The paper is concerned with efficient estimation of characteristics demand. We derive and estimate an inverse input demand system for quality characteristics by using 172,946 observations over 881 trading days in the Icelandic fish auctions. An improved estimation method based on an expanded random coefficient model is suggested as an alternative to the currently used two-stage method of Brown and Rosen (1982). The estimates demonstrate the improved efficiency of the suggested method. A number of empirical results emerge, including a general increase in the demand for quality.