Notes on demand: Conceptual and empirical benefits of applying Rachlin's discounting equation to demand data

Overviewing the exponential model of demand and introducing a simplification that solves issues of span, scale, and zeros

One of the most successful models of describing the decay in commodity consumption as a function of cost across multiple domains is the exponential model introduced by Hursh and Silberberg (2008). This model formulates the value of a commodity by including a "standardized price" adjustment. This adjustment allows for a theoretically scale-invariant parameter to estimate a normalized decay (α, the sensitivity to changes in price) in commodity consumption that was detangled from an organism's consumption when a commodity is free (Q₀). This scale-invariant parameter is sometimes referred to as the essential value (EV), which is generally represented as the inverse of α. However, the Hursh and Silberberg (HS) model has various shortcomings, notably as a result of the span parameter k and its influence on interpretations of α and, therefore, of essential value. We present an overview of the standardized price/real cost adjustment and challenges of and potential remedies to k within the HS framework and propose a simplified exponential model with normalized decay (Equation 10). The simplified exponential equation does not include the span parameter k and allows for straightforward analytic solutions for conceptually relevant and common demand metrics. Parities between the Hursh and Silberberg model and the simplified exponential with normalized decay model are demonstrated by conversions of α values between both models. Statistical parities between the simplified exponential with normalized decay model and the exponentiated model of demand with multiple data sets are also demonstrated. This simplified model then allows for consistent interpretations of α across commodities while retaining the theoretical benefits of the Hursh and Silberberg formulation of demand and the essential value. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

Modular Maximization Theory: A functional account of economic behavior in laboratory animal models with applications to drug-seeking behavior

Substance abuse research depends on precise and sensitive assessments of reinforcer efficacy in animal models. However, conventional methods often lack theoretical rigor and specificity to support these assessments. To address these gaps, the Modular Maximization Theory (MMT) is introduced as a comprehensive framework for understanding instrumental behavior. Like earlier maximization theories, MMT posits that behavior is distributed across alternatives to maximize utility over time. This concept is structured through five foundational postulates that define alternative actions and rules for choosing between them as budget constraints and utility functions. A key innovation of MMT is its incorporation of reinforcer utilization-encompassing both consummatory and post-consummatory activities-into the budget-constraint function. A model of ratio-schedule performance is developed under the assumption that utilization is proportional to demand, with utility represented as an additive power function of reinforcer magnitude. This model, termed PURSPU (Proportional Utilization, Ratio Schedule, Power Utility), effectively explains how reinforcer magnitude, response effort, non-contingent reinforcement, and income influence demand curves, behavior-output functions, dose-response relationships, and progressive-ratio breakpoints, while accounting for rate-dependent effects. The model also offers novel insights into choice behavior, including concurrent-schedule performance, income dependency, and delay discounting, as well as post-reinforcement pauses and run rates. Variations in budget constraints and utility functions are proposed as alternative models. Potential theoretical advancements, more targeted assessments of drug abuse liability, and the broader role of MMT in understanding human drug abuse are explored.

Hyperbolic discounting and exponentiated demand: Modeling demand for cigarettes in three dimensions

Behavioral economics has been a fruitful area of research in substance use. Mathematical descriptions of how individuals temporally discount the value of a commodity have been correlated with substance use and mathematical descriptions of drug consumption decreasing as a function of price (i.e., demand) predict maladaptive substance use. While there is a logical assumption that temporal factors affect demand for a drug, little has been done to merge these models. Thus, the purpose of this study was to combine models of discounting and demand, extending Howard Rachlin's work and contributions to novel areas of study. Data from 85 participants from Amazon Mechanical Turk (MTurk) who completed a hypothetical cigarette purchase task that included price of and delay to cigarettes were analyzed. Multilevel modeling was used to determine descriptive accuracy of combined additive and multiplicative models of discounting and demand. Of the discounting models used in conjunction with the exponentiated demand equation, the Rachlin hyperboloid best described the delay dimension of consumption. The multiplicative version of the Rachlin equation applied to both delay and price outperformed other models tested. Therefore, existing models of discounting and demand can be extended to modeling consumption data from complex multidimensional experimental arrangements.