1 Introduction

The objective of any supply chain coordination is to extract maximum possible value from the supply chain, which may not be the case if all the players act individually. Supply chain contracts are one of the essential tools to coordinate the supply chain and maximize the total supply chain profit. A supply chain contract is an agreement between different players for parameters like price, delivery time, quantity, and quality of order, to name a few. Researchers have developed various supply chain contracts such as wholesale price, buyback, revenue sharing, quantity flexible, sales rebate, quantity discount, and options contract (Govindan et al., 2013; Zhao et al., 2010).

The supply chain contracts have been extensively used in various industries and businesses. In this work, we study the agribusiness industry with a focus on contract farming. Contract farming (CF) is an agreement between a firm and a farmer or group of farmers before the commencement of production. Often the firm provides the farmer services and resources to help in the production (Ton et al., 2018). These agreements can address the problem of market failures and improve technology adoption, productivity, and welfare (Ragasa et al., 2018). CF provides benefits to the firm and the farmers in various capacities; on the one hand, it ensures timely supply of high-quality raw material for the firms (Swinnen & Maertens, 2007), and on the other hand, it provides insurance against price risk, access to credit and necessary inputs to the farmer (Barrett et al., 2012). Huang et al. (2019) show that contractual agreements in agricultural produce mitigate the grower’s risk and achieves a win–win situation for the grower and the wholesaler.

CF arrangements can provide new marketing opportunities for small landholder farmers in developing countries (Ochieng et al., 2017). CF is seen as one of the preferred institutional devices to make the smallholders’ path to development (Oya, 2012). However, at the same time, CF arrangements face several challenges, more so in the developing countries. One major issue is related to the sustainability of contracts in the face of extreme market conditions. It is very common to observe huge fluctuations in the market prices of agricultural commodities. It has been reported that both the parties of CF breach the contract when the market prices are very high or low to gain extra benefits (Schipmann & Qaim, 2011; Barrett et al., 2012; Ton et al., 2018). Tang et al. (2016) suggested the need for a mechanism in the contract that would generate mutual benefit and prevent frauds like side-selling. The objective of this work is to develop a supply chain contract between the firm and the farmer, which encourages them to remain in the contract even in extreme market conditions.

We propose a contract termed as Violent Market Price (VMP) contract, which incentivises the firm and the farmer to remain in the contract in situations of extreme price deviations. We determine upper and lower limits for the market price between which the contract price remains fixed; as the market price goes beyond these limits, the contract price shifts in a corresponding proportion. These limits are based upon penalties or costs incurred by the two parties for deviation from the contract. Agri-business firms can potentially benefit from the proposed VMP contract, especially in developing countries, where market prices of agricultural produce fluctuate hugely. Food companies like PepsiCo, Nestle, Hindustan Unilever Ltd., United Breweries are increasingly using contract farming (CF) for sustainable supply chain management. Industries like sugar, natural fabrics, pharmaceutical, and oilseeds are recently seen to form long-term relations with farmers using CF. Generally farmers decide whether to opt for the CF crop or the conventional crop. In India, some of the major CF crops are herbs, soybeans, tomato, cotton, potato, and sugarcane, while rice, wheat, maize, pulses, and vegetables are considered as conventional crops.

Researchers have designed various supply chain contracts to share risk and profits between supply chain members efficiently. To achieve this, different mechanisms have been proposed, such as penalty based contract and dynamic pricing contract. VMP contract is closest to these two mechanisms and yet possesses very different structure and properties. Penalty based contracts are generally modelled as penalising the guilty party based on shortage in the agreed quantity. Dynamic pricing contracts are based on the idea of having different pricing scheme for different order quantities, time, and other factors. The VMP contract is designed for the situation when the parties can breach the contract under extreme market conditions.

In the price structure proposed in the VMP contract, the contract parameters are determined through a Stackelberg game with the firm as the leader and the farmer as the follower. Backward induction is used to obtain the optimal decisions. We prove that the players have no incentive to breach the VMP contract even in extreme market conditions. Conditions for the firm and the farmer to participate in the VMP contract are also derived. Sensitivity and numerical analysis based on a case-study for parameters values are carried out.

The rest of the paper is organized as follows. In Sect. 2, we review the existing literature and position our work. In Sect. 3, we describe the model, and in Sect. 4, derive its solution. In Sect. 5, we present numerical solutions based on case-study and conduct sensitivity analysis of the model parameters. In Sect. 6 we describe the managerial implications of this study. Conclusion and scope for future work are finally presented in Sect. 7.

2 Literature review

Researchers have extensively studied supply chain management to address several societal issues, these include humanitarian logistics and supply chain management (Nurmala et al., 2017; Banomyong et al., 2019; Behl and Dutta, 2019) and sustainable green supply chains (Lee and Tang, 2018; Modak et al., 2018; Taleizadeh et al., 2018; Sana, 2020). Agri-business supply chain is yet another important area which is given considerable attention due to its potential in food security, nourishment, and poverty alleviation (Behzadi et al., 2017, 2018; Manavalan et al., 2019). In this work, we study supply chain contracts to address a specific problem which arises in the context of contract farming.

We next begin with the review of current literature on contract farming (CF) to give an overview of its benefits and challenges and to position the problem addressed in this work. We then review supply chain literature related to the methodology adopted in the VMP contract developed in this work.

Contract farming (CF) has been studied extensively in the context of helping farmers and strengthening agri-business activities. Miyata et al. (2009) suggested that CF has increased farmers’ income due to the technical assistance provided by firms. Similarly, Bellemare (2012) showed an increase in farmers’ household income with CF participation. Along with the welfare of farmers, CF is also found to contribute to the development of agri-business supply chains. Mertens and Velde (2017) conclude that CF leads to the expansion of agricultural activities and upgrades in agri-business supply chains. On similar lines, Ochieng et al. (2017) conclude that fairer risk-sharing arrangements could enhance small farmers’ participation. These findings motivate the study of CF as an efficient mechanism for agriculture supply chain management.

Along with the promises of CF, researchers have also investigated the emerging challenges associated with it. Kumar and Kumar (2008) have identified the prospects and constraints of CF in India. Some of the firms’ problems in expanding CF include the violation of terms and conditions by the farmers and frequent price fluctuation. Barrett et al. (2012) show a similar phenomenon that it is common for CF arrangements to fail due to price and yield volatility. The firms frequently drop the smallholder farmers, and the smallholder farmers often opt-out of CF (Barrett et al., 2012). Ton et al. (2018) find that both the firm and the farmer face risks in CF; for example, the farmers may side-sell products after receiving the firm’s services. Stone and Rahimifard (2018) review literature on resilience in agri-food supply chains and identify various risk sources such as socio-political, economic, and environmental. In this study, we address such shortcomings in CF structure. In particular, violation of the CF contract in the event of drastic price fluctuation is the main problem addressed in this work.

