The authors investigate the structure of preferences and uncertainty that guarantees that prices are fully revealing even though asset markets are incomplete and there are more sources of uncertainty than assets in the economy. A sufficient condition for fully revealing prices is that investors have preferences of the (possibly state-dependent) linear-risk-tolerance class. Finally, the authors discuss how their result allows one to extend certain existing literature on demand aggregation, welfare analysis, and the pricing of contingent claims to the case in which markets are incomplete and investors have asymmetric private information.