Trading networks with bilateral contracts

We study production networks in which firms match and sign bilateral contracts. Firms can buy from and sell to one another directly or via intermediaries. It is well-known that in this case group-stable outcomes might not exist. We show that the problem of determining whether an allocation is group-stable is NP-hard. We define a new stability concept, called trail stability, and show that any network of bilateral contracts has a trail-stable outcome whenever agents' preferences satisfy full substitutability. Trail-stable outcomes rule out consecutive and consistent pairwise blocks that form trails of contracts. Trail stability is a natural extension of chain stability and is a stronger solution concept in general contract networks. Trail-stable outcomes may not be immune to group deviations or efficient. In fact, we show that outcomes satisfying an even more demanding stability property -- full trail stability -- always exist. Fully trail-stable outcomes also rule out trail blocks, but an intermediary is not required to choose all contracts in the trail -- only local upstream-downstream pairs. We pin down conditions under which terminal contracts in trail-stable and fully trail-stable outcomes have a lattice structure. We describe the relationships between all stability concepts. When contracts specify trades and prices, we also show that trail-stable competitive equilibrium outcomes exist in networked markets even when agents' utility functions are not quasilinear.