Several agri-business scenarios have been modeled using supply chain management, e.g., Hovelaque et al. (2009) and Mason and Villalobos (2015) propose supply chain models for agriculture cooperatives, while Perlman et al. (2019) analyze a competing agri-business supply chain offering conventional product and its organic version. Peng and Pang (2019) consider a three-level supply chain consisting of a farmer, supplier, and distributor with yield uncertainties and government-assisted subsidy. Leithner and Fikar (2019) develop a decision support system using simulations for organic fresh food supply chain including food quality data. Castaneda et al. (2019) provide behavioural insights into different risk-sharing contracts for agri-business supply chains in developing countries. Some of the supply chain contracts between firm and farmers which have been proposed recently include Huh et al. (2012) and Tang et al. (2016). While Huh et al. (2012) develop a contract for CF, which allows the farmer the possibility of retracting the contract, Tang et al. (2016) develop a partially-guaranteed prices (PGP) contract for CF under which the firm offers a guaranteed unit price for any fraction of produce. While various models to study agri-business supply chains exist in the literature, our study uniquely addresses the problem of the failure of CF arrangements due to market price fluctuations. To the best of our knowledge, no work addresses the problem of sustainability of the contract due to violent market price fluctuations. In this study, we attempt to solve this problem. In our study, the firm-farmer interaction is modeled as a Stackleberg leader–follower game which is similar to the work of Wang and Chen (2017), Nielsen et al. (2019), and Taleizadeh et al. (2020).

The methodology of the proposed VMP contract is based on the underlying principles of dynamic price contracts and penalty-based contracts. Tang et al. (2012) introduce a dynamic pricing policy to counter disruption in raw material supplies. Zhang et al. (2012) develop a dynamic pricing model based on the delivery time required by customers. Chen and Chang (2013) propose a dynamic pricing model for new and remanufactured products in a closed-loop supply chain. Penalties have also been found to be effectively used in supply chain management. Starbird (2001) study the effect of rewards and penalties on an expected cost-minimizing supplier. A menu of penalty contracts is proposed by Gan et al. (2010), which provides greater certainty of demand and supply for a supply chain. Sana (2016) develops a supply chain model in which both manufacturer and retailer share the promotional effort cost, and the manufacturer pays the penalty for the shortage. Li et al. (2016) study supply chain disruption and introduce penalty and financial assistance terms to encourage the supplier to recover its production quickly. More recently, Tian et al. (2018) propose a penalty contract in which the buyers pay the penalty for cancelling some portion of their orders. However, none of these studies consider the effect of fluctuations in market price on the sustainability of the contract. In this research, an attempt is being made to achieve this.

The proposed VMP contract applies notions of dynamic and penalty contracts to develop a novel price mechanism. Conditions of high fluctuations in market price prevalent in CF often lead to the failure of the contract. To address this unique problem, the VMP contract pricing structure is designed so as to sustain even under such conditions of the market price. In the next section, we present the contract developed in this study and derive the expected profits of the farmer and the firm under this contract.

3 Model

One of the main problems identified in the literature on contract farming is the violation of contract in the event of adverse market conditions. When the market prices are very low, then the firms are found to break the contract. Similarly, farmers often break the contract when market prices are high. Contract Farming and Services Act of India also hinted about the problem of deciding the sale-purchase price in case of violent movement (upswing or downswing) of the market price (Contract Farming and Services Act, 2018). To address this problem, we propose a pricing contract termed as violent market price (VMP) contract, which provides flexibility in adapting to violent market fluctuations. We show that our contract sustains even under conditions of violent market price fluctuation.

3.1 VMP contract

We consider a single risk neutral firm and a single risk neutral farmer who enter into a contract. We assume that the farmer has a fixed area of \(T\) hectares to cultivate two kinds of crops: one being the conventional crop and the other, the crop that the firm wants the farmer to sow, which is termed as the contract farming crop. We assume that the productivity of the two crops are deterministic and are denoted by \(\theta_{k}\) tons per hectare and \(\theta_{c}\) tons per hectare for the conventional and contract farming crop, respectively. Market prices of the two crops are stochastic in nature with prices \(k\) per ton and \(\user2{c }\) per ton for the conventional and contract farming crop, respectively. We assume that prices of CF and conventional crop are uniformly distributed in \(\left[ {0, \alpha } \right]\) and \( \left[ {0, \beta } \right]\), respectively. Prices of agricultural commodities, especially in developing countries, can go very low. For e.g., in India, sometimes prices go as low as 2–3 Rs/kg (0.03 $/kg) for some produce. Not only this, dumping of the produce due to high production and very less market price is also common (Thow et al., 2018). This happens mainly due to poor infrastructure and high storage cost for farmers. Such dumping of the produce can thus be associated with market price tending towards zero. Thus the lower limit of the market price of the CF and the conventional crop is theoretically considered to be zero.

The farmer’s decision is to find how much of the land area (say \(q\) hectares) should be allocated to the CF crop and how much (\(T - q\)) to the conventional crop. The firm’s decision problem is to decide the price \(p \) ªper ton that will be offered to the farmer for the contract farming crop under stable market conditions. This price is different from the market price of the contract farming crop and is termed as the contract farming price. We assume that the firm requires \(r\) tons of the contract farming crop for its operations. Often the firm’s requirement of the contract farming crop is such that it cannot be satisfied by a single farmer. Hence it is assumed that the firm will buy the entire contract farming crop produced by the farmer and purchase the remaining quantity from the open market. More than 86% of farmers in India are small landholders who own less than two hectares of farm land. So, in this context, we assume that a single farmer may not be able to satisfy all the requirements of the firm. For example, Nijjer Agro Foods Ltd. had a CF project in Punjab state for tomato and chilly, which required 250 hectares of land. In a similar project of herbs in Karnataka state, Natural Remedies Pvt. Ltd. required 150 hectares of land (Sunanda, 2005).

The contractual agreement between the firm and the farmer specifies the quantity offered by the farmer and the price offered by the firm. Penalties are incurred by both the players upon deviations from the contractual agreement. The firm’s and the farmer’s penalties are denoted by \(Z_{Fi}\) and \(Z_{Fa}\) respectively, and the penalties are assumed to be common knowledge. Actual payments are made as per the payment structure proposed in the contract, which specifies the payment as a function of the market price of the CF crop.

The sequence of decisions of the firm-farmer interaction follows.

  1. 1.

    At the beginning of the sowing season, the firm moves first and informs the farmer about the contract farming crop and offers the contract farming price, i.e., price under stable market conditions, denoted by \(p\).

  2. 2.

    The farmer moves next and, based on the offer made by the firm, decides the land allocation \(q\) and thus the production quantity of the contract farming crop.

  3. 3.

    After harvesting, the firm procures the agreed quantity of contract farming crop from the farmer according to the following VMP contract payment structure,

    $$ \left\{ {\begin{array}{*{20}l} {p - \left( {L - c} \right),} \hfill & {0 < c \le L} \hfill \\ {p, } \hfill & {L \le c \le U } \hfill \\ {p + \left( {c - U} \right),} \hfill & {U \le c \le \alpha } \hfill \\ \end{array} } \right. $$
    (1)

where \(c\) is the market price of CF crop, \(L\) and \(U\) are the lower and upper limits of the market prices beyond which contracts often break (these limits are derived below). To incorporate the stability of the contract, we determine these limits between which the CF price is kept constant. For market prices which are beyond the limits, the contract farming price will shift in a corresponding proportion. The upper and lower limits are the prices at which the farmer and the firm are indifferent; i. e., at these limits, the players are indifferent between violating the contract and honouring it. The proposed payment structure gives the assurance of fixed price, which is the main feature of any contract, and provides price flexibility in extreme cases, which prevents contract violation. We next determine the upper and lower limits of the proposed contract.

