Identifying the Distribution of Random Coefficients in BLP Demand Models Using One Single Variation in Product Characteristics

Recent advances on the identification of the Berry, Levinsohn and Pakes (BLP,1995) random coefficient demand models focus on the structural demand functions. Yet, this does not automatically imply the identification of the distribution of the random coefficients. The latter is often necessary for counter factuals where the new values of product characteristics do not belong to the support in the factual scenario (e.g. new prices after mergers) or the structural demand functions change (e.g. new products are added). This paper provides novel arguments to identify the distribution of the random coefficients using one single variation in product characteristics. In a leading case where the random coefficients only include a random coefficient on price and individual-and product-specific random intercepts, observing market outcomes at two different price vectors already suffices to identify the distribution of the random coefficients. In theory, these arguments greatly weaken the usual requirements on the regressors or the moments of the random coefficients. In practice, these results are particularly useful when there is little (or limited) variation in product characteristics across markets.