Because money dominates our lives, it's easy to forget that savings isn't actually about money. When you produce more than you consume, the difference is the amount you saved. Thomas Woods demonstrates this really well in his book, Meltdown. I'll summarize it briefly.
Imagine a simple economy with no money, and a baker who is able to produce 10 loaves of bread per day. He has a big family, so he needs to consume 8 of those loaves to feed his family. Now let's say he wants to buy shoes from a shoemaker. The shoemaker needs to eat, so he agrees to make a pair of shoes for 10 loaves of bread. The baker saves his extra 2 loaves until he has ten, and then gives it to the shoemaker who later gives the baker a pair of shoes. Here it's clear that savings is not money, but real goods.
Introducing money into the picture doesn't change anything. Let's say a loaf of bread is worth one dollar. The baker could then sell his loaves of bread to someone else, and then use the money to buy a pair of shoes. The shoemaker would then trade his dollars for the bread he needs. The shoemaker is still funded by the real bread savings of the baker. Money is just facilitating the transactions.
To really make this clear, imagine if a third party counterfeited an extra 10 dollars, and paid it to another shoemaker for a pair of shoes. Now there would be two shoemakers with 20 dollars total, but there still are only 10 loaves of bread savings. There is no way for both shoemakers to both buy 10 loaves of bread. This new counterfeiter has now consumed without producing, and it's pretty clear that this is not going to be beneficial to the economy.
None of this changes when you move to the much larger real economy. It also doesn't change when you replace our individual counterfeiter with the Federal Reserve printing money. Savings is not about the money, it's about producing more than you consume.
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