## Pranab Bardhan

Print publication date: 1991

Print ISBN-13: 9780198287629

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198287623.001.0001

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# Interlinkages and the Pattern of Competition

Chapter:
(p.243) 13. Interlinkages and the Pattern of Competition
Source:
The Economic Theory of Agrarian Institutions
Publisher:
Oxford University Press
DOI:10.1093/0198287623.003.0013

# Abstract and Keywords

In this chapter, the authors make explicit the assumptions required for the frequent presumption of an ‘interlocker's edge’; and, given one or more of these assumptions, try to find out if the theory of interlinking predicts certain patterns of competition between lenders.

# 1. Introduction

There is now a large literature on interlinkages in rural credit markets, where both theoretical and empirical issues have been studied.1 If the various strands are carefully pulled together, we have a fairly complete analysis of the form assumed by an interlinked contract in different situations. However, the vast majority of these studies proceed on the assumption that there is a single monopolistic lender.

The issue of central interest in this chapter is different. We ask if the theory of interlinkage can throw any light on the nature of competition between lenders. Specifically, are there certain patterns of competition that are predicted by the theory of interlinkages?

To illustrate what we mean, consider an example. Suppose that there are two dominant money‐lenders in a village. One of them is a trader in the major crop of that village; the other is a landlord. There are a number of borrowers in the village. Some are landless labourers; others own land of varying plot sizes. The question is, Is there a predictable manner in which borrowers will be divided among the two lenders? For instance, who will lend to the landless borrowers? Where will farmers with sizeable land holdings get their loans from? Such a division will be called a pattern of competition. Similar questions may be asked when we ‘index’ borrowers differently (such as by the relative importance of consumption‐loan demand, as opposed to production loans), and indeed for other types of lenders.

Intuitively, we do expect that certain patterns of competition will be predicted by a properly formulated theory of interlinkage. The reason is that one kind of interlocker (say, a trader‐lender) will possess certain advantages relative to another (say, a landlord‐lender) for some groups of borrowers, (p.244) and will be at a relative disadvantage for other groups.

However, we must first retreat to a basic starting point: the precise nature of the advantage that an interlocked contract yields. In particular, consider a lender who is capable of offering interlinked contracts, and compare this lender with a pure money‐lender who cannot offer interlinked contracts. It is commonly accepted (see most of the references in n. 1) that the former is capable of extracting a greater return for each level of the borrower's reservation utility.

Our first task is to examine carefully this supposed advantage (Section 2 and the Appendix). In Section 2 we argue that certain assumptions are necessary for the interlocker to have a clear edge over the pure money‐lender. The assumptions cannot be immediately justified a priori, though they may certainly be valid in particular empirical situations. Briefly, the following alternative assumptions are needed to generate an interlocker's advantage.

1. 1. The lenders are restricted to offer only linear contracts, that is, where prices are independent of quantities transacted.

2. 2. The interlocker‐lender has advantages in the market(s) that he is active in, over and above his ability to be active in such markets. For example, a trader must not only be capable of trading in the output market: his trading activities must fetch him an additional pure profit that the pure money‐lender or the borrower cannot obtain.

3. 3. The pure money‐lender, by virtue of his specialized occupation, can observe the realization of a smaller number of variables than the interlocker, and consequently cannot condition loan repayment on the unobserved variables.

These conditions are discussed in Section 2. If none of these assumptions holds, then we show that an interlocker actually possesses no advantages over a pure money‐lender. In the context of our original question, it follows that no clear pattern of competition emerges when interlockers compete. This result, that under some conditions the pure money‐lender and the interlocker are equally powerful, is proved in Section A1 of the Appendix.

The purpose of this exercise is to isolate and make explicit the assumptions required for an ‘interlocker's edge’ to emerge. Given one or more of these assumptions, an approach to the theory of competition and interlinkage can then be outlined. In Section 3, we analyse in some detail a two‐stage model of Bertrand competition between a landlord‐lender and a trader‐lender. The landlord‐lender places a higher valuation on land than does the trader; the trader‐lender can sell output for a higher price than can the landlord. (We are therefore invoking the second assumption stated above.) We show that the landlord‐lender will deal with ‘small’ farmers who possess a high loan demand relative to their assets, while the trader‐lender will offer credit contracts to ‘larger’ farmers. The ability to interlock yields differential advantages to the two lenders, depending on the characteristics of the borrower. (p.245) Consequently, for a group of heterogeneous borrowers, there emerges a predictable division of these borrowers among the various lenders.

Interlinkages are seen, then, as complementary to the usual arguments for the existence of monopolistic lenders. There may simply be a dominant lender in a village; or, if there is more than one, a limited availability of loanable funds may prevent each from competing with the others. The lack of personal ties, developed over time, may prevent a borrower from seeking credit from another lender. Lenders may collude in a repeated, dynamic relationship which is nevertheless fundamentally non‐cooperative.

None of these arguments indicates the pattern in which borrowers in a region are likely to be ‘allocated’ among the lenders in that region. Such a pattern is predicted by the theory of interlinkage. This pattern may harden into ‘personal ties’, strengthening the lender's advantage. It might be reinforced by dynamic collusion, or be further insulated by the limited availability of loanable funds for each lender. It is in this sense that the theory of interlinkage is complementary to theories of monopolistic advantage.

# 2. When It Pays to Be an Interlocker

If the theory of interlinkage is to provide a basis for the competitive theory discussed in the introduction, a basic requirement must be fulfilled. As a first step, one should be able to demonstrate that an interlocker has a clear advantage over a pure money‐lender in his dealings with the borrower. In fact, this point is often regarded as self‐evident in much of the literature. But a careful inquiry into the basis of the ‘interlocker's edge’ will reveal the assumptions on which such a statement is based. In the present section, we do precisely this. In the process, we also mention some of the recent literature on interlocked markets, and hope that this will serve as a partial introduction for those unfamiliar with the literature.2

Briefly, the theory of credit interlinkage highlights the additional advantages to be gained by a money‐lender who is capable of intervening in some other market that the borrower is active in.3 The total advantage, in a sense, should exceed the sum of the advantages in acting independently in each market. For example, consider a trader who lends money. In his dealings with a farmer, it will generally be optimal for him to lend money at a low rate of interest but buy up the farmer's output at a low price. (A precise statement of this will be made below.) This activity in the credit market cannot be fully understood without reference to his dealings in the product market, and vice versa.4

Our first step is to explore when such an interlocked contract yields advantages over and above that obtainable from a pure money‐lending contract. We focus on interlockers versus pure money‐lenders, although the arguments apply more generally to any pair of interlockers. We start with an example.

