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

Abstract

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.

Quantitative modeling of behavior has been a fruitful venture in many aspects of psychological research, notably with regard to substance use. Two dominant paradigms for quantitative modeling in substance use research and addictions are discounting (Odum, 2011) and demand (Aston & Cassidy, 2019; Koffarnus & Kaplan, 2018), both of which are important indicators of substance use liability (Amlung et al., 2017; Zvorsky et al., 2019) and is based in the operant tradition. Conceptually, discounting generally refers to the devaluation of an outcome or commodity based on some characteristic of that commodity (e.g., delay to receipt; Odum, 2011). Descriptively, discounting refers to a mathematical description of a pattern of behavior extended across some variable, usually taking a hyperbolic form (McKerchar & Renda, 2012). The original formula for hyperbolic discounting offered by Mazur (1987), takes the form:

$$V=\frac{A}{1+bY}$$ (1)

where V is the discounted value of a commodity based on the variable of interest, A is the undiscounted amount of the commodity, Y is the independent variable that serves to discount the value of that commodity (e.g., delay, probability), and b is the discount parameter to be estimated. Higher values of b indicate steeper discounting (i.e., increased devaluation of a commodity) whereas lower values of b indicate shallower discounting (i.e., lower devaluation of a commodity) as a function of Y. The parameter A is typically set to the maximum undiscounted value based on the data analyzed, but can also be fitted as a free parameter (e.g., Jarmolowicz et al., 2018). For example, when determining how the subjective value (V) of $1,000 changes based on the time to receipt, A would typically be set to 1000. However, the undiscounted value can be determined, rather than assumed, when treated as a free parameter. Howard Rachlin proposed an adjustment to the Equation (1) by adding a scaling free parameter s to Y that adjusts for nonlinear scaling of psychophysical perception of the discounted variable (Rachlin, 2006),¹ based on Steven’s power law (Stevens, 1957), of a given independent variable seen in Equation (2).

$$V=\frac{A}{1+b{Y}^s}$$ (2)

This added a conceptual benefit compared to Equation 1 by accounting for individual perceptions, and also introduced a descriptive benefit by improving fits to the data.

While discounting processes have been most extensively studied to describe how delay to an outcome or commodity decreases its value, Rachlin also pioneered extending the concept of delay discounting to the change in subjective value based on the probability of an outcome (Rachlin et al., 1991) and social distance (i.e., closeness of personal relationship) of the person receiving the outcome (Rachlin & Jones, 2008). Along with Rachlin’s work in extending discounting paradigms to different commodities, his equation has been extended to model other decision-making data, such as vaccine acceptability (Jarmolowicz, Reed, Francisco, et al., 2018) and medication adherence (Jarmolowicz, Reed, Bruce, et al., 2018). Along with modeling decision-making, delay discounting has also been associated with substance use (Amlung et al., 2017) as well as other health-related behaviors (Story et al., 2014).

By contrast, behavioral economic demand (henceforth demand) is the study of how consumption of a commodity (e.g., cigarettes, food, fuel) changes as a function of the price of the commodity (Hursh, 1980, 1984). There are several metrics that quantify demand (Bickel et al., 2000). Two common metrics used to quantify demand are intensity (i.e., consumption when the commodity is free) and sensitivity (i.e., relative decrease in consumption as a function of increases in price). Generally, a commodity is determined to be more ‘elastic’ when consumption of it decreases sharply as a function of increasing price.² Conversely, a commodity that is ‘inelastic’ is one where consumption of it is relatively stable as prices increase until the price becomes exceptionally high. For example, demand for discretionary goods may be considered relatively elastic as demand for them decreases more sharply as price increases, whereas demand for material necessities (e.g., food, shelter) will remain relatively constant as prices increase. Rachlin championed incorporating economic demand theory into psychological theory (Rachlin et al., 1976), developing his models of discounting alongside Hursh’s models of demand. Much like discounting, demand has been associated with measures of substance use severity. Higher intensity and lower price sensitivity of a substance have been associated with higher levels of substance use (González-Roz et al., 2019; Martínez-Loredo et al., 2021; Strickland et al., 2020) and are predictive of treatment outcomes (González-Roz et al., 2020; Schwartz et al., 2021). While a staple in substance use research, demand can also be used to model various behaviors such as likelihood to deposit money for alcohol treatment (Traxler et al., 2022), public health behaviors during a pandemic (Strickland, Reed, et al., 2022), and demand for various commodities (Schwartz & Hursh, 2022).

While there are several equations used in demand research (see Koffarnus et al., 2022), we will only describe the exponentiated model of demand (Koffarnus et al., 2015), whose equation is:

$$Q=Q_0*{10}^{k\left(e^{-\alpha Q_{0}X}-1\right)}$$ (3)

where Q is the estimated consumption at some price, Q₀ is the estimated consumption when price is zero (intensity), k is a free parameter (typically the log span of the data +.5) used for model fitting, α is the relative change in consumption due to price of a commodity (sensitivity), and X is, typically, the price of the commodity. The exponentiated model is simply an exponentiation of the original exponential model (Hursh & Silberberg, 2008).

Whereas demand and discounting are associated with substance use, little has been done to understand how delay to and price of an addictive commodity interact outside of limited research and conceptualization (Gunawan, 2020; Hursh & Schwartz, 2022; Schwartz & Hursh, 2022). 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, 2020), including Rachlin’s equation (Rzeszutek et al., in press). 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). Therefore, the purpose of the present study was to integrate mathematical models of demand and discounting to describe hypothetical demand for cigarettes across a variety of prices of and delays to cigarettes. This was done in two ways, (1) analyzing demand for cigarettes at the lowest delay (i.e., typical demand) and demand at the lowest cost (i.e., consumption decreasing as a function of time) and (2) combining models of demand and discounting to accommodate the three-dimensional nature of the data. For analysis purposes, only the exponentiated equation (Koffarnus et al., 2015) and three popular discounting equations (Mazur, 1987; Myerson & Green, 1995; Rachlin, 2006) were used. We chose the exponentiated model of demand as it does not require response data to be excluded or otherwise transformed into other scales (e.g., the zero-bounded exponential model using an inverse hyperbolic sine transform; Gilroy et al., 2021). We chose the three discounting equations because of their popularity in studies of delay discounting in behavior analysis.

Methods

2D Demand by Price and Delay

To assess demand in a more conventional manner, 2-dimensional (2D) subsets of the entire data set were taken for consumption at the lowest delays (i.e., demand by price) and for consumption at the lowest prices (i.e., demand by delay). For the 2D analyses, only two equations were used primarily for illustrative purposes; these were the exponentiated model (hereafter EXPD; Eq. 3) and the Rachlin model (hereafter Rachlin; Eq. 2). For these analyses, both models were fitted to demand by price and demand by delay. This resulted in EXPD_Price and Rachlin_Price being compared, as well as EXPD_Delay and Rachlin_Delay being compared, where the subscripts refer to the single independent variable to which the equation is being applied. This was done primarily because demand data is usually analyzed in two dimensions and therefore lead into the more complex 3D analysis.

