Supply Chain Analysis of Contract Farming

Awi Federgruen
Graduate School of Business, Columbia University, New York, New York 10027, af7@columbia.edu

Upmanu Lall
Columbia Water Center and The Department of Earth and Environmental Engineering, Columbia University, New York, New
York 10027, ula2@columbia.edu

A. Serdar Şimşek
Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX 75080, serdar.simsek@utdallas.edu

Contract farming is a growing practice in developing countries and first-world economies, alike. It generates
necessary guarantees to sustain the continued operations of vulnerable farmers while enabling the manufac-
turers to manage the aggregate supply and price risk. We consider a single manufacturer who owns several
manufacturing plants, each with a random demand for the crop. The manufacturer selects a set of farmers
to offer a menu of contracts, which is exogenously specified or endogenously determined. Each “selected”
farmer chooses a contract from this menu in advance of the growing season. After the growing season, under
known demands and supplies, the manufacturer minimizes the distribution costs from the selected farmers
to the production facilities. We formulate this problem as a Stackelberg game with asymmetric informa-
tion, where the manufacturer is the leader and the farmers are followers. The manufacturer’s problem is a
two-stage stochastic planning program for which we develop two solution approaches. We have applied our
model to problem instances anchored on data from a large manufacturer of potato chips contracting with
thousands of small farmers in India. We report on the performance of the solution methods compared to
a lower bound based on the Lagrangean dual of the problem and show that the optimality gap is below
1%, for problem instances with 1,000 potential farmers. We also show how our model can be used to gain
various managerial insights. As an example, when constructing the contract menu endogenously, often a
small number of contract options suffices, depending on the degree of heterogeneity among the farmer pool.
Thus, relatively simple menus often suffice.

Key words : contract farming; food and water security; optimization under uncertainty; stochastic model
applications; agriculture/food industry

1. Introduction and Summary

Contract farming is an emerging practice in developing countries such as India, Brazil, and Turkey.
Food processing companies, hereafter referred to as manufacturers, contract with a large number of
relatively small and financially challenged farmers. The industry distinguishes between two types
of contracts: under production contracts, the buyer owns the product while it is being produced
and the farmer is compensated for the services he provides; in a marketing contract, the farmer
maintains ownership over the crops but, in its simplest form, the manufacturer guarantees to buy
(up to) a given quantity of the farmer’s crop in a season, for a set guaranteed price. More specifically,
the manufacturer buys either the farmer’s realized yield or the specified contract volume, whichever
is lower. In this paper, we confine ourselves to marketing contracts.

1
2 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

A manufacturer often contracts with hundreds or thousands of farmers. These farmers are dis-
persed across different regions, often with vastly different climate and soil conditions, water supplies
and farm sizes.
Contract farming protects the farmer against the many risks he is facing, in particular the risk
of volatile crop prices on the commodity spot markets, yield risks and the difficulties of finding
buyers for their crops. Indeed, commodity prices are very volatile; for example, in 2013, the spot
price for potatoes more than doubled compared to its value a year earlier; see Table 6 in Appendix
F for another example. (Appendices A–F can be found in the Online Appendix which is published
as supplementary material. Appendices G–K can be found in Federgruen et al. 2018.) Yield risks
arise due to factors such as uncertain weather conditions and high volatility in the water supply.
However, contract farming is also highly desirable, if not outright critical, for the manufacturer,
and there are several reasons for this. First, contracts are essential to enable or incentivize farmers
to remain in the market, given the daunting risks they experience otherwise. Second, the high
volatility of spot markets presents a major problem for the manufacturer as well. Third, while
airlines, for example, are able to hedge against the risk of fluctuating fuel prices by investing in
call options, such hedging instruments are far less prevalent for agricultural commodities, where
the option markets have limited liquidity. In India, for example, option contracts on commodities
are not yet permitted, even though the Security Exchange Board of India has considered their
introduction (see Rajib 2014, 2015).
Fourth, and perhaps most importantly, manufacturers typically need their input commodity (e.g.
potatoes) to comply with a series of detailed specifications. These specifications, for example the
use of specific seeds, can be built into a contract, but, on the spot market, it is very hard to find
(sufficient) supplies that conform with them (see MacDonald et al. 2004, p. 29). The contracts also
enable the buyer to provide equipment, raw materials, and financing to the typically cash-strapped
farmers. Finally, quality inspections are hard to perform in the context of spot market transactions,
but are naturally built into the farm contracts.
Take, as an example, the Starbucks company. Its most important input is unroasted coffee beans.
In 2014, Starbucks bought more than half a billion pounds of these beans, contracting with more
than 300,000 farmers worldwide in Latin America, Africa, and Asia; see Gruley and Patton (2014).
In its contractual arrangements, Starbucks often specifies bean varieties, including specific new
hybrid seeds. It also trains the farmers it contracts with, to standardize their methods in efficient,
sustainable ways under the company’s Coffee And Farmer Equity (C.A.F.E.) Practice program.
Monitoring and enforcing compliance is a challenge in all of contract farming, but no more than
in other industrial contracts with upfront price or quantity commitments. If anything, monitor-
ing of yields in open fields is easier than in industrial settings where products are manufactured
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 3

behind closed doors; it is often done using satellite remote sensing or drones. Much thought has
been given to best practices in this area. As an example, a “Legal Guide on Contract Farming”
was developed, in 2013 – 2015, by the International Institute for the Unification of Private Law
(UNIDROIT), in partnership with the FAO. The FAO also put out a document “Contract farming
– partnerships for growth”; see Chapter 6, “Monitoring performance”, for detailed and effective
monitoring procedures. Thus, contract compliance can be and is monitored effectively. Beyond
the formal justice system, this creates a natural deterrence for a farmer to violate the agreement,
thereby jeopardizing or foregoing future contracts in upcoming seasons.
Thus, contract farming is a novel supply chain arrangement that benefits all parties. In particular,
it generates necessary guarantees to sustain the continued operations of very vulnerable suppliers
(farmers) – a major socio-economic goal for developing countries.
Moreover, contract farming is a common and growing trend in first-world countries as well. It
covered no less than 39% of US agricultural production in 2008, up from 11% in 1969 and 28% in
1991; no less than 90% of sugar beet and tobacco production was conducted via contract farming;
see MacDonald and Korb (2011) and Table 9 in Appendix G. As in emerging countries, American
farmers are exposed to production risks resulting from droughts, frost, hail, diseases, and insect
infestations, among other unusual events, and the above severe volatility in commodity prices on
the spot market, providing a major incentive for contract farming.
However, the devil is in the tactical details. To ensure an adequate supply of the crops, how
many farmers should the manufacturer contract with? How should specific farmers be selected?
What is the impact of the offered menu of price-quantity contracts and how should the menu be
chosen? If the manufacturer has a network of production facilities, which farmers should supply
which of these facilities?
To answer these interdependent questions, we analyze the following model and develop effective
solution methods: There is a single manufacturer who owns several manufacturing plants, each
with a specific, albeit random demand for the crop. The manufacturer incurs two types of cost: pro-
curement and distribution costs. She offers a specific menu of contracts to a selected set of farmers
in advance of the growing season. A contract specifies the unit price and quantity (among other
terms and specifications). The actual purchase amount is the minimum of the committed quantity
and the realized yield. Offering a menu of contracts is essential under information asymmetry, i.e.,
when the manufacturer has less than perfect knowledge of all of the farmers’ characteristics, for
example their yield characteristics.
Each “selected” farmer chooses a contract from this menu which results in the highest expected
profits for him, among those that satisfy certain participation or risk aversion constraints. Our
model assumes that the water supply is the principal random factor affecting a farmer’s total yield,
4 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

primarily due to uncertainty about the rainfall process. In some applications, farmers have access
to surface water (from sources such as lakes and rivers) as a second source of water supply. In
addition, some farmers are able to supplement the water supply by drawing ground water from
a capacitated well, incurring additional energy and water costs. As with the rainfall and surface
water quantity, the well’s capacity in any given season is, often, random as well. The yield of the
fields is a function of the total water supply, i.e., the aggregate of the rainfall, surface and drawn
ground water. With minor adjustments, the model can be applied to other –or multiple– yield risk
factors.
During the season, the various random factors become known: first, the rainfall, surface water,
and the well capacity of each of the farmers become revealed. Thereafter, each contracted farmer
determines how much water to draw from a capacitated well (if available). This determines the
farmer’s total yield, a (non-linear) deterministic function of the farmer’s combined water supply.
The manufacturer observes, by the end of the season, its product demand at each plant and the
realized supplies of all contracted farmers. She then determines an optimal distribution plan to
supply its plants from the contracted farmers, drawing from the spot market or an external source
in case of a nationwide shortage of farmers’ supplies.
In selecting how many and what specific collection of farmers to contract with, the manufacturer
wishes to minimize expected aggregate costs subject to a coverage constraint ensuring, with a given
high likelihood, that the nationwide aggregate supply matches or exceeds aggregate nationwide
demand. The importance of this constraint follows from the fact that, for all the reasons outlined
above, the manufacturer cannot rely on the spot market as a supply source other than as a rare
emergency backup solution, at worst to be used for a small part of the overall supply.
We formulate this problem as a Stackelberg game, where the manufacturer is the leader and the
farmers are followers. The manufacturer has imperfect knowledge of part of the parameters in the
distributions that describe the farmers. In other words, some of the farmers’ information is private,
giving rise to a game with asymmetric information.
The farmers only need to know the parameters pertaining to their own farm and, at first, select a
contract from the menu. Various NGO’s and government agencies assist farmers with their contract
decisions. An example is SNV, working together with the Food and Agriculture Organization (FAO)
of the United Nations; see e.g., Sango et al. (2016) reporting on their contract farming support
program in Zimbabwe.
In our base model, the menu of contracts is exogenously specified. (A different menu may be
offered in different regions and to farmers of different land sizes.) However, we also show how
the manufacturer may choose the menu of contracts, endogenously, along with her other decision
elements, so as to minimize her expected costs.
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 5

Even for a given menu of contracts, the manufacturer’s problem resulting from the farmers’ best
responses to the offered menu is a complex two-stage stochastic program: she first has to select the
set of farmers to work with in the upcoming growing season, and after the growing season, under
known demands and supplies, she determines the distribution plan from the selected farmers to the
production facilities; this to minimize the aggregate of the expected procurement and distribution
costs. An important feature of this two-stage stochastic program is the aforementioned coverage
constraint. As mentioned, we also show how an optimal menu of contracts may be selected along
with the manufacturer’s remaining decision elements. In doing so, we follow a standard paradigm
in contract theory, see e.g., Bolton and Dewatripont (2005), Lovejoy (2006) or Lutze and Ozer
(2008).
Under a given menu of contracts, the manufacturer’s two-stage stochastic planning problem –
incorporating the farmers’ best response functions– may be formulated as that of minimizing a
supermodular set function subject to the above coverage constraint as a single side constraint. In
other words, we view the outcome of the stochastic program as a function of the set of selected
farmers. A set function is supermodular if the cost savings resulting from the addition of a given
farmer decrease when this farmer is added to a larger collection of farmers. We develop two algo-
rithmic approaches to solve the manufacturer’s problem, as well as a lower bound for the optimal
cost value, based on the Lagrangean dual resulting from the Lagrangean relaxation of a pair of
quadratic constraints that are equivalent to the above probabilistic coverage constraint.
We have applied our model to problem instances that are calibrated to data from a large man-
ufacturer of potato chips contracting with thousands of small potato farmers in India.
In summary, this paper makes the following contributions:
(1) To our knowledge, ours is the first paper to propose a farmer selection model to be used by a
manufacturer contracting with hundreds or thousands of farmers, to be chosen from an even larger
set of potential suppliers. The model assumes that each of the selected farmers is offered a menu of
contracts, which may be exogenously specified or endogenously determined along with the selection
of the farmers. The model is analyzed as a Stackelberg game with asymmetric information, with
the manufacturer as the leader and the selected farmers as the followers, each selecting the contract
which maximizes his expected profit subject to specific risk constraints. The manufacturer’s selec-
tion problem incorporates the end-of-the-season problem of optimal distribution from the selected
farmers to the plants as a two-stage stochastic program, since the farmers’ locations represent an
important criterion for this selection.
(2) We develop two approximate algorithmic approaches to solve realistic, large-scale problem
instances in a very modest amount of time: instances with a thousand potential farmers can be
solved in a few minutes of CPU time, when implemented in a standard laptop platform. The average
6 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

value of (an upper bound for) the optimality gap is less than 1%. This means that the heuristics
are indeed close to optimal and the lower bounds are close to exact. Moreover, the optimality gap,
always less than 1.5% for the best of the two algorithms, declines as the problem size increases.
(3) Our numerical studies have identified several interesting qualitative insights, the most impor-
tant of which are summarized in the Conclusions section.
The remainder of this paper is organized as follows. In Section 2, we review the relevant lit-
erature. Section 3 develops our model. Section 4 characterizes the solution to the farmers’ best
response problem in response to the offered menu of contracts. Section 5 shows how the menu(s)
of contracts may be endogenously determined. Section 6 develops the two algorithmic approaches
for the manufacturer’s optimization problem. Section 7 reports on a large scale numerical study.
Our major conclusions are summarized in Section 8.