The market price at which the farmer is indifferent between violating the contract and honouring it, determines the upper limit \(U\) beyond which the contract farming price will increase proportionally. The utility of the farmer by honouring the contract is given by \(\theta_{c} qp + \theta_{k} \left( {T - q} \right)k\) (which is the sum of revenue from contract farming crop and conventional crop). The utility from breaking the contract is \(\theta_{c} qc + \theta_{k} \left( {T - q} \right)k - Z_{Fa}\) (where \(\theta_{c} qc\) is the revenue of the contract farming crop obtained from sale in market, \(\theta_{k} \left( {T - q} \right)k\) is the revenue from the conventional crop and \( Z_{Fa}\) is the penalty incurred by the farmer due to violation of contract). Hence to determine the upper limit, we equate the profit of the farmer when he violates the contract with the profit when he honours it.

$$ \theta_{c} qp + \theta_{k} \left( {T - q} \right)k = \theta_{c} qc + \theta_{k} \left( {T - q} \right)k - Z_{Fa} $$
(2)

The market price at which these two profits are the same is defined as the upper limit of the VMP contract.

$$ U = p + \frac{{Z_{Fa} }}{{\theta_{c} q}} $$
(3)

Similarly, the market price at which the firm is indifferent from violating the contract and honouring it determines the lower limit \(L \) on the market prices below which contract farming price reduces proportionally. Hence \(L\) is determined by solving the equation in which the cost of the firm when it violates the contract equals its cost of honouring it.

$$ \theta_{c} qp + \left( {r - \theta_{c} q} \right)c = rc + Z_{Fi} $$
(4)

The market price at which these two costs are the same is defined as the lower limit of the VMP contract.

$$ L = p - \frac{{Z_{Fi} }}{{\theta_{c} q}} $$
(5)

The expressions for calculating the upper and lower limit are stated in Eqs. 3 and 5, respectively. These limits provide VMP contract the flexibility to sustain drastic market price fluctuations. As the market price of contract farming goes beyond these limits, the contract farming price is increased/ decreased by the difference in market price from the upper/ lower limits derived. This price shifting ensures that there is no incentive for the farmer and the firm to violate the contract even when the market price goes beyond the upper and lower limits, respectively. This ensures the firm’s ability to offer flexible price in the event of huge price fluctuations. The entire payment structure upon substituting for \(L \) and \(U\) is as follows.

$$ \left\{ {\begin{array}{*{20}l} {c + \frac{{Z_{Fi} }}{{\theta_{c} q }} , } \hfill & {0 < c < p - \frac{{Z_{Fi} }}{{\theta_{c} q}}} \hfill \\ {p ,} \hfill & {p - \frac{{Z_{Fi} }}{{\theta_{c} q}} < c < p + \frac{{Z_{Fa} }}{{\theta_{c} q}}} \hfill \\ {c - \frac{{Z_{Fa} }}{{\theta_{c} q}} , } \hfill & {p + \frac{{Z_{Fa} }}{{\theta_{c} q}} < c < \alpha } \hfill \\ \end{array} } \right. $$
(6)

Expression (6) describes the VMP contract payment structure. Players move as per the sequence of events and optimally determine contract parameters \(p^{*}\) and \(q^{*}\).

Lemma 1

The expected profit of the farmer in the VMP contract is given by

$$ \frac{{(Z_{Fa}^{2} - Z_{Fi}^{2} )}}{{2\alpha \theta_{c} q}} + \frac{{p\left( {Z_{Fa} + Z_{Fi} } \right)}}{\alpha } + \frac{{\theta_{k} \beta \left( {T - q} \right)}}{2} + \frac{{\theta_{c} q\alpha }}{2} - Z_{Fa} . $$

Proof

The expected profit \(E\left[ {\pi_{Fa}^{vmp} } \right] \) of the farmer from the VMP contract is derived using the payment structure given by (6). Since the market price of CF crop is distributed uniformly in \(\left[ {0,{\varvec{\alpha}}} \right]\),

$$ E\left[ {\pi_{Fa}^{vmp} } \right] = \mathop \int \limits_{0}^{L} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} q\left[ {p - \left( {L - c} \right)} \right]} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{L}^{U} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qp} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{0}^{L} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} q\left[ {p + \left( {c - U} \right)} \right]} \right]I_{k} dkI_{c} dc $$
(7)

where \(I_{k}\) and \(I_{c}\) are the probability density functions of the uniformly distributed market prices of the conventional and CF crop, respectively. For the production quantity \(\theta_{k} \left( {T - q} \right)\) of the conventional crop at market price \(k\), the farmer gets a profit of \( \theta_{k} \left( {T - q} \right)k\). When the market price of CF crop remains below the lower limit of the VMP contract, the farmer gets a price of \(\left[ {p - \left( {L - c} \right)} \right]\) for a quantity of \( \theta_{c} q\). Hence the farmer gets a total profit of \(\theta_{k} \left( {T - q} \right)k + \theta_{c} q\left[ {p - \left( {L - c} \right)} \right]\) when the market price of the CF crop is below \(L\). Similarly, the profit from the CF crop when its market price is between the lower and upper limits is \(\theta_{c} qp\), and \(\theta_{c} q\left[ {p + \left( {c - U} \right)} \right]\) for market price higher than \(U\).

Substituting for \(L\) and \(U\), the expected profit is given by

$$ E\left[ {\pi_{Fa}^{vmp} } \right] = \mathop \int \limits_{0}^{{p - \frac{{Z_{Fi} }}{{\theta_{c} q}}}} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + Z_{Fi} + \theta_{c} qc} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{{p - \frac{{Z_{Fi} }}{{\theta_{c} q}}}}^{{p + \frac{{Z_{Fa} }}{{\theta_{c} q}}}} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qp} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{{p + \frac{{Z_{Fa} }}{{\theta_{c} q}}}}^{\alpha } \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qc - Z_{Fa} } \right]]I_{k} dkI_{c} dc $$

On simplification, the expected profit reduces to

$$ \frac{{(Z_{Fa}^{2} - Z_{Fi}^{2} )}}{{2\alpha \theta_{c} q}} + \frac{{p\left( {Z_{Fa} + Z_{Fi} } \right)}}{\alpha } + \frac{{\theta_{k} \beta \left( {T - q} \right)}}{2} + \frac{{\theta_{c} q\alpha }}{2} - Z_{Fa} $$
(8)

⬜

Since the firm purchases crop from the farmer as well as the market, the profit function of the firm is expressed as the negative value of the total cost incurred. We use this convention throughout the paper.

