In experiments on operant behavior, other activities, called “background” activities, compete with the operant activities. Herrnstein's () formulation of the matching law included background reinforcers in the form of a parameter rO, but remained vague about the activities (BO) that produce rO. To gain more understanding, we analyzed data from three studies of performance with pairs of variable‐interval schedules that changed frequently in the relative rate at which they produced food: Baum and Davison (), Belke and Heyman (), and Soto, McDowell, and Dallery (2005). Results sometimes deviated from the matching law, suggesting variation in rO. When rO was calculated from the matching equation, two results emerged: (a) rO is directly proportional to BO, as in a ratio schedule; and (b) rO and BO depend on the food rate, which is to say that BO consists of activities induced by food, as a phylogenetically important event. Other activities unrelated to food (BN) correspond to Herrnstein's original conception of rO and may be included in the matching equation. A model based on Baum's (Baum, 2012) concepts of allocation, induction, and contingency explained the deviations from the matching law. In the model, operant activity B, BO, and BN competed unequally in the time allocation: B and BO both replaced BN, BO replaced lever pressing (Soto et al.), and key pecking replaced BO (Baum & Davison). Although the dependence of rO and BO on food rate changes Herrnstein's () formulation, the model preserved the generalized matching law for operant activities by incorporating power‐function induction.