Optimal Contracting in Networks

We study optimal contracting between a firm selling a divisible good that exhibits positive externality and a group of agents in a social network. The extent of externality that each agent receives from the consumption of neighboring agents is privately held and is unknown to the firm. By explicitly characterizing the optimal contract when the utilities are quadratic, we demonstrate how inefficiency in one agent’s trade propagates through the network and creates unequal, downward distortion in other agents’ trades. We further characterize the effect of propagation of the distortion on the firm’s profit in terms of agents’ influence and centrality in the network. We identify influential agents whose inefficiencies incur maximum loss on the firm’s profit. By explicitly solving for the simpler yet suboptimal bilateral contracts, we characterize the optimality gap between these and multilateral contracts in terms of the underlying network structure. When the network is balanced in the sense of homogeneity of influences, we show that the network structure will have no impact on the firm’s profit when contracts are bilateral. On the other hand, when the influences are heterogeneous and have high dispersion (as in star-like networks) we show that the restriction to bilateral contract can result in profit losses that grow unboundedly with the size of the network. Our result highlights the role of influential agents and network structure in design of optimal contracts and determines when knowledge of the network structure is beneficial to the firm.