Avoiding the production slowdown

The biggest production buildings that are necessary in any/every city (in my opinion) are: granary and forge, built in that order. The granary is needed in high food cities (ie. a city near flood plains, pigs, corn, rice, fish, or other food resources) to quickly recover population after you crack the whip. Some cities don't have much for hills or forests but still have lots of food - this is the perfect place to whip production because your citizens will grow back quickly.

The granary is also important in low food cities or production cities because they speed up growth so much. If a city is only making 1 or 2 food surplus a turn, they'll take a long time to get beyond size 6 or 7. A granary cuts any growth cycle time in half, so you can get working all those mines and mills sooner.

The forge is huge because it not only gives happiness bonuses with gold and silver, but +25% hammers applies to everything, not just buildings like Organized Religion, or units, like Police State (or military academies if you have Warlords).

By getting both these buildings up as soon as possible you can make any city more productive and then specialize from there with grocers, markets and banks in wealth cities, libraries and monastaries in science cities, and barracks, theatre and walls in border cities (where culture/invasion is an issue).

Too many of the specialization buildings offer too little yield in small cities or cities of the wrong type. In a typical early city running 70% research you might only be producing 4 or 5 gold. Is it worth spending 20 turns to get a market place for 1 extra gold? No, of course not. Get the stuff that will help ~immediately~, like workers, granaries, and forges, and then build other improvements as necessary. This is particularly true of things like the aqueduct, which offer absolutely no benefit unless they're sorely needed. Keep an eye on what cities are approaching happiness/health caps and build appropriately.

Also don't forget that often these caps can be overcome by trading resources with AI or settling on resources you don't already have, not just by putting up more city improvements.

Another thing to recognize is that units can be built in any city, whereas buildings have to be started in the city that needs them. This is pretty obvious, but the implications are widespread. If you get one or two good production centers with a barracks, heroic epic, ironworks, westpoint, etc, then your other cities don't need to produce nearly as much military and instead can focus on needed improvements. Thus, don't make your pop 1 city build a musketman because it needs another defender - build a granary there instead and get one of your powerhouses to pump out the military you need. Not only will your city be defended sooner, but your empire will do better as a whole because each city is doing what it does best: The production city making military, the small city setting up growth infrastructure, the science city scienteering, etc.
 
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