TD n. 641 2015

Vitor Farinha Luz, Paulo Monteiro, Vinicius Nascimento Carrasco, Humberto Moreira.

We consider the problem of a seller who faces a privately informed buyer and only knows

one moment of the distribution from which values are drawn. In face of this uncertainty,

the seller maximizes his worst-case expected profits. We show that a robustness property

of the optimal mechanism imposes restrictions on the seller’s ex-post profit function. These

restrictions are used to derive the optimal mechanism. The optimal mechanism entails

distortions at the intensive margin, e.g., except for the highest value buyer, sales will take

place with probability strictly smaller than one. The seller can implement such allocation by

committing to post prices drawn from a non-degenerate distribution, so that randomizing

over prices is an optimal robust selling mechanism. We extend the model to deal with the

case in which: (i) M goods are sold and the buyer’s private information is multidimensional

and (ii) the seller and the buyer interact for several periods. In the case of multiple goods,

there are several optimal mechanisms. With multiple goods full bundling is optimal, as

well as selling the goods in a fully separable way. In the dynamic model, we show that

repetition, period by period, of the static-optimal mechanism is optimal.