Integrating Models for both Delay and Price

To combine models of discounting and demand, we were influenced by Vanderveldt et al. (2015), who asked participants discounting questions for larger delay and probabilistic outcomes and compared an additive model of discounting (i.e., absolute effect of a factor) versus a multiplicative model of discounting (i.e., relative effect of a factor). While the original models proposed by Vanderveldt et al. used a model with the entire denominator raised to the power, (commonly referred to as Myerson-Green [MG]; Myerson & Green, 1995), we will present these equations using the Rachlin version (i.e., the denominator of Eq. 2) in keeping with the theme of the special issue. All final model formulations are displayed in Table 4.

Table 4.

Formulas Used for 3D Model Comparisons

Price	Delay	Operation	Short Form	Full Form
Exponentiated	Mazur	Addition	EXPDPrice + MazurDelay	$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Price}}}-1\right)}-Q_0\left(1-\frac{1}{1+\mathit{\text{bDelay}}}\right)$
Exponentiated	MG	Addition	EXPDPrice + MGDelay	$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Price}}}-1\right)}-Q_0\left(1-\frac{1}{{(1+\mathit{\text{bDelay}})}^s}\right)$
Exponentiated	Rachlin	Addition	EXPDPrice + RachlinDelay	$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Price}}}-1\right)}-Q_0\left(1-\frac{1}{1+{\mathit{\text{bDelay}}}^s}\right)$
Exponentiated	Mazur	Multiplication	EXPDPrice * MazurDelay	$Q_{pd}=\frac{Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Price}}}-1\right)}}{1+\mathit{\text{bDelay}}}$
Exponentiated	MG	Multiplication	EXPDPrice * MGDelay	$Q_{pd}=\frac{Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Price}}}-1\right)}}{{(1+\mathit{\text{bDelay}})}^s}$
Exponentiated	Rachlin	Multiplication	EXPDPrice * RachlinDelay	$Q_{pd}=\frac{Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Price}}}-1\right)}}{1+{\mathit{\text{bDelay}}}^s}$
Mazur	Exponentiated	Addition	MazurPrice + EXPDDelay	$Q_{dp}=Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Delay}}}-1\right)}-Q_0\left(1-\frac{1}{1+\mathit{\text{bPrice}}}\right)$
MG	Exponentiated	Addition	MGPrice + EXPDDelay	$Q_{dp}=Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Delay}}}-1\right)}-Q_0\left(1-\frac{1}{{(1+\mathit{\text{bPrice}})}^s}\right)$
Rachlin	Exponentiated	Addition	RachlinPrice + EXPDDelay	$Q_{dp}=Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Delay}}}-1\right)}-Q_0\left(1-\frac{1}{1+{\mathit{\text{bPrice}}}^s}\right)$
Mazur	Exponentiated	Multiplication	MazurPrice * EXPDDelay	$Q_{dp}=\frac{Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Delay}}}-1\right)}}{1+\mathit{\text{bPrice}}}$
MG	Exponentiated	Multiplication	MGPrice * EXPDDelay	$Q_{dp}=\frac{Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Delay}}}-1\right)}}{{(1+\mathit{\text{bPrice}})}^s}$
Rachlin	Exponentiated	Multiplication	RachlinPrice * EXPDDelay	$Q_{dp}=\frac{Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Delay}}}-1\right)}}{1+{\mathit{\text{bPrice}}}^s}$
Mazur	Mazur	Multiplication	MazurPrice * MazurDelay	$Q_{pd}=\frac{Q_0}{\left(1+\mathit{\text{cPrice}}\right)\left(1+\mathit{\text{bDelay}}\right)}$
MG	MG	Multiplication	MGPrice * MGDelay	$Q_{pd}=\frac{Q_0}{{(1+\mathit{\text{cPrice}})}^t{(1+\mathit{\text{bDelay}})}^s}$
Rachlin	Rachlin	Multiplication	RachlinPrice * RachlinDelay	$Q_{pd}=\frac{Q_0}{\left(1+{\mathit{\text{cPrice}}}^t\right)\left(1+{\mathit{\text{bDelay}}}^s\right)}$
Exponentiated	Exponentiated	Multiplication	EXPDPrice * EXPDDelay	$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_0\mathit{\text{Price}}}-1\right)}*{10}^{k\left(e^{-\beta Q_0\mathit{\text{Delay}}}-1\right)}$
Exponentiated Scaling	Exponentiated Scaling	Multiplication	EXPDtPrice * EXPDsDelay	$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_0{\mathit{\text{Price}}}^t}-1\right)}*{10}^{k\left(e^{-\beta Q_0{\mathit{\text{Delay}}}^s}-1\right)}$

Note. 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 (log₁₀ span +0.5). EXPD: Exponentiated demand equation. MG: Myerson-Green discounting equation.

Additive

To create an additive model of demand and delay, the additive model of discounting initially proposed by Killeen (2009) and adapted by Vanderveldt et al. (2015), when only including delay is

$$V=A-f\left(d\right)$$ (4)

Where V and A are the same as in Equation 1, and f(d) is the decrease in A attributed to the effect of delay. Based on Vanderveldt et al., assuming a hyperbolic function, this would result in

$$f\left(d\right)=A\left(1-\frac{1}{1+b{Y}^s}\right)$$ (5)

where all parameters are the same as described as above for Equation 2. However, because the purpose is to determine the effect of delay on demand, and to account for consumption changes across price points, Equation 4 can be converted to

$$Q_{pd}=Q-f\left(d\right)$$ (6)

where the effect of delay is applied to the estimated consumption of Q and providing an estimated consumption based on the effects of price of and delay to a commodity, Q_pd. The rationale for replacing A with Q₀ is that when A is treated as a free parameter it simply becomes the intercept in the nonlinear regression (i.e., the origin and highest point of the curve). In the case of Equation 3, Q₀ also acts as the intercept estimated from demand data, and therefore is also the origin and highest point of the curve, thereby granting equivalency when A and Q₀ are both free parameters. That is, when A is treated as a free parameter it estimates the intercept (i.e., the predicted consumption/value when Y is equal to 0) and it is conceptually equivalent to Q₀, therefore we tried

$$f\left(d\right)=Q_0\left(1-\frac{1}{1+b{Y}^s}\right)$$ (7)

as the additive version to model the effect of delay on consumption. From Equation 6, Q is then expanded to Equation 3, where estimated consumption based on price is subtracted by the decrease in consumption based on delay to the commodity, yielding

$$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_{0}X}-1\right)}-Q_0\left(1-\frac{1}{1+b{Y}^s}\right)$$ (8)

.³ When the delay (i.e., Y) is set to zero this reduces to Equation 3. When price (i.e., X) is set to zero, this reduces to Equations 4 and 5. It should be noted that many different equations can be placed into the additive model, including other exponential versions of discounting.

