האוניברסיטה העברית בירושליםThe Hebrew University of Jerusalem

המחלקה לכלכלה חקלאית ומנהל המרכז למחקר בכלכלה חקלאיתThe Center for Agricultural

Economic Research The Department of Agricultural

Economics and Management

Discussion Paper No. 16.09

Contracting with Smallholders under Joint Liability

by

Jonathan Kaminski

Papers by members of the Department can be found in their home sites:

מאמרים של חברי המחלקה נמצאים :גם באתרי הבית שלהם

http://departments.agri.huji.ac.il/economics/indexe.html

76100רחובות , 12. ד.ת P.O. Box 12, Rehovot 76100

Contracting with Smallholders under Joint Liability ∗

Jonathan Kaminski

Hebrew University of Jerusalem, Dept of Agricultural Economics,

P.O. Box 12, Rehovot 76100, Israel.

Email: kaminski@agri.huji.ac.il

December 22, 2009

Abstract

This paper examines the performance of contract farming when agents are groups

of jointly-liable farmers who receive input credit from a monopsonistic agribusiness.

Accounting for group mechanisms in credit repayment through joint liability and

peer monitoring, we derive the optimal monopsonistic contract under moral hazard

on production effort. The principal takes into account price incentives not only

on farmers’ effort but also on peer monitoring. Then, we show that the optimal

pricing rule is not monotonic with respect to the group’s characteristics. Imper-

fect information implies a distortion on pricing for low-efficient groups, which is

Pareto-improving from a social point. Groups of intermediary size and heterogene-

ity provide the best effort and peer-monitoring incentives.

JEL Codes: D82, L14, 013, Q13

Keywords : contract farming, moral hazard, joint liability, peer monitoring

∗I am grateful to participants of Toulouse Lunch Seminar, to Philippe Bontems, Pierre Dubois,

Stéphane Straub, Jean-Paul Azam, Shanta Devarajan, Vincent Requillard and François Salanié for valu-

able comments and useful advices. This work has benefited from the comments of Xavier Gine (World

Bank) at the NEUCD conference in Harvard University for a former version of this paper. I am also

indebted to participants of the conference on agricultural cooperatives held in Rehovot, at the Hebrew

University of Jerusalem, to Ayal Kimhi for financial support, and to Israel Finkelshtain, Ziv Bar-Shira,

Ethan Ligon, and Richard Sexton for technical advices.

1

1 Introduction

Contract farming has become one of the major tools for agricultural development in many

developing countries over the past three decades1. It typically involves an agribusiness

to provide quality inputs, a set of extension and other agricultural services, and pos-

sibly credit to individual or farmers’ groups. In exchange, farmers have a marketing

arrangement with a fixed price and outlet for their output on high-value markets (mostly

cash crops, organic and horticultural products) and must follow a particular production

method. Reciprocal obligations entail the provision of agricultural services and com-

mitment on purchase prices on the firm’s side, and quality requirements and exclusive

purchase rights on the farmers’ side.

The expansion of contract farming in the developing world has been driven by changes

in consumer demands, international trade, technology and policies. Contracting between

farmers and agribusinesses has enabled to tighten commodity chains and strengthen ver-

tical coordination2, which are increasingly required by the supermarket procurement sys-

tems or by manufacturing/retailing importers. In the course of commodity reforms (see

Akiyama et al., 2001), provision of inputs, rural credit, or extension services have become

problematic for small-scale farmers facing several market failures. Contract farming is

often one of the only channel that links smallholders to input and to output markets (Key

and Runsten, 1999) through the interlinking of markets. This is an efficient solution in

an incomplete market environment, as shown by the theoretical literature. Interlinked

agreements among rational economic agents arise in a very fragmented and incomplete

market environment (Mitra, 1983) where agents are isolated with barriers from entry to

some markets (Basu, 1983). Braverman and Stiglitz (1982) show that the creation of

efficient surplus is improved when interlinking occurs. Interlinkages make enforce trans-

actions easier and allow for information, enforcement and monitoring costs savings (see

1For a review and a complete list of references, see Bijman (2008). Little and Watts (1994) provide an

historical survey of contract farming in Sub-Saharan Africa. Glover (1984) focuses on outgrower schemes

in poor countries. Most of the papers in the development literature try to assess the impacts of contract

farming on smallholders’ welfare and when contract farming takes place. They provide an assessment of

advantages and drawbacks for contractors and farmers. This paper does not tackle all of these positive

issues. We are rather interested in efficiency and normative concerns.2Contracts reduce transaction costs through repeated interaction and a better control of the production

process (for instance see Winter-Nelson and Temu, 2002). They also provide better risk allocation for

farmers and lower coordination costs for contractors. The latter can better align with their customers’

demands by a greater regularity of agricultural product supplies and quality (with technical assistance),

bargain for more profitable prices for farmers, benefit from the low production costs of family farms, and

more easily access to credit and subsidies (vertical differenciation).

2

also Bell, 1989). Yet, this literature does not explore the endogenous mechanisms through

which contracts are enforced and production is monitored.

In this paper, we consider the interlinked nature of these contracts when the agribusi-

ness has a dominant position and contract with smallholders with no access to credit and

extension services. The outside option of smallholders is supposedly of limited scope,

because of market incompleteness (as for the case of outgrower schemes). We also account

for group mechanisms-namely joint liability3 and peer monitoring-whereby the agribusi-

ness elicit credit repayment of production inputs, due to several advantages. There is a

natural advantage to rely on peer-monitoring and social mechanisms through joint lia-

bility in many rural communities of the developing world, where many smallholders lack

appropriate collateral to secure a loan. Because of local informal institutions and social

ties, joint liability acts as an effective social collateral inducing delegated monitoring that

ensures workable credit repayment rates when groups are well designed4. Then, high

transaction costs are often prohibitive for the contractors who can rely on group mech-

anisms for monitoring, contracting, and enforcement. In addition, since credit is linked

to a productive output on which the contractor gets exclusive purchase rights, then it

provides an additional physical collateral.

Peer monitoring and joint liability are borrowed from the rich literature on group lend-

ing and collective credit agreements. Most of the theoretical literature has shown that

joint liability in credit agreements was efficient in several circumstances and may induce

higher profits for the contractor than under individual liability. First, group formation

allows positive assortative matching by affinities (Ghatak and Guinnane, 1999), thereby

reducing ex ante moral hazard (the risk type of the project or individuals: probability of

success). According to Besley and Coate (1995), the joint liability provides two opposite

repayment incentives, one positive coming from the mutual insurance of the group and

one negative because expected marginal profit depends not only on individual effort but

also on the effort of others (free rider’s effect). This negative effect from joint liability can

be mitigated by the use of credible social sanctions if farmers’ actions can be observable ex

post with a costly monitoring effort from the group. This will enforce cooperative behav-

iors within groups and will discourage opportunistic behaviors5. Armendariz de Aghion

3In case of one member’s defaulting, each member of the groups repays her share of the debt. Other-

wise, the group will be denied access to credit in the future, or the contract could not be renewed.4These collective credit agreements are widespread in the developing world, as witnessed by the suc-

cessful Grameen Bank in Bangladesh, and other microfinance institutions (see Morduch, 1999, for a

review).5They involve strategic defaulting due to ex post moral hazard (after the project’s outcome is realized),

that is, the strategy leading a borrower to default on credit repayment while being able to reimburse.

