Optimal Random Taxation

We consider the Ramsey problem of optimal government policy and give examples where lotteries are essential for the optimal solution. When markets are complete and the representative agent's coefficient of relative prudence is bounded, lotteries are typically not optimal. For the incomplete markets case, we prove analytically that lotteries improve welfare for some calibrations with CRRA utility, and we give a simple numerical example for this claim. When the agent's coefficient of relative risk aversion is unbounded for small consumption, lotteries typically improve welfare and can be used to approximate the first-best solution arbitrarily well in both the complete and the incomplete markets cases. However, in the complete markets case, when debt constraints are not too tight and the representative agent is sufficiently patient, lotteries can be purified in the sense that there is a deterministic solution that gives the same value.