It Doesn’t Need to be Too Much Like Money
As economists have observed for decades: if charging money to provide goods and services is such a perfect system that solves all problems, why don’t corporations do it internally? Why don’t the different departments charge each other for their services, or outsource everything? Apparently, the free market has determined that firms running on a command economy are superior to firms that run on the free market!
But there are still several reasons you might want something like money.
Allocating Resources
Various countries have experimented with using primitive “cybernetics” to make a command economy work. As Francis Spufford wrote in Red Plenty, one of the problems this ran into in the Soviet Union—beyond the fact that id didn’t have nearly enough computing power in the middle of the twentieth century for it to work—is that the equations kept saying that the optimal way to run the economy looked more like capitalism than they were willing to accept.
One of the most prominent examples was, the equations involved a vector, labeled “shadow prices,” because they functioned essentially the same way as prices under capitalism.
In general, there’s going to be some limiting factor on how much you can produce of each good. If you’re a doctrinaire Marxist, like the ones in the Soviet Union and a few I’ve met after it fell, you might feel compelled to insist that this must be labor. Since there’s no reason to be a doctrinaire Marxist, though, it can be lots of things: energy, the ability of the atmosphere to absorb greenhouse gases without destroying the climate, the number of atoms in the solar system.
In effect, how much of these limiting factors you are willing to allocate to each given purpose is a kind of price. Trying to fit the cost of an electric car, a hybrid or a gas-guzzler into your carbon budget is a lot like putting a price on carbon and making the carbon budget part of the money budget.
Preferences
Humans like to feel in control. Their revealed preference is to play a lottery that lets them pick their lucky numbers instead of one that just tells them if they won, for example, even though it makes no difference to the odds. So, while the AIs probably shouldn’t take this too far and hand out hydrogen bombs to anyone who wants them, there are definitely lots of examples where everybody would be happier picking out the things they want than having the algorithm always choose for them. Even if the algorithm knows them better than they know themselves, they’d get some psychic value just out of having the power to choose.
If everybody’s happiness is considered equal in value, this would work a lot like ration cards: everybody starts out entitled to a certain amount of the world’s energy, a certain amount of its platinum, a certain amount of its food, and so on, and can choose how to spend their share. But then some people would want to use more energy and no meat, so you’d want to allow them to trade. And then, the ratio of the trade value of any two goods would be the same, as demonimated in any other good, or else there would be arbitrage that made the discrepancy vanish. So, as soon as you allow trading, you’ve brought back money. Every MMORPG ever made has found this out, as players have decided on some in-game commodity, often one that’s harder to mine than in-game “gold” or “credits,” as its de facto currency. This system would be no different: whatever’s most convenient as the medium of exchange or the store of value will function exactly like money, and there will be markets where you can trade some of your unneeded food allocation for someone else's excess electricity.
Incentives
People aren’t motivated only by money, but probably no system can depend entirely on moral appeals for people to work harder, either. You might want to encourage them to install more solar panels and trade the excess energy by offering to trade something for it, and it would be something they can trade for whatever they want, so you’ve created money.
Eventually, the AIs might get so much better at humans at everything that it’s no longer important whether humans do anything productive or not. (Indeed, there’s really nothing humans are needed to do any more.) In that case, though, the AIs would still need to allocate and trade resources between themselves. And this would involve creating something a lot like money.