2. Literature Review
This paper contributes to the recent literature on supplier selection and diversification. Almost
all of this literature has been motivated by industrial or public health applications. The initial
literature on procurement problems under random yields confined itself to models with a single,
albeit unreliable supplier. See Grosfeld-Nir and Gerchak (2004) for a survey of the literature up
to 2004. Several papers, in the nineties and the new millennium, addressed procurement strategies
when there are two or an arbitrary number of potential suppliers, respectively. See Federgruen and
Yang (2011) for a recent survey on this literature steam.
There are a few papers with endogenously determined yield distributions, in particular: Liu et al.
(2010) consider a newsvendor model and a single supplier with a stochastically proportional yield;
the authors investigate the impact of improving the yield factor distribution and replacing it by
a stochastically superior one. Federgruen and Yang (2009) address yield factor distributions that
are endogenously determined by any number of suppliers as part of a competition game. Wang
et al. (2010) assume, within a two-supplier random capacity model, that the suppliers’ capacity
loss distributions may be controlled by selecting an effort level from a continuous spectrum. In
a two-stage process, the purchasing firm first decides how much effort to invest to improve the
capacity loss distribution of both suppliers; after observing whether these efforts have succeeded,
the purchasing firm decides on order quantities to cover a single season demand volume.
Our model differentiates itself from the above existing literature, by addressing several important
complications and generalizations: (i) each farmer/supplier is offered a menu of possible contracts
rather than a single order quantity at a single per unit cost price; (ii) after selecting a target
delivery quantity from the menu, the farmer can, at least partially, control what fraction thereof is
generated and sold to the buyer, by selecting the water supply from accessible wells to complement
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 7

his random rainfall (and surface water, if applicable); (iii) rather than servicing a single location
with random demand, we consider settings where random demands occur at an arbitrary number
of locations, and the purchased supply quantities need to be distributed to these demand points so
as to minimize aggregate distribution costs. Another important distinction in the contract farming
context is that the number of potential (and actually retained) suppliers is in the hundreds or
thousands, while in the above industrial applications, the number of potential suppliers is typically
below 10. The qualitative implication of this distinction is that, in our context, supplier selection
cannot be undertaken by a brute force enumeration of all possible sets of suppliers. At the same
time, our model addresses a single season, while several of the above papers deal with models with
multiple procurement opportunities and inventories carried from one period to the next.
Several papers have addressed procurement strategies for food processors. Devalkar et al. (2011)
consider a multi-period setting in which the manufacturer procures its input commodity entirely
from the spot market where it is subject to a spot price process which is stochastic but of a known
distributional form. The authors were motivated by ITC, one of India’s largest private sector food
processing companies, and its procurement of soybeans for the production of soybean oil and meal.
Since all inputs are purchased on the spot market, the Devalkar et al. (2011) model does not
involve the selection of supply sources and associated contracts and the paper does not need to
contend with yield uncertainty and distribution costs, as in our model addressing contract farming.
Similarly, Tan and Çömden (2012) present a planning methodology for a manufacturer wanting
to match the random supply of premium fruits and vegetables with random retailer demand for a
given set of contracted farms, operating under a given price contract.
Similar to our paper, Chaturvedi and de Albeniz (2011) consider a supplier selection problem
for a manufacturer, where the suppliers face uncertain yields and there is information asymmetry
between the manufacturer and the suppliers. However, the yield uncertainty is assumed to be of a
binary nature: either the supplier succeeds in delivering the full order size, or he delivers nothing.
This representation of the yield factor does not fit the farmers’ yield uncertainty, where it varies
continuously and depends, in an intricate manner, on the rainfall and other sources of water supply.
Rather than ensuring, with a coverage chance constraint, that the total supply covers a total random
demand volume, the authors assume that the manufacturer wishes to maximize the expected value
of a concave increasing function of the realized supply. Moreover, in Chaturvedi and de Albeniz
(2011) only the aggregate realized supply matters, while in our model, given the importance of
distribution costs, its geographic dispersion across the country is of essential importance as well.
Boyabatli et al. (2011) assume that the manufacturer may procure her input commodity either
from upfront contracts or from the end-of-the-season spot market; both sources are viewed as
equally desirable except for possible price differentials. The demand volume is deterministically
8 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

given. The (uniform) unit contract price is linearly dependent on the end-of-the-season spot price,
however with an upper and lower bound cap. (This is apparently a common practice in the beef
industry.) The quantity bought in an upfront contract is assumed to become available without any
yield uncertainty.
Huh and Lall (2013) endogenize the stochastic yield rate by incorporating irrigation decisions
under rainfall uncertainty, similar to our treatment of the farmers’ yield. Their model considers
a single farmer producing for a single market, who can allocate his land among several possible
crops, with uncertain commodity prices. A forward contract is offered to the farmer for a subset of
the possible crops. Similarly, Huh et al. (2012) address contract farming by a single location manu-
facturer who has access to a pool of identical local farmers, all experiencing the exact same rainfall
per acre; as opposed to a single farmer. de Zegher et al. (2017) present a contract farming model
in the wool industry, with a given set of contracted farmers. Each farmer targets an exogenously
given quantity of wool, but only a random fraction thereof is materialized. The farmers choose
between two growing techniques with different yield factor distributions and associated payments.
Finally, Mendelson and Tunca (2007) and Pei et al. (2011) are examples of a supply chain model
in which the agents can use the spot market as an alternative outlet to buy or sell items; see the
references therein.

3. The Model
The manufacturer starts by selecting the set of farmers to contract with, from a set of I potential
farmers, who are differentiated by their geographical location, their probability distributions of
seasonal rainfall (and surface water, if applicable), their local well capacity as well as production
cost and efficiency parameters. Information is asymmetric in the sense that some of the parameters
are privately known to the farmer, and only distributionally known to the manufacturer.
Each of the selected farmers follows by choosing a contract from a given menu of contracts. The
menu may be exogenously specified, or it may be designed endogenously. A contract specifies the
guaranteed unit price the manufacturer commits herself to pay per unit of supply, as well as the
corresponding quantity the manufacturer will buy at the end of the season, of course capped by
the supply the farmer is able to generate. Typically, a contract with a larger quantity commitment
comes with a lower per unit price, thus generating a trade-off for the farmer: if he chooses a contract
with a low quantity volume, but higher per unit price, there is a significant chance of oversupply,
the excess of which may be sold at the spot market at an uncertain and typically lower price.
Conversely, if he selects a contract with a higher quantity volume, the risk of oversupply is reduced,
but so is the profit margin.
The menu includes the “no-supply option” which is selected when the maximum expected profit
among all offered supply contracts falls below a given minimum participation threshold (or when
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 9

all contracts violate a risk aversion constraint). When evaluating a contract, the farmer takes
into account how rainfall and surface water can optimally be supplemented with well water, once
uncertainty about the rainfall, surface water, and well capacity have been revealed.
The best response of each farmer to the offered menu of contracts generates a distribution of
his supply, were he to be selected, along with the contract choice and associated price-quantity
combination. In assessing the supply quantity that would be obtained when selecting any given
farmer, the manufacturer faces two sources of uncertainty: first, the farmer, himself, faces a volatile
supply quantity due to the intrinsic randomness of various factors (such as rainfall, surface water,
and well capacity) determining the yield. The manufacturer’s uncertainty is further compounded
by the fact that, as mentioned earlier, she has incomplete, i.e., only distributional information
regarding some of the farmers’ parameters. Thus, the manufacturer faces a mixture of the possible
supply random variables that prevail under given values for the uncertain farmer parameters.
The manufacturer selects a set of farmers to contract with, in order to assure, with a high
likelihood, that the aggregate supply is sufficient to cover the aggregate demand which arises at the
end of the season at the different manufacturing facilities. (More generally, one may want to trace
out the optimal cost - coverage efficient frontier.) This coverage constraint plays a fundamental
role in the manufacturer’s problem. The manufacturer, typically, needs to secure its supplies from
contract farming, for all the various reasons listed in the Introduction. Thus, the spot market
is typically only used to cover limited shortages –with imperfect products– in the rare scenarios
permitted by the probabilistic coverage constraint.
In selecting the set of farmers, the manufacturer aims to minimize along with the expected pro-
curement costs, the expected distribution costs between the farmers and the production facilities.
Since the optimal distribution plan can be determined after the supply and demand realizations
are observed, the manufacturer faces a two-stage stochastic program, incorporating the farmers’
best responses to the offered menu of contracts: in the first stage, the farmers are selected; in the
second stage, the distribution plan is determined. The distribution costs are linear in the shipment
volumes between the farmers and the facilities. This assumption is satisfied when the shipments
either (i) are made in full truckloads, or (ii) are carried out by outside shipping companies. (In
our application described in Section 7, both assumptions are satisfied.) Distribution costs often
represent a major component of aggregate costs. (In our application, they amounted to at least
one third of aggregate costs.) As a consequence, the geographic location of each potential supplier
is an important selection criterion, along with many others. This fact is confirmed in our numerical
studies; see (41) and (42).
We complete this model section with a formulation of the manufacturer’s two-stage stochastic
program arising under a given menu of contracts. In Section 5, we describe how the design of an
10 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

optimal menu of contracts can be integrated with this planning problem. We use the subscript i
(j) to distinguish among the I farmers (J manufacturing plants). Subscript i = 0 refers to the spot
market (or an alternative emergency source, e.g. an import from a different country), assumed to
have an ample supply source S0 .
Parameters and Input Variables:
• Dj = the (random) demand at manufacturing plant j, j = 1, . . . , J
PJ
• Dtot = j=1 Dj : the aggregate (random) demand among all manufacturing facilities, with
mean µtot and standard deviation σtot
• Si = the (random) supply of farmer i, if selected, resulting from that farmer’s best response
to the offered menu of contracts, i = 1, . . . , I
• vi = the expected payment to farmer i, if selected, i = 1, . . . , I
• γij = the distribution cost per unit of crop, dispatched from farmer i to manufacturing facility
j, i = 1, . . . , I and j = 1, . . . , J
• γ0j = the (random) purchase, transaction and distribution cost per unit, obtained at the spot
market (or emergency source) and shipped to facility j, j = 1, . . . , J
•  = the maximum permitted probability of an aggregate shortfall at the end of the season
In addition to the expected payments to the selected farmers, captured by the v-vector, there
may be fixed transaction costs –e.g., monitoring or quality control costs– incurred for each farmer
the manufacturer contracts with. In Appendix K.2, we clarify how such fixed transaction costs can
easily be incorporated.
The joint distribution of the demand variables {Dj } is an input to the model. The random
variables {Si } and parameters {vi } are to be derived from the solution of the farmers’ best
response problems, as described at the end of Section 4. The distribution cost parameters {γij , i =
1, . . . , I, j = 1, . . . , J } are deterministic cost coefficients, which are inputs to the model. In contrast,
the parameters {γ0j , j = 1, . . . , J } are random at the first stage of the stochastic program since
they involve the end-of-the-season spot price of the commodity. In Appendix H, we discuss several
approaches to derive a probability distribution for the latter.
Decision Variables:
• Yi = 1 if farmer i is selected, i = 1, . . . , I; and 0, otherwise; Y0 = 1
• νij = the volume shipped from source i to manufacturing plant j, i = 0, . . . , I and j = 1, . . . , J
The manufacturer’s two-stage stochastic program may be formulated as:
I
X
(M ) min g(Y ) ≡ vi Yi + E{Si ,Dj ,γ0j } Ψ(Y ) (1)
Y ∈{0,1}I
i=1
" I
#
X
s.t. P Si Yi − Dtot < 0 ≤  (2)
i=1
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 11

where the recourse function Ψ(Y ) is defined as

I X
X J
Ψ(Y ) = min γij νij (3)
ν
i=0 j=1
J
X I
X
s.t. νij ≤ Si Yi i = 0 . . . I; νij ≥ Dj j = 1 . . . J; νij ≥ 0 i = 0 . . . I, j = 1 . . . J.
j=1 i=0

In the above formulation, it is assumed that any excess in the aggregate supply beyond the aggre-
gate demand, has no salvage value. In Appendix K.1, we show that our analyses can be adapted
to allow for such salvage opportunities, via the spot market or other channels. (Similarly, fixed
transaction costs per farmer could be added to the model, without any additional complexity; such
fixed costs are simply added to the coefficients in the first term of (1).)
In our base model, we assume that the farmers’ rainfall (and surface water, if applicable) vari-
ables, and hence the {Si } variables are independent of each other, as well as from the aggregate
demand variable Dtot or the future spot price. Clearly, seasonal rainfall quantities are correlated,
in particular among farmers in the same general region as well as with the future spot price. In
Appendix K.3, we therefore show how our results can be generalized to allow for such interdepen-
dencies. Our independence assumption impacts only the specification of the coverage constraint
(2). In evaluating the second term in the objective function (1), an arbitrary joint distribution of
the random variables {Si , Dj , γ0j } may be employed.
To appreciate the complexity of the manufacturer’s problem (M ), it is useful to consider a
greatly simplified (yet still NP-hard) version in which all of the manufacturer’s decisions may be
made under perfect information about all random variables, i.e., the supply volumes {Si } and the
demand levels {Dj }. This perfect information case is covered in Appendix I.
The coverage constraint, while essential, adds a significant complication to the farmer selection
problem. Note that even a single test whether a given set of farmers, i.e., a given vector Y , satisfies
the constraint is very complex and time consuming: it involves the calculation of the convolution of
potentially hundreds or even thousands of random variables, each with a complicated distribution,
to be determined by an extensive simulation study; see the discussion at the end of Section 4.
Instead, we show that the coverage constraint (2) may be approximated closely by one in which
only the first two moments of the random variables {Si } and Dtot are needed. More specifically,
we replace constraint (2) by:
   
µ tot − Y E Y E − µ tot
 ≥ Φq  = 1 − Φq  (4)
2
P I 2
P I
σtot + i=1 V ar(Si )Yi σtot + i=1 V ar(Si )Yi
PI
where YE = i=1 E(Si )Yi and with Φ(.) the cdf of the standard Normal distribution. This substi-
tution is justified by the following Central Limit Theorem result (proof of this lemma can be found
in Appendix A):
12 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

hP i  
I x−YE +µtot
Lemma 1. For any x, limI→∞ P i=1 Si Yi −D tot
<x =Φ √ 2 +PI
σtot i=1 V ar(Si )Yi