Lemma 2

The expected profit of the firm in the VMP contract is given by

$$ \frac{{(Z_{Fi}^{2} - Z_{Fa}^{2} )}}{{2\alpha \theta_{c} q}} - \frac{{(Z_{Fi} + Z_{Fa} )p}}{\alpha } - \frac{r\alpha }{2} + Z_{Fa} $$

Proof

The expected profit of the firm \(E\left[ {\pi_{Fi}^{vmp} } \right] \) in the VMP contract is derived as.

$$ E\left[ {\pi_{Fi}^{vmp} } \right] = - \mathop \int \limits_{0}^{L} [\theta_{c} q\left[ {p - \left( {L - c} \right)} \right] + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{L}^{U} [\theta_{c} qp + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{U}^{\alpha } [\theta_{c} q\left[ {p + \left( {c - U} \right) + (r - \theta_{c} q} \right)c]I_{c} dc $$
(9)

The firm purchases \(r - \theta_{c} q\) quantity from the open market at a price \(c\). If the market price is in the range \(\left[ {0, L} \right]\) then the firm purchases the quantity \(\theta_{c} q\) at unit price of \(p - \left( {L - c} \right)\) from the farmer. Similarly, the firm purchases the quantity \(\theta_{c} q\) at unit prices of \(p\) and \(p + \left( {c - U} \right)\) if market prices range in \(\left[ {L, U} \right]\) and \(\left[ {U, \alpha } \right]\) respectively. Substituting for \(L\) and \(U\), the expected profit of the firm is given as

$$ E\left[ {\pi_{Fi}^{vmp} } \right] = - \mathop \int \limits_{0}^{{p - \frac{{Z_{Fi} }}{{\theta_{c} q}}}} [\theta_{c} qc + Z_{Fi} + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{{p - \frac{{Z_{Fi} }}{{\theta_{c} q}}}}^{{p + \frac{{Z_{Fa} }}{{\theta_{c} q}}}} [\theta_{c} qp + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{{p + \frac{{Z_{Fa} }}{{\theta_{c} q}}}}^{\alpha } [\theta_{c} qc - Z_{Fa} + (r - \theta_{c} q)c]I_{c} dc. $$

Simplifying, we get the firm’s expected profit as

$$ \frac{{(Z_{Fi}^{2} - Z_{Fa}^{2} )}}{{2\alpha \theta_{c} q}} - \frac{{(Z_{Fi} + Z_{Fa} )p}}{\alpha } - \frac{r\alpha }{2} + Z_{Fa} $$
(10)

4 Model solution

We next derive the optimal model parameters \(p^{*}\), i.e., the price offered by the firm under stable market conditions and \(q^{*}\), the farmer’s optimal land allocation.

Proposition 1

In the subgame perfect equilibrium of the VMP contract, the farmer’s optimal land allocation \(q^{*}\) and the firm’s optimal CF price \(p^{*}\) are \(\sqrt {\frac{{Z_{Fi}^{2} - Z_{Fa}^{2} }}{{\theta_{c} \alpha \left( {\theta_{k} \beta - \theta_{c} \alpha } \right)}} }\) and \(\frac{{Z_{Fi} }}{{\theta_{c} q^{*} }}\) respectively. Furthermore, at optimality, the lower and upper limits of the payment structure of the VMP contract satisfy \(L = 0\) and \(U = \frac{{Z_{Fi} + Z_{Fa} }}{{\theta_{c} q^{*} }} \) , respectively.

Proof

The firm moves first and offers CF price \(p^{*}\) (price under stable market conditions), based on which the farmer decides the optimal land allocation \(q^{*}\). This situation is modelled as a Stackelberg game or a leader–follower game where the firm is the leader and the farmer the follower. Applying backward induction, we first derive the optimal land allocation for the farmer as a function of price \(p\). From first order conditions, we have.

$$ \frac{{\partial E\left[ {\pi_{Fa}^{vmp} } \right]}}{\partial q} = \frac{{Z_{Fi}^{2} }}{{2\alpha \theta_{c} q^{2} }} + \frac{{\theta_{c} \alpha }}{2} - \frac{{Z_{Fa}^{2} }}{{2\alpha \theta_{c} q^{2} }} - \frac{{\theta_{k} \beta }}{2} = 0 $$

The optimal land allocation \(q^{*}\) is thus obtained as

$$ q^{*} = \sqrt {\frac{{Z_{Fi}^{2} - Z_{Fa}^{2} }}{{\theta_{c} \alpha \left( {\theta_{k} \beta - \theta_{c} \alpha } \right)}}} $$
(11)

The firm’s expected profit is maximum when the CF price is 0. This can be deduced by observing that the expected profit function given by (10) is linear in \(p\) and has a negative slope of \( - \frac{{(Z_{Fi} + Z_{Fa} )}}{\alpha }\). However, there is a constraint on \(p\) that the lower limit \(L \) cannot be negative, hence the optimal value of price \(p^{*}\) should satisfy

$$ L = p^{*} - \frac{{Z_{Fi} }}{{\theta_{c} q^{*} }} = 0 $$
(12)

which implies that

$$ p^{*} = \frac{{Z_{Fi} }}{{\theta_{c} q^{*} }} $$
(13)

The optimal \(p^{*}\) and \(q^{*}\) derived by backward induction constitute the subgame perfect equilibrium of the VMP contract.

The lower and upper limits of the contracts are hence determined to be

$$ L = 0 $$
(14)
$$ U = p^{*} + \frac{{Z_{Fa} }}{{\theta_{c} q^{*} }} = \frac{{Z_{Fi} + Z_{Fa} }}{{\theta_{c} q^{*} }} $$
(15)

⬜

To gain insights into the performance of the proposed contract, it needs to be analysed for various scenarios. The motivation behind the proposed contract is to mitigate the effect of violent market price fluctuation. While the firm has scope to deviate from the contract when market prices are below the lower limit, the farmer has scope for violation of the contract when market prices are above the upper limit. Hence it is necessary to establish that there is no incentive for the players to violate the contract even under extreme market prices.

Proposition 2

Even under extreme market conditions, no player has any incentive to deviate from the contract.

Proof

The farmer’s violation of the contract happens when the market price of the CF crop goes beyond the upper limit of the VMP contract. In such a situation, the farmer sells the CF crop in the open market and obtains revenue of \( \theta_{c} qc\). Hence the farmer’s expected profit when he violates the contract denoted by \(E\left[ {\pi_{Fa}^{vfa} } \right]\) is given by.

$$ E\left[ {\pi_{Fa}^{vfa} } \right] = \mathop \int \limits_{0}^{L} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} q\left[ {p - \left( {L - c} \right)} \right]} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{L}^{U} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qp} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{U}^{\alpha } \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} q - Z_{Fa} } \right]I_{k} dkI_{c} dc $$
(16)

The third term of the above equation explains the penalty \(Z_{Fa}\) incurred by the farmer due to contract violation. The expected profit of the farmer when he honours the contract \(E\left[ {\pi_{Fa}^{c} } \right] \) is provided through Eq. (8). The difference in these profits is given by

(17)

The difference between the profits is zero since the VMP contract payment structure incentivises the player to honour the contract by shifting CF price according to market conditions. Similarly, the expected profit of the firm when honouring or violating the contract is found to be equal. Hence, both players have no incentive to violate the VMP contract in extreme market conditions.

Proposition 2 establishes that no player has any incentive to deviate from the VMP contract. In the following proposition, we compare scenarios of the players honouring versus violating the contract. We show that honouring the contract increases their total profit.

Proposition 3

The sum of the expected profits of the firm and the farmer is higher when both honour the VMP contract than when both violate it.

