A Modified Exponential Behavioral Economic Demand Model to Better
Describe Consumption Data
Mikhail N. Koffarnus, Christopher T. Franck, Jeffrey S. Stein, and Warren K. Bickel
Behavioral economic demand analyses that quantify the relationship between the consumption of a
commodity and its price have proven useful in studying the reinforcing efficacy of many commodities,
including drugs of abuse. An exponential equation proposed by Hursh and Silberberg (2008) has proven
useful in quantifying the dissociable components of demand intensity and demand elasticity, but is
limited as an analysis technique by the inability to correctly analyze consumption values of zero. We
examined an exponentiated version of this equation that retains all the beneficial features of the original
Hursh and Silberberg equation, but can accommodate consumption values of zero and improves its fit to
the data. In Experiment 1, we compared the modified equation with the unmodified equation under
different treatments of zero values in cigarette consumption data collected online from 272 participants.
We found that the unmodified equation produces different results depending on how zeros are treated,
while the exponentiated version incorporates zeros into the analysis, accounts for more variance, and is
better able to estimate actual unconstrained consumption as reported by participants. In Experiment 2, we
simulated 1,000 datasets with demand parameters known a priori and compared the equation fits. Results
indicated that the exponentiated equation was better able to replicate the true values from which the test
data were simulated. We conclude that an exponentiated version of the Hursh and Silberberg equation
provides better fits to the data, is able to fit all consumption values including zero, and more accurately
produces true parameter values.
Keywords: behavioral economics, demand analysis, cigarette consumption, data simulation, exponential
Behavioral economic demand analyses describe the relationship
between the price (including monetary cost and/or effort) of a com-
modity and the amount of that commodity that is consumed. Such
analyses have been successful in quantifying the reinforcing efficacy
of commodities including drugs of abuse, and have been shown to be
related to other markers of addiction (Bickel, Johnson, Koffarnus,
MacKillop, & Murphy, 2014;MacKillop & Murphy, 2007). Hursh
and Silberberg (2008) proposed a now widely used equation (Equa-
tion 6 in the source paper) to be fitted to consumption data across a
range of prices:
where Qis consumption of a given commodity at price C,Q
derived consumption as price approaches zero, ␣is demand elas-
ticity, and kis the span of the function in log
units. This equation
has a number of attractive features for the analysis of behavioral
economic demand data, and has become widely used as a result. It
allows for the independent measure of demand intensity (Q
demand elasticity (␣) for inferential and descriptive statistics.
Generally, this equation also describes demand data well and
accounts for a high proportion of the variance of consumption data
across a variety of contexts, procedures, and species (Hursh &
Silberberg, 2008;Koffarnus, Hall, & Winger, 2012;Koffarnus,
Wilson, & Bickel, 2015;Roma, Kaminski, Spiga, Ator, & Hursh,
2010). Demand intensity is often assessed in hypothetical purchase
task data without curve fitting by asking participants their level of
consumption without cost or other constraints. Outside hypothet-
ical purchase task assessments, however, unconstrained consump-
tion data are often unavailable and must be estimated from the
available data. Demand elasticity can be assessed on a point-to-
point basis without curve fitting, but these analyses are highly
sensitive to outliers in the data and do not provide a single measure
of overall demand elasticity. Furthermore, nonlinear regression
models allow for the inclusion of all consumption data in statistical
models, accounting for within-subject variability and consistency
in any statistical conclusions that are made.
The treatment of zero consumption values is one issue that has
arisen in our own and others’ research (e.g., Galuska, Banna,
Willse, Yahyavi-Firouz-Abadi, & See, 2011;Koffarnus et al.,
2012;Koffarnus et al., 2015;MacKillop et al., 2012;Yu, Liu,
Collins, Vincent, & Epstein, 2014). Fitting Equation 1 necessitates
log-transforming consumption values, as represented by the log Q
This article was published Online First August 17, 2015.
Mikhail N. Koffarnus, Virginia Tech Carilion Research Institute, Virginia
Tech; Christopher T. Franck, Virginia Tech Carilion Research Institute, Vir-
ginia Tech and Department of Statistics, Virginia Tech; Jeffrey S. Stein and
Warren K. Bickel, Virginia Tech Carilion Research Institute, Virginia Tech.
Supported by the National Cancer Institute (Grant U19 CA15734502).
None of the authors have any real or potential conflict(s) of interest,
including financial, personal, or other relationships with organizations or
pharmaceutical/biomedical companies that may inappropriately influence
the research and interpretation of the findings.
Correspondence concerning this article should be addressed to Mikhail
N. Koffarnus, PhD, VA Tech Carilion Research Institute, 2 Riverside
Circle, Roanoke, VA 24016. E-mail: email@example.com
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
Experimental and Clinical Psychopharmacology © 2015 American Psychological Association
2015, Vol. 23, No. 6, 504–512 1064-1297/15/$12.00 http://dx.doi.org/10.1037/pha0000045