Suppose that the price of an imported consumer good rises significantly. This could be due to a decrease in the global supply or else an increase in foreign demand. Further, suppose there is no import competing industry for this particular good.
Suppose demand for the imported good is unit elastic. Quantity demanded falls in proportion to the increase in price, so that total spending on the product remains unchanged. Foreign exchange markets continue to clear. The total amount spent on imports is unchanged as is the total amount earned from exports. Spending on domestic product is unchanged as well. Spending on domestic product generates an equal nominal income.
Real income has fallen. The price level, including the imported good, is now higher. And so, deflated by that index, real income is lower. Less of the imported good is enjoyed.
With nominal GDP targeting, this is all that happens. Spending on domestic product and nominal income were not impacted.
On the other hand, if the consumer price level is targeted, then the increase in the price of the imported good pushes the price level above target. This would trigger a monetary contraction. For the most part, the result would be reduced prices of domestically produced goods and services, with their prices falling enough to offset the inflation in the price of the imported consumer good. Of course, since falling demand for any particular good often signals the need to contract production, it is likely that output and employment would fall as well. Hopefully, in the long run, wages and other nominal incomes would shift to a lower growth path, leading to lower unit costs, and so a recovery of production and lower prices of domestic goods.
To the degree that the monetary contraction triggers higher interest rates and a net capital inflow, the exchange rate will rise. This could directly dampen the increase in the price of the imported consumer good, lower the prices of other imported goods, and reduce the foreign demand for exported goods. One the prices of exported goods and imported goods fall, the appreciation of the currency will be justified by purchasing power parity.
With inflation targeting, if the increase in the price of the imported consumer good comes as a surprise, (like the realization of a random variable,) then it is ignored. Bygones are bygones. There has been a temporary spike in inflation. Unfortunately, if the price of the consumer good is foreseen, and part of an ongoing process in real time, then inflation targeting would require a monetary contraction as above.
Of course, with "flexible" inflation targeting, the central bank can allow inflation when it thinks it best. But then, that isn't much of a rule.
Could it be that a large run up in oil prices, partly due to supply problems in the Middle East and Africa, but also due to growing gasoline demand from India and China, combined with "flexible inflation targeting," has led to an economic contraction that aims to force down other prices, including wages and other nominal incomes, enough to keep inflation on target?