Efficient Mechanisms for Bilateral Trading with Cooperative Investment

This paper analyzes a bilateral trade problem where the seller makes hidden investment that influences the buyer's private valuation. Complete contracts can be written on observable trade decisions. It is shown that efficiency level in investment and trade is negatively related to the scope of trade: if the first-best efficient trade levels are positive for sufficiently many buyer types, then it is impossible for budget-balanced incentive compatible trading mechanisms to achieve first-best efficient investment \emph{and} efficient levels of trade, even if no (ex-ante, interim, or ex-post) participation constraints are required. This result differs from the Myerson-Satterthwaite inefficiency theorem for bilateral trade without investment, where interim participation constraints are required. To fully characterize the second-best trading mechanism, I solve a constrained-Pareto program with moral hazard on seller side and adverse selection on buyer side. I show that the first-order approach to characterizing the seller's incentive constraint is valid. I then derive the second-best nonlinear trading schedule. It displays several distinctive features, in comparison to standard adverse selection and moral hazard models: when there is no bunching at the bottom types, both the highest type and the lowest type receive first-best levels of trade; bunching can occur for the bottom types, and when it does, for a range of the lowest types trade levels are \emph{above} first-best, and both the highest type and some middle type receive first-best levels of trade. Furthermore, the second-best investment can be either below (under investment), above (over investment), or equal to the first-best level, so hidden investment does not necessarily mean under investment; when renegotiation is permitted, there is always under investment. These results shed light on bilateral trade problems when investment decisions are important.