All-units discounts and double moral hazard

An all-units discount is a price reduction applied to all units purchased if the customer's total purchases equal or exceed a given quantity threshold. Since the discount is paid on all units rather than marginal units, the tariff is discontinuous and exhibits a negative marginal price (“cliff”) at the threshold that triggers the discount. This paper shows that all-units discounts arise in optimal agency contracts between upstream and downstream firms that face double moral hazard. I present conditions under which all-units discounts dominate two-part tariffs and other continuous tariffs. I also examine these tariffs when the upstream market faces a threat of entry. In the case considered, all-units discounts deter entry by less efficient rivals without distorting price and investment, whereas continuous tariffs either accommodate such entry or deter it by distorting price and investment.