Well. There are two factors involved here:
1. Support costs. Under despotism, every single city pays for 4 units on the board. Under monarchy, it's 2 for towns (1-6), 4 for cities (7-12), and 6 for metropolises (13+). I don't remember the exact numbers for republic (2 per city no matter what size?). I think democracy is zero support per city.
2. Commerce improvements. Gads! I'd need to go read the civilopedia again (at work atm). IIRC, both republic and democracy give noticable increases to commerce. I don't remember if it's calculated per city or applied after the fact to your entire civ. I don't remember what commerce adds monarchy gives. I don't think they get one directly.
So if you're going from despotism to monarchy, you are pretty much always going to take a cash hit. Unless you are getting monarchy very late in the game for some reason, odds are you've got lots of cities that are still size 6 or lower (most of them in fact). It's not that you make less money, but that you now have to pay more for your army.
The benefits of monarchy is that the -1 food production limit under despotism is lifted. Also, you get an extra military garrison (can use 3 instead of 2). Your workers are also more productive, allowing you to build roads and improvements faster. You don't go to monarchy to make more money. You go to monarchy to allow your cities to grow faster and to build up your empire a bit faster. The asumption is that you've expanded and now need to consolidate. Even though you're not making as much money as before, you gain more benefit from having larger cities under monarchy, so the overall commerce effect will be positive. Basically, if you aren't already expanded about to where you want to be, and you don't have at least half your cities at 6 or higher size, you should probably stay in despotism for a bit longer.
Republic is a bit trickier. You get a commerce boost, but unless you've got a lot of cities, you're going to pay a lot for your military. Ok. You're going to pay a lot anyway, but the point of a republic is that you don't need a huge army. Like all things in Civ, it's a tradeoff. You can either have a large army, or build faster. The trick is figuring out the right time to switch from one government form to another. If you are at peace, and have good relations with your neighbors, and have a good sized civilization, republic is probably a good idea. You get even more worker productivity and greater growth potential then in monarchy. You can't field nearly the size of military under republic as under monarchy though.