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Mallesh Pai, Associate Professor of Economics, Rice University
How Informed Do You Want Your Principal to Be?
Abstract: In many bilateral trade settings of interest, at the time of contracting, both parties, i.e., the buyer and the seller, have private information. However, most existing work restricts attention to one-sided private information to avoid the complications of an informed problem. Hence there is little understanding of the implications of two-sided private information on optimal mechanisms, and resulting seller profit and buyer welfare. Motivated by regulatory questions in the personalized pricing of products such as health insurance, we pose and answer the following question: how much (if any) private information would the buyer want the seller to have in a bilateral trade setting?
In our setting, the buyer's value for a good depends on his private type and the quality of the good that is unknown to him. The seller learns about the quality via a (noisy) private signal whose realization is the seller's private information. This quality affects both the buyer's value and the seller's cost, i.e., this is an interdependent values setting. We characterize the buyer-optimal outcome: this is the information structure and the corresponding seller-optimal equilibrium of the informed principal game that yields the highest consumer surplus. We show that there are conditions under which private information for the seller can lead both to greater profits and higher consumer surplus; that is, compared to being uninformed, seller private information can lead to Pareto gains. Additionally, maybe surprisingly, there are also conditions under which the seller does not benefit from having private information. These are settings where a privately informed seller leads to a lemons problem a la Akerlof (1970) albeit one where the information is two-dimensional (buyer and seller private information). The information structure for the seller that yields the highest welfare for the buyer is typically not fully informative, i.e., the seller is only partially informed about the quality of the good. We interpret this as a justification for buyer privacy regulations.