It is well known that the Normal distribution provides a very good approximation for sums of even
a modest number of independent random variables. Note that the number of farmers I is in the
hundreds or thousands, so the Normal approximation is virtually exact.
Let z be equal to Φ−1 (1 − ). Federgruen and Yang (2008) show that constraint (4) may be
replaced by a set of three linear or quadratic constraints. The manufacturer’s problem may thus
be formulated as:
I
X
0
(M ) min g(Y ) ≡ vi Yi + E{Si ,Dj ,γ0j } Ψ(Y ) (5)
Y ∈{0,1}I
i=1
I
X
s.t. (YE − µtot )2 − z2 σtot
2
− z2 V ar(Si )Yi ≥ 0 (6)
i=1
YE ≥ µtot + z σtot (7)
XI
YE = E(Si )Yi (8)
i=1

4. The Farmer’s Best Response Strategy

In any given season, each farmer’s yield is random, primarily because of volatility in the water
supply. (It is easy to adjust our model and analysis when there are additional or alternative random
yield factors.) The latter consists of natural rainfall and surface water, possibly complemented with
irrigation water, drawn from a well, if available. Water supply is the primary random yield factor
for many crops (e.g., rice, maize, cotton, soybean, and potatoes) and in many countries; see e.g.,
FAO (2007), Palma (2004), and Dawande et al. (2013).
In many developing countries, the farmers may face a third source of uncertainty beyond the
season’s rainfall, surface water, and his well capacity: even the availability of an electricity source
to enable irrigation from the well may be uncertain; when unavailable, the water may need to be
pumped with higher diesel costs. The water volume to be drawn from the well may be determined
after the season’s actual rainfall and surface water quantity, the well capacity, and the availability
of an electricity source become known.
Each farmer is offered a menu of L contracts, indexed by a subscript `, where a contract specifies a
unit price and an associated potential purchase quantity by the manufacturer. The actual purchase
quantity is the lesser of the potential quantity and the realized yield. If the realized yield at the
end of the season exceeds the potential purchase quantity, the farmer can possibly sell the excess
quantity on the spot market, where the unit price is (highly) random. In this section, we analyze
the farmer’s best response problem to a given menu of contracts. In the next section, we show how
the best response problem can be used in the design of an optimal menu of contracts.
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 13

The end-of-the-season spot price is exogenous to the farmers’ contract selection problem. Con-
sider, as an example, the Indian potato market—the primary input market for our industrial liaison.
Total potato production in 2012 – 2013 amounted to well over 40 million tons. As shown in Table
1, the average mid-size farm has less than 3 hectares of cultivated land. Even in the highest yield
regions, the average yield is no more than 25 tons per hectare. Thus, even if our manufacturer
contracts with 1,000 farmers, the total expected supply is less than 0.2% of the national supply,
hence with a negligible impact on the spot price. See www.potatopro.com/india/potato-statistics.

Small Medium Large

Avg. farm size (ha) / Percentage (%) 1.43 / 46.34 2.76 / 31.70 7.95 / 21.95
Table 1 Farm size distribution

We mentioned, in the Introduction, that it is very difficult to predict the spot price several months
or a year ahead of its realization. Moreover, farmers, particularly small farmers in developing
countries, can ill afford to take significant risks with their livelihood. When considering the spot
market as an outlet for any excess supplies, we therefore assume that the farmer considers a safe
lower bound for the next year’s spot price. (This lower bound may be based on recent price behavior
and media based information as to expectations.) However, more sophisticated farming companies
may use the actual predicted distribution of the spot price, in lieu of this lower bound. Whatever
spot price estimate the farmer is comfortable using, it has an important impact on his contract
selection, no less than the salvage value in the classical newsvendor model does.
The farmer’s problem is thus specified by the following list of notations.
Parameters and Input Variables for Each Farmer i = 1, . . . , I:
• Ri : the random rainfall and surface water, in cubic meters, at farmer i, with cdf Gi (·)
• Ci : the random capacity of farmer i’s well, in cubic meters
• Xi : farmer i’s random yield volume
• Si : farmer i’s random sales volume to the manufacturer
• π i : farmer i’s minimally acceptable profit level, to be met with a minimum likelihood (1 − i )
• p` : the per unit purchase price associated with contract `, ` = 1, . . . , L
• q` : the potential purchase quantity associated with contract `, ` = 1, . . . , L
• psi : the (lower bound) spot price considered by farmer i
• ci : farmer i’s variable production cost, exclusive of irrigation cost
• δiT : farmer i’s cost of drawing one cubic meter of water from the well ∈ {δiL , δiH }, (δiL is the
cost rate when an electricity source is available, δiH is the higher cost rate in the alternative case)
We assume that each of the farmer’s parameters {π i , ci , δiL , δiH } as well as the distributions of
the random variables {Ri , Ci } are drawn from finitely many types, even as I ↑ ∞.
Decision Variables:
14 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

• xi : the water volume, in cubic meters, drawn from farmer i’s well, if available, i = 1, . . . , I
• Zi` = 1 if farmer i chooses contract `, i = 1, . . . , I and ` = 1, . . . , L; and 0 otherwise
Xi , farmer i’s yield, is a (non-linear) function of his aggregate water supply, i.e.,

Xi = fi (αi , Ti ) with Ti = Ri + xi (9)

where αi denotes a parameter that is privately known to the farmer. The manufacturer has a
prior distribution for the parameter value, with cdf Fi (·). For notational simplicity, we assume that
the information asymmetry pertains to a single parameter αi . Generalizations to settings where
multiple parameters in the yield function fi (·, ·) are privately known, can be readily accommodated.
We make the following assumption regarding the shape of the yield function.

Assumption 1. The yield function fi (·, Ti ) is concave in Ti = Ri + xi .

The following are four examples of yield functions that satisfy Assumption 1.
Structure 1: αi represents the ideal yield under optimal water supply conditions. The yield Xi
is a linear function of the shortfall of the water supply vis-a-vis an ideal water quantity Wi , i.e.,

Xi = αi − β(Wi − Ri − xi )+ (10)

Structure 2: Identical to Structure 1, except that the yield declines (linearly) if the water supply
is in excess of the ideal water quantity Wi , i.e.,

Xi = αi − β − (Wi − Ri − xi )+ − β + (Ri + xi − Wi )+ (11)

Structure 3: Identical to Structure 1, except that the maximum yield αi is maintained as long
as the aggregate water supply is between Wi and W
ci , and declines linearly thereafter, i.e.,

Xi = min{αi , αi − β − (Wi − Ri − xi ), αi − β + (Ri + xi − W

ci )} (12)

Structure 4: This structure has been suggested in the agricultural literature, see e.g., Christensen
and McElyea (1988), Rao et al. (1990), Palma (2004), and Dawande et al. (2013).

Xi = αi (1 − exp(−βi (Ri + xi ))) (13)

Equation (9) represents Xi , the farmer i’s yield, as a function of the aggregate water supply. In
reality, Xi may depend on the aggregate of the daily shortfalls and surpluses vis-a-vis an ideal daily
water quantity, and therefore depend on the stochastic process of daily rainfalls. More specifically,
let Rit be the rainfall during day t at farmer i and assume the growing season consists of T days,
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 15

numbered t = 1, . . . , T . Then, under Structure 1, for example, the yield equation (10) should be
replaced by
T  +
X Wi xi +
Xi = αi − β − Rit − ≤ αi − β (Wi − Ri − xi ) (14)
t=1
T T
Many stochastic processes have been proposed to represent the rainfall process {Rit }, see e.g.,
Waymire and Gupta (1981) and Rodriguez-Iturbe et al. (1987). While we will proceed with the
aggregate yield model (9), its replacement by (14) (or other such yield models) is easily accommo-
nP o
L
dated. Note that, Si = min `=1 q` Z i` , Xi , i = 1, . . . , I.
Let Πi` (αi , Ri , Ci , δiT ) denote the random profit earned by farmer i when choosing contract `:
n o
Πi` (αi , Ri , Ci , δiT ) ≡ max (p` − ci ) min{q` , fi (αi , Ri + xi )} + (psi − ci )(fi (αi , Ri + xi ) − q` )+ − δiT xi
xi
(15)
s.t. 0 ≤ xi ≤ Ci

Each farmer i’s best response problem may be formulated as the following two-stage stochastic
program.
L
X h i
(F ) max Zi` E{Ri ,Ci ,δT } Πi` (αi , Ri , Ci , δiT ) (16)
Zi` i
`=1
XL
s.t. Zi` ≤ 1 (17)
`=1
XL
Zi` P{Ri ,Ci ,δT } Πi` (αi , Ri , Ci , δiT ) < π i ≤ i
 
(18)
i
`=1

In other words, we formulate the farmer’s problem as selecting the contract which maximizes
his expected profits subject to a risk aversion constraint. For the latter, we choose a maximum
probability i with which the farmer’s profits fall below the minimum profit level π i . Constraint
(18) may be replaced or complemented with an upper bound on the standard deviation of the
P h i
L T
profit level `=1 Z il stdev{R ,C
i i ,δ T } Πi` (Ri , C i , δ i ) or any other risk measure.
i

We now show how, for any of the contracts ` = 1, . . . , L, the profit value Πi` (αi , Ri , Ci , δiT ) may be
efficiently evaluated for any given quartet (αi , Ri , Ci , δiT ). Recall that the farmer has full knowledge
of the parameters in (15), including the structural form of the function fi (αi , ·).
Under Assumption 1, fi (αi , Ti ) is concave, hence differentiable in Ti everywhere, with the possible
exception of a countable set of values, where, in any case, the left and right hand partial derivatives
∂ − fi /∂Ti and ∂ + fi /∂Ti exist. Let φi` (αi , Ti ) ≡ (p` − ci ) min{q` , fi (αi , Ti )} + (psi − ci )(fi (αi , Ti ) − q` )+
and note that the optimization problem (15) can be written as

Πi` (αi , Ri , Ci , δiT ) = max φi` (αi , Ri + xi ) − δiT xi


(19a)
0≤xi ≤Ci

φi` (αi , Ti ) − δiT Ti + δiT Ri


= max (19b)
Ri ≤Ti ≤Ri +Ci
16 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

Lemma 2. Assume the yield function fi (·, ·) satisfies Assumption 1.

(a) Let Ti`∗ (αi , Ri ) denote the smallest optimizer of (19b). Then

Ti`∗ (αi , Ri ) = min Ri + Ci , min Ti ≥ Ri : ∂ − φi` (αi , Ti )/∂Ti ≤ δiT .

  

(b) Ti`∗ (αi , Ri ) is increasing in Ri .

(c) If the yield function fi (·, ·) is supermodular, as in all structures 1-4, then Ti`∗ (αi , Ri ) is
increasing in αi .
(d) Assume the yield function fi (αi , Ti ) is strictly increasing and differentiable in Ti , then
Ti`∗ (αi , Ri ) may be determined as follows:
Let t∗i (αi , q` ) denote the water supply quantity which generates a yield equal to the contract volume
q` , i.e., t∗i (αi , q` ) is the unique root of the equation

fi (αi , Ti ) = q` , if lim fi (αi , Ti ) ≥ q` , and t∗i = ∞, otherwise. (20)

Ti →∞

If t∗i < ∞, ∂t∗i (αi , q` )/∂q` = [∂fi (αi , t∗i )/∂Ti ]−1 . (21)

(I) Ri ≤ t∗i (αi , q` ):

(i) If (p` − ci )∂fi (αi , Ri )/∂Ti ≤ δiT , then Ti`∗ (αi , Ri ) = Ri .
(ii) If Ri + Ci ≤ t∗i (αi , q` ) and (p` − ci )∂fi (αi , Ri + Ci )/∂Ti ≥ δiT , then Ti`∗ (αi , Ri ) = Ri + Ci .
(iii) If Ri + Ci > t∗i (αi , q` ) and (psi − ci )∂fi (αi , Ri + Ci )/∂Ti ≥ δiT , then Ti`∗ (αi , Ri ) = Ri + Ci .
(iv) If (i)-(iii) do not apply and (p` − ci )∂fi (αi , t∗i )/∂Ti < δiT , then Ti`∗ is the unique root of
the equation (p` − ci )∂fi (αi , Ti )/∂Ti = δiT on the interval [Ri , t∗i ].
(v) If (i)-(iii) do not apply and (psi − ci )∂fi (αi , t∗i )/∂Ti > δiT , then Ti`∗ is the unique root of
the equation (psi − ci )∂fi (αi , Ti )/∂Ti = δiT on the interval [t∗i , Ri + Ci ].
(vi) If (i)-(iii) do not apply and (p` − ci )∂fi (αi , t∗i )/∂Ti ≥ δiT and (psi − ci )∂fi (αi , t∗i )/∂Ti ≤
δiT , then Ti`∗ (αi , Ri ) = t∗i .
(II) Ri > t∗i (αi , q` ):
(i) If (psi − ci )∂fi (αi , Ri )/∂Ti ≤ δiT , then Ti`∗ (αi , Ri ) = Ri .
(ii) If (psi − ci )∂fi (αi , Ri )/∂Ti ≥ δiT , then Ti`∗ (αi , Ri ) = Ri + Ci .
(iii) Otherwise, Ti`∗ is the unique root of the equation (psi − ci )∂fi (αi , Ti )/∂Ti = δiT on the
interval [t∗i , Ri + Ci ].