Multiplicative

Also influenced by Vanderveldt et al. (2015), we attempted a multiplicative model of demand for price and delay originally introduced by Ho et al. (1999), which conceptually is the estimated consumption divided by the effect of delay, or

$$Q_{pd}=\frac{Q}{f\left(d\right)}$$ (9)

Expanded, Equation 9 simply replaces A in Equation 2 with all of Equation 3, yielding

$$Q_{pd}=\frac{Q_0*{10}^{k\left(e^{-\alpha Q_{0}X}-1\right)}}{1+b{Y}^s}$$ (10)

Equation 10 then represents the proportional decrease of consumption based on price and time. Lastly, because of previous explorations between Rachlin’s equation and demand data (Rzeszutek et al., in press), we have also extended a multiplicative discounting model both price and delay,

$$Q_{pd}=\frac{Q_0}{\left(1+c{X}^t\right)\left(1+b{Y}^s\right)}$$ (11)

For Equation 11, the effect of price and delay on consumption is assumed to be multiplicative and hyperbolic, rather than price being exponential and delay being hyperbolic. We also extended this logic to the exponentiated equation, by creating a multiplicative version via the form

$$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_{0}X}-1\right)}*{10}^{k\left(e^{-\beta Q_{0}Y}-1\right)}$$ (12)

In Equation 12, consumption is reduced as a function of price and delay, where ${10}^{k\left(e^{-\alpha Q_{0}X}-1\right)}$ represents the decrease in consumption as a function of X, then ${10}^{k\left(e^{-\beta Q_{0}Y}-1\right)}$ should represent the decrease in consumption as a function of Y. Equation 12 can also be adapted to include scaling parameters on the independent variables

$$Q_{pd}=Q_0*{10}^{k\left(e^{-\alpha Q_0{X}^t}-1\right)}*{10}^{k\left(e^{-\beta Q_0{Y}^s}-1\right)}$$ (13)

This is similar to the logic of the Rachlin model but extended to exponential demand equations as described in Schwartz and Hursh (2022). Inclusion of this model is to help determine if the functional form of the data is better described by a hyperbolic form, or instead by inclusion of additional scaling parameters relative to existing demand equations.

Participants and Demographics

Workers from Amazon’s Mechanical Turk (MTurk) were recruited to complete a Human Intelligence Task (HIT) by the way of a Qualtrics survey. Workers were compensated a base of $1.50 for completion of the survey and up to a bonus of $2.00 based on correctly responding to the four attention checks embedded within the survey. Estimated time to completion was 30 minutes, resulting in an expected compensation of $7.00/hr. While the survey had differing branches based on participant screening, only results for participants who completed the delayed demand task will be reported. Participants were also required to have reported smoking at least five or more cigarettes a day to be able to access the experimental portion of the survey. Other than typical demographics (e.g., age, income, ethnicity), the Fagerström Test for Nicotine Dependence (FTND; Heatherton et al., 1991) was included as part of the survey. The FTND is used to assess the relative importance of cigarettes to an individual. This is useful as a snapshot for molar decision-making towards addictive substances as described by Rachlin (2007). Demand for cigarettes has been previously correlated with the FTND (González-Roz et al., 2019), and so comparing how different models detect these associations can help lend credibility to the analyses. Data were collected in October 2018. Procedures were approved by the Virginia Tech Institutional Review Board.

Delayed Demand Cigarette Purchase Task

Participants completed hypothetical cigarette purchase tasks (hereafter purchase tasks) that combined escalating prices for cigarettes as well as delays to access of those cigarettes being delivered from a local cigarette store or an online cigarette store. For example, the purchase task incorporating the smallest delay was,

“Imagine that you are out of cigarettes and need to order more. Imagine that you can order cigarettes and get them delivered from The Local Cigarette Store, which currently has a Delivery Delay of 5 minutes. For the questions that follow, we will ask you how many cigarettes from The Local Cigarette Store you would be willing to purchase and use within 24 hours of receiving them.”

with prices per cigarette of $0.01, $0.03, $0.06, $0.13, $0.25, $0.50, $1.00, $2.00, $4.00, and $8.00 (see Appendix A). Participants could then input the number of cigarettes they would purchase at each price point. For cigarette purchases with a delay greater than 5 min, cigarettes were sold and delivered from the online cigarette store. An example of a delayed access to cigarettes is as follows,

“Imagine that you are out of cigarettes and need to order more. Imagine that you can order cigarettes and get them delivered from The Online Cigarette Store, which currently has a Delivery Delay of 3 months. For the questions that follow, we will ask you how many cigarettes from The Online Cigarette Store you would be willing to purchase and use within 24 hours of receiving them.”

Prices for delayed cigarettes were identical to the ones listed above. Delays to cigarettes were 5 min, 6 hr, 1 day, 1 week, 1 month, 3 months, 1 year, and 5 years. On a page, all prices were presented at once in ascending order for a single delay, with the presentation of delays being randomly ordered. Therefore, participants completed eight purchase tasks, one for each delay.

To aid in task comprehension, additional instructions were presented prior to the purchase task that included a visualization of delays (see Appendix B). Participants were instructed to imagine that they must make the purchases now, the cigarettes are their usual brand, they have no access to other nicotine products, they must consume the cigarettes within 24 hr of receiving them, and they cannot give away the cigarettes. These instructions are based on best practices used in demand research (Reed et al., 2020). Prior to each purchase task, participants were required to indicate the delay to cigarettes as part of a comprehension check. A participant was required to select the correct delay prior to moving on to the purchase task for that delay. The was to help participants attend to the delay as the order of delay was randomly presented.

Data Organization and Analyses

All code used for data organization and analyses are available online at https://github.com/brentkaplan/delayed-demand. All data organization and analyses were conducted in R (R Core Team, 2022). The data.table (Dowle & Srinivasan, 2020) package was used to organize data, ggplot2 (Wickham, 2016) for two-dimensional figures, and plot3D (Soetaert, 2021) for generating three-dimensional figures. Prior to model fitting, datasets of individual participants were removed if they responded with purchasing 200 or more cigarettes. This is because it is virtually impossible to consume 200 cigarettes in a 24-hr period and therefore indicative of inattention to the experimental parameters. After this, datasets were excluded if they demonstrated nonsystematic responding (Johnson & Bickel, 2008; Stein et al., 2015). However, because there is no standard for determining nonsystematic data for three-dimensional demand and discounting data, datasets were excluded if reported consumption for the most expensive cigarettes, most delayed cigarettes, or most delayed and expensive cigarettes was higher than reported consumption at minimum price and minimum delay. It should be noted that this exclusion allows for values between the minimums and maximums to be higher than the minimums (i.e., the bounce criterion was not applied). While mixed effects models can handle some nonsystematic data (Kaplan et al., 2021) we chose to exclude data that may cause convergence issues since we are attempting a novel analysis of demand data that incorporated price of and delay to a commodity.

