Empirical evidence suggests that decision makers often weight successive additional units of a valued attribute or monetary endowment unequally, so that their utility functions are intrinsically nonlinear or irregularly shaped. Although the analyst may impose various functional specifications exogenously, this approach is ad hoc, tedious, and reliant on various metrics to decide which specification is “best.” In this paper, we develop a method that yields individual-level, flexibly shaped utility functions for use in choice models. This flexibility at the individual level is accomplished through splines of the truncated power basis type in a general additive regression framework for latent utility. Because the number and location of spline knots are unknown, we use the birth-death process of Denison et al. (1998) and Green's (1995) reversible jump method. We further show how exogenous constraints suggested by theory, such as monotonicity of price response, can be accommodated. Our formulation is particularly suited to estimating reaction to pricing, where individual-level monotonicity is justified theoretically and empirically, but linearity is typically not. The method is illustrated in a conjoint application in which all covariates are splined simultaneously and in three panel data sets, each of which has a single price spline. Empirical results indicate that piecewise linear splines with a modest number of knots fit these data well, substantially better than heterogeneous linear and log-linear a priori specifications. In terms of price response specifically, we find that although aggregate market-level curves can be nearly linear or log-linear, individuals often deviate widely from either. Using splines, hold-out prediction improvement over the standard heterogeneous probit model ranges from 6% to 14% in the scanner applications and exceeds 20% in the conjoint study. Moreover, “optimal” profiles in conjoint and aggregate price response curves in the scanner applications can differ markedly under the standard and the spline-based models.
markov chain monte carlo,utility theory,splines,single price spline,flexibly shaped utility function,price response,piecewise linear spline,choice models,latent utility,conjoint study,conjoint application,reversible jump method,bayesian methods,bayesian spline approach,scanner application,heterogeneity,capturing flexible heterogeneous utility,aggregate price response curve,piecewise linear,probit model,decision maker,bayesian method,panel data,birth death process
Spline (mathematics),Econometrics,Monotonic function,Mathematical optimization,Probit model,Conjoint analysis,Markov chain Monte Carlo,Birth–death process,Piecewise linear function,Mathematics,Bayesian probability