Quadratic quantity discount contract under price-dependent demand and consumer returns

• We introduce a new supply contract, the quadratic quantity discount contract. • It adopts the quadratic function form for the average discount function. • By comparing it with other contracts, we reveal its unique characteristics. • The investigation is done under basic and closed-loop supply chain situations. • It coordinates the supply chain without fairness and bargaining power issues. This study introduces a new type of supply contract, the quadratic quantity discount contract, whose average transfer payment is a quadratic function. Thus, both the total and average quantity discounts increase nonlinearly in order quantity, unlike the typical linear quantity discount contract, whose total discount is nonlinear but average payment is a linear function of order quantity. We reveal the quadratic quantity discount contract’s unique properties by comparing it with linear quantity discount and wholesale price contracts. Our investigation, conducted in basic and closed-loop supply chain situations, reveals the potential of the quadratic quantity discount contract to be considered a serious choice for supply chain managers in various situations. Specifically, we reveal that the quadratic quantity discount contract always coordinates the supply chain, achieving the first-best supply chain profit while always enhancing the profits of all players. This is the most important difference from the typical linear quantity discount contract that entails a profit distribution problem and bargaining power issues. Further, unlike other contracts, its profit improvement performance is predictable; moreover, it offers the acceptable wholesale price with an additional nonlinear average discount. However, when the supplier can exercise its power and only considers its own profit, the typical linear quantity discount contract can be considered the better option, as it produces superior profits for the supplier, but the profits of the retailer and supply chain are sacrificed in this case. If the supplier considers the supply chain’s long-term health and wants voluntary compliance from other players, the quadratic quantity discount contract is the optimal choice.