3

(1999) focuses on ex post moral hazard (strategic defaulting) through the lens of a group-

lending repayment game with a peer-monitoring decision level stage. For a monopolist

lender, it is proven that joint liability brings even more profit than individual creditors,

on condition that social sanctions are sufficiently credible and monitoring costs are not

too high. These insights also apply to the case of contract farming in developing coun-

tries with smallholders, where collective agreements are widespread (e.g. input credit).

This is notably the case for outgrower schemes whereby credit repayment is proceeded by

deducting its value from the output sales to the contractor. In this literature though, the

probability of the project’s success (and hence, of credit repayment) is exogenous and the

moral hazard problem is only explored through the project choice or the ex post behavior

of agents. Yet, peer monitoring must also affect actions of the agents ex ante through

individual and collective incentives for effort and project choice.

In this paper, we solve for a group-repayment game that is linked to a production

activity (the project) under the supervision of a Principal. The production activity is

undertaken according to an endogenous effort which can be only observable by other peers

(at some cost) and that determines the probability of success. Hence, both production

effort (and credit repayment probability) and the level of peer monitoring are endogenous

and simultaneously chosen by farmers. We restrict ourselves to the case of no strategic

defaulting, that is, when it is not possible for agents to contract or to market their

output with another agribusiness ex post. The group repayment game is linked to a prior

contracting stage between a Principal and a group of farmers (the agent).

The main contribution of this paper is to account for group mechanisms in the design

of contract farming, bridging the gap between the theoretical literature on interlinkages,

and the one on collective credit agreements. This is done by treating the problem of ex

ante moral hazard as endogenous on one hand. On the other hand, we consider contract

farming as an interlinked agreement with groups of smallholders within which strategic

interactions occur. We do not examine why contract farming with joint liability exist

or what is optimal scale of these arrangements in agricultural production. Rather, we

study the efficiency of such contracts and their optimal design from the agribusiness

perspective and from a social welfare point of view6. We show that the sufficient level

of peer monitoring needed to yield the cooperative effort equilibrium is less likely to be

undertaken by the group when individual and group production incentives are higher.

However, this is more likely to hold when the group’s marginal return on peer monitoring

6We assume that joint liability is favored by a Principal. We consider indeed the cases where the

Principal faces a disuasive contract monitoring cost, and farmers are endowed with no physical (or not

verifiable) collateral.

4

increases. Hence, the Principal faces a trade-off between price incentives on production

on one hand, and on peer monitoring on the other hand. Then, the optimal pricing rule

is not monotonic with respect to the group’s capacity of enforcing cooperative behaviors.

There are several implications for agricultural development. First, the efficiency of

contracting with smallholders relies on both individual and group characteristics. There

is thus a need for collecting data about individual and their related groups, which means

information on individual incentives and social dynamics. Second, information problems

are double-sided and have direct consequences on contract design. Imperfect information

on the group’s ability to peer monitor induces the Principal to adjust non-cooperative

pricing above or below the optimal level desired by the agribusiness, thereby inducing a

sub-optimal non-cooperative effort. Imperfect information on the group and individual’s

effort costs induce the Principal to move away from efficient pricing for the higher-cost

group types. Last, the efficiency of such contracts is affected by group design (size and

composition), with implications in terms of the regulation of co-operative formation.

The remainder of this paper is organized as follows. Section 2 presents the conceptual

framework with a group of two symmetric farmers (the agent) and an agribusiness (the

principal) under perfect information. Section 3 deals with imperfect information and

efficiency concerns. Section 4 explores the robustness of the model and the implications

of group’s size and heterogeneity. Section 5 concludes and discusses the policy implications

of the model.

2 The model

2.1 Game-theoretical framework and basic assumptions

In the basic framework, a group-the agent-is composed by two symmetric farmers who

contract with a monopsonistic agribusiness (hereafter denoted as the Principal). First,

the Principal sets the terms of the contract by defining the purchase price of the cash

crop p and providing one unit of input credit. The cost of one physical unit of input

is normalized to unity and the Principal faces a marginal receipt at farm gate ep7. Weassume that all farmers who participate to contract farming are cash-constrained and

cannot access rural credit markets, so they have no alternative to access inputs: seeds,

fertilizers, and pesticides. In the basic model, we also assume that the Principal gets

all the bargaining power and is a monopsony over crop purchases (at least at the local

7This is basically the FOB price of crop production from which transport and processing costs are

deducted.

5

scale), making a take-it-or-leave-it offer. Relaxing this assumption entails contractual

enforcement problems and possibility of side-selling. This will be discussed at the end

of the paper. The Principal is assumed to have perfect information about farmers and

group characteristics (relaxed in the next section). Second, the two farmers play a credit-

repayment game in group, that is, they are jointly-liable for the repayment of their peers.

If the group repays its whole credit at the end of the game, then contract farming can

be repeated in the future. Otherwise, the group is denied credit access in the repeated

stages of the game. The structure of this sub-game is the same as Armendariz de Aghion

(1999) except that the probability of repayment is endogenous in our model, and depends

upon an effort production variable. In addition, there is no strategic defaulting (ex post

moral hazard). Assuming no parallel market, there is no opportunity for side-selling the

cash crop in the case of contract farming with a monopsony. While peer monitoring is

a device to deter strategic defaulting in Armendariz de Aghion (1999), we use it here to

reduce the scope of ex ante moral hazard. The timing of the game is represented in the

below figure. Note that this one-shot game can be repeated n times, which would lead

to endogenize the value of contract renewal. Implication of repeated interactions will be

discussed at the end of the paper.

Principal-agent Simultaneous monitoring and effort Returns Learning Social sanctions contracting decisions: realizations: peer’s effort and refunding p*, and input credit γ*(e) and ec(γ)/ enc(γ) /Y Y ec/ enc W, V

Credit repayment sub-game

Figure 1: Principal-group game under joint-liability

After the contracting stage, farmers simultaneously choose a level of individual pro-

duction effort e and of peer monitoring γ, which are assumed to lie on the interval [0, 1].

The individual effort is private information but can be observed ex post with probability

γ. The equilibrium choice of γ is the minimal desired level of peer monitoring among the

group’s members. The level of effort represents a labor effort and the quality of input

application (that can be diverted or resold on the black market). Effort is assumed to

have a quadratic cost C(e) = ce2/2 and peer monitoring a linear one C(γ) = dγ.

