Shaping the Menu: Information Design under Screening

How should consumers learn when a seller chooses a menu of product qualities and prices? We study information design when the seller observes the learning policy, but not consumers' private signals, before choosing its menu. Under cost-curvature conditions, we characterize optimal learning when consumer surplus receives greater weight than profit. Even with two valuations, every optimal posterior distribution combines an interval of posterior types with an efficiently served atom at the highest valuation. The induced menu features a continuum of differentiated products. Strikingly, optimal learning can also induce a lower posterior atom, separated from this interval by a support gap. Buyers in this atom either purchase an isolated low-quality product or are excluded, even when trade with every buyer creates gains. Both configurations can be optimal for the same prior and welfare weight. Served buyers with distinct posterior types are never bunched at the same quality. General finite priors can yield several separating regions and trading atoms. We also characterize the optimal joint design of learning and valuation-dependent segmentation, showing, among other results, that every posterior type with positive gains from trade must be served. Joint design can raise both consumer surplus and profit relative to buyer-optimal private learning alone. Even random assignment to segments with different learning policies and separate menus can give consumers more surplus than any single market yielding the same profit. Additional price discrimination can therefore benefit consumers, even when privacy restrictions require segmentation to reveal nothing about valuations to the seller.