Optimal testing and social distancing of individuals with private health signals

In an epidemic, the regulation of social distancing and testing is critical for the large group of individuals who are possibly infected, but have not developed clear, distinct symptoms. Each individual’s reaction to a regulation scheme depends on its private probability assessment of being infected. Assuming no monetary transfers, we identify a simple class of schemes for welfare maximization: all individuals who ask for a test are tested with the same probability, independently of their infection probabilities, and the social distancing regulation depends on who asks for a test. Social distancing has a double role: to provide incentives so that the right people get tested, and to curb the spread of the disease. If testing capacities are scarce it can be optimal to test only a randomly selected fraction of those who ask for a test, and require maximal social distancing precisely for those individuals who ask unsuccessfully. If public costs and benefits are small, laissez faire is optimal.