Just to make things concrete, here are some rough starting budgets, working from some notes of a hard campaign, medium units sized game:

Ducal budgets

Note:
[4]=[1]-[2]-[3]
[5]=[4] in first period; fixed thereafter for 5 turns
[6]=[4]-[5]
[9]=[6]+[7]-[8]

Imperial budget

The numbers don't match exactly with my game (I'm vague on agent costs) - I'll check things add up tonight.

But basically, according to these figures, the HRE is only making a profit of 129 florins. Not much of an income, once we've blown the starting balance of 6000 florins.

On the first turn, Duke's could each spend up to their 1000 share of the starting pot of 6000 to purchase buildings or troops. The "carried forward" column assumes they purchase nothing.

By changing taxes or disbanding troops, Dukes could increase their net profits this turn.

The Chancellor would revise the table on the basis of any purchases and any changes to net profit. This net profit, together with anything left of the 1000, would be carried forward to next period. It would be available for them to spend or save next period.

Note also that gross profits are very unequal across Duchies. Innsbruck (& Staufen), as a castle, gets much less and has much less potential for profitable investments. This is why we need to think about the balance between castles and cities. In a solo game, I would invest nearly all my money in upgrading Innsbruck and Staufen to get good troops. We currently don't have good incentives for that in a decentralised game. Letting castle owners "sell" troops for above cost price might be one mechanism.