If the central bank is interested in keeping prices stable and there is a decrease in the velocity of money, what should the central bank do?
I know that the decrease in velocity of money shifts the LM-curve to the left and what I first thought of was that it should increase the money supply and that this would offset the decreased output and keep income and employment stable, but increasing money supply would increase the price level wouldn't it?
In this situation, the central bank should increase the supply of money. You have expressed concern that this would lead to inflation. However, the fact that the velocity of money has decreased will offset the increase in the supply of money and therefore prevent a rise in price levels.
We can see this if we use the simplified equation for the quantity theory of money. It holds that
M*V = P*Q
Where M is the total amount of money in circulation, V is the velocity, P is the price level and Q is the quantity of output.
If V goes down and we want P and Q to remain constant, we have to increase M. If we increase M by just the right amount, it will offset the decrease in V and there will not need to be any change in the value of either P or Q.