We model effort as a moral hazard variable, that is, the observed production outcome

is realized with a probability increasing with effort. For sake of simplicity, we assume that

farmers can reach two levels of production, when using one unit of input:

6

Y with probability e (1)

Y with probability 1− e

Note that although production outcome is random, there is no exogenous source of risk

in the model. Adding exogenous risk would not add much to our analysis, except when

considering risk-averse farmers.

The lower production outcome does not allow the principal to recover the input loan,

while this is the case for the higher one. In the latter case, the farmer gets a positive

profit, while he is defaulting in the second case:

(p− p)Y > 1 > (p− p)Y and pY > 1 > pY (2)

Since farmers have a limited liability, they get zero profit in case of defaulting (no side-

selling). If the group defaults, we assume that accessing credit is lost for the future and

any produced cash crop is seized8 Since we are interested in the case of joint liability, we

allow for a group of two symmetric farmers to be able to repay if one of the two farmers

defaults, such that:

pY + pY > 2 (3)

When playing in groups, farmers can choose their individual-maximizing effort, which

drives an individual optimum level of effort. However, they can attain a better level of

effort that would make every farmers better off. This cooperative effort ec that they want

to induce (maximizing the joint profit of the group) cannot be supported as an equilibrium

since everybody has individual incentives to deviate. Deviation involves savings on cost

effort while benefiting from other’s higher efforts (less expected individual and group’s

defaulting). Let us now formally describe how cooperative effort can be enforced through

peer monitoring.

With peer monitoring, farmers’ efforts are now possibly observable with some proba-

bility. Setting the cooperative level of effort as a group’s objective (or an informal rule or

norm), anyone who is observed cheating can be punished. If the group observes ex post

that a farmer has not respected this commitment and played its individual maximizing-

profit level of effort enc(non-cooperative), she will be imposed a social sanction W. This

social sanction can be a reputational loss in the farmer’s community or an exclusion from

the group for instances. Incentives to invest in peer monitoring thus depend on its effi-

ciency, that is, the relative level ofW compared to d, and on the additional profit moving8We assume limited liability of farmers as no more than the produced cash crop can be seized by the

monopsony.

7

from non-cooperative to cooperative production efforts. Simultaneous monitoring and

effort decisions are thus endogenous and related to one another. According to Armen-

dariz de Aghion (1999), full joint liability and implementation of the whole social sanction

are always optimal for the group. We then derive the optimal solutions under full joint

liability and punishment.

Once production occurs, farmers learn their peers’ effort with some probability and

punish cheaters. If the group repays its global debt, then it will be refunded for the next

period. The exogenous value of accessing credit in the next periods is denoted V > 09.

All parameters are common knowledge within the group.

2.2 The two-symmetric credit-repayment game

Assuming risk-neutrality, we can write the objective function of the group of two sym-

metric farmers under a full joint-liability agreement (if same efforts are undertaken by

players). This contains individual and group incentives:

Eπ = e2[pY − 1] + e(1− e)[p(Y + Y )− 2] + V e(2− e)− ce2/2 (4)

where e2 is the probability that both farmers repay their credit and e(1− e) is the prob-

ability that the individual farmer repays but has to finance the debt of her peer. Finally,

e(2− e) is the probability that at least one farmer repays its credit, so that the group can

be refunded in the next period. The two first terms in the RHS are individual profit incen-

tives, and the third one is a group incentive: the whole group repayment and motivation

for contract renewal or reputation-building.

We solve the game by backward induction. First, we compute the cooperative, and

then, the non-cooperative production effort levels (maximizing individual objective), then

we look at equilibria, according to peer-monitoring decisions. We notably define parame-

ter intervals where cooperative and non-cooperative effort levels are equilibria. Second,

according to the cooperative and non-cooperative effort levels, we derive the optimal

peer-monitoring investment. We finally derive simultaneous effort and peer-monitoring

equilibria according to the model’s parameters.

To get the optimal cooperative effort level, we maximize (4) with respect to effort

under the participation constraint that expected profit is non-negative and treating the

9V can be interpreted as a reputational loss for the farmer with respect to the Principal, but also as

earnings losses from not accessing future credits. In the latter case, V can be endogenized in a repeated

game framework, as mentioned earlier. However, an exogenous V is robust to our results (see final

discussion).

8

peer’s effort as endogenous. Note that this is sufficient since farmers are symmetric.

First-order conditions entail:

ec =p(Y + Y )− 2 + 2Vc+ 2pY − 2 + 2V (5)

To have interior solutions, we assume that

c > p∆Y = p(Y − Y ) => c/2 > 1− pY (6)

because of (3). To make sure that this assumption is restrictive, we need to consider cases

where c/2 < 1. The non-cooperative effort is the optimal individual effort, treating the

peer’s effort as exogenous, such as:

maxe

ee0[pY − 1] + e(1− e0)[p(Y + Y )− 2] + V (e+ e0(1− e))− ce2/2 (7)

where e0 stands for the exogenous effort of the other player. First-order conditions yield

the best-response function of each farmer:

enc(e0) =p(Y + Y )− 2 + V

c− e0

(pY − 1 + V )

c(8)

Note that for low values of V , efforts are strategic complements, while they become

strategic substitutes when V is higher.

Now, we look at the effort equilibria according to peer-monitoring levels. In the credit

repayment game, there are two pure strategies: play cooperatively or not. Here is the

table of all possible effort equilibria

Strategies Cooperative Non-cooperative

Cooperative (ec, ec) (ec, enc(ec))

Non-cooperative (enc(ec), ec) (enc(enc), enc(enc))

Table 1: Strategies of the credit repayment game

(ec, ec) is an equilibrium if the expected profit of playing cooperatively is greater than

deviating minus the expected social sanction, that is Eπ(ec, ec) > Eπ(enc(ec), ec) − γW.

Similarly, (enc(enc), enc(enc)) is an equilibrium if and only if the return from cooperative

behavior is not offset by the additional profit of deviation minus the expected social

sanction. In other terms, Eπ(enc(enc), enc(enc)) − γW > Eπ(ec, enc(enc)). After some

calculations, we find that the cooperative effort level is a symmetric equilibrium if and

only if:

γ > γc = min(c(∆e)2

2W, 1) (9)

9

and that the non-cooperative one is a symmetric equilibrium as soon as:

γ < γnc = (min(c(δe)2

2W, 1) (10)

where ∆e = ec−enc(ec) > 0 and δe = ec−enc(enc) > 0 because of (5) and (7) when effortsare interior solutions. When V < 1 − pY , δe > ∆e => γnc > γc, and this is the reverse

for the other case. For large V , when the peer-monitoring level is between these two

thresholds, only (ec, enc(ec)) can occur10 because γnc < γc. However, it can be shown that

in-between levels of peer monitoring cannot support this equilibrium. Two conditions are

necessary:

Eπ(ec, ec) < Eπ(enc(ec), ec)− γW => γ < γc

Eπ(ec, enc(ec)) > Eπ(enc(enc), enc(enc))− γW => γ > γnc,c

Only the second condition is relevant for in-between levels of peer monitoring. The equi-

librium exists for in-between level of peer monitoring whenever γnc < γnc,c = γnc + (δe−∆e)(ec(pY − 1 + V ) − V )/W < γc. Because V is large, then δe < ∆e and this con-

dition cannot be ever satisfied. Asymmetric equilibrium cannot be supported by given

peer-monitoring levels. The asymmetric equilibrium only exists for peer-monitoring lev-

els larger than γnc,c, that is, in the region where cooperative equilibrium exists and is

Pareto-dominating.

