We consider Bayesian estimation of a stochastic production frontier with ordered categorical output, where the inefficiency error is assumed to follow an exponential distribution, and where output, conditional on the inefficiency error, is modeled as an ordered probit model. Gibbs sampling algorithms are provided for estimation with both cross-sectional and panel data, with panel data being our main focus. A Monte Carlo study and a comparison of results from an example where data are used in both continuous and categorical form supports the usefulness of the approach. New efficiency measures are suggested to overcome a lack-of-invariance problem suffered by traditional efficiency measures. Potential applications include health and happiness production, university research output, financial credit ratings, and agricultural output recorded in broad bands. In our application to individual health production we use data from an Australian panel survey to compute posterior densities for marginal effects, outcome probabilities, and a number of within-sample and out-of-sample efficiency measures.