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In its original sense, seigniorage refers to the difference between what it costs to produce currency and what that currency’s value is, or its net revenue. If a government is able to produce new money at a lower cost than what the money is valued at, the net revenue is positive and the government has made a profit. If the cost of producing money is greater than the money's worth, the government takes a loss. In nations that operate on a fiat money system, however, seigniorage takes on a somewhat different definition. Fiat money is used in most countries today, and basically means any form of currency a country's government deems legal tender. In a fiat system, seigniorage can be defined as the interest the government earns off of securities that have been pulled in exchange for the production of more paper money. It is a currency's exchange value, in other words.
Seigniorage can be relatively easy to calculate with governments that primarily use metal coins for currency. Add up how much it costs to hammer out each piece of copper, gold or silver, and subtract that from the value assigned to the coin. Governments who successfully mint new coins worth more than they cost to produce typically make an economic profit.
This definition of seigniorage also applies to governments that issue paper money backed by precious metals, typically gold. Each bill represents an actual piece of gold held within a bank vault. The cost of minting paper money, contrasted with the worth assigned to it according to the value of the metal backing it, would be that currency's seigniorage.
In a fiat money system, paper money isn't backed by gold or any other precious metal. This is what is meant when it's said that paper money has no intrinsic value. The only value the money has is what the government and the nation's people assign to it. In other words, the value of money is upheld purely by collective faith. If the nation suddenly stops considering a bill to be worth something, it no longer is. This type of money system changes how seigniorage is derived.
Since the value of money is made up in a fiat system, there has to be a new method for accurately assessing the cost of producing money versus its value. This is accomplished by the government taking interest off of various securities—such as government bonds—in exchange for issuing new money. The interest derived is the seigniorage, or the net revenue earned for issuing new money.