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

Abstract

Objective: Howard Rachlin wrote extensively on how value diminishes in a hyperbolic form, and he contributed to understanding choice processes between different commodities as a molar pattern of behavior. The field of behavioral economic demand has been dominated by exponential decay functions, indicating that decreases in consumption of a commodity are best fit by exponential functions. Because of the success of Rachlin’s equation at describing how hyperbolic decay affects the value of a commodity across various factors (e.g., delay, probability, social distance), we attempted to extend his equation to behavioral economic demand data for alcohol and opioids. Method: Rachlin’s discounting equation was applied to estimate consumption on alcohol purchase task data and nonhuman drug demand data. We compared results of his equation to the exponentiated demand equation using both a mixed-effects modeling approach and a two-stage approach. Results: Rachlin’s equation provided better fits to consumption data than the exponentiated equation for both mixed-effects and two-stage modeling. We also found that traditional demand metrics, such as Pmax, can be derived analytically when using Rachlin’s equation. Certain metrics derived from Rachlin’s equation appeared to be related to clinical covariates in ways similar to the exponentiated equation. Conclusions: Rachlin’s equation better described demand data than did the exponentiated equation, indicating that demand for a commodity may decrease hyperbolically rather than exponentially. Other benefits of his equation are that it does not have the same pitfalls as the current exponential equations and is relatively straightforward in its conceptualization when applied to demand data.

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

“…Lastly, because of previous explorations between Rachlin's equation and demand data (Rzeszutek, Franck et al, 2022), we have also extended a multiplicative discounting model for both price and delay,…”

“…Because of the complexity of the data and models, fitting demand data using log 10 -scaled parameters allows for achieving lower tolerances and better fits to the data (Kaplan et al, 2021;Rzeszutek, Franck et al, 2022;Rzeszutek, Gipson-Reichardt, et al, 2022;Traxler et al, 2022). This does not transform the response variable but only the scale in which the parameter is expressed (and thus, optimized).…”

“…Table of equations used by factor (i.e., price or delay), mathematical operation used, shortform equation name, and long-form (i.e., full) equation used in the analyses. k was set to 1.65495 (log descriptively better fits to data than the exponentiated for some demand data (Rzeszutek, Franck et al, 2022), but descriptively better fits are not without limits. Versions of exponential decay models of demand have been successful at identifying associations between different substance use metrics (Gonz alez-Roz et al, 2019;Martínez-Loredo et al, 2021;Strickland et al, 2020;Zvorsky et al, 2019) as well as being predictive of treatment outcomes (Gonz alez-Roz et al, 2020;Schwartz et al, 2021;Secades-Villa et al, 2016).…”

“…Also note that for the 2D Rachlin equation, there is an analytic solution to P max (seeRzeszutek, Franck et al, 2022) which does correspond with the point at which the slope of the curve becomes -1 in log-log space.…”

“…However, this provides a precedent for using exponential demand models with delay as a variable. There is also a precedent for applying discounting equations to demand (Killeen, 2019(Killeen, , 2020, including Rachlin's equation (Rzeszutek, Franck et al, 2022). Factors for demand of a commodity may be affected by delay in an additive manner (i.e., delay has an absolute effect on consumption by price) or in a multiplicative manner (i.e., delay has a proportional effect on consumption by price).…”