During the late nineteenth century, the united states experienced a period of sustained deflation, or a falling price level. explain in terms of the quantity theory of money how a deflation is possible. is it necessary for the quantity of money to decline for deflation to occur?

It is not necessary a decline in quantity of money for deflation to occur.

The quantity theory of money states that if money supply and velocity of circulation don't change economic growth (positive change in GDP) will result in declining price levels

Explanation:

The quantity of money theory states that

where M is the money supply, V is the velocity of circulation, P is the price level and Y is the GDP

We can put this equation in terms of percentage changes, which gives

where the denotes the percentage change in the money supply, and similarly for the other variables.

Then for the percentage change in prices to be negative we have that

since

So if the there's no change in circulation velocity or gdp, then inflation can occur if there's a decline in money supply (percentual change in M is negative).

But it also could be other scenarios:

1.  money supply or output did not change and velocity of circulation decreased

2. money supply and velocity remained constant but GDP grew

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