In this paper, I refine the random utility model (RUM) of Block and Marschak (1959) to represent stochastic choice data with quasi-linear types. In my framework, choices are observed across pairs of goods and money. The random quasi-linear utility function is identified uniquely in my model. This identification implies a unique social welfare aggregator that is consistent with the Pareto efficiency criteria. In general, the uniqueness of the quasi-linear tie-breaking rule is not guaranteed, but it can be obtained in a special case where the tie-breaking is uniform. I also characterize a special case where the set of possible types is binary. (A continuation under the same name was published as a joint paper, but there is still some novel material in this version that might be used later in a separate paper.)