Lemma 2(d) provides an analytical characterization of Ti`∗ (αi , Ri ), the farmer’s optimal total
water supply when adopting contract ` from the menu, and given a rainfall value Ri . This optimal
water supply value is either given as a constant or as the unique root of a simple monotone
function. The most general case where fi (αi , ·) may fail to be increasing or differentiable everywhere,
can be handled analogously, except that more cases need to be distinguished. Note also that the
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 17

dependence of Ti`∗ on q` is very simple: ∂Ti`∗ /∂q` = 0 in all but case (I)(vi) where it is given by the
simple expression in (20).
The expected profit under each of the L contracts, as well as the likelihood of this profit value
meeting the minimum level π i , can thus be evaluated by unconditioning over the joint distribution
of the natural rainfall (and surface water, if applicable), the well capacity, and a binary variable
indicating whether a local electricity source is available. The farmer then chooses, among all con-
tracts which satisfy the specified risk bound, the one with the highest expected profit value. (If
none of the contracts are feasible, farmer i rejects the complete menu of contracts.) Under the
optimal contract, the distribution of the supply quantity Si (αi ) can be evaluated, most simply, by
generating a sample of K realizations from the above joint distribution of the random variables.
We conclude this section with a specification of the random variables {Si } and the expected
payment values {vi } which are required for the specification of the manufacturer’s problem (1)-(3):
The {Si } variables can be constructed as a mixture of the {Si (αi )} variables above, with the cdf
Fi as the mixing distribution. Finally,
Z
vi = pα(i) E(Siα )dFi (α),

with pα(i) the unit price in farmer i’s optimal contract, assuming farmer i faces a specific parameter
(vector) αi .
It is clear that, under asymmetric information, the supply variables are more volatile than when
information is symmetric. It stands to reason that this results in a larger pool of contracted farmers
and a larger expected aggregate supply value, since informational uncertainty now compounds on
the intrinsic supply and demand risks. Our numerical studies confirm this conjecture.

5. Selecting the Menu of Contracts

In the previous section, we analyzed the farmer’s best response problem to an exogenously given
menu of contracts. In this section, we show how this menu of contracts may be selected endogenously
by the manufacturer. In doing so, we follow a standard paradigm in contract theory, see e.g., Bolton
and Dewatripont (2005) or Lovejoy (2006), however applied to a complex operations setting.
Typically, a single uniform menu needs to be offered to all farmers in a broad category, for
example all farmers in a given district that are of medium size, say, see Table 1.
For notational simplicity only, we formulate the menu design problem assuming a single menu is
offered to all farmers. The generalization to multiple menus –one offered to each of several broad
categories of farmers– is straightforward.
Assume, therefore, that there are K types of farmers, differentiated by their privately known
parameter value αi . In other words, there is a list of K α-values, α1 < α2 < . . . < αK , such that,
18 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

for all i, αi = αk for some k. The manufacturer’s prior knowledge about each farmer i’s parameter
value αi is captured by a discrete cdf Fi (·) with support on {α1 , . . . , αK }. A menu of K contracts
{(p` , q` ) : ` = 1, . . . , K } is to be designed such that, in accordance with the revelation principle,
farmers of type k are incentivized to select the k th contract. The K contracts are to be chosen
on a decreasing curve p = P (q). Without loss of generality, the contracts are numbered such that
q1 ≤ q2 ≤ . . . ≤ qK and hence p1 = P (q1 ) ≥ p2 = P (q2 ) ≥ . . . ≥ pK = P (qK ).
When specifying the menu design problem, we address a somewhat simplified version of the
manufacturer’s problem, where in the objective (1) of (M ) (or (5) of (M 0 )) the second, distribution
cost related term is omitted, as are any risk bounds in the farmer’s problem, see (18). With
the menu specified as the optimal solution to this contract design program, below referred to as
(M cont ), the manufacturer solves the complete integrated farmer selection and distribution planning
problem based on the anticipated fully optimal contract choices of the farmers that incorporate
their risk bounds. In other words, the manufacturer solves the complete problem (M 0 ) without
any simplifications (or relaxations). In doing so, the manufacturer uses the methods described in
Section 6 and Appendix B.
For the sake of notational simplicity, we treat the well capacities {Ci } and the water drawing
cost rates {δiT } as deterministic. Let
• Ui (`|k, q): the expected profit value for farmer i assuming he is of type k and selects contract
`; 1 ≤ k, ` ≤ K.
• E(Si |q): the expected supply volume for farmer i, i = 1, . . . , I, if selected.
• V ar(Si |q): the variance of the supply volume for farmer i, i = 1, . . . , I, if selected.
• vi (q): the expected payment to farmer i, i = 1, . . . , I, if selected.
• π ik : the minimally acceptable profit level for farmer i assuming he is of type k; i = 1, . . . , I,
1 ≤ k ≤ K.
Treating the volume quantities {q1 , . . . , qK } as decision variables, we respecify the manufacturer’s
problem (M 0 ) as:
I
X
(M cont ) min vi (q)Yi (22)
Y ∈{0,1}I ,q
i=1
I
X
s.t. (YE − µtot )2 − z2 σtot
2
− z2 V ar(Si |q)Yi ≥ 0 (23)
i=1
YE ≥ µtot + z σtot (24)
XI
YE = E(Si |q)Yi (25)
i=1
q1 ≤ q2 ≤ . . . ≤ qK (26)

Ui (k |k, qk )Yi ≥ π ik Yi , i = 1, . . . , I and k = 1, . . . , K (IR) (27)

Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 19

Ui (`|k, q` )Yi ≤ Ui (k |k, qk )Yi , i = 1, . . . , I, k 6= ` (IC) (28)

Constraint set (27) represents each farmer’s individual rationality constraints (IR) ensuring that
his expected profit meets the minimum participation level. Constraint set (28) represents the
incentive compatibility constraints (IC) ensuring that farmer i selects contract k if of type k.
Appendix C provides analytical expressions for the various quantities Ui (`|k, q), E(Si |q), V ar(Si |q),
and vi (q) required in the model formulation.
In some contract design problems, it is possible to replace the full set of (IC)—incentive compat-
ibility constraints (28)—by a subset of so-called “local IC constraints”, where for any ` = 1, . . . , K
only the constraints corresponding with the values ` = k + 1 and ` = k − 1 need to be incorpo-
rated, see e.g., Bolton and Dewatripont (2005) or Lovejoy (2006). This reduction of the set of
IC-constraints would be possible if the maximand in the farmer’s objective function (19a) were
supermodular in (q` , αi ). However, the maximand fails to be supermodular, even if the yield func-
tion fi (αi , Ti ) is.
The menu design problem (M cont ) is a highly non-linear mixed integer problem. The natural
solution method is to alternate between
• (Mq ) identifying the optimal set of farmers, i.e., the optimal Y -vector, for a given menu of
contracts, i.e., a given q-vector, and
• (MY ) identifying an optimal contract menu, i.e., an optimal q-vector, for a given set of selected
farmers, i.e., a given Y -vector.
This iterative algorithm is guaranteed to converge in finitely many iterations; if completed till
convergence, it generates an optimal solution.
Problem (Mq ) is an integer problem, and a special case of the problem discussed in Section 6
and Appendix B; any of the methods proposed there can be applied to solve this problem. Problem
(MY ), on the other hand, is a continuous non-linear program with a small number (K) of continuous
variables, but O(IK 2 ) constraints. With the exception of the supply coverage constraints (23) –
(25), all other constraints, as well as the objective function are separable functions. All of the (IR)
and (IC) constraints, (27) and (28), involve only one or two of the decision variables.

6. The Manufacturer’s Problem: Approximate Solution Methods

In this section, we develop approximate solution methods for the manufacturer’s problem (M 0 )
under a given menu of contracts. These fall into two broad categories covered in Subsections 6.1
and 6.2, respectively.
a) Constructive improvement heuristics: These are iterative methods in which, in each iteration,
the current set of suppliers is modified by the addition or elimination of a supplier or the replace-
ment of a supplier by one outside the current set. The method terminates if none of the relevant
set perturbations results in a cost improvement.
20 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

b) Mathematical programming based heuristics

In Appendix B, we derive a lower bound for (M 0 ) via the Lagrangean dual of one of its formu-
lations; moreover we show how this Lagrangean dual can be computed efficiently.

6.1. Constructive Improvement Heuristics

The formulation of the stochastic program (M ) in (1)-(3), or its approximation (M 0 ) in (5)-(8),
projects the optimization problem onto {0, 1}I , the space of selection variables {Yi }. The vector
Y specifies the set of selected farmers. We first show that the objective function g(Y ) in (1) is a
supermodular set function. This means that for any pair of vectors Y 1 < Y 2 ,

g(Y 2 + ei ) − g(Y 2 ) ≥ g(Y 1 + ei ) − g(Y 1 )

with ei the ith unit vector in RI . In other words, the addition of a new farmer to a given set of
farmers results in a smaller expected cost saving, hence, a larger expected cost increase, compared
to when the same farmer is added to a subset thereof. Appendix A shows the proof of Theorem 1.

Theorem 1. The objective function of problem (M ) is a supermodular set function.

Supermodular set functions are important in the optimization and economics literature. For
example, many fundamental problems in a variety of supply chain models, including network
design problems, stochastic inventory problems, and pricing models can be posed as a problem of
minimizing a supermodular set function. (An example is the uncapacitated plant location problem,
see Cornuejols et al. 1977). This class of problems -even unconstrained- was shown to be NP-hard
even if the evaluation of the set function can be performed in polynomial time, as is the case in
our model, where it reduces to the solution of a series of independent transportation problems.
The complexity results follow from the fact that the general class of problems generalizes many
well-known NP-hard problems, including the special case discussed in Appendix I (with K = 1
scenarios) and the aforementioned uncapacitated plant location problem.
However, Cornuejols et al. (1977) and Nemhauser and Wolsey (1978) showed that, at least for the
unconstrained (supermodular) set minimization problem, simple heuristics such as the greedy-add
or greedy-drop heuristics work remarkably well. (The simplest version of the greedy-add heuristic
starts from a feasible solution and adds, at each iteration, a single new element that results in
the largest cost saving among all possible choices; the heuristic terminates when no addition of
any outside element results in a cost decrease. The greedy-drop heuristic starts with the full set
of elements, and in each iteration, drops whatever element results in the smallest cost increase.)
In numerical studies, these heuristics have performed remarkably well. Moreover, Cornuejols et al.
(1977) derives an impressive worst case optimality gap of e−1 ≈ 37%, albeit that a somewhat
unconventional optimality gap measure is used. More recently, the theory has been generalized to
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 21

include side constraints that are linear in the binary variables that describe the composition of the
selected set; see Lee et al. (2009) and Kulik et al. (2013).
Because of the fixed costs of engaging with the farmers, i.e., the first term in (1), g(Y ) is
not a monotone function of the selected set of farmers, which makes solving this problem more
complex (see Kulik et al. 2013). Nevertheless, these results bode well for the performance of similar
constructive heuristics for our problem, in particular:
Greedy-Add Heuristic, version 1: In this heuristic, we add, in each iteration, a new supplier to
the current set of suppliers. As long as the coverage constraint fails to be satisfied, we add any
supplier whose addition results in the smallest expected cost increase; in subsequent iterations,
we execute the addition only if the resulting expected cost increment is negative, and terminate
otherwise. As the initial set of suppliers, we select the solution to the perfect information problem
(P IM ), see Appendix I, with a single scenario (K = 1) and all random variables {Si } and {Dj }
replaced by their expected values.
The determination of the initial set of suppliers reduces to the solution of a fairly simple Mixed
Integer Linear Program (M ILP ). The complexity of the subsequent iterations amounts to at most
O(I 2 ) evaluations of the set function g(·) and the same number of tests whether the coverage
constraint is satisfied. Appendix J presents two variants of this greedy heuristic. Their performances
are comparable to the Greedy-Add heuristic, version 1, but, in our numerical experiments, are
dominated by the latter.

6.2. A Mixed Integer Programming Method

The manufacturer’s problem (M 0 ), i.e., the problem of minimizing (5), subject to (6) - (8) would
reduce to a Mixed Integer Linear Program (M ILP ), and thereby solvable for instances with thou-
sands of binary variables, were it not for the single quadratic constraint (6). This suggests a
heuristic approach where the quadratic term (YE − µtot )2 in (6) is replaced by a piecewise linear
approximation. Note that by (7) and (8):
I
X I
X
d0 ≡ µtot + z σtot ≤ YE = E(Si )Yi ≤ E(Si ) ≡ dN
i=1 i=1

Our M ILP -based heuristic thus consists of selecting a grid {d0 , d1 , . . . , dN } on the feasible interval
[d0 , dN ], replacing the quadratic function h(YE ) = (YE − µtot )2 by a piecewise linear function h̃(YE )
which coincides with the quadratic function h(.) at the grid points and is linear in between. Thus,
representing YE as a weighted average of the grid points, with the help of the auxiliary variables
{ηn : n = 0, 1, . . . , N }, we replace problem (M 0 ) by the M ILP :
I
X
min g(Y ) ≡ vi Yi + E{Si ,Dj ,γ0j } Ψ(Y ) (29)
Y ∈{0,1}I
i=1
22 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

N
X I
X
2
s.t. ηn (dn − µtot ) − z2 σtot
2
− z2 V ar(Si )Yi ≥ 0
n=0 i=1
XN N
X I
X
ηn = 1; ηn dn = E(Si )Yi
n=0 n=0 i=1

along with so-called SOS2 constraints to ensure that at most two of the η-variables are nonzero and
that these non-zero variables have consecutive indices. As before, we employ the sample averaging
method to evaluate the second term in (29).
The full formulation of the M ILP is thus given by:
I
X K X
X I X
J
min vi Yi + δk γij νijk (30)
Y,ν,η
i=1 k=1 i=0 j=1
N
X I
X
s.t. ηn (dn − µtot )2 − z2 σtot
2
− z2 V ar(Si )Yi ≥ 0
n=0 i=1
XN N
X I
X
ηn = 1; ηn dn = E(Si )Yi
n=0 n=0 i=1
J
X I
X
νijk ≤ Sik Yi i = 0...I k = 1 . . . K; νijk ≥ Djk j = 1...J k = 1...K
j=1 i=0

νijk ≥ 0, Yi ∈ {0, 1}, η ∈ SOS2

where δk , Sik , Djk , and νijk are defined as in Appendix I. This program is linear in all the decision
variables Y , ν, and η. CPLEX, as well as many other optimization software systems, have built-in
functions to represent SOS2 constraints.

