I'm new to the civilization series, and noticed that my units disband when my gold drops to negative.
How can I improve my gold production to prevent this from happening again?
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There are two factors at work. One is that you could be making more money, and the other is that you could be spending less.
One major source of income is your cities. You can build tile and city improvements that generate gold per turn, like Trading Posts and Markets. It helps to specialize one or more of your cities for gold production by placing the city in an area with a lot of gold-per-turn bonus tiles like Gold, Silver, or Gems, and then focus heavily on building things like Trading Posts and Markets.
The Great Merchant can also build the "Customs House" tile improvement that produces a lot of gold per turn. Alternatively, he can be used to conduct a trade mission to a city-state for a lump sum gold bonus.
Remember that by default, the city will decide which tiles to work and which to ignore. You can change the automatic focus in the city screen, or you can manually specify which tiles to work if you're confident you know better than the automated system does.
At a Civilization level, you can generate more gold by establishing road or railroad connections between your cities. With BNW, you can establish profitable trade routes with other Civilizations as well. Finally, you can sell excess luxury and strategic resources to your neighbors for gold per turn or lump-sum gold. This can be a staggeringly large amount of gold per turn for a well-connected empire.
Buildings in your cities are generally in the top 2 reasons why you are spending money. Not all buildings cost gold per turn, but most do. If you have every building in every city, chances are you're going to have a bad time.
You can sell buildings in the city view, which gives you a small gold boost and reduces the gold per turn cost. Think carefully about what a particular city is good for (usually production, science, or gold - perhaps two of those in a very large, well-placed city) and only build the relevant buildings in that city. For example, while every city can produce units, generally it makes more sense to only produce them in a small number of cities that are optimized for production and have the relevant military/production boosting buildings.
The other thing in the top 2 tends to be military. The larger your military, the more you spend per turn. On most of the average difficulty levels, you'll want to field a small army of front line troops and siege units. There should be enough of both to defend your border cities (1 siege + 1 infantry is generally sufficient) while waging a war with whatever neighbor you deem worthy of destruction (3-5 infantry + 3 siege is a good start). Trying to fill the whole map with units is going to kill your finances very quickly.
If you have a surplus of units, you can disband them. You can also gift them to city-states in return for a small relations boost.
You can view your sources of income and the costs of running your empire by hovering over the "gold" part of the status bar at the top of the screen. There will be a detailed breakdown of where gold is earned and spent.
Strategically placing your cities, specializing them with the proper buildings and constructing the proper tile improvements pays massive dividends. Conversely, if you dump a bunch of low quality cities in bad areas, they'll drain your gold and happiness without being productive enough to make up for it.
Lowering the difficulty can also help, as you'll need a smaller military, more units will be free, and the AI will be more intimidated by you and more likely to give you advantageous trade deals.
If you're into spreading your religion, there are a few religious tenets that can increase gold production.
Finally, there are several social policies (and a whole social policy track, Commerce) dedicated to producing and conserving gold. Similarly, there are wonders that can do wonders (lulz) for your gold production, either by saving you money on buildings or improvements you would normally pay for, or by helping you earn gold directly.
Gold is mostly produced by cities. Working certain tiles such as resources, tiles next to rivers, or coastal tiles will cause that city to produce gold. Once you have the correct technology, you can also build Trading Posts on tiles near your cities, increasing the gold production of that tile.
In Vanilla and in Gods & Kings, Gold is also produced by city connections. If a city you control is connected to your capital by roads (or connected over the sea using harbors), you will receive gold per turn based on its population. Roads do cost gold per turn to maintain but the city connections usually more than pay for them.
Certain improvements, such as markets, will further increase a city's gold output. So can social policies, especially those in the commerce tree.
You could also find ways to cut the costs of your empire. Avoid building roads that you don't need- usually you want as few as possible to connect all your cities. Also, disband any units that aren't useful. If they're obsolete and not worth upgrading, they're not worth keeping.
Finally, lowering the difficulty might help. This will, among other things, increase the number of free units you can get.