cost and of changing the seniority status of the lenders.
5.1
Formal mobilisation costs
Analysing what happens when the bank’s mobilisation cost, c, falls may
yield some interesting insights into the consequences of a government subsidy
to the formal sector; a policy that has been pursued in many developing
countries in connection with the forming of micro…nance institutions and
development banks.
For given equilibrium strategies, a fall in c increases bank pro…ts. Since
the e¤ect of a fall in c depends on the amount which the bank lends, the
e¤ect is largest in a pooling (type 1) equilibrium and smallest in a segmented
(type 2) equilibrium. Hence, continuously decreasing c will eventually cause
an existing co-funding equilibrium to break down, as the bank’s incentive to
deviate from a co-funding equilibrium by o¤ering a pooling contract at ra
becomes too attractive. This implies a discrete drop in bank pro…ts as the
economy must shift from the co-funding equilibrium to a pooling equilibrium
M
(which can only be sustained at the interest rate ra0
) or to a situation where
no equilibrium at all exists.
Figure 1 shows a numerical example of the former case. The Figure depicts the maximum pro…ts to the bank in a co-funding equilibrium, max B
co ,
as a function of c. The Figure also contains bank pro…ts at the pooling equiM
librium strategy, K; ra0
, the segmented equilibrium strategy, (K; rb ), and
the pooling contract, (K; ra ), when the latter contract is accepted by both
types. To the left of point A, a co-funding equilibrium cannot be sustained,
as the bank can increase its pro…t by o¤ering the pooling contract, (K; ra ),
which will be accepted by both types. This contract can, however, not be
sustained in a pooling equilibrium, and bank pro…ts are therefore pushed
M
down to the solid line, B K; ra0
.
[Figure 1 about here]
Turning to the moneylender, we have that as the economy moves from
a co-funding (type 3) to a pooling (type 1) equilibrium, the moneylender is
crowded out. Hence, his pro…ts must be reduced. Borrowers, however, might
gain from this since in a pooling equilibrium they always earn a rent. In the
co-funding equilibrium, on the other hand, type a borrowers will never earn
a rent.
We summarise the result in a Proposition:
Proposition 6 A reduction in formal sector mobilisation costs, c, may cause
a co-funding equilibrium to break down, thereby reducing pro…ts to both lenders.
18
More generally, the costs of informal funds, m, could well depend on the
costs of formal funds, c. In this case, m would also decrease when c is reduced.
This would make it more tempting for the moneylender as well to deviate
from a given co-funding equilibrium. In a type 3b equilibrium, a reduction
in m would furthermore reduce the rate, rbB , which the bank can charge type
b in a co-funding equilibrium. Both e¤ects would only reinforce the above
result.
5.2
Seniority
In the following, we investigate what happens if the bank has seniority in
the failure state such that the bank can seize (part of) X. Note that X
can be interpreted as the return on the investment in the failure state or,
more broadly, as the (liquid) value of the borrower in the failure state. As a
consequence, the right to X can be interpreted as a …rst-mover advantage in
seizing the returns or as a claim on collateral put forward by the borrower.
Assuming seniority by the bank can thus be interpreted as a possibility for
providing collateral in connection with formal loans.
A change in seniority will not a¤ect pro…ts on full loans since seniority
is not an issue here. That is, it will not a¤ect pro…ts in type 1 and type 2
B
equilibria. Consider instead what happens at a given partial loan, LB
a ; ra .
If X can be seized completely by the moneylender, as in the original model,
residual rent to the moneylender is:
R i LB
a
pi 1 + raB LB
a
(11)
is given by:
where Ri LB
a
R i LB
a = pi X i + (1
pi ) X
U
(1 + c) LB
a
(1 + m) K
LB
a : (12)
Changing to a situation where the bank has seniority, residual rent to the
moneylender at the same loan size changes to:
Ri LB
a
pi 1 + raB LB
a
(1
pi ) X B ;
(13)
provided that 1 + raB LB
X B , where X B is the amount of X for which
a
the bank has seniority. That is, X B is the value of the collateral put forward
by the borrower.
