The equation MV=PT originates from classical economics in the 19th century. In a given period the money supply, multiplied by the number of times it is used, equals the price level multiplied by an index of the quantity of goods and services produced. In other words, the total value of money that changes hands (MV) is the same as the money value of goods and services that changes hands (PT).
Close, T is the actual number of transactions rather than an index, and P is the transaction weighted average price of those transactions. Therefore the equation forms a trivial identity of two equal numbers but split into their interesting components. Eg. If there's £10 of money in the economy and it's spent twice a year (velocity of 2) then it can support 2 transactions a year of £10 each, 4 of £5 each etc. (£10/£5 being the price of those transactions.)
T at some point became Q to fit in with the income/expenditure/output model of the economy and it still holds conceptually so I don't have an issue with that as long as the price level P is defined to match the way Q is calculated.
It's worth noting here that Q represents real output while PQ represents nominal output (think nominal GDP). This is why you will hear a lot of talk about targeting nominal GDP rather than real, as that's the RHS of the equation. To actually target either component requires some control of the other. Eg. If you want to increase real output (so what we actually consume rather than just its value at current price levels that fluctuate) then it's not much use to increase M if P then goes up the same. If you could keep P the same though then you will increase real output. Unfortunately P is determined by markets so you can't really control it (price controls just leading to market distortions and supply/demand imbalances and prolonged inefficient disequilibrium). So the balance is really letting the market and government combined production keep Q at a desired and attainable rate of growth and managing M to keep P steady. (Note that this isn't saying that govt spending should be reduced, that ties into crowding out which is outside the scope of this basic equation. Also should mention, another development of the early equation was to change P to inflation rather than price level, I think Q became GDP growth for the units to balance.)
So you can see that if one of the variables changes, say V increases, then money is being spent more often and that will be reflected in either or both of a rise in the price level and and a rise in goods and services produced.
Monetarists use an assumption, that V (the number of times money changes hands) and T (the quantity of goods and services) are fixed. This is because they assume that an economy is always at full employment. So an increase in the money supply will only have one effect, it will increase the price level. Hence, reduce the money supply to reduce inflation.
They don't assume V is fixed but that it is measurable, therefore at any time the authorities can know what money supply is needed to control inflation. They certainly don't assume T (or Q) is fixed as GDP growth fluctuates, it's about managing M to keep inflation low. Nor do they assume full employment as there is a natural level of unemployment in the system (eg people moving job or companies/industries contracting to keep the economy efficient). This was before the NAIRU came into being but is a similar concept without the inflation expectations element.
Of course, monetarist economies always have mass unemployment so the assumption is invalid and therefore monetarism is invalid, because once you allow for goods and services to be variable, then an increase in money supply increases output. This is typically the case where there is unemployment. The government increases the money supply / issues currency to employ unemployed people, they spend their wages in Tesco who sell more goods so need to employ more people and so on. The money supply is increased and inflation is not the result.
I'm not sure where the "always suffers from mass unemployment" comes from. The natural rate is not "mass" if the economy is efficient and at equilibrium. Unemployment grows when there are shocks and monetarists say it's important to limit contraction in the money supply as this can make output (and employment) fall more than otherwise. Monetarism is about managing the economy with awareness of money supply and inflation not about minimising public spending.
I think there is some conflation of traditional monetarism with other theories of free market economics in the criticisms I've read on here. The reason that the equation isn't so useful is that the authorities only control a narrow measure of the money supply, but credit expansion also causes inflation. Also that V may not be stable or as easily measured as thought.
There also seems to be some assumption in the above that the government just has to increase spending to reach full employment, but there's no mention about what other inputs are needed to produce something. Eg. A nurse requires years of training, and equipment and a hospital when they are actually working. But a hospital building program increases demand for building materials and medical equipment so causes inflation. If there are no new resources available for these new workers to do something useful, will they just be diverted to counting pigeons in Trafalgar Square like the famous example from the New Deal? That will get unemployment down but won't increase wellbeing in the economy.
