In this article we develop a finite mixture negative binomial count model that accommodates unobserved heterogeneity in an intuitive and analytically tractable manner. This model, the standard negative binomial model, and its hurdle extension are estimated for six measures of medical care demand by the elderly using a sample from the 1987 National Medical Expenditure Survey. The finite mixture model is preferred overall by statistical model selection criteria. Two points of support adequately describe the distribution of the unobserved heterogeneity, suggesting two latent populations, the healthy and the ill whose fitted distributions differ substantially from each other.