## (p.246) Example 1: Landlord–labourer interlocking

The model that we discuss is drawn from Bardhan (1984).5 A labourer must see himself through a lean season and a peak season. In the peak season he can find employment for a given wage w > 0. He possesses one unit of labour power which he supplies inelastically. In the lean season he is unemployed, so that he must borrow to finance ‘lean consumption’, paying back the loan with his peak‐season income of w. He has a strictly quasi‐concave utility function u(c 1,c 2), where c 1 is lean consumption and c 2 is peak consumption. In the absence of any credit, he receives a utility u(0, w). This is his reservation utility.

Now introduce a landlord. He can provide loans to the labourer at an opportunity interest rate of r. He is also in need of labour in the peak season, which is available at the same wage rate w facing the labourer. The landlord can offer any combination of interest rate i and wage rate w′ to the labourer. The loan is then chosen by the latter. Which contract should he offer? It is easy to establish the following proposition.

1. PROPOSITION 1. There is a maximizing contract involving (a) an interest rate i * equal to the landlord's opportunity interest rate r, and (b) a wage rate w * less than w, the ‘market’ wage rate.

We omit the proof (see e.g. Bardhan 1984), although a brief statement of the underlying intuition might help. The landlord wishes the tenant to choose an efficient (Pareto‐optimal) level of the loan, provided that he can remove from the tenant the entire surplus above reservation utility by a suitable ‘tax’. So the landlord offers loans at his own opportunity cost r. And, given that labour is inelastically supplied, the reduction of the wage rate serves as a perfect lump‐sum tax to remove the surplus.6 Readers unfamiliar with interlocking models should observe that the credit activity alone (with i * = r) provides no clue regarding the totality of the contract, or the ‘implicit interest rate’ involved in the transaction.

Now replace the landlord with a pure money‐lender. This is an individual who is active only in the credit market. He hires no labour and so cannot demand repayment in terms of (underpaid) labour power. Suppose that he has the same opportunity cost of r for loanable funds. Can he earn the same return as the landlord?

Of course, he cannot offer the same contract: he is incapable of hiring labour. He can choose an interest rate. But in this case he cannot do as well as the landlord, for he must charge a rate greater than r in order to make any profit at all. However, this rate will not lead to a Pareto‐optimal choice of loan on the part of the borrower, and readers can see that this observation is enough to clinch the issue in favour of the landlord.

But an implicit assumption has been made to obtain the result. This is that (p.247) the money‐lender cannot condition the interest rate on the amount of loan borrowed. If he can, it is equally easy to see that he can replicate the return of the interlocker! Observe, first, that the optimal w * chosen by the landlord can be readily calculated given the parameters of the system. Now define a schedule i(L) by

$Display mathematics$
(1)

Readers can easily check that this indeed yields the pure money‐lender the same return as the landlord.

This result is simple; yet in an important sense it runs counter to the usual statements made regarding interlocking. If a money‐lender can charge a suitable nonlinear contract, it is not possible for an interlocker to earn an extra surplus in this model. We shall presently examine the generality of this observation.

Two remarks must be made concerning the money‐lender's optimal contract. First, it involves an interest rate that decreases in loan size. This is certainly not at variance with empirical observation. Second, we could have obtained the same result if we had allowed the money‐lender to ‘force’ the Pareto‐efficient loan size by offering an all‐or‐nothing contract. However, this second route does not lead to general results in models with incomplete information, whereas nonlinear contracts survive this extension (as we shall see).

Now, how general is this observation of equivalence? It certainly applies to the interlocking model that we have just described, as well as to its extensions. It applies to trader‐interlocking models, such as Gangopadhyay and Sengupta (1987a), to models of land–credit‐interlocking, such as Braverman and Stiglitz (1982) and Braverman and Srinivasan (1981), and even to models of incomplete information, where some of the borrower's characteristics are private information and unknown to the lender.

In the Appendix to this chapter we sketch a general model, which has all the above‐mentioned models as special cases, and which yields the equivalence of interlocker and pure money‐lender. However, our purpose is setting up the model is not to argue that interlocking confers no advantages. It is to make explicit the assumptions that are used in making the claim that interlockers are at an advantage vis‐á‐vis their pure money‐lending counterparts. In the rest of this section we discuss the assumptions with the aid of further examples.

1. ASSUMPTION 1. Nonlinear contracts are disallowed. This is a straightforward restriction that immediately yields advantages to an interlocker relative to a pure money‐lender. Indeed, this is the case in all the models we have mentioned. We should state, though, that we find it extremely difficult to understand the basis of such an assumption. It is certainly hard to defend it (p.248) a priori except possibly on the grounds of complexity of nonlinear contracts. And as we have already mentioned, it is not hard to quote empirical instances where the interest rate varies with loan size.

2. ASSUMPTION 2. Interlockers face better terms in the market in which they are active. To see the implications of this, consider the following example.

This example is based on Gangopadhyay and Sengupta (1987a). A farmer produces output using working capital and labour. The latter is inelastically supplied, so we write his (concave) production function as f(L), where L is the quantity of working capital. It must be borrowed. Loans may be obtained at an interest rate of R > 0. The price that the farmer obtains for his output is unity. The farmer seeks to maximize his sales net of production costs (repayment of loan plus interest).

Now introduce a middleman or trader, who has access to loanable funds at an opportunity interest rate of r < R. The trader can also obtain a price $p ~$ for every unit of output that he sells, with $p ~ ≥ 1$. A contract to the borrower involves an interest rate i at which loans are offered, and a price p at which he will buy the farmer's output.7 What contract should the middleman offer the farmer in order to maximize his own return? Here, one can establish a second proposition.

1. PROPOSITION 2. There is a maximizing contract which involves (a) an interest rate i * strictly less than the opportunity cost of funds r, and (b) a price p * that is strictly less than $p ~$, the price that the trader can obtain.

Readers might find it of interest to examine the intuition underlying Proposition 2. It is only a little more complicated than Proposition 1. Our purpose, however, is to examine what a pure money‐lender can achieve under these circumstances.