Proof

When the two players honour the contract, the expected profit of the farmer \(E\left[ {\pi_{Fa}^{vmp} } \right]\) and the firm \(E\left[ {\pi_{Fi}^{vmp} } \right]\) are as given by Eqs. (8) and (10), respectively. The farmer violates the contract when the market price of the CF crop goes above the upper limit, and the firm violates when the price goes below the lower limit. The expected profit of the farmer under these violations, denoted by \(E\left[ {\pi_{Fa}^{v} } \right]\), is given by.

$$ E\left[ {\pi_{Fa}^{v} } \right] = \mathop \int \limits_{0}^{L} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qc} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{L}^{U} \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qp} \right]I_{k} dkI_{c} dc + \mathop \int \limits_{U}^{\alpha } \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qc - Z_{Fa} } \right]I_{k} dkI_{c} dc $$
(18)

Similarly, we compute the expected profit of the firm \( E\left[ {\pi_{Fi}^{v} } \right] \) when violations happen.

$$ E\left[ {\pi_{Fi}^{v} } \right] = - \mathop \int \limits_{0}^{L} \left[ {rc - Z_{Fi} } \right]I_{c} dc - \mathop \int \limits_{L}^{U} [\theta_{c} qp + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{U}^{\alpha } \left[ {rc} \right]I_{c} dc $$
(19)

To prove the proposition, we require

$$ E\left[ {\pi_{Fa}^{vmp} } \right] + E\left[ {\pi_{Fi}^{vmp} } \right] - E\left[ {\pi_{Fa}^{v} } \right] - E\left[ {\pi_{Fi}^{v} } \right] \ge 0 $$

The difference in the profits for the farmer is given by

$$-$$

At optimality, \(L = 0\) (from Proposition 1). Hence the first term in \(E\left[ {\pi_{Fa}^{vmp} } \right]\) and \(E\left[ {\pi_{Fa}^{v} } \right]\) vanish. The second term is the same for the two profits. Hence the difference in their profit is

$$ \left\{ {\mathop \int \limits_{U}^{\alpha } \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} q\left[ {p + \left( {c - U} \right)} \right]} \right]I_{k} dkI_{c} dc} \right\} - \left\{ {\mathop \int \limits_{U}^{\alpha } \mathop \int \limits_{0}^{\beta } \left[ {\theta_{k} \left( {T - q} \right)k + \theta_{c} qc - Z_{Fa} } \right]I_{k} dkI_{c} dc} \right\} = 0 $$
(20)

The difference in the profits for the firm is given by

$$ \begin{aligned} E\left[ {\pi_{Fi}^{vmp} } \right] - E\left[ {\pi_{Fi}^{v} } \right] & = \left\{ { - \mathop \int \limits_{0}^{L} [\theta_{c} q\left[ {p - \left( {L - c} \right)} \right] + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{L}^{U} [\theta_{c} qp + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{U}^{\alpha } [\theta_{c} q\left[ {p + \left( {c - U} \right)} \right] + (r - \theta_{c} q)c]I_{c} dc} \right\} - \left\{ { - \mathop \int \limits_{0}^{L} \left[ {rc - Z_{Fi} } \right]I_{c} dc - \mathop \int \limits_{L}^{U} [\theta_{c} qp + (r - \theta_{c} q)c]I_{c} dc - \mathop \int \limits_{U}^{\alpha } rcI_{c} dc} \right\} \\ & = \left\{ { - \mathop \int \limits_{{p + \frac{{Z_{Fa} }}{{\theta_{c} q}}}}^{\alpha } [\theta_{c} qc - Z_{Fa} + rc - \theta_{c} qc]I_{c} dc} \right\} - \left\{ { - \mathop \int \limits_{{p + \frac{{Z_{Fa} }}{{\theta_{c} q}}}}^{\alpha } rcI_{c} dc} \right\} \\ & = \mathop \int \limits_{{p + \frac{{Z_{Fa} }}{{\theta_{c} q}}}}^{\alpha } Z_{Fa} I_{c} dc \ge 0 \\ \end{aligned} $$
(21)

The difference in the profits is non-negative for the firm. Hence Eqs. (20) and (21) establish the Proposition. ⬜

4.1 Participative Conditions

We next discuss the feasibility of the VMP contract. It may be possible that the firm is better off by purchasing all the required quantity from the open market, and the farmer can be better-off by selling both the crops to the open market completely. Hence, we need to derive conditions under which the firm and the farmer are better-off through participating in the VMP contract. Propositions 4 and 5 discuss the participatory conditions (individual rationality condition) for the firm and farmer, respectively. Thus, the firm and the farmer enter into the contract if and only if their respective participation conditions are satisfied. Failure of any of these participatory conditions leads to the situation of the contract becoming infeasible. Proposition 6 (which is proved using Propositions 4 and 5) gives the necessary and sufficient condition for the firm’s and the farmer’s participation.

Proposition 4

It is profitable for the firm to participate in the VMP contract if and only if \( \frac{{Z_{Fi}^{2} - Z_{Fa}^{2} }}{{2\alpha \theta_{c} q^{*} }} - \frac{{\left( {Z_{Fi} + Z_{Fa} } \right)Z_{Fi} }}{{\alpha \theta_{c} q^{*} }} + Z_{Fa} \ge 0. \)

Proof

In the absence of the VMP contract, the firm purchases quantity r from the open market. Then, the firm’s expected profit \(E\left[ {\pi_{Fi}^{wc} } \right]\) is given by.

$$ E\left[ {\pi_{Fi}^{wc} } \right] = - \mathop \int \limits_{0}^{\alpha } rcI_{c} dc = - \frac{r\alpha }{2}. $$
(22)

The expected profit of the firm in the VMP contract \(E\left[ {\pi_{Fi}^{vmp} } \right] \) is given in (10). The difference in profits is computed as

$$ E\left[ {\pi_{Fi}^{vmp} } \right] - E\left[ {\pi_{Fi}^{wc} } \right] = - \frac{{(Z_{Fi} + Z_{Fa} )p*}}{\alpha } + \frac{{\left( {Z_{Fi}^{2} - Z_{Fa}^{2} } \right)}}{{2\alpha \theta_{c} q^{*} }} + Z_{Fa } - \frac{r\alpha }{2} + \frac{r\alpha }{2} = - \frac{{(Z_{Fi} + Z_{Fa} )Z_{Fi} }}{{\alpha \theta_{c} q^{*} }} + \frac{{\left( {Z_{Fi}^{2} - Z_{Fa}^{2} } \right)}}{{2\alpha \theta_{c} q^{*} }} + Z_{Fa } $$
(23)

Applying the given condition of the Proposition in Eq. (23) we get

$$ E\left[ {\pi_{Fi}^{vmp} } \right] - E\left[ {\pi_{Fi}^{wc} } \right] \ge 0 $$

This implies the firm’s participation in the VMP contract and establishes the proposition. ⬜

We next derive the condition for the farmer’s participation in the VMP contract.

Proposition 5

The farmer participates in the VMP contract if and only if \( {-}\frac{{\left( {Z_{Fi}^{2} - Z_{Fa}^{2} } \right)}}{{2\alpha \theta_{c} q^{*} }} + \frac{{\left( {Z_{Fi} + Z_{Fa} } \right)Z_{Fi} }}{{\alpha \theta_{c} q^{*} }} - Z_{Fa} \ge 0\).

