For over two decades, current account imbalances have been an essential issue in the global policy debate as they threaten the world economy's stability. More recently, the government debt crisis of the European Union shows that internal current account imbalances of a currency union may also add to these risks. Moreover, oil price fluctuations and a contracting monetary policy that reacts to oil prices, previously discussed to affect the current account, may threaten the currency union by increasing internal imbalances. Therefore, this paper analyzes the oil price shock's impact on current account imbalances of a currency union with asymmetric labor market institutions. In this context, we show that oil price shocks can have a long-lasting effect on internal balances that the common monetary policy authority can reduce by choosing a core inflation target. Targeting core inflation, however, comes at the cost of lower production and higher unemployment. We show that these costs can be significantly reduced by increasing labor market flexibility.