This paper studies optimal fiscal and monetary policy under sticky product prices. The theoretical framework is a stochastic production economy without capital. The government finances an exogenous stream of purchases by levying distortionary income taxes, printing money, and issuing one-period nominally risk free bonds. The main findings of the paper are: First, for a miniscule degree of price stickiness (i.e., many times below available empirical estimates), the optimal volatility of inflation is near zero. This result stands in stark contrast with the high volatility of inflation implied by the Ramsey allocation when prices are flexible. The finding is in line with a recent body of work on optimal monetary policy under nominal rigidities that ignores the role of optimal fiscal policy. Second, even small deviations from full price flexibility induce near random walk behavior in government debt and tax rates. Third, sluggish price adjustment raises the average nominal interest rate above the one called for by the Friedman rule. Finally, an interest-rate feedback rule whereby the nominal interest rate depends on inflation and output fits well the data emanating from the Ramsey economy. However, the fitted rule is not of the Taylor type, for the implied inflation coefficient is less than one and close to zero and the output coefficient is negative.