We develop a model of consumption and income that allows for pervasive heterogeneity in the parameters of both processes. Introducing codependence between household income parameters and preference parameters, we also allow for heterogeneity in the impact of income shocks on consumption. We estimate the parameters of the model using a sample from the Panel Study of Income Dynamics, covering the period 1968–2009. We find considerable codependent heterogeneity in the parameters governing income and consumption processes. Our results suggest a great deal of heterogeneity in the reaction of consumption to income shocks, highlighting the heterogeneity in the self-insurance available to households.