Hence, a change in seniority decreases the residual rent to be extracted
by the moneylender. More importantly, this e¤ect is more pronounced in
connection with the risky b types. It thus makes it less attractive for the
moneylender to co-…nance type a, but it makes it even less attractive to co…nance type b. By lowering raB to compensate the moneylender for the loss in
19
seniority, it is always possible to sustain a co-funding solution with a higher
bank share, LB
a , in equilibrium. A change in seniority thus unambiguously
increases the scope for co-funding solutions in the sense that it allows for
higher total rent as well as higher pro…ts to both lenders. The result is
stated formally in the following Proposition:
Proposition 7 For any initial co-funding equilibrium, it is always possible
to …nd a new co-funding equilibrium after an increase in X B which involves
higher total rent and higher pro…ts to both lenders.
Proof. See Appendix.
Proposition 7 illustrates an important bene…cial e¤ect of improving the
scope for collateral requirements by the formal sector in undeveloped credit
markets.8
6
Summary and Discussion
In this paper, we have studied strategic interaction in undeveloped credit
markets between an informal moneylender and an institutional formal lender.
It was shown that the co-funding equilibrium is robust to two-sided strategic
interaction in both simultaneous and sequential settings.
Our model is consistent with the stylised facts outlined in the Introduction. Recall that moneylenders in informal credit markets tend to charge high
interest rates compared to …nancially viable institutional lenders; that they
only lend to relatively safe borrowers; and that they only provide relatively
small loans.
In the co-funding equilibrium, the risky types are …nanced by the bank.
Hence, the bank will experience high default rates. Also, in the co-funding
equilibrium, both lenders have an interest in reducing the size of the loan
provided by the moneylender. The intuition is that since the bank has a cost
advantage, total rent is decreasing in the moneylender’s co-funding share.
Moreover, a small loan requires a high interest rate in order to be pro…table;
and for the moneylender to transmit his information on types to the bank,
i.e. screen borrowers, a small loan is su¢ cient. Thus, the co-funding equi8
An immediate corollary of Proposition 7 is that if the initial equilibrium is of type 3a,
the new co-funding equilibrium will always Pareto dominate the initial co-funding equilibrium, since in a type 3a equilibrium, borrowers are always pushed to their reservation
utilities.
20
librium is consistent with stylised facts. Yet, by highlighting the importance
of strategic behavior, our explanation di¤ers from the usual suspects.9
The segmented equilibrium, where the moneylender …nances safe borrowers and the bank …nances risky borrowers, is also consistent with the observed
low default rates among informal lenders.
It is worth stressing that we have abstracted from transaction costs. Several papers emphasise that borrowers have to incur signi…cant transaction
costs when approaching the formal lender (see Giné, 2001; and Key, 1998).
If this is the case, we might get equilibria in which lending would only be
undertaken by the moneylender, and only to the safe types. Hence, transaction costs would allow for the possibility of a shut down of the formal lender
in some situations.
Turning to an often pursued policy in developing countries, viz. subsidies
to the formal sector, our model also yields interesting insights. A fall in
formal sector banking costs, say as a result of a subsidy, will, ceteris paribus,
increase bank pro…t. This e¤ect is largest in a pooling equilibrium, where the
bank fully …nances all borrowers. Moreover, lower formal sector banking costs
will tend to reduce the scope for co-funding equilibria as the bank’s deviating
incentive becomes stronger. However, since a shift from a co-funding to a
pooling equilibrium could imply strictly lower pro…ts to the bank, it may well
be that the bank will become less viable as a consequence of the subsidy.10
In the context of subsidies it is also interesting to gauge to what extent both
subsidised and fully pro…t-making institutions can coexist; a question posed
by Morduch (1999b). In our model, subsidies tend to inhibit co-existence.
The pooling equilibrium simply drives the moneylender out of the market.
Finally, the model sheds some light on the e¤ects of strengthening …nancial market institutions in order to facilitate collateral requirements by
the formal sector; a policy advocated by for instance Schreiner and Colombet (2001). Debt seniority raises the willingness of the moneylender to co…nance borrowers since it increases the moneylender’s return in the default
state. This e¤ect is largest with respect to risky types since they have a
higher default probability. Seniority thereby limits the scope for co-funding
contracts to safe borrowers. A collateral requirements by the formal lender
circumvents this problem by eroding the debt seniority of the informal lender,
and it brings about a new co-funding equilibrium with higher total rent and
higher pro…ts to both lenders. Hence, in dealing with the problem of …nancial sustainability of micro…nance institutions, expanding the scope for
9
These include high default rates, low geographic mobility, low income, and low education among borrowers, or some sort of market power per se (Robinson, 2001).