To compare different versions of the models, we report Akaike Information Criterion corrected for small sample sizes (AICc; calculated using the AICcmodavg package; Mazerolle, 2020), Bayesian Information Criterion (BIC), root mean squared error (RMSE), mean absolute error (MAE), log likelihood, and proportion of variance accounted for (R²). The formula used to calculate R² was 1–(SSE_model/SSE_mean) where SSE is the sum of squared errors of either the model (SSE_model) or of the mean (SSE_mean). Because of issues with R² and nonlinear regression, values span from -Infinity to 1 rather than being bounded from 0 to 1 (Johnson & Bickel, 2008; Koffarnus & Kaplan, 2018). While R² has its issues for nonlinear models, it is still relatively intuitive as a value of 1 indicates a perfect fit to the data and as such we decided to include it due to its ubiquity in discounting and demand research. In the case of models with increased parameters, RMSE, MAE, log likelihood, and R² are expected to improve relative to models with fewer parameters. AICc and BIC by contrast account for increased model complexity and penalize for number of estimated parameters, thereby allowing for model comparisons between models of differing complexity. We also included weighted AICc; w(AICc) and BIC; w(BIC) to compare probabilities between likely models of those tested. This was done via the equations proposed by Wagenmakers and Farrell (2004). All of these metrics are reported for purposes of transparency.

Multilevel Modeling

To fit the additive and multiplicative equations, multilevel modeling (also called mixed-effects modeling) was used to estimate parameters for both the group and individual. No parameters were bounded as part of the model fitting process. A detailed explanation of multilevel modeling for demand can be found in Kaplan et al. (2021) and for discounting in Young (2017). Briefly, multilevel modeling has benefits to traditional approaches of dealing with demand and discounting which involve fitting a nonlinear regression to the group/mean data, or by fitting each individual’s data (i.e., fit-to-mean and two-stage approach; Kaplan et al., 2021). Some issues with these approaches are that they assume independence of observations and can overparameterize the individual results. The multilevel approach allows for including all data in the model while providing individual estimates that are weighted against the group. The ultimate benefit of this approach is that it also allows for estimating the overall group estimate (i.e., fixed effect) while also providing predictions for each individual (i.e., random effect). Another benefit of multilevel modeling is that effects of covariates on different parameters can also be estimated, rather than correlating estimates from individual regressions to variables of interest in a second stage after demand parameters are obtained.

Because of the complexity of the data and models, fitting demand data using log₁₀-scaled parameters allows for achieving lower tolerances and better fits to the data (Kaplan et al., 2021; Rzeszutek et al., 2022; Rzeszutek et al., in press; Traxler et al., 2022). This does not transform the response variable but only the scale in which the parameter is expressed (and thus, optimized). The rationale for this is expressing parameters in log₁₀ space when model fitting, notably for α, is that a lower tolerance is more easily achieved, and it decreases convergence issues. Note that for models that include the s and t scaling parameters, these were not fitted in log₁₀ space because they approximate a normal distribution when b and c are being estimated in log₁₀ space.

Multilevel Modeling Process

Because fitting nonlinear regressions requires determining start values for the algorithm, we have found a process that increases the likelihood of convergence for multilevel demand models. The first step is fitting nonlinear regression via a grid search to find optimal starting values. This was done using the nls.multstart (Padfield & Matheson, 2020) package with 250 combinations of starting values, with the best-fit parameters of those values being retained. After fitting this model, parameter estimates were used as starts for the multilevel model, fitted with the nlme (Pinheiro et al., 2020) package. Because this approach also allows for the estimation of a covariate on a parameter (e.g., the effect of FTND on Q₀), the above process can be adapted by including an intermediary step using generalized nonlinear least squares to estimate parameters to use in the multilevel model. Because the algorithm will sometimes fail to achieve the desired tolerance when attempting to converge, tolerances were increased to allow for convergence to occur. Values of k were determined using code adapted from the beezdemand (Kaplan et al., 2019) R package but generalized to three dimensional demand. Thus, the log₁₀ span of all the data included to determine k was 1.15495. Because of standards of adding +.5 to the log₁₀ span of the data for the k parameter, we conducted comparisons between these for all 2D and some 3D fits to determine if the addition of +.5 was warranted in exponentiated models. For models that included the FTND as a covariate, FTND scores were normalized using the scale() function prior to model fitting.

Results

Data Organization

There were 20 participants whose data were removed because they reported a cigarette purchase above 200. Relative to reported purchasing at the lowest price and delay, 18 had higher consumption at the maximum price and delay, 31 had a higher consumption at maximum delay, and 14 had a higher consumption at maximum price. Therefore, for purposes of analyses there were a total of 45 data sets removed based on the 3D systematic criteria.

Demographics

Following data organization there were 85 participants who completed the purchase task and were included for analyses. Median time of survey completion was 21.8 min (min = 8.0, max = 52.6) indicating a median compensation of $9.63 per hr assuming all attention checks were correctly completed. Table 1 contains the demographics of the study sample that were included for the analyses. There were four participants who did not have FTND data, therefore all analyses involving the FTND covariate consist of 81 data sets.

Table 1.

Participant Demographics

	Total/Mean (SD/%)
Age	36.96 (±10.61)
Income	$47,562.14 (±33,530.09)
Income Distribution
0 to $24,999	20 (24.7%)
$25,000 to $49,999	26 (32%)
$50,000 to $74,999	18 (22.2%)
$75,000 to $99,999	7 (8.6%)
$100,000 to $149,999	9 (11.1%)
$150,000 to $199,999	1 (1.2%)
$200,000 or more	0
Marital Status
Single	16 (19.8%)
Not married	7 (8.6%)
Living with another	11 (13.6%)
Married	38 (46.9%)
Divorced	6 (7.4%)
Widowed	3 (3.7%)
Race
African American	2 (2.5%)
Asian	3 (3.7%)
Hispanic	3 (3.7%)
White/Caucasian	72 (88.9%)
Mixed	1 (1.2%)
Education
Less than high school	1 (1.2%)
High school/GED	10 (12.3%)
Some college	20 (24.7%)
2-year college degree	12 (14.8%)
4-year college degree	30 (37%)
Master’s degree	8 (9.9%)
Employment Status
Employed	69 (85.1%)
Unemployed	11 (13.6%)
Retired	1 (1.2%)
FTND Score
No dependence (0)	8 (9.9%)
Low dependence (1–2)	16 (19.8%)
Low to moderate dependence (3–4)	17 (20.9%)
Moderate to high dependence (5–7)	33 (40.7%)
High dependence (8+)	7 (8.6%)

Note. FTND: Fagerström Test for Nicotine Dependence. The FTND is scored from 0 to 10, with higher scores indicating greater dependence.