7. Numerical Analysis
We have applied our model to a large global manufacturer of potato chips contracting with thou-
sands of potato farmers in India. More specifically, we have created a numerical study with instances
that are anchored on data provided by the manufacturer, as well as data that are publicly available.
In Appendix D, we describe in detail how the problem instances were created. The exact numbers
are withheld in order to preserve confidentiality.
We have evaluated problem instances with seven different values for the targeted total number
of potential farmers I: I =50, 100, 150, 250, 500, and 1,000. These farmers are distributed over the
set of 19 districts, in six of India’s states, in which the manufacturer contracted during the two
years (2009 and 2010) used for our analysis. Appendix D describes the precise procedure employed
to distribute the total number of farmers, I, over the 19 districts listed in Table 10 in Appendix G.
For each value of I, we have evaluated 20 problem instances by considering five problem instances
for each of four demand level / coverage level combinations: high and low demand scenarios at a
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 23

coverage level of 95% and 99%, i.e.,  = 0.05 and  = 0.01, respectively. These five problem instances
use different realizations of the random parameters that specify an instance: the precise distribution
of the I farmers over the 19 districts, the farm sizes assigned to the farmers, as well as the sample
of realizations drawn from the joint supply and demand volume distribution {Si , Dj }; see (1)-(3).
In Table 2, we report the performance of the Greedy-Add heuristic (version 1 –the other greedy
heuristics presented in Appendix J were generally dominated by this Greedy-Add procedure) and
the mixed integer programming approach, described in Sections 6.1 and 6.2, respectively. (We
used 5e-4 as the relative M ILP gap / optimality tolerance for the M ILP method and when
solving the perfect information problem to determine the initial set of farmers in the Greedy-Add
heuristic.) Table 2 reports the average of the performance measures across the five instances that
were generated for each of the demand level / coverage level combinations. By computing the
Lagrangean dual as described in Appendix B, we determined an upper bound for the optimality
gap of the heuristic methods.

Greedy-Add MILP
TPFS DL CL R1 RT CNF OG R2 RT CNF OG R2
H 99% 1.386 2.6 42.4 1.23% 1.238 111.9 43.4 0.63% 1.232
95% 1.400 1.6 40 1.62% 1.175 5.9 39.6 0.54% 1.165
50
L 99% 1.873 2.7 36.8 2.48% 1.245 61.0 36.8 1.09% 1.231
95% 1.832 1.8 28.4 2.37% 1.172 17.3 29.2 0.70% 1.165
H 99% 1.374 4.9 88.8 0.81% 1.229 496.7 88.4 0.27% 1.225
95% 1.445 10.3 81 1.13% 1.160 804.2 83.6 0.26% 1.158
100
L 99% 1.849 10.9 70.8 1.33% 1.229 251.6 69.8 0.63% 1.224
95% 1.977 8.9 54.8 1.96% 1.162 66.2 52.8 0.36% 1.156
H 99% 1.548 26.5 124.2 0.57% 1.225 47.8 124.8 0.28% 1.223
95% 1.441 15.7 121.8 0.65% 1.158 94.7 118.6 0.19% 1.156
150
L 99%∗ 1.783 28.0 104.2 1.00% 1.225 130.6 102.2 0.40% 1.222
95% 1.907 22.6 85.4 1.00% 1.158 306.7 86.6 0.34% 1.156
H 99% 1.476 89.0 217 0.40% 1.223 50.3 213.8 0.14% 1.221
95% 1.419 55.4 199.4 0.72% 1.156 136.3 204.2 0.20% 1.155
250
L 99% 1.834 134.2 169.4 0.81% 1.221 199.2 161.4 0.29% 1.219
95% 1.766 66.5 161.2 1.11% 1.156 136.5 162 0.19% 1.154
H 99% 1.437 482.7 445 0.24% 1.221 263.1 424.6 0.08% 1.219
95% 1.430 354.4 392.8 0.59% 1.155 266.5 396.6 0.08% 1.154
500
L 99% 1.904 847.1 321 0.59% 1.218 269.3 317.4 0.17% 1.217
95%∗∗ 1.879 469.5 270.6 1.21% 1.155 484.3 290 0.10% 1.153
H 99% 1.455 23,723.5 866 0.28% 1.219 595.5 814.2 0.18% 1.219
95% 1.438 15,217.8 758.2 0.45% 1.154 412.7 775.2 0.08% 1.154
1,000
L 99% 1.926 34,253.6 617.8 0.59% 1.217 408.6 616.4 0.21% 1.216
95% 1.875 14,941.7 528.4 0.86% 1.153 406.7 580.2 0.08% 1.152
Table 2 Performance of Greedy-Add and M ILP heuristics
TPFS: Targeted Potential Farmer Size, DL: Demand Level, CL: coverage level, RT: Runtime (sec), CNF:
Contracted Number of Farmers, OG: Optimality Gap

PI
Column 4 reports the supply-demand ratio R1 , defined as i=1 E(Si ), the expected supply if
all I farmers were contracted, divided by the expected aggregate country-wide demand of the
24 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

manufacturer. Columns 5-8 and 9-12 report four performance measures for the above two heuristic
methods: (a) the runtime in CPU seconds, (b) the number of farmers contracted, (c) the upper
bound for the optimality gap, defined as
the expected cost of the heuristic solution − the value of the Lagrangean dual
Optimality Gap = ,
the value of the Lagrangean dual
PI PJ
and (d) a supply / demand ratio R2 , defined as ( i=1 E(Si )Yi )/ j=1 E(Dj ). Thus R2 reflects the
expected contracted supply, as opposed to the expected total supply from all potential farmers, as
in R1 .
We conclude that both solution methods are remarkably close to optimal and that the
Lagrangean dual generates a remarkably close-to-accurate estimate of the optimal cost value. The
average optimality gap for the Greedy-Add heuristic is 1.09% and that for the M ILP based
method is 0.35%. Indeed, the latter performs uniformly better. Moreover, the worst case optimality
gap across all scenarios is 5.02% for the Greedy-Add procedure and 1.53% for the M ILP based
approach. Even more encouraging is the fact that the optimality gap decreases with I, to zero,
suggesting that both heuristics are asymptotically optimal. Clearly, the results also imply that the
average accuracy gap of the Lagrangean dual based lower bound is no more than 0.35%, as well.
The Greedy-Add heuristic is faster, when I < 250. When I = 250 or I = 500, the CPU times are
comparable. However, for I = 1, 000, the M ILP method is significantly faster. Most importantly,
even instances with I ∼ 1, 000 can be solved in a few CPU minutes.
The extremely low optimality gap for the Greedy-Add heuristic (1.09%, on average) outdoes the
expectations raised by Theorem 1 and the subsequent discussion in Section 6. Beyond confirming
that this simple heuristic is a competitive alternative solution method, it also indicates that robust
farmers’ selections can be made in an expanding environment: if the buyer’s demand volumes grow
from one season to the next, so that the optimal number of contracted farmers increases from I 1
to I 2 , very little is lost by adding (I 2 − I 1 ) new farmers to the initial (close-to-) optimal set of I 1
farmers. Thus, absent major changes in the model parameters, the contracted farmer base may
be gradually expanded, without any significant optimality loss, allowing for stability and loyalty
towards the selected farmer pool.
The results show that with a modest number of suppliers (I = 50), the supply-to-demand ratio R2
is approximately 1.23 when pursuing a coverage level of 99%. The safety margin can be expected to
be significantly larger when, in contrast to our scenarios, there is more significant demand volatility,
or as the degree of information asymmetry is larger. Clearly, the required safety margin is larger
for scenarios with a coverage level of 99% as opposed to those with a coverage level of 95%. For the
latter, the required safety margin varies between 17% and 15% as the number of potential farmers
I is increased.
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 25

An interesting question is whether one or a few farmer characteristics can be identified that
determine whether a farmer is selected by the manufacturer or not. To answer this question,
we explored the instances associated with the categories marked with a ∗ and ∗∗ in Table 2, in
detail. Of the many considered farmer characteristics, the following three stood out when running
logistic regressions: (i) the expected payment to the farmer divided by his expected supply volume
(vi /E(Si )), (ii) the coefficient of variation of his supply volume (CV (Si )), and (iii) his distance to
the nearest manufacturing plant (∆i ). See Appendix E for a specification of the logit model and
its estimated coefficients.
While insightful, the findings also show that the selection problem involves complex trade-offs
among several farmer characteristics. Our Stackelberg game and the associated mathematical pro-
gram (M 0 ) efficiently determine the best among all 2I subsets of the I farmers. In contrast, assume
we confined ourselves to the I sets with the k lowest (vi /E(Si )) values, k = 1, . . . , I, and identified
the best such feasible set. 16% of the (close-to-) optimal farmer set found by (M 0 ) fails to be part
of this heuristic set.
The above instances assume, as in our base model, that the supply random variables {Si } are
independent and that the {γ0j }-coefficients are independent of the aggregate rainfall. In Appendix
K.3, we have explained how our model and solution methods can be extended when either one of
these two independence assumptions is relaxed. We have investigated how correlations among the
{Si } variables impact the optimal choices for the manufacturer. (For a given menu of contracts,
these correlation structures do not affect the farmers.) More specifically, we have re-evaluated the
eight sets of problem instances with I = 250 and I = 500 farmers, now assuming that the rainfall
quantities and the {αi } (intercept in the yield equation) parameters for farmers in a given state
are drawn from a multivariate Normal distribution with the same means and variances as in the
base model, but a common correlation coefficient κ = 0.25, 0.5, and 0.75. The main managerial
conclusions from this study are summarized in Section 8, under Conclusion (j).
Impact of the Menu Design
We have generated a small second set of scenarios to illustrate how our model can be used to
evaluate alternative contract menus and how the magnitude of various risk factors impacts the
system performance. To this end, we used a base scenario, with a coverage level of 95% and a
“high” demand scenario, targeting I = 150 potential farmers, this time assuming all farmers are
small-size farmers; see Table 1. (Because of the above described distribution procedure across the
19 districts, the actual number of potential suppliers I = 158.)
Here, we report on six sets of exogenously specified contract menus. Appendix F reports on a
parallel study in which contract menus are determined endogenously, based on the methodology of
Section 5.
26 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

The six sets of menus are differentiated by (i) the number of contract options: three vs. five;
(ii) the slope of the menu line: -10/3, -5/3, and -2/3. Each set consists of six menus, one for each
state. As in our first numerical study (see Table 2), all contracts pertaining to the same menu are
characterized by price-quantity pairs that lie on a line. See, however, below for a study where the
contracts are positioned on a non-linear curve. In each state, all six menus share the same central
contract (ps , 1.43q s ), as the central contract choice in the large scale numerical study reported
in Table 2; see Appendix D and Table 11 in Appendix G for details. In each of the menus, the
price-quantity pairs representing the different contracts are equidistant (as in our first numerical
study in Table 2). More specifically, the menus with three (five) options have quantity values
{q s − 6, q s , q s + 6} ({q s − 6, q s − 3, q s , q s + 3, q s + 6}). Each scenario is evaluated by drawing 100
realizations of the vector of all random variables.

MC CNF Percentage of CNF by Contract Option (%)

# of Contract Options: Five Three Five Three Five Three
O1 O2 O3 O4 O5 O1 O3 O5
D. CV = 0.15 Slope 1 606,764 590,203 124 123 13.7 33.9 21.8 27.4 3.2 24.4 61.8 13.8
R. CV= x Slope 2 555,655 554,445 117 118 12.0 9.4 40.2 35.0 3.4 11.9 55.9 32.2
Slope 3 549,689 539,072 116 116 5.2 12.1 30.2 36.2 16.4 7.8 54.3 37.9
D. CV = 0.15 Slope 1 659,109 666,461 138 142 10.9 31.9 29.7 23.9 3.6 33.8 47.2 19.0
R. CV = 3x Slope 2 598,221 592,582 131 132 6.1 9.2 44.3 38.2 2.3 7.6 69.7 22.7
Slope 3 595,429 585,797 130 129 6.2 6.2 32.3 37.7 17.7 6.2 48.1 45.7
D. CV = 0.3 Slope 1 692,986 705,191 140 140 12.1 31.4 18.6 35.7 2.1 21.4 62.9 15.7
R. CV = x Slope 2 661,500 679,119 134 134 10.4 9.0 38.8 39.6 2.2 10.4 53.0 36.6
Slope 3 662,380 641,916 132 132 4.5 11.4 28.0 37.1 18.9 6.8 51.5 41.7
D. CV = 0.3 Slope 1 728,059 Infeas. 157 Infeas. 9.6 29.3 31.8 26.1 3.2 Infeasible
R.CV = 3x Slope 2 699,212 727,830 150 151 5.3 8.0 42.7 41.3 2.7 6.6 64.2 29.1
Slope 3 750,188 721,211 148 148 5.4 5.4 29.7 39.9 19.6 5.4 43.2 51.4
Table 3 Impact of menu design on manufacturer’s performance measures
D.CV: Demand Coefficient of Variation, R.CV: Rainfall Coefficient of Variation, MC: Manufacturer Cost, CNF:
Contracted Number of Farmers, Oi: Contract Option i

In the first segments of Tables 3 and 4, we report various performance measures under each of
the six sets of menus. In Table 3, Columns 3 and 4 specify the expected costs to the manufacturer
under each of the six sets of menus. The fifth and sixth columns specify the number of selected
farmers, and the remaining columns display what percentage of farmers, out of the set of “selected”
farmers, opt for each of the contract options, again under the six sets of menus. (Recall, the options
are numbered in increasing order of their quantity volume.)
Across all six sets of menus, we have observed that an expected supply, roughly 16% higher
than the expected demand needs to be targeted. While the expected aggregate supply is virtually
invariant with respect to the menu choice, Table 3 shows that the number of contracted farmers
varies between 116 and 124, i.e., a 7% variation in the supplier base. The expected cost to the
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 27

