Alternative approaches for modeling concave willingness to pay functions in conjoint valuation

Standard approaches for modeling concave functions used in conjoint studies are unlikely to be suitable when attempting to value both small and large improvements. For instance, a concave quadratic function in the utility of an environmental attribute implies that at some point total WTP for the attribute will decline, violating standard monotonicity assumptions regarding preferences (e.g., more is better). Quadratic functions may be useful as local approximations as in Translog functions, but they are not useful as globally monotonic functions. Another approach is to use a standard log function, but in some instances, the attributes may take on zero or negative values which cannot be handled using a log transformation without ad hoc adjustments. Although one can often transform the data to remove zero observations, this may not be the way that respondents actually considered the attributes in the first place. For instance, a 0% change from a nonzero baseline would be a nonzero number in absolute terms, but respondents may be thinking in terms of positive or negative percentage improvements, and the associated model may fit better (as it does in the data we use below). A third linear in parameters approach to modeling nonlinear WTP is to use a piecewise linear (PWL) function by estimating a coefficient associated with each design point. As the number of points in the design increases, this function becomes more flexible, but without monotonicity restrictions it becomes likely that the function will exhibit at least one negative incremental value. In this article, we suggest the standard inverse hyperbolic sine (ISH) transformation (Burbidge, Magee, and Robb) as an alternative to log transformations when some of the explanatory variables take on zero or negative values, and the use of the nonlinear in parameters IHS function as an alternative to the Box-Cox when the explanatory variables take on zero or negative values. We compare these models with a PWL model for a conjoint style SP dataset for valuing improvements in fish populations in Washington State. On the basis of a nonnested model selection criterion, we find that the standard ISH model fits as well as the PWL model even though it uses 18 fewer parameters and the ISH model yields a more precise estimates of WTP.