In most nations, paths of monetary aggregates and prices consistently depart from stationary trends. This paper shows that this is a fundamental implication when monetary authorities of interdependent countries seek to smooth their home output and prices in the presence of incomplete world output-market integration and structural asymmetries. Using a two-country model with interdependent output supply schedules, we show that this conclusion holds whether the exchange rate floats or is fixed. It also holds if monetary policies are coordinated. Therefore, optimal monetary policy choices by central banks yield stationary paths for money and prices only under very specific conditions.