A zero-bounded model of operant demand

Contemporary approaches for evaluating the demand for reinforcers use either the Exponential or the Exponentiated model of operant demand, both derived from the framework of Hursh and Silberberg (2008). This report summarizes the strengths and complications of this framework and proposes a novel implementation. This novel implementation incorporates earlier strengths and resolves existing shortcomings that are due to the use of a logarithmic scale for consumption. The Inverse Hyperbolic Sine (IHS) transformation is reviewed and evaluated as a replacement for the logarithmic scale in models of operant demand. Modeling consumption in the “log₁₀-like” IHS scale reflects relative changes in consumption (as with a log scale) and accommodates a true zero bound (i.e., zero consumption values). The presence of a zero bound obviates the need for a separate span parameter (i.e., k) and the span of the model may be more simply defined by maximum demand at zero price (i.e., Q₀). Further, this reformulated model serves to decouple the exponential rate constant (i.e., α) from variations in span, thus normalizing the rate constant to the span of consumption in IHS units and permitting comparisons when spans vary. This model, called the Zero-bounded Exponential (ZBE), is evaluated using simulated and real-world data. The direct reinstatement ZBE model showed strong correspondence with empirical indicators of demand and with a normalization of α (ZBEn) across empirical data that varied in reinforcing efficacy (dose, time to onset of peak effects). Future directions in demand curve analysis are discussed with recommendations for additional replication and exploration of scales beyond the logarithm when accommodating zero consumption data.