Proof

For a land allocation \( q^{*}\), if the farmer does not enter in the VMP contract, then both crops are to be sold in the open market at their respective market prices. Then the expected profit of farmer \(E\left[ {\pi_{Fa}^{wc} } \right]\) is given by.

$$ E\left[ {\pi_{Fa}^{wc} } \right] = \mathop \int \limits_{0}^{\alpha } \theta_{c} q^{*} cI_{c} dc + \mathop \int \limits_{0}^{\beta } \theta_{k} \left( {T - q^{*} } \right)kI_{k} dk = \frac{{\theta_{c} q^{*} \alpha }}{2} + \frac{{\theta_{k} T\beta }}{2} - \frac{{\theta_{k} q^{*} \beta }}{2} $$
(24)

The expected profit of the farmer under the VMP contract \(E\left[ {\pi_{Fa}^{vmp} } \right] \) is given by (8). The difference in the expected profits is calculated as

$$ \begin{aligned} E\left[ {\pi_{Fa}^{vmp} } \right] - E\left[ {\pi_{Fa}^{wc} } \right] & = \frac{{Z_{Fi} p^{*} }}{\alpha } - \frac{{Z_{Fi}^{2} }}{{2\alpha \theta_{c} q^{*} }} + \frac{{p*Z_{Fa} }}{\alpha } + \frac{{\theta_{c} q^{*} \alpha }}{2} + \frac{{Z_{Fa}^{2} }}{{2\alpha \theta_{c} q^{*} }} + \frac{{\theta_{k} T\beta }}{2} - \frac{{\theta_{k} q^{*} \beta }}{2} - Z_{Fa } - \left\{ {\frac{{\theta_{c} q^{*} \alpha }}{2} + \frac{{\theta_{k} T\beta }}{2} - \frac{{\theta_{k} q^{*} \beta }}{2}} \right\} \\ & = - \frac{{\left( {Z_{Fi}^{2} - Z_{Fa}^{2} } \right)}}{{2\alpha \theta_{c} q^{*} }} + \frac{{\left( {Z_{Fi} + Z_{Fa} } \right)Z_{Fi} }}{{\alpha \theta_{c} q^{*} }} - Z_{Fa} \\ \end{aligned} $$
(25)

Applying the given condition of the Proposition in Eq. (25) we get \(E\left[ {\pi_{Fa}^{vmp} } \right] - E\left[ {\pi_{Fa}^{wc} } \right] \ge 0\). This implies the farmer’s participation in the VMP and establishes the Proposition. ⬜

Proposition 6

The firm and the farmer participates in the VMP contract if and only if \(\frac{{Z_{Fi}^{2} - Z_{Fa}^{2} }}{{2\alpha \theta_{c} q^{*} }} - \frac{{\left( {Z_{Fi} + Z_{Fa} } \right)Z_{Fi} }}{{\alpha \theta_{c} q^{*} }} + Z_{Fa} = 0.\)

Proof

The proof follows directly from the Propositions 4 and 5. ⬜

4.2 Free services

If the hypothesis of Propositions 4 and 5 were not satisfied, then the firm and the farmer may not participate in the VMP contract. To mitigate the impact of such restrictions, which hinder the participation of the firm and the farmer, a major feature usually considered in CF is the free services offered by the firm to the farmer. These services can range from purely advisory services to physical services like providing fertilizers, high-quality seeds (Ochieng et al., 2017). It has been found in most of the contract farming arrangements that these free services provided by the firm lead to an increase in productivity of the contract farming crop. Tang et al. (2016) considered free advisory services provided by the firm in their PGP contract. Often the free services lead to an increase in productivity of the contract farming crop. However, the farmer also incurs losses because the increase in productivity may not be adequate. Hence a farmer should opt for contract farming only if his productivity increases beyond a certain threshold from the free services. To capture the increase in productivity in CF crop due to such free services, the farmer’s participatory condition derived in Proposition 4 may also be expressed in terms of an increase in productivity. The following Corollary illustrates this aspect.

Corollary 1

The farmer has an incentive to participate in the VMP contract provided he receives services from the firm, which increases the productivity of the contract farming crop by at least \(\frac{{2Z_{Fa } }}{{\alpha q^{*} }} - \frac{{(Z_{Fi} + Z_{Fa} )^{2} }}{{\theta_{c} \alpha^{2} q^{*2} }}\)\(.\)

Proof

Since CF increases the productivity of crop by \({\varvec{\delta}}\), we assume that the productivity of CF crop in the contract is \({{\varvec{\theta}}}_{{\varvec{c}}}\) and in its absence is \({{\varvec{\theta}}}_{{\varvec{c}}}-{\varvec{\delta}}\). The expected profit in the absence of a contract is

$$ E\left[ {\pi_{Fa}^{wc} } \right] = \mathop \int \limits_{0}^{\alpha } \left( {\theta_{c} - \delta } \right)q^{*} cI_{c} dc + \mathop \int \limits_{0}^{\beta } \theta_{k} \left( {T - q} \right)kI_{k} dk = \left( {\theta_{c} - \delta } \right)q^{*} \frac{\alpha }{2} + \theta_{k} \left( {T - q^{*} } \right)\frac{\beta }{2} ) $$
(26)

The expected profit of the farmer \(E\left[ {\pi_{Fa}^{vmp} } \right]\) in the VMP contract is given by Eq. (8). The difference in the expected profits is

$$ E\left[ {\pi_{Fa}^{vmp} } \right] - E\left[ {\pi_{Fa}^{wc} } \right] = \frac{{(Z_{Fi} + Z_{Fa} )p^{*} }}{\alpha } + \frac{{\left( {Z_{Fa}^{2} - Z_{Fi}^{2} } \right)}}{{2\alpha \theta_{c} q^{*} }} - Z_{Fa } + \delta q^{*} \frac{\alpha }{2} $$

The farmer is better off in the contract if \(E\left[ {\pi_{Fa}^{vmp} } \right] - E\left[ {\pi_{Fa}^{wc} } \right] > 0 .\)

$$ \Rightarrow \delta > \frac{{2Z_{Fa } }}{{\alpha q^{*} }} + \frac{{\left( {Z_{Fi}^{2} - Z_{Fa}^{2} } \right)}}{{\theta_{c} \alpha^{2} q^{*2} }} - \frac{{2p^{*} (Z_{Fi} + Z_{Fa} )}}{{\alpha^{2} q^{*} }} $$

Simplifying and substituting for \( p^{*}\), we get the required condition as

$$ \delta > \frac{{2Z_{Fa } }}{{\alpha q^{*} }} - \frac{{(Z_{Fi} + Z_{Fa} )^{2} }}{{\theta_{c} \alpha^{2} q^{*2} }}. $$
(27)

⬜

On the other hand, firms often need to take into account their operational cost arising due to factors outside the contract to decide their participation in the contract. Such operational costs may be absorbed into the firm’s costs and penalty parameter \(Z_{Fi}\) of the contract. Proposition 4 provides the condition under which the firm eventually participates in the contract.

5 Numerical analysis

In this section, a numerical analysis of the model is conducted for the different parameters based on a case study. The optimal values of land allocation, CF price, and the expected profits of the players are calculated. Sensitivity analysis for each parameter is performed in which the effect of the parameter on the optimal values and the profits are discussed.

