I chose this topic because I couldn't pronounce the word, and so I was quite intrigued. So first off, for anyone else who may be curious, it is pronounced "sane-yur-ij".
Basically, it is the difference in what it costs to make a coin and what that coin is worth, face value. If it costs 5 cents to make a quarter, then the seigniorage is 20 cents. When the money is worth more than it costs to produce, as it does in this example, the government profits. Of course, the government will lose money if the reverse occurs.
The way that it works with dollar bills is slightly different. Today, the Federal Reserve buys and sells treasury bonds from banks, and replaces those bonds with bills. But since there's interest to be collected from bonds, when the bills go out of circulation and the bonds are returned, the government will profit.
A great example of when the government profited from seigniorage is with the collector's series of state quarters. It only costs about 5 cents to make a quarter, so since people don't intend to spend these, the government profits when, for instance, they're purchased from TV for more than their face value of 25 cents.
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