Denote by L * the level of the loan taken under the optimal trader contract.8 We can then write the interlocker's total profit as the sum of the implicit pure trading profits involved $( ( p ~ − 1 ) f ( L * ) )$ plus a residual term which captures credit market activities and the effect of interlinkage. Denote this latter term by D. Readers can now verify that, by offering a nonlinear interest schedule i(L) of the form

$Display mathematics$
(2)
where $r ~$ solves
$Display mathematics$
(3)
the pure money‐lender can earn precisely a profit of D; that is, he can pick up (p.249) the entire surplus of the interlocker except for pure trading profits. In particular, if $p ~ = 1$, so that the interlocker has no extra advantages in the output market (except his ability to be active in it), the money‐lender can obtain the same profit as the interlocker.

Return to the case $p ~ > 1$. This is precisely what we mean by our statement that ‘interlockers face better terms in the market in which they are active’. For the farmer can obtain only a price of 1. In this case, there is no way for the pure money‐lender to internalize the higher price $p ~$ into the contract that he is offering. Nevertheless, he can capture all the surplus barring ‘pure trading profits’. So even here, the statement that an interlocker can achieve more than the sum of his separate trading and money‐lending activities does not hold water.

But it is true that an ‘interlocker's edge’ appears. And, despite the fact that the edge may be small (especially if $p ~$ is near 1), it is capable of being translated into much larger advantages in a competitive model. On this, see Section 3 below.

There are, of course, other ways in which interlockers can command better terms in the market in which they are active. To see this, let us extend Example 1 to capture the possibility of involuntary unemployment. Specifically, suppose that in that example, there is unemployment even during the peak season. Put another way, suppose that there is a positive probability p that the labourer will receive a wage of 0 (unemployment), while with probability (1 − p) he will be employed at the wage w. The landlord, on the other hand, is on the ‘short side’ of the market, and his opportunity cost of hiring labour is always w. In this case, it can be shown that, whenever p > 0, the landlord can achieve a higher surplus than a pure money‐lender. Involuntary unemployment in the labour market leads to the interlocker's edge in this example.

1. ASSUMPTION 3. Differential observability. The equivalence result in the Appendix also leans on the assumption that both interlocker and pure money‐lender observe (and therefore can condition contracts upon) exactly the same set of variables. Often, such an assumption is unwarranted. In such situations, the role of differential observability becomes critically important if the lender's return under the optimal interlocked contract is necessarily conditioned on more variables than just the loan itself. Observe that this is not the case in situations such as those described by Examples 1 and 2.9 Once uncertainty is introduced, however, matters are entirely different.

## Example 3: Farmer‐lenders

Consider a farmer, or landlord, granting a fixed consumption loan L to a small farmer in the lean season preceding the harvest.10 The farmer produces output using labour; output is uncertain owing to exogenous considerations (p.250) such as weather. The production function is f(e,θ), where e is labour and θ is a random variable. In what form should the ‘repayment contract’ be drawn up?

Assume that the farmer has a utility function u(y,e) where y is his income (net of loan repayment). Assume further that the borrower must give at least a reservation expected utility û, to avoid defaulting on the loan.

The ‘repayment scheme’ is, then, a function R(Y), where Y is the output produced. A risk‐neutral lender will choose R(·) to solve

$Display mathematics$
(4)
$Display mathematics$
(5)
$Display mathematics$
(6)
where ê solves
$Display mathematics$
(7)

Now observe that the problem (4)–(7) describes a standard principal–agent problem. It is well known (see e.g. Mirrlees 1975 and Holmstrom 1979) that the optimal solution R(·) involves a function that is not the constant function. This observation leads to an immediate conclusion: that interlinkage between output and credit markets will be observed in the form of output ‘shares’ being released in repayment for loans taken.

The important exception to this rule is when the utility function of the borrower is linear in income (risk neutrality), in which case a fixed rate of interest is optimal.

Now consider a pure money‐lender who is not in a position to observe the output of the borrower. In this case, he cannot replicate the surplus of the interlocker, for exactly the same reason that fixed‐rent tenancy is not generally optimal. The assumption of differential observability does matter here, unless the borrower is risk‐neutral.

Examples of this sort can be multiplied. We only mention a variant that relies on an important but little studied feature: incomplete information regarding the borrower's characteristics (such as the productivity of his land or his labour).11 Such lack of information (even without uncertainty, in the sense of Example 3 above) will lead to the necessary conditioning of an optimal interlinked contract on variables other than the loan itself. If a pure money‐lender cannot observe these variables (such as the borrower's output), the interlocker who can is at an advantage.

1. ASSUMPTION 4. Price uncertainty and imperfect observability. Suppose that the interlocker operates in a market with price uncertainty. For instance, consider Example 1, and suppose that the wage rate ruling in a peak season is a random variable. In other words, there is a set of possible wages from which one will prevail: this is both the wage that would be paid to the labourer in the (p.251) peak season, and the wage that must be paid by the landlord if he hires labour in the market.12 All agents have a common prior regarding the distribution of wages.

Now, it can be easily checked that there is still an optimal contract just as in Proposition 1, where the landlord offers an interest rate on loans and a fixed wage (less than the expected value of the wage distribution). In this contract, the landlord does not require to condition any payments (or receipts) on the actual realization of the market wage. So it does not matter whether he gets to observe the realization of the market wage or not.

Suppose that he does not observe the realization of the market wage. Assuming no differential observability (that was discussed in Assumption 3), nor does the pure money‐lender. But then, it can be easily seen that the pure money‐lender cannot offer an equivalent contract to the labourer. Note that neither Assumption 1 nor 2 nor 3 holds, so that this is an additional case of non‐equivalence. It is driven by imperfect (or incomplete) observability of all the variables in the system, including a price that is uncertain.

This completes our discussion of the assumptions needed to get around the equivalence problem. In the Appendix, we demonstrate in a general model that, when none of these assumptions can be made, the interlocker's edge does not exist.

Of Assumptions 1–4, which are the most plausible? Assumption 1 is generally made on the grounds of analytical or expository convenience. But in this case it is clear that the assumption plays more than an expository role. When some other condition for non‐equivalence is satisfied, one might additionally invoke Assumption 1 to simplify the analysis.13 Taken by itself, we see no justification on either a priori or empirical grounds. At best, a case might be made to rule out nonlinear contracts on the grounds of complexity.