10
See Bose (1998) and Ho¤ and Stiglitz (1998) for related results on subsidies.
21
formal sector collateral is likely to be a better policy than relying on subsidies. At the same time, increasing the scope for formal-sector collateral
probably comes with an added bene…t, viz. the increased outreach of formal
institutions. This should be of interest to both donors and policymakers.
It is reassuring that the notion of a co-funding equilibrium in undeveloped
credit markets and the incentives described in this paper bear some resemblance to results from the corporate …nance literature on the coexistence
of direct lending (securities markets) and intermediated lending (banks) to
…rms operating in more mature capital markets.11 In that context, bank
loans are a more costly source of …nance than corporate securities, but nevertheless …rms use both. Due to the private nature of debts and the role of
banks as delegated monitors, a …rm can signal its creditworthiness to securities markets by a certain amount of intermediated loans. Moreover, a bank’s
repayment probability is increasing in the amount of direct lending, so there
exists a complementarity between intermediated and direct lending. As such,
the idea of a co-funding equilibrium is a rather general construct that seems
to be a natural outcome in economic settings characterised by asymmetric
information.
11
See Seward (1990) for a theoretical model and Dowd (1992) for an excellent survey.
22
A
Proofs of Propositions
This Appendix contains proofs of the Propositions in the text.
B
M
Proof of Proposition 1. First, since ra0
< ra0
< ra , type a will always
be …nanced in equilibrium. Suppose not, then the moneylender could always
M
increase his pro…t by o¤ering the contract LM
= (K; ra ) to type a,
a ; ra
which a would accept.
Second, the bank will always fully …nance type b in equilibrium. Suppose
not, then either: i) b is not …nanced at all; or ii) b is co-funded in equilibrium.
B
The moneylender will never fully …nance type b. In case of i), since rb0
< rb ,
the bank could increase its pro…t by o¤ering the contract (K; rb ) which type
b, and only type b, would accept. Case i) can therefore not be an equilibrium.
Consider then case ii), and note that since c < m overall mobilisation costs
must be higher in a co-funding solution compared to full …nancing by the
bank. This implies that the total surplus from the project to be divided
between the lenders and the borrower will be larger in case of full …nancing
by the bank. Hence, if type a does not receive …nancing from the bank, the
bank can increase its pro…ts from b by only o¤ering the full funding contract
(K; rb ). On the other hand, if type a is receiving a full funding contract from
the bank, the rent to type b from the partial funding contract must be higher
than from the full funding contract, since the latter is also available to b. In
that case, the bank can increase its pro…t from type b by, e.g., o¤ering only
the full funding contract to both a and b, which will be accepted by b since
rb > ra . Finally, if type a is also co-funded, the bank can increase its pro…t
by o¤ering a full funding contract with an interest rate that just gives b (or
a) the same rent as the co-funding contract. Thus, in a Nash equilibrium, b
will always be fully …nanced by the bank.
Third, type 1 and type 2 equilibria are mutually exclusive. In a type
1 equilibrium, the bank fully …nances both types. Hence, it must charge a
M
rate which the moneylender has no incentive to undercut, i.e. ra0
. In a type
2 equilibrium, the bank only …nances type 2 and hence charges the highest
interest rate consistent with not loosing type b clients, i.e. rb . Now, in a
type 1 equilibrium, the bank always has the option to switch to the contract
(K; rb ) which gives the same pro…t as in a type 2 equilibrium. Similarly, in
M
a type 2 equilibrium, the bank can always switch to the contract K; ra0
which gives the same pro…t as in a type 1 equilibrium. Thus, if a type 2
equilibrium exists, type 1 cannot exist, and vice versa.
23
Proof of Proposition 2.
The proof of Proposition 2 proceeds in
M
two steps. First, it must be shown that the contracts: (LB ; rB ) = (K; ra0
),
M
M
M
M
M
M
M
M
(La ; ra ) = (K; ra0 ), and (Lb ; rb ) where b Lb ; rb
0, constitute a
Nash equilibrium if and only if:
B
M
)
(K; ra0
B
(K; rb )
(14)
holds. Second, it must be shown that no other pooling equilibrium exists.
