A SIR macro model: Comparing the decentralized economy and the optimal policy

We present a Simple SIR Macro Model to study the economic impact of an epidemic in a model were agent types are unobservable. We solve for the decentralized economy equilibrium and for optimal solution (subject to the constraint that the planner cannot differentiate between agent types). We find that decentralized economy produces an endogenous lock-down in which economic activity decreases. We find that the decentralized economy will begin its lock-down sooner than what the optimal policy prescribes because agents have an incentive to selfishly avoid infection while waiting for others to get infected, recover and contribute to herd-immunity. The optimal policy in this model is not necessarily about forcing people to stay at home but to force people to get infected early on.