Assumption 4, while certainly valid at a logical level, is probably of little significance. In the example, for instance, we have both landlord and pure money‐lender not observing the realization of the market wage. It is very likely that the former will indeed observe its realization. If the latter does not, we still have non‐equivalence, but this is really due to differential observability (Assumption 3). In short, one would expect the interlocker to observe the realizations of all prices in the markets in which he is active. If the pure money‐lender does not observe them too, we really have a case of differential observability.

We feel on the other hand, that Assumptions 2 and 3 are important and relevant. Consider Assumption 2. It says that an interlocker faces better terms in the markets in which they are active. In the context of the trader‐lender example (Example 2), this simply says that the trader can fetch a higher price for output than the farmer. He can make pure trading profits. And this, as we have shown, gives him an edge over the pure money‐lender or, (p.252) indeed, over a different kind of interlocker. In fact, the extent of his advantage is precisely his pure trading profits.

This does not mean, of course, that in a competitive scenario both trader and the pure money‐lender can coexist, with the trader acting simply as a pure trader vis‐à‐vis the farmer. The trader will have to offer interlocking contracts to retain his advantage. See Section 3 for a detailed example.

Another instance of an interlocker facing better terms in his active market is that of a landlord who takes land as collateral for loans. With an imperfect land market, the landlord will clearly possess an edge over a pure money‐lender (or a trader) who does not value land as highly. We develop this idea, too, in Section 3.

Assumption 3, dealing with differential observability, is also of great relevance. It is natural to imagine that different interlockers will possess different sets of observable variables when dealing with a particular borrower. A trade who specializes in the buying and selling of a particular crop is far more likely to be better informed about the price of that crop and the amount of it produced by the farmer he is dealing with, compared with a trader who specializes in a different crop. In turn, both can be expected to have more information than a pure money‐lender. Consequently, the domain of variables on which an interlocker can condition is likely to vary across interlockers, and in all probability will be larger than the domain available to a pure money‐lender. As we have already seen, these differences are likely to lead to advantages for the interlocker, especially in models of incomplete information.

# 3. Interlinkage and the Pattern of Competition

In the previous section, we discussed conditions under which an interlocker may gain an advantage over a pure money‐lender, or another interlocker active in a different market. Specifically, the discussion was conducted by studying the monopoly return to an interlocker, and comparing that with the monopoly return available to a pure money‐lender. We now permit the agents to compete with each other.

In this section, we briefly indicate how the theory of interlinkage might be used to construct a theory of competitive patterns. By the term ‘competitive pattern’, we mean a description of the division of various types of borrowers among the different money‐lenders/interlockers who are actively interacting with them. For instance, suppose that we have, in a region, farmers growing either of two kinds of crops, A and B. Suppose that there are two trader‐lenders, each specializing in one of the two crops. Each offers credit‐cum‐trade contracts to the farmers, and competes with each other for the farmers. It is to be expected that a ‘reasonable’ non‐cooperative equilibrium in this scenario would involve the trader who specializes in crop A offering interlinked contracts with farmers who grow crop A, and similarly for the trader (p.253) who deals in crop B. This is a particular pattern of competition.

Perhaps of greater analytical interest are the patterns of competition that might emerge when a trader‐lender and a farmer‐lender (or landlord‐lender) compete. For instance, if potential borrowers can be indexed in terms of the ratio of consumption loan requirement to production loan requirement, which types are likely to gravitate to the trader‐lender, and which to the farmer‐lender?

Here, we provide a simple example that illustrates a pattern of competition between a trader‐lender and a farmer‐lender. Specifically, we try to highlight the monopolistic powers that might accrue to each as a result of their ability to interlock.

At the very outset, it should be pointed out that we are not claiming that interlocking is the basic source of monopolistic advantage. In a rural context, there might be many others. First, there could simply be a ‘natural monopoly’ in many villages.14 Indeed, this is perhaps the major assumption underlying most models of interlinkage, which proceed on the premise that there is one lender. Second, there might be more than one lender (in a village or region), but each might command a segregated chunk of the market owing to historically developed ‘personal ties’.15 No doubt such ties may be strong, and they might preclude a borrower from seeking a better deal elsewhere. Lenders with little knowledge of a new borrower could be unwilling to lend to him.16 Third, lenders might have access to a limited stock of loanable funds. This immediately confers on each lender a certain degree of monopoly power, for his rivals may not possess the extra wherewithal to undercut him.

Finally, what about two lenders who interlock in exactly the same way? The theory of interlinkage has little to say regarding the competitive behaviour of such agents. Far more relevant here is the theory of repeated games, which attempts to explain collusive behaviour in a backdrop of dynamic competition.17

It is for this reason that we emphasize the patterns of competition that may emerge from interlocked behaviour. These patterns might, in turn, harden into ‘historical ties’ and strengthen the surplus‐acquiring capacity of the lender. They might be reinforced by dynamic collusion, permitting each lender a greater profit margin in his sphere of activity. They might be further insulated by limited availability of credit. We submit that in all these cases the original pattern that emerges from a study of competing interlockers will stay the same.

Now to the specifics of an example. We consider two borrowers (farmers) who are characterized by (1) the amount of land (collateral) they possess, K j (j = 1,2), with K 1 < K 2, and (2) a production function18 θ f(K,E), where K is land input, E is labour input, and θ is a random variable with a distribution function G on [a,∞) where a > 0.

The farmers can sell their output on the market for a price of p. We take it (p.254) that each places on his land collateral a valuation of z per unit.19 Each farmer has a fixed demand for loan, L j.20

There are two potential lenders, with the same opportunity interest rate of loanable funds, r. Lender 1 is also a landlord. Lender 2 is also a trader. Two features distinguish them from each other.

First, the landlord values land more than the trader. Specifically, we assume that the farmer‐lender places the same unit value z on land as the other farmers. The trader's valuation is z * < z. This reflects (a) the imperfection of the land market, and (b) the fact that the landlord has a direct use for land.

Second, the trader can obtain a higher price for output. Specifically, we assume that the landlord can obtain the same price p as the borrowers, while the trader can get p * > p.

Lenders offer contracts to the borrowers, who are assumed to have no other source of loanable funds. The triplet c ≡ (q,α,i) describes a contract, where q is the price at which the lender offers to buy output from the borrower, α is the maximum fraction of collateral to be seized in case of a ‘large enough’ default (see below), and i is the rate of interest on the loan.