2D Demand Models of Price and Delay

Table 2 contains the model comparisons metrics the EXPD_Price, Rachlin_Price, EXPD_Delay, and Rachlin_Delay models. For 2D price (i.e., consumption at the lowest delay), the Rachlin_Price model was the more likely model that described the data. Depending on the randomization of the algorithm, for 2D delay (i.e., consumption at the lowest price), the EXPD_Delay model was the more likely model for cigarette consumption as the additional parameters of the Rachlin_Delay model were not justified in this case. Random effect fits to six participants who displayed “typical” demand data based on price can be found in Figure 1 for both price and delay models. Fits to all participants for 2D data can be found in the online repository.

Table 2.

Goodness-of-Fit Metrics for Different 2D Model Comparisons without Covariates

Model	df	AICc	w(AICc)	BIC	w(BIC)	RMSE	MAE	R 2
EXPDPrice	6	5012.40	1.32e-85	5040.77	1.61e-81	2.885	1.827	.951
Rachlin Price	10	4621.52	1.000	4668.71	1.000	1.906	1.104	.979
RachlinDelay	10	5259.50	.998	5304.39	.060	8.444	4.711	.783
EXPD Delay	6	5271.90	.002	5298.90	.940	8.202	3.987	.795

Note. Model summaries of Eq. 2, the delay discounting equation and Eq. 3, the demand equation, fitted to the minimal price and minimal delay respectively. Models are indicated by formula and factor (i.e., price or delay). df: Degrees of freedom. AICc: Akaike Information Criterion corrected. BIC: Bayesian Information Criterion. w(AICc) and w(BIC): Akaike and Bayesian weights calculated as per Wagenmakers and Farrell (2004). RMSE: Root mean square error. MAE: Mean absolute error. R²: Proportion of variance explained based on sum of squared errors (SSE), R² = 1-(SSE_model/SSE_mean). For AICc, BIC, RMSE, and MAE lower values indicate better model fits. For R² higher values indicate better model fits.

Figure 1.

Random Effect Fits from 2D Data Analyses for the EXPD Model (Dashed Line) and Rachlin Model (Solid Line)

Note. Y-axes are number of cigarettes purchased at each X-axis, either price of cigarettes (left side) or delay to cigarettes (right side). Participants were chosen as they displayed typical patterns of demand at the lowest delay and are meant to provide illustrative fits for the EXPD and Rachlin equations to 2D data. Each panel represents a participant. X-axes are log₁₀ scaled. Open circles are cigarettes purchased for 2D price, open squares are cigarettes purchased for 2D delay.

Estimates and model comparisons the FTND covariate analysis are in Table 3. For 2D price, FTND scores were related to Q₀ in both models. For 2D delay, all parameters were significant at the .05 level, with b, s, and α being lower based on higher FTND scores. That is, cigarette consumption was generally higher at later delays for those with higher FTND scores. Relative to EXPD_Delay, Rachlin_Delay was the more likely model based on model comparison metrics, although BIC values were extremely similar.

Table 3.

Fixed Effects Estimates with FTND as a Covariate from the 2D Delay only and Demand only Models for both Exponentiated and Rachlin Equations

Model/Parameter	Estimate	SE	df	t	p	AICc	BIC	Log Lik.
EXPD Price						4792.48	4829.88	−2388.15
Q 0	1.293	0.038	726	33.984	<.001
FTND Q 0	0.119	0.038	726	3.120	.002
α	−1.875	0.067	726	−27.842	<.001
FTND α	−0.057	0.068	726	−0.845	.398
RachlinPrice						5463.47	5524.07	−2718.51
Q 0	1.331	0.049	724	27.238	<.001
FTND Q 0	0.106	0.049	724	2.158	.031
c	0.184	0.143	724	1.280	.201
FTND c	0.003	0.142	724	0.022	.982
t	1.435	0.109	724	13.225	<.001
FTND t	0.228	0.108	724	2.102	.036
Rachlin Delay						4998.30	5055.88	−2485.86
Q 0	1.328	0.032	562	41.379	<.001
FTND Q 0	0.076	0.034	562	2.263	.024
b	−2.195	0.267	562	−8.226	<.001
FTND b	−0.924	0.257	562	−3.603	<.001
s	0.599	0.070	562	8.525	<.001
FTND s	0.184	0.067	562	2.755	.006
EXPDDelay						5023.09	5058.65	−2503.43
Q 0	1.303	0.028	564	46.602	<.001
FTND Q 0	0.114	0.028	564	4.019	<.001
α	−5.724	0.132	564	−43.311	<.001
FTND α	−0.345	0.134	564	−2.566	.011

Note. Model summaries of Eq. 2, the delay discounting equation and Eq. 3, the demand equation, fitted to the minimal price and minimal delay, respectively. Models are indicated by formula and factor (i.e., price or delay). Note that for model fitting, the parameters were estimated in log₁₀ space except for s and t. Therefore, all estimates but s and t are in log₁₀ units. Q₀: Estimated cigarette consumption at zero price and zero delay. α: Sensitivity to change in price/delay. b: Estimated effect of delay on cigarette consumption. s: Scaling parameter for delay. c: Estimated effect of price on cigarette consumption. t: Scaling parameter for price. FTND Parameter: Effect of Fagerström Test for Nicotine Dependence on parameter estimates (normalized). Italics: Best model for delay or demand based on model comparisons metrics. Bold: Estimated parameter is significant at the < .05 level. AICc: Akaike Information Criterion corrected (lower is better). BIC: Bayesian Information Criterion (lower is better). Log Lik.: Log likelihood (higher is better).

3D Demand Models

For all combinations of models tested, which equation was used to determine the decrease in consumption as a function a given factor, and the mathematical operation used, see Table 4. From here on, models will be referred to using the short form introduced in Table 4. Briefly, these indicate the equation used, the mathematical operation, and the factor to which the equations were applied. For example, Equation 10 becomes EXPD_Price * Rachlin_Delay, Equation 11 becomes Rachlin_Price * Rachlin_Delay, and Equation 12 would be EXPD_Price * EXPD_Delay. Because there are different ways to calculate or replace k in the literature, was also tested select models based on using the k span parameter, or adding 0.5 to the k span parameter.⁴ These adjustments of k were only applied to the EXPD_Price * Rachlin_Delay, EXPD_Price * EXPD_Delay, and EXPD^t_Price * EXPD^s_Delay models as to not otherwise double the number of models already being assessed. Otherwise, k was set to the log₁₀-span of the data +.5 which was 1.65495 for models that included the EXPD equation. Other than the EXPD_Delay model, the addition of +.5 improved fits involved the EXPD equation and therefore are what are reported.

