Why do insurers choose to exclude medical providers, and when would this be socially desirable? We examine network design from the perspective of a profit-maximizing insurer and a social planner to evaluate the welfare effects of narrow networks and restrictions on their use. An insurer may engage in exclusion to steer patients to less expensive providers, cream-skim enrollees, and negotiate lower reimbursement rates. Private incentives for exclusion may diverge from social incentives: in addition to the standard quality distortion arising from market power, there is a "pecuniary" distortion introduced when insurers commit to restricted networks in order to negotiate lower rates. We introduce a new bargaining solution concept for bilateral oligopoly, Nash-in-Nash with Threat of Replacement, that captures such bargaining incentives and rationalizes observed levels of exclusion. Pairing our framework with hospital and insurance demand estimates from Ho and Lee (2017), we compare social, consumer, and insurer-optimal hospital networks for the largest non-integrated HMO carrier in California across several geographic markets. We find that both an insurer and consumers prefer narrower networks than the social planner in most markets. The insurer benefits from lower negotiated reimbursement rates (up to 30% in some markets), and consumers benefit when savings are passed along in the form of lower premiums. A social planner may prefer a broader network if it encourages the utilization of more efficient insurers or providers. We predict that, on average, network regulation prohibiting exclusion has no significant effect on social surplus but increases hospital prices and premiums and lowers consumer surplus. However, there are distributional effects, and regulation may prevent harm to consumers living close to excluded hospitals.