Implicit in a contract is what happens in the case of a default. For a borrower with inputs (K,E), loan L, and a contract (q,α,i), a default takes place if qθ F(K,E) < L(1 + i). In this case, we assume that the lender will take away only that much of the collateral that is required to settle the debt at the valuation of the borrower, up to a maximum of α K.21

The borrowers are assumed to have a utility function

$Display mathematics$
(8)
where W is wealth (collateral plus output net of loan repayment) and v is a ‘disutility function’ of labour effort with v′(·) > 0 and v″(·) ≥ 0. Borrowers maximize the expected value of utilities.

Given a contract c = (q,α,i), borrower j chooses E j to find

$Display mathematics$
(9)
where θ is defined by
$Display mathematics$
(10)

The expected income of the landlord‐lender from the contract c is then given by

$Display mathematics$
(11)
(p.255) where $θ ∧$ solves
$Display mathematics$
(12)
and E j solves (5), given the contract c.22

The expected return of a trader‐lender, Y 2(c), is just the same as the expression above, with (z *,p *) substituted for (z,p).

We model competition between the two lenders as taking place in two stages. In Stage 1, each lender chooses the borrower he will deal with. There is a small (but positive) cost of dealing with each borrower. In stage 2, if a borrower has been chosen by both lenders, the latter compete in contracts, and if the borrower has been chosen by one lender, that lender behaves monopolistically with him. Or he may be chosen by neither, in which case he is excluded from all credit sources. We look at the subgame perfect equilibrium of this two‐stage game.23

At this stage, a further assumption will be made, purely for expository purposes. Consider the farmer with more land, farmer 2. We assume that, if he were offered a contract at the lender's opportunity rate r, q set equal to the market price of the borrower p and α = 1, then he would (using his optimal effort Ê2) repay the loan even in the worst state without using collateral. Formally, we assume that

$Display mathematics$
(13)
where Ê2 is the optimal effort choice for borrower 2 under the contract (p, 1, r).

Put another way, this assumption states that the loan needs of the larger borrower are ‘small enough’ relative to his land holdings. We note, though, that this assumption is much stronger than necessary for our results. All we need is a statement that the loan of the smaller borrower relative to his land ownership is larger; that is,

$Display mathematics$
(14)
However, we use (13) for simplicity.

We may now state the third proposition.

1. PROPOSITION 3. In equilibrium,

• (a) no borrower faces more than one lender in the second stage;

• (b) if both lenders are active, then the landlord deals with the ‘small’ farmer (borrower 1) and the trader deals with the ‘large’ farmer (borrower 2);

• (c) the trader is always active. Sufficient conditions on the parameters of the system can be given so that the landlord is also active.

• (For proof, see the Appendix.)

The main thrust of the proposition is that landlord‐lenders are more likely to deal with small farmers, while trader‐lenders are more likely to interact (p.256) with middle and large farmers. Small farmers are more likely to default on loans, so that lenders who place a higher valuation on the collateral these farmers can offer are in a better position to offer them loans. Similarly, larger farmers have a smaller default probability and also produce a higher output, so that trader‐lenders with an advantage in the commodity market clinch credit deals with these farmers. This is the pattern of competition that is predicted by the theory of interlinkage. As a by‐product, it also provides ‘spheres of influence’ for each lender in which he has potential monopoly power. In our formulation, this comes out starkly in the form of part (a) of the proposition.24

The example described here can be used to obtain additional insights regarding the interlinkage mechanism. For instance, it can be shown that the trader will demand collateral from the borrower even if the former has no use for the collateral (z * = 0). The reason is that the threat of removing the borrower's assets forces him to put in more labour in order to repay the loan. This enables the trader to charge a higher interest rate while keeping the borrower's effort level high.25 Collateral is not asked for because the lender values the collateral: it is because the borrower values it, and will work harder to repay the loan. A ‘tougher’ loan can then be charged. Similarly, it is perfectly possible that the landlord might pay a price q that exceeds the price he can obtain on the market, p. Again, the idea is to elicit effort by providing a more suitable incentive structure. This surplus can then be removed by a higher interest rate.

The approach that we have outlined to describe patterns of competition can be extended to different contexts. In the particular context of landlord–trader competition, similar models can be constructed to obtain other patterns. Two examples are: (1) trader‐lenders will be more likely to lend to borrowers demanding production loans, while landlord‐lenders will deal with borrowers demanding consumption loans; and (2) landlord‐lenders will lend to landless labourers (using their labour as collateral), while traders will interact relatively more with borrowers who own land and therefore engage in production.

Our purpose in this paper has been to indicate a way of modelling these differential relationships.

Appendix

A1. An Equivalence Result for Models of Interlinkage

In this appendix, we demonstrate that, when (a) nonlinear contracts are allowed, (b) interlockers do not face different terms (relative to other inter‐lockers or pure money‐lenders) in the markets in which they are active, (p.257) (c) there is no differential information, and (d) there is no market price uncertainty: all interlockers are equivalent in terms of the surplus they can extract. In particular, a pure money‐lender can do just as well as an interlocker.

The proof of this result is intuitive, once the basic concepts are formally presented. To do this, we need some unavoidably cumbersome notation, which we now develop.

Consider an agent whom we shall call the borrower. Denote by N = {;1, . . . , n}; the set of all commodities. IN is the set of non‐marketed commodities (e.g. family labour in a model with no wage labour); J is the (remaining) set of marketed commodities. Each commodity is an input and/or output and/or consumption good. Each commodity in J has a price. Denote by p a typical price vector.26

The borrower is endowed with a non‐negative vector w of non‐marketed commodities, and a production technology represented by f, where $f : R + n × Ω → R + n$. That is, for each input vector $x ∈ R + n$ and the realization of a random shock $z ∈ Ω , f ( x , z ) = y ∈ R + n$ is the output vector.27 Let F be the class of all conceivable production functions that the borrower might have.

The borrower has a von Neumann–Morgenstern utility function u: R nR, defined on consumption. So in this situation, the borrower chooses inputs $x ∈ R + n$ and a ‘consumption function’ c(y) to solve:

$Display mathematics$
(A1)
$Display mathematics$
(A2)
$Display mathematics$
(A3)
$Display mathematics$
(A4)
where the notation x J for some vector xR n denotes the projection of x on the coordinate subsystem J. For ease of exposition, we permit consumption to be negative, although readers can easily check that this makes no difference, for all the examples in the paper continue to be special cases.