Note also that for low V, the intermediary peer-monitoring levels induce the occur-

rence of both cooperative and non-cooperative equilibria. The cooperative one Pareto-

dominates the latter. This is represented in Figure A.1 (see appendix). From (9) and

(10), we can discuss the pattern of peer-monitoring thresholds that define the areas of

equilibrium occurrence. The slope and shape of these thresholds are examined in the

appendix. The main insights can be summed up in the following lemma.

Lemma 1 When 2 symmetric farmers play a credit repayment game under full joint

liability and moral hazard on production effort, the peer-monitoring level γc that is re-

quired to enforce cooperative equilibrium increases with V. The level of peer-monitoring

γnc needed to deter the non-cooperative equilibrium first increases then decreases with V.

Mixed-strategy equilibria with the probability to play cooperatively

β =γnc,c − γ

γnc,c − γc

are relevant in the regions where γ belongs to [γnc,c, γc] and [γc, γnc,c]. In [γnc, γc], neither

pure-strategy nor mixed equilibria exist.10This equilibrium would be an asymmetric one.

10

Proof. See appendix.

We found how effort equilibria are related to levels of peer-monitoring. Now, we need

to derive what are the levels of peer-monitoring equilibria, according to effort levels and

to the parameters of the model. Peer monitoring is profitable when it enables farmers

to undertake cooperative effort instead of non-cooperative one. As it is costly, optimal

peer monitoring is the minimum level of peer monitoring that can trigger a profitable

shift in effort equilibria. The solutions of the credit-repayment game can be stated in the

following proposition and represented in the below graph.

Proposition 2 In the credit repayment sub-game with two symmetric farmers under

moral hazard, simultaneous individual effort and collective peer-monitoring decisions en-

tail:

• Farmers play cooperatively with γ∗ = γc when V and d/W are low

• Farmers play non-cooperatively with γ∗ = 0 when V and d/W are high

• Farmers play mixed strategies with γ∗ = γ < γc when V is low and d/W is inter-

mediary

• All effort levels increase with V and p and decrease with c

Proof. See appendix for solving, computational details, and comparative statics.

V

d/W

1-pY

1

Non-cooperative effort and no peer monitoring

Cooperative effort and peer monitoring γc

Mixed strategy with

γ and β

Figure 2: Effort equilibria and optimal peer-monitoring decisions

Intuitively, when V becomes large, cooperative and non-cooperative effort become so

close that it is no longer profitable to peer monitor since the marginal profit is too low.

11

Even a lower peer-monitoring cost (associated with mixed strategies) is not profitable to

invest for intermediate expected profits. In contrast, mixed strategies are profitable when

V is low since it is associated to lower peer-monitoring costs and intermediate expected

profits.

Note that the occurrence of cooperative equilibrium first increases with V for its lowest

values since efforts are strategic complements when V is low (see figure 2). Figure 3 below

displays effort equilibria and their related levels along V and according to d/W . Note

also that an increase in p will correspond to a leftward shift of the curves of Figure 2.

Cooperative equilibrium can be more likely for low values of V but then is less likely. The

likelihood of non-cooperative equilibrium increases for larger values of V. In brief, a price

change has non-monotonic effects and should be accounted for by the Principal. That is

the aim of the next subsection. For sake of simplicity, we will only consider pure-strategy

equilibrium occurrence in what follows.

V0

1

1/2

1 -p Y

ec

enc (enc)

d/W=γcc/γc

Note : The black line is the effort equilibrium when d/W is very low. This trajectory is partly replaced by the grey portions when d/W is of intermediary values and below 1. For higher values, this is replaced by both grey and red portions

d/W=γcc/γc

Figure 3: Sequence of effort equilibria along V

2.3 The contracting stage

Assuming that the Principal is a monopsony having perfect information about farmers’

characteristics, it will optimize its own profit accounting for farmers’ reaction within

their groups, as it has been solved for the credit-repayment game. In a first step, the

12

monopsony maximizes its own profit, according to effort levels. We first solve this op-

timization problem for both effort levels, cooperative and non-cooperative ones. Then,

we look at the output price for which farmers are indifferent between playing coopera-

tively or not. Comparing the indifferent price bp to the optimal pricing under cooperativeand non-cooperative farmer’s behavior enables us to define an optimal pricing rule for a

monopsonistic Principal.

According to the effort reaction of farmers ex, then the monopsony

maxp

Πx(p) = (ep− p)(Y + ex∆Y )− 1 + (e2 + 2e(1− e)) (11)

where the x upperscript stands for the type of effort equilibrium (either x = C, or x =

NC). We assume that the participation constraint of farmers is not binding at optimum

(ep is sufficiently high). We obtain the following first-order condition:[ep− p∗

p∗]x =

1

εEY x/p− 2(1− ex)

p∗∆Y(12)

which is the optimal margin for the monopsonistic Principal, and where EY x is the

expected production level when effort is x and εEY x/p is the individual price elasticity of

supply. Note that the monopsonistic margin is below the standard Ramsey rule since the

monopsony internalizes farmers’ credit repayment and the associated complementarity

between interlinked input and output markets. When V is not too small, the optimal

margin is unambiguously higher under cooperative effort since effort is less reactive to

price. Then, under lower price elasticity and higher effort, the monopsony can propose

a lower price to cooperative farmers compared to non-cooperative ones. For low V the

above expression is also lower under cooperative effort reaction though ∂ec

∂p> ∂enc

∂p.

