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

Abstract

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).