We apply classical statistical decision theory to a large class of treatment choice problems with partial identification. We show that, in a general class of problems with Gaussian likelihood, all decision rules are admissible; it is maximin-welfare optimal to ignore all data; and, for severe enough partial identification, there are infinitely many minimax-regret optimal decision rules, all of which sometimes randomize the policy recommendation. We uniquely characterize the minimax-regret optimal rule that least frequently randomizes, and show that, in some cases, it can outperform other minimax-regret optimal rules in terms of what we call profiled regret. We analyse the implications of our results in the aggregation of experimental estimates for policy adoption, extrapolation of Local Average Treatment Effects, and policy making in the presence of omitted variable bias.

To determine the welfare implications of price changes in demand data, we introduce a revealed preference relation over prices. We show that the absence of cycles in this relation characterizes a consumer who trades off the utility of consumption against the disutility of expenditure. Our model can be applied whenever a consumer’s demand over a strict subset of all available goods is being analysed; it can also be extended to settings with discrete goods and non-linear prices. To illustrate its use, we apply our model to a single-agent data set and to a data set with repeated cross-sections. We develop a novel test of linear hypotheses on partially identified parameters to estimate the proportion of the population who are revealed better off due to a price change in the latter application. This new technique can be used for non-parametric counterfactual analysis more broadly.