Mostly through exceptional measures. They received some regular and recurring income from various sources but this was far from being sufficient. AACW gives you three sets of options to simulate this.
Issuing War Bonds (several interest rates proposed):
This was a popular decision. The government promised more money in the future in return of some money now. The higher the interest rate, the more people were attracted. They do have to believe a bit in your chance to repay their loan though!
In game terms, War Bonds don't cost you National Morale as this is not a mandatory measure, but as you are selling a bit of your future (if you have one, for the Confederacy this is not so evident) the game will make you pay in Victory Points, the milleage used to ascertain how successful you were in a given game.
You can only use this option once a year.
Exceptionnal taxes (several taxes proposed):
Not very popular as this was a mandatory measure and something that the Americans of the times were unused to. These don't please the people but you can harvest quite a bit of money with them. The amount collected is based on how many population points you have, modified by you current victory point total.
Inflation will also increase slightly (all your units and investments will cost more).
You can only use this option once a year.
Printing paper money:
Even if it appears as an easy solution that you can use at will (every month if you want!), it is the "desperate man's" response to a lack of money. Inflation will soon go through the roof, so use this one with care lest you totally cripple your economy and thus your war effort by printing tons of worthless paper.