We then look at the indifferent price level bp such that farmers are indifferent betweencooperative and non-cooperative strategies. According to our computations in the previ-

ous subsection, it means that dW= γcc

γc. This must satisfy the following equation:

(c+ bpY − 1 + V )2d

W= c(c+ 2(bpY − 1 + V ) (13)

such that 1 > c/2 > 1− bpY > 0 (14)

(13) admits solutions and at least one when dW

< 1. This is a necessary condition to have

a non-zero probability of playing cooperatively among farmers, as already derived in the

previous subsection. These solutions are such that :

1− bpY = V − c

d/W[1−d/W ±

p1− d/W ] = {V −V2, V −V1} = {1−pY , 1− pY } (15)

13

where V1 = 1 − d/W −p1− d/W > V2 = 1 − d/W +p1− d/W . (15) satisfies the

constraints stated in (14) if {V − V1, V − V2} ⊂ [0, c/2]2 for the two solutions to be

feasible. For at least one to exist, one of the two elements of {V − V1, V − V2} must becontained in [0, c/2]. We then derive the ranges of parameters (V, d/W ) where solutions

exist. This can be represented in the below figure:

V

d/W

c/2

1

Play NC: p < p

Play C: : p > p

Play C: p < p

Play NC: p > p

Play NC: p < p

Play C: p <p< p

Play NC: p > p

Play C always

Play NC always

Play NC always

Figure 4: Existence of p and p and farmers’ behaviour

The optimal solution for the monopsony would then account for the change in the

effort type of producers x, according to the parameters of the group: d/W and V . The

decision set can be illustrated in the below figure. We denote pc < pc the two bounds on

the Principal profit curve under cooperative effort such that:

maxΠnc(p) = Πnc(pnc∗) = Πc(pc) = Πc(pc) (16)

Since Πnc is always lower that Πc for every p, then pc < pc∗ < pnc∗ < pc. From Figure 4

and our previous computations, we know that the optimal pricing rule for the Principal

is pc∗ when d/W is low and pnc∗ when d/W > 1 or V is very large. The intermediary

case is displayed below. Note that an increase in ep makes shift all the four price curvesupwards. Then the sequence of optimal prices with respect to V would shift leftward. We

see that the sequence of optimal prices is discontinuous with respect to V , decreasing by

parts but non-monotonic. Then, the following proposition can be stated:

Proposition 3 Optimal pricing for a monopsonistic Principal entails (assuming ep issufficiently high):

• When d/W > 1 or when V is very large, p∗ = pnc∗(non-cooperative pricing)

14

• When d/W is close to 0, p∗ = pc∗ < pnc∗(cooperative pricing)

• When d/W is in between, p∗follows this sequence (possibly left-truncated when ep islarge) according to V : pnc∗, p, pc∗, p, pnc∗ (discontinuous pricing)

• pc∗ and pnc∗ decrease with V and increase with epProof. See appendix

V

p=1/Y

p=(1-c/2)/Y

p p

cp

*ncp

*cp

cp

Π nc

Π c

cp *cp *ncp cp p

Π Π c*

Π n c*

V increases

Figure 5: Optimal pricing rule when d/W is intermediary

Hence, according to parameters related to farmers and group’s characteristics, optimal

pricing can change and will induce different effort equilibria in the credit repayment

game. Note that the monopsony can achieve optimal profits along the [Πnc(pnc∗);Πc(pc∗)]

interval. In addition, an increase in ep does not necessarily ensure the monopsony toachieve higher profits since it will shift the two profit curves to the right (figure 5 above)

and cooperative effort might be less likely to hold in equilibrium. Last, note that a change

15

in d/W from 0 to 1 will make decrease the range of V s for which p∗ = pc∗ between p and

p. The assumption of perfect information now needs to be relaxed.

3 Imperfect information and welfare analysis

A fully-informed Principal has the ability to achieve optimal profits according (i) to the

group’s capacity to peer monitor and enforce cooperative equilibrium, and (ii) to the

level of farmers’ effort as a function of prices. Complete information about V , d/W ,

and c enables the Principal to know which pricing would be associated to which effort

equilibrium and its level in the credit repayment game. Let us first relax the assumption

that the Principal has perfect information about d/W , which means that the monopsony

cannot determine which equilibrium will be enforced by the group but knows both levels

of cooperative and non-cooperative effort equilibria.

We start by assuming that there is a probability µ that the Principal contracts with

a group endowed with d/W < 1, and we consider the case where it is optimal for the

Principal to induce a cooperative effort equilibrium by pricing pc∗ (for intermediary V ).

With probability 1 − µ, d/W > 1 and the group will play non-cooperatively for sure.

Assuming participation constraint is not binding at optimum, we only need to ensure the

contract to be incentive-compatible for the most efficient group (Laffont and Martimort,

2001). It can be written:

Eπ(pc, ec(pc))− dγc(pc) ≥ Eπ(pnc, ec(pnc))− dγc(pnc) (17)

= Eπ(pnc, enc(pnc)) +Ψ(δe(pnc), pnc)− dγc(pnc)

where Ψ(δe(pnc), pnc) = (δe(pnc))2[(c/2 + pncY − 1 + V )] = Wγcc(pnc), such as defined

in (A.4). Ψ(δe(pnc), pnc) − dγc(pnc) is the information rent that the Principal must let

to the efficient group (with a fixed transfer for instance). Since we assume non-binding

participation constraints, the maximization program of the Principal can be written:

maxpc,pnc

µΠc(pc) + (1− µ)Πnc(pnc) such that Eπ(pc, ec(pc))− dγc(pc) ≥Wγcc(pnc)− dγc(pnc)

(18)

Denoting λ as the Lagrange multiplier associated to the incentive-compatibility constraint

of the efficient group, we therefore obtain the following first-order conditions:

Πc0(pc) = 0 => pc = pc∗ (19)

Πnc0(pnc) =λµ

(1− µ)[∂Ψ(δe(pnc), pnc)

∂pnc− dγc0(pnc)] (20)

16

We obtain that second-best solutions rely on the sign of the expression in brackets. It can

be shown (see appendix) that the expression may be positive for low V s and is always

negative for high V s. The information rent induces an inefficiency for the Principal on

non-cooperative pricing, accounting for strategic complementariness when V is low and

strategic effort substitutability when V is high. According to the increasing value of

V , it turns out that non-cooperative pricing may be first distorted downward and then

(unambiguously) upward. Adjusting the non-cooperative price enables the Principal to

minimize the informational loss on profits. Conditions are summarized in the below

proposition. Note that, according to parameters, the pooling contract with pnc = pc = pc∗

is possible under specific values of the parameters, yielding no informational rent to the

efficient group, but a high distortion on non-cooperative pricing.

We now assume that the Principal has perfect information on d/W but cannot deter-

mine the effort reactions of farmers because of imperfect information on c for instance.

Consider the Principal faces an efficient group with a low cost of effort and probability υ

and an inefficient one with probability 1− υ. Then it can be shown by the same reason-

ing that the Principal has to let an information rent to the efficient group and to adjust

pricing for the cost-inefficient group in the same direction and for both cooperative and

non-cooperative prices.

Proposition 4 Under imperfect information on d/W , pc = pc∗ and pnc > pnc∗ if d/W

and V are not too low. In the reverse case, pc = pc∗ ≤ pnc < pnc∗. Imperfect information

on the individual cost of effort also entails the same kind of adjustments for cost-inefficient

groups on both cooperative and non-cooperative pricing.

Proof. See appendix

This makes sense with our former results. When d/W and V are very low, cooperative

effort is an equilibrium and its likelihood increases with price under the lowest values of V

because of strategic complementarity of efforts. So, it is better to decrease non-cooperative

price in this specific case in order to make the contract incentive-compatible since it will

reduce the opportunity for efficient farmers to choose the contract for the inefficient ones

by lower profits and a less likely cooperative equilibrium.

