A spline function class suitable for demand models

A function class suitable for estimating cost preferences in demand models is presented. The function class is applicable to any positive cost variable and is designed to be: (i) monotonically decreasing, (ii) to have decreasing marginal sensitivity with respect to cost, and (iii) to be differentiable at every point. It is shown how suitable functions can be formed from sequences of tailored functions in a manner that ensures their continuity and differentiability at the knot points. The proposed functions are well suited for demand models where price elasticities exhibit a damped pattern as the values of their argument increase. The usual linear-in-parameter functions or non-linear functions, such as the Box-Cox function, do not have an equally flexible way of accounting for such a pattern. This can be relevant when estimating transport demand models where the sensitivity of demand with respect to transport costs is known to decline as the cost increases, i.e. the phenomenon of “cost-damping”. However, it may also be relevant as a means to capture the marginal return of investments or declining marginal utility of income. To provide an illustration, the functions are incorporated in a multinomial logit model that is estimated from synthetically generated data by maximum likelihood. A Monte Carlo simulation study shows that the estimator is able to recover the true parameters.11The programs for generating the synthetic data and for estimating the models (in R and SAS software) are available as supplementary material to the electronic version of the paper. The practical application of the function class is also considered within the new large-scale Danish National Transport Model.