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You may remember that I once cooked an example called "gold for soybeans" where I showed that, if you are sitting at a point of the Production Possibility Frontier of Gold and Soybeans, the slope of the normal direction is the relative price of the two, and the intercepts of the tangent line with the axes are the GDP expressed in units of gold or of soybeans. Assuming that the PPF is convex (mild assumption) you get that moving along the PPF will increase one of the GDP values but decrease the other. This fails the change-of-numeraire sanity check. But if you can move the PPF outwards you can increase the GDP regardless of which numeraire you're using.
If you keep making more stuff, GDP grows by any numeraire, but if you can not make more stuff some numeraires will show a growing GDP and some a shrinking? Would that be a fair summary?
The discussion of fiat money leads me a bit of topic: I have theory on the relatively succesful fiat moneys of today.
Sometimes you hear statements about todays fiat moneys not being based on anything tangible (at least since the Bretton-Woods collapsed), but only on how much you trust the government. I think this needs expanding upon.
Let me first start with some historical examples of fiat money. The american and french revolutions printed money to finans their wars. The trust was low and as soon as trust went away in an area, huge inflation kicked in as people started to use other numeraires.
Today you can not stop using the governments numeraire. Why? Because of taxation. Of course people payed taxes in the 18th century to, but often as work or in kind. Today you need to pay your taxes and the taxes are in the governments currency. So even if you switch to exchanging beans for gold with your neighbour, you still need some euros, dollars, renminbis to pay the government. Which means that at some point you need to accept fiat money for real products.
So the currencies of today rests on taxation and the governments ability to punish you if you do not pay your taxes.
Assuming I was right on the origins of GDP in my comment on Colmans diary the other day:
I have long suspected that GDP/capita becomes an important measure at about the same time as you can get that number from the tax offices.
Then both fiat money and GDP stems from the same source: modern taxation.
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