AFP AFYM (AFYSD) AFSM (AFSSD)

# of contract options: Five Three Five Three Five Three
D. CV = 0.15 Slope 1 2,418 2,365 19.32 (1.59) 19.24 (1.56) 18.22 (1.23) 18.31 (1.21)
R. CV= x Slope 2 2,335 2,331 19.52 (1.53) 19.56 (1.57) 19.22 (1.39) 19.13 (1.39)
Slope 3 2,352 2,316 19.68 (1.57) 19.55 (1.57) 19.57 (1.50) 19.40 (1.49)
D. CV = 0.15 Slope 1 2,290 2,265 17.66 (4.37) 17.45 (4.41) 16.58 (3.67) 15.95 (3.52)
R. CV = 3x Slope 2 2,211 2,185 17.72 (4.41) 17.64 (4.43) 17.30 (4.09) 17.11 (4.04)
Slope 3 2,209 2,176 17.73 (4.48) 17.68 (4.47) 17.49 (4.29) 17.41 (4.26)
D. CV = 0.3 Slope 1 2,570 2,517 19.23 (1.51) 19.11 (1.49) 18.24 (1.19) 18.23 (1.16)
R. CV = x Slope 2 2,503 2,493 19.27 (1.45) 19.38 (1.50) 19.00 (1.33) 19.00 (1.33)
Slope 3 2,524 2,502 19.48 (1.49) 19.42 (1.49) 19.38 (1.43) 19.28 (1.42)
D. CV = 0.3 Slope 1 2,382 Infeas. 17.32 (4.37) Infeas. 16.36 (3.74) Infeas.
R.CV = 3x Slope 2 2,336 2,333 17.43 (4.40) 17.49 (4.40) 17.06 (4.12) 17.03 (4.06)
Slope 3 2,352 2,337 17.54 (4.44) 17.52 (4.43) 17.33 (4.27) 17.29 (4.25)
Table 4 Impact of menu design on contracted farmers’ performance measures
D.CV: Demand Coefficient of Variation, R.CV: Rainfall Coefficient of Variation, AFP: Average Farmer Profit,
AFYM: Average of Farmer Yield Means, AFYSD: Average of Farmer Yield Standard Deviations, AFSM: Average
of Farmer Supply Means, AFSSD: Average of Farmer Supply Standard Deviations

manufacturer exhibits an even larger variation of 12%. More specifically, the slope of the contract
menu line, hence the price sensitivity with respect to quantity variations in the contract, has a
significant impact on overall costs. Slope 3 allows for a 9.5% reduction in overall expected costs,
compared to Slope 1, assuming the menu consists of five contract choices. An additional 2% cost
saving can be achieved by limiting the number of contract options to three.
As we move from Slope 1 to Slope 3, the price sensitivity of quantity variations in the contract
diminishes. This induces more and more farmers to select higher quantity contracts, since the per
unit price sacrifice in doing so, is progressively smaller. Indeed the contract choice distribution
under Slope 1 (2) majorizes that under Slope 2 (3), both when the menus offer three or five contract
options. This shift towards the lower unit price / higher volume contracts explains the above cost
savings as we move from Slope 1 to Slope 2 and Slope 3. It also explains why the number of selected
farmers can be reduced as we move from Slope 1 to Slope 3.
Table 4 displays the implications the menu choice has for the contracted farmers. The third and
fourth columns exhibit the average expected profit level per contracted farmer. The fifth and sixth
columns show the average expected yield per contracted farmer, under the six sets of menus; the
numbers within the parentheses represent the average value of the yield standard deviations. The
seventh and eighth columns, similarly, display the average mean and standard deviation of the
expected supply provided per contracted farmer. Recall that the latter is given by the lower of the
yield and the contract volume.
Not surprisingly, the average profit per farmer is lower when the number of options in the contract
menu is reduced to three, under each of the menu slopes 1–3. More noteworthy is the fact that the
profit reduction is less than 2%. The impact on the manufacturer’s cost is similarly small, and not
28 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

always to the manufacturer’s benefit. Thus, little is lost or gained for either party, when offering
simple menus with a small number of options. We reach the same conclusion when designing the
contract menus endogenously; see below.
As we move from Slope 1 to Slope 3, the average yield and average supply increases. This increase
is enabled by the selected farmers choosing higher quantity contracts, when the associated unit
price reduction becomes smaller; see Table 3. It allows the manufacturer to contract with fewer
and more conveniently located farmers, maintaining (approximately) the same aggregate expected
supply at a significantly lower cost (by 12%). Since the average farmer loses no more than 4% of
his profits, a win-win situation can be created by an additional uniform, modest increase of the
unit prices under all contracts.
For the same set of problem instances, we have also explored how the different performance
measures are impacted by several risk factors. The second segments in Tables 3 and 4 exhibit
the performance measures when the standard deviations of the various rainfall distributions are
tripled. The third segments exhibit the performance measures when the volatility of the demand
distributions at the plants is doubled, and the last segments exhibit these measures when both risk
factors are amplified.
Doubling the coefficient of variation of the demand distributions results in an approximately 13%
increase in the targeted expected aggregate supply, to continue to meet the coverage constraint. In
other words, the safety supply needs to be doubled, with R2 increasing from 1.16 to 1.31. To enable
this, the manufacturer needs to expand the supplier pool by a similar percentage. Focusing on the
most efficient set of menus –with Slope 3 and 3-contract options– the increased demand volatility
results in an even larger increase (19%) of the manufacturer cost. On the other hand, an increase
of the volatility of the rainfall distributions impacts the manufacturer’s cost less severely (by about
8.5%), with similarly sized decreases of the farmers’ expected profits, yields and supplies. To hold
the cost increase to no more than 8.5%, it becomes optimal to expand the contracted farmer pool
by some 11%, under the most cost efficient sets of menus –with Slope 3 and 3-contract options.
In other words, it is optimal for the manufacturer to face the increased risks, whether supply or
demand risks, by diversifying the supply over a significantly expanded supplier pool.
When the volatility of both the rainfall and demand distributions is increased simultaneously,
the above effects are compounded. Compared with the base case, the expected minimal cost for
the manufacturer, under the set of menus with Slope 3 and 3-contract options, now increases by
34%, and the required number of contracted farmers grows from 116 to 148. Under the set of
menus with Slope 1, it becomes necessary to select all but one of the farmers, when the menu offers
five contract choices. When it offers only three, the coverage constraint cannot be met even when
contracting with all of the farmers. When comparing among the six sets of menus, we note that
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 29

the above described patterns for the base case, in the first segment of Table 3, carry over to all
subsequent segments.
We have also experimented with contract menus in which the price-quantity pairs lie on a non-
linear curve. To this end, we have adapted the three menu lines to parabolae which share the
same midpoint and tangent of that midpoint. Table 11 in Appendix G exhibits the contracts under
each of the three parabolae, in each of the six states. In Table 12 of Appendix G, we report both
the optimality gaps and the computational times using the M ILP heuristic under each of the six
sets of menus, for the base setting, as well as settings with doubled standard deviations of the
demand distributions, tripled standard deviations of the rainfall distributions, or both. The bounds
for the optimality gaps remain, invariably excellent, always below 0.5% and, on average, 0.088%.
Computational times are comparable to the times we observed under linear contract menus; they
are of the same order of magnitude.

8. Conclusions
We have developed a tactical planning model for a manufacturer contracting with part of a general
list of potential farmers as suppliers of a given agricultural commodity. The model allows the
manufacturer to select an optimal number and set of farmers to contract with, while determining
how supplies generated from the contracts are optimally distributed to meet the demands of each of
the manufacturer’s production facilities. The manufacturer selects a set of farmers to offer a menu
of contracts, which is exogenously specified or endogenously determined. The model is a Stackelberg
game with asymmetric information, with the manufacturer as the leader and the selected farmers
as the followers. A farmer, when selected, chooses the contract which maximizes his expected profit
under an optimal irrigation scheme, by solving a two-stage stochastic program, possibly declining
all contracts, if a minimum income level cannot be assured. The manufacturer uses the farmers’
random supply quantities, under their best response strategies, to determine his farmer selection
and end-of-the-season distribution plan, again by solving a stochastic program.
We have shown how both the farmers’ and the manufacturer’s problem can be solved to optimal-
ity and close to optimality, respectively, even when the number of potential suppliers is 1,000 or
larger. To this end, we have developed two approximate solution methods and a Lagrangean dual
lower bound. We have also shown how our model can be used to gain various managerial insights.
Here are our “top ten” insights:
(a) The extremely low average optimality gap of the Greedy-Add heuristic, one of the two
algorithmic approaches, indicates that robust farmer selections can be made in an expanding envi-
ronment. If the manufacturer’s demand volume grows from one season to the next, so that the
optimal number of contracted farmers increases, very little is lost by adding a new set of farmers
30 Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming

to the selected set of the prior season. This allows for robust and long-term relationships with the
selected farmers, as long as the farmers’ characteristics remain stable.
(b) Among many considered farmer characteristics, three stood out as primary selection criteria.
More specifically, the likelihood of a given farmer being selected for contracting increases, as (i)
the ratio of the expected payment to the farmer and his expected supply volume decreases, (ii) the
coefficient of variation of the farmer’s supply volume –a measure of his supply volatility– decreases,
and (iii) the distance between the farmer and the closest manufacturing plant decreases. These
characteristics are identified with logistic regression models.
(c) An individual farmer can, therefore, improve his chance of being selected for contracting by
reducing the volatility of his yield. The latter can be accomplished by implementing standardized
cultivation methods and by assuring efficient access to a well to supplement rainfall and surface
water when needed. The farmer’s attractiveness also increases with an increase in his expected
yield per acre.
(d) The above findings show that the selection problem involves complex trade-offs among sev-
eral farmer characteristics. Our Stackelberg game and associated mathematical program efficiently
determine a (close-to-) optimal set of farmers among all 2I subsets of the I potential farmers.
In contrast, if we confined ourselves to simple selection rules, e.g., all suppliers with an expected
payment per expected ton supplied below a given threshold, the resulting “best” set of farmers will
differ from the above (close-to-) optimal set for a substantial percentage of the required number of
farmers.
(e) Designing a contract menu around a given central contract option may impact the total
expected cost for the manufacturer as well as the number of contracted farmers significantly. We
report on instances where rotating a given contract menu line around a fixed central contract
option results in a 12% reduction of aggregate costs and a similar reduction of the number of
contracted farmers. In some cases, such rotation of the contract menu may imply the difference
between feasible and infeasible instances.
(f) From the farmer’s perspective, these rotations have a major impact on the percentage of
farmers that opt for the higher volume contracts as opposed to the lower volume ones.
(g) Under the same menu change, the expected profit per contracted farmer in these instances
changes much more modestly allowing for win-win opportunities for the manufacturer and the
contracted farmers. At the same time, as mentioned, fewer farmers are selected for contracting.
(h) When constructing the contract menu endogenously, often a small number of contract options
suffices, depending on the degree of heterogeneity among the farmer pool. In other words, little
is gained by constructing menus with more than three contract options. The same holds under
exogenously specified contract menus. Thus, relatively simple menus often suffice.
Federgruen, Lall, and Şimşek: Supply Chain Analysis of Contract Farming 31

(i) Folklore predicts that expected system-wide costs increase when demand and yield volumes
become more volatile. However, in this setting, an increase in the rainfall volatility may result in
a cost reduction for the manufacturer, as increased yield risks induce farmers to opt for higher
volume/lower price contracts to meet their minimum profit level constraints.
(j) As correlations among the farmers’ supply variables increase, the manufacturer always needs
to increase her total expected supply value. Typically, this implies an increase in the number
of contracted farmers, as well. Even more dramatically, feasible instances may become infeasible
under increased supply correlations. Increased correlations result in significant substitutions in the
optimal farmer pool, often replacing more reliable farmers for more conveniently located ones.

Acknowledgments
We thank Hau Lee, the Associate Editor and referees for their guidance and constructive suggestions, and
Mustafa Şahin for his help with the computer codes.

