To what extent is monetary policy effective in controlling the rate of inflation?

Monetary policy is a policy used by the central banks which involves increasing and decreasing the level of interest rates within an economy so as to control the level of inflation. Inflation is the rate at which the general level of prices for goods and services is rising. Increasing the level of interest rate causes the cost of borrowing to increase along with the reward for saving. Thus, consumers will choose to save money rather than spend it causing level of consumption within the economy to fall. Since consumption is part of aggregate demand from the equation AD= C +I+G+(X-M) this will lead to a fall in the level of aggregate demand within the economy. A shift to the left of the aggregate demand will cause price levels within the economy to fall thus achieving the aim of controlling the level of inflation. However, this is not always the case since private banks may be reluctant to pass on the increase in the level of inflation towards their customers thus causing no change to the level of inflation. In addition, it depends what has caused this high level of inflation in the first place since it may have been caused by cost-push factors such a rise in the oil prices. This is beyond the central banks control thus nothing could be done by the central bank so as to be able to control the level of inflation since increasing interest rates in such a case will do more damage than good to the economy.