M
) by the bank, the
Step 1: First, given the contract (LB ; rB ) = (K; ra0
M
moneylender can never earn pro…ts in the market. Hence, (LM
a ; ra ) =
M
M
M
M
M
M
(K; ra0 ) and (Lb ; rb ) with b Lb ; rb
0 are automatically best responses by the moneylender.
M
M
M
M
Second, given (LM
a ; ra ) = (K; ra0 ) and (Lb ; rb ) by the moneylender,
B B
M
(L ; r ) = (K; ra0 ) is a best response by the bank if and only if the pro…t
M
associated with this contract, B (K; ra0
), is positive and greater than the
pro…t associated with: i) o¤ering a full funding contract at another interest
rate; ii) o¤ering only a partial funding contract; and iii) o¤ering a full funding
contract and a partial funding contract. Obviously, with only two types of
agents, o¤ering more than two contracts can never increase pro…ts to the
bank, and if it o¤ers two contracts, at least one of them should be partial.
M
Re i): The best alternative interest rate is rb : Raising the rate above ra0
means that type a borrowers switch to the moneylender, and pro…ts from
type b are maximised at rb . Hence, the relevant condition becomes the one
B
in (14). Since rb > rb0
; this also ensures that pro…ts to the bank are positive
in equilibrium.
Re ii): If the bank is to o¤er only a partial funding contract, it must be
aimed at co-funding type b together with the loan from the moneylender,
M
(LM
b ; rb ). However, since total surplus from b’s project becomes smaller
with co-…nancing (due to m > c), and since the moneylender takes part of
M
it ( M
LM
0), the bank will necessarily get a smaller pro…t than by
b
b ; rb
fully …nancing b itself at the rate rb . Hence, this response is inferior to the
one in i).
Re iii): If o¤ering a full and a partial funding contract should be better
than i) and ii), the interest rate on the full funding contract must be less than
M
or equal to ra0
, so that type a borrowers are not lost. This implies that for
the partial funding contract to be accepted, it should leave b with a higher
M
rent than under (K; ra0
). The total surplus from b’s project becomes smaller
with co-…nancing, and the moneylender takes part of it, so the pro…t to the
M
bank will be smaller than under one pooling contract at ra0
.
B B
M
M
M
M
In sum, the contracts (L ; r ) = (K; ra0 ), (La ; ra ) = (K; ra0
), and
M
M
M
M
M
(Lb ; rb ) where b Lb ; rb
0; constitute a Nash equilibrium if and only
24
M
if (14) holds. Since ra0
< ra < rb , both types of borrowers earn a strictly
positive rent.
Step 2: A pooling situation where the bank charges an interest rate above
M
ra0 can never constitute an equilibrium, since the moneylender will have an
M
is the lowest rate that the
incentive to o¤er a cheaper loan to type a. ra0
moneylender is allowed to o¤er. However, if the moneylender does not o¤er
M
M
(LM
a ; ra ) = (K; ra0 ) to type a, the bank will raise the interest rate charged.
M
Hence, a pooling equilibrium must involve the strategies (LB ; rB ) = (K; ra0
),
M
M
M
M
M
M
M
M
(La ; ra ) = (K; ra0 ), and (Lb ; rb ) where b Lb ; rb
0.
Proof of Proposition 3. The proof of Proposition 3 also proceeds in
two steps. First, it must be shown that the contracts: (LB ; rB ) = (K; rb ),
M
M
M
M
M
(LM
LM
0 constitute a Nash
a ; ra ) = (K; ra ), and (Lb ; rb ) where b
b ; rb
equilibrium if and only if:
B
B
(K; rb )
(K; ra )
(15)
holds. Second, it must be shown that no other segmented equilibrium exists.
Step 1: First, consider the moneylender. Given the full funding contract
M
, whereas
by the bank, he cannot possibly earn a pro…t on type b since rb < rb0
he is extracting maximum pro…t from type a by pushing him to his reservation
M
M
M
M
M
LM
0
rate, ra . Hence, (LM
a ; ra ) = (K; ra ) and (Lb ; rb ) with b
b ; rb
are best responses by the moneylender.
M
Second, as in the proof of Proposition 2, given (LM
a ; ra ) = (K; ra ) and
M
B B
(LM
b ; rb ) by the moneylender, (L ; r ) = (K; rb ) is a best response by the
bank if and only if the pro…t associated with this contract, B (K; rb ), is
positive and greater than the pro…t associated with: i) o¤ering a full funding
contract at another interest rate; ii) o¤ering only a partial funding contract;
and iii) o¤ering a full funding contract and a partial funding contract.