Comparisons between all models can be found in Table 5. Predicted consumption from the top four performing models, the Rachlin_Price * EXPD_Delay, EXPD_Price * Rachlin_Delay, EXPD^t_Price * EXPD^s_Delay, and Rachlin_Price * Rachlin_Delay are in Figure 2 along with their respective R², MAE, and AICc values. Random effect fits for the same participants from Figure 1 are available for the Rachlin_Price * Rachlin_Delay in Figure 3. In all cases but one, additive and multiplicative models that included EXPD_Price and Rachin_Delay provided the best descriptive fits to the current data for their respective operations. The model that best fit the data based on model metrics was Rachlin_Price * Rachlin_Delay. For all versions of additive equations, they generally performed worse than multiplicative models and negative consumption was predicted at higher prices and delays, which is similar to results of additive models of discounting (Cox & Dallery, 2016; Vanderveldt et al., 2015), except where the models were explicitly limited to positive values. Individual fits for all models and participants can be found in the online repository.

Table 5.

Goodness-of-Fit Metrics for Different Demand Equations

Model	df	AICc	w(AICc)	BIC	w(BIC)	RMSE	MAE	R 2
EXPDPrice + MazurDelay	10	45704.76	.000	45772.97	.000	6.410	3.472	0.737
EXPDPrice + MGDelay	15	45680.51	.000	45782.81	.000	6.636	3.642	0.718
EXPDPrice + RachlinDelay	15	45656.33	.000	45758.63	.000	6.338	3.478	0.743
EXPDPrice * MazurDelay	10	43518.76	4.38e-216	43586.97	8.33e-200	5.420	2.529	0.812
EXPDPrice * MGDelay	15	43525.84	1.27e-217	43628.14	9.61e-209	5.390	2.537	0.814
EXPDPrice * RachlinDelay	15	42822.75	5.99e-65	42925.05	4.52e-56	5.085	2.184	0.835
MazurPrice + EXPDDelay	10	45693.84	.000	45762.05	.000	6.403	3.54	0.738
MGPrice + EXPDDelay	15	45654.16	.000	45756.46	.000	6.351	3.499	0.742
RachlinPrice + EXPDDelay	15	45530.53	.000	45632.83	.000	6.260	3.337	0.749
MazurPrice * EXPDDelay	10	43656.82	4.60e-246	43725.03	8.74e-230	5.482	2.620	0.808
MGPrice * EXPDDelay	15	43396.87	1.29e-189	43499.17	9.73e-181	5.424	2.506	0.812
RachlinPrice * EXPDDelay	15	43074.92	1.05e-119	43177.22	7.90e-111	5.191	2.139	0.828
MazurPrice * MazurDelay	10	43648.59	2.82e-244	43716.80	5.36e-228	5.482	2.671	0.808
MGPrice * MGDelay	21	43651.71	5.93e-245	43794.89	5.93e-245	6.145	3.436	0.758
RachlinPrice * RachlinDelay	21	42527.00	1.000	42670.18	1.000	4.849	1.855	0.850
EXPDPrice * EXPDDelay	10	43520.93	1.48e-216	43589.15	2.81e-200	5.499	2.480	0.812
EXPDtPrice * EXPDsDelay	21	42653.00	4.35e-28	42796.18	4.35e-28	4.903	1.949	0.846

Note. Bold indicates the best four models. See Table 4 for full forms of model equations. df: Degrees of freedom. AICc: Akaike Information Criterion corrected. BIC: Bayesian Information Criterion. w(AICc) and w(BIC): Akaike and Bayesian weights calculated as per Wagenmakers and Farrell (2004). RMSE: Root mean square error. MAE: Mean absolute error. R²: Proportion of variance explained based on sum of squared errors (SSE), R² = 1-(SSE_model/SSE_mean). For AICc, BIC, RMSE, and MAE lower values indicate better model fits. For R² higher values indicate better model fits.

Figure 2.

Fixed Effect Predicted Cigarette Consumptions of the Four Best Models Rachlin_price * Expd_delay (Top-Left), Expd_price * Rachlin_delay (Top-Right), Expd^t_price * Expd^s_delay (Bottom-Left), and Rachlin_price * Rachlin_delay (Bottom-Right) Models

Note. The surface indicates the predicted group consumption at a given Y (delay to cigarettes) and at a given X (price per cigarettes). Both X and Y axes are in log₁₀ units. Transparent circles at the lowest delay indicate 2D consumption by price, transparent squares at the lowest price indicate 2D consumption by delay. Each point is an individual response. Darker points indicate more responses at that value. Note that these 2D consumption points are identical in each panel. Individual consumptions above 50 are not shown. R²: Overall model proportion of variance accounted for (higher is better). MAE: Mean absolute error (lower is better). AICc: Akaike’s information criterion corrected (lower is better).

Figure 3.

Random Effects Predictions of Individuals from Figure 1 Using the Rachlin_price * Rachlin_delay Model

Note. Each panel is a participant. Each point on the Z-axis (vertical axis) represents cigarettes purchased at a given Y (delay to cigarettes) and at a given X (price per cigarettes). Both X and Y axes are in log₁₀ units. Vertical lines indicate the residual from each individual fit to the data. Note the z-axis scaling differs based on the individual.

3D Covariate Demand Models

The Rachlin_Price * Rachlin_Delay and EXPD^t_Price * EXPD^s_Delay (Eqs. 11 and 13, see Table 4 for full equations) were both assessed for relationships between FTND scores and estimated parameters as they were the models that best described the data. Table 6 contains the results of the inclusion of the effect of FTND on all estimated parameters from both models. In both cases, only Q₀ was related to FTND scores, with higher FTND scores being associated with higher Q₀s. For model comparisons, the Rachlin_Price * Rachlin_Delay model was the more likely model based on AICc/BIC relative to the other models tested for these data.

Table 6.