Now for some remarks on the constraints of the maximization problem. First, we allow consumption choices to be functions of realized output. This clearly must be true of some consumption goods, but it applies only to those bought after the realization of output. Consumption items that precede the ‘harvest’ cannot be conditioned. This particular specification is allowable in our model. Second, (A2) refers to a standard budget constraint for marketed commodities, while (A3) writes down commodity‐by‐commodity constraints on each non‐marketed commodity. Finally, (A4) refers to the connection between inputs and output for every state of nature.

Now we focus on loans, which we define to be a marketable commodity — say commodity 1 — described by the following characteristics. It is an ‘input’ and/or a ‘consumption good’ (but never an ‘output’ produced by the (p.258) borrower). If it is an input (a consumption good), we take it that a positive amount of it is used in production (in consumption) at the market prices.28

Next, we introduce the lender. He is defined by three characteristics.

1. 1. He can supply loans (commodity 1) at a price $p ∧ 1 < p 1$. But the prices faced by him in all other markets are the same as those faced by the borrower.

2. 2. He can be active in a certain subset MJ of the set of marketable commodities. Of course, 1 ∈ M. This means that he can offer the borrower price schedules different from market prices for these commodities. This, of course, necessitates some conditions on what he can observe in these markets.

3. 3. Let A = {;(x,y,c) triplets that can conceivably be chosen by the borrower};, and Q a space of observables. What the lender can observe is given by a mapping θ: AQ.29 In particular, some function g(x,y,c) is observable if, whenever g(x 1,y 1,c 1) ≠ g(x 2,y 2,c 2), θ (x 1,y 1,c 1) ≠ θ (x 2,y 2,c 2).

We now impose a condition on θ derived from M, the markets in which the lender can be active. For each iM, we assume either that (c i + x i,y i) is observable or that (x i,y ic i) is observable. (Formally, for each i, either the function g i(x,y,c) ≡ (c i + x i,y i) is observable or gi(x,y,c) ≡ (x i,y ic i) is observable.) In other words, total purchases and sales of commodities in M are observable. We then say that M is consistent with θ.

Our interest lies in keeping all the characteristics of the lender fixed (including the observability function θ), but varying M, the markets that he can interlock in.30 In particular, a pure money‐lender is a lender with M = {;1};.

We allow price contracts (possibly nonlinear) that are conditioned on the observables of the system. For a lender who is active in the set of markets M (an M‐lender), we can define a M‐contract as a mapping P:QR M. Its interpretation: for every observed θ ∈ Q, P(θ) is the price‐vector charged for the commodities in M.31

What is the borrower's maximization problem if he abides by an M‐contract? It is: find x and c(y) to

$Display mathematics$
(A5)
$Display mathematics$
(A6)
$Display mathematics$
(A7)
and
$Display mathematics$
(A8)

Denote by R(f) the maximized value in problem (A1)–(A4). The borrower of type f will accept the M‐contract if the maximized value in (A5)–(A8) is at least as great as R(f).

Our main definition is: an M‐contract and an M′‐contract are equivalent if (p.259) each borrower of type f, fF, faces exactly the same constraints (A6)–(A8) under the M‐contract and the M′‐contract.

Note that this is a strong definition of equivalence. In particular, it does not require us to specify the lender's maximization problem or the nature of his incomplete information about f. If equivalence in the sense of our definition holds, an M‐lender and M′‐lender can accomplish exactly the same outcomes.32

THEOREM. Fix the observability map θ. Then, for any M, M′ consistent with θ, and any M‐contract P, there exists an M′‐contract P′ such that P and P′ are equivalent.

Proof. A contract that alters only the price of the loan as a function of θ is clearly an M′‐contract, because 1 ∈ M′. So we now proceed in the following way.

Define P′: QR M as follows: for each iM′, i ≠ 1, Pi(θ) = p i for all θ ∈ Q. For i = 1, define first, for each (x,y,c), given the M‐contract P,

$Display mathematics$
(A9)

Note that y 1 = 0, so because 1 ∈ M, x 1 + c 1 is observable. Also, for each iM, i ≠ 1, given that either (x i + c i, y i) is observable or (x i,y ic i) is observable, clearly (y ic ix i) is observable. So p(x,y,c) varies only when θ varies.

We may therefore define P1(θ) by

$Display mathematics$
(A10)

This completes our definition of the M′‐contract. Readers should now check that, given our assumptions,

1. 1. the borrower of type f will accept the M′‐contract if and only if he accepts the M‐contract;

2. 2. under both the contracts, x 1 + c 1 > 0; and

3. 3. the feasible sets of the type (A6)–(A8) are identical under the given M‐contract and M′‐contract we have constructed. Q.E.D.

Remark. In particular, when M′ = {;1};, we have shown that an interlocker and a pure money‐lender can achieve exactly the same outcomes under the model described.

A2. Proof of Proposition 4 (outline)

1. (i) Suppose that more than one lender elect to deal with the same borrower. One can then prove that there is a unique Bertrand equilibrium payoff vector (p.260) for each firm in stage 2, and by standard arguments it can be seen that at least one lender earns an equilibrium payoff of zero. Knowing this, and given an additional positive cost of choosing to deal with a borrower, it is not a best response (in the Nash equilibrium sense) for that lender to choose to deal with the borrower, given that the other lender is doing so.

2. (ii) It suffices to show that the trader deals with borrower 2. For then, by the presumption that both lenders are active and (i) above, the landlord must deal with borrower 1.

Recall the definitions of U j(c) and Y i(c), j, i = 1, 2, given in the body of the paper. Now consider borrower 2. Fix a number U, and define

$Display mathematics$
(A11)

Define $U * = max { U / Y ~ 1 ( U ) ≥ 0 }$. Then, by the continuity of $Y ~ 1 ( U )$ (details omitted), $Y ~ 1 ( U * ) = 0$. For the rest of the proof, it suffices to show that $Y ~ 2 ( U * ) > 0$. (One can verify that this suffices by using the properties of Bertrand competition, the notion of a two‐stage equilibrium, and the fact that the cost of dealing with each borrower is positive, though infinitesimally small.)

Now, given the assumption (13), it is easily checked that the optimal contract to borrower 2, given a ‘reservation utility’ U *, is given by (q,α,i) = (p,1,r) = ĉ, say. But, again using (13), and the fact that p * > p, it is easy to see that Y 2(ĉ) > 0. So $Y ~ 2 ( u * ) ≥ Y 2 ( c ∧ ) > 0$.