Re i): The best alternative interest rate is ra where the bank attracts
both types of borrowers. Hence, for (K; rb ) to be optimal, it must be the
case that (15) holds.
Re ii): This action is inferior to (K; rb ) by the same argument as in the
proof of Proposition 2.
Re iii): This action is inferior to the strategy in i), again by an argument
analogous to the one used in the proof of Proposition 2.
M
M
M
In sum, the contracts (LB ; rB ) = (K; rb ), (LM
a ; ra ) = (K; ra ), and (Lb ; rb )
M
where M
LM
0 constitute a Nash equilibrium if and only if (15)
b
b ; rb
holds. It follows that both types of borrowers are pushed to their reservation
utilities.
Step 2: A segmented situation where the bank charges an interest rate
below or above rb can never be optimal since pro…ts from type b are max25
imised at rb . Similarly, the moneylender must charge ra to type a to maximise
pro…ts from type a.
Proof of Proposition 4. In a co-funding equilibrium, the contracts,
B
M
M
M
M
B B
B
LB
a ; ra , (Lb ; rb ), (La ; ra ), and (Lb ; rb ), must by de…nition satisfy: La <
M
K LB
K, LB
a . The proof will then proceed by deriving
b = K, and La
the necessary and su¢ cient conditions for these contracts to constitute a
co-funding equilibrium.
B
B B
Consider the moneylender: Given LB
with LB
a ; ra
a < K and (Lb ; rb )
B
with Lb = K, the optimal contract to o¤er to type a is either: i) a full
funding contract at ra , which yields positive pro…t by assumption; or ii) a
co-funding contract of size K LB
a , in which case the optimal interest rate,
B
raM , is set so as to extract all rent from type a given LB
a ; ra , i.e.:
(1 + raB )LB
a
pa X a
raM
=
U
pa
Xa
(K
(1 + raM ) K
(1 + raB )LB
a
LB
a)
LB
a
=U ,
(16)
1;
which is identical to the expression in (8). Note that type a will have no
B
B
incentive to switch to the contract (LB
b ; rb ) in this case since rb must exceed
B
rb0
(and hence ra ) in equilibrium. Now, the moneylender prefers the cofunding contract if and only if it yields higher expected pro…t than a full
funding contract to type a at the interest rate ra :
M
a
K
M
LB
a ; ra
M
a
(17)
(K; ra ) ;
where raM is given by (16). The condition in (17) is the …rst condition on the
set from Proposition 4.
With respect to type b, the moneylender can never earn a positive pro…t
B
on a full funding contract. Given LB
a < K and Lb = K, the only potentially
viable strategy is to o¤er type b a partial funding contract of size LM
b =
K LB
.
In
this
case,
the
lowest
pro…table
interest
rate
the
moneylender
can
a
M
o¤er, rb;low , satis…es:
M
pb (1 + rb;low
) K
LB
a + (1
M
pb ) min (1 + rb;low
) K
LB
a ;X
(1 + m) K
LB
a = 0 (18)
which gives:
M
rb;low
(
(1 + m) K LB
(1
a
= max m;
B
(K La ) pb
26
pb ) X
)
1 ;
(19)
which is identical to the expression in (10).
In a co-funding equilibrium, the moneylender must not have an incentive
to o¤er a partial funding contract to type b. Hence, in equilibrium it must
M
hold that type b has no incentive to accept a contract of the form (LM
b ; rb ) =
B B
M
and
(K LB
a ; rb;low ) given the contracts o¤ered by the bank. Given La ; ra
B
B
Lb = K, this places a condition on rb :
pb X b
(1 + rbB )K =
max U ; pb X b
which gives:
(
rbB = min rb ;
(1 + raB )LB
a
M
rb;low
K
M
(1 + rb;low
) K
B B
LB
a + r a La
K
)
LB
a
(20)
(21)
where rb = X b U =pb =K 1. This is exactly (9) from Proposition 4.
B
Given LB
and (9), the moneylender cannot possibly earn a strictly
a ; ra
positive pro…t on type b in equilibrium, and is therefore willing to o¤er any
M
potentially pro…table contract (LM
b ; rb ) in equilibrium.