We now want to look at the effect of imperfect information on welfare. We first define

an overall objective function for the whole industry (agribusiness Principal and producers).

We then derive the social optimum pricing rule and compare it to the solutions derived

earlier. The optimal effort (assuming that it yields positive profits for the agribusiness

and the farmers’ group) of the supply chain solves:

17

maxeepEY − pY + e(2− e)(pY + V )− ce2/2 (21)

which yield the following fist-order condition:

ePO =ep∆Y + 2pY + 2V

c+ 2pY + 2V(22)

Note that the social Pareto-optimal level of effort cannot be a cooperative equilibrium.

Indeed, equalizing this expression with (5) yields: ePO = 1 + (ep− p)∆Y/2 > 1 to ensure

that the agribusiness will get positive profits. The only feasible Paretian effort is thus

non-cooperative. By the same reasoning, we find that ePO = enc(enc) = (p−p)∆Y+2+VpY+1+V

,

which cannot be an interior solution either. Because of different objectives and vertical

differentiation, there does not exist any pricing rule that can yield a Pareto-optimal effort

level in the credit repayment game11. The Pareto-optimal effort is thus not achievable.

We need to look at the Pareto-optimal pricing rule, given the fact that e is a solution of

the credit repayment game. As earlier, denoting x = C,NC, we look at the solution of:

maxp

epEY x − pY + ex(2− ex)(pY + V )− cex2

/2 (23)

yielding:∂ex

∂p=

Y (1− ex(2− ex))ep∆Y + 2(1− ex)(pY + V )− cex(24)

*cp *ncp *cp *ncp ncpop c

pop p Note : Imperfect information on the group type makes non-cooperative pricing move following the arrows. The dash lines represent two social and Principal optimal rules

cep

∂∂

*ep

∂∂

poep

∂∂

ncep

∂∂

Figure 6: Comparison of two social optimum and Principal rules (ep increases)11It can be shown that the social optimum level of effort can be induced under individual liability.

18

The Pareto-optimal rule can be compared to the one of the monopsony. It can be repre-

sented in the above figure. The optimal pricing rules under Principal-agent contracting

are below the social Pareto-optimal ones. The largest inefficiency applies to cooperative

pricing because the optimal pricing rule for the Principal always entail that pc∗ < pnc∗.

Imperfect information is then Pareto-improving for non-cooperative pricing when V is not

too low, because of Proposition 4. Note that increases in ep are Pareto-improving sincePrincipal optimal and social optimal rules become closer.

4 Robustness

In this section, we discuss the validity of our results when we generalize the group design

by allowing larger or heterogeneous groups of farmers. We particularly study how size

and heterogeneity impact on effort and peer-monitoring incentives. We restrict ourselves

to the case of cooperative ec and non-cooperative enc(enc) equilibria. We also discuss the

issue of repeated interactions between the Principal and the farmers’ group, and how it

would affect our results.

4.1 Group size

Increase in the size of the group would change incentives for effort as the outcome would

largely depends on the behavior of others but not in the cooperative case as everybody

endogenize the same behavior. There are more people to share the deficit of defaulting

farmers with but it is possible to have more defaulting farmers. One’s own contribu-

tion to the likelihood of accessing future credit is reduced. Size changes the probability

distribution of ex ante expected profits.

We first look at the effect of size on cooperative and non-cooperative effort levels, then

we discuss the effect on peer monitoring. We assume groups are formed by an even number

of members. With n symmetric risk-neutral farmers, the cooperative effort (joint-profit

maximizing) is solved for:

maxe

k=n/2Xk=0

µn− 1k

¶[en−k(1− e)k][pY − 1 + nV − k(1− pY )

n− k]− ce2/2 (25)

An intuitive way to identify the channel whereby size influences effort incentive is to

look at the cumulative probability distribution of profits according to e, holding it fixed.

Then, effort incentives would be higher if the cumulative distribution exhibits first-order

stochastic dominance. The below graph shows that the answer is not obvious and that

there is no first-order stochastic dominance. However, according to our parameters, there

19

are distributions (for intermediary sizes) that may second-order stochastically dominate

the others and provide the optimal incentives.

When farmers play non-cooperatively, larger groups not only decrease incentives to get

V but also individual profits since the cumulative distribution of profits is less affected

by own actions. Strategic effort substitutability and complementariness are of smaller

scopes. As a consequence, the non-cooperative effort level decreases with group size.

V ( ) 2p Y Y V+ − + 1pY V− + Note : the cumulative distribution in bold represents profits when n=2 and the one in dash is when n is large

Prob 1 1-e² (1-e) (1-e)²

Figure 7: Cumulative probability distribution of profits when x = C

These discussions allow us to conjecture what changes in terms of peer monitoring

incentives. It exists a range of group size where cooperative effort incentives are higher,

meaning that γnc and γc take on higher values. Because of the curvature of the peer-

monitoring threshold with respect to V , then the frontier curve of parameters for which

cooperative effort can be supported as an equilibrium would shift upward, thus increasing

the occurrence to reach the cooperative equilibrium. For these specific parameter values,

then it exists an optimal size where effort and farmers’ profits are the largest. However,

when the group has a low ability to enforce the cooperative equilibrium, smallest groups

are desirable.

20

4.2 Group heterogeneity

Consider one homogeneous credit group in which the two farmers have the same bV . To ac-count for heterogeneity in our model, we study the two-player game with two asymmetric

farmers, endowed with two different V such that:

bV = V + V

2and ∆V = V − V (26)

We aim to compare effort and peer-monitoring incentives in the homogeneous group with

those provided by the heterogenous one. The cooperative effort remains the same because

the credit group maximizes the joint profit of farmers, i.e. ec(bV ).When farmers maximizetheir own profit, then each farmer will exert enc(V ) and enc(V ). It can be obtained:

enc(V ) =(p(Y + Y )− 2 + V )(1− pY − V ) + c(p(Y + Y )− 2 + V )

c2 − (1− pY − V )(1− pY − V )> enc(bV )

enc(V ) =(p(Y + Y )− 2 + V )(1− pY − V ) + c(p(Y + Y )− 2 + V )

c2 − (1− pY − V )(1− pY − V )< enc(bV ) (27)

The average non-cooperative effort in an heterogenous group is below the level of an

homogenous one if and only if enc(V )+enc(V )2

< enc(bV ), which is equivalent (after computa-tions) to

(1− pY − bV )22

[(∆V )2 − 4bV ] + (∆V )2

4((p(Y + Y )− 2 + 2V − 2c) > 0 (28)

This is true whenever heterogeneity is high for a given level of bV . The ratio (∆V )2

Vmust

be sufficiently high to ensure this condition.

