The zero lower bound on the interest rate and a Neoclassical Phillips curve

I derive the exact analytical solution for optimal monetary policy given a Neoclassical Phillips curve and a zero lower bound on the nominal interest rate. There is a particular range of interest rate rule parameters that may close the output gap. One way of closing the output gap involves stable but high inflation (the divine coincidence). In the general case inflation is variable and potentially lower. Thus, one can achieve stable OR low inflation, but not both. When the productivity shock has an unbounded support, only the variable inflation version of optimal policy is implementable. Optimal policy then involves a lagged interest rate response to shocks and a random walk price level.