Can a central bank react differently to demand-pull and supply-push inflation?

Monetary policy is as effective at fighting inflation whether it is caused by a surge in demand (for goods and services) or by a fall in supply: in both cases, pushing down demand reduces inflation. If there is inflation because of a surge in demand (strong credit growth, fiscal deficit, etc.), stamping it out merely requires getting demand and production back to normal. But if there is inflation because of a fall in supply (rise in commodity prices, etc.), stamping it out requires reducing production even more than what results from the fall in supply, and therefore driving up unemployment significantly. So could central banks adopt a behaviour rule where they combat inflation when it results from an abnormal surge in demand for goods and services, but not when it results from a fall in supply? The answer is ambiguous: it depends on the comparison between the social cost of unemployment and the social cost of inflation (inflation tax, loss of real income, financial instability). In the case of a positive demand shock, in contrast, because the economy is “overheating”, the social cost of the unemployment that results from the fight against inflation is low.