Fixed Effects Estimates with FTND as a Covariate from the Best 3D Models

Model/Parameter	Estimate	SE	df	t	p	AICc	BIC	Log Lik.
EXPDtPrice * EXPDsDelay						40802.30	40978.27	−20375.04
Q 0	1.456	0.034	6390	42.348	<.001
FTND Q 0	0.095	0.034	6390	2.774	.006
α	−1.865	0.074	6390	−25.206	<.001
FTND α	−0.021	0.075	6390	−0.279	.780
t	1.421	0.097	6390	14.648	<.001
FTND t	−0.070	0.099	6390	−0.706	.480
β	−5.435	0.441	6390	−12.320	<.001
FTND β	−0.318	0.443	6390	−0.717	.473
s	1.007	0.130	6390	7.730	<.001
FTND s	0.005	0.130	6390	0.037	.970
RachlinPrice * RachlinDelay						40729.09	40905.06	−20338.43
Q 0	1.480	0.037	6390	39.670	<.001
FTND Q 0	0.093	0.038	6390	2.443	.015
c	0.425	0.108	6390	3.944	<.001
FTND c	0.086	0.109	6390	0.793	.428
t	1.792	0.131	6390	13.709	<.001
FTND t	−0.083	0.134	6390	−0.620	.535
b	−4.082	0.561	6390	−7.273	<.001
FTND b	−0.101	0.556	6390	−0.182	.856
s	1.286	0.164	6390	7.844	<.001
FTND s	−0.026	0.162	6390	−0.161	.872

Note. Model summaries of the two best fitting 3D models used for covariate analyses. Note that for model fitting, the parameters were estimated in log₁₀ space except for s and t. Therefore, all estimates but s and t are in log₁₀ units. Q₀: Estimated cigarette consumption at zero price and zero delay. α: Sensitivity to change in price. β: Sensitivity to change in delay. s: Scaling parameter for delay. c: Estimated effect of price on cigarette consumption. t: Scaling parameter for price. FTND Parameter: Effect of Fagerström Test for Nicotine Dependence on parameter estimates (normalized). Italics: Best model for delay or demand based on model comparisons metrics. Bold: Estimated parameter is significant at the < .05 level. AICc: Akaike Information Criterion corrected (lower is better). BIC: Bayesian Information Criterion (lower is better). Log Lik.: Log likelihood (higher is better).

Discussion

The results of the current analysis appear to indicate that quantitative models of demand and discounting can be combined to model more complex and ecologically valid scenarios in the case of hypothetical demand data that contain aspects of price and delay. Typically, hypothetical purchase tasks involve determining demand of a commodity based on a single parameter (e.g., price). Modeling demand in such a way is thought to be a snapshot of molar processes that affect decision-making for a given commodity (see Kaplan et al., 2018 & Reed et al., 2020). While this aligns well with Howard Rachlin’s conceptualizations of addiction (Rachlin, 2007), there are many factors that could otherwise be missed when tasks are constructed in such a fashion. Although the devaluation of a commodity based on its delay has been well studied, demand for a commodity based on its delay is only in nascent stages of development (e.g., Gunawan, 2020; Schwartz & Hursh, 2022). While demand based on price did not see a relation with the FTND in this sample, demand based on delay identified significant relations between cigarette consumption and the FTND for both the exponentiated and Rachlin equations.

Price and Delay Appear to be Multiplicative

Based on the equations explored in this manuscript, models fitted using a version of the multiplicative demand and discounting, such as the Equations 10, 11, and 13, provided better descriptive fits to the data than models based on assumptions that factors interact additively. These results are similar to work in discounting (e.g., Cox & Dallery, 2016; Vanderveldt et al., 2015). A challenge with additive models is that without bounding predicted values, they will estimate values below zero (Cox & Dallery, 2016). This leads to interesting challenges for both the experimental manipulation as well as the methods of analysis. The first is, what exactly does a negative consumption value imply? Potentially, this would mean that a participant would be willing to sell cigarettes at that price. Or they would potentially pay to avoid experiencing a condition as described (i.e., they would do whatever they could to avoid waiting to consume some number of cigarettes if they had to wait some time to do so). To our knowledge, no such research exists that explores this possibility. Rather than discounting additive models of utility, studies that allow for the assessment of “negative” consumption are warranted. Ignoring the potential implications of negatively predicted consumption values, multiplicative models are both conceptually reasonable extensions of previous work and provide better descriptive fits to the data in the present case.

Behavioral Economic Demand: Exponential or Hyperbolic?

While only a single dataset was analyzed here, the model that provided the best descriptive fits to the data was the Rachlin_Price * Rachlin_Delay model. This is by no means definitive, as we will discuss different considerations for demand data, the value of “best fits”, how the data are scaled, and additional considerations when fitting complex multilevel models.

Distinctions Between the “Best” Model and the “Most Useful” Model

Inclusion of the extra parameters was justified based on the model comparisons for the 3D models with and without covariates. When analyzing 2D data without covariates, Rachlin outperformed EXPD for 2D price, but this was not the case for 2D delay, where EXPD_Delay was more likely than Rachlin_Delay. When including FTND scores as a covariate, EXPD_Price and Rachlin_Delay were the more likely models, although with caveats due to model fitting, discussed below. Previous work has indicated that the Rachlin model provides descriptively better fits to data than the exponentiated for some demand data (Rzeszutek et al., in press), 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 (González-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 (González-Roz et al., 2020; Schwartz et al., 2021; Secades-Villa et al., 2016). While Rachlin’s hyperboloid may provide better descriptive fits to some demand data, parameters estimated from it are yet to be established regarding their usefulness in understanding addiction processes relative to other models. The most practical issue with Rachlin’s equation is that the decay parameter (e.g., b or c) is dependent on the scaling parameter (e.g., s or t). This leads to interpretability issues due to the high correlation between these (r >.9), which in discounting research has resulted in the necessity of measures such as Effective Delay 50 (ED50; Franck et al., 2015; Yoon & Higgins, 2008) to produce more interpretable metrics that avoid issues of collinearity. This is the same reason Hursh and Silberberg (2008) proposed α as a measure of price sensitivity, as it converted models with multiple parameters to describe sensitivity into a single parameter. Another consideration is the theoretical backing for what α is meant to represent: an “essential value” that is independent of Q₀ due to the “real cost” adjustment in the exponent of the exponential equations (i.e., Q₀ * C). This “real cost” adjustment does not occur in the hyperbolic models. Whether there is some such thing as “essential value” that can be extracted from these models is a more detailed discussion than can occur here (see Strickland et al., 2022 for more detailed discussion this topic). Further research is required between models of demand to determine which equations produce the most meaningful parameter estimates relative to goodness-of-fit metrics.

Psychophysical Scaling Considerations

Part of the reason why Rachlin’s model may have been able to outperform the EXPD model is due to the addition of the scaling parameters, rather than the functional form of the model (i.e., exponential vs. hyperbolic). This is a possibility as the EXPD^t_Price * EXPD^s_Delay model was extremely close in fit relative to the Rachlin_Price * Rachlin_Delay model (also see the Additional Considerations section below). Given that these models were both the most performant on all metrics, had similar fits, and had similar outcomes with clinical covariates, stating that one model is definitively better is not possible here. However, one of the benefits of the exponential demand equations is that of α, a parameter theoretically independent of Q₀. Inclusion of scaling parameters to the exponential demand equations results in the same issue of b and s, in that they are correlated and dependent on each other. Thus, much like discounting models with scaling parameters, adding scaling parameters to demand models may nullify the very benefits of that model. The exponential equation was designed to specifically reduce the two sensitivity parameters from the linear-elasticity model of demand (Hursh et al., 1988) to a single sensitivity parameter. Much like what was described regarding the “best” model and “most useful” model, a decision must be made between the theoretical justification for any model and the variables of importance to which they are related.