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 Online Appendix for “Supply Chain Analysis of Contract Farming”
Awi Federgruen, Upmanu Lall, A. Serdar Şimşek1
Appendix A: Proofs
Proof of Lemma 1: By assumption, each of the farmer’s parameters {Wi , π i , ci , δiL , δiH } and the distribu-
tions of the random variables {Ri , Ci } are drawn from finitely many types. Since there are finitely many
contracts (L), all random variables {Si } are uniformly bounded by qmax = max`=1,...,L q` and their variances
are uniformly bounded from below. Finally, the random variable Dtot has a finite third moment. Let
I I
X X
rI3 = E |Si − E(Si )|3 + E |Dtot − µtot |3 and s2I = 2



Var(Si )Yi + σtot .
i=1 i=1
√
Since all random variables {Si } have uniformly bounded supports, rI3 = O(I), so that rI = O( 3 I). Similarly,
√
s2I = Ω(I), so that sI = Ω( I). It follows that the Lyapunov condition, limI→∞ srII = 0, is satisfied and the
Lyapunov Central Limit Theorem applies.
Proof of Lemma 2:
(a) The function φi` (αi , Ti ) is concave in Ti , since it is the composition of an increasing (piecewise linear)
concave function and the concave function fi (αi , ·). It follows that the maximand in (19b) is concave in Ti
and the characterization of Ti`∗ (αi , Ri ) follows immediately.
(b) Note that the interval [Ri , Ri + Ci ] is increasing in Ri : Let Ri1 < Ri2 , for any Ti1 ∈ [Ri1 , Ri1 + Ci ] and Ti2 ∈
[Ri2 , Ri2 + Ci ], Ri2 + Ci ≥ max(Ti1 , Ti2 ) ≥ Ti2 ≥ Ri2 , i.e., max(Ti1 , Ti2 ) ∈ [Ri2 , Ri2 + Ci ]. Similarly, min(Ti1 , Ti2 ) ∈
[Ri1 , Ri1 + Ci ]. The maximand in (19b) is independent of Ri , and therefore supermodular in (Ti , Ri ). It follows
from Lemma 2.8.1 in Topkis (1998) that arg maxRi ≤Ti ≤Ri +Ci {φi` (αi , Ti ) − δiT Ti }, the set of optimal Ti values
in (19b) is increasing in Ri . It follows that Ti`∗ (αi , Ri ), as the smallest element of this set, is increasing in Ri
as well.
(c) The proof is analogous to that of part (b). (d) Immediate from part (a).
Proof of Theorem 1: With I be the set of potential farmers, let I ∗ the set of selected farmers, and J the
set of manufacturing plants. Then the optimal cost of problem (3), denoted by g(I, J ) for given realizations
of supplies {Si } and demands {Dj }, is supermodular in (I\I ∗ , J ) by Theorem 3.4.1(a) of Topkis (1998) and
so is E(Ψ(Y )) by the fact that the expectation operator preserves supermodularity. The first term in (1)
is a modular set function and, in particular, a supermodular function. Since supermodularity is also closed
under summation, the complete objective function (1) is also supermodular as a function of (I\I ∗ ).
Appendix B: Lower Bound Computation – Lagrangean Dual
In this appendix, we derive a lower bound for the manufacturer’s problem, (M 0 ), which can be computed
with modest effort even for large size problem instances, with hundreds or thousands of farmers. The lower
bound is obtained by dualizing the constraints (6) and (8) with Lagrange multipliers λ ≥ 0 and −∞ < ρ < ∞.
We then obtain the following Lagrangean dual of the manufacturer’s problem:

max min L(Y, YE , λ, ρ) s.t. YE ≥ µtot + z σtot , where

λ,ρ Y,YE
I
" I
# " I
#
X X X
2 2 2 2
L(Y, YE , λ, ρ) = vi Yi + E{Si ,Dj ,γ0j } Ψ(Y ) + λ −(YE − µtot ) + z σtot + z V ar(Si )Yi + ρ YE − E(Si )Yi
i=1 i=1 i=1

1
Contact author email: serdar.simsek@utdallas.edu

1
2 Federgruen, Lall, and Şimşek: Online Appendix for “Supply Chain Analysis of Contract Farming”

For any fixed pair of Lagrange multipliers λ and ρ, the Lagrangean relaxation decomposes into an optimiza-
tion problem in the vector of Y and a separate problem in the single continuous variable YE . In particular,
for fixed λ and ρ, we can write L(Y, YE , λ, ρ) = L1 (Y, λ, ρ) + L2 (YE , λ, ρ) where
I
X I
X I
X
L1 (Y, λ, ρ) = vi Yi + E{Si ,Dj ,γ0j } Ψ(Y ) + λz2 V ar(Si )Yi − ρ E(Si )Yi
i=1 i=1 i=1

L2 (YE , λ, ρ) = −λ(YE − µtot )2 + ρYE + z2 σtot

2

L1 (Y, λ, ρ) represents a supermodular set function, since it equals g(Y ) plus two linear functions of Y , see
Theorem 1. Hence, L(Y, YE , λ, ρ) is also a supermodular function of Y .

B.1. Solution of the Lagrangean Relaxation for Fixed (λ, ρ)

We now show how the Lagrangean relaxation can be solved for fixed λ and ρ. We first show how we can
compute the optimal value of YE . Note that L2 (YE , λ, ρ) is a concave function of YE , so it achieves its minimum
in one of the two end points of its feasible interval [d0 , dN ] derived in Section 6.2. (Recall d0 = µtot + z σtot and
PI
dN = i=1 E(Si )) Therefore, we can find the optimal value of YE by comparing L2 (d0 , λ, ρ) and L2 (dN , λ, ρ):

YE∗ (λ, ρ) = d0 ⇐⇒ −λ(d0 − µtot )2 + ρd0 ≤ −λ(dN − µtot )2 + ρdN

ρ
⇐⇒ λ[d2N − d20 − 2(dN − d0 )µtot ] ≤ ρ(dN − d0 ) ⇐⇒ [dN + d0 − 2µtot ] ≤
λ
ρ
Conversely, λ
< [dN + d0 − 2µtot ] if and only if YE∗ (λ, ρ) = dN .
Thus, substituting YE by YE∗ (λ, ρ), the Lagrangean relaxation can be written as:
K I J I
X X X X
z ∗ (λ, ρ) ≡ min δk γij νijk + [vi − ρE(Si ) + λz2 V ar(Si )]Yi + λ[−(YE∗ − µtot )2 + z2 σtot
2
] + ρYE∗
Y,ν
k=1 i=0 j =1 i=1
(31)
X J

s.t. νijk ≤ Sik Yi i = 0...I k = 1...K

j =1
I
X
νijk ≥ Djk j = 1...J k = 1 . . . K; νijk ≥ 0, Yi ∈ {0, 1}
i=0

which is linear in the decision variables Y and ν; hence it is a M ILP .

B.2. Solution Methods for zD = maxλ,ρ z ∗ (λ, ρ)

It is well known and easily verified that the function z ∗ (λ, ρ) is jointly concave in (λ, ρ); however, as the
optimal value of a mixed integer program, it fails to be differentiable everywhere. Nevertheless, a steepest
ascent subgradient method can be employed to find the maximizing pair of Lagrange multipliers (λ∗ , ρ∗ ).
However, to ensure convergence, very small step sizes need to be chosen in the steepest ascent method.
We have observed that the following tâtonnement scheme converges considerably faster: at odd (even)
numbered iterations, the scheme fixes the last obtained value of ρ (λ) and finds the corresponding optimal
value of λ (ρ) via a standard bisection method. (In view of the joint concavity of the function z ∗ (λ, ρ), the
tâtonnement scheme is guaranteed to converge to an optimum solution.)
Solving the Lagrangean dual zD generates a very useful lower bound against which various heuristics
may be benchmarked. In addition, the optimal solution vector Y ∗ (λ∗ , ρ∗ ) in the Lagrangean relaxation (31)
provides another promising initial set of farmers to start the Greedy-add heuristic with.
Federgruen, Lall, and Şimşek: Online Appendix for “Supply Chain Analysis of Contract Farming” 3

Appendix C: Analytical Expressions for the Coefficients in (M cont )

In this appendix, develop analytical expressions for the coefficients in the mathematical program (M cont )
used to determine optimal contract menus. For notational simplicity, we assume, as in Lemma 2(d), that
the yield functions fi (αi , Ti ) are increasing and differentiable, so that the analytical characterization of the
optimal aggregate water supply Ti`∗ (αi , Ri ) provided there, can be employed. Since, under all yield functions,
Ti`∗ (αi , Ri ) is an increasing function of Ri , see Lemma 2(b), define Ri∗ (αi , q` ) as the unique root of the equation
−1
Ti`∗ (αi , Ri ) = t∗i (αi , q` ), i.e., Ri∗ (αi , q` ) is the inverse function Ti`∗ (αi , ·) evaluated at t∗i (αi , q` ) and denotes
the break-even rainfall quantity under which the optimal total water supply suffices to yield the quantity
value of contract `.
Z Ri∗ (αk ,q` )
Ui (`|k, q` ) =δiT E(Ri ) + (P (q` ) − ci ) fi (αk , Ti`∗ (αk , Ri ))dGi (Ri ) + (P (q` ) − ci )q` Gi (Ri∗ (αk , q` ))
0
Z ∞ Z ∞
fi (αk , Ti`∗ (αk , Ri )) − q` dGi (Ri ) − δiT Ti`∗ (αk , Ri )dGi (Ri )
 
+ (psi − ci ) (32)
Ri∗ (αk ,q` ) 0
(Z )
K
X Ri∗ (αk ,qk )
k k ∗ k ∗ k
E(Si |q) = P[αi = α ] fi (α , T (α , Ri ))dGi (Ri ) + qk Gi (R (α , qk ))
ik i (33)
k=1 0
(Z )
K
X Ri∗ (αk ,qk )
2 k 2 k ∗ k 2 ∗ k
E(Si |q) = P[αi = α ] fi (α , T (α , Ri ))dGi (Ri ) + qk Gi (R (α , qk ))
ik i (34)
k=1 0

V ar(Si |q) =E(Si2 |q) − E2 (Si |q) (35)

(Z )
K
X Ri∗ (αk ,qk )
∗
vi (q) = k
P[αi = α ]P (qk ) fi (αk , Tik (αk , Ri ))dGi (Ri ) + qk Gi (Ri∗ (αk , qk )) (36)
k=1 0

Appendix D: Numerical Analysis: Scenario Generation

The potato chips manufacturer contracts with several thousands of farmers located in 19 districts across six
states of India. (India is divided into 36 states or territories each of which is subdivided into districts.)
Indian farmers are typically very small. Table 1 in Singh et al. (2002) shows that in 1991, country-wide, a
total of 165.6 million hectares were cropped, by no less than 106.6 million farmers. (The total cropped area
barely grew between 1971 and 1991; however, the number of farmers grew from 70.5 million in 1971 to 106.6
million in 1991. This, of course, implies that the average farm size decreased significantly in this time span.)
The table partitions the farms into five categories, according to the size of the cropped area: sub-marginal,
marginal, small, medium, and large farms.
We only had access to aggregate data for the 19 districts, rather than data for each of the contracted or
potential farmers. When generating our problem instances, we assumed that the manufacturer only consid-
ered small, medium, and large farms, which, nationwide, represent 41% of all farms. We assumed that the
distribution of farm sizes among the list of potential farms, in each district, reflected the national distribution
as in Table 1. For any given farmer, in any of the districts, we therefore assigned a land size that is drawn,
independently, from this national three-point distribution.
4 Federgruen, Lall, and Şimşek: Online Appendix for “Supply Chain Analysis of Contract Farming”

D.1. Geographic distribution of the farmers

We were given data for the total contracted cropped area in each district, for the years 2009-2010. For
any problem instance with I country-wide potential farmers, we distributed them across the 19 districts in
proportion to the districts’ average cropped area in the above two calendar years. (The resulting district
numbers were adjusted by a factor uniformly chosen from the interval [0.9, 1.1] and then rounded to the
nearest integer.) Table 10 in Appendix G lists the 19 districts and the percentage of the total cropped area
in each district. In view of the above distribution procedure, we have that, in each of the country-wide
instances in Section 7, the percentages listed in Table 10 approximately mirror the distribution of the total
of I farmers among the districts.

D.2. The farmer’s supply quantity distribution

In terms of the yield functions, fi (αi , Ti ), we chose structure 1 see (10), specified as follows:
6
X
Xi = Λi αs Iis − β[Wi − Ri − xi ]+ , where (37)
s=1

(
1, if farmer i is located in state s, s = 1, . . . , 6; i = 1, . . . , I
Iis =
0, otherwise,

Λi is the land size of farmer i (in hectares), β > 0 is a non-negative parameter, and {α1 , . . . , α6 } are random
variables. Thanks to data provided by the Columbia Water Center, we had access to a 100-year time series,
for each of the 19 districts, of the total rainfall (in millimeters –mm) over the course of the 90 day growing
season. We assumed that each farmer experiences a random rainfall quantity generated from a Normal
distribution whose mean and standard deviation match the 100-year distribution pertaining to the district
the farmer resides in. The Center also provided an estimate of the ideal water supply W Sd (in mm) for
each district d = 1, . . . , 19. However, this number represents the ideal water supply, assuming constant daily
supplies. In practice, rain falls periodically, with many dry days in between rainy episodes, generating major
deficits even when the aggregate season-wide rainfall is well in excess of the ideal aggregate water supply
volume. (Recall the alternative deficit specification in equation (14) based on the daily rainfall process.)
Since we had no access to daily rainfall data, we scaled the numbers {W Sd , d = 1, . . . , 19} up by a factor
1
P19
f = 1.2/( 19 d=1
W Sd /ηd ) with ηd , the mean season-wide rainfall in district d, computed from the above
100-year distribution, resulting in adjusted numbers {W d S d , d = 1, . . . , 19}. In each problem instance, we
specified each farmer’s well capacity as 25% of his rainfall quantity.
Since Wi = Λi Wd S d(i) and Ri = Λi ρd(i) , with d(i) the district to which farmer i belongs, we get the following
final specification of the yield equation (37):
6
X
Xi = Λi αs Iis − β[Λi W
d S d(i) − Λi ρd(i) − xi ]+ (38)
s=1

We estimated the distributions of the six state specific intercepts {αs : s = 1, . . . , 6} and the value of the
slope β by running a regression based on the 38 observations of supply volumes per district, for a unit
(1 hectare) of land, in the two 2009 and 2010 contract years. (When calculating the deficit quantities in
Federgruen, Lall, and Şimşek: Online Appendix for “Supply Chain Analysis of Contract Farming” 5

(38), we assumed no water was drawn from wells, i.e., xi = 0.) The regression equation had an adjusted
R2 value of 0.92. The point estimates and standard errors for the intercept values are reported in Table 5;
they show significant differences in the farm productivity across the six states, with Uttar Pradesh roughly
2.5 times as productive as the state of Karnataka. In our problem instances, we assigned to each farmer i,
the yield function fi (·, ·) in (38), with an α-value drawn from a 5-point distribution, anchored on a Normal
distribution with mean and standard deviation given by the estimated mean µs and standard error σs of
the α−parameter pertaining to the state in which the farmer resides. (The 5-point distribution used the
values {µs − 2σs , µs − σs , µs , µs + σs , µs + 2σs } as its support, with probabilities {Φ−1 (−1.5), Φ−1 (−0.5) −
Φ−1 (−1.5), Φ−1 (0.5) − Φ−1 (−0.5), Φ−1 (1.5) − Φ−1 (0.5), 1 − Φ−1 (1.5)} to match the underlying distribution;
a discrete distribution was used to be consistent with our model assumption of finitely many farmer types.)

