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We consider the ability of monetary policy to fully stabilize pure demand shocks in a monetary union with strategically acting fiscal authorities. We show that when one national fiscal authority enjoys a strategic advantage over the other and fiscal policy can directly affect inflation, monetary policy cannot fully stabilize pure demand shocks at the union level, unless they are common. Moreover, we characterize a situation where country-specific fiscal policies diverge, being counter-cyclical for one country and pro-cyclical for the other, for high enough values of the direct effect of fiscal policy on the inflation parameter. The coordination of national fiscal policies becomes desirable for the union central bank.