Optimal Long-Term Contract for Fertilizer Supply Under Farmer's Dynamic Private Information

An agribusiness corporation designs an optimal long-term contract for chemical fertilizer supply to a farmer who holds private information on the quantity of organic fertilizer. This important private information, which affects the farmer's incentive to purchase the chemical fertilizer, is dynamically changing over time. We derive the optimal long-term contract and provide practical guidance on the contract implementation for fertilizer supply. We find that the optimal payment endogenously has a linear payment scheme from the second period onward. Such a contract is desirable because of its convenience of execution in practice. Specifically, the agribusiness corporation offers a menu of contracts and each contract consists of a membership fee and a sequence of wholesale prices. The wholesale prices increase as the complementary effect of the two fertilizers intensifies. We further analyze how the farmer's dynamic private information affects the optimal contract and its performance. Efficiency distortions and marginal payment increase as the private information becomes more correlated across periods. However, the correlation degree of the private information has a nonmonotonic impact on the farmer's expected payoff. Finally, we consider two model extensions to study the impact of the delayed effect of chemical fertilizer and endogenous organic fertilizer quantity on the optimal contract implementation. We prove the optimality of linear contracts under the two model extensions, more generalized crop yield functions, and substitute organic and chemical fertilizers, which provides a theoretical foundation for the prevalent linear contracts.