The tug-o-war for supremacy between inflation targeting and monetary targeting is a classic, yet timely topic, in monetary economics. In this paper, we revisit this issue within the context of a pure-exchange, overlapping generations model in which spatial separation and random relocation create an endogenous demand for money. We study AR(1) shocks to both real output and the real interest rate. Irrespective of the nature of the shocks, the optimal inflation target is always positive. Under monetary targeting, shocks to output necessitate negative money growth rates; for shocks to real interest rates, money growth rates may be either positive or negative depending on the elasticity of consumption substitution. Also, for output shocks, monetary targeting welfare-dominates inflation targeting but the gap between the two vanishes as the shock process approaches a random walk. In sharp contrast, for shocks to the real interest rate, we prove that monetary targeting and inflation targeting are welfare-equivalent only in the limit as the shocks become i.i.d. The upshot is that persistence of the underlying fundamental uncertainty matters: depending on the nature of the shock, policy responses need to be either more or less aggressive as persistence increases.