Q. Determine price level from the quantity theory of money?
The price level
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The price level is determined from the quantity theory of money: P = (M.V)/Y
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In the classical model, money supply M is an exogenous variable (therefore, growth rate in the money supply pM is exogenous). It is concluded by the central bank. In the same way V is an exogenous variable in agreement with quantity theory of money. So M.V is exogenous and given.
Remember that Y is determined by labour market and production function. If we combine this with quantity theory of money, we can determine the price level P: P = (M.V)/Y.
Now, assume that GDP is constant over time. Because V is stable (let's say it too is constant), percentage change in P is equal to percentage change in M. Which is, inflation is equal to the growth rate of money or p = pM.
Remember that we have removed the trend in Y that means Y cycles around some average over time. So Y isn't constant over time though there is no growth in Y. Thus p = pM will still be approximately true even when Y isn't constant (it will be true on average and in long run).
If we don't remove the trend in Y, result would instead be that inflation is equal to growth in money supply minus the growth in real GDP.