Informed seller in a Hotelling market

We consider the problem of a monopolist seller who is selling a good to a potential buyer and is privately informed about some of the good’s attributes. We focus on the case where goods with di¤erent attributes are horizontally di¤erentiated: they appeal di¤erently to di¤erent types of the buyer. Such a market is represented via a standard Hotelling (1929) model, where the seller’s and the buyer’s private information are locations on the Hotelling line and the buyer’s utility from consuming the good is given by a base consumption value minus a cost depending on his distance from the seller. We derive the optimal selling mechanism as the solution of an informed principal problem. We show that if the base value is su¢ ciently high, then it is optimal for the monopolist not to disclose any information, if the base value is su¢ ciently low, then it is optimal for her to reveal her private information. For intermediate base values, the monopolist gains by price discriminating the buyer’s types over their value for information. A crucial feature of the optimal mechanism is that di¤erent information content is disclosed by the seller to di¤erent types of the buyer. When the cost is linear or concave, the optimal mechanism can be implemented as a two-item menu: buy an opaque good without disclosure, or buy the information about the good with the option of purchasing the good afterwards at a predetermined exercise price. When the cost is convex, a continuum of o¤erings is optimal: buy the good with full disclosure; buy the opaque good without disclosure; or buy one of a continuum of options, each providing the right to return the good with probabilities that depend on the good’s type.