(Un)conditional distribution of compensating variation in discrete choice models

For a large class of additive random utility discrete choice models with income effects, we compute the probability distribution of the compensating variation. We show that the cumulative distribution function only depends on the choice probabilities. Our results are used to compute the distribution of equivalent variation. The moments of the compensating variation are a onedimensional integral of the choice probabilities. Using the expected compensating variation, we extend Shephard's Lemma to the probabilistic demand systems. Both conditional and unconditional (on the individual choice) distributions of compensating variation are considered.