A case study is performed using real data to analyze the proposed VMP contract. The geographical location for the case study is Central India. The conventional crop is taken as rice which the farmers in Central India usually sow. Cotton is the contract farming crop used in the case study. Cotton is mainly used as raw material by textile firms located in the region. The price of cotton at the time of the case study is 75000 Rs/Ton, and hence we choose the parameter \( \alpha\) of the uniform distribution as 150,000. Similarly, the price of rice is 20,000 Rs/ Ton, and hence \(\beta\) is chosen as 40,000. The average values of productivity of cotton and rice in the region are 1 Ton/Hectare and 4 Tons/Hectare respectively; hence, we take \(\theta_{c}\) = 1 and \(\theta_{k}\) = 4. Penalties of firm and farmer (denoted by \(Z_{Fi} \;{\text{and}}\; Z_{Fa}\)) incurred upon violation of contract are taken as 550,000 Rs and 100,000 Rs, respectively. The total land available for farming is 20 hectares, and the quantity of CF crop required by the firm is 20 tons.

For the above set of parameters, the optimal solution of the VMP contract is determined. The optimal land allocation to CF crop by the farmer is 13.77 hectare out of the total 20 hectares of land available. The remaining land is allotted to the conventional crop. The optimal CF price for the stable market condition is 39400 Rs/Ton. The upper and lower limits of the VMP contract are 46,660 Rs/Ton and 0, respectively. The expected profit of the farmer and the firm under the VMP contract are 1,531,107.80 Rs and − 1,500,000 Rs, respectively. The expected profit of the farmer without opting for the VMP contract is 1,531,107.80 Rs. Similarly, the expected profit of the firm without the VMP contract is − 1,500,000 Rs.

Table 1 provides the results of numerical experiments that were performed for different sets of parameter values. We next perform a sensitivity analysis of each parameter on the VMP contract variables choosing the first row of Table 1 as the base value of the parameters.

Table 1 Numerical Experiments
Full size table

5.1 Sensitivity analysis for \(Z_{Fa}\) and \(Z_{Fi}\).

Figure 1a-d are plots for sensitivity analysis of \(Z_{Fa}\) the penalty of the farmer. Figure 1a indicates that as the farmer’s penalty of violating the contract increases, the optimal land allocation for the CF crop decreases.

Fig. 1
figure 1

Sensitivity analysis of \({\text{Z}}_{{\begin{array}{*{20}c} {Fa} \\ \\ \end{array} }}\)

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This is evident since reducing the land for CF will reduce the chances of the violation, which is costlier because of the higher penalty. Figure 1b shows an increase in CF price for stable market conditions when the farmer’s penalty increases. This happens because of higher incentive given by the firm in the form of higher CF price to compensate for a higher penalty for the farmer. In Fig. 1b, we observe that the marginal increase in \(p^{*}\) is more for higher values of \(Z_{Fa}\). This is reflected in Fig. 1c, where the marginal decrease of the farmer’s expected profit is less for lower values of \(Z_{Fa}\) and marginal increase in the expected profit is more for higher values. Similar behaviour of the expected profit of the firm with respect to the farmer’s penalty is observed in Fig. 1d in which marginal increase in the expected profit is less for lower values of \(Z_{Fa}\) and the marginal decrease in the expected profit is more for higher values. Figure 2a-d show the sensitivity analysis of \( Z_{Fi}\), the penalty of the firm. Figure 2a shows that as the firm’s penalty increases, the optimal land allocation \(q^{*}\) increases. This can be explained as the farmer’s willingness to invest more in CF as there are less chances of the firm’s violation of the contract due to the higher penalty. This increase in the land allocation for CF leads to the firm offering lower CF price \(p^{*}\), as observed in Fig. 2b. This, in turn, decreases the expected profit of the farmer as observed in Fig. 2c. An increase in the firm’s penalty initially increases the expected profit of the firm due to the sharp decline in \(p^{*}\) but the expected profit decreases later, as seen in Fig. 2d.

Fig. 2
figure 2

Sensitivity analysis of \(Z_{Fi }\)

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5.2 Sensitivity analysis for \(\theta_{c}\) and \(\theta_{k}\).

Figure 3a-d show the sensitivity analysis of \(\theta_{c}\), the productivity of CF crop. As \(\theta_{c}\) increases, the optimal allocation of land for CF increases since the farmer gets more revenue from the CF crop on the same quantity of land (Fig. 3a). The increase in \(q^{*}\) leads to a decrease in the CF price \(p^{*}\) (Fig. 3b). The decrease in \(p^{*}\) leads to a decrease in the expected profit of the farmer (Fig. 3c) and an increase in the expected profit of the firm (Fig. 3d). Similarly, the analysis for the productivity of the conventional crop \(\theta_{k}\) is presented in Fig. 4a-d. As \(\theta_{k}\) increases, \(q^{*}\) decreases (Fig. 4a) since the farmer can get more profit from the conventional crop. This, in turn, leads to higher CF price \(p^{*}\) offered by the firm (Fig. 4b). The increase in \(p^{*}\) leads to higher expected profit for the farmer (Fig. 4c) and lower expected profit for the firm (Fig. 4d).

Fig. 3
figure 3

Sensitivity analysis of \(\theta_{c}\)

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Fig. 4
figure 4

Sensitivity analysis of \(\theta_{k}\)

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The rate of change of \(q\) with respect to \(\theta_{c}\) at optimality is observed to change direction at \(\theta_{c}^{*} = \frac{{\theta_{k} \beta }}{2\alpha }\). This explains plot 3.1 (in Fig. 3a), in which \(q^{*}\) initially decreases and later increases with \(\theta_{c}\). One can assume that as productivity of CF crop (\(\theta_{c}\)) increases, optimal land allotted to CF (\(q^{*}\)) should also increase. However, the increase in \(\theta_{c}\) also changes upper and lower limits, which may result in decrease in \(q^{*}\). On the other hand, the rate of change of \(q\) with respect to \(\theta_{k}\) does not change sign at optimality. This can be observed in plot 4.1 (in Fig. 4a) in which \(q^{*}\) decreases with an increase in \(\theta_{k}\). This decrease in \(q^{*}\) is attributed to the farmer’s profit function. As the productivity of conventional crop (\(\theta_{k}\)) increases, farmers has incentive to allocate more land to conventional crop and hence less land for contract farming (\(q^{*}\)).

5.3 Sensitivity analysis for \(\alpha\) and \(\beta\)

Figure 5a-d show the analysis of the upper bound of the uniform distribution of market price of CF crop (\(\alpha\)). The optimal allocation \(q^{*}\) to CF crop increase as \(\alpha\) increases since the farmer has higher chances of getting a higher price for CF crop (Fig. 5a). This results in a decrease in CF price \({{p}}^{\user2{*}}\) offered by the firm (Fig. 5b), which in turn decreases the expected profit of the farmer (Fig. 5c). The firm’s expected profit decrease since it needs to purchase the CF crop from the open market also where the price is increasing (Fig. 5d). Similarly, Fig. 6a-d show the results for the upper bound of the uniform distribution of the market price of the conventional crop \(\beta\). The optimal land allocation for CF crop decreases with increase in \(\beta\) since the farmer has higher chances of larger profit by allocating more land for the conventional crop (Fig. 6a). A decrease in \(q^{*}\) will cause the firm to offer a higher CF price \(p^{*}\) (Fig. 6b). The higher profit from conventional crop and the higher CF price leads to the higher expected profit for the farmer (Fig. 6c). The higher CF price will lead to a decrease in the expected profit of the firm (Fig. 6d).