Data Scaling Considerations

The current data were optimized in an untransformed response scale (i.e., number of cigarettes purchased). Exponential models of demand are conceptualized to be in log- (Hursh & Silberberg, 2008; Koffarnus et al., 2015) or log-like space (Gilroy et al., 2021), and are optimized in either log (Hursh & Silberberg, 2008), linear (Koffarnus et al., 2015), or log-like (Gilroy et al., 2021) space. The importance of log space for the conceptualization and interpretation of the elasticity parameter lies in the value of assessing proportional changes in consumption rather than absolute changes in consumption. Note that the space in which an equation is optimized does not necessarily affect this interpretation of the parameter or the ability to identify the point at which the slope of the predicted curve becomes -1 on a log–log scale (the price point at which expenditure is the highest, or P_max).⁵ However, because in the present analyses we focused on untransformed data, partially due to extra ease of combining demand and discounting equations, further comparisons in this realm are required.

Additional Considerations

When fitting any model, complexity should be taken into consideration. This is for two main reasons. The first is that increased model complexity may not be warranted and so study of model parsimony (e.g., using AICc, BIC) should be included to determine if the added benefit of a more complex model is worth the extra complexity. The second is that more complex models can be more challenging to fit via nonlinear multilevel modeling. Making tolerances more lenient may allow for a multilevel model to converge, rather than the algorithm aborting or oscillating between parameter estimates and never achieving convergence tolerance, but this leads to less accurate estimates. In R, it is possible to use the set.seed() function to create reproducible analyses, where the results of any analyses are the same so long as the code is executed in same order after running the set.seed() function.⁶ But changes to the order in which analyses or simulations are run can change the results.

For example, while preparing the code for reproducibility, changing the order of the way the models were run resulted in the Rachlin_Price with covariates being the more likely model, while the current analyses indicate the EXPD_Price with covariates is the more likely model. This is also true with the EXPD^t_Price * EXPD^s_Delay model, whereby one version had an AICc that was two points lower than the Rachlin_Price * Rachlin_Delay model. These inconsistencies may be alleviated via increasing the number of starts as determined by the researcher, as well as having more extensive grid searches, but this then comes at the cost of computational power and time. Because of this, researchers should be transparent in their analyses and aware of potential challenges. How start values for multilevel nonlinear models, and nonlinear models more broadly, are determined should be included when discussing analysis methodologies. Different methods of model fitting (e.g., fit to mean, two-stage, and multilevel modeling) may also produce different results on which model is the “best” model for these data. Indeed, there have been calls for open and reproducible science within behavior analysis (Gilroy & Kaplan, 2019) which is why we have made the code and data publicly available online. As analyses become more complex to help model individual behavior, transparency is necessary to identify best practices.

Limitations

An obvious limitation is that the data were generated from a novel hypothetical purchase task using an online convenience sample (i.e., MTurk). While crowdsourcing is commonly used in behavioral economic demand research (Mellis & Bickel, 2020; Strickland & Stoops, 2019), verification of cigarette use is not possible outside of participant self-report. While price sensitivity (i.e., α and b/s) did not correspond with the FTND in 3D models, it did correspond with Q₀ (i.e., intensity) in all models and was associated with price sensitivity for both 2D delay models of EXPD_Delay and Rachlin_Delay. The inclusion of multiple factors that affect consumption may affect previously established relations between drug demand and other assessments of drug use.

Another limitation was the determination of systematic versus nonsystematic data. While there are methods for assessing presumed patterns of discounting (Johnson & Bickel, 2008) and demand (Stein et al., 2015), there are not yet any accepted method for determining nonsystematic 3D data that includes characteristics of both. For purposes of exploratory analyses using this novel hypothetical purchase task, we used a very lenient criteria where consumption at the lowest price and delay was not above the highest prices and delays. It is possible that better fits could be obtained with more strict criteria, but we did not want to (1) artificially improve model fits by excluding data that did not meet a priori assumptions and (2) decrease the sample size too drastically given the novel nature of the task. Even though the criteria were relatively lenient, there were still a substantial number of participants’ data removed for purposes of model fitting. Part of this could have been how the task was presented, with delay being presented in a random order (across blocks) while price was always presented at once and in an ascending order (within blocks; i.e., each page had costs from lowest to highest with random delays). This may have been why the number of nonsystematic paths removed for delay “violations” were double that of those removed for price “violations” (i.e., 31 participants had a higher consumption at maximum delay, while 14 had a higher consumption at maximum price). This is also a likely reason for the relatively worse fits for 2D delay data by both the EXPD and Rachlin equations. Take for example the 2D demand and 2D delay data for participants A113 and A204. Visually, their consumption based on price appears to be systematic without any purchasing increases as price increases. However, for consumption based on delay, both have at least one “bounce” (i.e., an increase in cigarette purchases above a previous smaller delay). Procedural issues regarding multidimensional demand data are a necessary part of future research. Perhaps presenting purchase tasks in increasing order of delays would have decreased the relative number of nonsystematic datasets. Note that we are not advocating for removal of data as a result of nonsystematic responding strictly for purposes of improving model fits. The goals of this paper were to identify issues and considerations for 3D demand data while extending behavioral economic models of delay and demand, but also to address procedural limitations that could impact researchers interested in studying multidimensional demand.

Conclusion

Howard Rachlin was profoundly influential in behavioral economic conceptualizations of behavior, theories of addiction, and in understanding how the value of a commodity changes across different parameters. Because of this, we felt it fitting to extend his work into behavioral economic demand as a potential way to better describe demand data, explore behavior that changes in lawful ways, and further the quantitative analysis of behavior. Equations used for discounting were extended to demand, while existing models of demand were also generalized to allow for analysis of three-dimensional data. Equations that incorporated some version of Rachlin’s discounting equation for 3D demand data were the best fitting and favored models based on a variety of model comparison metrics. While this study has its limitations, we believe this is a good first step to better understand decision-making for addictive substances involving multiple factors as well as provide a foundation for analyzing complex decision-making more broadly, that honors Howard Rachlin’s legacy to the quantitative analysis of behavior.

Appendix A. Example of Purchase Task

Appendix B. Example visualization of the online store attending question presented to participants. Red indicates wait time from purchase to delivery, blue indicates consumption period.