State Mean Standard Error State Mean Standard Error

West Bengal 18.02 1.16 Uttar Pradesh 29.36 3.82
Bihar 18.30 1.99 Maharashtra 12.62 1.59
Gujarat 14.85 3.32 Karnataka 11.49 1.29
Table 5 Estimations of the parameters in equation (38)

In this numerical study, we assumed that the farmer’s intercept value α is his private information while the
manufacturer only knows the distribution from which it is drawn. No other sources of asymmetric information
are assumed. For the slope β, we uniformly used the number β = 0.0009, which corresponds to the lower
point of the 95% confidence interval of the estimate of this parameter.

D.3. The farmer’s cost parameters and menu of contracts

We obtained state-specific values for the cost parameters ci , the variable cost of producing a metric ton (MT)
of potatoes, exclusive of irrigation costs, and the irrigation cost per cubic meter of water. The former cost
value includes the cost of seeds, planting, ploughing, fertilizers, plant protection chemicals, harvesting, and
packing costs.
We specified a menu of contracts, differentiated by state and farm size category. (This represents a total of
18 menus.) Each menu is characterized by a linear price-quantity relationship. The menus were constructed
as follows: For each state s, we were given the average price, ps , paid per MT of potatoes, as well as the
average quantity, q s , procured per hectare. As an example, consider the contract menu for small farmers:
The point (ps , 1.43q s ) was selected as the midpoint on the menu, since an average small farmer has a land
size of 1.43 hectares, see Table 1. When quantities are displayed on the X−axis and prices on the vertical
Y −axis, the line with slope −3/2 was drawn through this point, and 5 equidistant points were selected on
either side of the midpoint. For the “medium” and “large” farmers, the same procedure was used except that
the midpoint was selected as (ps , 2.76q s ) and (ps , 7.95q s ), reflecting their respective land sizes, see Table 1,
and the slope of the contract line as -1 and −3/5, respectively.

D.4. The manufacturer’s demands and transportation costs

The manufacturer has three production facilities. In our study, we drew the product demands in these facilities
from a Normal distribution with a coefficient of variation of 0.15. For each problem instance, we generated a
“high demand version” and “low demand version” in which the expected aggregate demand is specified as 65%
6 Federgruen, Lall, and Şimşek: Online Appendix for “Supply Chain Analysis of Contract Farming”

and 50% of the aggregate supply across all I potential suppliers, respectively, assuming each farmer supplies
the quantity corresponding with the midpoint of his contract line. Finally, we generated transportation
cost rates {γij } by multiplying a given average cost rate per MT per mile, with the relevant, Google Maps
calculated, road distance between the centroid of the farmer’s district and the relevant production facility.
We used the following specification of the {γ0j } coefficients.

γ0j = 10 ∗ [max γij ] + E(ps ), j = 1, . . . , 3 (39)

i,j

The first term represents an upper bound for the per unit distribution cost. When modeling an explicit
linkage between the {γ0j }-value and the aggregate rainfall quantity, (39) was modified to
 tot
R − E(Rtot )

γ0j = 10 ∗ [max γij ] + E(ps ) − 0.084 ∗ ST D(ps ) ∗ , (40)
i,j ST D(Rtot )
adopting the estimated correlation coefficient in Bhanumurthy et al. (2013).
When evaluating the set function g(Y ) in the manufacturer’s problem, see (1), we evaluated the second
term in (1) by drawing a sample of 100 {Si , Dj : i = 1, . . . , I j = 1, . . . , J} vectors. To solve each of the
farmer problems, we drew a sample of 10,000 realizations for the vectors {Ri , Ci }, when evaluating each of
the possible contracts in the offered menu, see problem (F ) and equation (15).
All reported computation times refer to a laptop with Intel(R) Core(TM) i7-3537U CPU @ 2.00 GHz and
2.50 GHz processor, 8GB RAM and 64-bit Windows operating system.
Appendix E: A Logit Model to Identify Farmer Attributes as Selection Criteria
The following logit models provided the best fit to explain which farmers are selected as a function of several
attributes. Let (
1, if ti + i > 0
Y0 =
0, otherwise
with ti = β0 + β T Xi , where Xi is a vector of farmer characteristics, and i an unobserved error term. The
following models provided the best overall fit:

ti =117.6 − 0.36 (vi /E(Si )) − 11.86 CV (Si ) − 0.36 ∆i instance in * (41)

(0.01) (0.01) (0.56) (0.0006)

ti = 23.2 − 0.01 (vi /E(Si )) − 22.9 CV (Si ) − 0.03∆i instance in ** (42)

(<0.0001) (<0.0001) (<0.0001) (0.1)

where (vi /E(Si )) denotes the expected payment to the farmer divided by his expected supply volume,
CV (Si ) denotes the coefficient of variation of his supply volume, and ∆i denotes his distance to the nearest
manufacturing plant. The numbers within parentheses represent the logistic regression’s equivalent of the
p-values in ordinary least-squares regression. Beyond the generally low “p”-values, the goodness-of-fit in (41)
and (42) is further demonstrated by a percent concordants of 92% and 90%, respectively.
Appendix F: Numerical Analysis with Endogenously Determined Contract Menus
In parallel to the numerical study reported in “Impact of the Menu Design” part of Section 7, we have
conducted a parallel study in which the contract menu is designed endogenously, based on the methodology
of Section 5. More specifically, based on a set of I = 157 potential farmers and their characteristics, we have
evaluated what optimal price-quantity pairs should be offered in each of the 18 districts considered in Section
Federgruen, Lall, and Şimşek: Online Appendix for “Supply Chain Analysis of Contract Farming” 7

7 on the contract line with Slope 2. (The above farmer distribution procedure ended up generating farmers
in only 18 of the 19 districts.) In other words, instead of using five arbitrary and exogenously given contracts
on the menu line, we determine the optimal combination of contracts by solving the mathematical program
(M cont ), for each district separately. (For the purpose of this experiment, we assigned each district a fraction
of the countrywide (random) demand volume in proportion to the district’s total expected yield among all
farmers.) For this experiment, we also incorporated the spot market option for the farmers’ excess yields
by choosing a state specific (lower bound on) spot market price, psi , in the farmer’s problem (F ) as follows:
psi = max{0.8 ∗ min[average spot price in 2009, average spot price in 2010], 1.1ci }; see Table 6 for the spot
price data. In terms of the individual rationality constraints (IR) in (27), we selected the minimum expected
profit levels for each of the five types as follows: starting with the type-1 farmers (with the stochastically
smallest yield distributions), we selected a minimum profit level below the maximum achievable profit (under
any contract on the menu line). After specifying the minimum profit level for type i, i = 1, . . . , 4, we selected
a minimum profit level for type i + 1, above that of type i, but below its maximum achievable profit level,
see Figure 1 in Appendix G. Table 7 displays the minimum profit levels and optimal contract quantities for
all five types in each of the 18 districts. In many cases the optimal solution uses less than five distinct points;
the average optimal number of contracts is 3.17.

avg. spot price avg. spot price

S State 2009 2010 psi S State 2009 2010 psi
($/M T ) ($/M T ) ($/M T ) ($/M T )
($/M T ) ($/M T )
1 West-Bengal 236∗ 104 84 4 Uttar-Pradesh 177 93 131
2 Bihar 234 150 120 5 Maharashtra 258 154 123
3 Gujarat 145∗ 160∗ 130 6 Karnataka 258 216 173
Table 6 State summary statistics. S: state index. *: Based on incomplete spot price data

Finally, we have explored how increased volatility of the rainfall and demand distributions impacts the
optimal menu choices and associated expected manufacturer’s cost. We report the results in Table 8 for
District 1 and District 16, when the coefficient of variation of all demand and rainfall distributions are
increased by 50% and 250%. For District 1, we observe an interesting phenomenon: the increased volatility
benefits the manufacturer resulting in a (modest) cost saving, because majority of the farmers can now be
induced to accept higher quantity values, and hence lower prices (58% of the farmers are of types 2 and 3).
The reason for this upward shift of the contract volume is that, in order to continue to meet the minimum
expected profit level, these farmers need a higher contract volume to take advantage of better than average
yields. Increased rainfall volatilities do result in the farmers being worse off, but the average profit reduction
is strongly mitigated by the upward adjustments of the contract quantities. Increased demand volatility
results in higher costs for the manufacturer. When the volatility of both the demand and rainfall distributions
are amplified there is no feasible contract menu, given the coverage and all minimum expected profit level
constraints.
8 Federgruen, Lall, and Şimşek: Online Appendix for “Supply Chain Analysis of Contract Farming”

π ik : min. acceptable profits optimal quantities (prices) in the menu

S D π i1 π i2 π i3 π i4 π i5 q1∗ q2∗ q3∗ q4∗ q5∗
1 1 1700 1750 1800 1850 1900 23.13(151.1) 23.88(149.9) 23.88(149.9) 26.70(145.2) 29.44(140.6)
1 2 1700 1750 1800 1850 1900 21.75(153.4) 24.00(149.7) 24.00(149.7) 28.01(143.0) 28.01(143.0)
1 3 1700 1750 1800 1850 1900 22.34(152.4) 22.54(152.1) 25.19(147.7) 25.19(147.7) 28.30(142.5)
1 4 1700 1750 1800 1850 1900 21.42(154.0) 23.72(150.1) 23.72(150.1) 27.06(144.6) 27.06(144.6)
1 5 1700 1750 1800 1850 1900 23.37(150.7) 24.08(149.5) 24.13(149.5) 27.01(144.6) 27.01(144.6)
1 6 1700 1750 1800 1850 1900 22.03(152.9) 22.74(151.8) 25.45(147.2) 25.45(147.2) 28.06(142.9)
2 7 1600 1700 1850 2000 2100 19.11(157.8) 19.11(157.8) 19.11(157.8) 19.11(157.8) 19.11(157.8)
3 9 75 300 600 800 950 15.85(197.3) 12.80(185.7) 12.80(185.7) 12.80(185.7) 17.76(177.4)
3 10 200 400 600 800 1000 18.54(176.1) 18.54(176.1) 18.54(176.1) 18.54(176.1) 18.54(176.1)
4 11 1050 1150 1200 1250 1350 22.71(172.5) 23.54(171.1) 24.06(170.2) 24.06(170.2) 24.06(170.2)
4 12 1100 1200 1250 1300 1350 20.56(176.1) 26.38(166.4) 26.38(166.4) 26.38(166.4) 26.38(166.4)
5 13 2900 3325 3750 4150 4550 13.35(323.1) 15.55(319.4) 17.86(315.6) 21.29(309.9) 22.49(307.9)
5 14 2800 3225 3650 4050 4425 13.32(323.1) 15.53(319.4) 18.92(313.8) 20.13(311.8) 22.34(308.1)
5 15 2050 2500 2950 3350 3750 11.85(325.6) 16.74(317.4) 16.74(317.4) 18.53(314.5) 20.89(310.5)
6 16 1350 1500 1650 1750 1900 13.66(210.9) 14.16(210.1) 14.81(209.0) 14.81(209.0) 14.81(209.0)
6 17 1300 1450 1600 1750 1850 13.98(210.4) 13.98(210.4) 14.78(209.0) 14.78(209.0) 14.78(209.0)
6 18 1200 1350 1500 1650 1750 10.75(215.8) 13.79(210.7) 13.79(210.7) 13.79(210.7) 13.79(210.7)
6 19 1350 1500 1640 1750 1900 13.67(210.9) 14.23(210.0) 14.78(209.0) 14.78(209.0) 14.78(209.0)
Table 7 Minimally acceptable profits and the associated optimal contract quantities for five types of farmers in
each district. Numbers in parentheses are the associated price values. S: state index, D: district index.

District 1
opt. quantities (prices) in the menu
q1∗ q2∗ q3∗ q4∗ q5∗ Manuf. Cost
D.CV = x, R.CV = y 23.13 23.88 23.88 26.7 29.44 44,003.5
(151.1) (149.9) (149.9) (145.2) (140.6)
D.CV = x, R.CV = 1.5y 22.13 24.3 24.3 26.3 28.53 43,837.3
(152.8) (149.2) (149.2) (145.8) (142.1)
D.CV = 3.5x, R.CV = y 21.58 25.04 25.04 26.83 29.44 44,505.4
(153.7) (147.9) (147.9) (144.9) (140.6)
D.CV = 3.5x, R.CV = 1.5y Infeasible N/A
District 16
opt. quantities (prices) in the menu
q1∗ q2∗ q3∗ q4∗ q5∗ Manuf. Cost
D.CV = x, R.CV = y 13.66 14.16 14.81 14.81 14.81 12,157.0
(210.9) (210.1) (209) (209) (209)
D.CV = x, R.CV = 1.5y 13.34 13.49 15.56 15.56 15.56 12,369.6
(211.4) (211.2) (207.7) (207.7) (207.7)
D.CV = 3.5x, R.CV = y 13.66 15.11 16.82 18.14 18.14 13,387.6
(210.9) (208.5) (205.6) (203.4) (203.4)
D.CV = 3.5x, R.CV = 1.5y Infeasible N/A
Table 8 Impact of risk measures on manufacturer performance measures. Numbers in parentheses are the
associated price values. D.CV: Demand Coefficient of Variation, R.CV: Rainfall Coefficient of Variation

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Topkis, D. M. 1998. Supermodularity and Complementarity. Princeton University Press, Princeton, NJ.