3. (iii) Clearly, by (13) and the argument in (ii) above, the trader is always active. We now provide sufficient conditions for the landlord to be dealing with farmer 1 in equilibrium.

To do so, consider the monopoly problem for the landlord: to maximize Y 1(c), subject to the relations (9)–(12) (in Section 3). Let Ŷ1 denote this maximum value. Now consider an artificial problem, which is to perform this maximization exercise, with the additional constraint on contracts that α = 0. Let $Y 1 *$ denote this maximum value. Our condition is:

$Display mathematics$
(A12)

Readers can easily check that (A12) can be translated into a condition that L 1/K 1 must be ‘high’ (so that collateral is necessary for a monopoly lender to deal with the borrower) but not ‘too high’ (so that, even with collateral, it is not worth dealing with him).

We can now establish the following lemma.

LEMMA. There exists B > p such that, if p * ∈ (p,B], the landlord lends to the small farmer in equilibrium.

To show this, fix a number U and define, for p * ∈ [p,B],

$Display mathematics$
(p.261) These are the maximum returns of lenders 1 and 2 when dealing with borrower 1 with ‘reservation utility’ U.

Start with p * = p. Look at the monopolistic solution of the trader‐lender (with no reservation utility constraint for the borrower). If the lender's return is strictly negative, there is B > p such that it continues to be negative for all p * ∈ (p,B], by continuity of the trader's return. So we are done, because, by (A12), the landlord can certainly deal with the borrower.

If the trader‐lender's return is non‐negative, then by (A12), α > 0. In this case, define U by the condition

$Display mathematics$
(A13)

Let ĉ be the contract offered by the trader when U = Û. As we have observed, ĉ = (q,α,i) with α > 0. Now let the landlord offer exactly the same contract ĉ to borrower 1. Then, because p = p * but z * < z (and α > 0), it must be the case that Y 1(ĉ) > 0. So

$Display mathematics$
(A14)
That is, when p = p *, the landlord‐lender is active with the borrower 1 (by using the same reasoning as in part (ii) above). Now use continuity to define B > p such that, for all p * ∈ (p,B], conditions of the form (A13) and (A14) hold, with Û suitably defined. This completes the argument. Q.E.D.

# Notes

(1.) For surveys of the literature, see Bardhan (1980). Theoretical studies include Bharadwaj (1974), Bardhan (1984), Gangopadhyay and Sengupta (1987a, 1987b), Basu (1983), Braverman and Stiglitz (1982), Braverman and Srinivasan (1981), Mitra (1983), and many others. Among the empirical studies are Kurup (1976), Bandyopadhyay (1984), and Floro (1987). Floro (1987) and Das and Gangopadhyay (1987) also explicitly consider the competition between different interlocking money‐lenders, in independent work.

(2.) For an excellent survey of the literature during the 1970s and earlier, see Bardhan (1980).

(3.) We are focusing on credit markets and interlinkage, but the arguments apply more generally.

(4.) There is plenty of empirical evidence supporting such an observation, which is derivable on theoretical grounds. See references in n. 1.

(6.) Bardhan (1984: 91) notes—quite correctly—that the optimal contract takes the form of a two‐part tariff.

(7.) Of course, more general contracts can be considered. But it can be shown that the optimal contract derived in the text is optimal in the class of all contracts.

(p.262)

(8.) Readers should check that $L * = argmax { P ~ f ( L ) − ( 1 + r ) L }$.

(9.) The optimal contract described in Proposition 2 (for Example 2) does involve a return that is conditional on output, but it can easily be translated into a contract that depends only on the loan.

(10.) There are many models of interlinked contracts between landlord and tenant, although the simple version here is somewhat different. See Braverman and Srinivasan (1981), Braverman and Stiglitz (1982), and Mitra (1983), for example.

(11.) Newbery and Stiglitz (1979), as well as Braverman and Guasch (1984), deal with incomplete information in the somewhat different context of screening models.

(12.) We mention this in detail to distinguish a random wage from the randomness of the wage induced by unemployment. (See above, discussion following Assumption 2: there, the wage is random for the employee but not for the employer.)

(13.) Indeed, this is what we do below in Section 3.

(14.) For example, Bardhan (1984: 86) notes that ‘many village economies are characterized by a dominant landlord who, because of the size of his assets and urban connections, is able to obtain credit more cheaply than other local agents’.

(15.) For a discussion of this point, see Basu (1984).

(16.) However, we have reservations regarding the use of this observation as a long‐run explanation of monopoly power. Large and persistent profit opportunities open to a rival lender would eventually induce him to acquire information about the borrower and to lend to him.

(17.) For developments in this theory, see e.g. Friedman (1981), Green and Porter (1984), and Abreu (1984a).

(18.) We take it that the production function satisfies standard Inada‐type condition. This eases the exposition.

(19.) There is no market for land in this example. Accordingly, all valuations of land are to be interpreted as personal valuations (see e.g. Bhaduri 1977). We are also assuming for simplicity that unit valuations are insensitive to the amount of land owned.

(20.) The assumption that loan demands are fixed simplifies the exposition. Dropping it requires an analysis along the lines of the examples studied in Section 2.

(21.) We are deliberately ruling out collateral underpricing to ease the exposition. However, the example here can be extended to yield a model of collateral underpricing in a one‐lender, one‐borrower context. This might be of independent interest, as existing models of collateral underpricing (see Bhaduri 1977 and Basu 1983) are not very robust (see Gangopadhyay and Sengupta 1987b).

(22.) If there is more than one solution of (9), we choose the one most favourable to the lender.

(23.) Alternatively, a ‘one‐stage game’ could be studied, where both lenders simultaneously offer contracts to both borrowers. The equilibrium pattern of competition is unchanged, although the equilibrium profits of the lenders are affected. Our approach embodies (to some extent) a notion of ‘personalized ties’ in the second stage, should only one lender choose a borrower.

(24.) This part is a consequence of the two‐stage modelling, the positive cost associated with dealing with each borrower, and the Bertrand‐type competition that has been postulated.

(p.263)

(25.) Note that this line of reasoning is absolutely the opposite of that in Bhaduri (1977). There, a lender induces default to get at the collateral. Here, a lender asks for collateral to prevent default and obtain a higher interest rate.

