An exact solution for unit elasticity in the exponential model of operant demand

Shawn P. Gilroy, Brent A. Kaplan, Derek D. Reed, Donald A. Hantula, Steven R. Hursh

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


Research applying the behavioral economic demand framework is increasingly conducted across disciplines. With respect to psychopharmacology and substance abuse, real and hypothetical purchase tasks are regularly used to evaluate the demand for various substances and reinforcers, such as alcohol. At present, a variety of methods has been introduced to solve for the point of unit elasticity, or Pmax, in the exponential model of demand; however, these methods vary in their potential for error. Current methods for determining Pmax are presented here and a novel exact solution for Pmax in the exponential model of demand is introduced. This solution provides an exact calculation of Pmax using the omega function, as algebraic solutions are not possible. This novel approach is introduced, discussed, and systematically compared to earlier methods for determining Pmax using computer simulations and reanalyses of published study data. Systematic comparison indicated that this new approach, an exact analytic solution for Pmax, provides results that are identical to computationally intensive Pmax methods that directly evaluate the slope of the demand function. The exact analytic Pmax approach is reviewed, its calculations explained, and an easy-to-use web tool is provided to assist researchers in easily performing this calculation of Pmax. Implications for reducing potential sources of error are reviewed and future directions are also discussed.

Original languageEnglish
Pages (from-to)588-597
Number of pages10
JournalExperimental and Clinical Psychopharmacology
Issue number6
StatePublished - Dec 2019

Bibliographical note

Publisher Copyright:
© 2019 American Psychological Association.


  • Behavioral economics
  • Computer software
  • Exact solution
  • Reinforcer efficacy
  • Simulation

ASJC Scopus subject areas

  • Pharmacology
  • Psychiatry and Mental health
  • Pharmacology (medical)


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