Now, for the bank to …nd the co-funding equilibrium optimal, it must
be that it yields higher pro…ts than any of its feasible alternative strategies:
i) a segmented contract at rb ; ii) a pooling contract at ra ; iii) a co-funding
strategy with alternative interest rates; and iv) partial funding contracts to
both types.
Re i) and ii): They give rise to the last condition(s) on the set from
Proposition 4:
B
B
B
LB
a ; ra ; (K; rb )
max
B
(K; rb );
B
(K; ra )
(22)
where rbB is given by (21).
Re iii): Raising raB causes type a to drop out and is thus not optimal.
However, if (21) implies rbB < rb , the bank will have an incentive to raise
M
rbB in equilibrium unless the moneylender o¤ers the contract (LM
b ; rb ) =
B M
(K La ; rb;low ). This is case b in Proposition 4. If, on the other hand,
rbB = rb , the bank has no incentive to change rbB , and the moneylender can
o¤er any potentially pro…table contract to type b. This is case a.
Re iv): This is only relevant in case of a type 3a equilibrium. However,
since total surplus from b’s project becomes smaller with co-…nancing (due to
M
m > c), and since the moneylender takes part of the surplus ( M
LM
b
b ; rb
0), the bank will necessarily get a smaller pro…t than by fully …nancing b at
the rate rb . Hence, this strategy is not optimal either.
27
Proof of Proposition 7. Consider any initial co-funding equilibrium,
B
(LB
a ; ra ), satisfying (7), and consider an increase in formal lender seniority
from zero to some level X B < X.
B
(K; rb ),
First, the pro…ts associated with the outside options, M
a (K; ra ),
and B (K; ra ), are una¤ected by the change in X B .
B
^aB ,
Second, for an unchanged LB
a , it is possible to …nd a new ra , call it r
B
such that bank pro…ts from type a, a , and moneylender pro…ts from type a,
M
a , are unchanged. To see this, note that bank pro…ts from type a initially
are given by:
B
B
B
(23)
(1 + c) LB
a
a = p a 1 + ra La
since in a co-funding equilibrium, the bank always gets zero from type a in
the failure state. This expression should be set equal to bank pro…ts after
the change in X B , which are given by:
pa 1 + r^aB LB
a
pa ) X B
(1 + c) LB
a + (1
Hence, r^aB becomes:
1
r^aB = raB
(24)
pa X B
pa
LB
a
(25)
Third, consider how this in‡uences the second argument in (9). We know
that initially, the second argument in (10) must dominate, since (1 + m) (K
M
is thus given by:
LB
X in equilibrium. After the change in X B , rb;low
a)
M
rb;low
=
(1 + m) K
LB
a
(K
(1 pb ) X
LB
a ) pb
XB
1
(26)
which is strictly larger than m. Inserting this in the second argument in (9)
together with the expression for r^aB yields:
rbB = min rb ;
+
1
K
1
pb
pb
1
pa
pa
XB
(27)
equals the initial
where is the part that does not depend on X B , i.e.
value of the second argument in (9). Since pa > pb , the second argument in
(27) unambigously increases, which implies the following:
If the initial equilibrium was of type 3a, the new equilibrium, LB
^aB
a ;r
will still be of type 3a. Furthermore, since the second argument in (27) has
B
increased, it is possible to further increase LB
a and reduce ra such that both
B
M
B
B
a and a are increased, without reducing rb and hence b . Since outside
options are unchanged, a co-funding equilibriumwill be sustainable at this
higher LB
a , and pro…ts will have increased to both lenders.
28
If the initial equilibrium was of type 3b, rbB and hence B
b will have increased in the new equilibrium. This makes it possible to further increase LB
a
M
B
and reduce raB such that both B
and
are
increased,
without
reducing
a
a
b
below its initial level. Since outside options are unchanged, a co-funding equilibrium will be sustainable at this higher LB
a , and pro…ts will have increased
to both lenders.
29
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31
π B ( K , ra )
π B ( K , raM0 )
A
max(π coB )
Bank profit
π B ( K , rb )
0
Type 3 equilibrium
Type 1 equilibrium
Type 2 equilibrium
Bank’s mobilisation cost, c
Figure 1: Bank profits as functions of the mobilisation cost, c. See text for definitions.