Fig. 5
figure 5

Sensitivity analysis of α

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Fig. 6
figure 6

Sensitivity analysis of β

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We next conduct a probabilistic sensitivity analysis to derive further insights.

5.4 Probabilistic sensitivity analysis (PSA)

PSA helps in understanding the impact of uncertainties present in input parameters on the model output and its robustness. Monte Carlo simulation (MCS) and perturbation methods are some of the techniques used to perform PSA. In this study, we apply MCS to conduct PSA. Randomness in each input parameter is assumed to be uniformly distributed with a five per cent variation about the mean value. The distribution of the six input parameters are hence taken as: \(Z_{Fa} \sim U \left[ {95000, 105000 } \right],\) \(Z_{Fi} \sim U \left[ {475000, 525000} \right]\), \(\theta_{c} \sim U \left[ {0.95, 1.05} \right],\) \(\theta_{k} \sim U \left[ {3.8, 4.2} \right],\) \(\alpha \sim U \left[ {142500, 157500} \right],\) \(\beta\) \(\sim U \left[ {47500, 52500} \right].\) MCS is performed using these six random input variables to analyse the following outputs: optimum land allocation for CF by the farmer (\(q^{*}\)), optimum CF price (\(p^{*}\)), farmer’s expected profit, and firm’s expected profits. Coefficient of Variation (CV) is used to study the effect of uncertainties in the model and to compare variations in different data series. Figure 7a-d show the CV of outputs with respect to the number of simulations. It is observed that 10,000 simulations are required for the convergence of the CV in the four outputs under consideration. Hence we perform 10,000 simulations. Figure 8a-d represent the histograms of the outputs. CV for the six input variables are approximately 0.029. The output CV are 0.104, 0.107, 0.044, and 0.023 for \(q^{*}\), \(p^{*}\), farmer’s expected profit and firm’s expected profit, respectively. It can be observed that the proposed model is robust enough to handle input uncertainties. The maximum variation in output is 10.7%, even with the introduction of 2.9% uncertainties in each of the six input variables. These results of PSA should be considered while implementing the VMP contract.

Fig. 7
figure 7

CV of outputs with respect to the number of simulations

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Fig. 8
figure 8

Histograms of the outputs

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6 Managerial Implications

The model developed in this work is applicable in agri-business environments in which farmers and firm managers decide on the optimal quantity and price through the application of contracts. Contractual agreements mutually benefit the farmer and the firm. Farmers can plan important aspects of farming such as labor, fertilizers, and machines, while firm operations managers can plan their production and finances based on contractual price and quantity. The participatory conditions derived in the model enable managers and farmers to decide whether to opt for the contract or the open market for trade. This model also enables managers and farmers to evaluate the impact of additional services provided to farmers.

The above-mentioned benefits of the contracts are applicable if and only if the two parties honor the contract. Literature indicates evidence of violation of agri-business contracts, particularly in developing economies, the primary reason being violent price fluctuation. The VMP contract developed in this study overcomes this problem by incentivizing the two parties to honor the contract even under extreme price fluctuations.

The VMP contract aids in safeguarding the interests of farmers. Many agri-business firms have higher bargaining power than farmers; thereby, farmers become the weaker party in the event of contract failure. In developing countries, most farmers are smallholders, which mean legal fights in the event of contract failure are almost unthinkable. Any contract that incentivizes the two parties to honor it even under adverse conditions will benefit such farmers, and the VMP contract has been established to achieve this. The VMP contract provides farmers the benefits of assured income along with risk coverage during extreme events. This assurance can have a huge implication on the farming practices of small farmers. The contract is also beneficial to the firms. For example, firms often require specific quality of raw material from farmers. This requirement has become very important since firms want to standardize production processes and finished products. A long-term relation with farmers allows the firm to involve farmers in quality control process of the raw material. Having specific quality raw material on time is crucial for any operations manager to plan production and other business activities assuredly.

From the broad supply chain perspective, the VMP contract provides various benefits. The contract encourages long-term relationships, which is vital for any supply chain. It allows supply chain members to coordinate and overcome different challenges such as disruptions, market failures, and disasters. Strong supply chain linkages promote investment, research, and development which are also beneficial to the economy.

7 Conclusion

In this study, we have developed a novel supply chain contract to address the problem of violent price fluctuations due to extreme market price fluctuations and established the effectiveness of the contract in addressing this problem. While the literature acknowledges the issue of price fluctuation as a threat to CF in coordinating the supply chain, to the best of our knowledge, no contract has been developed to address this aspect. The VMP contract developed in this study uniquely addresses this problem and provides an effective solution. In the VMP contract developed, the firm- farmer interaction is modelled as a Stackelberg game. Our results show that there is no incentive for any player to violate the contract. Participatory conditions for the firm and the farmer to enter into the contract are also derived. The model determines the optimal contract parameters of price and quantity. Such pre-agreed price and quantity of raw material help the operations manager plan production. Other parameters of the contract include the upper and lower bounds of market prices beyond which the CF price shifts by the corresponding proportion. After the design of the contract, the two parties monitor the market conditions and activate the pricing mechanism at the time of payment.

The present work contributes to the practice of agri-business supply chain management as well as the supply chain contract literature in the form of the VMP contract. The proposed price mechanism provides a solution to the unique problem of CF, in which the contract fails due to drastic price fluctuations. Further, the proposed solution methodology contributes to modelling techniques in agri-business supply chains. Findings of this work show that it is possible to incentivize the firm and the farmer to honour the contract. VMP contract can prove to be an effective method for long term firm- farmer relations. The contract ensures a stable supply of raw material to the firm and stable income for the farmer. This work is an attempt to tackle the problem of sustainability in the CF arrangement and is also applicable to supply chains with sustainability challenges. Particularly, this work finds applications to supply chain contracts in which market price fluctuations are high.

We discuss some of the model limitations and scope for future work. The yield of crops was chosen to be deterministic; a more appropriate assumption would be to assume yield to be stochastic. Another variation of this model is when the farmer is risk-averse, and the firm is risk-neutral. The model assumes a uniform distribution of market prices. Assuming the probability distribution appropriate to the particular crop under study also provides scope for future work. A firm contracting with more than one farmer is another direction of future work. When a firm contract with multiple farmers, we observe that each farmer’s objective function and hence his decision will be the same as the model developed in this study. However, the firm’s objective function will be different since it negotiates with several farmers. The firm maximizes its profit which is a function of CF prices, subject to the individual rationality constraints of each farmer. Future research can be undertaken to develop such a model in which a firm contracts with multiple farmers. Another interesting scenario may arise when these farmers come together and collectively bargain with the firm for a higher CF price. Cooperative game theory approaches may be applied to study the coalition form game so induced. Firms bargaining with a collective of farmers may also be considered as scope for future research work.