Since farmers have heterogeneous preferences, their motivations to peer monitor each

other are different. The rule of the game is that the chosen peer-monitoring level is the

lowest among the desires of farmers. Meanwhile, the needed level of peer-monitoring to

enforce cooperative equilibrium are different among farmers. The first consequence is

that the cooperative equilibrium will be less likely to occur for low V s since it would

require more costly peer monitoring. Second, the cooperative equilibrium would be less

likely to hold for higher V s. Third, the two farmers might be located in different areas

of Figure 3, and this eventuality is more likely when heterogeneity increases. If they

are in the same area, nothing changes: each play either cooperative or non-cooperative

with the (higher) level of peer monitoring required to support the equilibrium. If not,

the equilibrium level of peer monitoring is the lowest of farmers’ desires. It can induce

both farmers to play non-cooperatively for low V s instead of asymmetrically (one wants

cooperative, and the other one do not). This is inefficient when it induces the high-V

21

type to play a lower effort level (non-cooperative instead of cooperative). This is more

likely when heterogeneity increases.

Overall, some restricted level of heterogeneity can be efficient for the group to increase

the average incentive for non-cooperative effort but over a certain threshold, the effect on

the most motivated farmers is worse than the positive effect on the less-motivated one.

In addition, cooperative equilibrium is less likely to hold.

It is possible to derive several implications, knowing that size and heterogeneity are

positively correlated. When the group has a high d/W then small groups are desirable.

When d/W is lower, intermediary groups are desirable since cooperative equilibrium can

be enforced under higher effort levels and low heterogeneity is consistent with cooperative

equilibrium. However, the optimal size as if the group was purely homogeneous may entail

inability to enforce higher cooperative effort equilibrium since heterogeneity might be too

high. Hence, the optimal size may be consistent with some level of heterogeneity which

allow farmers to play cooperatively with higher level of cooperative equilibrium (according

to the probability distribution of profits in the group). That is why larger groups exhibit

more free-riding and less cooperative behavior.

If group formation is under a free-adhesion principle, then farmers will accept new

members up to the point that it will no longer contribute to increase their individual

profits. It is not likely that free association of members will lead to the Pareto-optimal

group size and heterogeneity. Matching by affinities induces the most performing and

motivated farmers to go together while groups of less-performing ones would like to be

more heterogeneous to provide better incentives and expected profits to their members.

As a result, groups can be too small or too large. Starting from the optimal group size

and composition, the structure might not be coalition-proof with the willingness of the

most-performing farmers to establish their own group. The final matching outcome will

depend on the initial distribution of individual types. It is then likely that the optimal

group design is just a framework that is not sustainable as an equilibrium of a matching

process. However, any regulation of cooperative formation could be worse because it

will define membership through rigid exogenous parameters (such as origin, geographical

location, village residence...). Many contract farming schemes with smallholders took

place with village groups where low incentives for production were observed. Allowing

free association of members could clearly be a channel of welfare improvement.

4.3 Repeated interactions

If the credit-repayment game is repeated n times, then it is possible to endogenize the

value of V. Then, further implications can be derived. The probability to get play the

22

nth time is [e(2− e)]n. Let us just state that the value of contracting and accessing credit

is the value of expected profit minus an opportunity cost Z, which is the value of the

alternative option, assumed as less valuable than contracting. Hence, if the conditions

(parameters) of the game do not change over time

V =nX

k=1

[e(2− e)]k{e2[pY − 1] + e(1− e)[p(Y + Y )− 2]− ce2/2− Z} (29)

when e is the cooperative effort. Otherwise:

V =nX

k=1

[e+ e0(1− e)]k{ee0[pY − 1] + e(1− e0)[p(Y + Y )− 2]− ce2/2− Z} (30)

when e is the non-cooperative effort reaction as a function of e0, the other farmer’s effort.

All our conclusions from the above sections about the impact of V on contracting can then

be reinterpreted in terms of the parameters of the model. From the comparative statics

derived in the appendix and the above equations, it can be established that V reinforces

the positive effects of price p and ∆Y and the negative one of the effort cost c on effort

incentives. This also amplifies the effects of price and cost on the probability of effort

equilibrium. As a consequence, since price p is a decreasing function of V in the optimal

decision of the Principal, then there exists a unique crossing point where p∗ is such that

it is an optimal price for the given V (p∗). Optimal pricing is lower when c increases since

the price elasticity of supply is higher for both effort types. The combination of lower

pricing rules and lower V means that the effect of cost on optimal pricing is ambiguous

(increase effort incentives on one hand, but higher margin on the other hand).

With repeated interactions, the issue of imperfect information becomes less relevant

since the Principal could assess the parameters of the group as time goes by. Hence, if V

is large, repeated interactions would not be Pareto-improving from a social point of view.

5 Conclusion

This paper explored the design of contract farming with groups of smallholders, addressing

the problem of ex ante moral hazard on agricultural production through joint liability and

peer monitoring. The scope of ex ante moral hazard problem can be overcome by the use of

internal peer monitoring. However, we showed that effort incentives and peer-monitoring

ones go in opposite directions according to individual agents and group’s parameters.

Hence, the optimal contract for a monopsony entails specific pricing rules which can be

23

discontinuous as a function of these parameters. Imperfect information on individual and

group’s parameters may be welfare-enhancing.

More importantly, this paper provides several policy implications. First, any price-

regulation policy should be made accordingly to the available information on both indi-

vidual and groups (cooperatives) of smallholders. Second, group design will affect both

individual and group incentives and regulation of cooperative formation should then en-

able farmers to match by affinities through free association and co-option.

Our results are valid in the case where enforcement problems are only linked to ex

ante moral hazard issues and incomplete markets. However, peer monitoring is also a

useful device to reduce the scope of ex post moral hazard through strategic defaulting

on credit or side-selling the output. This enforcement problem occurs when there exist

alternative options for farmers, leading to less incentives for the agribusiness to provide

inputs. As shown by Delpierre (2008), competition would lead to a decrease of input or

credit provision, thereby justifying the observed competition-coordination trade-off by the

empirical literature (Poulton et al., 2004). Several solutions exist from a social welfare

point of view. First is to regulate the market structure so as to minimize side-selling

while promoting a certain degree of competition. Second is to promote capacity-building

for farmers’ associations so as to constitute a significant bargaining power when facing

a monopsonistic or duopsonistic downstream industrial structure. These two solutions

could be examined in future research.

Last, we should also consider policies that aim to overcome market incompleteness and

smallholders’ liquidity constraints by reducing transaction costs and information asymme-

tries. In this case, contract farming is less relevant when input use is provided to groups.

Second, the partial slackening of the liquidity constraint has ambiguous effects on input

use and income (Delpierre, 2008). Hence, these two sets of policies-industrial regulation

and more accessible rural markets-should be carefully designed and related to one another

during their respective implementation.

Our theoretical results might be translated in terms of testable hypotheses if empirical

studies for contract farming with smallholders are designed to capture all individual and

group-relevant information. This is left for future research.

