Transition choice probabilities and welfare analysis in additive random utility models

We study the descriptive and the normative consequences of price and/or other attributes changes in additive random utility models. We first derive expressions for the transition choice probabilities associated to these changes. A closed-form formula is obtained for the logit. We then use these expressions to compute the cumulative distribution functions of the compensating variation (CV) conditional on ex-ante and/or ex-post choices. The unconditional distribution is also provided. The conditional moments of the CV are obtained as a one-dimensional integral of the transition choice probabilities. This framework allows us to derive a stochastic version of Shephard's lemma, which relates the expected conditional CV and the transition choice probabilities. We compute the CV for a simple binary linear in income choice model and show that the information on the transitions leads to better estimates of the CV than those obtained when only ex-ante or ex-post information on individual choices is used. For the additive in income logit, we compute the conditional distribution of CV, which generalizes the logsum formula. Finally, we derive a new welfare formula for the disaggregated version of the representative consumer CES model.