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Easy Integration

The only way you are going to get anything done as an intergalactic bureaucracy is to make things as streamlined and efficient as possible. Otherwise your Hegemony is going to collapse under the weight of its own red tape. Therefore, it is actively in the Hegemony's best interest to leave the low-level running of a member state hands-off, and only regulate the integration points with the rest of the empire.

Think of your member state as a black box. You don't care what happens inside of the box, you just care that the interface with it does what you expect. In the case of your Hegemony you want to make sure that taxes are collected and goods are bought and sold using the standard credits currency. As long as that is true, what goes on inside the box doesn't matter to you (mostly).

What this looks like for each member state is that the local economy works almost exactly as it always has. No strict need to remove the old currencies or do anything else to shake things up. The local government will still collect taxes from it's citizens, and some of those taxes will go towards the Hegemony. All they need to do is exchange that share from Local Currency Dollars (LCD) to Hegemony Credits (HC), using some standardized rate that the Hegemony manages. The same thing happens with inter-state imports/exports. Some good or service costs X HC, and people will know what that translates to in their LCD.

Practical Example

If I want to import a good that costs 10 HC, and the exchange currently says that 1 HC = 10 LCD, then obviously I need 100 LCD to make that purchase. If I get paid in LCD then I would first need to exchange my currencies to HC (which may have extra costs associated).

This is really no different than the kind of currency exchanges that we have here on Planet Earth, except that we have a handful of major currencies (USD, Euro, Yen, etc.) instead of a single universal currency that everything can be mapped to.

A similar process would happen for the local government paying taxes to the Hegemony. If the government owes 1 million HC, then it would need to collect the equivalent of 10 million LCD (or whatever order of magnitude makes sense for a planet-wide tax).

Lego Empire

The nice thing about this set up is that it makes it incredibly easy to add new member states to the Hegemony. All you need to do is figure out what the exchange rate is between HC and LCD. After that everything is pretty boilerplate. Imports get paid for in HC, exports get bought using HC, Hegemony taxes are paid in HC, and everything else is left as-is. There is a little more work for the government to perform managing things, but the average person wouldn't have too much in the way of adjustments.

This also makes it really simple to travel between member states because you always have a standard currency to fall back to. I can turn my hard earned LCDs in HCs and then use those anywhere in the intergalactic empire.

The Hegemony also gets a very strong leverage tool in being the sole decider of every exchange rate. If you need to apply pressure to a member state then even just the threat of devaluing their LCD could be enough to keep them in line. If your LCD is suddenly worth less then your entire planet is now poorer and has less spending power compared to other member states.

Favorable exchange rates could also be used as an incentive for new members. Join up early and get locked into a low rate that gives your entire economy a boost. Take too long to decide or try to play hardball and instead you could be given a higher rate that devalues your LCD and causes a planet-wide depression. Don't join at all and you won't have access to trade goods at all. A black market would still exist for any group that didn't join, but it would be inherently more dangerous and expensive and less reliable than the legal route. And the more pieces you add to your empire, the more incentive the next guy has to join up as well.

Benefit: the Hegemony can use this to get its vassal states to bend the knee and keep it bent.

Argentina won't pass the law to allow the Hegemony to draft citizens from it? No problem! The hegemony will spend six days buying every other peso available in the market, causing its value to soar. That may sound cool to Argentina at first but quick variations in value are bad for economy. Then, on the seventh day, the Hegemony floods the market by selling all the pesos it bought and then some more for a ridiculous low value. Watch the country crash.

Even better, that wouldn't be the first time the Hegemony would use that tactic. If it has a history of doing so, then by the time the Hegemony announced their plan to buy pesos, every single person with more than a couple bills in the wallet and a sense of self preservation would know to move to another country more aligned with the Hegemony ASAP.

You can't use this tactic if everybody is on the same currency.

Speed.

Charles Stross has an extreme example of this in Saturn's Children/Neptune's Brood, with "fast" and "slow" money.

If you don't want money to be bound to physical tokens, you need to avoid the same money to be spent twice at the same time, so you always need confirmation from the bank or other institution that maintains the account, and this cannot be done sensibly over only a few lightseconds distance, so local economies need to be decoupled from interstellar trade.

Over vast distances, there is also a need to keep the value of the transferred money stable during transit, which is another reason to separate the local economies.

Currencies can serve as a form of investment, since the inflation rate of the individual currencies can be controlled independently. For example, should the local currency be doing badly, people might invest in the global currency before buying back the local currency. That is, unless the local currency's value is tied to the global currency.

They can also allow the local authorities to have a level of independence with their economic policies without having 'tourists' needing to exchange.

The currencies may also have a historical or cultural value which may be a result of conservatives or the goodwill of the ruling nation.

A more practical use would be for finer currency, since a unit of global currency may be worth hundreds of a local currency. This would allow locals to trade in sub-credit numbers, potentially allowing to make a profit from selling sub-credit goods for a full credit on the global market.

I would recommend doing some more detailed research on currency exchange should you find the time, some interesting facts may give you additional inspiration.

Such a system would be very much akin to how most online stores and F2P games work. What it gives you is a single value to assign to certain key goods. Think military hardware, munitions, critical electronic components, fusion reactors etc.

These are key things the Hegemony wants to remain in control over. They'd ban or heavily tax local production to bring it in line with their Credits. There might be a thriving black market of knock off goods. Think what key components does the Hegemony need to keep their people in check? Monopoly on FTL drives?

The local currency allows you to run an alternate economy. You want to do this if the region has a much lower GDP. You want to bring things in line with the local wealth. Think of food, medical supplies, etc. This could be seen at frontier planets that lack a sophisticated industry. They can't economically compete with the core worlds.

So dual currencies allow for a gap in wealth without outpricing basic goods for the poor.

You know how to easy win people over? Pay them.

The conversion toward "credit" may always be in favour of local currency. People love when they get THEIR faces on notes and eagles or other aviaries on coins. and the conversion is always in their favour. It's so good to have such stupid hegemony that allows to scam it. Like taking pens from your corporate jobs.
But it's the hegemony that dictate the conversion rate. Your planet, colony, continent get rough? Good luck buying stuff from outside when you are pushed into super-extra-mega-hyperinflation.

Also you want to leave your planet? You change your money to credit and you leave the trace. Like a bitcoin but in reverse. The owner is not known but the money is always traceable and can be simple tied to any operation they do. Like buy ticket.

While using local currency don't clutter the system and leave local authorities to deal with such things on their own.