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26

6 Appendix

V

γ

γ c

1-pY

γ nc A

B

D C

Figure A.1: Areas of effort equilibrium in the (γ, V ) space

6.1 Proof of Lemma 1

Proof. We have to analyze the shape of the two functions γc(V ) and γnc(V ). We find

that∂γc

∂V=

∆e

W

c(c− p∆Y )

(c+ 2pY − 2 + 2V )2 > 0 (A.1)

because of the assumption made in (6). The second derivative is:

∂2γc

∂V 2=

c(c− p∆Y )2

W (c+ 2pY − 2 + 2V )4 −4c∆e(c− p∆Y )

W (c+ 2pY − 2 + 2V )3 (A.2)

which is negative when V is sufficiently high. This is notably the case when V > 1− pY .

It is possible than the function is convex for the lowest values of V according to the values

of parameters c, p, ∆Y . For the other peer monitoring level, we find:

∂γnc

∂V=

cδe

W{ 2(c− p∆Y )

(c+ 2pY − 2 + 2V )2 −(1− enc)

c+ pY − 1 + V} (A.3)

which is always negative when V > 1 − pY . However, it can be positive for the lowest

values of V if c is high. Recall that γnc > γc when V < 1 − pY and γnc = γc when

V = 1− pY . For high values of V,the curve is convex. Note that the slope of γnc does not

really matter for low values of V since γc determines the frontier of equilibrium occurrence

according to peer-monitoring levels. Then the curves displayed in Figure A.1 are one

possibility. In any cases, γc is an increasing function and γnc decreases for high values of

V , driving the results stated in Lemma 1. For in-between levels of peer-monitoring, there

is no Pure-strategy Nash equilibria, only mixed ones exist. A mixed-strategy equilibrium

implies that a player is indifferent between the two strategies with the probability β to

27

play cooperatively and (1− β) to play non-cooperatively. At the equilibrium, the below

condition should be satisfied:

βEπ(ec, ec) + (1− β)Eπ(ec, enc(ec)) = βEπ(enc(ec), ec) + (1− β)Eπ(enc(enc), enc(enc))− γW

β =γnc,c − γ

γnc,c − γc

yielding the result of Lemma 1.

6.2 Proof of proposition 2

Proof. Note that mixed-strategy equilibria is only relevant in the regions [γnc,c, γc] and

[γc, γnc,c] because of Lemma 1. For large V , it means that mixed-strategy equilibria exist

in the region of cooperative equilibrium, which is Pareto-dominating, and supported by

lower peer-monitoring costs. Hence, mixed-strategy equilibria are only relevant for low

V . Denoting: γcc = Eπ(ec,ec)−Eπ(enc(enc),enc(enc))W

= min(γnc[1+2(pY −1+V )/c], 1), farmers

choose between:

• γ∗ = 0 if Eπ(enc(enc), enc(enc)) > Eπ(ec, ec) − dγc since peer-monitoring invest-

ment acts as a sunk cost, players choose the minimal amount to ensure a Pareto

improvement by a shift in effort equilibria.

• γ∗ = γc if Eπ(ec, ec)− dγc > Eπ(enc(enc), enc(enc))

• γ∗ = γ when Eπ(β)−dγ > Eπ(enc(enc), enc(enc)) and Eπ(β)−dγ > Eπ(ec, ec)−dγc.We obtain the indifference curves that delimit the parameters ranges within which

effort and peer-monitoring equilibria occur. Indifference between cooperative and

non-cooperative behavior is defined by the indifference curve:

d

W=

γcc

γc(A.4)

Let us check whether and when mixed strategies can Pareto-dominate the pure ones.

We know that it only occurs for low V. Indifference between mixed strategy and the

cooperative one is defined by the curve:

d

W=

γcc + γnc,c

γc − γnc,c(A.5)

This curve is below the one described in (A.4) for the lowest values of V and there-

fore imply that the mixed strategy will be preferred to the cooperative one up to

the point where γnc,c > 0. Doing the same for the indifference between mixed strat-

egy and non-cooperative one implies that the mixed strategy is preferred to the

28

non-cooperative one when the latter is Pareto-dominated and the former Pareto-

dominates the cooperative one, such that:

d

W=

β(γcc + γc)

γ− 1 < γcc + γnc,c

γc − γnc,c<

γcc

γc(A.6)

Then, only (A.5) is relevant to determine the region of Pareto-dominating mixed-

strategy equilibria, such as depicted by Figure 2. This curve is strictly increasing for

low V and crosses the curve of (A.4) when V < 1− pY . Then, the mixed strategy

is an equilibrium for the lowest values of V and intermediary levels of dW

< 1.

6.3 Comparative statics

Parameters of the model

Endogenous variables p c ∆Y V d/W

ec + - + + 0

enc(enc) + - + + 0

enc(ec) + - + + 0

δe +/- - - +/- 0

∆e + +/- - + 0

Pr(ec) +/- -/+ +/- +/- -

Pr(enc) -/+ +/- -/+ -/+ +

6.4 Proof of proposition 3

Proof. According to feasible equilibria of the credit-repayment game, the two first bullets

are trivial and are depicted in Figure 4. They are just an application of the equations

regarding the existence of indifferent prices for the farmers’ effort reaction. In the in-

termediary case, farmers play cooperatively only if price is between p and p. So, it is

optimal for the monopsony to propose pc∗ when the price that maximizes profit under

cooperative equilibrium is lying on [p, p]. But the monopoly can afford higher profits that

the non-cooperative one when p < pc by proposing p and when p > pc by proposing p.

Otherwise, nothing else better than pnc∗ can be done. An increase in ep induces a transla-tion of optimal pc∗and pnc∗ as well as for pc (larger effect) and pc (smaller effect) so that

the sequence of optimal prices in the discontinuous case can be left-truncated with respect

to V. Because both effort equilibria increase with V and their price elasticity decrease,

then optimal pricing rules can be lower for the monopsony, meaning a higher marketing

margin on the output.

29

6.5 Proof of proposition 4

Proof. We need to study the sign of ∂Ψ(δe(pnc),pnc)∂pnc

−dγc0(pnc). After several computations,we find that

∂Ψ(δe(pnc), pnc)

∂pnc=

(Y + Y )(2− pncY )− 2V Y − 2Yc+ 2pncY − 2 + 2V (A.7)

− (pncY − 1 + V )

(c+ pncY − 1 + V )2{(Y + Y )c+ V Y −∆Y }

dγc0(pnc) =cd

W∆e

∂∆e

∂pnc< 0 (A.8)

Expression (A.7) has an ambiguous sign, but is undoubtedly positive when V is low and

negative when V is high. Note that (A.7) is already negative when V = 1−pncY . It meansthat the distortion on non-cooperative pricing involves a move along the non-cooperative

profit curve of the monopsony along its decreasing slope when V is not too low. That

means that the difference between cooperative and non-cooperative pricing will increase

with V (as well as the information rent). For the very low values of V, the distortion may

be of opposite sign provided dWis low.

30

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