When we’re able to synthesize gold and silver with no cost at all, what do you think will be used as the standard for money? As in, if we were to still have paper money, it would be backed by the _______ standard…?
What would we synthesize them with? If the input material could be used to synthesize many different things, assumably that would become a medium of exchange.
Platinum and Rhodium? Palladium?
I’m assuming you mean to include all precious metals into your question. As per usual its unknown what will happen in the future, but a part of me says virtual money might constitute a sizeable portion of the money supply. I don’t care for the game myself, but I’m a bit interested in the purchasing power gold from World of Warcraft has for goods in the physical world. Cigarettes and alcohol will, in my opinion, most likely gain status as pseudo-money during any such event when the value of precious metals fall infinitely.
I think a regime of competing fiat currencies (electronic or else) would do well enough.
Gotta love science fiction. 
First question is, how the synthesizing works - if it’s e.g. by thermonuclear fusion, it could be easy to create basic elements like gold, but there could be always elements that are hard to synthesize (stable super-heavy elements?), which could make interesting money material. It could be still hard to get more complicated compounds and chemicals, so some of those could be of interest too (like anagathics, longevity drugs; or room-temperature superconductors).
If any element or material can be easily produced out of anything, the question becomes even more interesting. One could use energy in some easy to store form (energy is still needed to make the stuff). Should even that condition drop one day, if people would become effectively able to wish things into existence for free, humanity will be so different that it would be hard to even imagine how their life and trade would look like.
Whatever is sufficiently rare and valuable in given conditions can become money.
I once had the idea for a sci-fi society basing their money off of negative matter, due to the inability to manufacture it as well as its very important role in cheap wormhole fabrication.
I go with alchohol and cigarettes, or maybe wampum
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