Inflation and the Equation of Exchange

Let's start with a big question: What reasons are there for inflation to occur? One way of answering this question is to take the monetarist approach and focus on the so called Equation of Exchange. It will help us to easily identify the culprit.
First we take a look at the quantities necessary to understand this equation step by step and using an example. One quantity is the money supply M. It's simply the total amount of money present in the economy. For introductory purposes, I'll set this value to M = 100 billion $.
Also important is the velocity of money V. It tells us, how often each dollar (bill) is used over the course of a year. This quantity depends on the saving habits of the people in the economy. If they are keen on saving, the bills will only pass through a few hands each year, thus V is small. On the other hand, if people love to spend the money they have, any bill will see a lot of different owners, so V is large. For the introductory example, we'll set V = 5.
Note that the product of these two quantities is the total spending in the economy. If there are M = 100 billion $ in the economy and each dollar is spend V = 5 times per year, the total annual spending must be M · V = 500 billion $. This conclusion is vital for understanding the Equation of Exchange.
There are two more quantities we need to look at, one of which is the price level P. It tells us the average price of a good in the economy. If there's inflation, this is the quantity that will increase. Let's assume that in our fictitious economy, the average price of a good is P = 25 $.
Last but not least, there's the number of transactions T, which is just the total number of goods sold over the entire year. We'll fix this to T = 200 billion for now and make another very important conclusion.
The product of these last two quantities is the total sales revenue in the economy. If the average price of a good is P = 25 $ and there are T = 200 billion goods sold in a year, the total sales revenue must be P · T = 500 billion $. It is no accident that the total sales revenue equals the total spending. Rather, this equality is the (reasonable) foundation of the Equation of Exchange.
For the total spending to equal the total sales revenue, this equation must hold true:
M · V = P · T
which is just the Equation of Exchange. Now think about what will happen if we increase the money supply M in the economy, for example by printing money or government spending. We'll assume that the spending habits of the people remain unchanged (constant V). Since we increased the left side of the equation, the total spending, the right side of the equation, the total sales revenue, must increase as well.
One way this can happen is via an increase in price level P (inflation). Indeed empirical evidence shows that in the US every increase in money supply was followed by a rise in inflation later on.
Luckily there's another quantity on the right side which can absorb some of the growth in money supply. A rise in the number of transactions T (increased economic activity) following the "money shower" will dampen the resulting inflationary drive. On the other hand, a combination of more money and less economic activity can lead to a dangerous, Weimar-style hyperinflation.
At some point of your life, you probably thought to yourself: If governments can print money, why the hell don't they just make everyone a millionaire? The answer to this question is now obvious: The Equation of Exchange, that's why. If the government just started printing money like crazy, the rise in price level would just eat the newly found wealth up. Each dollar bill would gain three zeros, but you couldn't buy more with it than before.
Of course there can be much more trivial causes for inflation than a growing money supply. Prices are determined by an equilibrium of supply and demand. If demand drops, the retailers have to lower their prices to sell off their stocks. Similarly, if demand suddenly increases, the retailer will be able to set higher prices, resulting in inflation. This happens for example when a new technology comes along that quickly rises in popularity. Appropriately, this kind of price level growth is called a demand-pull inflation.

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