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Some fun facts and figures with granaries. I had a look at some simplified models, just to see what kinds of factors are important.
First time around, I assumed that all cities get to work tiles producing 2 food, 1 shield, 1 trade. One scenario has the first city building a granary, the other has the city doing the size 1-3 settler oscillation.
The city without the granary is simple. It has a 20 turn cycle, producing 20 shields worth of units (2 warriors, 1 spearman or 1 archer, depending on your techs), 1 settler and 50 trade. So after 60 turns you have accumulated 60 shield of units, 3 settlers, and 150 trade.
With the granary, I decided to produce 3 warriors to start with (that 3rd one only delays granary production by 3 turns in practice). I then oscillate between sizes 3-5 for settler production (you could get a second settler out more quickly, but this is just illustrative). Settler no. 3 appears in turn 61, only 1 turn later than in the no-granary situation, at which point you have also prouced 100 shields of units (in batches of 30, 20, 20, 10, 20). In the first 60 turns, you produce 210 trade (and that's after allowing for trade lost to luxury spending, assuming you have no luxury resources to hand).
Second scenario: assume we have 1 shielded grassland available, producing 2/2/1, and all other sqaures are 2/1/1 as per the original scenario.
Without the granary, the situation is the same, but you get to produce more shields for units. Ironically, some of your settlers may turn up one turn later, depending on how you want to waste shields (either going to wealth for a turn or two when you can't fit anything in before the settler build begins, or fitting a warrior in at the expense of delaying the setller by one turn). Settler 3 still comes along in turn 60 (or 61, possibly 62, depending on how you play it), but you have 100 shields worth of units (in blocks of 40, 30, 30, so a religious civ could fit a temple build in there as well).
With the granary, settler number 3 appears in turn 52, but with only 70 shields of units (30 at the start for warriors to explore, then 20, 20). In 60 turns you would generate 195 trade and 90 shields.
Results part 1: Using a single city as a settler pump, with 2 excess food at all times, you are equal or ahead with settler generation by the 3rd settler, and have considerably more trade (gold or science), even if you have to use the luxury slider to control happiness. The amount of shields that can be spent on units is comparable. So you 'break even' in settler production in 50-60 turns.
On the other hand, you lose out in terms of production from the first new city, by 20 turns in the first scenario, or nearly 10 turns in the second. That's quite a lot of shield production you are giving up. In particular, if you use all cities to churn out settlers as fast as possible, (assuming no granaries in the subsequent cities), then you don't catch up in terms of number of cities until you get to 15-20 (depending on how long you assume it takes a settler to move and found a new city). The REX phase is usually over before you get to that many cities (in my experience, you are doing very well to get 10 cities down without warfare on a large map, Emperor level). Of course that kind of rapid expansion leaves you over exposed to AI invasion, since you don't have resources to spare for much else aside from a few warriors, 1 spearman per city and lots of settlers.
So, my quick look suggests that the effectiveness of granaries depends to some extent on the excess food to shield ratio. More shields and less food makes granaries more effective. Fewer shields and plentiful food make them comparatively useless in the REX part of the game (since you are limited by the number of shields, not by the amount of food). If you are going for an all out (badly defended) REX then granaries don't help. If you are using one city as a settler pump, then they may, but the most natural settler pumps (high food) don't need them. If you have to use a +2 food city as a settler pump, then it is probably best to build a granary. If you can afford to rush buy a granary, all the better; you get the best of both worlds. It is rather situation dependent though, so get out and explore the world before committing yourself.
Granaries, of course, are useful after the REX phase too... (and don't forget, the *only* time you can grow too fast is when your city growth is outstripping your ability to develop squares (I've overlooked worker production in this analysis); as long as all your citizens can work developed tiles, the worst you do by having to use entertainers/luxury slider to control happiness is break even, and most of the time you gain some benefit (when compared to keeping your cities below the happiness limit)).
First time around, I assumed that all cities get to work tiles producing 2 food, 1 shield, 1 trade. One scenario has the first city building a granary, the other has the city doing the size 1-3 settler oscillation.
The city without the granary is simple. It has a 20 turn cycle, producing 20 shields worth of units (2 warriors, 1 spearman or 1 archer, depending on your techs), 1 settler and 50 trade. So after 60 turns you have accumulated 60 shield of units, 3 settlers, and 150 trade.
With the granary, I decided to produce 3 warriors to start with (that 3rd one only delays granary production by 3 turns in practice). I then oscillate between sizes 3-5 for settler production (you could get a second settler out more quickly, but this is just illustrative). Settler no. 3 appears in turn 61, only 1 turn later than in the no-granary situation, at which point you have also prouced 100 shields of units (in batches of 30, 20, 20, 10, 20). In the first 60 turns, you produce 210 trade (and that's after allowing for trade lost to luxury spending, assuming you have no luxury resources to hand).
Second scenario: assume we have 1 shielded grassland available, producing 2/2/1, and all other sqaures are 2/1/1 as per the original scenario.
Without the granary, the situation is the same, but you get to produce more shields for units. Ironically, some of your settlers may turn up one turn later, depending on how you want to waste shields (either going to wealth for a turn or two when you can't fit anything in before the settler build begins, or fitting a warrior in at the expense of delaying the setller by one turn). Settler 3 still comes along in turn 60 (or 61, possibly 62, depending on how you play it), but you have 100 shields worth of units (in blocks of 40, 30, 30, so a religious civ could fit a temple build in there as well).
With the granary, settler number 3 appears in turn 52, but with only 70 shields of units (30 at the start for warriors to explore, then 20, 20). In 60 turns you would generate 195 trade and 90 shields.
Results part 1: Using a single city as a settler pump, with 2 excess food at all times, you are equal or ahead with settler generation by the 3rd settler, and have considerably more trade (gold or science), even if you have to use the luxury slider to control happiness. The amount of shields that can be spent on units is comparable. So you 'break even' in settler production in 50-60 turns.
On the other hand, you lose out in terms of production from the first new city, by 20 turns in the first scenario, or nearly 10 turns in the second. That's quite a lot of shield production you are giving up. In particular, if you use all cities to churn out settlers as fast as possible, (assuming no granaries in the subsequent cities), then you don't catch up in terms of number of cities until you get to 15-20 (depending on how long you assume it takes a settler to move and found a new city). The REX phase is usually over before you get to that many cities (in my experience, you are doing very well to get 10 cities down without warfare on a large map, Emperor level). Of course that kind of rapid expansion leaves you over exposed to AI invasion, since you don't have resources to spare for much else aside from a few warriors, 1 spearman per city and lots of settlers.
So, my quick look suggests that the effectiveness of granaries depends to some extent on the excess food to shield ratio. More shields and less food makes granaries more effective. Fewer shields and plentiful food make them comparatively useless in the REX part of the game (since you are limited by the number of shields, not by the amount of food). If you are going for an all out (badly defended) REX then granaries don't help. If you are using one city as a settler pump, then they may, but the most natural settler pumps (high food) don't need them. If you have to use a +2 food city as a settler pump, then it is probably best to build a granary. If you can afford to rush buy a granary, all the better; you get the best of both worlds. It is rather situation dependent though, so get out and explore the world before committing yourself.
Granaries, of course, are useful after the REX phase too... (and don't forget, the *only* time you can grow too fast is when your city growth is outstripping your ability to develop squares (I've overlooked worker production in this analysis); as long as all your citizens can work developed tiles, the worst you do by having to use entertainers/luxury slider to control happiness is break even, and most of the time you gain some benefit (when compared to keeping your cities below the happiness limit)).
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