(26.) The model may be easily generalized to allow for prices that depend on the quantities of purchases and sales. However, we do not permit uncertain prices.

(27.) We could easily allow consumption to influence production, to capture nutrition‐efficiency relationships, for example. We do not do this here for ease of notation. Also note that the endowment of marketed commodities can be subsumed in the production function.

(28.) One can easily make assumptions on the primitives of the model (technology utility function) to ensure that this is the case for all production functions fF.

(29.) Equivalently, it is given by a partition of A, but our formulation is a bit more explicit.

(30.) Of course, throughout, M must be observable (relative to θ) in the sense described above.

(31.) The notation R M stands for Euclidean space with dimensionality equal to the number of elements in M. Note that the lender does not price‐discriminate between inputs, consumption, and output. Introducing this additional feature makes no difference to the analysis.

(32.) That is, if for every conceivable M‐contract, there is an equivalent M′‐contract, and, vice versa, the M‐lenders and M′‐lenders may be regarded as equivalent.

(p.264)

## Notes:

We are grateful to Bhaskar Dutta and Subhashis Gangopadhyay for helpful comments. Ray wishes to record special thanks to Sergy Floro, whose forthcoming Ph.D. dissertation (Floro, 1987) independently explores (theoretically and empirically) some of these issues as well as many others. Numerous conversations with her during 1984 and 1985 have led to a better appreciation of this particular topic.

(1.) For surveys of the literature, see Bardhan (1980). Theoretical studies include Bharadwaj (1974), Bardhan (1984), Gangopadhyay and Sengupta (1987a, 1987b), Basu (1983), Braverman and Stiglitz (1982), Braverman and Srinivasan (1981), Mitra (1983), and many others. Among the empirical studies are Kurup (1976), Bandyopadhyay (1984), and Floro (1987). Floro (1987) and Das and Gangopadhyay (1987) also explicitly consider the competition between different interlocking money‐lenders, in independent work.

(2.) For an excellent survey of the literature during the 1970s and earlier, see Bardhan (1980).

(3.) We are focusing on credit markets and interlinkage, but the arguments apply more generally.

(4.) There is plenty of empirical evidence supporting such an observation, which is derivable on theoretical grounds. See references in n. 1.

(6.) Bardhan (1984: 91) notes—quite correctly—that the optimal contract takes the form of a two‐part tariff.

(7.) Of course, more general contracts can be considered. But it can be shown that the optimal contract derived in the text is optimal in the class of all contracts.

(8.) Readers should check that $L * = argmax { P ~ f ( L ) − ( 1 + r ) L }$.

(9.) The optimal contract described in Proposition 2 (for Example 2) does involve a return that is conditional on output, but it can easily be translated into a contract that depends only on the loan.

(10.) There are many models of interlinked contracts between landlord and tenant, although the simple version here is somewhat different. See Braverman and Srinivasan (1981), Braverman and Stiglitz (1982), and Mitra (1983), for example.

(11.) Newbery and Stiglitz (1979), as well as Braverman and Guasch (1984), deal with incomplete information in the somewhat different context of screening models.

(12.) We mention this in detail to distinguish a random wage from the randomness of the wage induced by unemployment. (See above, discussion following Assumption 2: there, the wage is random for the employee but not for the employer.)

(13.) Indeed, this is what we do below in Section 3.

(14.) For example, Bardhan (1984: 86) notes that ‘many village economies are characterized by a dominant landlord who, because of the size of his assets and urban connections, is able to obtain credit more cheaply than other local agents’.

(15.) For a discussion of this point, see Basu (1984).

(16.) However, we have reservations regarding the use of this observation as a long‐run explanation of monopoly power. Large and persistent profit opportunities open to a rival lender would eventually induce him to acquire information about the borrower and to lend to him.

(17.) For developments in this theory, see e.g. Friedman (1981), Green and Porter (1984), and Abreu (1984a).

(18.) We take it that the production function satisfies standard Inada‐type condition. This eases the exposition.

(19.) There is no market for land in this example. Accordingly, all valuations of land are to be interpreted as personal valuations (see e.g. Bhaduri 1977). We are also assuming for simplicity that unit valuations are insensitive to the amount of land owned.

(20.) The assumption that loan demands are fixed simplifies the exposition. Dropping it requires an analysis along the lines of the examples studied in Section 2.

(21.) We are deliberately ruling out collateral underpricing to ease the exposition. However, the example here can be extended to yield a model of collateral underpricing in a one‐lender, one‐borrower context. This might be of independent interest, as existing models of collateral underpricing (see Bhaduri 1977 and Basu 1983) are not very robust (see Gangopadhyay and Sengupta 1987b).

(22.) If there is more than one solution of (9), we choose the one most favourable to the lender.

(23.) Alternatively, a ‘one‐stage game’ could be studied, where both lenders simultaneously offer contracts to both borrowers. The equilibrium pattern of competition is unchanged, although the equilibrium profits of the lenders are affected. Our approach embodies (to some extent) a notion of ‘personalized ties’ in the second stage, should only one lender choose a borrower.

(24.) This part is a consequence of the two‐stage modelling, the positive cost associated with dealing with each borrower, and the Bertrand‐type competition that has been postulated.

(25.) Note that this line of reasoning is absolutely the opposite of that in Bhaduri (1977). There, a lender induces default to get at the collateral. Here, a lender asks for collateral to prevent default and obtain a higher interest rate.

(26.) The model may be easily generalized to allow for prices that depend on the quantities of purchases and sales. However, we do not permit uncertain prices.

(27.) We could easily allow consumption to influence production, to capture nutrition‐efficiency relationships, for example. We do not do this here for ease of notation. Also note that the endowment of marketed commodities can be subsumed in the production function.

(28.) One can easily make assumptions on the primitives of the model (technology utility function) to ensure that this is the case for all production functions fF.

(29.) Equivalently, it is given by a partition of A, but our formulation is a bit more explicit.

(30.) Of course, throughout, M must be observable (relative to θ) in the sense described above.

(31.) The notation R M stands for Euclidean space with dimensionality equal to the number of elements in M. Note that the lender does not price‐discriminate between inputs, consumption, and output. Introducing this additional feature makes no difference to the analysis.

(32.) That is, if for every conceivable M‐contract, there is an equivalent M′‐contract, and, vice versa, the M‐lenders and M′‐lenders may be regarded as equivalent.