Consumer's surplus and a second-order approximation to compensating variation are compared to the exact compensating variation for a subsystem of several commodities. An empirical application to the relative retail price distortions created by the U.S. dairy program is presented. In the application, a new specification for weakly integrable incomplete demand models is developed and estimated. The integrable demand model has sufficient flexibility that it is not rejected by the data. Both welfare approximations are found to introduce large errors in the measurement of deadweight losses.